Advances in applied mechanics

Advances in Applied Mechanics Volume 25

Editorial Board T. BROOKE BENJAMIN Y. C. FUNG PAULGERMAIN

RODNEYHILL L. HOWARTH

C.4. YIH(Editor, 1971-1982)

ADVANCES IN

APPLIED MECHANICS Edited by Theodore Y. Wu

John W. Hutchinson

ENGINEERING SCIENCE DEPARTMENT CALIFORNIA INSTITUTE O F TECHNOLOGY PASADENA, CALIFORNIA

DIVISION OF APPLIED SCIENCES HARVARD UNIVERSITY CAMBRIDGE, MASSACHUSETTS

VOLUME 25

1987

W

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Contents PREFACE

vii

Slow Variations in Continuum Mechanics Milton Van Dyke I.

Introduction

11. Two-Dimensional Shapes 111. Three-Dimensional Slender Shapes

IV. Three-Dimensional Thin Shapes V. Closer Fits VI. Concluding Remarks References

1

3 24 35 37 42 43

Modern Corner, Edge, and Crack Problems in Linear Elastodynamics Involving Transient Waves

Julius Miklowitz Introduction Elastic Waveguides Elastic Pulse Scattering by Cylindrical and Spherical Obstacles 1V. The Two-Dimensional Wedge and Quarter-Plane References

1. 11. 111.

47 48 81 132 177

The Derivation of Constitutive Relations from the Free Energy and the Dissipation Function

Hans Ziegler and Christoph Wehrli 1. Introduction 11. Thermomechanical Theory

183 186

Contents

vi 111. Heat Conduction

IV. V. VI. VII. VIII. IX. X.

Elastic Solids Fluids Plasticity Soils Viscoplasticity Viscoelasticity Conclusion Appendix References

194 196 200 207 216 225 228 233 234 236

Creep Constitutive Equations for Damaged Materials A. C. F. Cocks and F. A . Leckie Nomenclature I. Introduction 11. Thermodynamic Formalism 111. Mechanisms of Void Growth IV. Nucleation of Cavities V. The Use of Average Quantities VI. Damage Mechanisms in Precipitation-Hardened Materials VII. Theoretical Constitutive Equations for Void Growth VIII. Experimental Determination of Constitutive Laws IX. Life Bounds for Creeping Materials X. Discussion Appendix: Mean Strains References

239 240 242 247 258 26 1 263 267 270 282 288 290 293

INDEX

295

Preface This volume contains four comprehensive articles. Milton Van Dyke’s article, “Slow Variations in Continuum Mechanics,” is a systematic approach to a wide range of flow problems where the motion is predominantly onedimensional. The simplest one-dimensional approximation to each such problem is derived as the lowest order contribution in an appropriate perturbation expansion. Julius Miklowitz has prepared a detailed survey of three areas of wave mechanics: (1) elastic waveguides, (2) elastic pulse scattering by cylindrical and spherical obstacles, and (3) the two-dimensional wedge and quarter plane. His article is entitled “Modern Corner, Edge, and Crack Problems in Linear Elastodynamics Involving Transient Waves. ” Hans Ziegler and Christoph Wehrli have made an ambitious attempt to unify a wide class of constitutive laws for elastic-plastic solids within the framework of thermodynamics in their article, “The Derivation of Constitutive Relations from the Free Energy and the Dissipation Function. ’’ Professor Ziegler died before this article went to press. It is fitting that this last article of his, written with his colleague Dr. Wehrli, is in an area which concerned him all his life. The fourth article is by Alan C. F. Cocks and Frederick A. Leckie and is entitled “Creep Constitutive Equations for Damaged Materials.” This article covers both the continuum mechanics of creep damage and materials science aspects of high-temperature material failure. It formulates creep constitutive equations consistent with each. This is an area which has seen intense development in the past few years, and the Cocks-Leckie article is most timely. I am indebted to my co-editor, Theodore Y. Wu, for his assistance in putting together this volume of the Advances. JOHN W. HUTCHINSON

vii

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ADVANCES I N A P P L I E D M E C H A N I C S , V O L U M E

25

Slow Variations in Continuum Mechianics MILTON VAN DYKE Departments of Mechanical Engineering and Aeronautics & Astronautics Stanford University Stanford, California 9430.5

I. Introduction Any good plumber can tell us that the velocities across a 1-in. pipe are just four times those in a 2-in. pipe to which it i s connected (or at any rate they would be if those nominal diameters were the actual ones). He might concede, however, that this common sense concliision becomes questionable in the vicinity of the juncture of the two pipes. This hydraulic approximation is a simple and useful idea which, as the quasi-cylindrical approximation, has its counterpart in other branches of mechanics. It has long been applied to specific problems in many research papers and textbooks, but no general exposition of it can be found. Here we undertake to discuss it from a unified point of view, with illustrations drawn mostly from fluid mechanics and the linear theory of elasticity. We recognize that the approximation is founded on the assumption that the boundaries of the region being considered vary much more slowly in some directions than in others. We call this a slow variation. (We could equally well regard it as a rapid variation in the transverse direction, because what matters is only the relative rate of variation of the geometry.) In three dimensions we can distinguish two classes of slowly varying geometry. The more common type is that of a .rfender object, exemplified by a needle, whose two transverse dimensions vary slowly in the longitudinal direction. The other type is that of a thin object, such as a razor blade, whose thickness varies slowly in the other two dimensions. Even in a slowly varying domain the solution could vary rapidly, as for waves traveling along a slowly varying channel. We exclude this important 1 Copynght 0 1987 by Academic Press, Inc All rights of reproduction m any form reserved

2

Milton Van Dyke

situation, to which a great deal of interesting work has been devoted. Thus we consider problems of equilibrium, governed typically by elliptic partial differential equations such as Laplace’s equation. (Note that from this point of view, steady fluid flow is an “equilibrium” situation.) We also disregard ends, edges, junctures, and other discontinuities. These exert only a local influence, which in most cases decays exponentially and so becomes negligible beyond a few widths. Viscous flow is an exception: although a disturbance can propagate only a few stream widths upstream, it is swept downstream and so extends in that direction a distance of the order of the width times the Reynolds number (Van Dyke, 1970). Local solutions for such discontinuities can be joined to the slowly varying solutions that we shall consider, using the method of matched asymptotic expansions, to render them uniformly valid. Thus Keller and Geer (1973, 1979) have connected thin slowly varying streams of a heavy inviscid fluid in a free jet, along a solid wall, or in a channel. We seek to embed the intuitive idea of slow variations into a systematic scheme of successive approximations. The result is a regular perturbation expansion of the solution in powers of a parameter that characterizes the slow variation. We can in principle extend the series to arbitrarily high order. In practice, of course, the computational labor grows so rapidly that only a few terms can ordinarily be calculated by hand; but it has increasingly been found possible to delegate that labor to a computer, using either purely arithmetic programs or the newer ones that carry out symbolic manipulation. The question then arises whether the series actually converges for at least some range of the perturbation parameter or is merely asymptotic with zero radius of convergence. We endeavor to answer this question both analytically and by study of numerical results. The key to treating a slow variation is to rescale the coordinates in different directions so that the variation formally becomes “normal.” This transfers the perturbation parameter from the boundary conditions to the differential equations, which can accordingly be simplified by approximation. It is remarkable that this simplification also renders the approximate solution uniformly valid. That is, without the rescaling, the perturbation solution is in many cases valid only for slight variations and breaks down where the variation is appreciable. With rescaling, on the other hand, arbitrarily large variations are acceptable, provided that they are slow. In this way our plumber can accurately treat a 1-in. pipe that expands to a 10-in. pipe, provided that it does so slowly. Thus a slow variation may in its original form be regarded as a singular perturbation problem, and the rescaling as the simplest technique for transforming a singular to a regular perturbation. This procedure was perhaps first systematically applied by Blasius (1910) to the steady plane laminar flow through a slowly varying symmetric channel. It is remarkable that within fluid mechanics it has subsequently been extended to other viscous flows, whereas the simpler potential flows have

Slow Variations in Continuum Mechanics

3

been largely neglected. We take advantage of this gap by first solving Laplace’s equation in many of our illustrative examples.

11. Two-Dimensional Shapes

The method of slow variations has been applied mostly to thin twodimensional shapes in the plane or slender axisymmetric ones in three dimensions. The governing partial differential equations can then be approximated by a succession of ordinary differential equations, and in fact in most applications the problem is reduced to mere quadratures. We consider such examples here, deferring fully three-dimensional problems to subsequent sections.

A. SYMMETRIC PLANESTRIP Consider a n infinite symmetric plane strip of slowly varying width, such as that sketched in Fig. 1. By slow variation we mean that although the width of the strip may change considerably, it does so slowly, because the slope of the boundary is small, say, of order E << 1. This may be expressed formally by writing the equation of the boundary as y = *f(Ex). Here x and y have been made dimensionless by reference to some characteristic transverse dimension of the strip, such as its minimum or average width. We assume that the derivative f’is of order unity, as are the combinations ff”, f 2 , f”’, and so on. 1. Potential in Hyperbolic Strip

To illustrate the essential simplification associated with slow variation, we consider the thin symmetric strip that is bounded by the hyperbolas y = *(l + E ~ x * ) ” * . In this example the width actually tends to infinity in both directions, but it does so slowly. Suppose that we seek within the strip

FIG. 1 .

Slowly varying symmetric strip.

Milton Van Dyke

4

a solution 9 of the Laplace equation lcIXx + $ ,, = 0 that assumes the values *l on the upper and lower edges. Here could be the steady temperature distribution in the strip with its edges maintained at different constant temperatures or the stress function for irrotational flow through a hyperbolic channel and so on. The exact solution of this problem can be found in elementary closed form using elliptic coordinates (Morse and Feshbach, 1953, pp. 1195-1196). Instead, we seek an approximate solution when E is small. We carry out a perturbation solution in powers of E in two different ways. First, we ignore the slowness of the variation of width and calculate a conventional perturbation expansion leaving the longitudinal and transverse coordinates unaltered. Second, we recognize the slow variation by rescaling those coordinates relative to each other before introducing the perturbation expansion. It is illuminating to compare these two different approximations, which we characterize as slight versus slow variations. a. Slight Variations In the conventional first method, in terms of the original coordinates x and y, the strip approaches a strip of constant width 2 as E tends to zero. For that the solution is simply (I, = y. Then expanding the equation of the boundaries in Taylor series for small E yields only even powers, which suggests that the perturbation solution will also proceed in powers of E ' , in the form (I,(% y ;

=Y

+ E 2 9 2 ( X , y ) + E493(X, Y ) + . . . .

(2.1)

Substituting this expansion into the Laplace equation and equating like powers of E' shows that each term separately must satisfy that equation. It is a familiar feature of perturbation methods (Van Dyke, 1975, Section 3.8) that when, as here, the position of a boundary varies with the perturbation quantity E , it is necessary, in order to carry out a systematic expansion scheme, to transfer the boundary condition by series expansion to the basic position of the boundary corresponding to E = 0. Here that transfer can be effected by Taylor series expansion. Thus the requirement that (I, = 1 on the upper edge of the strip is expressed in terms of the successive (I,, and their derivatives evaluated at y = 1, and similarly for the lower edge. Then like powers of E~ can be equated to give the boundary conditions

and so on. Plane harmonic functions satisfying these conditions are readily found as polynomials in x and y (or more easily as the real parts of odd polynomials in the complex variable x + iy). The resulting perturbation series is (I, = y [ 1 - i E Z ( 1

x (43

+2

+ 3x2

-

y2) +

A&" +

1 0~ 70y2 ~ + 1 3 5 -~ 2~ 7 0 ~ ~ 27y4) ~ '

+

*

. .].

(2.3)

Slow Variations in Continuum Mechanics

5

If this is interpreted as the stream function for inviscid flow in a channel, the derivatives of CC, give the velocity components. Thus the axial velocity is U =

$!ly

+ 3X2 - 3y’) + + 2 1 0 ~ ’- 2 1 0 +~ 1~3 . 5 -~ 810x2y2 ~ + 1 3 5 ~+~ .) . . .

= 1 - ;&*(I

x (43

j&E4

(2.4)

This approximation evidently fails for large x. Whether we examine CC, itself or its derivatives, the second term becomes as large as the first when x is of order 1 / ~ and , that difficulty is compounded in the third and higher approximations. It is clear physically why the approximation breaks down , boundary has departed at such great distances. When x is of order 1 / ~the significantly from the basic uniform strip, and the boundary condition has to be transferred over too great a distance to remain valid. b. Slow Variations

The preceding procedure is not the usual practical way of calculating the flow in a channel. Our intelligent plumber would instead apply the hydraulic approximation, assuming that at any station the velocity is parallel to the axis and constant across the channel. This gives the familiar quasi-onedimensional result that the speed u varies inversely as the cross-sectional area, in order to satisfy the requirement of continuity. In our special case of the hyperbolic channel the result is

(2.5) This hydraulic approximation has the great advantage over our preceding result that it is not restricted to slight variations of channel width. It is accurate even for enormous variations, provided that they take place slowly enough. Furthermore, it soon recovers its accuracy outside any local zone of rapid variation, such as a juncture or discontinuity. This is a special simple example of the quasi-cylindrical approximation, which is familiar in many other problems and fields. Some examples are current flow in a slowly varying wire and bending or torsion of a slowly varying beam or shaft. On its face the quasi-cylindrical approximation is an ad hoc rather than a systematic one. That is, it is not immediately obvious how it can be embedded into a rational scheme for successively calculating higher approximations. In fact, however, this is easily accomplished by first shifting the perturbation from the boundary conditions to the differential equation. This is accomplished simply by stretching the coordinates differentially; that is, we “square up” the geometry by introducing the contracted abscissa X = E X . Then the small parameter e disappears from the equation of the boundaries, which becomes y = f ( 1 + X 2 ) ’ ” . [Only the relatiue stretching of longitudinal and transverse coordinates is significant. Hence instead of contracting the longitudinal coordinate, we could alternatively magnify the transverse one by introducing y / e in place of y , and we would prefer this

6

Milton Van Dyke

choice when the original problem is scaled such that the region is “thin” rather than “long”-for example, if we had defined the hyperbolic strip on a smaller scale according to y = f ~ ( 1+ x2)’/’.] With this distortion of the coordinates, the perturbation parameter appears in the differential equation rather than the boundary conditions, as the problem becomes* E2+hxx

+ Gyy = 0,

*

at

= Zt 1

y

=

*(l

+X2)1/2.

(2.6)

As E tends to zero the boundaries now remain hyperbolic rather than degenerating to straight lines as in the preceding approximation. On the other hand, the differential equation does not remain Laplace’s equation, but reduces to simply t,byy = 0. Hence elementary quadratures yield the first approximation $1

=-

Y

G-2’

In the application to inviscid flow through a channel, this stream function gives the longitudinal velocity (2.5) of the hydraulic approximation. Furthermore, working with the stream function in this way has the advantage, because continuity has been satisfied exactly, that it also provides a first approximation to the transverse velocity, which, being small, is absent from the hydraulic approximation

(Here in the last form we have restored the original abscissa, the contracted one having served its purpose.) Had we treated the flow problem using the velocity potential rather than the stream function, this result for the transverse velocity would have been delayed to the second approximation. We can now systematically calculate higher approximations by assuming a regular perturbation series. Again the appearance of .s2 in the problem suggests an expansion in even powers, as

*=Jr +&2*,(x,y)+s4$3(x,y)+.... + x2

(2.9)

Substituting into (2.6) gives the problem for the nth approximation as

Thus at every stage, instead of seeking a solution of Laplace’s equation, as in the previous approximation of slight variations, we need merely perform * A mathematician would also change the notation for $ to indicate that it is not the same function of X and y as it was of x and y, but we follow the usual engineering practice of using the same symbol when no confusion can result.

Slow Variations in Continuum Mechanics

7

two quadratures. Thus we easily find the second approximation as

There is now no symptom of nonuniformity in this or any higher approximation; the perturbation is regular. For example, far downstream our hyperbolic channel approaches asymptotically the wedge-shaped channel y = + EX = + X . There our second approximation (2.1 1) becomes s=v[,+iE2(l

-$)+

1

0(E4,x-2)],

(2.12)

X

and this is just the expansion to order F * of the exact solution for a wedge of semivertex angle tan-' E, which may be regarded as produced by a source at the vertex:

*

=

tan-'(y/x) tan-'( & y / X ) tan-' E tan-' E

(2.13)

Expanding this in Taylor series for small E reproduces our approximation of slow variations (2.12) and shows that it converges for E < 1, that is, up to a wedge half-angle of 45". The contracted abscissa X having served its purpose, we can now restore the original coordinate x in o u r approximation (2.11) for the hyperbolic channel to obtain

*=d- 1 +

1 - 2E'X' y2 E2X2 [ l - E ' 6(1 E ~ x ' ) ( '-IfE2x?)

+

+ O(c4) .

]

(2.14)

As indicated by the final order symbol, our result in this form can still be regarded as an expansion in powers of E'. Now, however, the coefficients also depend explicitly on E. Hence this is no longer a power series or even an asymptotic expansion in the classical sense. ErdClyi (1961) calls such a series a generalized asymptotic expansion. If we further expand this result formally for small E, we reproduce our previous result (2.3) of slight variations. This process makes clear how the validity of the solution is thereby destroyed for large x. More precisely, it shows that whereas our result (2.14) of slow variations converges for E < 1, that of slight variations converges only for EX < 1 . It may seem remarkable that the approximation of slow variations is not only much simpler to calculate than that of slight variations-involving only quadratures rather than integration of partial differential equationsbut at the same time yields more complete results. It must be understood, however, that these twin advantages arise only when the variation is actually slow. When the shape varies on a normal scale rather than slowly (Fig. 2), only the first method can be applied.

8

Milton Van Dyke

FIG. 2.

Slightly varying symmetric strip.

If we interpret t,b as the stream function for potential flow, the variation of speed along the axis of the hyperbolic channel is as shown in Fig. 3, with E = 0.8. Even for this rather large value of the perturbation parameter, the first approximation of slow variation (the hydraulic approximation) yields reasonable accuracy throughout, and the second approximation lies considerably closer to the exact result. The latter is found, using elliptic coordinates, as u(x, 0) = &(tan-' &)-'[I

+~

~

+ (x ~1) ] - ' ' ~ .

(2.15)

This shows that, as for the limiting wedge, the approximation of slow variation converges for E < 1. 2 . Potential in General Symmetric Strip The approximation of slow variations is so simple for a symmetric strip that it is actually easier to treat the general shape than any particular one.

FIG.3.

Speed on axis of hyperbolic channel in potential flow.

Slow Variations in Continuum Mechanics

9

Thus for the strip of Fig. I , described by y = f f ( ~ x ) we , easily find the third approximation of our potential problem as

cc, = (Y/f)+ &'(l/f)"Y(f'

Y') + &4{&[f2(1/f)"1"Y(f2- y 2 ) - M / ~ ) ( ~ w ~- y4)) + o(E6). (2.16) -

This expansion terminates at the first term for the unrealistic shape with f ( X ) = (A + BX)-' but is otherwise an infinite series. In the corresponding viscous problem discussed later, in Section 4, Lucas (1972) caries out the indicated differentiation and replaces the transverse coordinate y by the fractional distance 77 = y / f across the strip. Here that gives the more explicit but lengthier form

cc, = 77 +;&'(2f2-ff)(7/

+ 4f'f'f"' + 8f2f'p

+

- f3f'4')(

lOff'S"+ 77 - v3)- &(24ff4 - 36ff"f"

- f3f4')(

7 - T ~ ) ] O( E

-

773)

&4[&(4f'4-

+

4f2p

+ 6f2Y2 (2.17)

~ ) .

This illustrates that the term in E " (where n is even) involves the first n derivatives of the shape function f: More precisely, it is a polynomial in odd powers of 77 of degree n + 1 multiplied, as Lucas (1972) observes for the corresponding laminar-flow solution, by a linear combination of all possible n-tuple products o f f and its first n derivatives f, f',f", . . . ,f'"' that involve n differentiations. 3 . Elastic Strip in Tension We consider now a slightly more complicated problem, from the linear theory of elasticity. Let the symmetric strip of Fig. 1 be subjected to tension at its distant ends. It is in a state of plane stress if it is very thin and plane strain if it is very thick. The problem for the stresses is the same in either case. It is convenient to work with Airy's stress function 4, according to which the three components of stress are given, in the notation of Timoshenko and Goodier (1951), by ax

=

4,,,

fly

=

A x ,

Txy

= -4xy.

(2.18)

Then 4 satisfies the biharmonic equation. In terms of the contracted abscissa X = E X , this becomes

,+,

2

+ 2.5 4xx,, + E 4 4 X X X X

=

0.

(2.19)

We see that again the approximation of slow variations reduces the partial differential equation to a succession of quadratures. The conditions that the edges of the strip y = * f ( X ) are free of stress become

Milton Van Dyke

10

If all variables are made dimensionless by reference to the tension force and a characteristic transverse length, the condition that the prescribed tension acts across each section of the strip is

4y

= i2

at Y = f ( X ) ,

(2.21)

and 4 is an even function of y by symmetry (so that there is no bending moment in the strip). For E = 0 the differential equation degenerates to ~ y v y 4 .= 0, and then quadratures yield the first approximation for the stress function

4

= t[Y2/f(X) + A X ) ] .

(2.22)

Hence the stress components are

The first of these is the simple quasi-cylindrical approximation according to which the tension is uniform across each section; but again the use of an auxiliary function has yielded a bonus, giving first approximations also for the other two smaller components of stress. Seeking corrections of order E’ and E~ to the stress function yields, after some calculation, the third approximation

4

= t(Y’/f+f)

+

kE2(l/f)”(f2 - Y2I2 - -ii[fz(i/f)”]pt + &,(1/f)(4)(2f2

-

E4{-&f(4)

+ y2))(f2 - y2)2+ q E 6 ) . (2.24)

This has a structure resembling that of the potential solution (2.16) and reflects the kinship of the harmonic and biharmonic equations. That resemblance is enhanced by writing this result in terms of the fractional distance 7 = y/f across the strip (and carrying out the differentiations), which gives

4

= tf{(l

+ 7’)

- iE2(2f2 -ff”)(l

-

2T2 + q4)

+ - =&(8f4 - 20ff”f” + 8f2y2 + 8 f 2 f , y + f’f‘“’) + A(24f” - 36ff”f” + 6f2f,r2 + 8f’fP - f’f‘”’, E ~ [

x (2 + q2)](1 - 72)2)+ 0 ( E 6 ) .

(2.25)

We find it informative to compare the solutions of our various examples for similarities of structure in their dependence on the perturbation parameter E, on the widthf of the strip and its derivatives, and on the fractional ordinate 7.Here, comparison with the corresponding form (2.17) for Laplace’s equation shows that, aside from a factor of 7 there and o f f here, the series solutions of the harmonic and the biharmonic equation have the same structure in that they both proceed in powers of e2, and the nth terms

Slow Variations in Continuum Mechanics

11

contain the same combinations of the ordinate f and its derivatives, multiplied by polynomials in v 2 that are of degree n for the harmonic equation but degree ( n + 1) for the biharmonic equation. The special case of a wedge (f = X ) can be solved in simple closed form (Timoshenko and Goodier, 1952, Section 35). In polar coordinates the stress function is

4

=

( r e sin 0)/(2a

+ sin 2 a ) ,

(2.26)

where the semivertex angle is a = tan-’ E. Expanding this in powers of E checks the result from our third-order slowly varying approximation (2.24)* and shows that again it converges for E < 1. 4. Laminar Flow in Channel

The approximation of slow variations has been carried farthest in the problem of steady laminar flow through a symmetric plane channel. Two different schemes of successive approximation have been proposed, depending on the magnitude of the Reynolds number R. If the Reynolds number is of order unity, each approximation requires only quadrature of polynomials, as in the preceding problems. However, if the Reynolds number is large, one must in each approximation solve a partial differential equation, which in the first approximation is Prandtl’s boundary-layer equation, but with unconventional boundary conditions. It is again convenient to work with the stream function rather than the velocity components and to eliminate the pressure from the Navier-Stokes equations by cross-differentiation. Then the flow is governed by the vorticity equation; in dimensionless form

+

(2.27)

Here the coordinates are again referred to a characteristic transverse dimension, and the stream function to the volumetric flow rate per unit thickness in half the channel. The Reynolds number R equals that flow rate divided by the kinematic viscosity. Then the boundary conditions are* *=+1,

+y

=0

at

y = EX).

(2.28)

As before, we contract the abscissa x, replacing it with X = E X , which transfers the perturbation parameter E from the boundary conditions to the

* Here we must recognize that linear terms in X do not contribute to the stress. * W e could require that #x vanish at the wall instead of # y , but that would complicate the calculations.

Milton Van Dyke

12

differential equation: QYYYY

+ 2E2Qxxyy+ E4Qxxxx = ER

6 = *l,

Qy = 0

at y

=

*f(X).

(2.29)

a. Reynolds Number of Order Unity If the Reynolds number remains fixed as E + 0, the differential equation (2.29) reduces to simply Qy;yy = 0, and the first approximation is then

lp)= i [ y / f ( X ) ]- i ( y 3 / f 3 >= ;7)

-

5q3.

(2.30)

This gives at any station the quasi-cylindrical approximation for the longitudinal velocity-the parabolic Poiseuille profile appropriate to flow with the prescribed flux rate through a parallel channel having the local width of that station. Again, however, because we have worked with the auxiliary function Q, which satisfies the continuity equation, this also yields the correct first approximation for the small transverse velocity component. To proceed systematically to higher approximations, we expand the stream function in powers of E . Equation (2.29) shows that, except in creeping flow, when the Reynolds number is negligibly small, odd as well as even powers of E now appear in the expansion (2.31)

Then substituting into the full problem (2.29) and equating like powers of E yields the sequence of problems

Blasius (1910) calculated the second approximation and used it to predict the separation in an exponentially growing channel. (He actually worked with the “primitive” variables u, 0,and p rather than Q and did not explicitly introduce the small parameter E, except in explaining the difference between what we have called slight and slow variations.) Abramowitz (1949) considered a special case of the third approximation. The channel-wall slope and the Reynolds number appear in the second approximation, and the

Slow Variations in Continuum Mechanics

13

wall curvature and the square of the Reynolds number in the third, giving

+ = -21( 3 7

-

3 v 3 )+ -~Rf'(577 280

- llv3

+ 77'

-

7')

+ 4488q7+ 1 1 5 5 ~ 987") ~ - T ( 1 2 1 3 ~- 3 2 7 9 + ~ 323477' ~ - 1518v7+ 3 8 5 ~ 357")] ~ x [f2(28757 - 8 2 2 2 ~ ~ 87787'

-

-

By delegating the mounting labor to a computer, Lucas (1972) calculated the 13th approximation for the general shape illustrated in Fig. 1 and additional terms for special shapes. He verified that the pattern apparent in (2.33) continues: the term + ( " + ' ) that multiplies E " contains alternate powers of R u p to R", odd powers of 77 up to 774"+3, and all possible n-tuple products o f f and its first n derivatives that involve just n differentiations. This last pattern is encountered in the potential solution (2.17), where only even powers of E appear. Lucas has calculated 25 terms of the series for a wedge with f ( X ) = X and tested it against the exact solution of Jeffery and Hamel (Rosenhead, 1963, p. 144). That solution in terms of elliptic functions has an infinite number of branches; but the Blasius series singles out the "principal" branch that includes plane Poiseuille flow and describes symmetric profiles with at most a single region of flow reversal near each wall. When the Reynolds number is negligibly small, the Blasius series (2.33) reduces to an expansion in powers of E ' . For the wedge, Lucas finds that it converges for E < 1. He extends its convergence by recasting it in powers of the semivertex angle tan-' E . By analyzing the coefficients, he finds that convergence is then limited by a square-root singularity at a semivertex angle of 129". This is the angle at which Fraenkel (1973) showed that the Jeffery-Hamel solution becomes singular, its derivative with respect to angle being infinite. When the Reynolds number is large compared with unity (say greater than 10 for practical purposes), the Blasius series reduces to an expansion in powers of ER. For the wedge, Lucas finds that it converges beyond the onset of separation (at E R = 4.712, according to Fraenkel), until it reaches a square-root singularity (at E R = 5.461). For other channels the convergence is not easy to judge. Among various shapes, Lucas shows the skin friction along a channel of sinusoidally varying width (Fig. 4) at high Reynolds number (that is, in the limit R -+ co, E + 0, with E R fixed). For ER = 4, the flow does not separate, and successive approximations appear to converge when recast into Pad6 approximants. For E R = 8, however, there appears to be a considerable extent of separated

14

Milton Van Dyke

flow; and Fig. 4 shows that the Pade approximants formed from 7, 9, and 11 terms of the series have not yet settled down (if indeed they ever will). b. High Reynolds Number Suppose that the Reynolds number is so large that the product E R is not small. We have seen that the Blasius series may converge; if so, an accurate solution can be found from sufficiently many terms. To attack the nonlinear problem directly, however, we must retain terms of order E R in the problem (2.29). With an error of order E * it becomes (1)

= ER($y

L ; : $ $(I)

=

*I,

(1)

$x,y

(1)

- $x

(1) $yyy)r

$y=0

at y

=

*tf(X).

(2.34)

This approximation was first proposed by Williams (1963), who observed that the differential equation is that of Prandtl's boundary-layer theory. However, it satisfies zero-velocity boundary conditions at both walls rather than being matched to an outer inviscid flow, and the pressure (which was here eliminated by cross-differentiation) is not imposed, but is found in the course of solution. Williams searched for self-similar solutions in symmetric channels and found only the slow-variation version of flow in a wedge. Later Blottner (1977) used a marching finite-difference scheme to calculate non-self-similar solutions of (2.34) for various channels. In particular, he treated a family of symmetric channels expanding slowly from a parallel upstream section through a cosine variation to a parallel downstream section of double the upstream width. For the shortest of his transitions he found separation at the walls followed by a short region of reversed flow. Similarly, Eagles and

Cf

0

FIG. 4. Skin friction in periodic channel y approximants formed from 2 N + 1 terms of series.

=

* 2 sin E X for E R = 8. [ N / N ] Pad6

Slow Variations in Continuum Mechanics

15

Smith (1980), as a basis for studying the stability of nonparallel flows, solved numerically the flow through a channel expanding to three times its upstream width according to + y = 1 + tanh( E X ) . Their forward-marching scheme showed separation starting at ex = 0.35 when E R reaches 15.5, and extending over 0.2 < E X < 1.5 when e R = 18.

B. AXISYMMETRIC SHAPES Axisymmetric problems of slow variation can be treated in exactly the same way as symmetric plane problems, and the results have a remarkably similar structure. We illustrate this by examining the solutions of three different problems. Our region of interest (Fig. 5) is the axisymmetric counterpart of the strip considered previously. We describe it by using the same slowly varying function f(e x ) , but replace the transverse Cartesian coordinate y by the cylindrical radius r. 1. Potential Flow through Pipe of Varying Radius

Of the various physical interpretations that could be given to the problem of the potential in a plane strip, that of temperature distribution has no realistic counterpart here, and so we consider potential flow. Then it might seem natural to work with the velocity potential, but we have seen that we gain half a step by using the stream function $ instead. Then the velocity components are given in cylindrical coordinates by u = (l/r)(d$/dr),

u = -(l/r)(d$/dx).

(2.35)

In axisymmetric flow the stream function no longer satisfies the Laplace equation, but instead the equation D2$ =

ILrr

- (W r ) +

In terms of the contracted abscissa X

=

$xx

= 0.

(2.36)

x, this becomes

r ( $ r / r ) r = -&xx.

FIG.5 . Slowly varying axisyrnrnetric shape.

(2.37)

16

Milton Van Dyke

Here we have combined the two r-derivatives in (2.36) to make it evident that again only quadratures are required at any stage of the approximation. As boundary conditions we set = 0 on the axis r = 0 and, in appropriate dimensionless variables, CC, = on the wall of the pipe r = f ( X ) . Then the first approximation is seen to be simply @ = r2/2f2(X), which gives the axial and transverse velocities

+

u = 1/f2(X),

2,

(2.38)

= E(Tf/f3).

Again the first of these is the hydraulic approximation of constant axial velocity at each cross section, and the second is our bonus obtained by working with Carrying the approximation to third order gives

+.

+ = (r2/2f2) +

&E’(

l/f2)”r2(f2 r2)

+ ~ ~ { ~ [ f ’ ( l / ” ) ’ ’ l ’ ’-r r~2() f-~& ( ~ / f ~ ) ( ~ b-~r4)} (f+ ~ o(E~). (2.39)

This expansion has a structure similar to that of (2.16) for the corresponding plane flow. The resemblance is increased by recasting into Lucas’s form, carrying out the indicated differentiations, and replacing the radius r by the fractional distance 7) = r/f to the wall. This gives $=L

(3y2- f f ” ) ( v 2 v4) + E 4 x [&( 18y4- 32ff”f” + 7f2T2 + 8f2fY- f 3 f ‘ 4 ’ ) (

2 7

-

2 + ’ 2

sE

&(,Of4

- 72ff”f”

+ 9f2Y2+ 12f2fY

-

v 2 - v4)

f3f‘”)(77’

-

v6)]. (2.40)

Here + / q has exactly the same form as I) itself in the corresponding solution (2.17) for plane flow, with only the coefficients being different. We can check this result and its convergence in the case of a conical pipe. The exact solution is given by a point source at the vertex, so that for a cone of semivertex angle a = tan-’ E it is 1 1-COS~ 1 1,21 - ( 1 + E2r2/X2)p’/2 *== $1 + E 1 ( 1 + &2)1/2 - 1 . (2.41) 2 1 - cos ff Expanding this for small E reproduces the result of setting f ( X ) = X in (2.40). This shows that the series converges for E < 1 in this case. 2. Torsion of Shaft of Varying Radius Let the axisymmetric shape of Fig. 5 now represent a solid elastic shaft that is twisted by couples applied at its distant ends. According to the principle of Saint-Venant, the precise way in which they are applied will significantly affect the deformation only within “boundary layers” that extend a couple of radii from the ends. Elsewhere each particle moves only

Slow Variations in Continuum Mechanics

17

in the tangential direction during twist (Timoshenko and Goodier, 1951, Section 104). Then the two nonzero components of stress can be expressed in terms of Michell's stress function 4 according to Tre =

-4

x 1 r2,

=

+r/r2.

(2.42)

The stress function satisfies the differential equation (2.43) 4rr- 3 ( 4 J r ) + 4xx= 0. The boundary conditions require that 4 increase by a constant amount

from the axis to the surface, say in dimensionless form at at

4=0

4=1

r=0, r=f(Ex).

(2.44)

Proceeding just as before yields for the third approximation

4

+

= ( r 4 / f 4 ) &E'(

1 / f 4 ) " r 4 ( f 2- r 2 )

+ ~ " ( ~ [ f ( l / f " ) " ] " r " -( f r' 2 ) - L(1/f")'"'r"(f4 - r")}. 384

(2.45)

In terms of the fractional radius 77, this becomes

4

=

+ i E 2 ( y-jy)7744(1 2 + E4[~(i00y4 - 112f2y+ 1 3 f 2 y 2 + 16f2f'f"' - f 3 ~ ( ~ 9 ~ 4 ( 1 - 772)

774

772)

-

&(210y4- 180f2y

+ 15f2y2 + 20f2f'f"'

+ O(E6).

- f 3 f ( 4 ) ) 7 7 4 ( 1 - q4)1 (2.46)

+

Here 4/773has exactly the same form as the of (2.17) for plane potential flow or the + / T of (2.40) for axisymmetric potential flow, but with different coefficients. This similarity is not surprising, because all three problems are members of the family of "generalized axisymmetric potential theory," of the form at r = 0, (2.47) at r = EX), with N being 0, 1, and 3, respectively. The approximate solution of this generic problem for slow variations is, to second order,

4

=

N+l [ ( N + 2)f" - f f " ] 7 7 N + 1 ( 1 - T ~ ) . (2.48) 2( N 3 )

$J+'+ & 2

+

Foppl found the exact solution for torsion of a conical shaft (Timoshenko and Goodier, 1951, Section 104) as 2 - 3 cos 6 + C O S ~0 = 2 - 3 cos a + cos3 a 2 - 3(1 E2r2/X2)-1/2 (1 E ~ ~ ~ / X ~ ) - ~ / ~ (2.49) 2 - 3(1 + E ~ ) - " + ~ (1 + E ~ ) - ~ / ~

'

+

+ +

18

Milton Van Dyke

Expanding for small E checks (2.46) whenf(X) the series converges for E < 1 .

=

X and shows that again

3 . Laminar Flow in Tube Blasius concluded his pioneering paper of 1910, which is devoted mainly to plane flow, by treating the corresponding axisymmetric problem of laminar flow through a straight pipe of slowly varying circular cross section. It is again convenient to work with the vorticity equation. In cylindrical polar coordinates and dimensionless variables, the Stokes stream function $ satisfies

(2.50) where the elliptic operator D’ is given by (2.36). The boundary conditions are =

O(r’)

*=’’,

I+!I~

as at

=O

r + 0, r=f(~x).

(2.51)

Again contracting the axial coordinate to X = E X transfers the perturbation parameter E from the boundary conditions to the differential equation, which becomes

x

(a’* ar2

a* r ar

+ E’%).

ax

(2.52)

a. Reynolds Number of Order Unity If the Reynolds number remains fixed as F + 0, the differential equation (2.52) reduces to (a*/ar’ - r-’a/ar)’+ = 0. This gives again at each station the local Poiseuille flow described by

9 = [ r ’ / f ’ ( ~ ) -] ;[r4/f4(x)] = 7’

- $17“.

(2.53)

Blasius (1910) found the second approximation, and Manton (1971) has calculated the third. Kaimal (1979) has generalized Manton’s results to a dilute suspension of solid particles and in so doing has tacitly corrected an error in one of Manton’s coefficients.* (Unfortunately, he has at the same time introduced typographical errors into two of the other coefficients.) We *The expression below Manton’s Eq. (4.7) should be a , = ‘f&

+ 13Q).

Slow Variations in Continuum Mechanics

19

may write Manton’s corrected result as $=

1

(

’

v2- -v4 + -~ERf-(477’ :

)

+ ~’{:(5f’

1

8

f

-9v4

””

-ff”)(~’- 2q4 + q6)+ - -(81877’ R 2 f’ 5400

+ 2 6 5 0 -~ 1~ 4 2 5 +~ 39017”~ f”

+ 6776 - 7’)

+

- - ( 1 5 6 ~~ 4 3 5 ~ 450$ ~

- 2395q4

387”) -

22577’

+ 6077”

- 677”)]}

+ O(E~).

f (2.54)

Chow and Soda (1972) independently undertook exactly the same analysis, but their result for the third-order terms that are multiplied by R bears little resemblance to that of Manton (which we have checked and corrected); in particular, they do not include the effect of the longitudinal curvature, as represented by f”. This form of Manton’s expansion is the axisymmetric counterpart of Lucas’s solution in the form (2.33). Comparison shows that the pattern is remarkably similar in the two cases, especially for the terms independent of the Reynolds number. Aside from an extra factor of 77 in the axisymmetric case, the plane and axisymmetric expansions for creeping flow at R = 0 have identical structure with respect to the perturbation parameter E, the radius f and its derivatives, and the powers of the fractional radius 7.The terms involving the Reynolds number have a different structure, however. In fact, the axisymmetric form is obtained from that for plane flow by replacing R throughout by R / f and then the two expansions differ only in their coefficients. b. High Reynolds Number When the Reynolds number is so great that the product ER is of order unity, the largest nonlinear terms on the right-hand side of (2.52) must be retained in the first approximation. Again this gives the equation of classical boundary-layer theory. Williams (1963) studied that approximation by replacing the radius with the fractional radius 77 = r / f ( EX) and discovered a reduction to an ordinary differential equation when the tube radius f ( EX) is an exponential. Thus self-similar solutions exist for laminar flow through a tube whose radius varies slowly and exponentially. Williams solved the resulting ordinary differential equation numerically and plotted the family of velocity profiles for exponentially converging and diverging tubes, showing reverse flow near the wall in the more divergent cases. Daniels and Eagles (1979) investigated the problem in greater detail and discovered multiple solutions. However, they believe that only the solutions on the branch computed by Williams, which includes Poiseuille

20

Milton Van Dyke

flow in a straight tube, are physically realistic. That is the branch represented by Manton's series. This self-similarity can be regarded as the axisymmetric counterpart of the Jeffery-Hamel similitude for plane flow, except that it is only a first approximation for a slowly varying exponential pipe, whereas the JefferyHamel solution is exact for any wedge angle. Nor has the solution been found in closed form, but only by numerical integration. This limited similitude can be seen in Manton's series (2.54). When f ( X ) = exp(aX) the factor ERf '/f in the second term becomes independent of X and equal to Daniels and Eagle's similarity parameter y = EAR. Likewise the factor & ' R 2 Y / f , as well as E 2 R 2 f t 2 / f 2in, the third term becomes the square of that parameter; and higher powers would appear in subsequent terms. On the other hand, the terms in E~ that do not contain R represent the start of a correction to the self-similar solution, of order ~~u~exp(2aX), which Daniels and Eagles discuss briefly. For a tube with other than exponential variation, nonlinear partial differential equations would have to be solved numerically. This has not yet been done, as it has been for plane flow. Instead, Eagles and Muwezwa (1986) have proposed expanding in powers of A = E R as well as E'. This evidently yields just a reordering of the expansion (2.54) of Blasius and Manton. Eagles and Muwezwa actually treat only the leading term of the expansion in powers of E ~ so , that they are solving the classical boundarylayer equations by expanding in powers of ER. Thus their first two terms are those in (2.54), and their third term is the part of (2.54) proportional to s 2 R 2(which confirms our correction of Manton's solution). They break new ground by giving also the term in E ~ RExamining ~ . the coefficients of this four-term series for a divergent and a convergent-divergent tube, they suggest that the series has a finite radius of convergence rather than being merely asymptotic, and they use it to compute velocity profiles up to E R = 10. C . ANTISYMMETRIC PLANESTRIP

So far we have considered only shapes having a straight central axis of symmetry. We turn now to unsymmetric shapes, beginning with the simplest case of a plane strip having a curved centerline. In general, both the width and the direction of the strip will vary slowly along its length (Fig. 6a). However, having already treated varying width for a straight centerline, we can illustrate the new features associated with curvature of the centerline by restricting attention to a strip of constant width (Fig. 6b). We could then easily treat the general case by combining the techniques used for the two special cases. Cartesian coordinates are no longer appropriate to the problem, for although the strip changes direction only slowly, it may eventually reverse

Slow Variations in Continuum Mechanics

21

FIG. 6. (a) General case of slowly varying strip with curved centerline; (b) special case of constant width.

its course. Instead, we employ as coordinates the curvilinear distance s along the centerline and the distance n normal to it (Fig. 6b). Then we describe the shape of the centerline implicitly by giving its curvature K (or radius of curvature 1 / ~ as ) a function of s. We choose to reckon the curvature positive if the center of curvature lies at negative n [which is the convention used in boundary-layer theory (Rosenhead, 1963, p. 202), but the opposite of that used by Wang (1980a) and Van Dyke (1983)], and we choose the length scale such that the width of the strip is 2.

1. Potential in a Meandering Strip Again we seek first a solution rC, of the Laplace equation. We require it to assume the values k1 on the upper and lower boundaries. The line element is ( 1 + Kn)ds2 + dn2, so that Laplace’s equation in our curvilinear coordinates system becomes v2*

=-

(2.55)

If we assume merely that the curvature K is small, we are again making the approximation of slight variations. The result does not fail for large s, as did the corresponding approximation for the symmetric strip, because for our special case of a strip of constant width the coordinates conform to the boundaries and we do not have to transfer the boundary conditions. However, this approach still has the disadvantage that beyond the first approximation we must at each stage solve a Poisson equation, which can be done only for a specific shape. This difficulty disappears if we assume that the curvature is not only small, but also slowly varying. We express this by introducing the contracted

22

Milton Van Dyke

longitudinal coordinate S = E S and setting K(S)

=

Eg(s).

(2.56)

Then the Laplace equation (2.55) gives

a

-(1+ an

a+

Egn)-+ an

E

a

1

a+

as1 + EgnaS

= 0.

(2.57)

This evidently reduces to (Clnn = 0 in the first approximation and to the same with a known right-hand side at each subsequent step. Thus again only quadratures are required to treat a general shape. In contrast to the previous problem for a symmetric strip, the expansion here evidently proceeds in odd as well as even powers of E. We readily calculate the fifth approximation as

+ = n + $&g(l- n2) - fs2g2(n - n’) - &e3[(4g3 - 6g”)(l - n2) + ( g ” - 6g3)(1 - n‘)] + &‘[&(4g4 - 7g” - 13gg”)(n - n’) + &7gf2 + llgg” - 24g4)(n - n’)] + O ( E ’ ) .

(2.58)

Here the N t h term, proportional to E ~ - ’ , contains alternate powers of n up to the Nth; and (except in the first term) each of these is multiplied by a linear combination of all products of the function g and its derivatives that contain N - 1 - 2i factors and 2i derivatives, where i = 0, 1 , . . .. If the function g ( S ) is taken to be a constant, the strip becomes a thin annulus, with g = 1 in the notation of Fig. 7. Then the exact solution is

FIG. 7. Thin annulus as a meandering strip.

Slow Variations in Continuum Mechanics

23

given by a vortex at the center as $=1-2

1~n)] In[(l + ~ ) / ( + In[(l + & ) / ( I - E ) ] '

The expansion of this result in powers of case and converges for E < 1.

E

(2.59)

checks our series (2.58) in this

2. Laminar Flow in a Meandering Channel Consider now the steady plane laminar flow through a channel having the shape of Fig. 6b. The Navier-Stokes equations yield for the stream function $ the vorticity equation (2.60) where the Laplacian V2 is given by (2.55). Here the Reynolds number R is again the volumetric flow rate per unit thickness in half the channel divided by the kinematic viscosity and the stream function is normalized to $ = *l at the channel walls n = *l. Wang (1980a) has treated this problem by assuming that the curvature is small, with K = Ek(s). In order to approximate for small E, he finds it necessary to assume also that the Reynolds number R is small, of order E. Then at each stage beyond the first he must solve a nonhomogeneous biharmonic equation. For the periodic channel with centerline curvature K = E cos As, he is able to proceed as far as the nonperiodic terms in the third approximation. a. Reynolds Number of Order Unity

As with the potential flow of the previous section, the process is greatly simplified by assuming that the curvature is not only small, but also slowly varying, as in (2.56). If the Reynolds number remains fixed as E + 0, the vorticity equation (2.60) reduces to simply a4$/dn4 = 0 at the first stage and to its nonhomogeneous counterpart in higher approximations, so that again only quadratures of polynomials are required. The fourth approximation is readily found (Van Dyke, 1983) as 1 1 $ = -(3n - n') + - E g ( S ) ( 1 2 4

1

x (67 - 2n2 - n") (1 - n2)2- E' 1 + -(189n 3150

- 13n' - 5n5)Rgg'

x (4663 - 2940n2

R

- n2)' - E'

1 + 240 + 3n2)g"

- 33n2)g3 -(1

1 + 1,478,400

1

+ 842n4 + 4n6 - 9ns)R2g" (1 - n2)' + O ( E ~ )(2.61) .

24

Milton Van Dyke

For creeping flow, at negligible Reynolds number, this series has the same structure as (2.58) for potential flow, except that each polynomial in n is of two-higher degree in order to accommodate the no-slip condition at the wall. For an annulus the curvature K = Eg is constant, and the exact solution has the form $ = A(1+ Egn)2 ln(1

+ Egn) + B ( 1 + Egn)2 + C ln(1 + Egn) + D. (2.62)

Here the coefficients A, B, C, D are rather complicated functions of Eg that are determined b y imposing the boundary conditions (2.28) that $ = f 1 and $n = 0 at n = *l. Expanding for small E checks our series (2.61) and shows that, as in the potential problem, it converges for E < 1 in this case. b. High Reynolds Number When the Reynolds number is so great that the product E R is of order unity, the leading nonlinear terms in (2.60) must be retained in the first approximation. With an error of O( E ) it becomes $nnnn

=

ER($n$snn

-

$s$nnn)*

(2.63)

This is again the classical boundary-layer equation. It is identical with the equation (2.34) for a symmetrical channel (with X replaced by S and y by n ) , because longitudinal curvature has no first-order effect in Prandtl’s boundary-layer theory. Therefore the self-similar solutions of Williams (1963) for narrow wedges and the non-self-similar solutions of Blottner (1977) and Eagles and Smith (1980) for symmetrical channels can be applied to channels with slowly curving centerlines. The only change is that neglect of centrifugal pressure then imposes an error of O ( E ) compared with O( E ’ ) in the symmetric channel.

111. Three-Dimensional Slender Shapes

The extra freedom that arises from allowing extension into a third dimension enormously increases the variety of possible slowly varying shapes. Only a few special configurations have yet been calculated, and those mostly for laminar flow. Much further study of three-dimensional slowly varying shapes is to be expected. We first direct attention to slender shapes, those whose two transverse dimensions are both small compared with their longitudinal dimension. The centerline can meander through three dimensions so that (as for a helix) it has torsion or twist as well as curvature, and the cross section can vary slowly in shape, proportion, size, and orientation. In unpublished work

Slow Variations in Continuum Mechanics

25

Todd (1978, 1979, 1980) has studied steady laminar flow through a slender pipe of such general slowly varying shape. He derives the governing equations and then describes the scheme of successive approximations that yield in turn the basic Poiseuille flow in the axial direction, the first approximation to the secondary motion in the cross section, and the first correction to the axial flow. He devotes special attention to the axial pressure gradient and is able to draw various conclusions without further specifying the shape of the pipe. To obtain more definite results, however, he considers special cases, mostly with circular or elliptic cross sections.

A.

STRAIGHT CENTERLINE

We choose to classify slender shapes first according to whether the centerline is straight. This is an essential distinction because, as we have already seen for plane strips, we can often use conventional coordinates if the centerline is straight, but most otherwise use curvilinear (and in general nonorthogonal) coordinates. 1 . Flow through Pipe of Varying Elliptic Section

Our first example of a nonplanar solution was the potential flow (2.38) through a straight pipe of slowly varying circular cross section. We now consider the generalization to a pipe of elliptic cross section, whose major and minor axes may vary slowly and independently. Thus it is described, in terms of the contracted axial coordinate X = E X , by y ’ / a 2 ( X )+ z’/b’(X)

=

1.

(3.1)

a. Potential Flow We used the Stokes stream function for the circular pipe, but no single stream function exists for a three-dimensional flow. We therefore treat potential flow through the elliptic pipe by using the velocity potential 4. It satisfies Laplace’s equation in the form +y,,

+ 4 z z + E 2 4 x x = 0,

(3.2)

and the flux through the pipe is prescribed, in dimensionless variables, as

The hydraulic approximation, with uniform axial velocity across each section, gives u =

c $ ~=

l/a(X)b(X).

(3.4)

26

Milton Van Dyke

The second approximation for the velocity potential was calculated by Olson (1971), and his analysis is reported by Sobey (1976), who uses it as the basis for treating inviscid flow with slight shear. With an error in sign corrected in Sobey’s expression for it gives the three Cartesian velocity components 1 1 ab 8 a’ E2Y a b

u = - - - E’ =

[(-$)

’ ( a 2- 4y2) +

b‘ + o ( & ~ ) , = E2Zab +

(3.5) 0(&3).

Rewritten in terms of the fractional transverse coordinates 7 = y / a ( X ) and 4‘ = z / b ( X ) ,these are

=

[

a2(L)’(1 ab 8 a2b a’ &-? + 0(&3), ab

= -- - & 2

- 477’)

=

+ b2($)’(1

+O(E~),

- 412)]

(3.6)

b’ ab

&-l+ o ( & ~ ) .

It is not clear that the next approximation will have the simple form of the first and second, a polynomial in y and z with coefficients depending on X . We see that having to work with the velocity potential has left us half a step behind working with the stream function, where the second approximation provides the secondary terms in the transverse as well as the axial velocities. However, we are ahead of working with the primitive variables, as we must in the next example. b. Laminar Flow The corresponding laminar flow has been calculated by Wild ef a/.(1977). Since neither a velocity potential nor a single stream function exists, they work with the Navier-Stokes equations for the three Cartesian velocity components and the pressure. The first approximation for the axial velocity u is Poiseuille flow, which has a simple closed-form solution for an elliptic pipe. In terms of the fractional coordinates 77 and 5, it is u = ( 2 / a b ) ( l- 9’-

6’).

(3.7)

Next, using the continuity equation and eliminating the pressure between the y - and z-momentum equations by cross-differentiation yields the first approximations for the transverse velocity components: 2a‘ u = &--77(1 -7 ab

2 -

52),

2b’ ab

w = s-6(1

-

77

2

-

62 ).

(3.8)

These exhibit a kinship with their counterparts (3.6) in potential flow.

Slow Variations in Continuum Mechanics

27

In the third step, a correction to u of order E R is calculated from the axial momentum equation. This gives L

u = - ( l - v 2 - 5 2) ab x [ I + E R ( C ~+ c , q 2 + c2v4+ c,c2 + c414 + c5v2l2) + O ( E ~E ,~ R ~ ) ] . (3.9)

We refer the reader to Wild, Pedley, and Riley’s equations (2.13) and (2.14) for the coefficients co through c 5 , which depend on the contracted axial coordinate X in fairly complicated fashion through the functions a ( X ) and b(W. 2. Flow through a Twisted Elliptic Pipe

Todd (1977) has treated the fully developed laminar flow through a straight pipe whose cross section is constant, but rotates slowly about its centroid in the axial direction. He considers a general cross section, but derives detailed results only for the ellipse. That cross section is described by y 2 / a 2 z 2 / b2 = 1, where x, y, z form a slightly nonorthogonal coordinate system that twists with the pipe. As shown in Fig. 8, those coordinates are

+

FIG. 8. Coordinate systems for slightly twisted elliptic pipe.

Milton Van Dyke

28

related to the Cartesian system 3, j j , Z by x = 2,

y

=y

cos 8 + Z sin 8,

z

= Z cos 8

-

j sin 8.

(3.10)

Here the angle 8, shown in Fig. 8 for right-handed twist, is a slowly varying function of the axial distance x. We make that slow variation explicit by setting

x=

8(x) = @ ( X ) ,

(3.1 1 )

EX.

Here 0 is a normally varying function of its argument and is simply X for a constant twist angle E. a. Potential Flow Again we examine first the simpler problem of potential flow. The velocity potential 4 satisfies the Laplace equation, which in our twisting transverse coordinates y, z and contracted axial coordinate X = E X becomes 4yy

+ 4 z z + E 2 [ 4 X X + 2@’(z4xy- Y 4 z x ) + @”(z4, - Y 4 Z ) + @’2(z24yy - 2YZ4yz + Y24*z- Y 4 y - z4,)l = 0.

(3.12)

The condition of tangent flow at the wall of the pipe is b2y4,

+ u ’ z ~ ,= E’@’(u’ - b2)yz[4x + @’(z+, at

(y’/la2)

-

y4,)]

+ (z’/b2) = 1 ,

(3.13)

and the condition of prescribed flux through the pipe is, in normalized form, (3.14) If the twist is negligible, the solution is simply a uniform parallel flow with 4 = x = X / E .Then the governing equations [(3.12)-(3.14)] show that for small twist angles the perturbation expansion proceeds in powers of E ~ Carrying out the second and third approximations gives

4

=

U’ - b2 E @ ’ ( X ) ,yz P + &3[(4@’3- (3’”) a2+ b x (c,a2yz + c2y3z+ c3yz3)+ Or@” x (c4a4 c5a2y2 + C6U2Z2 + c,y4 + c*y’z’ + c9z“)] + * * .

X -+ E

+

-

(3.15)

Here c,, c2, . . . , c, are algebraic functions of the thickness ratio p = b / a of the cross section. The first two terms of this expansion show that the streamlines twist more slowly than the pipe itself, in the ratio ( a ’ - b2)/(a2+ b2) = ( 1 - p 2 ) / ( 1 + p’). This factor agrees with one’s physical perception that the streamlines will not twist at all through a round pipe ( p = l ) , but must twist nearly as fast as the elliptic section when it becomes flat ( p << 1 ) .

.

Slow Variations in Continuum Mechanics

29

b. Laminar Flow Todd has treated the corresponding problem for laminar flow by using the Navier-Stokes equations for the three velocity components and the pressure. He shows that those equations assume rather complicated forms in the twisted coordinate system x, y, z of Fig. 8. The basic solution, for Poiseuille flow through an untwisted elliptic pipe, gives with our normalization (3.14) the axial velocity ~ = 2 [ -y’/a’-~’/b’]. 1

(3.16)

Twist of the pipe induces a crossflow. Todd finds that for small twist the velocity components in the directions of the major and minor axes of the cross section are given by 2,

=

EO‘(X)

3

+ 2p2 + 3p4

( 1 - P2)(3+ P 2 ) w = &O‘(X) 3 2p2 3p4 yu,

(3.17)

+

+

where u is given by (3.16). These results show that the streamlines again twist more slowly than the pipe itself, now in the ratio (1 -

3

+ 2p + 3p2 +

2p2

(3.18)

+ 3p4‘

This is smaller than the ratio for potential flow; evidently the no-slip condition at the wall inhibits the spiraling of streamlines. Powers of the Reynolds number enter into high approximations, which contain (in contrast to the potential flow) both even and odd powers of the twist parameter E. Thus Todd shows that the correction to the basic axial velocity (3.16) is in general proportional to FRO’;however, in the special case of the ellipse that secondary term vanishes, leaving only tertiary correction terms proportional to E ~ O E’O’’, ‘ ~ , E ~ R ’ O ’ ’and , E’R‘O’’ to be added to (3.16). These are multiplied by polynomials in y and z of degree as high as 10. The crossflow velocities of (3.17) have secondary corrections proportional to E’RO’’ and E’RO”, times polynomials in y and z. The coefficients of all these polynomials are rational functions of the thickness ratio 0, which Todd finds much too complicated to calculate except numerically by computer.

3. How through a Spiraling Circular Pipe Kotorynski (1979) discusses the steady laminar flow through any slowly varying three-dimensional channel and then treats as a specific example the slowly spiraling circular pipe shown in Fig. 9. This pipe can be viewed as being generated by translating a circle of unit radius so that its center moves along a helix of radius a while its plane remains normal to the axis of the

30

Milton Van Dyke

FIG.9. (a) Slowly spiraling pipe and (b) associated coordinates. Shown dotted is the nearly circular straight pipe that is tangent at station xo.

helix. The cross section of the pipe is then not quite circular in a plane normal to its own helical axis, which is what is usually understood as a coiled circular pipe, but the difference affects only third-order terms. As in the preceding example, we introduce a slightly nonorthogonal coordinate system x, y, z that spirals with the pipe, as shown in Fig. 9b. It is related to the Cartesian system 2, j , i by x = 2,

+

y = jj cos( E Z ) ,? sin( E X ) - a, z = 2 cos( &a)- j j sin( & a ) .

(3.19)

a. Potential Flow Again we consider first the simpler problem of potential flow. In the spiraling coordinate system just introduced, with the axial coordinate contracted to X = EX, Laplace's equation for the velocity potential becomes 4yy

+ 422 + E"4XX + 3 2 4 , - (Y + aMxJ + z24yy- 2(Y + a)z&,, (3.20) + (Y + a)*4,, - ( Y + a M y - 2423 = 0.

The condition of tangent flow at the wall of the pipe is y&

+ z4, = a z [ ~ +$ ~z4,, - ( y + a ) h ]

at y 2 + z 2 = 1 , (3.21)

and we prescribe the flux through the pipe in dimensionless form as

11

4,dydz

=E

11

4 x d y d z = T.

(3.22)

Slow Variations in Continuum Mechanics

31

When E is negligibly small, the solution is a uniform stream along the axis of the helix, with 4 = x = X / E . Higher approximations proceed in powers of E’, and one easily calculates the third approximation as 1

=-X E

+ E U Z + -81E ~ U [ Y ~ +Z z 3

-

~ U Y Z-

( 3 + 8 a 2 ) 2 ]+ . . .. (3.23)

Here the first two terms give the velocity components to order E as u = 1 , u = 0, w = &a.This represents at each axial station a stream that is uniform and parallel, with unit speed, but is slightly inclined to the x axis, at an angle w / u = &a. We see from Fig. 9 that this is just a uniform flow directed along the spiraling centerline of the pipe, rather than along its straight central axis. b. Laminar Flow Kotorynski treats the corresponding laminar flow by using coordinates that differ slightly from ours in orientation. (His y and z have the same spiraling origin as ours, but remain parallel to themselves rather than rotating.) His basic solution is, of course, the Poiseuille flow through a straight circular pipe; with our notation and normalization this gives u=2(1-r2)=2(1-y2-z2).

(3.24)

After some analysis Kotorynski finds the first approximation to the transverse elocities; in our notation again (and with a missing factor a restored) these are u=o,

w=2m(l-y

2

2

- 2 ) .

(3.25)

We notice that this flow field again represents at each axial station simply a parallel stream slightly inclined to the central axis. Rather than a uniform stream, it is now a Poiseuille flow with parabolic profile, but it is again inclined at an angle w / u = E U . Hence the solution to this order is nothing more than the basic Poiseuille flow in a straight circular pipe, but one that is tangent at each axial station to the actual curved pipe, as indicated in Fig. 9a. True effects of curvature would appear in subsequent approximations (together with the effects of the slight deviation from circularity of the tangent pipe). It seems likely that the next approximation could be calculated without great labor. Even in this problem, however, where the axis of the pipe never departs far from the central x axis, we would probably simplify the analysis b y recognizing that the centerline is curved and using the curvilinear distance s along it as the axial coordinate (as we did for plane flows in Section 11, C and will do for three-dimensional shapes in the following sections).

32

Milton Van Dyke B. CURVEDCENTERLINE

We turn now to slender three-dimensional shapes with curved centerlines. As in the corresponding plane problem, we restrict attention to unchanging cross sections, and it appears that in fact only such shapes have so far been treated in the literature. 1. Twist of an Elastic Ring Sector Gohner (1930, 1931a) has considered the elastic properties of a helical spring in tension or compression. He shows that if the pitch of the helix is small, it can be neglected with only very slight error. Thus the spring is modeled by a sector of an annular ring whose ends are subjected to equal and opposite forces directed along the axis of rotational symmetry,.as shown for a circular cross section in Fig. 10. The stress field is the same in each cross section around the ring. If the spring is loosely coiled, the stresses are, to a first approximation, those due to torsion in a straight shaft of the same cross section. Gohner finds the effects of curvature by expanding in powers of the ratio E = a / L of a typical cross-sectional dimension a to the coiling radius L. He computes the corrections of order E and E’ for circular, elliptic, and rectangular cross sections. Timoshenko and Goodier (1951) present a concise and accessible account of the analysis for a circular section. The only nonzero components of stress are the shearing stresses T~~ and reyacting over the cross section. The equation of equilibrium in which they appear can be satisfied by introducing a stress function 4 such that G L ~d 4 ( L + x ) a~y ’

(3.26)

rex = ~-

where G is the modulus of rigidity. Substituting these into the two relevant

t

X

I F FIG. 10. Loosely coiled ring sector of circular cross section.

Slow Variations in Continuum Mechanics

33

compatibility equations and eliminating T~~ and T~~ by cross-differentiation yields the equation to be solved for the stress functions; in dimensionless form it is

$4

a2d, -+----ax2

dy

3.5 1+ Exax

2c

=

(3.27)

0.

Here c i s a constant that is at the end of the calculation evaluated in terms of the applied torque FL by integrating the shear stress over the cross section. Here our x differs in sign from Gohner’s (and from Timoshenko and Goodier’s equivalent 5) because we measure it away from, rather than toward, the axis of rotational symmetry. The condition that the shear stress be tangent to the boundary of the cross section is satisfied by requiring 4 to vanish on the boundary. When the coiling ratio E is negligible, the third term in (3.27) disappears, and the solution is 4 = ;c(l - x2 - y’), which is the stress function for torsion of a straight shaft. The effects of coiling are introduced by expanding in powers of E ; and thus the third approximation is found to be

4

= ;c(l - X’ - y2)[1

- $EX

+

&E’(X~

+ 5 y 2 - 15) + . .

a].

(3.28)

The shearing stress is greatest at the innermost point of the cross section, where it exceeds the value for a straight shaft by the factor 1 + dE + ;&2 - . . .

(3.29)

It would be interesting to calculate further approximations if only to check a remarkable closed-form expression for the greatest shearing stress, quoted by Timoshenko and Goodier (1951), that was “communicated to S. Timoshenko in a letter from 0. Gohner.” In a subsequent paper Gohner (1931b) treats a ring sector subject to a general loading at its ends. Timoshenko and Goodier summarize the analysis for the case of a circular cross section and for pure bending in the plane of the ring. 2. Laminar Flow through a Coiled Pipe Much attention has been devoted to flow through a coiled pipe because of its practical importance. The simple potential flow has apparently never been analyzed, but steady, fully developed laminar flow through a loosely coiled pipe has been calculated for several cross-sectional shapes as a perturbation of the Poiseuille flow through a straight pipe. Just as Gohner neglected the pitch of a helical spring, so most investigators have approximated the coiled pipe by a torus. In creeping motion, when the Reynolds number is negligibly small, there is no secondary motion, the flow being directed normal to the cross section. For a circular section Larrain and Bonilla (1970) expand the solution in

34

Milton Van Dyke

powers of the coiling ratio u = (1

- r2)[1 - ~

E

=

a / L to find the velocity as

cos 8 - &’(3

E T

- l l r ’ - lor2 cos 28)

+ - -1. 1

(3.30)

A quantity of interest is the flux ratio, the flow rate through the coiled pipe divided by that in a straight pipe under the same pressure gradient. This is given by Q c / Q s= 1 + Z E

+ . . ..

1 2 - 1 3 10248~

(3.31)

Thus the flux is actually increased by slight coiling because, as Larrain and Bonilla pointed out, the velocity distribution (3.30) shifts toward the inner wall of the pipe and so shortens the fluid path. When the Reynolds number is not small, Dean (1928) showed that to a good approximation the flow depends not upon the coiling ratio and Reynolds number separately, but only upon the combination K

= RE’/’ = ( 2 f i a / ~ ) ( a / L ) ’ / ~ .

(3.32)

This (or one or another related parameter) has become known as the Dean number. Here R = 2ua/ v is the Reynolds number based on the mean axial velocity U and pipe diameter. Dean worked instead with a fictitious Reynolds number based on the maximum velocity that would arise in a straight pipe under the same pressure gradient; when recast in terms of the actual Reynolds number, his approximation for the flux ratio is

Q c / Q s= 1 - 0 . 0 3 0 5 8 ( ~ ’ / 2 8 8 )+~ 0 . 0 0 8 1 9 ( ~ * / 2 8 8-) ~* * * .

(3.33)

Thus when the Reynolds number is not small, slight coiling reduces the flux. The author (Van Dyke, 1978) has extended this series to ten more terms by computer and by analyzing it deduced that the flux ratio decays for large Dean number as 2 . 1 2 ~ - ’ / ~However, . numerical solutions and several approximate boundary-layer suggest instead a decay as ~ O K - ” ~ . This disagreement remains unresolved. Covering the whole range of Reynolds number requires a double expansion in powers of coiling ratio and Dean number. Larrain and Bonilla have carried that calculation as far as the tenth power. For the first effects on flux ratio they find -Qc =

03

(3.34)

Here the first and third coefficients of E~ appear in (3.31) and (3.33), but the intermediate term is new. Wang (1981) has considered the fact that a coiled pipe is a helix rather than a torus. He includes the first effects of torsion, or pitch, by using a nonorthogonal coordinate system. He finds that the secondary flow is affected significantly at low Reynolds number, but the flux ratio is unchanged to the order of (3.31).

Slow Variations in Continuum Mechanics

35

Earlier, Wang (1980b) used the same coordinate system to calculate the temperature distribution inside an electrically heated circular wire that is loosely coiled into a helix. Perturbing the solution for a straight wire shows that curvature adds corrections of order E = a / L and E’, but torsion again affects only subsequent terms.

IV. Three-Dimensional Thin Shapes We remarked in the introduction that a three-dimensional shape with disparate dimensions may be either slender, like a needle, or thin, like a razor blade; and we have so far considered only slender shapes. Thin three-dimensional shapes have been less studied, except for constant thickness as in the theory of elastic plates and shells. We now consider shapes of slowly varying thickness, restricting attention for simplicity to the case of a flat midplane. Then in the notation of Fig. 11 the two faces are described by z = EX, ~ y ) . We saw that for a two-dimensional shape the approximation of slow variations always reduces the mathematical problem to a succession of quadratures. For a slender three-dimensional shape, on the other hand, it is reduced only to partial differential equation in two rather than three variables. Now we shall see that for thin three-dimensional shapes either of these reductions may arise, depending on the nature of the boundary conditions.

FIG. 11.

Symmetric region of slowly varying thickness.

