Theoretical Acoustics of Underwater Structures

1 r drz<^ dz <9
2 r 1 + r +

Y

d2ur dt2 ' d2u(p a/ 2 <9 2 u z

^7 r ^ - a T r - ^ = '"aF-

(6L6)

In spherical coordinates the equations are much more complicated, do-RR +

1 droR +

1

STR*

1 5^^ # 50

dr^R

cot0

<92Ufl

1

2

^^ ^"aT ^sin^ ^ + - ^ ^ + f l ( ^ - ^ - ^ ) + ^ = P"air. 2 oR

1 Sr^ 3 #sin0 d<j> R

cot0 R

<9 t/0 dt2

Acoustics of Underwater Structures

136

dTRt , 1 drH

,

1

dau

, 3

2cot
,

F

_ /

2

^

(6.1.7) The stress equations are independent of whether or not the elastic solid is isotropic or anisotropic. The matrix relation between stress and strain, viz. {a} = [D]{e}} completes the description of a linear elastic solid. The 6x6 matrix [D] is the constitutive stiffness matrix, considered in Sections 6.5 and 6.6, whose elements depend on whether or not the elastic solid is isotropic or anisotropic. 6.2

Isotropic Displacement Equations

In cartesian coordinates, for an isotropic elastic solid, the relations between stress and strain are particularly simple. They are &xx = Ae + 2fieXX}

ayy = Xe + 2ficyy, T

Txy = fJ-Jxy j

yz — fJ'Jyz >

azz = Xe + 2fiezz, T~zx — f^Jzx ?

ox ay oz where A and // are the Lame constants and e is called the dilatation. The stresses are replaced in Eq. (6.1.5) by strains, by making use of Eq. (6.2.1), and then the strains are replaced by displacements by using Eq. (6.1.1). The resulting equations are written in vector form as (A + /i)V(V.u(x,*)) + /iV 2 u(*,<) + F(x,t) = P ^ f f i 0 ,

(6.2.2)

where u(x,tf) = (uX) wy, uz) is the displacement vector and F(x, t) = (FXi FyyFz) is the body force vector. In cylindrical coordinates, the relations between stress and strain are
a^ = Xe + 2//e^,

Tr = fl>Jri

T^z = flj^

e = V.u,

,

azz = Xe + 2fiezz, Tzr = fljzr

,

(6.2.3)

Elastic Equations

137

and in spherical coordinates they are CRR

= Ae + 2fi€RR,

(TQQ = Xe + 2pLCeg,

7ft0 = HJR9,

T~o = ^70<£>

< T ^ = Ae + 2/ie<^, 7-0J* = ^7<£il>

e = V.u.

(6.2.4)

The displacement equations in cylindrical and spherical coordinates are also given by Eq. (6.2.2), provided the interpretation V 2 u = V(V.u) - V x (V x u)

(6.2.5)

is strictly applied. Recall that V 2 u is not equal to V 2 operating on the separate components of the vector u, except in cartesian coordinates, as demonstrated in Section 1.4. The three components of the displacement equations can be expanded out, by making use of the relations for V, V. and V 2 given in Section 1.4, but it is generally not profitable to do so in cylindrical and spherical coordinates. The re­ arranged displacement equation, which is independent of the coordinate system, is (A + 2/i)V(V.u) - /iV x (V x u) + F(x) = p ^ .

(6.2.6)

This equation can be reduced by the substitutions u

=

Vtf + V x P ,

F

=

V^/4-VxPy,

(6.2.7)

to V ^ (A + 2//)V 2 tf + * / - p-^-j

+ V x ^ * V 2 P + P , - p-^j

= 0. (6.2.8)

Particular solutions of this equation are <92\P (A + 2 A i ) V 2 * - / > - ^ / i V ' P - ^

=

-*/.

=

-Pf,

(6.2.9)

which are the scalar and vector wave equations of linear isotropic elasticity. Because there are four unknowns, ^ and P , for finding the three displacements,

Acoustics of Underwater Structures

138

u, a supplementary condition is required. This is the commonly used gauge condition V.P = 0. (6.2.10) Separable solutions of the homogeneous vector wave equation, the second of Eq. (6.2.9) with P / = 0, are straightforward in cartesian coordinates. Got­ tlieb [1] has shown that separable solutions are possible in cylindrical coor­ dinates. The authors are unaware of general separable solutions in spherical coordinates. If the equations of elasticity are to be solved in cartesian coordinates in a region V bounded by a surface S with normal n = (nXl ny,nz), where nx, ny) and nz are direction cosines, then the boundary conditions are those of prescribed surface stress tractions A-

—

i^x^xx

\ ilyixy

\

'*'Z'ZX)

i

—

HxTxy

v Tly&yy

i

^z^"yzj

Z

=

nxrzx +nyTyz +nzcrzz,

(6.2.11)

over all S, prescribed surface displacement u 5 over all 5, or prescribed trac­ tions on part of S and prescribed displacement on the remaining part. Similar boundary conditions are obtained for the equations of elasticity in cylindrical and spherical coordinate systems. In the absence of body forces, and for time-harmonic excitation, the standard substitutions and resulting equations are written as Vf + V x P

=

u,

V tf + fc?¥ =

0,

2

2

2

V P + fc P = V.P

=

0, 0.

