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ADVANCES IN APPLIED MECHANICS VOLUME 7

This Page Intentionally Left Blank

ADVANCES IN APPLIED MECHANICS Editors

TH.

H. L. DRYDEN

VON

K ~ R M ~ N

Managing Editor

G. KUERTI Case Institute of Technology, Cleveland, Ohio

Associate Editors

F. H.

VAN DEN

DUNCEN

L. HOWARTH

VOLUME 7

1962 ACADEMIC PRESS

NEW YORK AND LONDON

COPYRIGHT

0 1982,

BY

ACADEMIC P R E S S INC.

ALL RI GHTS R E S E R V E D

NO PART O F T H I S BOOK MAY B E R E P R O D U C E D I N A N Y FORM B Y PHOTOSTAT, MICROFILM, OR A N Y QTHER MEANS, W I T H O U T W R I T T E N PERMISSION FROM T H E P U B L I S H E R S

ACADEMIC PRESS INC. 111 FIFTHAVENUE

NEW YORK3, N.Y.

United Kingdom Edition Published by

ACADEMIC PRESS INC. (LONDON) LTD. BERKELEY S Q U A R E HOUSE,LONDON W. 1

Library of Congress Catalog Card Number: 48-8503

PRINTED IN THE UNITEDSTATES OF AMERICA

CONTRIBUTORS TO VOLUME 7 G. I. BARENBLATT, Institute of Geology and Development of Combustible Minerals of the U.S.S.R. Academy of Sciences, Moscow, U.S.S.R.I RAYMOND HIDE,Physics Department, King’s College (University of Durham), Newcastle-upon-T yne, England2 HAROLDMIRELS, Lewis Research Center, National Aeronautics and Space Administration, Cleveland, Ohios W. OLSZAK,Institute of Fundamental Technical Problems, Polish Academy of Sciences, Warsaw, Poland PAULH. ROBERTS,Physics Department, King’s College (University of Durham), Newcastle-upon-Tyne, England4 J. RYCHLEWSKI, Institute of Fundamental Technical Problems, Polish Academy of Sciences, Warsaw, Poland W. URBANOWSKI, Institute of Fwdamental Technical Problems, Polish Academy of Sciences, Warsaw, Poland

Present address: Present address: of Technology, a Present address : Present address: Wisconsin. l

a

Institute of Mechanics, Moscow State University, Moscow, USSR. Department of Geology and Geophysics, Massachusetts Institute Cambridge, Massachusetts. Aerospace Corporation, El Segundo, California. Yerkes Observatory (University of Chicago), Williams Bay,

V

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Preface The seventh volume of Advances in Applied Mechanics includes two extensive reviews of topics in solid mechanics and an account of recent analytical results obtained in the field of hypersonic obstacle flow. A detailed presentation of the basic physical principles and problems of phenomenological magneto-hydrodynamics concludes this volume ; it may serve as an introduction into this comparatively new branch of hydrodynamics.

THE EDITORS July, 1962

Vii

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Contents ....................... ...............................

CONTRIBUTORS TO VOLUME 7 PREFACE.

v vii

Hypersonic Flow o ~ e rSlender Bodies Associated with Power-Law Shocks

.

BY HAROLD MIRELS.Lewis Research Center National Aeronautics and Space Administration. Cleveland. Ohio I . Introduction . . . . . . . . . . . . . . . . . . . . . . . I1 Hypersonic Slender-Body Theory . . . . . . . . . . . . . I11. Flows Associated with Power-Law Shocks . . . . . . . . . IV . Flows Associated with Slightly Perturbed Power-Law Shocks V Integral Methods . . . . . . . . . . . . . . . . . . . . . VI . Validity of Self-similar Solutions . . . . . . . . . . . . . VII Further Discussion of Integral Methods . . . . . . . . . . VIII Concluding Remarks . . . . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . Addendum . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . . .

. . . . .

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. . . . .

. . . . .

. . . . .

. . . . . . . . . .

..... . . . . . . . . . .

2 4 8 26 36 43 41 49 51 52 311

The Mathematical Theory of Equllibrium Cracks In Brittle Fracture

.

BY G. I BARENBLATT. Institute of Geology and Development of Combustible Minerals of the U.S.S.R. Academy of Sciences. Moscow. U.S.S.R.

. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Development of the Equilibrium Crack Theory . . . . . . . . . . . The Structure of the Edge of an Equilibrium Crack in a Brittle Body . . Basic Hypotheses and General Statement of the Problem of Equilibrium

I I1 I11 IV

Cracks . . . . . . . . . . . . . . . . . . . . . . . V . Special Problems in the Theory of Equilibrium Cracks VI Wedging; Dynamic Problems in the Theory of Cracks . References . . . . . . . . . . . . . . . . . . . . . . . .

.

56 62 69

....... 76 . . . . . . . . 90 . . . . . . . . 114 ....... 125

Plasticity Under Non-Homogeneous Conditions

.

.

BY W OLSZAK.J RYCHLEWSKI A N D W . URBANOWSKI. Institute Of Fundamentat Technical Problems. Polish Academy of Sciences. Warsaw

. Physical Foundations . . . . . . . . . . . . . . . . . . . . . . . . . Plane Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particular Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . Elastic-plastic Non-homogeneous Plates . . . . . . . . . . . . . . .

I 11 I11 IV

ix

.

132 151 183 190

CONTENTS

X

V . Limit Analysis and Limit Design . . . . . . . . . . . . . . . . . VI Propagation of Elastic-plastic Waves in a Non-homogeneous Medium . V I I . Other Problems . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

191 201 203 206

Some Elementary Problems in Magneto-hydrudynrtmics

BY RAYMOND HIDE A N D PAULH . ROBERTS.Physics Department. King’s College (University of Durham) Newcastle.upon.Tyne. 1. England

.

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Basic Equations of Magneto-hydrodynamics . . . . . . . . . . . . . . 219 Electromagnetic and. Mechanical Effects ; Dimensionless Parameters . . . 224 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 233 Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 Alfvbn Waves in Systems of Finite Extent . . . . . . . . . . . . . . 261 Gravity Waves : Rayleigh-Taylor Instability . . . . . . . . . . . . . 267 Gravitational Instability: Jeans’ Criterion . . . . . . . . . . . . . . 270 IX . Steady Flow between Parallel Planes . . . . . . . . . . . . . . . . 274 X . Flow due t o an Oscillating- Plane: Rayleigh’s Problem . . . . . . . . . 286 XI . Steady Two-dimensional Inertial Flow in the Presence of a Magnetic Field . 300 Appendix A : The Hydromagnetic Energy Equation . . . . . . . . . . . . 305 Appendix B : Relativistic Magneto-Hydrodynamics . . . . . . . . . . . . . 311 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 AUTHORINDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 SUBJECTINDEX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

11. 111. I V. V. VI . VII . VIII .

Hypersonic Flow over Slender Bodies Associated with Power-Law Shocks

BY HAROLD MIRELS Lewis Research Center+ National Aeronautics and Space Administration Cleveland. Ohio Page I . Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

I1. Hypersonic Slender-Body Theory

I11. Flows Associated with Power-Law Shocks 1. Basic Equations . . . . . . . . . . . 2. Alternative Formulations . . . . . . . Stream-function formulation . . . . Lagrangian formulation . . . . . . Sedov formulation . . . . . . . . 3. Analytic Solutions . . . . . . . . . . Blast wave . . . . . . . . . . . . Newtonian theory . . . . . . . . . “Sharp-blow’’ solution . . . . . . . Approximate solutions . . . . . . . 4 Nature of the Flow . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . Power-Law Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

IV Flows Associated with Slightly Perturbed 1 . Basic Equations . . . . . . . . . . . 2. Applications . . . . . . . . . . . . . Boundary-layer effects . . . . . . . Angle-of-attack effects . . . . . . . Effect of blunting the nose of very slender wedges and cones 3. Effect of #0 . . . . . . . . . . . . . . . . . . . . . V. Integral Methods . . . . . . . . . . . . . . . . . . . 1. Continuity Integral . . . . . . . . . . . . . . . . General case . . . . . . . . . . . . . . . . . . Hypersonic slender body approximations . . . . . Slender blunt-nosed bodies a t infinite Mach number 2. Momentum Integral . . . . . . . . . . . . . . . . General case . . . . . . . . . . . . . . . . . . Hypersonic slender body approximation . . . . . .

. .

.

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . VI . Validity of Self-similar Solutions . . . . . . . . . . . . . . . . . 1. Infinite Mach Number . . . . . . . . . . . . . . . . . . . . . . 2 . Finite Mach Number . . . . . . . . . . . . . . . . . . . . . . .

*

2

. . . . . . . . . . . . . . . . . .

Present Address: Aerospace Corporation. El Segundo. California

1

. . . . . .

4 8 8 13 13 14 16 16 16 18 19 21 23 26 26 29 29 31 33 34 36 37 37 38 39 40 40 41 43 43 46

2

HAROLD MIRELS

. . . . . . . . . . . . . . . V I I I . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Addendum.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

VII. Further Discussion of Integral Methods

References.

49 51

317

I. INTRODUCTION The steady-state equations of motion for hypersonic flow over slender bodies can be reduced to simpler form by incorporating the “hypersonicslender-body approximations” (e.g., Hayes [l] and Van Dyke [?]). The reduced equations are valid provided S 2 < 1 and ( M 6 ) - 2 is not near one, where M is the free.stream Mach number and S is a characteristic shock slope. If the streamwise coordinate is considered as time, these reduced equations are identical with the full (exact) equations for a corresponding unsteady flow in one space variable less. Forebody drag on a hypersonic slender body is equivalent to the net energy perturbation (from the undisturbed state) in the corresponding unsteady flow. Taylor [3, 41 has treated the unsteady constant-energy flow field behind the spherical “blast” wave which is generated when a finite amount of energy is released instantaneously at a point. The analysis assumes a very strong wave and is valid (for a perfect gas) until the decay of shock strength is sufficient to violate the strong shock assumptions. The problem of planar, cylindrical, and spherical blast waves was treated in a unified manner by Sakurai [5,6] and the flow-field modifications associated with more moderate shock strengths were found by a perturbation analysis. The solution for the cylindrical blast wave was obtained, independently, by Lin [7]. References [7] to [O] have pointed out that, within the framework of hypersonic slenderbody theory (in the limit (M6)-2--+0),the hypersonic flow over a bluntnosed flat plate, or circular cylinder, may be considered as the steady-state analog of the constant-energy planar, or cylindrical, blast-wave problem, respectively. The nose drag in the steady problem is equivalent to the finite energy which is instantaneously released in the blast-wave problem. The steady-flow solution is not correct near the nose (where the hypersonic slender-body approximation S2 << 1 is violated) and far downstream of the nose (where the strong-shock assumption, ( M 6 ) - 2-* 0, is violated). However, useful results are expected for the intermediate region. The blast-wave problems all exhibit flow similarity. That is, the flow fields at different times are similar, except for a scale factor on both the dependent and independent variables. Lees and Kubota [9,10] have observed that such similarity exists for hypersonic flows whenever the shock shape follows a power-law variation with the streamwise distance, provided the

HYPERSONIC FLOW OVER SLENDER BODIES

3

hypersonic slender-body equations are considered in the limit as (Mb)-2-p 0. This led to numerical solutions of the hypersonic flow over slender powerlaw bodes. The effect of nonvanishing values of (MS)-2 was also found by Kubota [lo] using a numerical perturbation analysis similar to that of Sakurai [5, 61. As noted by Hayes and Probstein [ll], all these authors were apparently unaware of the extensive early work of various Russian authors. Unsteady self-similar motions were studied in great generality by Sedov [12], Stanyukovich [ 131, and others. The application to hypersonic flows was developed by Grodzovskii [14], Chernyi [15, 161, and co-workers. A historical account of these developments is given in [ll]. More recently, Mirels [17, 181 has obtained additional numerical results and approximate analytical solutions for hypersonic flow over slender power-law bodies. In [18], the flow fields and bodies associated with slightly perturbed power-law shocks were investigated. Studies of the singular points of the equations of motion have been presented in [12, 19-21]. These indicate the conditions under which streamlines through a power-law shock terminate a t a limit line. Integral methods for finding the hypersonic flow past arbitrary blunt nosed slender bodies have been developed by Chernyi [22] and Cheng et al. [23]. The validity of the use of hypersonic slender body theory for slender bodies with blunted noses has been discussed in papers by Cheng [24] and Sychev 1251. The ionization of the flow field associated with hypersonic blunt-nosed bodies has also been investigated by Feldman 126, 271 in connection with communication with and detection of bodies entering the earth’s atmosphere. The purpose of the present paper is to give a unified discussion of the application of hypersonic slender-body theory, in the limit (M8)-2 << 1, for finding the self-similar flows associated with slender power-law shocks and related bodies. (“Related bodies” refers to any body associated with a power-law shock or a slightly perturbed power-law shock.) Integral methods for finding the non-self-similar hypersonic flow over more general body shapes are also discussed. The specific topics considered are as follows. Chapter I1 outlines hypersonic slender-body theory which is valid for 6 8 << 1 and (MS)-2not near one. The resulting equations are applied in Chapter 111, in the limit ( M S ) - 2 - 0,to find the self-similar flows associated with powerlaw shocks. Several alternative formulations of the equations of motion are noted, some analytical solutions are given and the general features of the flow field are discussed. In Chapter IV, the effects of slightly perturbing a power-law shock are considered. Numerical results are presented which permit the determination of the effects of thin boundary layers, small angles of attack, nose blunting and small but nonvanishing values of (MS)-2. Chapter V develops continuity and momentum-integral methods for finding the hypersonic flow over arbitrarily shaped slender bodies. The continuity

4

HAROLD MIRELS

integral is applied in Chapter VI to examine the validity of the self-similar solutions of Chapter 111. In particular, the violation of the hypersonic slender body assumption of small shock slope d2 << 1 at the nose of a powerlaw shock is investigated. Criteria are developed which define the downstream influence of nose bluntness. Chapter VII simplifies the integral methods for the case where y , the ratio of specific heats, is near 1 and most of the drag occurs a t the nose of the body. In Chapter 1’111, the validity of the blast wave solutions of Chapter I11 for describing the flow over bluntnosed cylinders is examined in more detail by comparison with exact solutions obtained by the method of characteristics (e.g. [26, 281). The importance of nose shape, which is neglected in blast wave theory, is demonstrated. The present discussion is an elaboration and extension of [as]. Nondimensional variables are used in Chapters 11 to IV. Here dimensional variables have a superscript bar. However, in Chapters V to VIII, dimensional variables are used exclusively, and the superscript bar is omitted. The notation is summarized after Chapter VIII.

11. HYPERSONIC SLENDER-BODY THEORY The equations of motion for hypersonic flow over slender bodies (e.g. [a]) are summarized herein. Two dimensional and axisymmetric flows are considered with (Z,?)and (G,d) being the streamwise and transverse coordinates and velocities, respectively. The superscript bar indicates a dimensional quantity. The equations of motion for an inviscid, non-conducting fluid are : continuity:

apil

FIG.1. Flow across oblique shock.

--

(2.la) ati

(2.lb)

x-momentum: 6 -

(d.lc)

r-momentum :

(2.ld)

az

energy :

a0

G-

ax

(c a

+ d- aa + 1 apaa = 0 a?

ag 1 ap +da? + ar = 0, Ty

+ c P)(k) = 0, ar

py

--

a p +7 + o re = 0,

HYPERSONIC FLOW OVER SLENDER BODIES

5

where a = 0 , 1 for two dimensional and axisymmetric flows, respectively. Equation (2.ld) assumes a perfect gas and is equivalent to stating that entropy is constant along a streamline (except for possible jumps across shock waves). Boundary conditions across an oblique shock (Fig. 1) may be expressed as

(2.2b) (2.2c) (2.2d) where subscripts 00 and s represent conditions upstream and downstream of the shock, respectively, M is the free-stream Mach number, and t is the angle which the shock forms with the free-stream direction. No assumptions regarding 6 and M are made in (2.1)and (2.2). Attention is now restricted to flows for which d2 << 1 (where 6 is a characteristic shock slope) and M 2 >> 1. The prod-L r,v uct 6M can be interpreted as the ratio of shock slope to Mach angle, 1/M, and is always greater than 1. Consider (Md) not very close to one (i.e., exGPD clude bodies which are sIender 0 L relatil’e t o the Mach angle, FIG.3. Physical quantities for study of hyperMdb << 1, and thus can be sonic flow over slender bodies. treated by linearized supersonic flow techniques). Eqs. (2.2) then suggest the following non-dimensional dependent variables for studying hypersonic flow over slender bodies :

p

(2.3a)

=

fi/prnZimw, p =p / p m ,

u = (a - G m ) / d m P , 21 = 77/27,6.

These quantities remain of order 1 as a2+ 0 and M + 03 provided ( M 6 ) 2 is not close to one. The independent variables may be non-dimensionalized according to x = %/L,

(2.3b) where

L

r

=qs1,

is a characteristic streamwise length (Fig. 2).

6

HAROLD MIRELS

The equations of motion can now be written, after substituting (2.3) into (2.1) and neglecting terms of order S2 compared with one, (2.4a)

continuity:

(2.4b)

r-momentum:

(2.4~)

energy:

aPv

aP

Z+F+ av -+ ax

=o,

a!?

r

1 ap --=O, p ar

av

v-+

a7

($+ v "(2) ar 0. py

=

Equations (2.4) can be solved independently of the x-momentum equation which is therefore omitted. After solving (2.4) the value of u can be found from Bernoulli's equation, which to the present order, becomes

If rb(x) and R ( x ) denote the non-dimensional body and shock ordinates, respectively, the boundary conditions associated with (2.4) are : (2.6a)

a t body surface:

vb

= rb',

(2.6b)

upstream of shock: urn= vm = 0,

p m = 1,

p m = l/yM2d2,

on the downstream side of shock: (2.6~) (2.6d)

ps =

(2.6e)

21,

[

2

7 + 1+ YY-

=-

2

Y + l

I'

(M6R')-2

,

R'[1 - ( M S R ' ) - 2 ] .

Equations (2.3) t o (2.6) constitute the hypersonic small disturbance equations as originally described by Van Dyke [2]. Hayes [ 13 has shown that the hypersonic small-disturbance equations correspond to the full equations (no restrictions regarding 6 or M) for unsteady flow in the transverse plane. This follows from the fact that the streamwise velocity ti, remains unchanged, to order d2, in hypersonic slender-body theory. The flow in the plane of an observer moving with then appears as an unsteady transverse flow. Formally, this velocity iim corresponds to replacing ti by ti, and 2 by zZ,l in (2.la), (2.lc), and (2.ld)

HYPERSONIC FLOW OVER SLENDER BODIES

7

or to replacing x by t in the non-dimensional equations. The quantity 6M can be eliminated from ( 2 . 6 ~to ) (2.6e) by noting a, G 1/M6. The quantity R'la, is then the shock Mach number in the unsteady case. Equivalent steady and unsteady non-dimensionalized flows are illustrated in Figs. 3 and 4. The unsteady flow in Fig. 4 is due to the expansion of a plane (a = 0) or cylindrical (u = 1) piston bounded by two parallel walls.

=o

v,=u,

'q p,=

I

FIG.3. Nondimensional hypersonic flow over slender body.

FIG.4. Unsteady (piston driven) flow which is equivalent t o steady flow of Fig. 3. (a) Flow at t = 1,. (b) Piston and shock displacement versus time.

The forebody drag on a slender body can be determined by a pressure integration along the body surface or by considering the momentum flux across an appropriate control surface. These give, respectively (employing the hypersonic slender-body approximations) x

=

where D ( x ) D ( f ) / p , ~ 7 ~ ~ 6 ~ ( L is 6 )the " + ~forebody drag up to station x , D(0)is the finite drag addition (if any) a t x = 0 and p b is the surface pressure. For u = 0, (2.7) gives the drag of a symmetric two-dimensional body (rather than half this value). The integration in (2.7b) is conducted at constant x . Forebody drag in a steady hypersonic flow corresponds to energy addition in the equivalent unsteady flow. Thus, the integral in (2.7a) may be interpreted as the energy addition by a piston (note, t'b = d r b / d x ) and (2.7b) may be interpreted as the net energy perturbation up till time t = x for unsteady flows of the type indicated in Fig. 4. The derivation of (2.7) is discussed in more detail in Section V.2.

8

HAROLD MIRELS

A single solution of the non-dimensional equations of motion (2.4)-(2.6) for a given value of M S and of y and a given body shape corresponds to an infinite number of physical flows. The physical flows are those having the same values of MS and y and body shapes which are reduced by (2.3b) to the non-dimensional body whose solution is known. This law of similitude has been applied to the case of real gas flows (in particular, flows in local thermodynamic equilibrium) by Cheng [24]. In [24], Cheng shows that similitude exists (in terms of the non-dimensional quantities defined by (2.3)) for flows in local thermal equilibrium provided the free-stream atmosphere (composition, pressure and density) is the same. The latter requirement is replaced by the condition that y be constant in the case of ideal gas flow. Additional discussion of the law of similitude for hypersonic flow over slender bodies is given in [ll]. Except for the discussion of Chapter VIII, the present paper is restricted to ideal gas flows with constant y .

111. FLOWS ASSOCIATEDWITH POWER-LAW SHOCKS The hypersonic slender body equations of Chapter I1 are now used to investigate the hypersonic flow field associated with slender power-law shocks. The limiting case ( M c ? ) - ~ -0 (i.e., M - a)is considered. It is shown that the motion is self-similar. Several formulations of the equations of motion are given, some exact and approximate solutions are indicated, and the general nature of the flow field is discussed. 1 . Basic Equations

In physical coordinates, the equation for a power-law shock is (3.1)

R ( f )= C f " ,

where C and m are constants. To nondimensionalize, let 1 be the streamwise length of interest and R ( L ) be the corresponding shock ordinate. The characteristic shock slope is then (Fig. 2) (3.2)

L

Note that 6 and L are not independent and that the characteristic geometric bL(l-"). Thus S and L always appear in the parameters are m and C combination SL('-") in the treatment of power-law shocks. I t is assumed that S2 << 1 so that hypersonic slender body theory is applicable.*

* The assumption P<< 1 or (M6)-2 + 0 is violated at the nose of power law bodies when vn # 1. The effect of this violation is discussed in Chapter VI.

HYPERSONIC FLOW OVER SLENDER BODIES

9

In nondimensional variables the shock shape is

R

(3.3)

= xm.

New independent variables are introduced, namely (3.4)

f

7 =rlxm,

= x,

so that

-.a

(3.5)

av

The variable y is the ratio of Y to the local shock ordinate and equals 1 a t the shock. Equations ( 2 . 6 ~ to ) (2.6e), with ( M 6 ) - 2 = 0, suggest that the dependent variables be expressed as (3.6)

p

= ,2re(m

- 1) F,

p=$,

v=rnfm-ly,

where F, I), and y are functions of 7 . Substitution into (2.4) then yields (3.7a)

(a, -

(3.7b)

(3.7c)

(9-y)

;(

--y-

);

-(a+l)B=O.

The boundary conditions a t the shock, y = 1, (from (2.6) and (3.6)) are ~ ( 1=F(l) ) =

The parameter (3.9)

p

p

2 ~

y+l’

*(1) = y+l. Y-1

is here a function of m, namely - 2 [RR”/(R’)2] = ___ ( 1 - 1). a+l m a+l

E-

Equations (3.7) and (3.8) completely define the flow associated with a power-law shock. Any streamline through the shock may be considered as defining a body, and the flow between this streamline and the shock can be found from a numerical integration of (3.7) and (3.8). Unless otherwise stated, the body will be taken to be that corresponding to the streamline through the origin (Fig. 5 ) . This body shape, for E > 0, is given by y = r / b or (3.10)

Yb

= qbEm,

10

HAROLD MIRELS

where

'qb

is the value of q at which

fP=v (3.11) (so that (2.6a) is satisfied). Comparing (3.3) and (3.10) shows that the body is similar to the shock and the ratio of their ordinates is q b . Both follow a power-law variation with 6,

FIG.5. Power-law shock with body corresponding to streamline through origin (0< p < 1).

The pressure distribution on the body can be found from (3.0) and (3.3) as (3.12)

p b = (dR/dE)'Fb = (drb/dt)'Fbl'qb',

where Fb = F ( q b ) . The drag up to station and (2.7b), respectively) as

6 may be expressed

(from (2.7a)

where 1

(3.13~) The quantity Ib = I(qb)is a function of IJ, y and p. Equations (3.13a) and (3.13b) give the drag on the body defined by the streamline through the origin. The limitation on in (3.13b) is due to the appearance of limit lines for u = 1 and p > 1 as discussed in Section 111.4. For u = 1, (3.13b) applies for a limited range of p greater than one (i.e. 1 < p PSB) as can be deduced from the discussion in 111.4. The drag on the upper surface of a streamline, up to the point (E,q), is given by

-=

(3.13d)

TABLE1. QUANTITIES DEFINING BODYSHAPEAND SURFACE PRESSURE ASSOCIATEDWITH

Y

P

POWER-LAW SHOCK (ZERO ORDERPROBLEM)

Approximate solution (Eqs. (3.41))

Numerical solution (Eqs. (3.7))

a=o

A

u=1

Approximate solution (Eqs. (5.16))

a = l

a=O

a=O

a = l _

rlb

%'b

Fb

0.947 0.774 0.682 0.608 0.531 0.453 0.410

0.930 0.886 0.843 0.789 0.698 0.514 0

0.964 0.941 0.918 0.888 0.835 0.717 0

0.930 0.764 0.680 0.618 0.555 0.493 0.430

~

a=O,l _

qb

F b

rlb

F b

co

Do

0 1/3 1/2 5/8 3/4 7/8 L

0.930 0.891 0.852 0.803 0.716 0.535 O

0.930 0.761 0.675 0.611 0.546 0.481 0.41.5

0.965 0.945 0.924 0.898 0.845 0.735 0

0.948 0.775 0.688 0.621 0.553 0.484 0.411

0 2.93 5.18 7.16 8.52 6.16 0.465

1.255 1.535 1.800 2.13 2.68 3.80 7.67

0.930 0.930 0.891 0.766 0.852 0.672 0.801 0.598 0.710 0.520 0.513 0.442 0 0.412

-2.53 2.05 0.965 +1.636 1.317 0.945 3.64 1.642 0.924 5.53 1.981 0.897 7.11 2.53 0.846 0.724 5.68 3.62 0.465 7.13 0

1.4

0 1/3 1/2 5/8 3/4 7/8 1

0.833 0.833 0.915 0.759 0.666 0.875 0.695 0.581 0.839 0.623 0.518 0.796 0.513 0.454 0.725 0.333 0.390 0,589 0 0.325 0

0.875 0 0.696 0.962 0.607 1.240 0.538 1.296 0.467 1.167 0.392 0.817 0.311 0.417

1.556 1.678 1.779 1.888 2.04 2.30 3.50

0.833 0.833 0.760 0.679 0.695 0.584 0.619 0.504 0.499 0.415 0.284 0.320 0 0.316

-1.019 $0.383 0.757 0.930 0.949 0.757 0.417

2.11 1.228 1.485 1.631 1.798 2.03 2.92

0.915 0.872 0.875 0.704 0.839 0.611 0.795 0.529 0.719 0.438 0.561 0.337 0 0.302

0.833 0.755 0.688 0.613 0.501 0.322 0

0.913 0.869 0.829 0.783 0.708 0.567 0

0.833 0.667 0.583 0.521 0.458 0.396 0.333

1.67

0 1/3 1/2 5/8 314

0.749 0.749 0.870 0.658 0.587 0.819 0.585 0.507 0.776 0.509 0.446 0.727 0.404 0.386 0.652 0.248 0.326 0.518 0 0.264 0

0.811 0 0.634 0.535 0.544 0.644 0.474 0.655 0.403 0.608 0.326 0.504 0.241 0.375

1.788 0.749 0.749 1.788 0.660 0.605 1.770 0.586 0.512 1.762 0.505 0.432 1.752 0.385 0.340 1.738 0.186 0.239 2.49 0 0.250

-0.625 +0.170 0.365 0.452 0.483 0.459 0.375

2.16 1.161 1.405 1.450 1.457 1.436 1.867

0.870 0.819 0.776 0.726 0.644 0.480 0

0.749 0.660 0.588 0.513 0.408 0.252 0

0.865 0.812 0.767 0.717 0.639 0.502 0

0.749 0.582 0.499 0.437 0.314 0.312 0.249

1.15

7/8 1

Reference

~ 7 1

POI

%'b

~1

F b

co

Da

Tlb

~ 7 1

Fb

0.805 0.645 0.554 0.469 0.370 0.257 0.255

_

m X

bz I

0

12

HAROLD MIRELS

Equation (3.13d) gives the drag of a symmetrical profile when a = 0. The integral I ( q ) is a measure of the energy of the transverse flow between (6,~) and the shock. Equations (3.13a) and (3.13b) show that I b and Fb are related by (3.14) Equations (3.7) were integrated numerically for various values of P and y by Kubota [lo] and Mirels [17], for G = 1,0, respectively. The numerical results for ?]b and F b are given in Table 1. These define the body shape and surface-pressure distribution corresponding to a given power law shock. Plots showing the variation of pl, F and #, with q, for u = 1, y = 1.4 and various values of ,f?, are also presented in [lo] and 1171. TABLE2. EVALUATION OF I b

Y

(0.

FOR

+

BLASTWAVE, p = 1 - 1)Ib

Numerical integration (Eqs. (3.7, 3.13))

1.0 1.15 1.2 1.a 1.4 1.667 1.67

o=o

o = l

1.0 1.088 1.109 1.140 1.164 1.213 1.206

1.0 1.098 1.134 1.170 1.203 1.249 1.278

Approximate solution (Eq. ( 5 . 1 5 ~ ) )

Ref.

o = 0.1

-

1.0 1.064 1.082 1.111 1.133 1.167 1.167

~ 7 1 [51 [51 15, 171 [51 ~ 7 1

When P = 0, the body is a wedge (u = 0) or cone (a = 1). Values of P in the range O < ,4l < 1 correspond to power-law bodies with infinite slope at [ = 0. For p = 1, it is found that qb = 0 and all the drag is added a t 6 = 0 (3.13b). This is the "blast wave" or "constant energy" case which can be used to estimate the shock shape and pressure distribution associated with a flat plate of semi thickness TN or a blunt-nosed circular cylinder of radius TN. If D(0)is the drag addition a t 6 = 0, the shock shape corresponding to P = 1 is (from (3.13b) and noting C = Sl('-")) (3.15a)

_ -_ '?N

2

13

HYPERSONIC FLOW OVER SLENDER BODIES

where CD, is the nose drag coefficient

(3.15b)

CD, 3

D(O)/[7fpm6m2v,~+1/20].

Numerical values of I b for use in (3.15a)are given in Table 2. The surface pressure distribution is found from (3.12). An exact analytical solution for the 3/ = 1 case was found by Sedov [la]and, independently, by Latter [30]. The Sedov solution is discussed in Section 111.3. The nature of the flow field, particularly for /I> 1, will be discussed further in Section 111.4. When no confusion results, the variables 6 and x will be used interchangeably. 2. Alternative Formulations The differential equations describing hypersonic flow through power-law shocks (3.7)and (3.8),are now reformulated from other viewpoints. The advantages of these new approaches are noted.

Stream-function formulation. The continuity equation (2.44 is satisfied by a stream function $ such that

(3.16) Considering a power-law shock, a suitable form for

$ is

(3.17) where 0 is a function of 7 and e(1) = 1. The quantity p/p' is constant along a streamline, downstream of the shock, and may be written as PlpY = w ,

(3.18) where w is a function of

6.

Considering conditions at 7

=1

yields

(3.19) Eliminating

(3.20)

6 in (3.19)gives

o as a function of

$,

and (3.18)becomes

14

HAROLD MIRELS

Equations (3.16), (3.17), and (3.20) permit p , v , and functions of 6 and 8. Thus

p

to be expressed as

(3.21a)

(3.21b)

(3.21c) and the momentum equation (2.4b) becomes

with the boundary conditions (3.22b)

e(1) = 1,

eyl) = (a

+ l ) ( y+ l)/(v - 1).

The streamline through the origin corresponds to 8 = 0. The streamlines downstream of the shock are defined, parametrically (with q as parameter), by (3.23) where (xi,Ri) are the coordinates of a streamline a t its intersection with the shock (Fig. 5 ) . If x / x , , r / R j and I(q) are tabulated as functions of q, in the course of integrating (3.22), each streamline and its drag are defined. Equations (3.21) and (3.22) are somewhat simpler than (3.7) and (3.8) in that the problem has been reduced to finding a single dependent variable 0 in terms of q. Equations (3.21) and (3.22) were used by Mirels [17] as a guide to find approximate analytical solutions. These are discussed in Section 111.3.

Lagrangian formulation. In the present section 0 and 6 are the independent variables. Equation (3.23) shows (3.24)

e =(R~/R)#+~

so that 0 can be considered as a Lagrangian variable. Note that for [ fixed, the local value of 0 is the ratio of the mass flow between the local streamline

HYPERSONIC FLOW OVER SLENDER BODIES

15

and the streamline 8 = 0 to the mass flow between the shock and the streamline 8 = 0. The present formulation may also be considered in effect to be a Mises transformation in that a stream function is used as an independent variable. The dependent variable considered herein is (3.25)

f

&!LeF 2

so that f = 1 at 0 = 1. If is fixed, f is the ratio of the local pressure to the value directly downstream of the shock. Equations (3.21c), (3.21b), and (3.2la) show, respectively,

(3.26a)

(3.26b)

(3.26~) Noting from (3.21~)that (3.27a) it follows that (3.27b) Equation (3.22a) can now be written

16

HAROLD MIRELS

Substitution of (3.26a) into (3.28a) reduces it to an integro-differential are found from equation for f in terms of 0. The quantities y, !~,t, and from (3.26). The main advantage of (3.28) is that the role of y is explicitly indicated. In particular, the nature of the flow for y near 1 can be readily deduced. The variation of f with 0 is nearly linear for < 1 and y near 1 so that a Taylor-series expansion of f about 0 = 1, which can be obtained directly from (3.28), represents an accurate solution. Reference [21] discusses the variation of f with B in great detail, for 0 = 0, and presents the Taylor expansion for f up to and including terms of order (0 - l)4. Sedov formulation.

Sedov [12] introduced the variables V E rnqlq,

z = ym2F/q2t,h and reduced the equations of motion to the form

V - m dz (3.29a) ~2 dV - [2(V - 1)

+ a(y - l ) V ] (V - m)2+ ( y - 1)(1- m)V(V - m ) - [2(V - 1)+ ~ V(V - 1)(V - m ) - [(l+ a ) V -

( -y 1))

K]Z

(3.29b) (3.2%)

dlny z - (V - m)2 V ( V - 1 ) ( V - m ) - [(l a ) V - 4 2 ’ dV

+ - (a + 1)V dl n (V - m)gL d In y

V-m

+

where = 2(1 - m)/y. The shock location corresponds to V = 2m/(y 1) and z 2y(y - l ) m 2 / ( y 1)2.The streamline through the origin is defined m for ,8< 1 and V = m l y for ,8 = 1. by V Equations (3.29) are particularly useful for studying the singularities and the general properties of the self-similar flows associated with-power law shocks. For details, the reader is referred to Sedov [12]. General properties of self-similar flows are discussed in 111.4.

+

3. Analytic Solutions

Blast wave. Sedov obtained an analytic solution of (3.29a) for the case of a blast wave (/I = 1 or m = 2/(a 3)). His solution is

+

(3.30)

z = ( y - 1)V2(m - V)/2(V - m/y).

Equation (3.30) was deduced by noting that the energy between a similarity line (i.e., q = constant) and the shock is constant when fi = 1 (see (3.13d)). Thus the transfer of energy across a similarity line is zero. This can be

17

HYPERSONIC FLOW OVER SLENDER BODIES

+

+

expressed in the form [ F / ( y- 1) ($p2/2)](p- q) Fp, = 0, which leads to (3.30). Following.the notation of [ l l ] , the blast wave solution may be expressed as

(3.31~)

0 =m [ y

+ 1 + u(y - l)],

@ =m

2y - 0

The solution is tabulated, for various y , in [12]. Analytic solutions to the blast wave problem were obtained independently by Latter [30] for (T = 2 and Kubota [lo] who generalized Latter's solution for u = 0,1, 2. Equations (3.31) can be considerably simplified for small values of E (y - l)/(y 1). First, note that F , varies between 0 and 1. Neglecting terms of order E compared with 1 it can then be shown that F , = F, = 1, F, = 2 - F , and that equations (3.31) become

+

(3.32a)

F3 em---2 - F3

[ ye]

2e( o+l)

(3.32b) (3.32~) (3.32d)

q~m q M F3W(u+1)

~

p/(a+l),

18

HAROLD MIRELS

A similar expansion is given by Freeman [31] and Brocher [32]. Equation (3.32b) shows that q essentially equals 1 for all values of 0 except 0 very nearly zero so that most of the disturbed mass is located at the shock wave. The pressure on the body (0 = 0) is half of that directly behind the shock.

Newtonian theory. Hypersonic flow in the limit of large M and small E is termed Newtonian flow (cf. e.g., [ l l ] ) . The Newtonian approximation for hypersonic flow associated with slender power-law shocks is now discussed by considering the solution of (3.26) and (3.28) for E = 0 and for E + 0. When E = 0, (3.26) and (3.28) show (3.33)

f

=1 - p(i

- ey2,

= 7 = 1,

*/*(i)= eq.

These equations are valid for all p and constitute an exact solution. Since q = 1 for all 8, the disturbed flow has zero thickness and the shock wave coincides with the body (or effective body) surface (i.e., q b = 1). There is a pressure variation across this infinitesimal layer due to centrifugal force. Now consider E - + 0 and p < 1. It can be shown that q = 1 O(E)and d / / d e = (/3/2)[1 O ( E ) ]so that (3.26) gives

+

+

(3.34a)

(3.34c) (3.34d)

The parameter q b is found by setting 0 = 0 in (3.34b). Equations (3.34) converge, uniformly over the flow field, to (3.33) as a limit when E - 0 . Equations (3.34) also represent the solution for ,8 = 1, in the limit E -+ 0, as may be seen by comparison with the exact solution (3.32). In particular, integration of (3.34b) yields (3.35) in agreement with (3.3213). Comparison of (3.35) and (3.33)shows that, for = 1, the solution for E + 0 does not converge uniformly over the flow field

p

HYPERSONIC FLOW OVER SLENDER BODIES

19

to the solution for E = 0. While both solutions indicate that most of the mass is concentrated at the shock wave, (3.35) gives l;lb = 0, while (3.33) gives l;lb = 1. This nonuniform convergence was discussed in [31], by a similar comparison of blast-wave and Newtonian theory. The nonuniform convergence, at 8 = 0, also occurs for fi > 1 as may be deduced from the discussion in Section 111.4. For small E and into (3.13~))

p < 1, (3.32) and (3.34) give (after substituting

(3.36) Interpreting I b as a measure of the sum of the net internal and kinetic energy perturbation a t a fixed x (i.e., unsteady viewpoint), it is found that the ratio of internal to kinetic energy is y / ( y - p), to the present order. The ratio is 1 for p = 0 and approaches infinity for fi = 1 and E + 0. “Sharp-blow” solutiovt. A closed-form solution of (3.26) and (3.28) can be obtained for the case a = 0, y = 715, ,8 = 413 (i.e., m = 3/5). This solution was obtained in [33-351 (as noted in [21]). It is an example of a flow which Russian authors refer to (according to the translation of [21]) as the “sharp-blow” problem. The solution is

(3.37a)

The streamline through the shock point (xi,Ri) is (from (3.23) and (3.37a)) (3.37b) where R, = x : ‘ ~ . This streamline is indicated in Fig. 6. For x/xi large, ) . streamline the streamline approaches the straight line r = - X / ( ~ X ; / ~The through the origin coincides with the negative Y-axis. The pressure, along each streamline, decreases as x increases and approaches zero as x + 00. The drag on the upper surface of a streamline, D(x)/2, is (from (3.13d) and (3.37))

20

HAROLD MIRELS

-

(3.38b)

3 1 32 xi1/‘

for

x+w.

The drag is finite except for the streamline going through xi

= 0.

8-

3FIG.6. Streamline through (xi, Ri) for “sharp blow” problem (Eq. 3.37b).

The flow is illustrated in Fig. 7(a). An infinite drag is imposed a t (0,O) to flow in the upper half plane (Y 0) and a lateral expansion (into a vacuum occupying Y < 0) is permitted for x > 0. I t is qualitatively similar to

>

FIG. 7. Flow associated with “sharp blow” problem (u = 0). (a) Steady flow described by (3.37). (b) Qualitatively similar steady flow. ( c ) Equivalent unsteady flow.

unseparated flow about the edge of a plate which is normal to a hypersonic stream (Fig. 7(b)). The corresponding unsteady flow is illustrated in Fig. 7(c) and was referred to as the “sharp-blow’’ problem in [21]. This unsteady flow may be described as follows (for u = 0 and all 7). Consider a massless piston a t Y = 0 to separate a semi-infinite column of gas (r > 0) from a semi-infinite

21

HYPERSONIC FLOW OVER SLENDER BODIES

vacuum (r < 0). The piston is unrestrained, a t time t = 0, and an infinite amount of energy is added to the gas column at the piston face. A shock is generated, which propagates into the gas and the piston is accelerated, instantaneously, to r = - 00. The resulting shock-shape parameter, designated PsB, varies from 2 for y = 1, to 1.117 for y- w . Corresponding values of PsB and y , from [21] and [34], are given in Table 3. TABLE3. SHOCK-SHAPE PARAMETER FOR SHARP-BLOW PROBLEM ~~

Y

1

1.1

1.4

5/3

2.8

00

PSB

0.5 2

Ref.

-

0.56888 1.5157 [34]

0.6 4/3 [33, 34, 351

0.61073 1.2748 [34]

0.626704 1.1913 [341

0.6416 1.117 [211

mSB

From physical considerations, it is expected that the sharp-blow flow has no net momentum in the r-direction. This can be readily demonstrated. Equating surface pressure to rate of change of v-momentum gives, in general (for u = 0) 1

(3.39)

For the sharp-blow problem, F6 is zero, and therefore the integral (a measure of the net v-momentum at any section) must be zero. Substitution of (3.37) into the integral verifies this property for y = 7/5. Thus, there must be no net v-momentum downstream of x if the flow in Fig. 7(b) is to be qualitatively similar to the sharp-blow problem. The “free layer” flow discussed in [ l l ] (for u = 0 , l ; y = 1 ; M = 00, = unrestricted) approaches, with increasing x, a power-law shock with p = 2 and is thus a special case (for u = 0) of the sharp-blow problem. (However, for the u = 0 free layer flow, there is finite v-momentum addition

6

at x = 0 and the v-momentum remains constant for x

> 0.)

Approximate solutions. An approximate analytical solution of the blast wave problem ,8 = 1 was obtained by Taylor [3] for u = 3 and by Sakurai [5] for u = 1, 2. These have been generalized by Mirels [17] for all /3 1 using the following approach.

<

The asymptotic form for 8 , near 8

= 0,

is (from (3.22a))

22

HAROLD MIRELS

where

This suggests the following approximate expression for 8

where C,, Do,and qb are, as yet, unknown constants. The corresponding values of F , # and 9 are (from (3.21)) (3.41b)

Equation (3.41a) satisfies the boundary conditions 8 = 0 a t q = q b and 8 = 1 at q = 1 and has the proper form near 8 = 0. The constants C,, Do and qb are determined by satisfying the boundary conditions on O', 8" and 8"' a t q = 1 as found from (3.22). For p = 1, it is known that qb = 0 so that C, and Docan be found by satisfying only 8' and 8" a t q = 1. The latter procedure gives, for P = 1,

which agrees with [ 5 ] . Numerical results for C,, Do, r j b and Fb are presented in Table 1. Comparison of these values of qb and Fb with those obtained from a numerical integration of the equations of motion shows that the approximate analytical solution is accurate when the shock is relatively close to the body (P near 0

HYPERSONIC FLOW OVER SLENDER BODIES

23

and y near 1). Sufficient results are presented to evaluate the accuracy of the approximate method for various values of 3f, and y . The constants C,, Do and q b presented in Table 1 can be used in (3.41) to obtain the variation of 8, F , t,h, and q~ with q. 4. Nature of the Flow

The physical nature of flow field associated with a power-law shock is now considered in more detail. First flows with a = 0, y = 1.4, and

FIG. 8. Schematic representation of flow fields and surface pressures for power law shocks (a = 0, y = 1.4). (a) Flow fields.

various P-values are discussed. These are generalized to a = 0 , y > 1. Finally, flows with a = 1 are considered. The discussion is based primarily on the results of Adamski and Popov [21]. The flow fields for a = 0, y = 1.4 are indicated in Fig. 8(a). The corresponding pressure variation, as a function of 8, is given in Fig! 8(b)

24

HAROLD MIRELS

and represents the integration of (3.28a). Referring to Fig. 8, case (1) has convex shock and body shape, with zero slope a t x = 0; case (2) corresponds to a wedge; and case (3) is a slender blunt-nose power-law body. Case (4) is the blast-wave problem. Finite drag is added a t (0,O). Note, Y b = 0. Case (5) represents a value of p between the blast-wave case, p = 1, and the sharp-blow case, pss = 413. Here q b is a finite negative number. The streamlines of the disturbed flow are concave

t'

1

'\

.2

-4

OSAWLE POINT

OCRlTlCAL POINT

.6

I

I

.8

1.0

1

b

FIG. 8 concluded. Schematic representation of flow fields and surface pressures for power law shocks (a = 0, y = 1.4). (b) Surface pressure.

downward near the shock and concave upward near the body. Thus, at fixed x , the pressure first decreases and then increases in going from R to rb (Fig. 8(b)). When p = pSs = 413, two flows are possible as noted in [21]. These are designated cases (6) and (7) in Fig. 8. The two possibilities arise because (3.28) has a singular point (in particular, a saddle point) at O S p = (1/7)3/2. The corresponding value of q is qsp = - 112. In both cases the solution for O s p 6 1 is f = 6 as indicated in Fig. 8(b). However, when continuing the solution for 0 6 O S p , two paths are possible. For one, case (6), f increases with decreasing 6 to the value f = 0.1101 at 0 = 0 [21]. The other possibility is case (7), for which f decreases with decreasing 8 according to f = 0. Case (7) then corresponds to the sharp-blow problem. The locus of the saddle point, in the physical plane (Fig. €!(a)) is indicated by the dashed curve r S p = qspxm. Cases (8) and (9) correspond to p slightly greater and much greater than Pse, respec-

< <

< <

25

HYPERSONIC F L O W OVER SLENDER BODIES

tively. For p > pss, the solution for f (starting at 0 = 1) first decreases with decreasing 8 but then reaches a critical point (dfld8 = co) beyond which f becomes a multivalued fundtion of 8 (Fig. 8(b)). This multivaluedness is not physically admissible, and the solution starting at 8 = 1, f = 1 cannot be continued past the critical point, as is indicated by the dashed curves in Fig. 8(b). If the value of q at the critical point is designated by qc, the curve rC = qcxmis a limit line in the physical flow. The solution for flow through a power-law shock cannot be continued past this line. A physical interpretation of the limit line can be readily given. I t is a line of constant q which is also a Mach line. This is shown as follows. The slope a t any point along a line of constant q is

drldx

(3.43)

= mqxm- l .

The slope of the Mach line a t any point is (only the upper family need be considered) (3.44)

dr _ - ~ dx

+ a a m x ~ - ~

where a is the local speed of sound. Equating (3.43) and (3.44), a line of constant q will be a Mach line when (3.45)

qc

= plc

+ l/yFcI*c.

The subscript c is used since (3.45) defines the critical point of the flow, as may be seen by writing (3.45) in the form (using (3.26b) and (3.26~)) (3.46) and noting that the denominator of (3.28a) is zero at this point. Thus, defining a power-law shock uniquely determines the flow up to but not beyond the limit line. The flow can be continued past this line only when it corresponds to a singular point (as in cases (6) and ( 7 ) of Fig. 8). Figure 8 is representative of the flow field for u = 0 and y > 1. For 1 < p < pss, q b is finite and negative, while for p > pss there occur limit lines. Note that the shock shape is relatively insensitive to the body shape when is negative. For example, when y = 715, m varies only over the small range 213 to 3/5 as q b goes from 0 to - 0 0 . For (T = 1 and y > 1, the flow associated with a power-law shock is confined to q 2 0 and a limit line occurs in the flow field for all > 1 (e.g., [19]). Cases (1) to (4), represent the flows for CT = 1, -. 1 < p 1, while case (9) represents the u = 1, p > 1 case. There is no flow to compare with the cr = 0 “sharp-blow’’ problem (except when y = 1).

<

26

HAROLD MIRELS

Studies of the integral curves and singular points of the differential equations considered herein have been presented in [lo, 12, 19, 20, 21, 34, 361 and also by other investigators. The discussion in [12] is particularly detailed. IV. FLOWS ASSOCIATED WITH SLIGHTLY PERTURBED POWER-LAW SHOCKS

The effect of a power-law perturbation of a power-law shock is now considered. The assumption (MIJ)-~-+ 0 is retained except in Section IV.3 wherein the effect of small but nonvanishing ( M 6 ) - a is evaluated. The material is based primarily on [17, 181. 1. Basic Equations

The flow associated with a power-law shock, found in Chapter 111, is considered to be a zero-order solution and is denoted by subscript zero. Thus the zero-order shock and body shapes are denoted by Ro = xm, Yb,O = rb,0xrn where q b , O is tabulated (as q,) in N R=xm(l+r2a2x 1 Table 1, etc. The effect of a small Ro8 x powerlaw shock perturbation is denoted by a subscript 2 in agreement with b#a9b$m the notation of [18]. The shock shape may then be expressed as (Fig. 9) '0

Fp:xm R/HM

.

R4

(4.1)

X

R

+E ~ u ~ x ~ ) ,

= zm(l

FIG. 9. Power law perturbation of power-law shock and corresponding body shape (0 < p < I ) .

where E2 iS Small.* The problem iS to find the flow field associated with this shock. The body streamline (corresponding to 0 = 0) can be shown to have the form (4.2a) (4.2b)

rb = Xm(qb,O

f

E2XN)

= p ( & z X N ) l ! ( o + 1)

(p < I), (p = 1).

The quantity e2 is a measure of the magnitude of the body perturbation. The ratio of the shock perturbation to the body perturbation is u2 except for the 3/ = u = 1 case where the ratio is u,(E,$'"''~. The independent variables ([,q), as defined in (3.4), are again used. Substitution of (4.1)into (2.6) with (MS)-2 = 0 suggests that the dependent variables be expressed as

* However, E~ should be larger than 6* since terms of order 6*have been neglected in obtaining the hypersonic slender body equations used herein.

27

HYPERSONIC FLOW OVER SLENDER BODIES

(4.3a) (4.3b) (4.3c) where po,p2,F,, etc. are functions of 7. The quantities v0,Fo, and t,bo are the same as v, F, and JI of Chapter 111. Substituting (4.3) into (2.4) and collecting terms of order cZo and eZ1 yield, respectively, the zero-order equations (3.7) and the perturbation equations. The latter are [18] U

(4.4a)

(4.4b)

P2’

~-

v - To

- .$2’ +

V2’

~-

71 - Po

$0

1

-

($+17

- Po +Pi

0)

P2

V-Po

I

v

+ +1h (0

*2

- 0,

v - Po

*O

F _o F_2 ’ +

(17 - Po)2$0 Fo

v -‘Po

71 - Po

+

where p = N / [ m ( u l)]. It is convenient to satisfy the shock boundary conditions a t the zero-order shock location, v = 1. The boundary conditions are

Equations (4.4) can be integrated numerically, starting a t 17 = 1, if p2/a,, F2/a2,and $Ja2 are considered as the dependent variables. The value of a2 is then determined by satisfying the tangent-flow boundary condition a t the body surface. This condition shows that a t r]b,O (from [18])

28

HAROLD MIRELS

TABLE4.RESULTS OF NUMERICAL INTEGRATION OF PERTURBATION EQUATIONS (EQ.(4.4)) (Data from ref. [la] Y

P

o=o

Case

1.15

0 1/3

3/4

1 1.4

0 1/3

3/4

1 1.67

0 1/3

3/4

1

a

p

@a

o=l Fb,2

P

aa

1.16 1.07 1.29 1.12 1.04 1.34 1.17 0.974 1.40 1.15 0.693 0.325 0.188 1.44

0.798 2.00 1.18 1.71 2.09 1.39 1.58 2.03 1.53 1.45 1.56 0.506 0.483 0.869

- 1/4

1.20 1.77 1.32 1.14 1.86 1.44 1.02 1.95 1.44 0.660 0.443 0.217

2.00 1.39 1.76 2.03 1.64 1.67 1.90 1.77 1.53 1.36 0.611 0.655

0 0

1.74 1.33 2.28 1.52 1.25 2.40 1.71 1.10 2.48 1.74 0.701 0.546 0.244

0.934 2.00 1.55 1.78 1.99 1.82 1.73 1.84 1.91 1.59 1.31 0.670 0.605

Effect of boundary-layer development. of small angles of attack. Effect of wedge and cone nose blunting,

* Effect

Case

Fb, 2

0 0

1.05 1.04 1.06

0.863 1.96 1.64

0 1/8

1.08 1.02

1.49 1.99

0 5/16

1.18 0.734

1.31 1.96

0.0518

0.503

1.13

0.843

1.09 1.14

1.91 1.59

0 1/8

1.19 1.06

1.45 1.87

0 5/16

1.38 0.655

1.29 1.65

0.0389

0.643

1.20 1.15 1.22

0.823 1.86 1.55

0 1/a

1.29 1.11

1.40 1.80

0 5/16

1.53 0.645

1.24 1.53

0.0294

0.828

1

- 1/4

1 - 1/4 0 0

1

HYPERSONIC

FLOW OVER SLENDER BODIES

29

(4.6a) (4.6b) Analytic procedures are required near l;)b,o, since this is a critical point of the differential equations. Numerical results for a2 and Fb,2 are given in Table'4 for a variety of physical problems which are discussed later. If a2 is known, the shock shape corresponding to a given body perturbation, and vice versa, is found from (4.1) and (4.2). The pressure distribution on the perturbed body is (4.7)

+

p b = (dRo/dx)2[Fb,0

E2Fb,2xN].

Expressions for drag are given in [ 1 8 ] . For P = 1 , (4.2b) shows e2xN = ( Y ~ / R ~ )so" +that ~ the perturbation analysis is valid for (rb/RO)'+l << 1 . The perturbed shock shape can be written (4.8)

R/Ro= 1

+ a2(rb/Ro)"+1.

Eq. (4.8) indicates that for 0 = 1 a body perturbation rb/Ro causes a shock perturbation of order ( Y ~ / RThus ~ ) ~ the . shock is relatively insensitive to body changes for the u = P = 1 case.

2. Applications Boundary-layer effects. If the boundary layer on a power-law body induces only small perturbations in the zero-order flow field, the magnitude of these perturbations can be found using the equations of the previous section. Expressions applicable for finding the boundary-layer displacement thickness 6* on a slender power-law body at hypersonic speeds are derived in [lo, 18, 23, 371 and other papers. Following the method of ClS], the non-dimensional boundary-layer displacement thickness S* is (assuming a linear variation of viscosity with temperature)

where (4.9b)

30

HAROLD MIRELS

TABLE5. BOUNDARY-LAYER PARAMETERS (FROM [18] BASEDO N NUMERICALDATAO F [38, 391.)

gw

J2

JI + J s

Js

0

-0.14 0 0.60 2.00

0.495 0.470 0.4235 0.383

0.134 0 - 0.257 - 0.538

0.2

-0.14 0 0.50 1.60

0.504 0.470 0.409 0.366

0.692 0.619 0.199 - 0.083

0.6

- 0.20

2.00

0.554 0.470 0.380 0.294

2.034 1.556 1.185 0.759

1

0 0.05 0.10 0.20 0.40 0.60 0.80

0.470 0.452 0.435 0.408 0.370 0.336 0.312

2.69 2.53 2.48 2.41 2.33 2.27 2.24

2

-0.10 0 0.30 0.50 1.00

0.543 0.470 0.333 0.274 0.1765

5.67 5.19 5.49 6.01

0 0.50

7.85

2(1 - m )

y

2m(o+ 1) - 1 '

Here R,L = fiwL/Pw, o = TP,,iie/Tep,=constant, g, is the ratio of the fluid enthalpy at the wall (assumed constant) to the free-stream stagnation parameters which enthalpy, and J 1 , J z , and J 3 are boundary-layer profile are tabulated in Table 6 . For m = 1 or y = 1, v 2 ( J , Jz J 3 ) = 0.664 1.73gw. Equation (4.9) was derived for Prandtl number 1 and is valid for /3 < 1, and for p = 1, ~7 = 0. The case p = ~7 = 1 is excluded. For y + 1, (4.9b) simplifies to

+

+ +

HYPERSONIC FLOW OVER S L E N D ER BODIES

31

If 7b.O is a power-law body and d* is the boundary-layer displacement d*. Using (4.0a) for 6*, this thickness, the effective body is Yb = Yb,O body corresponds to (4.2) with

+

(4.11a)

p = - p3 4

2(a

+ 1)

The perturbation procedure is valid provided E~ is small.* (It is small when the boundary layer is thin compared with the body radius, for p < 1, and with the shock radius, for p = 1.) Note that F~ contains the ratio of the hypersonic boundary-layer interaction parameter M3vo)/Re,i to the hypersonic similarity parameter (Md)2and may also be interpreted as the ratio of the boundary-layer induced pressure to the zero-order shock-induced pressure. Numerical results for u2 and Fb,2are given in Table 4 for p defined by (4.11a) and various values of y and /I.These determine the shock and surfacepressure perturbations induced by boundary-layer development on a powerlaw body. For m = 314, the perturbed shock and body shape are similar. I t is then possible to solve for the flow field without assuming small perturbations. The power-law exponent m = 314 corresponds to the strong interaction case for hypersonic flow over a flat plate [ l l ] .

Angle-of-attack effects. The equations of Section IV.l can be used directly to find the effect of very small angles of attack on two-dimensional power-law bodies. If the body is a t angle of attack a the equation of the upper surface becomes (4.12a)

7 b = yb,O

f

EzX,

where (4.12b)

E2

= -up.

Assuming E~ << 1, the resulting flow (in the upper half plane) is found from the equations with N = 1 - m, or (4.13)

p

=pi2

* The boundary layer effect is of interest provided it is larger than order 6a and larger than the entropy layer effect discussed in Section VI.l.

32

HAROLD MIRELS

The flow in the lower half plane is found by symmetry (F2(q)= - F 2 ( - q ) , etc.). Numerical solutions for u2 and F b , 2 have been obtained and are also noted in Table 4. The lift problem for a = 1 and p < 1 is formulated in [18] but no numerical results are given. A cylindrical coordinate system (x,r,8) is used where x is in the stream direction and (r,8) are polar coordinates in the transverse plane. The body and shock shapes are given by (4.14a)

r b = x"(qb,O

(4.14b)

R

= xm(l

+

E

~

-X sin ~ 8),

+ ~ ~ u ~ ~ ~ 6-) ." s i n

The dependent variables are expressed in the form indicated by (4.3) but with the perturbation terms multiplied by sin 8. In addition, the velocity in the 8 direction is w = ma,R, cos 8

(4.15)

where R, is a function of q. The differential equations defining the motion are the same as (4.4) except that a term Q2/q(po-- q ) must be added to the left hand side of (4.4a) and the &momentum equation (4.16) must be considered. The boundary conditions a t the shock are given by (4.5) with the additional condition (4.17)

+ 1))

Qz(l)/% = - 2/(r

and the boundary condition at the body surface is given by (4.6~3). The latter equations must be solved numerically to find the effect of small angles of attack on axially symmetric power-law bodies. The case corresponding t o p = 1 has not been formulated. The surface pressure distribution for both a = 0 and u = 1 may be expressed as (4.18)

f i b = m2z+

- l) [ F b , O

+

&2Fb,2X1 -

sin 81,

where 8 = n/2 and 3x12 for the upper and lower surfaces, respectively, when a = 0. If L ( x ) is the net lift up to station x , (4.18) shows (4.19a) (4.19b)

33

HYPERSONIC FLOW OVER SLENDER BODIES

The location of the center of pressure depends only on u and m, for given x . The lift of slender axisymmetric bodies (not necessarily power-law bodies) has been treated by Cole [40] using Newtonian theory. Gonor [41] has considered the special case of a lifting cone. The effect of power-law perturbations of the centerline of power-law bodies is also discussed in [18]. Effect of blunting the nose of very slender wedges and cones. The effect of blunting the nose of very slender wedges and cones is now considered. It is assumed that the wedge or cone is sufficiently slender so that the major

FIG. 10. Hypersonic flow past blunted slender wedge or cone. (a) Actual flow. (b) Hypersonic slender body approximation.

contribution to the drag is a t the nose. The zero-order flow is then a constantenergy (,d = 1) flow. The divergence of the body downstream of the nose induces a small perturbation in this zero-body flow, which can be found from the equations of Section IV.l. Fig. 10 illustrates the flow under consideration. The body shape, in dimensional variables, is (4.20)

fb = fN

+ 663,

where Sb is the vertex semi-angle of the wedge or cone. An equivalent body which has zero area a t 3 = 0 but has the same area divergence downstream of 3 as does (4.20) is given by f { + l = (6b%)'+1 2ufNdbR. In nondimensional variables, the latter becomes

+

(4.21b)

-6bx 6

for

0

=

o

or u

= 1,

YNdIXdb<<

1.

Equation (4.21b) is the hypersonic slender body approximation of a blunted slender cone or wedge. Both the physical flow and the approximation are illustrated in Fig. 10. The approximation is poor near x = 0.

34

HAROLD MIRELS

Comparison of (4.21b) with (4.2b) shows (4.22)

&2 = (6b/6)a+1,

= ((T

+ 1)/2.

Numerical solutions have been obtained for (T = 0, 1 and y = 1.15, 1.4, and 1.67. The results for a, and Fb,2 are tabulated in Table 4. Surface pressure and shock shape can then be found from (4.7) and (4.8). Chernyi (15, 161 has studied the hypersonic flow past blunted wedges and cones using an integral method which is described in V.2. The integral method is not restricted to flows wherein the major drag contribution is a t x = 0. 3. Effect of (M6)-2 # 0 Up till now the limiting case (MS)-2-* 0, which corresponds to infinite free-stream Mach number, has been considered. In this limit, the flow past a slender power-law body gives rise to a power-law shock. If the Mach number is decreased, the shock will depart from its power-law shape and the pressure distribution on the power-law body will be modified. This effect is studied herein. In particular, small but non-vanishing values of the parameter (4.23)

s1 G

(MS)-2

will be assumed and (2.4) and (2.6) will be solved by a perturbation procedure. The body shape is given by yb = r&rm and the zero-order solution is the same as discussed in Chapter 111. This problem is sometimes referred to as the effect of "counter pressure" in Russian literature and as "second order" blast wave theory (in the case of a blast wave). Let subscript 1 represent perturbation quantities (as in [17]). The shock boundary conditions (2.6) suggest that the perturbed shock has the form

R

(4.24)

= tm [l

+

ela1E2(l-

"'1,

and that the dependent variables can be expressed as = mtm-

(4.25a) (4.25b)

p

(4.25~)

p = $bo

[po

+~

= m 2 [ 2 ( m - 1) [F,

+

~2 0 -p"'I, ~

+ E1F1t2(1-

I,

E1$b1t2(m - 1 ) .

Equations (4.25) are in the form of (4.3) with N (4.26)

m,

t

p

= 2(1 - m),or

equiyalently,

=P.

Thus, substitution of (4.25) into (2.4) yields perturbation equations which are identical to (4.4) with p = /? and with the subscript 1 replacing the

HYPERSONIC FLOW OVER SLENDER BODIES

subscript 2 therein. The shock boundary conditions to be satisfied at 7 are [lo, 171

-2 v

1

=

m

+ q 2 - ( U + v + 2

y f l

4uy Y+l

]

35 =1

a,,

(4.27a)

(4.27b)

(4.27~) The flow at the body surface must satisfy (4.28)

P)l(Vb) = O*

since the body is unperturbed from its zero order power-law shape. For a given body, a, is determined such that (4.4) (with p = P ) and (4.27) yield a solution satisfying (4.28). To avoid trial-and-error choices for a, in a numerical integration of (4.4) and (4.27) it is advisable to decompose each dependent variable and its boundary condition into two parts, one independent of a, and the other proportional to a, (4.29)

(

)1 = ( 11.1

+(

)1,2a,.

+

For example, v, = pl,, v1,2a,, etc. The solution for ( )1,1 and ( )1,2 can then be found. The final solution is given by (4.29) with a, determined from (4.30)

= - 9l31(~b)/~l,2(~b).

Numerical results for a, and Fb,l are given in Table 6. Similar results are presented in [lo] for u = 1. (The latter appear to be somewhat in error, particularly for a,, as noted in [17].) The surface pressure and shock shape are found from

The pressure perturbation is of the same order as the free-stream pressure, = E J ~ ,so that the latter must be included when computing local depar-

pa

36

HAROLD MIRELS

tures from free-stream values. Equations (4.30) and Table 6 indicate that the surface pressure and the shock ordinate are increased by non zero values of E ~ . The surface pressure perturbation is independent of x . TABLE6. NUMERICAL RESULTS FOR EFFECTOF ( M 6 ) - a # 0. Y

1.15

1.4

1.67

P

0 1/3 1/2

FROM [17] a = l

a=O

a1

Fb.1

a1

Fb,l

1.000 1.34 1.43

1.800 2.79 3.16 2.77 0.910

0.489 0.885 1.08 1.17 1.18 1.10 1.23

1.10 2.50 3.37 3.84 3.79 2.93 1.32

0.476 0.807 0.932 0.976 0.976 0.964 0.992

0.918 1.97 2.49 2.67 2.51 1.95 1.14

0.465

0.783 1.63 2.00 2.11 1.98 1.57 0.931

5/8

-

314 718 1

1.31 1.03

0 1/3 1/2

1.000 1.21 1.21

518

-

3/4 718 1

1.07 0.965

1.548 2.17 2.26 1.78 0.799

0 1/3 112 5/8 3/4 7/8 1

1.000 1.14 1.11 0.991

1.348 1.77 1.78 1.39

0.930

0.707

-

-

0.762 0.863 0.900 0.911 0.922 0.969

V. INTEGRAL METHODS Hypersonic slender body theory, which assumes d2 << 1 and (Md)-2 not near one, was used in Chapters I11 and IV to find the flow associated with power law and perturbed power law shocks and bodies. In order that the equations of motion yield similarity solutions it was necessary to further ) - ~ either zero or very small. The treatment of more assume that ( M C ~ was general bodies and Mach numbers requires either a numerical integration of the full equations of motion or the use of integral methods. Some integral

37

HYPERSONIC FLOW OVER SLENDER BODIES

methods are outlined herein. The object is to provide a means for finding the flow over more general body shapes than considered in Chapters I11 and IV. Also, the integral formulation will be used in Chapter VI to estimate the error introduced by disregarding the violation of the hypersonic slender body assumptions at the nose of power-law shocks. The development in the remainder of this article is in terms of dimensional quantities and the superscript bar is omitted. 1. Continuity Integral

Conservation of mass is expressed in integral form for supersonic and hypersonic flow over an arbitrary body. The resulting expressions are then simplified by introducing approximations.

General case. Consider the streamtube which crosses the shock at (xi,Ri) and has a lateral width dRi (Fig. 11). At some downstream station x the streamtube has an ordinate r and a width dr. Conservation of mass yields (in physical variables) bum

p

(2nRJ"dR; = pw (2nr)"dr. X

(5.1)

XI

X

Integrating (5.1) at a fixed value of x and noting d8 (a 1) R t d R j / R U f gives 1

FIG.11. Notation for continuity integral.

Equation (5.2) relates r and 8. For 8

= 1,

= +

r

= R, (5.2)

becomes

If p / p , and u / u , could be expressed as functions of 8, (5.3) would relate the shock and body shape in a given problem. This might be done as follows. The flow along a streamline is isentropic downstream of the shock so that PIP' = PiIpiY or (5.4a)

38

HAROLD MIRELS

where subscripts i and s represent conditions directly behind the shock at xi and x , respectively, and f E p / p , is the pressure at ( x , ~ divided ) by the pressure behind the shock at ( x , R ) (Fig. 11). Similarly, Bernoulli's equation gives (5.4b)

(5)=I(+)

2 -,"@if-s]. 2

a

Y-

m

Ps

P

1

PWUW

All the variables on the right hand sides of (5.4) can be expressed in terms of shock shape except v and f . If a reasonable variation of v and f with 8 is assumed, (5.3) can be solved to find the body shape corresponding to a given shock, and vice versa. No other assumptions are involved. For example, it may be possible to assume that at each x the variation of f with 0 depends only on the local value of p and is given by the zero-order similarity solutions. In problems of hypersonic flow over slender bodies the term involving z' is negligible.

Hypersonic slender body approximations. If the hypersonic slender-body approximations are introduced, (5.4b) becomes u / u m= 1 and (5.4a) becomes

+

where E = ( y - l ) / ( y 1) is not necessarily small. Equations (5.5)and (5.3) yield (3.26a) if it is noted that for power law shocks, Ri' = 19-p/~R', and if the limit of infinite Mach number is taken. The Newtonian theory of Cole [42] assumes E<< 1, (MRi')-2/(y- 1)=0(1), E ( M R ' ) -<< ~ 1, E ( M R ~ ' )<<- ~1. U'ith these assumptions (5.5) and (5.3) become 2

0

+

To the same order f = 1 - p(1 - 0)/2, where ,d = - 2 [ R R " / ( R ' ) 2 ] / ( o 1). Equation (5.6) gives the body shape corresponding to a given shock shape. The shock and body ordinates, R and rb, are nearly equal. Thus if r b is given and R is t o be found, the derivatives of R and Ri can be replaced by derivatives of rb and Yb,,, respectively, in (5.6) and in the expression for f . The latter form is the one used by Cole [42] who in addition takes the exponent l / y in (5.6) to be 1.

HYPERSONIC FLOW OVER SLENDER BODIES

39

If (5.6) is used to study blunt-nosed slender bodies with finite drag addition a t the nose (i.e., p = 1 a t x = 0 ) , the derivatives of R,Ri cannot be replaced by the derivatives of rb and rb,). This is because the shock separation from the body is large. In addition it is necessary to retain the exponent as l/y.* Mirels [29] derived (5.6) as a generalization of Cole’s Newtonian solution. Although its validity for describing the flow over blunted (p = 1) slender bodies may be questioned (due to the violation at the nose of the hypersonic slender-body assumptions) it is certainly valid for the equivalent unsteady flow of a perfect gas. Slender blunt-nosed bodies at infinite Mach number. In the present section the assumption of body slenderness is not applied a t the nose. The effects of nose bluntness are therefore retained and can be evaluated. The limiting case M - , 00 is considered and the development follows that of Sychev !25]. Recall that t is the local shock angle relative to the free stream (Fig. 1). No assumption regarding the magnitude of t is made. Note that tan t = R’, sin2t = (R’)2/(1 R12),etc. The boundary conditions a t the shock, for M + 00, then become [cf. (2.41

+

Ps -pmum2

(R’)2

2

-

5

1

+( ~ ’ 1 ~ ’

1 _ ps -_, Pm

E

(5.7) % _ -I-um

2 R12 y + 1 1+(R’)2’

vs

um

-

2 R’ y + l 1+(R’)2’

and Eqs. (5.4) become

_ p - -1 -

pm

E

[(_

+ (R,’)2f]”’ + (R‘)2

R’)’ 1 Ri‘ 1

Equations (5.8) can be further simplified if it is assumed that the bluntness of the body is confined to the nose region and that x is sufficiently far ) ~ 1, and (5.8) and downstream so that (R’)2<< 1. Then ( Y / U ~ <( (5.2) yield

* If the exponent l/y is replaced by 1 in (5.6) the integral diverges for p = 1 and therefore does not exhibit the correct limiting behavior for y near 1 . Replacing l/y by 1 is equivalent to assuming isothermal rather than isentropic flow along a streamline as was discussed b y Freeman [31].