Milton Van Dyke

36

A. STEADYTEMPERATURE I N PLATEOF VARYING THICKNESS The simpler situation-reduction to quadratures-is exemplified by the three-dimensional counterpart of the first problem that we considered in this article: find a solution of Laplace's equation that assumes the values 1 on the two faces of a plate of slowly varying thickness. Again can be interpreted as the steady temperature distribution in the plate when its faces are maintained at different constant temperatures. However, it has now lost its other interpretation as a stream function for irrotational fluid motion, because a three-dimensional flow cannot be described by a single stream function. We now contract y as well as x by introducing X = E X , Y = EY. Then the problem becomes

+

*

+

22

=-

+ +YYL

E"+XX

+

+

=

*1

at

z

=

*f(X, Y).

(4.1)

Quadratures and iteration yield the second approximation $ = ( z / f ) + d."(l/f)xx + (l/f)YYlz(f' - z')

+ O(E4),

(4.2)

and further terms can readily be found. This, of course, reduces (with z replaced by y ) to the two-dimensional result (2.16) when the thickness varies only with x. For comparison, the exact solution for the doubly concave cone having f ( X , Y ) = ( X 2 + Y2)"' is found to be

+ = tanh- ( z / J x 2 + y 2 + z')/tanh-'(E/Jl + Expanding for small E

E

E').

(4.3)

shows that in this case the series (4.2)converges for

< 1. B. POTENTIALFLOWTHROUGH VARYINGGAP

The more complicated situation-reduction to partial differential equations in two variables-is exemplified by potential flow through the slowly varying gap between the two nearly parallel walls of Fig. 11. To a first approximation the velocity components u and ZJ are constant across the gap at any value of x and y, and the transverse component w is negligible. Then the continuity equation becomes

a

-(fu) dX

+ -a( f v ) aY

= 0.

(4.4)

Although we cannot describe the full three-dimensional flow by a single stream function, we can satisfy this quasi-two-dimensional approximation

Slow Variations in Continuum Mechanics

37

by introducing $ according to

fu

fu = +y,

=

(4.5)

-$x.

If the flow is irrotational, all three components of vorticity must vanish, but to a first approximation only the principal component u, - uy normal to the plane of symmetry need vanish. This gives the equation governing $: ($x/fL

+ (Gy/fIy = 0.

(4.6)

This equation is exact for both plane flow (when f = const) and axisymmetric flow (when f = y , the cylindrical radius from the x axis of rotation), . polar coordinates, but in general it involves a relative error of order E ~ In with x = r cos 8 and y = r sin 8, it becomes

As an application we consider uniform potential flow past a sphere tangent to a plane, which is equivalent to flow normal to the line of centers of two equal spheres in contact. This problem was treated by Latta and Hess (1973), who inverted the sphere into a plane and then solved an integral equation involving a Bessel function by a series of transformations. In particular, they evaluated the asymptotic behavior of the flow near the point of contact. Our approximation of slow variation becomes exact as that point is approached, and the spheres can be replaced by the osculating paraboloids. This corresponds to choosing our thickness function asf = X 2 + Y 2= (Er)’, and then (4.7) becomes @rr

-($r/r)

+ ($ee/r2>

= 0.

(4.8)

With the uniform flow directed along 8 = 0, that line can be taken as $ = 0. This suggests trying a solution of the form rC, = r k sin 8, and substituting shows that k = 1 &‘. We choose the less singular solution, which means that $ is locally a multiple of r’+&sin 8. Hence equations (4.5) show that the velocity diverges as rJS-’ = r-0.586approaching the point of contact. This is the result that Latta and Hess found b y more sophisticated means. Sobieczky (1977) has discussed compressible inviscid flow in a slowly varying gap. He considers in particular the transonic small-disturbance approximation for flow through narrow Lava1 nozzles of special form.

*

V. Closer Fits The slow variations discussed so far start from the quasi-cylindrical approximation, according to which the solution at any longitudinal station is that for an infinite cylinder of the local cross section. It is natural to seek

38

Milton Van Dyke

a better approximation by starting with a local solution that fits the boundary more closely. For a boundary of varying width, the next step beyond the quasi-cylinder requires a shape that is at each station tangent to the boundary. For example, in a plane symmetric or axisymmetric problem the solution at each longitudinal station can be approximated by that for the tangent wedge or cone, as indicated in Fig. 12. This approximation requires that only the slope of the boundary be changing slowly rather than its width-in dimensionally correct terms, that the product of width and curvature be everywhere small. In elasticity theory, Massonet (1962) has suggested that the simple solution for the stress field in an infinite wedge loaded at its tip can be used in this way to estimate the distribution of stresses in a beam of variable height. In viscous flow theory, Fraenkel (1962) has selected from the infinite family of self-similar Jeffery-Hamel solutions for laminar flow through a wedge the single one relevant to more realistic channels (the “principal” branch mentioned in our previous discussion of laminar flow in a channel). He has used it (Fraenkel, 1963, 1973) as the basis for an asymptotic theory of flow through a symmetric plane channel with slowly curving walls. His examples show that the higher-order corrections are remarkably small, even when the flow is separated, with regions of reverse motion near the walls. This implies that the first approximation for slowly curving walls is a good one. We observed earlier, in Section I1 (p. 19), that the Jeffery-Hamel solution for a wedge has no axisymmetric counterpart in general, but that self-similar solutions d o exist for an exponential pipe if it is slowly growing. Eagles (1982) has devised a scheme, somewhat analogous to Fraenkel’s, for applying those solutions as a first approximation to a wide variety of “locally exponential” pipes. Again, examples show that the higher-order corrections are very small. However, although reverse flow is included in the basic solutions, it cannot be admitted in the approximation for more general variations of radius.

FIG. 12. Tangent wedge or cone

Slow Variations in Continuum Mechanics

39

The possibility of making an approximation of this sort, starting with a closer fit than the cylindrical one, depends upon the existence of a family of basic solutions, preferably in simple closed form. We have seen that the cylindrical solution can be found for a wide variety of problems. It has a trivial form for the Laplace or biharmonic equation, and for the NavierStokes equations it is known for many cross sections, including ellipses. By contrast, the wedge and cone solutions needed to fit the slope of a boundary are much more limited. For example, not only does the Jeffery-Hamel solution for laminar flow through a wedge have no axisymmetric counterpart for a cone, but even the solution for the wedge involves elliptic functions, which greatly complicates the applications.

A. PLANE PROBLEMS

Closer fits are so little developed that in illustrating them we restrict attention to just three plane problems that we have already treated as slow variations. 1. Potential in Symmetric Strip

Consider again the potential in a symmetric strip (Fig. 1). The boundary is no longer restricted to be slowly varying, and so we describe it simply by y = * f ( x ) . (However, a different description might be helpful in higher approximations.) Then according to Fig. 12 the slope f ' ( x ) at any station x is to be identified with the slope E of the wedge in (2.13). Likewise y/(f/y)is to be identified with y / x . Thus the tangent-wedge approximation is found as

+ = tan-'(yf'/f)/tan-'y.

(5.1)

For potential flow through the channel, this gives the velocity along the centerline as u ( x , 0) = +.y(x, 0) = (f'/f>/tan-'f'.

(5.2)

In Table I we compare this approximation with the exact result for a hyperbolic channel of 30" half-opening angle. We also show for comparison the first approximation of slow variations, from (2.5). We might have anticipated that the closer fit would always prove more accurate. However, the result of slow variations is seen to be the more accurate near the throat of the channel, where the slope of the boundary is small, the tangent-wedge approximation being superior only for x > 2, where the product of wall curvature and channel width is small. We saw in Fig. 3 how the solution of this problem is improved by extending the method of slow variations to higher order. It would be desirable in the

Milton Van Dyke

40 VELOCITY

AXIS

ON

X

Slope

0 0.5 1.o 1.5 2 3 4 6 10

0 0.160 0.289 0.378 0.436 0.500 0.530 0.555 0.569

TABLE I POTENTIAL FLOW THROUGH HYPERBOLIC CHANNEL y = + ( I + x2/3)"' FOR

Curvature x width

Tangentwedge

Slow variation

Exact

0.666 0.569 0.384 0.235 0.144 0.060 0.029 0.010 0.002

1.000 0.969 0.890 0.791 0.694 0.539 0.432 0.304 0.188

1.000 0.961 0.866 0.756 0.655 0.500 0.397 0.277 0.171

0.955 0.926 0.854 0.764 0.675 0.530 0.427 0.302 0.187

same way to embed the tangent-wedge method into a systematic scheme of successive approximations. However, it is not clear how this is to be done. For plane laminar flow in a symmetric channel Fraenkel (1963), starting with the Jeffery-Hamel solution for flow in a wedge, has carried out the third approximation. However, he requires that the channel be mapped conformally onto a strip, which in our present example of potential flow means knowing the exact solution. 2. Potential in Meandering Strip

We now reconsider the meandering strip of constant width shown in Fig. 6 b . Whereas we previously assumed that the curvature K of the centerline was small as well as slowly varying by setting K = E g ( E s ) , we now assume it to be slowly varying but not necessarily small, with K

=

G(Es).

(5.3)

Then Laplace's equation ( 2 . 5 3 ) assumes, instead of (2.55), the only slightly more complicated form

a

-(1+ an

With terms in

E*

a+

Gn)-+ an

a 1 a4 = 0. aSl+GnaS

E*---

(5.4)

neglected, quadratures yield the first approximation +=1-2

ln[(l

+ G)/(1 + G n ) ]

ln[(l + G ) / ( 1- GI1

(5.5)

Comparison with (2.59) shows that this is the exact solution for a circular annulus that at each station fits the local curvature of the strip, as indicated in Fig. 13. We may call it the osculating annulus approximation.

Slow Variations in Continuum Mechanics

41

FIG. 13. Osculating annulus approximation for meandering strip.

In this case it is clear how higher approximations can be found. Substituting our first approximation (5.5) into the neglected terms in (5.4) yields an iteration equation that can again be solved by quadratures. The resulting correction of order E’ is found in terms of elementary functions-powers of (1 + G n ) and its logarithm (and this would appear to be true of all subsequent approximations). However, the results is too complicated to be given here. 3. Laminar Flow in a Meandering Channel

This osculating annulus approximation can be applied also to plane laminar flow (Van Dyke, 1983) because the required basic solution exists in simple closed form (2.60). (By contrast, we are unable to treat in this way the three-dimensional flow through a meandering pipe of circular cross section, because except at zero Reynolds number the required basic solution for laminar flow in a torus can be found only approximately, as shown in Section III,B,2.) a. Reynolds Number of Order Unity If the Reynolds number is fixed, introducing the slowly varying centerline curvature G ( E sof ) (5.3) into the vorticity equation (2.60) shows that the first approximation satisfies

a

-(1+

an

a i a alC, Gn)--(1 + Gn)- = 0. an dn 1 + Gn an

Milton Van Dyke

42 Four quadratures yield

+ = A( 1 + G n ) 2In( 1 + G n ) + B(1 + G n ) 2+ C In( 1 + G n ) + D.

(5.7)

this is just our previous solution (2.62) for laminar flow in an annulus except that now the coefficients A, B, C, D, which are determined by imposing the boundary conditions at the walls, depend on the axial coordinate through the slowly varying curvature function G (E S ) . The neglected terms are of order E R and E ~ and , higher approximations would proceed in powers of those two quantities. Of course, the analysis is too complicated to be carried much farther by hand, but the author has suggested that it could be continued by using a computer program that manipulates symbols, such as MACSYMA. b. High Reynolds Number When E R is of order unity, the first approximation is governed, with an error of order E’, by the nonlinear equation [;(l+Cn)--ER an a

-----

(::a:

::a:)]

a a* (1 + G n ) 1 + Gn a n an i

~-

= 0. ( 5 . 8 )

This represents those terms in the Navier-Stokes equations that contribute to second-order boundary-layer theory, with its effects of longitudinal curvature. This equation, like its counterpart (2.63) for the classical boundary layer, is parabolic, with no upstream influence, and can therefore be integrated numerically along the channel (at least so long as no reverse flow occurs). Blottner (1977) has carried out the calculations for a diverging channel with a semicircular centerline. When the divergence is enough to produce a region of reverse flow at the inner wall, the numerical integration becomes unstable shortly beyond the separation point.

VI. Concluding Remarks

The approximation of slow variations has not received the attention that it deserves. Our survey of the method shows that it has been only sporadically developed, except perhaps for steady laminar flows. Further exploitation will surely yield many informative and useful results in the various branches of continuum mechanics. Higher approximations will be needed, both to refine the accuracy and to establish limits of convergence. Hand calculation becomes infeasible at an early stage, even in simple problems; but the routine labor involved can be readily delegated to a computer. Lucas’s (1972) treatment of laminar flow through a channel shows how even numerical computation can be

Slow Variations in Continuum Mechanics

43

used effectively. Further advances are to be anticipated from the application of symbol-manipulation programs. No doubt the majority of useful results will be based on the simple quasi-cylindrical approximation, because that can be found in simple form for a wide variety of problems in many fields. However, serious consideration must be given to the suggestion of Fraenkel (1962) for laminar flow that greater accuracy will result when a basic solution can be found that fits the geometry more closely. The technique for calculating higher approximations on that basis will have to be developed, for it was not apparent to us how to improve systematically even the tangent-wedge approximation for plane potential flow through a symmetric channel. Similarly, a systematic technique must be developed for calculating higher approximations in thin three-dimensional regions. We saw that for potential flow through the gap between two nearly parallel walls a quasi-twodimensional stream function provides a convenient first approximation, but it is not obvious how that is to be refined. A lot of good hard thinking needs to be devoted to the idea of slow variations ! ACKNOWLEDGMENTS The writing of this article was supported by the National Science Foundation under Grant No. MSM-8300537. The author is indebted to Andreas Acrivos for helpful discussion. REFERENCES Abramowitz, M. (1949). On backflow of a viscous fluid in a diverging channel. 1.Marh. Phys. (Cambridge, Mass.) 28, 1-21. Blasius, H. (1910). Laminare Stromung in Kanalen wechselnder Breite. 2. Marh. Phys. 58, 225-233. Blottner, F.G. (1977). Numerical solution of slender channel laminar flows. Comp. Methods Appl. Mech. Eng. 11. 319-339. Chow, J. C . F., and Soda, K. (1972). Laminar flow in tubes with constriction. Phys. Fluids 15, 1700- 1706. Daniels, P. G., and Eagles, P. M. (1979). High Reynolds number flows in exponential tubes of slow variation. J. Fluid Mech. 90, 305-314. Dean, W. R. (1928). The stream-line motion of fluid in a curved pipe. Philos. Mag. 5(7), 673-695. Eagles, P. M. (1982). Steady flow in locally exponential tubes. Proc. R. Soc. London Ser. A 383, 231-245. Eagles, P. M., and Muwezwa, M. E. (1986). Approximations to flow in slender tubes. J. Eng. Math. 20, 51-61. Eagles, P. M., and Smith, F. T. (1980). The influence of nonparallelism in channel flow stability. J. Eng. Math. 14, 219-237. ErdClyi, A. (1961). An expansion procedure for singular perturbations. Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. 95, 651-672. Fraenkel, L. E. (1962). Laminar flow in symmetrical channels with slightly curved walls. I. On the Jeffery-Hamel solutions for flow between plane walls. Proc. R. SOC.London Ser. A 267. 119-138.

.(

44

Milton Van Dyke

Fraenkel, L. E. (1963). Laminar flow in symmetrical channels with slightly curved walls. 11. An asymptotic series for the stream function. Proc. R. SOC.London Ser. A 272, 406-428. Fraenkel, L. E. (1973). On a theory of laminar flow in channels of a certain class. Proc. Cambridge Philos. SOC.73, 361-390. Geer, J., and Keller, J. B. (1979). Slender streams. J. Fluid Mech. 93, 97-115. Gohner, 0. (1930). Schubspannungsverteilung im Querschnitt einer Schraubenfeder. Ing. Arch. 1, 619-644. Gohner, 0. (1931a). Schubspannungsverteilung im Querschnitt eines gedrillten Ringstabs mit Anwendung auf Schraubenfedern. Ing. Arch. 2, 1-19. Gohner, 0. (1931b). Spannungsverteilung in einem an den Endquerschnitten belasteten Ringstabsektor. Ing. Arch. 2, 381-414. Kaimal, M. R. (1979). Low Reynolds number flow of a dilute suspension in slowly varying tubes. Inf. J. Eng. Sci. 17, 615-624. Keller, J. B., and Geer, J. F. (1973). Flows of thin streams with free boundaries. J. FZuid Mech. 59, 417-432. Kotorynski, W. P. (1979). Slowly varying channel flows in three dimensions. J. Insf. Mafh. Its Appl. 24, 71-80. Larrain, J., and Bonilla, C. F. (1970). Theoretical analysis of pressure drop in the laminar flow of fluid in a coiled pipe. Trans. SOC.Rheol. 14, 135-147. Latta, G. E., and Hess, G. B. (1973). Potential flow past a sphere tangent to a plane. Phys. Fluids 16, 974-976. Lucas, R. D. (1972). A perturbation solution for viscous incompressible flow in channels. Ph.D. dissertation, Stanford Univ.; Univ. Microfilms, order no. AAD72-30664. Manton, M. J. (1971). Low Reynolds number flow in slowly varying axisymmetric tubes. J. Fluid Mech. 49, 451-459. Massonet, Ch. (1962). Elasticity: Two-dimensional problems, In “Handbook of Engineering Mechanics” (W. Fliigge, ed.), pp. 37-1 to 37-30, especially p. 37-23. McGraw-Hill, New York. Morse, P. M., and Feshbach, H. (1953). “Methods of Theoretical Physics.” McGraw-Hill, New York. Olson, D. E. (1971). Fluid mechanics relevant to respiration: Flow within curved or elliptical tubes and bifurcating systems. Ph.D. thesis, Imperial College, London. Rosenhead, L., ed. (1963). “Laminar Boundary Layers.” Oxford Univ. Press, London. Sobey, I. J. (1976). Inviscid secondary motions in a tube of slowly varying ellipticity. J. Fluid Mech. 13, 621-639. Sobieczky, H. (1977). Kompressible Stromung in einer ebenen Schicht variabler Dicke. Z. Angew. Mafh. Mech. 57, T 207-T 209. Timoshenko, S., and Goodier, J. N. (1951). “Theory of Elasticity.” McGraw-Hill, New York. Todd, L. (1977). Some comments on steady, laminar flow through twisted pipes. J. Eng. Math. 11, 29-48. Todd, L. (1978). Steady, laminar flow through non-uniform, curved pipes of small cross-section. Tech. Rept. No. 78-19, Inst. Appl. Math. Stat., Univ. British Columbia, Vancouver. Todd, L. (1979). Steady, laminar flow through curved pipes of small, constant cross-section. Tech. Rept. No. 79-1, Dept. Math., Laurentian Univ., Sudbury, Canada. Todd, L. (1980). Steady, laminar flow through non-uniform, thin pipes. Tech. Rept. No. 80-1, Dept. Math., Laurentian Univ., Sudbury, Canada. Van Dyke, M. (1970). Entry flow in a channel. J. Fluid Mech. 44, 813-823. Van Dyke, M. (1975). “Perturbation Methods in Fluid Mechanics,” Parabolic, Stanford, California. Van Dyke, M. (1978). Extended Stokes series: Laminar flow through a loosely coiled pipe. J. Fluid Mech. 86, 129-145. Van Dyke, M. (1983). Laminar flow in a meandering channel. SIAM J. Appl. Math. 43,696-702.

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Wang, C.-Y. (1980a). Flow in narrow curved channels. J. Appl. Mech. 47, 7-10. Wang, C.-Y. (1980b). The helical coordinate system and the temperature inside a helical coil. J. Appl. Mech. 47, 951-953. Wang, C. Y. (1981). On the low-Reynolds-number Row in a helical pipe. J. Fluid Mech. 108, 185-194. Wild, R., Pedley, T. J., and Riley, D. S. (1977). Viscous Row in collapsible tubes of slowly varying elliptical cross-section. J. Fluid Mech. 81, 273-294. Williams, J . C., 111 (1963). Viscous compressible and incompressible Row in slender channels. A I M J. 1, 186-195.

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A D V A N C E S I N A P P L I E D MECHANICS, VOLUME

25

Modern Corner, Edge, and Crack Problems in Linear Elastodynamics Involving Transient Waves JULIUS MIKLOWITZ Division of Engineering and Applied Science California Institute of Technology Pasadena, California 91 125

I. Introduction The solution of corner, edge, and crack problems, based on the equations of motion from the linear theory for a homogeneous, isotropic elastic material, is an important topic of long-standing interest, difficulty, and challenge. Three important subtopics are (1) elastic waveguides, (2) elastic pulse scattering by cylindrical and spherical obstacles, and (3) the twodimensional wedge and quarter-plane. We will focus our survey on work done in these subtopics. The basic difficulty in these problems is exhibited in the waveguide example of Love’s treatment of the free longitudinal vibration of a finitelength, circular section, cylindrical rod (Love, 1927). The rod here has a stress free lateral surface and edges (or ends), hence stress free corners. Its edge conditions are of the nonmixed type, that is, stress components (present case) or displacement components specified. Love shows that attempts to treat this problem with a classical separation technique, aided b y Pochhammer’s frequency equation for the infinite rod (Pochhammer, 1876) failed (hence nonseparability), leading to a solution in which the normal stress on the rod edges vanishes but not the shear stress. This led to the classical approximation that for a long, thin rod (radius << length), this shear stress could be assumed to be zero because its mate on the neighboring lateral surface is zero. Indeed, this type of approximation, with the simplifying features brought forth by the low-frequency long waves it represents, has led to a great deal of work on the creation and use of approximate 47 Copynght 0 1987 by Academic Press, Inc All nghts of reproduction In any form reserved

Julius Miklowitz

48

one-dimensional theories for treating waveguide and vibration problems in the rod, plate, and shell. This earlier work in waveguides is discussed in detail in Miklowitz (1978). Our survey in this area will focus on modern edge load problems for the semi-infinite waveguide. Our second topic is the scattering of an elastic pulse by cylindrical and spherical obstacles. The present class of problems has its background in acoustics and seismology, where in the latter a good share of the work has been on time-harmonic excitation. Gains in the related, more difficult problems of transient excitation have been made only in recent years, due primarily to the quest for information in protective construction problems (see Miklowitz, 1966a). The book by Pao and Mow (1973) presents a comprehensive survey and treatment of the subject. Our survey in this area will focus on three modern edge problems, the scattering of an elastic pulse by a circular cylindrical cavity and an elastic inclusion and the diffraction of an elastic pulse by a spherical cavity. Our last topic is the general problem of wave propagation in a twodimensional elastic wedge and its special case the quarter-plane, with their mixed or nonmixed edge conditions. Again, these problems are of considerable difficulty but of long-standing interest and importance in mechanics and seismology. The basic difficulty in getting solutions is due to the nonseparability of such problems. In an excellent survey published in Knopoff (1969), the various analytical methods that have been tried on these problems are reviewed. The survey ends on the sad note that none of the techniques tried (images, integral transforms, Weiner-Hopf method, iterations, and others) were successful. Knopoffs paper, however, is a very valuable contribution on this problem and its solution. Our survey in this area will focus on recent works in the two-dimensional elastic wedge and its special case the quarter-plane with their mixed or nonmixed edge conditions. Most of the work has been done on the quarter-plane.

11. Elastic Waveguides

A. EDGE LOAD PROBLEMS FOR

THE

SEMI-INFINITE WAVEGUIDE

1. Introduction and the Governing Equations for the Plate ( o r Layer) in Plane Strain The literature shows that contributions to the present subject have been mostly on the problems involving mixed edge conditions, that is, a mixture of stress and displacement components, stemming primarily from their separable nature. In the following sections first a review is presented of methods that have been used to study transient wave propagation in

Modern Corner, Edge, and Crack Problems

49

waveguide problems of this type and the information they have produced. Then a similar discussion is given of methods and contributions in problems involving nonmixed edge conditions. Advances have been made only quite recently in these nonseparable problems. It can be shown that when the formal solution (inversion integrals) for a waveguide problem can be written, inversion procedures based on modal integrals are available to obtain the solution, which can then be analyzed numerically or through approximations. For the semi-infinite waveguide, with its edge or end, the formal solution can be difficult to derive. Some procedures have been established for these cases and a discussion of them, and approximate solutions they have yielded, is worth a bit of our time. We draw on the displacement equations of motion, for it will become evident that they offer a certain convenience in treating the edge boundary conditions. We treat the semi-infinite plate in plane strain. Other waveguides, at least for mixed edge conditions, can be treated similarly. The displacement equations of motion for the plane strain case are u,(x, y, t ) + k - 2 ( k 2- l ) ~ , ,+ k-2u, u,,(x,

=

~d’ii

y, t ) + ( k 2 - l ) ~ ,+ k ’ ~ , , = , cF’V

x > 0,

- h < y < h,

t > 0,

(2.1)

where we are using y as the thickness coordinate and v as the corresponding displacement. The stress-strain relations are

+

+

u , ( x , ~ ,t ) / ( A 2 p ) = U , k-’(k’ - 2 ) ~ , , u,,(x, y, t ) / ( A + 2 p ) = k-’(k’ - 2 ) u , + u ~ , U,,(X,Y,

t ) / P = ux

+ uy,

Uz

= 4 a x

(2.2)

+ Uy).

Subscripts attached to displacements indicate differentiation with respect to that coordinate, but when attached to stress indicate the component in the usual way. Initial conditions are taken as u ( x , y , 0) = U ( X , y, 0) = u(x, y, 0) = 7qx,y, 0) = 0, x 2 0, - h 5 y S h,

(2.3)

and conditions at x + 03 as

2 . Plate ( o r Layer) in Plane Strain with Fixed Edge Conditions: Formal Solutions The formal solutions for the two problems of this type involving symmetric excitation can be written with the aid of the Fourier sine and cosine transforms. They are discussed in the following sections.

50

Julius Miklowitz

a. Longitudinal Impact Problem Figure 1 depicts the problem. The plate edge (x = 0) is subjected to a step uniform normal velocity u under zero shear stress a,... These edge conditions are given by 4 0 , Y, t ) = v o H ( t ) %.(O, Y, t ) = /l[VX(O, y, t ) + UJO, y, t ) l

=0

I

- h 5 y 5 h,

t > 0,

(2.5)

where use has been made of the third of (2.2). First we apply a Lapiace transform on t, parameter p , to (2.1), using (2.3). Then since we have a displacement type input for u, and noting the form of

-

g(2)s(K)

=

-K2$(K)

+

-

and

Kg(0)

g‘2’c(K)

=

-K2gc(K)

- g‘(o),

we further apply a sine transform to the first and a cosine transform to the second of the Laplace transformed equations (2.1). There result the doubletransformed equations ii,;”(x, y, p ) - k 2 v i i i - s - ( k 2- ~ ) K V - ‘ = - k 2 4 0 , y, PI, VJX, y, p ) - k-2v:V-C k-2( k2 - 1 ) ~ i i y ’

+

(2.6)

= k-2[t?,(0,Y, P ) + ( k 2- 1 ) i q O 7 Y,P)17

+

where vd = ( k i K ~ ) ” ’ and vs = ( k : + K ~ ) ” ~with , kd = p / c d , k, The Laplace transforms of the edge conditions (2.5) are

G(0, Y , P) = PG(0, Y , P) = V O / P ,

=p/c,.

(2.7)

since u(0, y, 0) = 0 from (2.3), and &.(O,

Y , P) = /l[fix(O,

Y , P) + U,(O, Y , PI1 = 0.

(2.8)

Now noting from (2.7) that U(0,y, p ) is independent of y , it follows that ii,,(O, y, p) in (2.8) vanishes, and hence Vx(O, y, p ) there also vanishes. Therefore, (2.7) and (2.8) reduce to

u(0, Y , P) = U 0 / P 2 ,

FIG. 1.

q o , Y , P) = G(0, Y , P) = 0,

Longitudinal impact problem

(2.9)

51

Modern Corner, Edge, and Crack Problems

and substitution of these into (2.6) determines their right-hand sides. Thus we see the choice of spatial transforms here was made so that the given edge conditions would be asked for. The solution of these coupled ordinary differential equations is easily obtained through substitution of the forms C"S(K, C-'(K,

U,P) Y,P)

= =

A n ( K , P) exP[-n(K, P ) Y l + B n ( K , P) exP[-n(K, P ) Y l +-

GpS(K, y, p ) ,

(2.10)

C i C ( K , y,

where U p s and 17;' are particular integrals needed for the now simple right-hand sides of (2.6). The n ( K , p ) are found to be + " I d , * T ~ The . particular integrals are found to be "' = u o ~ / p 2 ~and :, 6 ; ' = 0. It follows that the general solution is

_-

'(K,

z)-'(K,

y, P ) = A ( K , P ) cash 7d.Y -k B ( K , P) cash "Id'-I-u o K ( P " I d ) - 2 , y , p ) = - [ K - ' ~ ~ A ( K , p ) sinh qdy + K T ; ~ B ( K , P ) sinh q s y ] ,

(2.11)

stemming from the symmetry of the loading and the fact that the two algebraic equations, which yielded n(K, p ) = * " I d , * q s , must hold for all these values of n(K, p ) . The traction free conditions at the plate faces a,,and axyin (2.2) vanish at y = * h

are transformed with an eye on the fact that u involves a sine and u a cosine transform. Substitution of (2.11 ) into these transformed conditions, that is, 6yC(K,

h, p ) / ( A + 2 p ) = k p 2 ( k 2- 2)[KU-'(K, h, p ) - Uop-'] _a , ; ( ~ h, , p ) / p = - K Z ) - ' ( K , h, p ) C J K , h, p ) = 0,

+ fiy' = 0,

+

determines A and B and hence the transformed solution to the problem. The formal solutions (inversion integrals) can then be written by making use of the fact that u" = 2iu"",and 6 = 2 P . The formal solutions for the compressional strains are

where

Julius Miklowitz

52

The technique is basically due to Folk et al. (1958) and Curtis (1956), the first of these references being on the circular cylindrical rod and the second on the plate. b. Mixed Pressure Shock Problem Consider the problem shown in Fig. 2. The plate edge is subjected to a step uniform normal stress vX,under zero thickness displacement u. These conditions are given by

where uois the magnitude (a positive constant) of the a; input. The method of reducing (2.1) to a set of ordinary differential equations is the same as that leading to (2.6), except that the sine and cosine transform are interchanged so that the given information in (2.13) is asked for. Instead of (2.6), here the analogous transformed equations are

G;,(K, y , p ) - k V d U + ( k 2 - I ) K G =~ k2ii,(0,y, p ) + ( k 2 - l)V,,(o,y , p ) , G y 7 s ( ~ ,y, p ) - k-2V:G-S - k - 2 (k2 - ~ ) K U ,=~ -k-*Kfi(O,y, p ) . (2.14) 2

2--c

The Laplace transform conditions at the edge (x

ex(O,Y , P ) / ( A + 2

=

0) are

~ = )&(O, Y , p ) + k - 2 ( k 2- 2)V,(O, Y , P) = -uo/(A

+

V(0,Y, P) = 0,

(2.15)

where we have used the first of (2.2). From the second of (2.15) it follows that G,,(O, y, p ) = 0, and therefore the first reduces to &(O, y , p ) = -cr,,/(A + 2p)p. Hence the right-hand sides of (2.14) are determined. From here on, the procedure is the same as it was in the foregoing case, and a formal solution similar to (2.12) would be obtained. c. Mixed Edge Conditions: Problems with Antisymmetric Excitation Similar procedures exist for plate problems involving mixed edge conditions and antisymmetric excitation. An example is Nigul's (1963/64)

FIG. 2.

Mixed pressure shock problem.

Modern Corner, Edge, and Crack Problems

53

treatment of the semi-infinite plate subjected to an edge step moment, under zero transverse velocity, that is, a pinned edge. His formal solution is obtained by essentially the same technique as that devised for the rod in flexure by De Vault and Curtis (1962), which is a generalization of that in Folk et al. (1958). d. Formal Solutions for Other Waveguides The reader will be interested in some brief comments on formal solution techniques that have been devised for other waveguide problems involving mixed edge conditions. By developing exterior domain Hankel transforms for the infinite plate with a circular cavity at the plate center, Scott and Miklowitz (1964) solved the mixed problem of sudden uniform radial displacement of the cavity wall. Jones and Ellis (1963a), drawing on the Rayleigh-Lamb frequency equation for symmetric waves and the plane stress-plane strain analogy, extended the use of the technique in Folk et al. (1958) and Curtis (1956) to the mixed pressure shock problem in a semiinfinite, wide, rectangular bar in plane stress, with stress-free lateral sides. Rosenfeld and Miklowitz (1965) also found the technique in Folk et al. (1958) and Curtis (1956) useful in their work on wave propagation in a rod of arbitrary cross section. 3 . Plate ( o r Layer) in Plane Strain with Mixed Edge Conditions: Approximate Solutions a. Long-Time and/or Far-Field Approximations It is known that the formal solutions for the present mixed edge condition plate problems (2.12) for the longitudinal impact problem and the analogous one for the mixed pressure shock both have in them the lowest branch-type integral. Here the integrand is singular at w = 0. So, the method of stationary phase and the completion of the pieced-off part of the integral here (singular at w = 0) reduces the formal solution of mixed pressure shock problem to the long-time, far-jield approximation

where the dimensionless variables 5 = x / h, 9 = y / h, and T = c,t/ h have been introduced, a' = ( T - 5 / b ) / ( 3 6 ( / b ) " 3 ,b is the dimensionless plate velocity cp/c,, 6 = ( k 2- 2)2/6k4, and Go is the constant k2mo/4p(k2- 1 ) . The solution (2.16) is based on the approximation to the lowest

Julius Miklowitz

54

Rayleigh-Lamb frequency branch KI(P)

=

(iP/C,)[l - w W / c , ) * +

. . .I.

(2.17)

The second equation in (2.16) expresses the Poisson's ratio coupling of the plate. The corresponding approximation for the longitudinal impact problem with formal solution (2.12) differs only trivially from (2.16) in the constant Go. This supports the fact that the long-time, far-field solution is basically independent of the nature of edge compressional loading. It is not surprising that this type solution is the same for the rod and plate, since (2.17) has the same form as in the approximate theories. We should point out that these exact rheory solutions have a first approximation, which corresponds to the elementary theory solution shown in Fig. 3. This is the step function [in (2.16) it would be H ( T - [ / b ) instead of the integral of the Airy function appearing there]. The second approximation (2.16) shows, however, that the first should be used with caution, since having the former we see the finite jump in the latter cannot be valid. The first is useful for extremely low frequency, long waves. Skalak (1957) derived his long-time, far-field solution for the rod by inverting the time transform first and using the stationary phase method. Folk et al. (1958) and Curtis (1956) derived their solutions for the rod and plate by inverting the spatial transform (Fourier) first and then approximating the Bromwich integrals by the extended saddle point method. In Jones and Ellis (1963b) the extended saddle point technique was used as in Folk et al. (1958) and Curtis (1956), so as to get second-order terms in (2.16) that involve derivatives of the Airy function. These terms account for warping of the cross section of the bar, which is not represented by the plane section nature of (2.16). It was shown by Jones and Ellis (1963b) that the terms accounted for warping of plane sections exhibited in fringe patterns occurring in dynamic photoelasticity shock tube experiments on the rectangular bar. Jones and Norwood (1967) applied this type of analysis to the semi-infinite, circular section rod and assessed the lateral strains and

0

"

Z/C,

r

FIG.3. Long-time, far-field solution of rod longitudinal impact problem (ti axial velocity, uz axial stress).

Modern Corner, Edge, and Crack Problems

55

stresses [the latter corresponding to the rod equivalent of (2.16) are zero]. They showed that their analysis for the surface radial strain had closer agreement with the corresponding response record in Miklowitz (1958) than the rod equivalent of the second of (2.16). In Scott and Miklowitz (1964) the time transform was inverted first, but the approximation (2.16) does not occur, basically because of the spatial decay introduced by the axially symmetric nature of the problem. However, Scott and Miklowitz extended a stationary phase analysis in an earlier related work by Miklowitz (1962) to the problem in Scott and Miklowitz (1964) and obtained long-time, far-field, lowest-branch approximations for the radial and thickness displacements. The techniques in Miklowitz (1962) and Scott and Miklowitz (1964) include a time of occurrence-predominant period criterion for restricting the time region of validity for the approximation at a station. This so that higher lowest-branch frequencies and the first few thickness branches do not contribute. A comparison of the stationary phase approximation and (2.16) will also be of interest to the reader. The particular problem solved in De Vault and Curtis (1962) was that of the semi-infinite, circular section rod subjected to sudden pressure over half the edge ( - 7 ~ / 25 6 5 7r/2), while this edge begins to move laterally (along 6 = 0) with constant velocity. Inversion, evaluation, and experiments followed the theme in Folk et d. (1958) and Curtis (1956) and the work in Fox and Curtis (1958). De Vault and Curtis (1962) showed that for long time, or the far field, the flexural strain disturbance is composed predominantly of lowest-branch action, the head stemming from a group velocity maximum at moderately high frequency and the tail from the low-frequency, long-wave domain. Amplitudes for the head are proportional to the Airy function and for the tail to the Fresnel integral. A long-time approximation, involving the integrals for some higher antisymmetric real wave number segments, was carried out by Nigul. This work is quite detailed, pointing out what segments are needed for various quantities (midplane displacement, moment, etc.) for all long times at a station. Jones (1964) contributed similarly on a closely related problem to that of Nigul’s, although not an edge load problem. Important, however, was the fact that he evaluated, for a long time, contributions from the first four real segments for regions on these close to the dilatational wavefront [associated with maximum and minimum group velocities cg( K ) ] .

b. Short-Time, Near-Field Wavefront Approximations Rosenfeld and Miklowitz (1962) have applied the Cagniard-deHoop technique to the formal solution (2.12) and the analogous solution for the mixed pressure shock problem, and they established the amplitudes and locations of all wavefronts emanating from the edge source in these problems. We can therefore be brief here. Removing the Bromwich integrals

56

Julius Miklowitz

from (2.12) leaves solutions for Gx(x,y, p) and U,,(x, y, p ) , that is, the Laplace transforms of these strains. When the transformation K = kdl, with p real, is introduced into the remaining integrals in (2.12), Gx(x,y , p ) , for example, then takes the form

where now

The first integral is easily inverted. It corresponds to a leading plane step wave of u,, traveling with speed cd, and is due to the uniform (in y ) nature of the edge source. The inversion problem lies in the second integral. The first term in N corresponds to the dilatational and the second to the equivoluminal waves in u,. By focusing on the integral involving the first term (process is similar for the second), the hyperbolic terms in the integrand are expanded into exponentials. After some algebra, and consideration of the larger exponential terms in R, through use of the binomial theorem the exponential terms of R may be brought into the numerator o f the integrands in the form of a series of terms involving exponentials. Further algebra separates this series into one that involves terms with a single exponential. The result is a series of integrals, each of which represents the disturbance following a single wavefront. The integrals are of the form

I-, m

Gxd(x, y , p ) =

ud(<)

exp[-pgd(5)1 d5,

(2.19)

where the subscript d denotes dilatational, and gd(l) = g ( l , 'Y,

x,

where y = (2n

=

(l/cd)[(ixl + 'YvA(l))+ 677:(5)],

+ 1 ) h * y,

S

=

2mh,

and U,, an involved function of 5, also depends on n and m, which are determined with the aid of geometric ray theory. It is well known that discontinuities in either the geometry or load are sources of wavefronts,

Modern Corner, Edge, and Crack Problems

51

LINE LOAD SOURCE

HEAD WAVE

P - DILATATIONAL S- EOUIVOLUMINAL

REGULAR P- WAVEFRONT

FIG. 4.

Basic wavefronts in a plate.

and rays, along which wavefronts, normal to the rays, propagate. The corners x = 0, y = +h are such points. Figure 4 shows the basic system of wavefronts that are generated at the upper source x = 0 , y = h (at x = 0, y = - h there would be another system, symmetric to those in Fig. 4). Note that the system is that in Lamb’s line load problem for the half-space. One would expect this. Figure 5 shows a family of multiply reflected rays, these paths being generated by mode conversion at the plate faces y = +h, for a pair of P (dilatational) and S (equivoluminal) wavefronts emanating from the source. Letter n measures the number of times a front crosses the plate as a P front, and rn the number of times as an S front, ending up (in Fig. 5) at either P, or P, with coordinates (x, y ) . In Fig. 5, n = 1, rn = 2 for any path. Letters y and 6 correspond to the propagation distances along the multiply reflected ray path in the transverse direction y as P and S waves, respectively. The minus sign is taken in y in (2.19) for definition at P, in Fig. 5, because the last reflection is at a point on the lower plate face (plus sign for upper). With the aid of Snell’s law, sin (Y = k sin p, Rosenfeld and Miklowitz (1962) give expressions for the total distance in the x direction along any path and the wavefront arrival time. These can be used to find the wavefront positions. Clearly, (2.19) can be inverted by Cagniard’s technique. The requirements establish the analyticity of g ( 5) in the 5 plane by suitable cuts for the branch points of q & ( l )and ~ : ( lThen ) . the real path integral in (2.19) is traded for one along a path in this region of analyticity, which is determined by g d ( J )= t, real and positive. It follows using l ( t ) , defined implicitly by

i-SOURCE OF R A Y 5

FIG. 5 .

A family of rays.

58 &(()

Julius Miklowitz = t,

that (2.19) can be written as (2.20)

where T~ is the dilatational wavefront arrival time at a particular station (x, y ) . Since 2 Re{ } in (2.20) is the inverse Laplace transform, we have (2.21)

A time integration then gives u,d(x, y , t ) , and this can be approximated near 7d to give the wavefront time behavior and its amplitude. Results similar to (2.21) exist for the equivoluminal waves, including the head waves. Figure 6 shows the wavefronts that have developed by time t = 8h/cd and 16h/c,, to geometric scale. The wavefronts for the strains, for the edge load problems being discussed here, were all, except for the leading step, found to have the time behavior ( t - T ) ” ~ , where T is the arrival time of the particular front. The line load problem also treated in Rosenfeld and Miklowitz (1962) showed ( t - T ) - ” ~ for the regular wavefonts. Since the time input for this problem was the step function, this time behavior is in agreement with that for the regular wavefronts in Lamb’s line load problem. This would be expected. Numerical data are also given for the longitudinal impact problem for the amplitude coefficients of these wavefronts at t = 8h/cd and t = 16h/cd (see Figs. 6 and 7 there). It was established by these numerical data that at the surface the strongest wavefronts were created directly by the reflection of the head waves. Comparison of wavefront arrivals at the surface, with those in the near-field records in Miklowitz and Nisewanger (1957) and those in Meitzler ( 1 9 5 9 , showed definite agreement in the case of the head waves. Other waves were weaker and, since they were poorly defined in the records, were hard to identify positively.

FIG. 6 . Wavefronts

(t =

8 h / c , above, r

=

1 6 h / c , below, v = 0.3).

Modern Corner, Edge, and Crack Problems

59

A method for obtaining wavefront approximations that is applicable to a broader class of waveguide problems than the Cagniard-deHoop method was presented by Randles and Miklowitz (1971). The technique is based on high-frequency approximations to the branches of the Rayleigh-Lamb frequency equation, but not in their usual representation. That representation, with its complicating terraces at the dilatational wave speed and superposition of the branches at the equivoluminal wave speed, does not lend itself to high-frequency approximation. In Randles and Miklowitz (1971), the Rayleigh-Lamb spectra are mapped into a new space, defined by the change in variables

where Td, 7,are the usual thickness wave numbers. The x - q space offers a more direct approach to the high-frequency response of the plate. In the present method, the infinite plate, subjected to an impulsive normal line load on one of its faces, is treated. The x,71 variables lead to a new form of the formal modal solution. The wavefronts are extracted from this solution by exploiting the 7-x branches of the Rayleigh-Lamb equation and in particular their branch points, about which analytical continuations are made, which uncouple the dilatational and equivoluminal motions. An equivalent modal solution over the dilatational and equivoluminal branches is then derived, and this is approximated by using series representations for the branches. Summation over the modes corresponding to these branch approximations then gives the wavefronts. Comparison is made of the latter with the known wavefronts from Lamb’s line load problem. Randles (1973) has extended the work to anisotropic plates and produced some interesting information on certain cusp-type wavefronts. 4. Plate ( o r Layer) in Plane Strain with Nonmixed Edge Conditions:

Formal Long-Time Solutions and Their Inversions Because of the nonseparability in the present class of problems, direct application of integral transforms is not indicated. The right-hand sides of (2.6) and (2.14) point this out clearly; that is, they do not ask for the given edge conditions which are both stress or both displacement components. No analytical procedure has yet been devised that can set down the general formal solutions [like (2.12) or in another form] for both types of problems in this class. However, Miklowitz (1969) presented a method that handles such problems for nonmixed edge displacements for long time. It was extended to the case of nonmixed edge stresses later by Sinclair and Miklowitz (1975). The following sections present the technique, application to some of the problems, and results found.

Julius Miklowitz

60 a. Inversion Integral Forms

We return to the displacement equations of motion (2.1) and again apply a Laplace transform on t to them by using the initial conditions (2.3). Now, critical in the method, to these a Laplace transform on x, parameter s, is applied. The result is UJS,

Y, P) + ( k 2 - lbe;(s, Y, p ) + ( k 2 s 2- kf)u-(s, Y , P) = k2[sU(0,Y , P) + f d o , Y , PI1 + ( k 2 - I)vy(0,Y , P) =

f(s, Y , P),

(2.22)

U Y y ( ~ , y , p+) k p 2 ( k 2- l)sUY(s,~,p)+ kp2(s2- k , Z ) v " ( s , y , p )

k - * [ s W , Y , P ) + G(0, Y , PI1 + kp2(k2- l)U,(O, Y , P) = A s , Y, PI, =

where a tilde over quantities indicates the Laplace transform on x. Note that f ( s , y, p ) and g ( s , y, p ) are composed of the time-transformed edge unknowns U, U,, V,,, and i7, fix, and Uy,respectively, in a manner similar to the right-hand sides of (2.6) and (2.14). However, (2.22) alone asks for all these edge quantities, a basic factor in why the present technique can handle nonmixed edge problems. Use of terms like the first in (2.10), with K replaced by s, determines the characteristic roots and the complementary functions U; and V;. A further Laplace transform on y, along with a convolution, gives the particular integrals U p and 6;. Although certainly not restricted to them, particular interest will be symmetric loadings. In addition to edge loadings, we will also be interested in face loadings in these problems. The face conditions, when the loading there is normal, are u y ( x ,zkh, t ) = u 0 F ( x ) G ( t ) ,

u X y ( x *h, , t ) = 0,

(2.23)

where F ( x ) , G ( t ) are Laplace transformable functions and where uois an inherently negative magnitude constant of dimensions force-unit length. We, of course, could have symmetric shear-type loadings on the faces as well by interchanging the right-hand sides of (2.23). Now with symmetric loadings,

v ( x , 0, t ) = 0,

UJX,

0, t )

=

0,

(2.24)

for x > 0, t > 0, and hence from here on we need only consider the upper half of the plate, x 2 0,O 5 y 5 h. Making use of (2.23) then leads to the inversion integral statements for the displacements

where

U-(s, y, p ) = U,

+ Up,

e-(s, y, p ) =

v, + vp.

Modern Corner, Edge, and Crack Problems

61

where

_-

u , = 2[A(s,p ) cosh a y

+ B(s, p ) cosh P y ] ,

0; = 2[as-’A(s, p ) sinh a y - s p - ’ B ( s , p ) sinh P y ] ,

ii,(s, y, p ) = k;’

ijp(s, y, p ) = kd2

5,’

{ [ s 2 a - ’sinh a ( y - y’) + P sinh P ( y - y ’ ) ] f ( s y, ’ , p )

IOy

{ [ asinh a ( y - y ’ ) + s 2 P - ’ sinh P ( y - y ’ ) ] g ( s ,y ‘ , p )

and

N s , P) = B,(s,

‘4% P) = A N ( % p ) l R ( s , P),

p ) l R ( s , P),

2A,(s, p ) = -s[ k2(2s2- k f ) sinh p h . I + 2sP cosh p h . J ] , 2B,(s, p ) = -P[2k2sa sinh a h . I - (2s’ R ( s ,p )

=

(2s2 - k:)’ cosh ah sinh p h

I = T ( s , p )+ k;’

-

k:) cosh a h . J ] ,

+ 4s’aP

sinh ah cosh

Ioh

{k-’s[a-’(2s2 - k:) sinh a ( h - y ‘ )

+ 2P sinh P ( h - y ’ ) ] f (s, y ’ , p ) + [( 2s2 - kf) cosh a ( h - y ’ ) -2s’cosh P ( h - y ’ ) ] g ( y~ ’, , p ) }dy’ + k p 2 ( k 2- 2)U(O, h, +I),

J

=

ki2

loh

{[2s2cosh a ( h - y ’ ) - (2s2 - k:) cosh P ( h - y ’ ) l f ( s ,y ’ , p )

+ k 2 s @ - ’ [ 2 a Psinh a ( h x

a

=

-

y ’ ) + (2s’ - kz) sinh P ( h - y ’ ) ]

Y ’ ,PI} dY’ - V(0, h, P),

(ki - s~)’’~,

P

=

(k:

-

s*)”~,

T ( s , p )= a $ ( s ) G ( p ) / ( A

+ 2p),

and where Br, and Br, are, respectively, the Bromwich contours in the p and s planes. It should be noted at this point, however, that (2.25) is not yet the formal solution to the problem, since it contains the edge unknowns, throughf(s, y , p ) , g ( s , y , p ) , U(0,h, p ) , and V(0, h , p ) . Once these are determined for a specific problem, (2.25) becomes the formal solution. Then it can be inverted by modal techniques, assuming also the inverses of UF and ijp can be found. b. Boundedness Condition: Integral Equations for the Edge Unknowns It is well known that the complex segments of branches of the RayleighLamb frequency equation have images in all the coordinate planes of the

62

Julius Miklowitz

W - K space (Onoe et al., 1962). This makes a total of four such segments for a branch K , ( - ~ w ) for w 2 0, which are also known to pierce the w = 0 plane, with vertical tangents. There are an infinite set of such segments and ,those r) segments their images. Assuming a harmonic wave form of A e '( K x - w satisfying Im K , ( - ~ w )> 0, which are K ; ( - ~ w ) (superscript c denotes complex) and - K ~ ~ " ( - ~ W we ) , can show exponential decaying (in x ) edge waves. A few of these segments, in the low-frequency domain, are shown as dashed lines in the sketch to the left in Fig. 7. The two other segments satisfy Im K J ( - i o ) < 0. They are - K ; ( - I ' W ) and Kf(-iw), which are the solid lines in the figure. These lead to exponentially unbounded waves as x grows. Since R ( s , p ) in (2.24), set as 0, is another generalized form of the RayleighLamb frequency equation, which if s = iK and p = -iw, becomes the latter, R ( s , p ) = 0 must have the roots s,(p) = k,(p ) on analytical continuation arguments. It follows that there are four segments corresponding to those just discussed, two of which would give, through residue evaluation of the inner integral in (2.25), exponentially unbounded waves at x -+ 00, hence violating condition (2.24). These segments satisfy

Re S , ( P ) > 0,

(2.26)

that is, complex sJ(p ) and S;( p ) (where we have now dropped the superscript c), which are shown in the sketch to the right in Fig. 7 (the edge waves - s J ( p ) , and - f , ( p ) , lying in the left half-plane, are not shown there). It follows that these unbounded waves can be eliminated by requiring the residues in the Br, integral in (2.25) associated with them, to vanish, that is, A N ( s , P)lr=r,(p)

=

B N ( s , P ) I s = ~ I ( p )=

O,

(2.27)

where s , ( p ) are the two sets of roots of R ( s , p ) in (2.25) satisfying (2.26). Substitution of A , and BN from (2.25) into (2.27) gives two equations for I and J for each s , ( p ) . Compatibility of these equations requires that

J, - 41, = 0,

FIG. 7.

Frequency spectra and related s,( p ) for unbounded waves.

(2.28)

Modern Corner, Edge, and Crack Problems where

Ji and I j

are J and I in (2.25) for s

63

s j ( p ) and

=

< =2 k 2 ~ J ~ J A-Jk2(2sf - k:)BJ =

2s: - k:

{3

=

sinh{

z}

1

2 siPi

h/cosh{ z ’ h ,

where we have denoted s j ( p ) by sJ. Now arguing on the basis of Lerch’s theorem (see Widder, 1941) that for the present p can be assumed to be real and noting the conjugate nature of the roots sj( p ) satisfying (2.26), the boundedness condition (2.28) can be expanded into (2s; - k : ) ‘Osh PJY} . f ( ’ J ? y, P) cosh a,h cosh PJh sinh aJy + (2s; - k : ) sinh P J y ] g ( s J ,y, p ) ] dy cosh aJh cosh P,h ~

{ z ] { k ; 2 { o h [{2s;*-

PJ -C(O, h, p ) -

I

P,{ T(sj,p ) + k - 2 (k2 - 2)U(O, h, p ) }

=

0.

(2.29)

Equations (2.29) are two coupled integral equations for the edge unknowns for each set of parameters p and sj( p ) , j = 2 , 4 , 6 , . . . .* For a particular problem the edge conditions reduce the number of edge unknowns in (2.29). A solution of these equations for the remaining edge unknowns completes the formal solutions given by (2.25). Evidently a boundedness condition like (2.27) was first used in a simple strip problem by Doetsch (1937). (It was learned that M. Picone also used one quite early.) Later, Bentham (1963) used it effectively in the solution of static problems in elastic strips governed by the biharmonic equation for the Airy stress function for plane stress. The work by Miklowitz (1969) and Sinclair and Miklowitz (1975) extended this thinking to elastodynamics. The applications in the following sections demonstrate that (2.29) can be solved for the edge unknowns in problems involving nonmixed edge conditions. c. Problem A-Nonmixed

Pressure Shock: Formal Long-Time Solution

The problem is illustrated in Fig. 8. The plate edge is subjected to a step-uniform normal stress a, under zero shear stress mXy.These conditions are given by

where vAis the magnitude of the stress input. From (2.2), the corresponding

* These values of j j = 3,5, 7 , .

are consistent with the branch numbering we have used here. Note that

. . , for the present complex segments enter only before the conjugation that has

now been completed.

Julius Miklowitz

64

FIG. 8.

Nonmixed pressure shock problem.

Laplace transforms on time t for these edge conditions are 5X(O,

@X.”(O,

Y , P) = PCL[k2Ux(0,Y, P) + ( k 2 - 2)V,(O, Y , PI1 = -U*/P, (2.31) Y , P) = CL[%(O, Y , P) + G(0, Y , P)1 = 0.

Note that the edge unknowns Gx, V y , iiy, and fix occur explicitly in (2.31). The long-time formal solution for the present problem can be derived from (2.25) by approximating the double transforms t i - ( s , y, p ) and f i - ( s , y , p ) appearing there for small p . It follows that a small p determination of the edge unknowns in (2.31) through the boundedness condition (2.29) will consistently define U- and 6- for the analogous approximation of the latter. To accomplish this we must find suitable forms for the edge unknowns, for small p , that we can substitute into (2.29) to “open up” this condition, that is, to reduce these integral equations to algebraic ones. As a guide in constructing these forms, we first make use of the elementary theory for compressional waves in a plate to enable us to estimate those components of the edge unknowns corresponding to the long-time dependence of the displacements and displacement gradients at the edge. The governing equations for the elementary theory may be written as

c,2iie(x, t ) , u ~ ( xt ,) = - k - 2 ( k 2 - ~ ) u ‘ , ( xt ,) , u ~ ( x t, ) = 4 / ~ L k - ’ ( k-~ l)u:(x, t ) ,

U:,(x,

1) =

(2.32)

where the superscript e indicates elementary theory, and c, = c,[4( kZ - 1)/ k2]’’2.For the analogous problem to that we are treating here (see Fig. 8), based on the elementary theory (2.32), we take the semi-infinite plate (of unit width) subjected to a step edge force of magnitude 2 U ~ hThe . Laplace transform easily produces the solution for this problem giving at the edge x = 0, U - ‘ ( O , p ) = 6A/pk,,,

G:(O, p ) = - k 2 ( k 2 - 2)-’,

q o , p ) = -&at)-’, for - h 5 y 5 h, R e p > 0, where k,

(2.33) =p/c,

and

is the dimensionless

Modern Corner, Edge, and Crack Problems

65

stress defined by &A = a A k 2 / 4 p ( k ' - 1). To the estimates in (2.33) we add a supplementary set of edge unknowns. We represent these additional contributions by Fourier series in y , with the p dependence incorporated in the coefficients of the series. Guided by the symmetric nature of our problem, we choose for the displacement gradients*

for 0 5 y 5 h, R e p > 0. These quarter-range Fourier seriest are a good choice since they preclude the possibility of Gibb's phenomena at the end points of the interval of representation [0, h ] . This in turn ensures that the coefficients A , , B, must decay faster than n-' as n += 03. A proof of this is given by Sinclair ( 1 9 7 3 ) . Such decay will be of value in any subsequent numerical calculations. We now integrate ( 2 . 3 4 ) , which, invoking the first of the symmetry relations ( 2 . 2 4 ) , gives

(2.35)

for 0 5 y 5 h, Re p > 0, where we have introduced U'( p ) for the transformed supplemental corner displacement in the x direction, a , ( p ) = - ( 2 h / n v ) A , ( p ) and b , ( p ) = ( 2 h / n 7 - r ) B n ( p )Wenotethat . asaconsequence of the large n behavior of A , ( p ) , B , ( p ) noted earlier, that is, that they decay faster than n-', the coefficients in ( 2 . 3 5 ) must obey the order conditions a,(p)

=

o(n-2),

b,(p)

=

o(n-')

as

n + co.

(2.36)

Now these additional terms ( 2 . 3 5 ) are combined with those from the elementary theory. To do this we ( 1 ) integrate the last of (2.33), again invoking the first of ( 2 . 2 4 ) , which yields C'(0, Y, P) = & A ( k 2 - 2 ) Y / k 2 P ,

(2.37)

* Benthem first used series like these in his work on related static problems. There the coefficients were not p dependent. t It should be noted that these series representations can be obtained from the half-range series o n [ 0 , 2 h ] in the same way the half-range series on [ 0 , h ] is obtained from the full Fourier series on [-h, h]. Therefore a Fourier theorem holds true for the series of (2.34), and quarter-range is an appropriate name for them.

66

Julius Miklowitz

and ( 2 ) take u", u: so that when added to the corresponding elementary theory terms, satisfaction of the edge conditions (2.31) is assured. It follows then, from (2.33), (2.35), and (2.37),that the edge unknowns for the present problem can be represented by

for 0 5 y 5 h, R e p > 0. We note that differentiation of the first two equations here gives Cy(O,y, p ) and fiy(O, y, p ) . Substituting the right-hand sides of (2.22) into the boundedness condition (2.29) (with T = 0), and in turn (2.38), and carrying out the simple integrations, reduces (2.29) to the infinite set of linear algebraic equations

where

and a: = af + (nrr/2h)', p: = pi' + (nrr/2h)', for U ' ( p ) and a , ( p ) , b , ( p ) ( n = 1,3, 5 , . . .). We seek a solution of this set for small p , considering first the s , ( p ) as p + 0. These limiting roots of R ( s , p ) in (2.25) are the zeros of lim [ R ( s , p ) / k ; = - i ( k 2 - l ) s 2 r ( s ) ,

(2.41)

P+O

where r ( s ) = sin 2sh + 2sh. According to (2.39), we select the one infinite set of these complex zeros of r ( s ) that corresponds to the piercing points of K f ( - i ~ ) in the plane w = 0, and hence the first quadrant s j ( p ) , shown

Modern Corner, Edge, and Crack Problems

67

in Fig. 7. Robbins and Smith (1948) list (in order of increasing real part) the first 10 of these constant values s J ( p ) ,(i.e., 2sh) satisfying r ( s ) = 0. Sinclair (1973) shows that lim [ d s , ( p ) / d p ]= 0,

j = 2 , 4 , 6 , . .. ,

(2.42)

P+O

corresponding to the fact that the K ; ( + i w ) are normal to the w = 0 plane in Fig. 7. It follows that the zeros of r ( s ) in (2.41) are a good approximation to the s j ( p ) for a range of p , small, but greater than 0. This is obviously important to the validity of the long-time solution we are attempting to derive with the present technique. We now must determine the behavior of the unknowns a', a , ( p ) , and b,( p ) for p + 0. On the basis of the premise that the elementary theory will describe the dominant time variation for very long time, we require ord[ a"( p)] 5 ord[ p-'I, ord[ a,( p ) ] 5 [ p - ' 1 and ord[b,( p ) ] 5 ord[ p-'1 for p + O.* Moreover, for these terms to have significant contributions to the long-time solution, the orders of all three quantities have to be greater than 1 . We therefore seek i i c ( p ) , a , ( p ) , and b , ( p ) such that

Expanding the terms in (2.40) for p -+ 0 and substituting the results into the equations (2.39) for % ' ( p ) , a , ( p ) , and b , ( p ) yield, in view of the order requirements (2.43) and the retaining of only the largest compatible terms,? m

=O

for j = l , 2 , 3, . . . ,

as p + O (2.44)

where s', = sf - ( n r / 2 h ) ' . Clearly (2.44) admits the solution a , ( p ) = b,( p ) = 0 for p + 0. This solution insists that for the present small-p approximation any contributions to the edge unknowns other than those derived from the elementary theory, must be confined to % " ( p ) .Further, since the boundedness condition (2.44) is free of %'( p ) , this remains an unknown at this point. This indeterminacy can be attributed to the problem in the near-field, long-time domain asymptotically approaching a second boundary value problem in elastostatics (stresses prescribed), since this type of static

* Here ord [ ] for p + 0 is being used in the sense of large order 0. For example, the first of(2.43)means B " ( p ) = O ( P - ~ ) , O
68

Julius Miklowitz

problem admits an arbitrary rigid displacement field. Since, however, the elastodynamic problem of the second type has no such arbitrary displacement field, we turn to the far-field response and consider the boundedness condition as s and p + 0 together. For this limit, R ( s , p ) = 0 gives sl(p) = + k ,

+ O(p3)

as p + 0,

(2.45)

the lowest branch of R, corresponding to ~ ~ (inp (2.17) ) (note the agreement). The positive sign defines an s,( p ) that for Re p > 0, and a range of p small but not equal to zero, satisfies (2.26). Hence the boundedness condition applies for this s l ( p ) = k, + O ( p 3 ) ,and we substitute it into (2.39) and (2.40) and then let p -+ 0 and find, in view of the order conditions (2.43) and the solution of (2.44), m

U'( p ) =

1 (-)(n-l)/* n= 1,3,5, ...

2 n.rr

-a , ( p ) = 0

as p + O .

(2.46)

Equation (2.46) completes the determination of the small-p edge unknowns for the present problem. With the unknowns determined, it is an easy step to the formal solution. Substituting the reduced version of (2.38) [ Uc( p ) = 0, a , ( p ) = b , ( p ) = 01 into the inversion integral statements (2.25) for u and u, through f(s, y, p ) , g(s, y, p ) given in (2.22) and performing the simple integrations indicated there yields the double transforms

with 0"= ( k 2 - 2 ) k i s ( P 2 / k , ) [ y sinh Ph cosh a y + 2 4 sinh ah cosh B y ] and similar expressions for a",O", and a",which are not given here in the interests of brevity.* Equation (2.47) completes the formal long-time solution (2.25) for the present problem. It holds for small p. d. Nonmixed Pressure Shock: Long-Time Solution In tackling the inversion of (2.47) we treat three ranges of s separately: s / p + 00 as p + 0; s / p + c as p + 0, with c a nonzero constant; and s l p 0 as p + 0. -+

* These quantities are given, respectively, by the last three of equations (1.20) in Sinclair (1973), except for multipliers similar to ( k 2 - 2 ) k $ s p 2 / k pin 0"here.

Modern Corner, Edge, and Crack Problems

69

For the first of these ranges, which corresponds to the near field, (2.47) becomes

for 0 5 y 5 h, Re p > 0. Inversion of (2.48) then gives u(x,y, t ) v ( x , y, t )

- GA[cpt - K2(k2

-

as x] 2 ) G y as

t + co, t -+ co,

(2.49)

for 0 < x < X , 0 5 y 5 h, where X demarks the extent of the near field. In view of the forms found for the edge unknowns [see (2.38) with I?( p ) , a , ( p ) , b , ( p ) set equal to zero therein] the region of validity for (2.49) may be extended to include x = 0. Further, due to the uniformity in s and y of the asymptotics on p for this case, (2.49) may be differentiated with respect to x and y to produce the near-field, long-time strains. Turning- to the second case where s and p tend to zero concurrently, which corresponds to being in the vicinity of the wavefront, (2.47) subject to this limiting procedure gives

for 0 5 y 5 h, Re p > 0. Inversion of the terms in (2.50) then produces u(x,y, t ) v(x,y, t )

- GA[CPt x ] H ( c , t - x) - k - 2 (k 2 - 2)GyH(cPf- x ) -

as as

t+m,

t

+ co.

(2.51)

Comparison of (2.51) with (2.49) demonstrates that the former applies everywhere in the plate, x 2 0, - h 5 y 5 h. The analogous asymptotics on su and dv-lay furnish _I

UAX,

Y, t )

- -[k2(k2

-

2)-'1vy(X, Y , t ) -GAH(cpt - x) as t -

+ co,

(2.52)

again applying everywhere in the plate. Equations (2.51) and (2.52) constitute our first far-field approximation for problem A and exhibit waves that are nondecaying in space and time with (2.52), thus attesting to the long-time Poisson's ratio coupling of the longitudinal and thickness strains. Note that although (2.51) and (2.52) could have been obtained from the elementary theory directly, this is not what we have done in the present treatment. Rather, we have shown that the elementary theory solution is

70

Julius Miklowitz

FIG.9.

Long-time, far-field response of longitudinal and coupled thickness strains,

obtained as the first long-time approximation of the exact theory for the present problem. A higher-order approximation than (2.52), based on a second approximation to R ( s , p ) , is available for both s and p tending to zero. It can be derived from the lowest Rayleigh-Lamb branch, resulting in (2.16) again replaces Go there. This result was anticipated based on (2.17),where now earlier but was first proved in Sinclair and Miklowitz (1975) by the present method. Again, the nature of both approximations is exhibited in Fig. 3. The computation of these approximations for the plate (mixed and nonmixed edges) appears in Miklowitz (1969, Fig. 4); see Fig. 9 here. To conclude the inversion of the small-p formal solution for problem A, we consider the third s range, namely, p small with s + 0. Under such a limiting procedure, (2.47) establishes that ij-, 6- are 0 ( 1 ) in s. Hence we have no contributions to the displacements for t large, x + 00. Similarly, it may be shown that the strains are zero as x + 00 and thus our solution is in accord with the conditions at infinity (2.4). e. Problem B-Nonmixed

Line Load: Formal Long-Time Solution

Following the procedures adopted in Section 11, A, 4, c, we now seek long-time information for the line load impact on the edge of the waveguide depicted in Fig. 10. The edge conditions for this problem are ux(0, Y , t ) = - u d ( y ) H ( t ) ,

UX,(O,Y,

t ) = 0,

(2.53)

Modern Corner, Edge, and Crack Problems

71

FIG. 10. Nonmixed line load problem

where U~ is the magnitude of the stress input (force per unit length). Using (2.2), the Laplace-time transforms of (2.53) give 6xx(o,

Cxy(0,

y , p ) = p*.[k2ux(0, y, p ) + ( k 2 - 2)cy(o, y, p ) ] = -UEp-'b(y),

Y, P) = P[U,(O,Y, P) + fiX(0, Y , PI1

=

(2.54)

0,

for 0 5 y 5 h, Re p > 0. As in Section II,A,4,c, we must now postulate forms for the edge unknowns in order to open up the boundedness condition (2.29). The only differences encountered between the pattern established for problem A and that required for problem B will be due to the singular nature of the latter problem. To appraise such singular nature we resolve problem B for the long time into three problems, as depicted in Fig. 11. The first of these, problem B1, is the line load on the elastostatic half-space, sometimes referred to as the Flamant problem. Problem B1 will provide the long time singular parts ofthe edge unknowns. The second problem, problem B2, is the residual associated elastostatic problem. The stresses applied on the plate faces here

FIG. 11. Decomposition of nonmixed line load problem.

12

Julius Miklowitz

(a: and *a:,) are of the same magnitude as those acting on the corresponding sections of the half-space in problem B1, but opposite in sign. Problem B2 is made self-equilibrating by the introduction of a uniform normal stress, u B / 2 h , acting on the plate edge. The attendant edge values for this problem will contribute to the regularparts o f t h e edge unknowns. The third problem, problem B 3 , is the uniform normal load applied to our waveguide, recognizable as problem A with a modified stress input. Problem B3 will furnish the dominant time dependence of the edge quantities. in the long time, thereby completing the selection of the representations for the regular parts of the edge unknowns. We now consider each problem in this decomposition individually; first, the singular problem. For the Flamant problem at x = 0 we have*

u-s(o,y, = - ( G B h / p ) In(y/ h ) , u;(o, y, p ) = -6”,O, y, p ) = -GBh/py,

.3 (0,y, p ) = - ( n G B / 2 p k 2 ) sgn(y),

(2.55)

--s

6”0, Y , P) = G(0, Y,P) = - ( & 3 h / P k 2 ) S ( Y ) ,

for 0 < y 5 h, R e p > 0, with extension to include y = 0 in an integrable sense whenever this is asked for by the boundedness condition ( 2 . 2 9 ) . Here sgn( y ) = 2 H ( y ) = 2 H ( y ) - 1 is the signum function [sgn(O) = 0 by definition] and GB is the dimensionless stress input for problem B defined by GB =

(2.56)

u B k 2 / 2 r r p h ( k 2- 1 ) .