(6.2.12)

In these equations kj = CJ/C/ and ks = u>/c5, where Q = >/{(A + 2fi)/p} and cs — y/{fj>/p) are the phase velocities of longitudinal and shear waves in unbounded media. Stresses and strains can be written in terms of the functions \P and P by using definitions given in previous sections. Alternative substitutions, which have proved profitable in the analysis of the acoustics of isotropic layered media, considered in later chapters, find useful

Elastic Equations

139

application. In Eq. (6.2.12), let P = (0,0, G) + V x (0,0, H),

(6.2.13)

and replace the operator V 2 in the third equation by that of Eq. (6.2.5), giving V 2 (0,0,G) + fc2(0,0,G) =

V x (V 2 (0,0,//) + A : 2 ( 0 , 0 , ^ ) - V ( V . ( 0 , 0 , ^ ) ) ) V x (V 2 (0, 0, H) + fc2(0,0, H)) .

=

(6.2.14)

In cartesian and cylindrical coordinates, particular solutions of this equation are V2G + k2sG 2

2

V H + k sH

=

0,

=

0,

(6.2.15)

because V 2 (0, 0, G) = (0, 0, V 2 G), as can be deduced from the expressions com­ prising Eq. (1.4.6) and Eq. (1.4.7). However, this is not the case in spherical coordinates, as can be deduced from Eq. (1.4.8). Thus, for both cartesian and cylindrical coordinates, when u

=

Vtf + V x P ,

P

=

(0,0,G) + V x ( 0 , 0 , # ) ,

(6.2.16)

solutions of the reduced wave equations V 2 tf + fc,2tf = 2

2

V G + ib G 2

2

VH +kH

0,

=

0,

=

0,

(6.2.17)

give values of u which are solutions of the time-harmonic displacement equations, in the absence of body forces. Stresses and strains can be written in terms of the functions \£, G and H by using definitions given in previous sections. For spherical coordinates, when field quantities are independent of the angle , appropriate substitutions are u

=

V^ + V x P ,

P

== (0,0, G),

(6.2.18)

Acoustics of Underwater Structures

140

giving =

0,

= . 0,

i t 2 sin 0

(6.2.19)

since, for the case of (j) independence, V 2 ( 0 , 0, G)= (0,0, V 2 G - G/R2 sin 2 $) as can be deduced from Eq. (1.4.8). 6.3

Finite Element Variational Principle If a volume V of an elastic solid is bounded by a closed surface 5 , then

the elastic potential energy V is the strain energy minus the work done by body forces and surface tractions. Thus, with the vectors x and s denoting field points in a volume V and on a surface 5 , respectively, V

=

i^{e*(x,t)}TMx,*)}dV -±Jv{u*(x,t)}T{F(x,t)}dV-lJv{u(x,t)}T{F*(x,t)}dV

- /

{u*(s,t)}T{T(s,t)}dS-

i y " { U ( s , < ) } T { n s , < ) } d S , (6.3.1)

where {u(x, t)} is a column vector of displacements; {F(x, t)} is a column vector of body forces per unit volume; { T ( s , / ) } are surface tractions, being forces per unit area whose components are defined as positive if they are acting in the pos­ itive directions of their axes; {e(x,tf)} and {&(x,t)}

are column vectors of strain

and stress. T h e complex conjugate on vectors, and an arrangement of surface and volume force components as the sum of terms of the form a*6-f aft*, is necessary to make V real when field quantities are represented by complex variables. The arrangement is not necessary for the strain energy because {e*} T {
{^}T{o'*}

by virtue of stress and strain vectors being related by a real symmetric matrix, as later described. W h e n all field quantities are real the expression for V reduces to the s t a n d a r d definition for potential energy of an elastic solid. T h e kinetic energy is T

= \J

where p is material density. {ii*}T{u}

= {ii}T{u*}.

^«*(x'<)}T{«(x,t)}dV,

(6.3.2)

T h e function T is both real and positive, and

Here, when displacement and force column vectors are

Elastic Equations

141

assumed of length three and stress and strain vectors of length six, field quantities are those defined in previous sections. In certain circumstances, when analyzing thin shells for instance, displacements and forces can also include rotations and moments, respectively. Thus, generally, displacement and force vectors are of length Ni and stress and strain vectors are of length 7V2. By following a standard dynamical procedure, a suitable functional is the Lagrangian, C.= T — V, where

c =

5X p { "* ( x ' t ) } T { " ( x ' < ) } d v ~5X { c * ( x , < ) } T [ D ] { £ ( x ' t ) } d v 1

-jju*(xyt)}T{F(X,t)}dV+±Jv{u(x,t)}T{F*(x,t)}dV

+

+ \ Js{u*(s, t)}T{T(s, t)}dS + \ Js{u(s, t)f{T*(s,

t)}dS. (6.3.3)

The stress has been eliminated by using the constitutive matrix relation, {cr(x, t)} — [D]{e(x,tf)}, where the matrix [D] is defined in Sections 6.5 and 6.6 for isotropic and anisotropic elastic solids, respectively. There is a requirement that [D] is both real and symmetric to make real the function {e*(x,t)}T[D]{e(x,t)}. A further constraint on the stiffness matrix [D] is that it should be positive definite to make the strain energy positive. The differential equations of motion and the traction boundary condition can be obtained from Hamilton's variational principle 6 I* £dt = 0,