40

HAROLD MIRELS

(5.9)

-l+!+(

($+l

The term involving (R')2(y-1)/v is retained in (5.9) so that the equation is correct to order R12. If r is replaced by R and the limit f3 is replaced by 1, (5.9)relates shock and body shape. A reasonable estimate for f in the integral of (5.9) is to use its variation with 0 and P as given by the zero-order similarity solutions as is discussed later. Sychev 1251 used (5.9) to find the body shapes associated with a powerlaw blast wave and compared the result with the zero order similarity solution. He also investigated the validity of the similarity solutions for various degrees of nose bluntness. This work is described in Chapter VI. 2. M o m e n t u m Integral

General case. The net forebody drag up to station x can be found by considering the flux of momentum across the upstream and downstream faces of the control surface indicated in Fig. 12. The result gives

axI(# (5.10)

D(x) 2~

- (Pm

rb(x12

0

+ pm~m2)

Ra+1

FIG.12. Control surface for momenturn integral.

*

--

O

f

1

+ pu2)Pdr,

'b

where u = 0 corresponds to a symmetric two-dimensional body. Conservation of energy across these surfaces yields R

'b R

(5.11)

HYPERSONIC FLOW OVER SLENDER BODIES

41

Replacing the integral in (5.10) by the appropriate expression from (5.11) gives the final result

(5.12)

Rafl y ( y - 1)M2 fJ 1 1

-

+

'

Equation (5.12) gives the net drag up to station x in terms of an integration with respect to 7 at that station. Hypersonic slender body approximation. For hypersonic slender bodies (5.12) becomes

where terms of order d2 have been dropped and D ( x ) has been replaced by the sum of the finite drag addition at x = 0 (if any) and the surface pressure drag downstream of x = 0. Equation (5.13) is the dimensional form of (2.7). This equation was used by Chernyi to study non-self-similar hypersonic flows over blunt nosed bodies [15, 161. Chernyi's method, described in [22], is now outlined. Chernyi assumes that the pressure and velocity may be expressed in the form (6.14a)

V

Hm

-

2 R'[1 - (MR')-2] y+l

+ O(&Rf),

(5.14b) Equations (5.14) are correct in the limit valid directly behind the shock for all E .

E

<< 1. However, they are also Chernyi lets p be constant, a t

42

HAROLD MIRELS

its wall value (0 = 0 ) , when evaluating the integral on the right hand side of (5.13). This procedure is valid for ,B small since the term containing 0 is then negligible in (5.14b). I t is also valid for P = 1 and E << 1 since most of the fluid is concentrated at the shock and the region from the base of the shock to the wall is essentially at the wall pressure. W'ith fi = fib, the integral becomes (for M do)

-

i

(5.15a)

[. . . ] Y " ~= YR"+'RI2Ib,

*h

where

Here qb = rb/R and I b is defined by (3.13~).Substitution of (5.14) and (5.15) into (5.13) yields an integral equation relating rb and R. The surface pressure is found from (5.14b). The accuracy of this method can be estimated by solving for the flow associated with a power-law shock, in the limit M - t 00, and comparing with the zero-order solutions obtained in 111. The result for qb and is [22]

(5.16b)

F b

= 2 / ( y $- 1) - P/2.

Numerical values for qb and F b are listed in Table 1 for various values of u, y , and P. These agree within about 4% with the exact numerical solutions of (3.7). Hence (5.16) provides simple and relatively accurate estimates of 156 and Fb for the entire range of u, y , and P of interest. Values of I b obtained from (5.15~)are listed in Table 2. These are correct within 4% for u = 0 and 10% for u = 1. Since z b is raised to the l/(u 3) power in (3.15a) an error of less than 3O/, results from the use of (5.15~)therein. Chernyi numerically integrated (5.13), with (5.14) and M - do, to find the hypersonic flow over blunt-nosed slender wedges and cones [15, 161. His solutions are not limited to small departures from a blast wave flow as are the more exact perturbation solutions of Section IV.2. The parameters evolved in [lB, 161 have been used in [43] to correlate numerical data obtained from characteristic solutions of the hypersonic flow over blunt nosed plates at angle of attack.

+

43

HYPERSONIC FLOW OVER SLENDER BODIES

VI. VALIDITYOF SELF-SIMILAR SOLUTIONS The hypersonic slender body equations of motion were integrated in Chapter 111 in the limit (Md)-z+ 0, to find the self-similar flows associated 1). I t was with slender blunt-nosed power-law shocks and bodies (0 < p recognized that the solutions are not valid near the nose, where the assumption of slenderness is violated, but it was expected that the solutions become valid downstream of the nose. The validity of the similar solutions, for various degrees of nose bluntness, has been studied by Cheng [24], for u = 0, and by Sychev [25]. Their results are discussed herein.

<

1. Infinite Mach Number The indirect problem is considered first. Given a power-law shock R = Cxm, in the limit M - 00, what is the corresponding body shape and surfacepressure distribution? The shock is illustrated in Fig. 13. The point (x,,R,) is the point on the shock at which the local shock slope is unity. The (x,, R ; & ~ ~ ~ streamlines crossing the shock above R, can be treated by hypersonic LAYER slender-body theory. The flow below (xi'Ri)-X 0 X Re crosses a nearly normal shock and is not correctly described by that FIG. 13. Entropy layer, Re' = 1. theory, The latter flow undergoes a relatively large entropy increase when crossing the shock and is termed the entropy layer. Station x , in Fig. 13, is a downstream station at which << 1. The streamline through (xe,Rc),bounds the entropy layer and has the ordinate re a t x . Hypersonic slender-body theory neglects terms of order compared with one. The present problem is to determine the range of y and p for which the entropy layer at x creates effects which are larger than terms of order (R') and therefore must be correctly evaluated. From the assumption of a power law shock and R,' that at x

- -

=1

it follows

-

Equation (6.1) gives the fraction of the perturbed mass flow which is in the entropy layer. Using the estimates zd urn, v vbR'urn, pu(rc - rb) prnu,R8,1qba, and PbNprnu,2R'2 in ( 2 . 1 ~ )shows that the pressure differential across the entropy layer at x is of order (6.2)

(pe -pb)/pb

(R')2'P.

44

HAROLD MIRELS

< <

for 0 p 1 and The pressure differential is less than or equal to order therefore can be ignored. Thus, the pressure is uniform across the entropy layer and equals the wall surface value given by the zero order similarity solution corresponding to the power-law shock. The zero-order solution (which is valid outside the entropy layer) gives the correct variation of pressure with 8 throughout the entire flow. The thickness of the entropy layer at x can be estimated from (5.2) and (5.9). Considering u u,, f 1, and (1 Ri'2)/Ri'2 1 (for 0 8 8,) gives

- -

< <

-

+

where E is not necessarily small. Equation (6.3) is a measure of the crosssectional area of the entropy layer relative to the area enclosed by the shock. This area ratio is O(R')2for fi in the range

<

(6.4)

O
1).

In this range of ,8 the body shape associated with the power-law shock is given correctly, to order R12, by the zero-order similar solution and is a power-law body. However, for ,8 in the range y / ( y 1) < p 1, the calculation of body shape must take the entropy layer properly into account The latter range of p decreases as y if it is to be correct to order increases. I t includes blast-wave and nearly blast-wave flows. The body shape can be found from a numerical integration of (5.9) or by approximate methods based on that equation. Equation (6.3) may be compared with the corresponding entropy-layer thickness obtained from the zero-order similar solution (3.26a)

+

<

The ratio (6.5) t o (6.3) is

The entropy-layer thickness indicated by the similar solution is considerably larger than the correct value when and y are near 1. This is due to the fact that similarity theory considerably underestimates the fluid density near the wall. Equations (6.2) and (6.3) were obtained in [24] (for cr = 0), and the remaining equations where obtained in [25] (for u = 0,l).

45

HYPERSONIC FLOW OVER SLENDER BODIES

Sychev [25] has found the body shape associated with an axisymmetric blast wave by numerically integrating (5.9). He chose R = 2/g,y = 1.4, and used Sedov’s exact solution [12] to relate f and 8. The resulting body shape is given in Fig. 14(a), and the ratio Y b / R is given, as a function of x , in Fig. 14(b). The latter may be contrasted with the zero-order similarity result rblR = 0. The actual body shape associated with a blast wave has considerable thickness when compared with the similar solution. When blast-wave theory is used to estimate hypersonic flow over a blunt-nosed circularcylinder (with axis parallel to the stream direction) the shock shape R = VZx corresponds to a cylinder radius rN = l . 6 / 2 / G 1.6 for y = 1.4 (3.15a). The difference between the body shape indicated in Fig. 14(a) and

-

rl 12

-

0

8

16

24

32

a -

b

FIG.14. Body associated with blast wave R = v 2 x , u = 1, y = 1.4, M -+ m (from ref. [25]). (a) Shock and corresponding body. (b) Ratio of body to shock ordinates.

-

= rN 1.6 may not be very significant with regard to surface pressure and shock shape since blast-wave theory indicates very little mass flow in the region near the body and is relatively insensitive to small changes in body shape (4.8). The direct problem is now considered. Given a power-law body, in the limit M-+ bo, what is the corresponding shock shape and surface-pressure distribution ? When L,? y / ( y l), the zero order similar solutions are applicable. But when p > y / ( y I), these solutions are in error. One method for finding the shock shape and pressure distribution is to perturb the zeroorder similar solution to account for the excessive size of the entropy layer. This may be done by solving the perturbation equations of IV.l, using an “equivalent body” which is somewhat smaller than the given power-law body. The equivalent body to be used with the equations of IV.l should equal the cross-sectional area of the actual body minus the difference between the area of the entropy layer as given by the zero-order similar solution (6.5) and the actual entropy-layer thickness (6.6). This gives Yb

<

-

+ +

where E~ (- 1)[ ( y - l ) / ( y - ,L?)]d2(y-B)’YB, Yb.eq is the equivalent body, Ro is the zero-order shock and q b , O is the zero-order body to shock radius ratio.

46

HAROLD MIRELS

Equation (6.7) can be reduced to the same form as (4.2) and the procedure of IV.l can be used to find the shock shape and surface pressure corresponding to the original power law body. (Note that R,’/6 in (6.7) corresponds to mx*’l in (4.2).) The shock and surface-pressure perturbations are of the form given in (4.1) and (4.7) with N = 2(y - P)(m - l)/yP. These perturbations cannot be evaluated exactly since E~ is known only to within an order of magnitude. Inasmuch as E~ is negative, the perturbed shock is closer to the body and the surface pressure is lower than indicated by the zero-order solution. The above discussion regarding power-law shocks and bodies in the limit bo may be summarized as follows. When 0 p y / ( y 1) the similarity solutions describe the flow downstream of the nose to order (R’)2. The shock and body shapes are similar. When y / ( y 1) < p 1, the body associated with a given power-law shock (indirect problem) is thicker than indicated by similarity theory, due to the excessively low fluid density near the wall in the similarity solution, but the surface pressure is correctly given. For the direct problem and y / ( y 1) < p 1 the shock associated with a given power-law body is closer to the body* and the surface pressures are somewhat lower than indicated by the similarity solution. For all p, the flow asymptotically approaches the similarity solution as x + bo (R’+ 0).

< < +

M+

+

+ <

<

2. Finite Mach Nwmber In order that the zero-order similarity solutions be valid for describing hypersonic flow associated with a slender power-law shock or body it is also - ~1. This condition assures a strong shock and was necessary that ( M c ~ )<< automatically satisfied in the previous section because of the assumption of M-, 00. However, for M large but finite, the similarity solutions are not valid far downstream of the nose due to violation of the condition (M6)-2<<1. (The first departures from similarity theory have been described in Section IV.3.) Hence, for M # bo, the flow associated with a power law shock or body does not asymptotically approach the similarity solution as x-+ bo. It may be concluded that for 6 small and M large, the similarity solutions are valid (subject to the limitations discussed in the previous section) in the intermediate region between the nose and far downstream locations.

* In an actual flow-problem, the nose of the shock is upstream of the body. In regions relatively near the nose this effect tends to increase the shock ordinate relative to the similarity value.

HYPERSONIC FLOW OVER SLENDER BODIES

VII. FURTHER DISCUSSION OF

INTEGRAL

47

METHODS

The direct problem of determining the hypersonic flow associated with an arbitrary blunt nosed body by integral methods requires further discussion. The validity of the integral methods, in the presence of entropy layers, is discussed herein. Some properties of the entropy layer are noted. Finally, a simplification of the integral methods is introduced which is valid when y is near 1 and when most of the drag addition occurs a t the nose. The hypersonic slender-body form of the momentum integral (5.13) is obtained from the general form (5.12)by assuming u = u, in the integration with respect to r. The assumption u = u, is correct to order ( R ’ ) 2outside the entropy layer but requires

-

inside the entropy layer. Eq. (7.1) is found from ( 5 . 8 ) by assuming (Ri’)2/[1 (Ri’)2] 1 and (R’)2<< 1 . If the local value of R‘ is sufficiently small so that (7.1) is satisfied, the hypersonic slender-body form of the momentum integral is valid regardless of the degree of nose blunting. If (7.1) is satisfied, the continuity integral can be written

+

Outside the entropy layer, R i t 2 is negligible compared to unity but inside the entropy layer it must be retained. An interesting property of the entropy layer can be deduced from ( 7 . 2 ) . For r = Y , and f3 = OL, (7.2) shows Re

where use is made of f3 F (Ri/R)“+’and f = f b (in the entropy layer). The right-hand side of (7.3) is a constant which depends on the upstream shape of the shock. Equation (7.3) is a direct consequence of the fact that the streamwise velocity is constant to the present approximation (7.1). The cross-sectional area of the entropy layer a t any section is then inversely proportional to the average density in the entropy layer which, in.turn, is proportional to the local pressure to the l / y power.

48

HAROLD MIRELS

Cheng [23] has simplified the procedure for finding the flow over arbitrarily shaped blunt-nosed bodies. He assumes (7.1) is satisfied, M-+w and E << 1. In addition, he assumes that the cross-sectional area of the entropy layer is large compared with the cross-sectional area of the disturbed flow between the entropy layer and the shock (Fig. 15). The latter assumption may be expressed, from (7.2)

The numerator is obtained by assuming (Ri')2<< 1 and (R'/Ri')2~w 1 in the integral of (7.2). The inequality noted in (7.4) is satisfied for R' small provided 8, is not too small (i.e., the ROPY LAYER mass flow in the entropy layer is not X small relative to the total disturbed mass flow). FIG.16. Entropy layer which occupies With y = 1, R = re, and f b = major portion of disturbed flow. 1 - p/2 (7.3) becomes

Equation (7.6) provides a relatively simple relation between R and yb. However, the constant on the right-hand side can not be estimated since the upstream shape of the shock is not known. In order to evaluate the constant in ( 7 4 , Cheng resorts to the momentum integral (5.13) which can be written for M + 00 and E <( 1

The left-hand side assumes that most of the drag is added at the nose. The right-hand states that most of the energy of the transverse flow is in the form of internal energy corresponding to the pressure Pa. (The first error term on the right hand side is the error due to the pressure departing from pb at points outside the entropy layer and the second error term is the ratio of the net transverse kinetic energy to the net internal energy, which is small for E << 1.) Neglecting the error terms, (7.6) is the same as (7.5) except that the constant is evaluated, viz

49

HYPERSONIC FLOW OVER SLENDER BODIES

Equation (7.7) is relatively simple and has been solved in [23] for a variety of problems involving blunted flat plates. Studies of shock-boundary layer interactions were also made using this equation. In (7.7) the lateral displacement between the shock and the body is due entirely to the entropy layer. The cross-sectional area of the entropy layer is inversely proportional to the local pressure in order to conserve mass (continuity approach) and to conserve the internal energy (momentum integral approach). Rut the local pressure depends on the shock shape parameter B. Hence a consistent solution relating shock shape, pressure, and cross-sectional area is obtained from (7.7). This is similar to the strong interaction between shock waves and boundary layers wherein a consistent solution is also required ( e g , 11111). The effect of the downstream body shape rb is to laterally displace the entropy layer and thus effect the final self-consistent solution. Since (7.7) is derived directly from the momentum integral equation, recourse to the continuity integral was unnecessary. However, the continuity integral provided additional insight into the flow process. Equating (7.7) and (7.3) also provides an estimate of Re, namely (RI/YN)u+lM CD,. I t should be noted that the continuity integral is more sensitive to the details of the upstream shock shape than is the momentum integral. In order to integrate (7.2) across the entropy layer, it is necessary to estimate both the value of Re and the variation of R,' in the range 0 R, Re. The latter estimates define the variation of density across the entropy layer and thereby its thickness. However, in the momentum-integral method, the internal energy in the entropy layer depends only on the pressure f i b and on the cross-sectional area. The details of the density distribution are not required. Hence, the momentum integral is relatively insensitive to the upstream shock shape and provides more information (assuming the nose drag is known) concerning the downstream flow than does the continuity integral.

< <

VIII. CONCLUDINGREMARKS Several investigators [26, 28, 43, 441 have computed the inviscid hypersonic flow over blunt-nosed plates and circular cylinders using the method of characteristics and have compared the results with blast-wave theory. Since the method of characteristics provides an exact solution (within the accuracy of the computational procedure) it can be used to gauge the accuracy of blast-wave theory. The results of Vaglio-Laurin and Trella [28] are discussed herein. Reference [28] determined the axisymmetric flow about three configurations with long cylindrical afterbodies namely a sphere-cylinder

60

HAROLD MIRELS

(Fig. 16(a)) and two sphere-cone-cylinder combinations (Figs. 16(b), 16(c)). Several different flight conditions were assumed. The fluid was taken to be either a perfect gas (with y = 1.4) or air in equilibrium dissociation. The following was found with regard to the pressure distribution on the cylindrical afterbody. For the spherical nose configuration, f i b / + , - 0.40 was correlated versus M 2 C Z i rN/x, as indicated by "second" order blast wave theory (4.30 a) with y = 1.4, provided the origin for x was determined such that the pressure at the upstream end of the cylindrical afterbody matched the pressure obtained from a consideration of the hypersonic flow over a hemisphere. Good pressure correlation also resulted if the origin of the shock was arbitrarily located at a distance 2rN upstream of the body, regardless of Mach number and gas behavior (i.e., y = 1.4 or thermodynamic equilibrium). However, in order to cor&* ,..I relate the afterbody surface pressures associated with the less blunt spherecone-cylinder combinations, the value of rN used in M2Cg;rN/x had to be reduced. A procedure for obtaining the effective value of y N was suggested in [28] which resulted in good pressure correlation.

I

t

FIG.16. Nose configurations studied in [28].

The blast wave predictions for shock shape and for the flow between the body and the shock were less satisfactory. Only the spherical nose configuration resulted in the parabolic shock indicated by the zero order blast wave theory. Moreover, the latter correlation required that the x coordinate be measured from the apex of the nose, that the y coordinate be displaced outward from the axis by (2/5)rN and that the coefficients in the blast wave prediction be evaluated for gas conditions appropriate to the shock point at which the streamline is deflected approximately 20°. This is the streamline which [28] considers as bounding the entropy layer. The shock shape associated with each sphere-cone-cylinder combination could be correlated for different flight / ~ xo)"' ( X where conditions by an equation of the form y - yo = ( C O , / I ~ ) ~ xo,yo and m depend on nose shape and Ib, which expresses the effect of gas behavior, is selected consistent with conditions at the shock point where the streamline is deflected 20°.

HYPERSONIC F L O W OVER SLENDER BODIES

51

The disturbed flow between the 20° streamline and the shock satisfies the requirements of hypersonic slender body theory and was correlated for different flight conditions by non-dimensionalized flow variables such as those noted in (2.3). The correlation existed for the thermal-equilibrium flows as well as the flows with constant y in accordance with the hypersonic similitude discussion of Cheng [24]. The flow in the entropy layer has gone through a strong shock and does not satisfy the hypersonic slender body requirements. As a consequence, the entropy layer flow did not exhibit similitude. It may be concluded that blast-wave theory provides a good guide for estimating the inviscid hypersonic flow over the afterbodies of blunt nosed bodies particularly for distances far downstream of the nose. However, refinements are necessary for a more accurate determination of the flow a t stations which are not far downstream of the nose. The nose shape plays a greater role than accounted for in blast wave theory. The influence of the nose shape is most strongly felt in the entropy layer. In future developments it is expected that this influence will be taken into account to a greater extent and that reliable analytic estimates of the complete flow field about blunt nosed bodies will be possible. Experimental studies have been made of the hypersonic flow over slender blunt-nosed bodies [e.g., 10, 23, 45-48]. Viscous boundary layer effects complicate the problem of relating the experimental shock shapes and surface pressure to the predictions of blast wave theory. A full discussion is beyond the scope of the present study. Reference [23] is recommended for the reader who is interested in the experimental correlations of blast wave theory.

NOTATION In Chapters I1 to IV, barred quantities are dimensional and unbarred quantities are non-dimensionalized according to (2.3). (2.7),etc. However, in Chapters V t o V I I I only non-dimensional quantities are used and the superscript bar is omitted. a Speed of sound a,, a, Constants, (4.1). (4124) C Constant, (3.1) CD, Nose drag coefficient, (3.15b)

D(n) F(7)

Drag up to station n, (2.7) Pressure similarity variable, (3.6) Ratio of local pressure to value behind shock Energy integral, function of y , u, p , 7, ( 3 . 1 3 ~ ) Characteristic streamwise length Free-stream Mach number Power-law exponent, (3.1) Perturbation power-law exponent, (4.1) Pressure Shock ordinate

52

HAROLD MIRELS

Body ordinate Semi thickness or radius (a) a t x = 0 or (b) of plate or circular cylinder T Temperature 1 Time Streamwise and lateral velocities, respectively (u,v) Streamwise and lateral coordinates, respectively (%,I) (%$,Re) Shock point where slope is 1 (xi,Ri) Coordinates of streamline a t intersection with shock - 2 RR” ___ - 1 1 for power-law shock P .+ 1 R’a ’ Y Ratio of specific heats Characteristic shock slope, equals R ( L ) / Lfor power-law shock, (3.2) 6 Vertex-semi angle of wedge or cone ab E (Y - l ) / ( Y 1) rb(%)

VN

[ ]

81

a+l

[

+

(M8)-Z

Body perturbation parameter, (4.2) Similarity variable, r / x m Ratio of body to shock ordinate, Yb/R (R,/R)u+*,function of 11 for power-law shock ( j = P N / [ W 1)1 E x P Density 0,l for two-dimensional or axisymmetric bodies, respectively a Similarity variable for 3, (3.6) v Similarity variable for p, (3.6) Subscripts : b Evaluated a t body surface c Critical point, dflde = w e Associated with streamline bounding entropy layer S Downstream side of shock SB Sharp blow SP Singular point (saddle point) 0 Zero order solution (slender power-law shock) Perturbation due to el # 0 1 Perturbation due t o 8% # 0 2 rn Free-stream conditions Prime indicates differentiation with respect to independent variable ( )’ (usually q or n ) Pa

7 76

+

*

References 1. HAYES,W. D., On hypersonic similitude, Quart. Jour. Appl. Math. 5, 105-106 (1947). 2. VANDYKE,M. D., A study of hypersonic small-disturbance theory. N A C A R e p . 1194 (1954). (Supersedes N A C A T N 3173.) 3. TAYLOR, G. I., The formation of a blast wave by a very intense explosion, Pt. I, Theoretical discussion. Proc. Roy. Soc. ( L o n d o n ) , ser. A , 201, 159-174 (1950). 4. TAYLOR, G. I., The formation of a blast wave by a very intense explosion, Pt. 11. The atomic explosion of 1945. Proc. Roy. Soc. ( L o n d o n ) , ser. A , 201, 175-188 (1950).

HYPERSONIC FLOW OVER SLENDER BODIES

53

5. SAKURAI, AKIRA,On the propagation and structure of a blast wave, I, Jour. Phys. Soc. (Japan) 8, 662-669 (1953).

6. SAKURAI, AKIRA, On the propagation and structure of a blast wave, 11, Jour. Phys. Soc. ( J a p a n ) 9, 2 5 6 2 6 6 (1954). 7. LIN, S. C., Cylindrical shock waves produced by instantaneous energy release, JOUY.APPl. Phys. 26, 54-57 (1954). 8. CHENG,H. K., and PALLONE, A. J., Inviscid leading-edge effect in hypersonic flow, Jouv. Aero. Sci. 23, 700-702 (1956). 9. LEES, L., and KUBOTA,T., Inviscid hypersonic flow over blunt-nosed slender bodies, Jour. Aevo. Sci. 24, 195-202 (1957). 10. KUBOTA,T., Investigation of flow around simple bodies in hypersonic flow, Mem. 40, Guggenheim Aero. Lab., C.I.T., June 25, 1957. (Contract DA04-495-ORD-19.) 11. HAYES,W. D., and PROBSTEIN, R. F.. “Hypersonic Flow Theory”. Academic Press, 1969. 12. SEDOV,L. I., “Similarity and Dimensional Methods in Mechanics” (English Translation). Academic Press, 1959. 13. STANYUKOVICH, K. P., “Unsteady Motion of Continuous Media”. (English Translation). Pergamon Press, 1960. 14. GRODZOVSKII, G. L., Certain peculiarities of the flow around bodies a t high supersonic velocities, Izv. Akad. N a u k , S S S R , Otd. Tekhn. 1957, no. 6, 86-92. (Translated by Morris D. Friedman, Inc.) 15. CHERNYI,G. G., The effect of slight blunting of the leading edge of a profile on flow a t high supersonic speeds, Appl. Sci. Sec., Academy Sci. U S S R . 114, 197-200 (May-June 1957). 16. CHERNYI, G. G., Flow around a thin blunt cone a t high supersonic speeds, Appl. Sci. Sec., Academy Sci. U S S R 116, 263-265 (July-Aug. 1957). 17. MIRELS,H., Approximate analytical solutions for hypersonic flow over slender power-law bodies, N A S A T R R-15, 1959. P. R., Effect of body perturbations on hypersonic 18. MIRELS,H., and THORNTON, flow over slender power-law bodies. N A S A T R R-45, 1959. S. S., Cauchy’s problem and the problem of a piston for one-dimen19. GRIGORIAN, sional, nonsteady motions of a gas (automodel motion). P M M Jour. Appl. Math. and Mech. 22, 244 (1958). N. N., and MEL’NIKOVA,N. S., On the steady motion of gas driven 20. KOCHINA, outward by a piston, neglecting the counter pressure, P M M Jour. Appl. Math. and Mech. 22, 622 (1958). 21. ADAMSKI, V. B., and POPOV, N. A., The motion of a gas under the action of a pressure on a piston, varying according to a power law, P M M . Jour. Appl. Math. and Mech. 28, 3 (1959). 22. CHERNYI,G. G . , Application of integral relationships in problems of propagation of strong shock waves, P M M J O U Y . A p p l . Math. and Mech. 24, 1 (1960). 23. CHENG,H. K., HALL,J. G., GOLIAN, T. C., and HERTZBERG, A., Boundary-layer displacement and leading-edge bluntness effects in high temperature hypersonic flow, Jour. Aevo. Space Sci. 28, 5 (1961). 24. CHENG,H. K., Similitude of hypersonic real-gas flows over slender bodies with blunted noses. Jour. Aevo./Space Sci. 26, 9 (1959). 25. SYCHEV, V. V., On the theory of hypersonic gas flow with a power-law shock wave, P M M Jour. Appl. Math. and Mech. 24, No. 3 (1960). 26. FELDMAN, S., Numerical comparison between exact and approximate theories of hypersonic inviscid flow past slender blunt-nosed bodies, A R S Journal 80, No. 5 (1960).

64

HAROLD MIRELS

27. FELDMAN, S., On trails of axisymmetric hypersonic blunt bodies flying through the atmosphere, Jouv. Aero. Space Sci. 28, No. 6 (1961). R., and TRELLA,M., A study of flow fields about some typical 28. VAGLIO-LAURIN, blunt-nosed slender bodies, Polytechnic Institute of Brooklyn, PIBAL REP O RT No. 623, AFOSR 2, Dec. 1960. 29. MIRELS,H., Similarity solutions for inviscid hypersonic flow over power-law and related bodies, Amer. Rocket SOC.,Preprint 1111-60, 1960. 30. LATTER,R., Similarity solution for a spherical shock wave. Jour. AppZ. P h y s . 26, 954-961 (1955). 31. FREEMAN, N. C., A note on the explosion theory of Sedov with application t o the Newtonian theory of unsteady hypersonic flow, Jour. AerolSpace Scz. 27, 7 7 (1960). 32. BROCHER,E. F., Comments on the behavior of Sedov's blast wave solution as y -+ 1, Jour. AerolSpace Sci. 27, 12 (1960). 33. VON HOERNER,S., Losungen der hydrodynamischen Gleichungen mit linearem Verlauf der Geschwindigkeit, Z s . Natuvfor. 10a. 687 (1955). 34. HAFELE, W., Zur analytischen Behandlung ebener, starker, instationarer StoDwellen, 2s. Naturfor. 10a, 1006 (1955). 35. ZHUKOV.A. I., and KASHDAN,IA. M., Dvizhenie gaza pod deistviem kratkovremennogo impul'sa (The Motion of a Gas Under the Action of a Short Lived Impulse), Akust. Z h . 2 , 352 (1956). 36. GUDERLEY,G., Starke kugelige und zylindrische VerdichtungsstbDe in der Nahe des Kugelmittelpunktes bzw. der Zylinderachse, Luftfahrtforschung 19, 302 (1942). 37. BERTRAM,M. H., and FELLER,W . V., A simple method for determining heat transfer, skin friction, and boundary-layer thickness for hypersonic laminar boundary-layer flows in a pressure gradient, N A S A M E M O 5-24-59L, 1959. E., Similar solutions for the compressible laminar 38. COHEN,C. B., and RESHOTKO, boundary layer with heat transfer and pressure gradient, N A C A T R 1293, 1956. 39. SMITH,A. M. O., Rapid laminar boundary-layer calculations by piecewise application of similar solutions, Jour. Aero. Sci. 23, 901-912 (1956). 40. COLE, J. D., Lift of slender nose shapes according to Newtonian theory, Rep. P-1270, Rand Corp., Feb. 1958. 41. GONOR,A. L., Hypersonic flow around a cone at a n angle of attack, Zzv. A N U S S R , O . T . N . , No. 7 , 102 (1958). (Translated b y Morris D. Friedman, Inc.) 42. COLE, J . D., Newtonian flow theory for slender bodies, Jour. Aero. Sci. 24, 448 (1957). D. L., and BERTRAM, M. H.. The blunt plate in hypersonic flow. N A S A 43. BARADELL, T N D-408 (1960). 44. CASACCIO, A., Theoretical pressure distribution on a hemisphere cylinder combination, Jour. Aero/Space Sci. 28, No. 1 (1959). A., JR., and JOHNSTON,PATRICK J., Fluid-dynamic properties of 45. HENDERSON, some simple sharp- and blunt-nosed shapes a t Mach numbers from 16 to 24 in helium flow, N A S A M E M O 5-8-59L, 1959. 46. CREAGER, M. O., The effect of leading-edge sweep and surface inclination on the hypersonic flow field over a blunt flat plate, N A S A M E M O 12-26-58A. 1959. M. H., and HENDERSON, A., JR., Effects of boundary-layer displacement 47. BERTRAM, and leading edge bluntness on pressure distribution, skin friction, and heat transfer of bodies at hypersonic speeds. N A C A T N 4301, 1958. 48. VAS, I. E., and BOGDONOFF, S. M., Mach and Reynolds number effects on the flows over blunt flat plates a t hypersonic speeds, Princeton University Rept. 529, ARL Tech. Note 60-164, 1960.

The Mathematical Thsory of Equilibrium Cracks in Brittle Fracture

BY G . I . BARENBLATT Institute of Geology and Development of Combustible Minerals of the U.S.S.R. Academy of Sciences. Moscow. U.S.S.R.'

Page

. I1. The I11. The

I Introduction

............................

56

. . . . . . . . . .

62

Structure of the Edge of an Equilibrium Crack in a Brittle Body . . 1. Stresses and Strains Near the Edge of an Arbitrary Surface of Discontinuity of Normal Displacement . . . . . . . . . . . . . . . . . 2 . Stresses and Strains Near the Edge of an Equilibrium Crack . . . . . 3 . Determination of the Boundaries of Equilibrium Cracks . . . . . . .

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Development of the Equilibrium Crack Theory

69 73 74

.

IV Basic Hypotheses and General Statement of the Problem of Equilibrium Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Forces of Cohesion; Inner and Edge Regions; Basic Hypotheses . . . 2 . Modulus of Cohesion . . . . . . . . . . . . . . . . . . . . . . . 3 . The Boundary Condition a t the Contour of an Equilibrium Crack . . 4 . Basic Problems in the Theory of Equilibrium Cracks . . . . . . . . 5 . Derivation of the Boundary Condition a t the Contour of an Equilibrium Crack by Energy Considerations . . . . . . . . . . . . . . . . . . 8. Experimental Confirmation of the Theory of Brittle Fracture; Quasi-Brittle Fracture 7 . Cracks in thin Plates . . . . . . . . . . . . . . . . . . . . . . .

...........................

V. Special Problems in the Theory of Equilibrium Cracks . . . . . . . . . 1. Isolated Straight Cracks . . . . . . . . . . . . . . . . . . . . . . 2 . Plane Axisymmetrical Cracks . . . . . . . . . . . . . . . . . . . 3 . The Extension of Isolated Cracks Under Proportional Loading; Stability of Isolated Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . Cracks Extending to the Surface of the Body 5. Cracks Near Boundaries of a Body; Systems of Cracks . . . . . . . 8 Cracks in Rocks . . . . . . . . . . . . . . . . . . . . . . . . .

.

VI . Wedging; Dynamic Problems in the Theory of Cracks 1 Wedging of an Infinite Body . . . . . . . . . . 2 . Wedging of a Strip . . . . . . . . . . . . . . . . 3 . Dynamic Problems in the Theory of Cracks . . .

.

References

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

................................

* Present address: Institute Moscow. USSR .

of

Mechanics.

56

Moscow

State

76 76 80 81 82 84

85 89

90 90 98 97 103 107 112 114 114

119 121

125

University.

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G . I . BARENBLATT

I. INTRODUCTION In recent years the interest in the problem of brittle fracture and, in particular, in the theory of cracks has grown appreciably in connection with various technical applications. Numerous investigations have been carried out, enlarging in essential points the classical concepts of cracks and the methods of analysis. The qualitative features of the problems of cracks, associated with their peculiar non-linearity as revealed in these investigations, makes the theory of cracks stand out distinctly from the whole range of problems in the present theory of elasticity. The purpose of the present paper is to present a unified view of how the basic problems in the theory of equilibrium cracks are formulated, and to discuss the results obtained.

FIG.1.

FIG.2.

The object of the theory of equilibrium cracks is the study of the equilibrium of solids in the presence of cracks. Consider a solid having cracks (Fig. 1) which are in equilibrium under the action of a system of loads. The body, able to sustain any finite stresses, is assumed to be perfectly brittle, i.e. to retain the property of linear elasticity up to fracture. The possibility of applying the model of a perfectly brittle body to real materials will be discussed later. The opening of a crack (the distance between the opposite faces) is always much smaller than its longitudinal dimensions; therefore cracks can be considered as surfaces of discontinuity of the material, i.e. of the displacement vector. Henceforth, unless the contrary is stated, plane cracks of normal discontinuity are considered, i.e. cracks are pieces of a plane bounded by closed curves (crack colztozlrs), at which only the normal component of the displacement vector has a discontinuity. The case when the tangential component of the displacement vector is discontinuous a t the discontinuity surface can be treated in the same manner. One might think that the investigation of the equilibrium of elastic bodies with cracks can be carried out by the usual methods of the theory of elasticity in the same way as it is done for bodies with cavities (Fig. 2).

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57

However, there exists a fundamental distinction between these two problems, The form of a cavity undergoes only slight changes even under a considerable variation in the load acting upon the body, whereas cracks, whose surface also constitutes a part of the body boundary, can expand a good deal even with small increase of the load, to which the body is subjected. (In Figs. 1 and 2, dotted lines indicate additional loads and the corresponding positions of the body boundaries.) Thus, one of the basic assumptions of the classical linear theory of elasticity is not satisfied in problems of the theory of cracks, namely the assumption about the smallness of changes in the boundaries of a body under loading, which permits one to satisfy the boundary conditions at the surface of the unstrained body. This fact makes the problem of the equilibrium of a body with cracks, unlike traditional problems of the theory of elasticity, essentially non-linear. In the theory of cracks one must determine from the condition of equilibrium not only the distribution of stresses and strains but also the boundary of the region, in which the solution of the equilibrium equations is constructed. Non-linear problems of this type (“problems with unknown boundaries”) have long been known in various fields of continuum physics. Suffice it to mention the theory of jets and the theory of finite-amplitude waves in hydrodynamics, the theory of flow past bodies in the presence of shock waves in gas dynamics, Stefan’s problem of freezing in the theory of heat conduction, etc. The main difficulty in all these problems lies in the determination of the boundary of the region in which the solution is sought. Likewise, the basic problem in the theory of equilibrium cracks is the determination of the surfaces of cracks when a given load is applied. The differential equations of equilibrium and the usual boundary conditions of the theory of elasticity cannot in principle give the solution of this problem without the introduction of some additional considerations. This may be seen from the fact that we can construct a formal solution of the equations satisfying the usual boundary conditions no matter how we prescribe crack surfaces. The analysis of these solutions shows that in general the tensile stress u normal to the surface of a crack is infinite a t the crack contour. More exactly, near an arbitrary point of the crack contour

N

u =-

VS

+

finite quantity,

where s is the distance of a point of the body lying in the plane of a crack from the crack contour, N is the stress intensity factor, a quantity dependent on the applied loads, the form of the crack contour, and the coordinates of the point considered, but independent of s. The form of a normal section of the deformed crack surface near the contour appears in such cases unnaturally rounded (as in Fig. 3 or somewhat different; see details below).

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Generally speaking, however, there exist such exceptional contours of cracks for which stresses at the edges are finite ( N = 0) under a given load; at the same time the opposite faces of cracks close smoothly at the edges. The form of a section of the crack surface near the edge appears then as a cusp, cf. Fig. 4. It can be shown that for such contours, and only for them, the energy released by a small change in the contour of a crack is equal to zero. It follows that only such contours can bound equilibrium cracks.

FIG.3.

FIG.4.

Thus, when all loads acting upon a body are given, the problem of the theory of equilibrium cracks may be formulated as follows: for a given position of initial cracks and a given system of forces acting upon the body one requires the determination of the stresses, the strains, and the contours of cracks so as to satisfy the differential equations of equilibrium and the boundary conditions, and to insure finiteness of stresses (or, which is the same, a smooth closing of the opposing faces at the crack edges). If the position of the initial cracks is not given, then, since according to our model the body can sustain any finite stress, the solution of the problem formulated above is not unique. This is only natural because at one and the same load in one and the same body there need not be any cracks, or there may be one crack, or two, and so on. In the general case of curved cracks, the shape is determined not only by the load existing at a given moment but also by the whole history of loading. If however, the symmetry of the body and the applied monotonically increasing loads assure the development of plane cracks, then the contours of cracks are determined by the current load alone. All the results at present available in the theory of cracks correspond to particular cases of this simplified formulation of the problem. A given system of forces acting upon the body should in general include not only the loads applied to the body. The following example illustrates what is meant. Let us attempt to determine the contour of an equilibrium crack in the case of the loads represented in Fig. 1. If, in accordance with the usual approach in the theory of elasticity (as in the case of the cavity shown in Fig. 2), the surface of the crack is considered to be free of stresses, the result will be paradoxical: whatever contour of the crack we would

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take, the tensile stresses at its edge are always infinite. Consequently, there cannot exist an equilibrium crack; however small the force of extension may be, the body that has a crack breaks in two! Such an obvious lack of agreement with reality can be easily explained: simply using the model of an elastic body, we have not taken into consideration all forces acting upon the body. It appears that - and this is also one of the main distinctions between the problems of the theory of cracks and the traditional problems of the theory of elasticity - for developing an adequate theory of cracks it is necessary to consider molecular forces of cohesion acting near the edge of a crack, where the distance between the opposite faces of the crack is small and the mutual attraction strong. Although consideration of forces of cohesion settles the matter in principle, it complicates a great deal the analysis. The difficulty is that neither the distribution of forces of cohesion over the crack surface nor the dependence of the intensity of these forces on the distance between the opposite faces are known. Moreover, the distribution of forces of cohesion in general depends on the applied loads. However, if cracks are not too small, there is a way out of the difficulty: with increasing distance between the opposite faces the intensity of forces of cohesion reaches very quickly a large maximum, which approaches Young’s modulus and then diminishes rapidly. Therefore two simplifying assumptions can be made. The first is that the area of the part of the crack surface acted upon by the forces of cohesion can be considered as negligibly small compared to the entire area of the crack surface. According to the second assumption the form of the crack surface (and, consequently, the local distribution of forces of cohesion) near the adges, a t which the forces of cohesion have the maximum intensity, does not depend on the applied load.* The intensity of the forces of cohesion has the highest possible value for a given material under given conditions. This happens for instance a t all edge points of a crack formed at the initial rupture of the material as the load increases. For most real materials cracks are irreversible under ordinary conditions. If an irreversible crack is produced by an artificial cut without subsequent expansion or is obtained from a crack that existed under a greater load by diminishing the load, then the intensity of forces of cohesion a t the crack contour will be lower than the maximum possible one. The forces of cohesion that act a t the surface of a crack compensate the applied extensional loads and secure finiteness of stresses and smooth closing of the crack faces. With an increase in extensional loads the forces of cohesion grow, thus adjusting themselves to the increasing tensile stresses, and the crack does not

* Sh. A . Sergaziev very neatly compared cracks for which these assumptions are satisfied with “zippers”.

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expand further until the highest possible intensity of forces of cohesion is reached. The crack starts expanding+ only upon reaching the highest possible intensity of forces of cohesion at the edge. Successive expansion of the crack edge under increasing extensional load is represented schematically in Fig. 5.

FIG. 5. 1.2. The intensity of forces of cohesion is less than the maximum.

3,4. The intensity of forces of cohesion is equal to the maximum.

If use is made of the first of the above assumptions, molecular forces of cohesion will enter in the conditions that determine the position of crack edges only in the form of the integral d

where G ( t ) is the intensity of the forces of cohesion acting near the crack edge, t is the distance along the crack surface taken along the normal to the crack edge, and d is the width of the region subject to the forces of cohesion. For those contour points, at which the second assumption applies, this integral represents a constant of the given material under given conditions (temperature, composition, and pressure of the surrounding atmosphere, etc.), which determines its resistance to the formation of cracks. It can be shown that the quantity K is related to the surface tension of the material To, the modulus of elasticity E , and Poisson's ratio v by means of the simple equation

Furthermore, for all points of the crack edge at which the intensity of forces of cohesion is a maximum, the stress-intensity factor N , entering in + Quite a similar situation arises when a body moves over a rough horizontal surface under the action of a horizontal force. The motion of the body begins only after the force exceeds the highest possible value of the friction for the given body and the given surface.

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(1.1) and calculated without taking into account the forces of cohesion should be equal to Kln. For all points of the edge at which the intensity of forces of cohesion is below the maximum, the stress intensity factor calculated without considering forces of cohesion is smaller than Kln. The foregoing considerations elucidate sufficiently the nature of the forces of cohesion involved in this problem, and it is now possible to formulate the fundamental problem of the theory of equilibrium cracks.* When the symmetry of the body, of the initial cracks, and of the monotonously increasing forces insures development of a system of plane cracks, this problem can be stated as follows. Let a system of contours of initial cracks be given in a body. I t is required to find the stress and displacement fields corresponding to a given load as well as a system of contours of plane cracks surrounding the contours of the initial cracks (and perhaps coinciding with them partly). Mathematically the problem consists in constructing such a system of contours that the factor of intensity of the tensile stress, calculated without taking into account the forces of cohesion, should be equal to K l n at all edge points, not lying on the contours of the initial cracks, and should not exceed K l n a t all points of contours, lying on the contours of the initial cracks. The foregoing formulation of the problem eliminates from our direct consideration the molecular forces of cohesion (they enter only through the constant K ) . Therefore stress and strain fields furnished by the solution of this problem will not be realistic in a small neighbourhood of the crack edges. When cracks are reversible, or when the applied load is great enough to cause the contours of all the cracks to expand beyond the contours of the initial cracks, the form of the latter evidently is no longer of any importance. The equilibrium state corresponding to the highest possible intensity of forces of cohesion a t least at one point of the crack contour can be stable or unstable. Accordingly, further extension of the crack with increasing load proceeds in essentially different ways. In the case of stable equilibrium, a slow quasi-static transition of the crack from one equilibrium state to another takes place, when the load is increased gradually. If the equilibrium is unstable, the slightest excess over the equilibrium load is accompanied by a rapid crack extension that has a dynamic character. In some cases, when there exist no neighbouring stable states of equilibrium, this leads to the complete rupture of the body. The theory of cracks developed in such a way that problems of this latter type were mainly treated until recently.

* Such general formulations of problems seem advisable to us despite the fact that their general solution in effective form exceeds by far the possibilities of present mathematics. General statements of problems are a help in realizing the meaning of specific solutions and difficulties arising in developing the theory.

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Sometimes the condition for the onset of crack extension is therefore identified with the condition for complete fracture of the body. It should be clearly understood, however, that this is only true in special cases, the practical significance of which must not be exaggerated. Below, following a brief outline of the development of the mathematical theory of cracks, the fundamentals of the theory of equilibrium cracks are given as well as the results for the most typical special problems treated hitherto. A t the end of this review dynamic problems in the theory of cracks are discussed briefly. When writing this article the author endeavoured to avoid the repetition of available presentations of some aspects of brittle fracture. Thus the review deals with the theory of cracks proper, i.e. with the mathematical theory of brittle fracture. The numerous available experimental investigations are referred to only inasmuch as they are necessary for confirming the theory presented and establishing the limits of its applicability. Experimental investigations of brittle fracture, unlike the mathematical theory, were discussed more than once in reviews and monographs. At the same time, questions concerning exclusively mathematical techniques of solving the problems of elasticity theory are discussed only briefly, if a t all. Also the question of the formation of the initial cracks will not be touched. Trying to preserve a unified point of view in discussing certain results of other investigators, the author permitted himself sometimes a deviation from the original treatment.

11. THE DEVELOPMENT OF THE EQUILIBRIUM CRACKTHEORY

Investigations in the field of the theory of cracks were started by C. E. Inglis [ l ] about fifty years ago. His paper presents the solution of a problem within the classical theory of elasticity concerning the equilibrium of an infinite body with an isolated elliptical cavity (in particular, with a straight-line cut) in a uniform stress field. N. I. Muskhelishvili [2] - also within the classical theory of elasticity - obtained in a simpler and more effective form the solution of a problem concerning the equilibrium of an infinite body having an elliptical cavity in an arbitrary stress field. However, in spite of their outstanding significance for subsequent investigations, papers [l, 21 did not prepare the foundations for the theory of cracks proper. The fact is that the solutions obtained in these papers possess two properties which were difficult to explain. First, the length of a crack was found to be indefinite at a given load so that it was possible to construct a solution with an arbitrary value of this parameter. Everyday experience suggests nevertheless that the dimensions of cracks existing in a body should be connected somehow with the extensional loads applied to

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

03

the body. As the load increases, cracks existing in the body do not expand at first when the load is small; upon reaching a certain load they begin to expand, the expansion depending on the manner in which the load is applied. In some cases cracks expand rapidly up to complete rupture of the body with the load maintained constant, in other cases they expand slowly, stopping as soon as the increase of the load is suspended. Since the opening of a crack is usually small compared to its longitudinal dimensions, it is natural to represent a crack as a cut; but then the tensile stresses at the crack edges in Inglis’ problem are infinite, and in general the same thing happens in the problem treated by Muskhelishvili. Clearly solutions with infinite tensile stresses at the edges of a crack are unacceptable in a physically correct model of a brittle body. Thus, direct application of the classical scheme of the theory of elasticity to the problem of cracks leads to a problem which is incomplete and yields physically unacceptable solutions. A. A. Griffith’s papers [3, 41 are rightly considered fundamental for the theory of cracks of brittle fracture. The important idea, first advanced in these papers is that an adequate theory of cracks requires the improvement of the model accepted for a brittle body by the consideration of molecular forces of cohesion acting near the edge of a crack. Griffith treated the following problem : An infinite brittle body stretched by a uniform stress Po at infinity has a straight crack of a certain size 21. I t is required to determine the critical value of Po at which the crack begins to expand. The molecular forces of cohesion were considered as forces of surface tension being internal forces for the given body; their effect on the stress and strain field was neglected. Under this condition the change dF of free energy (“total potential energy” in Griffith’s terminology) of a brittle body with a crack, compared to the same body under the same loads but without a crack, is equal to the difference between the surface energy of the crack U and the decrease in strain energy of the body due to formation of the crack W . For the crack to expand, the change in free energy of the body must not grow with an increase in the size 21 of the crack. Thus, the parameters of the critical equilibrium state are obtained from the condition

a(u- w) = 0. ai

But the surface energy of the crack U is equal to the product of the surface area of the crack and the energy To required to form the unit surface of the crack. Under certain sufficiently general assumptions, the quantity To, the surface tension, can be considered constant for a given material under given conditions. Therefore, according to Griffith, the determination of the critical load reduces to the determination of the quantity aWla1, “the elastic energy release rate”. Analysing the simplest case, Griffith calculated

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G . I . BARENBLATT

this quantity by using Inglis’ results [l] and obtained relations determining the critical values of tensile stress in the forms

for plane strain and plain stress, respectively. The theoretical part of Griffith’s paper contains also the results of the investigation of the structure of a crack near its ends. This is carried out on the basis of the classical solution of elasticity theory, constructed without considering forces of cohesion, hence with infinite tensile stresses at the ends of the crack, if it has the shape of a cut. Griffith made an attempt to improve this description of a crack by considering it as an elliptical cavity with a finite radius of curvature p at the end (Fig. 3). However, according to his own estimate the magnitude of the radius of curvature at the end of the crack was of the order of the intermolecular distance, which clearly indicates the incorrectness of the approach: in any investigation based on the concept of a continuous medium distances of intermolecular order of magnitude cannot be considered as finite. This part of Griffith’s work is inadequate for the following reason. In determining the equilibrium size of a crack, the effect of molecular forces of cohesion on the stress and strain fields can be neglected, but this cannot be done in analysing the structure of a crack near its ends. The distance at which the effect of forces of cohesion is appreciable is comparable to the distance over which the form of a crack varies essentially. Therefore, to a considerable part, Griffith’s analysis of the structure of crack edges cannot be accepted as correct, and in particular his conclusion concerning the rounded form of cracks near the ends is wrong, as will be shown in detail later. This aspect of the matter, obviously of prime importance, remained unclarified until recently and led in a number of cases to misinterpretations of Griffith’s results [B]. In addition t o the basic shortcoming pointed out here, there were some errors in calculations in the theoretical part of the paper [3]. Shortly after it had appeared, A. Smekal [6] published a detailed comment on it, containing also quite an interesting general discussion of the problem of brittle fracture and correcting the errors. In a subsequent paper by K. Wolf [7] a more precise and simpler account of Griffith’s results was given, and similar calculations were made for somewhat different (but also uniform) states of stress. In [7] the relation of Griffith’s theory of fracture to previously proposed theories of strength was also discussed. In connection with his experiments on the splitting of mica I. V. Obreimov investigated [8] the tearing-off of a thin shaving from a body by a splitting wedge that slides over its surface and has a single point of contact with the

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shaving. Using the approximate methods of thin-beam theory, Obreimov established the relation between the form parameters of a crack and the surface tension by means of an energy method similar to that used in Griffith’s paper. The method of paper [8] was continued later by many investigators [9-121. The determination of the elastic energy release rate awlal for tensile stress fields more complex than a uniform one, as well as for other configurations of cracks encountered considerable mathematical difficulties. The investigations of H. M. Westergaard [13], I. N. Sneddon [14, 151, I. N. Sneddon and H. A. Elliot [16], M. L. Williams [17] clarified the distribution of stresses and strains near the discontinuity surfaces of the displacement. Together with the classical papers by Muskhelishvili [ 2 , 18, 191 the investigations of Westergaard and Sneddon constitute the mathematical basis of subsequent works on the theory of cracks. However, the conditions of equilibrium for new particular cases and, still less, for a somewhat more general case of loading were not obtained in these papers. In the papers by R. A. Sack [20], T. J . Willmore [21], and 0. L. Bowie [22] the conditions of equilibrium were obtained for some new special cases of loading and position of cracks. The energy method was applied directly in these papers, and thus considerable difficulties in the calculations had to be overcome. In view of the fact that the equilibrium states in the problems treated in [20-221 are unstable and unique, the conditions of equilibrium are identical with those for complete fracture of the body. The papers by G. R. Irwin [23] and E. 0. Orowan [24], in which the concept of quasi-brittle fracture was developed, represent an important stage in the theory of cracks. Irwin and Orowan noticed that a number of materials, which behave as highly ductile in standard tensile tests, fracture by a quasi-brittle mechanism when cracks are forming. This means that the arising plastic deformations are concentrated in a very narrow layer near the surface of a crack. As was shown by Irwin and Orowan, it is possible in such cases to employ Griffith’s theory of brittle fracture, introducing instead of surface tension the effective density of surface energy. This quantity, in addition to the specific work required to produce rupture of internal bonds (= surface tension), includes the specific work required to produce plastic deformations in the surface layer of a crack; it is sometimes several orders of magnitude larger than the surface tension. The idea of quasi-brittle fracture extended considerably the range of applicability of the theory of brittle fracture and was undoubtedly one of the main reasons for reviving interest in this problem. Irwin, Orowan and other authors published a series of papers [23-321 devoted to the development of the generalized theory of brittle fracture, to the investigation of the limits of its applicability, and to the analysis of experimental data from the view point of this theory. Special notice deserves the paper by

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H. F. Bueckner [33] in which a quite general energy analysis of brittle and quasi-brittle fracture was carried out on the basis of the Griffith-IrwinOrowan scheme. In all the foregoing papers the question of the structure of a crack near its edge remained without clarification. In a very interesting paper [34] devoted to the physico-chemical analysis of deformation processes, P. A. Rebinder first expressed the thought about the wedge-like form of a crack at its ends and about the necessity of a corresponding development of Griffith’s theory. H. A . Elliot [35], N. F. Mott [36], and Ya. I. Frenkel [5], in analysing the form of a crack, proceeded from the idea of a crack of infinite length between two solid blocks of the material, which were at normal intermolecular distance from each other before formation of the crack. In [35] the blocks were considered to be semiinfinite. Starting from the classical solution for a straight-line crack [l] and a disk-shaped crack [20] having a diameter 2c in a uniform tensile stress field fi, the distributions of normal stresses cry and lateral displacements v were determined in [35] for points of the planes distant half the normal intermolecular distance from the crack plane. The function ay(2v)containing fi and c as parameters was identified with the relation between molecular forces of cohesion and the distance ; by integrating this function, the surface tension was determined, which thus was found to be connected with p and c. The author identified this relation with the condition of fracture, which of course differed from Griffith’s condition. The distribution of the lateral displacements so obtained was identified with the form of the crack. Such an approach is inadequate for the following reasons. The formal application of the apparatus of classical elasticity for the determination of stresses and deformations near the edge of a crack is unjustifiable, since in applying this apparatus all distances (even those which are considered small) must be large compared to the intermolecular distance. Moreover, forces of cohesion act not only inside the body but also on a part of the crack surface. If this fact is taken into account, the edges of a crack have a pointed rather than a rounded shape, and there is no infinite stress concentration at the ends. This will be shown below in detail. Thus, stress and displacement distributions near the edge of the crack surface differ essentially from the corresponding distributions obtained according to the solutions of Inglis [11 and Sack [20], in which the surface of cracks was supposed to be free of stress. Note also that the decrease of a,(2v) with increasing v is very slow in paper [35], much slower indeed than the natural velocity of diminution of the intensity of forces of cohesion, Ya. I. Frenkel [5] treated the problem of a crack of infinite length cutting through a thin strip in longitudinal direction. The use of the approximate theory of thin beams, which is unsuitable for analysing the form of a crack

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

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near its ends, did not permit him to obtain an adequate result. Incidentally, the comments on Griffith’s theory contained in this paper cannot be accepted as well justified either. Frenkel criticizes Griffith because of the instability of the equilibrium in the case of a straight crack in a uniform tensile stress field (considered by Griffith) and he ascribes this instability to Griffith’s wrong idea about the form of the crack ends. This is not true. The conclusion about stability or instability of equilibrium of a crack does not depend on considerations concerning the structure of the crack ends. As will be shown later, instability of a crack in a uniform field occurs even when allowance is made for smooth closing of cracks a t the ends; it is a part of the problem itself rather than a consequence of the peculiar crack shape assumed. Frenkel’s conclusion concerning the existence of a stable state of equilibrium in addition to the unstable one is due to his incorrect replacement of the uniform state of stress by another one.* In a paper by A. R. Rzhanitsyn [37] an attempt was made to solve the problem of a circular crack in a body subjected to a uniform tensile stress under consideration of the molecular forces of cohesion distributed over the crack surface and with smooth closing of the crack. Unfortunately the application of inadequate methods (averaging stresses and strains) did not allow the author to obtain the correct conditions of equilibrium. An idea first suggested by S. A. Khristianovitch [38] was of great importance for the proper understanding of the structure of cracks near their ends. Khristianovitch considered, in connection with the theory of the so-called hydraulic fracture of an oil-bearing stratum, an isolated crack in an infinite body under a constant all-round compressive stress a t infinity, maintained by a uniformly distributed pressure of a fluid contained inside the crack. The problem was treated in the quasi-static formulation. In solving it, Khristianovitch hit upon the indefiniteness of the crack length. He noticed, however, the following circumstance. Under the assumption that the fluid fills the crack completely, tensile stresses a t the end of the crack are always infinite, whatever the size of the crack. But if the fluid fills the crack only partially, so that there is a free portion of the crack surface which is not wetted by the fluid, then at one exceptional value of the crack length tensile stresses a t the ends of the crack are finite. I t turned out that for this value of the crack length (and only for this one) the opposite faces of the crack close smoothly at its edges. Khristianovitch advanced a hypothesis of finiteness of stresses or, which is the same, of smooth closing of the opposite faces of a crack a t its edges as a fundamental condition determining the size of a crack. The use of this hypothesis made it possible to solve a number of problems concerning formation and expansion of cracks in rocks [38--431.

* Besides these basic shortcomings there are some errors in calculations in [5] indicated in [37].

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In all these papers, however, molecular forces of cohesion were not taken into account directly. Now in dealing with cracks in rock massifs it is quite permissible to neglect forces of cohesion. The estimates show that the effect of rock pressure is far greater here than the action of forces of molecular cohesion, particularly if the natural fissuring of rocks is taken into consideration. Under other conditions (in particular, in many cases when massifs are simulated in laboratories) forces of cohesion play an important part and their consideration is of great significance in analysing the conditions of equilibrium and expansion of cracks. A very interesting early work by H. M. Westergaard [44] should be mentioned in connection with these investigations (see also [13]). On the basis of the analogy with the contact problem noted by the author, it is stated that there is no stress concentration at the end of a crack in such brittle materials as concrete. The same paper gives formulas which describe correctly stresses and strains near the ends of equilibrium cracks of brittle fracture in the absence of forces of cohesion. However, Westergaard did not connect the condition of finiteness of stress with the determination of the longitudinal dimension of a crack, which he assumed to be given. I n papers [45, 461 by G. R. Irwin (see also [47, 48, 49, 331) an important formula was established that correlates the strain-energy release rate with the stress intensity factor near the ends of a crack in a problem of the classical theory of elasticity. On the basis of this formula the strain-energy release rate was determined, and the conditions of fracture were obtained for several new cases of loading and position of cracks [47, 50, 32, 51, 521. Beginning with the work of Griffith, in most of the theoretical investigations problems of a similar type were treated: the equilibrium state, .,in which the intensity of forces of cohesion at the contour is a maximum, turns out to be unstable, and the condition for the onset of expansion of a crack coincides with the condition for the beginning of complete fracture of the body. Thus the condition for onset of the expansion is identified in some papers with the onset of rapid crack propagation and fracture for all cracks. In general, that is not true. Cracks actually may be stable so that the beginning of crack development is not necessarily connected with the fracture of a body; and one should not imagine that stable cracks are rare, that they are not encountered in practice and are difficult to produce experimentally. As the experimental investigations carried out by numerous authors beginning from I. V. Obreimov [8] show, the extension of cracks is stable in many cases throughout the greater part of the process of fracture. A. A. Wells [30] obtained stable cracks over a certain range of extensional forces in steel plates under combined external tensile stresses and internal stresses due to welded seams. F. C. Roesler [53] and J. J. Benbow [54] investigated stable conical cracks in glass and silica. The same authors [ Q ] obtained stable cracks in wedging a strip of organic glass. Recently

69

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

J. P. Romualdi and P. H. Sanders [52] obtained stable cracks within certain limits of loads for a tensile plate stiffened by riveted ribs. References to other investigations in which stable cracks were obtained and analysed can be found in a monograph by B. A. Drozdovsky and Ya. B . Fridman [65]. All these papers confirm strongly the possibility of using the concept of brittle and quasi-brittle fracture for stable cracks. Consideration of stable cracks greatly extends the problems that can be formulated in the theory of equilibrium cracks. Indeed, in the case of unstable cracks, only the determination of the load a t which a crack begins to expand is of interest, since the process becomes dynamic upon reaching this load. In the case of stable cracks, however, one has to investigate the quasi-static expansion of cracks with change in loads. In papers [56-611 the formulation of problems in the theory of equilibrium cracks of brittle fracture was improved and supplemented in accordance with the foregoing considerations. In these papers a new approach to problems of the theory of cracks was proposed, which is based on the general formulation of problems concerning elastic equilibrium of bodies in the presence of cracks, as it was given in [40]. The further discussions in this review are based on this approach, which is presented in the following chapter. A number of new problems of the theory of cracks were formulated and solved on that basis.

111. THE STRUCTURE OF THE EDGEOF AN EQUILIBRIUM CRACK I N BRITTLE BODY

A

1. Stresses and Strains Near the Edge of an Arbitrary Surface of Discontinuity of Normal Displacement

As has already been pointed out, one can construct a formal solution of the differential equations of the theory of elasticity, which satisfies the boundary conditions corresponding to the applied load, if one prescribes arbitrarily a surface of discontinuity of the displacement. In the present section the behavior of the solutions of the equations of elasticity near the edge of a surface of discontinuity of displacement is investigated. For simplicity we shall restrict ourselves here to surfaces of discontinuity of normal displacement, appearing as plane faces bounded by closed curves (contowrs). Near an arbitrary point 0 a t the contour of such a surface, let us take a vicinity whose characteristic dimension is small compared to the radius of curvature of the contour at the point 0. Deformation in this vicinity can be considered as plane and corresponding to a straight infinite cut in

70

G. I. BARENBLATT

an infinite body subjected to a system of symmetrical loads (see Fig. 8; the plane of deformation is a plane normal to the contour of the discontinuity surface at the point 0; the trace of the cut in the drawing is the intersection of that plane with the discontinuity surface). Loads can be applied a t the surface of the cut and inside the body; the loads a t the surface can be assumed to be normal without losing the generality of the further analysis. Consider now this configuration in more detail.

FIG.8.

The stress and displacement fields can be presented as the sum of two fields (Fig. 6), the first of which corresponds to a continuous body under loads applied inside the body; the second belongs to a body with a cut, symmetrical loads being applied at the surface of the cut only. The shape of the deformed surface of the cut is determined by the second state of stress, since normal displacements at the place of the cut for the first state of stress are equal to zero by symmetry.* The analysis of the first state of stress can be carried out by the usual methods of the theory of elasticity and is of no special interest; we shall consider this state of stress as given. Let us assume that the line of the cut corresponds to the positive semi-axis x ; the normal stresses, g ( x ) , applied at the surface of the cut in the second state of stress, represent the difference between the stresses applied a t the surface of the cut in the actual field, G ( x ) , and the stresses at the place of the cut, P ( x ) , corresponding to the first state of stress. Applying Muskhelishvili’s method [18] to the analysis of the second state of stress, we obtain the relations determining stresses and displacements (34

(3.2)

+

ax(2) a,(2)= 4 Re @(z),

u p - ia:;

= @(z)

+ Q(2) +

(2

-2 ) @ 3 ,

This convenient method of reducing the load to a load distribution over the discontinuity surface was developed in the most general form by H. F. Bueckner [33].

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

x =3

71

- 4v,

+

where z = x i y ; U,(~),U~(~),CT~,! are the components of the stress tensor of the second state of stress; d2),d2)are the displacement components along the x and y axes corresponding to the second state of stress; ,u = E / 2 ( 1 Y) is the shear modulus, E is Young's modulus, and v is Poisson's ratio. The analytical functions p,u,@,Qare expressed by formulas

+

(3.4)

[VFy

@ ( z ) = . n ( z ) = p'(z) = o ' ( z ) = w 1 ~, 2ni v z 0

(3.5)

A t the cut ( x 2 0, y following relations hold :

= 0)

and its prolongation ( x

< 0,

y = 0) the

Using known formulas for limiting values of a Cauchy-type integral a t the ends of the contour [19], we obtain an expression for the tensile stresses near the end of the cut along its prolongation,

(3.7)

where s1 is the small distance of the point considered from the end of the cut (Fig. 6 ) . Similarly, we have for the distribution of normal displacements of points a t the surface of the cut near its end

where s2 is the distance of a surface point of the cut from its end, and negative and positive signs correspond to the upper and lower faces of the cut, respectively (Fig. 6 ) .

72

G . I. BARENBLATT

This result also fully elucidates the distribution of normal tensile stresses and normal displacements near the contour of an arbitrary surface of normal discontinuity. Indeed, the following formulas are readily obtained from relations (3.7) and (3.8):

where uy is the tensile stress at a point of the body a small distance s1 away from the contour of the discontinuity surface, lying in the osculating plane to the contour of the discontinuity surface through the point 0; N is the

4"

a

f"

b

f"

0

C

X

FIG. 7.

stress intensity factor, a quantity dependent on the acting loads, on the the configuration of the body and of the discontinuity surfaces in it, and on the coordinates of the point 0 considered; G(0) is the magnitude of the normal stress applied to the discontinuity surface at 0 (Fig. 6 ) ; s2 is the small distance of a point of the discontinuity surface from its contour. Depending on the sign of N , there are in general three possibilities. If N > 0 , an infinite tensile stress acts a t the point 0. The shape of the deformed discontinuity surface and the distribution of normal stresses uy near the point 0 are represented in Fig. 7a. If N < 0 , then an infinite compressive stress acts at the point 0; the shape of the deformed discontinuity surface and the distribution of stresses near 0 are represented in Fig. 7b. The opposite faces of the crack overlap in this case, and it is quite evident that this case is physically unrealistic. Finally, if N = 0 , the stress acting near the contour is finite and tends to the normal stress applied at point 0 of the contour if 0 is approached. Thus the stress ay is continuous at the contour, and the opposite faces of the discontinuity surface close smoothly (Fig. 7c).

MATHEMATICAL T H E O R Y O F E Q U I L I B R I U M CRACKS

73

The investigation of the stress and strain distribution near the edge of the surface of normal discontinuity was begun by Westergaard [44, 131 and Sneddon [14, 151 and continued later by the author [40], by Williams [17], and by Irwin [45-471. In view of the character of the stress states considered in [14, 15, 45-47] results were obtained only for the case N > 0.

2. Stresses and Strains Near the Edge of an Equilibrium Crack The results obtained in the preceding section pertain to an arbitrary surface of discontinuity of normal displacement. We now show that, for an equilibrium crack, N = 0 a t all points of its contour.

FIG. a.

Consider a possible state of the elastic system, which differs from the actual state of equilibrium only by a slight variation in the form of the crack contour in a small vicinity of the arbitrary point 0 (Fig. 8). The new contour is a curve that encloses the point 0 lying in the plane of the crack. This curve is tangential to the former contour of the crack at points A and B close to 0 ; everywhere else the contours of all the cracks remain unchanged. In view of the closeness of the points of tangency A and B to the point 0, the initial contour of the crack at the portion A B can be considered as straight. The distribution of normal displacements of the points of the new crack surface and the distribution of tensile stresses a t these points prior to the formation of the new crack surface are, according to the above, given, to within small quantities, by (3.10)

v = T

4(1 - y 2 ) N

E

~

Vh-y,

a,=

N

VY - 1

where N is the stress intensity factor a t the point 0. The energy released in the formation of the new crack surface, which is equal to the work required to close this new surface, is given by

74

G . I . BARENBLATT b

(3.11)

h

- 2(1 - va),N2 -

hdx

E

=

2(1 - v ~ ) T c N ~ ~ S

E

a

where 6s is the area of the projection of the new crack surface on its plane. The condition of equilibrium of the crack requires that 6 A vanishes; this together with (3.11) implies that N = 0. Thus we arrive a t a very important result characterizing the structure of cracks near their contours : 1. T h e tensile stress at the contour of a crack i s finite. 2. The o#posite faces of a crack close smoothly at its contour. I t appears, therefore, that contrary to Griffith’s conception the form of a crack near its edge is as represented in Fig. 4. Since the only acting forces at the surface of a crack near its contour are forces of cohesion, it follows from (3.9) that the tensile stress at the crack contour is equal to the intensity of forces of cohesion at the contour. In particular, if there are no forces of cohesion, the tensile stress at the crack contour is equal to zero. The condition of finiteness of stresses and smooth closing of the opposite faces a t the edges of a crack was first suggested as a hypothesis by S. A. Khristianovitch [38], to serve as a basic condition that determines the position of the crack edge. The proof of this condition given above follows [60] mainly. Formula (3.11) for the case of plane stress was first proved by Irwin [45, 461 irrespective of finiteness of stresses and smoothness of closing (see also the review by Irwin [47] and the paper by Bueckner [33]). The early paper by Westergaard [44] contains a statement concerning the absence of stress concentration at the end of a crack in brittle materials like concrete, but the condition of finiteness of stress that appears in this work was not connected with the determination of the size of the crack. We have confined ourselves here to the examination of cracks of normal discontinuity only for simplicity of treatment. Analogous reasoning, in particular the proof of finiteness of stress a t the crack edge, can be extended without any substantial changes to cover the general case in which also the tangential displacement components have a discontinuity a t the crack surface. 3. Determination of the Boundaries of Equilibrium Cracks

The conditions of finiteness of stresses and smooth closing of a crack a t its contour permit us to formulate the problem of equilibrium cracks for a given system of loads acting upon the body: for a given position of initial

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

75

cracks and a given system of forces acting upon the body, it is required to find stresses, deformations, and crack contours in the elastic body so as to satisfy the differential equations of equilibrium and the boundary conditions, and to insure finiteness of stresses and smooth closing of the opposite faces at the crack contours. We shall illustrate the solution of this problem by an elementary example of an isolated straight crack in an infinite body under all-round compressive stress q a t infinity and with concentrated forces P applied at opposite points of the crack surface (Fig. 9). The solution of the equilibrium equations satisfying the boundary conditions can be obtained by Muskhelishvili's method [18] for an arbitrary crack length 21. Stresses and displacements are expressed by formulas (3.1)-(3.3) with @(z) =

4c2 (3.12) z

='(c4 +

+).

FIG.9.

Evidently, equilibrium equations and boundary conditions do not determine the length of the crack. The distributions of stresses uy at the prolongation of the crack and normal displacement v of points of the crack surface near its edge are given by

(3.13)

Finiteness of stress and smooth closing of the crack at its ends are assured simultaneously by the condition (3.14)

1=-,

P n4

which determines the crack size under given loads P and q. Let us now attempt to determine the size 21 of an isolated straight crack in an infinite body stretched by uniform stress Po at infinity in the direction perpendicular to the crack. If the crack surface is assumed to be free of

76

G . I . BARENBLATT

stress, then one can easily show that the tensile stress a t the prolongation of the crack near its edge depends on the distance s1 as follows:

Po v 1 .

(3.15)

UY =--

V2S, ’

hence it appears that for no 1 the stress uy will be finite a t the crack end and there does not exist an equilibrium crack! This paradoxical result is due to the fact that we did not take into account the molecular forces of cohesion acting near the crack edges and thus did not completely account for the loads acting upon the body. The consideration of these forces and the definitive formulation of problems in the theory of equilibrium cracks of brittle fracture are discussed in the following section.