Note that U’(0, y , p ) in ( 2 . 5 5 ) is determined to within an arbitrary rigid-body displacement; such indefiniteness is consistent with a second boundary value problem in elastostatics and will be incorporated in a U “ ( p ) term in the same manner as in the previous section. One should bear in mind that currently we are merely postulating the forms for the singular parts of the edge unknowns. Accordingly, ( 2 . 5 5 ) is only a reasonable guess as to what these terms might be, based on the thesis that in the near field the long-time singular nature of a problem involving exclusively outward propagating disturbances is the same as the singular behavior of the corresponding elastostatic problem. For use in problem B2 we set y = h in the stress distribution associated with the Flamant problem to obtain X =-

h

2UB =rrh [ l

(x/h)2

+ (x/h)’]’

(x 2 0).

(2.57)

In choosing edge forms for problem B2 we observe the similarity of the

* The appendix at the end of the paper by Sinclair and Miklowitz (1975) gives a sketch proof of (2.55).

Modern Corner, Edge, and Crack Problems

73

prescribed stresses at x = 0 for this problem with those in problem A. This similarity suggests that suitable representations for the edge unknowns in problem B2 would be the expressions in (2.38) with &A therein replaced by -7rGB/4. However, problem B2 has in addition to its edge stresses the equilibrating plate-face stresses and u:,,.*This fact further suggests that we modify the forms arrived at subsequent to the replacement by dropping the -&B/4pkp term. Now turning to problem B3, we note its complete equivalence to problem A, subject to a B / 2 h being equal to uA.It follows that the appropriate forms for problem B3 are obtained from (2.38) on exchanging &A there for 7rGB/4. Combining the edge unknown representations for these last two problems will cancel the elements in 6 ( 0 , y, p ) , U x ( O , y, p ) , which give rise to TUB/^^ and thus produce the compact forms

for 0 5 y 9 h, R e p > 0. Here, the superscript r on quantities denotes their regular nature; the Uc( p ) is once again the transformed supplemental corner displacement. Contingent upon the validity of the thesis that the forms in (2.55) will describe the singular contributions to the edge unknowns in their entirety, the terms in (2.58) will in fact be regular and consequently the a n ( p )and bn(p ) there will comply with the large-n order condition of (2.36). Moreover, such regularity then guarantees the validity of term-by-term differentiation of U', 6' in (2.58). Thus, on adjoining (2.55) to (2.58) and the terms U:, fif obtained therefrom, the postulation of the edge unknowns for problem B is concluded.? We are now in a position to open up the boundedness condition and evaluate the edge unknowns for small p . To achieve this we closely follow the method established for problem A in Section 11, A, 4, c. The forms for the edge unknowns for problem B are substituted into the boundedness condition (2.29) and the expanded (2.29) integrated. This results in an

* Since the plate-face stresses u),uky of (2.57) decay as x + CO, problem B2 is amenable to a finite-element treatment. Such a treatment is outlined in Appendix 1 of Sinclair (1973). f Details of the forms for the edge unknowns for this problem and of the ensuing opening up of the boundedness condition utilizing the forms are given in Sinclair (1973).

74

Julius Miklowitz

infinite system of linear equations for the unknowns a'( p ) , a,( p ) , and b,( p ) , which is similar to (2.39), the only differences encountered being contained in an FB(s,p ) term. The system has one equation for each s j ( p ) satisfying (2.26). For p small, s j ( p ) + s j ( j = 1 , 2 , . . .), the roots of r ( s ) in (2.41), and U"( p ) , a,( p ) , and b,( p ) obey the order requirements (2.43). In view of these order requirements, the boundedness condition for problem B reduces to

(2.59) for p + 0, s^ = $ ( j = 1 , 2 , . . .), wherein all quantities have been rendered dimensionless by the introduction of 6,,= pa,( p ) / v B h ,6, = p b , , ( p ) / a B h , s^ = sh, s^: = s:h2, with sin 2 4 + 2< = 0. Here si(s^) is the sine integral. As in Miklowitz (1969) and Benthem (1963), we now employ the method of reduction* to evaluate 6,, 6,. Results for the first 10 6, and 6, found, using 24 roots 4, are displayed in Table I. TABLE I FOURIERCOEFFICIENT

1 3 5 7 9 11 13 15 17 19

-0.1129 -0.0242 0.0077 -0.0030 0.0014 -0.0007 0.0004 -0.0003 0.0002 -0.0001

VALUES

0.2891 0.0036 -0.0026 0.001 1 -0.0005 0.0003 -0.0001 0.0001 -0.0001 0.0000

The numerical decay of the a*,, 6, values in the table is faster than l / n 2 for n > 3, in agreement with our large-n order condition (2.36). Such numerical decay supports our thesis that in the long-time near field the singular nature of problem B is the same as for the corresponding static problem. By using the 6,,6, values of Table I, the edge displacements associated with problem B2 can be evaluated, enabling comparison with the finiteelement treatment given in Appendix 1 of Sinclair (1973). Such a comparison shows agreement to within 1 '/o .

* That is, solving the finite 2 N x 2 N system of linear equations associated with the first N roots i, and then increasing* the size of the finite sysstem solved until stable estimates of the desired number of 8, and b, have been found.

Modern Corner, Edge, and Crack Problems

75

The edge displacements associated with problem B, however, cannot be evaluated at this juncture since Uc( p ) is an unknown. To ascertain U'( p ) we proceed as in problem A and consider the boundedness condition for the case of s and p tending to zero together. This limiting process defines sl as given in (2.45), and the boundedness condition associated with s1 then furnishes the necessary additional equation for the determination of Uc( p), name 1y , m

P

nrr

n=1,3,5,..

as p

+

0.

(2.60)

Substituting our numerical values for 6, into (2.60) then gives Uc(p) = GB,u^'h/pwith 2" = 0.935. We have now completed the determination of the edge unknowns for problem B. f. Problem B-Nonmixed

Line Load: Formal Long-Time Solution

For the purposes of exhibiting our results we remove the rr6,/4pkP term, carry out the simple inversion of the remaining terms, and then define the static edge displacements (i.e., time-independent displacements) and their gradients as follows:

m

CY =

nn nrry -cos-, n=1,3,5 , ... 2 2h

for 0 < y 5 h, with 6,,having a symmetric delta function at y = 0. Using , and setting k2 = afford a means of calculating our numerical values of i n6. 2, 6,G,, and 6, of (2.61). The results of this calculation are plotted in Fig. 12. In proceeding to the small-p formal solution for problem B, and thus to the long-time solution, we substitute the now determined edge unknowns into the formal solution (2.25), with the aid off and g in (2.22). Then the inversion process is undertaken for the pertinent three ranges of s in exactly the same manner as in Section 11, A, 4, d. For the near jield this process produces the edge displacements and their gradients. In the far jield, the process gives

4

u(x, y, t , V(X, y , t ) U,(X, y, t ) = -k2(k2 - 2)-'UY(X,y, t )

-

(r6B/4)[Cpt

- xlH(cpt

-

- [ ( k 2- 2)T&~y/4k~]H(C,t (2.62) - - r ( 6 B / 4 ) H ( C p t - X), - X),

76

Julius Miklowitz

I

f5

> Y

0 ,Thickness Coordinate y / h

FIG. 12. Static edge displacements and their y derivatives for line load problem

for x, t + 03, as the first approximation. The second approximation (for x, t + a)is again (2.16) with -C?o there now - 7 ~ & ~ / Comparison 4. of (2.51) and (2.52) with (2.62), and the analogous comparison for the second approximations, demonstrates that the long-time, far-field approximations for problems A and B are the same if equal normal forces act on the edge, x = 0, in both problems, that is, if uA= uB/2h or equivalently = 7rGB/4. g. Problems Involving Nonmixed Edge Displacements: Comments In the work of Miklowitz (1969) the problem of the plate with a built-in edge, subjected to two symmetric suddenly applied normal loads on its faces, a short distance from the edge, was treated by the foregoing methods. Interestingly, the head of the pulse solution (2.16) was found, but here it represented the long-time, far-field response of the reflection of the incident pulse from the edge. The magnitude of this response [instead of &o in (2.16)] involved the Fourier coefficients [a*, in (2.59)] and another coefficient Bo

Modern Corner, Edge, and Crack Problems

77

for a term in the edge unknowns representing the singularities at the corners (0, + h ) of the built-in edge. Cooper and Craggs (1966) treated a related problem with a finite diff erence-numerical method. The edge conditions were

a velocity shock problem. The results for u, on the plate faces were exhibited in two figures. One shows the time response at a station in the near field (x = h ) , which has some resemblance to the grosser features of a corresponding response record in Miklowitz and Nieswanger (1957) (see axial strain record for x = 1 in Fig. 13 here). The second is a spatial response record that exhibits features like those of (2.16). The work of Bertholf (1967) is significant. He used an integration method on the nonmixed pressure shock problem for the rod. He shows some

FIG. 13. Shock tube response records for (a) surface radial displacement and (b) surface axial strain at various x stations along I-in.-diameter rod of 24s-T Al alloy.

78

Julius Miklowitz

interesting results for near-field, radial and axial strain, response, which compare favorably with corresponding records from the work of Miklowitz and Nieswanger (1957) (again see Fig. 13 here).

CANTILEVERED B. TRANSIENT RESPONSEOF TWO-DIMENSIONAL TO BASE MOTIONS PLATESSUBJECTED In the work by Miklowitz and Garrott (1978), the foregoing general ideas have been extended to the finite waveguide and base motions. Here the essential differences are that a finite Laplace transform on the propagation coordinate replaces the one-sided Laplace transform for the semi-infinite waveguide, and a related entirety condition on the transformed solution replaces the above-mentioned boundedness condition. To solve the problem of the cantilevered finite length plate, the solution of the similar problem for a semi-infinite plate is needed. So the first case solved here is the problem of a cantilevered semi-infinite plate, subjected to a step transverse velocity at the base, where the normal displacement is assumed to be zero. The integral equations resulting from the boundedness condition were solved for the Laplace-transformed edge unknowns, which yielded the shear and normal strains at the base, with the latter becoming singular at the corners. These strains are evaluated numerically. The exact theory solution and the Bernoulli-Euler approximate theory solution are shown to agree for the long-time, near-field region away from the base. For the finite-length cantilevered plate, the solution obtained from the Bernoulli- Euler approximate theory is used to reduce the entirety condition to the same set of equations that resulted from the boundedness condition for the semi-infinite plate. The strains at the base are shown to be the strains at the base for the semi-infinite plate multiplied by a reflection function. The traveling wave and vibrational forms of the solution are found for the interior of the plate, away from the base. These are evaluated numerically. 1. Long-Time Response of a Finite Cantilever Plate to Antisymmetric Dynamic Surface Loading This contribution is due to Lotfy and Leipholz (1984a). Their work in this area treats the transient response of a finite, isotropic, homogeneous, elastic cantilever plate in a state of plane strain to an antisymmetric surface line load, which is assumed to be a step function in time. This analysis is based on the method given by J. Miklowitz for solving nonseparable waveguide problems by using a double Laplace transform and an entirety condition on the solution. The comer stress singularities are considered in the evaluation of the stress distribution at the fixed end. Then the near-field

Modern Corner, Edge, and Crack Problems

79

solution is found by means of asymptotic expansion. Moreover, the transverse displacement along the plate is obtained in the traveling wave form as well as in the vibrational one, which is evaluated numerically and discussed. It is concluded that the engineering methods in which a “dynamic load factor” is used in conjunction with the static solution tend to underestimate the values of the deflections beyond the point of load up to the free end of the cantilever plate.

2 . Long-Time Response of Finite Cantilever Plates to Dynamic Surface Loadings This is a further contribution from Lotfy and Leipholz (1984b) in this area. The work is concerned with the long-time analysis of the response of a finite, isotropic, homogeneous, elastic cantilever plate to different dynamic surface loadings while the plate is assumed to be in a state of plane strain. This is an extension to the solution of the antisymmetric loading case given by the authors. The analysis is based on a method given by J. Miklowitz for solving nonseparable waveguide problems in which an entirety condition is used on the solution. The effect of the material properties on the stress singularities at the corners of the.fixed end is considered for a calculation of the singularity exponent for a realistic engineering material with its corresponding Poisson’s ratio being taken into account. Moreover, a change of the loading function is considered in the general solution of the problem. It is concluded that, for the finite plate, the results calculated for antisymmetric loading are good approximations to the unsymmetric case. This work may be considered as a step toward the assessment of the importance of stress wave propagations in finite plates, since such waveguides have some practical significance in structural engineering.

C . AXIALLYSYMMETRIC INFINITEPLATE( O R LAYER)PROBLEM 1. Mixed Edge Condition Problem: Sudden Normal Displacement

on Circular Cavity Wall Scott and Miklowitz (1964) extended the work in the preceding section to a problem of the infinite plate with a circular cylindrical hole or cavity. The plate is excited by a uniform-step radial displacement on the cavity wall along with zero shear stress. As such, the problem is one of mixed edge conditions, related to the longitudinal impact problem of Section 11, A, 2 , l . The displacement equations of motion for cylindrical coordinates ( r , z) were employed. To write the formal solution extended Hankel transforms were developed for the exterior domain a 5 r < a,a being the radius of the hole. These transforms can be obtained from the following expansion

80

Julius Miklowitz

formulas given in Titchmarsh (1962):

where CO(X, Y , a ) = JO(XY)Y , ( x a )- J , ( x a )YO(W>, Cl(X, Y , a )

=

J , ( X Y ) Y , ( x a ) - J , ( x a )Y I ( X Y > ,

and f(r ) is a suitably restricted arbitrary function. An alternative foimal derivation of (2.63) is given by Scott (1964). Defining the zero and first-order transforms by

PW

=

"m) =

Im

r f ( r ) C o ( k ,r, a ) dr, (2.64)

r f ( r ) C , ( k ,r, a ) dr,

respectively, the corresponding inverse transforms are, from (2.63),

(2.65)

Again, as in our other transforms, integration by parts produced transforms of derivatives and derivative combinations of f ( r ) (analogous to those of the Hankel transform and their relations to the zero and first-order transforms ?( k ) , ?'( k ) , and f ( r ) and d f / d r at the cavity wall r = a. Two mixed edge problems can be solved, namely, the conditions %(a, z, t ) = u o d t ) ,

crrz(a, z,

t ) = 0,

(2.66)

or a,,(% 5 t ) = U O A t ) ,

&(a, 5 t )

=

0.

(2.67)

The conditions (2.66) governing the problem treated in Scott and Miklowitz (1964) excite compressional waves and (2.67) flexural waves. It should be pointed out that a mixed pressure shock problem, like that solved in Section 11, A, 2, b, did not separate in the present coordinates. The inversion, far-field, long-Time approximation and numerical evaluation was carried out by essentially the same methods used in the preceding section. Further detail is left to Scott and Miklowitz (1964), Titchmarsh (1962), and Miklowitz (1962).

Modern Corner, Edge, and Crack Problems

81

D. ELASTODYNAMICMODELING OF ELECTRICAL SIGNALPROCESSING DEVICES Progress has been made in elastodynamic wave problem modeling for certain of the subjects. It is well known, for example, that thin strips of material on a substrate of another material can guide surface waves. The great advantage in using such strips and the surface waves they guide, instead of electromagnetic waves, is the extremely large reduction in size of surface-wave devices compared to their electromagnetic counterparts. Tiersten (1969) discusses elastic surface waves guided by thin films. Freund (1971) treated guided surface waves on an elastic half-space as a WienerHopf problem. Auld (1973) has a good discussion of strips and other types of waveguides on a substrate. The scattering of Love waves or Rayleigh waves by the edge of a thin layer on a half-space was studied by Simons (1975, 1976). Both are Wiener-Hopf problems in which use is made of Tiersten's boundary conditions (1969) to approximate the effect of the surface layer on the half-space.

111. Elastic Pulse Scattering by Cylindrical and Spherical Obstacles

A. SCATTERING OF A N ELASTICPULSE BY A CIRCULAR CYLINDRICAL CAVITY 1. Line Load Source: General Features of the Wave System Consider the infinite elastic solid with a circular section cylindrical cavity as shown in Fig. 14. The cavity is infinitely long, of radius a, and has its axis along r = 0. We assume a line source S to the right of the cavity at x = xo(xo> a). The problem is one of plane strain with coordinates ( r , 8, t ) . In Fig. 14 it is assumed that one wave system is active with a wavefront velocity c. The wavefronts involved are depicted for two different fixed times t l , f Z , where tl < t Z . In the figure the numbers 1,2, and 3 indicate, respectively, the incident reflected and diffracted wavefronts (solid lines). p e associated rays (dashed lines) are indicated by the numbers ?, 2, and 3, respectively. Since there is symmetry with respect to the x axis, only the wavefronts and rays in the upper half of the figure have been numbered. The shadow zone, as Fig. 14 shows (crosshatched in the figure) is bounded by the rays ?, outward from their point of tangency with the cavity wall at 2 2 1/2 ct' = (xo - a ) , and the part of the cavity wall defined by cosC'(a/xo) I 8 5 27r - cos-'(a/x,). The illuminated zone covers the rest of the plane for r 2 a. For the time t l ( t , < t ' ) it may be seen that the wavefronts and rays are 1, ? and 2,?, the incident and reflected pairs just to the left of S. Note

82

Julius Miklowitz

FIG. 14. Wavefronts, rays, and wave regions in the scattering of an elastic pulse from a circular cylindrical cavity.

these are confined to the illuminated zone because time t , is too short for a disturbance to be created in the shadow zone (note the incident wavefront 1 is a complete circle), but only that part to the left of S has been shown for the sake of simplicity in the figure. For time t2 ( t 2 > t ’ ) , we have the outer system of wavefronts and rays 1, i, 2, i,and 3,? to the left of S and 1, ? and 2 , i to the right (the latter are really continuations of those to the left, the portions between again not being shown for simplicity). The new wavefronts and rays 3 , j created here are due to diffraction by the cavity. The source of these diffracted wavefronts and rays is the point ct’ on the cavity wall, that is, where 1 , 2 touch ct’. The diffracted wavefront 3 is the involute of the disturbed portion of th? cavity wall between ct’ and the leading edge of this wavefront. The rays 3, of course, must be perpendicular to the wavefront and hence the disturbed portion of the cavity wall is the envelope of these rays. Since the exciting source of the diffracted wavefront is the line source at the point ct’, one would expect it to experience two-dimensional spatial decay. It is of interest to point out further that as time grows the edge of 3, and its lower half-plane mate, progress further along the cavity wall, out into the illuminated zone and back into the shadow zone periodically ad injinitum. This creates a continuous spiraling diffracted wave with its edge always on the cavity wall and its tail approaching r infinite. These are the fundamentals of wave propagation in the cylindrical cavity scattering problem. In the elastic case two basic wave systems are at

Modern Corner, Edge, and Crack Problems

83

play, but fundamentally they both behave as the system we have just discussed. Two important limiting cases for the position of the line source S are indicated in Fig. 14, namely, when xo + a or xo -+ 00. The first case is the line source on the cavity wall at x = a. This is basically a Lamb-type problem for the exterior space r 2 a. It was treated by Miklowitz (1963) and will be discussed in detail in the sequel. The case of xo + 03 is equivalent to the plane incident pulse, traveling toward the cavity. This problem was treated by Baron and Matthews (1961) and Baron and Parnes (1962) and later by Miklowitz (1966b) and Peck and Miklowitz (1969). It will be discussed in detail later, including comparisons of the methods and results used in the quoted works. The general case where S is at xo, but the obstacle is the rigid cylinder, has been treated by Gilbert and Knopoff (1954) for wavefront approximations. We will discuss this in a later section on approximations in the incident plane wave case. 2. Friedlander's Representation of Solution It is known for harmonic wave diffraction problems involving a circular cylindrical or spherical scatterer, that in the shadow zone a solution based on Fourier series converges more and more slowly as the frequency is increased. The related difficulty in transient excitation occurs in the response at early and out to moderately long times that are associated with high frequencies. A method of solution suitable for short-time response was developed by Friedlander (1954) and Friedlander (1958). His representation of the solution can be found through an application of Poisson's summation formula. This formula can be written as

As proved in Titchmarsh (1948), sufficient conditions for the formula are that ( 1 ) g ( v ) is of bounded variation in (-m, a), (2) g ( v ) tends to 0 as 7 + fa,and (3) the integral

exists. The condition of bounded variation on g ( v ) can be relaxed to functions of square integrability, that is, functions that satisfy

J

-a,

as proved in Morse and Feshback (1953).

Julius Miklowitz

84

Applied to the Fourier series representation of the response function f ( r , 8, t ) , (3.1) gives m

j ( r , e, t ) =

F ( r , n, t)ei"e = n=-m W

=

m=-m

[=

F ( r , q, t)eiv(o+2mm)d7)

J-cc

f " ( r , 8 + 2m7r, t ) ,

C m=-m

(3.4)

where 00

F ( r , q, t)elq0dq.

f " ( r , 8, t ) = -m

We refer to f " as the wave form off and to the sum on rn in (3.4) as the wave sum. The wave form of the response f " has a clear physical interpretation. This response is the disturbance propagating outward in 8 with the wave fronts behaving geometrically as discussed earlier; that is, the diffracted fronts of f " wind around the cavity. From his wavefront expansions, Friedlander found that f" is identically zero for 8's beyond the wavefront; therefore, for finite t, the sum on rn is finite. Thus, as we noted earlier, f " overlaps itself as it winds around the cavity, and the wave sum on rn is simply the sum of the overlapping responses. Both the wave sum f and the wave form f" are defined on --CO < 8 < 00, but f" is not periodic in 8; however, the wave sum on rn gives the total solution the 277 periodicity in 8 that is physically required. This may be seen by asking for the value of f at a 8 that is outside the usual physical range, say 8 = 477 instead of 8 = 0. All this means is that in (3.4) the terms that contribute to the total solution differ (from the solution for 8 = 0) by 2 in their value of rn, but the number of terms and the values of the individual terms are identical to the 8 = 0 case. It is of further interest to note that one can interpret (3.4) as giving definition to solution f on a Riemann surface having the origin as branch point with sheets (2rn - l)7r < 8 < (2rn + l ) ~rn, = 0, *l, *2,. . . . In this the source, say at ( a , O), is represented by an infinity of sources at ( a , 2rn7r) that corresponds physically to the already-discussed fact that a single source can signal to a receiver not only through a direct ray, but also by rays corresponding to the diffracted waves that wrap themselves around the cavity. Finally, it is important to note that to solve for the total response f one needs to solve only for f" corresponding to the physical plane rn = 0. Simple substitutions for the higher rn terms, together with (3.4) , then give J: This will be demonstrated in the sequel. 3. Normal Line Load Source on Cavity Wall: Formal Solution Consider in Fig. 14 the case of a normal line load PF( t ) applied suddenly, at time t = 0, to the cavity wall at r ( = x ) = a, 8 = 0; P is a magnitude

Modern Corner, Edge, and Crack Problems

85

constant of dimensions force per unit length and F ( t ) prescribes the time behavior of the input. The problem is one of plane strain ( u 2 = a / a z = 0, where u, is the displacement component in the z direction). The boundary value problem may be stated as V2*(r, 0, t ) = t,i/c:, V 2 + ( r ,0, t ) = $/& r < a , -co<~0, 4 ( r , 090) = * ( r , 0,O)

=

d r , 0, 0) = $ ( r , o,o> = 0, rZa,

cr(a, 0, t ) =

-Wt)&(O)/a,

-~,
u r d a , 0, 1) =

lim

[ + ( r , 0, t ) , +IJ!, u,,

u8,.

. .I = 0,

(3.6)

0,

t>O

(3.7)

t > 0,

(3.8)

--co
(3.5)

where V2 is the Laplacian. Equations (3.5)-(3.8) represent, respectively, the governing wave equations on the potentials, initial conditions, boundary conditions for the cavity wall, and the radiation condition. The associated displacement-potential relations are u, = a + / a r

+ a$/rd0,

uo = d 4 / r d O

-

d$/dr,

(3.9a)

and the stress-potential relations u,=

cd

$ + 2p

[$+ $ (31, (3.9b)

ffz =

v(u, + re).

Noting that we have the radius of the cavity a as a characteristic length here, the problem (3.5)-(3.8) can be solved with the double integral transform methods for two-dimensional waveguide problems. In particular, we employ the Laplace transform pair and the real argument exponential Fourier transform pair by v and 0, respectively. The procedure then produces the transformed governing equations (3.5) as d2$-"( r, v, p ) d$-" + rdr2 dr 2 ---w d$-" (r' v 3 + r r2 dr2 dr

r2

*

-

( r 2 k i+

-

( r 2 k t+ v2)$-w

Y ~ ) $ - =~ 0,

(3.10a) = 0,

where the superscript w implies that these quantities are the wave forms of

86

Julius Miklowitz

( ). General solutions of the Bessel equations (3.10a) are I,(kd,,r),Ku(kd,,r), but the radiation condition (3.8) requires we select the K , functions. Hence we have the general solutions

where kd = Pledr k, = p / c , . Making use of (3.9) then produces the formal solution for the displacements in the physical plane ( m = 0)

where (3.11b)

Crw(r,v , p ) = T ( v , p ) L k , A K ' ( k d r )+ - UB K . ( k , r ) ] , r Ciw(r, v,p) = -iT(v,p)

where

+ k,BKl(k,r)

1

,

(3.11~)

and where f ( p ) is the Laplace transform of F ( t ) , u = kda, u = k,a, k = cd/c,, and the prime denotes differentiation with respect to the arguments of the K,, functions. Note that the form of the transformed solutions (3.11b) and ( 3 . 1 1 ~ restricts ) K , ( u ) and K , ( u ) to nonvanishing values. 4. Normal Line Load Source on Cavity Wall: Exact Inversion

Since the K , functions are entire functions of their order u (Erdelyi et al., 1953), (3.1 l a ) can be inverted exactly through residue theory and contour integration a la the method for inverting the spatial transform first. Since the problem is one of symmetry about 0 = 0, it suffices to consider 0 > 0. We then complete the integration path of the inner integral in (3.11a) in the lower half v plane, hence restricting the roots or branches of C ( v, p ) to Im v j ( p )< 0 for R e p > 0 to ensure convergence. Residue theory then

Modern Corner, Edge, and Crack Problems

87

reduces (3.11a) to

(3.12)

+

where vj( p ) = (v, ivi)j in terms of its real and imaginary parts and N, = CU;", No = CUT".It is known that residue series, such as that in (3.12), have usefulness in the present class of diffraction problems only in the shadow zone. Friedlander (1954), for example, shows in the related general source problem in acoustics that the terms of an analogous series, when approximated for large p with 0 in the illuminated zone, do not exist as Laplace transforms. Friedlander treats this case then, not by residues, but by approximating the Laplace transform asymptotically and using the method of steepest descents to get the reflected wavefront. Gilbert and Knopoff (1959) did similarly for their elastic wave problem. Restricting (3.12) to shadow-zone response, we can invert the Bromwich integral there by termwise contour integration, as we did in waveguide problems earlier, by completing the Bromwich contour to the left. Since the K , functions and their derivatives in the integrands of (3.12) have a common (logarithmic) branch point at p = 0, and since the branches v j ( p ) of the frequency equation C ( v, p ) = 0 also have a similar branch point there and no other singularities in the p plane (as will be shown shortly), we can cut this plane along the negative real axis. Then the Bromwich contour can be completed up the imaginary axis ( p = jziw). Here, however, with a branch point only at p = 0, the integration is like that in Miklowitz

I

FIG. 15. Contour integration in the p plane for cylindrical cavity source problem.

88

Julius Miklowitz

(1978) (the pressurized cylindrical cavity problem) and the contour is that of Fig. 15. Noting that conjugation for the paths L 1 , L2 can involve only the branches --vj(-iw) and Cj(iw),we find using the basic relations K-,(Z) = K J z ) ,

K,(4

=

K,(z),

(3.13a)

which follow from inspection of the integral representation for K , ( z ) (see Copson, 1935), and K,(-ix)

=

i(n./2) e x p ( i n . v / 2 ) ~ ~ ' ( x ) ,

(3.13b)

which (3.12) reduces to

X

exp{i[v,(-iw)O

aC la v I

Y = - ",(

-

wt]}

dw

(3.14a)

- IW ), p = - LW

where the roots v j ( - i w ) may be found from

and where 77 = w a / c d and 5 = w a / c , . The integrands of (3.14a) correspond to component diffracted and radiated harmonic waves, one for each mode of propagation pair [ w , v j ( - i w ) ] traveling in the positive 0 direction, and outward in r, the latter stemming from the H y ) ( k , r ) , H v ) ( k d r )character of the functions N,, No. Figure 16 shows the position of two such diffracted waves (i.e., the negative 0-traveling wave from the 0 < 0 solution is also shown) corresponding to a time t when these waves have already begun their second trip around the cavity. It is clear that the Riemann surface sheets m = rtl in addition to m = 0 are involved here, hence the corresponding terms in (3.4). As we have seen in our waveguide problems, a solution such as (3.14a) can be evaluated by numerical integration once the lower branches of C in (3.14b) have been found numerically. This permits a detailed study to be made of the various waves in the problem with an eye on both wave number v and frequency w. In a later application to the problem of diffraction of a plane dilatational pulse by the circular cylindrical cavity, we shall see that these numerically evaluated branches are the key to very accurate responses in the shadow zone for the relatively early times. Further, they are the key to long-time, far-field (in 0 ) approximations, which will also be demonstrated shortly. The frequency equation (3.14b) was first studied by Viktorov (1958). He noted that with 77 and 5 real, C could have only complex roots, but offered

Modern Corner, Edge, and Crack Problems

89

FIG. 16. Wave propagation from a surface line load source in a circular cylindrical cavity.

no proof of this. He did disclose there was one branch for large Ivl, large o,and vanishingly small imaginary part vi,that had the limiting real value vR = w a / c , corresponding to the Rayleigh surface wave on the free halfspace. Later, Gilbert (1958) obtained approximations for an infinity of roots having the limiting velocities cd as p approaches infinity on the real p axis. In his thesis Peck (1965) made a thorough analytical and numerical study of the branches of C. An abstracted version of this work appears in Peck and Miklowitz (1969). With the aid of (3.13a) it is easy to prove that if C ( v, p ) in (3.1 1) has the root (or branch) vj( p ) , then it also has the roots - v j ( p ) and .tV,(p). It follows therefore that all the branches for real 77 (imaginary p ) can be found from the branches having Im v > 0 and 71 > 0. Numerical results for seven of the branches, taken from Peck (1965) and Peck and Miklowitz (1969), are shown in Fig. 17. They were calculated for Poisson’s ratio 114. The three roots designated P1, P2, and P3 are the first three of an infinity of roots whose phase velocities approach cd as 77 + co. Their asymptotic approximation, to two terms, is

- 77 + aj(t7/2)

1/3

vjh)

e

-2rri/3 9

(3.15a)

where the a, are the roots of the Airy function (a, = -2.338.. .). The three roots designated S1, S 2 , and S3 are the first three of an infinity of roots

90

Julius Miklowitz

FIG. 17. Projections of branches of frequency equation.

whose phase velocities approach c,, their asymptotic approximation being (3.15a) with 7 replaced by 5. The branch marked R is a single branch whose phase velocity approaches the Rayleigh wave velocity c R . The asymptotic approximation for this branch is vR(T)

-

(cd/cR)T

+ yl + i?2v exp(-ysv),

(3.15b)

where yi are real positive functions of cd and c,. The derivations of (3.15) as well as expressions for the yi may be found in Peck (1965), the latter also being given by Viktorov (1958). It is important to note that all the branches v j ( ~are ) complex, as the lower half of Fig. 17 shows. As Re v, hence w grows, Im v also grows for the P and S branches. For the R root, Im v first grows but then decays to zero. It follows, observing (3.14a), that root R will contribute the predominant disturbance for large 8, that is, for large 8 + 2rn7r. It may be observed in Fig. 17 that as 77 + 0, the P roots approach v = -1, and the S roots and R root approach v = + l . This behavior was analyzed by Peck (1965) by first expressing C ( v, p ) in (3.1 1) as a power series in p . This series was then used to generate the approximations for the roots found numerically, which show that v + k l as p + 0 along the imaginary axis of p . Through the symmetry properties of the roots, it is only nezessary to

Modern Corner, Edge, and Crack Problems

91

) with E + 0, it is investigate the region v = 1. Setting v = 1 + ~ ( p then, found that the low-frequency approximations for the roots are contained in

+ ( 2 j + 1)ni](

In

:)-’

+ O[(ln p)-’].

(3.16)

For small p , In p is approximately real and negative, therefore canceling the negative sign in (3.16). It follows the roots v j ( p ) leave v = 1 with the slope ( 2 j + l)n/ln[(k2 + l ) / ( k 2 - l ) ] in the v plane (see lower part of Fig. 17). For j = 0 (root R ) and j > 0 ( S roots), Im vj > 0, in agreement with these roots in Fig. 17. Denote these as vj( p ) . The P roots correspond to those in (3.16) having j < 0 and hence Im v, < 0. These then are i i j ( p ) and do not appear in Fig. 17. Through the symmetry of the roots in v = 0, however, the P roots appearing in Fig. 17 must be - Y j ( p ) with j < 0. These roots leave v = -1 with Im vj > 0. It may be seen that these results are consistent with the low-frequency character of the vj in Fig. 17. This information on the branches v j ( - i o ) of the frequency equation C in (3.14b) enables one to evaluate the solution (3.14a), or like solutions, numerically or through approximations, as we shall demonstrate in the following sections. 5. Normal Line Load Source on Cavity Wall: Rayleigh Waves and the Long-Time, Far-Field Solution It has already been pointed out that one would expect the Rayleigh waves from the high-frequency, short-wave limit of branch R to predominate in a far-field solution, since the limiting wave numbers on this branch are the only real wave numbers in the spectra. Such a solution was derived in Miklowitz (1963). The derivation of the approximate solution and its evaluation form an instructive example in the present class of problems. The time-dependent Rayleigh waves are obtained by first picking out of (3.14a) the term corresponding to the branch v j ( - i w ) containing the Rayleigh wave real pair (v,v ) as a high-frequency, short-wave limit (7,v >> 1). The pertinent branch of C in (3.14b) is defined by this limit, that is, R. The desired approximation of C and other compatible terms in (3.14a) were found by using the appropriate Debye asymptotic expansion for the HI‘’ functions in (3.14a) and (3.14b). The general expansion needed is one in which both order v and argument 7 (or 5) are large and positive. The present work further imposes that v / v = CJCR > v/{ =

CJCR

> 1.

(3.17)

The general expansion may be found in Erdelyi et al. (1953). Equation (3.17) means that through the substitution v / 7 = cosh a, this expansion for the first-order term can be written as

HI‘)(7)

= -i ( 2 /

tanh (y)1’2e-v(tanh (1 + O ( 1 , v)), ol-a)

(3.18)

92

Julius Miklowitz

where (Y = tanh-' q, q = (1 - b2K2)'/2,K~ = ci/c:, and b2 = c,'/ci. Here HV'(5) is just like (3.18) except that fl is substituted for a, where fl = tanh-' s and s = (1 - K ~ ) ' ' ~ . Making this first-order approximation to (3.14b) yields (2/b2K4)[(K2- 2)2- 4qS]

= 0,

0 < K < 1,

(3.19)

the well-known equation for the speed of a Rayleigh wave on the free surface of an elastic half-space. Taking into account the continuity of (3.18) in v and w, the derivation of (3.19) proves that the branch of C containing the Rayleigh wave pair has the real asymptote =

cdv/cRI

(3.20)

in agreement with (3.15b). The latter equation shows that the branch approaches this asymptote (for 7, v >> 1) through a vanishingly small Im v. We have already pointed out that 6 + 2 m ~ can r ultimately be taken large (the far-field nature of the present approximation) by invoking the Friedlander representation of the solution. Then if we argue that the contributions of the branches of the frequency equation for 7 + 0, vj + *I (hence real vj) will give the static solution (which will be accounted for later), (3.14a) reduces to

x exp( i {

[w+i Im vR(Re v)] e - w t )) dw,

(3.21)

CR

where the subscript R denotes Rayleigh wave response of the displacements, w L is an arbitrarily large but finite positive frequency, and Im vR(Re v ) is given by the third term in (3.15b). The behavior of Im vR is shown in Fig. 17. Note that the first term in the exponential is the first term in (3.15b), the real part of vR. The bracketed expressions containing N,, No in (3.21) are approximated by using (3.18) and related expressions. Then setting F ( t ) = s ( t ) ,the delta function, (3.21) reduces to

where PCRK

M=TpaL'

8[2-(1+b*)~~]+4(K*-2)~+(K~( K 2 - 2)2

Modern Corner, Edge, and Crack Problems

93

A ( r ) = Q - q - tanh-' Q + tanh-' q, B ( r ) = S - s - tanh-' S

+ tanh-'

Q ( r ) = (1 - b2K2r2/u2)'/2,

s,

~ ( r=) (1 - K2r2/a2)'l2,

in which r has been restricted to the neighborhood of the cavity wall ( a 5 r < U / K ) where the major effects occur, rendering A ( r ) and B ( r ) both real and 0 or greater. Both approach zero as r + a. Note that D - 8 = 0 gives the Rayleigh surface wave arrival time t = a 8 / c R . From (3.22), taking into account the exponential decay of Im vR(Re v), we find the bounds on the magnitudes of uf,W and u:: are given by

<

1:

u"f,"( r, v ) sin[( D - 0 ) v ] r v ) cos[(D - 8) v ]

{ut?R( -sw

(3.24)

3

where Im vRL = Im vR(Re v ) = Im vR(vL). Since vL is an arbitrarily large but finite number, Im vRL can be a corresponding arbitrarily small but nonvanishing number (see Fig. 17). Therefore, the upper bounds in (3.24) correspond to spatially ( 8 ) nondecaying waves and the lower bounds in general spatially ( 0 ) decaying waves. From (3.22) and (3.24) we can write (3.22) as the approximation

(3.25) where Im vRais an intermediate value of Im vR(Re v ) between Im vRL and 0. Note that Im vRa can approach but cannot equal zero. Noting (3.23), we see the integrals in (3.25) are simple and can be found in most integral tables. Equations (3.25) therefore reduce to ufz(r, 8, t ) --- M exp(-Im v,,0)

{

exp[-vLA(r)l VA(r) A'(r) ( D - 8)'

x [ A ( r ) sin(D - 8)v,

x [ B( r ) sin( D

-

+

+ ( D - 8) cos(D - 8)vL]

0 ) vL + ( D - 0 ) cos( D

-

I

0 ) vL] , (3.26a)

Julius Miklowitz

94

x [ A ( r )cos(D - B)vL - ( D - 0 ) sin(D - e)vL]

1

x [ B ( r )cos(D - O)vL- ( D - 0) sin(D - O)vL] , (3.26b) based on A ( r ) and B( r ) > 0. Equations (3.26) represent Rayleigh waves in the near surface interior a < r 5 U / K traveling with the velocity cR. It is clear that they are continuous through the arrival time ( D - 0). Note that they decay exponentially with r and 0. Since vL is a large real number, the r decay is severe for r > a. The station 0 + 2m7r (for fixed r ) can, of course, be large, but it must be finite since it must always be behind the dilatational wavefront. It follows, since Im vRn can approach zero, that the exponential decay with 0 is much less severe than with r in the present approximation. The maximum responses in (3.26) occur at the surface as one would expect from our study of the Rayleigh waves in the strongly related Lamb’s line load problem. To find these responses we let A ( r ) and B ( r ) +- 0 + (as r + a + ) in (3.26), which results in

(3.27a)

x

C O ~ ( D-

e)v, -

sin(D - e ) v L D-8

x

COS(D -

e)v, -

sin(D - B)vL D-0

}].

(3.27b)

Noting that

:1

A(r) d ( D - 0) = T A ( r ) 2+ ( D - 0)’

for

A ( r )> 0

and that the integrand function here is zero for A( r ) + 0 + and D # 8, we conclude that the limits in (3.27b) are 7r times the symmetric delta function 6 , ( D - 0). Imposing ( D - 0) small then, (3.27) reduce to the limiting

Modern Corner, Edge, and Crack Problems

95

singular surface responses

where uo = P / p . The work of Miklowitz (1963) shows that the corresponding stress &(a+, 8, t ) behaves as Sb(D - 8), where the prime indicates differentiation with respect to ( D - 8 ) . It should be pointed out that these singular displacements are essentially (as Im vRa += 0+) those found in Lamb’s line load problem, in effect then showing that they represent highfrequency, short-waves that cannot “see” the curvature of the cavity wall. Identifying the displacements in (3.28) with f ” ( r , 8, t ) in (3.4) and then substituting them into the latter equation extend the solution to all of finite 6 + 2mn-, giving then the wave sums u:~,u”,. The observer, standing at a station 8 in the physical plane, sees each of the waves in (3.28)as a periodic phenomenon in t (of period D = c R t / a = 27r). This agrees with the diffracted wavefronts 3 depicted in Fig. 14, noting now that (3.28) represent two-sided discontinuities. The far-field, long-time solution in the present problem for 8 + 2m7r > 1 is obtained by imposing 8 + 2mn- >> 1 and D >> 1 , along with ID ( 8 + 2mn-)l < 1 , on the extension ( 3 . 4 ) of (3.28), and projecting the result into the physical plane. It follows that we have the responses shown in Fig. 18 for the positive 8-traveling waves. The solid lines in the figure represent the approximations (3.28). The dashed line connection (in the case of the radial displacement) represents an assumption that the heads and tails of these Rayleigh waves cancel away from ID - ( 8 2rnn-)l small. Note that the head associated with a D - ( 8 + 2 m r ) = 0 event can extend out to the dilatational wavefront, and similarly the tail to time infinite. Clearly then, near neighboring events are chiefly responsible for the dashed portions of the wave sum u:~. The only further consideration necessary for the long-time solution is the static solution. Since the input is the delta function 6(t ) , the static solution must be zero. To complete the solution it is only necessary to have a system like that in Fig. 18 for the negative &traveling waves, that is, from the 8 < 0 solution. The total response at the station 8 = 7r/2, for example, would have the same figure for ufR as that in Fig. 18, except the period would be 7r, and u i R would change its sign every 7r. The infinite discontinuities in (3.28) and Fig. 18, of course, are directly dependent on the nature of the input function F ( t ) [here 6( t ) ] . It is therefore of interest to get the response to the step H ( t ) , which offers a much less severe high-frequency input [i.e., ( S ( i o ) l = 1 , \l?(io)l= 1/03. A solution

+

96

Julius Miklowitz

FIG. 18. Response (wave sum) for long time at cavity wall due to positive @-traveling Rayleigh waves (cavity wall line load delta function source).

corresponding to (3.28), and its extension by (3.4), can be obtained for the step case through the use of the Duhamel integral operating on a series of the first terms on the right-hand side of (3.28). This yields the Rayleigh positive 6-traveling wave contributions to the long-time solution as

(3.29a) U r ~ ( a +6, ,

UO

t ) - - K2(2 - K 2 ) 2R x H[D- (6

C

exp - Im vRa(6 + 2rn.rr)l

m = ~ o

+ 2rnr)],

(3.29b)

where M and M o are large and are determined by the number of periodic waves occurring in a certain domain of large time and H [ D - ( 6 + 2 r n r ) l is the step function 0, H [ D - ( 6 + 2 r n r ) ] = $, 1,

D < 6 + 2rn.rr, D = 6 2rn77, D>6+2rn.rr.

+

Modern Corner, Edge, and Crack Problems

97

The terms in (3.29a) are valid in the vicinity of D - ( 8 + 2m77) = 0. The full curve can be obtained by numerically integrating [with time away from D - ( 0 + 2rn77) = 01 the u : curve ~ in Fig. 18. Again here the negative 0-traveling waves must be superposed on (3.29) for the full dynamic contribution to the long-time solution in the physical plane. At 0 = 7r/2 in the present case the oddness of u & / u , (wrt 0 ) leads to an alternating series of square waves of duration 77 and decaying (with 8 + 2rn77) amplitude and occurring with periodicity 277. In the present case the static solution has existing constant (wrt time) terms that can be neglected with respect to the terms in (3.29a); hence the latter are again the predominant disturbances in the long-time solution. Note that urR are still represented by infinite discontinuities. In the case of uYR the static solution would have to be added to the dynamic for the total long-time solution. The reader will be interested in the extension of the present approximation to’the linear viscoelastic case given in Miklowitz (1963). In the work a correspondence principle leads to similar results at the infinite Rayleigh wave discontinuities, but to further spatial decay away from these times. 6. Difraction of Plane Compressional Pulse by Cavity: Formal Solution Figure 14 depicts the problem. Recall that this is the case that is equivalent to the line source S at xo + 00, resulting in the incident plane pulse shown in the figure. Here the incident plane pulse is an elastic compressional one, which we can write as

corresponding to a step in stress (T, of amplitude (T, (inherently negative). The governing wave equations (3.5), where t > 0 is replaced by t > -a/cd (the time at which the incident wave strikes the cavity), hold here as well as the displacement- and stress-potential relations (3.9). The solution is separated into incident and scattered parts so that the total solution is given by (3.31) It follows, since the incident stresses are known, that the boundary conditions at the cavity wall for the scattered wave solution are

so that the total stress there is zero. Since the incident wave strikes the

98

Julius Miklowitz

cavity at t

=

- a / c d , we require that

&(r, 0, - a / c d )

=

+s

=

t,b,

=

6, = 0,

r 2 a,

-a< e < a. (3.33)

Radiation conditions for the scattered waves can be stated as lim

r+w

a nd / o r O+*m

[ 4 s ( r , & t ) , ccrs, u,,, . . .I

= 0,

t

'- a / % ,

(3.34)

completing the statement of the problem. The formal solution of the problem can again be written in the wave sum form (3.4). We argue that since the only given function in the problem is the incident potential, once its expression in wave-sum form has been found, we need only require that each term of the wave sum for 4sand 6, satisfy the wave equations (3.5). Correspondingly, from (3.32)-(3.34) we have c+Xa, 4 t ) = -;(a,

4t),

c 9 s ( a , 8, t ) = - a x a , 0, t ) ,

4r(r, 8, - a / c d )

= qjy =

t,bF = & = 0,

(3.35) (3.36) (3.37)

that is, the boundary conditions, initial conditions and radiation conditions are also satisfied term by term. Since action in the present problem begins at - t = a / cd, we use the bilateral Laplace transform on time t, and again the exponential Fourier transform (with real argument) on the circumferential coordinate 8. We find again the transformed governing equations (3.10a) and their admissible general solutions (3.10b), but both now for &-"(r, v, p ) and $""(r, v, p ) . To complete our statement of the general transformed solutions for &-" and $-", we must derive 6;"(r, v , p ) . We derive it in the following by applying the Poisson summation formula to the Fourier series [see (3.4)] of the Laplace transform of c$~.From (3.30), ii= & ( p ) exp(kdr cos e ) , where 6 0 ( p )= c+0cz/2(h + 2 p ) p 3 .Noting that the exponential function in &, is a generating function for a Fourier-Bessel series, we find (3.39) n=-w

(Erdelyi et al., 1953). The absolute value sign is permissible because I-, ( z ) = Zn(z) for integral n. We now apply the integral for f " in (3.4) to and take its Fourier exponential transform, with the result

6,

m

6iw(r, v, p ) = &O(p)

m

qv((kdr)eis('+v) d v do.

(3.40)

-m

Since I , approaches zero exponentially as v + +a, the inner integral here is uniformly convergent and we may interchange the integrations by writing

99

Modern Corner, Edge, and Crack Problems (3.40) as

&Fw(r,

v, p ) = 6 O ( p )

zlr)l(kdr)

-m

[

do] dv.

elo(r)+v)

(3.41)

The inner integral here may be recognized as the inverse of the Fourier exponential transform of the delta function 27r6( 7 + v). Hence,

&Fw(r,

v, p ) = 2r&O(P)qvl(kdr),

(3.42)

and from this equation and (3.10b) our present general transformed solution is

6-w(r,v, p ) = 2T&O(P)qu](kdr)+ a ( v , P ) K v ( k d r ) , &-"(r, v, P) = P ( v , P ) K " ( k J ) .

(3.43)

Substituting (3.43) into the double transforms of the boundary conditions (3.35) and (3.36) determines a(v, p ) , p( v, p ) . They are a(v,p ) = 2 T & O ( p ) [ D ( v p, ) 1 - ' { [ ( 2 v 2 + kfa2)qul(kda)- 2kda1/vl(kda)l

x [ ( 2 v 2+ k:a2)K,(k,a) - 2 k s a K : ( k , a ) ]

-

4v2[11ul(k,a) - k d a l i , l ( k d a ) ] [ K y ( k s a-) k,aK:(k,a)l),

(3.44)

P ( v , p ) = i47r&oo(P)[D(v,p)l-'v[k~a2+ 2 ( u 2- I ) ] ,

where D ( V , p ) = - { [ ( 2 V 2 -I-k : a 2 ) K , ( k d a )- 2kdUK:(k,U) x [ ( 2 v 2+ k f a 2 ) K , ( k , a ) 2k,~K:(k,a)] -4V2[ K , ( kda) - kdUK'( k d a ) ] [K , ( k,a) - ksUK:(ksU)]}. The Wronskian K,( z ) I : ( z ) - K I( z )Iy(z ) = z-' has been used to simplify the expression for P ( v, p ) . The function D( v, p ) , which is used in Peck and Miklowitz (1969) and Peck (1969, is another form for C ( v, p ) of (3.11). Their zeros are identical. We can now write the formal solution for the displacements u r , u y , which are again given by the double inversion integrals (3.11a) except that now U;", U B w are given by U;w(r,

=

+

r-l[kdra(v,p)K:(kdr)

rkd&O(

p

I[ul(

kd

- ivp(v,p)Ku(ksr)l

r ,,

UBw(r, v , P ) = - r - ' { [ i v a ( v , P ) K , ( k d r ) + k,rP(v, p)K:(kr)l - i2r&O(

p )vz

(3.45)

kdr)).

The differences in the expressions for p( v, p ) in (3.44) and the bracketed terms in (3.45), and the corresponding terms in Peck and Miklowitz (1969), stem from the use of the Fourier transform pair instead of the alternate pair used here; that is, this introduces a change in sign for the quantities that are odd in v.

Julius Miklowitz

100

7. Diffraction of Plane Compressional Pulse by Cavity; Exact Inversion The diffracted displacement waves in the present problem, as in the preceding line load case, can be obtained by inverting the bracketed terms in (3.45) in the manner of Section 111, A, 4. However, there is an important difference in the inversion of the Fourier transform. It stems from Avl(kdr) in the double transform of the incident potential &FW(r,v, p - in (3.42). The Bessel function ~ . I ( Z )is not an analytic function of complex v by itself, so that continuation off the real axis of v (hence contour integration) for terms involving this function [see a ( v , p ) in (3.44)] would not be possible. As pointed out in Peck (1965) and Peck and Miklowitz (1969), however, the sum of the incident and scattered dilatational potentials, the first of (3.43), is analytic, since it is even in real v ; hence the absolute value signs on order v may be dropped. It is clear that this statement also applies to other transformed response functions, for example, the displacements in (3.45), since they are associated with a ( v, p ) and the transforms of the incident pulses there. Then, since I,, K , are entire functions of their order v, inversion proceeds as in the line load case (see Section 111, A, 4). Again considering 8 > 0, the analog of (3.12) is

based on the branches of D ( v, p ) satisfying Im vj( p ) < 0 for Re p > 0 for convergence, where v j ( p ) = (v, + i v i ) j ,vj > 0 [as in (3.12)] and N,, = D,j-W N - D,j-W Bs ,with the subscript denoting scattered.* Contour integraTs

,

Bs -

tion in the p plane follows that in Section 111, A, 4 reducing (3.46) to the analog of (3.14a):

(3.47)

where again the roots v j ( - i w ) are those depicted in Fig. 17. Recall that they are those of C ( v, - i w ) = 0 in (3.14b) [or equivalently of D( v, - i w ) = 0 of (3.44)]. Note that the last terms in (3.45) represent the incident wave and are easily inverted through the known pair (3.30), (3.42).

* Again here we will ultimately be interested only in the diffracted wave parts of these scattered wave representations for the reasons pointed out in the discussion following (3.12).

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Again here it may be seen that the integrand of (3.47) corresponds to component diffracted and reflected harmonic waves, one for each mode of propagation pair [ w , v j ( - i w ) ] , traveling in the positive 8 direction, and outward in r, the latter stemming from the H v ) (k d r ) and H v ) (k,r) character of the functions Nrs,N o s .These waves are generated by incident wave-cavity interaction. Figure 19 shows the position of two such diffracted and reflected waves (i.e., the negative 8-traveling waves from the 8 < 0 solution are also shown) corresponding to a time t when the incident wavefront has already enveloped the cavity and gone past it. It is clear that the Riemann surface sheets rn = *l, in addition to rn = 0 , are involved here, hence the corresponding terms in (3.4). Figure 19 points out that no diffraction starts until the incident wavefront reaches the vertical line, 8 = *7r/2, through the cavity center; that is, difiracted waves have their origin at 8 = * 7 r / 2 , r = a, propagating into the shadow zone. This occurs when time t = 0 (note that the incident wavefront reaches the cavity at 8 = 0 at time t = - u / c d ) . The reflected harmonic wave and wavefront, only partially shown in the figure, surround the cavity except in the shadow zone, where they do not occur. It should be pointed out that singularities occur in the integrands of the Bromwich integrals of (3.46) at p = 0 and cause the small circular paths in the direct contour integration in the p plane to give infinite contributions.

FIG. 19. Scattering of a plane compressional pulse by a circular cylindrical cavity.

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Peck (1965) studied these. The results are also given in Peck and Miklowitz (1969). Noting that our interest is restricted to the shadow zone, where there is quiescence for t < 0, Peck made use of the convolution theorem of the one-sided Laplace transform, which led to zero contributions from the small circular paths and associated path integrals that were proper at p = 0. Details are given in Peck and Miklowitz (1969). 8 . Diffraction of Plane Compressional Pulse by Cavity: Numerical Evaluation of Solution

In Peck (1965) and Peck and Miklowitz (1969), the solution (3.47) and associated velocities were evaluated by numerical integration. The results presented here (Figs. 20-23), taken from Peck and Miklowitz (1969), are for the velocities since they have the most interesting pulse behavior. The velocities were evaluated at the cavity wall r = a, at the points 8 = (3/4)7r and 8 = (5/4)7r for 0 < c d t / a < lo.* The integrals for the modes of propagation (modes for short) associated with the frequency branches P1, P2, P 3 , R, S1, and S2 (see Fig. 17) were summed. Poisson's ratio was taken to be 1/4. The normalization constant ti, = aocd/(h+ 2 p ) used in the figures is the particle velocity behind the incident step-stress dilatational pulse. Figure 20 shows the radial velocity in the wave form of solution, that is, u" (our u;), at 8 = (3/4)7r (both individual mode contributions uy and the mode sum ti" are shown). The largest contribution comes from the P1 mode, which also exhibits strong impulsive behavior at the arrival time of diffracted P waves. Note that the P1 branch in Fig. 17 has the smallest Im v except for R. The second-largest contribution comes from the R mode. It is noteworthy that no significant impulsive behavior occurs at the diffracted Rayleigh ( R )wave arrival time. As one would expect from their higher Im v (Fig. 17), contributions from the higher P modes (P2 and P 3 ) are seen to decrease rapidly as the mode number is increased. The S l mode has a very small contribution, and the S2 mode response is too small to be plotted. The mode sum is essentially zero ahead of the arrival time of the diffracted P wave. The plot in Fig. 20 is too crowded to show this, but it can be seen in Fig. 22. The slight oscillations about zero ahead of the P-wave arrival time (visible in Fig. 20) are probably caused by truncation of the infinite integrals at 7 = 40, since the convergence becomes quite slow as c d t / a is decreased. In Fig. 21, the mode contributions and mode sum are shown for the positive 8 propagating u" wave at 8 = (5/4)7r. The additional propagation of 7r/2 in 8 has effected some striking changes. The most obvious change is the decrease of amplitude of the wave. Second, the mode convergence

* Results for velocities at 0 in Peck (1965).

= n

and displacements at 0 = ( 3 / 4 ) ~and 0 = m are presented

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FIG. 20. Modal response 1'; and mode sum u" at r = a, 0 Miklowitz (1969).]

= (3/4)7r.

[From Peck and

a, 0

= (5/4)7r.

[From Peck and

FIG. 21. Modal response u; and mode sum uw at r Miklowitz (1969).]

=

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Julius Miklowitz

has become even more rapid than at 8 = (3/4)7~.Third, the mode sum is clearly zero ahead of the P arrival (the infinite integrals converge much more rapidly at higher 8's because of the imaginary part of vj) and the slow, smooth rise of the pulse from zero at the front, which is characteristic of diffracted waves (Friedlander, 1954), has become apparent in the numerical results. Finally, a pulselike behavior has begun to emerge at the R wave arrival time. This delayed emergence of the Rayleigh-type pulse is similar to the behavior in the half-space problem with a buried source disturbance. This behavior is also an early indication of the long-time dominance of the Rayleigh pulse shown by Miklowitz (1966). To obtain the total response at the 8 = (3/4)7r point, the wave sum of the ti" waves is obtained from (3.4). This is illustrated in Fig. 22. The first

FIG. 22. Waves ti" and wave sum ti at r = a, 0 = (3/4)7r.[Fourier series result after Baron and Parnes (1962). From Peck and Miklowitz (1969).]

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wave to arrive is the rn = 0 wave, which is just the ti" wave at 8 = (3/4)7r given in Fig. 20. The second wave to arrive is the m = -1 wave, propagating in the negative 8 direction. By virtue of the symmetry of ti" in 8, this second wave is identical to the m = 0 wave at 8 = (5/4)7r, given in Fig. 21. The m = 1 and m = -2 waves also arrive at 8 = (3/4)7r for c d t / a < 10, but their contributions are negligible for this time interval. Thus, the wave sum is essentially the sum of the ti" waves for m = 0 and rn = -1, as shown in Fig. 22 by the solid curve. The correspondence with the long-time solution is seen to be good; the slight deviation for c d f / aapproaching 10 is probably due to the neglect of the P 4 mode in the m = 0 wave. The waves and wave sum for the circumferential velocity tj" (our ti;) at 8 = (3/4)7r are shown in Fig. 23. Some comparisons with the radial velocity

FIG. 23. Waves 0"' and wave sum d at r = a, 0 = ( 3 / 4 ) ~[Fourier . series result after Baron and Parnes (1962). From Peck and Miklowitz (1969).]

Julius Miklowitz

106

results in Fig. 22 worthy of comment are as follows: (1) a much larger disturbance is contributed by the rn = -1 wave at the P-l arrival time, ( 2 ) a barely detectable Rayleigh pulse occurs in the m = -1 wave, and (3) the short-time oscillations in the m = 0 wave, caused by truncation of the infinite integrals, are slightly larger than they were for the radial velocity. 9. Difraction of Plane Compressional Pulse by Cavity: Fourier Series Solution, Comparison of the Two Methods, and Results Let us assume that at time t = 0 the incident wave 4t in (3.30) strikes the cavity at r = x = a. Then on the basis of the initial conditions (3.33), taken now at t = 0, we apply the Laplace transform to the wave equations (3.35). These transformed partial differential equations have as solutions the classical separable forms for the scattered potentials

&&, 0, P ) = &(r, P ) cos no, 4ns(r,0, P ) = G n k P ) sin no, where &, (cln must be solutions of the equations d2& d$ r2-+r-!-(r2k2+ dr2 dr 2 -

r2%+ dr2

(3.48)

n2)& = 0 ,

r-dGn - (r2k; + n2)qn = 0. dr

(3.49)

Noting that these equations are of the same form as (3.10a), having solutions (3.10b), it follows that (3.48) become cos Gns(r,8, P) = Bn(n, P ) K n ( k J )sin nd,

$ns(r,

0, P ) = A n ( n ,

p)Kn(kdr)

(3.50)

making use of the radiation condition (3.34) once again, but just for r + 00. The general transformed solutions for the scattered potentials &, qSare then written as m

c GAr, 6, P ) = 6 d P ) c

&(r, 6, P )

=

&(P)

n =O

enAn(n, p)Kn(k,r) cos no, (3.51)

m

n=l

enBn(n, P ) K n ( k J )sin no

through superposition, where en = 1 for n = 0 and en = 2 for n 2 1. The transformed incident potential can be expressed similarly as

&(r, 0 , p ) = & ( p ) exp(kdr cos 8 ) = & ( p )

C enIn(kdr)cos no. (3.52)

n=O

Making use of the Laplace transforms of (3.31), from (3.51) and (3.52) we have the transformed general solutions &( r, 0, p ) , &( r, 8, p ) . The coefficients

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A n ( n ,p ) , Bn(n,p ) are determined by using the conditions of a stress-free boundary (3.32) now with -rr < 0 < rr and t > 0, leading to the formal solutions for the displacements (3.53) where

(3.54) n=l

where

a ( v, p ) , D( v, p ) being given in (3.44). This is the Fourier series representation of the solution. Inversion of (3.53) is accomplished by termwise contour integration of the bracketed terms in (3.54), corresponding to the scattered waves. The terms involving the 1,’s there correspond to the incident wave and can be inverted by inspection. We note that when n = 0 in (3.54), u, is the only displacement, and this term corresponds to axially symmetric deformation. Substitution of n = 0 into the An term of (3.54) shows, as we might expect, that P, takes the form of P(r, p ) in Miklowitz (1978), the transformed solution of the pressurized cylindrical cavity problem in which D(0, p ) here equals F( p ) there. Recall that the possible zeros of F ( p ) were of concern in our inversion and that we proved that F ( p ) had no zeros for Re p 2 0, leading to the contour integration over the contour shown in Fig. 15. It follows that the same contour can be employed for the other n values in the present problem; that is, ( 1 ) since the integer order In ( p ) functions are entire functions of p, we have only the branch point at p = 0, the one common to all the K n (p ) functions, and (2) there are no zeros of D( n, p ) in Re p 2 0 because the present problem has a static solution. The technique would therefore yield a series of line integrals from the integration up the imaginary axis, plus a series of residue terms from the corresponding paths about p = 0. This solution can be evaluated by integrating the line integrals numerically. Baron and Matthews (1961) used essentially the foregoing technique* for solving the present problem. In their work they completed the path of

* Instead of the Laplace transform on time transform with complex argument.

t,

they used a half-range exponential Fourier

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the Fourier transform inversion integral in the upper half-plane, which necesitated locating the zeros of the denominator function [corresponding to our D( n, p ) ] there for each value of n. Completion of the path along the real axis in their technique would have eliminated the need for these zeros, as it did in the technique leading to the inversion of (3.53) (with its analogous integration along the imaginary axis in the p plane). Since the number of these zeros increases with increasing n, it is not a trivial numerical problem to locate these zeros; the problem is aggravated further by the complicated representations of the Bessel functions making up D( n, p ) . For the present step-incident stress pulse, Baron and Matthews (1961) found that only n = 0,1, and 2 (a three-term Fourier series) were needed for the moderately longer time response of the circumferential (hoop) stress u, at a station ( a , 0 ) where the maximum of the stress was found. This is because the contributions from the circumferential modes for n > 2 are important for early time, but negligible for the later times. In Baron and Matthews (1961), numerical results are presented for the u, response at stations ( a , 0) and ( a , 7r/2), restricted away from early time (up to approximately the time it takes the incident pulse to cross the cavity). For a Poisson’s ratio of 1/4, peak stresses were found to be about 10% higher than the long-time static values. In a later paper Baron and Parnes (1962) made a similar analysis for the response of the radial and circumferential velocities tir, u,. Again with n = 0, 1,2, they evaluated these velocities for several cavity wall stations (a, 0 ) selected from the range 0 5 8 5 n-. The response curves appear in Figs. 7 and 8 of their work. Comparison of these results at the shadow-zone station [a, (3/4)7r] with those of the wave sum method is made in Figs. 22 and 23 here. Comparing the curves marked wave sum and Fourier series result, one sees that the three-term Fourier series results obtained by Baron and Parnes are in fairly good correspondence except for the early times, where the three terms are not enough to give even qualitatively correct results. The close correspondence of the wave sum representation with the physics of the problem is brought out by the fact that pulselike disturbances at the P - , and R - , arrival times are a natural part of the wave sum representation. In Fig. 22 for the radial velocity, note also that the results are in better agreement just a bit after the Po arrival time. In his thesis Peck (1965) made further comparisons of results at other cavity wall stations. Also, using a three-term Fourier series for the incident wave, instead of the exact one used by Baron and Parnes, Peck found that their results were in better agreement with those of the wave sum method. Conceivably this may be due to cancellations of higher-order n contributions to the incident and scattered waves. The reader will also be interested in the discussion of the Fourier series technique in the book by Pao and Mow (1973). They also apply it to the present problem, throwing further light on the influence of the higher-order n values through numerical evaluations of these modes.

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For sharper inputs than the step, of course, the higher n modes would certainly be needed. In conclusion we emphasize the following on the two methods: the approach based on the wave-sum method provides better physical insight into the early-time, near-field wave motions than does the Fourier series method. The wave-sum form of solution converges rapidly at short times, where the Fourier series solutions are ineffective. The convergence at long time is still fast enough so that seven modes of propagation provide an accurate solution. It must be kept in mind that, for general loading functions, the comparative convergence properties of the two methods depend on the time constants of the load. For load histories. that are “more impulsive” than the step function, the wave-sum method would be even more rapidly convergent; however, for more gradually applied loads, the Fourier series method would become more advantageous. It is also important to recognize the disadvantages of the wave-sum method. First, numerical evaluations of the type presented using this method are restricted to the shadow zone. Second, the relatively difficult mathematics of Bessel functions of complex order come heavily into play. Finally, the roots of a complicated transcendental equation must be evaluated before the evaluation of the inversion integrals themselves, making the overall numerics for the present approach considerably more involved than those for the Fourier series approach. 10. Difraction of Plane Compressional Pulse by Cavity: Approximations and Comments It is the purpose here to discuss briefly approximation techniques that have been applied to the present problem. Gilbert (1959) contributed high-frequency information for both normal (present case) and oblique incidence of the plane compressional pulse on the cavity. He considered only the portion of the cavity in the illuminated zone, restricted to 101 < ~ / 3 . Gilbert used a geometric optics method with further ray theory considerations to determine the reflected wavefronts (see Friedlander, 1958). He found for the present problem (incident plane step in a,)that the reflected stresses also behaved as a step at their wavefronts. He also found that the presence of the cavity produced a maximum amplification of the field (total) stresses of approximately two and that the maximum stress was the circumferential stress uo.Grimes (1964) also studied the wavefronts in the illuminated zone by using the method of steepest descents applied to the Friedlander representation of the solution ( m = 0). The approach, of course, is of interest here; however, Grimes finds a linear rise in time for the total u, at r = a that does not agree with the findings of Gilbert (1959) discussed earlier. Finally, the work by Gilbert and Knopoff (1954) on the rigid cylindrical cavity will be of interest to the reader in the present context since they

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FIG.24. Long-time radial velocity response at the cavity wall due to the positive &traveling Rayleigh waves (plane step wave source). Note: =.,:u

derived wauefront approximations for both the illuminated and shadow zones. They use Friedlander’s representation of the solution. Wavefront approximations for the illuminated zone are then written through the method of steepest descents. For the early motion in the shadow zone, integration was accomplished through a method developed for high-frequency approximations in analogous scattering problems in acoustic and electromagnetic wave theory. Approximations for long time in the present problem have also been contributed. Miklowitz (1966a) studied the Rayleigh waues for long time and the far field a la the method he used on the line load problem (see Section 111, A, 5). In the present case of plane wave impingement on the cavity, the Rayleigh surface waves are not singular at their arrival time, but they are still dominant in the dynamic long-time solution. The fact that they are nonsingular at their arrival times makes them experience spatial decay with 0 at these times, too, and further gives them heads and tails about their arrival time that must be summed to get an accurate description of their periodic (in 0) behavior (see the radial displacement in Fig. 18). Figures 18 and 24-27 show the qualitative nature of these waves ( n in these figures is the m we have used in the wave sum here; co = cR here).* The resultant response for the radial velocity uyR(a,0, t ) u i , where ui = c R u i / p and ui is minus the uoinput in (3.30), is shown by the solid line in Fig. 24. It represents the sum of the component waves, where because the heads and tails of these waves level off fairly rapidly and are of opposite sign, only one or two neighboring waves need be accounted for in addition to the main wave at a certain arrival time for a reasonable approximation, for

* The analysis underlying these 111, A, 5 .

figures is essentially the same as that described in Section

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111

example, to the right of the kth wave arrival, in addition to this wave, the k - 1, k + 1, and k + 2 are used in the sum. A similar procedure yields the circumferential velocity Rayleigh surface wave in the response to the plane step wave source. Here, however, the evenness of v R with respect to the Rayleigh wave arrival time requires consideration of the positive @traveling waves (from the 0 + 2n7r > 0 domain) and negative &traveling waves (from the 0 + 2n7r < 0 domain) together. In this manner the oddness of u#, with respect to 6 assures convergence. The situation is shown in Fig. 25a for the station 0 = ~ / 2 ,

FIG. 25. (a) Component Rayleigh surface waves of circumferential velocity at long time shown in their relative positions of time at 0 = 7r/2 (plane step wave source). Note: u& = u",. (b) Rayleigh surface waves of circumferential velocity at 0 = 7r/2 at long time (plane step wave source). Note: u& = ti:,.