(6.3.4)

when the strains {e(x)} are replaced by the displacement derivatives obtained from Eq. (6.1.1), displacements and their complex conjugates being assumed independent variables. The procedure closely follows that of the field problem considered in Section 5.1. Let the volume V surrounded by a surface 5 be a typical finite element with iV3 nodes. In what follows, the meaning of the "symbolic" operations will be made clearer in later chapters which consider actual finite element problems. Following a standard procedure, let {«(x,0}

=

{«(x,-,i)}

=

{u(x,t)}

=

[iV(x)]{a(f)}, [A]{a(t)}, [tfWp^Hute,*)},

(6-3.5)

Acoustics of Underwater Structures

142

where [JV(x)] is a matrix of polynomial functions, {a(t)} is a column matrix of unknowns, and {u(xi,t)} is a column vector of the values of {u(x,t)} evaluated at the nodes. Further, by assuming that the strains can be put into the form {e(x,i)} = [BWRA-^Mxiti)},

(6.3.6)

the Lagrangian can be re-written as £

=

l^ixi^flA-1]*71

p[N*(x))T[N(x)]dV[A-1]{u(xiJ)}

- ^ ( x ^ f i A - ' r

7

J^B^x^mBix^dVlA-^Mxij)}

+{u*(xi,t)}T[A-lrT\Jv[N*(x)}T{F(x,t)}dV +Mxi,t)}T[A-1)T\

Jv{N(x)f{F*(x,t)}dV

+{«*(*,, t)}T{A-'rT\ +{u(xi,t)}T[A-1]Tl-

j[N\s)]T{T{s,

t)}dS

J [N(s)f{T*(S,t)}dS.

(6.3.7)

When u(xi,t) and u*(xi,t) are considered independent coordinates, the functional is minimized by applying Lagrange's equations d dC dt du*(xi)

8C du*(xi)

±_dC_

dC

dtdu(xt)

du(xi)

—

u,

'

"'

{b 6 }

-*

to each separate displacement, w*(xt) and w(x,), of the vectors {u*(x,, <)} and {u*(xi,t)}, respectively. On applying Lagrange's equations using the u*(x,) coordinates, the matrix equation [M]{u(xi,t)}

+ [S]{u(xi,t)}

= \Fv(t)} + {F.(t)}

(6.3.9)

is obtained, where the mass matrix, stiffness matrix, body force excitation vector and surface traction excitation vector are [M]

=

[A-1]^

f Jv

p[N*(x)f[N(x)}dV[A-%

Elastic Equations

143

[S] =

[A-i]*T [ Jv

[B*(x)f[D}[B(x)}dV[A-^

{Fv(t)}

=

[A'1}*?

{Fs(t)}

=

[A-1]*7 [[N*(s)]T{T{s,t)}dS. Js

Jv

f[N*(x)]T{F(x,t)}dV, (6.3.10)

For time-harmonic motion, Eq. (6.3.9) becomes [[S]-u>2[M]] N rN 3 x N XN 3

Wxt)W)} = N XN 3 x 1

{FV(LJ} + {F,(a;)} N XN3 x 1

(6.3.11)

for a finite element with N3 nodes and N\ unknowns at each node. Following the procedure described in Section 5.2, an assemblage of elements gives a system matrix equation of the same form as Eq. (6.3.11). The assembly of the elements reflects the fact that the assemblage functional is a sum of the element functional. Displacement constraints at nodes are inserted by appro­ priate modification of assembled matrix rows and columns. For example, if all elements in row and column i are set to to zero, except for unity on the diago­ nal, and the ith element in the excitation vector is set to zero, then this a forced condition of zero displacement for coordinate number i. On applying Lagrange's equations using the U(XJ) coordinates, the matrix equation of motion [Mf{ii'(x,-,0} + [ S f K ( x ^ ) } = {Ft(t)} + {Ft(t)}

(6.3.12)

is obtained. When the complex conjugate of this matrix equation is taken, and the mass and stiffness matrices assumed Hermitian, viz. [M*]T = [M], [S*]T = [S],

(6.3.13)

then Eq. (6.3.9) is recovered. When damping is included in the stiffness matrix [D], as described in a later section, then the stiffness matrix [S] is not Hermitian, so this pragmatic approach to include material damping in the matrix equations is only conceptually possible following on from Eq. (6.3.9). Other properties of Hermitian matrices relevant to dynamics problems can be found in Section 1.10.

144

Acoustics of Underwater Structures

Table 6.1: Illustrative constants for fluids.

iFluid [Air Water Machine Oil jSilicone Oil 6.4

/»(kgm 3 ) 1.21 1000 900 800

c (m s ) 343 1500 1600 2000

B (Pa) 1.42xl0 U5 2.25 xlO 0 8 2.30 xlO 0 9 3.20 xlO 0 9

\i (Pa s) 1.80xl0-° 5 LOOxlO" 03 l.OOxlO- 01 4.0

v (m s *)| 1.49x10-°^

l.OOxlO-H, l.llxlO-H 6.25xlO-°J

Fluid Constants T h e constants of interest for fluids and gases are density p, sound speed

c, bulk modulus B = pc2, dynamic coefficient of viscosity p and the kinematic coefficient of viscosity v — p/p.