IV. BASICHYPOTHESESAND GENERALSTATEMENT OF EQUILIBRIUM CRACKS

THE

PROBLEM OF

1. Forces of Cohesion; Inner and Edge Regions; Basic Hypotheses

In order to construct an adequate theory of cracks of brittle fracture, it is necessary t o supplement the model of a brittle body by considering the molecular forces of cohesion acting near the edge of a crack at its surface. It is known that the intensity of forces of cohesion depends strongly on the distance. Thus, for a perfect crystal the intensity f of forces of cohesion acting between two atomic planes a t the distance y from each other is zero if y is equal to the normal intermolecular distance b. With y increasing up to about one and a half of b, the intensity f grows and reaches a very high maximum f, VETo/b E/10; after that it diminishes rapidly with further increase of y (Fig. 10). Here E is Young’s modulus, and To is the surface tension related to f ( y ) by the formula

-

N

m

r

b

The maximum intensity fm defines the theoretical strength, i.e. the strength of a solid if it were a perfect crystal. The actual strength of solids is usually several orders of magnitude lower because of defects of crystal structure. For amorphous bodies the relation between the intensity of forces of cohesion and the distance has qualitatively the same character. Data at present available, which confirm the above character of the relation between the intensity of forces of cohesion and the distance, lead

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

77

to the following conclusion. I t has long been known that the strength of thin fibers exceeds considerably that of large specimens of the same material [62, 631. Experiments carried out recently with filamentary crystals of some metals revealed an exceptionally high strength approaching the theoretical value [63]. I t is supposed that this phenomenon is due to the relatively small amount of structural defects in thin fibers and filamentary crystals. Furthermore, numerous direct measurements of the intensity of molecular forces of cohesion for glass and silica [64-661 were made recently. The

FIG.10.

FIG.11. I-inner region, ZZ-edge region.

very elegant method used in this kind of measurements is based on the application of a regenerative microbalance and was suggested and employed by B. V. Deryagin and I. I. Abrikosova [64, 651. However, these direct measurements deal with very great distances y compared to the normal intermolecular distance and thus determine only the end of the falling branch of the curve f ( y ) . A macroscopic theory for forces of cohesion at such distances was developed by E. M. Lifshitz [64] and was found in good agreement with the results of these aforementioned measurements. The relation f ( y ) , if y equals several normal intermolecular distances, seems to be beyond any strict quantitative theory and difficult for the direct experimental determination at present. A description of available attempts to estimate the relation f ( y ) at such distances and, consequently, the theoretical strength can be found in [67, 63, 681. The distance between the opposite faces of a crack varies from magnitudes of the order of the intermolecular distance near the crack edge to sometimes rather great magnitudes far from the edge. I t is therefore convenient to divide the crack surface into two parts (Fig. 11). The opposite faces in the first part - the inner region of the crack - are a great distance apart, hence their interaction is vanishingly small, and the crack surface can be considered free of stresses caused by the interaction of the opposite faces. The opposite faces of a crack in the second part are adjacent to the crack contour - the edge region of che crack - and come close to each other so that the intensity

78

G. I. BARENBLATT

of the molecular forces of cohesion acting on this part of the surface is great. Of course, the boundary between the edge and inner region of the crack surface is conventional to a certain extent. For very small cracks there may be no inner region of the crack a t all. Since the distribution of the forces of cohesion over the surface of the edge region is not known beforehand, a substantial part of the loads applied to the body is not known. I t is thus impossible to handle the problem of cracks directly in the way it was stated in Chapter 111. But the following method of solving problems of cracks is possible in principle: the distance between the opposite faces of a crack is found at each surface point as a function of the unknown distribution of forces of cohesion over the surface. Assuming the relation f ( y ) between forces of cohesion and distance as given, a relationship can be obtained which determines the distribution of forces of cohesion over the crack surface. Such an approach is not practicable. First, the relation f ( y ) is not known to a sufficient extent for a single real material. Even if it were known, the problem would constitute a very complex non-linear integral equation, the effective solution of which presents great difficulties even in the simplest cases.* Attempts were made to prescribe the distribution of forces of cohesion over the crack surface in a definite manner, but these attempts cannot be considered sufficiently well founded. For sufficiently large cracks, consideration of which is of principal interest, the difficulty connected with our lack of knowledge of the distribution of forces of cohesion can be avoided without making any definite assumptions concerning this distribution. In this case the general properties of the relation between forces of cohesion and distance allow the formulation of two basic hypotheses which not only simplify essentially the further analysis, but permit the determination of contours of cracks, although the forces of cohesion are finally altogether excluded from consideration as loads acting upon the body. First hypothesis: The width d of the edge region of a crack i s small compared to the size of the whole crack. This hypothesis is acceptable because of the rapid diminution of forces of cohesion with the increase in the distance between the opposite faces of

* In papers of M. Ya. Leonov and V. V. Panasyuk [69, 701 the relation f ( y ) is approximated by a broken line, and on the basis of this approximation a linear integral equation for the normal displacements of the crack surface points is derived. It is solved approximately, the representation of the solution being not quite successfully selected SO that the form of the crack at its end appears wedge-shaped with a finite edge angle. In fact, as was shown above, the edge angle must be zero. The shortcoming of these papers lies also in t h e application of the results obtained by the methods of mechanics of continua to cracks whose longitudinal dimensions are only of the order of several intermolecular distances.

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

79

a crack. Of course, there exist micro-cracks to which this hypothesis cannot be applied. However, as the width d of the edge region is quite small, the hypothesis is already valid for very small cracks and certainly for all macrocracks. Nevertheless, the width d is considered to be sufficiently great compared to micro-dimensions (for instance, compared to the lattice constant in a crystalline body), so that it is permissible to employ the methods of continuum mechanics over distances of the order of d.

Second hypothesis: The form of the normal section* of the crack surface i n the edge region (and consequently the local distribution of the forces of cohesion over the crack surface) does not depend on the acting loads and is always the same for a given material under given conditions (temperature, composition and pressure of the surrounding atmosphere and so on). When the crack expands, the edge region near a given point, according to the second hypothesis, moves as if it had a motion of translation, and the form of its normal section remains unchanged. This hypothesis is applicable only to those points of the crack contour where the maximum possible intensity of forces of cohesion is reached; an expansion of the crack occurs then a t this point with an arbitrarily small increase in the loads applied to the body. Equilibrium cracks, on whose contour is a t least one such point, will be called mobile-equilibrium cracks to distinguish them from immobile-equilibrium cracks which do not possess this property, i.e. do not expand with an infinitesimal increase in the load. Thus the second hypothesis and all conclusions based on it are applicable to reversible cracks as well as to irreversible equilibrium cracks, which formed a t the initial rupture of a brittle body in the process of increasing the load. I t is not applicable to cracks which result from equilibrium cracks existing at some greater load by diminishing that load; nor can it be applied, to artificial cuts made without subsequent expansion. The second hypothesis is suggested by the fact that the maximum intensity of the forces of cohesion is so very great and exceeds by several orders of magnitude the stresses which would arise under the same loads in a continuous body without a crack. Therefore it is possible to ignore the change of stress in the edge region when loads vary and, consequently, the corresponding variation of the normal sections. These two hypotheses reformulate the results of the qualitative analysis of the brittle-fracture phenomenon carried out by a number of investigators beginning with Griffith. They are the only assumptions concerning the forces of cohesion which underlie the theory presented below and appear in this explicit form in [56,571. = intersection '

with a plane normal to the crack contour.

80

G. I . BARENBLATT

2. Modulus of Cohesion

The body considered is assumed to be linearly elastic up to fracture. The elastic field in the presence of cracks can then be represented as the sum of two fields: a field evaluated without taking into account forces of cohesion and a field corresponding to the action of forces of cohesion alone. Therefore the quantity N entering in formulas (3.15) and, as was proved, equal to zero can be written as N = N o N,, where the stress intensity factor N o corresponds to the loads acting upon the body and to the same configuration of cracks without considering forces of cohesion, and the stress intensity factor N , corresponds to the same configuration of cracks and forces of cohesion only. According to the first hypothesis the width d of the edge region acted upon by forces of cohesion is small compared to the crack dimensions on the whole and, in particular, to the radius of curvature of the crack contour a t the point considered. In determining the value of N , we may thus assume that the field belongs to the configuration discussed in Section I I I , l , i.e. to an infinite body with a semi-infinite cut, with symmetrical normal stresses being applied to the surface of the cut. Hence it follows from (3.7) that

+

m

d

where G(t) is the distribution of forces of cohesion different from zero only in the edge region 0 t d . According to the second hypothesis, the distribution of forces of cohesion and the width d of the edge region at those points of the contour, where the intensity of forces of cohesion is a maximum, do not depend on the applied load; the integral in (4.2) represents then a constant characterizing the given material under given conditions. This constant will be denoted by K :

< <

.=IT. d

G(t)dt

(4.3)

0

I t was termed the modulzls of cohesion since this quantity characterizes the resistance of the material to an extension of its cracks, caused by the action of forces of cohesion. As will be shown below, the quantity K is the only characteristic of the forces of cohesion, that enters in the formulation of the problem of cracks. The dimension of the modulus of cohesion is: (4.4)

[ K ] = [ F ][ L ] - 3 / 2= [MI [ L ] - ' / 2 [ T ] - 2 ,

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

81

where [I;],[I.],[MI, and [TI denote the dimensions of force, length, mass, and time, respectively. Constants of a similar dimension are encountered in the contact problem of the theory of elasticity [71, 72, 731. It is no coincidence, that there exists a profound connection between the contact problem and problems in the theory of cracks of brittle fracture; it seems that this was first pointed out in the papers of Westergaard [44, 131.

3. T h e Boundary Condition at the Contour of a n Equilibrium Crack For points of the contour of an equilibrium crack, a t which the maximum intensity of cohesion is reached, the second hypothesis is applicable, and (4.2) may be written as

N,

(4.5)

considering that N

1

= --

K;

7c

= 0,

we obtain

The boundary condition a t contour points of an equilibrium crack, a t which the intensity of forces of cohesion is maximal, can also be formulated as follows: on approaching these points, the normal tensile stress a,, a t the points of the body lying in the crack plane, if calculated without taking into account forces of cohesion, tends to infinity according to the law (4.7)

cry = -

K

+ 0(1),

Vs

where s is the (small) distance from the contour point considered. Satisfying (4.6) a t least a t one point of the contour is the condition that the crack is in the state of mobile equilibrium. One should not connect, in general, the reaching of the state of mobile equilibrium by the crack with the onset of its unstable rapid growth and still less with complete fracture of the body. A mobile-equilibrium crack may be either stable or unstable. Only in case of instability is the condition for the onset of rapid crack propagation given by (4.6). However, not even in this case is complete fracture of the body unavoidable, since the transition from the unstable state of equilibrium to the other, stable one, is possible. Numerous examples illustrating various possibilities will be discussed in the following chapter.

82

G. I. BARENBLATT

If a crack is irreversible and there are points on its contour where the intensity of forces of cohesion is less than maximal,* then the second hypothesis is not applicable a t such points. Since cohesive forces that act in the edge region of the crack surface are smaller near such points than those acting near points of the type considered above, it follows from (4.2) that - N , < K l h ; and since N o = - N,, we have for these points (4.6 a)

K No<--. n

As the load increases, forces of cohesion in the edge region grow; they compensate the increase in the load and insure finiteness of stress and smooth closing at the crack contour. However, the crack does not expand at a given contour point until the forces of cohesion become maximal. The second hypothesis now becomes applicable, and condition (4.6) is satisfied. In determining the form of contours of equilibrium cracks, conditions (4.6) and (4.6a) permit us to exclude the forces of cohesion altogether from the consideration of the loads acting upon the body. Instead, we work with their overall integral characteristic, the modulus of cohesion. Special estimates show [57, 581 that the influence of molecular forces of cohesion on the stress and displacement field is essential only in the neighbourhood of the edge in a region of the order of magnitude d . Forces of cohesion thus determine the structure of cracks near their ends, and the forms of crack contours depend on them only through the integral characteristic K .

4. Basic Problems in the Theory of Equilibrium Cracks The basic problem in the theory of equilibrium cracks can be stated in its most general form as follows. A certain system of initial cracks and a process of loading the body, i.e. a system of loads acting upon the body, dependent on one monotonously increasing parameter A, are given. The value of I for the initial state may be assumed as zero. I t is required to determine the form of the crack surfaces and to find the distribution of stresses and strains in the body corresponding to any il > 0. The process of varying the load is supposed to be sufficiently slow so that dynamic effects need not be considered. When the symmetry of body, loads, and initial cracks insures the possibility of developing a system of plane cracks and the extensional loads grow monotonously with increasing A, the configuration of cracks in the body is determined by the current load only and not by the whole history of the

* For instance, contour points of non-expanded cuts or of cracks formed from cracks which existed under a greater load when the load is diminished.

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

83

process of loading, as it is in the general case. In this case the problem is formulated as follows (it will be called problem A ) . In a body bounded by a surface Z contours of an initial system of plane cracks I', are given (Fig. 12; the plane of the drawing is the plane of the cracks). It is required to find the enclosing the elastic field and the contours of a system of plane cracks contours To (and perhaps coinciding with them partially) corresponding to a given load, i.e. to a given value of 2.

r

This problem reduces mathematically to the following one. I t is required to construct the solution of the differential equations of equilibrium of elasticity theory in the regon bounded by plane cuts with contours 'I and by the body boundary Z under boundary conditions corresponding to the given load. The contours I' must be determined so that condition (4.6) is satisfied at points of these contours not lying on and condition (4.6a) at points of lying FIG. 12. on I',.

@

r,, r

If the cracks are reversible or if the applied loads are sufficiently great so that the contours T do not coincide with I', at any point, then the form of the initial contours is of no importance. It is then possible, without prescribing the initial cracks, to formulate directly the problem of determining the contours I' of equilibrium cracks of a given configuration so that condition (4.6) is satisfied at each point of Here we assume that the initial cracks are such that they are compatible with the realization of the given configuration of cracks when the load increases. This problem will be called problem B.

r.

I t may happen that a solution of either of the above stated problems does not exist. If this happens, it has quite a different significance for the problems A and B. If no solution of problem A exists this means that the applied load exceeds the breaking load, hence its application causes fracture of the body. The limiting value of the parameter 2 up to which the solution of problem A exists, corresponds to the breaking load. The determination of the breaking load for a given configuration of the initial cracks and a given process of loading presents an important problem in the theory of cracks. Non-existence of the solution of problem B signifies that, whatever initial cracks may be within a given configuration, they will not expand under a given load, hence the applied load is too small. In such cases the conventional description of the state would be that mobileequilibrium cracks do not form under the given load.

84

C. I. BARENBLATT

5. Derivation of the Boulzdary Condition at the Contour of a n Equilibrium Crack

b y Energy Considerations Molecular forces of cohesion so far have been considered as external forces applied to the surface of the body. This was necessary for analysing the structure of cracks near their ends. ‘If only boundary conditions are to be obtained, another approach can be employed which considers the forces of cohesion as internal forces of the system. On the basis of this approach, the idea of which goes back to Griffith [3,41, a relation between the modulus of cohesion and other characteristics of the material will be obtained. As before, let there be a certain configuration of equilibrium cracks in a brittle body and consider as in Section II1,2 a possible state of the elastic system, which differs from the real one only by a variation in the crack contour near a certain point 0 (Fig. 8). However, unlike Section 111,2, the characteristic size of the new area of the crack surface is assumed to be large compared to the dimension d of the edge region, though small compared to the size of the crack as a whole; according to the first hypothesis (Section IV,1) such an assumption is permissible. Under this assumption forces of cohesion can be considered merely as forces of surface tension, and a certain amount of work must be done to overcome these forces in increasing the crack surface. The influence of forces of cohesion on the stress and strain fields can be neglected since it is essential only in the neighbourhood of the crack edge, whose dimension is of the order of the width of the edge region. The work SA required for the transition from the actual state to a virtual one is equal to the difference between the corresponding increment in surface energy SU and released elastic energy SW: (4.8)

SA

= SU - SW.

For the actual state of an elastic system to be an equilibrium state, 6 A must vanish, hence (4.9)

6U

= 6W.

Quite similarly to Section III,2 an expression for SW is obtained: (4.10)

sw = 2(1 - v2)nNo2SS E

where N o is the value of the stress intensity factor at the point 0 calculated without taking into consideration the forces of cohesion. Formula (4.10) in a somewhat different form was established by Irwin [4547]. If the form of the edge region near a given point of the contour corresponds to the maximum intensity of forces of cohesion, then, according

85

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

to the above, in forming a new crack surface the edge region is displaced without deformation; the work against the forces of cohesion per unit of newly formed surface is then constant and equal to the surface tension To. Therefore, 6U = 2T0SS, because two surfaces form in rupture. Together with (4.9)and (4.10),we have

No =

(4.11)

ETo n(1 - v2) *

Comparing (4.11)and (4.6),we obtain a relationship correlating the modulus of cohesion K , defined independently by (4.3),with the surface tension To and the elastic constants of the material E and Y :

(4.12)

6 . Experimental Confirmation of the Theory of Brittle Fracture: Quasi- Brittle Fracture After Griffith's work [3, 41 many investigators attempted to carry out experimental verifications of the theory of brittle fracture. We cannot Spherical bulbs 21 inches

D inches

Po psi

0.15 0.27 0.54 0.89

1.49 1.53 1.60 2.00

864 623 482 366

Cylindrical tubes

POv'

237 228 251 244

21 inches

D inches

psi

0.25 0.32 0.38 0.28 0.26 0.30

0.59 0.71 0.74 0.61 0.62 0.61

678 590 526 655 674 616

240 232 229 245 243 238

analyse all this work here in detail and shall dwell only on several of the most characteristic papers, referring for details and discussion of other numerous investigations to the special publications [62, 55, 74-78]. Griffith's paper [3]gives descriptions and results of the following experiments. Cracks of various length 21 were placed on spherical glass bulbs and cylindrical tubes, whose diameter D was sufficiently great so that a special verification showed no influence of the diameter on the experimental results. After the tubes and bulbs had been annealed to relieve residual stresses

86

G. I. BARENBLATT

produced by making the cracks, they were loaded from the inside by hydraulic pressure up to fracture. The breaking stress Po corresponding to each crack length 21 was measured. According to the foregoing theory it appears that the breaking stress Po at which a given crack becomes unstable (onset of mobile equilibrium) can depend only on the crack length 21 and the modulus of cohesion K . From dimensional analysis [79] it follows that Po = a K / v c where a is a dimensionless constant. Consequently, +,Vdmust be constant for a given material (in full accord with (2.1)). " Griffith's experiments, which are tabulated here, confirm the constancy of this quantity and thus the foregoing theoretical scheme. FIG. 13. The remarkably elegant experiments of 1. Steel indentor. 2. SDecimen. Roesler [53] and Benbow [54], in which 3. Steel support. stable conical cracks were produced, are of special interest for the confirmation of the theory of brittle fracture. The scheme of these experiments is presented in Fig. 13; the photograph of conical cracks in fused silica, borrowed

'

FIG. 14.

from Benbow's paper [54], is given in Fig. 14. The cracks were formed by the penetration into a specimen of glass [53] and fused silica [54] of a cylindrical steel indentor with a flat end. In accordance with the

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

87

above, the diameter s of the base of a conical crack can depend only on the diameter do of the indentor base, the force P pressing the indentor, the modulus of cohesion K , and Poisson’s ratio v. Since the correct formulation of the corresponding problem of elasticity theory does not include Young’s modulus, it should not be included in the number of determining parameters of the crack problem. Dimensional analysis yields (4.13)

where p is a dimensionless function of its arguments.

I 1x10~

I

10x10~

I

100~ 107p

FIG. 15.

Experiments carried out with indentors of three diameters on eleven glass specimens [53] confirm well the existence of the universal relation (4.13). A t large values of P, when the first argument of the function q becomes vanishingly small, self-similarity takes place, and the following relationship holds : (4.14)

Fig. 15 represents a graph, taken from Benbow’s paper [54], of the s ( P ) relation according to data from experiments with fused silica carried out under conditions corresponding to the self-similar regime. As can be observed, these experiments give a conclusive proof of the validity of relation (4.14) and confirm thereby the above scheme. The experiments described were carried out with materials which can be considered as perfectly brittle. This refers especially to fused silica. Benbow [54] presents certain facts indicating that the mechanism of crack

88

G . I . BARENBLATT

formation in fused silica is closer to being perfectly brittle than it is in glass: cracks in glass grow for a long time under constant load, whereas in fused silica their size is established quickly and then remains unchanged; after removal of the load, cracks in glass remain distinctly visible, but in silica they are imperceptible, etc. However, the significance of the theory of brittle fracture greatly exceeds what should be the limits of its applicability to those comparatively rare materials that are perfectly brittle. Experimental investigations show that when cracks appear some materials, which behave as highly plastic bodies in common tensile tests, fracture in such a way that plastic deformations, though present, are concentrated in a thin layer near the crack surface.

D. K. Felbeck and E. 0. Orowan [28] carried out experiments on fracture of low-carbon steel plates with a saw-cut crack under conditions corresponding to Griffith’s scheme of uniform extension. Experimental results are in good agreeFIG.18. ment with Griffith’s formula, but the surface-energy -- density exceeds by about three orders of magnitude the surface tension of the material investigated. It was found in good agreement with the specific work of plastic deformations in the layer near the crack surface, which was determined by independent measurements. On the basis of this and similar experimental results Irwin [23] and Orowan [24] advanced the concept of quasi-brittle fracture, which permitted an important extension of the limits of applicability of the theory of brittle fracture. Here the theory of brittle fracture covers the case when the plastic deformations are concentrated in a thin layer near the crack surface. The energy T required to form the unit surface of a crack is expressed as the sum of the specific work against the forces of molecular cohesion To (= surface tension) and the specific work of plastic deformation T,: (4.15)

T

= To

+ T,.

A formal extension to quasi-brittle fracture is made as follows (Fig. 16, the plastic deformation zone near the surface is shaded). Imagirfc the whole plastic region cut out and shift the crack end to the end of the plastic region. This can be done, if the forces exerted by the plastic zone upon the elastic zone are considered as external forces applied to the crack surface. After that the previous reasoning remains unchanged, if the plastic zone is assumed as thin and use is again made of the hypothesis concerning the invariability

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

89

of the edge region (which here includes the boundary of the elastic and plastic zones). The modulus of cohesion is now expressed as

(4.16) where G,(t) is the distribution of normal stresses acting on the boundary of the elastic and plastic zones. When the contribution of molecular forces of cohesion to integral (4.16) can be ignored in comparison to the contribution of stresses that act in the region ahead of the actual crack end and have the order of magnitude of the yield point stress a,, we obtain an estimate for the modulus of cohesion:

(4.17) Note that the value of a, at the yield point near the crack end may differ from that a t the yield point obtained in tensile tests with large specimens. The concept of quasi-brittle fracture is somewhat related to the concept of the “plastic particle” a t the ends of notches with a zero radius of curvature, advanced in a classical monograph by H. Neuber [80]. In the following we shall speak of cracks of brittle fracture, bearing in mind the possibility of extending the results to the case of quasi-brittle fracture. Of course, in this latter case it is necessary to take into consideration the irreversibility of cracks of quasi-brittle fracture. 7. Cracks in Thin Plates

If the state of stress can be assumed to be plane, then all relations derived previously for the case of plane strain hold also for thin plates, if only E is replaced by E(l - v2) and the modulus of cohesion is assumed to have some other value K,. Repeating the derivation of formula (4.12)for the plane stress state we obtain

(4.18)

K I 2= n E T .

The experiments show that the surface energy density T in the case of quasi-brittle fracture increases with a reduction in the plate width [48], which is due to a broadened plastic-strain zone near the crack surface. An approximate theoretical analysis of this phenomenon was attempted by I. M. Frankland [81]. Bearing in mind the complete analogy of the analysis of plane stress and plane strain we shall in the following consider only plane strain.

90

G. I. BARENBLATT

V. SPECIALPROBLEMS IN

THE

THEORY OF EQUILIBRIUM CRACKS

This Chapter deals with solutions of special problems in the theory of cracks available at present. A few of the examples have illustrative character, but most problems presented are interesting in themselves. 1. Isolated Straight Cracks

In this and the following section isolated mobile-equilibrium cracks are examined, and all along the contour the maximum intensity of forces of cohesion is assumed to prevail. The problem reduces here to the determination of crack contours corresponding to a given load so that condition (4.6) is satisfied at these contours, and it represents a particular case of problem B formulated above. I t is supposed that the initial cracks guarantee the possibility of producing such cracks ; the necessary requirements for the initial cracks in the cases of reversible and irreversible cracks follow readily from the solutions obtained. Let us consider an isolated straight mobile-equilibrium crack extending along the x-axis from x = a to x = b in an infinite body subject to plane strain. Let p ( x ) be the distribution of normal stresses, which arise a t the place of the crack in a continuous body under the same loads. This distribution is computed by the usual methods of elasticity, and we may consider it as given. I t may be shown by using Muskhelishvili’s solution [2, 181 that tensile stresses near the crack ends calculated without taking into account forces of cohesion become infinite according to the law ay = N / V s + . . ., where h

I.

are the values of the stress intensity factors for points a and b, respectively. Satisfying condition (4.6) at these points, we obtain relations that determine the coordinates of the crack ends a and b : b

(5.2) 1p(d a

ve

-~

x - a dx = K v b - a,

[p(x)

v E

d x =K

V G .

In particular, if the applied load is symmetrical with respect to the crack middle, where we place the origin of coordinates, then - a = b = 1, and Eqs. (5.2) become one relation determining the half-length of the crack 1 :

MATHEMATICAL T H E O R Y OF EQUILIBRIUM CRACKS

91

(5.3) Note that (5.2) and (5.3)represent finite equations, since f i ( x ) is a given function. These equations determine the position of the ends of an isolated straight-line mobile-equilibrium crack under a given load, if this load guarantees that such a crack can exist.

FIG.17.

A method to calculate the strain-energy release rate aWlal for a symmetrical isolated crack was indicated by K. Masubuchi [82]. He proposed a trigonometrical representation of stresses $(x ) and displacements v of points of the crack surface, m

(5.4) f i ( x ) = E 41

2

m

sin d l n A n x

I

v

n=l

=4

2AnsinnB,

x =lcosO.

n= 1

As was shown by Masubuchi,

(5.5)

aw - -ai

m

nE

2 (nAn)2.

qi -t q z n= 1

Equating this expression to 4 T , where T is the surface energy density, a relation between the applied stresses and the crack size can be obtained, though in a form far more complicated than (5.3). Let us now look a t a few examples. A crack may be kept open by a uniform tensile stress Po applied a t infinity. As already pointed out, this

92

G . I . BARENBLATT

problem was first treated by Griffith [3, 41. equation (5.3) yields

In this case p ( x )

Po

and

Relation (5.6) appears in Fig. 17 as the dotted line. One sees that the size of a mobile-equilibrium crack diminishes with increasing tensile stress, which is indicative of the instability of mobile equilibrium in this case. Despite this instability the size I defined by (5.6) has a physical meaning: If there is a crack of length 21, in a body, to which constant tensile stress Po is applied at infinity, then a t I,< I this crack does not expand (and closes if it is a reversible crack) while at I, > I it grows indefinitely. Thus, the equilibrium size is in a certain sense critical (this will be discussed in more detail in Section V,3). It is obvious that instability of mobile equilibrium in this case fully corresponds to the substance of the matter and, contrary to the opinion expressed by Frenkel [5], is not connected with Griffith’s incorrect ideas about the geometry of the crack ends. If stresses vanish at infinity, and if a crack is maintained by a uniformly distributed pressure applied over a part of its surface (0 x I,) while the remaining part of the crack surface (I, x I ) is free of stress, then the half-length of the mobile-equilibrium crack I is given by the relation [58]

< <

< <

(5.7)

This relation is shown in Fig. 17 by the solid lines which may be obtained from each other by a similarity transformation. It is evident that the opening of a crack, i.e. the appearance of a free segment, is possible provided 1, is not less than the corresponding size of a mobile-equilibrium crack kept open by a uniform tensile stress a t infinity, Po, which is determined b y (5.6). Therefore all the solid lines (Fig. 17) must start from the dotted line. A limiting case of (5.7) is of interest. It occurs when p , tends to infinity Const = P. This corresponds to a crack and I, tends to zero, while 2p,l, kept open by concentrated forces applied at opposite points of its surface. The half-length of the crack is then given by

Note that (5.6) and (5.8) may be obtained, disregarding the value of the numerical factor, by dimensional analysis. For example, the size of a crack maintained by concentrated forces is determined only by the magnitude P of these forces and the overall characteristic of the forces of cohesion, K . It is obvious that the modulus of elasticity and Poisson’s ratio do not enter

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

93

in the number of determining parameters, since the corresponding problem of the theory of elasticity is naturally formulated only in terms of stresses. Considering the dimensions of P and K , we see that it is possible to set up only one combination with the dimension of length from these quantities, namely the ratio P 2 / K 2 ,and no dimensionless combination exists. Thus the length of a mobile-equilibrium crack must be proportional to P 2 / K 2 , and the coefficient of proportionality a universal constant [cf. 791. Let now a crack be maintained by two equal and opposite concentrated forces P , whose points of application are separated by L along the common line of action of the forces; the crack is supposed to be perpendicular to the line of action of the forces and located symmetrically [ 5 8 ] .

FIG.18.

The distribution of tensile stresses at the place of the crack in a continuous body is in this case given by (5.9)

(the origin of coordinates is taken in the middle of the crack). Using (5.3), we obtain the relation determining the crack size in the form (5.10)

A plot of PIKVZ versus the relative length of the crack IIL for v = 0.25 is shown in Fig. 18. As can be seen, at P > Po two lengths of a mobileequilibrium crack correspond to each value of P , the smaller decreasing and the greater increasing with increasing P. States of mobile equilibrium corresponding to the smaller equilibrium length are unstable ; the corresponding branch of the load-length diagram in Fig. 18 is shown by the dotted line. States corresponding to the greater length are stable (solid line in Fig. 18).

94

G. I . BARENBLATT

The smaller size I, is the critical size at a given load P ; initial cracks present in the body and smaller than 21, do not expand under the action of applied loads of magnitude P (in case of reversible cracks they close), and those which are greater expand until the crack reaches the second (stable) equilibrium size.* At P < Po equation (5.10) has no solution. This means that, whatever length of the initial crack we take, it will not develop into a mobileequilibrium crack at the given load. The size of a mobile-equilibrium crack lo different from zero corresponds to the critical value of forces Po.

FIG. 19.

An interesting problem concerning the influence of riveted stiffeners on crack propagation was treated by J. P. Romualdi and P. H. Sanders [ 5 2 ] . This problem is schematized by the authors as follows (Fig. 19). An infinite plate is stretched by a uniform stress Po in the direction perpendicular to a crack. The action of the rivets and the stiffeners is represented by two symmetrically located pairs of opposite concentrated forces equal in magnitude to P ; they are considered as given. Substituting the corresponding stress distribution in (5.3)and working out? the elementary though somewhat cumbersome integrals, we obtain the relation between the applied load and the half-length of an equilibrium crack 1:

+ + A z ( A +12(1 B-2)vA-B+2

1-v AvA--2 (5.11)

2(1

+ v ) ( 2 B- A -___4)

Aq/A-B+2

9 --,Y 0 - L

1

l=z,

;

4y02

+

yo

+

( 1 + v ) ( B A ) ( 2 B- A - 4) A 3 ( A B - 2) V A - B T ]

+

B = y o z + l z + 1,

+-nv% .’

A =VBZ-412.

* Note that, because of dynamic effects accompanying the expanding of the initial cut, the crack actually may “overshoot” the stable equilibrium state to %me extent. This will be discussed later in more detail. t Computation of the integrals and numerical calculations for the graphs in Fig. 20 were made by V. Z. Parton and E. A. Morozova.

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

95

The results of the calculations are plotted in Fig. 20 for Y = 0.25, P / K V z = 0.2 and for several values of the parameter yo/L. As is seen, mobile-equilibrium cracks are unstable in the absence of stiffeners. The influence of the stiffeners shows itself first of all in an increase of the size of a mobile-equilibrium crack at a given load and, as an especially important feature, in the appearance of stable states of mobile equilibrium at sufficiently small yo/L, i.e. when rivets are spaced closely enough. The appearance of stable states of mobile equilibrium changes considerably the character of the crack expansion (see details below).

1.0

L

05 -

t FIG.20.

The authors observed experimentally the transition of cracks from unstable mobile-equilibrium states to stable ones; their experiments, carried out with aluminium alloy plates in the presence and absence of stiffeners, reveal a considerable increase in size of mobile-equilibrium cracks in the presence of stiffeners at the same value of Po. In [52] the stress intensity factor a t the crack ends was also determined experimentally for several stable and unstable mobile-equilibrium states. In the absence of stiffeners, measurements of the stress intensity factor were made by the direct method, i.e. by diminution of tensile stresses near the crack ends (at distances obviously large compared to the size of the crack-edge region). In the presence of stiffeners the stress intensity factors were measured indirectly. The values of these factors were found to coincide except in two cases when they were smaller by approximately 15 per cent. However, these two tests carried out with one and the same specimen, with a stable crack in one case

96

G . I . BARENBLATT

and unstable crack in the other, gave values of the stress intensity factor close to each other. (A somewhat lower value of this factor at the end of the stable crack can be explained by the considerable dynamic effects which, according to the authors, occur in the transition from the unstable state to the stable one.) Thus it may be supposed that the deviation observed is due to some peculiarity of the specimen. Altogether, these experiments confirm directly the proposed general scheme. This discussion can be readily extended to straight cracks in an anisotropic medium, placed in the planes of elastic symmetry of the material. The problem of a straight crack in an orthotropic infinite body subjected to a uniform stress field was treated by T. J. Willmore [21] and A. N. Stroh [83]. In [83], the results of [16] were also extended to cover the case of a straight crack in an anisotropic body under an arbitrary stress field, and the stress intensity factors at crack ends were found for this problem. Paper [84] brings the solution of the general problem concerning a straight mobileequilibrium crack in an orthotropic body subjected to an arbitrary stress field symmetrical with respect to the line of the crack.

2. Plane Axisymmetrical Cracks

If a disk-shaped mobile-equilibrium crack of radius R is maintained in an infinite body by an axisymmetrical load, tensile stresses near the crack contour calculated without taking into account forces of cohesion tend to infinity according to the law R

(5.12)

where p ( 7 ) is the tensile-stress distribution at the place of the crack in a continuous body subjected t o the same loads. According to the general condition (4.7),the equation determining the radius of a mobile-equilibrium crack R is R

(5.13)

This equation was established in [56, 571. Its derivation is based on the application of the method of Fourier-Hankel transforms, developed by I. N. Sneddon [14, 151 for solving axisymmetrical problems of elasticity. In particular, if a mobile-equilibrium crack is kept open by a uniform tensile

97

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

stress at infinity is given by

Po, then

p(r)

Po

and the radius of an equilibrium crack

(5.14)

This problem was first solved by R. A. Sack [20] by the energy method; his method is quite similar in principle to Griffith's [3, 41 treatment of the corresponding plane problem. If there is no tensile load at infinity, and if the crack is kept open by a uniformly distributed pressure Po over a part of its surface (0 Y ro) while the remaining part of the crack surface (ro Y R ) is free, then the radius of the mobile-equilibrium crack is found from the relation

< <

< <

(5.15)

Here, just as in the plane case, the radius of the loaded part of the crack surface ro must not be less than the critical radius for a given pressure Po, which is defined by (5.14). In particular, if a disk-shaped crack is maintained open by equal and opposite concentrated forces P applied a t its surface, then the radius of a mobile-equilibrium crack is determined by the formula (5.16)

Relations (5.14) and (5.16) can be obtained, except for the numerical factor, from dimensional analysis (cf. (5.6) and (5.8)). If a disk-shaped crack is kept open by equal and opposite forces P whose points of application are 2L apart along the common line of action, then the radius of a mobile-equilibrium crack R is determined from the equation (5.17)

P KL3I2

-

L2

The above solutions were obtained in [56, 571 ; the interpretation of the relations obtained is quite similar to the corresponding cases for a straight crack. 3. T h e Extension of Isolated Cracks Under Proportional Loading; Stability of Isolated Cracks The problem of this section is a special case of problem A. A complete investigation is carried out for symmetrical loading and initial cracks, straight and disk-shaped cracks being considered simultaneously. An example of a problem concerning the growth of an unsymmetrical initial

98

G . I. BARENBLATT

crack is given, which illustrates the general procedure of solving this problem. Under proportional loading the tensile stresses at the place of the crack, but in a continuous body subjected to the same load, are proportional to the loading parameter I ; hence p ( x ) = A / ( x ) and p ( r ) = A/(,f(r)in the cases of straight and disk-shaped cracks, respectively. Introducing the dimensionless variable 5 equal to x/1 and r / R in these cases, respectively, one obtains relations (5.3) and (5.12) in the form

p

=q(c),

(5.18)

where q ( c ) is defined respectively by

and c denotes, respectively the half-length 112 or the radius R. Thus the relation of the crack length to the parameter I of proportional loading is completely determined by the length of the initial crack and by the function q ( c ) , corresponding to a given load distribution. Certain properties of the function q ( c ) can be obtained under the most general assumptions. Omitting the case of a crack maintained by concentrated forces applied at its surface, let us suppose that the crack is kept open by any loads, in particular, by concentrated loads applied inside the body and perhaps by distributed loads applied a t the crack surface. In this case the functions p ( x ) , p ( r ) , and, consequently, / ( c E ) are obviously bounded. For small c we obtain from (5.19), respectively:

Suppose that the tensile loads applied to the body on each side of the crack are bounded and, for definiteness, equal to AP. Then the following relations are valid : m

5

-m

(5.21)

m

m

@(x)dx= IP,

5

0 m

P

f(c6)dE = 2c '

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

99

Eqs. (5.21) and (5.19) yield asymptotic representations for the functions ~ ( c ) when c 00 : -+

(5.22)

Thus, under the assumptions made, p(c) tends to infinity when c + O and c- 00. Owing to the boundedness of f(c[), the integrals in expressions (5.19) do not become infinite a t any c, therefore p(c) vanishes nowhere and, consequently, has at least one positive minimum, one falling branch, and one rising branch. If the forces applied to the body on either side of the crack are not bounded, then the function p(c) may not have rising branches and, consequently, minima. This happens in particular in case of a uniform tensile stress field when p = lfio and (5.23)

for a straight and axisymmetrical crack, respectively. By definition, an equilibrium crack is stable if no (sufficiently small) change in its contour produces forces which tend to move the crack further away from the disturbed state of equilibrium. It is evident that immobileequilibrium cracks are always stable. For stability of a mobile-equilibrium crack it is necessary that its size should grow with an increase of the loading parameter I . Suppose indeed that the corresponding size of a mobileequilibrium crack c grows with increasing load. If the crack size is diminished without changing the load ( I = const), the crack extension force will be greater than it was in equilibrium. Therefore the equilibrium is disturbed, and the crack tends to widen under the action of the excess force. Conversely, if the crack size is slightly increased compared with its equilibrium size, then the equilibrium is disturbed in the opposite direction, and the crack tends to close, if it is reversible.* If near a given equilibrium state the equilibrium crack size c diminishes with an increase of I , then it is obvious that its small change under a constant load will produce forces favouring further departure from the equilibrium state. The corresponding equilibrium state will be unstable. Hence the equilibrium state of a crack is stable, if for given c and 1 the following condition is satisfied: (5.24)

adc> o .

* If the crack is irreversible, then with an increase in its size no reverse closing takes place, but no further expansion of the crack takes place either. Equilibrium is attained in this case because of diminution of forces of cohesion acting in the edge region of the crack.

100

G . I . BARENBLATT

Differentiating (5.18) with respect to I , we find (5.25) Thus the condition for stability of the state of mobile equilibrium is

y'(4 > 0, (5.26) and only those states of mobile equilibrium are stable which correspond to rising portions of the curve y(c).

4

cz

c

0 Go

GI

c2

FIG.21.

Now we have everything that is necessary for the complete investigation of the extension of an isolated symmetrical crack under proportional loading. Let a function y(c), such as shown in the graph of Fig. 21, correspond to a given system of loads applied to the body and consider first the case when p(c)+ 00 as c+ OQ (Fig. 21a). Such a case occurs in particular when the loads applied on both sides of the crack are bounded. Let the dimension of the initial crack 2c, correspond to an unstable brach of y(c). Then the crack length remains constant with increase of I , until I reaches the magnitude, for which the initial crack of size 2c, becomes one of mobile equilibrium. Since the mobile equilibrium is unstable, the crack begins to expand under constant load, until it reaches the nearest stable mobile-equilibrium state. With further increase of I the crack size grows continuously, until the load corresponding to a maximum of y(c) is reached, then changes again in a stepwise manner when the transition to another stable branch takes place, after which it grows continuously with increasing A. The path of the point representing the change of the crack is indicated by the number 1 in Fig. 21a.* Let now the size of an initial crack 2c, correspond to a stable

* Owing to dynamic effects that occur in this transition, the crack may overexpand a little beyond the size of the stable mobile-equilibrium crack corresponding to the given load (apparently that happened in the experiments described in paper [ S Z ] ) . In this case, a further increase in the load leaves the length unchanged up to reaching mobile equilibrium, after which the crack starts t o lengthen further. Naturally, the purely static theory considered here cannot describe these dynamic effects; the corresponding parts of the graph in Fig. 21a are dotted and designated by the number 1'.

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

101

branch of ~ ( c ) . The crack size now remains unchanged up to the load a t which it reaches mobile equilibrium, after which it increases continuously. The path of the representative point is indicated by the number 2 in Fig. 21a. In the case considered, no fracture of the body occurs for any values of the parameter 1. If 1 is less than its critical value (corresponding to the lowest of the minima of ~ ( c ) ) then , great as the size of the initial crack may be, it does not expand under a given load. The size of the mobile-equilibrium crack corresponding to this critical value of A is finite. This means in particular: if a crack is kept open by forces applied inside the body and perhaps by loads distributed over the crack surface, and if the forces applied on each side of the crack are bounded, then there exists a critical value of the parameter 1; for all values of 1 greater than the critical one there exists a t least one stable and one unstable state of mobile equilibrium. Let us now turn to the case when ~ ( c-+) 0 as c + 00 (Fig. 21b). If the size of an initial crack 2c, corresponds to a stable branch of ~ ( c )then , the crack does not expand until a load is reached at which its state becomes a mobile equilibrium. After that, the crack grows continuously with increasing A, until a value of 1 is reached that corresponds to a maximum. If this I-value is exceeded, the solution of the problem does not exist any longer, and fracture of the body occurs. The path of the representative point is indicated by the number 1 in Fig. 21b. If the size of an initial crack 2c, corresponds to the right-hand unstable branch of p(c), then no expansion of the initial crack occurs with increasing 1,until a value of 1 is reached for which the state of the initial crack becomes a mobile equilibrium. The slightest exceeding of this value of 1 causes complete fracture of the body. If the size of an initial crack 2c3 corresponds to the left-hand unstable branch of the curve ~ ( c ) then , for c3 < co the crack develops in the same manner as in case 2 ; for c3 > co the development of the crack is similar to case 1 in Fig. 21a before reaching a maximum, after which the body fractures. The investigation of other forms of the curve ~ ( c can ) easily be carried out by combining the cases considered. We see that the knowledge of the ) it possible to describe completely the behavior of a function ~ ( c makes symmetrical isolated crack in an infinite body under proportional loading. In the case of reversible cracks, a change in the crack size can be traced by means of the graph of p(c) also for a non-monotonous variation in the load. I t is of interest to note that in this case a decrease in the load produces a stepwise diminution of the crack size, but this happens, in general, when critical equilibrium states are passed that are different from those corresponding to an increase in the load. Recently, L. M. Kachanov [84a] carried out an investigation generalizing the previous treatments so as to cover the case of the time-dependent modulus of cohesion. This investigation is of basic importance in connection with the problems of so-called “stress rupture”.

102

G. I . BARENBLATT

The analysis carried out in the present section is based on [59]. Consider now the solution of a problem concerning the extension of an unsymmetrical initial crack in one simple case. Let a straight initial crack with the end coordinates x = - a, and x = bo be given in an infinite unloaded body (for definiteness assume b,< a,) and let equal and opposite concentrated forces P be applied at opposite points of the crack surfaces, say, at x = 0. The magnitude of the force P plays the role of the loading parameter. According to (5.l ) , the values of the tensile-stress intensity factors No at x = - a and x = b are, respectively,

When

P < PI,where

5.28)

both factors N , and Nb are less than K l n so that the crack expands neither to the right nor to the left. A t P = P, the factor Nb becomes equal to K / n ,

I

FIG.22.

mobile equilibrium is reached and the end b begins to move to the right. The advance depends on the magnitude of the applied force according to the relation (5.29)

As long as P

< P,,

where 2

(6.30)

5 = 2a,, K2

we have N a < Kln, and the left end does not move. At P = P2,we have b = a,, a symmetrical crack in mobile equilibrium, and a t P > P, the

-

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

103

development of the crack continues according to (5.8). The development of the initial crack with changing P is plotted in Fig. 22. 4. Cracks Extending to the Surface of the Body

If a crack extends to the surface of the body, it becomes difficult to obtain effective analytical solutions. Mapping of the corresponding region on a half-plane cannot be carried out by means of rational functions, and Muskhelishvili’s method does not make it possible to obtain solution in finite form. Therefore it is necessary to resort to numerical methods in analysing such problems. A number of numerical solutions have been derived up to now; the mobile-equilibrium states are unstable in all analysed cases.

FIG.23.

0. L. Bowie [22] treated the problem of a system of k symmetrically located cracks of equal length extending to the free surface of a circular cut in an infinite body (Fig. 23). The body is stretched at infinity by the allround stress p,. Bowie employed Muskhelishvili’s method for calculating stresses and strains. To obtain the solution in effective form, the author used a polynomial approximation to the analytical function mapping the exterior of the circle with adjacent cuts on the exterior of the unit circle. For the determination of the dimensions of mobile-equilibrium cracks Bowie used directly Griffith’s energy method and computed the strain-energy release rate. Numerical calculations were made for cases of one crack and two diametrically opposite cracks. To obtain sufficient accuracy of calculations it proved necessary to retain about thirty terms in the polynomial representation of the mapping function. The numerical results for the cases k = 1 and R = 2 obtained by Bowie are shown in Fig. 24. It follows from these computations that at LIR > 1 the tensile stress for two cracks with a circular cavity is very close to the tensile stress for one crack of length

104

G . I . BARENBLATT

2(L + R),so that the influence of the cavity proper is almost unnoticeable. Furthermore, in the case of small crack lengths the conditions of mobile equilibrium are obviously determined by the tensile stresses directly a t the

3-

I

I I I

2 -\

I 0 FIG.24.

1

I

1

2

uniaxial tension,

3 a 1

---

all-round tension.

surface of the circular cavity. As is known, in case of uniaxial extension the highest tensile stress a t the boundary of the cavity is equal to 3p0 and in case of all-round extension 29,. Thus the ratio of equilibrium loads in these cases a b Y should approach 213, and this is found in agreement with Bowie's calculations. The problem of a straight crack ending on a straight free boundary of the half-space (Fig. 25) was treated independently by L. A. Wigglesworth [86] and G. R.Irwin [51] using different methods. Wigglesworth [85] investigated the case of an arbitrary distribution of norFIG.25. mal and shearing stresses over the faces of the crack. For a symmetrical distribution of stresses he reduced the problem to an integral equation for the complex displacement w ( x ) = zc(x) + i v ( x ) of points of the crack surface:

'w I

(5.31)

X

MATHEMATICAL T H E O R Y OF EQUILIBRIUM CRACKS

105

+

Here L ( x , t ) is a singular integral operator and p ( x ) = a ( x ) it(%);a(x) is the distribution of normal stresses; t ( x ) is the distribution of shearing stresses. Equation (5.31)is solved in the paper by an integral-transform method. Detailed calculations are made for the case when the surfaces of the crack and boundary are free of stresses, the tensile stress Po being applied a t infinity parallel to the boundary of the half-space. For stresses near the crack end the author obtains in this special case the following relations :

hence we find a t the prolongation of the crack (@ ax = a,, = 0.793 Po

(5.33)

PI

= z)

ax,,= 0,

which together with (4.6)gives the expression for the length of the mobileequilibrium crack in the form 1=

(5.34)

K2

1.61,.

K2

n2(0.793)Po Po Irwin [51] investigated only the last special case. He represented the unknown solution as the sum of three fields. The first field corresponds to I, y = 0) in an infinite body subjected to constant a crack (- 1 x tensile stress Po a t infinity, the second field corresponds to the same crack under normal stresses Q ( x ) symmetrical with respect to the x and y axes and applied at the crack surface, the third field corresponds to a half-space x 0 without crack, a t the boundary of which ( x = 0) the distribution of normal stresses P ( y ) , symmetrical with respect to the x axis, is given. Satisfying the boundary conditions at the free boundary and the crack surface, Irwin obtained for P ( y ) and Q ( x ) the system of integral equations

< <

I

(5.35) m

xy2 0

d y =Q(x),

106

G. I. BARENBLATT

which he solved by the method of successive approximations. The first approximation yields a relation for the length of the mobile-equilibrium crack 1 : (5.36)

K2

2K2

I= 7c2

= 1.69 7

1.0952P02

Po

,

which differs, as is seen, insignificantly from the more exact relation (5.34). H. F. Bueckner [50] treated a problem of one straight crack reaching the boundary of a circular cavity in an infinite body. No stress is applied at infinity and at the boundary of the cavity, the surface of the crack is free of shearing stresses, normal stresses are applied symmetrically and vary according to a given law - p ( x ) . Such a form of the problem arises in the analysis of rupture of rotating disks. Like Wigglesworth [85], Bueckner proceeds independently from a singular integral equation for the lateral displacements of points of the crack surface. He considers a one-parameter family of particular solutions of this equation, corresponding to certain special distributions p , ( x ) . In the general case it is recommended to represent P ( x ) as a linear combination of P , ( x ) : n=m

n=o

the coefficients a, are determined by the least-square method or by collocation. The factor of stress intensity at the crack end No is expressed in terms of the coefficients a,. If the length of the crack is far less than the radius of the circular cavity, then we have in the limit the previous particular case of a straight boundary. As it follows from Bueckner’s calculations in this particular case when P = Po = Const, the expression for the length of a mobile-equilibrium crack is (5.38)

1=

7c2

2K2 a, = 0.159 P 0 1.132$02 K2

which is in good agreement with (5.34) and (5.36). In [50] Bueckner also treated a problem of a crack reaching the surface of an infinitely long strip of finite width under an arbitrary load, symmetrical with respect to the line of the crack (Fig. 25b). He showed that it is possible to replace with a high degree of accuracy the integral equation occurring in this case by one with a degenerated kernel. The numerical solution obtained by Bueckner in the special case when the load is produced by couples M , applied on both sides of the crack at infinity, gives the relation between the length of a mobile-equilibrium crack and the load; it is represented by the curve in Fig. 26.

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

107

As has already been pointed out, in all cases discussed in the present section mobile-equilibrium cracks are unstable. Thus, when loads increase, extension of an initial crack does not take place until it reaches mobile equilibrium, after which the body fractures. In these problems the load a t which an initial crack reaches mobile equilibrium coincides with the breaking load, which is in general not true. In the paper by D. H. Winne and B. M. Wundt [32] some of the solutions presented in this section were employed for the analysis of fracture of rotating notched disks, and of notched beams in I I I I bending. The experiments conducted 0 a5 I by Winne and Wundt, analysed on L the basis of these calculations, revealed FIG.26. close coincidence of the values of surfaceenergy density T (or, which amounts to the same, of the moduli of cohesion K ) determined from the angular speed, a t which fracture of rotating notched disks occurs, and from the loads a t which fracture of notched beams in bending occurs. This confirms that the quantities T and K are characteristics of the material and do not depend on the nature of the state of stress. I

I

5 . Cracks near Boundaries of a Body; Systems of Cracks

Crack development in bounded bodies possesses some characteristic peculiarities. Difficulties of a mathematical character do not allow us to

a

b

i

I

f l t 0 1 0

t

I

h

FIG. 27.

carry out here as complete an investigation as in the case of isolated cracks. However, the qualitative features and some of the quantitative characteristics of this phenomenon can readily be elucidated in connection with the

108

G . I. BARENBLATT

simplest problems that yield to analytical solution. Let us examine first of all the problem of a straight crack in a strip of finite width (Fig. 27a). The crack is assumed to be symmetrical with respect to the middle line of the strip, and the direction of its propagation is normal to the free boundary. The load keeping the crack open is considered symmetrical with respect to the line of the crack and the middle line of the strip. In solving the problem we use the method of successive approximations developed by D. I. Sherman [86] and S. G. Mikhlin [87]. As the first approximation we take the solution of a problem in the theory of elasticity for the exterior of a periodical system of cuts (Fig. 27b). Denoting again by $(t) the distribution of tensile stresses, which would be at the place of the cracks in a continuous body under the same loads, we obtain the equation determining the half-length of a mobile-equilibrium crack 1 in the form

(5.39)

m-t -m

where t = sin (nt0/2L),m = sin (n1/2L). In the particular case represented in Fig. 27, when the crack is maintained by equal and opposite concentrated forces P with points of application 2s apart along their common line of action, (5.39) becomes

+ 1) sin (nl/L)

V8(a2

(5.40)

u(2a2

+ 1) cosha

+

%(a2 l)m,

where a = sinh a/m, u = ns/2L. When s a t the crack surface), (5.40) reduces to

=0

I’

(concentrated forces applied

(5.41)

Let us also quote the relation between the size of a mobile-equilibrium crack and the load for the case of a uniform tensile stress at infinity, P/2L, (5.42)

__

-

K?Z -

v

nl 2 -cot n 2L

Relation (5.40) for various u is presented in Fig. 28. The solid and dotted lines denote, as usual, stable and unstable branches. As is seen, for u 2 ucw 0.5 there are no stable branches, hence for distances between points of application of forces exceeding 2Lln m 0.64 L mobile-equilibrium cracks are always unstable. Quite similarly to the analysis in Section V.3 (extension of an

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

109

isolated crack under proportional loading) the graph in Fig. 28 makes it possible to describe completely the extension of any symmetrical initial crack when the load increases. The present analysis is based on papers [58, 881. The solution of the corresponding problem in the theory of elasticity for the case s = 0 was obtained by Irwin [45]. The problem of a periodical system of cracks under uniform loading a t infinity was solved by Westergaard [13] and independently by W. T. Koiter [89].

t 0

\ N

\

\

0.5

1 10 T L

FIG.28.

In the first approximation only the shearing stresses vanish at the lines of symmetry (shown by the dotted lines in Fig. 27b, which correspond to the boundaries of the strip); the normal stresses are different from zero. To obtain the second approximation, the first approximation is addcd to the solution for an uncracked strip, at the boundaries of which the normal stresses are given; their distribution is chosen in such a manner as to compensate the normal stresses at the boundary obtained in the first approximation. Now the boundary condition is no longer satisfied at the crack surface. To obtain the third approximation, the second approximation is added to the solution for the exterior of a periodical system of cuts, a t the surface of which the distribution of normal stresses is equal to the difference between the given stresses and those obtained in the second approximation, and so on.

110

G. I. BARENBLATT

Special estimates obtained in [88] show that for stable mobile-equilibrium states the considerations of the second and subsequent approximations leads to corrections of the order of 2.5-3 per cent in the above relations. This permits us to confine ourselves to the first approximation. In addition to these problems (the periodical system of cracks and the system of radial cracks ending in a circular cavity), several other problems of systems of cracks have been treated; they deal with straight cracks located along one straight line. Mathematical methods developed by Muskhelishvili [go, 181, D. I. Sherman [91], and Westergaard [13] permit

FIG.29.

the reduction of any such problem to quadratures. Let us here consider the simplest example: it is the problem of the extension of two collinear straight cracks of the same length in an infinite body, stretched by a uniform stress p at infinity (Fig. 29). This problem was treated by Willmore [21]; it also occurs in a paper by Winne and Wundt [32] (the authors refer to a private communication by Irwin). According to the solution presented in [21], the sizes of the cracks remain unchanged at p < pl, where (5.43)

,

a

a=-
Here K’, E’ are standard notations of elliptic integrals. At p = 9, the cracks attain an unstable state of mobile equilibrium, after which the inside edges of the cracks join and form a crack of length 2b. The further extension of the crack depends on whether the bracketed expression in (5.43) is greater or less than unity. If it is less than unity, which happens for a < 0.027, the size of the crack resulting from the joining of the inside edges is less than the size of the mobile-equilibrium crack corresponding to the load p,. In this case the crack remains unchanged up to the load p , = VTK/n]/b,after which the body fractures. If it is greater than unity, complete fracture of the body occurs immediately upon reaching the load p,. Assuming b - a = 21 and making b - + 00 in (5.43), we obtain in the limit (5.6), as expected. The solution given in [32] leads to the same qualitative results. However, it cannot be accepted as correct because

MATHEMATICAL T H E O R Y OF EQUILIBRIUM CRACKS

111

it is based on the erroneous expressions of the stress intensity factors given in [47]. The case of two identical cracks maintained open by concentrated forces applied at their surface was treated in [88]. A complete investigation of the general case of symmetrical loading for a system of two cracks can be carried out quite similarly, with expressions for the stress intensity factors a t the crack ends x = a and x = b

h

h’, =

v2

x Va(b2- a*) b

[ f i ( t ) tt2 v-Ea2d t

- C]

b

(5.44)

As is seen from these examples, collinear cracks “weaken” each other and reduce their stability. Ya. B. Zeldovitch noticed that in the case

FIG.30.

of a “chess-board’’ pattern of cracks (Fig. 30) the inverse phenomenon occurs. As the calculations show, even for uniform normal loads a t the crack surfaces, mobile-equilibrium cracks may become stable for a certain mutual position. We consider briefly the so-called “size effect” in the brittle fracture of bounded bodies. Take similarly shaped bodies, which differ only in the characteristic size d and in the characteristic scale of the applied extensional load S (it is supposed that macroscopic cracks present in the bodies are also geometrically similar). In brittle fracture, the value S = So that corresponds to fracture depends only on the characteristic size of the body d and the

112

G. I. BARENBLATT

modulus of cohesion K . There is only one way to form a characteristic having the dimension S from the quantities K and d , and it is impossible to make any dimensionless combinations. Therefore the following simple rdations govern the magnitude of the breaking load:

where So has the dimension of a force, of a force distributed along a line (as, for instance, a concentrated force in plane strain), and of a stress, respectively. The quantities ei are constants for a given geometrical configuration of the body. About the fracture of geometrically similar bodies a great deal of experimental data is a t present available, which permits clarification of the limits of applicability of the theory of brittle fracture. Detailed information on this topic can be found in a paper by B. M. Wundt [92], and some new results have been presented by S. Yusuff [93].

6. Cracks in Rocks

The investigation of crack extension in rock massifs is of great interest in theoretical geology. Cracks can form in rocks because of various causes of tectonic character, but also because of some artificial actions (mining excavations, hydraulic fracture of oil-bearing strata, etc.), In connection with the theory of hydraulic fracture of an oil-bearing stratum a number of problems of the theory of cracks have been treated, among them the problem of the vertical crack: A crack in an infinite space subjected to all-round compressivepressure q at infinity is maintained open by a flowing viscous fluid injected into it (Fig. 31). Themain peculiarity of the FIG. 31. problem is that the fluid does not fill the crack completely: there is always a free part of the crack on both sides of the wetted area. The pressure Po in the flowing fluid troughout the wetted area of the crack can be considered constant in first approximation. Indeed, a t the end of the wetted area an abrupt narrowing of the crack takes place, and almost all of the pressure drop will occur there. The problem is called so, because the actual fissure, idealised by this problem, is located in a vertical plane, and q represents the lateral pressure of the rocks. In comparison with the action of lateral rock and fluid pressures the action of the forces of cohesion may be

“ 2

113

MATHEMATICAL T H E O R Y OF E Q U I L I B R I U M CRACKS

neglected, as estimates show.* Condition (5.3) determining crack sizes becomes 1

hence 1 =I,

(5.47)

21-l

[ sin . - .

The expression for the maximum half-opening of the crack vo is (5.48) As calculations show, for values 1,/1 close to unity which are usually encountered in practice, the opening of the crack is almost constant all along the wetted area of the crack; the crack closes rapidly along the free part. - This problem of the vertical crack was first stated and solved in a paper by Zheltov and Khristianovitch [38].

The problem of the horizontal crack [40] is stated as follows. In a heavy half-space at a certain depth N a horizontal disk-shaped crack is formed by injecting viscous fluid as before; the surface of the crack is again divided R,) and a free part (Ro< Y R ) , and the into a wetted part (0 r fluid pressure p in the wetted part may again be considered as constant. Forces of cohesion, as in the preceding case, are neglected. Under the assumption that the depth of the crack position H is sufficiently great, the boundary condition at the boundary of the half-space need not be taken into account. The condition of finiteness of stresses a t the crack contour yields in this case

<

< <

(5.49)

where y is the specific weight of the rock. For the volume of the injected fluid one obtains (5.50) T/

=

4(1 - y 2 ) # R 3

fp

(2) v(z) [ - ,

=z3

2 3

--

z -

3

3(1

+

z

v-)]'

* The condition that forces of cohesion be negligibly small is K/gvl<< 1. It is in general not satisfied in laboratory scaling.

114

G . I. BARENBLATT

In practice, z = Ro/R is close to unity so that it is possible to use the asymptotic form of (5.50) 4(1 - v2)$R9 v2(1 - z) [l 3E ~

(5.51)

L'

=--

+ v2(1 - z) - 3(1 -41.

The maximum half-opening of the crack is determined by the formula (5.52)

Thus, if the depth of the crack position, the fluid pressure, and the specific weight of the rock are known, Ro/R can be found according to (5.49). Then the crack radius is obtained from (5.51) and a knowledge of the total volume of the injected fluid V , after which the determination of the remaining parameters does not encounter any difficulties. In [40, 411 problems were also treated concerning horizontal cracks in a radially varying pressure field caused by the higher lying rocks. Under certain conditions a complete wetting of the crack surface (i.e. the absence of a free part) may in this case occur. Yu. P. Zheltov [43] proposed an approximate method for solving the problem of the horizontal crack in a radially varying vertical pressure field. A comparison between the results obtained by this method and the exact solutions for certain cases showed quite satisfactory agreement. By using the method of successive approximations Yu. A. Ustinov [94] estimated the influence of the free boundary in the problem of the horizontal crack. If the depth is larger than twice the crack radius, the influence of the free boundary is negligibly small. The problem of a crack formed by driving a horizontal wedge of constant thickness into a heavy space was treated in [39]. The solution of the problem of the vertical crack was extended by Zheltov [42] to cover the case when the rock is permeable and the injected fluid flows through the rock.

PROBLEMS I N THE THEORYOF CRACKS VI. WEDGING;DYNAMIC 1. Wedging of an Infinite B o d y

Wedging is formation of a crack in a solid by driving a rigid wedge into it. The most characteristic property of the wedging of a brittle body is that the wedge surface never comes in complete contact with the body: there is always a free portion in the front part of the wedge; ahead of the wedge a free crack forms, which closes at some distance from the edge of the wedge (Fig. 32).

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

115

It appears that the problem of wedging of an infinite body by a fixed wedge [39, 58, 951 is the simplest to formulate among problems of this kind; it yields to an effective exact solution by the methods of elasticity theory and gives a qualitative idea of wedging under more complex conditions. Let a uniform, isotropic brittle body be wedged by a thin, symmetrical, perfectly rigid semi-infinite wedge with thickness 212 a t infinity (Fig. 32). In front of the wedge a free crack forms, which closes smoothly a t a certain

- /

c/

Y

B'

"/////TI

FIG.32.

point 0; the position of the point 0 with respect to the front point of the wedge C is not known beforehand and must be determined in the course of solving the problem. If the wedge has a rounded front part (Fig. 32a). the position of the points of departure of the crack surface from the wedge, B and B', is not prescribed and must also be determined in the course of solving the problem. If the wedge has a truncated front part (Fig. 32b) as e.g. in the case of a wedge of constant thickness, the position of the points of contact is quite definite; they coincide with the corners of the wedge front. I t is evident that the stress at the points of departure is in this case infinite. We shall a t first assume that there is no friction a t the surface of contact between wedge and body. The field of elastic stresses and strains satisfies the usual equations of static elasticity in the exterior of the crack. In view of the assumed slenderness of the wedge, the boundary conditions may be transferred from the crack surface proper to the x-axis. Without considering forces of cohesion, the boundary conditions are (6.1)

uxy=o,

u,=O

- ZJ,

uxy

v = ff ( x

(O,<X
(12 Q x

y=0),

< 00,

y = 0);

116

G. I . BARENBLATT

here uy,uZyare the stress-tensor components; I, and 1, are the distances of the point 0 from the edge of the wedge and from the points of departure B, B ' ; f ( t ) defines the wedge surface in a system of coordinates with origin a t the front point of the wedge; the positive and negative signs correspond to the upper and lower faces of the cut, respectively. As is seen, the problem of wedging is a peculiar combination of the contact problem in the theory of elasticity [18,72,73] and the problem of the theory of cracks.

The position of the points of departure of the crack surface from the wedge in the case of a wedge with rounded edge, and the position of the point of closing with respect to the edge are determined from the following conditions: 1. Stresses at the points of departure must be finite. For the contact problem a similar condition was first suggested as a hypothesis by Muskhelishvili [96, 181 and independently by A. V. Bitsadze [97]; it was proved in [6l]. 2. Stresses at the crack edge are finite or, which i s the same, the opposite faces of a crack close smoothly at its end. Since the intensity of forces of cohesion at the crack edge is maximal, stresses near the crack edge calculated without taking into account forces of cohesion must tend to infinity according to (4.7).

The problem of wedging is a mixed problem of the theory of elasticity. For its solution it is convenient to consider the singular integral equation for the compressive force acting on the face of the wedge, uy = - + ( x ) . If + ( x ) is known, the determination of the elastic field obviously reduces to the solution of the first boundary-value problem in the theory of elasticity for the exterior of a semi-infinite straight-line cut, which can be found by Muskhelishvili's method ([18], 95). This solution gives an expression for the lateral displacements at points of contact between wedge and crack surface :

where 5 =.vG, and the root may assume positive and negative values for displacements of the upper and lower face. The second condition (6.1) yields the fundamental integral equation of the problem :

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

117

which can be shown to be equivalent to the singular integral equation obtained from (6.3) by differentiation with respect to c :

and the condition

where h = f ( b o ) . By using the methods for singular integral equations developed in the monogra.ph by Muskhelishvili ([19], Chapter 5) the solution of equation (6.4) can be found in the form:

where A is an indefinite constant. The integral in (6.6) does exist in view of the finiteness of f ( m ) = h, and it tends to zero as x-+ bo ; this together with (6.5) determines the value of the constant A : (6.7)

A =

Eh 2(1 - v2) *

For finiteness of stress at the points of departure x = I 2 in case of a wedge with rounded edge, it is necessary and sufficient that the bracketed expression in (6.6) vanishes at x = 1,. This gives one equation for the determination of 1, and I,:

Now the following expression for the tensile stresses a t the prolongation of the cut results from the solution:

t - x

118

G. I .

BARENBLATT

Together with (4.7) it leads to

Relations (6.8) and (6.10) are finite equations which determine the unknown constants I, and 1,. In the particular case of constant wedge thickness f ( t ) h, condition ( 6 4 , which is no longer valid, is replaced by the relation 1, = I,, and (6.10) gives the following expression for the length of a free crack in front of a “square” wedge : (6.11) In 1951 other special forms of the wedge are also treated such as a wedge rounded-off with a small radius of curvature and a wedge rounded-off according to a power law. Investigation of the first example shows that

I

I

I

FIG. 33.

roundness affects slightly the length of the free crack in front of the wedge. I n [95] also a case when Coulomb friction acts on the faces of the wedge is treated. In [84] wedging of an anisotropic body by a semi-infinite rigid wedge is studied.

I. A. Markuzon [98] treated a problem of wedging an infinite body by a wedge of finite length 2b (Fig. 33). In case of constant thickness of the wedge 2h, the relation between crack length 21 and wedge length 2b, other things being equal, is as represented in Fig. 34 (1, is the length of a free crack for an infinite wedge defined by (6.11))-

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

119

In [98] the influence of a uniform compressive or tensile stress at infinity on the length of a free crack, when the wedge is of finite length, was also investigated. Relation (6.11) can be used for the experimental determination of the modulus of cohesion K . For that purpose a wedge is driven into a plate of the testing material, the wedge being substantially more rigid than the plate. The length L of the resulting free crack is measured. The modulus of cohesion can then be found according to the formula (6.12)

The wedge must be sufficiently long in order to eliminate the influence of the plate boundary, and it should be driven in, until the distance between the wedge end and the crack end stops varying with further displacement of the wedge. The plate must be wide and sufficiently thick so that the state of stress essentially corresponds to plane strain. To insure a straight-line form of the crack, it is necessary to compress the specimen in the direction of crack propagation. This is recommended by Benbow and Roesler [9]. (It can be shown that (6.11) and (6.12) remain unchanged in this case.) 2. Wedging of a Strip

In strict formulation, problems concerning the wedging of bounded bodies are very difficult to solve. Up to now there are but a few approximate solutions, based on the application of the approximations of simple beam theory. The first of these solutions was obtained by I. V. Obreimov [8]; as a matter of fact, this work was the first investigation in which wedging was considered. In connection with his experiments on the splitting of mica, Obreimov examined the case when a strip being torn off has small thickness and only one-point contact with the wedging body (Fig. 35). In order to establish a relation between the surface tension of mica and the parameters of the crack shape, Obreimov applied to this problem the methods of strength of materials, considering a shaving as a thin beam. The theoretical part of the work of Obreimov is not free from shortcomings. Later, corrections were introduced into these calculations in the book by V. D. Kuznetsov [99] as well as by M. S. Metsik [lo] and N . N . Davidenkov [12]. In addition Metsik improved the experimental procedure of [8]. Application of the approximations of thin-beam theory for the determination of the crack length is justifiable in some cases. However, these approximations cannot be applied to describe the form of the crack surface in the immediate vicinity of its edge, even if the distribution of forces of cohesion in the edge region is explicitly

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considered, as it was done by Ya. I. Frenkel [5]. The fact is that the longitudinal dimension of the edge region cannot be assumed to be large compared to the shaving thickness; hence a shaving cannot be considered as a thin beam in the region where the forces of cohesion are acting. To illustrate the approximate approach based on the methods of simple beam theory, we discuss the paper by Benbow and Roesler [ 9 ] in more detail. Note that in this work possibilities and limits of applicability of the above approach are most clearly pointed out. The fdllowing statement of theproblem is considered (Fig. 36). A strip of finite width b is wedged symmetrically so that the crack passes along the

i FIG. 35. 1. a Body being wedged. 1. a Wedge.

FIG. 36. 1. a Body being wedged.

2. Grips.

middle line of the strip. At the end of the strip, compressive forces Q / 2 are applied to insure straight crack propagation; the wedging force P produces a crack length 1 and initial width h. Having obtained an expression for the strain energy from dimensional considerations, the authors write the equilibrium condition for the crack in the form (6.13)

T E

ha

l

so that for a given material the quantity h2/1 is uniquely determined by the quantity bll. The experiments made with specimens of two different plastics [ 9 ] give a conclusive proof of the existence of such a one-one relation. For small b/Z, i.e. for long cracks, it is possible to obtain an asymptotic form of relation (6.13) by considering both halves of the strip as thin beams

fixed a t the section corresponding to the crack end. The expression for the strain energy of the strip is in this case (6.14)

U

= 3h2B/13,

where B = E J is the stiffness of the beam, J = nb3/96, and n is the transverse thickness of the beam. The surface energy of the crack is, evidently, 2Tnl.

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

121

In the mobile-equilibrium state the variation of surface energy corresponding to a small variation of the crack length 61 is equal to the corresponding variation of strain energy of the strip. Hence it follows that (6.15)

- -au =2Tn

T - 3h2b3 ---. E 6414

or

ai

By comparing the second formula (6.15) with (6.13), an asymptotic expression for @(b/l) can be found as b/l ---t 0 : b 3

3

(6.16)

Q=&)

*

From (6.15) an expression for the length of an equilibrium crack is obtained : 4

4

(6.17)

vc

Thus in this case the length of the crack is proportional only to whereas in the wedging of an infinite body by a semi-infinite wedge the length of the crack is proportional to h2 (cf. (6.11)). Relation (6.15) was used by Benbow and Roesler for the determination of the surface-energy density of the plastics investigated. The careful experimentation and the scrupulous evaluation of the sources of possible errors and of their magnitude are remarkable. In the recent review of J. J. Gilman [ll] a detailed summary and a bibliography of experimental investigations on wedging can be found.

3. Dynamic Problems in the Theory of Cracks Considerable attention is nowadays given to questions of dynamics of cracks. A detailed consideration of these questions is beyond the scope of the present review; we shall confine ourselves here to a brief information about basic results achieved in theoretical investigations of dynamics of cracks. In the paper by N. F. Mott [36] the crack-expansion process is treated in the case of an isolated straight crack in an infinite body subjected to a uniform field of tensile stresses &* On the basis of dimensional analysis Mott obtained an expression for the kinetic energy of a body, (6.18)

*

d

= kpl2V2pO2/E2,

Unlike [36J, plane strain is here considered rather than the state of plane stress.