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Julius Miklowitz

where the component positive O-traveling waves are shown above and component negative &traveling waves below. The resultant v f R (a, 8, t ) / ui wave, obtained by summing the component waves in Fig. 25a, is shown in Fig. 25b. Here again because the heads and tails of the component waves level off rapidly, and the k and i waves are of opposite sign, only a few neighboring waves in addition to the main wave are needed in the sum, for example, in the 7r domain to the right of the kth wave arrival; in addition to this wave, the k - 1, k + 1, i + 1, and i - 1 waves are used. In both Figs. 24 and 25a the 27r period of these resultant waves as well as their spatial attenuation should be noted. For other 8 stations in the physical plane, similar figures can easily be constructed. The positive O-traveling component and resultant accelerations aYR(a,8, t ) / a , and arR(a,0, ?)/ai are shown in Fig. 26a and b, respectively, where ai = c i w i / a p . The figures show that only the neighboring wave to the right or left of the main wave, at a particular arrival time, need be considered in addition to the latter in the summation. Finally in Fig. 27a, b we show the circumferential stress weR(a, 8, t ) response at the cavity wall due to the component and resultant positive O-traveling waves for both the delta function (Fig. 27a) and step function (Fig. 27b) wave sources. Soldate and Hook (1960) derived the long-time response at the cavity wall ( I = a ) in the present problem. They used the Fourier series form of the solution (3.53), (3.54)and applied the asymptotics of the Laplace transform and its inverse to the transformed displacements (3.54)and the corresponding hoop stress for r = a. Specifically these Laplace transforms are expanded in power series in p and inverted term by term. With our step

FIG. 26. Long-time (a) radial and (b) circumferential acceleration response at cavity wall due to positive 0-traveling waves (plane step wave source). Note: a $ = up, and a R= uiR.

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FIG. 27. Long-time circumferential stress response at cavity wall to plane wave (a) delta function and (b) step function source due to positive &traveling Rayleigh waves.

input it is found that only a few terms are needed to give adequate results for the very long time approximation sought. As the input sharpens, more and more terms are needed, and further the accuracy of the results is harder to assess. Soldate and Hook found the leading terms of the long-time velocities to be the rigid-body velocities of the cavity wall u,He(a,e ) / u i = -(c;/cRcd) cos 8, u&( a, o)/ u, = ( c : / cRcd)sin 8,

(3.55)

and the leading term of the hoop stress to be

ays(a, @ ) / a=i - ( 2 / k 2 ) ( k 2- 1 - 2 cos 2 8 ) .

(3.56)

As it should be that (3.56)is the same as the solution for the elastostatic hoop stress that can be obtained through superposition of Kirsch’s classical solution for the rectangular plate with a circular hole subjected to uniform compressions on one set of the edges (see Timoshenko and Goodier, 1970, pp. 90-97). Of course, (3.55) and (3.56) are the limiting solutions for time

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Julius Miklowitz

infinite. As such they will be the only responses present, the periodic Rayleigh waves having long since died out through spatial decay. However, since the decay to the static solution is quite slow, one would expect, say for moderately large times, to have the effects of both the static solution and the Rayleigh surface waves acting at cavity wall stations. At the stations 8 = 0 and n, where ursin (3.56) is zero (for A = p, k2 = 3), one would still have the Rayleigh waves a& (depicted in Fig. 27b) acting there. This adds an interesting new consideration in the determination of dynamic stress concentrations in the cylindrical cavity problem.

B. SCATTERING O F A PLANECOMPRESSIONAL PULSE BY A CIRCULAR CYLINDRICAL ELASTICINCLUSION: A BRIEF DISCUSSION 1. The Problem

Figure 14 again depicts the problem with the incident plane pulse shown there, but now the cavity is replaced by the cylindrical elastic inclusion. The inclusion, which is in the interior region 0 5 r < a, is an elastic solid of different properties from those in the exterior region r > a ( r = a is the interface). To state the problem mathematically we first define the potentials 4a,$a, a = 1,2, where 1 corresponds to the inner solid and 2 the outer solid. The outer solid then is governed by the same equations as in the plane pulse cavity problem of Section 111, A, 6 , that is, governing wave equations (3.5), displacement- and stress-potential relations (3.9), and solution forms (3.31), all with subscript 2 on C#J and $ and related quantities. We note that 4iis given by (3.30). Added here then would be (3.5) and (3.9) with subscript 1 on 4 and )I to represent the inner solid. Boundary conditions at r = 0 would now be that this interface has continuous displacements and stresses. Quiescent initial conditions of the type (3.33) on the outer solid potentials, and now on the inner solid potentials, are assumed. In turn the radiation conditions (3.34) apply again to the outer solid potentials, and consistently, we require the inner solid potentials to be bounded. This completes the statement of the problem. This more general case of transient wave scattering from a cylindrical elastic inclusion is more complicated than the preceding cases treated for the cavity because of the refracted waves that are generated at the interface of the two solids. As Fig. 28 shows, for example, for an incident P-wave ray, the refracted P wave is responsible for a multitude of other rays that are due to its external refractions to the outer solid and reflections within the inclusion. The problem is of interest, for example, in the dynamic response of jiber-reinforced composite materials. In these materials one asks whether stress wave singularities arising from wave focusing can cause

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FIG. 28. Ray geometry of the refracted dilatational waves.

separation at the fiber (inclusion)-matrix (outer solid) interface, hence weakening the composite. 2. The Literature: Methods and Results

The problem has been attacked by Ting and Lee (1969), KO (1970), Achenbach et al. (1970), and Griffin and Miklowitz (1974). The objective in the first two of these works was to ultimately determine the dispersive effect of an array of inclusions on the incident stress pulse. The latter two are concerned mostly with focusing. Focusing occurs when a ray touches a caustic, a caustic being an envelope ofconverging rays. Figure 29 (taken from Griffin and Miklowitz, 1974) shows one example of a caustic that is generated within the inclusion. As one can see, the caustic is formed by the family of refracted P-wave rays in the interior, once reflected from the interface (the 1-pp rays in the figure). This case is the important practical one in which the fiber is a stiffer material than the matrix ( c = c d l / c d 2> 1 ) . The papers by Ting and Lee and by KO show that caustics can occur and have wavefront singularities there. However, they d o not bring out the important wavefront singularities that occur after focusing. Ting and Lee studied the interaction of an incident plane dilatational pulse with a circular cylindrical (or spherical) elastic inclusion using the wavefront analysis of geometrical acoustics (see Friedlander, 1958). They determine the pressure field for the times that include the incident wavefront’s reflection at the

116

FIG. 29.

Julius Miklowitz

( 1 - p p ) rays and caustic for c = 1.5, 0 5 f3 < 27r. [From Griffin and Miklowitz

(1974).]

interface, its transmission (refraction) into the inclusion, and its emergence into the outer medium. Curves are given for the dilatational (and equivoluminal) wavefront positions (with time) and for the corresponding magnitudes at the pressure wavefronts. Making use of integral representations of the Kirchhoff type, KO determined wavefront stresses and displacements for the interior, exterior, and interface fields. He presented the dilatational wavefront positions for the cases where the inclusion is either more stiff ( c > 1) or less stiff ( c < 1) than the matrix material. Further, he presented numerical results for the wavefront magnitudes of the stresses and displacements along the interface as a function of circumferential angle. In their work Achenbach et al. were mainly interested in focusing effects. They carried out experiments on dynamically edge-loaded (explosive charges of lead azide) Plexiglas sheets having a single circular aluminum inclusion and, interestingly, obtained photographic evidence of separation of the inclusion from the Plexiglas. As the magnitude of the charge was increased, the amount of separation increased, with total separation of the inclusion occurring for the largest charge used. Analytical work was carried out to try to correlate the regions of separation with the focusing of the first wavefront (dilatation). They used a geometric acoustics approximation for this wavefront up to its arrival at a focus point on a caustic. Then, since this approximation breaks down on the caustic, they followed Friedlander’s work in acoustics in which he used Poisson’s integral formula to carry the approximation beyond the caustic point. In the elastic case analogously Love’s integral representation for the displacement field [see Love (1944) and (1904) for derivation] was the tool. Correlation was found for the experimental results.

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The work by Griffin and Miklowitz corroborates, and considerably extends, the findings of Achenbach et al. through a more general method of analysis for treating singular wavefronts. In this method a Watson-type lemma is used. The lemma relates the asymptotic behavior of the solution at its wavefronts to the corresponding asymptotic behavior of its Fourier transform on time for large values of the transform parameter. Again, as in the foregoing work on the cylindrical cavity, the Friedlander representation is used to handle the 8 variable, here, however, finding application also to the interior region (inclusion) through the aforementioned Watsontype lemma. The lemma not only handles the first wavefront to arrive, but also the later arriving ones. This property is quite important in focusing problems since they often have later arriving waves that have focused and are singular. In Griffin and Miklowitz, both of the cases c > 1 and c < 1 are analyzed in detail (Fig. 30 shows the rays and caustic for the latter case). Careful studies of the ray geometry involved in focusing, aided by the Watson-type lemma, bring out the nature of this phenomenon. In the case c < 1 (fiber softer than matrix), it is shown that along a ray the incident step stress pulse remains a step pulse upon refraction. However, upon reaching the where caustic the wavefront singularity becomes of the type ( 1 - t d f d l is the dilatational wave arrival time. Further propagation takes the wavefront past the caustic, where it becomes logarithmic, that is, lnlt - t d l l . The case c > 1 shows that logarithmic singularities develop in the stress wavefronts here, too, much as in the manner of the previous case, that is, after the ray touches and goes beyond the caustic. As Fig. 29 shows, this happens to the once reflected family of refracted rays (1 p p ) . It is clear from this figure, therefore, taking into account the fact that there are two systems of wavefronts traveling in this problem (the positive- and negative-8 Y

Refracted dilatation

FIG. 30. Refracted dilatational rays and caustic for c

=

0.5, 0 5 0 < 27r.

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Julius Miklowitz

traveling systems), that every point on the interface r = a experiences a logarithmic singularity (from refracted waves that have reflected once). Other points of interest found in Griffin and Miklowitz are that (1) the interior refracted wavefronts have logarithmic singularities that are refracted into the exterior solid unchanged, (2) the interior wavefronts reflect n number of times, a process that results in a decay in the magnitude of their coefficients as ldl", d < 1, and (3) the effects of the diffracted waves in the problem were negligible with respect to those of the focused refracted waves.

C. DIFFRACTION OF A N ELASTIC PULSE BY A SPHERICAL CAVITY: A BRIEF DISCUSSION 1. The Literature

Important to the literature on the present topic were the relatively early paper by Nagase (1956) and the paper by Nussenzveig (1965). Nagase treated the problem of the diffraction of harmonic in time waves from the cavity, generated by an exterior point source, say at r, > r,, with r, the radius of the cavity. He treated both dilatational and equivoluminal sources and obtained important high-frequency approximations. With a similar

FIG. 31.

Problem of cavity surface normal point load sources.

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119

tz

S 4 Z S k I

GIDENT DILATATION WAVE FRONT

FIG. 32. Problem of incident plane dilatational pulse.

interest Nussenzveig treated the related case of the high-frequency scattering of an acoustic harmonic plane wave from an impenetrable sphere. Later work by Norwood and Miklowitz (1967) extended the results of Nagase and Nussenzveig to transient elastic wave (or pulse) diffraction from the spherical cavity. Figures 31 and 32 depict the two problems treated in Norwood and Miklowitz (1967), respectively, a sudden normal point load on the cavity wall and the impingement of a plane dilatation pulse on the cavity. Approximations at the cavity wall (in the shadow zone) for the displacements at the dilatational wavefronts, and for the Rayleigh surface waves, were obtained for both problems. The next section reviews briefly the essential features of the method of solution used by Norwood and Miklowitz and the results obtained. 2. Method of Solution: Results The problem of the diffraction of a pulse from a spherical cavity is closely related to that for the circular cylindrical cavity. Indeed, the geometrical optics of the former case is similar to that of the latter. In fact, the wavefronts for the circular cylindrical cavity case are the meridional sections of those in the spherical cavity case, the full fronts in the latter being obtained by rotation about their axes of symmetry. One would expect, then, that the present two problems would be closely analogous to the problems of Sections III,A,3 and III,A,6, with methods of analysis being analogous to those used on the latter. We bring out here therefore only the essential differences in the methods and features of the two classes of problems, further detail being left to Norwood and Miklowitz.

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The Laplace transform is again used on time t. This reduces the governing potential equations of motion to V 2 6 ( r ,8, p ) = k i 6 ,

V 2 x = k:x,

4= ax/ae,

(3.57)

4

6

where is the transformed dilatational; X, are equivoluminal scalar potentials; r, I3 are the radial and latitudinal angle coordinates (see Fig. 31); and kd = p/cd, k, = p / c s , p being the Laplace transform parameter. Further spherical harmonics separation of (3.57) and invoking the radiation condition produce the general solutions 00

6= nC=o An(kdr)-”2Kn+1,2(kdr)Pn(cos e),

(3.58)

where Kn+,,2(z ) is the modified Bessel function of the second kind of order n and Pn(cos 8 ) is the Legendre polynomial of order n.

+

a. Point Load Problem This problem is analogous to the line load problem of the circular cylindrical cavity. Consideration of the boundary conditions at the cavity wall produces the Laplace transformed solution 6, $ for the problem. The Bromwich integral then gives the formal solutions for + ( r , 8, t ) , $(r, 13, t ) . Singularities of 4 in the p plane are simple poles stemming from the frequency equation A(ro, n, p ) = 0 and the transform of the time input function of the point load. Such poles cannot be in the right half p plane, Re p > 0, since the problem has a static solution. It follows the Bromwich contour can be traded for a path up the imaginary axis. The dynamic solution for then becomes

6,

+

4 ( r , 0, iw)e”‘ dw, where

(3.59)

6has the form

with a similar expression for $(r, 0, I). As we have noted earlier, a series such as that in (3.59) would have slow convergence properties at the high frequencies. Our remedy for this in the cylindrical cavity and inclusion problems was the use of Poisson’s summation formula (3.1) and related theorem given in Section llI,A, 2. The analogous treatment here exploits Watson’s transformation, a technique introduced initially and used widely in the study of the diffraction of electric

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waves by the earth (see Watson, 1918). It was also used in Nagase (1956) and Nussenzweig (1965). Watson’s transformation is based on the formula (3.60) where C, is the contour shown in Fig. 33. The formula is easily checked by evaluating the residues at v = ( n + 1)/2. On applying (3.60) to 6 in (3.59) [i.e., identifying the series in (3.59) with that in (3.60)] and using the expression P,(cos 0 ) = exp(im)P,,[cos(.rr - O)], one finds

(3.61) where 4”-1)/2rro,

r, (.

-

11/21 = . f i r 0 3 iw, (.

v

=

wr/c,,

b

-

=

l)/21[K”(ibv)/(ibv)1/21,

c,/cd.

By substituting - u for u in the integrand of (3.61), using the identities P-(A--I)/z(cos 0 ) = P~A-l)/z(cos O), K-,(z) = K,(z), it can be shown that this integrand is an odd function of u. It follows that the lower half of C, in Fig. 33 may be replaced by its reflection in the origin, the dashed line in the figure. This contour and the upper half of C , are equivalent to a straight line located just above the real axis on which the expansion m 1 - 2 C ( - )“ exp[inu(2s cos u?? s=o

--

+ I)]

(3.62)

ReY

I FIG.33.

Integration of the v plane.

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Julius Miklowitz

is valid. Substitution of this result into (3.61) yields

c ( - 1” W

&(r, e, iw)

=

-

2i

O0

-v

4+1)/2

A

S=O

~(v-I)/2[COS(7T - 8)l

x e x p [ i ~ v ( 2 s+ l ) ] dv

(3.63)

with a similar result for &. Next in the method one exploits the zeros of A (and corresponding simple poles) in the integrand of (3.63). They are the branches of the frequency equation A [ r o , iw, ( v - 1)/2) = 0, which give the modes of propagation through residue theory. This is done through the sequence of paths C , passing between the zeros of A, as shown in Fig. 33. The paths C, can be shown to give a zero contribution to the contour integrations [see Norwood’s thesis (1967) for the proof]. It follows that (3.63) reduces to W

&(r, 8, i w )

= 47r

C

s=o

(-)‘

1

j=1,2,

(3.64) for 0 < 8 d T, vj being the zeros of A ( r o , iw, ( v - 1)/2) in the second quadrant. Similar expressions for $ and displacements a,, U, are given in Norwood and Miklowitz (1967). At this point it should be emphasized that Watson’s transformation was the important tool that enabled us to get a solution in terms of the frequency branches for the stress-free spherical cavity. Simply put, it has allowed us to trade one set of poles for a much more important set, just as Poisson’s summation formula did for us in the cylindrical cavity problem. In Norwood (1967) it is proved that the two techniques are equivalent for the present problems. The usefulness of (3.64) in obtaining the high-frequency wavefront and Rayleigh surface wave approximations will become apparent in the sequel. The dilatational wavefronts for the displacements u,, u, are derived in Norwood and Miklowitz for the step in time load. They are obtained from expressions like (3.64) by utilizing high-frequency, large-wave-number approximations to the P branches of A ( r o , iw, ( v - 1)/2) = 0, say v2j,which are given in Nagase’s work (1956). They are of the same form as (3.15a), which means the higher the branch, the larger its imaginary part. Note that this is indicated by one of the series of poles in the second quadrant of Y in Fig. 33. Seeking the response on the surface of the cavity, we first substitute the v2j into U,(ro, 8, iw), which gives -1/2

m

t(,(ro, 8, iw) = M S=O

j = 1,2,..

(3.65)

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123

valid for 0 < 0 < 77, where M is a constant. Also Ue is similar. Letting s = 0 we see that the terms in (3.65) correspond to two diffracted waves, one that reaches the station (ro, 0 ) directly from the point source at the north pole ( r o ,0) and the other through a reflection from the south pole ( r o ,T).This reflection process goes on ad injnitum at the north and south poles, times permitting, the reflecting waves being represented by s 2 1. They are waves that encircle the cavity 2 s times, so that the corresponding angular paths are increased by 277s. Note that here our wave sum is the sum on s in (3.65). Recall that the inversion path in the p plane that led to (3.65) was the imaginary axis, and this path is equivalent to the Bromwich path. This permits setting io = p in (3.65) and inverting it through the asymptotics of this inversion integral for large p and short time. Leaving the details to Norwood and Miklowitz, it was found to determine the wavefronts for u,, ue only the terms s = 0, j = 1 needed to be taken into account for 6 > 1. These wavefronts have the forms

where uo = P / r o ( A + 2 p ) ( P is the magnitude constant of the point load), T = tcd/rO- 6, H ( t ) is the Heaviside step function, and the inequalities 1 < 0 < T,

( t c d / r O< ) m i n ( k 0 , 2 ~- 6 )

FIG. 34. Radial displacement response to point load at cavity wall r and 3n-14.

= ro,

for 0

=

?r/3

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Julius Miklowitz

FIG.35. Tangential displacement response to point load at cavity wall r and 31~14.

= r,,,

for 6

=

1r/3

must hold, and M,, Me and b are constants. These wavefront responses for Poisson’s ratio 1/3 and the two stations ( r o , 7r/3), ( r o , 37r/4) are plotted in Figs. 34 and 35, respectively, for u, and ue. Their behaviors are typical of the diffraction of scalar pulses by curved boundaries. The Rayleigh surface waves were also calculated using (3.15b) by essentially the method given in Miklowitz (1963). The results are similar to those in Fig. 18. b. Incident Plane Dilatational Pulse This boundary value problem is analogous to the corresponding one of the cylindrical cavity. The technique for solving it is essentially the same as that just used to solve the point load problem. The resulting dilatational wavefronts for u,, ue at the station 7r/6 for Poisson’s ratio again 113 are similar to (3.66). They are shown in Fig. 36, where U1 = T ~ ~ ~+/ 2p), ( T T~ being the magnitude constant of the axial stress associated with the incident potential pulse. The Rayleigh surface waves here are similar to those found in Miklowitz (1966a). The latter are discussed in Section 111, A, 10 and exhibited in Figs. 24-21.

Modern Corner, Edge, and Crack Problems

FIG. 36. Response of radial and tangential displacement at cavity wall r plane pulse diffraction problem.

125

=

ro, and 8 = ~ / 6 ,

D. SOMERELATEDPROBLEMS OF INTEREST I N ELASTIC WAVE SCATTERING Datta (1977) has presented a survey that is of interest here. He discusses four topics: (1) the scattering of (a) dilatational waves by a liquid sphere or cylinder, (b) waves by rigid spheroids, and (c) waves by a rigid circular disk; (2) wave propagation in a half-space containing a cylindrical cavity. The nonseparable nature of this last problem has led to contributions only in recent years with pioneering work by Ben-Menaham and Cisternas (1963) on the spherical cavity and later important contributions by Thirwenkatachar and Viswanathan (1965) and Gregory (1967). Datta and Sangster (1977) point out that their work, based on matched asymptotic expansions, has applicability to the problem and they demonstrate this. Datta and El-Akily (1978) continued their interest in the problem of diffraction of two-dimensional elastic waves by inclusions and cavities in a half-space. First, a representation theorem for elastic waves in a half-space is given. This representation is in terms of multipole solutions for a half-space and can be used for numerical solution for arbitrary bodies. Now Datta

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and El-Akily confine their attention to the application of matched asymptotic expansions (MAE) to obtain the scattered field when the wavelength is large compared with the linear dimensions of the inclusion. The application is illustrated by solving two problems, that is, diffraction of a plane SH wave by an elliptical inclusion and diffraction of P or S waves by a cylindrical cavity in a half-space. Scheidl and Ziegler (1978) also contributed to the present general problem. They treated the problem of the interaction of a pulsed Rayleigh surface wave and a rigid cylindrical inclusion. The motion of the rigid circular inclusion and the unstationary stress field in the surrounding linear elastic matrix half-space are described analytically and numerically. The incident wave is a nondispersive surface wave pulse of prescribed shape. The method of solution makes use of the stationary case of loading by a periodic train of wave pulses and its time Fourier series representation. Wave reflections at the free surface of the half-space are considered for numerical reasons by approximating the plane surface by large convex and concave cylindrical srlrfaces, respectively. The results give detailed answers to questions having been raised in the design of composite materials and in earthquake engineering. Part of the numerical results, shown in the figures, give a clear picture of the motion of the rigid inclusion and the stress histories in the near field of the surrounding matrix. The influence of the free surface on these results is quite strong, depending on the depth-to-radius ratio n / a . Strong motion effects of multiple reflections at the free surface and the interface and a waveguide effect are clearly indicated from the time histories of displacement and stresses. Finally, the last figure of the work shows the wave field from diffraction of a pulsed Rayleigh surface wave traveling from the right to the left. The eight pictures in the figure were taken from a photoelastic s 15%. model in equal time intervals of 8 X

*

1. Wave Diflraction by a Finite Rigid Strip and Crack Of note also is work on wave diffraction by a finite rigid strip and crack. The early work by Ang and Knopoff (1964a) on the diffraction of a timeharmonic dilatational wave for both cases should be pointed out. Thau and Lu (1971) have treated the transient incident dilatational wave case for the finite crack to establish corresponding stress intensity factors at the crack edges. 2. Modal and Surface Wave Resonances in Acoustic Scattering from Elastic Objects and in Elastic Wave Scattering from Cavities Of further importance is the work of Uberall (1978) involving resonances and scattering. In Uberall's abstract he points out that the problems of

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127

scattering of waves from an obstacle, no matter in which field of physics they are being considered, exhibit an essential similarity. This similarity extends from the scattering of sound waves by an elastic object (acoustics) over elastic wave scattering from a cavity (geophysics and materials testing) to the problems of particle scattering from particles and nuclei (high-energy and nuclear physics). It follows that with progress in the various fields we find that more may be learned by exchanging methods of one field of physics in another one, and vice versa, so that the basic similarities of the scattering problem may be exhibited and exploited. Considering the above, after having studied surface wave phenomena in acoustic scattering by what is known in nuclear physics as Regge pole techniques, we apply the Breit-Wigner resonance scattering formalism of nuclear physics to both acoustic and elastic wave scattering (essentially another way of exhibiting the Regge poles) and relate the resonances to the eigenvibrations of the scattering object. 3 . Diffraction of a Plane Compressional Elastic Wave by a Semi-infinite Rectangular Boundary of Finite Width The problem is of basic interest in geophysics. The solution to it is developed in terms of matched asymptotic expansions by Viswanathan and Sharma (1978). The outer problems are solved by using the Wiener-Hopf method and the inner problems by the Kolosov and Muskhelishvili complex potentials. These solutions are then matched at appropriate regions, which, incidentally, leads to the behavior of the displacement field near the edge of the scattering boundary. Because these coefficients have bearing on the form of the scattering boundary, these are finally calculated numerically and presented in the form of graphs as functions of the angle of incidence of the plane wave.

E. THE SCATTERINGOF ELASTICWAVESBY CRACKS Of strong practical interest in the subject of fracture mechanics and related structural design and damage are the surface-breaking and subsurface cracks. Indeed the scattered wave fields generated by the interaction of incident surface or body waves with these cracks would be expected to yield most of the important information about the geometries of the cracks. It follows in the subject of quantitative nondestructive evaluation (QNDE) that there is considerable interest in scattering by surface-breaking and subsurface cracks as important steps in solving the inverse problems of obtaining the crack geometries from the scattered wave fields. However, having the solutions to the corresponding direct problems is a prerequisite to the inverse problems.

128

Julius Miklowitz

The present crack problems fall into a class that is nonseparable classically, and as such they are difficult to solve. The nonseparability stems from media corners and crack edges. Mathematically, however, the elastodynamic quarter-plane boundary value or boundary initial value problems insolved can be handled through integral transforms, integral equations, and asymptotic and numerical analysis. 1. The Case of Time-Harmonic Loads and Waves

a. The Surface-Breaking and Subsurface Crack Problems: Their Solutions and Numerical Results The present problems have been addressed in four important papers, three on the surface-breaking crack by Achenbach et al. (1980), Mendelsohn et al. (1980), and Kundu and Ma1 (1981). The fourth paper by Achenbach and Brind (1981) treated the subsurface crack. In Achenbach et al. (1981) the surface-breaking crack is assumed to be the two-dimensional normal edge crack of depth d in an elastic half-plane (Fig. 37). In Achenbach and Brind, the subsurface crack is assumed to be the two-dimensional crack normal to the free surface of the elastic half-plane with its tips at y = a and y = b, respectively, where b / a > 1 (Fig. 38). It was assumed also in these works that (1) the two faces of the cracks involved do not touch one another and hence these cracks never close completely and (2) the loads and waves were time harmonic in nature. The total field in the half-plane for each of the problems in Achenbach et al., Mendelsohn et al., and Achenbach and Brind is composed of the superposition of a specific incident field (say free surface or body waves) in the uncracked half-plane and the scattered field in the cracked half-plane generated by suitable surface tractions on the crack faces. Through equilibrium arguments in the plane of the crack these surface tractions are equal

FIG. 37. Waves incident on a surface-breaking crack of depth d. [After Mendelsohn, Achenbach, and Keer (1980).]

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129

FIG. 38. Two-dimensional subsurface crack. [After Achenbach and Brind (1981 ).]

and opposite to the tractions generated by the incident wave in the uncracked half-plane. Then by decomposing the scattered field into symmetric and antisymmetric fields with respect to the plane of the crack, one obtains a pair of elastodynamic quarter-plane boundary value or boundary initial value problems. Integral transform methods then reduce the two boundary value problems to two uncoupled singular integral equations, which are solved numerically. In turn, integral transform methods then reduce the two boundary initial value problems to two coupled integral equations, which can be solved through asymptotics and some algebra and by an array of constants. Numerical results presented graphically in Achenbach et al. for the surface-breaking crack and an incident arbitrary (to a constant) Rayleigh surface wave disturbance include variations of (1) normal and tangential crack-opening displacements with crack depth for three values of dimensionless frequency, (2) crack-opening displacements at the mouth of the crack with dimensionless frequency, and (3) mode I and mode I1 normalized dimensionless stress-intensity factors with dimensionless frequency. Considered here also were six different line-loading configurations (through symmetries) applied at (Fx,, 0) on the surface of the half-plane about the crack and directed to the left or right of the crack, respectively. Then assuming that k,~, >> 1, only the surface motions due to these loads interact with the crack, kR being the wave number of the Rayleigh surface waves. These loadings cause mode I or mode I1 deformations (for example, displacement fields). Tangential crack-face loadings produce mode 11 deformations with antisymmetric displacements. Numerical results were presented graphically in Mendelsohn et al. (1980) for the surface-breaking crack and the following three types of incident arbitrary (to a constant) waves: (1) Rayleigh surface waves originating at x = -a, y = 0, (2) plane longitudinal waves originating at r = a, 0 5 O0 < ~ / 2 and , (3) plane vertically polarized shear waves also originating at

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Julius Miklowitz

r = a,0 5 Bo < r r / 2 , Bo being the angle of incidence. See Fig. 37. For the incident Rayleigh surface wave disturbance, the numerical results included variations of the absolute values of the normalized horizontal and vertical displacements of the forward- and back-scattered surface waves with dimensionless frequency. For the incident plane longitudinally and vertically polarized shear waves, the numerical results included variations of the forward- and back-scattered displacement fields with (1) the dimensionless frequency, for the angles of incidence O,, = 0", 30°, and 60°, and (2) the angle of incidence, for a low and high frequency. Numerical results presented graphically in Achenbach and Brind (1981) for the subsurface crack and the case of an incident arbitrary to (constant) Rayleigh surface wave disturbance include variations of the dimensionless horizontal surface displacement in the ( 1 ) back-scattered, forward-scattered, and transmitted Rayleigh waves with dimensionless frequency, for three values of a / b, (2) back-scattered and transmitted Rayleigh waves with a / b, for three values of the dimensionless frequencies, and (3) phase shifts relative to the incident wave of the back-scattered and transmitted Rayleigh waves with dimensionless frequency, for the three values of a / b in (1) above. In addition, numerical results were given for the normalized mode I and mode I1 stress intensity factors as a function of a / b for the threedimensional frequencies in (2). The paper by Kundu and Ma1 (1981) is on the diffraction of elastic waves by a surface crack on a plate. The problem is that of a surface-breaking crack on one edge of the plate, one of the plane strain and of time-harmonic loads and waves, akin to the papers by Achenbach et al. and Mendelsohn et al. The incident waves are assumed to be either plane strain body waves (compressional P or shear S V ) of arbitrary angle of propagation or surface Rayleigh waves propagating at right angles to the crack. The complete high-frequency-diffracted field on the plate surface is calculated for each incident wave. The solution is obtained by the application of an asymptotic theory of diffraction. In Kundu and Ma1 the following numerical results were displayed for the normalized x component of the displacement on the plate surface y = 0, due to various types of wave incidence, as a function of the dimensional frequency k,l = w l / c , ( 0 , the frequency; c , , the P wave speed). Curves included (1) amplitudes of transmitted, reflected, and secondary shear converted Rayleigh waves for Poisson's ratio 1/3 and 1/4, (2) amplitudes of Rayleigh waves due to shear wave incidence at different angles for primary diffracted waves and secondary shear converted Rayleigh waves, (3) shear wave incidence at different angles, and (4) body and surface waves of significant amplitude produced by incident P waves at 30". A related interesting discussion of these data is presented.

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131

Further, the reader should be aware of an important general work on the present subject, the ASME monograph on “Elastic waves and non-destructive testing of materials,” edited by Y. H. Pao in 1978, AMD, Vol. 29. 2. The Case of Transient Loads, Waves, and Wavefronts: Wavefront Analysis in the Nonseparable Elastodynamic Quarter-Plane Problems

We now draw on the work by Miklowitz (1982), a two-part paper on wavefront analysis in the nonseparable elastodynamic quarter-plane problems. This work involves loadings and waves (wavefronts) that are transient in nature and therefore differs from the time-harmonic wave nature of the works of Achenbach et al. (1980), Mendelsohn et al. (1980), and Kundu and Ma1 (1981). However, the basic quarter-plane features of Achenbach et al., Mendelsohn et al., and Miklowitz (1982) are similar; that is, the boundary value problems of Achenbach et al. and Mendelsohn ef al. are, in Miklowitz, boundary initial value problems. The boundary (edge) conditions in Achenbach et al. and Mendelsohn et al. are like those in Miklowitz except for the latter’s time dependence. Integral transforms and related singular integral equations are a property of the solution techniques in both works, in which the solutions for these integral equations are found by numerical analysis in Achenbach et al. and Mendelsohn et al. and by direct simple integrations yielding algebraic equations with the following numeric evaluations in Miklowitz. It follows that the techniques in Miklowitz can be applied to the quarter-plane problems in the former two works, hence giving the transient wavefront fields at their fronts and just behind them for regular wavefronts, as well as two-sided wavefronts involving a precursor, for these problems [i.e., the counterparts (high-frequency pulses) of the solutions in Achenbach et al. and Mendelsohn ef al.]. The scattered transient wavefront fields obtained here then will be very valuable to the QNDE methods based on scattering of ultrasonic elastic waves by cracks, with interest toward solving the inverse problems of obtaining the crack geometries from these scattered transient wavefront fields. It has been pointed out that the techniques in Miklowitz (1982) can be applied to the quarter-plane problems in Achenbach et al. (1980) and Mendelsohn et al. (1980), hence giving the transient wavefront fields for these problems, that is, the counterparts of the solutions in Achenbach et al. and Mendelsohn et al. In order to reduce the text of Miklowitz (1982), Parts 1 and 2, for the reader, the following sections contain main point consultations providing the necessary background for treating the present problem (i.e., the surface-breaking crack problem). The surface-breaking crack problem and its solution with the nonseparable elastodynamic quarterplane are discussed in Sections IV,B,1-7. We have partial resulting wavefront

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Julius Miklowitz

fields in scattering by the surface-breaking crack for the quarter-plane interior; see Sections IV,C,l-2.

3 . Di’raction of Elastic Waves by Cracks, Analyzed on the Basis of Ray Theory Achenbach et al. (1978) contributed an important paper for this topic. Their work concerns application of elastodynamic ray theory to diffraction by cracks. The paper discusses three theories: geometrical elastodynamics, geometrical diffraction theory, and uniform asymptotic theory, which provide approximations of increasing accuracy. As an example, diffraction of a normally incident plane longitudinal wave by a plane crack of length 2a is discussed in detail. Further, Achenbach et al. have published a book on this general subject (1982). The title is Ray Methodsfor Waves in Elastic Solids, with applications to scattering by cracks (Pitman Advanced Publishing Program, Boston, Mass.)

IV. The Two-Dimensional Wedge and Quarter-Plane

A. INTRODUCTION Progress has been made in the study of wave propagation in a twodimensional elastic wedge. Most of the work has been on the special case of the quarter-plane. As in the analogous waveguide case, the separable problem with mixed boundary conditions (tractions specified on one edge and a mixture of stress and displacement on the other) can be handled with a double integral transform (see Section II,A,2). Wright (1969) treated four such problems involving uniform inputs for each of the two stresses and two displacements to one edge (see Section II,A,2). Brock and Achenbach (1970) treated similarly the longitudinal impact problem for two welded quarter-spaces. Both works present numerical evaluations of the solutions. Concerning the quarter-plane problem with its nonseparability and nonmixed boundary conditions on its edges (both having either the two stresses or the two displacements specified or on one edge the two stresses and the other the two displacements), we first note the early work of Lapwood (1961). He treated the problem of the sudden line load on one edge and studied, through a multiintegral transform and successive approximations, the behavior of the Rayleigh surface wave transmitted and reflected by the corner. Viswanathan (1966) treated the analogous but more complicated problem of two welded quarter-spaces. His analysis modeled Lapwood’s, which he used to study the various first-order events in the problem. These

Modern Corner, Edge, and Crack Problems

133

included the body waves radiated from the interface corner, transmitted Rayleigh pulse, and the Stoneley pulse traveling along the interface. The work of Alterman and Rotenberg ( 1969) on the quarter-plane with stress-free edges and an internal line compressional source using a finite difference method will also be of interest. Note also the work of Krau (1968a,b) on the rigid quarter-plane scatterer in the infinite elastic solid, a three-dimensional problem. He formulated it as a Wiener-Hopf problem (see Noble, 1958) in two complex variables and studied the scattered waves generated by an incident plane compressional pulse. On wave propagation in the two-dimensional elastic wedge of arbitrary angle, we note first the excellent SUNey by Knopoff (1969), who discusses a variety of analytical methods that have been tried for nonmixed edge conditions but have failed. Achenbach and Khetan (1977) have reported success in applying the method of self-similar solutions to such a problem, that is, a line load suddenly applied to one of the edges, which then moves with a constant velocity. Their method generates a system of coupled integral equations that they solve by series expansions and numerics. They show numerical results for the radial particle velocity on one edge of the wedge. Achenbach (1976) has reported further numerical results for the radial particle velocity along the wedge edges for the case of antisymmetrical traveling line loads on these edges. Wojcik (1977), also using self-similar solutions, has been able to reduce this type of wedge problem (no characteristic length) to a single unknown governed by a Fredholm integral equation of the second kind in one dimension. To date numerical evaluation of this equation needed to obtain wave information in the problem has not been carried out. Indeed, changes in the Rayleigh pulse, as it negotiates a general angular discontinuity (i.e., transmission and reflection), have been of great concern in the seismology and mechanics literature. The problem was studied early by Viktorov, de Bremaecher, Knopoff and Gangi, and others by analytical (through iteration methods) and experimental techniques. Knopoff (1969) cites these references. Still others may be found in the survey by Miklowitz (1966). Using the basic theme of Section II,A,4 in this article in nonseparable elastic waveguide problems (i.e., those with nonmixed edge conditions) by Miklowitz (1969), Sinclair and Miklowitz (1975), and Miklowitz and Garrott (1978) exploiting a boundedness condition on the solution, these co-workers have found similar boundedness conditions on solutions and related conditions that eliminate inadmissible events in such solutions for the nonseparable quarter-plane problems. The waveguide and quarter-plane problems are, of course, quite different, the former having a characteristic length and the latter not. This leads to certain distinct differences in the boundedness condition and other parts of the method for these two different types of

134

Julius Miklowitz

problems, one (the waveguide) being an eigenvalue problem and the other (the quarter-plane) not. To outline the present general technique for determining the wavefronts in nonseparable quarter-plane problems, we draw on the pressure shock problem depicted in Fig. 39. In-plane coordinates are x and y with u and u corresponding displacement components, respectively. Note the nonmixed stress-type edge conditions. The normal stress ax= a o f ( t )is suddenly applied to t = 0 to edge x = 0 under zero shear stress. Aside from being a vehicle for demonstrating the solution technique, the problem is a longstanding basic one in elastodynamics. The present work determines all the wavefronts for the waves in the problem. Figure 39 depicts these wave events, interior and on the edges (surfaces). Note that there are two critical shear regions, 0 = 0,,, and 0 = Ocry. They are associated with the x and y axes, respectively.

FIG. 39. The elastodynamic quarter-plane problem, its wavefronts, and critical shear regions, 0 5 0 5 ecryand e,, 5 0 z ~ 1 2 .

These problems, like the waveguide ones with their nonmixed edge conditions, cannot be solved by direct separation techniques. The technique here employs the plane-strain displacement equations of motion, transforming out time t with the one-sided Laplace transform (parameter p ) in the usual way. There remains the need for a procedure to reduce the coordinate x, leaving coordinate y and ordinary differential equations in this variable. However, no transforms on x exist that “ask for” the nonmixed edge displacements or stresses, and therein lies the problem. The present method proceeds by employing a one-sided Laplace transform on x (parameter s) that gives rise to time-transformed edge unknowns (on x = 0) for the displacements and their gradients, [e.g., U ( 0 , y, p ) , U J O , y , p ) ] . Then employing the traction-free conditions on the edge y = 0, one generates a

Modern Corner, Edge, and Crack Problems

135

+

generalized form of the Rayleigh function R ( s , p ) = (k: - 2 ~ ’ ) 4s2a/3, ~ where (Y = i ( s 2- k;)’” and p = i ( s 2 - kf)’’2, with p and s complex and kd = p/cd, k, = p/c,, cd and c, being the dilatational and equivoluminal body wave speeds. Here R ( s , p ) has the real roots s, = *kRJ = * p / c R , , j = 1 , 2 , 3 . The cRJare the three Rayleigh surface wave speeds. Speed cRJis that of the well-known Rayleigh wave that propagates along a traction-free elastic surface. The other two roots have also been studied at length, but in boundary initial value problems for the homogeneous elastic solid, they usually do not arise and indeed are inadmissible in the solution. They are commonly referred to as nonphysical since they can correspond to events propagating with speeds greater than the dilatational wave speed. It is these nonphysical roots that lead to residues that must be ruled out of the quarter-plane solutions. These residues generate four integral equations for the time-transformed edge unknowns. A special contour integration over Riemann surfaces, comprised of the s plane and adjacent sheets, is needed to generate these inadmissible Rayleigh roots. It follows that the quasi-formal solution for a problem has the form of a double inversion integral over the Bromwich paths Br, and Br, in the complex p and s planes, respectively (quasi since it contains the timetransformed edge unknowns). However, only inversion in the s plane is needed, since it turns out that in these problems inversion can be carried out with the Cagniard-deHoop method. For the dilatational wavefront and Rayleigh surface wave events the forms of the time-transformed unknowns on the loaded edge x = 0 are known (Miklowitz, 1978; Rosenfeld and Miklowitz, 1962), for example, ud(0, y, p ) = A ( p ) e xp(-kdy)/h, uR(o,

y, p )

=

c ( p ) exp(-kRIY),

Cd(0, y, p ) = B(p) exp(-kdy)/fi, fiR(O, y, p ) = D ( p ) exp(-kRlY).

This leaves only the four p-dependent coefficients [e.g., A( p ) ] to be determined. Substitution of these forms into the four integral equations and performing simple integrations reduce these integral equations to four algebraic equations for the coefficients A ( p ) , B ( p ) , C(p), and D ( p ) . Solution of these establishes the time-transformed knowns on the loaded edge x = 0. The latter are then substituted into the quasi-formal solution, which then becomes the formal solution, valid for high frequency (i-e., p large). The wavefronts are then deduced by applying the technique in Rosenfeld and Miklowitz, which makes use of the Cagniard-deHoop inversion of the Br, integral and the p-large asymptotics of the Laplace transform.

B. METHODOF SOLUTION FOR THE NONSEPARABLE Q A R T E R - P L A NE PROBLEM In this section the present method for solving nonseparable quarter-plane problems is developed.

Julius Miklowitz

136

1. General Quasi-Formal Solutions

First the governing equations for the general elastodynamic quarter-plane problem for plane strain will be set down. The displacement equations of motion are u,(x, Y, t ) + ( k 2 - l ) k - 2 ~ x+y K2uYY= c i 2 U ,

v,,(x, Y , t ) + (k’ - 1 ) ~+~k2UYy ~ ” = ci2V

(4.1)

for x > 0, y > 0, and time t > 0, where c i = ( A + 2 p ) / p and c: = p / p are, respectively, the squares of the dilatational and equivoluminal body wave speeds, A and p are Lame’s constants, p is the material density, and k2 = ci/ ct. Corresponding stress-strain relations are c x ( x , Y, t ) / ( A + 2 P ) = u,(x, Y , f ) + ( k 2 - 2)k-221y(x,Y , t ) , (4.2) my(% Y , t ) l ( A + 2 P ) = (k’ - 2)k-2u,(x, Y , t ) + v,(x, Y , t ) , and a, = 4 a x + u y ) , u x y ( x Y, , t ) I p = vx(x, Y , t ) + U , ( X , Y , t ) ,

where v is Poisson’s ratio. The subscripts in (4.1) and (4.2), and generally in this work, when associated with displacement, indicate partial diff erentiation, but when associated with stress, identify the component. Time t partial differentiation is indicated by an overdot. Initial conditions are taken as u ( x , y , 0 ) = U ( x , y , O ) = v ( x , y , O ) = rj(x,y,O) = 0

(4.3)

for x 2 0, y 2 0, and radiation conditions as lim [u, u,,

r-m

21,

u,, etc.]

=

0

(4.4)

for t 2 0, where r = (x’ + Y ~ ) ” Uniform ~. loadings along the edge and the finite amplitude plane waves they generate will, of course, not be required to satisfy (4.4). We now apply one-sided Laplace transforms on t, parameter p , and x, parameter s, to (4.1) with the result GYY(%Y , P) + ( k 2- 1bCY(.%Y , P) + ( k 2 s 2- k 3 J ( s , Y, P)

k 2 b f i ( 0 , Y ,P) + &(O, Y , P)1 + ( k 2 - 1 ) q o , Y , P ) = f(s, Y , P), ~ , y ( s , Y, P) + ( k 2 - W 2 s G y ( s , Y , P) + (s2- k W 2 u ’ ( s ,Y, P) = k - 2 [ s c ( 0 ,Y , P) + &(O, Y , P I ] + ( k 2- l)k-2iS,(0, Y , P ) =

=

(4.5)

g ( s , Y , P),

where a bar and a tilde over quantities indicate Laplace transforms on t and x, respectively.

Modern Corner, Edge, and Crack Problems

137

The general quasi-formal solutions based on (4.1), (4.3), and (4.5) are

where

and where

The terms in (4.6b) associated with A,, and A,, represent the complementary functions of d and 6,and the p * , and p*.pterms represent the particular integrals. The solutions (4.6a) are called quasi-formal solutions since they contain time-transformed edge ( x = 0) unknowns [displacements and displacement gradients through the f and g functions in (4.6d)l. These unknowns d o become known quantities for particular problems later in the method, which then makes (4.6a) a formal solution. Note that Br, and Br, are Bromwich contours in the p and s planes, respectively. Associated with (4.6a) through (4.6d) are the radiation conditions on y, according to (4.4), which can be met with suitable values involving the (Y and /3 functions in ( 4 . 6 ~ ) With . interest in an s-plane inversion, we fix p and require that it be real and greater than zero. Then, as shown in Fig. 40, taking cuts to the left from branch points s = fk,, fk d , we can represent analytic branches of a (s) and p (s) by

138

Julius Miklowitz

F I G . 40. Defining complex vectors for s, s i ( s 2 k;)”’, p = i ( s 2 -

e)’/’,

+

-

k d , s + k , , s - k,, s

+ k,,

and a

=

and where ~3

=

IS - kI,

p4 =

IS + ksl,

Hence (see Fig. 40), we have

on BrsL, the lower half of Br,, and

(4.10)

on Br,”, the upper half of Br,. It follows then from (4.9) that the multivalued functions a ( s ) and p ( s ) satisfy

on Br,, (and Brp), and hence, according to (4.6b), will satisfy (4.4) for y + co, provided that

Modern Corner, Edge, and Crack Problems 43,s)

= -P%s)(s,

- --1

2k:

139

P) [ P ( s ) f ( s , y ' , p ) - k 2 s g ( s ,y', p ) ] e - P ' s ' y dy'. ' (4.11b)

Substituting (4.11) into (4.6b) reduces the latter to d(s, Y , P) = [ A ( %P) - P,($,(S, Y , P ) l e - o l ( s ) y

+ [P-&% Y , P) - P : n ( s ) ( S , P ) l e a ( s ) y PWY, + "s, P) - P p ( ~ ) l e - p (+ s )[P--p(s) y - P:,(r)le 4s)

G(s, Y , P) = - s { " s ,

(4.12)

P) - P , ( s ) l e - m ( s ) y

- [P-,.Cs, - P.oou(s~leais)yl

+ -"(s, S

P(s)

P) - P.p(s)le-P(s)y - [P-P(s)

- P%le

PWY},

where we have used A ( s , p ) for A,,,) and B ( s , p ) for A,(,) and have, for simplicity, dropped the arguments for the p ' s after the first line. Similarly, from (4.10), the multiplied functions a ( s ) and P ( s ) satisfy

on Brsu (and Brp), and hence, according to (4.6b), will satisfy (4.4) for y + a,provided that m

A,,,) = P a ( s )

Julius Miklowitz

140

where we have used A(s, p ) for A - a ( , )and B ( s , p ) for A - o ( s )and again have, for simplicity, dropped the arguments for the p's after the first line. It is important to point out now that the branch functions a ( s )and P ( s ) defined in (4.7) and (4.8), respectively, are analytic all along Br,, having

on BrsL from (4.9) and

on Brsu from (4.10). Equations (4.12) for fi and 6 are bounded as y + co, the critical terms being those involving exp[a(s)y] and exp[P(s)y], where

and the coefficients of these exponential functions vanish as y += CO. Similarly, (4.14) for u" and v" are bounded as y += co,since the critical terms are those involving exp[ - a (s)y] and exp[ - P ( s)y], where

and the coefficients of these exponential functions vanish as y + 00. It is of of (4.9) and (4.10) that according further interest to note from the results to the principle of reflection a ( s ) = - a ( s ) and P ( s ) = - P ( s ) on Brsu, so that (4.14) forms a natural conjugate of (4.12).This leads to a real formal solution for the displacement over BrsL, as well as other conjugations in the s plane. 2. The Pressure Shock Problem

The pressure shock problem, shown in Fig. 39, is now introduced. The double-transformed boundary conditions for this problem, obtained from (4.2), are for y = 0

eY(s,O , p ) / ( A + 2 p ) = (k2 - 2)k-*[su"(~, 0, p ) - U(O,O,p)] + z?~(s,0, p ) = 0, GYX(S,

and for x

0,P ) / P =

u"y(s,

0, p ) + SC(S, 0 , P) -

0, P) = 0

(4.15)

=0

ax(O,.Y,P)I(A + 2 ~ =)%(O, @xY(o> Y , P ) / P = fix((), where we have takenf(t)

=

Y , P) -I-(k2 - 2)k-2fiY(0, Y , P) = g o / ( A + 2~u)P,

Y , P) + qA0, Y , P) = 0 ,

(4.16)

H ( t ) the Heaviside step. Now from (4.12) we

Modern Corner, Edge, and Crack Problems

141

derive ii,,(s,y, p) and i y ( s ,y, p ) , and substituting these and (4.12)into (4.15), we find A(s, p) and B(s, p) to be

where

+

q ( s , p ) = ( k : - 2s2)pYU- 2s2p?p ( k 2- 2)sU(O, O , p ) , r ( s , p ) = --[2aPpmU+ ( k : - 2s2)pTP+ PU(O,O, p ) ]

and

R ( s , p ) = ( k t - 2 ~ + 4s2aP ~ ) ~ is the generalized Rayleigh function. Further, using (4.16) in f ( s , y, p) and g ( s , y, p), the latter reduce to

f(s, Y , P) = k2sG(0,Y , P) + q o , y, P) + ~ O I P P , As,Y, P > = k-2[sfi(0,Y, P) + ( k 2 - 2)Oy(O,Y , P I ] .

(4.18)

Substituting (4.17) and (4.18) into (4.12) defines the doubly transformed displacement solutions G(s, y, p) and i( s, y, p), from BrsL. Similarly, from (4.14) we have -

(4.19)

where

Julius Miklowitz

142

where it should be noted, in comparing (4.19) and (4.20) with (4.12) and (4.17), that important changes from a ( s ) and p ( s ) to (yo and p(s)have resulted in the exponential functions and the subscripts of all the p functions, the latter functions having a sign change as well. Equations (4.19) and (4.20) together with (4.12), (4.17),and (4.18) lead to the formal inversion through (4.6a) as discussed earlier. 3. Indicated Contour Integration for Solution As pointed out in the introduction, one needs a somewhat special contour integration in the s plane to solve the present class of quarter-plane problems, one in which the so-called nonphysical roots and associated poles and residues of the generalized Rayleigh function are generated and, hence, can be exploited in the form of conditions on the solution ruling out unbounded and inadmissible events. A general contour that can d o this is depicted in Fig. 41. It is composed of a nest of component contours, based on four branch cuts taken to the left of the branch points, s = * k d and k k , . It may be seen that the cuts, and contours along them, are in a rotated (by a small counterclockwise angle 6 ) position below the real axis. Indeed the separation of the cuts and related contours in this nest was convenient in keeping track of the vectors s T kd , s T k, as the contours were negotiated. More important, however, was the fact that this nest of contours did its job of generating the nonphysical zeros (and corresponding poles) of the generalized Rayleigh function. As the large arrow in the clockwise direction across the cuts and contours in Fig. 41 indicates, these cuts and contours are limiting ones that, as 6 + 0, move to superposed positions along the real axis.

FIG. 41.

Limiting nest of contours as S + 0.

Modern Corner, Edge, and Crack Problems

143

Certain other features of the cuts and contours in Fig. 41 should be pointed out. It turns out that oppositely directed lineal path contributions along inside contours cancel, for example, the whole of parts of the paths L s , L6, and L, and their mates I , , 12, and I,, respectively. To show that this is the case we proceed as follows. Consider first integration along the first contour (in Fig. 41) L , , L 2 , L ? , L,, and 1,. It is straightforward with the four vectors s k k , , s + k d in expected positions. Now, however, s - kd leaves path 1, to begin its excursion along paths 13, 12, and I , . Note though, in the final position of these contours (i.e., when 6 + 0), they will be just an E above the real axis. This position for lj, 12, and I , has taken s - k, across its cut to the beginning of the next sheet,* arg( s - k d ) = -( 7~ + E ) , of the Riemann surface for the function ( s - kd)’/’.We see then that from the end of path I , , s - k, can rotate counterclockwise a distance 2e back to the first sheet and begin its negotiation of paths L 5 , L6, L7, 17, 16, and I,. These paths all lie in the first sheet. Such a process gives cancellation of the oppositely directed lineal path contributions from the respective pairs of the whole or parts of the paths L s , L6, L 7 , and I,, 12, 13. It should be noted from Fig. 41 that a similar process involves the complex vectors s + k d , s k , , their cuts and related contours. The complex vector s - k, does not leave the first sheet of ( s - ks)ll2. As noted earlier, this integration generates the nonphysical as well as the physical zeros of the generalized Rayleigh function. These zeros are s = s * , ( p ) = +kR, = + p / y,c,, -yJ being positive constantst with j = 1 ,2 ,3 . The nonphysical roots are s = s L J ( p )= * k R J ,j = 2 ,3 , and as Fig. 41 shows (assuming 6 + 0) lie along the real axis between -kd and k d . As the figure indicates, these zeros lie on the line arg(s - kd) = -T, common to both the first and second sheets of the Riemann surface. It follows that the poles associated with these zeros generate the half-circular paths indicated in Fig. 41 for the s 2 ( p ) = kR2 case, along I, and L7. We see that the half-circle along L7 lies on the ending of the first sheet and the half-circle along I, on the beginning of the second sheet. Hence, their sum is a complete continuous counterclockwise circular path [about zero s z ( p ) = k,,] lying on the Riemann surface. The other nonphysical Rayleigh function zeros generate similar circular paths in the contour integrations. On the basis of the foregoing considerations, the contour integration reduces to that shown in Fig. 42. The contributing contours are just L , , L,, L,, L,, and l,, l,, 19, I , , plus six circular paths for the poles associated with the Rayleigh function roots s = s , , ( p ) = f k , , j = 1 ,2 ,3 . The paths around s + , ( p ) = *kR,, j = 2 , 3 , are all free of the cuts and lineal paths intersecting them, which is indicated in Fig. 42 by complete circles. Circular paths are

+

*The s plane in Fig. 41 can be taken as the first sheet -T < arg(s - kd) < T . ‘The y,’s depend on Poisson’s ratio. For Poisson’s ratio = 1/4, y , = 0.92, y z = 1.78, and y, = 2. These are the y,’s we have chosen for our problem.

144

Julius Miklowitz

FIG. 42. Remaining contours after S

--*

0.

also shown for skl(p ) = *kRl. As Fig. 41 shows, there must be two such paths for -kR1 associated with the pairs of paths L, and l,,, and L, and 1,. The radiation condition (4.4)reduces the number of contours in Fig. 42 still further. Figure 42 and the analysis leading to it show that the right-half s plane linear path integrals, parts of L, and 1, for 0 < s < kd, and the whole of L4 and l4 for kd < s < k,, all represent exponentially unbounded contributions through the kernel of the Br, integral exp(sx), for x > 0 and Re p > 0. The radiation condition (4.4)therefore rules out these contributions. They can also be identified with waves traveling in the negative x direction by noting the double-inversion integral kernel exp{ p [ t + ( s / p ) x ] } would then be exp{p[ t - ( x / - c ) ] } , since s has the dimensions of p / c , where c is the wave speed. Equation (4.4)would also rule out these waves. Further, consider the parts of L3 and I, integrals in the left-half s plane, where -kd < s < 0, also having the double-inversion integral kernel exp{p [ t + ( s / p ) x ] } . The s interval here may be written as -kd < s = p / c < 0,which corresponds to cd < c < CO; hence exp{p[t + (s/p)x]} = exp{p[t - ( x / c ) ] } for these integrals. These would be waves propagating in the positive x direction, but they have speed c > c d and hence are inadmissible in a linear elastic solid. We are thus left with the contours depicted in Fig. 43. It is important to point out that this analysis, ruling out unbounded contributions

FIG.43. Remaining contours after imposing solution boundedness and wave speed restriction ( c 5 cd) to path integrals L , , L,, I,, and I,.

Modern Corner, Edge, and Crack Problems

145

and inadmissible events, is related to the procedure usually used in separable elastodynamic wave problems. It is, of course, much simpler in the latter, usually done. early in the analysis by setting the coefficients of the two unbounded kernels, like e n y and e P v here, equal to zero. The related procedure here is more complicated and has come along later in the analysis. Consistently, we now require that the contributions (residues) from the nonphysical poles, generated by the zeros of the generalized Rayleigh function S * ~ ( P ) = * k R j , j = 2,3, and s l ( p ) = k R 1 ,must be ruled out of the solution ( k R j= p / c R j ,where cRj is the wave speed). Of these, the residues at the poles s j ( p ) = k,, j = 1,2,3, are ruled out by the boundedness condition on the solution. The residues at xi(p ) = - k R j , j = 2,3, are ruled out because the corresponding speed of these Rayleigh wave events are greater than cd. Hence, as the next section points out, the residues for these five poles are set equal to zero. This eliminates the circular paths in Fig. 43 for these contributions, generating four coupled integral equations that guarantee boundedness for the solution and the elimination of inadmissible events. Solution of these integral equations determines the edge unknowns in our problem, at least for wavefront (high-frequency) events. 4. Conditions for Solution Boundedness: Inadmissible Events Derivation of the four integral equations for the edge (x = 0) unknowns begins with first establishing that s+,(p) = *k,, j = 2,3, and s l ( p ) = k R I are zeros of the generalized Rayleigh function. Consider for example, s 2 ( p )= kR2= p / y 2 c , , which lies along 13, L, in Fig. 41. Using the limiting values of s, s + k d , s k k,, a, and p on these paths, with Poisson’s ratio 1/4, hence y 2 = 1.78, one finds by substituting these values in cfR(s, p ) / p ‘ that the latter will vanish since

R ( s ,P)I.r=kRZ

[ ( k : - 2 s 2 ) 2 f 4s2a(s)P(s)1s=kR2 = [k: - 2 ~- ~~ S ~) ~ ~( S ) P ( S ) ] ~ = ~ ~ ~

=

= =

[ ( k : - 2s2)’ - 4s2(plp2”’2(pjpq)’’21~=kR, ( k z - 2ki2)’ - 4k:,(kf, - k2R2)’I2(k:- kZR2)1’2

(4.21)

vanishes. Note that p l , p 2 , p 3 , and p4, defined generally in (4.7) and (4.8), are at S = k ~ 2kd, - kR,, kd -I- k ~ 2k,, - kR,, and k, i- k ~ 2respectively. , Note also that a ( s ) = ( p l p 2 ) 1 ’ 2and p ( s ) = ( ~ ~ p ~= )-p(s) ’ ’ ~ in (4.21) are continuations from Br,, and Brsu, respectively, and that these forms hold for s = - k R 2 , + k R 3 as Well. Similarly one can establish that s = k R 1and the admissible s = - k R l are zeros. The residues at s = kR, , k,, j = 2,3, can be shown to be nonvanishing through the usual calculations.* These residues can then be set equal to

*

* These calculations are like those in the later section dealing with the physical Rayleigh waves on the free edge y = 0.

146

Julius Miklowitz

zero by requiring that A N ( S , P)ls=kR,,+kR,

=

BN(% P ) I s = k R t , * k R J

= 0,

j = 2, 3,

(4.22)

where A , ( s , p ) , B N ( s , p ) are given in (4.17). Now with s + ~ ( P=) *kRJ, j = 2,3, and sl(p) = k,, representing the five Rayleigh poles here, the two equations (4.22) are expanded using (4.17) and (4.6d). After some algebra and a simple integration on y for the coefficient of the input term a o / p p , the general integral equation is found to be

lom {:[(k:

X

- 2s:) exp(-a]y)

- 2 a j ~ exP(-P,y)l j

[k2sJU(0,y, p ) + fiy(o,Y, p ) ]

[ ( k t - 2s;) exP(-a,Y) + 2s; exp(-PJy)l

x [sJa(O,y, p ) + (k2 - W y ( 0 ,Y,PI]}

dy

+(k2-2)k:U(0,0,p)+ (4.23)

where sj = s j ( p ) ,j = 1, k2, *3. The s, are basic to the definitions of the aj and Pj. Consider a l and PI first. From (4.7) and (4.8) we have, when s1 = kR1 (see Fig. 41), a1= i(plp2)’/2= i ( k i l - k i ) 1 / 2 and PI = i(p3p,)’/2= i(k;, - k:)’l2. Likewise, when s , ~= f k R 2 , s + = ~ *kR3 along paths l3 and L7, (4.7) and (4.8) show that = ( p l p 2 ) I / ’ = (ki - k;2)1/2 and P+’ = -(~~p*)= l’~ -p+2 = -(k: - ki2)”’, with a,3 and P*3 having the same forms as ak2and P*2 except that kR3replaces kR2.Equation (4.23) represents five coupled integral equations for the two time-transformed edge displacement unknowns U(0,y, p ) , fi(0,y, p ) and their corner values U(0,0, p ) , f i ( O , O , PI. Now substituting the j = 1 terms, sl, a l , and PI in the foregoing into (4.23), one finds that it splits into two equations, the real and imaginary parts created by the imaginary nature of a I and P I .These two real integral equations provide the definition of the time-transformed corner displacements U ( O , O , p ) and fi(O,O, p ) , given by U(O,O,P)

-

P ( P ) K ( P )]=($:))

loffi

(~k~,(~’~~~;b‘[~~~Pnll,‘]

147

Modern Corner, Edge, and Crack Problems where K ( p ) = (2 and

- 2k2R1)/lQ1(21}00kR1/~p,

Q(p)

=

-~~RIIQI(/~;(~;

p(p)

= ( k 2 - 2)-1k,2

- 2ki1).

Now, if these integral expressions for U ( 0 , 0, p ) and V(0, 0, p ) are substituted into (4.23), then there are just four integral equations involving only the time-transformed edge unknowns U(0,y, p ) , 8(0, y, p ) , and related derivatives a,,(O, y , p ) and V,,(O, y , p ) . These four remaining equations are paraPh2, a * 3 ,and P k 3 . metric in s*2 and s * ~ ,respectively, with related This solution provides the information required to reduce the quasi-formal solution to the formal solution, which can then be inverted by known exact and approximate solution techniques. The next section shows how the four integral equations can be used to find these time-transformed edge unknowns for high-frequency events (dilatational wavefronts and Rayleigh surface waves) on the loaded edge x = 0 of this problem.

5 . Determination of Time-Transformed Edge (x = 0 ) and Corner (x = y = 0) Unknowns for Wavefront ( High-Frequency) Events and Their Inverses To derive the time-transformed edge (x = 0) unknowns representing wavefront (high-frequency) events we will use as a guide similar general features of related events from the elastic half-plane and waveguide problems. Indeed, we can construct forms for the time-transformed edge unknowns that will reduce the four integral equations to algebraic equations for the time-transformed p-dependent coefficients of these unknowns. From the wavefront analysis work of Rosenfeld and Miklowitz (1962) on the plane strain semi-infinite plate, involving mixed pressure edge conditions, we assume that at the dilatational wavefront the time-transformed displacements U(0,y , p ) , V(0, y, p ) are, respectively, y, p ) fid(0, y, p )

21

A(p)y-1/2exp(-kdy), B(p)y-'/2 exp(-kdy),

(4.25)

from which we also have Uyd(0, y , p ) Cyd(0, y, p )

-kdA(p)y-1/2exp(-kdy), r_r

(4.26)

-kdB(p)y-'12 exp(-kdy).

It may be noted the y - ' l 2 in these expressions is in agreement with the classical form for the propagation of two-dimensional surfaces of discontinuity in elastodynamics. The exp( -kdy) through its shift operator nature guarantees the expected one-dimensional wave nature of these quantities, re.g.7 ud(o, y, p ) = ud(o, y , - Y/cd)l. Consider next the Rayleigh wave disturbance on the edge x = 0. Here as in the half-plane problem one would expect a nondecaying in space ( y )

148

Julius Miklowitz

one-dimensional disturbance. Hence, it is assumed that

from which one also has UyR(O,

y, p ) = - k R I c ( p )

fiy R(0 ,

y , p ) = -kRID(p)

exp(-kRly), exp(-kRly).

(4.28)

It follows, then, neglecting the equivoluminal disturbance on the edge (x = 0) for the moment, that the right-hand sides of the expressions (4.25)-

(4.28) may be substituted into the four integral equations obtained from (4.23) and (4.24) as discussed after (4.25). The simple integrations these terms provide reduce the four integral equations to four algebraic ones that determine the time-transformed coefficients A( p ) , B ( p ) , C ( p ) , and D( p ) through Cramer’s rule. They are given by

A ( p ) = Ap?/*,

B ( p ) = B P - ~ ’ ~ , C ( p ) = C P - ~ , D ( p ) = Dp-’, (4.29)

where A, B, C, and D are constants given in (4.54) of the Appendix to this section. Substituting the right-hand sides of (4.29) into (4.25) and (4.27) and then inverting the latter, one finds the dilatational displacement wavefronts on the loaded edge (x = 0) to be

and the Rayleigh displacement wavefronts there to be

Note that the corresponding wavefronts for the velocities zid, v d and behave as ( t - y/cd)l/* and the step H ( t - y / c R 1 ) ,respectively. Accordingly, the accelerations behave as ( t - y / ~ ~ ) - and ~ ” 8 ( t - ylc,,), both singular. Concerning the two-sided equivoluminal wavefronts on the edge, x = 0, the present method yields them in terms of the coefficients A, B, C, and 0, as we shall see later. In effect, this says that the dilatational and Rayleigh wavefronts (on x = 0) generate these equivoluminal wavefronts on the edge x = 0, as well as the other wavefronts in the problem. Interesting is the fact that the two-sided equivoluminal wavefronts on edge x = 0 have the same time behavior as the dilatational wavefronts there, which is also true in Lamb’s plane strain problem (see Miklowitz, 1978). To determine the corner displacements u ( 0 , 0, t ) and u(0, 0, t ) , the righthand sides of (4.25) through (4.28) are substituted into (4.24), resulting in UR

Modern Corner, Edge, and Crack Problems

149

real integrals of the types

I, lo*

{sin m y ] e z dy cos my

=

[

2(

+ m’)”’

a2

+ m’)

7T

{

{sin cos m myy ] e - a y d y = : ] / ( a ’

] ’”[ a +

- a]112]

m2)1’2+a ] ’ / 2 ’

(4.32)

+ m’),

where a and m are greater than zero. The right-hand sides of (4.32) therefore reduce to algebraic equations for ii(0, 0, p ) and a(O,O, p ) involving A( p ) , B ( p ) , C ( p ) , and D ( p ) , which, using the right-hand sides of (4.29), yield

U(O,O,p ) =

V(0,0, p ) = Gp-*,

l y 2 ,

(4.33)

where u^ and 6 are given in (4.55) of the Appendix for this section. It follows that the corner displacements are

a,

u(O,O, t ) = Gt, v(O,O, t ) = (4.34) with corresponding velocities and accelerations being time steps and delta function singularities, respectively. 6. The Formal Solution for the Time-Transformed Displacements G(X, Y , P), a(x, Y, P ) With the time-transformed edge unknowns in (4.25) through (4.28) now determined by (4.29) and (4.54), and similarly the time-transformed corner displacements by (4.33) and (4.55), the quasi-formal solutions for the displacements u ( x , y, t ) , v ( x , y, t ) [(4.6a)-(4.6d)] become the formal solutions for high-frequency (wavefront) events for these displacements. It follows, using (4.7)-(4.10), (4.12), (4.17), (4.19), (4.20), and (4.33), that these formal solutions can be written as

I

+

- p.oCp]ePY}ds

+

- poO,]edy)

(4.35a)

150

Julius Miklowitz

where R(S, p ) u d R ( s , p ) = [(k: - 2s’)’

-

4 ~ ’ c ~ @ ] p-?4s2(k: ~ - 2S’)p:p

+ S [ ( k ’ - 2)(k%- 2s’)fi - 2~@V*]p-’, R ( s , p )UsR(s,p ) = -4a@(k: - 2s2)~COa- [(k: - 2 ~ ~-)4’~ ~ a @ ] p ? @ -@[2(k2 - 2)saU + (k: - 2s2)V*]p-*, (4.35b) R(f,p)ud,(f,p) = -[(k: - 2s’)’- 4.?Gp]pYs + 4S2(k: - 2S2)pYp +s[(k’-2)(k:-22S’)fi -2spiqp-2, R ( S , p ) U , R ( T , p p ) =4G@(ks-2Sz)p?e + [(k:-2S2)’ -4f2Gp]p?p - p[2(k2 - 2)sLyfi + (k: - 2S2)8]p-’, and where the arguments ( s ) of a, (Y, p, and 6 have been deleted for simplicity and R ( s , p ) , R(f,p) and the p functions are given in (4.17), (4.12), and (4.6d), respectively. The subscripts d and s indicate essentially terms associated with dilatational and equivoluminal motions, respectively, and subscript R indicates association with the Rayleigh functions R ( s , p ) and R ( f , p ) . With the aid of the principle of reflection in complex variables, (4.35) can be reduced to real equations for U and V through complex conjugation. Consider, for example, the conjugation of the first terms on the right-hand sides of u d R ( s , p ) and U d R ( f , p ) in (4.35b), respectively, on BrsL and Brsu in (4.35a). Analysis of these terms at the intersection point of Brsu and the real s axis, s = y, shows that

where *A( y, p ) are complex numbers, with A, and A2 being real. It follows from (4.36) by analytical continuation away from s = y that A,(s, P) + iAz(s, P)

3

(4.37)

along BrsL and Brsu, respectively. Equations (4.36) show that both cases of the principle of reflection are involved since both a real and an imaginary number make up A(y,p). According to (4.37), the first case has the form (4.38) where A,(s, p ) ds is imaginary at s = y. This is because A,( y, p ) is real and ds = - i ds, is imaginary over all of BrsL, hence at s = y, since s = y - is, there, s, being the imaginary part of s. It follows from the principle of

Modern Corner, Edge, and Crack Problems reflection that A , ( f ,p ) d f

=

151

-A,(s, p ) ds and therefore (4.38) becomes

(4.39) According to (4.37) the second case has the form 1

iA,(s, p ) ds

+

iA,(f, p ) d f ] ,

(4.40)

Brsu

where iA2(s,p ) ds is real at s = y, since A2(7, p ) and now ids are real. It follows from the principle of reflection that iA,(f,p ) dS = iA2(s,p ) ds and hence (4.40) become

‘I

[ i A , ( s , p ) ds - iA,(s, p ) d s ] = 7T

IBr,,

Im[iA,(s, p ) d s ] . (4.41)

Br,l.