These constants can vary with pressure, temper­

ature and time-harmonic frequency of excitation. Specialized textbooks, such as t h a t of Kaye & Laby [3], should be consulted for recognized numerical values. Typical constants are given in Table 6.1. W h e n a fluid, such as water, contains a small proportion of a gas, a low sound speed can be achieved. Let water have sound speed cw and density pw, and air have sound speed ca and density pa. Let the fractions per unit volume of air and water be k and (1 — fc), respectively. Simple mixture theory gives the effective density, sound speed and bulk modulus of this "bubbly" fluid as

P=(l-*)Ptn+*pa,

C=y/(B/P),

B-l

=

{l-k)/Pxvcl+{khP), (6.4.1)

where 7 = 1.4, for air, is the ratio of specific heats at constant pressure and constant t e m p e r a t u r e , and P ~ 9.81pw(h

+ 9.75) is the ambient pressure at

submergence depth h. These formula, valid for low frequencies well away from bubble resonant frequencies, give a small k value of c = ^(jP/kpw).

As an

example, take k = 0.01 and a depth of h = 50 m, to give the low sound speed c = 286 in "bubbly" water of just 1% air content.

6.5

Isotropic Elastic Constants A material is isotropic if its properties measured at any fixed reference po­

sition are independent of direction. This must be distinguished from the term homogeneous which means t h a t properties measured in a certain direction do

Elastic Equations

145

not change with location. T h e scale of any inhomogeneities is crucial.

Thus

consider, for example, an isotropic infinite elastic solid with periodically spaced isotropic circular fibres, which are parallel to the x axis. On length scales large compared with the periodic spacing, the composite can generally be regarded as anisotropic; on a length scale small compared with the periodic spacing the composite is regarded as comprising its isotropic component parts. For an isotropic elastic solid, the symmetric constitutive stiffness m a t r i x [D], relating the stress and strain matrices as {a} — [D]{e}, is

/ A + 2// A A A A + 2/J A A A A + 2p [D} = 0 0 0 0 0 0 \ 0 0 0

0 0 0 ^ 0 0

0 0 0 0 \x 0

0 \ 0 0 0 0 fi )

(6.5.1)

where

M =

[&XX ) Gyy , &ZZ i Ty z j Tztf) ^ F t / J

) j

{a}

=

[ 0 " r r , (Tffxf,) (Tzzi

{a}

=

[&RR,

W = U) = U} =

T(f>zi TZr*5 Tr\

ved,<7, Te, T ^ R , TRQ]

l^xx) £yy 5 ^zz •> "Jyz 5 ~Yzx ) lxy\

>

[€rr , 6 ^ 0 , €zz , y^z,

5

yzr 1 Jr\

[tRR, toe, £4>,le, 7^R> IfRe]

,

,

(6.5.2)

are the stress and strain column vectors in cartesian, cylindrical and spherical co­ ordinates. T h e subscript order, (xx,yy,

zz,yz,zx,xy)

for cartesian coordinates,

is a widely adopted standard. For other subscript orders, the corresponding [D] m a t r i x is obtained by interchanging rows and columns. For example, the sub­ script order (xx, yy, zz, xy, yz, zx) is frequently used: in this case, in Eq. (6.5.1), replace row 4 by row 6, row 5 by row 4 and row 6 by row 5, the same pro­ cedure being necessary for the columns. For this example, the [D] matrix of isotropic elasticity is unaltered. No particular choice of subscript order should be anticipated. T h e constants A and /i are the two independent Lame elastic constants. The constitutive stiffness m a t r i x [D] is independent of the orientation of the coordi­ nate axes. T h e inverse [D]'1

is called the receptance or compliance matrix. In

Acoustics of Underwater Structures

146

terms of the more familiar "engineering" constants, Young's modulus E} Poisson's ratio v and shear modulus G = E/2(l -f ^), the stiffness matrix elements are obtained by making use of the relations K3A + 2M)

p

-

E 2(1 + 1/)'

A = / (lI I +

A

) ; r o . , A + 2^, = / 1(lf g+- yi/)(l-2«/)" o.,•(«•»•»)

i/)(l-2i/)'

In some textbooks, the bulk modulus (hydrostatic pressure divided by de­ crease in volume per unit volume) and shear modulus are the basic pair. These can be interchanged with the standard pair E and v by using the conversion formulae _9G5_ (3B + GY

(3S-2G) 2(3£ + G ) '

E 3(1-2*/)'

E 2(1+ i/)"

l

'

Elastic constants can vary with pressure, temperature and time-harmonic frequency of excitation. Specialized textbooks, such as that of Kaye & Laby [3], should be consulted for recognized numerical values, but a rough guide to com­ monly used constants is given in Table 6.2, in which c\ — \/{(X -f 2fi)/p} and cs — \J(nlp) are the longitudinal and shear wave speeds in unbounded isotropic media. Hysteresis damping can be included in the constitutive matrix by setting the constants E, G, A and \i to complex values, viz. E = E(l — irj) etc. where rj is the hysteretic loss factor which ranges from 0.001 for lightly damped materials through 0.01 for moderately damped materials to values in excess of 0.1 for heavily damped materials. This assumes a time-harmonic response of exp(—iu;£); if the time-harmonic response is exp(-fiu;£), then E = E(l -f it}) etc. For "nonmathematical" solids it must be borne in mind that elastic constants and loss factors can vary in a complex way with temperature, pressure and frequency. Energy loss in a fluid may be simulated by a complex sound velocity, c = c(l—irj), where in this case the fluid loss factor is about twice the equivalent structural loss factor, because sound speed is proportional to the square root of the bulk modulus.