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where p is the density of the body, 1 the half-length of the crack, V the rate of crack expansion, and k a dimensionless factor which Mott considered constant and left indefinite. Adding to the static-energy equation (2.1) the derivative with respect to I of the kinetic energy (6.18) and assuming the remaining terms in (2.1) to be the same as in Griffith’s static problem, Mott found the rate of crack expansion (6.19)

where I , is the half-length of the mobile-equilibrium crack defined by (5.4). Thus, as the crack propagates, its extension rate increases, approaching the limit (6.20)

The ultimate rate constitutes, according t o Mott, a certain part of the longitudinal wave propagation velocity. In this reasoning the use of the static expression for the decrease of the strain energy W remains unfounded. Moreover, the quantity k in (6.18) and (6.19) need not be constant in general; it may depend on l/l*, V/cl and other dimensionless combinations. E. Yoffe [loo], using the exact formulation of dynamic elasticity theory, investigated the problem of a straight crack of constant length, moving with constant velocity in an infinite body stretched by uniform stress a t infinity. Notwithstanding the somewhat artificial character of the problem, an important result was obtained in this paper, which has quite a general meaning: If the crack propagation rate becomes greater than a certain critical rate, the direction of crack propagation is no longer the direction of maximum tensile stress, and the crack begins to curve. The magnitude of the critical rate Vl is about 0.4 cl, where c1 is the longitudinal wave propagation velocity in the given material (the ratio Vl/cl depends slightly on Poisson’s ratio v of the material). D. K. Roberts and A. A. Wells [ l o l l made an attempt to evaluate the constant k , which remained indefinite in [36]. Using the value of k obtained, they found the ultimate crack expansion rate close to that found by Yoffe. However, their estimate, based as it is on the solution of a static problem, is too rough; and since the straight-line direction of crack propagation in [ l o l l was assumed as certain, the close agreement between the critical rate found by Yoffe [loo] and the ultimate rate obtained in [ l o l l must be considered as incidental. If the straight-line direction of crack propagation is somehow insured (for instance, by a large compression of the body in the direction of crack propagation or by the anisotropy of the material), then the maximum rate

MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS

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of crack propagation coincides with the velocity of propagation of Rayleigh surface waves in the given material, which is about 0.6 cl. The fact that the ultimate rate of crack propagation coincides with the Rayleigh velocity was first stated by A. N. Stroh [l02]. The heuristic proof given in that paper amounts to the following. Stroh correctly notes that the ultimate rate of crack propagation does not depend on the surface energy of the body, and he assumes the surface energy to be zero. Proceeding from this, Stroh is led by energy considerations to the conclusion that the tensile stress near the crack end (on its prolongation) is equal to zero. Thus the crack may be thought of as a disturbance moving on a stress-free surface, which can propagate only with the Rayleigh velocity. In fact, from Stroh’s reasoning it may only be concluded that the tensile stress at the very contour of the crack is equal to zero. From this fact, however, it does not follow that the rate of crack propagation is equal to the Rayleigh velocity, as can be seen from the following simple example. Take a body subjected to allround compressive stress at infinity and wedged by a semi-infinite wedge as in Fig. 32, moving with infinitely small velocity. Forces of cohesion and, consequently, surface energy are assumed to be zero. In view of the infinitesimal velocity of the wedge, dynamic effects are insignificant, hence, according to Section 111,2, the tensile stress at the crack end must vanish. ,4t the same time, the rate of crack propagation is equal to the velocity of the wedge, i.e. it is also infinitesimal. By arguments based on the analysis of exact solutions of the dynamic equations of elasticity, the conclusion that the ultimate rate of crack propagation is equal to Rayleigh velocity was drawn independently and simultaneously by several authors. I. W. Craggs [lo31 considered steady propagation of a semi-infinite straight crack with symmetrically distributed normal and shearing stresses applied on a part of the crack surface adjacent to the edge. In a paper by Dang Dinh An [lo41 a non-steady field of stresses and strains was investigated, acting in an infinite body with a semi-infinite crack, along the surface of which symmetrical concentrated forces normal to the crack surface begin to move suddenly away from the edge with constant velocity. Paper [95] examines the wedging of an infinite isotropic brittle body by a semi-infinite rigid wedge of arbitrary form, moving with constant velocity. In [84] a similar problem is treated for a case of an anisotropic body. B. R. Baker [lo51 considers a non-steady distribution of stresses and strains in a solid with a semi-infinite crack, at the surface of which constant normal stress is applied at the initial moment, after which the crack begins to expand with constant velocity. From the various problems treated in these papers the following general result was obtained which led to our earlier conclusion: when the characteristic rate involved in the problem approaches the Rayleigh velocity, peculiar resonance phenomena arise. Note that the appearance

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of resonance when the Rayleigh velocity is approached is not specific for the problems of cracks: the investigation of the problem of a punch moving along the boundary of a half-space, carried out by L. A. Galin [72] and J. R. M. Radok [log], reveals [95] that the same resonance phenomena occur, when the velocity of the punch approaches the Rayleigh velocity. I t appears that the limiting character of the Rayleigh velocity is most directly illustrated by a problem of wedging. Obviously the maximum possible rate of crack propagation can be reached in wedging a body by a moving wedge. The analysis of this problem shows [95] that with increasing velocity of the wedge the length of the free crack in front of the wedge decreases and tends to zero when the Rayleigh velocity is approached. For larger wedge velocity a free crack does not form in front of the wedge. Hence the maximum rate with which a crack can expand is equal to Rayleigh velocity. K. B. Broberg [107, 1081 treated the problem of a uniformly expanding crack of finite length in an infinite body subjected to a uniform tensile stress field. The solution obtained by Broberg is an asymptotic representation for great values of time of the solution of the problem treated by Mott [36] and Roberts and Wells [ l o l l . However, unlike [loll, Broberg’s solution was obtained on the basis of the exact methods of the dynamic theory of elasticity. Independently of [102-104, 57, 95, 1051 and in full accord with the results of these investigations, Broberg obtained that the rate of crack expansion in his problem, equal to the ultimate rate of crack expansion in the problem considered in [36, 1011, coincides with the Rayleigh velocity. Note the papers by B. A. Bilby and R. Bullough [lo91 and F. A. McClintock and S. P. Sukhatme [110] which treat uniformly moving cracks of finite and infinite length, respectively, a t the surface of which symmetrical shearing stresses parallel to the crack edge were applied. Instead of plane strain we have in this problem what is often called anti-plane strain: one displacement component, parallel to the crack edge, is different from zero. The investigation of such cracks reduces to the solution of a single wave equation (reducing to Laplace’s equation for equilibrium cracks). Cracks under anti-plane strain conditions are of considerable interest, being the simplest model for which an effective solution is possible for many problems, which are intractable for cracks under plane-strain conditions because of the great mathematical difficulties. An analysis of the dynamics of crack propagation on the basis of the approximations of the simple beam theory was carried out by J. J. Gilman [ l l ] and J. C. Suits [ l l l ] .

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ACKNOWLEDGEMENT

The author is very grateful t o Prof. Ya. B. Zeldovitch and Prof. Yu. N. Rabotnov (USSR Academy of Sciences) and Dr. S. S. Grigorian for the invariable interest and attention given t o his work on cracks and for a number of valuable advices. He recalls with appreciation the valuable discussions with Prof. S. A. Khristianovitch (USSR Academy of Sciences). The author considers it his pleasant duty to express his sincere thanks t o Prof. G. Kuerti (USA) and Prof. G. G. Chernyi for the amiable assistance in writing this review. Credit is also given t o I. A. Markuzon who assisted the author in compiling the bibliography.

References 1. INGLIS,C. E., Stresses in a plate due to the presence of cracks and sharp corners, Trans. Inst. Nav. Avch. 66, 219-230 (1913). 2. MUSKHELISHVILI, N. I,, Sur l’int6gration de 1’6quation biharmonique, Izvestiya Ross. Akad. nauk 18, 6 ser., 663-686 (1919). 3. GRIFFITH,A. A,, The phenomenon of rupture and flow in solids, Phil. Trans. ROY. SOC.A 221, 163-198 (1920). 4. GRIFFITH,A. A., The theory of rupture, Proc. 1st Intern. Congr. A p p l . Mech.. Delft, pp. 55-63 (1924). 5. FRENKEL,YA. I., The theory of reversible and irreversible cracks in solids, Zhurn. tekhn. fiz. 22, 1857-1866 (1952) (in Russian). 6. SMEKAL,A,, Technische Festigkeit und molekulare Festigkeit, Naturwzss. 10, 799-804 (1922). 7. WOLF, K., Zur Bruchtheorie von A. Griffith, Zeitschr. ang. Math. Mech. 3, 107-112 (1923). 8. OBREIMOV, I. V., The splitting strength of mica, Proc. Roy. SOC.A 127, 290-297 (1930). 9. BENBOW,J. J., and ROESLER,F. C., Experiments on controlled fractures, Proc. Phys. Soc. B 70, 201-211 (1957). 10. METSIK,M. S., A photoelastic measurement of the splitting work of mica crystals, Izvestiya V U Z . ser. fiz. No. 2, 58-63 (1958) (in Russian). 11. GILMAN,J. J., Cleavage, ductility and tenacity in crystals, in “Fracture” (B. L. Averbach et al., eds.), pp. 193-221. Wiley, New York, 1959. 12. DAVIDENKOV, N. N., On the surface energy of mica, Prikladna mekhanika 6, No. 2, 138-142 (1960) (in Ukrainian). 13. WESTERGAARD, H. M., Bearing pressures and cracks, J . AppZ. Mech. 6, No. 2, A-49-A-53 (1939). 14. SNEDDON, I. N., The distribution of stress in the neighborhood of a crack in a n elastic solid, Proc. Roy. SOC.A 187, 229-260 (1946). 15. SNEDDON, I. N., “Fourier Transforms”. McGraw Hill, New York, 1951. 16. SNEDDON, I. N., and ELLIOT,H. A,, The opening of a Griffith crack under internal pressure, Quart. A p p l . Math. 4, 262-267 (1946). 17. WILLIAMS,M. L., On the stress distribution at the base of a stationary crack, J . Appl. Mech. 24, 109-114 (1957). 18. MUSKHELISHVILI, N. I., “Some Basic Problems of the Mathematical Theory of Elasticity”, 4th ed. Izd. AN SSSR, M.-L., 1954 (Transl. from 3rd ed., Nordhoof. Holland, 1953). 19. MUSKHELISHVILI, N. I., “Singular Integral Equations”. GTTI, M.-L., 1946 (Transl. Nordhoof, Holland, 1953).

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20. SACK,R. A,, Extension of Griffith theory of rupture to three dimensions, Proc. Phys. SOC. 68, 729-736 (1946). 21. WILLMORE, T. J., The distribution of stress in the neighborhood of a crack, Quart. Journ. Mech. Appl. Math. 2, 53-64 (1949). 22. BOWIE,0. L., Analysis of an infinite plate containing radial cracks originating a t the boundary of an internal circular hole, J . Math. and Phys. 26, 60-71 (1956). 23. IRWIN,G. R., Fracture dynamics, in “Fracturing of Metals”, pp. 147-166. ASM. Cleveland, Ohio, 1948. 24. OROWAN, E. 0.. Fundamentals of brittle behavior of metals, i n “Fatigue and Fracture of Metals” (W. M. Murray, ed.), pp. 139-167. Wiley, New York, 1950. 26. IRWIN,G. R., and KIES, J. A , , Fracturing and fracture dynamics, Weld. Journ. Res. Suppl. 81, 96s-100s (1952). 26. IRWIN,G. R.. and KIES, J. A., Critical energy rate analysis of fracture strength, Weld. Journ. Res. Suppl. 88, 193s-198s (1954). 27. OROWAN, E. O., Energy criteria of fracture, Weld. Journ. Res. Suppl. 84, 167s-160s (1955). 28. FELBECK, D. K., and OROWAN. E. 0.. Experiments on brittle fracture of steel plates, Weld. Journ. Res. Suppl. 84, 570s-575s (1955). 29. WELLS,A. A., The mechanics of notch brittle fracture, Weld. Research 7 , 34r-56r (1953). 30. WELLS,A. A., The brittle fracture strength of welded steel plates, Quart. Trans. Inst. Nav. Arch. 98, 296-311 (1956). 31. WELLS,A. A., Strain energy release rates for fractures caused by wedge action, N R L Rept. No. 4706 (1956). 32. WINNE,D. H., and WUNDT,B. M., Application of the Griffith-Irwin theory of crack propagation to the bursting behavior of disks including analytical and experimental studies, Trans. A S M E 80, 1643-1658 (1958). 33. BUECKNER, H. F., The propagation of cracks and the energy of elastic deformation, Trans. A S M E 80, 1225-1230 (1958). 34. REBINDER,P. A,, Physico-chemical study of deformation processes in solids, i n “Yubileinyi Sbornik, Posviashchennyi XXX-letiu Velikoi Oktyabr’skoi Revolutsii”, pp. 533-561. Izd. AN SSSR, M., 1947 (in Russian). 36. ELLIOT,H. A., An analysis of the conditions for rupture due to Griffith cracks, Proc. Phys. Soc. 69, 208-223 (1946). 36. MOTT,N. F.. Fracture of metals; theoretical consideration, Engineering 186, 16-18 (1948). 37. RZHANITSYN, A. R., On the fracture process of the material in tension, in “Issledovaniya PO Voprosam Stroitel’noi Mekhaniki i Teorii Plastichnosti” (A. R. Rzhanitsyn, ed.). pp. 66-83. Stroiizdat, M., 1956 (in Russian). 38. ZHELTOV, Yu. P., and KHRISTIANOVITCH, S. A , , On the mechanism of hydraulic fracture of an oil-bearing stratum, Izvestiya A N S S S R , O T N , No. 5 , 3-41 (1955) (in Russian). 39. BARENBLATT, G. I., and KHRISTIANOVITCH, S. A., On roof stoping in mine excavations, Izvestiya A N S S S R , O T N , No. 11, 73-86 (1955) (in Russian). 40. BARENBLATT, G. I., On some problems of the theory of elasticity arising in investigating the mechanism of hydraulic fracture of an oil-bearing stratum, Prikl. matem. i mekhan. 20. 475-486 (1956) (in Russian). 41. BARENBLATT, G. I., On the formation of horizontal cracks in hydraulic fracture of an oil-bearing stratum, Izvestiya A N S S S R , O T N , No. 9. 101-105 (1956) (in Russian). 42. ZHELTOV, Yu. P., An approximate evaluation of the size of a crack forming in hydraulic fracture of a stratum, Izvestiya A N S S S R , O T N , No. 3, 180-182 (1957) (in Russian),

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43. ZHELTOV, Yu. P., On the formation of vertical cracks in a stratum by means of a filtrating fluid, Izvestzya A N SSSR, O T N , No. 8, 56-62 (1957) (in Russian). 44. WESTERGAARD, H. M., Stresses a t a crack, size of the crack and the bending of reinforced concrete, J . Amerzc. Concvete Inst. 6 , No. 2, 93-102 (1933). 45. IRWIN,G. R., Analysis of stresses and strains near the end of a crack traversing a plate, J . Appl. Mech. 24, 361-364 (1957). 46. IRWIN,G. R., Relation of stresses near a crack to the crack extension force, Proc. 9th Intern. Congr. Appl. Mech., Brussels pp. 245-251 (1957). 47. IRWIN,G. R.. Fracture, in “Handbuch der Physik”, B. VI (S. Fliigge, ed.), pp. 551-590. Springer, Berlin, 1958. 48. IRWIN,G. R., KIES, J. A., and SMITH,H. L., Fracture strength relative t o onset and arrest of crack propagation, Proc. Americ. Soc. Test. Matev. 68, 640-657 (1958/1959). 49. KIES, J. A., SMITH,H. L., IRWIN,G. R., La mkanique des ruptures et son application aux travaux de l’ingbnieur, Mdm. scient. rev. mdtalluvgie 67, No. 2, 101-117 (1960). 50. BUECKNER, H. F., Some stress singularities and their computation by means of integral equations, in ”Boundary Problems in Differential Equations” (R. E. Langer, ed.), pp. 215-230. Univ. Wisconsin Press, 1960. 51. IRWIN,G. R., The crack-extension-force for a crack a t a free surface boundary, N R L Rept. No. 5120 (1958). J. P., and SANDERS, P. H., Fracture arrest by riveted stiffeners, 52. ROMUALDI, Proc. 4th Midwest. Conf. Solid Mech., Univ. Texas Press pp. 74-90 (1959/1960). 53. ROESLER,F. C., Brittle fracture near equilibrium, Proc. Phys. Soc. B 69, 981-992 (1956). J. J., Cone cracks in fused silica, Proc. Phys. Soc. B 76, 697-699 (1960). 54. BENBOW, 55. DROZDOVSKY, B. A,, and FRIDMAN, YA. B., “The Influence of Cracks on the Mechanical Properties of Structural Steels”. Metallurgizdat, M., 1960 (in

Russian). 56. BARENBLATT, G. I., On the equilibrium cracks due to brittle fracture, Doklady A N SSSR 127, 47-50 (1959) (in Russian). 57. BARENBLATT, G. I., On the equilibrium cracks due to brittle fracture. Fundamentals and hypotheses. Axisymmetrical cracks, Prikl. matem. i mekhan. 28, 434-444 (1959) (in Russian). 58. BARENBLATT, G. I., On the equilibrium cracks due to brittle fracture. Straightline cracks in flat plates, Prikl. matem. i mekhan. 28, 706-721 (1959) (in Russian). 59. BARENBLATT, G. I., On the equilibrium cracks d r e to brittle fracture. Stability of isolated cracks. Relation to energy theories, Prikl. matem. i mekhan. 28, 893-900 (1959) (in Russian).

G. I., On conditions of finiteness in mechanics of continua. Static 60. BARENBLATT, problems in the theory of elasticity, Prikl. matem. i mekhan. 24. 316-322 (1960) (in Russian). 61. BARENBLATT, G. I., On some basic ideas of the theory of equilibrium cracks forming during brittle fracture, an “Problems of Continuum Mechanics” (M. A. Lavrent’ev. ed.), pp. 21-38. SIAM, Philadelphia, 1961. 62. ALEXANDROV, A. P., ZHURKOV, S. P., “The Phenomenon of Brittle Fracture”. GTTI, M., 1933 (in Russian). 63. GARBER, R. I., and GUINDIN, I. A., Strength physics of crystalline bodies, Uspekhi fizicheskikh nauk 70, 57-110 (1960) (in Russian). 64. DERYAGIN, B. V., ABRIKOSOVA, I. I., and LIFSHITZ,E. M., Molecular attraction of condensed bodies, Uspekhi fizicheskikh nauk 64, 493-528 (1958) (in Russian). B. V., and ABRIKOSSOVA, I. I., Direct measurements of molecular 65. DERJAGUIN, attraction of solids, J . Phys. Chem. Solids 6 , No. 1/2, 1-10 (1958).

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66. JONGH,J. G. V., DE, “Measurements of Retarded van der Waals-Forces”. Thesis, Abels, Utrecht, 1958. 67. BORN,M. and KUN HUANG,“Dynamical Theory of Crystal Lattices”. Clarendon Press, Oxford, 1954. 68. LEIBFRIED, G., Gittertheorie der mechanischen und thermischen Eigenschaften der Krystalle, i n “Handbuch der Physik”, B. VII, I (S. Fliigge, ed.), p. 104-324. Springer, Berlin, 1955. 69. LEONOV,M. YA., and PANASYUK, V. V., Development of the finest cracks in a solid, Pvikladna rnekhanika 6 , 391-401 (1959) (in Ukrainian). 70. PANASYUK, V. V., Determination of stresses and strains near the finest crack, Nauchn. zapiski In-ta mashinovedenia i avtomatiki A N U S S R 7 , 1 1 4 1 2 7 (1960) (in Russian). 71. LANDAU, L. D., and LIFSHITZ,E. M., “Theory of Elasticity”. Pergamon Press, London, 1960. 72. GALIN,L. A , , “Contact Problems of the Theory of Elasticity”. GITTL, M., 1953 (in Russian). 73. SHTAERMAN, I. YA., “Contact Problem of the Theory of Elasticity”. GITTL, M., 1949 (in Russian). 74. PARKER, E. R., “Brittle Behavior of Engineering Structures”. Wiley, New York, 1957. 75. DAVIDENKOV, N. N., “Impact Problem in Metallography”. Izd. AN SSSR, M.-L.. 1938 (in Russian). 76. PASHKOV, P. O., “Fracture of Metals”. Sudpromgiz, L., 1960 (in Russian). 77. POTAK,YA. M., “Brittle Fracture of Steel and Steel Parts”. Oborongiz, M.. 1955 (in Russian). B. L., and 0th. (eds.), ”Fracture”. Wiley, New York, 1959. 78. AVERBACH, 79. SEDOV,L. I., “Similarity and Dimensional Methods in Mechanics”, 4th ed. GITTL, M., 1957 (Transl. from 4th ed., Acad. Press, New York, 1959). H., “Kerbspannungslehre. Grundlagen fur genaue Spannungsrechnung”. 80. NEUBER, Springer, Berlin, 1937. 81. FRANKLAND, I. M., lriaxial tension a t the head of a rapidly running crack in a plate, Paper A S M E , No. APM-11 (1959). 82. MASUBUCHI, K., Dislocation and strain energy release during crack propagation in residual stress field, Proc. 8th Japan Nath. Congv. A p p l . Mech. pp. 147-150 (1958/1959). 83. STROH,A. N., Dislocations and cracks in anisotropic elasticity, Phil. Mag., V I I I ser. 3, 625-646 (1958). 84. BARENBLATT. G. I., and CHEREPANOV, G. P., On the equilibrium and propagation of cracks in an anisotropic medium, Prikl. matem. i mekhan. 46, 46-55 (1961) (in Russian). 84a. KACHANOV. L. M., On kinetics of crack growth, Prikl. matem. z mekhan. 26. 498-502 (1961) (in Russian). 85. WIGGLESWORTH, L. A., Stress distribution in a notched plate, Mathematika 4, 76-96 (1957). 86. SHERMAN, D. I., A method of solving the static plane problem of the theory of elasticity for multiconnected regions, Trudy Seismologicheskogo Institutu Akademii Nauk S S S R , No. 54 (1935) (in Russian). 87. MIKHLIN,S. G., The plane problem of the theory of elasticity, Trudy Seisrnologicheskogo Institutu Akademii Nauk S S S R , No. 65 (1935) (in Russian). 88. BARENBLATT. G. I., and CHEREPANOV, G. P., On the influence of body boundaries on the development of cracks of brittle fracture, Izvestiya A N S S S R , O T N , ser. mekh. i mash., No. 3, 79-88 (1960) (in Russian). Correction. The same journal, No. 1 (1962).

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89. KOITER,W. T., An infinite row of collinear cracks in an infinite elastic sheet, Ingenieur-Archiv 28. 168-172 (1959). 90. MUSKHELISHVILI, N. I., Basic boundary-value problems in the theory of elasticity for a plane with straight-line cuts, Soobschenia A N Gruz. S S R 8, 103-110 (1942) (in Russian). 91. SHERMAN, D. I., A mixed problem in the theory of potential and in the theory of elasticity for a plane with a finite number of straight-line cuts, Doklady A N S S S R 2 7 , 330-334 (1940) (in Russian). 92. WUNDT, B. M.. A unified interpretation of room-temperature strength of notched specimens as influenced by their size, Paper A S M E , No. 59-MET-9 (1959). 93. YUSUFF,S., Fracture phenomena in metal plates, Paper presented at the X t h Intern. Congr. A p p l . Mech., Stresa (1960). 94. USTINOV,Yu. A., On the influence of the free boundary of a half-space on the crack propagation, Izvestia A N S S S R , O T N . ser. mekh. i mash., No. 4, 181-183 (1959) (in Russian). 95. BARENBLATT, G. I., and CHEREPANOV, G. P.. On the wedging of brittle bodies, Prikl. matem. i mekhan. 24, 667-682 (1960) (in Russian). 96. MUSKHELISHVILI, N. I., “Some Basic Problems of the Mathematical Theory of Elasticity”. 2nd ed. Izd. AN SSSR, M.-L., 1935 (in Russian). 97. BITSADZE, A. V., On local deformations of elastic bodies in compression, Soobschenia A N Gruz. S S R 3, 419-424 (1942) (in Russian). 98. MARKUZON, I. A., On the wedging of a brittle body by a wedge of finite length, Prikl. matem. i mekhan. 26, 356-361 (1961) (in Russian). 99. KUZNETSOV, V. D., “Surface Energy of Solids”. GITTL, M., 1954 (Transl., H. M. Stat. Office, London, 1957). 100. YOFFE,E., The moving Griffith crack, Phil. Mag., Vff ser. 42, 739-750 (1951). 101. ROBERTS, D. K., and WELLS,A . A,, The velocity of brittle fractures, Engineering 178, 820-821 (1954). 102. STROH,A. N., A theory of the fracture of metals, Advances i n Physics 6, 418-465 (1957). 103. CRAGGS, I. W., On the propagation of a crack in an elastic-brittle material, J . Mech. Phys. Solid5 8, 66-75 (1960). 104. DANGDINHAN, Elastic waves by a force moving along a crack, J . Math. and Phys. 38, 246-256 (1960). 105. BAKER,B. R., Dynamic stresses created by a moving crack, Paper presented at the X t h Intern. Congr. A p p l . Mech., Stresa (1960). 106. RADOK,J. R. M., On the solutions of problems of dynamic plane elasticity, Quart. A p p l . Math. 14, 289-298 (1956). 107. BROBERG, K. B., The propagation of a brittle crack, Paper presented at the X t h Intern. Congr. A p p l . Mech., Stresa (1960). 108. BROBERG, K. B., The propagation of a brittle crack, Arkiu for Fysik 18. 159-129 (1960). 109. BILBY,B. A,, and BULLOUGH, R., The formation of twins by a moving crack, Phil. Mag., V I I ser. 45, 631-646 (1954). F. A., and SUKHATME, S. P., Travelling cracks in elastic materials 110. MCCLINTOCK, under longitudinal shear, J . Mech. Phys. Solids 8, 187-193 (1960). 111. SUITS,J. C., Cleavage. ductility and tenacity in crystals. Discussion, i n “Fracture” (B. L. Anderson, and 0th.. eds.), pp. 223-224. Wiley, New York, 1959.

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Plasticity Under Non-Homogeneous Conditions

BY W . OLSZAK. J . RYCHLEWSKI

AND

W . URBANOWSKI

Institute ofFundamenta1 Technical Problems. Polish Academy of Sciences. Warsaw. Poland Page

.

I Physical Foundations . . . . . . . . . . . . . . . . . . . . . . . . 132 132 1. Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 2 . Plastic Non-homogeneity of Real Bodies . . . . . . . . . . . . . . 133 3. Definition and Classification of Non-homogeneous Elastic-plastic Bodies . 148 I1. Plane Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 1. Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 151 2 . Basic Assumptions and Equations . . . . . . . . . . . . . . . . . 151 155 3. The Possibilities of Solving . . . . . . . . . . . . . . . . . . . . 4. Equations of Equilibrium of a Non-homogeneous Body in Curvilinear 156 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . The Geometry of Slip Lines and Trajectories of Principal Stresses . . . 160 6 Biharmonic States of Equilibrium . . . . . . . . . . . . . . . . . 165 7. Analytical Solutions in Particular Cases . . . . . . . . . . . . . . 167 8. Approximate Solutions . . . . . . . . . . . . . . . . . . . . . . 174 9 Inverse and Semi-inverse Methods . . . . . . . . . . . . . . . . . 177

.

.

I11. Particular Solutions . . . . . . . . . . . . . 1. General Method . . . . . . . . . . . . . 2 . Axially Symmetric Problems . . . . . . . . 3. Spherically Symmetric Problems . . . . . 4. Torsion of Prismatic Bars . . . . . . . . . 5 Rotating Circular Disc . . . . . . . . . .

. . . . . . . . . . . . 183 ............ 183 ............ 185 . . . . . . . . . . . . . 189 ............ 189 . ............ 190 IV . Elastic-plastic Non-homogeneous Plates . . . . . . . . . . . . . . . . 190 V . Limit Analysis and Limit Design . . . . . . . . . . . . . . . . . . . 191 1. One-dimensional Structural Elements . . . . . . . . . . . . . . . . 191 195 2. Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 3. Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Minimum-weight Design . . . . . . . . . . . . . . . . . . . . . . 199 VI . Propagation of Elastic-plastic Waves in a Non-homogeneous Medium . . 201 203 VII . Other Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

. . . . . . . . . . . . . . . . . . . .

203

......................... . . . . . . . . . . . . . . . . . . . . . . .

205 206

1 Physically Non-Linear Bodies

2 Loose and Cohesive Granular Media . . . . . . . . . . . . . . . . 204 3. Assumption of Non-homogeneity as a Method of Solving Homogeneous

Plastic Problems References . . . . . .

131

132

W.

OLSZAK,

J . RYCHLEWSKI AND W. URBANOWSKI

I. PHYSICAL FOUNDATIONS 1. Introductory Remarks

'

The theory of plasticity represents one of the branches of Continuum Mechanics which is in a particularly rapid development. The progress consists in both a careful analysis of the basic equations and physical relations, and an extension of the range of problems and effective solutions covered by this theory. In recent years one of the points of interest has been formed by the problems connected with bodies, whose plastic properties are functions of the position of the point under consideration P , i.e. with bodies exhibiting plastic non-homogeneity ; the investigations take usually also the plastic anisotropy into account, i.e. the dependence of the plastic properties on the direction considered.* This progress is mainly due to the growing demands of engineering applications, which involve ever increasing parameters such as higher pressures, temperatures, velocities, and greater spans, the requirement of minimum weight of the structure playing also an important part. Properties of materials which have so far been disregarded must now be taken into consideration. The present survey deals with problems of theoretical and experimental research related t o plastically non-homogeneous media. In real media the non-homogeneity of mechanical properties may be caused by numerous phenomena and its nature may be very diverse. Consequently, it is expedient to state precisely what kind of non-homogeneity will be considered in this review. First of all it is evident that a universal property of bodies occuring in practice is their microscopic non-homogeneity. It is well known that there have been many successful1 attempts to include microscopic non-homogeneity in the structure of Continuum Theory. Our considerations, however, do not concern this type of non-homogeneity. Without going into more precise definitions and specifications, let us state that the object of our interest is the macrosco~icnon-homogeneity taking place in regions whose linear dimensions are comparable with the characteristic dimensions of the body under consideration. Moreover, it is known that in most materials used in practice a random distribution of mechanical properties is encountered; sometimes the variations are considerable and due to various phenomena, (as, for instance, the technological processes involved, etc.). This feature of materials is the subject of a developing new science - the statistical theory of strength of

* A t the International IUTAM Symposium in Warsaw, 1958, 18 papers were devoted to these problems, [74]. Some of them will be quoted on several occasions and their results taken into account in the present survey article.

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

133

materials - which deals with stochastically non-homogeneous media, their mechanical properties being random variables. Here, however, we shall consider well-defined distributions of non-homogeneity, the mechanical properties being definite functions of position. I t may be noted that the first attempts in this direction were made about twenty-five years ago (see W. Olszak, [73 e-j]) ; they represented a comparatively general approach, based on the assumption of both nonhomogeneity and anisotropic structure of the body. The scientific and engineering importance of these properties were emphasized ; moreover, these papers, in which, in particular, the static problems of thick-walled pressure pipes, gun barrels, tunnel linings, galleries, and mine shafts were examined [73 e-j], resulted in some effective solutions for comparatively simple systems in the case of non-homogeneous curvilinear anisotropy of the polar and cylindrical types. In these publications the postulate of simultaneous occurrence of the critical plastic state a t all points of the body (for instance, over the entire thickness of a pressure pipe) was also formulated; it was shown that this effect can be achieved by a suitable choice of the field of non-homogeneous properties in an isotropic or anisotropic body.*

2. Plastic Non-Homogeneity of Real Bodies Prior to rigorous definitions and equations, it is necessary to grasp the actual physical conditions connected with mechanically non-homogeneous bodies. As already mentioned, we are mainly concerned with the plastic non-homogeneity, The non-homogeneity of mechanical properties may be caused, for instance, by the following phenomena: the influence of flow of elementary particles; the action of temperature gradients; a non-homogeneous hardening of the material; different types of surface working; a non-homogeneity of the composition, etc. In certain cases, it may be created by ourselves (plastic working, surface and radiation working, etc.), in other case5 it occurs independently (for instance, in soils). We now proceed to consider some typical causes, devoting more attention to phenomena which are new and less familiar.

* A method was also indicated for a practical realization of this theoretical assumption [73 f], [73 h]. [73 i], [73 j]. Analogous problems under more general assumptions have been treated recently (see Chap. I11 of this survey). The following general remark m a y be of interest here. The plastic non-homogeneity can result in effects which, if compared t o those involved in plastic homogeneity, may exhibit essential quantitative and even qualitative differences. Many such results were obtained in the course of the investigations presented in t h e papers which are quoted in this survey.

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W . OLSZAK; J . RYCHLEWSKI A N D W . URBANOWSKI

A . Plastic non-homogeneity due to the action of neutron flow. The problems of plastic non-homogeneity become particularly important in connection with investigations on the influence of flow of elementary particles on the mechanical properties of materials. Numerous experimental investigations have proved that the action of neutron flow is especially strong. The sources of neutron flows of various intensities and energies are provided above all by nuclear reactors. The action of radiation of considerable energy on solid bodies is a very complicated phenomenon and depends on the one hand on the composition and energy of the radiation and on the other hand on the composition of the matter subjected to the radiation. A physical analysis of the mechanism of radiation can, for instance, be found in [17], [30],[6]. We shall present here a general description of phenomena that lead to changes of mechanical properties of materials undergoing neutron radiation. Neutrons, as particles without charge, bring about structural changes only by direct interaction with nuclei. Passing through the crystal structure, the neutron is subject to elastic impacts with atoms (ions) of the material, and a transfer of energy to the latter occurs. The energy needed for displacing an atom from its place in the cristal lattice depends on the kind of material and its structure, the average being 20-25 eV. The neutrons in flows we are interested in possess various energies, the average being of the order of 1-2 MeV. The probability of displacing an atom from the atom site of the lattice is therefore enormous. Moreover, the displaced atom has such a large energy that it displaces other atoms and a chain reaction occurs. The displaced atoms as a rule become interstitial. Thus, along the way of a neutron there arises a large quantity of pairs of defects of crystal structure of the type “vacancy-inclusion’’ (so-called Frenkel’s defects). I t was found that a radiation of sufficient intensity can result in displacing as many as l0-20% of atoms. Frenkel’s defects play an important role in the change of mechanical properties of materials. From the viewpoint of the theory of dislocations it can be stated that the above defects create additional obstacles on the motion of dislocations. This effect has a considerable influence on the plastic and strength properties of the material. In addition to the above effect the creation of displacement zones is of considerable importance, [127], [6]. In the final stage of the motion of a neutron, when it has lost a notable part of its energy, the displaced atoms have a smaller energy. Then the number of impacts among the atoms increases and is so large that practically all neighbouring atoms are set in motion. There arises a local thermal spike (covering several tens of A) estimated a t about a few thousand degrees. I t is accompanied by a violent evaporation and an instantaneous hardening of the structure (in about 10-10 sec), The resulting non-homogeneities of the crystal structure are called displacement zones. They obviously influence the mechanical properties, for instance by a dislocation mechanism.

PLASTICITY UNDER NON-HOMOGENEOUS

CONDITIONS

135

According to [el, in the case of light metals the creation of Frenkel’s defects is decisive, while in the case of heavy metals the displacement zones are of utmost importance. Other phenomena have also an influence on the mechanical properties. In some cases (for instance, for materials of large effective cross sections for some processes, if the neutron flux density is large) we have to take into account nuclear reactions caused by the neutrons, first of all nuclear changes. Quantitative changes of mechanical properties of materials depend on the quantity of radiation measured by the number of neutrons passing through 1 cm2 of the surface of the body. Denoting by n the number of neutrons per unit volume of the flow, by v its average velocity, the intensity of the neutron flow is nv (in neutron/cm2sec) and the quantity of radiation is nvt (in neutron/cm2). There already exists a considerable amount of experimental data concerning the influence of neutron flow on the mechanical properties of various materials. These continuously growing data can be found in various journals devoted to physics and metallurgy, and in review articles [52], [ l l l ] . We especially note the excellent review of V. S. Lensky [57 b], dealing wholly with changes of mechanical properties of materials (274 references) .* It should be observed that the experimental data are by no means complete and that there are no extensive generalizations. The investigations have been carried out under various conditions, and frequently cannot be compared. Moreover, for obvious reasons most of the investigations performed by physicists are devoted to pure elements and often to crystals. All these facts make it difficult to formulate general rules; however, certain most interesting conclusions seem to be already safe. a) Numerous investigations exhibited changes of elastic properties of materialsundertheinfluence ofneutron bombardment, [17], [52], [50], [59]. In some cases significant changes of the elastic modulus E were noted, for instance a three-fold increase in the case of graphite, when nvt = 1020 neutron/cm2, [126]. In the case, however, of structural materials, the changes of E , even for large quantities of neutrons, are of the order of 5% and can be neglected in static calculations. On the other hand, the damping factor undergoes a significant change [57 b]. b) Of much greater importance from our standpoint is the increase of the yield limit and the strength of metals, observed by many investigators. The reasons for this phenomenon have been mentioned before. The results of J. C. Wilson and R. G. Berggren are of particular value [125].

* Many valuable informations are to be found in [I1 I]. This book, however, appeared only when this survey article was already in print and, therefore, could not be reviewed here.

136

W. OLSZAK, J , RYCHLEWSKI AND W. URBANOWSKI

I I I I I 2

3

4

10 qaneutron/cm

5

6

7

FIG.1. Influence of neutron flux on the shear yield limit for monocrystals of copper, [46]. 100000

J

C

c

0 0

I

60000

I 0

t

60000

'I

Y

4

F

& 2

*

40000

20000

I

2

Jnkgra Ted Neutron Flux ().IMeV)

-

FIG. 2. Change of the yield limit due to neutron bombardement for four stainless steels, [125].

PLASTICITY U N D E R NON-HOMOGENEOUS C O N D I T I O N S

137

The increase of the yield limit of a copper crystal, [46], is shown in Fig. 1. Already when nvt = lox8 neutron/cm2, a tenfold increase of the yield limit was observed. Let us note the typical shape of the curve: first a large increase, and subsequently a kind of saturation. 425000

100000

0

r 2

I

4

I

6

Megrated Neutron Flux (nvt; > I MeV)FIG.3. Influence of neutron flux on the plastic properties of silicon-carbon steel, [125].

Evidently, the results for structural metals are more interesting. The results of investigations for a few types of stainless steels, [l25],are presented in Fig. 2. We note a 2 to 2.5-fold increase when nvt = 8.1Ox9neutron/cm2, and the same trend of the phenomenon. The increase of the yield limit and strength for a silicon-carbon steel A-212B is shown in Fig. 3. Plastics used in reactor engineering exhibit a far greater variety of phenomena under neutron bombardment. For instance, the yield limit for some plastics increases quite strongly, while for others it decreases or remains unaltered.

138

W.

OLSZAK,

J . RYCHLEWSKI AND W . URBANOWSKI

c) In some cases a change of the nature of the whole stress-strain diagram under the action of neutron flow was observed. A typical example is provided by the curve for nickel, [21], [7], shown in Fig. 4. The material with a hardening characteristic approaches, as a result of neutron radiation, a material with a rigid-plastic characteristic (curve B ) .

0

20

30

40

True stmin ,% ---+ FIG.4. Change of the stress-strain graph for nickel due to radiation in a reactor (A ordinary sample, B - radiated sample, N loaon/cm2, > 1 MeV, T m SO'C), [7], [21].

The results of investigations [125] for an annealed stainless austenite steel of type 347 (Fig. 5 ) are most interesting. Curve A corresponds to the material before the action of neutrons, the velocities of loading being 0.01 and 0.05 per min. Curves B (velocity of loading 0.01 per min) and C (velocity of loading 0.05 per min) correspond to the material which acquired the quantity of neutrons nvt = 7.8 neutron/cm2. We observe that the material, which originally exhibited strong hardening effects even without yield platform, approached a perfectly plastic model after the action of a neutron flow. d) Radiation effects on materials are, as a rule, of permanent nature. Some effects of this kind can only be eliminated by a thermal working. We have stated before that the changes caused by neutron flows are quantitatively dependent on the intensity and time of radiation. Since the distribution of neutrons in the radiated body is non-homogeneous, the neutron bombardment results in a body the properties of which are functions of position; the body, therefore, is non-homogeneous, [57 b], [44 a]. Thus there arises a necessity of determining this non-homogeneity. Suppose that we know from experiment the function which describes the variation of a property under consideration (1.1)

K = K(uzvt)

under conditions of a homogeneous radiation (e.g. Figs. 3, 4). We assume that the nature of this function is the same at all points in the case of non-

139

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

homogeneous distribution of neutrons*. I t is then evident that the mechanical non-homogeneity is known if the spatial distribution of neutrons in the body is known. We now devote some attention to this problem. 120000

c

80000

t

O

T

1ooooo

14 H

I-

6oom

@"

G

-

-

40000

looooo~-+

20000

0

4

8

Total Elongation, per cent

,I2

E

FIG.5 . Change of the tension graph for stainless austenite steel due to neutron bombardement. [125].

I t is known that the basis of the neutron transport theory is the Boltzmann equation. The relevant theory includes effective integration methods for this equation and various approximate procedures. In our case the essential point is to assume a sufficiently exact and sufficiently effective approximaThis assumption is not very accurate; we shall, however, make i t for the time being.

140

W . OLSZAK, J . RYCHLEWSKI AND W . URBANOWSKI

tion. This, obviously, depends on many factors, namely the distance of the sources, the nature of the medium, of the flow, et.c. We have to apply different approaches, for instance, to structural elements of a reactor located near the sources and to problems of radiation working. It seems, however, that many problems of interest can be treated by the constant cross-section approximation. Moreover, let us observe that in our approach the angular distribution is of no importance, the only relevant distribution of neutrons being their spatial density. Consequently, it is convenient to deal with the corresponding integral equation, instead of the integro-differential equation. If the scattering is isotropic and the body is homogeneous and non-re-entrant, the constant cross-section approximation in the case of stationary processes yields the relation, [15],

where the following notations have been adopted: c, mean number of secondaries per collision; I , total mean free path; r, position vector, p(r), neutron flow density, s(r), source strength. Solving this integral equation, we obtain the distribution of the mechanical non-homogeneity

(1.3)

K

=K

.

[p(r) t ] ,

For many, perhaps even for most of the cases, the diffusion approximation is entirely satisfactory. Making the same assumptions as before, and, moreover, assuming that the sources do not significantly depend on position, we have instead of the integral equation the partial differential equation (1.4)

where L is the diffusion length, depending on the neutron energy. Solving the appropriate boundary value problem for this equation, we obtain the density of the neutron flow and hence the distribution of the mechanical non-homogeneity from Eq. (1.3). In the case of non-stationary processes the solution of the appropriate equations leads to the following result: I

(1.5)

The fact that the energies of neutron flows we are interested in are of the order of 1-2 MeV calls for the expressions (1.2) + (1.5) to be handled

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

141

with some care. The numerical data should be taken from investigations concerning fast-neutron reactors. The foregoing considerations will now be illustrated by an elementary example, which, however, is of considerable importance. Suppose that the semi-space x 3 0 is subject to a parallel neutron flow the intensity of which is independent of time. We assume that no generation of neutrons occurs in the body (absence of nuclear reactions). Making use of the theory of diffusion we have

whence

Let us finally note that in view of the phenomenon of saturation, mentioned in connection with Figs. 1 and 2 , for large times of radiation the non-homogeneity of the medium should decrease. In some cases the non-homogeneity due to direct action of a neutron flow is superimposed on the non-homogeneity due to the associated increase of temperature. Let us also recall that the states of stress and strain are subject to other processes accompanying the neutron bombardment, first of all volume changes, [loo]. The great sensitivity of plastic and strength properties to the influence of neutron flows constitutes undoubtedly a really striking phenomenon for specialists in the field of plasticity. The practical aspects are also of considerable importance. Many modern structures are exposed to neutron radiation; their strength analysis has to take into account the resulting non-homogeneity. Moreover, there arise many interesting phenomena in connection with the possibility of applying radiation-working to improve the properties of the material. All these aspects will strongly influence the development of the theory of plastically non-homogeneous materials.

B. T h e action of tempevatuve gvadients. The influence of temperature gradients on the mechanical properties of materials is a well-known phenomenon. Most of the available data concern the behaviour of metals in variable temperature fields. Without going into details of the physical analysis, we shall present some facts concerning the influence of temperature on the yield limit upl and the strength uUlt. Fig. 6 represents the change of strength of some of the most frequently used Soviet structural steels, investigated by M. I. Zuyev and others, [128]. The dependence of the yield limit and strength of the steel O H N l M 0-65

142

W. OLSZAK, J. RYCHLEWSKI AND W. URBANOWSKI

FIG.6. Influence of temperature on the strength for some of the most frequently used structural steels, [128].

t°C

-

10

FIG.7. Influence of temperature on the yield limit and strength for steel O H N l M , [44 b].

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

143

on the temperature is shown in Fig. 7, 144 b]. Similar graphs for an aluminumalloy-clad sheet 2024- T3 is presented in Fig. 8, [as]. In the case of uranium, a decrease of the yield limit was observed, from upr= 25.10-s psi a t room temperature, to upz= 18.10-3 psi for 600" F, and a decrease of strength from uWrt = 90.10-3 psi at room temperature to uult= 32.10 psi for 600" F, [25 b]. The same results for Thorium are the following: upzfrom 27 to 12, uultfrom 38 to 22. The tensile strength in terms of temperature for materials extensively applied in reactor structures is given in Fig. 9, [lo].

TotFIG.8. Influence of temperature on the strength, yield limit and extension for aluminumalloy-clad sheet, [28].

It follows from the graphs 6 and 7 that in the range of temperatures, in which structural transformations do not yet occur, we observe significant variations of strength properties. Starting from 300-400" C we note a very fast decrease of the yield limit and of the stength. As a rule this is accompanied by a significant trend of these characteristics (cf. Fig. 7 and also Fig. 8) ; the properties of the material approach those of a perfectly plastic body. Incidentally, let us observe that alumothermic chromium makes an interesting exception from the above rules; its strength increases from 470 kg/cm2 for 20" C to 1000 kg/cm2 for 1100" C, [43]. The modulus of elasticity E of metals decreases with the increase of temperature, whereas the Poisson ratio v increases.

144

W. OLSZAK, J . RYCHLEWSKI AND W . URBANOWSKI

Many empirical formulae have been proposed to describe the dependence of mechanical properties on temperature. For some metals the formula (1.8)

K , = K,e-

a(Ta

- TL)

may be used. A decrease of temperature leads for numerous steels to an increase of the yield limit andtensilestrength, [124]. Certain other phenomena also occur, for instance the brittle fracture investigated by T. Pelczydski, [go]. According

1

Temperature OCFIG.9. Influence of temperature on the tension strength for reactor structural materials, [lo].

to the data presented by G. V. Uzhik for the steel 1010 the yield limit increased from 2300 kg/cm2 at the temperature 17" C to 7200 kg/cm2 at - 1'37" C ; this was accompanied by an increase of the tensile strength from 3700 kg/cm2 to 7300 kg/cm2. The influence of temperature on the mechanical properties of other structural materials may be different for various groups of materials. For

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

145

instance, ceramic materials, concretes etc., in the range of temperatures resulting in no structural changes (chemical processes, sintering, etc.) do not generally exhibit significant changes of the mechanical properties. In contrast to this, plastics are very sensitive to temperature variations. When the distribution of temperature in the body is inhomogeneous, the body with mechanical properties depending on temperature becomes elastically and plastically non-homogeneous. These non-homogeneities are determined similarly to the case of neutron flows. Solving the appropriate boundary value problem for the heat conduction equation, [70],

A P T - CP-

aT at

+ W = 0,

where A is the heat conduction coefficient, c the specific heat, p the density, and W = W ( P ) the intensity of heat sources per unit time and volume, the mechanical non-homogeneity is obtained in the form (1.10)

K

=K[T(P,I)].

Observe that in some cases it is also necessary to take into account the thermal non-homogeneity, [116] (for instance the dependence of A on P as the result of the action of variable T ) . I t should be noted that the influence of temperature can result in mechanical non-homogeneity of two kinds, namely: a) that existing while the temperature is variable; b) that arising after the temperature field has become homogeneous, due to temperature peaks which led to structural changes. Still a different topic is formed by problems in which high time temperature gradients occur (thermal shocks, etc.).

C.. Non-homogeneous strain-hardening of metals. Consider a body made of a material exhibiting strain-hardening. If the body be subjected to forces resulting in a plastic deformation, then in the case of a non-homogeneous stress field this process leads to a body of variable yield limit; this is a typical plastically non-homogeneous body. We do not dwell here on these problems but regard them as fairly well-known in literature. Non-homogeneities of this kind are very frequently encountered in practice. They occur as the result of technological processes of plastic working: cold rolling, drawing, forging, etc. The distribution of such nonhomogeneities is determined by considering the history of the loading pr6cess a t all points; this constitutes a problem of plasticity with strain-hardening. For torsion, this problem was examined by R. Hill, [34 b]. I. Berman examined the plastic non-homogeneity of a pipe manufactured by means of cold working from a sheet, [4]. W. Truszkowski found experimentally

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W. OLSZAK, J. RYCHLEWSKI AND W. URBANOWSKI

a plastic non-homogeneity in the “neck” of a rod subjected to large plastic deformations in tension, [123]. Test pieces subjected to a dynamic load may also exhibit a non-uniform distribution of plastic properties, which was confirmed by D. B. Taylor, [121], and J . D. Campbell, [8].

D. Surface working. In the field of machine elements and structures, a procedure is frequently applied which we call surface working; it consists in applying a hardening effect to the surface of a manufactured element. This working is carried out in many ways; for instance, the processes considered in A, B, and C can be applied. The diffusional processes (nitriding, cementation, cyaniding, metallization) lead to a change of the properties of the body up to a depth of 2-3 mm and even more. Sometimes the working by means of high frequency currents is being used. Another effective method is shot peening. These procedures of surface working produce, besides other effects, a change of mechanical properties, and hence effects of non-homogeneity caused by the elementary structure that varies with the depth. On the surface, a high strength layer is created, which gradually passes into a homogeneous weaker core (cf. M. T. Huber, [40a]). The non-homogeneity due to surface working concerns as a rule the plastic properties, the elastic properties remaining essentially unaltered. E. Complex structwres. Frequently the mechanical non-homogeneity is due to the non-homogeneous composition of the body. A typical example are soils which usually are media of variable properties (see Chap. VII). A required type of non-homogeneity may in practice also be obtained in another way, namely by designing “composite” structures, well-known in engineering practice and based on the co-action of two or more “components” possessing different mechanical properties. Typical examples are reinforced concrete structures with variable amounts of steel reinforcements, sandwich and layered structures, etc. An approximate analytic model of such a structure may be a body, whose mechanical properties vary continuously in such a way that the variable average properties of the actual material are imitated. The first suggestions of a theoretical nature and the related practical solutions employing the possibilities contained in this idea were given in the papers mentioned above, [73 f]- [73 h], [73 j]; the objects of applications were thick-walled high-pressure pipes of reinforced concrete, thick-walled linings for mine shafts, tunnels and galleries of variable density of reinforcement. In [73 i], the same author applied an analogous idea to gun barrels. Obviously, there exist other causes of non-homogeneity, not dealt with here. Thus the mechanical properties of many bodies depend, e.g., on the degree of humidity, on gradual chemical changes, etc.

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

147

One of the possible non-homogeneity parameters of averaged mechanical properties may also be a variable thickness of a plate or shell. In fact, the limit analysis of load-carrying structures of variable thickness is a particular case of the general theory of non-homogeneous bodies. In addition, it may be observed that actual engineering structures are, as a rule, characterized not only by the above considered non-homogeneity (artificial or not); in fact, another typical property is their anisotropy (material or structural). This concerns first of all the reinforced-concrete structures mentioned before, in which the steel reinforcement is always arranged in certain particular directions. However, the anisotropy may also be due to processes of plastic deformations or plastic working, as well as to structural arrangements (e.g. ribs), etc.* Thus in order to approach the real conditions as closely as possible, the majority of papers (by Polish authors in particular) take into account not only the non-homogeneity of the material but also its anisotropy. This is also true for the first of the above mentioned papers on these problems [73 f ] - [73 i]. The theory of plastically non-homogeneous media is faced with two groups of tasks. First, it is necessary to develop sufficiently effective methods of determination of the states of stress and strain, when the non-homogeneity of the medium is given. Secondly, we must be able to determine certain postulated distributions of non-homogeneity (e.g., optimum distribution), if the loading is given; in other words, it is necessary to create sound theoretical foundations of the methods in use (surface working, plastic or thermal working, complex structures) and future methods (e.g. radiation working) of improvement of the properties of structures. As regards machine parts, N. 1. Popov and Ya. B. Fridman, [93], have indicated a possibility of creating the required non-homogeneity (mainly in metallic elements) by means of a suitable mechanical or thermal treatment. Elastic-plastic bending of beams with surface effects due to such a treatment were investigated by A. T. Poletzky, [92]. N. D. Sobolev and Ya. B. Fridman, [114], considered the interaction of two scalar fields: the field of reduced stress and that of the local yield limit (more strictly, the local strength). They introduced coefficients

* Experimental investigations (for instance those of A . Krupkowski [53], P. M. Naghdi and J. C . Rowley [SS],L. Hu and J. Marin 1391) have proved t h a t even initially isotropic materials acquire anisotropic properties in the course of plastic deformation. Hence, if we extend the theory to include the class of anisotropic bodies, we should specify the type of anisotropy considered - whether only initial anisotropy or also anisotropy caused by plastic deformation is taken into account. The principal difficulty in taking into account the change of anisotropy during plastic deformation consists in the choice of the function Y determining the plastic potential. No ways of general solution of this problem have so far been proposed, our object therefore will not be quite as general, and the initial anisotropy will be the only one taken into account.

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characterizing the degree of utilization of the strength, both the ultimate one and that related to the reduced stress distribution. All these facts which prove that the non-homogeneity of mechanical properties of a material can have a strictly determined (and regular) nature, call for an investigation of its influence on the states of stress and strain of the systems under consideration. We note that as a consequence of the above considered processes there may, in the bodies considered, arise stresses which essentially influence their subsequent response to loadings. The problems related to this phenomenon are outside the scope of this survey and will not be discussed here.

3. Definition and Classification of Non-Homogeneous Elastic-Plastic Bodies The relations between the strain and stress tensors and their time derivatives are expressed by the so-called constitutive equations, the mathematical formulations of which contain also certain quantities (moduli) that characterize the mechanical properties of the material independently of the states of strain and stress. The material characteristics describing the behaviour of the body in the elastic range will be called its elastic moduli, the set of which will symbolically be denoted by Me'. It is known that the elastic moduli MeJcan be regarded as components of a certain tensor (tensor of elastic moduli). Such an approach has many merits, such as the possibility of expressing the constitutive equations in a form invariant with respect to coordinate transformations. Thus, in an elastically homogeneous body the invariants of the tensor of the elastic moduli are independent of the position P ( x j ) . Such bodies are described by the set of moduli Md'. Now, non-homogeneous elastic bodies are characterized by a tensor field of moduli, the invariants of which depend on position; the corresponding set of moduli will be denoted by Mpel (see [73 k], [73 11, [71], [82 h] and [82 i]. If elastic-plastic bodies are investigated the problem requires introduction of suitable additional concepts. The phenomena occurring in the material when the yield limit is reached will be described by means of certain quantities which, by analogy, will be called the plastic moduli of the material and will symbolically be denoted by Me'. Let the generalized concept of plastically non-homogeneous bodies be related to bodies the yield condition of which has the form (1.11)

F(Ji;MpP') = 0,

where 1,denote the invariants (i = 1, 2, 3) of the stress (or strain) tensor and MpPrthe set of the plastic moduli; in view of the assumed non-homogene-

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

149

ity the moduli M,P' depend on position P ( x i ) . In the simplest case of a perfectly plastic body the set of plastic moduli is reduced to a single function

K pP'. * Thus, the approach is similar to that for the case of elastic bodies. The plastic moduli Mp' may be regarded as the components of a tensor (the tensor of plastic moduli).+ If its invariants are independent of position P ( x , ) , we shall refer to plastically homogeneous bodies (the set of moduli being then denoted by MOP'),whereas plastically non-homogeneous bodies require introduction of the concept of a tensor of plastic moduli whose invariants are functions of the coordinates of P(xi) (the corresponding set of moduli being then denoted by MpP') The different types of bodies, which can be represented by theoretical models on the basis of the above concepts, may be classified according to various principles. In view of engineering applications it seems that the classification presented in the papers [73 k], [73 I], (the C-classification) may prove particularly useful; accordingly, the more important types of bodies may be classified as follows: (I) elastically and plastically homogeneous bodies (M;', and Mop! respectively) ; symbol C,, : (11) elastically homogeneous but plastically non-homogeneous bodies (M;' and MpP', respectively); symbol C o , p ; (111) elastically non-homogeneous but plastically homogeneous bodies (MPC'and MOPL,respectively); symbol C P , , ; (IV) elastically and plastically non-homogeneous bodies (Mpe'and Mpp', respectively) ; symbol Cp,p. The order of the groups is, generally speaking, such that each body is of a more general type than the preceding one. In the most general group (IV) two cases are to be distinguished: (IV,) the moduli Mpcland MpP' depend on position P and are independent of each other; the corresponding symbol is C p ~ ; (IV,) the moduli Mpc' and MpP' depend on position P but remain (as implied by experimental data and certain theoretical consideration, see,

* It is interesting to note t h a t there exists a possibility of representing the workhardening type of a n elastic-plastic body (for any form of the e,o-graph) by means of a model of a perfectly plastic non-homogeneous body; the same holds for the explanation of some phenomena known from experiment, for instance "the Bauschinger effect"; this problem has been treated by G. Colonnetti. [12], and H. F. Bohnenblust and P. Duwez, [ 5 ] . See also the footnote 15b in [73 k]. t However, this tensor must satisfy certain additional conditions which are discussed in the papers of W. Olszak and W. Urbanowski, [82 hJ, [82 j].

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W. OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI

e.g. [72], [73 k], [73 I], [82 h], [82 i]) cozqbled in a certain manner; this can symbolically be expressed by the relation (1.12)

MpPt = f(MPet) ;

the corresponding symbol is C p 7 ( = C,).* The theory of classes (1)-(111)can be deduced from the theory of class (IV) by introducing successively stronger limitations on the generality of the basic assumptions. Another classification (the S-classification) that holds essentially for work-hardening bodies is given in a paper of W. Olszak [73 k], [73 11, where it is proposed on the basis of the scalar function characterizing the workhardening properties (Prager’s function, [94a], [94b]). In addition to this, the paper [73 k] presents relationships between the classifications of the non-homogeneity in accordance with the (C) and (S) schemes. An essential point is the special role played by class I1 (symbol C,,p). The importance of this class is readily observed, if we bear in mind the physical foundations exposed in Sec. 2. Metals subjected to the action of neutron flow are typical examples of bodies possessing marked nonhomogeneity of plastic properties, non-homogeneity of the elastic properties being insignificant. The same holds for the non-homogeneity of metals as a consequence of work-hardening, surface working, and temperature variations in certain ranges. From the standpoint of mathematics, it is important to select properly the functions describing the non-homogeneity ; this selection is also strongly influenced by the physical considerations of Sec. 2. As a rule we shall assume that these functions are bounded and sectionally continuoust.

* The actual nature of the coupling evidently depends on the nature of the causes that lead to non-homogeneity of the properties. If a cause of measure A leads to a change of the elastic and plastic properties MpI = ,)@?I and, say, the functions q and

$-’

Me’ = $ ( A )

are single-valued, we have a one-to-one coupling

MPI = q[$-’(M6’)],

f = fp *

$-1.

Since, on the one hand, there is a great number of various bodies and of causes of non-homogeneity, and, on the other hand, these phenomena are very little investigated, it is difficult to state more definite views on this topic. f The condition of continuity (and also of differentiability of the functions M p etc.) will be used in subsequent considerations; they are specified in greater detail in the papers referred to.

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151

11. PLANESTRAIN 1. Introductory Remarks Problems of plane strain form a branch of the theory of plasticity with comparatively old traditions and numerous results. The theory of plane strain is particularly advanced for homogeneous perfectly plastic bodies ; the character of the basic equations has been analyzed, important general theorems have been proved, integration methods have been developed, and a number of exact and approximate solutions are available. This state is reflected in the treatment of the theory of plane strain in the monographs by A. A. Ilyushin, R. Hill, V. V. Sokolovsky, W. Prager and P. G. Hodge, and L. M. Katchanov. For a body with variable plastic properties the situation is quite different. The plastic non-homogeneity influences essentially both the mathematical side of the problem and its physical sense. The character of the fundamental equations changes and becomes complex, and some of the familiar theorems and notions cease to hold. These difficulties and also the circumstance that the interest in these problems is comparatively recent, result in the fact that even the theory of an incompressible perfectly plastic body is only in the first phase of development. Integration methods are still undeveloped, and there are not many solutions available. It would, therefore, be premature and even impossible to give such a systematic and deep analysis of these problems as was given by H. Geiringer [29] for the classical case.* Therefore, our survey will, of necessity, have a somewhat fragmentary character. I t will discuss prospective possibilities of development rather than the developments themselves. A number of problems, among them essential ones, will be omitted or only touched upon. Thus, for instance, we shall confine ourselves to the analysis of the statical side of the problem, with questions related to deformations being only mentioned. Our attention will mainly be focused on the differences between plastically non-homogeneous and homogeneous bodies. 2 . Basic Assumptions and Equations

In principle, a rigid-plastic, incompressible, anisotropic, plastically non-homogeneous medium without strain-hardening will be considered. Several remarks will also be devoted to plastically non-homogeneous bodies having other properties. By classical case we mean that of homogeneity.

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W. OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI

Plane strain, which is the object of this Chapter, will be defined starting from the two-dimensional case which is the more general one. In the case of non-homogeneous (isotropic or anisotropic) bodies, the twodimensional problem ceases to be trivial, as was shown in Ref. [79]. A two-dimensional state may be defined in four ways. Starting from the most general, they are as follows: 1) dEii depend on the coordinates x1,x2 only, the stress tensor Ta being still a function of all the three variables, T,, = T,(x,,x2,x3); 2 ) in addition to what was said under 1, we have

3) in addition to what was said under 1 and 2, we have

4) in addition to the above conditions, the non-vanishing components of the stress tensor are exclusively the normal and shear components parallel to the basic plane, and u33. The plane strain problem is thus seen to be a special case of the more general two-dimensional strain problem. Therefore a clear distinction will be made between these two definitions in the non-homogeneous theory. The fundamental equations may be specialized for various types of anisotropy (monoclinic, orthotropic, transversally orthotropic, cubically anisotropic, and isotropic). In the case of two-dimensional stress, analogous definitions can be given for all the possible particular cases (and various anisotropy types).* In Cartesian coordinates the familiar fundamental equations for the plane strain are as follows. The tensor field of stresses uii = aii(x,y) is determined by the relations (2.1)

= a,

0;x

=0

o6a = B k J X X

+

.yy)

;

the equations of equilibrium (body forces being omitted) are (2.2)

0xx.x

+

uxy.y

= 0,

uxy,x

+

oyy,y

= 0;

the yield condition is (2.3)

( u x x - ayJ2

+42,

The tensor field of strain rates dij (2.4)

= 4K2(x,y).

= dii(x,y)

d,, = dlY = dg*

satisfies the relations

=0

* An interesting case of quasi-plane stress in an elastic semi-space subject to the action of temperature on the boundary has been given by W. Nowacki, [70].

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

153

and the incompressibility condition

+

ixx iyy = 0.

(2.5)

Its relation to the vector field of displacement rate is (2.6)

8xx

= vx,x,

8yy

= vy,y,

gxy

= S(vx.y

+

vy,x).

The stress tensors and strain-rate tensors are related by the condition 2i.q

(2.7)

8yy

-

- 8xx

2 u ~ =~ctg 28, uyy -

uxx

where 8 is the angle between the tangent to the trajectory of maximum stresses (the slip line) and the x-axis. Assuming the five functions uxxruyy,uxy,v,,vy to be unknown, we have five equations for their determination. They are the Eqs. (2.2), (2.3) and (2.8) (2.9)

(vx,y

+

vy,x)(uyy

-uxx)

+

2(Jxy(vx,x

- %y)

vx,x

+

yy,y

= 0, = 0.

This system of equations was obtained first by B. de Saint-Venant, [lo4 a ] ; the only difference in our case is that we have a function of x,y instead of a constant on the right-hand side of (2.3). This circumstance, however, will be seen to have serious consequences, thus making the entire problem quite different from the classical one. The boundary conditions are formulated, in general, in terms of stresses and velocities. If the boundary conditions in the stresses are such that we can determine the field uii from Eqs. (2.2) and (2.3), the vector field v will be found from (2.8), (2.9), and from the boundary conditions in the displacement velocities. The form of the latter enables us to formulate the following simple theorem : Let us consider two bodies, geometrically identical but mechanically different, i.e. characterized by plastic non-homogeneities of different types, K,(x,y) and K,(x,y). Let these two bodies become entirely plastic under the action of two different systems of forces. As a result the stress field a,,(') will be realized in the body (1) and us?)in the body (2). If now (a) the boundary conditions in the displacement velocities stresses are the same for both bodies, and (b) the relation (2.10)

holds (i.e. the slip lines coincide), then the same velocity field will be obtained in both bodies, i.e. v(') = v ( ~ ) .

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W . OLSZAK, J . RYCHLEWSKI A N D W . URBANOWSKI

In the case of homogeneous bodies, this theorem reduces to the familiar possibility of superposition of a hydrostatic pressure.* Let us add that the system ( 2 4 , (2.9) is linear and the same as in the classical case. The differences between the non-homogeneous and the classical case lie in the first group of our equations. Thus, the statical side of the problem i.e. the system of equations (2.2) and (2.3) will be the main object of our interest. In other words, the object of our discussion will be statically admissible stress fields in plastically non-homogeneous bodies. There are various types of approach to a system of two partial differential equations of the first order (2.2) and an algebraic condition of the second order (2.3). As is known, the two classical methods for homogeneous bodies are : (a) (B. de Saint-Venant, [lo4 b]) : The equilibrium equations (2.2) have to be satisfied identically by the Airy function, thus leading from (2.3) to the following equation for its determination (2.11)

(a,yy

-Q

xJ2

+ 4Qpy = 4K2.

This is a difficult non-linear equation. An ingenious solution was obtained in this way by L. A. Galin [27]. (b) (M. LCvy, 1581): The yield condition is satisfied identically by the trigonometric substitution (2.12)

uXx= u - K sin 20,

uYy= u

+ K sin 20,

uxy= K cos 20,

thus leading from the equilibrium equations to two partial differential equations for the two functions u,0 which are now sought. This way has led to remarkable achievements in the theory of plane strain of a perfectly plastic, incompressible, homogeneous body. There are also a number of other methods (see for instance H. Geiringer ~91). For the time being, we shall follow the best explored way (b). Other possibilities will be mentioned later. Substituting (2.12) into (2.2),we have u,.

(2.13)

+

- 2K(0,, cos 20 + O,y sin 20) = K,xsin 20 - K,ycos 20,

u , ~ 2K( - O,x sin 20

+ O,y cos 20) = - K,xcos 20 - K , ysin 20.

These equations have been given by A. Kuznetzov [55a]. Treated as equations for a,@ with a given function K(x,y), these equations are quasi-

* In the theory of plane strain of elastically non-homogeneous bodies, the same stress fields may be, in general, related to an infinite number of different strain fields (cf. [80 c]).

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

155

linear, non-homogeneous. Clearly, the terms containing the derivatives of K do not influence the form of the characteristics, which are still identical with the slip lines. The system of equations (2.13) differs in an essential way from its particular case when K = const. I t constitutes an irreducible system [55 a]. In other words, no change of variables of the type (2.14)

is capable of reducing this system to a linear one. The proof has been given by A. I. Kuznetzov (see also Sec. 4). Let us write the characteristic system corresponding to (2.13):

+

du - 2KdO - K,%dy K,,dx

= 0,

(2.15)

(2.16)

dy

= tgOdx;

da

+ 2KdO + K,,dy - K,,dx = 0,

dy

= - CtgOdx.

on the slip line of the first family on the slip line of the second family

I t can be shown that these equations have no integrable combinations (cf. Sec. 3). If, however, it is assumed that the change of variables (2.14) reduces the system (2.12) to a linear system and that x,y, say, are its characteristic variables, then it follows that (2.17)

fl(x,y,u,O)= const fz(x,y,u,O)= const

constitute integrable combinations of (2.15), (2.16). This contradiction proves the theorem. Therefore, the plane-strain problem of a plastically non-homogeneous body is essentially non-linear (irreducible to a linear one). The solution of such a problem encounters serious mathematical difficulties. In conclusion of these introductory remarks let us observe that Eqs. (2.13) are also valid, of course, for a strain-hardening body, in particular for a primarily non-homogeneous strain-hardening body. In this case K is not a function given beforehand, but depends on the deformation process. 3. The Possibilities of Solving As already noted, the system of equations describing the plane strain equilibrium of a plastically non-homogeneous body offers serious difficulties in the forms (2.2), (2.3) as well as (2.13) or (2.11). In particular, the technique

156

W. OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI

developed for the homogeneous case cannot be applied to the case of the Eqs. (2.13). Thus, for instance, the Kotter-Hencky relations are no longer valid, no solution in the form of simple waves can be obtained etc. The question arises, what are the prospects and actual possibilities of solving problems of non-homogeneous bodies.

Analytical methods, Analytical integration methods of irreducible nonlinear partial differential equations have not yet been developed. The difficulties encountered in the considerably simpler homogeneous case do not seem to predict that exact integration methods for actual boundaryvalue problems will be found soon, simple particular cases being of course excluded. It may be expected that even if appropriate methods are developed in the future the solutions will have a rather complex form (cf. the integration by the Riemann method in the homogeneous case), Afifiroximate meth.ods. The most important of these methods is of course the numerical (or graphical) method of characteristics, which is very versatile and has an “algorithmic” character. However, it gives no possibility of analysing the influence of each particular parameter on the solution. Therefore other approximate methods have been devised, especially for the case of weak non-homogeneity (B. A. Druyanov, A. I. Kuznetzov). Variational methods have not yet been applied, but their application is not expected to be different from the homogeneous case. Inverse and semi-inverse methods. I t seems that one of the typical features of non-homogeneous problems is at present the preponderance of inverse and semi-inverse methods, which offer remarkable possibilities of obtaining closed-form solutions in a relatively simple manner. These methods are important, first, because they enable us to examine the qualitative influence of plastic non-homogeneity ; second, because they may constitute a measure of efficacy of the approximate methods; and, third, because in view of the hyperbolic character of the system of equations they can represent parts of the solution in more complex cases. In Secs. 6 and 7 the results hitherto obtained by means of approximate and inverse methods will be discussed in greater detail. 4. Equations of Equilibrium of a Non-homogeneous Body in Curvilinear

Coordinates In what follows it will be more convenient to use the equilibrium equations of a plastically non-homogeneous body referred to an arbitrary system of orthogonal curvilinear coordinates, in particular to the system of slip lines and trajectories of principal stresses.

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

157

Preparatory to this section and the next, let us briefly recall the most important facts of the geometry of nets. Consider a curvilinear orthogonal net, related to a fixed Cartesian reference frame by (2.18)

y

x = x(a,P)

=

r(a,P).

(The expression d i n e will be applied to a line the labels correspond to the lines considered.) The Lam6 parameters (2.19)

dsa

H --

' - da

2

P

= const

and vice-versa;

+ Y , ~ ) " ~ , Hp =-las, = (x,p + y;p)l/' dP

=( x , ~

2

2

are functions of a,P and satisfy the relations

+

(2.20)

= 0.

&),p

They determine the net within the indeterminacy of displacements and reflections. These quantities have no direct geometrical sense, because they depend on the way in which the parameters of the net are chosen. For (2.21)

a = a(a*),

P

= P(P*)

we have (2.22)

Ha* = a,a** Ha,

H p = P,p * Hp.