Summing the right-hand sides of (4.39) and (4.41), we have, using the first of (4.37), Im{[A,(s,p)

+ i A , ( s , p ) ] ds} = -1

Im[A(s, p ) d s ] . (4.42)

‘ l r [Br.,

Inspection of all the other pairs of terms in (4.35b) and those in (4.35a) (the p functions) also conjugate to give (4.42), including the slightly different pairs of terms in (4.35b) involving u^ and 6.In the case of the latter pair of terms, (4.42) results through a direct conjugation of the pairs without appeal to the principle of reflection. I t follows, therefore, that use of (4.42) reduces (4.35a) to (4.43) where A, and A, are the integrands of the first integral on the right-hand side of (4.35a) for U and 6, respectively. 7. Inversion of the Time-Transformed Displacements U(x, y , p ) , V(x, y, p ) by the Cagniard-deHoop Technique The formal solutions (4.43) for U ( x ,y , p ) , V(x,y , p ) can be inverted by the Cagniard-deHoop technique (see Miklowitz, 1978), since all of the terms in A,(s, p ) and A , ( s , p ) in (4.43) exhibit exponential terms in which the transform parameter p occurs homogeneously linear. The integrands of the first integral on the right-hand side of (4.35b) will show this, and hence (4.43) will, once the Cagniard-deHoop transformation is invoked.

152

Julius Miklowitz

The Cagniard-deHoop method of inverting (4.43) begins by introducing the real variable 6 = s/ kd into the integrals in (4.43); that is, with =

kdl,

a = ikd(12- 1)’l2,

these integrals become

where

ds = kd dl,

p

=

ikd(12- k2)’12,

(4.44)

Modern Corner, Edge, and Crack Problems

153

Inversion of the integrals in (4.45) by the Cagniard-deHoop method proceeds by introducing transformations of the type g ( { ) = t, where t is real and positive. These transformations deform the paths of integrations off the path BrsL. Figure 44, which stems from Fig. 43, depicts the { plane with its singularities, branch cuts, contours, and complex vectors. Branch points are at { = *1 and { = *k, with the four related branch cuts being on the real axis from k, 1, -1, and - k to -a, respectively. The Rayleigh and -{il generate two simple poles there. This was function zeros at pointed out earlier, in the discussion of Fig. 42. These zeros correspond to the -kR1 pairs shown in Fig. 41, which are associated with paths L1 and l,o and L8 and 1 5 , respectively The superscripts 1 and 8 on -LR1 in Fig. 44 indicate the two contributing paths L , and L8 in the present conjugated

FIG.44. contours.

C-plane singularities, complex vectors, and related Cagniard-deHoop integration

Julius Miklowitz

154

form of solutions (4.45). Later we will show these two contributions are equal and that they add to give the Rayleigh surface wavefronts. Now from (4.7) and (4.8), we have (y'((Y) = j ( i - 2 - 1 ) W = ( r I )1/2el(*+../2) 1 2 cc, = (4,+ 4*)/2, (4.46) p ' ( 5 ) = i((' - k 2 ) l I 2= (r3r4)1/2e('?r+-rr/2)77 = (43+ 44)/2, 3

3

where the r, = pl/kd ( i = 1 , 2 , 3 , 4 ) are defined in (4.7) and (4.8). From (4.46) it is not difficult to show that (4.47)

in the third quadrant of the S plane (see Fig. 44) and that Re 5 and Im 5 are 5 0 there. These facts establish the convergence of the integrals associated with exp[ -pgd( 5)] and exp[ -pgs( l ) ] in (4.45), written for paths in the third quadrant. First, for example, we have to show that the integrals for U(x, y , p ) , involving these exponentials over the paths C R - and CR+in Fig. 44 vanish [V(x, y, p ) is only trivially different]. We have from (4.45) rcdp2u(x, y , p ) l C R -

=

1

$2

Im{LFd(l) exp[-pgd(l)l

Cn

+ Fd5) exP[-Pgs(5)lldS1,

(4.48)

where Fd(5) = GdR(s) - A ( s ) , F,(O = cvR(c)- C L I P ( ( ) , and R = 151. The contribution of the integral in (4.48) is assessed by employing Jordan inequalities in the usual way. Hence from (4.48)

+ F,( R exp( @)]i exp{ i[ kdR(x sin 4 x exp[kdR(x cos 4 + y sin 4)] .

1

The last integral in (4.49) reduces to

-

y cos 4 ) + +}}I (4.49)

Modern Corner, Edge, and Crack Problems

155

where K and K (both real) define the order of, and constant associated with, the component terms of Fd(Re'') + F,(Re'"), respectively. The order index K satisfies the requirement K > 0 since it takes on the values $, 1, i, and 2. It follows from (4.50),therefore, that the integral in (4.48)also vanishes. Note that for x > 0, y > 0 (4.50)holds without restriction. For x > 0, y = 0 or y > 0, x = 0 (4.50)holds; however, it reduces to the usual one-dimensional form in each case. Finally, when x = y = 0 (the vertex of the quarter-plane), (4.50) becomes indeterminate and does not hold. However, U ( O , O , p ) and V ( O , O , p ) are given by (4.33),and u^ and 6 there are defined by (4.55)and (4.54)of the Appendix for this section. Similarly, we can show the counterpart integral of (4.48)over the path CR+ also vanishes. Except for a difference in integration limits [here, between -cos-'( y / R ) and -7~121, we again have the equation (4.49)but now for the path C R + .Through the transformation q = cos 4, the last integral in (4.49)for the present case becomes

and since q << 1 when R

+ 00,

this integral reduces to the approximation

(4.51) and it follows from (4.51)that the integral in (4.48)for the C,, path also vanishes. The limit (4.51)holds for all cases (i.e., x > 0, y > 0 ; x > 0, y = 0 ; y>o,x=o; x=y=O). The contributions of the integrals in (4.45)associated with exp[ phd(l)] and exp[ph,(l)] and over the paths C R - and CR+also vanish. The analysis involving Eqs. (4.48)-(4.51)can be used to show this. First we note that F d ( 6 ) exp[phd(l)l + F s ( l ) exp[phs(l)l in (4.48), where Fd(6) = &-a(lJ- L?m(l)and F , ( l ) = & ( f ) - & Y P ( l ) are defined through (4.49, reduces to 1 l{cd[(k212/(Y')- ( k 2- 2)cdcR:Ic - l[(Cd/cRIa') - 1ID) exp(-kRIY) 2k2 (Y' + cdcii Cd{l[k2P'+ (k2 - 2)cdCR:Ic - [cdc,:P'+ 5'30) eXp(-kRly) p' cdc,:

--{ +

+

Julius Miklowitz

156

It follows, using this expression for the absolute value of the integral over C R - , that (4.50) governs, giving the result

[

.IrK

lim

R-w

exp(-kR 1y ) (1 - exp(- k,xR)) 2kdxR"

1

-+ 0,

(4.52)

where K = 1 and K contains the multiplier C. Because of the result (4.52), the integral over C,- also vanishes. It also follows for the absolute value of the integral over CR+that (4.51) holds in the present case, giving the result (4.53) where K and K are as stated in (4.52). The result (4.53) proves that the integral over C,, also vanishes. Appendix

Modern Corner, Edge, and Crack Problems

with

157

158

Julius Miklowitz

Definitions of the constants u^ and ments are given by (4.33),

6 in the p-dependent corner displace-

6 = c%(k2- 2 ) - ’ [ - f I + a,A + b,B + c,C + d , D ] , v* = - 2 c ~ a ~ ( C ~ ~ S ~ ~ ) -+l [b,B a , A-k C , c -k d , D ] , a,, where a,, a,, b,, b,, c,, c,, d,, d,, A, B, C, 0, given in (4.54) and c, and k are defined after (4.1).

(4.55)

c R ~ , and s R 1

are

C. WAVEFRONTEVENTS FOR THE DISPLACEMENTS, VELOCITIES, AND ACCELERATIONS I N T H E QUARTER-PLANE INTERIOR 1. The Dilatational Wavefronts Let gd(5)

=

t(real), which from (4.45) gives f =

gd(5) = (1/ cd)[ i ( l2- 1)’”y - 5x1.

(4.56)

Transformation (4.56) deforms path Br,, into the equivalent path - C ( C for Cagniard-deHoop), shown in Fig. 44, which is defined by solving (4.56) for 5 = h ( t ) , where 5 d ( t ) = -[(cdt/r) sin

e + i((cdt/r)’ - I}”* cos el, r/cd 5 t < 00,

0 5 0 5 7r/2

(4.57a)

in which x = r sin 8, y = r cos 8, and r = (x2 + Y’)’’~. It is easy to show that (4.57a), representing path C, is one-half of the branch of the hyperbola (4.57b) in the third quadrant of the 5 plane. The earliest time in (4.57a) is f = r/cd (the wavefront arrival time), which when substituted into (4.57a) gives sin 6, the point of intersection of the beginning of path c and 5d the negative 5 axis, as Fig. 44 indicates. Equation (4.57a) shows that as time grows from t = r/ cd , 5 d moves on C outward into the third quadrant, approaching for large time the asymptote defined by lim(Re ld/Im gd) = tan 8. The dilatational wavefronts are extracted from the first integrals in (4.45) on path -C,

Modern Corner, Edge, and Crack Problems

Now substituting gd(l) = t from (4.56) and transforms the latter into

ld(t)

159

from (4.57a) into (4.58)

where we have used d l d ( t ) / d t = l/gb[fd(t)], the prime on gd indicating first derivative, and we have dropped the argument t of ld. By inspection of (4.59) we note that it yields the dilatational acceleration components

(4.60) H being the Heaviside step function. To approximate the dilatational acceleration components in (4.60) near their fronts, we note that there r = gd( l d ) = r/ cd + E, where 0 5 E << 1. The point l d = -lo= -sin 0 satisfies gh(-fo) = 0, and hence l d = -lo,corresponding to minimum time r/cd is a saddle point of the relief of exp[-pgd(ld)] in (4.58), where now p is large. The method of steepest descents could be employed at this point, but the technique in Rosenfeld and Miklowitz (1962) is much simpler. The latter begins with a Taylor series expansion of g d ( l d ) at (d = -loalong the path C with the result =

gd(ld)

=

gd(-lCI) + [gi(-l0)/21(ld

=

r/Cd

+ [g:(-10)/21(ld +

+ +*

+. . * *

‘

’

(4.61)

(4.63)

Julius Miklowitz

160

where

t?&(-lO)

A(-lo)

=

{ [ k 2- 2 sin2 0)’ - 4 sin’ 0 cos 0(sin2 e - k 2 ) ” ’ ] f i ~ ~ , ( - [ o ) + 4 sin2 0( k’ - 2 sin’ 0 ) i T O - cd sin 0 [ ( k 2- 2 ) ( k Z- 2 sin’ 0 ) x u^ + 2 sin 0 ( k 2 - sin2 e ) ’ l % ] } / ~ ( - [ ~ ) ,

=

-2 sin 2 k 0{ k 2cos sin2 0 6 +k2-2)A+sin0(&+

=

s i n e { [ A ( -k2- (sin2 k 2 -02 ) ) 7 cos 0 2k

[(

1)B]

and *a3

p-A-lo)

+ B sin O(L cos e - 1)](

+ [c-(kLzsnj 9

( k’

-

)‘I2

cos 8 + 1 2)CdC:R:)

+ D sin 8 X

[ A { k 2 ( k 2- sin’ 0)”? - (k’ - 2) sin 0) -

B{sin’ 0

+ ( k’

R( -lo)= ( k2 - 2 sin2 0)’

-

sin’ O)”’}]

+ 4 sin2 0 cos 0( k2 - sin2 0)’12

The velocity and displacement wavefront components are obtained by simple successive time integrations of (4.63). The results are (4.63) except for a replacement of ( t - r / cd)-‘12there, by 2( t - r / cd)’/’ for the velocities &(x, y, t ) and &(x, y , t ) and by (4/3)(t - r/cd)3/2 for the displacements uci(x, y, t , and vd(x, y , It is important to note that the approximation (4.63) for iid(x, y, t ) (as well as L i d , ud) breaks down at the free edge y = 0 ( 0 = ~ / 2 through ) an

Modern Corner, Edge, and Crack Problems

161

infinite value caused by the last term in $ Z a ( - l 0 ) . Also, at the loaded edge x = 0 ( 8 = 0), this approximation fails through a vanishing value. On the other hand, approximation (4.63)for &(x, y , t ) (as well as i r d , u d ) does hold on both of the quarter-plane edges.

2 . The Equivoluminal Wavefronts The process for deriving the present wavefronts is basically the same as that used for the foregoing dilatational wavefronts. Here it is applied to the g s ( l ) integrals in (4.5).We now let t = g,(l) =

(1/Cd)[i(c2

-

k 2 ) ' l 2 y- 5x1.

(4.64)

Transformation (4.64)deforms path Br,, into the path - C in Fig. 45,which is defined by solving (4.64)for 5 = f , ( t ) , where

l , ( t ) = - [ ( c d t / r ) sin 8 + i{c,t/r)'

- 1}'/2kcos 81, r / c , 5 t < co, 0 5 8 5 ~ 1 2 .

(4.65)

Again (4.65)is one-half of a branch of a hyperbola [similar to (4.65)]in the third quadrant of the 4' plane with the same asymptote as (4.57b).The earliest time in (4.65)is t = r / c s , which is both the wavefront arrival time for the regular part (no precursor involved) of the two-sided equivoluminal wave and the final time associated with the precursor part of this wave. Figure 45 shows that t = r / c , , with lo= - k sin 8, the corresponding saddle point, is at the corner of path C. As we shall see, the two-sided wavefront is obtained through expansions along C from t = r / c,, outward from the corner along the hyperbola ( a short time, say to t = r / c , + E ) and along the horizontal line (a short time, say to t = r / c , - e ) . The first of these

1.8 -5R,

8 domain:

-k

-5,

I < k s i n 8 < k or I < kcos8 < k

FIG.45.

I

-I

or

I

e c r x < 8 <7 r / 2

0 < 8 < ecry

Contours for the head and two-sided equivoluminal wavefront approximations.

162

Julius Miklowitz

expansions gives the regular part and the second the precursor part of this wavefront event. As pointed out earlier and shown in Fig. 45, equivoluminal waves in the quarter-plane problem occur in two overlapping regions that are associated with the edges (surfaces) y = 0 ( x axis) and x = 0 ( y axis). These regions are found from Snell’s law for the reflection of S V wavesat criticalAangles of incidence, which for the x- axis edge is sin & = k sin p, & and /3 being angles of incidence and reflection for SV and P waves, respectivelx (see Miklowitz, 1978). The critical region izeasily seen to be ( l / k ) < sin p < 1 , which can be written here, replacing p with 0, as Box = sin-’( l / k ) < 0 < ~ / 2 or

1 < k sin 6 < k.

Similarly, for the y axis, the Snell’s law is cos & the critical region 0<8<

ecry= c o s - ’ ( l / k )

or

=

(4.66a)

k cos b, which leads to

1 < k cos 0 < k.

(4.66b)

Note that the second inequalities in (4.66a,b) agree with the position of the saddle points in Fig. 45 for the two cases. Note also that the saddle points are on the branch cut from 5 = -1 to -a; hence the path C has to include a part of path L2 just below the cut from -loto -1 in Fig. 45. This part of path C contributes the head wave and is typical of this event in critical region analysis for equivoluminal waves. The head wavefronts will be treated first and then the two-sided equivoluminal wavefronts. The head wavefronts are derived by first noting that on the L2 part of path C in Fig. 45, 5 becomes real and negative in the limit as the branch cut is approached. Further, 151decays along this line from 5 = -lo= -k sin 0 toward its end at the branch point 5 = - 1 . From (4.65), then, it follows that

l , ( t ) = lds(t) = [ I

- ( ~ , t / r ) ~ ]cos k

e - (Cdt/r) sin e,

(4.67)

showing that t 5 r / c , along this part of path C. The minimum time tds is reached at the branch point = -1; hence substituting the latter into (4.64), we find td,

=

( l / c d ) [ i ( l - k2)’l2y+ X I = (r/cd)[(k2 - I ) ’ / ~cos 8

+ sin el

(4.68)

since p ‘ ( - l ) = i ( 1 - k2)1’2= i[-i(k2 - 1)’12]= (k2 - 1)”2, in agreement with its positive value on L2 [i.e., p ( l ) = (p3pq)’/2]. Now integrals analogous to (4.58) and (4.59) are written for the g,(C) integrals in (4.45), through the transformations (4.64), (4.65), and dld,(t)/dt = l/gi[ldb,( t ) ] , which leads to

(4.69)

163

Modern Corner, Edge, and Crack Problems

To approximate the head equivoluminal acceleration components in (4.69) near their fronts, we first note that there t = tds + 8, where 0 S E << 1. Now the point 5ds corresponding to the minimum time fds is 5ds = -1, which in fact led to (4.68). So we expand gs(&) at 5ds = -1 along L2 (horizontal line part of C) via = gs(5ds) =

gs(-1) + g;(-l)(lds fds + g ; ( - l ) ( l d s +

+ 1) +

’ ’ ‘

(4.70a) (4.70b)

and from (4.70b) we have g:(-I) =

(f

-

?ds)/(lds

+ 1).

(4.71)

Then using (4.64) and the fact that j3‘ is a positive number on L 2 , we find that gi(-l) = -(r/cd)[sin 8 - ( k 2 - 1)-’12 cos 81,

a negative quantity in the present region 8,., < 8 < rr/2. Now near &s from (4.70) we have a’(5ds) = =

i[(
- 1)(5ds

-i[2(t -

(4.72) =

-1

+

fds)/\gi(-1)\]’’2>

(4.73)

in agreement with the value of a‘ on path L, [i.e., a = - - ( ~ ~ p ~ ) ~It/ ~ ) ] . follows from (4.70a) that g:(ldh,) = gi(-l), and substituting this and the - ,.iP(-1) 1 ) into (4.69) reduces approximation t?sR(&S) - &({ds) -- ~ ~ ~ ( the latter to

(4.74) where we have used lds = -1 and pi = ( k 2 - 1)”’. Now substituting given by the right-hand side of the second equation of (4.73), and ( k 2 - 1)1’2 for p’, into cSR(-l) - &(-l), and making use of (4.72) for gi(-l), (4.74) reduces to the real results for the acceleration head wavefront components

The complete time-order term in (4.75) is easy to derive if needed. Again here, the corresponding velocity and displacement head wavefront cornponents can be obtained through successive time integrations of (4.75). The

164

Julius Miklowitz

results are (4.75) except for a replacement of ( t - td,)-”’ there by 2 ( t - t d s ) ” > for the velocities hda and ud, and by (4/3)(t - f d a ) 3 ’ 2 for the displacements udc and v d s . It should be noted that the wavefront approximations (4.75) break down, having very large values near the critical value 0 = O,., = s i n - ’ ( l / k ) and an infinite value at I9 = O,,,, through f(I9). In Miklowitz (1978) it is shown that for Lamb’s line load problem the analogous spatial singularity in the critical region near Box can be traded for one in time, which in turn can be removed through a convolution with a slower-rising input function. The head wavefronts for the critical region associated with the y axis given by (4.66b) are easily obtained from (4.75) by just an interchange of sin I9 and cos I9 in f ( I 9 ) . It follows, a s f ( 0 ) shows, that (4.75) holds on both of the quarter-plane edges. The nature of the two-sided equivoluminal wavefronts has been discussed earlier. It remains to derive these events. Considering the x axis critical region again, given by (4.66a),we now have the inverses, analogous to (4.69),

where (4.76) and (4.77) represent, respectively, forms that will ultimately yield the precursor and regular parts of the acceleration two-sided equivoluminal wavefront components. We note that f d s and f , , which appear in (4.76) and (4.77),respectively, are given by (4.67) and (4.65), respectively. Superscripts B (for before) and A (for after) in (4.76) and (4.77) indicate the precursor and regular parts of this event. Subscript 2 indicates the two-sided nature of the event. To reduce (4.76) and (4.77) to the desired results as indicated earlier, we expand g , ( f ) along path C away in both directions from the corner at the saddle point - f o = - k sin I9 or t = r / c s . Consider (4.76) first. The time domain is r / c , - E 5 t 5 r / c s , which is indicated in Fig. 45 by the small portion b of path C to the right of t = r / C, . Note that the earlier time here is t = r / c , - E , with 0 5 E << 1, and that time grows along b toward the saddle point -loat t = r / c , . Expanding g , ( f ) at -So along b gives 2 =

gs(fds)

r / c s + [g:(-
(4.78)

Modern Corner, Edge. and Crack Problems where gS(-&,)

=

i2rc,/c: cos’ 0

=

165

-rc,/cs cos’ 0. From (4.78) we have

Noting, however, that on path h, tds + lois a real positive vector, from (4.79) we must have

Then using (4.78) again, and (4.80), we have

For (4.77) we proceed similarly. The time domain is r / c , 5 t 5 r / c s + F , 0 5 E << 1, which is indicated in Fig. 45 by the small portion a of path C below the corner at t = r / q . With t = r / c , at the corner being the initial time in this event, time grows downward and to the left along a. Expanding g , ( [ ) at -lo along a gives t

=

g,(J,)

2

r / c , + [ g t ( - L o ) / 2 1 [ ~+ , cn12,

(4.82)

and from (4.82)

Noting, however, that on path a, from (4.83) we must have

5, + 50 = - [ a t

5, + -

is a negative imaginary vector,

~/c,)/lg:(--5~~)111’2.

(4.84)

Then using (4.82) again, and (4.84), we have

g:(5,) = g’Y-50NL + 5”) = i[21g:(-50)l(t

- r/c,)I’”.

(4.85)

Now by approximating each of fi,,d&J - &(Sds) and fiSR(L) - &(L) by fisR(-l0) - & ( - f b ) in (4.76) and (4.77), and making use of (4.80) and (4.85) there and that now a’ = - i ( k 2 sin2 0 - 1)’12 and p‘ = k cos 0, (4.76) and (4.77) reduce to the acceleration two-sided equivoluminal wavefront components

r/c, - t)-”’H(r/c, - 1 ) ) - r / c , ) - ‘ / * H ( t- r / c J ’

{it

(4.86)

166

Julius Miklowitz

where

and

~ ? a ( - l=o-(sin ) 8 / 2 k ) { ( ~ ~ ~ ) ’ / ~ { [ i8k( k~2ssin’ i n ~8 - 1)-’12 -

( k’ - 2)]A

+ k sin 8 [ i(k’ sin’ 8

-

l)-”’ - 1]B}

x [ I - i(k’ sin2 e - I ) ’ / ~ ] - ’ ’ ~

+ cd{[ik4sin’ B(k’ sin’ 8 - I)-”’ - ( k 2 - 2)cdc,’]c + k sin e[icdci’,(kZsin’ 8 I)-’/’ - 130) -

x [-i(k’ sin’

e - I)’/’

+ cdci:]-’ + kc2v0 sin 8 / p ( k 2sin’ 8 - I)}, ,.i‘?fl(-lo) = (1/2k2){{-k sin 8 [ k 3cos 8 + k 2 - 2]A l)]”’ - k[cos 8 + k sin’ BIB}[ red/( k COS 8 + Cd{-k Sin 8 [ k 3COS 8 + (k’ - 2)cdci:]c k[CdCi: cos 8 + k sin2 e ] ~ } x [ k cos 8 + cdci:]-’ + c i a , / p } , -

R(-c,)

=

k4( 1 - 2 sin’ 8)’ - i4k3 sin2 8 cos 8( k2 sin2 8 - 1)’12.

Here again the corresponding velocity and displacement two-sided wavefront components can be obtained through successive time integrations of (4.86). The results are (4.86) except for a replacement of the pair ( r / c s t ) - ’ / 2 and ( t - r/cJ’” there by 2 ( r / c , - t ) ’ / ’ and 2(t - r/c,)”’ for the velocities us, and vs2, and by (4/3)(r/c, - t ) 3 / 2 and (4/3)(t - r / c J 3 / ’ for the displacements uS2 and us’. The wavefront approximations (4.86) break down at the critical 8 = O,, = sin-’(l/k) and at the free edge surface 8 = v / 2 , the extremes of the present critical range. Specifically, at 8 = Ocrx, us’, us’, and us’ and us’, U s 2 and 0,’ all become infinite due to the last term in ,.i?m(-&,), and at 0 = v / 2 they vanish due to the cos 0 multiplier in (4.86). As in the case of the head wavefronts, the two-sided equivoluminal wavefronts for the critical region associated with the y axis are easily obtained from (4.86) by interchanging sin 8 and cos 8 there.

Modern Corner, Edge, and Crack Problems

167

D. WAVEFRONTEVENTS FOR THE DISPLACEMENTS, VELOCITIES, ACCELERATIONS O N T H E QUARTER-PLANE EDGES (SURFACES)

AND

1 . The Rayleigh Surface Wavefronts Of prime importance on the quarter-plane edges are the Rayleigh wavefronts. With the loaded edge ( x = 0) Rayleigh wavefront approximations already given by (4.31) for the displacements uR, vR; velocities tiR, tjR, and accelerations iiR, ,i their behavior being, respectively, t - y / c R 1 ,H ( t y / c R l ) ,and S ( t - y / c R , ) ,we need only derive the corresponding approximations for the traction-free edge ( y = 0). As discussed after (4.45), the Rayleigh surface waves on y = 0 stem from the residues associated with the simple poles at -lk; shown in Fig. 44. These poles are zeros of R ( ( ) in (4.45). It follows, therefore, because of there are involved these zeros and since y = 0, that only 6 d R ( l ) and 6sR(l) in the Rayleigh wavefronts, and gd(6) = g s ( l ) = -xl/cd.

(4.87)

Substituting the right-hand side of (4.87) into (4.45) then reduces the latter to

Now from (4.57a) and (4.64) we have

5 = (d([) = = -Cdf/X, (4.89) which are the negative 5 values that hold along paths L , and L 8 , shown in Fig. 41. In Fig. 44 the semicircular path along L , , about the pole - l k , , is shown. Another such path about the pole -li, is contributed by L g . Both are indicated by -52; in Fig. 44. To obtain the contributions of the Rayleigh poles at -52; we use a contour integration for the integrals in (4.88) over the path

r = CR++ C R -+ CklL+ C i I L= BrsL,

(4.90)

which may be seen in Fig. 44. The contributions of C,, and CR- in (4.90) vanish according to the discussion after (4.45) involving (4.46)-(4.51). It follows from (4.88) and (4.90) then that

168

Julius Miklowitz

in which 6 = -lfdt + A e i s = - 5 + ~ Ae”, ~ where l~~ = cd/cR1 and A and 6 are, respectively, the modulus and argument of the complex radius vector 5 + lR1 on paths Cfd:L.It remains to carry out the limit indicated in (4.91). We consider the CLlLintegral first. On L , , a’ = -i(l*- 1)’12 and p’ = -i(12 - k2)’/*;hence these are the values of a’ and p‘ in the neighborhood of l =-lRl. Using these values of 5; a‘, and p’ in the CklLintegral and obtaining the limiting value indicated in (4.91), we have

Modern Corner, Edge, and Crack Problems

169

and where

[

c0s(w’2)] sin(w/2)

= {[ 1 f

{lil

-

( k 2 - 1)}-1/2]/2}1’2.

We now consider the C”,,, integral. On L , , a’ = i(5’- 1)”* and p’ = i(L* - k2)1’2;hence these are the values of a‘ and p’ in the neighborhood of 5 = lR1.We note also that these values of a’ and p’ are the conjugates of those in the preceding analysis of the CklLintegral that led to (4.92). Hence, our present case gives the conjugates of the expressions (for UR and G R ) within the inner brackets of (4.92), which have i as a multiplier. It follows that the sum of the two bracketed expressions leads to real results that are twice the value of the real parts of the inner bracketed expressions in (4.92). Therefore, multiplying the right-hand side of (4.92) by 2, and inverting it, gives for the acceleration Rayleigh wavefront components on the edge y = 0,

The corresponding velocity and displacement Rayleigh wavefront components on the edge y = 0 are obtained through successive time integrations of (4.93). The results are (4.93) except for a replacement of the S ( t - x/cR1) there by H ( t - x ) / c R I )for the velocities URand VR and by ( t - X / C R I ) H ( f x/cR1) for the displacements uR and uR. It should be noted that these wavefronts have the same time behavior as those on the loaded edge x = 0 as (4.31) and the discussion following that equation point out. 2. The Dilatational Wavefronts The dilatational wavefronts on the unloaded edge y = 0 are derived from (4.45). Substituting y = 0 into (4.45) and using (4.87) again gives

(4.94)

Julius Miklowitz

170

Now noting from Fig. 44 that the path - C ( y = 0) also contains path L 2 , on which 5 = -cdt/x from (4.89), we see the saddle point -Lo has moved to the branch point 5 = -1. Hence, an expansion from 5 = -(l+), along L2 to the left in the figure, will provide the wavefront approximations we seek. so transforming (4.94) with 5 = -cdt/x and d5 = -(cd/x) dt and noting that 5 = -1 corresponds to t = x/cd, (4.94) becomes

Now on L', and hence in the neighborhood of

5 = -1,

( ~ ' ( 5=) i(12 - I)'/* = [ 2 ( 5 + I ) ] " ~ = -i[2(cdr/x p'(5) = i(5'- k')'/'

- 1)]"',

= (k' -

(4.96)

in agreement with a = - i ( p I p 2 ) ' I 2 ,p = ( ~ ~ p ~ the ) " basic ~ , values on L 2 . Substituting the values (4.96) for a' and p' into (4.95) and then inverting the latter and retaining the leading wavefront terms there, the acceleration dilatational wavefront components are found to be - [ 2 ( k 2 - 1 ) ] ' " / ( k 2 - 2) 7 ~ k2 ( -

2)

x [x( t - t ) ] p ' ' 2 H ( t

-

t)]+

x-'O[ H ( t

-31. (4.97)

Again the corresponding velocity and displacement wavefront components can be obtained from (4.97) by successive integrations. The results are a replacement of ( t - x/ cd)-1/2 in (4.97) by 2( t - x/Cd)'I2 for tid, tjd and by (4/3)(t -x/cd)3/2 for u d , ud.

3 . The Equivoluminal Wavefronts It was pointed out in the discussion after the head wavefront approximations for the interior (4.75) that these hold also on the edges of the quarterplane. Hence, here we need only derive the two-sided equivoluminal wavefronts for the edges of the quarter-plane. We consider the y = 0 edge first, where (4.94) holds, and note that the saddle point -lo= - k sin 6 has moved to the branch point 5 = -k. Hence, expansions from 5 = - ( k + ) to the left along path Ll and from 5 = - ( k - ) to the right along path L2 will provide the wavefront approximations we seek. We again have 5 = -cdt/x and d5 = -(cd/x) dt, and 6 = - k corresponds to t = x/c,. Along L2 in the neighborhood of 6 = - k then, (4.94) becomes

Modern Corner, Edge, and Crack Problems

where 0 5


E

171

t = -k,

Again as in (4.76) and (4.75) for the interior, the superscripts B (for before) and A (for after) in (4.98) and (4.99) indicate the precursor and regular parts of this two-sided event. Treating the precursor (4.98) first, on L 2 , near 5 = -(k-),

p’

=

i(5’-

1 ) , / 2 = -i(k2

=

i(12- k2)1’2= i[-2k((

1)W,

-

+ k)]”2 = k{2[1 - ( c , ~ / x ) ] } ” ~ ,

(4.100)

in agreement with the basic values a = - i ( p l p 2 ) ’ 1 2 and p = ( p 3 p 4 ) ’ I 2on L 2 . Then substituting the values (4.100) for a’and /3’ into (4.98), inverting the latter and retaining the leading wavefront terms there, one finds the precursor parts of the acceleration two-sided equivoluminal wavefront components to be

{ ii!(~, v&,

[{

0, t ) } - (k2 - 1)’12 2(2~Jx)’”/k’} 0, t ) vkx -1 Efj(x/c, - t)‘” H(x/c,{ E ; , + E;,(x/c, - ? ) ‘ I 2

I1

where

Efj = 2(2k2 + 1) Re $ T n ( - k ) E:,

=

-{2 Re &Ta(-k)

E:2

=

-(2=

t),

(4.101)

+ Cdk(k2 - 2)$,

+ 4$?p,[-(k-)]

+ c,(k2 - 2)u^/k}

Cd/k)6,

and where Re bZa(-k)

=

-(2k)-’{(m,)’/’{[-(k2

- 2) cos q

+ k4(k2 - 1)-’I2 sin q]A + k[-cos - cd(c:/ciI

q

+ (k2 + k2

-

-

1 ) - 1 / 2 sin q]B} l)-’{[k2

-

+ (2kcd/cRl)D} + [c:kaO/p(k2 b?p,[-(k-)]

2)c:/ci, -

+ k4]C

1)1}9

- ( 2 k ) - ’ [ ( ~ , ~ ) ’ / ’ {-( k2)A ~ + kB} + Cd( k2 - 2 ) C + kCR I D - c : u ~ / kp],

and q = tan-’[-(k2 - 1)’/2/2]. The subscript 1 on b?J-(k-)] indicates that b?p here represents the coefficient of its leading wavefront term. It remains to treat (4.99) similarly, which will yield the regular wavefronts for the two-sided equivoluminal events. Here on L , , near 5 = -(k+), a ’ does not change from its value on L2 given by the first of (4.100). However

Julius Miklowirz

172

p’ changes its nature, that is, p’

=

i(5’

-

k 2 ) 2= [2k(5 + k)]’12 = -ik[2(cSt/x

-

(4.102)

These values of a‘ and p’ are in agreement with the basic values a = - - i ( p l p 2 ) ’ / * and p = - i ( ~ ~ p ~on) ”L ~, . Therefore, substituting the righthand sides of the first of (4.100) and (4.102) for a ’ and p’ into (4.99), inverting the latter, and retaining the leading wavefront terms there, one finds the regular parts of the acceleration two-sided equivoluminal wavefront components to be

where

Et = (2k2 + 1) Im ;Ta,(-k) + Im ;_”,,[-(k+)] + 3c,k36, Et

=

Re ; Y m ( - k )

-

2 Re ;l‘ar[-(k+)]

- c,(k’-

2)[(k2 - 1)’12 - 2k-’]u^,

=

- ( 2 k ) - ’ { ( . r r c , ) ’ / 2 { [ k 4 (-k 2 cos q + ( k 2- 2) sin q]A + k [ ( k 2- 1)-’12 cos q + sin q]B} + C , ( C : / C ’ , , + k’- 1)-l{(C,/CR])(k4(k2- 1) - ( k 2- 2)(k2- I)’”]C + k [ ( ~ : / ~ i ~ ) ( k ’ - ( k 2 - l)”2]D}},

and where Im ; T m ( - k )

Im ;?,J-(k+)]

=

- ( 2 k * ) - ’ { k ( . r r ~ , ) ’ / ~ {+ k [(kk 2- 2)/2]A - (1 - k2/2)B}- k2~RI{2(k2 -l)C

+ (cd/cRI

-

kCR1/Cd)D)}r

and where Re ; T U ( - k ) and q are given in (4.101), while Re C;TO1[-(k+)] is equal to ;‘?pl[-(k-)] in (4.101). The complete acceleration two-sided equivoluminal wavefront components us2(x,0, t ) and us2(x,0, t ) are formed by each of the pairs us2and Us2 obtained from (4.101) and (4.103). Note that the time behaviors of the pairs u!* and U $ are the same [i.e., (x/c, - t ) ’ ” and ( t - x/c,)~’*].Similarly, B us2 and are H ( x / c , - t ) and H ( r - x/c,), respectively. Important is the fact that these y = 0 edge (surface) wavefronts are much less singular than the corresponding interior wavefronts given by (4.86). This type of behavior is well known in Lamb’s problem (see Miklowitz, 1978). As in our earlier wavefront events, the velocities U s 2 ( x ,0, t ) and GS2(x,0, t ) and the displacements us2(x,0, t ) and uS2(x,0, t ) are obtained from (4.101) and (4.103) by successive integrations.

02

Modern Corner, Edge, and Crack Problems

173

Consider now the two-sided equivoluminal wavefronts on the loaded edge x = 0. From (4.45)these events are governed by

where fiSRsR, &, R ( l ) , P ' , and a' are defined in (4.45).Now in the critical region 0 < 8 < Ocry, we have for the y axis (x = 0) edge from (4.64)and (4.69, gs(5)= P ' ( ~ ) Y / c , ,

5,(1) = - i k [ ( c , t / ~ ) ~11"2, -

(4.105)

respectively. From the second of (4.105)we see that our 5 contour includes the lower half of the Im 5 axis, the origin and the positive Re l part of path L3 (see Fig. 45).This equation also shows that now the origin corresponds to the time t = y/c,. The next step is a contour integration involving the 5 contour shown in Fig. 45. We find, using the first of (4.105),that (4.104) can be traded for

Now from the first of (4.105),

In (4.107)P' is positive, in agreement with its basic values /3 = ( p 3 p 4 ) ' l 2on the lower half Im 5 axis and L 3 .Also a' has the same character there [i.e., a = ( p l p 2 ) 1 / 2 ]From . the second of (4.105)we see that the wavefronts here stem from small 5. Indeed, the paths b' and a' near the origin of 5 in Fig. 45 will be shown to yield from (4.106)the precursors and regular parts, respectively, of the two-sided equivoluminal wavefronts. We treat first the precursors from the L, integral in (4.106),noting on path b', with small 5, that

where 0 5

E

<< 1.

Julius Miklowitz

174

Compatible with the small value of & ( t ) indicated in (4.108) are the relations a’= i ( L 2 - 1) = i ( - i ) = 1,

p’

=

c d t / y = k,

5,( t ) = k [ 1 - ( ~ , t / y ) ~ ] ’k/[ 2~ ( 1 - C ~ C / Y ) ] ’ ” , d5,( t ) = -k&-’[ 1 - ( ~ , t / y ) ’ ] ~ dt, ’/’ ( 1 - p’)I/’= [ i 2 ( k - 1)]’”= i ( k - 1)’/’. 5

(4.109)

From (4.107) and (4.109) we then have from (4.108),

where

S:l

=

Sfi, = (2c,)’”(k3 - k2 + 2)A,

-B,

Sfl = -(2~,)”’l?,

S:2

6

=

= cS(k3- k2

+ 2)A,

~~y/2c,k~.

From (4.110) then, one finds the leading wavefront terms for the precursor parts of the two-sided equivoluminal acceleration components to be

t),

(4.111)

where y / c , - 6 5 t 5 y / c , . The regular parts of the two-sided equivoluminal wavefronts are obtained from the first integral (along the negative Im 5 axis) in (4.106), noting on path a’, with small 5, that

{fisR[ls(t)l

-

dls(r)]. (4.112)

Now compatible with small values of 1 6 ( t )in (4.112) are the relations for a’,P’, and ( 1 - P‘)”’ in (4.109), and

l s ( t ) = - i k [ ( c , t / y ) 2 - 1]’/’= - i k [ 2 ( c s t / y - 1)]’12, dL,(t) = -ikc,y-’[(c,t/y)’ - 1]-1/2 dt.

(4.113)

175

Modern Corner, Edge, and Crack Problems

Using (4.107), the values for a ' , p', and ( 1 - p')"' in (4.109) and the relations (4.113), we then have from (4.112)

(4.114)

where

Stl = -2-'/'[(TCd/(k

+ 1))'"B

S t z = -(c,/2)'/'[k3 - ( k 2- 2)][ncd/(k

- 2CiC,:(k2 -

-

CiC,:)-'D

- CdV^],

l)]'"A,

St, = -[TCdCs/2(k - l)]1'2B,

St2 = 2 ~ , { { ( ~ ~ d ) ' / z [ 2 ' / 2-(2)/k k 2 - (k'+ k2 - 2)/2(1 + k)"']} + {2(k2- 2)c:c,;/k(1 + cdc,:)

+ Cd[k3 + ( k 2 - (Cd/2)[k3 -

-

2)CdC,;]/[k

+ CdCR:]

( k 2 - 2)cdc,:]/[k - CdC,:]}c -2Cd(k2-2)s/k}.

From (4.114) then, one finds the leading wavefront terms for the regular parts of the two-sided equivoluminal acceleration components to be

- y/cs)-'/2 + y-112 S"I A + y-'St2(r - y / c , ) ' / 2

St'( t x(y-l/2

where y/c, d t d y / c , + F , with 0 5 F << 1 . The complete acceleration two-sided equivoluminal wavefront comy, t ) and us2(0,y, t ) are formed by each if the pairs us2 and ponents us2(07 iis2, obtained from (4.111) and (4.113). Note again here that the time behavior of the pairs are the same [i.e., (y/c, - t ) - ' / * and and ( t - y / ~ , ) - ' / ~ ]Similarly, . i:2 and ii$ are H ( y / c , - t ) and H ( t - y / c , ) , respectively. Important here is the fact that the usZwavefront singularities are the same as those of corresponding interior results given in (4.86). The same is not true of the vs2 wavefronts, which we have already noted behave as H functions, whereas the corresponding vs2 wavefronts in (4.86) have the infinite singularities ( y / c , - t ) - ' / ' and ( t - y / ~ , ) - " ~Further, . it is interesting to note that the us2(0,y, 1 ) singularities match the singularities of the dilatational wavefronts given by (4.30). Also observe that the us2(0,y , 1 ) wavefronts are much more singular than those of usz(x70, t ) . Perhaps the most important comparison of the two-sided equivoluminal wavefront singularities stems from the symmetries of these events relative to the x and y axes. That is, along the x axis iis2(x,0, t ) (x/c, - t ) ' / ' and ( t - x/c,)'/~,

-

176

Julius Miklowitt

-

whereas along t h e y axis its mate, ijs2(0,y, t ) H ( y / c , - t ) and H ( t - y/c,). Similarly, the normal acceleration along the x axis, ijs2(x,0, t ) H (x/c, - t ) and H ( t - x/c,), whereas the normal acceleration along the y axis, us2(0,y, t ) ( y / c , - t)-"2 and ( t - y / ~ , ) - " ~ .Clearly, these wavefronts show the stronger singularities along the loaded edge x = 0. Finally, it is important to note that our use of the positive Re 5 part of path L3 in the preceding analysis for the x = 0 edge does not violate unboundedness via the reasoning leading to the vanishing of L, in Fig. 43. Clearly, this is so, since with x = 0, exp(sx) = exp[kd Re(l)x] = 1 along L3; hence we have boundedness there.

-

-

4. The Dilatational Plane Wavefront To obtain this wavefront we return to (4.45) and draw on the terms there exhibiting the singularity (5 + 1)-' at 5 = -1, shown in Fig. 44. The terms that contribute are contained in

where

-$Ta(5) = ~ i a , 5 * / 2 k ' p ( 5-~ l ) ,

-$Te([)

=

(4.1 16b) ~ i c ~ ~ ~ ~- 1). / 2 k ~ p ( ~

The contributions follow from contour integrations for each of the integrals of (4.116a), using (4.116b). It can be seen from (4.1 16b) that the contour integrals replacing (4.1 16a) are the lower-half circular path about 5 = -1, C-,, for the first integral in (4.116a) (see Fig. 44), and the upper half circular path C l U (not shown in Fig. 44) for the second integral there. The second integral in (4.116a) over Br," is an alternative representation to that of BrrL, useful here because of the second of (4.116b). Indeed, because of the simple pole nature of both terms in (4.116b), ( 5 + 1)-' and l)-', we are able to integrate fully around the circular path about 5 = -1 for each of these terms. This evidences itself in (4.116a) by the multiplicative factor 2 out front in (4.1 16a). Mathematically we have then

(c+

(4.117)

Modern Corner, Edge, and Crack Problems

177

c+

Then, substituting 5 + 1 = Ae’” and 1 = Ae-“‘ into the first and second integrals of (4.117), respectively, and carrying out the integrations (in which moduli A on C-,,,, + 0) and inversion, gives the complete plane displacement wave

which also serves to represent the wavefront. It follows by successive time differentiations of (4.1 18) that u d p and iid, are, respectively, represented by H ( t - x/cd) and 6 ( t - x/cd). From the first of (4.2), and the fact that v does not contribute to the present event, (4.118) yields the stress

It follows from (4.1 19) that on the loaded edge x in agreement with the first of (4.18).

=

0, u,(O, y , t ) = a O H (t ) ,

REFERENCES Achenbach, J. D. (1976). Wave propagation, elastodynamic stress singularities, and fracture. In “Theoretical and Applied Mechanics” (W. T. Koiter, ed.), pp. 77-88. North-Holland, Amsterdam. Achenbach, J. D., and Brind, R. J . (1981). Scattering of surface waves by a subsurface crack. J. Sound Vibration 76, 43-56. Achenbach, J. D., and Khetan, R. P. (1977). Elastodynamic response of a wedge to surface pressures. In[. J. Solids Struct. 13, 1157-1171. Achenbach, J. D., Hemann, J. H., and Ziegler, F. (1970). Separation at the interface of a circular inclusion and the surrounding medium under an incident compressive wave. J. Appl. Mech. (Trans. A S M E ) 31, 298-304. Achenbach, J. D., Gautesen, A. K., and McMaken, H. (1978). Application of elastodynamic ray theory to diffraction by cracks. In “Modern Problems in Elastic Wave Propagation.” Int. Union Theor. Appl. Mech. Symposium held at Northwestern Univ., Evanston, Illinois, Sept. 12-15, 1977 (J. Miklowitz and J . D. Achenbach, eds.), pp. 219-238. Wiley-Interscience, New York. Achenbach, J. D., Keer, L. M., and Mendelsohn, D. A. (1980). Elastodynamic analysis of an edge crack. J. Appl. Mech. (Trans. A S M E ) 41, 551-556. Alterman, Z. S., and Rotenberg, A. (1969). Seismic waves in a quarter plane. Bull. Seismol. SOC.Am. 59, 347-368. Ang, D. D., and Knopoff, L. (1964a). Diffraction of vector elastic waves by a clamped finite strip. Proc. Natl. Acad. Sci. U.S.A. 52, 201-207. Ang, D. D., Knopoff, L. (1964b). Diffraction of vector elastic waves by a finite crack. Proc. Natl. Acad. Sci. U.S.A. 52, 1075-1081. Baron, M. L., and Matthews, A. T. (1961). Diffraction of a pressure wave by a cylindrical cavity in an elastic medium. J. Appl. Mech. (Trans. A S M E ) 28, 347-354. Baron, M. L., and Pames, R. (1962). Displacements and velocities produced by the diffraction of a pressure wave by a cylindrical cavity in an elastic medium. J. Appl. Mech. (Trans. A S M E ) 29, 385-395. Benthem, J. P. (1963). A Laplace transform method for the solution of semiinfinite and finite strip problems in stress analysis. Q. J. Mech. Appl. Math. 16(4), 413-429. Bertholf, L. D. (1967). Numerical solution for two-dimensional elastic wave propagation in finite bars. J. Appl. Mech. (Trans. A S M E ) 34, 725-734.

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Brock, L. M., and Achenbach, J. D. (1970). Wave propagation in two joined quarter-spaces. Dev. Theor. Appl. Mech. Proc. Souiheasi Conf: 5rh, pp. 449-476. Cooper, G . J., and Craggs, J . W. (1966). Propagation of elastic waves. J. Austr. Math. SOC.6 , 55-64. Copson, E. T. (1935). “An Introduction to the Theory of Functions of a Complex Variable,” p. 344. Oxford University Press, London. Curtis, C. W. (1956). Propagation of elastic and plastic deformations in solids, OOR Rep. Contr. DA-36-034-ord-1456, Suppl. 2, Proj. TB2-0001 (187), Lehigh University, Bethlehem, Pennsylvania. Datta, S. K. (1978). Scattering of elastic waves. In “Mechanics Today” ( S . Nemat-Nasser, ed.), Vol. 4, pp. 149-208. Pergamon Press, New York. Datta, S. K., and El-Akily, N. (1978). Diffraction of elastic waves in a half space. I: Integral representations and matched asymptotic expansions. In “Modern Problems in Elastic Wave Propagation.” Int. Union Theor. Appl. Mech. Symposium held at Northwestern Univ., Evanston, Illinois, Sept. 12-15, 1977 (J. Miklowitz and J. D. Achenbach, eds.), pp. 197-218. Wiley-Interscience, New York. De Vault, G. P., and Curtis, C. W. (1962). Elastic cylinder with free lateral surface and mixed time-dependent end conditions. J. Acoust. SOC.Am. 34, 421 -432. Doetsch, G. (1937). “Theorie und Anwendung der Laplace-Transformation,” pp. 378-383. Springer-Verlag, Berlin. Erdelyi et al. (1953). “Higher Transcendental Functions 11,” Bateman Manuscript Project, McGraw-Hill, New York. Folk, R., Fox, G., Shook, C. A,, and Curtis, C. W. (1958). Elastic strain produced by sudden application of pressure to one end of a cylindrical bar: I. Theory. J. Acoust. Soc. Am. 30, 552-558. Fox, G., and Curtis, C. W. (1958). Elastic strain produced by sudden application of pressure to one end of a cylindrical bar: 11. Experimental observations. J. Acoust. SOC.Am. 30. 559-563. Freund, L. B. (1971). Guided surface waves on an elastic half space. J. Appl. Mech. (Trans. A S M E ) 38, 899-905. Friedlander, F. G. (1954). Diffraction of pulses by a circular cylinder. Comm. Pure Appl. Math. 7, 705-732; see also Chapter 6 of “Sound Pulses,” Cambridge Univ. Press (1958). Gilbert, F. (1959). Elastic wave interaction with a cylindrical cavity. Report to WDIE, AFBMD, Contract AF04(647)-342. E. H . Plesset Assoc. Inc., Los Angeles, California. Gilbert, F. (1960). Scattering of impulsive elastic waves by a smooth convex cylinder. J. Acoust. SOC.Am. 32, 841-857. Gilbert, F., and Knopoff, L. (1959). Scattering of impulsive elastic waves by a rigid cylinder. J. Acoust. SOC.Am. 31, 1169-1175. Griffin, J. H., and Miklowitz, J. (1974). Wavefront analysis of a plane compressional pulse scattered by a cylindrical elastic inclusion. Inf. J. Solids Srruct. 10, 1333-1356. Grimes, C. K. (1964). Studies on the propagation of elastic waves in solid media. Ph.D. thesis, California Institute of Technology, Pasadena, California. Jones, 0. E., and Ellis, A. T. (1963a). Longitudinal strain pulse propagation in wide rectangular bars: Part 1, Theoretical considerations. J. Appl. Mech. (Trans. A S M E ) 30, 50-60. Jones, 0. E., and Ellis, A. T. (1963b). Longitudinal strain pulse propagation in wide rectangular bars: Part 2, Experimental observations and comparisons with theory. J. Appl. Mech. (Trans. A S M E ) 30, 61-69. Jones, 0. E., and Norwood, F. R. (1967). Axially symmetric cross-sectional strain and stress distributions in suddenly loaded cylindrical elastic bars. J. Appl. Mech. ( Trans. A S M E ) 34, 718-724. Jones, R. P. N. (1964). Transverse impact waves in a bar under conditions of plane-strain elasticity. Q. J. Mech. Appl. Math. 17, 401-421.

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Knopoff, L. (1969). Elastic wave propagation in a wedge. I n : “Wave Propagation in Solids” (J. Miklowitz, ed.), pp. 3-43. American Society of Mechanical Engineers, New York. KO, W. L. (1970). Scattering of stress waves by a circular elastic cylinder embedded in an elastic medium. J. Appl. Mech. (Trans. A S M E ) 37, 345-355. Kraut, E. A. (1968a). Diffraction of elastic waves by a rigid 90” wedge, Part I. Bull. Seismol. Soc. Am. 58, 1083-1096. Kraut, E. A. (1968b). Diffraction of elastic waves by a rigid 90” wedge, Part 11. Bull. Seismol. Soc. Am. 58, 1097-1115. Kundu, T., and Mal, A. K. (1981). Diffraction of elastic waves by a surface crack on a plate. J. Appl. Mech. (Trans. A S M E ) 48, 570-577. Lapwood, E. R. (1961). The transmission of a Rayleigh pulse round a corner. Geophys. J. R. Astron. SOC.4, 174-196. Lotfy, A. A., and Leipholz, H. H. E. (1984a). Long-time response of a finite cantilever plate to antisymmetric dynamic surface loading. J. Sound Vibration 94, 161-173. Lotfy, A. A,, and Leipholz, H. H. E. (1984b). Long-time response of finite cantilever plates to dynamic surface loadings. J. Sound Vibration 94, 175-182. Love, A. E. H. (1927). I n “Mathematical Theory of Elasticity,” 4th Ed., pp. 289-291. Dover Publications, New York. Meitzler, A. H. (1955). Propagation of elastic pulses near the stressed end of a cylindrical bar, Ph.D. thesis, Lehigh University, Bethlehem, Pennsylvania. Mendelsohn, D. A., Achenbach, J. D., and Keer, L. M. (1980). Scattering of elastic waves by a surface-breaking crack. Waue Motion 2, 277-292. Miklowitz, J. (1958). On the use of approximate theories of an elastic rod in problems of longitudinal impact. R o c . 3rd U. S. Narl. Congr. Appl. Mech. pp. 215-224. ASME, New York. Miklowitz, J. (1962). Transient compressional waves in an infinite elastic plate or elastic layer overlying a rigid half space. J. Appl. Mech. (Trans. A S M E ) 29, 53-60. Miklowitz, J. (1963). Pulse propagation in a viscoelastic solid with geometric dispersion. I n : “Stress Waves in Anelastic Solids, IUTAM Symposium” (H. Kolsky and W. Prager, eds.), pp. 255-276. Springer-Verlag. Berlin. Miklowitz, J. (1966a). Elastic wave propagation. In “Applied Mechanics Surveys” (H. N. Abramson, H. Liebowitz, J. N. Crowley, and S. Juhasz, eds.), pp. 830-836. Spartan Books, Washington. Miklowitz, J. (1966b). Scattering of a plane elastic compressional pulse by a cylindrical cavity. Proc. 11th Inf. Congr. Appl. Mech., pp. 469-483. Springer-Verlag, Berlin. Miklowitz, J. (1969). Analysis of elastic waveguides involving an edge. I n “Wave Propagation in Solids’’ (J. Miklowitz, ed.), pp. 44-70. ASME Publications, New York. Miklowitz, J. (1978). “The Theory of Elastic Waves and Waveguides,” North-Holland Series in Applied Mathematics and Mechanics, Vol. 22, pp. 367-409. North-Holland, Amsterdam. Miklowitz, J., and Garrott, W. R. (1978). Transient response of two-dimensional cantilevered plates subjected to base motions. I n “Modern Problems in Elastic Wave Propagation,” Int. Union Theor. Appl. Mech. Symposium held at Northwestern University, Evanston, Illinois, Sept. 12-15, 1977 (J. Miklowitz and J. D. Achenbach, eds.), pp. 373-400. WileyInterscience, New York. Miklowitz, J., and Nisewanger, C. R. (1957). The propagation of compressional waves in a dispersive elastic rod. Part 11: Experimental results and comparison with theory. J. Appl. Mech. (Trans. A S M E ) 24, 240-244. Morse, P. M., and Feshbach, H. (1953). “Methods of Theoretical Physics, Part I,” pp. 466-467. McGraw-Hill, New York. Nagase, M. (1956). Diffraction of elastic waves by a spherical surface. J. Phys. Soc. Jpn. 11, 279-301.

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Nigul, U. K. (1963/1964). The application of the three-dimensional theory of elasticity to the analysis of flexural waves in a semi-infinite plate acted on by a short-time boundary loading. frikl. Mat. Mekh. 27, 1602-1620. Norwood, F. R. (1967). Diffraction of transient elastic waves by a spherical cavity. Ph.D. thesis, California Institute of Technology, Pasadena, California. Norwood, F. R., and Miklowitz, J. (1967). Diffraction of transient elastic waves by a spherical cavity. J. Appl. Mech. (Trans. A S M E ) 34, 735-744. Nussenzveig, H. M. (1965). High-frequency scattering by an impenetrable sphere. Ann. P h J x ( N .Y.)34, 23-95. Onoe, M., McNiven, H. D., and Mindlin, R. D. (1962). Dispersion of axially symmetric waves in an elastic rod. J. Appl. Mech. (Trans. A S M E ) 29, 729-734. Pao, Y.-H., and Mow, C . C. (1971). “Diffraction of Elastic Waves and Dynamic Stress Concentrations.” Crane Russak Co., New York, 391. Peck, J . C. (1965). Plane strain diffraction of transient elastic waves by a circular cavity. Ph.D. thesis, California Institute of Technology, Pasadena, California. Peck, J. C., and Miklowitz, J . (1969). Shadow-zone response in the diffraction of a plane compressional pulse by a circular cavity. Int. J . Solids Strucr. 5, 437-454. Pochhammer, L. (1876). J. Math. 81, 324-336. Randles, P. W. (1973). Cusped wavefronts in anisotropic elastic plates. Int. J. Solids Struct. 9, 31-52. Randles, P. W., and Miklowitz, J. (1971j. Modal representations for the high-frequency response of elastic plates. Inr. J. Solids Struct. 7 , 1031-1055. Robins, C. I., and Smith, R. C. T. (1948). A table of roots of sin z = --i. Philos. Mug. Ser. 7 39, 1004-1005. Rosenfeld, R. L., and Miklowitz, J. (1962). Wavefronts in elastic rods and plates. Proc. 4th U. S. Natl. Congr. Appl. Mech. I , pp. 293-303. ASME, New York. Rosenfeld, R. L., and Miklowitz, J. (1965). Elastic wave propagation in rods of arbitrary cross section. J. Appl. Mech. (Trans. A S M E ) 32, 290-294. Scheidl, W., and Ziegler, F. (1978). Interaction of a pulsed Rayleigh surface wave and a rigid cylindrical inclusion. In “Modern Problems in Elastic Wave Propagation.” Int. Union Theor. Appl. Mech. Symposium held at Northwestern University, Evanston, Illinois, Sept. 12-15, 1977 (J. Miklowitz and J . D. Achenbach, eds.), pp. 145-169. Wiley-Interscience, New York. Scott, R. A. (1964). Transient wave propagation in elastic plates with cylindrical boundaries, studied with aid of multi-integral transforms. Ph.D. thesis, California Institute of Technology, Pasadena, California. Scott, R. A,, and Miklowitz, J. (1964). Transient compressional waves in an infinite elastic plate with a circular cylindrical cavity. J. Appl. Mech. (Trans. A S M E ) 31, 627-634. Simons, D. A. (1975). Scattering of a Love wave by the edge of a thin surface layer. J. Appl. Mech. (Trans. A S M E ) 42, 842-846. Simons, D. A. (1976). Scattering of a Rayleigh wave by the edge of a thin surface layer of negligible inertia. J. Acoust. Soc. A m . 59. 12-18. Sinclair, G. 9. (1973). On nonmixed symmetric end-load problems in elastic waveguides. Ph.D. thesis, California Institute of Technology, Pasadena. California. Sinclair, G. B., and Miklowitz, J. (1975). Two nonmixed symmetric end loadings of an elastic waveguide. Int. J. Solids Struct. 11, 275-294. Skalak, R. (1957). Longitudinal impact of a semi-infinite circular elastic bar. J . Appl. Mech. (Trans. A S M E ) 24, 59-64. Soldate, A. M., and Hook, J . F. (1960). A theoretical study of structure-medium interaction. Report to AFSWC, Kirtland Air Force Base, Contract AF29(601)-2838, National Engineering Science Co. Report AFSWC-TN-61-6. Thau, S. A,, and Lu, T.-H. (1971). Diffraction of transient horizontal shear waves by a finite crack and a finite rigid ribbon. Int. J. Eng. Sci. 8, 857-874.

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Timoshenko, S. P., and Goodier, J . N. ( 1970). “Theory of Elasticity,” 3rd Ed. McGraw-Hill, New York. Tiersten, H. F. (1969). Elastic surface waves guided by thin films. J. Appl. Phys. 40, 770-789. Ting, T. C. T., a n d Lee, E. H. (1969). Wavefront analysis in composite materials. J. Appl. Mech. (Trans. A S M E ) 36, 497-504. Titchmarsh, E. C . (1948). “Introduction to the Theory of Fourier Integrals,” 2nd Ed. Oxford University Press, London. Titchmarsh, E. C. (1962). “Eigenfunction Expansions, Part I,” 2nd Ed. Oxford University Press, London. Uberall, H. (1978). Modal and surface wave resonances in acoustic-wave scattering from elastic objects and elastic-wave scattering from cavities. I n “Modern Problems in Elastic Wave Propagation.” Int. Union Theor. Appl. Mech. Symposium held at Northwestern University, Evanston, Illinois, Sept. 12-15, 1977 (J. Miklowitz and J. D. Achenbach, eds.), pp. 239-263. Wiley-Interscience. New York. Viktorov, I . A. (1958). Rayleigh-type w‘ives on a cylindrical surface. Sou. Phys. Acoust. (Engl. Trans.) 4, 131-136. Viswanathan, K . (1966). Wave propagation i n welded quarter-spaces. Geophyr. J. R. Astron. Soc. 11, 293-322. Viswanathan, K . , and Sharma, J. P. (197X 1. On the match-asymptotic solution o f t h e dilfraction of a plane elastic wave by a semi-infinite rigid boundary of finite width. I n “Modern Problems in Elastic Wave Propagation.” i n t . Union Theor. Appl. Mech. Symposium held at Northwestern University, Evansion. Illinois, Sept. 12-15, 1977 (J. Miklowitz and J. D, Achenbach, eds.), pp. 171-195. Wiley-Interscience, New York. Watson, G. N. (19181. The diffraction of electric waves by the earth. Proc. R. SOC.Lotidon Ser. A 95, 83-99. Wojcik, G. L. ( 1977). Self-similar ela\tritlynamic solutions for the plane wedge. Ph.D. thesis, California Institute of Technology, Pasadena, California. Also available as a Graduate Aeronautical Laboratories Report, ( .I.T. Wright, T. W. (1969). Impact on an elahtic quarter space. J. Acousr. Soc. Am. 45, 935-943.

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A D V A N C E S I N APPLIEL? M E C H A N I C S , V O L U M E

25

The Derivation of Constitutive Relations from the Free Energy and the Dissipation Function HANS ZIEGLER AND CHRISTOPH WEHRLI Department of Mechanics Swiss Federal Institute of Technology CH-8092 Zurich. Switzerland

I. Introduction Mechanics may be based on two principles: the reaction principle and the principle of virtual power. The reaction principle (still cited in most books in its special Newtonian version, valid merely for pairs of material points) must be stated in the following general form: (1) Every force has its reaction. A force is called internal or external for a given system according to whether its reaction acts within it or without. (2) In any virtual state of motion corresponding to a rigid displacement of the system, the total power of the internal forces is zero. The principle of virtual power states that in any virtual state of motion (whether admissible or not) the total power of the internal, external, and inertia forces is zero. Applying these two principles to translations and rotations (particular cases of rigid motions), one obtains the theorems of linear and angular momentum. They are free of internal forces. In the case of a discrete system, composed of material points and/or rigid bodies, certain internal forces are usually known, whereas others (called reactions) are unknown. Applying the theorems of linear and angular momentum t o the entire system, we may not obtain sufficient differential equations to determine its motion. However, if we apply the theorems to the various parts of the given system, obtained by successive subdivision, more and more internal forces become external, and the same process increases the number of differential equations until a complete set is obtained to determine the motion and the reactions. Integration of this set may be difficult. It can be facilitated, however, by applying the principle of virtual power to the real motion and to the most general admissible state of motion. 183 Copynghr Q 1987 by Academic Press, Inc All nghts of reproduction in any form reserved

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In this manner one obtains, respectively, the energy theorem and Lagrange’s equations. Let us note in passing that to formulate Lagrange’s equations (and in consequence analytical mechanics as a whole), one starts by choosing a set of generalized coordinates q,, together with the corresponding velocities q,, and proceeds to define the generalized forces QIby means of the expression for the virtual work or power. There is no way to invert this process: velocities and forces are inexchangeable. In the case of a continuum the system to be considered is the mass element, and the internal forces are by definition unknown. In place of the q, we have the coordinates determining position and orientation of the element plus a set of strains determining its state of deformation. In place of the q1 we have the linear and angular velocities of the element plus the material strain rates, and the Q, appear replaced by the resultant force and moment plus the stresses defined by the expression for the virtual work or power of deformation. The theorems of linear and angular momentum are still valid; however, they are insufficient to determine the motion. Since the internal forces are unknown, so is the connection between strains and stresses. Subdivision of the element is of no avail since it procuces elements with the properties of the orignal one unless the subdivision is carried to the point where the continuum loses its character and disintegrates into its molecules. There are two ways out of this impasse. The first has been followed in the past by continuum mechanics. Here, the observed response of given materials is modeled by constitutive relations connecting the stresses with the strains, the strain rates, or even the strain history. In cases where the history comes into play, an alternative-which will be preferred in this article-can be based on the introduction of internal parameters in the form of strains, in the definition of the corresponding internal stresses and in the inclusion of these entities in the constitutive relations. These approaches have been quite successful. However, they are semiempirical, different for each material, and not tied together by a general principle. The other way out of the dilemma consists in the attempt to explain the macroscopic response of a material in terms of its microscopic structure. Purely mechanical efforts in this direction have not been very successful. However, the example set by the kinetic theory of gases appears promising. Here, the thermodynamic behavior of the macrosystem (the mass element of the continuum) is explained by the mechanics of the microsystem (the total of the particles of which the element is composed). Since the microscopic motion eludes the macroscopic observer in its details, statistics must be used to interpret it on the phenomenological level. As the particles are in continuous motion, the temperature has to be introduced as a macroscopic measure of the microscopic kinetic energy, and the energy flux by microscopic exchange between particles within and outside the element while it

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retains its shape must be interpreted phenomenologically as a heat flow. In short, the attempt to physically understand material response turns continuum mechanics into thermodynamics. The reverse is true as well. As long as thermodynamics confined itself to gases, its connection with continuum mechanics remained concealed. Once it is applied to other materials, the molar volume has to be replaced by the strains and the pressure by stresses, and it becomes clear that thermodynamics and continuum mechanics become inseparable, forming one single branch of science. As a first consequence of this insight, the mechanical energy theorem is to be dropped. True, it still holds for the microsystem, but on the phenomenological level it must be replaced by the first fundamental law of thermodynamics. The next consequence is the recognition of the second fundamental law. Moreover, it is imperative that both laws be applied not to the entire continuum, but to its material elements. In other words, thermodynamics is to be conceived as a field theory in the same manner as continuum mechanics. Once this is recognized, the timehonored restriction to extremely slow processes becomes obsolete, and it becomes possible to treat reversible as well as irreversible processes with the same ease. It is clear, on the other hand, that exceedingly fast processes, where the macro- and microvelocities become comparable, have to be avoided, but then they are exceptions within the framework of continuum mechanics. In the vast majority of cases, deformation of a continuum is an irreversible process. So is heat conduction. Here, the fundamental laws are not sufficient to establish constitutive relations. The gap has been partially closed by Onsager (193 1) with his reciprocity relations. They establish the symmetry of the matrix connecting velocities and forces. In this function they are clearly restricted to linear processes and hence lack the status of a physical law, not to mention the fact that many processes in continuum mechanics are nonlinear. About a generation ago one of the present writers (Ziegler, 1958) proposed an orthogonality principle for irreversible processes, based on the dissipation function and including Onsager’s theory as a special case. He later showed (Ziegler, 1961) that orthogonality (which, incidentally, requires regular or at least regularized dissipation functions) is equivalent to a number of extremum principles. The most appealing of these (applicable also in cases of irregular dissipation functions) postulates that the irreversible process, subject to certain side conditions, always maximize the rate of entropy production. During the ensuing decades these principles were discussed in various publications and restricted to purely dissipative processes of the elementary and the complex types. (The first restriction precludes the presence of gyroscopic forces such as Coriolis forces or magnetic fields. Elementary processes are defined by velocities in the form of components

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of a single vector or tensor; in the complex case more elementary processes may occur at the same time but must be coupled.) Continuum mechanics has proved to be the most fertile field of application of the principles mentioned. Here, they allow one to establish constitutive relations, deducing them from a single pair of scalar functions characterizing the material: its free energy and its idssipation function. The process and some of its results have been described in the first writer’s book on thermomechanics (Ziegler, 1977), and some questions still open in the first edition have been elaborated in the second (Ziegler, 1983). In the meantime it has been possible to generalize and supplement the few applications treated in this book. Besides, more applications have been supplied by Houlsby (1979, 1980, 1981a,b) and by Germain et al. (1983). The writers feel that the time has come to present a systematic account of the leading functions providing constitutive relations. Such an account is the topic of this article. The principle of maximal rate of entropy production is quite general and can be applied with the same ease to small and large deformations. In spite of this, the review will be restricted, wherever the strain tensor becomes part of the development, to its first approximation, that is, to infinitesimal strains. The authors hope that the results may motivate future work extending the method to finite deformations. The next section presents a brief account of the theory. For its justification and for more details the reader is referred to the book (Ziegler, 1983), which will be cited for brevity as Z, followed by the number of the relevant section. Some of the mathematical tools needed are collected in an appendix at the end of the article. The rest of the article is devoted to a variety of materials. It will show that the response of most of the models in use can in fact be derived from their free energy and their dissipation function. It will also show that (and why) a few models do not fit into the pattern.