Elastic Equations

147

Table 6.2: Illustrative elastic constants and wave speeds in solids. Anechoic and decoupling are nominal materials for noise control and matrix is a nominal material for a sandwich construction filling.

| Material I Steel Aluminium Alum. Bronze Titanium Lead Brass Copper I Rock seabed Fast seabed Slow seabed Anechoic Decoupling Matrix Hard rubber Soft rubber Neoprene Polyurethane Polystyrene Polyester resin Perspex || Ice

E 0.195xl0 12 OJOOxlO11 O.lOOxlO12 0.115xl0 1 2 0.160xlO n O.lOOxlO12 0.120xl0 12 0.420xlO n 0.260xl0 0 9 0.260 xlO 0 9 0.600 xlO 0 8 0.260xl0 0 7 0.300xl0 1 0 0.200xl0 1 0 0.500xl0 0 7 0.360 xlO 0 9 0.200xl0 0 9 0.340xl0 1 0 0.350xl0 1 0 0.570xl0 1 0 0.621xl0 1 0

V

0.290 0.330 0.330 0.320 0.440 0.340 0.340 0.340 0.490 0.480 0.480 0.460 0.400 0.400 0.499 0.480 0.485 0.350 0.330 0.340 0.330

G 0.756xlO Xi 0.263xlO u 0.376xlO n 0.436xlO n 0.556xl0 10 0.373xlO n 0.448xlO n 0.157xlO u 0.872xl0 0 8 0.878 xlO 0 8 0.203 xlO 0 8 0.890 xlO 0 6 0.107xl0 10 0.714xlO° 9 0.167xlO° 7 0.122xl0 09 0.673 xlO 0 8 0.126xl0 10 0.132xl0 10 0.213xl0 10 0.233xl0 10

P 7700 2700 8300 4500 11300 8500 8800 2500 1400 1400 1600 800 1200 1100 950 1400 1100 1000 800 1200 910

Cl

5760 6200 4230 6050 2140 4260 4580 5090 1780 1280 574 123 2310 1970 938 1500 1450 2340 2550 2700 3180

cs 3130 3120 2130 3110 701 2100 2260 2500 250 250 113 33 945 806 42 295 247 1120 1280 1330 1600

||

i 1

|

Acoustics of Underwater Structures

148 6.6

Anisotropic Solid

A material is anisotropic if its properties measured at any fixed reference position are dependent on direction; see, for example, the glossary provided by Winterstein [4]. For a generally anisotropic solid, with stress and strain vectors

w

~

[&xx 5 &yy > &zz > ^yz » ^xz > TXy\

,

—

l^xx) tyy) €zz, lyz) Ixzt lxy\

—

\&rr j &(f>i &zz > T
—

\£rr ) €4>-> €zz 5 7^2 5 Jrz j Yr0J >

=

[&RR,0'e6,0',Te(f),TR(}„TRo]

> ,

,

(6.6.1) in which the subscript order of Auld [5] and Jones [6] is used, the constitutive stiffness matrix relation relating stress and strain as the matrix equation {a} —

[D]{c} is

I dn

<*12

C?13

di2

C?22

di3

G?23

di4

[D]

1 dis \ die

di4

dis

^23

^24

rf-25

C?26

G?33

^34

C?35

^36

^24

^34

C?44

d 45

c/46

^25

^35

C?45

^55

^56

<^36

d 46

<*56

^66

C?26

^16

\ (6.6.2)

y

This is a symmetric matrix with 21 unknown constants, whose values depend on the orientation of the coordinate axes. See the textbook of Auld [5] for discussion on the constants of constitutive matrices for various symmetries imposed by the microscopic structure of the material. A special case is the stiffness matrix for orthorhombic symmetry for which /

[D]

dn

di2

e?i3

d\2

dii

^23

di3

d-iz

^33

0 0 0

0 0 0

0 0

V 0

0 0 0 e/44

0 0

0 0 0 0

\ '

<^55

0 0 0 0 0

0

^66

/

(6.6.3)

This corresponds to the important engineering case of orthotropic material, for which there are nine unknown constants. If an isotropic material is reinforced

Elastic Equations

149

by periodically spaced fibres in directions parallel to all three axes, then for wavelengths large compared with fibre diameters and spacings, the material is orthotropic. The elements of the constitutive matrix are, see, for example, the textbook of Jones [6], dn

= ( 1 - VyzVzy)Exx/A,

di2 = (l/yar + VzxVyz)Exx

^13 = (Vzx + VyxVzy)Exx/A,

d22 = ( 1 ~

d23 = {Vzy + VxyVzx)Eyy/&,

d33 = ( 1 - VxyVyx)Ezz

d±4 = Gyz,

c/55 = Gzx,

dee —

/ A,

VxzVzx)Eyy/A, / A,

Gxy,

A = ( 1 - VXyVyX - VyZVZy ~ VZXVXZ ~ 2VyXVZyVXZ),

(6.6.4)

where Exx, Eyy and Ezz are Young's moduli in the local z, y, and z directions; Gyz = Gzy, G^a; = G ^ and G r y = Gyx are shear moduli in the local y — z, z — x and x — y planes; i/xy, vyX) vyz, vzy, vzx and vxz are Poisson's ratios which are related by the symmetry relations vxy/Exx

= VyXjEVy,

vxz/Exx

= VZXJEZZ,

vyzjEyy

— vzyjEzz.