An isothermal net is a net with a certain parametrization (called isothermal parametrization), for which we have (2.23)

Ha = Hp

=H .

The necessary and sufficient condition for the net described by Ha,Hp to be isothermal is (2.24)

Ha = A (a) * B(P). -

HP

The geometric quantities are determined by Ha,Hp in the following way: for the angle of inclination rp of the vector tangent of an a-line with respect to the x-axis we have (2.25)

158

W. OLSZAX, J . RYCHLEWSKI AND W. URBANOWSKI

the curvatures of the net are

These quantities are positive for the configuration illustrated by Fig. 10. It can easily be seen that

X

FIG. 10. Positive directions of the lines a$.

Four coordinate systems will be introduced

C ; x,y

Cartesian coordinates ;

L ;ar,P

arbitrary (that is, not related to any physical phenomena) fixed orthogonal curvilinear coordinates;

P ; y,S

orthogonal curvilinear system of trajectories of principal stresses ;

S;

5 , ~ orthogonal curvilinear system of slip lines.

The equations describing the equilibrium of a plastically non-homogeneous body in plane strain, referred to the L-system, have the form [69]

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

159

Applying the substitutions of M. Lkvy a , = a - K sin 24,

where

app = a

+ K sin 24,

4 is the angle between the tangents

Gap = K

cos 2$,

to the a- and (-lines, we have

Ha K,p cos 2 4 - K,a sin 24 + -

= 0,

HP K , cos 2 4 + +Ha

= 0.

HP

(2.31)

K,p sin 2 4

If L coincides with C, we have, of course, the Eqs. (2.13), 4 8. If t,h ~ n / 4the , system L coincides with P, and Eqs. (2.31), referred to the trajectories of principal stresses, have the form

If t,b E 0, the system L coincides with S, and the equations (2.31),referred to the slip lines, are (2.33) a,t

+2

Ht,, K A Hrl

Ht + Hrl K,rl = 0,

a,,,

-

H = 0. + 2K;Hrlt Ht + -K t

Since, obviously, the net of slip lines constitutes a natural system of coordinates for plane strain, some remarks will be devoted to Eq. (2.33). In the homogeneous case when K = const, we obtain immediately, making use of (2.25) with pl = 8, the Kotter-Hencky integrals

and the relation (2.35)

8,crl

=0

together with (2.27) determines geometric properties of the slip lines known as the first and the second Hencky theorem.

160

W. OLSZAK, J. RYCHLEWSKI A N D W. URBANOWSKI

I t is seen from (2.33) that none of the above facts take place in the case of a plastically non-homogeneous body. Moreover, in the general case of K # const, no relations exist that constitute a generalization of (2.34). Certain integral relations along the slip lines could only be obtained by postulating additional relations. If there exists a function F such that

then

(2.37)

0

+F

=f(q),

0

-F

= g(5).

This can happen only in very special cases of the form of the slip lines and of K(c,q). Treating the relations (2.36) as partial differential equations for K , we can write the conditions, which determine admissible classes of slip lines. I t is found, for instance, that by requiring K to be determined up to a constant, we obtain among other relations [In (Hc/Hq)],cq = 0, which, in view of (2.24),is equivalent with the isothermal property of the net. 5 . The Geometry of Slip Lines and Trajectories of Principal Stresses*

I t is known that not every orthogonal curvilinear net can play the role of slip lines in a homogeneous body. Only Hencky-Prandtl nets satisfying the familiar Hencky theorems expressed by (2.35) and (2.27) are admissible. In a non-homogeneous body this problem becomes more complex. Let us write the compatibility condition of the system (2.31) with respect to u. On differentiating and subtracting we obtain

(2.38)

+

HP Ha cos 214 *K,uu sin 294 S K , ~-P cos 214 * K J ~ 2HU 2HP

~

+ [ A , cos 214 + A , sin 2 # ] K , , +

+

~

[B, cos 214

+ [(C, + C,) cos 2# + (C, + C,)

+ B, sin 21,hlK,~+

sin 2I41K = 0

where

*

In this Section, mainly the results of Ref. [80 a] are discussed.

161

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

This equation relates, in any fixed system L , the plastic non-homogeneity function K(cr,P) to a geometric quantity, #(cr,P), connected with the net of slip lines or, in view of # = n / 4 , with a geometric quantity connected with the net of trajectories of principal stresses. Thus it characterizes the geometric properties of the slip lines and trajectories of principal stresses. This equation must hold if the stress field is to satisfy, in addition to the equilibrium equations (2.28), the yield condition (2.29). I t will be called the geometric condition of plastic state. If the system L coincides with C, then $ f 8, and the geometric condition of plastic state has the form

x+

w,,, - K , ~ ,cos) 28 + 2 ~ , , sin , 281 +

(2.39)

-

-

4 K , , [COS 28 8,y - sin 28 O,,] 2 K { 2 cos 28

[e,,,

+ (e,,)'

- (8Jz]

+ 4 K , , [COS28 - 8,, + sin 28 - 8,y]+

+ sin 28 - [-

48,, 8,y

+ O,,,

- O,,,])

= 0.

The form of (2.38) is most interesting when expressed directly in the net of trajectories of principal stresses or the net of slip lines. If $ ~ n n / 4the , system L coincides with P, and we have

If $

= 0, the

(2.41)

system L coincides with S, and we have

+ 2 K [(G) HC .c

(%),?I

=O.

Eqs. (2.40), (2.41) relate the plastic non-homogeneity function K with the geometry of the net of trajectories of principal stresses and of slip lines expressed by means of Lam6 coefficients and subject to the restrictions (2.20).

162

W. OLSZAK, J. RYCHLEWSKI A N D W. URBANOWSKI

If the form of the non-homogeneity function K = K(a,P) is given, the equation (2.38) determines the class of nets admissible as slip lines and the class of nets admissible as trajectories of principal stresses. If we have K = K ( y , d ) , the admissible nets of trajectories of principal stresses are determined by (2.40); if we have K = K(C,q),the admissible nets of slip lines are determined by (2.41). Let us consider for instance K = const. I t can easily be seen that only the last terms remain in (2.38), (2.39), (2.40), (2.41). Therefore the nets admissible as trajectories of principal stresses in a homogeneous body are subject to the restriction (2.42)

[In(H,Ha)I,,d = 0,

whereas those admissible as slip lines must satisfy (2.43)

Upon confronting the latter result with (2.25), it is seen that this is equivalent to the Hencky theorems, as could be expected. Therefore, the latter constitute a particular case of the geometric condition of plasticity for K = const. If K # const, the direct geometric sense of the relevant restrictions is difficult to grasp (i.e. to express in terms of net geometry) even in the simplest cases. Another approach to the geometric condition of plasticity is also possible. Taking for instance (2.40),we can, for a given class of nets or a given net y,6, seek the class of non-homogeneity for which they are admissible. Thus, for instance, for nets or trajectories of principal stresses subject to (2.42) we have (2.44)

and, in addition to K = const, we have an enormous variety of other nonhomogeneity types. Every orthogonal net can be considered to constitute a net of slip lines (or trajectories of principal stresses) in a body of an appropriate nonhomogeneity type* or, even more, for a large class of such bodies. In other words, the set of nets admissible for slip lines (or trajectories of principal stresses) has no common geometric features to distinguish them from other orthogonal nets.

* In this section, not all the conditions for which the relations in question take place are considered (boundary-value problems are not discussed).

163

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

We proceed now to discuss certain particular cases of Eqs. (2.40), (2.41).

A. Trajectories of principal stresses. Eq. (2.40) can be written in the convenient form (2.45)

+ (In Hd),yK,df [In (HyHd)],&

K,yd -k (In Hy),dK,y

0.

If K = K(y,S) is given, we have for Hy,Hd the non-linear system of equations, (2.45), (2.20). The inverse problem is somewhat simpler, and some elementary examples will be given. A Cartesian system (H,= Hd = 1) may constitute a net of trajectories of principal stresses only when (2.46)

K

= K,(4

+ KAY).

This is a hint for the solution of boundary-value problems with rectilinear edges. The polar net (Hr = 1,H, = Y ) may be admitted as a net of trajectories of principal stresses only when (2.47)

Let us consider a class of nets with the property that one of the families, the a-lines for instance, is composed of straight lines. In view of x, = 0 from (2.26), we have Ha = A ( a ) . The parameters can be chosen so that Ha = 1. Then, HB = BI(P)a B,(P) from (2.20). For the Lam6 parameters we obtain

+

(2.48)

Hy

= 1,

Hd = y

+ D(a),

and by virtue of (2.26) we have (2.49)

Ra = Ha

=

f D(S),

D(S) = R d l y = 0.

The geometric condition of plastic state takes the form (2.50)

Hence (2.51)

We observe that in the class of nets for K = const only the polar net may constitute a set of trajectories of principal stresses.

164

W. OLSZAK, J. RYCHLEWSKI A N D W. URBANOWSKI

The geometric condition of plastic state takes a particularly simple form for isothermal nets in isothermal parametrization. Substituting

K

(2.52)

=

T*h

h = H-I,

where

we have

h T,yd - h,y5 T = 0,

(2.53)

where, from (2.20);

P h = 0.

(2.54)

We see at once that the following expression* is a particular integral:

T=C-h

(2.55)

or

K=H-2.

Another immediate observation is that

involves (2.57)

and vice-versa. The geometrical sense of (2.56) is evident.

From (2.26) we have

x y = h2’(d),

constant on the y-lines;

= hl’(y),

constant on the &lines;

xd

therefore the net is composed of two families of circles. Conversely, if the net is of that type, (2.56) follows from (2.26). Thus, if the trajectories of principal stresses are two families of circles, then (2.58)

K =H - l W

+ g(41

and vice-versa. Thus, for instance, for a bipolar net we have (2.59)

K

= (cos y

+ cash 4 [f(r)+ g(41.

For logarithmic spirals (2.60)

*

, p - l n - =r y y , a

p+ln-= rd a

This integral has been obtained by S. Zahorski directly from (2.40).

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

165

we have (2.61) and the equation determining the admissible non-homogeneity types reduces to the telegraphist's equation (2.62)

TP!3

+tT

=0

(a particular integral is, among others, the case of K = const, T = h-1).

B. Slip lines. No detailed analysis can be given here (cf. [80 a]), since this is a considerably more difficult problem than the foregoing case. The investigation of the existing possibilities is more essential, because the slip lines are the characteristics for the equations (2.13). Let us observe incidentally that the possibility of a generalization of the Kotter-Hencky equations considered in Sec. 3 exists only if zK,tt

+ K , c Hc [ p .t) + 2%] Hc + ZK(%) Ht

(2.63)

Ht - K,qq Hll

+ K,q[(") + 2

21+

2K

, t =0,

@)

Hll

= 0,

,rl

which is a serious restriction compared to the general case (2.41). An instrument for the analysis of the geometry of slip lines is available in the form of the geometric condition of plastic state (2.38). More detailed information on the questions* discussed in the present section can be found in [SO a].

6. Biharmonic States of EquilibriGmt A tensor field aii satisfying, in addition to the equilibrium conditions and the yield condition, the equation (2.64)

P2(Uxx

+

Uyy)

=

pv2n= 0,

will be called a bihannonic state of plastic equilibrium.

* Such considerations arise in connection with statical problems of originally homogeneous bodies exhibiting strain-hardening. Natural reference frames for the nonhomogeneity appearing in the course of the process of deformation of such bodies are the nets 5.17 and 7.6, f This section contains chiefly the results of Ref. [80 b].

166

W. OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI

Such states are, of course, not typical for problems of plasticity and must be of exceptional character (particularly so in the homogeneous case). However, the analysis of such states is useful from many points of view. Let us recall, for instance, that one of the most elegant solutions of the classical theory - that obtained by L. A. Galin - is connected with such a state. Biharmonic states of plastic equilibrium have interesting geometric features of the trajectories of principal stresses and of slip lines. To see this, let us write the biharmonic condition in the system L : (2.66) Substituting the derivatives of the equilibrium equations (2.31), we obtain the biharmonic condition in the form of a relation resembling (2.38), except that sin 2# and cos2# as well as cos 2# and - sin 2# are interchanged. By comparing these two relations it is seen that the geometric condition of biharmonicity of plastic equilibrium is expressed by the two relations

If the system L coincides with C, #

+

+

= 0 , we

find

+ qYy- e , % ~ 0,

K,,~ 2 ~ ,e,% , 2 ~ ,qY , K ( - 4 0 , qY ~ (2.67)

+

K , , ~- K , ~ , 4~,,e,,

=

+ 4 ~ , , e , , + 4~[e,,, + (e,,)~- (e,,)zi

= 0.

For the trajectories of principal stresses and for the slip lines (and also for any other net maintaining a constant angle with the principal direction) we obtain

(2.68) Thus, for biharmonic states of equilibrium the form of the equations relating K(a,P) with H,(a,P),Hp(a,P)is the same in all nets forming a constant angle with the net of trajectories of principal stresses.

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

167

In particular we obtain for K = const the following necessary geometric condition of biharmonicity: every net making a constant angle with the net of trajectories of principal stresses (including the net of trajectories of principal stresses and that of slip lines) must satisfy the following system of equations resulting from (2.66) and (2.20): (2.69) a)

[In (HaH/)],a)= 0,

b)

(%),a

= 0,

(%),u

c) = 0. These are therefore Hencky-Prandtl nets of a particular type, the additional restriction being a). I t can be shown, [80 b], that these nets constitute logarithmic spirals (with the polar and the Cartesian net as particular cases). For isothermal nets in isothermal parametrization the biharmonicity conditions have the form hT,aB - h,aBT = 0, (2.70)

h.( T,aa - T,pp) - (h.,aa - h,pp)T = 0.

It is easy to find a K satisfying these relations for each particular net. Thus, for instance, if a bipolar system is to be a net of trajectories of principal stresses, 12 = cosh u cos v , and if the state is to be biharmonic, this is possible only for the following non-homogeneity types :

+

+ cos v)(C, sin u + C2cos u + C, sinh v + C, cosh v ) , K = (cosh u + cos v)(C, sinh u + C, cosh u + C, sin v + C, cos v ) . K

(2.71)

= (cosh u

7 . Analytical Solutions in Particular Cases

The integration of the equilibrium equations may in the case of special distributions of plastic non-homogeneity be performed in a closed form. Several relatively simple cases will be given here. A, Two simple solutions in Cartesian coordinates have been obtained by A. I. Kuznetzov a) Let the semi-plane, bounded by the line y = 0, be acted upon by the forces, [55 a, b], uyy= - p = const,

(2.72)

uxy= 1 = const.

Assuming that K is a function of y only, we obtain an elementary selution of the Cauchy problem for the system (2.2) from (2.3) in the form (2.73)

uyy

=-p,

uxy

=t,

uxx =-

It should be assumed in addition that It1 including the edge.

p

f V P ( y )- t 2 .

< K ( y ) in the entire semi-plane

168

W. OLSZAK, J. RYCHLEWSKI AND W. URBANOWSKI

The slip lines can easily be found:

For t = 0, these are straight lines y = & x -t C; this possibility is foreseen in Eq. (2.41). If K increases with y , the lines (2.74) approach their rectilinear asymptotes. Observe that the occurrence of the plastic state in the entire region is an assumption (independent of the relation between the quantities P,t,K), which requires the realization of the stress pattern following from (2.73) in sections x = const (cf. for instance the case of t = p = 0).

FIG. 11. Compression of a non-homogeneous plastic layer between two rough rigid plates.

b) Let us consider, [55 c], the known problem of compression of a perfectly plastic layer, Fig. 11 (I >> h) between two rough rigid plates. It is assumed again that K = K ( y ) . The boundary conditions are

y = - It,

uxy= - n K ( - h),

0

< Iz < 1.

Assuming that uxyis independent of x , we obtain for the system (2.2), (2.3) the solution uxy= ay

4-b,

(2.76) oXx= - ax - c

uyy= - ax

- c,

+ 2 V K 2 ( y )- (ay + b ) 2 .

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

169

From (2.75) we have

and from the equilibrium condition of an element of width x , li'

c = - V K 2 ( y )- ( a y + b ) 2 d y . h

(2.78)

-h

The stress distribution is a generalization of the known solution found by Prandtl. Solving the system (2.3), (2.9) for the velocities with the boundary condition v y = 'f U

(2.79)

for

y

= ih,

we obtain vy =

Y u--, h

(2.80)

V

The constant C will be obtained from the incompressibility condition

(231)

- r v x d y =2U(l- x). -h

The velocity distribution (2.80) generalizes the known solution by Nkdai. Ref. [55c] contains a discussion of the possibility of preservation of rigid zones adjacent to the plates. Making use of the result obtained, the author, following A. A. Ilyushin, [42 b], studied an approximate approach to flow problems of a plastically non-homogeneous material between curved surfaces. B. In polar coordinates some results have been obtained in cases when K is a function of the radius only or of the angle o$y. a) For (2.82)

K

=K ( r )

170

W. OLSZAK, J. RYCHLEWSKI AND W. URBANOWSKI

the stress distribution around a hole has been discussed in detail in [82 a, b] (cf. Chap. 111) and generalized by A. I. Kuznetzov to the case when ur, # 0. b) Assuming that

K = K(p), the plastic equilibrium of a wedge (Fig.12) has been studied by A. Sawczuk and A. Stepieri, [lo5 d], [llO]. This is a one-dimensional problem. In the components of the stress deviator (2.83)

(2.84)

Str

= ~r - S,

spq

- S,

UP,

srq = ~ r q ,

+

s = &(urt a),

FIG.12. Wedge loaded by side pressure.

the equilibrium equations (2.28) in polar coordinates can be written in the form (since no quantity depends on Y ) (2.85)

%,,

+ 2s,,

=0,

s,p,p

+ 2st, +

S?, = 0.

Bearing in mind that s,,: :,s,

s,, = 1: - 1: 0, one obtains the yield condition

(2.86)

s,

2

+

2 Srp

=q

p).

The boundary conditions are seen from Fig. 12. Substituting sp, of (2.85) into (2.86), we obtain the non-linear ordinary partial differential equation (2.87)

Srp.,, =

f [K2(p)- s&]'/2.

The types of solution and the possibility of the appearance of discontinuities were discussed. The determination of strain rates was reduced to the integration UP the equation (2.88)

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

171

For linear non-homogeneity

m = 2 ( K , - K0)a-l

(2.89) a 2

K=m(a-p)+Ko,

-
the solution of (2.87) was found in the form of a power series: srp=-Ko(2p+ (2.90)

[

-m p2--

KO

sip = - KO 2(a - p)

4

3 p

3

m +(a KO

+

...) 4

p)2 - - ( a - p ) 3 +

* * -

(successive terms can be found from a recurrence formula). The remaining components are also expressed in the form of series. From the relevant equations it follows that the line p = a/2 is a discontinuity line. From the condition

+ + 4; = 2 ~ p q ,

(2.91)

urr

the ultimate load is found: (2.92)

p = 2 K 0 [ 2 ( i ) ' + "3. 2 KO ( E 72

-z(Ey 3 2

+...I.

Certain numerical results are shown in Fig. 13. The value of K = J b 2 will be discussed in Sec. 9.* C. A. I. Kuznetzov has generalized L. A. Galin's solution, [55 a, b], of the case of the infinite plate with a circular hole for the boundary conditions

*

In the paper [20d], B. A. Druyanov solved the case when K = K , eZnlpll

where K , and a stand for constants. I t may be noted that in both cases the nonhomogeneity is symmetrical with respect t o the axis of the wedge, which facilitates the solution.

172

W. OLSZAK, J . RYCHLEWSKI AND W . URBANOWSKI

Requiring that the state of equilibrium of the plastic part be biharmonic (the semi-inverse approach), he assumes that

(2.94)

(:T

K=K(cQ)+dK -

The admissibility of this non-homogeneity type may be verified in the light of the general considerations of Sec. 5 (cf. [80a]).

-n

FIG.13. Limit loading for wedge whose yield limit is a linear symmetric function of angle, [lo5 d].

The results obtained by means of L. A. Galin’s method are as follows. The plastic zone is bounded by an ellipse the semi-axes of which are (2.95)

c1,2 = 4 1 i P )

173

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

where (2.96) c = a e x p

__p+Q

(4K(m)

-1 +--

2

P

2K(m)

----, p=--.Q - - P 2 KA( m K )) 2K(m)

The stresses in the elastic region are determined by the relations

where (2.98)

z=x+iy

,

tmax I = 0 and tmax I homogeneous disc with circular hole, [55 a].

FIG.14. Boundary of the plastic zone and the graph

,

=0

The stresses in the plastic region are r u,,=-f~+2K(m)ln-+fK a

(2.99)

upp= - p

a,,

= 0.

+ 2K(m) l n j l +In:)

+

+

g),

for non-

174

W . OLSZAK, J . RYCHLEWSKI A N D W . URBANOWSKI

Fig. 14 shows the boundaries of the regions and diagrams of tmaX/K(m) on the x-axis and the y-axis in the following cases (2.100)

AK

AK

Roo)=-0.5,

AK

jqoo) = 1,

= 0,

for the conditions (2.101)

p

Q = 3K(m),

=0,

P

= 2.6K(m).

D. Problems of a non-homogeneous elastic-plastic strain-hardening body require a different approach and are not discussed here. Let us only mention the paper by W. Olszak, J. Murzewski and J. Golecki, [77 a], devoted to the problem of a semi-plane loaded by a concentrated force. The Epdiagram is assumed in the form of two intersecting straight lines (linear strainhardening). The three parameters determining this relationship are assumed to depend on the depth coordinate. Closed-form and approximate solutidns have been obtained for incompressible and compressible material, respectively. However, a characteristic discontinuity at the boundary between the elastic and plastic domains appears; this problem is discussed separately. 8 . Approximate Solzltions

For problems with a "weak" non-homogeneity approximate closed-form solutions may be sought on the basis of the known solution of the classical problem. This approach is founded on the intuitive expectation that a weak non-homogeneity should not considerably disturb the stress and strain-rate field. It should, however, be noted that this expectation has no mathematical foundation and recalled that for problems of perfect plasticity instability of solutions may occur, [42 a]. A. The problem of a punch pressed into a body of weak non-homogeneity has been studied by B. A. Druyanov by means of the perturbation method. He solved the problem of a smooth punch, [20 a], and a rough punch, [20 b], and a broad and narrow semi-strip [20 c]. Assuming that

K = Ko(l - &)Y/*,

(2.102)

H h =-, a

where 2H is the depth of the strip, 2a the breadth of the punch, y the depth coordinate ( y < 0), Eqs. (2.13) are obtained in the form*

w,=+ cos 29, * v,. (2.103)

+ sin 29, *

The author takes o

*

=

pl,.

+ sin 29, = ln(12h.- cos 29, =

u/2K, q

=

f3

E)

cos 29,,

In(1 - E ) (sin 29, - 2w). 2h

+ n/2.

175

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

Assuming that E = (KO- K h ) / K Uis sufficiently small, we seek the solution in the form m

m

(2.104)

w =

2

p=

W$,

0

2 pr&i, #

0

which leads to the system of equations (2.105)

+

+

.. pt,y= F Z , ( ~ , , W . . ,po,o~o), ,.

w , , ~ cos 2p0 p,,% sin 231, * pz,y= Fl,(pl,,w,. ,po~mo), 9

+ sin 2p0 p,,, - cos 29, *

where Flr,F2$are linear functions of

*

p),,w,.

f

X

FIG.15. Approximate characteristic net for E > 0 and the velocity distribution on the free boundary. _ _ - _ classical problem, ___ non-homogeneous problem h = 10) ( E = 0.5.

The systems (2.105) are hyperbolic and have common characteristics, known from the solution of the classical problem (i = 0). All systems are linear for i > 0. Starting out from the continuity condition for to and p on the characteristics, the author formulates the boundary-value problems for these equations. The equation of the characteristics is also sought in the form m

(2.106)

y

=

2

Y,(X)&i.

0

The problem of the velocities is stated in a similar way, and an analogous linearization is obtained. In Refs. [20 a] and [20 b] expressions are obtained for all the required quantities in the second approximation (order E ~ ) . Figs. 15, 16 give the

176

W. OLSZAK, J. RYCHLEWSKI AND W. URBANOWSKI

results of Ref. [20]. For the normal pressure under the punch (with E = 0.5, h = lo), the maximum correction of the first approximation in relation to the zero order approximation is S.9%, that of the second 4.9%. In a paper devoted to the problem of a punch pressed into a narrow strip on a rigid foundation, it is shown that the equations of all approximations may be reduced to the telegraphist’s equation and therefore integrated in principle by quadratures. In view of the necessity of heavy computations, the author stopped at the first approximation, expanding the Bessel functions

HA

-%

I

2

FIG. 16. Stress distribution under the punch 1 - ayo/K0, 2 - ~,t/Ko, 3 - 10t,y/Ko (E = 0.5,

h = 10)

under the integrals in power series. The computations for given values of the parameters were performed by means of the digital computer “Strela” in the numerical computation centre at Moscow University. The perturbation method could probably be applied to solve some problems. However, the work required in comparison to the simpler numerical method of characteristics may prove so large that it will no longer be justifiable by the desire to obtain a solution in analytic form.* B. A certain linearization of the problem of weak non-homogeneity has been proposed by A. I. Kuznetzov, [55 a, b]. His idea consists in superposing on the tensor field oiii(o), constituting the solution of the classical problem, a small correction in form of a tensor field aii(l), having the character of a

* In the paper [ZOe], the same author gave the solution for the punch indentation problem with a n exponential type of non-homogeneity (numerical solution by the method of characteristics).

PLASTICITY ‘UNDER NON-HOMOGENEOUS CONDITIONS

177

self-equilibrating system of stresses and satisfying a certain linearized “yield condition”. Within this framework the theory of equilibrium of a,?) is developed. However, the proposed method of linearization is open to objections (cf. [lo3 a]). C. For non-homogeneous elastic-plastic bodies satisfying the HenckyIlyushin equations, a generalization of the algorithm of the Ilyushin method of elastic solutions has been proposed (V. S. Lensky [57 b], cf. Chap. 111). This method should be particularly useful for weak non-homogeneity types. 9. Inverse and Semi-Inverse Methods

The survey by P. F. NemCnyi [67] was devoted to the results and prospects of the development of inverse and semi-inverse methods in continuum mechanics. For plastically non-homogeneous bodies these problems will be treated in detail in [lo3 b]. We shall confine ourselves here to a few remarks and examples. First let us formulate a remark which seems to be almost trivial. Consider a body with boundary conditions expressed in stresses. Any statically admissible field aii(0)(i.e. a field satisfying the equations of equilibrium and the boundary conditions) may serve as a statical solution for a certain type of plastic non-homogeneity. I t suffices to find the corresponding K from (2.3). Such a field aii(0)may, for instance, be taken over unchanged from any branch of continuum mechanics. Observe that a similar approach to problems of plane stress and plane strain of elastically non-homogeneous bodies is much less trivial, [SO c] ; Rather than to a definite non-homogeneity type, every field aii(o) corresponds to a vast class of these types. For a plastically non-homogeneous body, the solution of classical theory of elasticity may, for instance, be taken for ajj(0).From the viewpoint of the theory of plasticity, in this way only a very narrow class of solutions may be obtained, namely biharmonic solutions. There are still various approaches possible ; for instance Airy’s stress function Q may be used. Take SZ as a function satisfying at the edge ([69] for instance) the equation

J

(2.107) Q = - x pz(s)ds 0

S

S

f

+Y

S

0

py(s)ds

+

J

( p y ( s ) ~( ~P) z ( s ) ~ ( s ) ) d s .

0

Then K is obtained from Eq. (2.11). In some cases, it is more convenient to use directly the stresses (cf. the example A below). The above remarks are obvious and their importance is limited. However, in view of a great number of possible solutions, a number of valuable results

178

W.

OLSZAK,

J . RYCHLEWSKI AND W. URBANOWSKI

can be found in addition to the very special and less interesting ones (for instance, for K = 0, the incompressible liquid, in regions of hydrostatic states). The complete statical problem is formulated by giving a) external loads, b) geometric and mechanical properties of the body, c) the stress field. All these quantities may be prescribed with various degrees of definiteness. Thus, for instance, the geometric form may be left indeterminate to the extent of a continuous transformation, the load may be taken from a certain class, the stress field may be determined by prescribing some of its components or, for instance, the trajectories of principal stresses etc. This variety of possibilities is typical for the general structure of the inverse and semiinverse methods. We shall not endeavour to give a general definition of the inverse or semiinverse methods (cf. the remarks on this subject in [67]). For the problems, in which we are interested, the method must include at least the following two features: a) K is not given or is given only with a certain indeterminacy; b) certain properties of the field are assumed a priori. If the form of the trajectories of the principal stresses is assumed beforehand, then 0 = 0 ( x , y ) is known in Eqs. (2.13). The system of equations for u and K thus obtained is hyperbolic, and its characteristics coincide with the trajectories of principal stresses as can easily be seen.* I t is convenient to take the equations directly in this form, that is (2.32). The boundary conditions cannot be arbitrary here. A. An illustration of the efficacy of the simple approach discussed above is furnished by the results for a wedge, [105d], Fig. 12. Taking the elastic solution, we obtain the uninteresting result (2.108) K ( q ) = I d a= K 0 [ 2ctgaa(l - cos 2y)

+ 1 - 2 ctgor sin 21711.

Taking now, as a statically admissible system of stress fields in the inverse method, the expressions

(2.109)

a

s=-~(;z”+2z)d~+s(0), 0

with the function z satisfying the boundary conditions of Fig. 12, we have (2.110)

* This

K2(q) = i ( t ” ) 2

$22.

fact has been pointed out by W. Szczepihski.

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

179

For instance, for

I

- 2K,v,

(2.111)

we have (2.112)

FIG. 17. Eccentric cylinder subject to interior pressure and a force in the slit.

where

B. A more difficult problem is treated in a paper by W. Olszak and S. Zahorski, [83 c]. The object of the paper is the plane quasi-static problem of static flow of an eccentric ring (Fig. 17) as determined by two parameters, the ratio of the radii p ( t ) and the ratio ~ ( tof) the eccentricity to the outer radius : (2.114)

p=-

C

a

(O
e q=T

(O
It is assumed that the ring is made of a perfectly plastic material, showing non-homogeneity of certain definite types, and that it is under conditions

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W . OLSZAK, J. RYCHLEWSKI A N D W. URBANOWSKI

of plane strain. The “quadratic” yield condition is assumed (Huber-Mises) adapted to non-homogeneous bodies. The ring is always kept in equilibrium under the internal pressure # ( t ) , the external pressure q(t),and the force S(t) being the resultant of the normal forces in the slit. The one-parameter problem ( p = const or q = const) is a particular case of the two-parameter case under consideration. The assumptions are as follows (semi-inverse method) : a) the trajectories form a bipolar net; the two cylindrical surfaces (= circles) are coordinate lines; b) the state of plastic equilibrium is biharmonic. The problem is considered in a system of moving bipolar coordinates by applying the inversion transformation. The analytic function (2.115)

maps the eccentric ring in the original system of polar coordinates O(R,@) onto the concentric ring in the inverted system I(r,p),where the minus sign corresponds to the “internal inversion” and the plus sign to the “external inversion”. The corresponding relations between the stress field in the 0-system and the I-system [if the existence of the biharmonic stress function w ( Y , Q ) ) is assumed] take the form

(2.116) ZRT =

1 C2

+

- - tr;(r2 h2

2hr COST),

constituting a generalization of the relations obtained by J. H. Michell in 1902, [60]. The yield condition is assumed in the form (2.117)

(OR

+

- a d 2 4t&

= 4[K(P,t)I2.

where (2.118)

K(P,t) = K(t)R(P).

Five possible types of plastic non-homogeneity are considered, I, 11, 111, IV, and V, of which the last is a linear combination of the former four

K(P,t)= K(1) [-f x2(4R,(P) A W ) R , ( P )fp2(1)Rs(P)f Y2(1)R,(P)l (2.119)

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PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

and takes in the inverted system I a form depending on the radial coordinates rli and r, (for the “internal” inversion) and rIe and re (for the “external” inversion) :

By a judicious choice of the functional coefficients ~ ~ ( t ) , A ~ ( ( t ) and ,p~(t), v2(t)we can approach the real properties of the plastic material in a satisfactory manner. Starting from the stress function

which is bihannonic in ri and re, we are able to satisfy the yield condition (2.117) for a general plastic non-homogeneity type described by (2.119). The stress components will be obtained, according to (2.116), in the form

+ D,(2hi2 In

Y,

+

+

- ri2 + hi2) E0(2S2In re - re2 he2),

+ D,(2hi2 In + ri2 + 3hi2 - 4r;hi cos yi) + + EO(2he2In re + re2+ 3S2 + 4rchecos vC), Y;

ZRT

=0

(2 * 122) -

-

where the functions A,,~,,8,,. . .,Go, and also the pressure difference n(t) = p ( t ) - q(t), under which the ring is in equilibrium, are determined from the boundary conditions. Also the force S(t) in the slit is determined together with its moment about the inversion pole.

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W . OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI

In the case of the one-parameter problem ( p = 0.5), diagrams of all the mechanical quantities appearing in the paper have been made for a set of values of the parameter ~ ( t and ) of the ratio of the radii R / d . Thus, for instance, the diagram of K ( P ) = k(t)R,(P) is represented in Fig. 18. This case corresponds to a “practically” homogeneous material with timevariable yield point. The yield condition, expressed in terms of principal stresses (net of curvilinear coordinates), becomes linear. Using this, the authors were able

FIG.18. Klk plotted against the ratio Rld for different values of the parameter 77.

to superpose the several types of non-homogeneity and the corresponding solutions obtained, so as to adapt the final result to the actual conditions (e.g. homogeneity, or non-homogeneity of a specified character). In addition to the problem of continued plastic flow, it proved possible to solve the problem of a non-homogeneous eccentric cylinder, part of which remains elastic, the rest being already plastified (cf. W. Olszak, [73 u], and W. Olszak, 2. Mrbz [76]). Some singular properties of the corresponding closed-form solution were demonstrated.

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

183

111. PARTICULAR SOLUTIONS 1. General Method

In the Hencky-Ilyushin theory of small elastic-plastic strain the nonhomogeneity of elastic and plastic properties is manifested by the fact that the form of the a;,&,-curvedepends on the point considered (see Fig. 19). Various types of such non-homogeneities have been discussed on the basis of the character of the a;,q-curve by W. Olszak [73 p]. In the case when

FIG. 19. Curve ci, ei for two different points.

the non-homogeneity is caused by neutron flux, an effective computation method has been proposed by V. S. Lensky, [57 b] as a generalization of the method of elastic solutions derived by Ilyushin, [42 a]. The applicability of this method to cases of non-homogeneity due to other causes is obvious. In the case considered, the basic laws of the theory of small elasticplastic strain have the form (34

amn

- as,,

2a;

= - (Em,

3 ~ i

- Ed,*),

where a;,&;are the stress and strain intensities, respectively, p is the neutron flux, Go a certain value of the modulus G in the body e.g. the maximum

184

W. OLSZAK, J. RYCHLEWSKI AND W. URBANOWSKI

value, and K the bulk modulus. The meaning of the function R is similar to that in the case of homogeneity. If the curve OA'B' in Fig. 19 corresponds to the point where G is a maximum, then

AA' a=-.

(3.4)

CA'

ConsideringR (which makes the problem non-linear) as a small parameter, we shall generalize the method of elastic solutions in the following way. Substituting in the equilibrium equations omnof (3.1) and bearing in mind that

(3.5)

Emn

= +(um,n

+

Gn.m),

we obtain equations analogous to the Lam6 equations of the theory of elasticity, except that additional terms containing the functions R,v and their derivatives are involved. Transferring the latter to the right-hand side, we have (34

(A,

+ p0)grad div u + p 0 P u = @ + IL,

where &,po denote the maximum values of the Lam6 moduli in the body, @ the vector of the body force and R, a vector depending on !2 and and their derivatives. The boundary conditions are written in the form (3.7) (3* 8)

S,

= T,

+ T,

u=v

on

S,,

on

S,,

where T, denotes the vector of the surface load prescribed on the surface-part

S,,V the vector of displacement given on the surface-part S,, and T, a vector depending on 5 2 , ~and their derivatives. S, denotes the vector of stress computed from u on the basis of Hooke's law (fictitious elastic stresses) acting on a surface element of which the normal is v . For the zeroth iteration we take the solution of Eqs. (3.6) for R 0,rp 3 0. Solving this problem of classical theory of elasticity, we obtain u0,oin,E;,, and calculate ~,O(x,y,z,t) In the regions ci0 < E ~ ,where E, = ~,(x,y,z,t) is the limit of elastic strain, the function R is given (the distribution of the modulus G being known) and independent of E $ . In the regions E,O > E ~ , the function Ro is computed from (3.2). Having uo,Ro, we compute R,,T, and, substituting in (3.6), we face the next problem of the theory of elasticity. Solving this we obtain u',&,,~~,, and so on. The above algorithm is very general, but it is seen to lead to very cumbersome computations. Its convergence in the classical case, for plates and shells, has been proved by V. M. Panferov, [ 8 Q ] .

186

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

In conclusion of this section let us mention the paper by D. D. Ivlev, [45 b], where a plastically non-homogeneous body is considered under the

conditions of so-called total plasticity, corresponding to the state of stress on the edges of the Tresca prism. 2. Axially Symmetric Problems In what follows we give a survey of the results obtained for axially symmetric problems.

A. Isotropic, thick-walled cylinder. For the non-homogeneous case this problem has been solved by W. Olszak and W. Urbanowski, [82 a] and [82 b]. These authors assume that the modulus of elasticity in shear, G,

3

Q

b

r

Q

b

r

--L

Q

r

FIG.20. Various cases of propagation of the plastic region in a non-homogeneous thickwalled cylinder.

and the yield limit for shear, K , are functions of the radius only, G = G(r), K = K ( r ) . Further assumptions are: incompressibility of the material in the elastic and plastic ranges, plane strain with respect to the axis of the cylinder, and an ideally plastic material. I t was found that the necessary condition for the plastic region to start from the interior surface of the cylinder and to expand towards its exterior surface [case (l), Fig. 201 is that the function

186

W. OLSZAK, J. RYCHLEWSKI AND W. URBANOWSKI

(3.9)

be monotonically decreasing in the interval [a, b ] , where a denotes the interior radius, and b the exterior radius. If f(r) = const, the whole of the cross section becomes plastic simultaneously [case (a)]. For f ( r )monotonically increasing, the plastic region would start from the external surface of the cylinder [case (2)]. This region would expand with increasing load p ( t ) towards the internal surface of the cylinder. In Fig. 20, J,(r) denotes the second stress invariant.

If f(r) is not monotonic in the interval [u,b], the radial distance, a t which the first plastic deformation will occur, appears somewhere inside the wall [case (3)]. Two critical values of the internal pressure p are found, the first, p,, being characteristic of the appearance of the first plastic deformations in the cylinder, the second, p,, being related to the phenomenon of full plastification of the material in the entire system considered. The authors considered in greater detail the first case (plastic zone starts from the interior surface). They proved that the first critical pressure p , (corresponding to the appearance of the first plastic deformations) is equal to (3.10)

p

K(4

-2 G(a) a,[g(u)

- g(b)],

whereas the second critical pressure (3.11)

p,

p,

= h(b) - h(a),

In the particular case when f ( r ) =

g(r) = -

{Far.

is given by h(Y) =2

5

dr.

r = const, we have

If p , < p < p,, the radius n(a < n < b) of the interface separating the two regions (elastic and plastic) may be calculated as the root of the transcendental equation (3.13)

The state of stress in the elastic (outer) region is determined by the expressions

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

187

(3.14)

where (3.15)

The state of stress in the plastic (inner) region is determined by the expressions a, = h(r) - k(a) - p , (3.16)

+ 2K(r) - 9, a, = h(r) - h(a) + K ( r ) - p . at = h(r) - h(a)

A similar problem for non-homogeneous strain-hardening characteristics has been studied by Lee Ming-hua and Pei Ming-li, [ 5 6 ] . The problem of a quasi-static motion of a thick-walled circular cylinder made of a rigidplastic material and subjected to internal pressure p has been treated by E. T. Onat, [ 8 5 ] . He obtained the relation (3.17)

where 9’ stands for the rate of internal pressure, U for the boundary velocity, Yo and Yo‘ for the yield stress and the slope of the stress-strain curve a t the yield point, respectively. It follows that the quasi-static motion can only be maintained by decreasing the pressure, if the non-dimensional rate of hardening Yo’/Yois smaller than Experimental investigations of thick-walled cylinders under internal pressure have been carried out by M. C. Steel and J. Young [118] and M. C. Steel and L. C. Eichberger [119]. The results show the appearance of an irregular strain distribution over the cross-section, although its structure was initially carefully prepared to be uniform. This phenomenon may be explained as the effect of small non-homogeneities of another type than the axially symmetrical distribution.

v%

B. Orthotropic thick-walled cylinder. This problem has been solved by W. Olszak and W. Urbanowski, [82e] and [82f]. The authors assume

188

W. OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI

that the yield limits in the radial and the axial directions are the same and constant, that is Qr = Q, = Qo = const. Similarly it is assumed that the yield limit in shear Qrt in the plane normal to the axis of the cylinder is constant, whereas the variable plastic properties are assumed to depend on the variable yield limit in the circumferential direction Qt = Qt(r). Such assumptions provide a fair approximation to the properties of thick-walled reinforced concrete pipes with circumferential reinforcement. The state of stress in the plastic region under the condition of plane strain (E, = 0) is determined by the expressions (3.18)

where

The constant C and the signs in the above expressions should be determined from the conditions a t the boundary of the region. The practical application of the result for thick-walled non-homogeneous tubes is the object of the papers by W. Olszak [73 el, [73 f], [73 j], and [73 q] where reinforced concrete tubes with variable amounts of reinforcement, such as are used in coal mines for pressures of 20 to 50 atm, are considered. C . Transversally isotropic non-homogeneous cylinder. B. R. Seth, [112], takes the yield condition in the form (3.19)

Tll - T33

+ K l k )( TI1 + T,2 + T33)

= K,(r)

where T,, >, T,, 2 T33 are the principal stresses referred to the strained frame of reference. Assuming a linear stress-strain law for large deformations and rotations he found the yield stress for tension and compression.

D. The Boussinesq problem for the semi-infinite space. The problem of a non-homogeneous linear foundation subsoil was treated by I. BabuSka,

PI. The problem of the elastic semi-space loaded by a concentrated force has been treated by K. Hruban, [38 a] and [38 b]. The physical relatibn between the stress intensity oi and the strain intensity E~ is assumed in the parabolic form E~ = (ui/K)”, K and n denoting constants. Also Poisson’s ratio is assumed to be constant. The results for various n are collected in a table [38 a]. Under the provision of an “active” straining process these solutions hold also for a non-homogeneous elastic-plastic body (on the basis of Hencky’s “deformation” theory).

PLASTICITY U N D E R NON-HOMOGENEOUS CONDITIONS

189

3. Spherically Symmetric Problems Problems of spherical symmetry are treated in the literature in a manner analogous to axially symmetric problems ; therefore results will be indicated only briefly. W. Olszak and W. Urbanowski discuss in Refs. [82 c] and [82 d] the problem of a thick-walled spherical shell with interior radius a and exterior radius b made of an incompressible material ( v = 1/2), radially non, to the action of internal and homogeneous [G = G(r),Q = e ( ~ ) ]subjected external pressures p and q, the difference I7 = p - q being a monotonically increasing function of time t. Here Q = Q ( r ) denotes the yield limit in tension. The results obtained do not differ qualitatively from those obtained by the same authors for the thick-walled cylinder. The functions describing the states of stress and strain are of course different. The case of a thick-walled transversally isotropic sphere under internal pressure was studied by B. R. Seth, [112], the assumptions being those of Section 2. Some problems of thermoplastic strain of the spherical shell have been discussed by M. Rogoziliski, [1021.

4. Torsion of Prismatic Bars

For this problem A. I. Kuznetzov considered in the first part of his paper [55 b] the state of full plasticity of the cross section, assuming that the yield limit is an arbitrary point function and formulating the characteristics for the problem. It was found that in this case the characteristics are also slip lines, but, in general, are not straight lines. The function obtained by the author, a generalization of the “sand hill surface” to the case of plastically non-homogeneous bars, may be interpreted as a characteristic surface of the wave equation with variable velocity, inversely proportional to the yield limit of the material. The slip lines are the extremum lines* on this surface, its contour lines are the trajectories of the shear stresses, and its ridges are lines (or points) of discontinuity of the stress field. I t was also found that in the particular case when the characteristics are straight lines, the yield limit varies along the normal to the contour.

* The slip line passing through the points M , and M , is a n extremum line of the integral

190

W. OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI

The author succeeded also in effectively solving the problem in the particular case of partial plasticity of a circular cross-section, if the yield limit is a function of the radius only. 6. Rotating Circular Disc

The problem of determining the angular velocity o,for which the disc becomes entirely plastic, was studied by M. Zyczkowski, [129 a], [129 b]. The assumptions are as follows: the material of the disc is perfectly plastic; the yield limit Q is a function of the radius only, Q = Q(r),similarly the specific gravity y = y ( r ) . In addition, it is assumed that the disc is fully connected (without hole) Although the author has indicated how the problem may be solved in the case of variable thickness, the results are only given, in principle, for a disc of constant thickness. Assuming that the values of the functions Q(r) and y ( r ) do not deviate considerably from their mean values and introducing the yield condition of maximum shear stress, the author obtained 1

(3.20)

0

where R denotes the exterior radius, p = r / R and g is the acceleration due to gravity. For Q(r) and y ( r ) varying in an arbitrary manner, the author indicated a method for approximate solution of the problem based on the HuberMises yield condition.

IV. ELASTIC-PLASTIC NON-HOMOGENEOUS PLATES The basic equation for elastic-plastic bending of non-homogeneous plates of arbitrary form has been established in the papers by W. Olszak and J. Murzewski [76 a]-[76 c] for arbitrary boundary conditions and for various types of the yield condition. The analysis is concerned with those particular cases of bending, for which the principal directions of stress and strain coincide with those of orthotropy. The following equations express the relations between the bending moments m1,m2 and the curvatures a1,a2 of an elastic-plastic orthotropic plate :

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

191

The elastic rigidities 8,,8, and Poisson’s ratios v1,v2, appearing in these equations, are functions of the coordinates of the point P of the middle surface of the plate. Also the plastic moduli, appearing in the coefficients x,w1,w2 may vary with the position of the point P . The latter are functional coefficients defined by

x = hcl/h,

h

=h(P)

for a plate of variable thickness

are “left-hand” limit values of principal stresses a t the distance hcz/hfrom the middle plane of the plate. The quantities x,wl,w2 may be expressed as functions of the principal curvatures a1,a2. The corresponding values of the coefficients w1,w2 are given, and the existence of the plastic potential in the “quadratic” form is assumed (here the orthotropy of the material is taken into account). Making use of the results of [76 a]-[76 c] the same authors applied in [76 a], [76 d] the general equations of elastic-plastic bending of plates to the problem of axially symmetric bending of such plates. These references contain some numerical examples : (a) elastic-plastic bending of an orthotropic ring with variable thickness, clamped along the exterior periphery and acted on uniformly by a linear load on the interior edge; this problem may be reduced to the corresponding problem of a non-homogeneous ring. (b) elastic-plastic bending of a simply-supported densely meshed circular grid-work ; this consists of radial and circumferential beams and is uniformly loaded on the central platform; such a system is adequately represented by a continuous model of a non-homogeneous elastic-plastic plate with well-determined mechanical properties. (c) a circular reinforced concrete plate with variable radial and circumferential percentage of steel reinforcement. a;l,aaCl

V. LIMITANALYSIS AND LIMITDESIGN 1. One-Dimensional Structural Elements

A. Non-homogeneity function. For the purpose of limit analysis it is convenient to distinguish between the “transverse” non-homogeneity and the axial (longitudinal) non-homogeneity of a structural element. The transverse plastic non-homogeneity corresponds to the variable yield-point distribution across the thickness of a bar. Thus this type of non-homogeneity influences both the dimensions and shape of the yield locus I; in the stress-

192

W. OLSZAK, J. RYCHLEWSKI AND W. URBANOWSKI

resultant space. However, the most important thing in this case is the influence on the shape of the yield locus. If the yield point uo* is a continuous function of the bar thickness [ao* = ao(l f ) , say, with f ( z ) # 01, then the limit state of the rectangular cross section is defined in pure bending by the following relations

+

F

(54

= M = Mo*,

Mo* = uob

j+ [l

f(z)]zd~,

-H

where h stands for the width of the bar, and 2H for its depth. This type of transverse non-homogeneity was applied to simple cases of beams by

a)

I

n FIG.21. Bending of a rectangular cross section with a piece-wise linear non-homogeneity distribution.

A. S. Grigoryev, [31], assuming that f ( z ) = f ( - z ) . Similarly in the case of simultaneous bending and tension or compression, the parametric equation of the yield locus is simply H

I n

The elimination of the parameter zo from these equations yields the corresponding interaction curve, Thus the plastic interaction curve or, in general, the plastic interaction surface F is a function of the transverse non-homogeneity . If the non-homogeneity function is a step function, the corresponding yield locus can be obtained in a similar way. P. G. Hodge, [35 b], applied this approach to derive approximate interaction curves for a homogeneous material. For a rectangular cross section and a piece-wise linear non-homogeneity distribution as shown in Fig. 21 a the interaction curve given in Fig. 21 b

PLASTICITY U N D E R NON-HOMOGENEOUS C O N D I T I O N S

193

has been obtained in Ref. [35b]. The material is supposed to obey the Coulomb-Tresca yield condition. In Fig. 21 n and m stand for the properly defined dimensionless axial force n and bending moment m. The concept of step-wise transverse non-homogeneity is therefore useful in the linearization of limit analysis problems. The influence of non-homogeneity on the shape of the yield locus is evident. The behaviour of structures made of imperfectly plastic material was studied by J. Heyman, [33 d].

‘t

FIG. 22. Interaction surface for simultaneous bending, compression and shear of a step-wise transversally non-homogeneous element.

A composite structural element made of materials characterized by distinct yield properties (e.g. reinforced concrete) can also be considered as step-wise transversally non-homogeneous. For such a case the yield locus in simultaneous bending, compression, and shear has been derived by A. Sawczuk and M. Janas, (1061. The octant of the interaction surface is shown in Fig. 22. The details can be found in the reference. The difference between the interaction curve in the m,n-plane for a homogeneous and a non-homogeneous material becomes apparent simply by comparison of Figs. 21 and 22. Thus the non-homogeneity influences

194

W. OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI

the solution of limit-analysis problems because it influences the shape of the interaction surface F . Thc case, when only the size but not the shape of the yield locus varies with the longitudinal coordinate of a bar, corresponds to the axial (or longitudinal) non-homogeneity. To this group belong, for example, the cases of plastic beams of variable cross sections. Solutions of practical

FIG.23. Load carrying capacity of circular non-homogeneous arch with concentrated load applied a t the arch center.

problems concerning the “variable rigidity” can be found in standard textbooks on the limit analysis of beams and frames (cf. [3] and 135 c]), since it appears that the methods of analysis are the same as in the case of constant size of the yield locus. B. Arches. The load carrying capacity of circular arches of transverse non-homogeneous cross sections was studied by A. Sawczuk and M. Janas, [106]. For a simple concentrated load applied at the arch center results have been compared with the experimentally obtained values. The comparison is shown in Fig. 23.

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

195

2. Plates

A number of papers are concerned with the influence of non-homogeneity on the phenomena accompanying the exhaustion of load carrying capacity of different types of plates and shells. These are based on the general theory of plastically non-homogeneous bodies (cf. [73 k]-[73 13). In most cases the extremum theorems are used; in the kinematical approach, an upper bound of the load intensity, which cannot be lower than the actual intensity of the ultimate load, is determined or a lower bound of this intensity is determined by considering the statically admissible fields of internal forces. Statical methods, often giving valuable insight, but generally involving more complicated procedures, were applied to the analysis of systems exhibiting plastic non-homogeneity coupled with orthotropy in the papers by J. Murzewski, [65], and A. Sawczuk, [lo5 a] and [lo5 b]. The problems connected with the determination of the field of internal forces for non-homogeneous plates were treated on the basis of the HuberMises and Coulomb-Tresca yield conditions. As an example let us mention Ref. [lo5 a] by A. Sawczuk, who studies the problem of choosing the type of non-homogeneity so that the circular plate, subjected to axially symmetric load, passes into the fully plastic state a t once over the entire region. If the value of the limit moment is denoted by N ( r ) and the shear force by Tv(r),then the necessary condition may be expressed by means of the two inequalities

(5.3)

N>rT,,

dN dr

-<

T,.

The distribution of the limit moments for a circular plate of radius R without a hole, simply supported a t the outer edge and subjected to the load p = const whose the action is restricted to the annulus a r R, is

< <

N 0 -- PR2 -(1 - a2 + 2a21n a), 4 , (5.4)

O
where p = r / R and a = a/R. The distribution N ( r ) being known, we can choose in a suitable way the thickness of the plate

196

W. OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI

(5.5)

or the corresponding yield limit, or also the distribution of the steel reinforcement in the plate, because then M , = M , = N ( r ) . For some types of load distribution on circular isotropic plates whose non-homogeneity is determined from the postulate of equal circumferential and radial moments,

-I+

P

I

(5.6)

W )=

0

O
If(f)Pdf.

O
0

examples of non-homogeneity functions are shown in Fig. 24.

FIG. 24. Non-homogeneity functions for plates for different load distributions.

Ref. [13] by H. Craemer may, to some extent, also be regarded as related to the problem of plastically non-homogeneous plates. For a rectangular plate, the author assumes the absence of twisting moments, which is possible for a suitable choice of the non-homogeneity function. In this way he obtains the following equation relating the moments to the load +:

(g*

(5.7)

where m = m,

=

L- my.

$)m

=

P,

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

197

A number of other particular cases have been considered as, for instance, that of plates with linearly variable mechanical properties (A. Sawczuk, [lo5 a ] ) ; here, the influence of this type of non-homogeneity on the kinematics (system of facture lines) was examined. D. Niepostyn, [68], studied the problem of the load-carrying capacity of non-homogeneous plates of various types, based on the theory of fracture lines (kinematical approach) as proposed by K. W. Johansen, [47]. Problems of optimum design of orthotropic plates lead to prescribed non-homogeneity functions. This question was analyzed, among others, by 2. Mrbz, [64 a ] ; the optimum distribution of reinforcement has been found for circular reinforced-concrete plates. The limit analysis of ribbed-plate structures was taken up. These may be treated as structures with jump-like variability of non-homogeneity (cf. A. Sawczuk and M. Kwiecifiski, [l08]); solutions were obtained by means of both the kinematical and the statical approach. Some other problems of the same kind have been treated for various support conditions (cf A. Sawczuk, [lo5 a]) as, for instance, for simply supported rectangular plates with one additional support within the plate ; two-span plates; transversely ribbed plate strips; etc. In Ref. [lo5 b], A. Sawczuk is concerned with the problem of limit analysis and limit design of flat slabs, taking into consideration the conditions of both the idealized “point” support and the actual support (the reaction being distributed over a finite area). The existence of “voutes” adjacent to the supporting columns is also taken into account. For various geometrical parameters characterizing prismatic and conical voutes, formulae together with corresponding diagrams and tables are given.

3. Shells Some problems of limit states of shells have been solved as a result of further investigations. Thus, the paper by W. Olszak and A. Sawczuk, [81 a], is devoted to plastically non-homogeneous shells. The authors considered an axially symmetric membrane shell passing into the plastic state over its entire volume. The Coulomb-Tresca and Huber-Mises yield conditions were considered. Assuming constant ratio of interior forces directed along the parallel and the meridian ( N a and Np), that is N a / N , = k = const, they determined the variability of the yield limit necessary for those assumptions :

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W. OLSZAK, J. RYCHLEWSKI AND W. URBANOWSKI

where y1,yZ are the radii of the principal curvatures, and 121 is the load component normal to the middle surface of the shell. The component Z ( v ) is related to the component Y ( v )tangent to the meridian, by means of the equation (5.9)

2’

+ Z(l- K) cot + (1 + k)Y = 0.

Solutions are obtained for some typical loading conditions, Fig. 25. In addition, the field of displacement increments is determined under the assumption of uniform elongation increments. This assumption is discussed in detail in paper [41 a].

FIG.25. Variable thickness of a membrane shell passing to the plastic state over its entire volume.

Load carrying capacities of cylindrical tanks with axial non-homogeneity were studied by A. Sawczuk and W. Olszak, [log]. It was found that for linearly varying load intensity the total collapse of a cylindrical wall of a tank occurs only for a very limited range of dimensions. In general partial collapse occurs. However, by making a tank wall non-homogeneous in the direction of the generators total collapse can be assured. Thus an efficient way of design is found. If I Z ~stands for the properly defined circumferential yield force a t the tank bottom and n, for the corresponding value a t the tank top and if n varies linearly in-between, then total collapse is assuredif wl = 0. The corresponding stress and displacement rate field are shown in Fig. 26.

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199

Thus the influence of the longitudinal non-homogeneity on the mode of collapse of a structure is found to be very important. The collapse-load intensities for a non-homogeneous tank as shown in Fig. 26 is simply (5.10)

where mo stand for the dimensionless moment at the tank bottom and c2 = 2L2/HR refers to the tank dimensions. Because of the influence of non-homogeneity on the mode of collapse, the kinematical method of solution of limit analysis problems has to be applied very carefully.

FIG.26. Stresses and displacement rates in a cylinder tank with axial non-homogeneity.

Some other questions concerning non-homogeneous tanks are discussed in Refs. [lo91 and [lo?]. For stepwise transversally non-homogeneous shells, the yield locus, similar to that of limited interaction (cf. P. G. Hodge, [35 b]), has been derived by %. Mr6z, [64 b]. Here a discussion of types of failure depending upon the actual plastic regime is presented. 4. Minimum Weight Design

Problems of minimum-weight design are to some extent related to those of plastic non-homogeneity. Design with the requirement of lowest material

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consumption may in some cases lead to plastically non-homogeneous structures. This is, for instance, the case if we design structures of variable depth with the purpose of keeping the material consumption as low as possible. Even with plastically homogeneous material this may lead to point-variable integral quantities (taken over the cross section) which are characteristic of the plastic behaviour of the different cross sections. Some papers are devoted t o problems of economic design of one-dimensional types of structures such as beams, frameworks, and frames. Here belong the papers by M. R. Horne, [37], J. Heyman, [33], J. Foulkes, [22], and W. Prager, [94d]. As a rule they use the relation between the unit weight of a steel beam and its section modulus. The object of the papers by W. Prager, [94 d], and H. G. Hopkins and W. Prager, [36], is the minimum weight design of discs and plates the material of which is subjected to the Coulomb-Tresca yield condition and the associated flow-rule. Use is made of the following postulate: The structure designed for minimum weight is capable of failing in a number of different ways, and for any mode of failure the load does work that must be equal to the rate at which mechanical energy is dissipated in plastic flow. The problem of minimum weight of shells was treated by E. T. Onat and W. Prager, [86]. The problem of a cylindrical shell was analyzed by W. Freiberger, [23] on the basis of the condition that the shell (which satisfies the minimum-weight postulate) should be designed so that the number of modes of collapse is infinitely great. As P. G. Hodge has pointed out, [35 a], this postulate does not necessarily lead to a minimum-weight shell. Anisotropic minimum-weight shells were investigated by M. Sh. Mikeladze, The problem of minimum weight of a circular plate satisfying the HuberMises yield condition was studied by W. Freiberger and B. Tekinalp, [24], who formulated the relevant variational problem. The papers on minimum-weight design based on the general theorems of limit analysis are in another group. D. C. Drucker and R. T. Shield, [19], and D. C. Drucker, [18], discussed the criteria for determining the upper and the lower bounds of the weight of a structure carrying a definite load and subjected to definite boundary conditions. The condition of constant rate of dissipation per unit volume of structure of minimum weight (or per unit area of the middle surface of a plate) is of basic importance in the theory of limit design. These principles were used by E. T. Onat, W. Schumann, and R. T. Shield, [87], for the design of circular plates.

PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS

VI. PROPAGATION OF ELASTIC-PLASTIC WAVESI N

A

20 1

NON-HOMOGENEOUS

MEDIUM There are two principal reasons for the development of theoretical research on stress waves in non-homogeneous media.* The first is the necessity of finding a theoretical explanation of certain experimental results obtained for test-pieces subjected to several tests. The occurrence of permanent deformation was observed here, modifying the yield limit in different ways a t different stations and thus giving rise to non-homogeneity. The other stimulus for the study of the influence of variable mechanical properties on the propagation of stress waves came from dynamic tests of soils. A soil is as a rule non-homogeneous, and the difference between its mechanical properties a t different depths may be considerable. The non-homogeneity type of a test-piece is essentially different from that of a soil; it is the result of plastic strain and is called “strain-type non-homogeneity”, while the non-homogeneity of a soil is of primary nature. The first survey of the investigations concerned with the non-homogeneity of the first type has been made by Kh. A. Rakhmatulin and G. S. Shapiro, [99]. A general and broad analysis of the two trends has been given by N. Cristescu in his survey, Ref. [14 b]. Investigations of the first type were started by Kh. A. Rakhmatulin, [98 a], [98 b]. Two cases should be distinguished here: the yield limit can be an increasing or a decreasing function along the test piece. This depends on whether the test piece (a cylindrical bar, in general) has been subjected to repeated impacts a t opposite ends or a t the same end. A particular nonhomogeneity type was studied by Kh. A. Rakhmatulin, who assumed that the yield point a, = as(x)is the only point function, every other mechanical property being uniform. In the general case of a curvilinear CJ,&-characteristic, the solution in the loading region was determined by the method of characteristics. In the unloading region the possibility of using an inverse method has been shown: the form of the curve representing the unloading wave is assumed, and the deformation u, and the velocity ut are assumed to be known on this curve. Then, the solution in the unloading zone can be determined by solving a Cauchy problem. Assuming linear strain-hardening Kh. A. Rakhmatulin, [98 b] obtained a closed-form solution. The case of repeated loads acting on the end of a cylindrical test-piece was studied by V. S. Lensky, [57 a] assuming linear strain-hardening.

* The importance of the problem is well illustrated by the fact t h a t recently (September 1961) a special International Symposium on “La Propagation des kbranlements dans les Milieux Hetkrogknes” in Marseille was held, organized by the Centre National de la Recherche Scientifique (C.N.R.S.) and directed by Th. Vogel. About 20 papers were read (theoretical and experimental). These, of course, could not be reviewed here.

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The work of Kh. A. Rakhmatulin was continued for other physical relations in the papers of S. D. Mokhalov, [63 a], [63 b]. The paper by S. Kaliski and J. Osiecki, [48], is devoted to the investigation of the propagation of stress waves in soils. By examining the behaviour of the soil many authors observed that the unloading characteristic depends in a definite manner on the maximum stress that is followed by the unloading process. Assuming that the unloading is elastic, one should also assume that Young's modulus of the unloading characteristic is a function of the load ao(x) in the unloading wave. The stress-strain relation in the unloading zone has then the form (6.1)

a(x,t) = a&)

+ E(a0)[W) -

&0(4I,

where eO(x) is the strain in the unloading wave. For such a relation, the solution for the unloading region can be obtained by iteration in an inverse way by assuming beforehand the form of the unloading wave. The problem is reduced to the search for the Riemann function and the solution of a Cauchy problem. I t has been pointed out in Ref. [48]that in many practically important cases, for which the Riemann function can be obtained in an accurate manner, the solution of the problem under consideration can be obtained in closed form. P. Perzyna in Refs. [91 a] and [91 b] investigated the solutions in loading regions in the general case of non-homogeneity. In Ref. [91 a] it is shown that in the case of curvilinear physical a,&-relations the solution may be obtained by the method of characteristics. The case of linear strain-hardening has been considered in greater detail for the physical relations

where a(%)and P ( x ) denote the variable elastic and plastic modulus, respectively, and E,(x) is the variable strain at the elastic limit. In the case of sudden loading with a load increasing in time, two strong discontinuity waves propagate in the body, representing the front of the elastic wave and the front of the plastic wave, bounding two regions: the elastic region and the plastic region. By satisfying the kinematic and dynamic conditions and by making use of the relations along the characteristics, the solution for the discontinuity waves has been obtained. Ref. [91 b] is devoted to the determination of the solutions by iteration in the loading regions, elastic and plastic. In these regions the problem is reduced to the solution of two generalized Picard problems. In the general case of non-homogeneity, paper [32] by R. Gutowski, S. Kaliski and J. Osiecki is concerned with the analysis of the solutions in the unloading regions. The physical relations are assumed in the form

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203

The solutions are obtained by iteration in the inverse way, the form of the unloading curve being assumed beforehand. Certain general solutions concerning the problem of propagation of elastic-plastic waves in non-homogeneous slender rods are given in Ref. [14 a], and some generalizations to the case of finite strain are given in Ref. [91 c]. The paper by J. Osiecki, [88 a], is devoted to a broad tentative application of the results of theoretical investigations to the determination of the state of stress in a soil in the case of dynamic load on the surface of the soil. The second part, [88 b], brings a solution of the problem of reflection from a rigid and a deformable wall under various physical relations and general non-homogeneity of the soil. These results make it possible to apply the theory to the computation of the deflection of a plane floor of an underground structure. The propagation of longitudinal stress waves in visco-elastic bars was studied by V. N. Kukudyanov and L. V. Nikitin, [54]. The assumed stressstrain relation takes the influence of the strain rate into consideration, the yield point being assumed to be variable along the bar. With arbitrary time-variable stress or velocity being prescribed a t the end of a semi-infinite bar, or with the end of the bar being subjected to the impact of a body of finite mass, the solution of the boundary-value problem is reduced to a mixed problem for the telegraphist's equation. Only a particular type of non-homogeneity is considered, and the influence of the strain rate is taken into consideration by using the simplest model of a body with relaxation.

VII. OTHER PROBLEMS 1. Physically Non-Linear Bodies

Paper [84] by W. Olszak and M. Zyczkowski is devoted to physically non-linear, non-homogeneous bodies. The basic equations of this theory were derived*, and a suitable classification of such bodies into five different groups was proposed. Some simple cases of application of the derived relations were also discussed. The problem of torsion of a circular bar is solved in an effective manner. The physical non-linearity is assumed to be of the parabolic type, and its

* For "active" straining processes, these equations may also serve as basic equations for non-homogeneous elastic-plastic bodies according to Hencky's "finite" ("deformation") theory of plasticity.

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non-homogeneity is supposed to be a function of the coordinate Y (in cylindrical coordinates r,p,z). The non-homogeneity law is assumed in the form

(7.1)

f,(r) = 1

+ n -a, Y

or in the form

where n is a numerical coefficient and a the exterior radius. To a certain extent these assumptions are justified e.g. in the case of cast bars. This holds in particular for the hyperbolic type of non-homogeneity (7.2),because the variation of the properties of the material is more significant in the neighbourhood of the surface than inside the bar. Assumptions (7.1) and (7.2) make solutions in closed form possible. In addition, the equations of longitudinal vibrations of a rod of a physically non-linear and “longitudinally” non-homogeneous material were derived. 2. Loose and Cohesive Granular Media

In view of many practical cases in which the physical properties of loose and cohesive materials exhibit a marked variability depending on the position of the point considered, the necessity of analyzing non-homogeneous loose and cohesive granular media is evident. Take for instance the angle of internal friction @, which explicitly depends on the granularity. In loose materials, this angle decreases with decreasing soil grain, and the same applies to the water content (cf. Fig. 27). The value of the cohesiveness K can also vary in a similar manner. On the other hand, a definite connection between the granularity and the depth z below the surface has been observed in many particular cases (cf. e.g. [73 a]). Thus a theory which is to be applied to problems of real soils can also require essential corrections of the assumption of homogenity. The monograph by V. V. Sokolovsky, [117], contains a complete theory for homogeneous loose and cohesive materials. Problems of non-homogeneity were approached in the papers by W. Olszak [73 r], [73 y]. This nonhomogeneity may be “natural” or “artificial”. In addition, there are phenomena and processes which leave the mechanical properties of the

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205

material unchanged, but result in effects that are equivalent to a change of a homogeneous interior structure into a non-homogeneous one. The above papers, [73 r] and [73 y], are based on the formal analogies, which exist between the theory of plasticity and the theory of loose and cohesive media in limiting states of their equilibrium. According to the fundamental notions of the theory of non-homogeneous bodies, [73 k]

1

wFIG.27. Angle of internal friction in granular media as a function of moisture.

and [731], the basic equations of non-homogeneous loose and cohesive materials were established, their classification into appropriate groups proposed, and some plane problems discussed. Some plane problems of the theory of limiting equilibrium of loose and cohesive non-homogeneous isotropic media were discussed by C. Szymadski, [120],and 2. Sobotka, [115].

3. Assumption of Non-Homogeneity as a Method of Solving Homogeneous Plastic Problems The notion of mechanical non-homogeneity may sometimes be used as an auxiliary method for the effective solution of problems of elastic-plastic equilibrium and plastic flow. One of the main difficulties in problems of the theory of plasticity originates in the non-linear character of the basic equations, which can be directly integrated in rare cases only. On the other hand, special types of non-homogeneity functions can in certain cases be considered with the aim of obtaining exact solutions of the basic equations. If, in addition, free parameters are introduced into these functions, their

2 06

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OLSZAK,

J . RYCHLEWSKI AND W. URBANOWSKI

values may afterwards be fixed in such a manner as to approach the given real conditions as closely as possible (e.g., homogeneity or non-homogeneity of a specified type). It has already been observed by some authors (cf. e.g. A. M. Freudenthal, H. Geiringer, [26]) that it is often more desirable to know the closed-form solution for a sufficiently approximate physical model than to solve for an exact physical model for which the corresponding results may be obtained only for particular conditions only by elaborate numerical methods. Thus, instead of rigorous physical assumptions and approximate solutions, we introduce approximate physical assumptions for which we can obtain exact solutions. Here, e.g., also the inverse and semi-inverse methods should be mentioned (for plane strain problems, these were discussed in Chap. 11). Such an approach is possible in problems of elastic-plastic equilibrium, as shown, e.g. in papers [73 u], [73 XI, and in problems of quasi-static plastic flow (cf. papers [83 c], [83 d]). In an analogous way, some previously known approximate solutions have been verified, for instance, in the papers [129a], [129b] by M. Zyczkowski. Sometimes solutions can be sought in still another way, as exemplified in some of the papers discussed in this survey: particular assumptions for the form of the non-homogeneity function are introduced, chosen for their simplicity and without reference to experimental evidence that such a type of non-homogeneous model reflects actual mechanical properties of the material in question. Nevertheless, the results thus obtained may provide useful hints for different actual problems related to other mechanical properties*.

Ref erelzces 1 . B A B U ~ KI., A , On the state of stress in a non-homogeneous foundation soil (in Czech with Russian and German summaries), A p l i k . Matem. 5, 2 , 361-369 (1957). 2. BARTA,J., On the minimum weight of certain structures, Acta Techn. Acad. Scz. Hungar. 18, 67-76 (1957). 3. BEEDLE,L. S., “Plastic Design of Steel Frames”, Wiley, New York, 1958. 4. BERMAN, I., Expansion of thick-walled cylinders fabricated from cold bent plates, J . A p p l . Mech. 3, 87 (1960). 5. BOHNENBLUST, H . F., DUWEZ,P., Some properties of mechanical model of plasticity, J . A p p l . Mech. 16, 222-225 (1948). 6. BRINKMAN, J. A., On the nature of radiation damage in metals, J . AppZ. P h y s . 8, gS (1954).

* A similar approach may be found in other fields of mechanics of deformable bodies as, for instance, in the analysis of finite deformations (cf. [ l o l l ) , in some recent studies of mechanical behaviour of granular media (cf. [la]), or in basic theoretical research on plasticity (cf. [34 f]).

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7. BRUCH,C. A,, MCHUGH, W. E., HOCKENBURG, R. W., Variations in radiation damage to metals, J . Metals 8, 1362 (1956). J. D., The yield mild steel under impact loading, J . Mech. Phys. 8. CAMPBELL, Solids 1, 8, 54-62 (1954). F., MASSONET, C., Recherches sur le flambement de colonnes en acier 9. CAMPUS, A37, & profil en double t6, sollicitCes obliquement, Bull. C . E . R . E . S . . Lilge, No. 7, 119-338 (1955). 10. CAP, F., “Physik und Technik der Atomreaktoren”. Springer, Wien, 1957. 11. CHECHULIN, B. B., Study on the micro-nonuniformity of the plastic deformation of steel, NACA TN, No. 1411, Aug. 1957; A p p l . Mech. Rev. 8, 11, 431-432, Rev. 3106 (1958). G., a) Les ph6nomknes de coaction Clasto-plastique et l’adaptation 12. COLONNETTI, ?I la resistance des matkriaux. A n n . Inst. Techn. Bdt. Trav. Publ., No. 99 (1949) ;

13. 14.