11. Thermomechanical Theory

A. THERMODYNAMICS Let us consider a mass element of a continuum (Z.4.3), and let us provisionally assume that its state can be described by the absolute temperature 6 > 0 and a set of mechanical parameters, the components of a strain tensor E,,. Let us further assume that d’ is a stress tensor and that the expression 1 := (l/p)a”&,,,

where p is a density, the dot indicates material differentiation, and the summation convention is used, representing the specific power of deformation.

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Here and in what follows, “specific” will always mean “per unit mass.” In an expression like (2.1) it is customary in thermodynamics to refer to the d,] as velocities and to the a ” / p as the associated (specific) forces. Macvean (1968) has studied the various strain tensors proposed in the past. He has shown that only a few of them allow a representation (2.1) of the specific power and thus are thermodynamically acceptable. One of them is Green’s strain tensor, based on the so-called Lagrangean description of the deformation; it is associated with Kirchhoff’s stress tensor, and p has to be interpreted as the density in the reference state. It will be simpler for our purpose to use the so-called Eulerian approach, based on rectangular Cartesian coordinates, and to replace (2.1) by 1 = (l/p)a,d,,

(2.2)

where p is the instantaneous density; a,J is Euler’s stress tensor; and d , = (vl,l + v , , , ) / 2 is the deformation rate in the sense of Prager (1961), with subscripts following a comma denoting partial differentiation with respect to the corresponding coordinate so that is the velocity gradient. It is true that d , cannot generally be interpreted as the time derivative of a strain. However, if we restrict ourselves, where necessary, to small deformations, we may write d,J = d,J, where F,, is the engineering strain and u,]the engineering stress, the dot indicates partial differentiation with respect to time, and p may be considered constant. The entities E,, and 6 will be considered as independent state variables. Functions of them are dependent state variables, also denoted as state functions. Examples are the specific internal energy u ( E , , ,a), the specific entropy s ( E , , 6), and the specific free energy +(&,], 6) .= u - 6s.

(2.3)

If we exclude radiation, the specific heat supply per unit time is - q , . , / p , where q, is the vector of heat flow. The specific entropy supply per unit time is - ( q , / 6 ) , , / p . The first fundamental law =

( l / p ) ( a , , d ,-

(2.4)

%,I)

states that the rate of increase of the specific internal energy is the sum of the specific power of the stresses and the specific rate of heat supply. The second fundamental law may be written j=

r*(r)

+ s*(i)

(2.5)

7

where S*(r)

=

-(l/P)(qJO),,

and

S*(l)

2 0.

(2.6)

It states that the specific rate of entropy increase consists of a reversible and an irreversible contribution. The reversible contribution is the entropy

188

Hans Ziegler and Christoph Wehrli

supply from outside; the irreversible contribution, to be interpreted as an entropy production within the element, is nonnegative. From (2.4) through (2.6) we deduce p(

-

6s) = a . . d . .- 1 9.-9- pas* (' ) ,'I 6 ,

or

U..d..= p ( au - 6-as) d , 'J

V

a&,

a&,

+ p;(-

-

6-as) a6

4 + 9.

(2.7)

+p6~*(".

(2.8)

6

In the special case of pure heating (or cooling) the rate of deformation is zero and the element receives (or loses) heat, whereas no net flow of heat across the element takes place. Thus, d, = 0 and q, = 0 but ql,,# 0. Since the process is reversible and 8 is different from zero, it follows from (2.8) that a u / a 6 = 6(as/a6),

(2.9)

and since both sides of (2.9) are state functions and hence are independent of the particular process, this equation must be generally valid. Thus (2.8) reduces to a$, = p(a!b/aE,)dr,+ (41/6).9,,+ P 6 S * " ' ,

(2.10)

where (2.3) has been used. Equation (2.10) suggests the subdivision of the stress tensor into two parts, u,,=

UF)

+ cry,

(2.11)

in such a way that a ( '41)

=

p(~!bla&r,).

(2.12)

Since the u?' defined in this manner depend on a potential ~!,t which, however, is a function of the E , ~and 6, we call them quasiconservative stresses, and we note that, on account of (2.3) and (2.9), Eq. (2.12) may be supplemented by s=

-a$/as.

(2.13)

For the remainders a:) in (2.11), Eq. (2.10), together with the inequality (2.6), supplies the condition

u f ) d , - (a,,/6 ) q , = p 6 ~ * ( '2) 0,

(2.14)

that is, the so-called Clausius-Duhem inequality. A glance at (2.2) shows that the a y ) correspond to specific forces a f ' / p , which appear in irreversible deformations; they will be denoted as dissipative stresses. Heat flow q1 across the element (in contrast to heat supply, determined by qr,z)is another irreversible process; the corresponding force is - 6 . , / p 6 .

The Derivation of Constitutive Relations

189

The case just discussed is particularly simple. In addition to and 6, most materials also require internal parameters for the description of their state. Let us assume, for example, that a single internal strain tensor a, is needed, and let us denote the associated force by p , / p . Independent state variables are then E ~ aIJ, ~ , 6; the state functions depend on all of them, and the terms with d,J in (2.7) through (2.10) must be supplemented by the corresponding terms with hl,. Thus, (2.12) is to be supplemented by

py

(2.15)

= P(atCr/w,),

and (2.14) must be replaced by Upd,

+ pIp’ci,,

-

(6,,/6)q, = p6s*“’ 2 0.

(2.16)

However, since the alJare internal parameters, the p, do not appear in the first fundamental law, and if we exclude gyroscopic components (i.e., forces that depend on the velocities ci,, in such a manner that their power is always zero, as, e.g., the Lorentz force acting in a magnetic field or the Coriolis force in a rotating coordinate system), we have p, = 0 or equivalently

pip' + p y ’

=

(2.17)

0.

The generalization for more than one internal strain tensor is obvious.

B. ORTHOGONALITY The simplest materials dealt with in continuum mechanics are elastic. They may be defined (Z.5.1) by the conditions a , = 0 and u:’ = 0. Thus, the first two terms in (2.16) are zero. In the isothermal or adiabatic case, the last term vanishes, too, and the deformation becomes reversible process. It is entirely determined by the specific free energy $ ( E , ) : the specific entropy follows from (2.13), the internal energy subsequently from (2.3), and the only constitutive equation (2.12) connects the stresses with the strains. More general processes and those taking place in more general materials are irreversible and require more constitutive relations, connecting the dissipative forces a‘,d’/p, pbd’/p, and - b , , / p . i ) with the velocities d,, ci,, and q l . So far, the only condition at our disposal is (2.16),

‘+?)d, + P ( ? p ) ~-, ( 6 , , / 6 ) q=, Pcp 2 0,

(2.18)

where cp is the specific dissipation function, cp :=

&*(I’

2 0,

which may be considered as a function of d,, the state E,,, a,,, 6.

(2.19) q1 and generally also of

190

Hans Ziegler and Christoph Wehrli

The constitutive equations we are looking for have been established separately for many idealized materials, and the corresponding dissipation functions are easily derived from them. As stated in Section I, however, these relations are not connected by a general law, as should be expected, and attempts to deduce them mechanically from the microstructure of the various materials have not been very successful. Once the microstructure becomes important, however, it is inevitable that statics and hence thermodynamics come into play, and the mechanical reasoning just described can in fact be replaced by an approach that is more in the spirit of thermodynamics. Let us note that, in the chain of reasoning leading from the fundamental laws to the inequality (2.14), the dissipation function (2.19) appeared already in (2.7), long before the dissipative stresses were introduced in order to interpret cp as a specific power. It therefore seems reasonable to invert (2.14.2) the conventional approach: instead of starting from the forces and interpeting cp as a special expression of the specific power and hence as a function of secondary importance, we accept cp as the primary function and propose to derive the dissipative forces from it. This is equivalent to postulating (as already at the end of Section II,A) that the forces have no gyroscopic components. If it is possible to deduce the dissipative forces from cp in a similar manner as the quasi-conservative forces from I,!J, each material is characterized by two scalar functions I,!J and cp. In the case of pure heat conduction, the velocities q, are the components of a vector. The same is true for the dissipative forces Fld’ := -a,,/@. The dissipation function cp(q,), assumed to be regular, may be described in velocity space by the dissipation surfaces cp = const. (Fig. 1). Equation (2.18) reduces to Fld’q,= cp(q,).

(2.20)

It determines the projection of the vector Fld’ onto ql, leaving, however, its direction free. If this direction is to be given by rp, it must, as has been shown in (Z.14.3), be determined by the gradient acp/aql of cp in velocity space and hence by F (I d ) =

where v

= cp[(dcp/dqk)qk]-’

v(acp/ast),

ensures that (2.20) is satisfied.

pl=const

FIG. 1. Orthogonality in velocity space.

(2.21)

The Derivation of Constitutive Relations

191

Geometrically, Eq. (2.21) may be interpreted as an orthogonality condition: the dissipative force Fld’ corresponding to a velocity q, is orthogonal to the dissipation surface cp = const passing through the end point of 4,. Equation (2.21) maps the velocity space onto the space of the dissipative forces. If this mapping and its inverse are single valued, it can be shown (2.14.5) that the dissipation surfaces are star shaped with respect to the origin, strongly convex, and ordered in the sense that each of them encloses those with smaller values of cp. It can even be demonstrated (2.15.3) that the orthogonality condition implies that both sides of (2.20) are nonnegative, so that the second fundamental law is automatically satisfied and the vector Fjd’ has the direction of the outward normal of the dissipation surface. All these results remain valid in the case of pure deformation without internal parameters or heat flow. Here, the velocities d , are the components of a symmetric tensor and so are the dissipative forces F‘,d’ := ‘+‘,d)/p. Both tensors may be represented as vectors in a six-dimensional space, together with the dissipation surfaces cp(dtf)= const. In place of (2.20) and (2.21) we now have (2.22) (2.23) where v

=

cp“W/dd,l)

dkll-’.

The dissipation surfaces have the same properties as in the case of heat conduction. Certain cases where cp is not regular will be treated in Sections VI and VII. If, finally, the ci,, are the only velocities-that is, in cases where heat flow and deformation are absent, whereas the internal parameters are still in the process of approaching their equilibrium values-the dissipative forces are F (,d ) := P y ’ / p , and (2.22), (2.23) hold again provided that d,, is replaced by 4,.

C. MAXIMALRATE OF ENTROPYPRODUCTION In order to discuss a few more general results, let us denote the various velocities considered in Section II,B by ui and the dissipative forces by Fid’, where the number of velocity components is three or six according to whether ui is a vector or a symmetric tensor. It has been mentioned in Section II,B that, in general, the dissipation function depends not only on the velocities, but also on the independent state variables. If it is a function of the velocities alone, the results obtained in velocity space have their corollaries in force space. Let cp’(Fld’)be the

192

Hans Ziegler and Christoph Wehrli

dissipation function in terms of the dissipative forces. Then, the corollary (2.14.3) of the orthogonality condition states that the velocity u, corresponding to a dissipative force Fid’ is orthogonal to the dissipation surface cp’ = const passing through the end point of Fld’. A particularly interesting example is the theory of the plastic potential (Section VI,A). If, on the other hand, the dissipation function p also depends explicitly on the independent state variables, the corollary need not hold. In soil mechanics, fro example, the theory of the plastic potential breaks down (Section VII). It can be shown (Z.15.1) that the orthogonality condition is equivalent to various extremum principles. Of particular interest is the principle of maximal dissipation rate: Provided that the dissipative force Fld’ is prescribed, the actual velocity u, maximizes the dissipation rate I ( d ) = F ‘ d ’ ~ subject to the side condition I

cp(u,) = Fjd’u,=

> 0.

*

(2.24)

It is obvious on account of (2.19) that this principle may also be stated as a principle of maximal rate of entropy production: Provided that the dissipative force Fid’ is prescribed, the actual velocity u, maximizes the rate of entropy production s*“) subject to the side condition s*ll’

-

(l/-S)Fjd’u, > 0.

(2.25)

We note that these extremum principles are slightly more general than the orthogonality condition: they are still applicable in cases where cp exhibits irregularities corresponding to singularities like edges or corners in the dissipation surfaces. To treat such cases by means of orthogonality, edges or corners have to be smoothened; that is, the singularities must be considered as limiting cases of regular dissipation functions. It often happens that a dissipation function is quasi-homogeneous, satisfying a functional equation of the type (2.26)

where the function F(cp) is free except for the condition F ( 0 ) = 0. The corresponding dissipation surfaces are similar and similarly located with respect to the origin. Ziegler (1963) has proved that in this special case the dissipative forces may be derived from a potential, which is constant on dissipation surfaces in velocity space. It is easy to show that the inversion is true: representation of the dissipative forces by means of a potential requires that this potential (and hence the dissipation function) be quasihomogeneous. This result, together with mathematical contributions by Moreau (1970), gave rise to a theory of “pseudo-potentials” by Germain (1973), which includes singularities like nonexistence of the gradient dcplau, for certain velocities (but excludes dissipation functions dependent on velocities and explicitly also on the independent-state variables). Since

The Derivation of Constitutive Relations

193

quasi-homogeneous dissipation functions are special cases and since their potential has no physical significance, we will not pursue this line of thought. In (2.14.3 and 2.15.1) the orthogonality condition and the equivalent extremum principles have been established for velocities in the form of vectors or symmetric tensors. The corresponding applications in Section II,B were heat low, deformation, and relaxation of the internal parameters, each of these processes considered separately. In general, they occur simultaneously, and the question arises how the orthogonality condition or the extremum principles are to be applied in this case. We obviously have two options (2.14.4). Case (a): If the various processes are independent, each of them is governed by its own orthogonality condition. There are three dissipation functions, p'"( d v ) , ( P ( ~ )k,,), ( and ( P ' ~ 4,). '( Each of them may also depend on E ~ a,, , and 6. Using (2.21), (2.23) and inserting the proper dissipative forces, we obtain the constitutive equations

where

(2.28)

Since the entire dissipation rate is given by (2.29)

the superscripts of cp might be dropped in (2.27) but not in (2.28). Case (b): If, on the other hand, the various processes are coupled, we have no means of establishing the constitutive relations unless we postulate that the orthogonality condition holds in the 15-dimensional space of all velocities d,, ai,, q,. The decomposition (2.29) does not hold, and (2.27) has to be replaced by

where (2.31)

Cases (a) and (b) confront us with an alternative. As observed in the conclusion of (Z), there is no continuous transition between them. In either case, the orthogonality condition ensures (see 2.15.3) that the ClausiusDuhem inequality (2.16) is satisfied; the second law thus appears as a

194

Hans Ziegler and Christoph Wehrli

consequence of the principle of maximal rate of entropy production. With our present knowledge there is no way to decide between the possibilities (a) and (b). The generalization of (2.27) through (2.31) for more internal strains is obvious. Incidentally, in the linear case, where cp is purely quadratic in the velocities and hence homogeneous of degree 2, all of the v’s are $, and one does not make a mistake using (2.30) in place of (2.27). In the applications to be discussed in the next few sections, it is not necessary yet to distinguish between cases (a) and (b). Later applications will be treated both ways or, where this does not seem necessary, according to case (a). In order to obtain the entire stresses (2.11), cry =

uy

+ ohd’,

(2.32)

the dissipative stresses just discussed have to be supplemented by the quasi-conservative stresses (2.12), u ( y4 )

where $ is now a function of

= P(W/dEtI),

E ~ a,,, ,

(2.33)

and 6. Similarly, we have (2.15),

Pip' = P ( W / a a t l )

(2.34)

p ! ! ) = -p!@ V ’

(2.35)

and, according to (2.17),

Relations (2.27) through (2.35) permit us to derive the constitutive equations (or inequalities) of any material from its specific free energy rl, ( E , ~ cr,, , 6 ) and its specific dissipation function cp ( d l l ,CU,, qt, E,,, a,,, 6 ) . In many applications, particularly those where p may be considered constant, it is convenient to replace the functions $ and cp by ?If= P*,

a)= Pcpo,

(2.36)

that is, by the free energy and the dissipation function per unit volume.

111. Heat Conduction

Heat conduction without deformation is a purely dissipative process. Its velocity is the heat flow vector qi; the corresponding dissipative force per unit volume is pFjd’ = - 6,i/ 6.

(3.1)

Since the quasi-conservative force is identically zero, and the free energy is constant and may be assumed to be naught. As regards the dissipation function, it is convenient to set the factor 1 / 6 in evidence. Thus, the most

The Derivation of Constitutive Relations

195

general case is governed by the functions

*

=

0,

= P(P =

(1/8)y(qJ,

(3.2)

where y(q,) is a postiive definite function, possibly also dependent on 6. The orthogonality condition (2.21) supplies the constitutive equation

a,#= - y ( 4, ) [ ( a y / aqk ) qk ]-'(a

(3.3)

y/a41).

If CP is quasi-homogeneous, we have (ay/aqk)qk

=

F(y)

and hence

8.t = -[Y(qJ)/F(Y)l(dy/aq~)' (3.4)

If CP is homogeneous of degree r, F ( y ) is equal to ry, and the constitutive equation becomes 6.n = -(l/r)(aY/aq,).

(3.5)

In the case of an isotropic matrerial, @ has the form @ =

(1/fi)7(qd,

(3.6)

where y(q(l,) is positive definite and q ( l )= qZ1is the only basic invariant [see (A.l) in the Appendix] of the heat flow vector. On account of (A.6) the general constitutive equation (3.3) becomes 8.r = -CY(4~1,)/q~I,lqr.

(3.7)

In the linear case, the general dissipation function (3.2) reduces to

a) = (1/6)y,q,q,,

(3.8)

where y!, is a symmetric tensor, positive definite and possibly dependent on 6. The dissipation function (3.8) is homogeneous of degree 2, and the orthogonality condition supplies the well-known differential equation of heat conduction in an anisotropic body, =

Z , '

-y,qJ'

(3.9)

Here, it is particularly easy to see (2.15.4) that the orthogonality condition excludes gyroscopic forces, for an antimetric part of yo would represent gyroscopic terms. In a linear isotropic material we have =

(Y/6)q(l,,

(3.10)

where y is a positive scalar, possibly dependent on 6. The corresponding differential equation of heat conduction,

fi,, = - 791,

(3.11)

is equivalent to Fourier's law (2.15.4). Let us stress that even this simple case, where the vectors qz and a,, have opposite driections, relies on orthogonality. To be sure, one is tempted to explain the collinearity of q,

196

Hans Ziegler and Christoph Wehrli

and in an isotropic body by considerations of material symmetry. However, a gyroscopic term in the force - 6.i/6 (easily explained, e.g., in terms of the molecular motion with respect to the rotating earth or in its magnetic field) would be compatible with the isotropy of the material, but it would modify (3.11).

IV. Elastic Solids A. LINEAR ELASTICITY

As mentioned at the beginning of Section II,B, the elastic material may be defined by the absence of internal parameters and dissipative stresses. In the theory of elasticity it is customary to neglect thermal effects, assuming, for example, that the deformation are isothermal. Thus, not only the dissipative stresses but also the dissipative force (3.1) corresponding to heat flow are zero. It follows that the dissipation function is naught and the deformation is reversible. The density may be treated as constant (2.5.3). The independent state variables are the strains ey ;the velocities are the deformation rates d,, = i,,,and the corresponding forces per unit volume are the stresses aI,. The most general case is governed by the functions @ = 0,

q

= P4(F,,),

(4.1)

where 4 is positive definite. According to (2.12) the only constitutive equation is (4.2)

= av'/aE,J.

If the material is isotropic, v' assumes the form ?(&(I)

9

E ( 2 )3

%J,

(4.3)

where = E,,, E ( ~ = ) E ~ E , ~E, ( ~ = ) EyEJk&kl are the basic invariants (A.2) of the strain tensor. On account of (A.6) the general constitutive equation (4.2) becomes (4.4)

In the linear case, the general expression (4.1) for the free energy reduces

to (4.5)

q = LZCyklFtjFklr

where clJklis a postivie definite tensor of order 4 obeying the symmetry conditions Cykl

=

C,ikl

=

Ctjlk

=

ckly.

(4.6)

The Derivation of Constitutive Relations

197

Here, (4.2) yields the well-known stress-strain relations

In a linear isotropic material we have

where A and p are Lami's constants, satisfying the inequalities p

> 0,

3A

+ 2 p > 0.

(4.9)

The stresses are now given by the generalization (+I,

=

A&(l,aJ

+ 2WlJ

(4.10)

of Hooke's law. The assumption of isothermal deformations is an idealization, useful if they are sufficiently slow. If they are fast, as, for example, in elastic vibrations, it is more realistic to treat them as adiabatic. It can be shown [see, e.g., (Z.5.3)] that in this case equations (4.1) through (4.10) remain valid with modified values of C,,k[ and A, provided that I,!Iis interpreted as the specific internal energy, written as a function of E~ and s. Since, in the adiabatic case, entropy supply is zero as well as entropy production, s remains constant, and $ appears as a function of E~ alone.

B. THERMOELASTICITY If the deformation of an elastic body is neither isothermal nor adiabatic, the strain tensor has to be supplemented by the additional independent state variable 6. There also appears an additional velocity q, with the corresponding force (3.1) per unit volume. Both leading functions 9(&,], 6) and @ ( q l ) are now generally different from zero; the last one may also depend on and 8.The stress is still quasi-conservative and given by (4.2). Equation (2.13) may be used to obtain the entropy per unit volume, S

=

-av'/a6;

(4.1 1)

(2.3) supplies the internal energy,

u = 9 + ss,

(4.12)

and heat conduction is governed by (3.3), where y ( q j ) is possibly also a function of E~ and 6. If the material is isotropic, T has the form

W&(I),

E(21, & ( 3 ) ,

61,

(4.13)

and @ is given by (3.6), where y ( q ( , , )may also depend on the arguments of v'. The stresses follow from (4.4), and heat conduction obeys (3.7).

Hans Ziegler and Christoph Wehrli

198

In the linear case, it is convenient to start from a reference state where a, = 0, 6 = a0,and to measure E,, from this state. The free energy may be obtained by expanding q with respect to the small quantities E,,, 6 - 6, and by truncating the expansion after the second-order terms. We thus obtain

q

= 'PO-

so(6 - 60)

+ ;Ct,k[&,,&k[ - K,E,(

PC

6 - 6,) - -(

26 0

6 - 60)2,

(4.14)

where the linear term in E , ~alone is omitted for a reason that will be presently explained and where (2.7.4) the constants 'Po,S o , cgkl, K,,, c are in turn the free energy and the entropy in the reference state, the fourth-order tensor of elasticity constants introduced in Section IV,A, the tensor governing thermal stresses, and the specific heat capacity in the reference state. The dissipation function is given by (3.8). The stresses (4.2) become UZJ = CyklEkl

- K , J ( ~-

(4.15)

60).

Since (4.14) contains no linear term in E,, alone, a,, is zero as required in the reference state, and the two terms on the right of (4.15) represent the stresses due to deformation and to temperature increase, respectively. From (4.11) we obtain

s = so +

K,Ey

+(PC/fi0)(6

- 60),

(4.16)

and heat conduction obeys (3.9). In a linear isotropic material (4.14) has to be replaced by P ' = 'Po - So(6 - 8 0 ) +

A 2

+

,uE(~)

- (3A

+

~ , u ) K E ( , ) (~ 80)

- -(P C 6 - 60)2,

(4.17)

26 0

where K is the coefficient of thermal expansion and (3.8) is to be replaced by (3.10). The stresses (4.15) become Uij

=

[AE(l)

- (3A

2/.L)K(6 - 60)ISij + 2/.L&ij,

(4.18)

the entropy (4.16) assumes the form

s = so

(3h

~,U)KE(I)

PC -(a

-

601,

(4.19)

60

and heat conduction is governed by Fourier's law (3.11). In the special case where the deformation is isothermal, 6 = a0,and (4.15), (4.18) reduce as expected to (4.7) and (4.10), respectively. If the deformation is adiabatic, S = So, and elimination of 6 - 6, by means of (4.16) or (4.19) yields uij

=

[ Cijkl+ ( 60/P C ) KijKkf 1Ekl

(4.20)

199

The Derivation of Constitutive Relations

in place of (4.7) and Uij

=

[A

+ (3A + 2p)*(KZ6o/pC)]E(1)&j+ 2 p . 8 ~

(4.21)

in place of (4.10). The stress-strain relations are thus essentially the same as in the isothermal case, but the coefficients are modified as mentioned at the end of Section IV,A. For most elastic solids, the correction is of the order of a few percent (steel 2%, aluminum 5%). To shed some more light on Eq. (4.21), let us note that, on account of (4.17) and (4.19), the internal energy (4.12) of the linear isotropic material is

where Uo is its value in the reference state. With (4.22) and (4.21) the first fundamental law (2.4), written in the form

u = U..k.. V ‘ J - q. .

(4.23)

+ 2p)K6&(1)- (pC/60)8&.

(4.24)

A J 7

yields

q,,,

=

-(3A

Besides, it follows from (4.18) that (+(I)

= (3A

+ ~ c L ) [ E (I) 3

~ (6 6011.

(4.25)

The specific heat capacity of the material [see, e.g., (Z.8.1)] is defined as the ratio -q,,,/p&. With (4.24) it becomes - q j , j / p & = (c/eO)fi

+(

3 +~ ~ C L ) ( K ~ / P ) ( & ( ~ ) / & ) -(4.26)

If the volume of the element is constant, specific heat

= 0, and (4.26) supplies the

(4.27)

c, = (c/ 6,) 6.

If, on the other hand, the pressure is constant, we have account of (4.25), = 3 ~ and & thus the specific heat Cp =

[(C/Oo)

=

0, hence, on

+ 3(3A + 2 p ) ( ~ ~ / p ) =] 6C,[l + 3(3A + 2 p ) ( ~ ~ 6 ~ / p C(4.28) )].

Equation (4.21), valid in the adiabatic case, therefore becomes Uij

=

{A

+ (A + ~/.“,/c,)

-

11}.8(1)&j

+ 2Wij.

(4.29)

The factor cp/c, recalls acoustics, where it appears in the velocity of propagation of small disturbances. In fact, the velocities of irrotational and equivoluminal waves in an isotropic elastic solid [see, e.g., Kolsky (1963)l would be c1 = [(A

+ 2p)/p]”*

and

c2 = (

p / ~ ) l ’ ~ ,

(4.30)

200

Hans Ziegler and Christoph Wehrli

respectively, provided that the process was isothermal. In the more realistic adiabatic case A has to be replaced by the bracket in (4.29). Thus, c2 remains unchanged and c, becomes CI =

+ 2 p + ( A + 2g/3)(cP/c,, - 1)]/p}1’2.

{[A

(4.31)

The inviscid liquid, to be treated in Section V,B, is obtained from the elastic solid by setting p = 0. The only possible waves are irrotational, and their velocity (4.31) assumes the well-known value c1 = [ ( A / ~ ) ( c , / c , , ) l ~ ’ ~ .

(4.32)

V. Fluids

A. GASES The fluids to be treated in this section will be assumed to be isotropic. They may be defined by the absence of internal parameters and the condition that the quasi-conservative stress be isotropic. Absence of internal parameters and isotropy of the material suggest a free energy of the type (4.13). The quasi-conservative stress is thus given by (4.4), and since it is to be and E ( ~ ) It . follows that E ( , ) and 6 might isotropic, V, is indpendent of be used as independent state variables. In a gas, however, all deformations are possibly large; hence is not a convenient variable. It is customary to replace it by the volume v of a mole. If m is the molecular mass, the specific volume is v l m , and since the specific power of the pressure p is 1

=

-(p/m)v,

(5.1)

the quasi-conservative force associated with u is - p / r n . In the case of an inviscid gas, the dissipation function is independent of the deformation rates and hence of the form (3.6). The specific leading functions are therefore *(u,

a),

P

=

(l/P6)Y(q(iJ,

(5.2)

where the function y may also depend on u and 6. In analogy to (2.12) and according to (2.13), we have -

p/m

= a$/av,

s =

-a+/aa.

(5.3)

The stress tensor is fig =

-

P&,,

and heat conduction is governed by (3.7).

(5.4)

The Derivation of Constitutive Relations

20 1

The ideal gas is characterized (2.8.1) by the molar free energy m * = I c , , d 6 - 6 I ~ d a - R 4 l n - ,v m

(5.5)

where R is the gas constant and c,( 6) the molar heat capacity (in contradistinction to the heat capacities in Section IV,B, which were referred to the unit of mass). Applying the first equation (5.3) to $, we obtain the wellknown equation of state p

=

R6/v.

(5.6)

Heat conduction follows from the second equation (5.2) and obeys (3.7). The second equation (5.3) supplies the entropy per mole,

and (2.3) yields the molar internal energy, mu

=

1

cud$+,

dependent on temperature alone. In the case of a real gas, the molar free energy given in (5.5) has to be replaced by m$

=

I

c,db

-

8

I

v-b a C " d 6 - R6 In-6 m v'

(5.9)

where a and b are van der Waal's constants, measuring, respectively, the cohesion between the molecules and their proper volume. In place of (5.6) we now obtain the equation of state of van der Waals, p

=

R6/(v

-

b ) - a/v'

(5.10)

[compare, e.g., Hatsopoulos and Keenan (19691. The molar entropy becomes ms=I5d8+Rln-, V - b 6 m and the internal energy mu

=

1

c,d6

-

a v

(5.11)

(5.12)

is now a function of 6 and v. For viscous gases, the quasi-conservative stress (5.4) must be supplemented by a dissipative stress tensor, to be calculated as in the next section. For linear viscosity the total stress is given by (5.28) or (5.30), where d , is the deformation rate and p follows from (5.6) or (5.10).

202

Hans Ziegler and Christoph Wehrli

In thermodynamics, inviscid gases have long played the preeminent role. As a consequence, the significance of the equation of state is often overrated, and this raises probems as soon as inelastic materials are to be considered. Once the fundamental significance of the governing functions $ and cp is recognized, these problems disappear and the equation of state becomes a mere accessory. The inviscid gases just treated are defined by their free energies, (5.5) and (5.9), respectively; their equations of state, (5.6) and (5.10), follow from J!,I by means of the first relation (5.3). For viscous gases, (5.6) or (5.10) yield merely the quasi-conservative part - p S , of the stress, and Eqs. (5.28) or (5.30), dependent also on cp and supplying the total stress, are by no means equations of state.

B. LIQUIDS In contrast to gases, the volume changes of liquids are small. Thus, the molar volume may be replaced as an independent variable by and p may be treated as a constant. The quasi-conservative stress is still isotropic and hence of the form -pS,. Its specific power (2.1) is =

(l/P)(T#JE,J = -(P/P)slJEl]

=

-(P/P)&(l);

(5.13)

the only quasi-conservative force is thus -p/p, where p is the hydrostatic pressure. In an inviscid liquid, the leading functions are 6),

W & ( I ) ,

@ =

(5.14)

and 6. The stress, following from (4.2)

where y may also depend on and (A.6), is =

(1/8)Y(%l)),

p = -aW/a&,,,.

-PS,~,

(5.15)

Heat conduction, determined by @, obeys (3.7), and the entropy might be obtained from (4.11). In the linear case, the free energy follows, as observed at the end of Section IV,B, from (4.17) by setting p = 0. We thus have

9=

- So(6 -

A PC 6,) -k -.5;1) - 3hK&(,,(6- 60)--(0 2 2 60

- a,)*.

(5.16)

The dissipation function is given by (3.10). The stress (4.18) becomes u.. = -pa..rJ,

P

=

A[3K(6 - 6,) - &(,)I,

and heat conduction is governed by Fourier's law (3.11).

(5.17)

The Derivation of Constitutive Relations

203

The incompressible liquid is obtained by letting + 0 and A + co. Together with 9,the hydrostatic pressure becomes indeterminate. In a viscous liquid the velocity q1has to be supplemented by the deformation rate d,, where d ( l ,= & ( , ) . Apart from the basic invariants d ( l , , d ( 2 , , d ( 3 ) ,and q ( l ) ,defined by ( A . 2 ) and ( A . l ) ,one generally also needs the mixed invariants ( A . 3 ) , m ( l ,= q,d,]q, and m(,) = q,dlldJkqk.The free energy per unit volume is still given by the first expression (5.14); the dissipation function (Z.15.4), however, becomes @ ' ( d ( , ,d, w , d v , , q ( l ) ,m ( l , >

(5.18)

Two cases have to be distinguished, discussed already in Section II,C. If deformation and heat flow are coupled, the only way of establishing constitutive equations is to apply the orthogonality condition in the ninedimensional space of the velocities d,] and 4,.The dissipative forces are given by (2.30) and (2.31), where the terms with &, are to be dropped since internal parameters are absent. The constitutive equations, obtained by means of (2.32) and Section X,B,2, are

(5.19)

where p is given by the second equation (5.15) and

If, on the other hand, deformation and heat flow are independent, the orthogonality condition has to be applied to the two processes separately. The dissipative forces are given by (2.27) and (2.28), where the terms with CU, are again t o be dropped. The dissipation function, similar to (2.29), has the form @ =

The stresses become

@ l ( d , l , ,4 2 , , 4 3 , )

+ (1/8),Y(q(l)).

(5.21)

204

Hans Ziegler and Christoph Wehrli

where p is again given by the second equation (5.15) and

For heat conduction one obtains (3.7). It has been observed in Section II,C that there is no way to decide at present between the two cases just presented. Let us add a few results for the case where the two processes are independent. If the liquid last considered is free of bulk viscosity, @ has the form @ =

@,(d;,,, d{3J + ( 1 / f i ) Y ( q , J ,

(5.24)

where the primes designate the deviatoric part of the deformation rate. On account of (A.12) the corresponding stresses are

where, according to (A.13), (5.26) The hydrostatic stress is still given by the second equation (5.15) and heat conduction by (3.7). The Newtonian liquid may be obtained as the linear case of (5.19), corresponding to quadratic functions and @, that is, to (5.16) and @ =

W

I

,

+ 2P'42) + ( Y / 6 ) q ( l ) ,

(5.27)

where A ' and p ' , possibly dependent on F ( , ) and 8,determine the viscosity in a similar manner as LamC's constants determine the elasticity of a linear isotropic solid. Since the dissipation function (5.27) is of the type (5.21), deformation and heat flow are automatically independent. According to (5.23) v = and (5.22) reduces to

4,

(+I,

=

( - P + A'd(l,)a, + 2p'd,,,

(5.28)

where p is given by the second equation (5.17). Heat conduction is governed by (3.11). If the Newtonian liquid is free of bulk viscosity, (5.27) has to be replaced by (5.29) @ = 2P'dl2, + ( Y / f i ) % l ) , and (5.28) reduces to c,, = -pa,,

+ 2p'd:I,

with the second equation (5.17) and (3.11) still valid. Incompressible liquids will be treated in the next section.

(5.30)

The Derivation of Constitutive Relations

205

C. INCOMPRESSIBILITY There are two ways of dealing with incompressible materials. To discuss them, let us assume isotropy, absence of internal parameters, and indpendence of deformation and heat flow. Heat conduction can then be treated separately by the methods of Section 111. The first approach disregards incompressibility, that is, the conditions E ( , , = 0, d ( l )= 0 , as long as possible and introduces them only in the final results. Starting from the leading functions

W&(,), &(a,

Ei3),

a),

@(d,I,, 4 2 ) , d(3))

(5.31)

(where @ might also depend on the arguments of q ) and applying the methods expounded in Section II,C, together with (A.6), one obtains

and

where (5.34)

With + 0 the derivative d T / d ~ becomes ( ~ ~ indeterminate. The first term on the right of (5.32) may be written -p6, and represents a hydrostatic pressure. Since it is indeterminate, subtraction of a term ( N / ~ E ( ~ ) ) E does not affect the result. The modified equation (5.32) reads

where the terms containing the derivatives of reasoning, applied to (5.33), yields

w

are deviators. A similar

and (5.37) where an indeterminate hydrostatic term has been dropped since the principle of absent dissipative forces, established in (2.14.3) in connection with orthogonality, requires that dissipative forces whose corresponding velocities do not appear in @ are zero. Incidentally, (5.36) might also be obtained as the deviatoric part of (5.25) for d ( l ,+ 0.

Hans Ziegler and Christoph Wehrli

206

The second approach recognizes the incompressibility conditions from the beginning, starting from the leading functions W E ( * )9 E(3) 3

and introducing Thus,

Q,(h 43))

a),

9

(5.38)

0, d ( l )= 0 as side conditions in the differentiations.

E ( ~= )

u(4)

=

IJ

(a/d&ij)(Y

+ y’E(l)),

(5.39)

where y’ is a Lagrangean multiplier. By means of (A.6) we obtain

u9)=

+ 3(’3*/3&(3))Etk&k~+ y’sq.

2(’3W/’3&(2))&~

( 5.40)

The hydrostatic pressure becomes =

- i u3 ( 4 I1 )

=

-[Y‘

(a*/a&(3))&(*)I.

(5.41)

Solving this equation for y‘ and inserting the result in (5.40), we obtain (5.35). In a similar manner, the modified orthogonality condition u(d) IJ

=

4a/adl,)(Q, + Y”d(1))

( 5.42)

yields u (11 d)

= V[2(a@/ad(2))d~~ + 3(’3@/’3d(3))dtkdk~+ ?“6~1,

(5.43)

and the principle of absent dissipative forces, requiring u y )to be a deviator, leads back to (5.36) and (5.37). If the material just considered is a liquid, 9 is independent of E ( ~ and ) E ( ~ ) It . follows from (5.35), (5.36), and (2.32) that

where p is indeterminate and v is given by (5.37). Equation (5.44) represents a special liquid of the Reiner (1945)-Rivlin (1948) type, characterized by coefficients of d , and of the parenthesis that are coupled by the dissipation function. More general Reiner- Rivlin liquids, with independent coefficients, do not satisfy the orthogonality condition. As far as we know, there is no evidence for their existence. The often-used constitutive equation u, = - p a ,

+ 2P‘(d,,),

d,3))4

(5.45)

defines the so-called quasi-linear liquid (2.9.4). If (5.45) is to be a special case of (5.44), the coefficient of the parenthesis in (5.44) must be zero. The orthogonality condition thus requires that Q, and hence p f depend on d ( 2 ) alone. The incompressible Newtonian liquid is characterized by a quadratic dissipation function, that is, by @ = 2p‘d(,,.

(5.46)

The Derivation of Constitutive Relations

207

Here, (5.44) reduces to uy =

-P&, + 2PL'd,,,

(5.47)

where p is still indeterminate. Quasi-linear and Newtonian fluids are examples of materials whose dissipation functions depend on the second basic invariant d(2)alone. We will encounter more materials of this type and note that here (5.36) and (5.37) reduce to the simple equation u ( I,d )

=

[@(d~*J/ddd,,.

(5.48)

The process of specialization carried through in this section can be inverted: the incompressible Newtonian liquid can be generalized starting from the dissipation function =

2P'dO) + ( Y / 6 ) % , )

(5.49)

and adding terms of successively higher degree in d, and qt, which are expressible in the basic invariants. This has been done elsewhere for independent (Z.16.1) and for coupled (2.16.3) processes.

VI. Plasticity A. RIGID, PERFECTLYPLASTICMATERIALS In this and the remaining sections we will assume that deformation and heat flow are independent. The dissipation function then consists of two parts, dependent, respectively, on deformation and heat flow. The second term may be dropped provided that one uses the appropriate results of Section 111 (with y possibly dependent on deformation and temperature) in order to obtain heat conduction. It has been shown by Houlsby (1979, 1980, 1981a,b) that the treatment of plastic materials such as ductile metals and even of soils may be used on appropriate function W and @. The dissipation function of an arbitrary material can be represented geometrically by means of dissipation surfaces @ = const in the space R of the principal deformation rates d,, . . . , or by surfaces @' = const in the space R' of the principal stresses u l r .... If the mapping between the two spaces is one-one, both families of surfaces have the properties discussed for the cp surfaces in connection with the orthogonality condition in Section II,B. In the case of an incompressible liquid, d(l)= 0. Thus, @ is only defined in the deviatoric plane E (see Section X,B,4) of the space R. It may be represented by curves @ = const in E. Since a!:) is a deviator, (5.44) maps

208

Huns Ziegler and Christoph Wehrli

these curves onto the deviatoric plane E ’ of the space R ‘ , and addition of an arbitrary hydrostatic stress supplies surfaces @’ = const in the form of cylinders with axes perpendicular to E‘. The rigid, perfectly plastic material (2.17.2) may be defined as a special case of the incompressible liquid, characterized by a dissipation function (5.38) that is homogeneous of the first degree in d, (and independent of the state variables). Here, (5.37) yields v = 1, and (5.44) supplies the stress deviator a; = 2(a@,/ad,2,)dt,+ 3(a@,/W,,)(d,,d,,

-

fd(,,S,),

(6.1)

whereas the isotropic part of the stress tensor remains indeterminate. Let us consider the values of @ on an arbitrary ray s emanating from the origin 0 in the plane E. Since the dissipation function is homogeneous of the first degree, @ increases proportional to the distance from 0 on any such ray. On account of (6.1) a; is the same for all points on s. It follows that the curves @’ = const in E’ coincide and define a yield surface in the shape of a cylinder with axis perpendicular to E ‘ . Since @ has been assumed to be independent of the state variables, the corollary (Section II,B) of the orthogonality condition holds. Thus, the cylinder, considered as the limiting case of a dense layer of @ surfaces, is convex, and the vector d representing the strain rate lies in its outward normal in the end point of the vector cr but is of indeterminate magnitude since a;, on account of (6.1), is homogeneous of degree 0 in d,,. The vector d thus obeys the so-called normality condition, and this is sometimes expressed by saying that it is associated with the yield condition, that is, with the equation of the yield surface. Convexity of the yield surface and the normality condition together represent (Z. 10.2) what is usually called the theory of the plastic potential. The simplest case is the u. Mises material, defined by a dissipation function that is independent cf d ( 3 ,and hence of the form @ = k(2d(2,)1/2,

(6.2)

where k is a scalar (possibly dependent on 9).The dissipation function (6.2) is constant, according to (A.14), on circles in E around the origin 0 and proportional to their radii. The deviatoric stress (6.1) or (5.48) becomes

and it follows that

where (A.2) has been used. This equation evidently represents the yield condition. In the deviatoric plane of space R’ it is to be interpreted as a circle (Fig. 2 ) of radius k d , and the entire yield surface is the corresponding circular cylinder. Equation (6.3) (written in principal values) represents the

The Derivation of Constitutive Relations

209

F I G . 2. Yield loci of v. Mises and Tresca material.

normality condition, and from (6.4) we finally conclude that k is the yield stress in simple shear. In the case of a Tresca material the dissipation function depends on d(*, and d ( 3 ) .Since its structure is complicated, it is convenient to define it implicitly (2.17.2). However, in the deviatoric plane it allows the simple representation f

2kdI

*2kd,, *2kd111

(I, - I ) ,

(11, - 11),

(6.5)

(111, -III),

where the roman numerals refer to the six sectors subdividing the entire plane of Fig. 3. On account of (A.15) the function @ is constant on regular hexagons, one of which is outlined in Fig. 3. In order to obtain the stresses we recall that, on account of (5.42), Eq. (6.1) is equivalent to the orthogonality condition

r h = (a/adij)(@+ ~ ‘ d ( l ~ ) >

F I G . 3 . Sectors corresponding to the six definitions of Q in (6.5).

(6.6)

210

Hans Ziegler and Christoph Wehrli

where the Lagrangean multiplier y’ is to be determined so that the right-hand side is a deviator. In the open region I of Fig. 3, (6.6) supplies

the interior of region I is thus mapped onto a single point I’ on the projected axis U , in Fig. 2. The other open regions in Fig. 3 yield the remaining corners of a regular hexagon with center 0‘. On the line dividing regions I and -11 in Fig. 3 we have an additional side condition, d, + d,, = 0. Inserting it with a Lagrangean multiplier y” in the orthogonality condition (6.6), we obtain gi = f(4k

-t y”),

=

-f(2k - f’),

a;,, = -!(k

+ 7”)

(6.8)

in place of (6.7). Since these equations represent the straight line connecting the points I’ and -11’ in Fig. 2, the yield locus is a regular hexagon with sides parallel to the projected axes u,, . . . . If k is again the yield stress in simple shear, the Tresca hexagon, usually obtained by the condition that the maximal shearing stress be equal to k, circumscribes the v. Mises circle. The yield surface is the prism with the hexagonal cross section of Fig. 2. Tresca yield is an example of an irregular dissipation function. Its gradient is not defined on the boundaries between the six sectors of Fig. 3, and this corresponds to the corners of the hexagons on which @ is constant. However, any difficulties can be avoided by rounding the corners, that is, by considering them as limiting cases of smooth curves. The correspondence between the points in Fig. 3 and those of the yield locus then shows that the Bow rule associated with the Tresca yield surface is satisfied.

B. CONSTRUCTION OF

THE

DISSIPATION FUNCTION

The connection with analytical mechanics pointed out in Section I and the fact (Section II,B) that orthogonality need not hold in force space show that the dissipation function @ in velocity space deserves priority over @’ and, in the present case, over the yield locus. Experimentally, however, it is easier to determine the yield locus than the dissipation function. It has been established, for example, that certain ductile metals are neither exactly v. Mises nor Tresca materials [see, e.g., Hill (1950)l. However, the analytical formulation of a more reliable yield condition is not easy, and the construction of the corresponding dissipation function would present another problem. We will confine ourselves, therefore, to the demonstration that the theory of the plastic potential, based on a prescribed yield condition, allows one in principle to construct the corresponding dissipation function. Let F(U:,)= 0 [ H O )< 01 (6.9) be the equation of a yield locus in the plane E’, convex and star shaped

7 l e Derivation of Constitutive Relations

21 1

with respect to the origin, and let us forget for a moment that it might be expressed in the basic invariants cri2, and cri3). For a stress state at the yield limit the normality condition

d,

= V(c3F/d(T:I)

( v 2 0)

(6.10)

describes the corresponding strain rates. Since v is arbitrary, each point on the yield locus (6.9) is mapped onto an entire ray s emanating from the origin in E. Let us first assume that the yield locus is strongly convex. If we let the stress point move along it in a prescribed sense, the outward normal rotates nonstop in the same sense, and so does the image s in the plane E. The value of the dissipation function in a given point of s is the scalar product of its own radius vector d and the radius vector u'of the corresponding point on the yield locus. Since the yield locus is convex and star shaped with respect to the origin Of,the product u' d is nonnegative. It increases proportional to the distance from 0; the dissipation function obtained in this way is thus homogeneous of the first degree. Since each ray s is the image of a single vector u' (even if the yield locus has corners), the dissipation function is single valued. If the yield locus is merely weakly convex, it contains at least one straight section. The stress points lying on it correspond to a single ray s in E. However, since s is orthogonal to the straight section, the scalar product u' d, evaluated for a given point on s, is the same for all corresponding vectors a';thus, CJ is still single valued. Incidentally, that the @ surfaces are convex and star shaped with respect to 0 has been shown elsewhere (Z.14.5).

-

C. ELASTIC,PERFECTLY PLASTICMATERIALS If the initial response of an otherwise perfectly plastic material is elastic, we call it elastic, perfectly plastic. Its treatment requires a set of internal parameters in the form of an internal strain tensor aii. Provided that we identify it with what is usually called the plastic strain &$'I, the difference E 'J. - - E (?IP ) = 8 : ) is the elastic strain. If plastic volume changes can be excluded as in Section VI,A, aii = E $ ' ) is a deviator. Let us assume that the elastic part of the response is linear. The governing functions then follow from (4.8)and the second expression (5.38). They are (6.11)

where ( E is the second basic invariant (A.2) of the tensor E~ - a,,, and where CJ is homogeneous of the first degree in ci,. The external stress

Hans Ziegler and Christoph Wehrli

212

is quasi-conservative and given by u, = (a/as,,)(yr + Y’%)

+ 2P(E,, - %J),

= A&(,$,

where y’ is a Lagrangian multiplier. Since expected connection (4.10),

E(,)

= E::;,

(6.12)

(6.12) supplies the

+~PE?),

(6.13)

a,, = AE~;{S,

between stress and elastic strain. Equation (6.12) can be decomposed according to

a;

- @,),

=

V(1) =

(3A + 2P)E(1).

(6.14)

The quasi-conservative part of the internal stress is

P?)

=

(a/aa,)(*

+ y’ac,,)

=

-2P(E,, - a,)

+ 7’6,.

(6.15)

From (6.15) and the first equation (6.14) it follows that p(4),

,

=-

(6.16)

fl;,

and by analogy with (6.1) we get the dissipative internal stress

PP’

+

= 2(d@/d&,2,)&, 3(dQ/c3&(3))(&&k,

- :&(2)8,,).

(6.17)

Applying (2.35) to the deviatoric parts of p,,, we finally obtain the connection

+

a; = 2 ( 8 @ / 8 & ~ ~ / ) 3(d@/arjl,q’)(Et[’&L!) &~) - f&{2q)SlI)

(6.18)

between stress and plastic strain rate. For an elastic, perfectly plastic material of the v. Mises type, (6.2) suggests the dissipative function @ = k(242q))1’2,

(6.19)

while the free energy is still given by the first expression (6.11),

*

+

(6.20)

= (A/~)E;;;~

(Houlsby, 1979). Thus, (6.18) reduces to

a; = k ( $ & ; Z q ) ) - L / 2 & (‘IP )

5

(6.21)

which is analogous to (6.3). Yield surface is still the circular cylinder (6.4), and (6.21) represents the associated flow rule. In the case of a Tresca material the dissipation function is given by (6.5), provided that we replace d , , . . . , by iip), . . . . Yield surface is the prism with the hexagonal cross section of Fig. 2, and the plastic strain rate obeys the associated flow rule.

The Derivation of Constitutive Relations

213

D. LINEARHARDENING The simplest case of a hardening material corresponds to the governing functions

q

(A/2)~:1)+ P ( E

=

+ c~‘a(2),

- a)(2)

@(&(2),

&(jl),

(6.22)

where @ is still homogeneous of the first degree in &,J. The external stress is again quasi-conservative and given by (6.12), and the connection between stress and elastic strain is (6.13). The decomposition (6.14) is still valid, but in place of (6.15) we obtain

P F ’ = ( a / a a y ) ( q + ~ ’ a ( 1=) )- 2 ~ ( ~-ya y ) + 2 ~ ‘ a + y 7’6, (6.23) for the quasi-conservative part of the internal stress. Instead of (6.16) we now have P(Y)

= - u; + 2PI‘ylJ,

(6.24)

whereas the dissipative part of the internal stress is still determined by (6.17). By means of (2.35) we finally obtain the differential equation U;-

2p‘&‘,P’= 2(d@/aii,q’)i‘,P’+ 3(a@/ai.{,q’)(E$’iif) - f~129’6,~)(6.25)

connecting the stress and the plastic strain. For a hardening material of the v. Mises type, the functions (6.22) become

Y

=

(A/~)E;;,’* +

+ P’E$‘;,

@ = k(2&!5))”*

(6.26)

(Houlsby, 1979), and (6.25) reduces to - 2cLrEv) = k ( 2l &.(P) ( 2 ) ) - 1 / 2 i ( yP ) .

(6.27)

It follows that ((T’- 2/.~’~(’))(2) = 2k2.

(6.28)

The yield locus (Fig. 4) in the plane E ‘ is the circle of radius k f i with

FIG. 4. Yield locus of hardening material.

214

Huns Ziegler and Christoph Wehrli

center C given by the vector 2 p ’ ~ ‘ ”=’ 2 p ’ ( ~ \ ,”.’. .). Yield surface is the cylinder with the circular cross section of Fig. 4, and (6.27) represents the associated flow rule. During plastic Bow, the cylinder moves in the direction of This corresponds to Prager’s hardening rule (1955), which coincides in this special case with the one of Ziegler (1959). It is remarkable that the functions 9 and Q, (together with the condition E::) = 0) determine not only the yield surface and the associated flow rule but also the hardening rule. The hardening material of the Tresca type is governed by the free energy given in (6.26) and by the dissipation function Q, = * 2 k & p ’

(1,

-0,. . . 1

(6.29)

in connection with Fig. 3, where d,, . . . , is to be replaced by i;”’, . . . . The yield locus follows from Fig. 4 provided that the circle is replaced, as in Fig. 2, by a regular hexagon. The plastic strain rate is determined by the flow rule associated with the hexagonal prism, and the hardening rule corresponding to the functions and Q, used here is the one of Prager. It is doubtful whether a pair of governing functions can be found that yields Ziegler’s hardening rule. A slightly more general type of hardening has been mentioned by Germain et al. (1983).

E. RATE-DEPENDENT YIELD Experiments by Manjoine (1944) have shown that the response of certain materials is nearly elastic, perfectly plastic except that the yield stress depends on the strain rate. Materials of this type can be characterized by the free energy (6.20) and a dissipation function like (6.19) or (6.5) with d , , . . . , replaced by &!”I, . . . , and with a factor k that is a function of the plastic strain rate. A simple extension of (6.19) is the function Q, = A[1

+ B(E‘(29’)’/“](2&12)’’2,

(6.30)

where A, B,and n are constants. Since CP depends on it!; alone, the stress deviator is given by (5.48), where d, and v r )are to be replaced, respectively, by i f )and oh.We thus obtain the stress deviator (6.31) and the yield condition a(2) = 2A2[ 1

dependent on the plastic strain rate.

+ B(i124))’/“I2,

(6.32)

The Derivation of Constitutive Relations

215

In the case of uniaxial stress we have I

Furthermore,

=

i$,)= - E ( ’ , I / 2

-

-

2

(6.33)

3ffI.

and hence (6.34)

i.129’ = $ ( & i P ’ ) 2 .

Thus, (6.31) yields a, = *A&[l

+ B ( ~ ) ’ / n ( i i p ) ) 2 / n ] ( k i p ) S 0).

(6.35)

With the notations A a = : a,,

2 / n =: l/p,

B(:)’/“=: D-llp,

(6.36)

(6.35) becomes a, = ao[l+ ( i ‘ I ” ’ / D ) ’ / p ] ( i i p ) > 0)

or iip)= D ( c , / c T-~1)’

( f f ,

’uo).

(6.37) (6.38)

This is the relation deduced by Cowper and Symonds (1957) and Bodner and Symonds (1962) from an analysis of Manjoine’s test results. In Fig. 5 , a,/uOis plotted against ii”’/D for a few values of p . The parameter a. is the yield stress for vanishing plastic strain rate, and D is the value of E i P ) for which the yield stress becomes 2ao. Comparison of

FIG. 5. Rate-dependent yield in simple tension.

216

Hans Ziegler and Christoph Wehrli

Figs. 5 and 11 shows that the material considered here might be characterized as elastic, viscoplastic.

VII. Soils A. NONASSOCIATED FLOW

The response patterns described in the preceding section are useful models for the actual behavior of ductile metals. One is therefore tempted to assume that normality in force space and, in particular, the flow rule associated with the yield condition of a plastic material, are necessary consequences of the orthogonality condition formulated, as in Section II,B, in velocity space. However, this is not the case. Counterexamples like soil and concrete have been known for a long time. Experiments by Richmond and Spitzig (1980) have shown that certain steels and polymers subjected to high pressure contradict associated flow. These examples do not invalidate maximal rate of entropy production. In fact, Houlsby (l979,1981a,b) has demonstrated that, for certain materials, the orthogonality condition, as formulated in Section II,B, supplies yield conditions and flow rules that are not associated. As shown elsewhere (Z.14.3), where the orthogonality condition has been established in velocity space, it implies normality in force space only under the condition that the dissipation function depends on the velocities alone. If it also depends on the independent state variables, normality in force space is not to be expected and, as a consequence, yield conditions and flow rules need not be associated. As stated in Section I, there is no general duality between velocities and forces. The elastic, perfectly plastic material of Section VI,C is distinguished by a dissipation function dependent on the internal strain rates alone. It hence obeys the associated flow rule. Materials like soils, on the other hand, may be characterized by governing functions of the type

together with the conditions that the internal strain a,, = E:) be a deviator and that be homogeneous of the first degree in the internal strain rates. The only difference with respect to (6.1 1) is the dependence of the dissipation This does not affect the reasoning leading function on the dilatation from (6.11) to (6.18). On account of the second equation (6.14), however, the argument in the dissipation function (7.1) can be replaced by ( T ( ~ ) . It is therefore customary to talk of pressure-dependent yield. Quite a number of models have been proposed to describe the response of the materials in question. They have been collected, among others, by

The Derivation of Constitutive Relations

217

Chen and Saleeb (1982). A few of them will be presently discussed, others in Section VII,B, and it will be shown that their response follows from leading functions of the type (7.1). Case 1: The simplest case (2.17.6) is the material with the free energy (7.1) and the particular dissipation function @ = A[B -f i ( A

+ $ / L ) E , ~ ~ ]=C A~ [~ B~ -~ ( U ( ~ ) / & ) ] ( E ! ; ; ) ” * ,

(7.2)

where A, B are positive constants and the inequality

+ fip)Ei,,

&(A

= u ( ~ ~ /5&B

(7.3)

is to be repeated since @ is nonnegative. From (5.48) and (7.2) we obtain the stress deviator U; =

A[ B - u(,~/V?)( ii;’)p1’2EF)

(7.4)

c12)= A ~ ( B -

(7.5)

and the equation

of the yield surface. In the space R’ it is a circular semicone with axis g (Fig. 13 later in chapter) and the longitudinal section of Fig. 6. The vertex is determined by B, and A is the tangent of the semiaperture. The yield stress in hydrostatic tension is uil)= B&; in simple shear it is k = A B / a (compare Section V1,A). In uniaxial stress ul we have a(1) = u,and = 2u:/3. Thus, the yield stresses in simple tension or compression are u: = A B f i / ( f i

+ A),

u; = - A B & / ( f i - A ) ,

(7.6)

respectively. It follows that u: always exists and is smaller than B&, whereas a; only exists if A < The perfectly plastic material follows from (7.2) by letting B + co and A + 0 so that AB + k f i . The yield condition (7.5) is equivalent to the one proposed by Drucker and Prager (1952) for soils and confirmed by Richmond and Spitzig (1980) for certain steels and polymers under high pressure. Since = 0, the

a.

FIG. 6.

Longitudinal section of Drucker-Prager yield surface.

218

Hans Ziegler and Christoph Wehrli

tensor i?) is a deviator. The vector i(p) in Fig. 6 is therefore parallel to the deviatoric plane E’ and not normal to the yield surface as assumed by Drucker and Prager. In fact, the flow rule (7.4) is not associated with the yield condition (7.5). So far, the strongest support for normality in stress space have been Drucker’s postulates (1951). They are obviously not generally valid. At the vertex of the semicone, (7.3) holds as an equation, and (7.4) yields ah = 0. The corresponding vector i ( pis)still parallel to E ’ , but apart from this its direction is arbitrary. Case 2: Certain materials, such as cohesionless soils, respond similarly > 0. One to the one just treated but cannot sustain stress states with a(,) possibility of dealing with them, proposed by Houlsby (1979), is equivalent to using (7.2) with B = 0. The result is a yield cone with vertex at 0’ and again a nonassociated flow rule. Case 3: Another model, used extensively, is obtained by truncating the cone of Fig. 6 , keeping only the portion where u(,) < 0 and closing it by a circular area in the deviatoric plane. In this plane the normality condition breaks down since it requires, together with the deviatoric character of I?(’), that i ( p=)0. However, if we start from the functions 1I’ and @, restricting (7.2) to the domain a(1) 5 0 and setting A = 0 for a(,) = 0, Eq. (7.4) yields = 0 for a(,) = 0, and the corresponding vector i ( p represents ) an arbitrary deviatoric strain rate. Here, no problem with normality arises in the plane E’, for the yield surface appears as a truncated semicone open at either end and complemented by a single point at the origin.

VC

B. VARIOUSMODELS and @ used in A few models of soils are defined by the functions Section VII,A. In order to discuss additional models, we retain the expressions (7.1) for the governing functions, the assumption that @ is homogeneous of the first degree and the postulate that the plastic strain is a deviator. We thus have a yield surface in stress space, a vector that is always parallel to the deviatoric plane E’, and in general no normality in stress space. Case 1: Let us start with the dissipation function @ = { A [ B - (u(l)/J5)]i~!))}1/2,

where A, B are positive constants and

From (5.48) and (7.7) we obtain the stress deviator

(7.7)

The Derivation of Constitutive Relations

219

anc he equation 4 2 ,

=

A(B -

%/m

(7.10)

of the yield surface. In the space R’ (7.10) represents a paraboloid of revolution (Fig. 7). The yield stress in hydrostatic tension is c l ~ ( = ~ ) B&; in simple shear it is k = d m (compare Section V1,A). The cohesionless case of the material just considered can be obtained 5 0 and either by setting B = 0 or by restricting (7.7) to the domain setting A = 0 for a(,)= 0. Case 2: The dissipation function

CD

=

A { [ B- ( ( + ( 1 ) / ~ ‘ 3 ) ] ~

-

( B - C)2}1’2(i)2q))1/2,

(7.11)

where C < B is another positive constant beside A and B and where (+(I)

s cJ7

(7.12)

is a generalization of (7.2). The corresponding stress deviator (5.48) is 0;=

A { [ B-

-

( B - C)2}1’2($~2q))-1/2$~), (7.13)

and the equation of the yield surface is (+[2)

= A 2 { [ B-

(~(~ - () B/ - C)’}. fi)]~

(7.14)

In the space R’ (7.14) represents one of the two shells (Fig. 8) of a hyperboloid of revolution with the asymptotic cone of Fig. 6 . The yield stress in hydrostatic tension is a(,)= C&; in simple shear it is k = A(BC - C2/2)1’2. The cohesionless case is obtained by setting C = 0 or by restricting (7.11) 5 0 and setting A = 0 for to = 0. Case 3: Another dissipation function is

CD

=

+2[k

- ((+(I)/&)]${’)

(1, -0,. . .,

FIG.7. Longitudinal section of yield surface in case of VII,B.

(7.15)

Hans Ziegler and Christoph Wehrli

220

FIG. 8. Longitudinal section of yield surface in case of VII,B

(7.16) and I, - I , . . . , are the sectors obtained from Fig. 3 provided that the notations dl, . . . , are replaced by &ip),. . . . The reasoning following (6.5) now yields -4

I -

3[k - ( ~ ( i ) / f i ) I ,

uii =

alii

=

- $ [ k - ( ( ~ ( i ) / & ) l , (7.17)

that is, (6.7) with k replaced by the expression between parentheses. For ( T ( ~ )= 0, (7.17) reduces to (6.7); yield locus is therefore the Tresca hexagon of Fig. 2 . For nonvanishing values of ( T ( ~ ) , (7.17) supplies yield hexagons yield surface is whose linear dimensions are proportional to k - c(~)/&; thus the regular hexagonal pyramid considered by Drucker (1953). Its axis is g ; its intersection with the deviatoric plane E' is the hexagon of Fig. 2 ; and its vertex is the point u(l j / & = k on g, that is, the point with coordinates (k/fi)(l, 1,l). Case 4: Let us finally consider the dissipation function

(7.18)

where A', A", B are positive constants and where I, - I , . . . , are sectors (Fig. 9) containing the projected positive and negative axes P:"", . . . ,respectively. Since 0 must be positive definite, we require ff(1)

5

BJ3,

(7.19)

and we further assume that A' < A",

(7.20)

a condition that will be motivated in connection with (7.30). On account

The Derivation of Constitutive Relations

FIG. 9. Sectors corresponding to the six definitions of

Q,

22 1

in (7.18)

of (A.15) @ is constant on hexagons in the deviatoric plane E, having three axes of symmetry. Their convexity requires that a , / 2 < -u2, that is, ( a , / 2 ) + a2 < 0.

(7.21)

Since must be continuous along the boundaries between the various sectors, for example, on Of‘,we have A’&‘,”’ = -A”&‘,[’ along this boundary or, on account of (A.15), A‘a,

+ A”a2 = 0.

(7.22)

It follows from (7.21) and (7.22) that 2A’ > A”.

(7.23)

Applying the orthogonality condition rh = ( d / d & r ’ ) ( @

+ y’&ir/)

(7.24)

to the open sector I and determining the multiplier y’ by the condition that r : ,is a deviator, we obtain 1

- 2J ‘ ( B

a ; ,=

-U ( ~ I / ~ ) ,

gill = - i A ’ ( B

- u ( , , / A ) .(7.25)

In sector -I, A‘ must be replaced by -A”. In the deviatoric stress plane (Fig. l o ) , (7.25) yields the single point I‘ as the image of sector I in Fig. 9. According to (A.15), its distance from 0’ and the distance of the image -11‘ of sector -11 from 0’are b , = J;AIB,

-b,

=

J~AI~B,

(7.26)

respectively. These points and the corresponding ones on the other projected axes define a hexagon with three axes of symmetry, convex since hl

+ (bJ.2) > 0

(7.27)

222

Hans Ziegler and Christoph Wehrli

FIG. 10. Yield locus corresponding to Shield’s pyramid for u ( ~=)0.

on account of (7.26) and (7.23). In order to see that the sides of the hexagon in Fig. 10 are the images of the rays dividing the sectors in Fig. 9 , we note that (7.22) may be interpreted as the equation of the ray OP. Using it as an additional side condition, we replace (7.24) by a$ = (d/ail,“’)[@

+ y’i:!; + y”(A’E(IP)+ A ” i i f ’ ) ]

(7.28)

and obtain C T ~=

= afr1

+

$ A ’ ( B - a(l)/J5) f y ” ( 2 A ’ - A ” ) ,

-fA’(B - CT(~)/&)

= -iA’(B -

~(l)/&)

+iy”(2A”- A’),

(7.29)

+

- ;?“(A’ A”)

in place of (7.25). With a(,)= 0, (7.29) is the parametric representation of a straight line in Fig. 10, and by means of ( A . 1 5 ) it is easy to see that it contains points I’ for y” = 0 and -11‘ for y“ = -B. Yield locus for a(1) =0 is the hexagon of Fig. 10, having three axes of symmetry, and the yield surface for nonvanishing values of a ( l )is the pyramid intersecting E ‘ in this hexagon and with vertex at a(1) = B& on the axis g. Let the constants in (7.18) be determined by A’ =

2 d 3 sin cp’ 3 sin cp‘ ’

+

A” =

2& sin cp’ 3 - sin cp”

B

= &c

cot cp’,

(7.30)

where c is a positive constant and cp’ an acute angle. With (7.30), the inequalities (7.20) and (7.23) for A’ and A” are obviously satisfied, and the yield surface becomes the pyramid proposed by Shield (1955) as the only correct interpretation of Coulomb’s law. In fact, it is easy to see that, with (7.30), the total stresses corresponding to (7.29) satisfy the condition

+

+

aI- aII (ul uI,)sin cp’ = 2c cos cp’,

which represent Coulomb’s law r=c-utancp’

(7.31)

(7.32)

The Derivation of Constitutive Relations

223

on the straight line connecting the points I' and -11' in Fig. 10. For the other rays dividing sectors in Fig. 10 the proof is analogous. OF C. CONSTRUCTION

THE

DISSIPATION FUNCTION

It has been noted that the materials considered in the two preceding sections do not satisfy the normality condition in stress space. However, they obey a restricted normality condition. In the space R', the hydrostatic stress a(1)is represented by a point P' on g. Once a(,) is prescribed, the end point of the vector u lies in a plane E" passing through P' and parallel to E'. The deviator u'is the projection of u onto this plane. The yield surface intersects the plane E" corresponding to a(,) in a curve that may be considered the yield locus for the given value of In cases 1 through 3 of Section VII,A and cases 1 and 2 of Section VII,B, the yield surface is a surface of revolution with axis g; the yield locus in any plane E" is therefore a circle about P'. On account of Eqs. (7.4), (7.9), and (7.13), written in principal values, the vector i ( phas ) the direction of u'in all these cases and hence obeys the normality condition with respect t o the yield locus in the plane E". Comparison of Figs. 2 and 3 and of Figs. 10 and 9 shows that the same is true in the remaining cases 3 and 4 of Section VII,B. The models just mentioned are not the only ones proposed in literature. Other materials [see, e.g., Chen and Saleeb (1982)] pose the problem already discussed in Section VI,B: to find the dissipation function corresponding to a given yield condition. The solution is similar to the one given there. Let (7.33)

be the equation of the yield surface, with cross sections that are convex and star shaped with respect to their points P'. Postulating that the plastic strain rate is a deviator and that it obeys the restricted normality condition in any plane E", we have

,~ = v ( d F / d a b )

&(PI

(u2

0).