(6.6.5)

The elements of the orthotropic stiffness matrix depend on the orientation of the axes, for which transformations are considered in Section 6.9. The inverse of the stiffness matrix is the receptance matrix defined by {e} — [-D]-1{cr}. Thus, /

l/Exx ~VyxlEyy rni-i -^x/Ezz L J ^ ~ 0 0 \ 0

-VyxlEyy I/Eyy -vzy/Ezz 0 0 0

-vzxjEzz ~VZy/EZZ \/Ezz 0 0 0

0 0 0 1/Gyz 0 0

0 0 0 0 \/Gzx 0

0 0 0 0 0 \/Gxy

\

)

The physical meaning of the various elastic constants is examined in textbooks on elasticity theory, see, for example, those of Jones [6] and Smith [7]. For other subscript orders, the corresponding [D] matrix is obtained by interchanging relevant rows and columns. For example, the subscript order (xx,yy,zz,xy,yz,zx) is perhaps more natural: in this case, replace row 4 by row 6, row 5 by row 4 and row 6 by row 5, the same procedure being necessary for columns. It must be borne in mind that no particular choice of subscript order should be expected.

Acoustics of Underwater Structures

150

Figure 6.2: Cross-section of m a t r i x material with periodically spaced fibre rein­ forcement in x direction only.

6.7

Unidirectional Fibre Reinforcement A local cartesian axes set (x, y, z) is shown in Figure 6.2. In this axes set an

isotropic material, called the matrix, occupies all space. T h e matrix is reinforced everywhere by periodically spaced thin fibres which are all parallel t o the x axis. T h e ("'engineering" elastic constants can be obtained from t h e micromechanics equations of Chamis [8], for this special case of unidirectional fibre reinforcement in an isotropic m a t r i x material. Let t h e known constants of the isotropic m a t r i x be Young's modulus Em, Poisson's ratio vm, shear modulus Gm = Em/2(l

+ vm) and density pm. Let the

known constants of the fibre material in local coordinates be Young's modulus in t h e fibre direction EfXXt Young's modulus perpendicular to t h e fibre direc­ tion Efyy, Gfyz,

shear modulus in x - y plane Gjxy,

shear modulus in y - z plane

Poisson's ratio i/fxy a n d density pf. Other (not needed) elastic constants

of the fibre are obtained as follows: Efzz

- EJyy,

the remaining five Poisson's ratios i/fyXy vjyz, lation i/fX2 = vjxy,

vjzy,

Gfzx

= GJxy)

i/fzx, vjxz

vjxz

=

vfxy-

require the re­

t h e availability of i/jyz and use of the symmetry relations

given in Eq. (6.6.5). If the partial volumes of the m a t r i x and fibre are km and

Elastic Equations

151

Table 6.3: Effective material constants of carbon fibre reinforced plastic, E and G being in G P a .

1 */

■t-^xx

0.1 52.7 0.2 102.0 0.3 152.0 | 0.4 202.0

^"yy — ^zz

{-Jxy —

A.21 5.18 6.18 7.40

{Jzx

1.54 1.88 2.27 2.74

Gyz

Vxy — Vxz

Vyz

1.51 1.83 2.18 2.59

0.380 0.360 0.340 0.320

0.409 0.414 0.420 0.427

P II 1280 1360 1440 1520 |

kj, respectively, where km -f kj = 1, then according to Chamis [8], simplified micromechanics properties of the composite are as follows: &xx — kf&fxx

i

krn&mt

Eyy — EZZ = Em/ ( 1 — (1 — Em/Efyy)ykj\

,

GXy — Gxz — Gml ( 1 — (1 — Gm/GfXy)y/kf)

,

Gyz = Gml \l - (1 - Gm/Gfyz)yfkJ) Vxy — Vxz =: KfVfxy

Vyz = —14p = pjkf

,

T" ^m^rri)

Eyy/2Gyz, + Pmkm,

(6.7.1)

the other three Poisson's ratios being found from the symmetry relations of Eq. (6.6.5). For a numerical example, consider an isotropic plastic material with con­ stants given by the "matrix" material in Table 6.2. The plastic is reinforced with carbon fibre for which Ejxx vjxy

= 500.0, Ejyy

= 50.0, Gjxy

= 30.0, Gjyz

— 0.20 and pj = 2000, with E and G values being given in G P a .

= 15.0, Effec­

tive elastic constants of the fibre reinforced material are shown in Table 6.3 for various fibre proportions, kj.

These elastic constants can be used to construct

the stiffness m a t r i x [D], given by Eq. (6.6.3), of the fibre reinforced composite material.

6.8

Thin Layered Composite T h e dynamics of a thin composite plate or shell can be represented by dif­

ferential equations a n d / o r energy methods which contain stiffness constants Aij,

Acoustics of Underwater Structures

152

Figure 6.3: (a) Laminated composite comprising a stack of M thin fibre rein­ forced plates; and (b) definition of stacking angle relative to local (x,y) and global (xg,yg) axes. Bij and Dij, as described in later chapters. These constants can be obtained from simple formulae which are equally applicable to flat plates and curved shells. First, consider a laminated plate, comprising M thin layers, as shown in Figure 6.3. Each single layer comprises an isotropic matrix which is reinforced by unidirectional fibre reinforced material, with stacking angle £ relative to a global xg axis. The stacking angle can vary from layer to layer. In its local x — y plane a typical thin layer has "engineering" elastic constants Exx and Eyy the Young's moduli, Gxy the shear modulus, and i/xy and i/yx the Poisson's ratios which are related by the symmetry relation Exxjvxy — Eyyjvyx. These constants can be obtained from the simple mixture theory of Chamis, described in Section 6.7, provided the relevant elastic constants of the matrix and fibre materials are known. In the local (z, y) axes set, the stiffness matrix of a layer is defined by the standard plane stress matrix equation Qn Q12

Q12 Q22

0

0

0 0

\

/ \ [

Q66 / \

ex cy lxy

\ ,

)

(6.8.1)

Elastic Equations

153

where Q l l = EXX/{\ Ql2 = EXXVyx/(\ In the global {xg,yg,zg)

- VXyVyX),

Q22 = Eyy/(1 ~ VXyVyx),

~ VXyVyX),

Q QQ = GXy .