15. 16. 17. 18. 19.

20.

21. 22.

23. 24. 25.

b) Elastic equilibrium in the presence of permanent set, Quart. A p p l . Math. 7 , 353-362 (1950). CRAEMER,H., Ideal-plastische isotrope und orthotrope Platten bei Vollausnutzung aller Elemente, Ing.-Arch. 3, 28, 151-158 (1955). CRISTESCU.N., a) On the propagation of elastic-plastic waves in non-homogeneous rods, in [74], 429-430; b) European contributions to dynamic loading and plastic waves, i n “Proc. Sec. Symposium on Naval Structural Mech., Brown Univ., April 1960”, pp. 385-442. Pergamon Press, New York, 1960. DAVISON, B., “Neutron Transport Theory”, Univ. Press, Oxford, 1957. DERESIEWICZ, H., A review of some recent studies of the mechanical behavior of granular media, A p p l . Mech. Rev. 6, 11, 259-261 (1958). DIENES.G. J., VINEYARD, G. H., “Radiation Effects in Solids”, Intersciences Publishers, New York, 1957. DRUCKER. D. C., On minimum weight design and strength of non-homogeneous plastic bodies, i n [74], 139-146. DRUCKER. D. C., SHIELD,R. T., a) Design for minimum weight, i n 9th Congrks Intern. MCcan. Appl., Univ. Bruxelles 5, 212-222 (1957); b) Bounds on minimum weight design, Quart. A p p l . Math. 3, 15, 269-281 (1957). DRUYANOV, B. A,, a) Indentation of the thick plastically non-homogeneous plate by a rigid punch (in Russian), Izv. Akad. Nauk S S S R , Mekh. Machinoved. 8, 161-166 (1959) ; b) Indentation of thick plastically non-homogeneous plate by a rough punch (in Russian), Izv. Akad. Nauk S S S R , Mekh. Machinoved. 2, 129-131 (1960) ; c) Indentation of the thin plastically non-homogeneous plate by a rigid punch (in Russian), Izv. Akad. Nauk S S S R , Mekh. Machinoved. 4, 156-158 (1960) ; d) Limit equilibrium of the plastically non-homogeneous wedge (in Russian), Dokl. A N S S S R , 5, 127, 990-992 (1959); e) Izv. Akad. Nauk S S S R . Mekh. Machinoved., 3 (1961). FARIS,F. E., The effects of irradiation on structural materials, i n [97], p. 484 (Paper No. 747). FOULKES,J., a) Minimum weight design and the theory of plastic collapse, Quavt. A p p l . Math. 10,p. 347 (1953) ; b) The minimum weight design of structural frames, Proc. Roy. SOC.(London), Ser. A 228, p. 482 (1954). FREIBERGER, W., Minimum weight design of cylindrical shells, J . A p p l . Mech. 4, 28, 576-580 (1957). FREIBERGER, W., TEKINALP,B., Minimum weight design of circular plates, J . Mech. Phys. Solids 4, 294-299 (1956). FREUDENTHAL, A. M., a) “The Inelastic Behavior of Engineering Materials and Structures”, Wiley, New York-London, 1950; b) Thermal-stress analysis and mechanical design, i n “Nuclear Engineering”, 1959, Chapter 11, pp. 538-609.

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26. FREUDENTHAL, A. M., GEIRINGER, H., The mathematical theories of the inelastic continuum, i n “Handbuch der Physik” (S. Fliigge, ed.), Bd. 6. Elastizitat und Plastizitat, pp. 229-433. Springer, Berlin-Gottingen-Heidelberg, 1958. 27. GALIN,L. A., Plane elastic-plastic problem (in Russian), Prikl. Mat. Mekh. 3, 10, 367-386 (1946). 28. GATEWOOD, B. E., “Thermal Stresses”, McGraw-Hill. New York-London-Toronto, 1957. H., Some recent results in the theory of an ideal plastic body, in 29. GEIRINGER, “Advances in Applied Mechanics”, 3, Academic Press, New York, 1953. 30. GLEN,J. W., A survey of radiation effects in metals, Phil. Mag., Suppl. 4, 1955. 31. GRIGORYEV, A. S., On elastic-plastic deformations and the load-carrying capacity of rods and plates, non-homogeneous in the plastic range, i n [74], 181-190. 32. GUTOWSKI, R., KALISKI,S., OSIECKI,J . , Propagation of the unloading plane wave in a non-homogeneous soil (in Polish), Biuletyn WAT, 2(85), 3-18 (1959). J., a) Plastic design of beams and frames for minimum material con33. HEYMAN, sumption, Quart. Appl. Math. 8, p. 373 (1951); b) Plastic analysis and design of steelframed structures, Prelim. Publ., 4th Congr. Int. Assoc. Bridge Struct. Eng., Cambridge 1952; c) Plastic design of plane frames for minimum weight, Struct. Eng. 31, p. 125 (1953); d) Limit design of framed structures made of imperfectly plastic materials, in [74], 251-257. 34. HILL,R., a) “The Mathematical Theory of Plasticity”, Univ. Press, Oxford, 1950; b) On the problem of uniqueness in the theory of a rigid-plastic solid ( I ) ,J . Mech. Phys. Solids 4, 4, 247-255 (1956); c) (11) 1 , 6, 1-8 (1956); d ) (111) 3, 6, 153-161 (1957); e) (IV) 4, 6, 302-307 (1957); f ) A general theory of uniqueness and stability in elastic-plastic solids, J . Mech. Phys. Solids 3, 6, 236-249 (1958). 35. HODGE,P. G., JR., a) Discussion on paper by W. Freiberger “Minimum weight design of cylindrical shells”, J . Appl. Mech. 3, 24, 486-497 (1957); b) The linearization of plasticity problems by means of non-homogeneous materials, i n [74], 147-156; c) “Plastic Analysis of Structures”, McGraw-Hill, New York, 1959. 36. HOPKINS,H. G., PRAGER, W., Limits of economy of material in plates, J . AppZ. Mech. 3, 23, 372-374 (1955). 37. HORNE,M. R., Determination of the shape of fixed-ended beams for maximum economy according to the plastic theory, Prelim. Publ., 4th Cong. Int. Assoc. Bridge Struct. Eng., Cambridge, 1952. 38. HRUBAN,K., a) The semi-infinite solid with a non-linear deformation law (in Czech with Russian and English summaries), Rozpravy Ceskoslov. Akad. VBd, 1,51 (1958) ; b) The basic problems of a non-linear and non-homogeneous halfspace, i n [74], 53-61. 39. Hu, L., MARIN, J.. Anisotropic loading function for combined stresses in the plastic range, J . AppZ. Mech. 1, 22, 77-85 (1955). 40. HUBER,M. T., a) Some remarks on the mechanical properties of solids (in Polish), Arch. Mech. Stos. 1, 6, 1-20 (1953); b) Foundations of the rational computation of strength (in Polish), I n i y n . Budown. 11/12, 6, 506-507 (1948). 41. HUNDY, B. B., SINGER, A. R. E., Inhomogeneous deformation in rolling and wire drawing, J . Inst. Metals 9, 83, 401-407 (1954-1955). 42. ILYUSHIN,A. A , , a ) “Plasticity“ (in Russian), Moscow, 1948; b) Problems of the theory of plastic flow on surfaces (in Russian), Prikl. Mat. Mekh. 3, 18, 265-288 ( 1 954). A. A., LENSKY,V. S., “Strength of Materials” (in Russian), Fizmatgiz, 43. ILYUSHIN, Moscow, 1959. P. M., a) On the strength of thick-walled cylindrical 44. ILYUSHIN, A. A., OGIBALOV, and spherical shell subjected to radiation (in Russian), Znzh. Sbornik 28, 134-144

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45.

46. 47.

48. 49. 50.

51. 52.

53. 54.

55.

56. 57.

58. 59. 60.

61. 62. 63.

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(1960) ; b) “Elastic-Plastic Deformations of Thick-Walled Cylinders” (in Russian), Izdat. Moskov. Univ.. Moscow, 1960. IVLEV, D. D., a) On some works on the theory of plasticity by K. N. Shevtchenko (in Russian), Izv. Akad. Nauk S S S R . Otd. Tekhn. Nauk 2, 159-162 (1958); b) Some remarks on the theory of non-homogeneous plastic media (the space problem), Arch. Mech. Stos. 2, 13 (1961). JAMISTON. R. E., BLEWITT,T. H.. Some deformation characteristics of reactor irradiated copper single crystals at 78“ K and 300” K , Phys. Rev. 1, 91 (1953). JOHANSEN. I(. W., a) Bruchmomente der kreuzweise bewehrten Platten. MBm. AIPC, 1, Zurich, 1932; b) The ultimate strength of reinforced concrete slabs, 3rd Congr. Int. Assoc. Bridge Struct. Eng., Lihge, 1948, 565-570; c) “Pladeformler” (in Danish), Copenhagen, 1944. KALISKI,S., OSIECKI,J., The unloading wave in a body with variable unloading modulus, in [74], 403-413. KATCHANOV, L. M., “Mechanics of Plastic Bodies” (in Russian), 2nd ed., MoscowLeningrad, 1956. KLIMENKOV. V. I., ALEKSEYENKO, Yu. N., in “The Session of the Acad. Sci. of UdSSR on the Peaceful Uses of Atomic Energy (1-5 July 1955)” (in Russian), Izdat. Akad. Nauk SSSR. p. 322, Moscow, 1955. KOGLER,F., SCHEIDIG,A., “Baugrund und Bauwerk”, 3rd ed., Berlin, 1941. S. T., KUTAYCEV, V. I., PRAVDYUK, N. F., Effects of radiation KONOBEYEVSKY, on the structure and properties of structural materials (in Russian), in “The Investigations in the Field of Geology, Chemistry and Metallurgy”, p. 276, Izdat. Akad. Nauk SSSR, Moscow, 1955. KRUPKOWSKI, A,, The deformation of plastic metals by strain, A n n . Acad. Pol. Scz. Techn. 7, 113-120 (1939/1945). V. N., NIKITIN,L. V., Propagation of waves in non-homogeneous KUKUDZHANOV, bars of visco-elastic-plastic material (in Russian), Zzv. Akad. Nauk S S S R , Old. Tekhn. N a u k , Mekh. 4, 53-59 (1960). KUZNETZOV, A. I., a) Plane deformation of non-homogeneous plastic bodies (in Russian), Vestnik Leningrad. Univ., Ser. Mat.. Mekh., Astr. 8, 112-131 (1958); b) The problem of torsion and plane strain of non-homogeneous body, Arch. Mech. Stos. 4, 10, 447-462 (1958); c) The problem of non-homogeneous plastic layer (in Russian with Polish and English summaries), Arch. Mech. Stos. 2, 12, 163-172 (1960). LEE, MING-HUA,PUI, MING-LI, Thick-walled cylinder with non-homogeneous strain-hardening characteristics under internal pressure, in [74], 167-170. LENSKY,V. S., a) A problem of propagation of elastic-plastic waves (in Russian), Vestnik Moskovsk. Unio. 3, 17-22 (1949); b) Effects of radiations on the mechanical properties of solids (in Russian), Znzh. Sbornik 28, 97-133 (1960). LEVY, M., Compt. Rend. 16, 73. 1098-1103 (1871). M., Modifications produced in MAYER,G., PERIO,P., GIGON,J., TOURNARIE, non-metallic materials by radiation, in [97], p. 674 (Paper No. 362). MICHELL,J. H., The inversion of plane stress, London Math. Soc. Proc. 84, 134 (1902). MIKELADZE, M. SH., On the minimum weight of anisotropic shells (in Russian with Georgian summary), Soobshtch. Akad. Nauk Gruz. S S R 1, 19, 11-18 (1957). MISES, R. v., Mechanik der plastischen Formanderung von Kristallen. Ztschr. angew. Math. Mech. 3, 8, 161 (1928). MOKHALOV, S. D., a) On the problem of propagation of elastic-plastic waves along a bar with variable yield stress (in Russian), Utch. Zap. Tomsk. Gos. Univ. 26, 49-67 (1955); b) On the repeated shock on the end of a semi-infinite bar (in Russian), Utch. Zap. Tomsk. Gos. Univ. 26, 68-76 (1955).

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64. M R ~ z Z., , a) On the design of non-homogeneous technically orthotropic plates, in [74], 191-202; b) The load-carrying capacity of orthotropic shells, Arch. Mech. Stos. 1, 12, 85-107 (1960). 65. MURZEWSK~, J., Reinforced concrete slabs supposed on all sides and frameworks in elastic-plastic state (in Polish), Czasop. Techn. 82, 21-29 (1957). 66. NAGHDI,P. M., ROWLEY,J. C., An experimental study of bi-axial stress-strain relations in plasticity, J . Mech. Phys. Solids 1, 8, 63-80 (1954). 67. N E M ~ N YP. I , F., Recent developments in inverse and semi-inverse methods in the mechanics of continua, in “Advances in Applied Mechanics”, Vol. 2, 1951. D., Load-carrying capacities of non-homogeneous rectangular plates 68. NIEPOSTYN,

(kinematic approach), to appear in Arch. Mech. Stos. 69. NOVOZHILOV, V. V., “Theory of Elasticity” (in Russian), Sudpromgiz, 1958. W., “Thermoelasticity Problems” (in Polish), PWN, Warszawa, 1960. 70. NOWACKI, , ,I ,Note on the definition of homogeneity and heterogeneity of elastic 71. N O W I ~ ~ SJK

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v) Applications of the theory of plasticity to problems of non-homogeneous and anisotropic plates and shells, 4th Yougoslov. Congr. Theor. Appl. Mech., Opatija, 1958; W) Quelques applications techniques relatives B la th6orie de la plasticit6 des milieux non-homogenes, Bull. C.E.R.E.S. 10, 31 1-334 (1959); x) On an elastic-plastic problem of the eccentric circular cylinder, Bull. Acad. Polon. Sci., CZ. I V , 11, 7, 617-628. 629-639 (1959); y) On some basic aspects of the theory of non-homogeneous loose and cohesive media, Arch. Mech. Stos. 6, 11, 751-766 (1959). 74. OLSZAK,W. (Editor), “Non-Homogeneity in Elasticity and Plasticity”, R o c . I.U.T.A.M. Symposium, Warsaw. September 2-9, 1958; Pergamon Press, London-New York-Paris-Los Angeles, 1959. 75. OLSZAK,W., M ~ b z ,Z., Note on the completness of the elastic-plastic solution to certain boundary value problems for the eccentric ring, Arch. Mech. Stos. 3, 10, 441-444 (1958). 76. OLSZAK,W., MURZEWSKI, J., a) Elastic-plastic bending of non-homogeneous orthotropic circular plates, Arch. Mech. Stos. 4, 9, 46&485; 5, 9, 605-630 (1957); b) Elasto-plastische Biegung von nicht-homogenen orthotropen Kreisplatten, Ztschr. angew. Math. Mech. 7/8, 37, 277-278 (1957); c) Elastic-plastic bending of non-homogeneous orthotropic plates, Bull. Acad. Polon. Sci.. Cl. I V , 4, 6, 211-218 (1958); d) The general case of axisymmetric bending of elastic-plastic plates, Bull. Acad. Polon. Sci., CZ. I V , 4, 6, 219-228 (1958). 77. OLSZAK, W., MURZEWSKI, J., GOLECKI,J., a) Non-homogeneous elastic-plastic semi-infinite plate loaded by a concentrated force, Arch. Mech. Stos. 2, 8, 197-214 (1956); Bull. Acad. Polon. Sci., Cl. I V , 4, 4, 209-220 (1956). P., a) Extremum theorems in the theory of plasticity W., PERZYNA, 78. OLSZAK, of non-homogeneous and anisotropic bodies, Arch. Mech. Stos. 6, 9, 695-712 (1958); b) Bull. Acad. Polon. Sci.. Cl. I V . 2, 6. 95-102, 103-105 (1957); c) Criteria of validity of variational theorems in mechanics of inelastic nonhomogeneous anisotropic deformable bodies, Arch. Mech. Slos. 4, 10, 559-569 (1958). K I , a) Two-dimensional problems in the W., PERZYNA, P., S Z Y M A ~ SC., 79. OLSZAK, theory of plasticity of non-homogeneous anisotropic bodies, Arch. Mech. Slos. 3, 9, 335-385 (1957); b) Bull. Acad. PoZon. Sci., Cl. I V , 1, 7 , 7-19, 21-27 (1959). 80. OLSZAK, W., RYCHLEWSKI, J., a) Geometric properties of stress fields in plastically non-homogeneous bodies under conditions of plane strain, I.U.T.A.M. Symposium, Haifa, 1962; b) On biharmonic states of ideally plastic bodies, (in preparation) ; c) Nichthomogenitats-Probleme im elastischen und vorplastischen Bereich, asterr. Ing.-Archzv 1, 1961. A,, a) Some problems of the limit analysis and design R I . OLSZAK,W., SAWCZUK, of non-homogeneous axially symmetric shells, Proc. 2nd Symp. Concr. Shell Roof Structures, Oslo, 1957; b) On velocity fields in the theory of plastic flow of non-homogeneous shells, to appear in Arch. Mech. Stos.; c) Limit analysis of anisotropic and non-homogeneous structures (in Portuguese), Tkcnaca, 84, 305, 31-45: 306, 189-201 (1960); d) Theorie de la capacit6 portante des constructions non-homogbnes et orthotropes. Annales de Z’Institut Technique d u Bdtiment et des Travaux Publics, Mai 1960, No. 149. 82. OLSZAK, W., URBANOWSKI, W., a) Non-homogeneous thick-walled elastic-plastic cylinder subjected to internal pressure (in Polish; summary in English and Russian), Arch. Mech. Stos. 3, 7, 315-336 (1955); b) Bull. Acad. Polon. Sci., Cl. I V , 3, 4, 153-163 (1956); c) Non-homogeneous thick-walled elastic-plastic spherical shell subjected to internal and external pressures (in Polish: summary in English and Russian), Rozpr. I n t . 1, 4, 25-41 (1956); d) Bull. Acad. Polon. Sca., C1. I V , 3, 4, 165-171 (1956) (in English and Russian); e) The orthotropy

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and the non-homogeneity in the theory of plasticity (in Polish; summary in English and Russian), Arch. Mech. Stos. 1, 8, 85-110 (1956); f) The flow function and the yield condition for non-homogeneous orthotropic bodies (in English), Bull. Acad. Polon. Sci., Cl. I V , 4, 5, 191-203 (1957); g) Plane problems of the theory of plasticity of non-homogeneous and anisotropic bodies, J . Sci. Eng. Res., Kharagpur (India) 2, 2, 201-208 (1958); h) The plastic potential and the generalized distortion energy in the theory of non-homogeneous anisotropic elastic-plastic bodies, Arch. Mech. Stos. 4, 8, 671-694 (1956) ; i) Bull. Acad. Polon. Sci., C1. I V , 1, 8 , 29-37, 39-45 (1957); j ) 9th Intern. Congr. Appl. Mech., Brussels 1957, 8, 23-80; k) Quelques problhmes fondamentaux relatifs B la tbkorie des milieux elasto-plastiques anisotropes e t non-homogenes, Bull. C.E.R.E.S., Liige 10, 293-310 (1959); 1) Plastic non-homogeneity: A survey of theoretical and experimental research, i n [74], 259-298. 83. OLSZAK,W., ZAHORSKI, S., a) Elastisch-plastische Biegung des nichthomogenen orthotropen Bogenstreifens, tisterr. Ing.-Archiv 2, 13, 1 0 6 1 2 0 (1959) ; b) A nonhomogeneous orthotropic circular segment as a n elastic-plastic problem, Arch. Mech. Stos. 4, 11, 409-419 (1959); c) Some problems of continued plastic flow of the eccentric cylinder, Arch. Mech. Stos. 5/6, 12 (1960); d) The notion of non-homogeneity as applied to a problem of continued plastic flow, in “Problems of Continuum Mechanics”, Engl. ed., 311-326, SOC.Industrial Applied Methematics, Philadelphia, 1961. M., On the fundamentals of the theory of physically 84. OLSZAK,W., ZYCZKOWSKI, non-linear non-homogeneous bodies (in Polish; summary in English and Russian), Arch. Mech. Stos. 1, 7, 151-168 (1955). 85. ONAT,E. T., The effects of non-homogeneity caused by strain-hardening on the small deformations of rigid-plastic solid, i n [74], 171-180. 86. ONAT,E. T., PRAGER,W., Limits of economy of material in shells, De Ingenieur, 1955. W., SHIELD,R. T., Design of circular plates for minimum 87. ONAT,E. T., SCHUMANN, weight, Ztschr. angew. Math. Phys. 8, 499 (1957). 88. OSIECKI,J., Propagation of a plane stress wave in a non-homogeneous solid (in Polish), Biuletyn W A T 8(97), 20-44 (1960); b) Reflection of a plane stress wave in a non-homogeneous solid medium (in Polish), Biuletyn W A T 9(98), 3-26 (1960) ; c) Propagation of a plane stress wave in a non-homogeneous solid, Proc. Vibr. Probl. l ( 6 ) , 2, 67-96 (1961); d) Reflection of a plane stress wave in a non-homogeneous solid meaium, Proc. Vibr. Probl. 2(7), I, (1961). 89. PANFHROV, V. M., On the method of elastic solutions in the problems of plastic cylindrical shell (in Russian), Dissert., 1946. 90. P E L C Z Y ~ ~ T., S KThe I , ability of steel t o brittle cracking (in Polish; summary in English and Russian), Biuletyn W A T 1(84), 8. 13-28 (1959). 91. PERZYNA, P.. a) The problem of propagation of elastic-plastic waves in a nonhomogeneous bar, in [74], 431-438; b) Propagation of elastic-plastic waves in a non-homogeneous medium, Arch. Mech. Stos. 5, 11, 595-612 (1959); c) Analysis of plane elastic-plastic waves in a non-homogeneous medium. I. Finite strains. 11. Infinitesimal strains, Bull. Acad. Polon. Sci.. Cl. I V , 8 , 485-502 (1960). 92. POLETZKY, A . T., Bending of surface-treated beams (in Russian), Sborn. Stat. Tchelabinsk. Politekhn. Inst. 1, 23-36 (1954). 93. POPOV,N. I., FRIDMAN, YA. B., Raising of machine parts strength by the method of oriented hardening (in Russian), in “Problems of Strength and Deformation of Metals and Alloys”, 8, Moscow, 1954. 94. PRACER, W., a) J . Appl. Phys. 1, 18 (1955); b) J . A$@. Mech. 12, 16 (1945); c) Minimum weight design of plates, De Ingenieur 67, (1955) ; d) Minimum

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95. 96. 97. 98.

99.

100. 101. 102. 103. 104. 105.

106. 107.

108.

109. 110.

111. 112. 113.

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weight design of a portal frame, Proc. ASCE, J . Eng. Math. Div. Paper No. 1073, 1-10, 4 (1956); e) On a problem of optimal design, in [74], 125-132. PRAGER,W.. HODGE,P. G., JR.. “Theory of Perfectly Plastic Solids”, Wiley, New York-London. 1951. PRIMAK,W., Experimental evidence of thermal spikes in radiation damage, Phys. Rev. 98 (1955). “Proceedings of the International Conference on the Peaceful Uses of Atomic Energy”, 7, 1956. RAKHMATULIN, KH. A., a) On the propagation of unloading wave along a bar with variable yield stress (in Russian), Prikl. Mat. Mekh. 10, 333-346 (1946); b) The study of the propagation laws of elastic-plastic waves in bars with variable yield stress (in Russian), Prikl. Mat. Mekh. 1, 14, 65-74 (1950). RAKHMATULIN, KH. A., SHAPIRO, G. S., Propagation of perturbations in nonlinear elastic and inelastic medium (in Russian), I z v . Akad. Nauk S S S R , Otd. Tekhn. Nauk 2, 68-89 (1955). REMNEV, Yu. I., On the effect of radiation on stresses and small deformations in solid bodies (in Russian), Doklady Akad. Nauk S S S R 3, 124 (1959). RIVLIN,R. S.. Large elastic deformations, in “Rheology“ (F. R. Eirich, ed.). Vol. 1. Theory and Applications, Academic Press, New York, 1956. R O G O Z I ~ ~M., S K I ,Some problems of thermo-plasticity of a spherical shell, in [74], 215-226. RYCHLEWSKI, J., Some problems of plastic non-homogeneity, (in preparation). SAINT-VENANT, B. DE, a) J . Math. Pures Appl., Ser. 2, 16, 308 (1871); b) J . Math. Pures Appl., Ser. 2, 18. 373-382 (1871). SAWCZUK, A.. a) Some problems of load-carrying capacities of orthotropic and non-homogeneous plates, Arch. Mech. Stos. 4, 8, 549-563 (1956); b) Some problems of limit analysis of elements subjected to tension and bending (in Polish; English and Russian summaries), Rozpr. I n i y n . 2, 5, 255-284 (1955); c) Grenztragfahigkeit von Platten, Baupl. Bautechn. 11, 315-320, 359-364 (1957) ; d) Limiting equilibrium of a non-homogeneous plastic wedge, in [74]. 203-210. M.. Load carrying capacity of concrete arches (in Polish; SAWCZUK, A., JANAS, Summary in English and Russian), Arch. I n i y n . Lqd. 1, 1961. SAWCZUK, A., KONIG, J. A., Load-carrying capacity of orthotropic cylindrical tanks loaded by loose materials (in preparation). SAWCZUK, A., K W I E C I ~ ~ SM., K I ,Load-carrying capacities of slab-beam structures (in Polish; summary in English and Russian), Arch. I n i y n . Lqd. 5, 335-370 (1957). SAWCZUK, A,, OLSZAK, W.. Methods of limit analysis of reinforced concrete tanks, Symposium on Simplified Methods of Shell Analysis, Brussels, 1960 (in print). SAWCZUK, A., S T F P I E ~A,, ~ , Problem of non-homogeneous wedge under plane plastic flow conditions (in Polish), Zesz. N a u k . Politechn. Warszaw., Budownictwo (in print). SERGEYEV, G. YA., TITOVA,V. V., BORISOV.K. A,, “Metallurgy of Uranium and Some Other Reactor Materials” (in Russian), Atomizdat, Moscow, 1960. SETH, B. R., Non-homogeneous yield condition, in [74], 133-138. SHEVTCHENKO, K. N., a) Elastic-plastic state due to concentrated force acting on a semi-plane (elastic-plastic problem) (in Russian), Dokl. Akad. N a u k . 1, 61 (1948) ; b) Concentrated force acting on a semi-plane (elastic-plastic problem) (in Russian), Pvikl. Mat. Mekh. 4, 12, 385-388 (1948); c) Letter to the Editor (in Russian), Izv. Akad. Nauk S S S R , Otd. Tekn. Nauk 9, 151 (1958); d) On the method of solving some elastic-plastic problems (in Russian), I z v . Akad. Nauk S S S R , Old. Tekn. Nauk 9, 152-155 (1958).

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Some Elementary Problems in Magneto-hydrodynamics BY RAYMOND HIDE*

AND

PAUL H . ROBERTS

.

Physics Department. King's College (University of Durham) Newcastle-upon- Tyne. 1. England

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I1 Basic Equations of Magneto-hydrodynamics . . . . . . . . . . . 1 Continuity of Matter and Momentum . . . . . . . . . . . . . 2 . Thermodynamic Equations . . . . . . . . . . . . . . . . . . . . 3 . Electrodynamic Equations . . . . . . . . . . . . . . . . . . . . 4 . Range of Validity . . . . . . . . . . . . . . . . . . . . . . . .

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1.Introduction.

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I11. Electromagnetic and Mechanical Effects; Dimensionless Parameters . . . 1 Electromagnetic Effects . . . . . . . . . . . . . . . . . . . . . 2 . Mechanical Effects . . . . . . . . . . . . . . . . . . . . . . . . 3 Two-dimensional Theorem : Analogy with the Proudman-Taylor Result for Rotating Fluids . . . . . . . . . . . . . . . . . . . . . . .

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IV . Boundary Conditions . . . . . . . . . . . . . . . . 1 Methods of Deriving Boundary Conditions . . . . 2. The Electromagnetic Boundary Conditions . . . . 3 . The Mechanical Boundary Conditions . . . . . . 4 . Small Departures from a Steady State . . . . . 6. Boundary Conditions a t a Solid Insulating Surface

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. . . . . . . V . Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 1 . The General Effect of a Magnetic Field . . . . . . . . . . . . . . 244 246 2 . Alfvdn Waves . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Sound Waves . . . . . . . . . 4 The Effect of Dissipation . . . . 6 Shock Waves . . . . . . . . .

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VI . Alfvdn Waves in Systems of Finite Extent . . . . . . . . . . . . 1 Orders of Magnitude . . . . . . . . . . . . . . . . . . . . . . . 2 Standing Waves in a Fluid Bounded by Conducting Planes . . .

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. . 282 VII . Gravity Waves: Rayleigh-Taylor Instability . . . . . . . . . . . . . 267 1. Introduction: Choice of Model . . . . . . . . . . . . . . . . . . 267 2 . Solution of Model Problem . . . . . . . . . . . . . . . . . . . . 288 * Now at Department of Geology and Geophysics. Massachusetts Institute of Technology. Cambridge 39. Massachusetts . t Now a t Yerkes Observatory (University of Chicago). Williams Bay. Wis'consin

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RAYMOND H I D E AND PAUL H. ROBERTS

. . . . . . . . . . . . . . 270 Steady Flow between Parallel Planes . . . . . . . . . . . . . . . . 274 Flow due t o a n Oscillating Plane: Rayleigh’s Problem . . . . . . . . . 286 1. Definition of Problem : Dimensionless Parameters . . . . . . . . . . 286 2. The Formal Solution. . . . . . . . . . . . . . . . . . . . . . . 288 3. Discussion of Some Limiting Cases . . . . . . . . . . . . . . . . 292

VIII. Gravitational Instability : Jeans’ Criterion IX. X.

4. Rayleigh’s Problem

. . . . . . . . . . . . . . . . . . . . . . .

296

XI. Steady Two-dimensional Inertial Flow in the Presence of a Magnetic Field 300 Appendix A : The Hydromagnetic Energy Equation . . . . . . . . . . . . . 305 Appendix B : Relativistic Magneto-Hydrodynamics . . . . . . . . . . . . . 31 1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

I. INTRODUCTION Magneto-hydrodynamics - or hydromagnetics - is the study of the flow of an electrically conducting fluid in the presence of a magnetic field. Nowadays it is often regarded as a part of “plasma physics”. This article, which has developed largely out of a final honours course given to physicists, deals with a number of mathematically elementary problems in the subject which are nevertheless far from trivial physically, and which contain pitfalls for the unwary. Thus, our principal objective is to provide an introduction to hydromagnetics in which the physical wood is not obscured by mathematical trees. Although hydromagnetics is based on the equations of classical hydrodynamics and electromagnetism, no serious investigations were carried out in this field until comparatively recently. This is because the parameter which measures the strength of the coupling between the magnetic field and the fluid flow, the so-called magnetic Reynolds number (see Ch. 111),

(where a is the electrical conductivity and p the magnetic permeability of the fluid and L and U are, respectively, a length and a flow velocity characteristic of the system*) is so very small for most practical values of L , U , and ap that hydromagnetic phenomena cannot readily be produced in the laboratory. On the other hand, owing to the vast length scale associated with cosmical phenomena, even poorly conducting fluids moving quite slowly are strongly coupled to any magnetic field that is present. It is hardly surprising, therefore, that the origins of hydromagnetics are to be found in

*

Rationalized M.K.S. units are used throughout.

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

217

the study of the magnetism of celestial bodies and are not associated with any discovery, accidental or otherwise, made on the scale of a terrestrial laboratory.? The possession by the Sun of a general magnetic field was first inferred in 1899 from the appearance of the corona during an eclipse. In 1908 Hale discovered by direct measurement of the Zeeman splitting of lines in the solar spectrum that sunspots are the seat of strong magnetic fields, of order 10-1 volt sec m-2 ( = 10-l weber m-2 = lo3 gauss), and subsequent observations, by several workers, of the general solar field show that its strength at the surface is. about volt sec m-2, and somewhat variable during a typical solar cycle. It is now known that a number of stars possess quite strong surface magnetic fields (0.1 to 1 volt sec m-2), some of which vary quite rapidly, altering considerably, and in some cases reversing in direction, in periods as short as a few days. Nowadays it is known that cosmical magnetic fields are not restricted to celestial objects such as the Sun, Earth, and stars. There seems to be a general Galactic magnetic field of about volt sec m-2. This f ield possibly prevents the spiral arms from collapsing gravitationally and may also play a key r6le in the processes leading to the formation of stars. Moreover, hydromagnetic interaction between this field and rapidly moving gas clouds in interstellar space is the likely cause of the acceleration of cosmic rays to stupendous energies. The importance of the r61e of hydromagnetics in cosmical physics can be judged from the theoretical contributions stemming directly from work on the origin of the magnetic fields of celestial objects, heating of the solar corona, dynamics of solar rotation, theory of magnetic storms, to name but a few astrophysical problems. Details of this work and exact references can be found in several recent review articles [l, 2 , 3 , 4 , 5 , 5 6 ] . Although the early development of hydromagnetics took place as a branch of cosmical physics, in the past fifteen years or so there have been several investigations of laboratory systems. Thus, the fact that a pressure head results when a current is passed through a conducting fluid immersed in a magnetic field has been exploited, in the electro-magnetic pump, in order to circulate liquid metals in systems, such as nuclear reactors, where the presence of moving machinery would pose severe technical problems. Theoretical work on hydromagnetic flow along pipes (see Ch. IX) followed the invention of the electro-magnetic pump and so did experimental work using liquid metals [el. For practical reasons these experiments have been confined to low values of R, when the coupling between the motion and field is weak. I n such circumstances the principal effect of the field is to enhance energy dissipation, through the agency of ohmic heating associated t One wonders how ordinary fluid dynamics would have developed had no fluid less viscous than thick treacle, say, been available to the experimenter.

218

RAYMOND HIDE AND PAUL H. ROBERTS

with the induced electric currents. In practice, ohmic dissipation may exceed viscous dissipation by one or two orders of magnitude. I t is the high density of liquid metals that limits the value of U which safely can be attained in controlled experiments. However, by using media of lower density, such as an ionized gas moving at much higher speeds, it has been possible to work at higher values of R and thus secure tighter coupling between the magnetic field and fluid motion. Under the controlled coriditions of an ordinary shock tube [7] values of R of order unity have been attained, and even higher values are associated with gas discharges of very high current intensity, on which, unfortunately, it is notoriously difficult to carry out reliable measurements. Laboratory systems differ in an essential way from cosmical ones: fixed, rigid conductors capable of carrying currents without moving, and/or insulators capable of withstanding strong electrostatic potential differences are always present in the former, but always absent in the latter. As a consequence, there are in principle laboratory phenomena which have no counterpart in cosmical physics. By the same token, even if an indefinite range of R were available to the experimenter, many cosmical phenomena could not be reproduced in laboratory models. In the past ten years the study of high-current gas discharges and other low-density plasma systems has been strenuously promoted by workers aiming to produce the high temperatures required to induce thermonuclear reactions in light elements such as deuterium [8]. The concomitant increase in the number of physicists and electrical engineers, as well as mathematicians, engaged in hydromagnetic research has been considerable. Now, as a result of recent renewed interest in the problem of devising an economic method of direct conversion of thermal to electrical energy, mechanical engineers are beginning to show an interest in the subject [9], and there are aeronautical engineers who see applications of hydromagnetics to the problems of propulsion and boundary layer control [lo]. On the more academic level, according to current issues of their journals many applied mathematicians are now examining the effects of a magnetic field on the “standard solutions” of problems in ordinary fluid dynamics. The make up of this article is evident from the table of contents. The selection of topics arises largely from the authors’ personal interests and limitations of space, and is arbitrary to that extent. We have endeavoured, however, to ensure that the introductory sections, dealing with the basic equations, dimensionless parameters and boundary conditions, and an appendix dealing with the hydromagnetic energy equation, are reasonably complete. It is hoped that the present survey will provide an introduction to hydromagnetics for workers possessing some knowledge of continuum mechanics but who are otherwise unfamiliar with the subject.

ELEMENTARY- PROBLEMS IN MAGNETO-HYDRODYNAMICS

219

11. BASICEQUATIONS OF MAGNETO-HYDRODYNAMICS 1. Continuity of Matter and Momentum

The continuity and momentum equations governing the flow of a Newtonian fluid of density p, coefficient of shear viscosity py, and coefficient of bulk viscosity pc, relate the values of pressure ( p ) , fluid velocity (u) and body force (F) a t a general point in space, coordinates (x1,x2, x3) a t time t. They are, respectively, (2.1)

3 = 9 + u - g r a d p = - pdivu, Dt - at

and

1 (2.2)

In practice, fluid dynamicists distinguish between “compressible” and “incompressible” fluids, corresponding, respectively, to whether or not density changes associated with pressure variations in the fluid result in significant dynamical effects, [ l l ] . Such effects are insignificant only when the speed of fluid flow is much less than that of sound in the medium, and when accelerations are slow compared with those associated with sound waves. Under these conditions the continuity equation (2.1) simplifies to (2.3)

div u

= 0,

and the corresponding term of (2.2) vanishes. In the case of a conducting fluid carrying an electric current, whose density j is measured in amp m-2, in the presence of a magnetic field, whose intensity B is measured in volt sec m--2, to the usual body force we must add the Lorentz force j x B newton m-3. Thus (2.4)

F=pgrad@+jxB,

where @ is the potential of external forces, such as gravity or any other force causing gross acceleration. The seven scalar equations to which (2.1), (2.2) and (2.4) are equivalent contain fourteen unknowns, and we must therefore supplement them with further mathematical relations. These relations stem from thermodynamic and electrodynamic considerations.

220

RAYMOND HIDE AND PAUL H. ROBERTS

The thermodynamic relations (see Appendix A) comprise an equation of state together with statements concerning irreversible molecular transport processes, such as diffusion and thermal conduction, leading to entropy changes. The electrodynamic relations are Maxwell’s equations together with a statement concerning the dependence of the current on the electric field present (e.g. Ohm’s law). We shall assume in most of this article that the gravitational potential @ is a known function of the space coordinates. In one problem where variations of p affect ds (cf. Ch. VIII) to the equations of hydrodynamics, thermodynamics and electrodynamics must be added Poisson’s equation

VW

(2.5)

= - 4?Lcp,

where G is the universal gravitational constant. 2. Thermodynamic Equations To illustrate the thermodynamic relations required for a typical compressible fluid, consider a perfect gas for which

where W is the universal gas constant divided by the gramme-molecular weight. Two extreme cases will be cited to illustrate the range of possibilities as regards entropy variations, without having to write down the field equations governing molecular transport processes. These are the isentropic case, in which changes of state are so rapid that transport processes can be ignored, and the entropy per unit mass

s = cu log, (Pp-Y),

(2.7)

(where c, is the specific heat a t constant volume, q, is the specific heat at constant pressure, and y cp/c,) of a fluid particle remains constant, and the isothermal case in which heat conduction is so effective that the temperature of a fluid particle remains constant. I t is convenient to consider two types of incompressible fluid, namely, those which are barotropic and those which are baroclinic. Barotropic incompressible fluids have uniform density and in consequence there is no contribution of the gravitational part of F to the dynamic pressure field. In the absence of hydromagnetic effects, hydrodynamical flow of a barotropic fluid has to be generated by applying forces a t the bounding surfaces of the fluid. When p is kept constant in equations (2.1) and (2.2) there is no need to supplement them with thermodynamic relations. For this reason the study of the flow of incompressible barotropic fluids is the most highly developed branch of fluid dynamics.

=

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

22 1

Baroclinicity is associated with density variations, the action of gravity (or of any other general acceleration) on which gives rise to buoyancy forces (see the first term on the right-hand side of (2.4)).It is not generally possible to maintain hydrostatic equilibrium in the presence of these forces and hydrodynamic flow must ensue. Baroclinicity arises in a variety of ways; it may be due to variations in temperature, chemical composition, or both. Differential heating produces temperature variations from place to place in a fluid and, as a result of thermal expansion, these are associated with density variations. Thermal convection is one example of this type of system, but there are also situations in which baroclinic effects inhibit fluid motion. If the heated incompressible fluid has a volume coefficient of thermal expansion a, the equation of state is

where po is the density at T = 0. Entropy changes are taken into account by including the equation of heat conduction which, in the absence of heat sources within the fluid, has the form

(2.9)

pcp- Dt = - pcp

+u

*

grad

= div

grad TI,

[ K ~ c ~

where the thermal diffusivity is K , and cp is the specific heat a t constant pressure, ~ p being c ~ identical with the coefficient of thermal conductivity. When baroclinicity is due to variations in chemical composition, (2.8) has to be replaced by a relationship between density and composition, and (2.9) by the equation of diffusion of matter. A particularly simple example is that of two immiscible liquids of different density, where baroclinic effects arise only at their surface of contact and there is no diffusion of matter. Baroclinic effects of this type can arise in gravity waves at the free surface of a liquid (see, e.g. Ch. VII).

3. Electrodynamic Equations Now we must write down equations relating current density j, magnetic field B, electric field E (in volt m-l) and charge density 6 (in coulomb m-9. If (2.10a, b)

H

G

B/p amp m-l,

D E EE coulomb m--2,

222

RAYMOND HIDE A N D PAUL H. ROBERTS

where ,u and E denote, respectively, magnetic permeability and dielectric constant,* these are the following. curl H

(2.11) (2.12)

curl E

= j,

=-

aB/at,

div B = 0,

(2.13)

and divD = 6.

(2.14)

(2.11) is Ampere's circuital law relating the magnetic field to its basic source, the electric current, displacement current being neglected (see Sec. 11. 4 below and also equation (B. 1.)); (2.13)expresses the fact that the field is solenoidal (=source free), the isolated magnetic pole being amathematical fiction. (2.12)is Faraday's law of induction, which, in its differential form conceals many subtle difficulties of interpretation brought out clearly in relatively few standard texts [12]. (2.14) and (2.10b) relate the electric field to the volume density of electric charge 6; (2.14)includes a statement of the inverse square law of electrostatics. A unit electric charge moving at velocity u relative to a magnetic field B experiences in addition to a force E, a force u x B. Thus, if the conducting fluid satisfies Ohm's law with conductivity u (in ohm-l m-I), (2.15)

j

= a(E

+ u x B),

(cf. equation A . 21 and Sec. 11. 4 below) This completes our set of equations governing seven unknown vector quantities, u, F, E, D, H, B, j, and four unknown scalar quantities, p , p, T and 6, or twenty-five scalars in all. An integral energy relation based on the differential equations of this section is derived in the Appendix A. 4. Range of Validity

Now we must consider the range of validity of our equations. For this purpose it is convenient to introduce a typical flow speed, U , a speed V G B / ( ~ u p ) based ' / ~ on the magnetic field strength (see equation 5.1) and the ordinary sound speed, a,. If c is the speed of electromagnetic waves then U2/c2,ao2/c2and V2/c2are measures of ordered kinetic energy, thermal energy and magnetic energy respectively in terms of the rest energy of the fluid.

* For free space,

,u = p o = 4n * lo-' newton amp+ (henry m-l) E = E,, = 8.86 x lo-'* coulomba newton-' m-a (farad m-1) ; (poeJ-1/2 = 2.995 x lo8 m s-l, the speed of light.

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

223

When U/c << 1 the D/Dt terms in (2.1) and (2.2) are non-relativistic (see Appendix B) and, to the same approximation, although the effective electric field.depends on the local frame of reference in which it is measured, being (E u x B) in a frame relative to which the fluid moves with velocity 11, the magnetic field is frame independent. When ao/c << 1, that is to say, when the root mean square speed of thermal motion is much less then the speed of light, the relativistic correction P / c 2 to the density p (cf, equations B.20 and B.21) is negligible. If t is a characteristic period of time associated with a hydromagnetic system, L being a typical length (see Ch. I) then according to (2.10, 2.11, 2.12, and 2.14),

+

so that when the time taken for energy to be transmitted across the system is much greater then that taken for electro-magnetic waves to cross it, (more precisely, as we shall see below in Sec. V.2, when I/ << c ) the neglect of displacement current, aD/at, in equation (2.1l ) , of the electrostatic body force 8 E in the expression for the body force (%.4),and of the advective contribution 23u to j in equation (2.15) is justified. However, even when our equations are non-relativistic in the sense that U / c << 1, ao/c << 1, when I/ is not much less then c equations (2.4), (2.11), and (2.15) become

(2.4‘) (2.11’) (2.15‘)

F=pV@+j x B+6E, curlH

=j

+ aD/at,

j=6u+a[E+uxB].

The neglect of terms of order ( V / C )in~ equations (2.4’), (2.11’) and (2.15’) is a convenient way of filtering out of a mathematical theory terms representing electromagnetic waves. This is a useful procedure when an insight into other effects is being sought, and has been used in most of the problems dealt with below. In others, where the limit of very high magnetic field strength is of physical interest (see Sec. V.3) terms of order ( V / C )and ~ higher are retained. Plasma oscillations are automatically excluded by taking Ohm’s law (equation 2.15’) to relate the current to the electric field [13]. Because E does not appear in the expression (2.4) for the body force, some authors have erroneously ignored 6 in (2.14). This procedure misses an essential point, that because of the magnitude of E (see (2.10))only very low charge densities are needed to set up quite strong electric fields, and although there are systems having geometries such that 6 is zero everywhere there are others in which electrostatic charges play an essential rale.

224

RAYMOND HIDE AND PAUL H. ROBERTS

In addition to supposing that relativistic effects can be ignored, we have assumed implicitly that the fluid can be regarded as a continuum with isotropic transport coefficients Y, K and a. This is a valid procedure in sufficiently dense media, but in the case of a tenuous fluid - and much current work in the subject is concerned with low density plasmas continuum theory breaks down. It would take us too far away from our main topic even to summarize how criteria for the validity of continuum theory can be deduced by considering the forces on individual atomic particles and thus formulating equations which tend to our continuum equations in appropriate limits. A number of writers have considered this problem, and we refer here to the original papers [13, 14, 15, 161.

111. ELECTROMAGNETIC A N D MECHANICALEFFECTS ; DIMENSIONLESS PARAMETERS 1. Electromagnetic Effects

Operate on (2.15) with curl, make use of (2.11, 2.12, 2.13) and the vector identity curl (curl a) = grad (div a) - V2a, and thus find that

aB

- = curl (u x

(3.1)

at

B) + APB,

where

A

(34

3

(puo)-l m2 sec-l.

According to (3.1) the rate of change of B depends on two agencies, namely, motional induction, represented by curl (u x B), and ohmic dissipation due to electrical resistance, represented by the term AV2B (cf. (2.15)). The magnetic Reynolds number (see Ch. I, (1.1)) is (3.3)

R

= ayLU = LU/A-

(curl (u x B)(/(APB(.

This number is so called because it is analogous to the ordinary Reynolds number (3.4)

R'

EZ

UL/v

which measures the ratio of inertial force p(u * V)u to viscous force p v P u in the equation of motion (2.2). Indeed, if F is irrotational, the vorticity vector (3.5)

w

= curl u

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

225

satisfies the equation (3.6)

ao

- = curl (u

at

+

x o) v P o ,

(if the fluid is incompressible) which is mathematically identical with (3.1) and has an analogous physical interpretation. To illustrate the nature of the solution of (3.1) consider two extreme cases, R = 0 and R = 0 0 . In the first of these, for which (3.1) becomes

i3B at

-- - APB,

(3.7)

motional induction effects are absent. (3.7) is the well known diffusion equation, acceptable solutions of which represent a decaying magnetic field, with time constant t = L2/A. The circuit equivalent of such a system is an inductance ( I ) and resistance ( Y ) in series, the time constant of which is (I/Y)seconds. Observe that R is equal to t / ( L / U ) . Small R is the case when the motion is so slow that free decay of currents due to ohmic losses prevents any significant change in the magnetic field due to the fluid flow. In the other case R = 0 0 , there is no ohmic dissipation and thus

aB_ _ curl (u x B) = 0. at

This equation should be compared with that satisfied by the vorticity vector, w in a homogeneous inviscid fluid (in the absence of a magnetic field) namely (3.9)

a o_ _ curl (u x o)= 0 at

(see equation (3.6)). Eq. (3.9) is the basis of Helmholtz’s celebrated theorem in classical hydrodynamics [17], which states that in a homogeneous inviscid fluid the circulation around any circuit comprising material particles cannot change, and in consequence, the vortex lines move with the fluid. Equation (3.8) shows that the magnetic flux through a region bounded by any material circuit cannot change, and in consequence the lines of magnetic force move with the fluid [l]. The concept of magnetic lines of force being “frozen” into the fluid when R is infinite and, as we shall see when we consider mechanical effects, acting as “elastic strings”, leads to a valuable insight into hydromagnetic phenomena at high magnetic Reynolds number. Even when the conductivity is not perfect, this concept is still qualitatively valuable provided R >> 1. There will now be relative motion between fluid particles and magnetic lines of force but this will be slight,

226

RAYMOND HIDE AND PAUL H. ROBERTS

of order ulR [ 5 ] . By analogy, in high Reynolds number flow (R’ i> 1) satisfying (3.6), the order of magnitude of the speed with which fluid particles can slip across vortex lines is ulR‘.* Now we can be a little more precise about what is meant by an “inviscid fluid” and a “perfect conductor”, idealisations which we often make when effects of damping are of secondary interest. Respectively, these cases require that both R‘ and H shouldapproach infinity. However, in some such problems, it is important to consider the possibility that the final solution will depend on a further parameter, the ratio (3.10)

RIR’

= VIA.

For example, as is shown in Sec. IV.5, the boundary conditions to be imposed may depend on this quantity. The Prandtl number VIK[18], where K is the thermal diffusivity, resembles the parameter VIA in form and these dimensionless quantities have some analogous properties. v / K is the ratio between the Piclet number ULIK, [18], and the Reynolds number, ULIr. ULIK is a measure of the magnitude of the ratio of the convection term to the conduction term in the heat flow equation (2.9). In problems of forced convection, the ratio of the natural length scales of the velocity and temperature fields is determined by the value of VIK. In Table 1 are listed some typical values of o,A and t = L2/A together with that value U , of U for which R = 1. At higher values of U than this, motional induction effects are strong. While in cosrnical systems typical flow speeds are much in excess of U,, in only one of the laboratory examples cited (the last one) would it be practical for U to exceed U,. In many idealized problems the magnetic field B can be regarded as consisting of two parts, B, and b (say). B, is the inducing field which may be due to currents flowing in external circuits or to currents within the system; b is the induced field resulting from inductive interaction between u and R. When B, is due to currents within the system, if these are freely decaying with time constant to = L2/1, in dealing with the theory of b the concept of perfect conductivity is a valid approximation only when the time scale zl associated with b is much less than to. If tl is the time taken for AlfvCn waves (see Sec. 111.2 and Ch. 17, Eq. (5.1))to traverse a distance L , then the requirement that z, >> tl is equivalent to the requirement that the Lundquist number (3.16) should be very large (see below). In such problems, in which the effects of the free decay of B, are not of primary interest, this assumption of perfect conductivity is very useful because it automatically filters out these effects and thereby reduces mathematical complexity. It is clear,

*

For further discussion of the motion of lines of force, see [13].

TABLE1

Temperature T (OK)

Kinematic Viscosity v (m2s-l)

d

4 x 103

Magnetic Variable Stars

lo9

Solar Corona

10s

log

Inter-planetary Gas

10"

105

Interstellar Gas

10'6

104

E: 2

I

m

v

m tm

I

N

%

X

3

v

w

2 2 2 2 2 2

%

X

3

1 06

10-2

104 f

10'6

2

1 0'6 3

1017

For Ionized Hydrogen: 432

02

1

104

106 ?

10'2

0-6

102

10-4

10

10'0

0-4

10

10-1

103

1013

104

1

103 ?

108

10'8

10-9

l?

10-15

10-'6?

10'8

10-9 ?

10 ?

10-20

10-15 ?

lo21

10-10 ?

103 ?

10-21

10-14 ?

1029

10-13 ?

v

x 10W7 newton ampp2

o(

p-1T5/2,

1

cc

T-Y2

u~copper = 5 x 107 51-1 m-1

227

po =

10-3

n.

107

1

2

Sunspots

10

10-7

2

6 x 103

10

n.

log

1

3

Solar Granulation

10-1

n.

10-6 ?

105

3

104

103

P 2

log

10-1

3

Earth's Liquid Core

10

I

10

10-2

2 2

105

10-

107

3

10-1

Time L z / A (set)

104

1

3

Ionized Hydrogen

10-7

373

Decay

wv

= (pr7-1

I-

10-1

0,

Liquid Sodium

G.1

10-1

2

293

Mercury

Density

p (kg m-?

Fluid velocity corresponding to magnetic Reynolds number equal to unity U , (m sec-l)

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

Length Scale L (m)

E1ectr oMagnetic Viscosity 1 (m2s-l)

228

RAYMOND HIDE A N D PAUL H. ROBERTS

however, that a complete picture cannot be obtained in this way. For example, the theory of the setting up of B, in the first instance must take into account departures from infinite conductivity, since a perfect conductor completely excludes magnetic fields from its interior. When B, is due to currents flowing in external circuits it is presumed that these circuits are energized by some source (battery, dynamo, etc.). Within the region of fluid under consideration, there is no current associated with B, so that free decay of B, due to ohmic dissipation within the fluid does not arise. However, there is the possibility of interaction between the induced currents in the fluid with those in the external circuits, and this gives rise to energy exchange between the two systems. This should be borne in mind when dealing with energetics (see Appendix A). I n using (A.10) to obtain an energy balance equation for the whole system, the region of integration must include the external circuits.

Cowling’s Theorem. A simple but useful result follows immediately from equations (2.11) to (2.15). According to the last of these, a t any neutral point of B (i.e. where B = 0 ) ,j = aE.If the system possesses a closed line on which B = 0, then, if the component of j along this line always has the same sense, the line integral of E around this line is not zero. Hence the system cannot be steady; in fact lines of force collapse on the neutral line a t a rate which is independent of u. This result was first given by Cowling [19]. One system to which Cowling applied his theorem is the mechanism proposed by Larmor [20] to account for a centred axial dipole field external to the sun in terms of axisymmetric meridional motions in the solar interior. Because the magnetic field lines lie in meridian planes, the system possesses a t least one neutral line within the fluid, and this line will, of course, be a circle about the axis of symmetry. Moreover by (2.11) j is azimuthal everywhere and has a constant non-zero component round the neutral line. Thus, by Cowling’s theorem, Larmor’s mechanism cannot account for a sleady magnetic field. Therefore, in the search for self-exciting dynamos, it has been necessary to invoke more complicated forms of u and B (see for example, 1211). 2. Mechanical Effects

Having examined the extent to which the motion u modifies the magnetic field B, we now consider the extent to which B modifies u. Even when there is strong coupling between u and B (i.e. when R >> 1) mechanical effects will not be noticeable if the field is too weak. A useful measure of the relative magnetic field strength is S, where (3.11)

s2 = (+B2/p)l(4pU2)= B2IppU2,

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

229

which is the ratio of the magnetic energy to the kinetic energy of ordered mass motion of the system, if B is a typical field strength. Another parameter, (3.12)

where 2, is the average pressure in the fluid, measures the magnetic energy in terms of the thermal energy (i.e. kinetic energy of random molecular motion). Note that for a perfect gas, (2sz/y.S2)1/2 is equal to U divided by the speed of sound, which is, of course, the Mach number. is usually called “beta” by plasma physicists and S-2 is sometimes called “dynamic beta”. In the case of an incompressible fluid (for which the Mach number is zero) vanishes and is therefore a redundant parameter. In order to classify hydromagnetic phenomena in incompressible fluids we distinguish two general cases, R << 1 and R >> 1, corresponding, respectively, to weak and strong coupling between u and B. Because in the weak coupling case the externally impressed field B, is hardly affected by u, we can take B = B,. Except in the cases of certain especially simple flow fields and geometrical configurations of the boundaries, when E may practically annul (u x B) (see (2.15)),

s-2

s

j

(3.13)

N

a(u x B,).

Hence, the magnetic term in the body force (see (2.4)) (3.14)

This force retards the motion and degrades the kinetic energy through the agency of ohmic heating. If S is such that the so-called Hartmann number (3.15)

M

B , L ( U / ~ Y=) S(RR’)l12 ~/~

is much greater than unity, the dissipative force represented by (3.14) is more powerful than viscous friction ; otherwise viscosity is largely responsible for energy dissipation. In the field of a powerful electromagnet, waves on the surface of mercury are attenuated a t a rate much in excess of that due to viscosity acting alone and this effect is readily demonstrated. Some insight into the subtle interplay between viscous and magnetic forces is to be gained from analysing hydromagnetic flow between parallel planes as we shall see when this problem is discussed in Ch. IX. In contrast to the case of weak coupling ( R << 1) when the mechanical effect of B on u is essentially dissipative, when the coupling is strong ( R >> 1) the fluid effectively takes on “elastic” properties, the importance of which depends on S. When S is small these “elastic” forces are insignificant. (However, in the case of turbulent flow S may not remain small indefinitely because in general the lines of magnetic force will increase in length as they move with the fluid [22]. As a result B will increase a t the expense of U

230

RAYMOND HIDE AND PAUL H. ROBERTS

until a state of equipartition has been attained, and oscillations about this equilibrium state may occur.) When S is large, however, the magnetic field dominates. Because of the elastic properties of the magnetic lines of force oscillations can occur in which the inertia is provided by the fluid and the restoring force by the field. These oscillations give rise t o hydromagnetic waves which travel with velocity V = B/(,up)'12 (see Sec. V.2), which were first discovered theoretically by AlfvCn [23]. When R >> 1, the Lundquist number,

K

(3.16)

aLB(p/p)'12 = S R ,

which is a magnetic Reynolds number based on the AlfvCn speed, is the appropriate parameter determining the degree of mechanical coupling between the field and the motion. 3. Two-dimensional Theorem: Analogy with the Proudman-Taylor Result for Rotating Fluids In view of the number of papers in the recent literature which deal with nearly uniform flow in a nearly uniform magnetic field [24, 25, 261, it is instructive to consider whether any general statements can be made about b where B, is supposed to be a uniform magnetic such flows. Let B = B, field parallel to the z axis, and b << B,; and let u = U, ul, where U, is a uniform flow in the direction of unit vector 8 (say), and u1 << U,. For a steady system, t o first order of smallness (2.11) and (2.15) lead to

+

(3.17)

+

U,x b - A c u r l b - B , x

ul=-EE,,

+

the electric field E having been taken as E, E, where E, = - ITo x €3., On taking the curl of (3.17) and using the fact that for a steady system curl E, = 0 by (2.12), we find that (3.18) The equation governing the steady flow of an incompressible fluid may be written (see (2.2)) (3.19)

(U

*

P)u= - V I + Y P ~+U (pp)-l(B * V)B,

where (3.20)

=PIP

+ B212tLp.

Applying (3.19) t o nearly uniform flow in the presence of a nearly uniform field, to first order of smallness we have

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

23 1

(3.21)

where (3.22)

171 = P l l f

+ Bob*lPup.

Equation (3.21) may be rewritten in the form (3.23)

curl

[U, x u1 - Y curl u1 - (pp)-lB, x b] = ITl.

Now eliminate ITl from (3.21) by operating on it with curl and find (3.24)

(pp)-lB0a(cur1b)/az = p-lB0

= U, awlas - Y

P ~ ,

where j = p-l curl b (see (2.11)) and w = curl u1 is the vorticity (see (3.9.) and (4.34)). Finally, by eliminating one or the other of the two variables w and j between (3.18)and (3.24),it follows that both variables satisfy the differential equation

Written in this form, it is easy to see that when S2>> 1 and S2RR' >> 1 (see (3.3), (3.4), (3.11)),the right-hand side of (3.25) can be ignored, and to a first approximation (3.26)

a2

- (w,j) = 0. a22

S2>> 1 implies that the AlfvQ speed is very much greater than U,. S2RR'>> 1 is equivalent to requiring that the Hartmann number (3.15) should be much greater than unity; this is a weaker requirement than that of high conductivity ( R >> 1) and low viscosity (R' >> 1). Equations (3.25) and (3.26) are independent of the form of El and n1,. When I7n1= 0, that is, when the total pressure, fil B,b,/p, is constant, but El # 0, it can be shown that

+

(3.27)

a more restrictive condition than (3.26), which it replaces. When, in addition to VITl = 0, we require that El = 0, (3.28)

232

RAYMOND HIDE A N D PAUL H. ROBERTS

where I and I I designate, respectively, components perpendicular and # 0 and parallel to the direction of the magnetic field. Finally, when El = 0, it may be shown that

onl

(3.29) The foregoing results are reminiscent of the so-called Proudman-Taylor theorem governing slow, steady hydrodynamical flow of an inviscid, homogeneous, uniformly rotating (non-conducting) fluid [ 2 7 ] . This flow is twodimensional, having no variation in the direction of SZ, the basic rotation vector, Coriolis forces being the dynamical constraints operating in this case. If u is the flow velocity relative to a uniformly rotating frame of reference, (2.2) still holds provided 2pSZ x II is added t o the left-hand side, and centrifugal effects are included in F (see (2.5)). On taking the curl of the resulting equation, if SZ = (0, 0, Q), then, remembering that p is assumed unifom and, since u = 0, j = 0, (3.30)

2 Q a ~ I a z= vP(cur1 u )

+ curl (u x curl u ) ,

and in the limit of small viscosity (more precisely when the Ekman number (v/2RLz)”2 is very small) and slow relative flow (small Rossby number, UILR), where U is a typical flow speed and L a characteristic length, we have the result (3.31)

aulaz = 0.

Some writers have erroneously concluded that there is an exact parallel between the hydromagnetic case and the rotating fluids case. According to (3.26) and (3.31) this is not so. Although the Proudman-Taylor theorem has been amply verified by experiment, and work on the dynamics of rotating fluids now forms a large and fascinating chapter of hydrodynamics, the experimental verification of the hydromagnetic two-dimensional theorem has not been given. It is instructive to consider flows which satisfy the condition aulaz = 0. In the case of the flow caused by the uniform motion in the z direction of a solid object immersed in a fluid of indefinite extent, a whole column of fluid extending from the object to infinity in both upstream and downstream directions partakes of the motion of the object and in consequence, the total energy of the flow is infinite. In the absence of the constraint that au1a.z = 0 the total energy is finite (e.g. potential flow). Hence, while the latter flow can be set up from rest in a finite time by the application of finite forces, the former cannot. This can be important when one considers the mathematical uniqueness of solutions of steady state problems, and care has to be exercised in the interpretation of such solutions. As Stewartson [28]

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

233

has emphasized the solution, if unique, should be the limit as t+ bo and v + 0 (and/or A + 0 ) of some time dependent solution for a real fluid (v # 0, A # 0 ) irrespective of the order in which one proceeds to these limits. The danger of such pitfalls can be reduced by working not with t , v and A, but with the appropriate dimensionless parameters which measure their values (see above).

IV. BOUNDARY CONDITIONS* 1. Methods of Deriving Boundary Conditions

In many situations, two media ( M , and M,) of almost uniform composition are separated by a relatively thin layer Y (of thickness 1, say) in which there is a rapid and continuous transition between the two states. Also, some of the physical variables (such as, for example, the normal component D, of electric displacement D) change rapidly in 9.I t is clear that, rather than treat 9as being of finite thickness, it would be theoretically simpler to treat it as an abrupt interface S of zero thickness. However the basic differential equations break down on such an interface since some quantities (e.g. 6) are unbounded and likewise some derivatives ( e g the normal derivative of D,) do not exist. Nevertheless it is necessary to establish relations between the fields on either side of S before the problem can be solved uniquely. These are called “boundary conditions”. The form they take depends on the relative magnitude of 1 compared with other length scales of the system. In a viscous fluid, 1 must be small compared with the boundarylayer thickness, but, if viscosity is ignored, the boundary layer must be thought of as being contained in Y no matter how small 1 is. Similarly in a fluid of finite conductivity, 1 must be small compared with the thickness of the electromagnetic boundary layer (in which the eddy currents flow) but, if resistivity is ignored, the boundary layer must be thought of as being contained in Y no matter how small 1 is. The boundary conditionsmay be derived in one of two equivalent ways. Either the basic differential equations are integrated across Y before the limit 1+ 0 is taken, or the integral equations (from which the differential forms were originally derived) are used. We will adopt the latter approach. Displacement currents are retained in the first instance, but are neglected subsequently in taking the hydromagnetic approximations. We use the following notation. We will define (locally Cartesian) coordinates x+, on S and locate a point Q near S by its shortest distance x, to S Owing to the length of this section, in which it was felt desirable to discuss boundary conditions in more detail than is customary, the principal results are given in boxed equations.

234

RAYMOND HIDE AND PAUL H . ROBERTS

and the coordinates xq,x, of the foot of the normal from Q to S. We take the normal vector n = n(x,,xs) to be directed from M , into MI and the coordinates x,,xq,x, to be right-handed in that (alphabetical) order. It is convenient to introduce an abbreviation DIV A defined by DIV A = div A - [(n * grad) A] * n, (4.1) aAq/axq aA,/ax,. Consider a point P of S: let it be xq = 0, x , = 0 for convenience. The integrations we will perform are of two types: (i) Integrations over the interior dV and surface dS of a “penny-shaped” disk. The radius Y of the penny is small compared to the radii of curvature of S at P and small compared to all the physical length scales except,

+

T” / MI

/

Mz

FIG1.

FIG.2.

+

possibly, the boundary layer thicknesses. Its thickness I, 1, is small compared to Y . Its top surface lies in x, = 1, and its bottom surface in x, = - I, initially. Since the disk is considered to be fixed in space and the boundary may be moving (with velocity U , in the direction of its normal, say), a t a later time dt, the top surface lies in x, = 1, - U,&, and the bottom surface in x, = - I , - U,dt (see Fig. 1). (ii) Integrations over the surface d S and round the perimeter dr of a rectangle. The plane of the rectangle will be taken to be either x, = 0 or xq = 0 and, in the former case, the rectangle is defined by its intersections with x, = I,, xq = 0, x, = - I, and xq = M(>> 1, 12) (see Fig. 2). Again M is small compared with the radii of curvature and all physical length scales except, possibly, the boundary layer thickness.

+

2. The Electromagnetic Boundary Conditions (a) Normal component of B.

Apply the equation (cf. (2.13)) (44

I

Beds =0

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

236

to the penny-shaped disk. We find

(4.3)

&'[B,(') - B,(')]

+

5

B * dS

+ O(73) = 0.

b

Here

(4.4)

B,(l) = limit B,,(Zl,O,O),

B,(')

= limit

Zl+O

B,,(- Z,,O,O)

l,+O

and b denotes the curved surface of the disk. Now B is everywhere bounded; thus the integral over b can be made arbitrarily small compared with the remaining terms of (4.3)by choosing I , and I, sufficiently small. Hence (4.3) gives

(The left-hand side is an abbreviation for B,,(') - BL').) (b) Normal component of D. Apply the equation (cf. (2.14))

I

Dads

5

=

6dV

to the penny-shaped disk. The argument is essentially as in case (a), and gives

(4.7) where

7 = limit dV+O

~

'I

6dV

7C7'

is the surface charge density on S: dimensions coulomb/m2. (c) Normal component of j. From considerations of conservation of electric charge, we find

(4.9)

5

at

In applying this to the penny-shaped disk, we must recognize that (in the relativistic case) currents arising from convection of charge and (in the zeroresistivity case) eddy currents will flow in an infinitely thin layer a t the

236

RAYMOND HIDE AND PAUL H. ROBERTS

interface. Denoting this surface current by J = J(x,,x,) we can easily show that j * dS

(4.10)

=

- m2DIV J + O ( y 8 ) .

b

Thus, by (4.9), we have (4.11)

The first term on the right-hand side is negligible if displacement currents are ignored; the second if eddy currents are ignored. (d) Tangential components of E and H. Apply the equation (cf. (2.12)) (4.12)

to an elementary rectangle of the type described in the first section. If the rectangle lies in the plane x, = 0, we find that the value of the left-hand side of (4.12) is

Hence, since the right-hand side of (4.12) only differs from M[E,I12 by a second-order quantity, we have

I [E,]12 = U"[BSI,2. I

(4.14)

Similarly

I [ESl12 = - U,[B,]12.1

(4.15)

According to (4.14) and (4.15) the tangential components of E, measured in a frame of reference which is locally moving with S, are continuous. Similarly, from the equation (cf. (2.11')), (4.16)

$

1

H . dr = (j + aD/at) dS

ELEMENTARY PROBLEMS I N MAGNETO-HYDRODYNAMICS

237

we find (4.17)

I [Hq]12=

If p and

E

(4.18) (4.19)

Js

-

Ufi[Os]i2,

[Hs]12 =

- Jq

+

un

[Oq]i2.]

are continuous across S, (4.14), (4.15) and (4.17) require that (1 - un2/c2)[Es]12= - PunJs,

(1 - Ufi2/c2)[Eq]i2 = - PUnJq,

(1 - un2/c2)[Hs1i2= - J q .

(1 - Un2/c2) [Hq]I2= J s ,

If displacement currents are neglected, (4.18) and (4.19) reduce to (4.20) (4.21)

[Hs1i2 = - Jq,

[Hq1I2 = J s t

[Eq1I2= - P u n J q ,

[Es1i2 = - PUUnJs.

Equation (4.20) may be combined with (4.5) to give (4.22)

[HIl2 = J x n.

This completes the set of electromagnetic boundary conditions. They are still valid if differentiated with respect to xq or x s or differentiated with respect to t , following the motion of the boundary. They are therefore not independent. For example, if we differentiate (4.7) with respect to t following the motion of the boundary, or if we differentiate (4.6) with respect to t before applying it to the disk, we find (4.23)

-

U,, [DIV DIl2 = a7 - a

at

Differentiating the first of (4.17) with respect to xs and the second with respect to xq and subtracting, we find (4.24)

[(curlH),,Il2 = - DIV J

+ U,, [DIV DIl2.

By subtracting (4.23) and (4.24), and using (2.11’), we recover (4.11). Similarly, from (4.5) we find (4.25)

and from (4.14) and (4.15) we find, by differentiating with respect to x, and xq respectively, (4.26)

[(curl E)n]i2

=-

U,, [DIV BIl2.

Thus, using (2.12), we see that (4.25) and (4.26) are equivalent.

238

RAYMOND HIDE AND PAUL H. ROBERTS

In the zero resistivity case, (4.7) and (4.17) do not restrict the solutions in M , and M , : they merely serve to determine q and J. Also, since E = - u x B in this case, the two conditions which do restrict the solutions, namely (4.14) and (4.15), may be written: (4.27)

[(un -

Un)BqIi2= Bn[uq]i2,

[(un

- Un)Bsl,’ = Bn[21s112*

In the finite resistivity case, no eddy currents flow in 9 and the only surface current is that due to displacement of surface charge q, i.e. (4.28)

J

= +q[u(’)

+ u ( ~ -) ] &qn(n [u(’)+ u ( ~ ) ] ) . *

If displacement currents are ignored, J is negligibly small. Otherwise (4.28) determines J and, by (4.14), (4.15) and (4.17), four restrictive boundary conditions. 3. The Mechanical Boundary Conditions

(a) Normal component of u. Apply the equation (cf. 2. 1) (4.29)

at

to the penny-shaped disk. Assuming that p remains bounded everywhere within the disk (i.e. excluding fictitious mass surface densities), the righthand side of (4.29) may be written

and (4.29) gives

i.e. (4.32)

There are two main possibilities: either S is the contact surface between two “immiscible” media, i.e. media unrelated physically or chemically, or M , and M , are composed of the “same” fluid in two different thermodynamic or

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

239

chemical states, the fluid particles crossing S from one to the other ( e g S is a shock front or detonation front). In the former case, (4.32)simplifies to give (4.33)

(b) Tangential components of u. If we define the vorticity by (4.34)

w

= curl u,

we have, by Stokes' theorem, (4.35)

If we apply (4.35) to the rectangular circuits of Sec. IV.1 we find (4.36)

[21J12 =

w,,

[U5I12 =

-

w,,

0=

w,

Here W is the surface vorticity, dimensions m.sec-l. Its rBle may be clarified if we compare the viscous and electromagnetic boundary layers as v - 0 and A + 0. . If v # O ( A # 0 ) the viscous (electromagnetic) boundary layer is of finite thickness, but may be the seat of tangential shears (currents) which are so large that the vorticity (current density) integrated across the boundary layer does not vanish but tends to a limit W(J) as v + O ( A --+ 0). Then, in the case v = O ( A = 0 ) ,as we have seen, the viscous (electromagnetic) boundary layer is entirely contained within a, our rectangular path of integration shown in Figure 2. x contains a finite integrated vorticity W (current density J) no matter how small 1 may be. If however v # O ( A # O),a contains none of the viscous (electromagnetic) boundary layer as the limit I -,0 is taken, and the rectangular path of integration contains zero integrated vorticity (current) and we must, therefore set W = O(J = 0 ) , i.e. in this case we have (4.37)

(c) Normal components of the stress tensor. Let p:;"' be the total stress tensor in the fluid, that is the sum of the mechanical stress tensor and the electromagnetic stress tensor p,",:"'. The equation of conservation of momentum (cf. (2.2), (2.4), and Appendix A) may be written

pyh

(4.38)

Here EE x B is the momentum of the electromagnetic field and is negligible when displacement currents are negligible, as is the electrostatic part of Pfj"'. Apply (4.38) in the integrated form

240

RAYMOND HIDE AND PAUL H. ROBERTS

5

(4.39)

fit?t"'& *7

7

"5

- - - [PU at

to the penny-shaped disk.