(7.34)

From here on the reasoning following (6.10), with E replaced by E" and d by i ( pshows ), that and how a single-valued function @(E{2q), Elf!, a(,)), homogeneous of the first degree in the plastic strain rates, can be constructed. D. COUPLED ELASTICA N D PLASTICDEFORMATIONS In all of the models discussed so far in Sections VI and VII, it is possible to decompose the total strain into plastic and elastic contributions, E ? ) = a,,

Hans Ziegler and Christoph Wehrli

224

and = E,, - q,, respectively. In those cases where, as in (6.11), the free energy can be expressed in terms of the elastic strains and the dissipation function in terms of the plastic strain rates, it is obvious that the elastic and the plastic deformations are independent. In the case of the hardening material of Section VI,D the free energy (6.22) also contains the plastic strains. However, the connection between stress and elastic strain is still given by the generalized Hooke’s law (6.13), and (6.25) connects the stress with the plastic strain and its time rate. In the soils treated so far, the dissipation function contains also the (elastic) dilatation. However, Eq. (6.13) is still valid and allows one to replace by u(ll i n the dissipation function so that the orthogonality condition supplies relations like (7.4), (7.9), and (7.13) connecting the stress with the plastic strain rate. It follows that in all these examples elastic and plastic deformations can be obtained separately. Houlsby (1979, 1981b) has pointed out that, in certain geological materials, the plastic deformation alters the elastic properties, so that the two types of deformations are coupled. As an example, he mentions the case where the shear modulus is a function of the elastic strain. The simplest model of this type is defined by the modification

w = (A/2)41)

+ ( P + V%))(E

(7.35)

- .)[a

of the free energy (7.1), where v is a constant, and by the dissipation function (6.19). The external stress is quasi-conservative and given by a,, = A E ( l l 6 ,

(7.36)

+ 2 ( P + V a ( 2 J ) ( E y - alJ)

in place of (6.12). The generalization of Hooke’s law now reads

+

u,,= he‘(:;6, 2 ( p

+ ve~$:)E~~’;

(7.37)

besides, (7.36) yields u; = 2(P

+ v a ( > l ) ( E ; - a,).

(7.38)

The quasi-conservative part of the internal stress is

Plp’

=

- 2 h + va[Z))(&,, - alJ)+ 2

4 E

-a

h p y

+ 7‘6,

(7.39)

in place of (6.15). From (7.39) and (7.38) it follows that p(4h lJ

=-

a; + 2vF;;E:;),

(7.40)

and from (6.19) we obtain

Pip'

=

1 .(PI

-I/z€(P) y

k(F(2))

.

(7.41)

Equation (2.35) now yields the relation

+

u; = 2 v ~ ~ ~ ~k($€j24))-1’2i.Ip) ~ I p ’

(7.42)

The Derivation of Constitutive Relations

225

and finally the yield locus

The two constitutive equations (7.37) and (7.42) may be considered as generalizations of (6.13) and (6.21) or (6.27), respectively. Each of them contains a term that establishes coupling between the elastic and plastic deformations. Expression (7.35) seems to be equivalent to the free energy of Houlsby (1981b). In a former paper (1979) he used a slightly different expression

v = (A/2+ P / ~ ) & :-tI )( P

V ~ ( Z ) ) ( &-

a)(*).

( 7.44)

In another publication Houlsby (1980) also derived the so-called “modified Cam-Clay’’ model of Schofield and Wroth (1968) from appropriate functions V and @.

VIII. Viscoplasticity A. CREEP O F METALS

Let us return to the material defined by the governing functions (6.11), dropping, however, the condition that @ be homogeneous of the first degree. Here, the internal strain tensor a,, becomes what is usually called the viscous strain F:’, and the difference F:’ = E,] - E:’ is again the elastic strain. The externl stress is quasi-conservative and given by (6.12). Stress and elastic strain are connected by (6.13) or (6.14), and the quasi-conservative part of the internal stress is supplied by (6.15). Equation (6.16) is still valid; combined with (2.35) it yields u; = Since the dissipative internal stress obeys relations that are analogous to (5.36), (5.37), we have

By”.

=

v[2(a@/’3b(Z))b,,+ 3 ( d @ / a b ( 3 ) ) ( & & k j

-

3b(?.1fi,)1,

(8.1)

where I/

= @(2(a@/ab(,))b,,,

+ 3(a@/ab(l,)cy(l,)’.

(8.2)

The simplest special case is the one where the dissipation function depends on a ( 2 )= &$; alone. In the deviatoric plane E of the system E ; ” ) , . . . , the function @ is then constant on circles about the origin. Equations (8.1) and (8.2) reduce to a; = 2v(a@/a&j;j)&!:”,

v = @[2(a@/a&;;;)&;,;]-l

(8.3)

or, equivalently, to the single relation (5.48), u;,= ( @ / 6’1 y )F,, .(“I

f

(8.4)

Hans Ziegler and Christoph Wehrli

226

Let us specialize further, assuming that

(8.5)

@ = 2p'(&j;)y,

where p' is a coefficient (possibly dependent on 6).This dissipation function is homogeneous of degree 2n. With n = 1 it is analogous to the dissipation function (5.46) of the incompressible Newtonian liquid; with n = 4, a case we will exclude in what follows, it reduces to (6.19),that is, to the dissipation function of a plastic material of the v. Mises type. With (8.5) the deviatoric stress (8.4) becomes a:, = 2 c L ( ( & 3 - l'1& ( u )

(8.6)

and by comparison with (5.45) we see that the viscous response is the one of a quasi-linear liquid. From (8.6) we obtain

and hence the inversion of (8.6), &(U) tj

-

l-nj2n-l

,2

( u ; ~ , / ~1 c L

(q2CLf).

(8.8)

On account of (8.5) and (8.7), @' in the space R' is constant on circular cylinders with axis g, and (8.8), written in principal values, establishes orthogonality in R'. With (1 - n)/(2n - 1 ) =: rn

-

1

and

(2p')l'l-2n =: k

(8.9)

(8.8) assumes the form & (ZJ a )

=

k((T[2))m-1(Tk'

(8.10)

This is Odqvist's equation (1934; see also 1966) for secondary creep of incompressible materials, a generalization of the so-called Norton's law (1929). Another special case is obtained if one replaces (8.5) by @ = A(E(IU)2)n (I, -I), . . . ,

(8.11)

where A is a positive scalar (possibly dependent on 6) and where the roman numerals I, - I , . . . , refer to the six sectors of Fig. 3, provided that d l , . . . , are replaced by . . . . The function @ is constant on the regular hexagons mentioned in connection with (6.5). In the open sectors I and -I the deviatoric stress becomes

&iU),

gf = ~ A ( & $ u ) 2 ) n - ' & ~ ( ~uf ), = ,

gill= -fA(&(1u)2)"-1&iU). (8.12)

The images of I and -I are thus the projected axes ( T ~ , -vl,respectively, in Fig. 2, and similar statements hold for the other sectors. The line dividing the sectors I and -11 in Fig. 3 is mapped onto the open sector between the projected axes (T, and -uI1in Fig. 2, and in this sector, as well as the one

The Derivation of Constitutive Relations

227

between the projected axes -uIand vII,(8.12) has the inversions &!U)

= -i'"' =(3/2~)

(u;2)l

-n/2n-l

4,

i ( U )

111

- 0.

(8.13)

In the remaining sectors in Figs. 3 and 2 similar expressions are valid.

B. ELASTIC,VISCOPLASTIC MATERIALS The free energy and the dissipation function given by (6.11) may also be used to define elastic, viscoplastic materials. As shown in the preceding section, stress and elastic strain are connected by (6.13) or (6.14) and the stress deviator follows from (8.1) and (8.2). Let us first use the special form (8.5) of the dissipation function (2.17.1). It has been noted that it corresponds for n = 1 to a modified Newtonian liquid and for n = 4 to an elastic, perfectly plastic material of the v. Mises type. Elastic, viscoplastic materials may be characterized by an exponent n that is slightly larger than $. To show this, let us note that, in principal axes, (8.6) assumes the form u) n 1 . ( u ) u;= 2 p ' ( & 9 - & I ,. . . .

In the case of uniaxial stress uIwe have af= (2/3)aI and in analogy to (6.33) and (6.34). Thus, (8.14) yields

(8.14)

&$'] = (3/2)d1"'2

uI= 21.'")"(&'1"'')"-'&ID'.

Figure 1 1 , which, for negative values of

$Iu),

is to be reflected at the origin,

I

FIG. 11.

(8.15)

m 3 Response of viscoplastic materials in simple tension.

228

Hans Ziegler and Christoph Wehrli

shows u1/2p’ as a function of i;’)for a few values of n, including the Newtonian and the perfectly plastic cases. For f < n < 1 the curves leave the origin with a vertical tangent. For n + they approach the vertical axis and the horizontal cr1/2p’ = that is, the perfectly plastic response. Another choice of the dissipation function is

m,

+

(1’)

1/2

@ = 2aEr4’j’ k(2d(,,) ,

(8.16)

where a and k are scalars. The corresponding stress deviator (8.4) is u; = [2a

+ k(i&(Z))

1 .(I>) - 1 / 2

Id‘,”’.

(8.17)

This is the constitutive equation proposed by Hohenemser and Prager (1932) for viscoplastic materials of the Bingham type (1922). Let us finally note that the dissipation function (6.30) exhibits a certain similarity to (8.5). This similarity appears also in Figs. 5 and 11. The principal difference is that the curves of Fig. 5 start from the point ul/uo= 1, whereas those of Fig. 11 start at the origin. Materials with rate-dependent yield thus have a definite yield stress for vanishing strain rate, whereas the stress of the material considered in this section tends to zero with &:L’) + 0. It is questionable whether this difference is practically observable. Germain el al. (1983) mention a few more general viscoplastic materials and an application to damage of ductile materials. Returning to Fig. 11 we observe that, for n > 1, the curves (as the one displayed for n = 5 ) leave the origin with a horizontal tangent. With increasThus, ing n they approach the horizontal axis and the vertical E i ’ ) = the dissipation function (8.5) may also be used for materials that tend to the locking material described by Prager (1957).

m.

IX. Viscoelasticity A. LINEARVISCOELASTICITY The response in pure tension of the materials treated in Sections IV through VIII can be modeled by simple combinations of springs, dashpots, and other simple elements as described, for example, by Lee (1962). On the other hand, such models, suggested by the results of tension tests, may be used to establish the general response of more complicated materials. The description of viscoelastic materials can be based on the Maxwell grid (Fig. 12), where certain elements might be dropped but none of the Maxwell elements is to be replaced by a single dashpot if impact response is to be ensured (2.18.1). (The optional single spring allows modeling of solids as well as fluids.) To simulate the actual response in simple tension, it is usually necessary to introduce quite a number of internal parameters

The Derivation of Constitutive Relations

229

FIG. 12. The Maxwell grid. a(2)

, ' . . , a ' " )besides the external extension F . The generalization for arbitrary deformations requires a set of internal strain tensors a!," ( r = 1 , 2 , . . . , n ) beside the external strain E,,. The linear case corresponds to quadratic functions and @. If we confine ourselves to the isotropic case, readmitting thermal processes, the governing functions (2.18.2) are generalizations of (4.17) and (5.27). The free energy may be defined by a(l)

1

w

n+ I

-

(6-

PC 1 (3h'" f 2p'r')K'r)(& a ( ' ) ---(a -

)(I)

26 0

r= I

-

ad2, (9.1)

where p is constant and a!:+')= 0, and the dissipation function by (9.2)

The entropy might be obtained from (9.1) by means of (2.13) and the internal energy subsequently from Eq. (2.3), modified by addition of the arguments a;) besides q, and 19. Since (9.2) does not contain i,,, the external stress is quasi-conservative and given by

n+l

- ( 6 - 6,,)6, r=

The internal stresses are

1

(3h'"

+ 2pCL(r')K'r).

(9.3)

Hans Ziegler and Christoph Wehrli

230

(9.5) where the fact has been used that v = since @ is quadratic in b!,‘). Application of the orthogonality condition to q, yields -

6,l/6

=

; ( w a s l )= ( Y / 6 ) 4 ,

(9.6)

and hence Fourier’s law (3.11). Equation (2.17), applied to the various internal stresses (9.4) and (9.5), supplies the differential equations A ( r ) ’ b ( r ) 6 1 J+ 2p(r)r(y(r)= [ A ( r ) ( E - a ( r ) , - (3A‘” + 2 ~ ‘ ~ ) ) (1)

x K‘”(6

-

60)]6,

+ 2p(r)(E, - a!,‘’).

(9.7)

The problem that remains is the elimination of the internal parameters from (9.3) and the n equations (9.7). To solve it, it is convenient to decompose these relations into their deviatoric and isotropic parts, obtaining fl+l

u; = 2

c /P(&; - a:;)’),

r=l

and t L ( r ) ~ b t= ) r /dry&; - at)’), K ( r ) ’ b ( r) K ( ~ ) [ (-Ea ( r ) ) (-l 3) ~ “ ’ ( 6- 6 ~ 1 , ( r = 1 ~ 2 ,. . , n )

(9.9)

where K ( r )= A ( r ) +Z

(r)

3tL

and

~

(

r =) A ( r P

+ ;,,(r)i

(9.10)

are bulk moduli. If we differentiate the first equation (9.8) n times, expressing k(r)r after each step in terms of E ; - a!;)’ by means of the first equation (9.9), we obtain the time derivatives c+;, &, . . . , u:)’ as linear functions of the i;, i;,. . . , E:)’ and the various E ; - a:)’. Eliminating the n terms E ; - a!;)’ from the resulting n + 1 equations [the first equation (9.8) and its n derivatives], we are left with a single tensorial differential equation of the type u;

+ p(I)’c+L+

* * *

+ p(n)tF!,n)f =

q(0)tE. + q ( l ) t i ;

+

. . . + q ( n ) r E (, n ) ’

(9.11)

between the strain and stress deviators. If we further differentiate the second equation (9.8) n times, expressing b!:; after each step in terms of ( E - 3~(‘)(6 - a0)by means of the second equation (9.9), we obtain . . . , air] as linear functions o f i ( l )- 3 ~ ( ~ ).G(~) 8 , - 3 ~ ( ~ ) 8. .,,. the &(n) (1) - 3K(r)6(n) ( r = 1,2,. . . , n + 1 ) and of ( E - a ( r ) ) (l )31(“)(6 - a0) ( r = 1,2, . . . , n ) . Elimination of the last n expressions from the n + 1

The Derivation of Constitutive Relations

23 1

equations we dispose of leaves a single scalar differential equation of the type

+ p%(,) + . . . + p'"'a;;; =

+

u(l)

q(0)E(I)

+ r"'(6

-

+

f

*

.+ q(n)E;;))

8,)+ r ( ' ) &+ . . . + r(n)a(n) (9.12)

Except for the thermal terms, (9.11 ) and (9.12) are the differential equations commonly used in texts on viscoelasticity [e.g., Fliigge (1975)l. It is clear that the coefficients are not free but determined by the functions and @, that is, by A'", p ( r ) A"", , P ' ~ ) 'and , K ( ~ )A . simple example and a generalization for nonlinear response are given elsewhere (2.18.2). The case where the material is free of bulk viscosity is obtained by introducing the internal strains a:) as deviators. The second equation (9.9) must then be dropped since the left-hand side becomes indeterminate. The second equation (9.8) reduces to n+l U(1) =

3

c

K ' r ' [ E ( I) 3K"'(8 -

a,)],

(9.13)

,=I

a degenerate form of (9.12), connecting the isotropic parts of E~ and uii with the temperature increase. The remaining equations (9.8) and (9.9) become nfl

u; = 2

c

r=l

/dry&;

- a!;)),

p('"&";' - p ( r ) ( ~ - ;a!)),

(9.14)

and the process expounded following (9.10) supplies a differential equation of the type (9.11) for the deviatoric parts of E,, and uo. If thermal efects can be neglected, the terms with 6 - 6, in (9.1) and with q ( ] )in (9.2) are to be dropped. Thus, the terms containing 19 - 8, disappear from (9.8) and (9.9) as well as (9.13), and the coefficients r"), r(l) , . . . , r ( " )in (9.12) become zero. and the K ( ~ ) are ' zero, whereas the K ( ' ) In an incompressible material become infinite. Thus, (9.13) reduces to (+(I)

=

-3P,

(9.15)

where p is an indeterminate hydrostatic pressure, and Eqs. (9.14) can be written

Elimination of the internal parameters now yields the differential equation (9.11), where the primes after E , , and its derivatives may be dropped.

232

Hans Ziegler and Christoph Wehrli B. RIVLIN-ERICKSEN LIQUIDS

A class of isotropic materials proposed and discussed by Rivlin and Ericksen (1955) is defined by the condition that the stress depends alone on the gradients of the displacement, velocity, acceleration, and higher accelerations up to a certain order, so that thermal effects, in particular, are absent. In the special case where the material is an incompressible liquid, Truesdell and No11 (1965) present the constitutive equations in the form wv = -pa,

+ Fv(AF;, A:),

. . . ,A:'),

(9.17)

where A t ' ( r = 1,2,. . . , n ) are the Rivlin-Ericksen tensors

where v,,, is the velocity gradient and F, a deviatoric function. If we confine ourselves to small displacements and displacement gradients, the tensors A:' reduce to 2~:) ( r = 1, 2, . . . , n ) , and the constitutive equation (9.17) reduces to (9.15) and 0;=

Fv(F,,, i,,, . . . , &I,").

(9.19)

In view of Section IX,A it appears remarkable that the time derivatives of the strains are present in (9.19) up to the order n, whereas the stress derivatives are absent. In any event the question arises whether the RivlinEricksen liquid can be obtained by the approach that has been quite successful in the preceding sections. Let us note that, in the approximation leading to (9.19), the density is to be treated (2.5.3) as constant. If we neglect thermal effects but retain the notion of internal parameters, we have to start from a free energy of the form V C ( E ~ a:)) ,

( r = 1,2,. . . , n).

(9.20)

On account of the orthogonality condition and of (2.17), the dissipation function depends on the time derivatives of exactly those internal parameters that appear as arguments of 9.Besides, it may depend on e,,, a:) and possibly even on F y , so that @(iv, &!,'I,E , , a!,'))

( r = 1, 2, . . . , n ) .

(9.21)

The quasi-conservative stresses obtained from (9.20) by means of (2.33) and (2.34) are functions of the arguments of VC. The dissipative stresses depend on the arguments of a, no matter whether we calculate them by means of (2.27), (2.28) or (2.30), (2.31). According to (2.32) the external stress deviator assumes the form (9.22)

The Derivation

of

Constitutive Relations

233

and (2.35) yields n differential equations of the type

In order to arrive at (9.19) from (9.22) and (9.23) it must be possible to eliminate the internal parameters and their time derivatives. To do this, let us differentiate (9.22) and (9.23) n times with respect to time, obtaining a::, cib;‘, . . . , a K f l ) ( r ) altogether ( n + 1)2equations for F , , ~ ,i p q.,. . , EL+’), and (ib, . . . , u r ) ’ ,that is, for ( n + l ) ( n 3) unknowns. Eliminating the n ( n + 2) internal parameters including their derivatives, we are left with a single equation containing 2n + 3 unknowns, namely, ePq and a; with n + 1 and n derivatives, respectively. In other words, we obtain a differential equation of the type

uL,

+

(9.24) That the order of the operator G, is one higher in E~~ than in uPqis due to the fact that, in the interest of generality and in contrast to Section IX,A, E, has been included as an argument of the dissipation function (9.21). The result (9.24) can be obtained without use of the orthogonality condition. In fact, (9.22) and (9.23) and hence (9.24) follow from the mere assumption that the dissipative stresses depend on the (internal and possibly external) strain rates and strains. In the exotic case of dissipative stresses containing higher-order time derivatives of at;) and e,, an increased number of differentiations would lead to an equation of the type (9.24) with an operator of higher order. On the other hand, it is in general impossible to reduce (9.24) to (9.19). Even in the linear version (9.11) of (9.24) this is not possible. Already in the case n = 1, treated in (Z.18.2), the coefficient p“” of 6;does not vanish. It follows that the approach based on internal parameters and the constitutive equations proposed by Rivlin and Ericksen are incompatible whether the orthogonality condition be used or not.

X. Conclusion The materials treated starting with Section I l l confirm what was stated in the introduction (Section I ) : an amazing number of constitutive relations of practical significance can be deduced from appropriately chosen expressions for the free energy and the dissipation function. Of particular interest is the fact that orthogonality in velocity space, which is essentially responsible for the results, does not necessarily imply orthogonality in force space since there is in general, as noted in Section I, no duality between the two spaces. As a consequence, orthogonality in velocity space is apt to explain the actual behavior of soils (Section VII), where the

234

Hans Ziegler and Christoph Wehrli

theory of the plastic potential and its justification by Drucker (1951) break down. On the other hand, we have encountered a few cases where the orthogonality condition restricts the form of the constitutive equations generally used. They are (Section V,C) the Reiner-Rivlin liquid, where the scalar functions multiplying d, and dikdkj- ( d ( 2 ) / 3 )in (5.44) are not free but determined by the dissipation function, and the quasilinear liquid, where the viscosity function p' in (5.45) must be independent of d ( 3 ) . Another exception is the Rivlin-Ericksen liquid (Section IX,B). Here, the approach based on internal parameters supplies the constitutive equation (9.24) instead of (9.19) no matter whether maximal rate of entropy production is used or not.

Appendix 1. Let u, be a vector and t,, be a symmetric tensor of order 2 in an orthogonal Cartesian coordinate system. The only basic invariant of u, may be defined by U(I) =

('4.1)

u,u,,

the basic invariants of t,] by t(l)

=

tt,,

t(2)

=

'tj'p~

t(3)

=

tvqktki,

(A.2)

and the mixed invariants by m ( l ,= ultvuj,

rn(*)= Uitjjtjkuk.

('4.3)

The invariants of the deviator are

Differentiating the invariants, one obtains

of q, and djk depends only on 2. In an isotropic material a function the invariants of qi and djk (and possibly on certain independent state

The Derivation of Constitutive Relations

23 5

variables). Thus, @ has the form (A.7)

3. Let @ be of the form @(di2),di3)).

According to (A.6) we have

or, on account of (AS) and (A.4),

FIG. 13. Vectorial representation t of a symmetric tensor r,,.

(A.lO)

236

Hans Ziegler and Christoph Wehrli

I tm

FIG. 14. Deviatoric plane E Ig.

It follows that

(A.13) 4. A symmetric tensor t,, with principal values t i , . . . , can be represented as a vector t = ( t I , . . . ) in an orthogonal Cartesian coordinate system tI, . . . (Fig. 13). The istropic part t,,,6,/3 of t,] appears as the projection t , , ] / & of t onto the axis g including equal angles with the positive axes t , , . . . . The deviatoric part t ; is represented by the projection t’ o f t onto the plane E l g passing through the origin 0. Its magntiude is It’l =

(ti2

+ . . .)I12

= (t;2,>”*.

(A.14)

The unit vector e, (Fig. 14) in the projection of the axis tl onto the deviatoric plane E is equally inclined with respect to the axes t i , and t i l l . Hence, e , = (2, -1, -l)/&, and it follows that the projections a , , . . . , of t’ onto the projected axes t,, . . . , are -

a , = t . e , = J ; t i ,. . . .

(A.15)

REFERENCES Bingham, E. C. (1922). “Fluidity and Plasticity,” p. 215. McCraw-Hill, New York. Bodner, S. R., and Symonds, P. S. (1962). Experimental and theoretical investigation of the plastic deformation of cantilever beams subjected to impuslsive loading. J. Appl. Mech. 29, 719-728. Chen, W. F., and Saleeb, A. F. (1982). “Constitutive Equations for Engineering Materials,” Vol. 1. Wiley, New York. Cowper, G . R., and Symonds, P. S. (1957). Strain hardening and strain rate effects in the impact loading of cantilever beams. Tech. Rep. No. 28, from Brown Univ. to the Office of Naval Res., Contract Nour-562 (10). Drucker, D. C. (1951). A more fundamental approach to plastic stress-strain relations. Proc. 1st U. S. Natl. Congr. Appl. Mech., pp. 487-491. Drucker, D. C. (1953). Limit analysis of two and three dimensional soil mechanics problems. 1. Mech. Phys. Solids 1, 217-226.

The Derivation of Constitutive Relations

237

Drucker, D. C., and Prager, W. (1952). Soil mechanics and plastic analysis of limit design. Q. Appl. Math. 10, 157-165. Flugge, W. ( 1975). “Viscoelasticity,” 2nd Ed., p. 164. Springer-Verlag, Berlin. Germain, P. (1973). “Cours de Mtcanique des Milieux Continus,” p. 147. Masson, Paris. Germain, P., Nguyen, Q. S., and Suquet, P. (1983). Continuum thermodynamics. J. Appl. Mech. 50, 1010-1020. Hatsopoulos, G. N., and Keenan, J . H . (1965). “Principles of General Thermodynamics,” p. 232. Wiley, New York. Hill, R. (1950). “The Mathematical Theory of Plasticity,” p. 22. Clarendon, Oxford. Hohenemser, K., and Prager, W. (1932). Ueber die Ansatze der Mechanik isotroper Kontinua, Z. Angew. Math. Mech. 12, 216-226. Houlsby, G. T. (1979). Some implications of the derivation of the small-strain incremental theory of plasticity from thermomechanics. Rep. C U E D I D Soils TR 74, Univ. of Cambridge. Houlsby, G . T. (1980). The derivation of theoretical models for soils from thermodynamics. Rep. Cambridge Univ. Eng. Dept. Houlsby, G. T. (1981a). A study of plasticity theories and their applicability to soils. Ph.D. thesis, Univ. of Cambridge. Houlsby, G. T. (1981b). A derivation of the small-strain incremental theory of plasticity from thermodynamics. Soil. Mech. Rep. SM020//GTH/81, OUEL Rep. 1371/81, Univ. of Oxford. Kolsky, H. (1963). ‘‘Stress Waves in Solids,” p. 13. Dover, New York. Lee, E. H. (1962). Viscoelawsticity, in “Handbook of Engineering Mechanics” (W. Flugge, ed.), pp. 53.1-53.22. McGraw-Hill, New York. Macvean, D. B. (1968). Die Elementararbeit in einem Kontinuum und die Zuordnung von Spannungs- und Verzerrungstensoren. Z. Angew. Math. Phys. 19, 157-185. Manjoine, M. J. (1944). Influence of rate of strain and temperature on yield stresses in mild steel. 1. Appl. Mech. 11, 211-218. Moreau, J. J. (1970). Sur les lois de frottement, de plastici,ti et de viscositi. C. R. Hehd. Seances Acad. Sci. ( S e r . A , ) 271, 608-611. Norton, F. H. (1929). “Creep of Steel at High Temperatures,” p. 67. McGraw-Hill, New York. Odqvist, F. K. G. (1934). Creep stresses in a rotating disc, Proc. 4th Int. Congr. Appl. Mech., pp. 228-229. Cambridge, England. Odqvist, F. K. G. (1966). “Mathematical Theory of Creep and Creep Rupture,” p. 21. Clarendon, Oxford. Onsager, L. (1931). Reciprocal relations in irreversible processes. Phys. Rev. 37(11), 405-426; 38(11), 2265-2279. Prager, W. (1955). The theory of plasticity: A survey of recent achievements, James Clayton Lecture. Proc. Inst. Mech. Eng. 169, 41-57, Prager, W. (1957). On ideal locking materials. Trans. SOC.Rheol. 1, 169-175. Prager, W. (1961). “Introduction to Mechanics of Continua,” p. 64. Ginn, Boston. Reiner, M. (1945). A mathematical theory of dilatancy. Am. J. Math. 67, 350-362. Richmond, O., and Spitzig, W. A. (1980). Pressure dependence and dilatancy of plastic flow. R o c . 15th IUTAM Congr. Theor. Appl. Mech., Toronto, pp. 377-386. Rivlin, R. (1948). The hydrodynamics of non-Newtonian fluids. Proc. R. SOC.Ser. A 193, 260-281. Rivlin, R. S., and Ericksen, J. L. (1955). Stress-deformation relations for isotropic materials. J. Rational Mech. Anal. 4, 323-425. Schofield, A. N., and Wroth, C. P. (1968). “Critical State Soil Mechanics.” McGraw-Hill, New York. Shield, R. T. (1955). On Coulomb’s law of failure in soils. J. Mech. Phys. Solids 4, 10-16. Truesdell, C., and NOH, W. (1965). “Non-Linear Field Theories of Mechanics,” Vol. 111/3 of “Encyclopedia of Physics” (S. Fliigge, ed.), p. 481. Springer-Verlag, Berlin.

23 8

Hans Ziegler and Christoph Wehrli

Ziegler, H. (1958). An attempt to generalize Onsager’s principle, and its significance for rheological problems. Z. Angew. Marh. Phys. 9b, 748-763. Ziegler, H. (1959). A modification of Prager’s hardening rule. Q.Appl. Math. 17, 55-65. Ziegler, H. (1961). Zwei Extremalprinzipien der irreversiblen Thermodynamik. Ing. Arch. 30, 410-416. Ziegler, H . (1963). Some extremum principles in irreversible thermodynamics with application to continuum mechanics. In “progress in Solid Mechanics,” (1. N. Sneddon and R. Hill, eds.), Vol. 4, pp. 140-144. North-Holland, Amsterdam. Ziegler, H. (1977). “An Introduction to Thermomechanics.” North-Holland, Amsterdam. Ziegler, H. (1983). “An Introduction to Thermomechanics,” 2nd Ed. North-Holland, Amsterdam.

A D V A N C E S I N APPLIEI) M E C H A N I C S , V O L U M E

25

Creep Constitutive Equations for Damaged Materials A. C. F. COCKS* AND F. A. LECKIE Department of Theoretical and Applied Mechanics University of Illinois at Urbana-Champaign Urbana, Illinois 61801

NOMENCLATURE Elastic, plastic, and total strains Mean elastic, plastic, and total strains experienced by a grain-boundary element of material with outward normals n f and n: Mean elastic, plastic, and total strains of macroscopic element of material Stress Mean stress in grain-boundary element of material Stress applied to a macroscopic element of material Equilibrium stress field resulting from elastic analysis Residual stress field Thermodynamic stresses associated with different internal state variables Principal applied stresses Stress normal to a grain boundary Elastic compliance and stiffness matrices Constants in creep law of Eq. (3.1) Helmholtz free energy Internal displacement variables Thermodynamic force associated with ak Internal state variable used in the analysis of strain softening Thermodynamic driving forces for irreversible processes Volume Volume of material element with outward normals np and n: Area of grain boundary Volume fraction of grain-boundary elements of volume V, Measure of the density of grain-boundary cracks Void radius Critical void radius required for nucleation Void spacing

* Present address: Department of Engineering, Leicester University, Leicester LEI 7RH, England. 239 Copyright 0 1987 by Academic Press, Inc. ,411 rights of reproduction in any form reserved.

A. C. F. Cocks and F. A. Leckie

240

Number of voids in grain-boundary element Number of potential nucleation sites Outward normals to a grain boundary Strain-rate and damage-rate potential Temperature-dependent material properties Scalar measure of damage Creep damage exponent Creep damage tolerance Strain to failure Time to failure Time to initiate failure in a structure Convex function of stress appearing in damage growth rate equation Value of ~ ( a , ,at) yield for a perfectly plastic material Plastic multiplier

I. Introduction Constitutive equations used by engineers to describe the plastic behavior of materials have largely been developed intuitively, the main concern being that the equations should describe the material macroscopic properties while retaining sufficient simplicity to allow convenient structural analysis. A common and convenient approach is to describe the material behavior in terms of internal state variables and to assume the existence of a scalar convex potential from which the strain rate and rate of change of internal state variables can be derived. This allows the proof of uniqueness and the development of bounding theorems. This intuitive approach is consistent with a thermodynamic description expressed in terms of internal state variables that are obtained from an understanding of the microscopic mechanisms that lead to plastic flow. Rice (1971) proved the existence of a potential from the fact that plastic straining is due to the motion of dislocations through the material. Simple modeling of the dislocation mechanisms further permits the identification of a number of scalar and second-order tensor state variables (Cocks and Ponter, 1985a) (which relate to the shear yield strength of a family of slip systems and the residual stress field developed during plastic straining). The evolution of the state variables can be found from the potential, which can be shown to be convex. The advantage of such an approach is that it gives a framework for the development of more complex constitutive models and, perhaps more important, an indication of those loading conditions for which a single state variable model adequately describes the material and structural response. At high temperatures a material subjected to a constant stress can creep and eventually fail due to the growth of internal damage. The engineering approach is again largely intuitive. State variables that measure the amount of damage are introduced into the constitutive equations. Additional rules

241

Creep Constitutive Equations

are developed for the evolution of the state variables with time, and failure occurs when one of the variables reaches a critical value. This approach has proved successful for situations of constant load and in situations involving moderate levels of cyclic loading. Ashby and Dyson (1984) have examined all the presently known mechanisms of failure in creeping materials. A list of these mechanisms is given in Table 1. They are divided into three groups: geometric, environmental, and bulk mechanisms. Geometric instabilities can be explained by using constitutive equations developed for the bulk mechanisms taking into account changes in geometry resulting from large plastic deformation. Premature failure can occur in aggressive environments due to the loss in load-carrying capacity of a surface layer that increases in thickness with time. This load is shed onto the remainder of the body, which deforms and becomes damaged by one of the bulk mechanisms. Central to the understanding of all these failures is the understanding of the bulk mechanisms of damage to which this paper is devoted. TABLE I TYPES OF DAMAGING PROCESSES

Nucleation and growth o f voids Coarsening of precipitate particles Strain softening Necking Oxidation, internal and external

IDENTIFIED BY

ASHBY

AND

DYSON(1984)

Bulk processes Geometric effect Surface, environment

The object of the present paper is to combine this knowledge of the microscopic mechanisms with experimental observations to obtain constitutive equations for damaged materials. Initially we concentrate entirely on the mechanisms described above and obtain a general structure for the constitutive equations. This gives a framework for the development of constitutive equations for particular materials. I n the present paper we attempt to d o this for the copper and an aluminum alloy tested by Leckie and Hayhurst (1974, 1977). Throughout the development of this paper it should be remembered that any theoretical constitutive equations should be verifiable experimentally. This means that these equations should only contain a limited number of experimentally obtained quantities and state variables. The general structure for the constitutive equations is obtained by following the approach of Rice (1971) in identifying a number of internal state variables. The state of the material is then described in terms of its Helmholtz free energy from which we derive the thermodynamic forces associated with each state variable. When the rate of increase of the internal variables are only functions of their associated forces and the present state of the material,

242

A. C. F. Cocks and F. A. Leckie

it is possible to prove the existence of a scalar potential form which the inelastic strain rate and rate of increase of the state variables are derivable. We find this to be true for situations where the damage is in the form of voids, but, as we shall see in Section VII, this is not necessarily the most appropriate form for the constitutive laws. A situation for which it is convenient to express the material response in terms of a single potential occurs in precipitate-hardened materials when the damage is in the form of dislocation networks that grow around the precipitate particles. The thermodynamics is developed in Section 11. In Section I11 we examine each mechanism of void growth and in Section IV we briefly discuss the process of void nucleation. The analyses of these sections result in constitutive equations with a large number of state variables. This can be partially overcome by formulating the problem in terms of the distribution of damage in the material. We do this in Section V following a method proposed by Onat and Leckie (1984). In Section VI we examine the strain-softening mechanism proposed by Ashby and Dyson (1984) and Henderson and McLean (1983), which has been analyzed by Cocks (1986). Sections I1 through VI are concerned primarily with the basic structure of the constitutive equations. When the damage is in the form of voids we need to know the distribution of these voids within the body before being able to obtain the exact constitutive laws. Constitutive equations are presented in Section VII for two simple distributions of voids that result in zero and full constraint as defined by Dyson (1979). In Section IX we analyze the experimental data available to us to make decisions on the structure of the material laws. In general the amount of information available is quite limited and we must be content with sets of equations that contain a limited number of state variables. The type and physical nature of the state variables that prove appropriate in a given situation can change as the type of loading (monotonic, nonproportional, cyclic) is changed. The remainder of the paper is devoted to the application of the material models to structural problems. It is found that when the rate of increase of damage is mathematically separable in expressions for stress and of damage that it is possible to obtain upper bounds to the life of a component. 11. Thermodynamic Formalism

In this section we present a general thermodynamic description of material behavior in terms of internal state variables that are a measure either of the present dislocation structure or of the distribution and size of voids within the material. We identify the conditions under which it is possible to derive a scalar potential from which the strain rate and rate of change of internal state variables can be derived. In later sections it is shown that the proposed mechanisms for void nucleation and growth as well as that

Creep Constitutive Equations

243

for strain softening satisfy these conditions. The approach described here follows that used by Rice (1971). At a given instant in time we can define the state of the material by using the Helmholtz free energy, which is expressed in terms of the position of any dislocations; the size, shape, and distribution of any voids; and the applied load. This equation of state can only be obtained by postulating a reversible process by which the present state could be reached. The free energy is calculated by following this path to the present state. There may be a large number of such reversible paths, and the one chosen need not parallel the actual path that is followed during the irreversible process. For the situations considered in this paper the reversible paths followed are purely conceptual and cannot be followed in practice. When we define the state of the material in terms of the present dislocation structure, we introduce the dislocations into the material by a series of cutting, displacing, and resealing operations. When voids grow in the material by a diffusioncontrolled process, we form the voids by scooping material out and spreading this material evenly along the grain boundary. In each case the contribution to the free energy can be written as

cc, = f ( a k ) v

(2.1)

where each (Yk represents the position of a dislocation or the volume fraction of voids within an element of material and is either a scalar vector or second-order tensor and V is the volume of the element of material. The total free energy can be found by loading the material elastically to the present stress:

where c u k , is the elastic stiffness tensor, which may be a function of the ( Y k ’ s , and &ekl is the elastic strain resulting from the application of the applied stress IT^. Where a k represents the dislocation structure, the uncoupling of the contributions to the free energy from E: and ak follows from the fact that the stress field that results from the presence of the dislocations is a residual stress field. Here we have expressed the free energy in terms of the elastic strain E ; rather than the total strain E: and the plastic strain E $ . Use of E: and E ; implies that the history of loading is known and, since E : is a function of ai,that the final state is uniquely related to this history, whereas in practice a given dislocation distribution can be obtained by following a large number of different reversible and irreversible paths. Also we can follow two histories of loadings that give the same plastic strain, but completely different dislocation structures. For example, a material that is loaded monotonically to a uniaxial strain E could have a lower yield stress and a completely different dislocation structure to a sample that is cycled between E and - E for a number of cycles.

244

A. C. F. Cocks and F. A. Leckie

The thermodynamic forces associated with the state variables can be found by differentiating Eq. (2.2),

(lr = (gar&:/ + sk&k)

v,

F',

and a,

(2.3)

where

U,v = d$b/d&',

and

S k v

=

d$/dffk,

(2.4)

where s k has the same tensorial nature as f f k . Here, and throughout this paper, a repeating index implies summation over all the internal variables. We now accept the Clausius-Duhem inequality as the proper statement of the second law of thermodynamics for irreversible processes,

V ( d S / d t )+ ( a / a x , ) [ q t / T=] 6 Z 0,

(2.5)

where S is the entropy per unit volume, q, the rate of heat flux out of an element of material, T the absolute temperature, x, the distance from some origin, and 6 the rate of entropy production. For isothermal processes Eq. (2.5) becomes, after making the usual manipulations (Rice, 1971), .T

a,,",

v-

*

Z 0.

(2.6)

Substituting Eq. (2.3) into this equation gives fly&: - sk&k

2 0,

(2.7)

where &: = 6; - &', is the inelastic strain rate. This strain results from the motion of dislocations through the material or from the growth of voids, so that E; =

(2.8)

gk&kr

where g k is a second-order tensor if f f k is a scalar, a third-order tensor if ffk is a vector, and a fourth-order tensor if f f k is a second-order tensor. Combining Eqs. (2.7) and (2.8) gives (r&k

-

Sk)&k

20

or

Fk&k

2 0,

(2.9)

where FL = (a& - s k ) is the thermodynamic driving force for the process. To proceed further we need expressions for the &'s in terms of the state variables and their affinities. For each damaging mechanism examined in the following sections, we find that &k

=

(yk(Fk, a k ) ,

(2.10)

that is, that &k is only a function of the affinity associated with it, Fkr and the present state of the material. In such situations Rice (1971) has shown that it is possible to find a scalar potential from which the inelastic strain rate can be derived. We repeat that proof here and further show that in addition the rate of change of the internal state variables is derivable from the same potential.

Creep Constitutive Equations

245

If we multiply both sides of Eq. (2.8) by da,,, we find (2.11)

EEdu,I = g k b k d a , .

From Eq. (2.9) we note aFk/aUq

=

(2.12)

gk.

Substituting this and Eq. (2.10) into Eq. (2.11) gives =

&

b k ( F k 9 ah

)(a F k / a U y

day.

At a given instant in time we know the state variables ak.Treating these as constants in the above expression gives &;da,, = b k ( F kar k ) dFk = d @ .

(2.13)

The right-hand side of Eq. (2.13) is now an exact differential, (2.14)

and (2.15)

This is the result originally found by Rice (1971). Further, Eq. (2.13) gives (Yk

= a@/dFk = -a@/dSk.

(2.16)

We make use of Eqs. (2.15) and (2.16) in the following sections, where we analyze each of the bulk damaging processes. The results obtained in this section so far are central to the developments in the remainder of this paper. Because of the importance of these results, we summarize them briefly in the following. The internal structure of the material is described in terms of a number of internal state variables a h , such that the Helmholtz free energy takes the form $ = $(&&,,

ak

1-

The thermodynamic forces are then

aIJv= d $ / d E t l ,

s k v

=a$/dak.

If Eq. (2.10) is satisfied, the inelastic strain rate and rate of increase of the internal variables can be derived from a potential @, E;

=

a@/au,,,

bk = -d@./ask.

Next we consider a composite material, for which Eqs. (2.15) and (2.16) hold in each element. This analysis allows us to piece together a number of microscopic elements to give the overall response of a macroscopic element of material. For simplicity we will also assume that there is only

A. C. F. Cocks and F. A. Leckie

246

one state variable associated with each element. At a given instant the total strain rate in each element for a constant remote stress Z,, is &;

=

+

&;

E;,

which is compatible with the remote strain rate field k:",. If du,, is the increment of stress in each element for an increment of remote stress d Z , when the material responds elastically, application of the principle of virtual work gives

k:",dZ,, v = c (&', + &;) du, vk,

(2.17)

k

where E: is the remote inelastic strain rate, V the total volume of the composite, and v k the volume o f the kth element. Rearranging Eq. (2.17) gives

The elastic strain rate &; gives rise to a changing residual stress field p,, and associated with the stress field d u , is an elastic strain field ds;. Equation (2.18) then becomes, after making use of Eq. (2.15),

(2.19)

where DVkris the elastic compliance tensor and kth element, Q k = Q k ( u v , Sk, (Yk).

ak for constant

The increment of

dQk

=

(Yk

(aQk/da,)

the potential for the

Qk

is then

(aQk/ask)

duq f

dSk.

(2.20)

Substituting Eq. (2.20) into Eq. (2.19) gives

Since & is a residual stress field and de; a compatible strain field, the first term on the right-hand side of Eq. (2.21) is zero. Rearranging Eq. (2.20) and noting Eq. (2.16) gives E; d Zij v -

c

(Yk d S k v k

=

k

d@k v k

=

dQ

k

Therefore

Eg where

Uk

= vk/

= d@/aZq

v.

and

&k

= - ( 1/ v k ) ( a @ / d s k )

(2.22)

Creep Constitutive Equations

247

Equation (2.22) demonstrates that for the composite system it is possible to obtain a single macroscopic potential, which is the volume average of the microscopic potentials, from which the strain rate and rate of change of internal variables can be derived. The macroscopic potential @ contains information about the residual stress field pV within the material that results from the nonuniform accumulation of plastic strain. Cocks and Ponter (1985b) demonstrate that the rate of change of the residual stresses are also derivable from @. We do not include these expressions here so as not to cloud the general results of this paper. Specific forms for the function @ are given in Sections VI to VIII. The residual stress fields in these expressions remain constant as the damage increases, so that the evolution laws for the residual stress field are not required.

111. Mechanisms of Void Growth

As the material creeps, voids can either grow within the grains or on the grain boundaries. In structural situations, when the design life of a component is long, the most common mode of failure involves the growth of intergranular voids. The general situation we consider in this section is shown in Fig. 1, where we assume that the number and spacing of the voids remain fixed during the life. The influence of the nucleation of additional voids is examined in the next section.

BOUNDARIES

i

1

t

2.. E . . 11' 11 FIG.1. An element of material containing a number of cavitated grain boundaries subjected to a stress Z,,.

A. C. l? Cocks and F. A. Leckie

248

n

A

k

n

n

n

A

2 la-I

FIG.2. A typical grain-boundary slab of material, which contains voids of radius r: spaced a distance 21" apart. [Note: Underlining in figures is equivalent to boldface in text.]

The analysis of this problem is facilitated by isolating a volume of material surrounding a grain boundary and examining the response of this element. The behavior of the entire material is then found by combining all of the elements to form a composite system. A typical grain-boundary element is shown in Fig. 2. We characterize the position of this boundary in terms of its outward normals np and n;. The voids within an element of material can grow by one of a number of mechanisms: power law creep, grain-boundary diffusion, surface diffusion, or a coupling of any two or all three of these mechanisms. Cocks and Ashby (1982) have shown, however, that the materials' response can be adequately described if it is assumed that the dominant mechanism of the three simple mechanisms operates alone. These mechanisms are shown in Fig. 3, and we consider each in turn in the following subsections. The strain resulting from the growth of these voids is accommodated in the rest

(a 1

(b)

( C )

FIG. 3 . The rate of growth of the voids can be controlled by ( a ) power law creep, ( b ) grain-boundary diffusion, or (c) surface diffusion.

Creep Constitutive Equations

249

of the material by deformation due to power law creep and by grainboundary sliding. We examine grain-boundary sliding separately in Section III,D and the overall response of a macroscopic element in Section II1,E.

A. VOID GROWTHBY POWER LAW CREEP Cocks and Ponter (1985b) have analyzed creep deformation by lower law creep for a void-free material by using the thermodynamic approach described in the preceding section. For constant or slowly changing stresses it is possible to define a steady-state potential such that

i.,]

=

a4/aci1.

(3.1)

It is often assumed that

4

=

[g"g"/(n + l ) l ( u e / ~ o ) f l + l ,

where go,E,, and n are material constants and u, is an effective stress. We use Eq. (3.1) as the starting point for the analysis of this section. The state of the material can be described in terms of the volume fraction of voids that it contains. If it is assumed that all the voids are the same size and are uniformly spaced along the boundary, then we can further isolate an element of material of volume which contains a single void (Fig. 4), and assume that it experiences the stress applied to the grainboundary element of Fig. 1 . Analysis of this type give the mean strain rate E of the grain-boundary element. This strain rate can be obtained from the following energy balance:

fz

v",

(3.2) The second term on the left-hand side arises from the increase in surface energy as the surface area A, of the void increases. Now A, = 47rri, and the volume of the void v h = $7rr;.

Therefore

A, = 2 v h /

rh

= (2.f:/ r h )

va,

(3.3)

where f : = v h / V" is the volume fraction of voids. Substituting Eq. (3.3) into Eq. (3.2) gives

250

A. C. F. Cocks and F. A. Leckie

2 ija

?

,r

?

0 aij

VOLUME

va ~

/

3 . J . J . xija FIG. 4. A void of radius r,, embedded in an element of material of volume V", which is subjected to a stress Z,.

where Z, = 2y,/rh. Here Xc is simply the surface tension and Eq. (3.4) is a statement of virtual work, with Z, treated as an applied surface traction. The microscopic stress aU can be changed by varying the macroscopic stresses ZG and Z t . For increments of Z, and C, the virtual work expression becomes

Therefore E;

= a@"/aZ;

f:

=

(3.6a)

and -aQa/aZ:,

(3.6b)

where Q, is a scalar function of Z,; Zz, and fy". The result of Eq. (3.6a) is due to Duva and Hutchinson (1983). Inclusion of the surface tension term leads to the additional result of Eq. (3.6b). Surface tension does not strongly influence the form of the potential, but

Creep Constitutive Equations

25 1

its inclusion leads to a compact form of the constitutive equation. The effect of surface energy can be easily included in the work of Duva and Hutchinson (1983), who obtain expressions for Q, for dilute volume fractions of voids. Another method of finding Q, is to use extremum theorems (Martin, 1966; Ponter, 1969) to obtain bounds on a. This approach allows the analysis of concentrated as well as dilutely voided materials (Cocks, 1986). Although it has proved convenient to develop the constitutive model in terms of the volume fraction of voids in the grain-boundary elements, when comparing the different mechanisms of void growth it is more advantageous to express the equations in terms of the area fraction of voids in the plane of the grain boundary f;, (3.7a) We can define an associated internal stress

x; = 2 y J P

=

(;)'/3x:y3.

(3.7b)

Now combining Eqs. (3.6b) and (3.7) we find

The important point about the preceding result is that the internal damage variable is nonunique. We could choose any function of the volume fraction fl to describe the damage and still obtain the general form of result of Eq. (2.15).

B. VOIDGROWTHB Y GRAIN-BOUNDARY DIFFUSION At low stresses, deformation due to power law creep is slow and the mechanism of void growth changes to one directly controlled by the diffusion of material. A void grows by material diffusing along its surface by surface diffusion to the grain boundary. It then flows along the grain boundary, where it is uniformly deposited (Fig. 5). These are two sequential processes,

(a 1

(b)

FIG. 5. The void grows by material flowing along its surface to the tip and then along the grain boundary where it is uniformly plated. In this process the shaded region in (a) is transported to the shaded area of (b).

252

A. C. E Cocks and F. A. Leckie

and so it is the slower one that determines the overall rate of growth. In this subsection we consider the situation where the void growth is limited by the rate of grain-boundary diffusion, and in the next subsection we examine the conditions when surface diffusion controls the rate of growth. For simplicity we assume that the grain-boundary energy is the same as that of a perfect crystal, so that the voids remain spherical as they grow. Again we isolate a grain-boundary element of material (Fig. 2) and perform the analyses in terms of the local stress field. The conceptual reversible path we follow in defining the free energy requires making a cut along the grain boundary (Fig. 6 ) . Material is then scooped out to form the voids, and it is spread evenly along the grain boundary. The surfaces are then rejoined and the resulting change in free energy is

where A, is the grain-boundary area. Application of the stress Zg gives the total free energy

Duva and Hutchinson (1983) give CGkl for a dilute volume fraction of voids in an incompressible elastic material. Cocks (1986b) gives an approximate

H FIG.6 . Conceptual reversible process of forming a void on the grain boundary: ( a ) initial void-free material; (b) a cut is made along the grain boundary; (c) material is scooped out to form a void; (d) this material is spread evenly along the grain boundary; (e) the two pieces of material are rejoined.

Creep Constitutive Equations

253

result for any volume fraction, C;A/ = fE(1 - f?)&,~fi,/ + $[E(1 - f l ) / f l 1 6 $ k r ,

(3.10)

where 8, is the Kronecker delta and E Young's modulus. Differentiating Eq. (3.9) gives the affinities (3.11)

where C z is the von Mises effective stress C z 2 = $P;Sg, CE the mean stress C: = fC&, and S ; the deviatoric component of stress SP; = X; - CES,. The second law of thermodynamics, Eq. (2.6), then becomes

zp

where C: = 2 ys/ rh. The inelastic strain rate E results from the plating of material onto the grain boundary. The rate of thickening of the grain boundary 6, is directly proportional to the rate of increase of the volume of the voids,

ti, = 2j',"1*n~.

(3.14)

The inelastic strain rate can then be obtained from the kinematic relationship (see Appendix)

k'pIp= (1/4Z")(d,np+ tip:). Combining Eqs. (3.14) and (3.15) gives

gypin terms

(3.15) off:,

E;P = j;"n:nP.

(3.16)

This expression is equivalent to Eq. (2.8), where g,

= npnp.

Equation (3.16) can be substituted into Eq. (3.13) to give the rate of energy dissipation,

(CP;npnq - X:)

9 C$ + [Z-1 E ( lCZ2 - f : ) ' + 8 E ( l -f:)'

]}f:2 0. (3.17)

This mechanism tends to dominate at stress levels where C/E < lop3,so that the term in the square brackets, which scales as C'/ E, is always much smaller than the other terms, which scale as C, and can be ignored. Equation (3.17) then becomes (Ccnpnp -

~ r ) f :2 0.

(3.18)

A. C. F. Cocks and E A. Leckie

254

The component of stress ZGnPnp is simply the stress normal to the grain boundary. A detailed analysis of this mechanism of void growth gives a growth rate (Cocks and Ashby, 1982; Raj et al., 1977)

j;, = a(ZGnPnp - ~

3 2 1 In~ 1/E, '

(3.19)

where R is a temperature-dependent material parameter. This expression is of the form of Eq. (2.10), which allows us to prove the existence of a scalar potnetial @.," such that E;*

= a@.,"/aZ;

and

f:= -a@:/aZ:.

(3.20)

Following the analysis leading to Eq. (3.8), we can express the potential in terms of the area fraction of voids f; instead of fi and define a stress Z r = 2y,/l", such that

f; = -a@:/aZ;.

(3.21)

C . VOID GROWTHLIMITEDBY SURFACEDIFFUSION In the preceding subsection we assumed that surface diffusion was sufficiently fast for the void to maintain a spherical shape as it grew. When the rate of surface diffusions is slower thap grain-boundary diffusion, material cannot be supplied fast enough to the tip of the growing void. Material is then only removed from the void tip region, and the void adopts a cracklike profile as it grows. The detailed shape of the growing void is now a function of the history of loading, but the rate of void growth is only a function of the shape of the crack-tip region (Cocks and Ashby, 1982; Chuang et aL, 1979). Here we limit our attention to the steady-state process of void growth at constant stress and assume that the new steady state is soon reached when the stress is changed. We can now define the state of the material surrounding a given grain boundary by using two internal state variables: the volume fraction of voids in the grain boundary element E through which we relate the inelastic strain and the area fraction of voids f;: in the plane of the grain boundary, which we use as a measure of the surface area of the voids. What we have described so far is the material response to a tensile stress normal to the plane of the grain boundary. When this stress is compressive, the void profile is completely different (Fig. 7), and if the stress is changed from tension to compression, the transient response before reaching the new steady state can be significant. In the present section we limit our attention to tensile stresses, although similar expressions can be derived for compression.

255

Creep Constitutive Equations

L L L L

x

x

(a 1

(b)

FIG. 7. (a) Shape adopted by growing void for tensile stress across the grain boundary in the limit of surface diffusion-controlled growth. (b) Shape adopted by sintering void for compressive loading.

At a given instant the free energy can be obtained by again following the reversible process of Fig. 6,

+

$ = iCP;krErFE;eV, 2fEyYsAa.

(3.22)

This expression is similar to Eq. (3.9) of the preceding subsection. The second term on the right-hand side arises from the introduction of free surfaces in the material when the voids are formed. In practice the contribution from this effect is also a function off: and the history of loading, but in most practical situations the dependence on these variables is weak, and the simple form of Eq. (3.22) is sufficient. Differentiation of Eq. (3.22) gives the thermodynamic forces d$/aE;' d$ldf;:

= ZP; V,,

=

(3.23)

2YA.

(3.24)

In the derivation of the preceding equation the variation of C & with both

f: and fE has been neglected, since, as in the preceding section, the effect is small. The second law, Eq. (2.6), now becomes C;"E'pi"Va - 2y,A,fE 2 0. As before the inelastic strain rate EP;" is related to and Eq. (3.24) becomes X;npnpf:V, - 2ysA,f:

2 0

or

(3.25)

f:

through Eq. (3.16)

XGnpnpf: - 3Xij: 2 0 (3.26)

where XE = 2 y,/31*. Chuang et al. (1979) have analyzed this mechanism in detail. They show that for steady-state growth

f:

=

[fE"23XE/(1 -f:)3]A(X;npnp)2,

f~ = [ f ~ ' ' ~ / (-f;:)'i~(z;npn;)~, i

(3.27) (3.28)

256

A. C. F. Cocks and F. A. Leckie

where A is a temperature-dependent material property. If we substitute these expressions into Eq. (3.26), we find that

Ezn:nyf:

=

31$:,

(3.29)

that is, the work done by the applied load all goes into creating the new crack surfaces. Equation (3.27) is in the form of Eq. (2.10),

f := fw:,

X'hu,J'hn),

(3.30)

where F," = I t n p n p . Following the analysis of Section 11, we can prove the existence of a scalar potential 4" such that (3.31) and (3.32) When considering the life of a component,f: is a more important variable than f:, because failure occurs when the voids have linked to form a crack. This occurs when f; = 1, and the value off," at this instant is unimportant. Equation (3.29) gives a relationship between f: and f:. Combining this with Eq. (3.32) gives f : =f;(F,"/3E.,") =

-a@:/aI;,

(3.33)

and so the strain rate and rate of increase off: are again derivable from the same potential.

D. GRAIN-BOUNDARY SLIDING

In the preceding three subsections it was shown that when voids grow within an element of material it is possible to prove the existence of a scalar potential from which the inelastic strain rate and rate of growth of the voids can be derived. Also the strain rate of a void-free element of material is given by the potential form of Eq. (3.1). Another contribution to the inelastic strain rate arises from grain-boundary sliding in the material. Here we analyze this contribution to the deformation of the material. Any arbitrary amount of sliding in the grain boundary can be divided into components u," in the sa direction and u ; in the direction of t" (Fig. 2). There is no change of internal structure when sliding occurs, and so u: and up are not state variables, but their rate of change gives the speed of the process. If we assume that the size of the voids remains constant, then the only contribution to the free energy is the elastic stored energy, which results from deforming the material elastically, $ = ; c ; ~ , E ~E:? '

v,.

Creep Constitutive Equations

257

Differentiating the preceding equation and substituting the result into Eq. (2.6) gives the rate of energy dissipation

q E y 2 0, where f i y is the inelastic strain rate that is related to the rate of change of internal variables through the kinematic relationship

kzs = (1/41")[tirny + UPnP] =

(1/4l")[ti:(s:np

+ s p n r ) + u p ( t p n y + tpnp)].

This equation is equivalent to Eq. (2.8) of Section 11, although now the internal displacement rates are not state variables changing, they are a measure of the rate of the irreversible process (Ts,/21*)~: + (T,,/21")2ip 2 0,

where T,,

= $Zz(apnp

+ spn:)

and

T,,

= $c(rpnp

+ tpnp)

(3.34)

are the shear components of the remote stress Z; in the directions of ur and up, respectively. It is conventional to assume that the sliding rates tip and uip are linear functions of their resolved shear stresses. These relationships are of the general form of Eq. (2.10), so that it is possible to prove the existence of a potential Car [Eq. (2.15)] such that E;s where (75,

+

= T:,,)"'

= am:/ac;,

(3.35)

~ ~ , , / 4 1 and ' ~ ~ 77 is a material constant. Here T,,,,~ = is the maximum resolved shear stress in the plane of the grain

boundary. When grain boundaries slide freely, they are unable to support any shear stresses and complex stress states can develop in a material even under simple loading conditions. In Section VII we see that an effect of this is to magnify the stress transmitted across certain grain boundaries.

E. DEFORMATION RATE O F MACROSCOPICELEMENTOF MATERIAL The general situation considered in this section is shown in Fig. 1 . In each grain-boundary element there are two contributions to the inelastic strain rate: from void growth and grain-boundary sliding. The contribution from void growth is given by one of Eqs. (3.6), (3.20), or (3.31) and that from grain-boundary sliding by Eq. (3.35). By making use of Eq. (2.21) these equations can be combined with Eq. (3.1) to give the macroscopic

A. C. F. Cocks and F. A. Leckie

258 potential

6,

=

1

d ~ $dV

+ C d@:

u,

+ C d @ : u,

a

"8

=

d6,

(3.36)

a

where V . is the volume of material outside of the grain-boundary regions. Then

E;

= a6/axv

(3.37)

and each f z is given by

IV. Nucleation of Cavities In the preceding section we assumed that the number and spacing of cavities remained fixed throughout the life of a component. In practice cavities nucleate continuously during the lifetime, resulting in a gradual decrease of their spacing and an enhancement of the growth rate of the other cavities [Eq. (3.19)]. The processes by which cavities nucleate are not fully understood at the present time, but most recent studies of the subject follow the analysis of Raj and Ashby (1975) in assuming that cavities form due to the coalescence of vacancies in the material. Such analyses lead to a threshold stress for nucleation. Below this stress the rate of nucleation is so slow that it can be assumed that voids never nucleate, while above this stress the rate of nucleation is so fast that it can be assumed that voids nulceate as soon as the threshold stress is exceeded. All studies to date predict values of the threshold stress that are much greater than the applied stresses used in the majority of creep experiments (Argon et al., 1980). Attention has therefore focused on ways in which these high levels of stress concentration can be achieved in the material (Argon et al., 1980; Wang et al., 1985; Cocks, 1985). Argon et al. (1985) and Wang et al. (1985) show how the level of stress concentration can be achieved during grain-boundary sliding, while Cocks (1985) demonstrates how differences between the diffusion characteristics of the grain boundary and particle-matrix interface can lead to high stress concentration. Neither approach leads to satisfactory explanations of the nucleation process. It is evident that further experimental and theoretical studies are required to clarify the problem. In this section we outline the approach of Raj and Ashby (1975) and show that a potential exists from which the inelastic strain rate can be derived. Also, in the simplified situation considered here, the rate of nucleation and void-growth rate are derivable from the same potential.

Creep Constitutive Equations

259

Consider the situation for which a grain-boundary element of material contains a number n, of potential sites for the nulceation of voids. At a given instant in time the material contains an area fraction f ; of voids that are growing on n of the nucleation sites. Within an element of material we now have two state variables f ; and n. In calculations of the contribution to the free energy from the nucleation of the cavities, the reversible process outlined in Fig. 6 can again be followed. Material can be scooped from the nulceation side and spread onto the grain boundary. In the process, the stress normal to the grain boundary does work, and the internal surface, and hence the total surface energy, increases. Consequently, part of the volume of the void at a given instant can be assigned to the nucleation process and the remainder to the growth process. In the early stages of void growth the rate of growth is generally controlled by grain-boundary diffusion. Examination of Eq. (3.19) then reveals that a void must have a radius r," = 2y,/C:npnp

(4.1)

before it can grow. Problems arise in choosing a small number of state variables when we allow the voids to nucleate continuously. We will assume, however, that we can, at least approximately, characterize the material response in terms off;: and n, so that the free energy expression takes the form

*"

=

*"(Ey,fha, n )

(4.2)

=

x$;; + x;f; + z:ri",

(4.3)

and

where Zz

=

a$"/an".

Equation (2.6) then becomes

C ; q " - x;f;

-

Xti"

2 0.

(4.4)

Now part of the inelastic strain rate E:" results from the nucleation of the voids k:","and the remainder gh,Tfrom their growth. Equation (4.4) then becomes

(xf;Bh,a- x;f;) + ( x ; B ; m- C t i " ) 2 0,

(4.5)

and since the nucleation and growth processes can occur independently of each other, each of the fenced terms above must be greater than or equal to zero. We examined the first of these terms in the preceding section and showed that a potential exists from which the inelastic strain rate and rate of increase of area fraction of voids can be derived [Eq. (3.20)]. This potential contains the number of voids [n" = 1/1"'], which is now a variable,

260

A. C. F. Cocks and F. A. Leckie

that is, the potential

4:

K ( X P ; , Zr.Grn").

=

(4.6)

The approach used in analyzing thermally activated processes such as the nucleation of voids differs slightly from that described in Section 2. The change in entropy associated with the random nucleation of voids becomes important in determining the response of the material. It is conventional to omit the entropy term from the expression for the free energy and include its effect as a kinetic relationship for the rate of nulceation, which is largely phenomenological. The effect of this is that the rate of energy dissipation given by Eq. (4.5) is less than zero. This energy must be supplied through thermal fluctuations, which is incorporated in the entropy terms. The term

C G E F - X;,"ri" of Eq. (4.5) can be expressed as ( C ; B ; - C,")riU since Ey = BP;ri" results directly from the nucleation of the voids. As before, the term (XP;BP;- X;,") (which is now negative) can be interpreted as a driving force for nucleation; it is a measure of the additional energy required from thermal fluctuations that is incorporated in the entropy expressions. If this is a small negative quantity, then the rate of nucleation is fast, but if it is large, the rate of nucleation is slow. It is instructive to consider the nucleation process in slightly more detail. We are interested in the situation when a grain-boundary contains n, possible nucleation sites per unit area. When a void of critical radius r," forms at one of these sites, there is an increase in the free energy due to the formation of the new surface. The free energy is then (excluding the contribution from the entropy)

+ = $ c , , ~ , E ~E?? v + 4.rrrfySn,

(4.7)

where it has been assumed that the voids that form are spherical. Differentiation of Eq. (4.7) then gives

4 = X F E Yv + 4.rrrfySri,

(4.8)

where the change in stiffness due to the formation of the voids has been ignored since it is always small compared to the other terms and rc has been taken as a constant. The rate of energy dissipation, Eq. (2.7), then becomes

D = ZZE'"," v - 4.rrrfySri,

(4.9)

where the inelastic strain rate

E:"

= (rim/

Va)+:npnp.

(4.10)

Creep Constitutive Equations

261

Substituting this into Eq. (4.9) gives

D

=

(XFn,n,:.irr?

-

4.irr:yJri"

=

FEri".

(4.11)

Analyses of the nucleation process (Raj and Ashby, 1975; Cocks, 1985) give ria = ( n z

-

n")Aexp(FE/RT),

(4.12)

where A is a temperature-dependent material property and n,* is the number of potential nucleation sites. This equation is of the form of Eq. (2.10), and Eq. (4.10) is equivalent to Eq. (2.8), so that it is possible to prove the existence of a scalar potential @; from which the strain rate and rate of nucleation can be derived,

,

EP"

= a@:/aX;,

ri" = a@/aF, =

-a4/aX,",

(4.13)

where X,"= 4.irrf y s , In the analysis of Section I1 it was shown that the potential form applies if the preceding derivatives are made by assuming that the state of the material is constant. This implies that r, is constant. The derivative of Eq. (4.13) should therefore be obtained by assuming that rc is constant. Equation (4.1) can then be used to give the variation of r, with stress. The total strain rate and damage rates f; and r i m are derivable from the potential cp"

= (D,"

+ @,".