(6.8.2)

axes set, the stacking angle £ is defined by a right-

h a n d screw rotation of the global xg axis, about the global zg axis, until it is coincident with the local x axis. In the global axes set, the matrix relation is, see, for example, Jones [6], Qgll Qgl2 Qgl6

> gx T gy 1

gxy

Qgl2 Qg22 Qg26

Qgl6 Qg26 Qg66

z

gx gy Jgxy

(6.8.3)

e

where Qgll

Qu cos 4 £ + 2 ( 9 i 2 + 2 Q 6 6 ) sin 2 £ cos 2 £ + Q22 sin 4 £,

Qgi2

(Qu + Q22 - 4Q 6 6) sin 2 £ cos 2 £ + Q i 2 ( s i n 4 £ + cos 4 £),

Qgl6

(Q11 - Q12 - 2Q 6 6) sin£ cos 3 £ + (Q12 - Q22 + 2Q 6 6 ) sin 3 £ cos£,

Qg22

Q n sin 4 £ + 2(Qi2 + 2Q 6 e) sin 2 £ cos 2 £ + Q 2 2 cos 4 £,

Qg26

( Q u - Q12 - 2 g 6 e ) sin 3 £ cos£ + ( Q 1 2 - Q 2 2 + 2Q 6 e) sin^ cos 3 £,

Qg66

(Q11 + Q22 - 2 Q i 2 - 2Q6e) s i n 2 £ cos 2 £ + Q 6 6(sin 4 £ + cos 4 f ) . (6.8.4)

For the M layered stack of fibre reinforced layers, the constants A{j, and Dij, i,j

Bij

= 1,2,6, comprise the elements of the 6 x 6 constitutive m a t r i x

[Dg], relating stress and m o m e n t resultants to midsurface strains in the global (xg,yg,zg)

axes set. T h e m a t r i x relation, {
for a thin layered

composite is defined by the symmetric m a t r i x relation /

Nqx

\

(

gy gxy M,gx Magy

\

MgXy

\

An A12 A16

A12

Aie

flu

B\2

5l6

A22 A2Q

A2e A6e

B\2

B22 B26

B26 Be6

B11 B12

B\2 B22 B26

Bi6 B26

D12

D16 D2e Dee j

B16

Bee

Here, (NgX} Ngy, Ngxy, Mgx, Mgy,Mgxy) ment resultants, and {tgx,£gy,lgxy,

B\e Dn D12 Die

D22

A>6

\

(

e

gx

e

\

gy

Jgxy

(6.8.5)

K>gx K

\

T

gy

gxy

/

comprises the vector of stresses and mo­ Kgx,Kgy,Tgxx)

comprises the vector of mid-

surface strains, changes in curvature and twist, see, for example, reference [6].

154

Acoustics of Underwater Structures

Table 6.4: Stiffness values of layered composite.

| Subscript i,i 1,2 1,6 2,2 2,6

1

6,6

D 0.536 x 106 0.370 x 105 0.282 x 105 0.536 x 106 0.282 x 105 0.389 x 105

B 0.342 x 108 0.864 x 10"3 0.211 x 107 -0.342 x 108 -0.211 x 10 7 0.864 x 10-3

0.347 0.824 0.846 0.347 0.846 0.839

A | x 101U x 109 x 109 x 1010 x 109 x 109 [

T h e stiffness constants, comprising the m a t r i x [Dg]> and the effective density are obtained by s u m m i n g the layer constants as follows: M

1

Aij = /^V'rn —tm-l)Qgij m=l M V ~ 'X 2—<^

m

>

^m-l)Qgij J

&ij

=

M

9 Z - / ^ m ~^m-l)Qgij m=l 1

1

M

Ps = -^^2(tm-tm-i)p(m),

(6.8.6)

m=l where tm — zm — zc with zm and z m _ i being the z coordinates of the upper and lower surfaces of the m t h layer; zc is the z coordinate of the laminate's midsurface usually taken as zc = (ZM + ZQ)/2\ laminate; p is layer density.

h is the total thickness of the

T h e stiffness A{j values are related t o in-plane

stretching of the midsurface; the D{j values to bending of the surface; the Bij values t o coupling between bending and stretching.