+ E ( E x B ) ] d V-

I

pUiUjdSj

It is easily shown that

(4.40)

a5 z5 at

at

2 at

1

[pu

+ EE x B],dV

[pu

+ EE x B],dV = - nrzU, [

= - nrzU, [

+

p ~ , ,] ~nr2U, ~ [sE,B, - EE,B,],~ 0 ( r 3 ) ,

+

p ~- ~nr2U,B, ] ~ [~ E E , ] , ~

+

nrzU, [ E E , B ~ I , ~W 3 ) , [PU

+ EE x B],dV = - nrzU, [

+

p ~ ~~ G] Y ~~ U~ , B , [ E -E ~ ] ~ ~

+

. I

nr2U,[~E,Bq]12 O(r3). It therefore follows that

+ U,[EE,Bs - EEsB,11', - Un)11' + UnBn [~Esll'[~EnBsll', - [pus(u, - Un)]i' - Un&[~Eq11' + Un[~E,Bq11'.

WF'I1' = - [p%& (4.41)

total Ppnq 11 = total 2 [fins 11 =

-

U,)11'

[puq(Hn

These can easily be expressed in an alternative form which involves $yh, J and 7. We will suppose, for analytical simplicity, that E and ,u are continuous across S. The last term on the right-hand side of the first of (4.41) may be written

+

+

+

+ +Uw(Bs(l)+ Bs"))[~Eql,',

iUn&(EP(l) Eq('))[BSI1' - +U,E(E,(') ES('))[B,],'

(4.42)

- iUn(Bq(l) Bq"))[~Esl,'

which, on using (4.14), (4.15) and (4.17), is equal to

+

MEq(') Eq(2)) [EqI1'

+ M E s ( ' )+ El')) [EsI1' +

241

ELEMENTARY PROBLEMS I N MAGNETO-HYDRODYNAMICS

Similar results hold for the remaining equations of (4.41) and we find ~

~=

+

+

- [pun(un ~ 1 ~ n ) 1 ~1 2 qi(En(1) 2 - [ ~ ~ 9 ( u-nUn)l,2

[f~,",~= li~

+ q&(E9'')+

+ (arx

~ ~ ( 2 ) )

Eq''))

+ ~(z))),,,

+(~(1)

+ (J' x &(B(')+ B'"))9,

[pFhi1z = - [ p u , ( ~ ,- u n ) 1 1 2 + v+(E,(~) + EP)) + (J' x

+

i(~(1)

B(S)))~,

(4.44)

+

where J' = J qU,n = (qU,,J,,J,). In the absence of displacement currents, all these results assume simpler forms: the terms with E and B do not appear in (4.40) and the terms with q do not appear in (4.44). The method we have chosen to derive (4.44) may seem unnecessarily elaborate, but it introduces the mean fields (E(') E('))/2 and (B(') B(2))/2 appearing in the final result in a natural and unforced way. In the case in which M , and M , are inviscid, the mechanical stress tensor is diagonal and, if the fluids are "immiscible", the last two equations of (4.44) give

+

(4.45) B,J

=-

+ U,n

x {(E(l)

x B('))

+

+ (E(2)+ U,,n x B(2))}

If the fluids are perfectly conducting, (4.45) gives (4.46)

+

IBn[J - i q ( ~ ( ' ) u@))

+ qU,,nIl2 = 0.1

If B, is zero, (4.27) and (4.46) are identically true, and the first of (4.44) requires the total pressure P to be continuous across S. If B, is not zero, u is continuous by (4.27) and J = q(u - U,n) by (4.46). I t follows by (4.14), (4.15), (4.17) that B and E are continuous across S, and that therefore q and J are zero. Also, the first of (4.44) requires that the pressure is continuous across S. Summarizing we have: (a) If B, # 0, E, B, u and p are continuous, q and J are zero. These conditions are not independent and are all satisfied by making u and p continuous. (b) If B, = 0, u, and P must be continuous. 4. Small Departures from a Steady State

We are often faced by the question of whether a certain steady state is stable or not. A necessary condition for stability is that the system be stable against infinitesimal perturbations. Thus, denoting by B,, E,, the magnetic field, electric field, . . . in the steady system, we examine b, E = E, e, . . . . If the amplitudes of solutions of the form B = B, these perturbations grow in time, we conclude that the system is unstable.

...

+

+

242

RAYMOND HIDE AND PAUL H. ROBERTS

The perturbations will, in general, involve a motion of the interface S separating MI and M,, and, of course, B, E,.. . must satisfy on S the boundary conditions we have derived in the preceding sections. However, it is generally more convenient to apply conditions at So, the position of the interface before the system is perturbed. For this reason, we will briefly discuss the problem of translating the boundary conditions at S to equivalent conditions at So. Erect coordinates xq,xs on So, as before. Let the equation of S be (4.47)

xn = E ( x q , x s J ) *

Let P be the point (t,xq,xs) of S and Po the point (O,xqrxS),of So. The magnetic field at P is, by supposition,

+

B ( P ) = Bo(trXq,Xs)

b(EtXq,%),

(4.48)

to first order. The direction of the unit normal to S at P is given by (4.49)

N

= (1,

- allax,, - aEjax,).

Hence the normal component B N ( P )of B at P is

The unperturbed steady state satisfied the boundary conditions (4.5). Hence Bm(P0) is continuous. Thus, since B N ( P )must be continuous also, we require (4.51)

where we have used double brackets to emphasize that the condition is to be satisfied on So and not on S. To first order, N, Q = (a(/ax,, 1, 0) and S = (@/ax,, 0, 1) form a righthanded triad of vectors at P on s. By (4.14) we see

i.e. to first order (4.63)

ELEMENTARY. PROBLEMS IN MAGNETO-HYDRODYNAMICS

243

On expanding these quantities in terms of their values on So, we find

or

Since E, refers to a steady state, U,, = 0 and so, by (4.14), [ [Eoq]]12= 0. Further, by (2.12), curl E, = 0. Thus (4.55) may be written (4.56) again expressing that the tangential electric field, in a frame moving locally with S, is continuous. A similar result holds for eos. The other boundary conditions of Sections IV.2 and IV.3 may be transformed into conditions on So in exactly the same way. 6. Boundary Conditions at a Solid Insulating Surface

In determining the flow past insulating (or poorly conducting) surfaces a t high Reynolds and magnetic Reynolds numbers, it is often convenient to assume that, except in a thin boundary layer near the insulating surfaces, the viscosity and resistivity are negligible. The problem is then divided into two parts. First, the structure of the boundary layer is determined and the “jump conditions” across it are discovered. Second a solution of the equations governing the flow in the main body of fluid is chosen to satisfy these jump conditions. In the present section we will illustrate the first of these processes (see [28]). Consider the steady flow in a sufficiently small region of the boundary layer to be laminar and in the xq direction, say. Since the boundary layer is thin, we may consider that B and 11 vary much more rapidly with x, than with xq or x,; i.e. ajax, >> a/&,, alax,. We may also assume that B, is approximately constant and u, zero in the layer. I t follows that the basic equations (3.1) and (2.2) reduce to (4.57) (Here x, is measured out of the fluid.) The solution to these is

244

RAYMOND HIDE AND PAUL H. ROBERTS

(4.58)

Bq = A

+ C exp

P'xn/(h)1/2,

(4.59)

where (4.60) (4.59) satisfies the condition that ztq vanishes at the surface of the insulator Also (4.58) and (4.59) show that the thickness of the boundary layer is of the order of ( A I J ) ~ / and ~ / V that , the changes in B, and ug across it are x, = 0.

(4.61)

i.e. [ztg]12

(4.62)

);(

= B, 1 P'

1/2

[B,112

or more generally, (4.63)

When ( 2 1 ~is) large, as it is as a rule (except invery tenuous media, see Table 1) we may often (cf. Ch. XI) replace (4.63) by (4.64)

[n x BIl2 = 0.

However, in the general case it is important to realize that even though the limit R - 00, R'+ ce has been taken in the main body of the fluid, it is still necessary to specify R ' / R = A/v (cf. (3.10)).

V. PLANE WAVES 1 . The General Effect of a Magnetic Field

In the absence of a magnetic field, an ideal fluid cannot transmit shear waves. I t can, however, transmit compressional waves, and these travel with the same velocity in all directions. In the presence of a field, the situation is radically different. We have seen in Ch. I11 that the lines of

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

245

force are “frozen” in a perfectly conducting fluid; that is, particles of the fluid lying on P particular line of force at one time lie on that line of force for all times. Following Faraday and Maxwell we may picture the lines of force as mutually repelling elastic strings, whose tension per unit area of cross-section is B2/2p. By the arguments of Ch. 111, their mass per unit length and per unit area of cross-section is p. Thus, if they are stretched and then released, a transverse wave will travel down them with a velocity of vc(B2/2p)/p] = B / ( 2 p ~ p ) ~Thus, / ~ . in the presence of a field, a conducting fluid can transmit a shear wave in the direction of the field. These waves are often called “AlfvCn waves” after their discoverer AlfvCn [23]. A more precise argument, which makes allowance for the repulsion of neighbouring lines of force, shows that (neglecting displacement currents) their velocity V is actually greater than that derived in the approximate argument above. It is

v = B(pup)-”? The repulsion between neighbouring lines of force has an important effect on the transmission of compressional waves. Since this magnetic force effectively increases the pressure driving sound waves across the field, the velocity of compressional waves travelling across the field is enhanced. Moreover the velocity of compressional waves is no longer the same in all directions. In fact it will be shown that in a direction making an angle 8 with B their velocity is (neglecting displacement currents)

if their amplitude is infinitesimal. (a,,= velocity of sound for zero B ; see Sec. 11.4.) The velocity in the direction of the field is unaffected by the field, since the field does not resist the motion of particles of fluid along the lines of force. There are many accounts of the properties of AlfvCn waves in the nonrelativistic case: we mention a few: [l, 3, 4, 5, 23, 29, 30, 311. The relativistic case is treated in [32] an& the rotating case in [21, 33, 341. The influence of a magnetic field on sound waves is discussed in, among other places, [31, 351. The effects of dissipation upon the propagation are described in some of these references and also in [l, 4, 361. Magneto-hydrodynamic shock waves are studied in [37,38,39]. In Sec. V.2 below we discuss transverse AlfvCn waves and touch briefly on the effect of Coriolis forces and displacement current upon their propagation. In Sec. V.3 the effect of a magnetic field on sound waves in a perfectly conducting inviscid fluid is considered. In Sec. V.4 the effects of finite conductivity and viscosity on the results of Sec. V.3 are discussed, and shock waves travelling perpendicular to the magnetic field are considered.

246

RAYMOND HIDE AND PAUL H . ROBERTS

2. Alfvkn Waves

Consider a perfectly conducting inviscid fluid of infinite spatial extent pervaded by a uniform magnetic field B,. Let a disturbance be generated in this fluid, and let the magnetic field then be B

(5.3)

= B,

+ b.

First suppose that the fluid is incompressible; more precisely, if L and z are a length and a time characteristic of the disturbance, we consider that L / t << a,. It follows that div u

(5.4)

= 0.

(We do not suppose here that IbI is small compared with B,.) Suppose that wave motion exists for which the total pressure is constant:

p

(5.5)

+1

-

B2 = constant.

2P

Then, by (2.2) and (3.1), it may easily be shown that ikl

1

at

PP

- + (us grad)u - - (B * grad)B = 0 , (5.7)

ab at

-

+ (u - grad)B - (B - grad)u = 0.

From the second of these equations we see that ab,/at = 0 , where z is a coordinate measured in the direction of B,. By supposition, B = B, and u = 0 before passage of the wave. Thus, any parts of b and 11 which do not depend on t must be excluded. It follows that

(5.8)

b, = U, = 0.

The equations (5.6), (5.7) governing these transverse waves admit two types of solution (5.9)

b(x,y,z,t)= b(x,y,z - Vt,O),

(5.10)

b(x,y,z,t)= b(x,y,z

+ Vt,O),

u = Vb/Bo, u = - Vb/B,,

although they do not admit a linear combination of these solutions except in the case of waves of infinitesimal amplitude. Solutions (5.0) and (5.10) correspond to waves travelling parallel and antiparallel to the field B,, respectively. Even in the compressible case (when L / t << a, does not hold), the same results can be established for small perturbations, i.e. when u<< V and

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

247

b << B,. For then, if we suppose that solutions satisfying (5.4) exist, p is constant to first order, and, taking the divergence of the equations of motion, we find that the Laplacian of the total pressure vanishes everywhere, i.e. (5.5) must hold and by (5.8),R2 and RO2are equal to first order, and therefore, by (5.5),the kinetic pressure is constant to first order as, therefore, is p. This justifies the original supposition that solutions satisfying (5.4) exist. Similar solutions can be derived even if displacement currents are included in (2.11) and the space charge contribution 6E to the body force F is included in (2.4). Then it may be shown that if, in place of (5.5) we assume

p

(5.11)

1 1 +B2- -&E2= constant, 2P 2

the equation of motion may be written = - grad(&E2)+6E-

(5.12)

where E = - u x B is the electric field and 6=divD=-&div(u

(5.13)

x B)

is the charge density. Again ab,/at is zero and in consequence (5.8) follows. Assuming that solutions of the form b = xu exist, we find that the righthand side of (5.12) may be written as

:[

- &BO2-

(5.14)

+ (u

*

gradju]

.

Thus (5.12) can be written (5.15)

(1

+ g)[$ + (u - grad)u

I(,",

)

- - grad B = 0,

and this has the same form as (5.6) if we choose (5.16)

x = f (pp)'i2(1 + V2/c2)1/2.

Thus we have two sets of exact solutions (5.17)

b(x,y,z,t)= b(x,y,z - Ct,O),

(5.18)

b(x,y,z,t)= b(x,y,z

where (5.19)

+ Cl,O),

u = Cb/B,,

u = - Cb/B,,

248

RAYMOND HIDE A N D PAUL H. ROBERTS

The division of energy between the electric field energy B = &E2/2,the magnetic field energy 2? = B2/2p and the kinetic energy density 9 = pu2/2 is determined by (5.20)

2?=.9++,

(5.21)

B

= (C/C)29?.

In the relativistic case (V-• w ) , this gives the familiar equipartition of energy between the magnetic and electric fields in electromagnetic waves which, in fact, (5.17) and (5.18) become in this case. In the non-relativistic case (V/c<( l ) , (5.20) shows that the energy is equally divided between the magnetic and velocity fields. The waves are unattenuated and non-dispersive. In the presence of rotation, this is not the case. In the non-relativistic case, the equation of motion for the rotating system is 1 + (us grad)u -- (B. grad)B + 2 8 x u = P at PP

au

-

(5.22) where 8 is the angular velocity of the system. Suppose GI is parallel to B,. If we assume that all quantities depend only upon t and z (or if we assume that the system is only slightly perturbed from the equilibrium state), we find that transverse waves are governed by ab au -=B-, at O az

(5.23) (5.24)

au _ _ _B,- -ab at pp az

(5.25)

p

+

1 -

2P

2 8 xu,

B2 =constant.

These equations admit solutions in the form of undamped waves (5.26)

u

= (uux,uy,O) exp i(wt - kz),

provided (5.27)

w =

*R&

+ VW)1/2.

(a2

In highly rotating systems (Q >> V k ) , this dispersion relation requires (5.28)

w = f 2R,

0

=

*

1,'%2/2R.

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

249

The first of these corresponds to the usual behaviour in the absence of a field (the semi-diurnal tide in the case of the Earth). The other corresponds to an AlfvCn wave whose phase velocity V2k/2!2 has been greatly reduced (by a factor of VR/2!2) by the rotation. Both waves (5.27) have the same group velocity

+ V2k2)-1/2.

V , =kV2(Q2

(5.29)

The partition of energy between magnetic and kinetic energy for the AlfvCn wave solution is

93 = ( V k / ~ ) ~ 9 .

(5.30)

Thus, in a highly rotating system, the magnetic energy exceeds the kinetic energy by a factor of (2!2/Vk)2. In the case of the Earth’s core, for example, a realistic estimate of V is 40 cm sec-l (taking B = ,045 weber/m2).For this value V , = 5 cm sec-l for k = 2n/3,000 km-l. Also, for the AlfvCn wave solution 93/95 3 lo4. There are strong indications that 99/9>> 1 in the Earth’s core, although %?/9 + 100 might be a more realistic estimate.

-

The modifications necessary in order to include the effects of displacement currents are easily carried out, and lead to identical results except that V is replaced by V ( l V2/c2)-1/2.

+

3. Sound Waves We will assume that the compressional waves are of infinitesimal amplitude and that before the arrival of the wave

u =o, B = B,

= constant,

E =0,

j =0,

6 =0,

p

p = p,,

a = a,,

=

Po,

where a (= I / ( y P / p ) )is the velocity of sound* at density p and a t pressure p in the absence of B,. During the passage of the wave, we will suppose that

+ Here we are regarding a as a variable replacing T.

250

RAYMOND HIDE AND PAUL H. ROBERTS

Henceforth the primes will be omitted, and we will work to first order in b, f ~ p, , a, u, E,j, 6. We may divide u into its irrotational and solenoidal parts (5.33)

u = - grad #

+ curl A,,,

and express B in the form (6.34)

B

= B, f

curl At,.

I t is convenient to divide the solenoidal parts into their toroidal (F)and poloidal (9’)parts by writing* (5.35) (6.36)

where 1, is a unit vector in the direction of B, and Bo = IBol. Using (5.33) - (5.36) together with (5.7) and working to first order, we find (5.37)

Similarly, by (5.6), working to first order, we find

(6.39)

* The notion of toroidal and poloidal vector fields was first introduced by H. LAMB (Proc. Lond. Math. SOC.18, 61-66, (1881)). In essence, Lamb showed that rather than describe a vector field by its three scalar component fields V,, V y and V z , it is sometimes more convenient to represent it by scalar fields #I, T and P related to V,, J’y and V , by P#I= div v;

(P- a y a 9 ) T = - (curl v),;

P(P- a2/a$)P

=

- (curl2 v),

It is then possible to express V in the form V = grad4

+ curl TI, + curl2PI,

Here curl TI, is described as a toroidal field and curl2 PI, as a poloidal field. The curl of a toroidal field curl TI, is the poloidal field curl* TI,, and the curl of a poloidal field curl* PI, is the toroidal field curl (- V*P)l,. Similar expansions are possible, as Lamb showed, in which I, is replaced by r, the radius vector drawn from the origin of coordinates. In this case, the lines of toroidal fields lie entirely on surfaces of constant Irl and, in fact, this led to the terminology “toroidal” by which they are now described.

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

251

(5.40)

Also by the equation of continuity (2.1) we have

aP =poV2c#J.

(5.41)

at

I t is evident from (5.37) to (5.41) that the parts of u and B involving the toroidal fields 9 separate from the remainder involving B and 4. We will not discuss the toroidal parts further as they have been fully treated in the preceding section. From (5.38) to (5.41), it is easily shown that B b , B,,, V2c#J,and p satisfy

+

+ VZ)V2 + ao2v2

at4

(5.42)

If the waves travel in the z-direction, (5.42) separates into (5.43)

In fact, the poloidal and compressional parts of the solution separate: the compressional part corresponds to a sound wave travelling along the z-axis with velocity a,; the poloidal part to a shear wave of the type discussed in Sec. V.2. If the waves are propagated perpendicularly to B, (in the x-direction, say), B,, vanishes and P b , V2c#J,and p satisfy (5.44)

Thus, in this case the compressional waves travel with a velocity (5.45)

+ [-_lu2 + ao2

c

c2

V2 V2

=

[4'+ v2

(1 - Uo~/""~ (1 V2/c2)

+

.

'

this is greater than their velocity a, in the absence of a field. In the strong-field limit V >> c , V >> a,, (5.42) factorizes immediately to give (5.46)

($

- c2v2)($ - a,2-$)B

=o.

252

RAYMOND HIDE AND PAUL H. ROBERTS

I n fact, in this case, the waves are electromagnetic. In the non-relativistic limit c )> V , c >> a,, (5.42) reduces to (5.47)

This corresponds to two waves, whose phase velocities in a direction at any angle 19 to B, are (see Fig. 3)

(5.48)

4 [ao2

+ v2 + 2 a , ~cos

e y 2

g [ao2+ v2 - 2a,v

cos e y z .

For weak fields (V << a,) these velocities reduce to a, and V cos 8.

FIG.3.

Denoting by % the increment in internal energy density during the passage of the wave, it can be shown that the division of energy between internal, electric, magnetic and kinetic forms is such that (5.49)

9+&=3?+%,

(5.50) (5.51)

8 = (C/C)2@,

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

253

where C is the phase velocity of the wave. In the strong-field limit ( V + 00, C- c) 9 and are negligible compared to 8 and B, which become almost equal. In fact, as we have already remarked, the waves are electromagnetic in this case. In the non-relativistic case, the electrical energy is negligible and by (5.49) (5.52)

.9=37+@. 4. The Effect of Dissipation

We consider only the non-relativistic case, and adopt the formalism of the last section, together with Ohm’s law in a form suitable for moving conductors (2.15) (5.53)

j

= a(E

+ u x B),

and the relationship (see Appendix (A. 3) and (A. 4)) (5.54)

[p - p ([ - - v 3” ) divu] 6 i j - p ~(auis.i+a.i prh=

auj)

relating the viscous stresses to the rates of strain. The equations of electromagnetism (2.11) to (2.14) together with (5.53) give, working to first order, (5.55)

(5.56)

Similarly the equation of motion (2.2) gives, to first order, (5.57)

(5.58)

(5.59)

+

where v’ = [ 4v/3. These with the equation of continuity (5.41) complete the equations. Thus, it follows that F,and , Y bobey:

254

RAYMOND H I D E AND PAUL H . ROBERTS

(5.60)

while P,,, Pb, Vz$ and p satisfy (cf. (5.47))

(5.61)

Equation (5.60)is exact for solutions depending only upon 2 and t. Let us consider solutions proportional to exp i(kz - ot). Taking k to be real, we find o is determined from it by

+

(5.62)

w = - 8ik2{(A

Y)

f [(A - Y)' - 4V2k-2]1/2}.

Thus, for example [40],in an insulating box of side 1 (in the direction of B,) the standing waves which decay away most slowly (k = x / l ) have a decay time of

+ + Y) f [(A -

2P/x'(A ={2lznZ{(A

'

Y),

Y

if )~ 4V212n-2]1/2}-1, if

I >nil - 4 / 2 V , 1 < nil - v1/2V.

(6.63)

In the first case the waves die away as an oscillation of continually decreasing amplitude; in the second they die away aperiodically. If we take w to be real, we find k is determined from it by

k

[Vz - io(A1/2

= &i(Av)-1/2{&

+ y U 2 ) 2 ] 1 ( 2 f [v' - io(A1P - ~ 1 / 2 ) 2 ] 1 / 2 } .

(5.64)

For small I and v(i, e. <( V 2 / o )this gives

k

=i Vl ( h ) - ' / 2 [ 1- i W ( 2

k

=

+ Y)/2V'],

or

-+ wV-'[l + iw(A + v ) / 2 V 2 ] .

(5.65)

The first of these possibilities correspond to shear waves damped heavily by viscosity and resistivity. Their decay length, in which the amplitude diminishes by a factor e-l, is (cf. Sec. IV.5) (5.66)

do = (Av)1/2/V.

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

255

The second corresponds to AlfvCn waves. Their decay length is (5.67)

do = 2V3/02(1

+ v).

The result (5.67) may be understood by considering the decay of an AlfvCn wave in a coordinate system moving with it. In this frame, the current system is approximately stationary and its time of decay must be of the order of 12/A, where 1 is the spatial scale of the disturbance. If v exceeds A, the decay time will be governed by viscosity and 12/v would be a better estimate. I t is therefore plausible that, in the general case, the decay time is of the order of 212/(A v). In the present case I = V / w ,so this decay time is 2V2/w2(A v ) . Now, in this time, the wave has travelled a distance of 2V3/w2(A v) which must, therefore, be the decay length do of the wave in a frame fixed in the fluid. Further discussion of the attenuation of shear waves is given in Chs. VI. and X. For the compressional waves governed by (5.61) we will consider damping only in the special case of waves travelling in the x-direction. As in Sec. V.3, the B wave separates from the compressional wave in this case, and the sound wave is governed by

+

(5.68)

+

+

[$-( a o 2 + v f $ ) $ ] [ $ - A 5 ] .

=V2&9’.

Again consider solutions proportional to exp i ( k x - ot). Taking k to be real, we find that, for lightly damped sound waves (Ak,v’k << a, or V ) , w is determined from it by

1

Av’k2

+

w = (ao2 Y2)l/’k

+ . . . ] - i k 2 [v’ + (ao2AV2 + V 2 )+... -

(5.69)

Thus, for example, the standing waves in a box of side 1 which decay most slowly (k = n/l) have a decay time of

+

(5.70)

z = 12/n2[vf

AY2(ao2

+

V2)-1].

If we take o to be real, we find that, in the slightly damped case (Aw,v’w << ao2 or V 2 ) ,k is determined from it by

fk

+ V2)-1/2

= o(ao2

(5.71)

+

AV’d

iw2(a02

...I +

+ V2)- 312 [v’ + (ao2AV2 + V2)

The decay length do of these waves is (5.72)

do = (ao2+ V 2 ) 3 / 2 / W Z [ V ’

+ AV’2(ao2+

V2)-1].

+...I.

256

RAYMOND H I D E AND PAUL H. ROBERTS

5 . Shock Waves Now we examine compressional waves of finite amplitude in a particularly simple case in which the effect of non-linearity can easily be included. We no longer suppose that the medium is perfect but ignore dissociation and ionization processes, i.e. we consider the gas to be completely ionized before the passage of the shock. We also ignore all relaxation effects associated with the vibrational and rotational energy of the molecules of the gas. We consider plane shocks which are propagated in a direction perpendicular to the applied magnetic field. Gas ahead of shock

Temperature density p, pressure p,

TI

Gas behind shock

Shock front (inframein which it is a t rest)

Temperature Ta density pr pressure p2

FIG.4.

Suppose the shock wave moves with a velocity u, in the negative x-direction and that the field into which it moves is in the z-direction and of strength B,. Transform to a frame moving with the shock-wave (see Fig. 4). In this frame

u

(5.73)

= (u,O,O)

B

= (O,O,B),

where (5.74)

u+ul,

B - B,,

p-p,,

p-+pl, T - TI,

as x + -

p-+p,,

p+p,,

as x +

00.

We will suppose that (5.75)

u-u,,

B - B,,

T-+ T,,

+

00

For strong shocks the transition between states (5.74) and (5.75) takes place in a distance of the order of a few mean free paths, and (5.74) and (5.75) hold closely over large ranges of x .

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

257

Since, in the moving frame, the system is stationary, curl E must vanish. This implies, using (2.15), that E , is constant and therefore aB-1 _ - U B - u1B1,

(5.76)

pc a x

since E y - u,B, as x + - 00. As X + W, aB/ax- 0 . Hence

+

u1Bl = U ~ B , .

(5.77)

By the equation of continuity ( 2 . 1 ) , we have a(pu)/ax= 0,

(6.78)

i.e. (5.79)

Pl"1

=P2%

In a perfectly conducting fluid (see Sec. 111. 1) according to (5.76) and (5.78) (5.80)

Blp

= Bl/pl =L constant.

Thus in these circumstances the total pressure may be written (5.81)

It follows that the velocity of waves of infinitesimal amplitude is (5.82)

V(aPlap)s= V(.,2

+ V2),

in agreement with the calculations of Sec. V.3. (The suffix S in (5.82) means constant entropy.) Returning to the general case of non-zero transport coefficients, by (5.77) and (5.79), ~ ' ~ , u ~ , p ~ , p ~are , Brelated , , B , to the strength of the shock q by

(5.83) By the momentum equation (2.2),we have (5.84)

Integration, using (5.83), gives (5.85)

B2.

258

RAYMOND HIDE AND PAUL H. ROBERTS

Thus, (5.86)

Next consider the energy equation (A.18). We have

i.e.

Eliminate next dp/dx from the second term on the left-hand side of (5.88) by using (5.84), whence

(5.89)

Integrating, using (5.83), we find

(5.90)

plul

(u, +

+; q)- ufi2 P *

Hence

(5.91)

Equations (5.83), (6.86) and (5:91) are the necessary generalizations of the well known Rankine-Hugoniot relations. If we introduce the total pressure (5.81) and the total internal energy per unit mass

U* = U f

(5.92)

1

-B2,

2PP

(5.86) and (5.91) become (5.93)

Pl

+PP?

= p2

+

P2"a2,

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

259

P p2 1 u,*+ 2 + 2I u12 = u2*+ + - u22. P1 P2 2

(5.94)

which are formally identical to the corresponding Rankine-Hugoniot results. Solving equations (5.83), (5.86) and (5.91), using the fact that for an ideal gas

U

(5.95)

= W T / ( y- l),

p / p =W T ,

and a2 = Y P / P

(5.96)

we find (5.97)

where V12 = B12/ppl, V22= B22/pp2. We notice that (5.97) may be rewritten

Thus (5.101)

u12

2 a12 + v12

with equality only in the case of infinitesimally weak shocks (11 = 1) i.e. sound waves (cf. (5.45)). Thus the shock velocity is supersonic with respect to the undisturbed gas. Similarly it is easily shown that M~ is subsonic with respect to the gas behind the shock front. Ry (2.7), (5.83), (5.95) and (5.99), it is easily shown that the change in entropy across the shock is

AS

(5.102)

{

W log [{(Y+ 1 (Y- 1)

=-

+ V M Y - 1 ~ -7 1 ) 3 / 2 4 + 1) - T ( Y - 1)1

) -~ (Y- 1))

V"Y

260

For

RAYMOND HIDE AND PAUL H. ROBERTS

= 1,

A S is zero. For 7

>1

(5.103)

>

>

0, i.e. a(AS)/aq 0, with equality only if 7 = 1. It follows that A S with equality only if 7 = 1. The foregoing results relate only the conditions prevailing at 00 and - 00, and they are independent of the viscosity V I P , thermal conductivity K P C ~ ,and electrical conductivity u of the gas. There are three characteristic lengths based on the properties of the undisturbed gas, namely, I = vl’/al, the mean free path between collisions, I‘ = K1cp,/Wyla,, a thermal penetration depth, and I“ = ( ~ ~ , u u ~an ) - ~electromagnetic , penetration depth, which together with the shock strength, 17, determine the character of the transition between the states on either side of the shock. As can be seen directly from (5.76), (5.85) and (5.90), the structure of the shock depends on the relative magnitude of these three lengths. In practice, I’ >> I for an ionized gas. For a highly conducting gas (one for which al,uv, >> l ) , a strong shock has much the same general character whether a field is present or not: the velocity, density, pressure and field, change rapidly in a region whose width is of the order of 1. However, I‘ must govern the spatial variations of the temperature, and preceding this region is a layer (thickness -1’) in which the variables together with the temperature change more gradually. The field and density are closely proportional throughout. For a weakly conducting gas (I” >> I ) , strong shocks have a different character. The velocity, pressure, density and temperature change rapidly in a region whose thickness is of the order of 1’. However, I” must govern the spatial variations of field and preceding this region is a layer (thickness l”), in which these variables, together with the field, change more gradually [38].

+

-

I N SYSTEMSOF FINITE EXTENT VI. A L F V ~ WAVES N

1. Orders of Magnitude

In the case of an isolated body of fluid, dissipation of energy is associated with the attenuation of AlfvCn waves which happen to be present and, as we have seen in Sec, 17.4, this depends on the electrical conductivity and viscosity of the fluid. However, when the body of fluid is in contact with

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

261

solid conductors, currents associated with the AlfvCn waves crossing the fluid will generally penetrate the solid conductor and thus give rise to increased energy dissipation. In examining this effect, we will take v for the fluid to be zero, and we will suppose that the electromagnetic time constant tl = L2/A, associated with the ohmic dissipation within the fluid, is large compared to the time constants t 2= L2/A’, associated with ohmic dissipation within the conducting shell bounding the fluid, and t3= L / V , the time taken by AlfvCn waves to cross the body. Here A’ denotes the magnetic diffusivity of the conducting shell, and L a typical dimension of the fluid body. We may easily derive rough estimates of the order of magnitude of the effect of dissipation in the conducting shell in two extreme cases, depending on whether t 2>> t3,or t 2<< t3. First let t 2>> t3,and consider standing waves of wavelength comparable with L. We may regard these standing waves as the resultant of progressive waves and their reflections in the boundary. Since z2 >> t3,the reflected and incident waves are almost in phase. Thus, the amplitude of the standing wave will be of the same order of magnitude throughout the fluid, including regions near its boundary. Now let B be a typical field strength. At the boundary 3?, the incident waves produce an almost sinusoidal magnetic field of frequency w = 2izV/L and by the customary electrodynamic skindepth arguments, fields and currents will be attenuated exponentially with depth in the conductor, the skin depth being ( A ’ / 2 7 ~ w )= ~ /(~A ’ L / ~ ~ C ~ V ) ~ / ~ . The rate of power dissipation per unit area of B will therefore be ~ ~ , U - ~ B ~ ( A ’ joule/m2 V / L ) ~ ’sec. ~ But the total energy of the incident wave per unit area of B is of order p-lB2L joule/m2. Thus, the time constant of the decay process is approximately ( I U - ~ B ~ L ) / ~ ~ ~ - ~f B~(A’V/ (L3/A’V)1/2 = (tzt3)1/2. Comparing this with zl,we see that dissipation in the solid conducting shell cannot be neglected if z2 << z12/t3.

If t2<< t3,the amplitude of the standing wave is very small near the boundary of the fluid since the incident and reflected waves are nearly in antiphase. In fact, if B denotes a typical field strength far from the boundary, the field strength a t the boundary is approximately (LV/A’)1/2B.(see Sect. VI. 2 below) Thus, while the total energy of the incident wave lying above unit area of the conducting shell is still of order p-lB2L joule/m2, the rate of power dissipation per unit areaof shell is (LV/A’).~ ? C ~ - ~ B ~ ( A ’ V / L ) ~ / joule/m2 sec. Thus, the time constant of the process is approximately (A’L/V3)1/2 = (t33/t2)1/2. Comparing this with zl,we see that dissipation in the solid conducting shell cannot be neglected if z2 >> ~ ~ ~ / t ~ ~ . Observe that, in the present context, the decision as to whether the conducting shell is a “good” or “bad” conductor depends on whether t2>> ts or t 2<< z3,and not upon whether t2>> z1 or z2 << tl.

262

RAYMOND HIDE AND PAUL H. ROBERTS

2. Standing Waves in a Fluid Bounded b y Conducting Planes

We will consider a very simple model of the process discussed above. We will assume that the fluid transmitting the AlfvCn waves is inviscid, incompressible, and lies between two walls z = + a and z = - a. In the regions z > a and z < - a, we suppose that there lies a homogeneous conductor, and that the whole system is immersed in an initially uniform field B, in the z-direction (see Fig. 5 ) . We will examine the properties of standing AlfvCn waves in the fluid, i.e. we suppose that

B

(6.1)

u

= (b,O,B,),

z--a

4

= (u,O,O). z- +a

7

I/

/1

fluid: solid: density-p maqnetic diffurivity -A’ maqnetic diffurivity-h

solid: maqnetic diffurivity -h’

FIG. 5 .

The associated electric field is

E = (O,E,O),

(6.2)

where

(6.3)

E={

(B,u

+ Aab/az)

A’ablaz

in the fluid, in the solid.

I t is readily verified from Sec. 17.4, that in the fluid

and that in the solid

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

Let us suppose that b and we have

for IzI

< a, while by

(6.8) b = C exp [(z

for z

zl

are proportional to en'. Then, by (6.3) to (6.5),

(6.3) and (6.6),

+a ) v m ) ] ,

+a)Vm)],

E = C v m exp [(z

<-a

(6.9) b = C exp [ ( a - z ) V m ] ,

for z

263

> + a.

E

=-

C

Vm exp

[(a - z) V m ] ,

The continuity of b and E requires

(6.10)

Thus, the eigenvalues of n are determined by (6.11)

If the walls have zero conductivity, (6.11) gives (6.12)

2anl V

V(1

If s

+nW2)

= (2s

+ l)ni,

s = 0,1,2..

..

> ( 2 ~ V / n i l ) - ~the ' ~ ,mode is damped aperiodically in a decay time of

(6.13)

-&-

="[

2 v2 1

+

V W + } ] 742s

1)

#

while, if s < (2aV/nil)-1/2, the mode is damped periodically with a time constant of decay of

z = 8a2/ln2(2s+ 1)2.

(6.14)

These results (for s = 0, 2a = I, v = 0 ) agree with (5.63). If the walls have infinite conductivity, (6.11) requires (6.15)

2anlV

V(1

+nW2)

- asni,

s =1,2..

..

264

RAYMOND HIDE AND PAUL H. ROBERTS

If s > 2aV/n1, the mode is damped aperiodically with a decay time of (6.16)

otherwise (if s time of

< 2aV/n1), the

mode is damped periodically with a decay

The limit in which the conductivity of the fluid is infinite and all dissipation takes place in the walls is particularly interesting. Then for both A' = 0 and A' = 0 0 , the waves are not damped at all and there must be some intermediate value for which dissipation is a maximum and for which the maximum decay time of the wave is least. For 1 = 0, (6.11)may be written as coth (vzu/V)= - (Vz/~1')1/2

(6.18)

or as tanh (na/V)= - (~1'/Vz)l/~.

(6.19)

For brevity we shall write (6.20)

an

= V(- x

fiy) = a ( -

x fi y ) / t 3 ,

and (6.21)

= (av/a')l/2= (22/t3)1'2,

(Note:

tl = a2/A, t 2=

$/A',

t3= a/V.)

We will consider only that mode for which x is least. For small x, (6.18) gives by successive approximation

(6.22)

= 0.56419

= 1.57080

x - 0.40528 x2 - 0.11063 x3 - 0.09886 x6 + . . .,

+ 0.56419 x + 0.11063 x3 - 0.06744 x4 - 0.09886 2' + . . . .

(6.23)

For large

x, (6.19) gives by

successive approximation

265

ELEMENTARY PROBLEMS I N MAGNETO-HYDRODYNAMICS

x-' + 0.50000 xF2 - 1.26260x-3 - 1.68961x-' + . . . ,

= 1.25331

(6.24)

= 3.14159 -

1.25331

x-'

+

..

- 1 . 2 6 2 6 0 ~ - ~2 . 0 9 4 4 0 ~ - ~ + 1.68961~-~+ .

(6.25) TABLE2

x

x

Y

x

x

0 0.003 0.01 0.03 0.1 0.3 0.6 1.0 1.3 1.4 1.5 1.55 1.6

0 0.00169 0.00562 0.01674 0.05436 0.15029 0.26082 0.34863 0.37801 0.38255 0.38500 0.38554 0.38566

7~12= 1.57080 1.57249 1.57644 1.58772 1.62723 1.74034 1.90853 2.11898 2.25771 2.29954 2.33908 2.35800 2.37635

1.65 1.7 1.75 1.8 1.9 2 3 4 7 10 30 100 300

0.38541 0.38481 0.38390 0.38271 0.37957 0.37559 0.3 1752 0.26242 0.16512 0.11906 0.04118 0.01248 0.004 17 0

m

Y

2.39415 2.41140 2.4281 1 2.44428 2.47509 2.50394 2.70773 2.81812 2.95984 3.01522 3.09977 3.12906 3.13742 3.14159 = n

Returning to the original non-dimensionless units, we see that the time constant of the standing waves is (6.26)

266

RAYMOND HIDE AND PAUL H. ROBERTS

< < < <

in agreement with the qualitative discussions of Sect. I. For 0 x 3, a Newton’s method based on (6.18) converges rapidly. For 0.3 x bo, a Newton’s method based on (6.19)converges rapidly. The results, computed on the “Pegasus” computer of the computing laboratory of the University of Durham, are presented in the table above, and lox and y are plotted in Fig. 6. I t will be seen that x has a sharp maximum of approximately 0.38567 near x = 1.5916. Thus the longest free decay time t associated with the system is never less than 2 . 5 9 3 ~ ~ . 4r

3-

1-

I-

0.01

0.I

10 .

1 0

I00

1000

x

FIG.6.

In the case of the Earth’s core, we may take.

I‘ I 4.5 x 1

5

m/s (based on an average poloidal field of 5 x lo-‘ weber/m2),

2.6 ma/s (based on u

=3

x lo5mholm,),

1‘ = 7.9 x lo3m2/s (based on u = lo2 mholm.), a = 3.5 x 108m.

Thus x = 4.5 and so x = 0.25, and t I 9.8 years. Had we ignored dissipation in the mantle, we would have found by (6.14) that t = 1.2 x lo6 years. Note also that since x > 1, we must, in the present context, regard the mantle of the Earth as being a good electrical conductor.

ELEMENTARY PROBLEMS I N MAGNETO-HYDRODYNAMICS

267

VII. GRAVITYWAVES: RAYLEIGH-TAYLOR INSTABILITY

1. Introduction: Choice of Model

In general, the surface waves travelling along the interface between two conducting fluids will bend the lines of force of any magnetic field present 151. The reaction of the lines of force to this bending will affect the propagation of the surface waves. In this section, we consider the simplest case of two semi-infinite immiscible, incompressible fluids of densities p1 and p2 separated, in the undisturbed equilibrium state, by an infinite plane horizontal interface. We suppose the fluid of density p2 lies above the fluid of density pl. For simplicity, we ignore viscosity and treat the fluids as perfect electrical conductors. In the absence of a field, there is a discontinuity of tangential fluid velocity a t the interface between the fluids. For example, consider a wave travelling in the (horizontal) x-direction. By continuity of fluid mass, the fluid particles in the troughs are moving in the positive x-direction and those on the crests in the negative x-direction. But the crests in the waves of one fluid are the troughs in the waves of the other. Hence, there is a discontinuity of velocity across the interface between them. Clearly, in the presence of a field which has a non-zero component normal to the interface, this discontinuity of velocity implies the existence of a discontinuity in the tangential components of E,and this in turn implies the existence of a surface current, across which the tangential components of R are discontinuous. But this is impossible since it would imply a discontinuity of tangential stress, giving rise to infinite acceleration. I t follows, therefore, that the perfect fluid approximation is incompatible with gravity wave solutions of the type that arise in the absence of a magnetic field. More precisely, if o is the frequency of the wave and k its horizontal wave number, there are no such solutions for which 1 and are negligible, compared to both V 2 / w and u / k 2 . In a real fluid, there can be no discontinuities of 11 and B, but these quantities change rapidly within a boundary layer separating the two fluids. The relationship between the net change in B across this boundary to the net change in u can only be found by studying the structure of the boundary layer itself. This is found to depend, in an essential way, upon v/1. We will not enter into a full discussion of this problem here because it is treated in full elsewhere [41; see particulary 9 I11 A, B of this paper]. Instead we shall consider the simpler case [42] in which the prevailing (uniform) field B, is everywhere tangential to the interface and in the z-direction (say) and thus avoid the foregoing complications. We shall take the upward vertical to be in the y direction, and initially let the density be a general function of y . I)

268

RAYMOND HIDE AND PAUL H. ROBERTS

2. Solution of Model Problem

The basic equations of the problem are (2.2), (3.8), (2.3), 2.13)

au

(7.1) -

at

1 + (u - grad)u - (B PP

aB

-

at

(7.2)

+ (u

*

div B

(7.3)

*

grad)B = -

grad)B - (B * grad)u = 0, = 0,

DplDt = 0,

where 1, is a unit vector upwards and g is the acceleration due to gravity. The steady state which satisfies these equations is (7.4)

u = 0, B

= Bo = constant,

p

=

Po = constant - g

5

pay.

In the slightly perturbed state, write (7.5)

B

= Bo

+ b,

f~= $0

+ p’,

p = PO

+

~

‘

8

and neglect the squares and products of b,u,p’, and p‘ wherever they occur. We will henceforth omit primes. We then find from (7.1) to (7.3)

(7.7) (74

div b

= 0,

where

and 5 is the displacement of the fluid particle from its equilibrium position: i.e. (7.10)

u = ayat,

to first order. I t satisfies (7.11)

d i v 5 = 0.

269

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

By the equation of continuity, P = - EYaPolaY.

(7.12)

Thus, by (7.6), (7.7) and (7.11), we have (7.13)

a% ~

at2

1

= - -grad Po

6

Bo2_ a2c_ g +_ + -Eyly. aP0

PPO az2

Po ay

Suppose now that pa is a step-function (7.14)

Others of the physical variables will also have a step-function behaviour across y = 0, but, by integrating (7.11) across an infinitesimal path crossing y = 0, we see that (7.15)

[EY112 =

0.

As in Ch. IV, this notation means (7.16)

[tYl12

= limit

t y- limit Ey.

y++o

y+-0

Similarly, from (7.13) (7.17)

[all2= f Y t Y [P112 = g M P 2 - P l ) .

Supposing that all quantities are proportional to exp i(Zx we find, by (7.13) and (7.15),

+

I ~ Z-

wt),

(7.18)

where m12 = -

(7.19)

m22 =

(Z2

+

f12),

- (22 + n2),

Wm,< 0, Wm2> 0.

From the form of (7.18) and (7.19) it is evident that the amplitude of the waves dies exponentially with distance from the interface, the scale length of the attenuation being unaffected by the presence of a field. Condition (7.17) requires (7.20)

270

RAYMOND HIDE AND PAUL H. ROBERTS

where

+

(7.21)

VSZ= 2BOZ/P(fl Pz).

In the absence of gravity, (7.20) corresponds to A l f v h waves travelling along the interface with a velocity V s appropriate to the mean density (pl p z ) / 2 and with an amplitude which dies exponentially with distance into each media. Gravity waves travelling in the x-direction are unaffected by the field and it follows that the criterion for instability ( p z > PI) is unaltered by the presence of a field. The phase velocity of gravity waves travelling in the z-direction is increased by the field to

+

(7.22)

In astrophysical circumstances, the magnetic field may be associated with a strong rotation Q. It is well known that, if 0 is parallel to g, it tends to inhibit instability 1431. If S2 is perpendicular to g it promotes the instability of surface waves travelling in the direction of n and stabilizes waves travelling perpendicular to Jz and g. In the case in which B, is perpendicular to g and 51,it can be easily shown that, if 2QV > g, all waves travelling in the direction of B, are stable no matter what their wavelength and no matter what the difference in densities of the media may be: however, waves in the S2 direction are always unstable. In the case in which B, is parallel to S2 and the heavier fluid lies on top, waves in the direction perpendicular to SZ and g are stable provided their wave number is less that Q 2 ( p z - pl)/g(pz pl), and it is likely that a sufficiently high viscosity or resistivity would stabilize them completely. Waves travelling in a direction parallel to n are stable provided their wave number n exceeds g(Pz - P l ) / ~ S 2 ( P Z Pl).

+

+

VIII. GRAVITATIONAL INSTABILITY : JEANS' CRITERION

A problem of astronomical interest is that of establishing the physical conditions under which gravitational condensations of matter will arise in a large mass of gas [44]. Jeans [45] considered this problem first and put forward the criterion that the size L, of the condensation must exceed a certain value L j usually called the Jeans' wavelength:

where a, is the local speed of sound and p,, is the local density of the gas. Because of the importance of Coriolis forces and hydromagnetic forces in

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

27 1

cosmical physics, subsequent writers have extended Jeans’ analysis to include their effect, and have shown, in fact, that Jeans’ criterion is independent of them [46]. Because the exact circumstances under which Jeans’ method of analysis is acceptable are not immediately obvious in some of the treatments which have been given, we shall enumerate these circumstances before deriving their results. Imagine a large mass of gas of typical dimension Lo, in which the velocity is uo, satisfying

Dug = - grad Po

ZT where

Po is the

pressure and

@j0

+ po grad @,

is the gravitational potential satisfying

V2@o = - 4nGpo.

(8.3)

In order to discuss the gravitational stability of this motion, we consider small perturbations about the state characterized by po,uo,@o,~o.Let @, Po pl, the new values of these quantities be po pl, uo ul, Q0 respectively. By (8.2) and the equation of motion for (ug ul), we have, to first order,

+

+

+

+

+

gradp, - pograd G1 (8.4)

Jeans effectively considered a system in free motion under gravity, i.e. grad Po = 0, so that the right-hand side of (8.4) could be set equal to zero. If to is the shortest time scale associated with the basic motion and tlthat associated with the perturbation, when (8.5) we find

Thus (8.4) reduces to

(8.7)*

po au, = - grad 9,

at

+ po gradQ1.

Similarly, using (8.5), the equation of continuity (2.1) gives aPl - - po div u,. at ‘Writers who have followed Jeans’ approach have used (8.7) as a starting point.

272

RAYMOND HIDE AND PAUL H. ROBERTS

These, together with

and the appropriate thermodynamic relations, are the equations governing the perturbation. Jeans simplified the problem still further by considering one-dimensional isentropic disturbances in the z-direction, say, having the harmonic form exp i(Kz - ol). He found that their frequency is

(8.12)

[

-w_- &a, 1 2n

Ll

(EL)2]1/2,

where L , = (2n/k)is their wavelength. For L,< L j , w is real; in this case, the disturbances are propagated in the z directions with phase velocity o L J Z n , which reduces to a, when L , <( L,. For L , > L,, o is imaginary and the disturbance is aperiodic and increases exponentially with time; after a time of t,, where (8.13)

tl

=“?[($Y a0

- 1/2 - 11

,

its amplitude has increased by a factor of e. Clearly L, = L,.

tl is

infinite when

These results can be understood by the following rough argument. Suppose a slight condensation occurs in the gas in a region 9i? of typical dimension L,. Because of this condensation, any two halves of 92 attract each other with a gravitational force of the order of (8.15)

Thus, an increase in pressure of approximately (8.16)

is required to act across the interface between them, in order to prevent further condensation. If this exceeds the increase aO2pl in gas pressure caused by the condensation, the region 9 will condense further, i.e. the medium is gravitationally unstable if

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

273

(8.17)* GL12PoPlln> UO2Pl. This rough analysis makes it clear why Coriolis forces and a uniform magnetic fields do not affect Jeans’ criterion. The magnetic field does not influence and is not influenced by the motion of particles along the lines of force. Thus, provided the length scale of the magnetic field is large compared to L,, condensations of W along the lines of force are unaffected by the field. Similarly, in a rotating system, condensations along the lines of S2, the angular velocity, are not affected. In rederiving Jeans’ criterion for a conducting gas in the presence of a uniform magnetic field B,, we follow the notation of Sec. V.3, adding the effects of self-gravitation. We omit suffixes 1 hereafter. We find that (5.37), (5.39), (5.40) and (5.41) still hold. However,

replaces (5.38) and (neglecting displacement currents),

(8.19) replaces (5.47). Thus the expressions (5.48) for the phase velocitias are replaced by

(8.20)

where 0 is the angle between B, and the direction of propagation of the wave. When these wave velocities are complex, the amplitude of the wave increases without limit, i.e. the system is unstable for such waves. By (8.20), it is seen that this happens when (8.17) is obeyed. This, therefore, remains the criterion for instability, even in the presence of a field. In fact, for waves travelling in the direction of the field, (8.19) separates into (cf. (5.43))

* Since the physical argument leading to (8.17) is independent of the detailed model chosen, and is almost certainly correct, the conclusion, based on (8.13). is acceptable. However, Jeans’ model is probably not the best approach, since there are uncertainties in the value of t o that should be employed. The most serious difficulty is that the value of t oimplied by (8.2) with grad Po = 0, never satisfies (8.5) when tl is given by (8.13). These difficulties have not been discussed by writers who start with (8.7), but have led other writers (cf. e. g. McCrea, [44; to consider alternative models to avoid the present inconsistencies.

274

RAYMOND HIDE AND PAUL H. ROBERTS

As in the discussion of Sec. V . 3 , the vanishing of the second bracket is uninteresting, and we find (8.22)

(g-

a2

ao2-

a22

1

- 4nGpo

B = 0,

showing that waves travelling in the direction of B, are unaffected by the field. On the other hand, for waves travelling perpendicularly to the field, the criterion for instability is

(8.23)

k(ao2

+ V2)ll2< (4nGpo)'12,

proving that the field stabilizes these waves.

IX. STEADYFLOW BETWEEN PARALLEL PLANES In this section we consider steady laminar flow, in the x-direction (see Fig. 7), of an incompressible conducting fluid along a rigid pipe of length c and of rectangular cross-section 2a x 2b, in the special case when

C

-

I

FIG.7.

a << b and a << c and when the externally impressed magnetic field B, is uniform S and directed parallel to the z-axis. Under these conditions, except near the side walls a t y = b and at the entrance and the exit of the pipe, the flow velocity, u, the induced electric field, E, current density j, and magnetic field, b, will be independent of position coordinates x and y . This simplifies the problem enormously. As will be shown below, the parameter in terms of which the impressed magnetic field has to be measured is the appropriate Hartmann number (see (3.15))

(94

M

= B,a(a/vp)1/2.

When M = 0 there are no hydromagnetic effects, so that E = j = b The velocity is then given by

= 0.

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

u = (U,,O,O) =

(%(

1-

275

$) .o,o) ,

where - Po is the (constant) impressed pressure gradient along the pipe. In the absence of any external field of force, such as gravity, having a component in the z-direction, the pressure 9 is independent of z. The parabolic velocity profile given by (9.2) satisfies the only two boundary conditions arising in the absence of hydromagnetic effects, namely, that because there can be no slipping of the fluid relative to the pipe walls, (9.3)

u,(a) = ux(- a ) = 0 .

The mean velocity is

(9.4)

UoG-

2a

j

P,a 3pv

zC,dz=-.

-a

When M # 0, an induced current j = ( O , j , O ) flows and, because j , is a function of z only, the induced magnetic field b associated with j is in b) the x-direction (see (9.12)). Hence, the resulting body force j x (B, (see (2.4)) has two components, - jybx in the z-direction and jyBo in the x-direction. The first of these is offset by a pressure gradient in the z-direction which sets up elastic forces in the fixed rigid side walls of the pipe. The second force, j,,Bo, cannot be offset by any hydrostatic pressure field and in consequence it reacts on the flow field, causing a general retardation of the motion and modifying the velocity profile.

+

In the non-hydromagnetic case, M = 0, having supposed that b >> a, it is not necessary to be specific about the side walls because no boundary conditions on the differential equation governing the flow field remote from y = & 6 result from considerations of conditions near y = f b. It is immaterial, for instance, whether the situation is regarded as the limiting case of flow along a pipe of rectangular cross-section when alb tends to zero, a remaining finite, or of that of flow along a pipe of annular cross-section as the mean radius of curvature, (rl rz)/2, tends to infinity, the width (r, - rl) of the annular remaining finite and equal to 2a. However, when M # 0, although direct frictional effects of the side walls in y = f b can still be ignored, because j is in the y direction, the manner in which the current circuit is completed via the side walls and through conductors (if any) external to the fluid has to be specified. Otherwise, there are insufficient boundary conditions ta determine the mathematical problem uniquely.

+

If the side walls are not in electrical contact with one another outside the fluid, the total current,

2 76

(9.5)

RAYMOND H I D E AND PAUL H. ROBERTS

I =(j,dz -a

(ampere per unit length in the x-direction) must vanish. Hence, regions of positive j , will have to join with regions of negative j , via regions in the fluid near y = f b in which current flows parallel to the z-axis. Electric charges present on the side walls at y = f b are associated with an electric field E having a y-component only within the fluid.

Conductance /unit N a / b)

lenqfh

FIG.8.

A t the opposite extreme we have the case corresponding to perfect electrical contact outside the fluid between the side walls in y = f b. Then E , must vanish because otherwise I would be infinite. E , vanishes in the annulus problem; otherwise the line integral of E around (say) a circle parallel to both walls would not vanish, and according to Faraday’s law of induction this is inconsistent with the supposition that the system is steady. In general, if the side walls are connected externally via a conductor having conductance N-l[aa/b] per unit length in the x-direction (see Fig. 8) where N is a dimensionless parameter, by Ohm’s law applied to the external circuit,

5

b

(9.6)

N-l(aa/b) E , d y

+ I = 0,

-b

(see (9.5)). In deriving (9.6) use has been made of the fact that E , is independent not only of x and y, but of z also (see equation (9.9b) below). Now make further use of the fact that E , is independent of y and thus simplify (9.6) to (9.7)

2aaE,

+ N I = 0.

277

ELEMENTARY PROBLEMS I N MAGNETO-HYDRODYNAMICS

The two extreme cases considered above correspond, respectively, to N + 00, so that I = 0, E , # 0, and N = 0 so that E , = 0, I # 0. Having discussed the boundary conditions, apply the equations of Ch. 11. By (2.11)

by (2.12) (9.9a,b) and by (2.13) (9.10)

dB* = o ;

whence

~

dz

B,

= B,,

b, = 0.

Now combine (9.8) with (2.15) and make use of the fact that u, to find that (9.11a, b, c)

- - - paE,;

dz

db, - = p a [ E y - u, B,]; dz

=

0 =pu[E,

(u,,O,O)

+ u,b,].

At this point, we introduce the assumption that i, = 0, whence, by (9.8), b y is constant and this constant must vanish because by cannot be discontinuous at the side walls a t y = 3 b. Hence, by (9.8), (9.12) by (9.9), (9.11a) and (9.11~)

E = (O,E,,O),

(9.13)

where, by (9.9b) E , is independent of z, and by (9.10) (9.14)

B

= B,

+ b = (b,,O,B,).

Now combine (9.14) and (9.12) with the equation of motion, (2.2). The z-component leads to (9.15) if gravitational effects are ignored (i.e. a@/az = 0, see (2.5)); this is the hydrostatic pressure gradient which has to be offset by stresses in the side

278

RAYMOND HIDE AND PAUL H. ROBERTS

walls a t y = & b. The only other component of (2.2) of interest, the xcomponent, leads to d2u, 1 db, O = P o + p ~ ~ + - B ,u O d z

(9.16)

- t

where, as noted above, - Po is the imposed pressure gradient along the pipe, being equal to the pressure drop between the two ends of the pipe divided by the length c. Eliminate db,/dz between (9.11b) and (9.16), whence (9.17)

0

=

[Po

+ OBOE,] + pv d2uz -- aBo2ux. a22

'"n 0.6

0.4-

0.2 , M , 10

0

20

,

,

30

FIG.9.

Because the term in square brackets is independent of z , this equation can be integrated to give (9.18) where [ E z/a, and the no-slip boundary condition (9.3) has been introduced to evaluate the constant of integration. Now we must eliminate E , by making use of (9.7). First observe that by (9.12) and (9.llb), (9.19)

. IY =

cosh MC

Po aE,Bo coshMC

6{(7 )

-

('- cosh M )}'

279

E L E M E N T A R Y P R O B L E M S I N MAGNETO-HYDRODYNAMICS

so that

Hence, (9.7) leads to

where U , is the value of U when M = 0 (see (9.4)). E , vanishes when N = 0 ; otherwise, E , is always positive and opposes 11 x B,, as expected. The variation of E , with M when N = 00 is illustrated by Fig. 9. Now substitute for E , in (9.18) and find

+

cosh MC

2(N 1) M ( M N tanh M )

(9.22)

+

In order to study the limiting forms of (9.22) first note that cosh x = 1

+ -21 x 2

;

sinh x

+x X

(9.23) 1 cosh x = - e x 2

;

tanh x 1 1 - 2e-2x

;

sinh x coth x

1

+-ex

2

A

1 $- 2e-2x

x>> 1.

.

When M is close to zero, (9.2) is a close approximation to zt, irrespective of N . For the variation of u, with t at other values of M , see Fig. 10, which illustrates two cases, corresponding to N = 0 and N = 0 0 . (As in Figs. 12 and 14, the profile on only one side of the plane of symnietry is given.) In both cases, increasing M results in reduced u, everywhere, but the reduction is more pronounced near [ = 0 than elsewhere. This has the effect of flattening the velocity profile. Evidently, these effects are much more pronounced when N = 0 than when N = m, a result which is due to the lesser restriction on the current flow in the former case than in the latter. The average flow velocity is (9.24)

u = uO M3 2 ( M +( NN t a ’)n h M ) [M - tanh M I , -

+

280

RAYMOND HIDE AND PAUL H. ROBERTS

(see (0.4)); the variation of CT with M for N = 0 and N in Fig. 11. Observe that, according to (9.21) and (9.24), when N (9.25)

E,

= 00

is plotted

=m

= B,U.

Case (b):N-O

FIG.10.

1.0

E L E M E N T A R Y PROBLEMS I N MAGNETO-HYDRODYNAMICS

FIG.11

FIG.12.

281

282

RAYMOND HIDE AND PAUL H . ROBERTS

The current density (9.26)

-

I-

+ +

(1 1/N) coshM5 (1/N tanh M / M ) cosh M

is plotted for a number of typical cases in Fig. 12. j , is always negative on 5 = 0, and is negative everywhere when N = 0. When N # 0, there are regions near 5 = & 1 in which j , is positive; the higher the value of M the thinner these regions become, and when M = oa, this return current in the positive direction flows in a sheet of zero thickness at the wall.

FIG.13.

In Fig. 13 the variation of i,(O) with M is plotted for N = 0 and N = 00. It is noteworthy that - j,(O) approaches the value Po/Boas M - 00 and is 90% of this asymptotic value when M = 2.5 in the case N = 0, and when M = 4.2 in the case N = 00. - Po/Bois just that value of j , required for a static balance between the force j x B, and the impressed pressure gradient along the channel. The induced magnetic field b, can be found by integrating (9.26) with respect to 5 (see (9.12)), giving

where A is a constant of integration. further information.

To evaluate this constant requires

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

283

By a simple application of Ampere's circuital law, (9.28)

bX(1) - b*(- 1 ) = PI,

(see (9.5) and (9.27)),and as I , given by (9.29)

I

=-

tanh M

BaaE, ~

tanh M

N

(see (9.7) and (9.21)),only vanishes when N = 00, b , ( l ) is not in general equal to b x ( - 1 ) . Because bx must be continuous everywhere, and outside

Case (a): N-w

.20I

0 1.0

z/a

0.6

0.4

0

0.2

0

0.2

0.4

% Z I I I I l O0 z/a 1.0

3.6 0.6

FIG. 14.

the fluid, b , must be uniform, if bu and bl stand respectively for the uniform values of b, outside the fluid in > 1 and [ < - 1 , (9.30)

bu = b x ( +

I),

61 = b x ( -

1))

whence, by (9.28), (9.31)

b, - bi = pI

which shows that generally the system is not symmetrical in all respects about the plane 2' = 0.

284

RAYMOND HIDE AND PAUL H. ROBERTS

The constant A depends on the properties of the external circuit, conductance per unit length N-l(ua/b). Let this conductor be made up of two components in parallel, one in the upper space mainly a t >> 1 and the

0

5

1 0

15

20

30

asymptotic

1.0-

0

25

, 5

1 0

15

FIG.16.

20

25

M

value

, 30

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

286

other in the lower space mainly at 5 << - 1, having conductances aN-l(ua/b) and (1 - a)N-’(ua/b) respectively. Now b, = a p I ;

(9.32)

br = - (1 - a)pul;

and this together with (9.30) suffices to determine A in (9.27). The “annulus problem” corresponds to a = 0, so that b, = 0, bz = - p1. Fig. 14 illustrates b, as a function of 5 for two cases, (a) N = 00 and (b) N = 0, when a = 0.5. In both cases the profile is antisymmetrical about 5 = 0. Although varying a does’nothing to the induced currents and adds only a constant A to b,, this does not mean that a is unimportant. The magnetic energy density depends on the square of the magnetic field strength, so that the total energy in the system depends in general on the external circuits. For this reason, one expects a to be of the greatest significance in nonsteady problems, one example of which would be the generation from a state of rest (say) of the steady flow described here. In conclusion, it is of some interest to evaluate the magnetic Reynolds number (see (1.1)) defined as follows:

R

(9.33)

= 2upaU.

On substituting for U from (9.24) we find that (9.34)

R

= 2upaUo

(N -k ’)

(M - tanhM) \M2 ( M + N tanh M ) (M

+ N tanh M )

The variation of R with M is plotted in Fig. 15. When N = 0 0 , R increases indefinitely with M , in contrast to the behaviour when N = 0, when R tends asymptotically to unity as M goes to infinity. This is related to the difference in the distribution of current in the two cases (see Fig. 12). When N = 0 the bulk of the current flow is in the main body of the fluid and j v drops to zero at = & 1. On the other hand, when N = 0 0 , the return current j , near 5 = & 1 flows in a layer of ever decreasing thickness as M increases, and a magnetic Reynolds number based on this thickness tends to infinity more slowly - logarithmically in fact - with M , and is probably more significant physically than the one based on the distance 2a. For references to the early work on flow between parallel planes, see [3]. Globe [47] has treated the problem of flow along an annular pipe in a radial magnetic field and Shercliff [ 6 ] has considered the more difficult problem of flow along a circular pipe in a uniform transverse magnetic field.

286

RAYMOND HIDE AND PAUL H. ROBERTS

X. FLOW DUE

TO AN

OSCILLATING PLANE:RAYLEIGH'S PROBLEM

1. Definition of Problem: Dimensionless Parameters

In order to understand the effect of viscosity in modifying the motion of a fluid in contact with vibrating solids, Stokes examined a particularly simple case (see [48],p. 317). He supposed that an infinite plane located at z = 0 executes harmonic vibration in a direction ( x , say) parallel to itself, and that the (incompressible) fluid in contact with this plane at z = 0 occupies the whole of the region z > 0 and is a t rest at very large values of z. Assuming that v is constant, and that no slip occurs between the fluid and the vibrating surface, he showed that if the velocity of the vibrating plane is (U, cos o t , 0, 0), where U,, and w , which are assumed to be constant, are, respectively the velocity amplitude and angular frequency of the vibration, the velocity at any point in the fluid, u = (u,,uy,u,) is given by (10.1)

u, = U,exp

[

-z

($3

[

-

($7

cos wt - z -

,

uy = 0 ,

u, = 0.

According to this expression the velocity amplitude falls off exponentially with distance from the plate, having dropped to e-lU, = 0.3679 U , at a distance z = A where

A

(10.2)

(2v/~)l/~.

In addition to this variation of amplitude with z , there is also a variation in phase, due to the inertia of the fluid. The wavelength associated with this variation in phase is %A, at which distance from the plate the velocity amplitude is eW2*U,= 0.0018 U,. The flow generated by the vibrating plane can be regarded as a heavily damped plane shear wave, the coupling between different layers being due to viscous friction. The tangential force per unit area at any level can be obtained from the viscous stress tensor (10.3)

This gives

p , =d PVUO exp( (10.4)

5)[-

cos(ot -

$)+ sin (cot -

;)I

so that the tangential force per unit area acting on the plate is (10.6)

pxz(O,t) = (pvU,/d)(sin o t - cos wt).

,

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

287

The negative of the second term in this bracket is in phase with the motion of the plane and corresponds to a dissipative force tending to stop the motion. The other term is x12 out of phase and represents an effective increase in the inertia of the vibrating body due to the presence of the fluid. The rate a t which internal stresses do work at any surface parallel to the vibrating plane, in general p,,u, pz,,uy P,,z~, (see Appendix A), reduces to p,,u, in this problem. Denoting this quantity by Q , we find by equations (10.1) and (10.4) that

+

(10.6)

Q

= Q(z,t) = -

+

~

which oscillates at double the frequency of the vibration about a non-zero average value (10.7)

~ ( z= ) - __ PVUO2exp

24

(- g).

The power input per unit area required to maintain the motion, which will be denoted by P , must equal - Q(0,t) so that (10.8)

The direction of energy transfer between the vibrating plane and the fluid depends on the sign of P , which alternates, changing twice each half cycle. However, the average value of P , given by (10.9)

P

= pvUO2/2A

is essentially positive, corresponding to a net energy transfer from the vibrating plane to the fluid, where it is dissipated by viscous friction. Rayleigh [48]made use of Stokes' result for the drag on the plane (see Eq. (10.5)) in his investigation of the effect of the boundary layer on the propagation of sound in tubes, and it is of interest to ask how, in a conducting fluid, Stokes' result is modified by the presence of a magnetic field. The obvious extension of Stokes' problem is to the case of a conducting fluid in the presence of an impressed uniform magnetic field of strength B, in the z direction, the formal solution of which is presented below in Sec. X.2 and discussed in Sec. X.3 (see also [34,491). In Sec. X . 4 the corresponding aperiodic solution is derived (see also [36, 50, 511). The overall behaviour of the system now depends on three parameters (10.10)

288

RAYMOND HIDE AND PAUL H. ROBERTS

a,@, and y measure, in suitable units, the magnetic field energy, the electrical

resistivity, and the velocity amplitude of the oscillating plane. From their definitions we can relate a,@ and y to the more familiar dimensionless parameters of Ch. I11 provided we base these parameters on the characteristic length L = ( v / o ) ’ / ~which , is of the order of the boundary layer thickness in Stokes’ problem. Thus we find the Reynolds number (3.4)

R’

(10.11)

f UoL/v = y ,

the magnetic Reynolds number (3.3) (10.12)

R E U0L/I = y/p,

the Hartmann number (3.15) (10.13)

M =- B,L(a/pv)1/2=

and the Lundquist number (see (3.16)) (10.14)

K

E

B,La(p/p)’/2 = 0 1 1 / 2 / p .

The fluid motion is no longer of the form given by equation (10.1). I n the presence of the magnetic field, u, consists of two parts characterized by different attenuation and phase factors. The form of these parts suggests that they should be termed “velocity” mode and “magnetic” mode. The relative amplitudes of these modes, their associated attenuation and phase factors, and the induced magnetic and electric field depend on a,p and y. These quantities also depend on the electromagnetic boundary conditions, which in turn are determined by the electrical properties of the region z 0 not occupied by the fluid. We restrict attention to the case when this region is filled by an insulator. There are no additional difficulties associated with 0 is a conductor. the other case when the region z Explicit solutions can be found in a sufficient number of cases, corresponding to different limiting values of a and p, to cover most situations of physical interest. The results demonstrate quantitatively a complicated interplay between hydromagnetic and viscous effects (see Sec. X.3).

<

<

2. The Fovmal Solution

b

We divide B into a uniform field B, (b(z,t),O,O):

= (O,O,Bo)

and an induced field

=

(10.15)

B=B,+h.

The velocity field has only an x-component u = (u(z,t),O,O).According to the analysis of Sec. V.4, zc and b are governed, without approximation, by the linear equations

289

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

(10.16)

(10.17)

The boundary conditions necessary to determine the unique solution from these equations are the following. The fluid must be at rest at large distances from the plane and must not slip relative to the plane a t the plane itself; i.e. (10.18)

u(z+

0) =

0,

u ( z = 0) = u, cos ot.

The boundary conditions on the magnetic field require that 1, vanish a t infinity and, (since z < 0 is occupied by an insulator), at the plane itself; i.e. b(z+ )..

(10.19)

= 0,

a q z = oyat = 0.

In the steady flow problem of Ch. IX, it was necessary to specify the nature of the external electrical connection between the fluid at y = 00 and y = - 0. However, in the case of unsteady motion, the total current I flowing in the y-direction must vanish for it vanishes initially (by supposition) and, were dIldt non-zero, the induced e.m.f. LdIjdt (where L is the inductance of the external circuit) divided by J?,Eydy would be infinite. Thus (10.19) gives

+

b(z+

(10.20)

0) =

b ( z = 0) = 0.