It was stated earlier that this approach to void nucleation leads essentially to a threshold stress for nulceation. We can apply the preceding analysis to small regions of grain boundaries, where the local stress depends on the interaction of these individual elements. The stresses change continually and once the threshold stress is reached locally, voids can nucleate and grow; the stress is then shed onto other regions where further nucleation can occur. In this way it is possible to obtain a continuous nucleation of voids as the material creeps. As in the preceding section, the damage potentials @: can be combined with the sliding potentials @: and the potentials for the grain interiors to give a global macroscopic potential from which the strain rate and damage rate can be derived. V. The Use of Average Quantities

In the preceding sections we developed the thermodynamic description of the material in terms of discrete state variables. Rice (1971) describes how the thermodynamics can be expressed in terms of average quantities. Here, instead, we make use of a result due to Onat and Leckie (1984) in expressing the distribution of cavitated boundaries in terms of a series of even-order tensors. The potential can then be expressed in terms of these tensors. whose rates are derivable from it.

A. C. F. Cocks and F. A. Leckie

262

Following Onat and Leckie (1984)we now define a grain boundary in terms of the outward normals on either side of the boundary. We further assume that there is no anisotropy in the distribution of grain boundaries and that all boundaries with the same outward normal contain the same area fraction of voids f; (here we use f: instead off: so that we can combine the various mechnanisms of void growth) and for simplicity we assume that the voids are all present at the beginning of the life. If one end of an outward normal is placed at the origin of a linear coordinate system, the other end lies at a point on a sphere s of unit radius (Fig. 8). We can extend this line beyond s by an amount that scales as the area fraction of voids on the grain boundaries represented by the outward normal. The surface r formed by the tips of these lines is symmetric about any diameter since a mirror image is formed when the second normal is drawn in the opposite direction. Onat and Leckie (1984)show that the shape of r can be described by a series of even-order tensors

D

=-

'I

47r s

ft ds,

D,

(5.1)

and so on, where Kykl

=

ninjnknl- S(SVnknl+ Siknjnf+ ailnjnk + sjkninf+ 6jfnink+ Ljkfnink+ 6krninj)

+ &(6ij6k/ + 8ikajl + 8i/ajk).

(5.2)

All odd-order tensors are zero because of the symmetry of r. Onat and Leckie (1984)further show that the damage on each grain boundary can be written in terms of these tensors,

f;: = D + Dij(ninj - f6,)

+ DijkrKijkr + .

'

..

(5.3)

In the preceding three sections we made use of Eq. (2.21)to define a macroscopic potential @,

Eg d Z i j -

xf; dZ.,"u,

=

d@.

(5.4)

a

Substituting for

or Eg d Z , -

47r

fz and using Eq. (5.3)gives

Is [ + D

DG(ninj -

+ D,kl~,kl +

.

*

1

d Z t ds = d @ .

(5.6)

Creep Constitutive Equations

263

FIG. 8. The surface r is the locus of the end points of a series of radial lines of length f: in the direction of nu, which originate on a sphere of unit radius s.

We can now define a set of stress like quantities

and so on. The second of Eqs. (5.7) then becomes

and so on. Again we find the same general form of result as in the preceding sections. The inelastic strain rate and the rate of change of the state variabfes are derivable from a single scalar potential.

VI. Damage Mechanisms in Precipitation-Hardened Materials In the situations considered in the preceding sections it has been assumed that the tertiary stage of creep is a direct result of the growth of voids in the material. The nickel-based superalloys tested by Dyson et al. (1976) and the aluminum alloy tested by Leckie and Hayhurst (1974) exhibit

264

A. C. F. Cocks and F. A. Leckie

extensive tertiary regions and yet are found to be virtually void free until just before failure. Stevens and Flewitt (1981) suggested that the steadily increasing strain rate of the nickel alloy is a result of precipitates coarsening in the material, which thus become a less effective barrier to dislocation motion. Dyson and McLean (1983) found that a model based on precipitate coarsening is unable to explain the response of these alloys and proposed that tertiary creep is a result of an increase of the mobile dislocation density with increasing strain. Henderson and McLean (1983) observed that dislocation networks form around the coherent precipitates of the nickel-based superalloy I N738LC when the material creeps. They suggested that the presence of these networks aids the climb of dislocations over the particles either by acting as a source or sink for vacancies or by providing a fast diffusion path for the vacancies, resulting in an increased strain rate. This mechanism has been analyzed by Ashby and Dyson (1984) and by Cocks (1986a). A detailed analysis of the deformation mechanism is given by Shewfelt and Brown (1977) for particle-strengthened materials. The essential features of the process are shown in Fig. 9. For a material to deform plastically, dislocations must be able to glide along slip planes. If the material contains obstacles to dislocation motion, then the dislocation must bypass them either by bowing between them (Fig. 9a) or by climbing over them (Fig. 9b). The time for a dislocation to transverse a slip plane is determined by the climbing part of the process.

PART OF DISLOCATION CLIMBING OUT OF SLIP PLANE \

(b) FIG. 9 . ( a ) A section of dislocation gliding between two obstacles. ( b ) The section of dislocation between the two obstacles can only glide if part of it climbs over the particles.

Creep Constitutive Equations

265

Consider an element of material that sees a remote stress Xc. Within this material dislocations move on slip planes with outward normals n‘I, in the direction sa. We can identify two families of internal state variables at a given instant in time: vg,the area enclosed by a segment of gliding dislocation (Fig. 9a), and T,, the area enclosed by a segment of dislocation in the glide plane that requires climb over an adjacent particle in order for it to move (Fig. 9b). The free energy at any instant can be found by following the procedure outlined in Section 11, $ = 5C,,krE;,E;V + A T g ) + d

vc),

(6.1)

where f ( v g ) and g ( v c ) represent the free energy due to the presence of dislocation lines in the material. At this stage we are only interested in the deformation in the material, and Eq. (6.1) does not include any information about the damage in the material, which is in the form of dislocation networks around the precipitate particles. We include this damage later in the kinetic relationships for climb-controlled deformation and relate its growth to the glide part of the deformation process. This means that Eq. (6.1) does not give a full description of the state of the material, but by adopting this approach the physics of the process is clearly defined. Differentiating Eq. (6.1) gives

i = L,E“,V+f’(vg)7jg+g ’ ( v J 7 j c .

(6.2)

Substituting this into the second law [Eq. (2.6)] gives

z $ ; v -f(vg)7jg- g’(rlc)7j, 2 0.

(6.3)

Now the inelastic strain rate EE is related to 7jg and 7j, by (see Appendix for derivation of this expression)

g;

=

(7jg + 7 j c ) ( b / 2 V ) [ n , s+, n,snl.

(6.4)

Combining Eqs. (7.3) and (7.4) gives the rate of energy dissipation [Tb -fYvg)17jg + [ T h - g’(77c)lric 2 0,

(6.5)

where T = $Xv[n,s,+ n,s,] is the shear component of the remote stress in the direction of slip. To complete the thermodynamic description we need expressions for 7jg and 7j,. Conventional theorems of dislocation glide assume that 7jg

=

7jg[Tb- f ( v g ) I .

(6.6)

When climb is required for a section of dislocation to glide, Shewfelt and Brown (1977) demonstrate that

.ic= 7j,[Tb - g ’ ( v , ) ,geometry],

(6.7)

where the geometric terms are functions of the size, shape, and spacing of the particles and, as we shall see later, of the density of dislocation loops

A. C.E Cocks and F. A. Leckie

266

surrounding the particles, the growth of which is related to 7jg. These expressions are of the form of Eq. (2.10), and making use of the result of Eq. (2.15), we can define potentials for each process, and then = UgQg

+ (1 - U g ) Q c ,

(6.8)

where Qg is the potential for glide deformation and Qc is that for climb and u, is the volume fraction of material at a given instant in time where the deformation is controlled by glide. The forms of these potentials suggested by our understanding of the mechanisms of deformation (Cocks and Ponter, 1985b; Shewfelt and Brown, 1977) are Qg

=

+

Qc = [ A ( w ) / ( n 1 ) ] ( r- s,)"+'

F ( r - s,),

(6.9)

for the average mean rate of deformation of the material, where sg and s, are the maximum values o f f ( q g ) /b and g'( q,)/ b, respectively, as a dislocation bypasses the obstacle, and F ( r - sg) is a function of ( T - sg), the exact form of which need not be specified. The quantity w represents a measure of the density of dislocation loops around the precipitate particles. Cocks (1986b) demonstrates that the rate of growth of this damage is governed by the glide-controlled part of the deformation process; and our definition of this damage can be chosen such W =

7jg = -a&/as,

=

-a4/asg.

(6.10)

The inelastic strain rate obtained by differentiating Eq. (6.8) using the potentials of Eq. (6.9) is then (.;]"T ),s E G " = a): -" - ;- [ugf(7 - sg) + ( 1 - u ~ ) A ( w ) -

a7

a

(6.11)

I,

Cocks (1987) demonstrates that if ro is the stress at which all the particles are bypassed by an Orowan bowing mechanism, then for a stress r a fraction T / T ~of the strain can be assigned to the pure gliding process and the remainder to the climb-controlled process. Then ug =

TIT".

A further constraint we must impose on the deformation process is that the glide-controlled and climb-controlled processes must occur at matching rates. This ensures that the increase of dislocation line length during the deformation process is minimized. As a result the thermodynamic force sg must adjust itself such that

f(.

-

sg) = A ( w ) ( r - S J n ,

and the overall deformation rate becomes (6.12)

Creep Constitutive Equations

267

with the damage accumulating at a rate h

=

-a@/as,

= v , ~ ( T- s,)

=

A ( ~ ) (-Ts , ) n ( T / 7 0 ) .

(6.13)

In the preceding analysis we have assumed that there is only one slip plane operating. For the deformation of a grain in a polycrystal to be compatible, more than one slip system needs to operate. So the potential, which is expressed in terms of the local stress, is the sum of the contributions from each slip system. The macroscopic potential is then the volume average of the microscopic potentials for each grain. The stress within each grain must be in equilibrium with the applied stress, which is the only stress that appears in the macroscopic potential. In the steady state the strain rate in each grain is the same (Cocks and Ponter, 1985a; Stevens and Flewitt, 1981) and all the state variables are related to each other, and so a single state variable theory can be used to describe the material behavior. We further assume that each grain contains the same distribution of obstacles, and so s, for each slip system is the same. If the damage accumulates slowly, so that the strain rate remains uniform throughout the material, the same density of loops will form around each particle. In the multiple-slip situation it is this density of loops that is the damage in the material. The overall potential is then given by @ = v,F(Z.,

-

Z,)

+ (1 - v , ) [ A ( w ) / n+ l ] ( Z e- ZJn+',

(6.14)

with EP, = a@/aZ,

(6.15a)

and b = -a@/aX:,,

(6.15b)

where Z is an effective stress, Z, is a threshold stress related to s, of Eq. (6.9), Zg is related to S,, and w is a measure of the density of dislocation surrounding the particles. After evaluating the strain rate and damage rate, we must impose the constraints that vg = Z e / E o and f(Ze

- Z g ) = A(w)(Xe - &)",

(6.16)

where Zo is related to the Orowan stress T~ and Eq. (6.16) satisfies the condition that the two contributions to the strain rate must occur at matching rates. It does not matter how the dislocations approach the particles; they will always see the same density of dislocation surrounding the particles. In this instance the damage w is therefore a scalar quantity. VII. Theoretical Constitutive Equations for Void Growth

In the preceding sections we concentrated mainly on the structure of constitutive equations without saying much about particular forms for these

268

A. C. F. Cocks and E A. Leckie

equations. In the preceding section, however, we did end up with a specific form of equation. In this section we obtain forms for these equations for two simple distributions of cavitated boundaries. These two distributions of voids represent the extremes of behavior we would expect when the damage is in the form of voids. In the next section we adopt a different approach and try to obtain simple constitutive laws directly from the material data.

A. UNCONSTRAINED CAVITYGROWTH In the absence of any voids a material creeps at a rate [Eq. (3.1)]

where

We will assume that this is the strain rate measured experimentally, and so Eq. (7.1) includes the effects of any grain-boundary sliding. When grain boundaries slide freely, the stress across the grain boundary may not equal the resolved component of the remote stress. Analysis of the effects of grain-boundary sliding (Cocks and Ashby 1982; Rice, 1983; Tvergaard, 1984, 1985) suggest that stress normal to a given grain boundary is

where S, are the remote deviatoric stresses, Emis the mean stress, and c is constant, which is a function of the shape of the grains that ranges from 1 to 4. In this section we assume that the strain resulting from the growth of the voids on the grain boundaries is readily accommodated by grain-boundary sliding and the deformation of the grains, in such a way that the stress normal to the grain boundary is still given by Eq. (7.2). When void growth is controlled by grain-boundary diffusion, the additional strain rate due to the growth of the voids can be found by combining Eq. (7.2) with Eqs. (3.16), (3.19), and (3.7). The macroscopic potential is then

where 2: is given by Eq. (7.2) and C:ha is the associated thermodynamic internal stress. Differentiating Eq. (7.3) gives the strain rate and rate of

Creep Constitutive Equations

269

increase of damage,

(7.4)

The last term in Eq. (7.4) represents the contributions to the strain rate from void growth and from the additional grain-boundary sliding required to accommodate this growth. If, instead, the void growth is controlled by surface diffusion, the strain rate and void growth rate can be obtained by combining Eqs. (3.16), (3.27), (3.28), and (7.2); then (7.6)

and

with

B. CONSTRAINED CAVITYGROWTH

In obtaining the potentials of Eqs. (7.3) and (7.6) it was assumed that the presence of the voids does not affect the stress field within the material. For large area fractions of voids or small stresses the local strains resulting from the growth of the voids under the applied stress cannot be accommodated by grain-boundary sliding. The stresses in these regions must then relax. The limiting case is when the local stress in the vicinity of the grain boundary is much less than the applied stress. This local stress can then be ignored and the grain-boundary regions can be treated as penny-shaped cracks embedded in a creeping material (Rice, 1983; Tvergaard, 1985; Hutchison, 1983). For noninteracting cracks with no grain-boundary sliding, Hutchison (1983) shows that

270

A, C. E Cocks and F. A. Leckie

where N , is the number of cavities on a grain boundary and CI = T I / * ( n + I)( 1 + 3/ n)-"*. Differentiating @ gives the rates strain and damage rates

-Iiz)

n+l

n: n,"]

(7.10)

and

(7.11) The important result here is that the size of the grain-boundary facets plays a role in determining the rate of increase of damage. The larger the facet size, the larger the volume of material unloaded due to its presence and the faster the creep and damage rates. Tvergaard (1985) has suggested modifications to these equations to take into account the effects of grainboundary sliding based on computations of an axisymmetric problem and the analysis of Rice (1983). The stress Cijn:nq is replaced by C, of Eq. (7.2) and the magnitude of C , is increased by a factor ranging from 1 to 200, depending on the degree of constraint imposed by the surrounding grains. The strain rate and rate of change of internal state variables follow as before. In the following section we analyze a simplified form of these potentials, so that we can compare the predictions with the results of multiaxial stress state experiments. We assume that the most critical damage lies on grain boundaries that are normal to the maximum principal stress and ignore the damage accumulating on other grain boundaries. The consequences of this assumption are analyzed.

VIII. Experimental Determination of Constitutive Laws

A large body of experimental data has been collected on the fracture properties of a number of materials under both uniaxial and multiaxial states of stress. In general these experiments have been conducted at constant stress, or when the stress has been varied, this has been done in a proportional manner. An exception to this are the experiments of Trampczynski et al.

Creep Constitutive Equations

27 1

(1981) on thin-walled tubes of copper, an aluminum alloy, and a nimonic. In these experiments the axial load was kept constant while the torque experienced by the tube was cycled between two prescribed limits. First we consider the case of constant load and develop constitutive equations that can deal specifically with this situation. Then we consider the situation'of nonproportional loading. In particular we concentrate on developing equations for copper and an aluminum alloy, which have been tested extensively by Leckie and Hayhurst (1974,1977) over the temperature range 150-300°C.

A. GENERALFEATURESO F MATERIALBEHAVIOR AT CONSTANTSTRESS When materials are tested in uniaxial tension, it is often found that the time to failure tf and the uniaxial stress u are related by an equation of the form tf =

Au-"

(8.1)

over a range of stress, where A and v are material constants. In most materials it is found that Y is less than the creep exponent n. For the copper tested by Leckie and Hayhurst (1977), it was found that v = 5.6 and n = 5.9. For the aluminum alloy, however, it was found that v was greater than n,

FIG. 10. Unaxial creep curves for (a) copper and (b) an aluminum alloy.

272

A. C. E Cocks and F. A. Leckie

with u = 10 and n = 9. We explain this observation later through consideration of the strain-softening mechanism of Section VI. Any constitutive equations we develop for the material behavior, as well as reflecting the stress dependence of Eq. (8.1), must also reflect the shape of the uniaxial creep curve. Figure 10 shows two typical creep curves for aluminum and copper. The important characteristic of these curves is the value of the quantity A [the creep damage tolerance (Ashby and Dyson, 1984)1,

which is a measure of the material’s ability to redistribute stress in a structural situation. Here A 10 for aluminum and A == 4.0 for copper. The results of multiaxial stress state tests are conveniently plotted as isochronous surfaces in stress space (Fig. 11).These surfaces connect points that give the same time to failure. Figure 11 shows the isochronous surfaces found experimentally for copper and the aluminum alloy (Leckie and Hayhurst, 1977). Failure in copper is a function of the maximum principal stress, while aluminum fails according to an effective stress criterion. When the deformation of a structure subjected to proportional loading is analyzed, it is found that a single state variable theory adequately describes the response of the structure (Cocks and Ponter, 1985b). It is also found that a single state variable theory gives good predictions of the time to failure of structures subjected to constant and moderate levels of cyclic loading (Cocks and Ponter, 1985b). In Section VIII,B we reduce each of

FIG. 11.

Isochronous surfaces in plane stress space for copper and an aluminum alloy.

Creep Constitutive Equations

273

the void growth models of Section VII to a single state variable theory by assuming that most of the void damage lies on grain boundaries that are normal to the maximum principal stress. We obtain isochronous surfaces for each model and compare them with the experimental surfaces for copper and aluminum.

B. THEORETICAL SINGLESTATE VARIABLE THEORIES OF CREEPDAMAGE In Section VI we developed a single state variable theory for the creep deformation of a material that strain softens, where the damage is represented by a scalar quantity [Eqs. (6.13) and (6.14)]. For conditions of constant stress, Eq. (6.14b) can be integrated to give the time to failure tf,

C

-

A,,(C,,- CJnP

dt, 0

where of is the value of w at failure. If C, = ($,,S,,)''2 is the von Mises effective stress, then failure occurs in a uniaxial test conducted at constant stress Ec after a time

Failure occurs in the multiaxial test after the same time if (C,

-

Z,)"C,

=

(Cc - C.,)"C,

that is, if

Z J X C= 1. This equation represents the shape of the isochronous surface in stress space, which, in principal plane stress space, is simply a von Mises ellipse (Fig. 12). The stress dependence of the time to failure is given by Eq. (8.4), ffcc l / ( &

-

CJnCc.

(8.6)

The uniaxial creep curve can be obtained by integrating Eq. (6.11). The shape of this curve and the value of A depend on the exact form of the function f ( o ) . The way to choose f ( w ) would be to fit the shape of the creep curve. Ashby and Dyson (1984) show that for this mechanism one would expect large values of A, A > 10. Next we turn our attention to the theoretical models of void growth. We assume in each case that only one family of grain boundaries are cavitated, those that lie on the boundaries normal to the maximum principal stress. Equations (7.4) and (7.5) for void growth by grain-bc Jdary diffusion then

274

A. C. E Cocks and F. A. Leckie

FIG. 12. Isochronous surface for strain-softening mechanism.

becomes

where

C:

=

cS,.ninj + Z,,,

and n is in the direction of the maximum principal stress XI. In general ZT so that Eq. (8.7) simplifies to

&,f;;'/* <<

The time to failure can be found by integrating Eq. (8.8) between the limits fh

giving

=J;

at

lf;

t = 0,

f,l,l2h l / f h dfh

fh

=

= fc

at

(n/i3)Z?tp

t = tf,

(8.9)

(8.10)

The time to failure in uniaxial tension test is given by setting Z:" = +(1 + 2c)Zc in Eq. (8.10), and the shape of the isochronous surface is then represented

Creep Constitutive Equations

275

by 3 Z t / ( 1 + 2c)Zc = 1

(8.11)

Isochronous surfaces for c = 1 and c = 1.5 are shown in Fig. 13. The stress dependence of the time to failure is now tfE l / Z .

(8.12)

When void growth is controlled by surface diffusion, Eqs. (8.7) and (8.8) become

(8.14) Integration of Eq. (8.14) between the limits of Eq. (8.9) gives the same isochronous failure surface as for grain-boundary diff usion-controlled growth [Eq. (8.11)], but the stress dependence of the time to failure is now rfa1

1 ~ ~ .

(8.15)

In each of these situations where void growth is unconstrained, the first terms in the expression for the strain rate [Eqs. (8.6) and (8.13)] dominate and there is very little tertiary creep and so A = 1 .

I

I

-c = 1.0 --1.0 Oa5I

C = 1.5

FIG. 13. Isochronous surfaces for unconstrained void growth for two values of c.

A. C. F. Cocks and E A. Leckie

276

For constrained cavity growth the constitutive relationships for one family of cavitated boundaries become

(8.16) (8.17)

where X I = Z,,n,n, is the maximum principal stress. Again if Z, >> &,f;;”*, Eq. (8.17) can be integrated between the limits of Eq. (8.9) to give the time to failure (8.18)

In uniaxial tension (8.19)

and the shape of the isochronous surface is

( w ~ c ) n - l=(1.w ~ c ) A surface for n to failure is

=

(8.20)

5 is shown in Fig. 14. The stress dependence of the time tfK

1pn.

(8.21)

In this instance the value of f h does not appear in the expression for the strain rate [Eq. (8.16)], so the material creeps at the same rate up to the instant of failure, and A = 1.

C. CONSTITUTIVE EQUATIONSFOR A N ALUMINUM ALLOY A N D COPPER In section VIII,B we examined the experimental data available on an aluminum alloy and copper, and in the preceding subsection we examined the prediction of a number of simple constitutive relationships that resulted from our analysis of the damaging mechanisms. The question now is “do any of these equations adequately describe the response of these materials?” The predictions of the strain-softening model are consistent with the behavior of aluminum. The isochronous surface in each case is a von Mises ellipse and the value of A is quite large. The stress dependence of the time

Creep Constitutive Equations

211

FIG. 14. Isochronous surface for constrained void growth when n = 5.

to failure is slightly different, but over a small range of stress a power law relationship gives a good fit to the strain rate data:

: = f(w ) &"@Jv o ) (&/a1,

(8.22)

ij

and then cj

=f(w)do(Z,/ao)"+'

and

t,cc

l/En+'.

Therefore v = n + 1, which is consistent with the experimental observation on aluminum, when it was found that v = 10 and n = 9. There appears to be ample experimental evidence to support the predictions of the strainsoftening mechanism. In the next subsection on nonproportional loading, we find further evidence for the applicability of this mechanism. The isochronous failure surface for copper (Fig. 1 1 ) is the same as that for unconstrained diffusive growth with c = 1, but the stress dependence of the time to failure, v = 5.6 as opposed to 1 or 3, and the creep ductility, A = 4.0 instead of 1, are completely different. The stress dependence of the time to failure is similar to that for constrained cavity growth, n = v, but the shape of the isochronous surface is completely different and the prediction of the creep ductility A is again too small. These discrepancies are due, in part, to the fact that we have not included nucleation in our constitutive relationships. Also, for failure to occur more than one family of grain boundaries must fail. Metallographic examination

278

A. C. F. Cocks and E A. Leckie

of copper specimens before final failure reveals the existence of a number of grain-boundary fissures, the number of which increases as the failure time is approached. Thus the number of cracklike features increases with time. We could analyze this situation by using the potential form of Section IV with a large number of state varaibles. However, we would like to describe the behavior in such a way that we need only use a single state variable. When the voids are small, there is not much accumulation of strain as they grow, there is little contribution to the macroscopic strain, and their growth is essentially unconstrained. But as they get bigger, the local strain rate increases and the growth can become constrained or the voids can link to form a crack. In either case we can analyze the behavior as if a crack were present. When a cracklike feature forms, the stress locally is relaxed and this can have a strong influence on the overall creep rate. If we assume that these cracks all lie on grain boundaries that are nearly normal to the maximum principal stress, then we can use Eq. (8.16) for the creep rate, where p is a measure of the volume fraction of cracklike features, which increases as the material creeps, and is now our measure of damage in the material. The problem now becomes one of determining the rate of increase of v. If, as before, we consider proportional loading and ignore the effects of rotation of material elements, we can obtain an expression for the rate of increase of p. First consider what happens on a given grain boundary as the material creeps. The rate of growth of voids on the grain boundary is given by Eq. (7.5),

.if = [~/(l’fk’’ln

l/fh)l~n,

(8.23)

where En is the stress normal to the grain boundary. If the outward normal to this boundary makes a small angle 8 with the direction of maximum principal stress (Fig. 15), then to first order

xn= c,(i- 8’) + (xII+ ~ , ~ ~ / 2 ) 8 ~ ,

(8.24)

where XI, and Elrlare the other two principal stresses. For final failure to occur, all boundaries with 8 < 6 must fail where 6 is a material property. From Eqs. (8.23) and (8.24) we see that damage accumulates fastest on the grain boundaries that lie normal to the maximum principal stress. When

FIG. 15. The outward normal to the grain boundary makes an angle 0 with the direction of maximum principal stress.

Creep Constitutive Equations

279

these boundaries fail, the stress is redistributed onto other parts of the material, and so the stress increases in the unfailed parts of the material. Initially the strain rate is given by

there are no grain-boundary cracks, and p = 0. The time for the formation of the first cracks can be found by integrating Eq. (8.23), with Z, = El, between the limits of Eq. (8.9); where we now interpret f, as the value of fh at which constraint effects become significant or the point where the voids coalesce to form a physical crack,

(8.26) We saw in Section 111 that we could choose any function of fh or fv to represent the damage. It is convenient at this stage to define the damage as

(8.27) Then Eq. (8.26) becomes

and t, = *J3/fEI.

(8.28)

For loading times in excess of t, the damage, in the form of cracks, spreads to grain boundaries with larger values of 8. The density of cracks is then directly related to the value of 0 = 8, at which cracks have just nucleated, so that (8.29)

8, = BV,

where B is a constant. As v increases C,in the uncracked material increase so that

1; = Z T ( L v ) ,

(8.30)

where the asterisk signifies a local stress, but the stresses Z,, and Z,,, remain the same. The time t at which a crack just nucleates on boundaries with 8 = 8, can be found by combining Eqs. (8.23), (8.24), and (8.27) and integrating between the limits $ = 0 at t = 0 and tb, = at t = t. Then

+,

where CT is a function of p, which is a function of t. For a grain boundary

A. C. F. Cocks a n d F. A. Leckie

280

with O

= Bc

+ AO,

$ = $=- A$ after a time t, where

(8.32)

Combining Eqs. (8.31) and (8.32) we find

A$

lof[

ZT20,AO -

=

1

On the boundaries for which 8 time At, where A$

= '[xl(l

+ &I1 )20cA0] dt.

&I

(8.33)

+ AO, failure occurs after an additional

=

Oc

-

Of)

i3

(

+

(

+ Clll ) O : ] A t .

(8.34)

Combining Eqs. (8.30), (8.31), (8.33), and (8.34) we find

_ -- { W 1 - 0 3 + [(%I + ~ At

2&r*c13/fl

l l l ~ / ~ l- ~6:) : l ~ ~

- G I 1

+LI/2)1

9

(8.35)

and making use of Eq. (8.29) we obtain the rate of increase of damage

where it should be remembered that Zf is also a function of u. Now the damage has a stronger influence on the strain rate of the material and there is a more extensive, and more realistic, tertiary stage of creep. It should be noted, however, that the times to failure obtained from this model are not too different from the unconstrained situation with one set of damaged planes, but the important modification is the more detailed description of the creep curve, particularly the tertiary stage. The stress dependence of the time to failure is still, however

It would appear that higher values of v require the inclusion of the nucleation process in the constitutive law, although a value of v = 3 is obtained when the growth of damage on the grain boundaries is controlled by surface diffusion. The effects of including nucleation are examined elsewhere (Cocks and Leckie, 1986). The analysis of Section IV demonstrates that the nulceation of cavities is driven by the stress normal to the grain boundary. The rate of growth of damage in the material then varies in a nonlinear manner with stress (Raj and Ashby, 1975; Cocks, 1985b). Cocks (1985b) shows how the overall damage on a grain boundary can, in certain instances, grow at a rate proportional to C;, where n is the creep exponent. This

Creep Constitutive Equations

28 1

mechanism, combined with the preceding analysis, appears to have the potential of explaining the experimental observations made on copper.

D. NONPROPORTIONAL CYCLICLOADING

In the previous subsections we concentrated primarily on proportional, or constant, loading situations where we could describe the material behavior in terms of a single damage parameter. In some structural situations components experience nonproportional cyclic loading. Examples of this can be found in a number of components of the liquid-metal cooled fast breeder nuclear reactor, which experiences cyclic thermal loading as the reactor is shut down and started up. It is therefore important to understand how a material behaves under these types of loading conditions. Trampczynski et al. (1981) have performed nonproportional cyclic loading experiments on thin-walled tubes of copper and aluminum. In these experiments the axial stress was maintained constant while the shear stress was cycled between +T. The stress levels were chosen such that the direction of maximum principal stress rotated through 32". In the tests on the aluminum alloy it was found that the magnitude of the components of strain rate were the same before and after the reversal. It was also found that the time to failure was the same as when the stress state was held constant for the entire life. These results suggest that the damage in the aluminum can be treated as a scalar quantity. This conclusion is again consistent with the predictions of the strain-softening mechanism. In tests on copper it was found that the magnitude of the strain rates decreased after a single reversal of stress and the life was increased by a factor of 2 over that of a constant stress test. Metallographic examination of the failed specimen revealed that one set of damage grew at one end of the cycle and another set at the other extreme. The fact that the time to failure in the cyclic test is twice that in the static test indicates that there is no interaction between these two sets of damage. Two state variables are now needed in the constitutive law and the strain-rate potential becomes [Eq. (7.911

Evolution laws for each damage measure ( u , and u,) must be developed as in the previous subsection. When the material is loaded so that X.,n:n; is the maximum principal stress, damage will accumulate on the planes nearly normal to n' and u, will increase as u2 remains equal to zero. When the stress system is rotated so that C,,nfn: is the maximum principal stress,

A. C. F. Cocks and E A. Leckie

282

damage will accumulate on boundaries nearly normal to n2 while u1 remains constant. Immediately after the reversal, when u2 = 0, the magnitude of Eijn:nJ decreases in Eq. (8.37) and we would expect the magnitude of the strain-rates to decrease, although the presence of this damage does affect what these strain rates are.

IX. Life Bounds for Creeping Materials When the damage in the material is in the form of dislocation loops, or even in some situations where it is in the form of voids, the damage rate equation can be written in the form oj = f ( w ) x ( f l i j ) ,

(9.1)

where f ( w ) is a function of the damage w and x ( a )is a function of stress. For damage growth rates in the form of Eq. (9.1), it is possible to extend the results of Ponter (1977) to obtain upper bounds on the life of a component for situations of constant and cyclic loading. First we obtain bounds when the load remains constant and then in Section IX,B we examine cyclic loading.

A. BOUNDS FOR CONSTANT APPLIEDLOAD

As in Section VII1,C a convenient measure of damage in these situations is

where wi is the initial damage. Then

*

=

h/f(w)

and Eq. (9.1) becomes

q=

X(flij1.

Now consider a structure subjected to a constant load P (Fig. 16) and then as an element of material deforms and becomes damaged the stress it experiences changes. I f f ( w ) is a monotonically increasing function of w, then when the structure fails after a time tf,

where

$c

is the value of $ when an element of material fails. Integrating

Creep Constitutive Equations

283

FIG. 16. Structure subjected to a constant load P fails after a time

Eq. (9.3) from t

=0

to t

= tf

I,.

and integrating over the volume give

[o'rx(..) d t d V 5 g,Y

(9.5)

A further bound on the left-hand side of Eq. (9.5) can be obtained by considering the same structure composed of a model material. Here we consider two such model materials. 1. Model Creeping Material

Consider a material that creeps according to the relationship

where we further assume that ~ ( u , ,is) a convex function of stress. Then if a: and at are two arbitrary stress states, the convexity condition is

(d,- ~ t ) [ a x ( d , ) / a d+, ~~ ( u 2-, )~ ( d 2, )0.

(9.7)

If we identify a: with the actual stress field in the material uv and at with the solution for the model material u$,integrating Eq. (9.7) between t = 0 and t = tf and then over the volume gives

[

Y

5.

["~(u,.) d t d V r [ v ~ o " x ( u ; ) d t d V = x(u;)dVtf O

(9.8)

since both uij and a; are in equilibrium with the same load P. Substituting Eq. (9.8) into Eq. (9.5) gives

A. C. l? Cocks and E A. Leckie

284

2. Perfectly Plastic Material Model

Now consider a perfectly plastic material with a yield surface given by an equation of the form

x(u,)

- x c = 0,

(9.10)

where xc is a material constant equivalent to the yield stress. The value of xc is chosen such that a structure composed of the model material collapses at the applied load P. The inelastic strain rate is given by the associated flow rule

i.;

=

@.i[ax(c.,)/aar,l,

(9.11 )

where @ is a plastic multiplier. Multiplying both sides of Eq. (9.3) by @, integrating over the life and the volume, we obtain after noting Eq. (9.4), (9.12)

Similarly Eq. (9.7) can be multiplied through by @, integrated between t = 0 and t = tf, and after setting a ; to the stress field at the limit load for the perfectly plastic material and a: to the stress field in the real material, integrated over the volume to give

-

I,lo"

@ x ( a i )dt dV

2 0,

(9.13)

where EE' is the strain rate under a stress ah. The stress fields a; and a; are in equilibrium with the same load P,, and for the model material, plastic flow can only occur when ~ ( u f = , )xC. Equation (9.12) then becomes (9.14)

Substituting this into Eq. (9.12) we obtain tf5

lCIcIxc.

(9.15)

If it is not convenient to obtain the exact limit load for the model material, the above bound applies when an upper bound to the unit load is obtained. Then, if P is an upper bound to the limit load for a given yield function X c and is the exact solution for xc, x c

5xc

Creep Constitutive Equations

285

and Eq. (9.15) becomes (9.16)

tf 5 * r I X c .

B. CYCLICLOADINGBOUNDS

For situations where the cycle times are large compared to a characteristic time for stress redistribution, the results of the preceding section can be readily extended. Consider the global damage rate c

c

* = J V & d v =J v x ( u . , ) d v ;

(9.17)

then for a given constant stress state ,y(a,])we can define a time (9.18) and Eq. (9.17) becomes

*

=

*,v/I.

(9.19)

In the present context it can be readily seen that Eq. (9.9) and (9.16) are bounds on I for a constant equilibrium stress field uij. First consider the bound of Eq. (9.9); then Eq. (9.19) becomes (9.20) where a$ is in equilibrium with the same applied loads as uv.I f the applied load changes, then Eq. (9.20) can be integrated to give (9.21) where tf is the time to failure and W ris the value of V at failure, which is less than $,V. Therefore Eq. (9.21) becomes (9.22) If the loading is applied in a cyclic manner with a cycle time of t,, Eq. (9.22) can be expressed as (9.23) where

T

is a dimensionless measure of time,

T

= t/

tc; and ifis the beginning

286

A. C. F. Cocks and F. A. Leckie

of the cycle at which failure occurs. Rearranging Eq. (9.23) gives (9.24) If the number of cycles to failure is large, then ff will be close to Similarly, if Eq. (9.16) is used as the bound on f, we find

tf.

(9.25) where xc is a function of r and is the value of the yield function that will give plastic collapse at the instantaneous load. Again the bound is retained if xc is replaced by f c , where f c corresponds to the yield function that results from un upper bound limit load calculation. A more interesting and perhaps more important situation is when the load is cycled fast compared to tile characteristic relaxation time for the structure. The above bounds, Eqs. (9.24) and (9.25), also apply to this situation, but in certain situations it is possible to obtain better bounds in terms of rapid-cycle solutions. As before, rapid-cycle solutions can be obtained in terms of the deformation solution for a model creeping material or a model plastic material. Under rapid-cycle loading conditions, the only variation of stress during a cycle is that due to the elastic reponse of the material, so that c+ij(t) = q t )

+ Pij,

(9.26)

where 6e(r)is the elastic stress field at a given instant in time and pij is a residual stress field. We assume here that the elastic constants are unaffected by the presence of the damage so that ae(t ) is the same for each cycle. This assumption breaks down when the first element of material fails so that it cannot support any stress. The following are therefore strictly bounds for the time to initiate failure in the components. 1. Model Creeping Material

A bound on the life of a component can again be obtained by considering the result of Eq. (9.5). Let af of Eq. (9.7) be the actual stress field in the component ae under conditions of rapid cycling and af the distribution obtained from the solution for the model creeping material a?. Integrating Eq. (9.7) over a cycle and then over the volume gives, after noting Eqs. (9.6) and (9.26),

where A&$ = j& 6; dt is the compatible inelastic strain accumulated during

287

Creep Constitutive Equations

a cycle of duration 2,. The first term on the left-hand side of Eq. (9.27) is then identically zero. Substituting Eq. (9.27) into Eq. (9.5) then gives (9.28) where, as before, r

= t/t,.

2. Model Plastic Material Under conditions of rapid cycling, shakedown boundary solutions can be used to facilitate the construction of bounds on the time to initiate failure in a component. Ponter (1977) considers a general cyclic loading history. Here, however, we limit our attention to the class of problems where the load is cycled between two prescribed limits (Fig. 16). The results can readily be generalized to include the situations considered by Ponter (1977). Again we make use of the inequality of Eq. (9.3) and the convexity condition of Eq. (9.7). As before we identify mt with the actual stress field and a!, with the shakedown solution for an elastic perfectly plastic material of yield strength X(UB) = xo (9.29) where the magnitude of xc is chosen such that the structure composed of the model plastic material just shakes down. If we now write the associated flow rule in incremental form,

dE: = P Lax (

)/ uv

1

and apply Eq. (9.7) at each extreme of the cycle, we obtain P1x('b)

- pIX('F) - ~

1s

l [ a x ( a ~ ) / d a ~ - l(+lJ( a ~

O,

P 2 X b Z , ) - p 2 x ( u 3 - P2Eaxbi;")/a(+:I(u;- a:) 2 0,

(9.30)

where the first of Eqs. (9.30) applies when 0 5 T 5 A and the second when A < r 5 1. Here we will assume that A d 1 - A. Combining Eqs. (9.30) and noting that plastic straining can only occur when Eq. (9.29) is satisfied leads to the result (9.31) where p u = afi- a: = a; - a$ is a residual stress field and dEk is the increment of plastic strain experienced by an element of material during a cycle. The inequality of Eq. (9.31) is still retained if pl/A and p 2 / ( 1- A ) is replaced by p, where p is the maximum of p , / A and p 2 / ( 1- A ) . Integrating Eq. (9.31) over the volume then gives

Jvfi{Ax(gi)+ ( 1 - A ) x ( a ~ , dV ) ) 2 xc

I,

(cL1

+ p 2 )dv

(9.32)

A. C. F. Cocks and E A. Leckie

288

Multiplying both sides of Eq. (9.3) by /. and integrating over the time to initiate failure and the volume gives t;

1

/ . { A x ( g ; ) + (1 - A ) x ( a i ) )dV =

I":I

& ,I

dt dV.

(9.33)

Combining Eqs. (9.32) and (9.33) and noting that the inequality of Eq. (9.4) gives

The above bound still holds if xC is replaced by X,, the yield function resulting from a kinematic bound for the shakedown solution. As discussed by Ponter (1977), this bound can drastically overestimate the time to failure of a component. The bounds of Eqs. (9.28), (9.24), and (9.25) can then give more accurate estimates of the life.

X. Discussion The aim of the present paper was to try to obtain a structure for creep constitutive laws through an understanding of the microscopic mechanisms responsible for deformation and failure. This work extends the thermodynamic approach adopted by Rice (1971) and Cocks and Ponter (1985a) to include the effects of damage, which exists either in the form of microscopic voids or as dislocation networks that aid the dislocation climb process. For each mechanism analyzed it is possible to prove the existence of a potential from which the inelastic strain rate and the damage rate are derivable. This result follows from the fact that the rate of increase of the microscopic damage is driven by its associated thermodynamic force [Eq. (2.10)]. These kinetic relations have been obtained from more detailed analyses of the processes that are responsible for the creeping behavior. For the mechanisms of void growth, the kinetic relations are essentially exact, while for nucleation and the strain-softening mechanism they are semiempirical in that they are simplifications of the more complex behavior that retain the physics of the process. The approach adopted here is similar in some respects to that used by Lemaitre and Chaboche (1985). Their measures of damage and their kinetic relationships are, however, entirely phenomenological. The structure of their theory is based on the results of Rice (1971) for dislocation glide mechanisms of deformation. In this instance the rate of glide of a dislocation is only a function of the thermodynamic force associated with it, so that the resulting potential is only a function of the thermodynamic forces. When

Creep Constitutive Equations

289

considering the various mechanisms of damage, we find a slightly different result, Now, the rate of increase of an internal variable that is a measure of the damage is also a function of the present state of damage in the material. The resulting potential is therefore also a function of the present state of damage. When analyzing the various void growth mechanisms, we find that a large number of discrete state variables are required to describe the material response to general loading conditions (Section 111). This number can be reduced somewhat by representing the damage in terms of the distribution of cavitated boundaries (Section V). For situations where this distribution of damage is relatively smooth, the distribution can be represented by a scalar and a second-order tensor state variable. But this is rarely the case, and some grain boundaries generally suffer much more damage than the remainder. There is then a peak in the distribution of damage and higher even-order tensors are required to describe the state of the material. If one, however, is only interested in certain classes of loading, it is possible to identify simple measures of damage that can adequately describe the materials behavior (Section VIII). For example, in situations where the load is varied in a proportional manner, a single scalar measure of damage is sufficient. This damage measure relates to the density of grain-boundary fissures that lie normal, or nearly normal, to the direction of maximum principal stress. When nonproportional loading between two prescribed limits occurs, two scalar state variables are required, which relate to the accumulation of damage normal to the directions of maximum principal stress at the two extremes. Over the range of stress and temperature used in the experiments of Leckie and Hayhurst (1974,1977) and Trampczynski et al. (1981), the results of their tests on copper can be explained in terms of our understanding of the void nucleation and growth mechanisms (Section VIII). The aluminum alloy tested by them, however, exhibits a completely different type of behavior. They found that even for nonproportional loading histories that a single scalar measure of damage adequately describes the materials response. This is consistent with the analysis of the strain-softening mechanism described in Section V1. It is shown that Section IX that when the damage growth rate expression is of the form

where w is a measure of the damage and f(a)and g ( w ) are functions of stress and damage, respectively, that it is possible to obtain upper bounds on the life of a component for any loading history. At the present time we are unable to obtain similar results for the more general constitutive relationships obtained in Sections I1 through IX.

290

A. C. F. Cocks and E A. Leckie

Appendix: Mean Strains

In this Appendix we obtain expressions for the mean strain experienced by a body when portions of it undergo inelastic deformation. Results are obtained for an anisotropic elastic materials whose stiffness and compliance matrices satisfy the usual symmetry conditions:

The form and magnitude of these matrices is also allowed to vary with position in the body. First we consider the situation where a small element of material deforms plastically; then we examine the material's response when discontinuities occur across a plane within the material.

A. MICROSCOPIC INELASTIC DEFORMATION

Consider the situation of Fig. 17, where a volume V' of an element of material of volume V suffers a uniform inelastic strain d E : . We wish to calculate the mean strain suffered by the larger element. Each strain component must be calculated individually. By way of example we will obtain expressions for the component dEY2. Associated with the inelastic strain d E : is an elastic strain d E t , which together form a compatible field and a residual stress field p V , where Pv =

cljkl

dEekl.

('4.2)

Let 68 be the elastic stress distribution in the material for unit applied P

dEij

FIG.17. Macroscopic plastic strain dEP, resulting from an increment of plastic strain d.$ in a volume V .

Creep Constitutive Equations

29 1

stresses E12= E2,. Then application of the principle of virtual work gives r (dEy2+ d E ; , ) V = 2dEy2V = J GV(dE;+ d$) d V =

V

Iv

cVkltkl(dE; -k dE$) dV,

(A.3)

where &.I is the compatible elastic strain field resulting from the application of the unit remote shear stresses. Making use of Eqs. ( A . l ) and ( A . 2 ) the preceding expression becomes

The first term on the right-hand side of Eq. ( A . 4 ) is zero since pij is a residual stress field and is compatible. Therefore

I

ds5 dEP -GVd K l2 - 2 v v' Similar expressions are readily obtainable for other components of strain. The mean remote strain is related to the local strains through the solution of an elastic problem. For an isotropic elastic material subjected to the stresses E12= E2, = 1, G12 = 621 = 1, GI, = 622= 633= 6 3 2 = 6 2 3 = 0, and Eq. (AS) becomes dEY2 = ( V ' / V ) d&Y2

64.6)

dEIJ = ( V ' / V ) dE$.

('4.7)

or in general

B. INELASTIC STRESSRESULTINGFROM INTERNAL DISCONTINUOUS DEFORMATIONS Consider the body of Fig. 18, which contains a plane of area A with outward normals n+ and n-. If the positive side of this plane moves an amount u wrt the negative side, the body experiences a macroscopic strain d E $ . The object of this section is to calculate what these strains are for given internal displacements. Again we do this by considering the equilibrium stress field that results from the application of unit dummy loads in the direction o f the required strain. As an example we calculate the shear strains dEy2 and dE8,. We now treat the body as if it contained an internal surface S, that is subjected to tractions TT = -6.,nf along S: and T ; = -GVn,: along S;, where, as before, GV is the elastic stress field resulting

292

A. C. F. Cocks and F. A. Leckie

dEi j

P

FIG.18. Macroscopic plastic strain dEP, resulting from discontinuous deformation across a plane with outward normals n' and n-.

=

1. Application of

dV.

(A.8)

from the application of the remote stresses CI2= the principle of virtual work then gives

2 dEY, V +

T:u, dS

=

[

&,,E;

V

By making use of Eqs. (A.l) and (A.2) this becomes

The second term on the right-hand side of Eq. (A.9) is zero, and we have (A.lO)

For an isotropic homogeneous material, zero, and Eq. (A.lO) becomes

=

=

1 with all other stresses

dEy2 = ( A / 2 V ) ( n l u+ , n2u1).

Using the full range of dummy loads we find

dEf:= ( A / 2 V ) ( n , u + j nju,)

(A.ll)

for an isotropic homogeneous material. ACKNOWLEDGMENT The authors gratefully acknowledge the financial support of the Department of Energy through contract DOE 1198 with the Materials Research Laboratory at the University of Illinois at Champaign-Urbana.

Creep Constitutive Equations

293

REFERENCES Argon, A. S., Chen, I.-W., and Lau, C.-W. (1980). Intergranular cavitation in creep: Theory and experiments. In “Creep-Fatigue-Environment Interactions” (R. M. Pellouse and N. S. Stoloff, eds.), p. 46. Am. Inst. Min. Eng., New York. Ashby, M. F., and Dyson, B. F. (1984). Creep damage mechanics and micromechanisms. Nafl. Phys. Lab. U.K. Rep. DMA(A) 77. Chuang, T.-J., Kagawa, K. I., Rice, J . R., Sills, L. B. (1975). Non-equilibrium models for diffusive cavitation of grain interfaces. Acfa Mefall. 27, 265. Cocks, A. C. F. (1985). The nucleation and growth of voids in a material containing a distribution of grain-boundary particles. Acfa Metall. 33, 129. Cocks, A. C. F. (1986a). Creep constitutive equations for strain-softening materials. Submitted for publication. Cocks, A. C. F. (1986b). Creep deformation of porous materials. Submitted for publication. Cocks, A. C. F., and Ashby, M. F. (1982). On creep fracture by void growth. Prog. Mater. Sci. 27, 189. Cocks, A. C. F., and Ponter, A. R. S. (1985a). Constitutive equations for the plastic deformation of solids. I-Isotropic materials. Leicester Univ. Eng. Dep. Rep. 85-2. Cocks, A. C. F., and Ponter, A. R. S. ( 1985b). Constitutive equations for the plastic deformation of solids. 11-A composite model. Leicester Univ. Eng. Dep. Rep. 85-1. Duva, J . M., and Hutchinson, J. W. (1983). Constitutive potentials for dilutely voided nonlinear materials. Div. Appl. Sci., Harvard Univ. Rep. MECH-47. Dyson, B. F. (1979). Constrained cavity growth, its use in quantifying recent creep fracture studies. Can. Metall. Q. 18, 31. Dyson, B. F., and McLean, M. (1983). Particle-coarsening uoand tertiary creep. Acfa Mefall. 31, 17. Dyson, B. F., Loveday, M. S., and Rodgers, M. J . (1976). Grain-boundary cavitation under various states of applied stress. Proc. R. Soc. London Ser A . 349, 245. Hayhurst, D. R., Trampczynski, W. A., and Leckie, F. A. (1981). Creep rupture under non-proportional loading. Acfa Metall. 28, 1171. Henderson, P. J., and McLean, M. (1983). “Microstructural contributions to friction stress and recovery kinetics during creep of the nickel-base superalloy IN738LC. Acfa Mefall. 31, 1203. Hirth, J . P., and Nix, W. D. (1985). Analysis of cavity nucleation in solids subjected to external and internal stresses. Acta Metall. 33, 359. Hutchinson, J. W. (1983). Constitutive behavior and crack tip fields for materials undergoing creep-constrained grain-boundary cavitation. Acta Metall. 31, 1079. Leckie, F. A,, and Hayhurst, D. R. (1974). On creep rupture in structures. Proc. R. Soc. London Ser. A. 340, 324. Leckie, F. A,, and Hayhurst, D. R. (1977). Constitutive equations for creep rupture. Acta Mefall. 25, 1059. Lemaitre, J., and Chaboche, J. L. (1985). “MBcanique des Matenaux Solides,” Dunod, Paris. Martin, J. B. (1966). A note on the determination of an upper bound on displacement rates for steady creep problems. J. Appl. Mech. 33, 216. Onat, E. T., and Leckie, F. A. (1984). A continuum description of damage. T & AM Rep. No. 469. Univ. Illinois at Urbana-Champaign. Ponter, A. R. S. (1969). Energy theorems and deformation bounds for constitutive relations associated with creep and plastic deofrmation of metals. J. Mech. Phys. Solids 17, 493. Ponter, A. R. S. (1977). Upper bounds on the creep rupture life of structures subjected to variable load and temperature. In[. J. Mech. Eng. Sci. 19, 79. Ponter, A. R. S., Bataille, J., and Kestin, J. (1979). A thermodynamic model for the time dependent plastic deformation of solids. J. Mec. 18, 511.

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Raj, R., and Ashby, M. F. (1975). Intergranular fracture at elevated temperature. Acra Metall. 23, 653. Raj, R., Shih, H. M., and Johnson, H. H. (1977). Correction to intergranular fracture at elevated temperature. Scr. Metall. 11, 839. Rice, J. R. (1971). Inelastic constitutive relations for solids: An internal variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433. Rice, J. R. (1983). Constraints on the diffusive cavitation of isolated grain-boundary facets in creeping polycrystals. Actn Metall. 31, 1079. Shewfelt, R. S. W., and Brown, L. M. (1977). High-temperature strength of dispersion-hardened single crystals. I1 Theory. Philos. Mag. 35, 945. Stevens, R. A,, and Flewitt, P. E. J. (1981). Dependence of creep rate on microstructure. Acta Metall. 29, 867. Trampczynski, W. A., Hayhurst, D. R., and Leckie, F. A. (1981). Creep rupture of copper and aluminum under non-proportional loading. J. Mech. Phys. Solids 29, 353. Tvergaard, V. (1984). Constitutive relations for creep in polycrystals, with grain-boundary cavitation. Acta Metall. 32, 1977. Tvergaard, V. (1985). Effect of grain-boundary sliding on creep constrained diffusive cavitation. J. Mech. Phys. Solids 33, 447. Wang, J. S., Stephens, J. J., and Nix, W. D. (1985). A statistical analysis of cavity nucleation at particles in grain boundaries. Actn Metall. 33, 1009.

Subject Index

A Airy’s stress function, 9-1 1 Aluminum alloy creep constitutive equations, 276-281 isochronous surfaces in plane stress space, 272 nonproportional loading, 281, 289 uniaxial creep curves, 271 Annulus, as meandering strip, 22, 41 Axial velocity, 15-16 coiled pipe, 34 hyperbolic strip, 5 pipe of varying elliptic section, 25-26 twisted elliptic pipe, 29

B Bernoulli-Euler approximate theory, 78 Bessel equations, 85 Blasius series, 13 Bromwich contour, 87 Bromwich integrals, 101

Cagniard-deHoop technique, 55, 151-156 integration contours, 153 Cagniard’s technique, 57 Cantilever plate, 78-79 Caustic, 115-117 Channel, laminar flow, 11-15 295

meandering, 23-24, 41-42 Clausius-Duhem inequality, 188-189, 244 Compressional strains, 5 1 Constitutive equations, 193 gases, 200-202 heat conduction, 195 incompressibility, 205-207 linear elasticity, 196-197 linear viscoelasticity, 228-23 1 liquids, 202-204 plasticity, see Plasticity, constitutive equations Rivlin-Ericksen liquids, 232-233 soils, see Soils, constitutive equations viscoplasticity, 225-228 Continuity equations, varying gap, 36 Continuum mechanics, 184-186 Copper creep constitutive equations, 276-281 isochronous surfaces in plane stress space, 272, 277 nonproportional cyclic loading, 281 uniaxial creep curves, 271 Coulomb’s law, 222 Creep constitutive equations aluminum alloy, 276-281 constant stress, 271-273 copper, 276-281 experimental, 270-282 nonproportional cyclic loading, 281-282 single state variable theories, 273-276

Subject Index

296

Creep constitutive equations (cont.) theoretical, 267-270 damage, 272, 289 failure in multiaxial test, 273 global damage rate, 285 isochronous surface for strain-softening mechanism, 274 stress dependence of time to failure, 280 Creeping material, life bounds, 282-288 model, 283, 286-287

D Damage mechanisms, precipitationhardened materials, 263-267 deformation rate, 266-267 dislocations, 264-265 potentials, 266-267 Dean number, 34 Deformation coupled elastic and plastic, 223-225 rate, macroscopic element of material, 257-258 Dislocation glide, theorems, 265 networks, 264 Displacement gradients, 65 Displacement-potential relations, 85 Dissipation function, 193 construction plastic materials, 210-21 1 soils, 223 deviatoric plane, 209 heat conduction, 195 incompressible Newtonian liquid, 206-207 isotropic material, 235 linear viscoelasticity, 229 liquid free of bulk viscosity, 204 Newtonian liquid, 204 rigid, perfectly plastic materials, 207-209 Rivlin-Ericksen liquids, 232 sectors corresponding to definitions, 209, 22 1 soil models, 218-223 nonassociated flow, 217 specific, 189-190 thermoelasticity, 198 Tresca hardening material, 214 Tresca material, 209, 212 viscoplasticity, 228

viscous liquid, 203 v. Mises material, 208-209, 212 Dissipation rate, maximal, 192 Dissipative forces, 190-191 Dissipative internal stress, 212, 225 Drucker-Prager yield surface, 217

E Edge load problems, see Plate in plane strain Elastic compliance, 253 Elasticity, linear, constitutive equations, 196- 197 Elastic pulse scattering circular cylindrical cavity, 81-1 14, see also Cavity wall, normal line load source; Plane compressional pulse Friedlander’s representation of solution, 83-84 line load source, 81-83 wavefronts, rays, and wave regions, 82 diffraction by spherical cavity, 118-125 incident plane dilatational pulse, 119, 124-125 literature, 118-1 19 method of solution, 119-125 point load problem, 118, 120-124 radial displacement response, 123, 125 tangential displacement response, 124- 125 Elastic ring sector, slow variations, 32-33 Elastic strip, in tension, 9-1 1 Elastic wave scattering, 125-127 by cracks, 127-132 ray theory, 132 time-harmonic loads and waves, 128-131 transient loads, waves, and wavefronts, 131-132 diffraction by semi-infinite rectangular boundary of finite width, 127 modal and surface wave resonances in acoustic scattering, 126- 127 wave diffraction by finite rigid strip and crack, 126 Elastodynamic modeling, electrical signal processing devices, 81 Electrical signal processing devices, elastodynamic modeling, 81 Energy dissipation, rate, 253, 257, 260-261

Subject Index Entropy production, maximal rate, 191-194 thermoelasticity, 197-198 Eulerian approach, 187 Extended Hankel transforms, 79-80 Extended saddle point technique, 54 External stress, 224, 229-23 1

F Failure, mechanisms, 241 Finite difference-numerical method, 77 First law of thermodynamics, 185, 187 Flamant problem, 71-72 Fourier-Bessel series, 98 Fourier coefficient, values, 74 Fourier series solution, 106-109 Free energy linear viscoelasticity, 229 Rivlin-Ericksen liquids, 232 Friedlander’s representation of solution, 83-84, 92, 109-1 10

G Gap, varying, potential flow, 36-37 Gases, constitutive equations, 200-202 Generalized asymptotic expansion, 7 Generalized axisymmetric potential theory, 17 Geometric optics method, 109 Grain boundary cavitated, 247 diffusion, void growth, 251-254 slab, 248

H Heat conduction, 194-196 Helmholtz free energy, 243, 245, 256, 265 void, nucleation, 259-260 Hooke’s law, 197, 212-213, 224 Hoop stress, 113 Hydrostatic pressure, 206 Hyperbolic strip, potential flow, 3-8 slight variations, 4-5 slow variations, 5-8

297 I

Illuminated zone, 87, 110 Incompressibility, constitutive equations, 205-207 Inelastic deformation, microscopic, 290-291 Inelastic strain rate, 255-257, 260-261, 266, 269 constrained cavity growth, 276 grain boundary diffusion, 274 growth controlled by surface diffusion, 269, 275 remote, 246 without boundary cracks, 279 Inelastic stress, internal discontinuous deformations, 291-292 Internal energy, 201 Internal stress, 224 associated, 251 elastic, perfectly plastic materials, 212 hardening material, 213 linear viscoelasticity, 229-230 Inviscid gas, dissipation function, 200 Irrotational waves, velocity, 199-200

J

Jeffery-Hamel solution, 20, 38

L Lamb’s line load problem, 164 Lam6 constants, 197 Laminar flow channel, 11-15 boundary conditions, 11 high Reynolds number, 14-15 meandering, 23-24, 41-42 Reynolds number of order unity, 12-13 pipe coiled, 33-35 spiraling circular, 3 1 twisted elliptic, 29 varying elliptic section, 26-27 tube, 18-20 Laplace’s equation meandering strip, 21-22, 40 pipe of varying elliptic section, 25 spiraling circular pipe, 30 twisted elliptic pipe, 28

298

Subject Index

Laplace-time transforms bilateral, 98 edge conditions, 64,71 Laplace transform displacement equations of motion, 60 edge conditions, 50, 52 Life bounds, creeping materials, 282-288 constant applied load, 282-285 cyclic loading, 285-288 Liquids constitutive equations, 202-204 incompressible, constitutive equations, 206-207

M Macroscopic potential, 268-269 Manton’s expansion, 19-20 Maxwell grid, 228-229 Meandering channel, laminar flow, 23-24, 41-42 Meandering strip, potential flow, 21-23, 40-41 Mean strains, 290-292 Mechanical energy theorem, 185 Method of reduction, 74 Method of steepest descents, 87, 109 Michell’s stress function, 17 Molar entropy, 201 Molar free energy, 201 Molar internal energy, 201

N Newtonian liquid dissipation function, 204 incompressible, dissipation function, 206-207 Normality condition, 21 1 Normal line load source, on cavity wall Bromwich contour, 87-88 exact inversion, 86-91 formal solution, 84-86 frequency equation, 88-90, 92 governing equations, 85 Lamb’s line load problem, 94-95 positive @-travelingRayleigh waves, 96 Rayleigh waves and long-time, far-field solution, 91-97 shadow zone, 87-88

static solution, 95 wave propagation form, 88-89 Norton’s law, 226 Nucleation threshold stress, 258 voids, 258-261

0 Odqvist’s equation, 226 Orthogonality, 185, 189-193 linear viscoelasticity, 230 modified, 206 rigid, perfectly plastic materials, 209-210 soils, 221-222

P Pipe coiled, laminar flow, 33-35 spiraling circular laminar flow, 31 potential flow, 30-31 twisted elliptic laminar flow, 29 potential flow, 28 varying elliptic section laminar flow, 26-27 potential flow, 25-26 varing radius, potential flow, 15-16 Plane compressional pulse diffraction by cavity approximations, 109-1 14 bilateral Laplace transform on time, 98 Bromwich integrals, 101 circumferential velocity Rayleigh surface waves, 11 1-1 12 exact inversion, 100-102 formal solution, 97-99 Fourier series solution, 106-109 geometric optics method, 109 hoop stress, 113 long-time circumferential stress response, 112-1 13 long-time radial and circumferential acceleration response, 1 12 modal response, 102-103 numerical evaluation of solution, 102-106

Subject Index positive O-traveling Rayleigh waves, 110, 112-113 rigid-body velocities, 113 scattered wave solution, 97-98 wave sum, 104-106 scattering by circular cylindrical elastic inclusion, 114-1 18 literature, 115-118 problem, 114-115 Plasticity, constitutive equations, 207-216 dissipation function construction, 210-21 1 elastic, perfectly plastic materials, 211-212 linear hardening, 213-214 rate-dependent yield, 214-216 rigid, perfectly plastic materials, 207-210 Plastic material, model, life bounds, 284-285, 297-298 Plastic strain, macroscopic, 290, 292 Plate axially symmetric infinite, 79-80 cantilevered, 78-79 varying thickness, steady temperature, 36 Plate in plane strain antisymmetric excitation, 52-53 fixed edge conditions, 49-53 frequency spectra, unbounded waves, 62 governing equations, 48-49 longitudinal impact, 50-52, 54 long-time, far-field approximations, 53-55 mixed edge conditions, 53-59 mixed pressure shock, 52 nonmixed edge conditions, 59-78 boundedness conditions, 61-63 far-field approximation, 69-70 Flamant problem, 72-72 infinite set of linear algebraic equations, 66 inversion integral forms, 60-61 line load, 70-75 pressure shock, 63-68, 68-70 residual associated elastostatic problem. 71-73 shock tube response records, 77 static edge displacements, 75-76 uniform normal load, 72-73 short-time, near-field wavefront approximations, 55-59 wavefronts, 57-58 Poiseuille flow, 18, 26-27

299

straight circular pipe, 31 Poisson’s ratio, coupling of longitudinal and thickness strains, 69-70 Poisson’s summation formula, 83-84 Potential flow pipe spiraling circular, 30-31 twisted elliptic, 28 varying elliptic section, 25-27 varying radius, 15-16 strip hyperbolic, 3-8 meandering, 21-23, 40-41 symmetric, 8-9, 39-40 varying gap, 36-37 Power law creep, void growth, 249-251 Precipitation-hardened materials, damage mechanisms, 263-267 Principle of maximal dissipation rate, 192

Q Quarter-plane problem, 131-132, see ulso Two-dimensional elastic wedge equivoluminal wavefronts, 162 Quasi-cylindrical approximation, slow variations, 5

R Rayleigh-Lamb frequency branch, 54 Rayleigh-Lamb frequency equation, 62 Rayleigh-Lamb spectra, 59 Rayleigh surface wave circumferential velocity, 111-1 12 speed, 135 subsurface crack, 130 surface-breaking cracks, 129-130 Rayleigh waves near surface interior, 94 normal line load source, on cavity wall, 91-97 positive O-traveling, 96, 110, 112-1 13 time-dependent, 91 Rays caustic and, 115-1 17 family of, 57 geometry, refracted dilatational waves, 115

Subject Index

300

Ray theory, 132 Reaction principle, 183 Refracted dilatational waves, ray geometry, 115

Reiner-Rivlin liquids, constitutive equations, 206 Reynolds number coiled pipe, 34 high, laminar flow channel, 14-15 meandering channel, 24, 42 tube, 19-20 order unity, laminar flow channel, 12-13 meandering channel, 23-24, 41-42 tube, 18-19 Rivlin-Ericksen liquids, constitutive equations, 232-233

S Second law of thermodynamics, 185, 187, 244, 253, 255, 265 Self-similar solutions, 133 Shadow zone, 81-82, 87 Fourier series solutions, 109 wavefront approximations, 110 Shield’s pyramid, 222 Shock tube response records, 77 Signum function, 72 Skin friction, along channel, 13-14 Slow variations antisymmetric plane strip, 20-24 laminar flow in meandering channel, 23-24 meandering strip, 21-23 axisymmetric shapes, 15-20 laminar flow in tube, 18-20 pipe of varying radius, 15-16 torsion shaft of varying radius, 16-18 closer fits, 37-42 meandering channel, 41-42 meandering strip, 40-41 symmetric strip, 39-40 perturbation solution, 2 quasi-cylindrical approximation, 5 symmetric plane strip, 3-15 elastic strip in tension, 9-11 general symmetric strip potential, 8-9

hyperbolic strip potential, 3-8 laminar flow in channel, 11-15 three-dimensional slender shapes, 24-35 coiled pipe, 33-35 pipe of varying elliptic section, 25-27 spiraling circular pipe, 29-3 1 twisted elliptic pipe, 27-29 twist of elastic ring sector, 32-33 three-dimensional thin shapes, 35-37 Snell’s law, 162 soils, constitutive equations, 216-225 coupled elastic and plastic deformation, 223-224 dissipation function construction, 223 models, 218-223 nonassociated flow, 216-218 Specific free energy, 187 Specific heat, 199 Specific heat capacity, 199 State variables, 240-241 families, 265 Stationary phase method, 54-55 Strain rate, total, 246 Strain-softening mechanism, isochronous surface, 274 Stream function laminar flow, 11-13 meandering channel, 23-24 meandering strip, 22 pipe of varying radius, 15-16 plate of varying thickness, 36 potential flow, 4-8 Stress deviator, 208-209, 214, 218 Rivlin-Ericksen liquids, 232 viscoplasticity, 226, 228 Stress distribution, Flamant problem, 72 Stress function, 9-1 I elastic ring sector, 32-33 torsion shaft of varying radius, 17 Stress-potential relations, 136 semi-infinite waveguide, 49 Stress tensor gases, 200 linear elasticity, 196-197 liquids, 202-204 subdivision, 188 thermoelasticity, 198- 199 Subsurface crack, 129-130 Surface-breaking crack, 128-130 Surface diffusion, void growth limited by, 254-256 Symmetric strip, potential flow, 8-9, 39-40

Subject Index T Thermodynamic formalism, 242-247 Thermodynamics, 186-189 average quantity use, 261-263 Thermoelasticity, constitutive equations, 197-200 Transverse velocity, 15-16 hyperbolic strip, 6 Tresca material dissipation function, 209, 212 hardening, 214 Tube, laminar flow, 18-20 Two-dimensional elastic wedge boundedness condition, 133-134 dilatational plane wavefronts, 176-177 equivoluminal wavefronts, 161-166, 170- 176 formal solution for time-transformed displacements, 149-15 1 inadmissible events, 145-147 indicated contour integration for solution, 142-145 inversion of time-transformed displacements by Cagniard-deHoop technique, 151-156 limiting nest of contours, 142-143 plane-strain displacement equations of motion, 134-135 pressure shock problem, 134, 140-142 quasi-formal solutions, 136-140 Rayleigh surface wavefronts, 167-169 remaining contours, 144 self-similar solutions, 133 stress-strain relations, 136 time-transformed edge and cornel unknowns, 147-149 Wiener-Hopf problem, 133

V Virtual power, principle, 183 Virtual work, expression, 250 Viscoelasticity, linear, constitutive equations, 228-231 Viscoplasticity, constitutive equations, 225-228 creep of metals, 225-227 elastic, viscoplastic materials, 227-228 Viscous flow, 2 Viscous strain, 227

301

v. Mises material, 208-209, 212 hardening, 213 Void growth constrained cavity growth, 269-270 constrained isochronous surface, 277 deformation rate, macroscopic element of material, 257-258 discrete state variables, 289 grain boundary rate, 269, 278 mechanisms, 247-258 grain-boundary diffusion, 251-254 grain-boundary sliding, 256-257 limited by surface diffusion, 254-256 power law creep, 249-251 necessary void radius, 259 profile, 254-255 unconstrained cavity growth, 268-269 unconstrained isochronous surfaces, 275 Voids, nucleation, 258-261 Vorticity equation, 11, 23

W

Watson’s transformation, 120-122 Watson-type lemma, 117 Wavefronts dilatational, 122-123, 158-161, 169- 170 displacement, 148 plane, 176-177 time-transformed displacements, 147 equivoluminal, 161-1 66, 170- 176 acceleration two-sided, 171-175 head and two-sided approximation contours, 161 head wavefronts, 162-163 precursor, 170-171 quarter-plane problem, 162 two-sided, 164-166 nonseparable elastodynamic quarter-plane problems, 131-1 32 p-dependent coefficients, 156-158 Rayleigh displacement, 148 Rayleigh surface, 167-169 singularity, 117 Wiener-Hopf problem, 133

Y Yield stress, 214-215, 217

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