When the lamination is

symmetric with respect t o the midsurface, the B{j values are zero. For a numerical example, a thin laminated composite comprising four layers of carbon fibre reinforced plastic is considered for which constants are obtained from Table 6.3 for kf = 0.4. T h e thickness of each layer is h = 0.01 and stacking angles, t o p layer to b o t t o m , are £ = [0,30,60,90]. T h e effective stiffness constants are given in Table 6.4. For a thin layered axisymmetric shell, t h e m a t r i x relation, {ag} = [jD^]{e^},

Elastic Equations

155

is defined in cylindrical coordinates as /

N8 + N9S

\

( Mi Ml

A\2 Ml

Ms M§

A\§

^26

^66

#16

#26

#66

Jg4>s

#11

#12

#16

£>11

^12

^16

Kg

#12

#22

#26

£*12

^22

^26

#16

#26

#66

#16

#26

#66

N94>s Mg(f> Mgs

\

M

9* J

\

#11 #12

B\2 #22

Bie #26

\ /

€

9 \

(6.8.7)

Kgs /

\

T

94>* )

where s is distance along the cylinder generator, <j> is the angle around the cylinder in a direction defined so t h a t (r, <j),s) is a right-handed axes set in which the positive direction of r is outward. Thus, for a cylinder, identify the cylindrical ((f), s) directions with the cartesian (x,y)

directions in Eq. (6.8.1)

to Eq. (6.8.6). T h e stacking angle £ (or fibre winding angle) is a rotation of the <j) direction until it is aligned with the fibre direction. For circumferentially reinforced material £ = 0, and for axially reinforced material £ = 90°.

6.9

Local t o Global Transformation In Figure 6.4 are shown local, ( x , y , z), and global, (xg,yg,

zg),

coordinate

systems which are rotated with respect to one another. The elements of the generally anisotropic stiffness constitutive matrix, Eq. (6.6.2), in the local coor­ dinate system are assumed known. For the orthotropic case, for example, they could be found from the Chamis formulae of Section 6.7, for unidirectional fibre reinforcement in the x direction. Also note t h a t stress and strain order in the stiffness relation, {a} — [#]{e}, are assumed here as \0J \CJ

— —

[&xxi &yy j &zzi Tyz t ^zxi \€xx,

e

yyi

e

zzi

lyz-> Jzx,

TryJ > Jxyj

>

(6.9.1)

which is the s t a n d a r d subscript ordering. T h e transformation of the stiffness m a t r i x [D] from the local coordinate system (x, y, z) to a global coordinate system (xg,ygjzg)

is obtained by a minor

modification to the equations of Auld [5] as [D9] =

[N]T[D][N],

(6.9.2)

Acoustics of Underwater Structures

156

Figure 6.4: Local (x,y,z) where

and global (xg,yg,zg)

( Nn

cartesian coordinates.

N12

N13

N14

JV 15 N16

N2i N31 NA1 N51

N22 N32 NA2 N52

N23 N33 NA3 N53

N2A

N34 NAA N54

N25 N35 N45 AT55

N2e N36 NA6 N56

\ Nei

N62

NQ3 Net

iV 65

N6e J

[N] =

\ (6.9.3)

with N12 = aly,

Nil = «£*, iVi4 =

#21 = ^24

N15 =

Q>xyQ>xz j

#22 = aJ y ,

fly*,

N25 =

= yyyz>

N

3l =

N34 = N41 = a

yy zz

N51 =

a

azyazz,

2azxaXXy

N16 =

Q'xxQ'xy)

N2s =

a

N2e =

CLyxO-yy,

2

yz>

N33

o,zzazx,

N36

= -

N43

-

J,<2yZQ,ZZ

,

= Clyx zz ~r fly^^a?*

N*e -

Q>yyQ>Zx

1 QyxQ'Zy

iV 52 =

AT53 =

LQ,ZZ(XXZ,

N42 —

N4s

<4>

a22y)

N35 =

2ayxaZX) H~ Q>yz zy<>

Uyzdyx,

N32 =

L,

a

axzaxx,

N13 =

a

a

% yy zyi

a

2azyaxy,

«L. a

zxazy

j

y

Elastic Equations

N54 = o-xydzz + o,xzazy,

157

7V55 = axzazx

Nei — 2axxayx, NQ4 — axyayz

+ axzayy,

-f axxazz,

N62 — 2axyayy, Ne$ = axzayx

+ axxayZ}

JV56 = axxazy 7V63 =

-f

axyazx,

2axzayZi

NQQ = axxayy

+

axyayx.

(6.9.4) T h e constants a^- are found from the direction cosine m a t r i x [A] relating the local {x} and global {xg}

axes sets by the 3 x 3 m a t r i x relation {x} =

[A]{xg},

where the element a^ is the cosine of the angle between local axis i and global axis j . Several m e t h o d s have been proposed for determining the m a t r i x [A], two of these being presented below. They should suffice for most practical problems in which there is unidirectional fibre reinforcement along the local x axis; if the first m e t h o d of determining the directional cosines appears to be unduly difficult, then the second m e t h o d should be tried. T h e first m e t h o d of calculating the m a t r i x elements a^ is called the "three axes rotation angles" m e t h o d . Make coincident the global axes and the local axes by applying these consecutive rotations, in the sense of a right-hand-screw, t o t h e global axes set: \p about the global x axis; 6 about the new global y axis; about the new global z axis. T h e product rule of successive transformations gives the direction cosine m a t r i x as cos# cos cos\p sin<j)-\-smtp sin0 cos<j) simp siiuf) — cosip sin# cos cost/>cos^>—sin^ sin# sin^ sin?/>cos+cosi/>sin0sin<£ I . sin# — simp cos# cost/> cos# /

(6.9.5) T h e second m e t h o d of calculating the m a t r i x elements a^ is called the "Euler rotation angles" m e t h o d . Make coincident the global axes and the local axes by applying these consecutive rotations, in the sense of a right-hand-screw, to the global axes set: ip about the global z axis; 0 about the new global x axis;