Another way of looking a t this boundary condition is in terms of the magnetic energy outside the fluid due to the current flowing within the fluid. If I is non-zero, the magnetic energy outside the fluid is infinite and, indeed, in setting up the steady state, this energy would have to be supplied by the source. In an oscillating problem, if the current were non-zero, the power required to drive the plane would have to be infinite. To clarify the question further, we will consider the following simple example. An insulating cylinder of radius a immersed in a conducting fluid oscillates in a direction parallel to its axis and in the presence of a magnetic field which is radial in the region Y > a. I t is clear that the oscillating plane problem is a limiting form of this oscillating cylinder problem as a+ 0. To preserve this similarity, we take cylindrical polar coordinates (z,B,x), z being radial and x along the axis of the cylinder. The 8-increasing direction corresponds to the y-decreasing direction of the plane problem. The oscillation generates an x-component of u and b, and a y-component of j and E. The continuity of E at the surface of the cylinder requires that (10.21)

- nu2

ab(z = a) at

ab(z = a)

290

RAYMOND HIDE AND PAUL H. ROBERTS

If we assume that all quantities vary as eiWt in time (see (10.22)), according to (10,21), when w # 0, the value of b on the cylinder vanishes as a+ m . However, if o = 0, we can only conclude that ahlaz vanishes on the cylinder. More precisely, we may only assume the truth of (10.20) when wa2/11>> 1. If the criterion is not satisfied, the total current in the y direction need not be zero. Thus, care must be exercised in interpreting the results of idealized problems of the kind discussed in this section. When the moving surface is that of a solid conductor rather than that of an insulator, currents will be induced in this conductor and the behaviour of the system will be modified significantly. It is then necessary to specify carefully the relative motion between the source of the main magnetic field and the vibrating solid. In the literature there has been a certain lack of clarity on this point and some errors have been made. The difficulty seems related to the correct application of the law of induction which is often incompletely dealt with in standard texts of electromagnetism. The subtleties of this point have been considered by a few writers (see Sec. 11.3). Since (10.16) and (10.17) are linear, we seek solutions of the form (zc,b) oc exp (iwt - qz/L),

(10.22)

(L= (V/O)’/~),

where W ( q )> 0, by (10.18) and (10.20). On substitution, we find

(i - q 2 ) ( i - pq2) = uq2.

(10.23)

When u

= 0,

the roots of this equation are

(10.24)

q1 corresponds to Stokes solution in which the shear wave is attenuated in . corresponds to electromagnetic skin currents which a distance ( 2 ~ / w ) ’ / ~q2 are attenuated in a distance ( 2 2 / ~ ) ’ ’(although ~, since u = 0, they are not excited in this case). (10.23) has two roots q1 and q2 which tend uniformly to (10.21) as u-+ 0. We can therefore, without ambiguity, term them the “velocity mode” and “magnetic mode”, respectively. If u # 0 both these modes are excited. Let (10.25)

By equation (10.16), g, is related to f, by (10.26)

(i - P4l2k1 = - Y41flt

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

and a similar equation relates g, and and (10.20), we have (10.27)

fl

(10.28)

291

Also, by the conditions (10.18)

f2.

+ f z = 1, g2 = 0.

g1+

From (10.23) it follows that (10.29)

= ip-

QlQ2

112.

Solving equations (10.26, 27, 28) for f i , f 2 , g, and g, and using (10.29) as a means of simplifying the final results, we find (10.30)

fl

VF41 - 4 2

=

[1 (10.31)

g1 =

+ VPI (41 -

’

fz

VP42 - 41

=

[1

42)

+ VPI

(42

- 41) ’

1

- g2 = -

V F 11 + VPI(41 - 4 2 )

*

From these results,

and

Note that, on the plane itself, (10.34) and e”t

(10.35)

E,(z = 0) = (UoBo) 1 ~

+ va’

Thus, in the limit 8 - 0, there is a surface current on the plane, while the electric field there is given by (10.36)

E

=-

Uo x Bo.

In the limit p- 0 0 , both current and electric field tend to zero. of the force driving the plane is the average, The mean rate of work over a cycle, of (10.37)

P=-vp[U$]

E = O

9

292

RAYMOND HIDE AND PAUL H. ROBERTS

and, by (10.25),

+ +Uoe-ioo”,

U ( Z = 0)= +UOeiog

(10.38)

so that (10.40)

P

=

(pyu,21: L

-H(f1ql

+ f2q2) + periodic terms.

Hence, using equations (10.29) and (10.30) and averaging, (10.41)

3. Discussion of Some Limiting Cuses

Having given the formal solution, we now present the results in a number of limiting cases. We are interested in low, moderate and high conductivity ( p >> 1, p = 1, p << l), and these cases will be designated A, B, and C. In each of these cases we must consider first the effect of a weak magnetic field, and then the effect of a strong field. We put 5 = z / L . The results for case A are summarized in Table 3. Observe that a/P turns out to be the appropriate measure of the magnetic field. In the weak field case, although the magnetic mode of u, is associated with a slow fall-off of that of the with z , its amplitude is only a small fraction, velocity mode. In the presence of a strong field, the phase and amplitude factors of the velocity mode now depend strongly on a and Bo, the amplitude of this mode of u, being only slightly less than in the absence of a magnetiac field. The magnetic mode of u x is weak, the amplitude at z = 0 being ,8-1/2. According to the form of q8, this mode corresponds to an AlfvCn wave damped by electrical resistance (see Sec. V.4) Now consider case B, that of moderate conductivity, corresponding to p = 1. The results are summarized in Table 4. Now it is a that measures the strength of the impressed magnetic field. Observe that in the presence of a weak field, in contrast with cases A and C, q1 and q2 contain terms of order a1/2.The amplitude factors of each mode of u, are the same, namely, 0.5. In the strong field case, the phase factors of each mode correspond to an AlfvCn wave. The velocity mode is much more rapidly attenuated than the magnetic mode, and at moderate distances from the plane, the magnetic mode dominates. The form of the attenuation factor of this mode shows

-

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

TABLE3 Case A.

P

1 : (u

<< (pv)-I)

Low Conductivity

EY

UOBOeiot

293

294

RAYMOND HIDE AND PAUL H. ROBERTS TABLE

Case B.

p

=1

4 (I? =

(pv)-l)

Moderate Conductivity

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

TABLE5 Case C.

P<< 1

(n>> ( p 4 - l )

High Conductivity

1

i

2a3/2+.1/2

EY

UoBoeiWf

+T a .. q= 1 (l

.)

295

296

RAYMOND HIDE AND PAUL H. ROBERTS

that viscosity and electrical resistivity play equal parts in dissipating the energy of the wave. Finally, we consider the third case, C, that of high conductivity (fi << 1). The results are summarized in Table 5. When the magnetic field is weak, the velocity field is only slightly modified by it. The magnetic mode of 21, is weak, having a small amplitude a t z = 0 and a high attenuation factor, of order /?-1’2. There is no term in ul/’in the expressions for q1 and q2. In the presence of a strong field, the velocity mode of 21% is characterized by a small amplitude at z = 0 and rapid attenuation, the motion consisting almost entirely of an Alfven wave, which, from the expression for y2 can be seen to be damped by viscosity. In all cases, the mean power P required to maintain the vibration has to be increased in the presence of a magnetic field; in the case of a strong magnetic field P is proportional to B,. The possibility of detecting the effect of a magnetic field on the propagation of sound in a tube of mercury has been considered [34]. As p >> 1 for mercury, u/P is the appropriate measure of Bo. As 50/w,frequencies of vibration as low as a few cycles per second would be needed to produce any marked effect. In the kilocycle region the sound speed would be reduced by something of the order of one per cent. The situation should be rather more favourable if liquid sodium were used, because then it would be possible to work at much higher frequencies.

-

4. Ra yleigh’s Problem

Instead of forcing the insulating plane z = 0 to oscillate, we will now simply suppose that a t time 1 = 0 it is jerked into uniform motion with velocity U,, in the direction of the x-axis. In the absence of a magnetic field, this problem was first considered by Rayleigh [48] and is sometimes named after him. The simplest method of recovering his result is by the method of Laplace transforms (equivalent to Heaviside’s operational method), We replace a/at by p , (10.17) then gives

where the superimposed bar distinguishes transformed from untransformed quantities. This must be solved in conjunction with the boundary conditions (cf. (10.18)) (10.43)

n(z =)..

= 0,

n(z = 0)= uo/fi.

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

297

Thus (10.44)

Inverting the Laplace transform (see, for example, [52], p. 354, No. 29) (10.45)

where erfc x is the complement of the error function: m

(10.46)

In the case in which the magnetic field is non-zero, the operational solution may be easily derived by replacing io by p in the analysis of Sec. X.2. For example, by (10.25) and (10.30), we have

where

The method we use to invert (10.47) is a simple extension of a method employed by Roberts [36] to solve a modified form of the present problem. Express u(t) in the form

(10.49)

where

Then (see, for example, [52], p. 354, No. 27)

298

RAYMOND HIDE A N D PAUL H . ROBERTS

(10.51)*

{(AWl

+ ( v l k l - j11~2sz)es~z},

- v1/2sZ)esle

= O,(P)

+ %(P)’ say,

where v1 and v2 are the parts of v involving exp slz and exp szz, respectively, and

where ,8 = ~ V ( A V ) ’ / ~/ ( v). A I t follows that

Now, by elementary methods (see, for example, [52], p. 353, Nos. 7 , 8)

(10.54)

and also (see, for example, [52], p. 356, No. 53)

* The analytical advantages of this transformation are somewhat offset by the apparent dimensional inconsistencies it introduces. The reader should therefore take heed that, since O(p’/i) is related to a ( p ) by (10.51), the direct Laplace inversion of ti($) by a Bromwich integral involving ePt must lead to a function v ( t ) in which 1 has the dimension (time)’/%. (See 10.57 and 10.60 below.)

299

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

(10.56)

['P V (li(P" 2 -

"1'

- P2)

esla N

where I, and I , are modified Bessel functions of the first kind of orders zero and unity, respectively. Thus

where 1 6 = V(A1/2 - y1/2) (t' - zv- '/2),

(10.58) (10.59)

5

= p [(t- t ' ) 2

+ (A- 112 + Y- l l Z ) Z ( t - t ' ) ] l / Z .

In an exactly similar fashion

if

t

where C is defined as before and 5' is given by (10.61)

Having calculated vl(t) and v&) in this way, we can now calculate ~ ( twith ) the help of (10.49):

300

RAYMOND HIDE A N D PAUL H. ROBERTS

(10.62) #”-

l/Z

,A- 1/2

Solutions for short times (i.e. for t << zv-lI2 for ul,and t << z I . - ~ / ~for u2) may be quickly derived from (10.57), (10.60) and (10.62) or, indeed, directly from the operational form (10.47) itself. Solutions for large times mav be computed from (l0.57), (10.60) and (10.62) by a steepest descent approximation. (For details of this procedure applied to a similar problem, see Roberts [36]. Chang and Yen [all also discuss approximations in some detail .)

XI. STEADYTWO-DIMENSIONAL INERTIAL FLOWIN MAGNETJCFIELD

THE

PRESENCE OF A

We have seen in an earlier section (see (3.26)) that in the presence of a very strong uniform magnetic field B, = (0,0,B,) (say) slow (i.e. S = m ) steady motion of a perfectly conducting (i.e. R = m ) fluid will be twodimensional, in planes perpendicular to B,. To gain an insight into this theorem, this chapter deals with the problem of nearly uniform flow in the x direction of an incompressible inviscid fluid across a strong magnetic field in the z direction. For steady flow, by (2.2) (11.1)

p ( u . V)u = - V p

+ j x B.

Assume now that

\i.here U , and B, are constants and 21, v , w,b,, by, b,, e x , e y , e, are small quantities. If the motion is two-dimensional, having no dependence on y (i.e. ajay = 0 ) , we can introduce scalar functions 4 and +!I of x and z , where

which automatically satisfy (2.3) and (2.13). By (2.12) as aB/at (11.4)

=0

301

ELEMENTARY PROBLEMS I N MAGNETO-HYDRODYNAMICS

By (2.15), when

E,

=0;

E,

q--+0 0 ,

= U,B,

to first order of smallness

+ uB, + Uob, = E , + uB,+ U,b,;

E,

= 0.

(11.5) By (11.3) and (2.11)

j,

(11.6)

= V2$

so that to first order, (11.1) becomes (11.7a, b)

pU,- a2+ az ax

=--

ax

+ B,V2$;

- p U a24 o w=-aP a2

+

where V 2 denotes the operator (a2/ax2 a2/az2)). By (11.4), E , is independent of x and z and is, therefore, constant. According to (11.5),this constant must be U,B,, the undisturbed value of E,, and equal to E,, so that ey = uB,

+ U,b,

= 0,

whence (11.8) The undisturbed situation in which the electric field E has one component, in the y direction, equal to U,B,, and no electric current flows, is only possible when there are insulating surfaces at y = f 00 on which charges can be set up. Now eliminate p between (11.7a) and (11.7b) and thus find (11.9) and with the aid of this equation and (11.8), (11.10)

If (11.11)

S 2 = B 20 lPPUO2I

(see (3.11)), (11.10) may be re-written as

(11.12)

302

RAYMOND HIDE AND PAUL H. ROBERTS

Consider solutions which are harmonic in the x-direction, namely = eikx F ( z ) .

(11.13)

F ( z ) satisfies (11.14)

which has solutions

F ( z ) = A eka

(11.15)

+ B e - + Ceq, + De-

gr,

where

q

(1 1.16)

=ik/S

and A , B , C and D are constants to be determined by the boundary conditions of the problem. According to (11.3) and (2.11)

4 = ( A & + Be-kz + Ceqz + De-qx)eskx, $ = - ' t S ( p / p k 2 ) 1 / 2 [ k A e k-z k B c k P+ qCdl

+ qCeqa - qDe-q*)&, w = - ik(Aekz + Be-kz + Cd* + De-q*)etkx, b, = - iS(,uup/k2)1/2[k2Aek'+ k2Be-kE + qzCeq* + q2De-qz]eEkx, u

(11.17)

- qDe-qx]eEkx,

= ( A h k *-

Bke-k*

b, = - S(,u~p)l/~ [KAe" - k B e -

j,,

= v2$ = -

k2 (1

kz

+ qCeqz - qDe-q*]e'kx,

+ S-2)(p/,u)1/2[Ceg*- De-gl]eZkX.

Now consider the specific problem of flow along a channel with insulating walls. Three of these walls, in y = b and z = d are plane and the fourth is slightly wavy, occupying

+

(11.18)

z = f cos k x ,

the mean position being z = 0. If d << b, the flow can be regarded as twodimensional, and if / << d deviations from uniform motion in the x direction will be slight. As a result of the electric currents in the fluid, the magnetic field in the insulating regions outside the fluid will be distorted. Introduce magnetic "stream" functions, and 2$ in the upper and lower regions, z > d and z < f cos k x , respectively. These functions satisfy

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

303

(see (11.6)) since j = 0 outside the fluid. Solutions of (11.19), which remain finite a t z = w and vary harmonically in the x direction with wave number k , have the form

lt,b

(11.20)

= ae- kz elk, ,

2t,b

=

pekr erkx,

where a and p are constants. The induced magnetic field components in regions 1 and 2 are, respectively,

The constants A , B , C, D , a and p, are determined by the boundary conditions, which we must now consider. Since the fluid is assumed inviscid, the only requirement of u on z = d and z = f cos k x is that the normal component should vanish, Hence w = 0 on z = d, so that

+ Be- + Ceqd + De- = 0 and d(feikx)/dx= w / ( U 0+ u ) on z = f cos k x , which, to first order, (11.22)

Aekd

to ik/eikx= u / U o on z

kd

=0;

A

(11.23)

qd

leads

hence

+ B + C + D = -fU,.

Assuming that A/v >> 1 (see (4.64)) j, cannot be infinite, even a t the boundaries of the fluid, so that b has to be continuous. The continuity of the normal component of b on z = d gives b,(d) = ,b,(d), and of the tangential component, b,(d) = ,b,(d) ; whence, (11.24)

S(p/p)1/2{Akekd - Bke-kd

+ Cqeqd - Dqe-Qd}= ikae-kd,

and

+

(11.25) iS(p/,u)1/2{Ak2ekd Bk2e-kd

+ Cq2eqd+ Dq2e-qd} = k2ae-kd.

On the other surface, z = / cos k x , to first order of smallness, the continuity of b leads to b,(O) = ,b,(O); b,(O) = ,b,(O); whence (11.26)

S(p/p)1'2{kA- kB

+ qC - qD} = ikp,

and (11.27)

+

iS(p/p)'/2{K2A K'B

+ q2C + q2D} = - k 2 p .

Equations (11.22) and (11.27) suffice to determine A , B , C, D , wci'nd p. Eliminate a between (11.24) and (11.25) and p between (11.26) and (1.1.27) and thus find (11.28)

+ q(q + k)Ceqd+ q(q - k)De-qd = 0

2k2Aekd

304

RAYMOND HIDE AND PAUL H. ROBERTS

and 2k2B

(11.29)

+ q(q - k)C + q(q 4-k)D = 0.

Now eliminate B between (11.23) and (11.29), and, making use of (11.16) find (11.30)

2A

+ C(2 + i / S + 1/S2)+ D(2 - Z/S + 1/S2)= - ZfU,.

Eq. (11.28) may be re-written as (11.31)

+ (i/S - 1/S2)Cerkd/S- (i/ S -k l/Sz)De-ikd/S= 0.

2Aekd

Eliminate B between (11.22) and (11.23); whence (11.32)

A(ekd- e-kd)

+ C(e'kd/S- e-kd) + D(e-rkd/S- e-kd) = fe-kd U,.

A t this stage, it is convenient to simplify the problem by considering a deep system, for which d is very much greater than 2 4 k . Eq. (11.32) then reduces to Aekd + Ce'kd/s + De-

(11.33)

W S

= 0.

In order to illustrate the behaviour of the solution, we shall consider two extreme cases, S << 1 and S >> 1. In the first case, it may be shown that to first order in S 2 the velocity potential is (11.34)

4 = fU,

-

(1 - 2S2)e-k"+ 2S2

(reverting to real quantities). When S recover the well-known solution

4 = - f U,e-

(11.35)

-1

sin k(z - d)/S cos kx sin kd/S

=0

corresponding to B,

= 0,

we

kE cos kx,

which falls off quite rapidly with z. In the presence of a weak magnetic field (S << 1) there is an additional term which varies harmonically with z . Although the amplitude of this term at z = 0 is very small, being S2 of the other term, it predominates a t great distances in the case considered. This result is of limited physical interest because, in a real fluid, such rapid spatial variations (the wavelength in the z direction is 2nS/k) would be rapidly attenuated by friction. In the case S << 1

so that

( 11.36)

jy

= V2$ = - 2{k2Uo(f)

I/:!

cos k(z - d ) / S sin kd/S sin kx.

ELEMENTARY PROBLEMS I N MAGNETO-HYDRODYNAMICS

306

h'ow consider the case of a very strong magnetic field (S >> 1). Equations (11.30),(11.31),(11.33)and (11.23)then lead to

'="'

sin k(z - d ) / S cos k x sin kdIS

(11.37) to zeroth order in S-l.

If S

>> kd,

then

4 = - fUO(1- z / d ) cos k x

(11.38)

and as kd >, 1, 4 hardly varies at all between z = 0 and z = 2 n / k , in contrast to the rapid exponential decay in the other case, S <( 1. This result is in accord with Sec. 111.3. The flow field is completely altered in character as a result of the presence of the magnetic field, the energy of the disturbance arising at the wavy wall being transmitted to considerable distances from the wall. Corresponding to

d

given by (11.37)we have

* "):(

(11.39)

'''cos k ( z - d ) / S sin k x , sin kd/S

=

and

(11.40)

j

-

Y-

-k2fU

&

' ( p r

cos k ( z - d ) / S . sin k x . sinkd/S

ENERGY EQUATION APPENDIXA : THE HYDROMAGNETIC

We have seen (Ch. 11) that, except for the "degenerate" case of incompressible flow, it is necessary to supplement the hydrodynamic and electromagnetic equations by thermodynamic relations. However, strictly speaking a fluid in mass motion cannot be in thermodynamic equilibrium, and the application of relations which are valid only in thermodynamic equilibrium is, at first sight, questionable. Provided the mean intermolecular distance d is small compared with any macroscopic length scale L characteristic of the hydrodynamical flow, it is easy in principle to define density p, velocity of mass motion u, and internal energy per unit mass U as point functions of position. For example, to determine the value of p for a particular point P , we simply draw a sphere 9 ' centred at P whose radius Y is large compared to d but small compared to L ; we divide the mass A contained in 9 'by the volume 4nr3/3= V of 9.The resulting value, p, will

306

RAYMOND H I D E A N D PAUL H . ROBERTS

be a meaningful definition of the density at P since it must be insensitive to the value of Y chosen (provided d << r << L ) and will not be subject to significant statistical fluctuations. Similarly, to define u, we divide the total momentum of the particles in Y by A, and to define U we divide the energy of the particles in 9, measured in a frame moving with velocity u, by A. Having defined p, u and U , we may define a “temperature” T , and “entropy per unit mass” S, and “enthalpy per unit mass” W , and a “pressure” fi simply by applying the usual thermodynamic relations as though they were exactly valid for 9.For example, we can define T by

T

(A4

=(y

- l)U/g.

The values T , S, W , and fi obtained in this way are those which would be appropriate to Y if, moving in a frame with velocity u, the content of Y is isolated and allowed to come to thermodynamic equilibrium (keeping the volume constant). Provided L >> 1, the mean free path between collisions, we can choose Y so that 1 << Y << L. Then the thermodynamic state of Y before isolation is close to the equilibrium state arising after isolation and may be regarded as a small statistical fluctuation from it. We may, therefore, apply the results of classical fluctuation theory to Y and assign an entropy per unit mass S‘ to Y by means of the familiar (see, for example [53], p. 11) A S ‘ = k log R, (A4 where SZ is the number of complexions appropriate to the state of Y before isolation. S’ may be regarded as the “true” entropy at P ; it will in general differ from the entropy S defined from Y after isolation. However, for given p and U , the entropy of Y is a maximum when it is in thermodynamic equilibrium: i.e. S - S’ is positive and of order (1/L)2S:Similarly, after isolation, the mechanical stress tensor fiyh for Y assumes the form appropriate to thermodynamic equilibrium ; viz. fib,,. Before isolation the mechanical stress tensor will in general differ from fibii, and the difference,

(A.3)

fimech $7 - fib,,

=-

fiyp = viscous stress tensor

will be of order (l/L)fi.For Newtonian fluids, fi? will depend only on the instantaneous values of the space derivatives of u a t P and, since it is symmetrical, the only possible first order combinations are

Thus fi?

must be a linear combination of these:

E L E M E N T A R Y P R O B L E M S I N MAGNETO-HYDRODYNAMICS

307

Here, the constants pv and pC are termed “the viscosity” and “the bulk viscosity”. (The latter is zero for a monatomic gas; cf. [54], 5 2.5.) Throughout this paper, we use the quantities T , S, W , p etc. defined after the isolation of 9.As a consequence, relations such as

TdS = dU

+ pd(:).

1 TdS = dW - - d p ,

(-4.6)

P

hold true despite me fact that irreversible changes are taking place. This is a consequence of the fact that if, in following the hydrodynamical flow, a fluid element undergoes an expansion dV, the work it does on the surrounding fluid is, in a dissipative medium, less than PdV. I t follows that TdS must exceed the total energy supply. In applying relationships such as (A.5) and (A.6) it is, of course, not necessary to consider that d p , dW,dS, etc. refer to changes following the motion. For example, we may consider dW(dS) to be the difference between W ( S )a t two adjacent points of the fluid a t one particular time. (A.6) then determines the corresponding changes d p in 9, giving grad p

(A.7)

= p grad

W - pT grad S.

Similarly, applying (A.5) to the same fluid particle at two adjacent times,

We will use this relation in the discussion of the hydromagnetic energy equation below. The total energy of all forms contained in a volume V (fixed in space) is I[pU V

1 1 + %pu2 - p@ + B2+ 2P -

where grad Q, is the gravitational force prevailing. To avoid the difficulties that arise when one tries to treat the general case of a self-gravitating medium, we restrict attention to a gravitational field of external origin to the fluid and which is constant in time but not necessarily in space. Now let us consider the energy budget. Bodily transport is responsible for an outward flux of internal, gravitational and kinetic energy over the surface S of V , amounting to

308

RAYMOND HIDE AND PAUL H. ROBERTS

(dS outwards from V ) . Thermal conduction removes heat energy at a rate

-

1

grad T * dS.

S

The (Poynting’s) electromagnetic energy efflux is

5

(E x H) * d S .

S

This term must, of course, be evaluated in such a way that flux of electromagnetic energy between the fluid and external circuits is not overlooked. Finally the rate at which mechanical stresses do work on the surrounding medium is

5

ui

p y h dSj.

S

Thus

at V

pcpK(grad T ) * dS

(A.lO) S

s

S

s

If we apply the divergence theorem to the integrals on the right-had side of (A.lO) and shrink the volume V to a point, we find

(A.ll)

+ div (pcptc grad T)- ax,a (ui$Fh). ~

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

309

By (A.3), this may also be written

-aE_ - - div Q ,

(A.12)

at

where (A.13)

E

= pU

1 1 B2 + -&E2 + 1 pu2 - p@ + 2 2E.l

is the total energy density, and (A.14)

Qi=

u2

+ W -@

-

wjfir-

ar + -1 (E x B)i

PC~K-

8%

#u

is the total energy flux vector. From Maxwell's equations, it is readily verified that (A.15)

=-

1 div-(E

x B) - j . E .

#u

Also, since we are assuming a@/at = 0,

a

- (p@) = @-

at

aP

at

=-

@ div p~

+ pu

= - div (p@u)

grad @.

Thus ( A . l l ) may be written

P D t ( U + +uz - @)

=j

-E

a + pu .grad @ + div (pcpcgrad T ) - -(u$Yh). axi

(A.16)

The left-hand side represents the rate of increase of kinetic and internal energy in V , making allowances for that advected over the boundary. The first term on the right-hand side represents the difference between the rate at which the electromagnetic field does work on the material particles in V and the rate at which the mass motions generate electromagnetic energy in V , i.e. it is the rate at which electromagnetic energy is converted irreversibly into heat. The second term represents the rate a t which gravitational forces do work. The remaining terms on the right-hand side can be interpreted as before. Equation (A.16) may be cast into an alternative form by using the momentum equation

310 (A.17)

.

RAYMOND HIDE AND PAUL H . ROBERTS DU~

p-=-Dt

apyh+ (6E+ j x B)i + p grad @. axj

Multiplying this equation scalarly by u and subtracting the result from (A.16) we find (A.18)

p-

DU Dt

= -ei,pyh

- 6u E + j * (E+ u x B) + div (pcpK grad T ) .

Thus, by (A.3), (A.8) and the equation of continuity (2.1), pT-

DS Dt

=e,jpp+ (j - 6u).(E+ u x B)+ + T div (‘7 grad T).

PCPK (grad T ) 2

(A.19)

~

T

~

The first term on the right-hand side of (A.19) may be written (A.20)

and is essentially positive (or zero, in inviscid fluid). On assuming Ohm’s law (A.21)

j =6u

+ a(E + u x B),

the second term on the right-hand side of (A.19) may be written (A.22)

1 - (j - 6 (I

~ )or ~ a(E + u x B)2,

and is essentially positive (or zero, in a fluid of infinite electrical conductivity). Of the last two terms, the first is essentially positive (or zero, in a fluid of vanishing thermal conductivity) while the second is zero if divided by T and integrated over a region over whose boundary no heat flows. Thus, integrating over such a region (A.23)

1g P

dV

t 0,

in agreement with the second law of thermodynamics. If we ignore displacement currents, the term in E2 in equations (A.9, 10, 11, 13, 15) are negligible, as are the terms in 6 in equations (A.17, 18, 19, 21, 22). A full discussion of this non-relativistic case is given by Hide [55] for an incompressible fluid.

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

311

APPENDIXB : RELATIVISTIC MAGNETO-HYDRODYNAMICS In Section 111.3 and throughout most of this article, we have adopted the “quasi-equilibrium’’ approsimation to the electrodynamic equations in which displacement currents are neglected, and the space charge contribution 6E to the body force F is ignored in the equations of motion. In Sections IV and V.2, the full electrodynamic equations have been considered whenever the resulting complications did not give rise to obscurities. We have referred to such analyses as being “relativistic”, although this is not strictly correct; a full relativistic treatment must also allow for the possibility that the velocities, both random and ordered, of the material particles are comparable with c, the velocity of light. In this appendix we briefly consider this possibility for an ideal, perfectly conducting fluid. Nevertheless, it should be noticed that, as far as perturbation analyses are concerned, there is no contradiction in supposing that the frequencies are so large that displacement currents are not negligible, and yet that the macroscopic fluid velocity is small compared with c. Under these circumstances, the analyses presented in sections IV and V.2 are valid provided that the r.m.s. thermal velocities of the fluid particle are small compared with c. Even for a hot fluid, similar results hold if the density p is replaced by p p / c 2 (see below). The full electrodynamic equations are

+

Let 11s introduce the customary devise of representing space-time as a Euclidean &space in which the coordinators are (x,y,z,ict). Define the electromagnetic stress tensor Fli by the array

(B.5)

0

H,

- HY

- icD,

-H,

0

H,

- icD,

HY

-HH,

0

- icD,

icD,

icDy

icD,

0

and the current 4-vector J by

312

RAYMOND HIDE AND PAUL H . ROBERTS

Then equations (B.l) and (B.4) may be rewritten

j i = aFii/axi,

(B.7)

and equations (B.2) and (B.3) may be rewritten (B.8) Also, pFiiJi i s a four-vector closely related to the force density acting on a fluid element, since (J3.9)

PFajJj = (BE

(B.lO)

+ j X B)a,

(a = 1,2,3),

PFajJj = ij * E/c.

(In these equations, and in what follows, the summation convention has been adopted. Also, Latin letters have been used to denote suffixes which range over 1 to 4, and Greek letters for those which range over 1 to 3.) For a perfectly conducting fluid

E=-uxB,

(B.ll)

and (R.10) may therefore be rewritten as (B.12)

~FajJ= j iU * (jx B ) / c .

The electromagnetic energy 4-tensor is (cf. e.g. [55]) (B.13)

T!?m.) = / i FOiFaj - 7 dijFdFab $7

[

l

l

*

The electromagnetic stress tensor introduced in Section I V is c2T$jm.';the Poynting's vector E x H is cTti".'; the energy density in physical space is - T f i m . ) . The energy tensor [67], Ch. 8, eq. 178) (B.14)

of the material particles is of the form (cf.

Ti?'' = pUiUj - Sij/c2,

where Sii is the mechanical stress tensor, U, is the 4-velocity dx,/ds of the fluid and p is the rest density. In an ideal fluid Sij takes the form (cf. [55], Ch. 8, eq. 206) (B.16)

Sij = - fi(d;j

+ UiUj),

313

ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS

where fi is the pressure in the frame locally at rest with respect to the fluid. The total energy tensor for the fluid when an electromagnetic field is present is (B.16)

T$7. . T!e,m“.) + T!Yt), - :7

and the equations governing its motion is (B.17)

aTiilaxi = 0.

For the process of reducing this equation to more familiar forms, the reader is referred to [57]. The final results of reducing (B.16)and (B.17)to physical space are the equations

aP

1

aP

(B.19)

x

B),,

at

where 3/ =: (1 - u2/c2)-”9. These are the required relativistic generalizations of the equations of motion and continuity. Note that, if u is small compared with c, they reduce to (B.20)

j x B,

(B.21) These are identical to Euler’s equation and the equation of continuity plc2. provided p is replaced in these equations by p

+

ACKNOWLEDGEMENTS

We wish to record our thanks to Dr. I. D. C. Gurney for discussions relating to Appendix A and to Professor P. A. Sweet for discussion relating to Ch. V I I and for drawing our attention to a paper by McCrea [44].

References 1. ALFVI~N, H., “Cosmical Electrodynamics”. Clarendon Press, Oxford, 1950. 2. COWLING, T. G., Solar electrodynamics, in “The Sun” (G. P. Kuiper, ed), Ch. 8, pp. 532-591. University Press, Chicago, 1952. 3. COWLING, T. G., “Magnetohydrodynamics”. Interscience Publishers, New York, 1957.

314

RAYMOND HIDE AND PAUL H. ROBERTS

4. DUNGEY,J. W., “Cosmic Electrodynamics”. University Press, Cambridge, 1958. 5. LUNDQUIST, S., Studies in magnetohydrodynamics, Ark. Fys. 6, 297-347 (1952). 6. SHERCLIFP, J. A., The flow of conducting fluids in circular pipes under transverse magnetic fields, J. Fluid Mech. 1, 6 4 4 4 6 0 (1956). 7. DOLDER,K., and HIDE, R., Experiments on the passage of a shock wave through a magnetic field, Nature, 181, 1116-1118 (1958); Rev. Mod. Phys. 82, 770-779 (1960). S., and LOVBERG, R. H., “Controlled Thermonuclear Reactions”. 8. GLASSTONE, von Nostrand, New York, 1960. 9. LINDLEY, B. C., Large scale power by direct conversion, Engineering 189, 567-569 (1960). 10. RESLER.E. L., and SEARS,W. R., The prospects for magneto-aerodynamics, J . Aero. Sci. 4, 235-246 (1958). 11. HOWARTH, L., “Modern developments in fluid dynamics: high speed flow”. 2 vols., Clarendon Press, Oxford, 1953. 12. CULLWICK, E. G., “The fundamentals of electromagnetism“. University Press, Cambridge, 1939. 13. SPITZER,JR., L., “Physics of fully ionized gases”. Interscience Publishers, New York. 1956. 14. CHAPMAN, S., and COWLING, T. G., “The Mathematical Theory of Non-Uniform Gases”. 2nd Edition, University Press, Cambridge, 1952. 15. SCHLUTER, A., Ionized gas in a magnetic field, in “Electromagnetic phenomena in cosmical physics” (B. Lehnert, ed.), Proc. I.A.U. Symp. 6, pp. 71-80. 1957. 16. LEHNERT, B., Plasma physics on a cosmical and laboratory scale, Nuovo Cimento 18, Serie X, 56-110 (1959). 17. LAMB,H., “Hydrodynamics”. 6th Edition. University Press, Cambridge, 1932. 18. GOLDSTEIN,S., “Modern developments in fluid dynamics”. 2 vols., Clarendon Press, Oxford, 1938. 19. COWLING, T. G., The magnetic field of sunspots, Mon. Not. R . Astron. SOC.94, 39-48 (1934). 20. LARMOR, J., How could a rotating body such as the Sun become a magnet ? Rep. Brit. Ass. 159-160 (1919). 21. HIDE, R., and ROBERTS,P. H., The origin of the main geomagnetic field, in “Physics and Chemistry of the Earth” (Ahrens, Press, Rankama, Runcorn, ed.) vol. 4, Chap. 2. Pergamon Press, London, 1961. G. K., On the spontaneous magnetic field in a conducting liquid in 22. BATCHELOR, turbulent motion,. Proc. Roy. SOC. Lond. A, 901, 405-416 (1950). 23. A L F V ~ N H., , On the existence of electromagnetic-hydrodynamic waves, A r k . f . mat. astr. o. fysik. 29B, No. 2 (1942). The existence of electromagnetic-hydrodynamic waves, Nature 160, 405-406 (1942). 24. GREENSPAN, H. P., and CARRIER,G. F., The magneto-hydrodynamic flow past a flat plate, J . Fluid Mech. 6, 77-96 (1959). K., Motion of a sphere through a conducting fluid in the presence 25. STEWARTSON, of a strong magnetic field, Proc. Camb. Phil. SOC.62, 301-316 (1956). 26. RESLER,JR., E. L., Some exact solutions in linearized magneto-aerodynamics for arbitrary magnetic Reynolds numbers, Rev. Mod. Phys. 82, 848-854 (1960). LIN, C. C., Note on a class of exact solutions in magneto-hydrodynamics. Arch. Rational Mech. Anal. 1, 391-395 (1958). 27. PROUDMAN, J., On the motion of solids in a liquid possessing vorticity, Proc. Roy. SOC.A82, 408-424 (1916). TAYLOR, G. I., Experiments on the motion of solid bodies in rotating fluids, Proc. ROY. SOC.A104, 213-218 (1923).

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28. STEWARTSON, K., On the motion of a non-conducting body through a perfectly conducting fluid, J , Fluid Mech. 8, 82-96 (1960). Motion of bodies through conducting fluids, Rev. Mod. Phys. 82, 855-859 (1960). 29. ELSASSER,W. M., The hydromagnetic equations, Phys. Rev. 79, 183 (1950). 30. FRIEMAN, E. A,, and KULSRUD,R. M., Problems in hydromagnetics, Advances i n Applied Mechanics 6 , 195-231 (1958). 31. VAN DE HULST,H. C., Interstellar polarization and magneto-hydrodynamic waves, i n “Problems of cosmical aerodynamics”, p. 45. (Proc. Symp. Int. Un. of Theor. and Appl. Mech., Paris 1949.) Central Air Documents Office, Dayton, Ohio, 1951. 32. BAN&, JNR.,A., Magneto-hydrodynamic waves in incompressible and compressible fluids, Proc. Roy. SOC.A288, 350-366 (1955). WILLIAMS,W. E., Propagation of small amplitude magneto-hydrodynamic waves, Q. J . Mech. 6% Appl. Math. 18, 272-277 (1960). B., Magneto-hydrodynamic waves under the action of the Coriolis force I, 33. LEHNERT, Astrophys. J . 119, 647-654 (1954), - 11, Astrophys. J . 121, 481-490 (1956). 34. HIDE, R., and ROBERTS,P. H., Hydromagnetic flow due to an oscillating plane, Rev. Mod. Phys. 82, 799-806 (1960). 35. COLE, J. D., Magneto-hydrodynamic Waves, i n “The Magneto-hydrodynamics of conducting fluids” (D. Bershader, ed.), pp. 17-36. Stanford. G . S. S., The propagation of small disturbances in hydromagnetics, LUDFORD, J . Fluid Mech. 6 , 387-400 (1959). 36. ROBERTS,P. H., Hydromagnetic disturbances in a fluid of finite conductivity, I, Astrophys. J . lee, 315-326 (1955), - 11, Astvophys. J . 126, 418-428 (1957). Propagation of induced fields through the core, Ann. Geophys. 16, 75-86 (1958). 37. DE HOFFMAN, F., and TELLER,E., Magneto-hydrodynamic shocks, Phys. Rev. 80, 692-703 (1950). 38. MARSHALL, W., The structure of magneto-hydrodynamic shock waves, Proc. ROY. SOC.A288, 367-376 (1955). 39. HELFER,H. L., Magneto-hydrodynamic shock waves, Astrophys. j.117, 177-199 (1953). SHERCLIFF,J. A., One dimensional magneto-gas-dynamics in oblique fields, J . Fluid Mech. 9, 481-505 (1960). LOST, R., Magneto-fluid dynamic shock waves, Rev. Mod. Phys. 82, 706-709 (1960). GERMAIN, P., Shock waves and shock wave structure in magneto-fluid dynamics, Rev. Mod. Phys. 82, 951-958 (1960). LUDFORD, G. S. S., The structure of a hydromagnetic shock in steady plane motion, J . Fluid. Mech. 6, 67-80 (1959). 40. HIDE, R., Waves in a heavy viscous incompressible electrically conducting fluid of variable density, in the presence of a magnetic field, Proc. Roy. SOC.A288, 376-396 (1955). 41. ROBERTS,P. H., and BOARDMAN, A. D., The effect of a vertical magnetic field on the propagation of gravity waves along the plane surface of a semi-infinite viscous, electrically conducting fluid, Astrophys. J . , 186, 552-592 (1962). 42. TALWAR, S. P., Hydromagnetic stability of a conducting inviscid incompressible fluid of variable density., Zeit. jiir Astrophys. 46, 161-168 (1958). TALWAR, S. P.. Stability of a conducting rotating fluid of variable density, J . Fluid Mech. 9, 681-592 (1960). 43. HIDE, R., The character of the equilibrium of a heavy, viscous, incompressible, rotating fluid of variable density; I, General Theory; 11, Two special cases, Quart. J . Mech. 6.Appl. Math. 9, I, pp. 22-34, 11, pp. 35-50 (1956).

316

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44. MCCREA,W. H., The formation of population I stars. Part 1. Gravitational contraction, Mon. Not. R . Astr. SOC.117, 562-578 (1957). SPITZER,JNR., L., The birth of stars from interstellar clouds, J . Washington Acad. Soc. 41, 309-318 (1951). 45. JEANS,J . H., On the stability of a gaseous nebula, Phil. Trans. R . Soc. Lond. A199, 1-49 (1902). “Astronomy and Cosmogony”. University Press, Cambridge, 1929. S., and FERMI,E., Problems of gravitational stability in the 46. CHANDRASEKHAR, presence of a magnetic field, Astrophys. J . 118, 116-141 (1953). CHANDRASEKHAR, S., The gravitational instability of an infinite homogeneous medium when Coriolis force is acting and a magnetic field is present, Astrophys. J . 119, 7-9 (1954). CHANDRASEKHAR, S., The gravitational instability of an infinite homogeneous medium when a Coriolis acceleration is acting, in “Vistas in Astronomy” (A. Beer, ed.), pp. 344-347. Pergamon Press, 1955. FRICKE,W., On the gravitational stability in a rotating isothermal medium, Astrophys. J . 190, 356-359 (1954). 47. GLOBE,S., Laminar steady state magneto-hydrodynamic flow in an annular channel, Phys. of Fluids 2, 404-407 (1959). 48. RAYLEIGH, LORD,“The Theory of Sound”. Reprinted by Dover Publications, New York, 1945. RAYLEIGH, LORD,Scientific Papers, volume 6. University Press, Cambridge, 1920. 49. KAKUTANI, T., Effect of transverse’magnetic field on the flow due to an oscillating flat plate, I ; 11, J . Phys. Soc. Japan 18, 1504-1509 (1958); 15, 1316-1331 (1960). AXFORD, W. I., The oscillating plate problem in magneto-hydrodynamics, J . Fluid Mech. 8, 97-102 (1960). 50. LUDFORD, G . S. S., Rayleigh’s problem in hydromagnetics: the impulsive motion of a pole-piece, Arch. Rat. Mech. 6. Anal. 8, 1 P 2 7 (1959). 51. CHANG,C. C., and YEN, J . T., Rayleigh’s problem in magneto-hydrodynamics, Physics of F l t d s 2, 393-403 (1969). 52. CARSLAW, H. S., and JAEGER,J . C., “Operational methods in applied mathematics”. University Press, Oxford, 1941. G . S., “Introduction to statistical mechanics”. Clarendon Press, 53. RUSHBROOKE, Oxford, 1949. 54. LIGHTHILL, M. J., Viscosity effects in sound waves of finite amplitude i n “Surveys in Mechanics” (Batchelor and Davies, eds.) University Press, Cambridge, 1956. 5 5 . HIDE, R., Hydrodynamics of the Earth’s core, in “Physics and Chemistry of the Earth” (Ahrens, Rankama and Runcorn, eds.), vol. 1, Ch. 5 . Pergamon Press, London, 1956. 66. CHANDRASEKHAR, S., “Hydrodynamic and hydromagnetic stability”, Oxford : Clarendon Press, 1962. 57. SYNGE,J. L., “Relativity: the Special Theory”, North-Holland Publishing Co., Amsterdam, 1956.

ADDENDUM TO:

Hypersonic Flow over Slender Bodies Associated with Power-Law Shocks BY HAROLD MIRELS Note added in proof Reference [49] is an important recent contribution to the theory of hypersonic flow associated with slender power law shocks. The method of “inner and outer expansions” is used therein to obtain uniformly valid solutions far downstream from the blunt nose of slender bodies. The inner expansion describes the flow in the entropy layer, the outer expansion describes the flow external to the entropy layer and the two expansions are combined to give a uniformly valid description of the entire far downstream flow, Explicit expressions were obtained for the asymptotic shape of the body, as x -+ m, corresponding to a given one- or two-dimensional blast wave. The asymptotic body shape, corresponding to a power law shock, can also be found by the methods of the present paper. The procedure is indicated herein, not only for blast waves, but for all power law shocks where the entropy layer effect must be considered (ie., [ y / ( y l)]< jl 1). Dimensional variables are used. The limiting case M + 00 and (R‘)2<( 1 is considered. The method used is to find the difference between the body shape as indicated by zero order similarity theory (3.26a) and by the continuity integral (5.9). For << 1 and a power law shock, 0 = (R’/Ri’)z/p, the latter becomes

+

<

Let Vb.0 represent the body location according to zero order similarity theory. The difference between (3.26a) and (A) is then

where the higher order term in (A) has been omitted. For (R’)2small, the main contribution to (B) comes from the integration in the vicinity of 0 M 0

317

318

HAROLD MIRELS

(i.e., the entropy layer) and f can be replaced by the zero order wall pressure in each integral. Substituting z = r9-a(R')zinto the second integral, integrating by parts and taking the limit ( R ' ) 2 + 0 then gives

/b,o

where T( ) is the gamma function. The error in (C) is of the order of the second term or the right hand side of (A). Equation (C) gives the asymptotic body shape, as x --* (M,associated with a power law shock in the range where the entropy layer is important (i.e., y / ( y 1)< fi f 1 as discussed in Section VI) and is the desired result. The difference between the actual body location, rb/R, and the zero order similarity body location, r ] b , o , goes to zero as (R')2---r 0. For fi = 1, equation (C) becomes

+

This is in exact agreement with the results of [49] which were obtained by the method of inner and outer expansions. Thus Y b x2I3Y and N x'lzv for the one dimensional and two dimensional blast waves, respectively. The constants of proportionality are found from (D). The asymptotic body shape is not similar to the shock shape, except for y = 1, due to the factor y in the denominator of the exponents of x . Equation (C) can be used to find the asymptotic body shape associated with any given power law shock. The pressure distribution on the body is found from the zero order similarity solution. This equation can also be used to find the asymptotic shock shape and surface pressure distribution associated with a given power law body (ie., direct problem) in the manner described in Section VI.1. Some recent work [50-521 might also be briefly mentioned. Van Hise [50] has systematically studied the shock shapes and surface pressures associated with a series of long slender bodies of revolution having varying nose bluntness. The flow field was found, using the method of characteristics, for air (perfect gas assumption) and helium at Mach numbers from 5 to 40. Nose fineness ratios (base diameter to length) were varied from 0.4 to 4. Flow parameters obtained from zero order and first order blast wave theory were used as a guide to obtain good correlations of the surface pressure and shock N

HYPERSONIC FLOW OVER SLENDER BODIES

319

shape. The correlations of Van Hise [50], and similar correlations of Lukasiewicz [€ill, appear to be the most extensive presently available. Inger [52] has examined the similitude requirements for nonequilibrium dissociated diatomic gas flows over blunt nosed slender bodies a t hypersonic speeds. (Recall that Cheng [24] treated equilibrium real gas flows.) The free stream was taken to be in an arbitrary state of dissociative nonequilibrium, as may exist in high temperature hypersonic tunnels. The flow downstream of the shock was also taken to be out of equilibrium and the conditions under which different flows are similar was examined. Such studies are expected to become increasingly more important, both for correlation of ground tests with flight and for more accurate estimates of the disturbed flow field generated by high speed-high altitude vehicles.

Additional References 49. YAKURA, J. K., A theory of entropy layers and nose bluntness in hypersonic flow, Amer. Rocket SOC., Preprint 1983-61, 1961. 50. VANHISE, V., Analytic study of induced pressure on long bodies of revolution with varying nose bluntness a t hypersonic speeds, N A S A TR R-78, 1961. 51. LUKASIEWICZ, J., Blast-hypersonic analogy-theory and application, Amer. Rocket SOC., Preprint 2169-61, 1961. 52. INGER,G. R., Nonequilibrium hypersonic similitude in a dissociated diatomic gas. Douglas Report SM-38972, 1961.

This Page Intentionally Left Blank

Author Index Numbers in parenthese are reference numbers and are included to assist in locating references when the author’s names are not mentioned in the text. Numbers in italic refer to the paKe on which the complete reference is listed.

A Abrikosova, I. I., 77(64, 65), 727 Adamski, V. B.. 3(21), 16(21), 19(21), 20(21). 21(21), 23, 24(21), 26(21), 53 Alekseyenko, Yu. N., 135(50), 209 Alexandrov, A. P., 77(62), 85(62), 127 AIfvBn, H.. 217(I), 225(1), 230, 245(1, 23). 313, 314 Averbach, B. L., 85(78), 728

B BabuSka, I., 188, 206 Baker, B. R., 123, 124(105), 729 Banos, Jnr.,A., 245(32), 375 Baradell. D. L., 42(43), 49(43), 54 Barenblatt, G. I., 67(39, 40, 41), 69(40, 56, 57, 58, 59, 60, 61). 73(40), 74(60), 79(56, 57), 82157, 58), 92(58), 93(58), 96(56, 57, 84), 97(56, 57), l02(59), 109(58, 88), 110(88), 111(88), 113(40), 114(39, 40, 41). 115(39, 58, 95), 116(61), 118(84, 95), 123(84, 95). 1241.57, 95), 726, 128, 129 Barta, J.. 206 Batchelor, G. K.. 229(22). 374 Beedle, L. S., 194(3), 206 Benbow, J. J.. 65(9), 68(9), 86(54), 87, 119, 120, 125, 127 Berggren, R. G., 135(125), 136(125), 137(125), 138(125), 139(125). 214 Berman, I., 145(4), 206 Bertram, M. H., 29(37), 42(43), 49(43), 51(47), 54 Bilby, B. A,, 124, 129 Bitsadze, A. V., 116, 129 Blewitt. T. H., 136(46). 137(46), 209 Boardman, A. D., 267(41), 375 Bogdonoff, S. M., 61(48), 54 Bohnenblust, H. F., 149, 206 Borisov. K. A., 135(111), 213 Born, M. Kun Huang, 77(67), 128 Bowie, 0. L.. 65(22), 103, 726

32 1

Brinkman, J. A , , 134(6). 135(6), 206 Broberg, K. B., 124, 129 Brocher. E. F., 18, 54 Bruch, C. A., 138(7), 207 Bueckner, H. F., 66, 68(33, 50), 70, 74, 106, 726, 127 Bullough, R., 124, 729 Bupp. L. P., 135(125), 214

c Campbell, J. D., 146, 207 Campus, F., 207 Cap, F., 143(10), 144(10), 207 Carrier, G. F., 230(24), 314, 319(24) Carslaw, H. S., 297(52), 298(52), 376 Casaccio, A., 49(44), 54 Chandrasekhar, S., 271(46), 316 Chang, C. C., 287(51), 300, 316 Chapman, S., 224(14), 314 Chechulin, B. B., 207 Cheng. H. K., 2(8). 3, 8, 29(23), 43, 44(24), 48, 49(23), 51(23), 5 3 , 319(24) Cherepanov, G. P., 96(84), 109(88), 110(88), l l l ( 8 8 ) , 115(95), 118(84, 95). 123(84. 95), 124(95), 128, 129 Chernyi, G. G., 3, 34, 41, 42(15, 16, 22). 53 Cohen, C. B., 30(38), 54 Cole, J. D., 33, 38, 54, 245(35), 375 Colonnetti, G., 149, 207 Cowling, T. G., 217(2, 3). 224(14), 228, 245(3), 285(3). 313, 374 Craemer, H., 196, 207 Craggs, I. W., 123, 124(103), 129 Creager, M. 0.. 51(46), 54 Cristesen, N., 201, 203, 207 Cullwick, E. G., 222(12), 374

D Dang Dinh An, 123, 124(104), 729 Davidenkov, N. N., 65(12). 85(75), 119, 125. 128 Davison, B., 140(15), 207

322

AUTHOR INDEX

de Hoffman, F., 245(37), 375 Deresiewicz, H., 206(16), 207 Deryagin, B. V., 77(64, 65). 727 de Saint-Venant, B., 153, 154, 273 Dimes, G. J . , 134(17), 135(17), 207 Dolder, K., 218(7), 374 Drozdovskii, B. A., 69, 85(55), 727 Drucker, D. C., 200, 207 Druyanov, B. A., 171, 174, 175(20a, 20b). 176(20), 207 Dungey, J. W., 217(4), 245(4), 374 Duwez, P., 149, 206 Dzugutov, M. Ya., 141(128), 142(128), 274

E Eichberger, L. C., 187, 274 Elliot, H . A., 65, 66(35), 96(16), 725, 726 Elsasser, W. M., 245(29), 375

F Faris, F. E., 138(21), 207 Fehlbeck, D. K., 65(28), 88, 726 Feldman, S., 3, 4(26), 49(26), 53, 54 Feller, W. V., 29(37), 54 Fletcher, J. F., 135(125), 274 Foulkes, J., 200, 207 Frankland, I. M., 89, 728 Freeman, N. C., 18, 19(31), 39, 54 Freiberger, W., 200, 207 Frenkel, Ya. I., 84(5), 66, 67(5), 92, 120, 125 Freudenthal, A. PI., 143(25b), 206, 207, 208 Fricke, W., 271(46), 376 Fridman, Ya. B., 69, 85(55), 727, 147, 212, 274 Frieman, E. A,, 245(30), 375

Golian, T. C., 3(23), 29(23), 48(23), 49(23). 50(23), 5 3 Gonor, A. L., 33, 54 Greenspan, H. P., 230(24), 374, 319(24) Griffith, A. A , . 63, 64(3), 85, 92, 125 Grigorian, S. S., 3(19), 25(19). 26(19), 5 3 Grigoryev, A. S., 192, 208 Grodzovskii, G. L., 3, 5 3 Guderley, G., 26(36), 54 Guindin, I. A., 77(63), 727 Gutowski, R., 202, 208

H Hafele, W., 19(34), 21(34), 28(34), 54 Hall, J . G., 3(23), 29(23), 48(23), 49(23), 51(23), 53 Hayes, W. D., 2, 3(11), 8(11), 18(11), 21(11), 31(11), 49(11), 52, 53 Helfer, H. L., 245(39), 375 Henderson, A. Jr., 51(45, 47), 54 Hertzberg, A., 3(23), 29(23), 48(23), 49(23), 50(23), 5 3 Heyman, J., 193, 200, 208 Hide, R., 218(7), 228(21), 245(21, 34), 254(40), 270(43), 287(34), 296(34), 310, 312(55), 374, 375, 376 Hill, R., 146(34b), 206(34f), 208 Hockenburg, R. W., 138(7), 207 Hodge, P. G., Jr., 192, 193(36b),194(35c), 199, 200, 208, 273 Hopkins, H . G., 200, 208 Horne, M. R., 200, 208 Howarth, L., 219(11), 374 Hruban, K., 188, 208 Hu, L., 147, 208 Huber, M. T., 146(40a), 208 Hundy, B. B., 198(4la), 208

I G Galin, L. A,, 81(72), 116(72), 124, 728. 154, 208 Garber, R. I., 77(63), 727 Gatewood, B. E., 143(28), 208 Geiringer, H . , 151, 154, 208, 208 Germain, P.. 245(39), 375 Gigon, J., 135(59), 209 Gilman, J . J.. 65(11), 121, 124, 725 Glasstone, S., 218(8), 374 Glen, J . W., 134(30), 208 Globe, S., 285, 376 Goldstein, S., 226(18), 374 Golecki, J., 174, 271

Ilyushin, A. A., 138(44a), 142(44b), 143(43, 44b). 169(42b), 174(42a), 183, 208 Inger, G. R., 318(52), 319, 379 Inglis, C. E., 62(l), 64, 66(1), 725 Ivelev, D. D., 185, 209 Irwin, G. R., 65(23, 25, 26), 68(47, 48, 51). 73(45, 46, 47), 74, 84, 88, 89(48), 104, 105, 109, 111(47), 726, 727

J Jaeger, J. C., 297(52), 298(52), 376 Jamiston, R. E., 136(46), 209 Janas, M., 193, 194, 273

323

AUTHOR INDEX Jeans, J. Johansen, Johnston, Jongh, J.

H., 270, 316 K. W., 197, 209 Patrick J., 51(45), 54 G . V., 77(66), 728

K Kachanov, L. M., 101, 128 Kakutani, T., 287, 316 Kaliski, S., 202(48), 208, 209 Kashdan, Ia. M., 19(35), 54 Katchanov, L. M., 209 Khristianovitch, S. A., 67(38, 39), 74, 113, 115(39), 126 Kies, J. A., 65(25, 26). 68(48, 49), 89(48). 126. 127 Klimenkov, V. I., 135(50), 209 Kochina, N. N., 3(20), 26(20), 5 3 Kogler, F., 209 Koiter, W. T.. 109, 129 Konig, J . A., 199(107), 213 Konobeyevsky, S. T., 135(52), 209 Krupkowski, A., 147, 209 Kubota, T., 2(9), 3, 11(10), 12(10), 17, 26(10), 29(10), 35(10), 51(10), 5 3 Kukudzhanov, V. N., 203, 209 Kulsrud, R. M., 245(30), 315 Kultigin, V. S., l41(128), 142(128), 214 Kutaycev, V. I., 135(52), 209 Kuznetsov, V. D., 119, 129 Kuznetzov, A. I., 154, 155(55a), 167(55a, 55b), 168(55c), 169(55c), 171, 173 (55a). 176, 189(55b), 209 Kwiecinski, M., 197, 213

L Lamb, H., 225(17), 250(250), 314 Landau, L. D., 81(71), 128 Larmor, J., 228, 314 Latter, R., 13, 17, 54 Lee, Ming-Hua, 187, 209 Lees, L., 2(9), 53 Lehnert, B., 224(16). 245(33), 374, 315 Leibfried, G., 77(68), 128 Lensky, V. S., 135(57b),138(57b),143(43), 177, 183, 201. 208, 209 Leonov, M. Ya., 78, 128 LBvy, M., 154, 209 Liftshitz, E. M., 77(64), 81(71), 127, 128 Lighthill, M. J., 307(54), 316 Lin, C. C., 230(26), 314 Lin, S. C., 2(7), 53 Lindley, B. C . , 218(9), 314

Lovberg, R. H., 218(8), 314 Lubinskaya. M. A., 141(128), 142(128), 214

Ludford, G. S. S., 245(35, 39), 287(50),315 Lukasiewicz, J., 318(51), 319, 319 Lundquist, S.. 217(5), 226(5), 245(5), 267(5), 374 Lust, R., 245(39), 315

M McClintock, F. A., 124, 729 McCrea, W. H., 270(44), 313, 316 Mc Hugh, W. E., 138(7), 207 Marin, J.. 147, 208 Markuzon, I. A., 118, 119(98), 729 Marshall, W., 245(38), 261(38), 315 Massonet, C., 207 Masubuchi, K., 91. 128 Mayer, G., 135(59), 209 Mel’nikova, N. S., 3(20). 26(20), 5 3 Metsik, M. S., 65(10), 119, 125 Michell, J. H., 180, 209 Mikeladze, M. Sh., 200, 209 Mikhlin, S. G., 108, 128 Mirels, H., 3(18), 4(29), 11(17), 12(17), 14, 21, 26(17, l a ) , 27(18), 28(18), % ( l a ) , 30(18), 32(18), 33(18), 34(17), 35(17), 36(17), 39, 53, 54 Mokhalov, S. D., 202, 209 Mott, N. F., 66, 121(36), 122(36), 124(36), 126 Mr6z, Z., 182, 197, 199, 270, 211 Murzewski, J., 174, 190, 191(76a, 76d), 19.5, 210, 211 Muskhelishvili, N. I., 62(2), 65, 70, 71(19), 75, 90, 110, 116(18), 117, 125, 7 29

N Naghdi, P. M., 147, 210 NemBnyi, P. F., 177, 178(67), 210 Neuber, H., 89, 128 Niepostyn, D., 197, 210 Nikitin, L. V., 203, 209 Novozhilov, V. V., 158(69), 177(69), 210 Nowacki, W., 145(70), 152, 210 Nowinski, J., 148, 150(72), 210

0 Obreimov, I. V., 64, 65(8), 68, 119(8), 125 Ogibalov, P. M., 138(44a), 142(44b), 143(44b), 208

324

AUTHOR INDEX

Olszak, W., 132(74), 133(73e, 73f, 73h, 73i. 73j). 146(73f, 73g, 73h, 73i, 73j), 147(73f, 73g, 73h, 73i). 148(73k, 731, 82h, 82i), 149(73k, 731, 82h, SZj), 150(73k. 731, 82h. 82i), 152(79), 154(80c), 160(80a), 165(80a, 80b), 167(Sob), 170(82a, 82b), 172 (80a), 174, 177(80c),179(83c), 182, 183, 185, 187, 189, 190, 195173k, 731). 197, 198, 199(109), 203, 204(73a), 205(73k, 731, 73r, 73y), 206(73u, 73x. 83c, 83d), 210, 211, 212, 213 Onat, E. T., 187, 200, 212 Orowan, E. O., 65(24, 27, 28), 88, 126 Osiecki, J., 202(48), 203(88b), 208, 209, 212 Ostanenko, A . V., 141(128), 142(128), 214

P Pallone, A. J,, 2(8), 53 Panasyuk, V. V., 78, 128 Panferov, V. M., 184, 212 Parker, E. R., 85(74), 128 Pashkov, P. O., 85(76). 128 Pelczynski, T., 144, 212 Perio, P., 135(59), 209 Perzyna, P., 152(79). 201(9la, g l b ) , 203(91c), 211, 212 Poletzky, A. T., 147, 212 Popov, N. A., 3(21), 16(21), 19(21), 20(21), 21(21), 23, 24(21), 26(2l), 53 Popov, N . I., 147, 212 Potak, Ya. M., 85(77), 128 Prager. W., 150, 200, 208, 212, 213 Pravdyuk, N. F., 135(52). 209 Primak, W., 213 Probstein, R . F., 3(11), 8(11), 18(11), 21(11), 31(11). 49(11), 53 Proudman, J., 232(27). 314 Pui, Ming-Li, 187, 209

R Radok, J . R . M., 124, 129 Rakhmatulin, Kh. A., 201, 213 Rayleigh, Lord, 286(48), 296, 316 Rebinder, P. A., 66(34), 126 Remnev, Yu. I., 141(100), 213 Reshotko, E., 30(38), 54 Resler, Jr., E. L., 230(26), 314 Resler, E. L., 218(10).. 314 Rivlin, R . S., 208(101), 213 Roberts, D. K., 112(101), 124(101), 729

Roberts, P. H., 228(21), 245(21, 34, 36). 267(41), 287(34, 36), 296(34), 297, 300, 314. 315 Roesler, F. C., 65(9), 68(9), 86(53), 87(53), 119, 120. 125, 127 Rogozinski, M., 189, 213 Romualdi, J. P., 68(52), 69, 94, 95(52), lOO(52). 127 Rowley, J . C., 147, 210 Rushbrooke, G. S., 306(53), 316 Rychlewski, J., 154(80c), 160(80a), 165 (Boa, Sob), 167(80b), 172(80a), 177 271, 213 ( ~ O C103a), , Rzhanitsyn, A. R., 67(37), 126

S Sack, R. A., 65(20), 66(20), 97, 126 Sakurai, Akira, 2, 3, 12(5), 21, 22(5), 5 3 Sanders, P. H., 68(52), 69, 94, 95(52), 100(52), 127 Sawczuk, A., 170, 172(105d), 178(105d), 193, 194, 195, 197, 198, 199(107, 109). 211, 213 Scheidig. A., 209 Schliiter, A., 224(15), 314 Schumann, W., 200, 272 Sears, W. R., 218(10), 314 Sedov, L. I., 3(12), 13, 16, 17(12). 26(12), 45, 53, 86(79), 93(79), 128 Sergeyev, G. Ya., 135(111), 213 Seth, B. R., 188, 189, 213 Shapiro, G. S., 201, 213 Shercliff, J. A., 217(6), 245(39), 285(6), 314, 315 Sherman, D. I., 108, 110, 128, 129 Shield, R. T., 200, 207, 212 Shtaerman, I. Ya., 81(73), 116(73), 728 Singer, A. R. E., 198(41a), 208 Smekal, A , , 64, 125 Smith, A. MO., 30(39), 54 Smith, H. L., 68(48, 49), 89(48), 127 Sneddon, I. N., 65, 73(14, 15), 96(16), 125 Sobelov, N. D., 147, 214 Sobotka, Z., 205, 214 Sokolovski, V. V., 204, 214 Sokolowski, M., 145(116), 214 Spitzer, Jr., L., 223(13), 224(13), 226(13), 314 Stanyukovich, K. P., 3, 53 Steel, M. C., 187, 214 Stepien, A., 170, 213 Stewartson, K., 230(25), 232, 243(28), 374, 315

325

AUTHOR INDEX Stroh. A. N., 96(83), 123, 124(102), 728, 129 Suits, J. C., 124, 729 Sukhatme, S. P., 124, 729 Sychev, V. V., 3, 39, 40, 43, 44(25), 45(25), 53 Synge, J. L., 312(57), 313(57), 376 Szymaliski, C., 152(79), 205, 277, 274

T Talwar, S. P., 267(42), 375 Taylor, D. B., 146, 274 Taylor, G. I., 2, 21, 52, 238(27), 374 Tekinalp, B., 200, 207 Teller, E., 245(37), 375 Thornton, P. R . , 3(18), 26(18), 27(18), 28(18), 29(18). 30(18), 32(18), 33(18), 53 Titova, V. V., 135(111), 273 Tournarie, M., 135(59), 209 Trella, M., 4(28), 49(28), 50(28), 5 4 Truszkowski, W., 146(123), 274 Turski, S., 150(72), 270

U Urbanowski, W., 148(82h, 82i), 149(82h, 82j). 150(82h, 82i). 170(82a, 82b), 185, 187, 189, 277, 272 Ustinov, Yu. A., 114, 729 Uzhik. G. V., 144(124), 274

V Vaglio-Laurin, R., 4(28), 49(28), 50(28),54 van de Hulst, H . C., 245(31), 315 Van Dyke, M. D., 2, 4(2), 6, 5 2 Van Hise. V., 318(50), 319, 379 Vas, I. E., 51(48), 54 Vineyard, G. H.. 134(17). 135(17), 207

Vinograd, M. I., 141(128), 142(128), 274 von Hoerner, S., 19(33), 54 van Mises, R., 209

W Wells, A. A ,, 65(29, 30, 31). 68, 122(101), 124(lOl), 726, 129 Westergaard, H. M., 65, 68(13), 73, 74, 81, 109, 110, 725, 727 Wigglesworth, L. A., 104, 106, 728 Williams, M. L., 65, 73, 125 Williams, W. E., 245(32), 375 Willmore, T. J., 65(21), 96, 110(21), 726 Wilson, J. C., 135(125), 136(125), 137 (125), 138(125), 139(125), 214 Winne, D. H., 65(32). 68(32), 107, 110(32), 7 26 Wolf, K., 64(7). 725 Woods, W. K.,135(125), 274 Wundt, B. M., 65(32), 68(32), 107, llO(32). 112, 726, 729

Y Y a Kura, T. K. 317(49), 318(49), 379 Yen, J. T., 287(51), 300, 376 Yoffe, E., 122, 729 Young, J., 187, 274 Yusuff, S., 112, 129

Z Zahorski. S., 179(83c), 272 Zakharov, A. I., 134(127), 274 Zheltov. Yu. P.. 67(38, 42, 43), 74(38), 113, 114, 726, 727 Zhurkov, S. P., 77(62), 85(62), 127 Zhukov, A. I . , 19(35), 54 Zuyev, M. I., 141, 142(128). 274 Zyczkowski, M., 190, 203, 206, 212. 274

Subject Index A

E

Alfvbn waves, 245 ff. in systems of finite extent, 261 ff. Ampere’s law, 222 Angle-of-attack effects (hypers. fl.), 31 ff.

Effective electric field, 223 Energy considerations in crack-boundary conditions, 84 f. Energy equation in MHD, 305 ff. Entropy layer (hypers. fl.), 43 ff., 48 Equilibrium cracks, basic hypotheses, 76 ff. boundaries of, 74 boundary conditions, 81 stress and strain at, 73 f. structure of, 69 ff. Equivalent steady and unsteady flows, 7

B Baroclinic, 220 Barotropic, 220 “Beta” in MHD, 229 Biharmonic states of equilibrium, 185 ff. Blast wave, 12, 16 ff. Blunt-nose effect, 33 f. Boundary conditions in MHD, 233 ff. at solid insulating surface, 243 f. electromagnetic, 234 ff. mechanical, 238 Boundary-layer effects (hypers. fl.), 29 ff. Brittle fracture, 82 experimental confirmation of, 86 ff.

F Faraday’s law, 222 Finite Mach number, 46 Flow between parallel planes (MHD), 274 ff. Flow due to oscillating plane in MHD, 286 ff. limiting cases, 292 ff. Flow fields associated with power-law shocks, 23 ff. “Frozen” field lines, 225

C “C-classification”, 149 Cohesion, forces of, 76 ff. modulus of, 80 f . Continuity integral (hypers. fl,), 37 ff. “Counter pressure”, 34 Cowling’s Theorem, 228 Cracks extending to body surface, 103 f f . in rocks, 112 ff. in thin plates, 89 f. isolated, 90 ff. near body boundaries, 107 ff. plane axisymmetrical, 96 f. stability of, 97 ff. Crack systems, 108 ff.

G Granular media (plast.), 204 f. Gravitational instability in MHD, 270 ff.

H Hartmann number, 229 Hypersonic slender-body theory, 4 ff. Hypersonic slender-body approximation, 38 f., 41 ff.

I Immobile-equilibrium cracks, defined 79 Infinite Mach number (hypers. fl.), 39 f., 43 ff. Integral methods (hypers. fl.), 36 ff., 47 ff. Inverse methods (plast.), 177 ff. Isotropic cylinder (plast.), 185 ff.

D Disc with circular hole, 173 Dissipation effects in MHD, 253 ff. “Dynamic beta” in MHD, 229 Dynamic problems (th. of cracks), 121 ff.

326

327

SUBJECT INDEX

K Ktitter-Hencky integrals, 159

L Lagrangian formulation (hypers. fl.), 14 ff. Limit analysis (plast.), 191 ff. Lundquist number, 230

M Magnetohydrodynamics, basic equations of, 219 ff. Magnetic Reynolds number, 216, 224 Minimum weight design, 199 f. Mobile-equilibrium cracks, defined 79 Momentum integral (hypers. fl.), 40 ff.

N Newtonian flow, 18 f. Non-homogeneity function, 162 Non-homogeneity, macroscopic, 132 plastic, 133 ff. Non-homogeneous elastic-plastic body, 148 ff. equilibrium in curvilinear coordinates, 156 ff. plates, 190 ff. strain-hardening, 145 f.

0 Ohm’s law in MHD, 222 One-dimensional structural elements (plast.), 191 ff. Orthotropic cylinder (plast.), 187 f.

P Particular solutions (plast.), 167 ff. Physically non-linear bodies (plast), 203 f. Plane strain, 151 ff. possibilities of solving, 155 f. Plane waves in MHD, 244 ff. Plastic equilibrium of wedge, 170 ff. Plastic layer, 168 Plastic state, geometric conditions of, 161 Plates (plast.), 195 ff. Power law shocks, 8 ff. perturbed, 26 ff. Proudman-Taylor analogue in MHD, 230 ff. Punch pressed into body, 174 ff.

Q Quasi-brittle fracture, theory of, 65

R Rayleigh velocity, 123 Rayleigh’s problem in MHD, 296 ff. Related bodies, 3 Relativistic magneto-hydrodynamics, 311 ff. Rotating disc (plast.), 190

S “Second order” blast-wave theory, 34 Sedov formulation (hypers. fl.), 16 Self-similar solutions, validity of, 43 ff. Self-similarity (cracks), 87 “Sharp-blow’’ solution, 19 ff. Shells (plast.), 197 ff. Shock waves in MHD, 256 ff. Similitude in hypersonic flow, 8 Slip-line geometry, 160 ff. Small departures from steady state (MHD), 241 ff. Stable cracks, 68 Standing waves in MHD, 262 ff. Stream-function formulation (hypers. fl.), 13 f. Stress intensity factor, defined 57, 72 Stress-strain diagram changed by neutrons, 138 f. Stress trajectories, 163 ff.

T Temperature gradients, influence on mechanical properties, 141 ff. Three-dimensional problems (plast.), 183 ff. Torsion (plast.), 189 f. Two-dimensional state (plast.),definedl52 Two-dimensional inertial flow in MHD, 300 ff.

w Waves in a non-homogeneous medium (plast.), 201 ff. Wedging, 114 ff. of a strip, 119 ff.

Y Yield limit changed by neutrons, 136

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