A note on nonlinear elastic wave propagation

MECHANICS

RESEARCH

COMMUNICATIONS

0093-6413/83/010037-08503.00/0

Vol.

10(1),37-44, 1983. Printed in the U S A Copyright (c) 1983 Pergamon Press Ltd

A NOTE ON NONLINEAR ELASTIC WAVE PROPAGATION

Cohen and Marcelo Epstein Department of Civil Engineering,

H.

University of Manitoba, Winnipeg,

Canada.

(Received 22 July 1982; accepted for print 16 November 1982)

Introduction

Much of the effort and success in dealing with wave propagation in nonlinear materials has centered around the concept of a propagating singular surface of order two - an acceleration wave. Although this technique has led to new insights into nonlinear material behaviour it is, by its very nature, limited, in terms of the information which it can provide. The information as to the dynamical state of the medium is restricted to yield only the nature of the acceleration jump at the wave front. At best, if one is able to solve the propagation condition for the shape of the wave front, then the technique will yield, after solution of the growth-decay equation, the evolution of this jump as it moves with the wave front. Usually, these solutions to the propagation condition consist either of plane waves or waves with a high degree of geometrical as well as dynamical symmetry. For these cases the growth-decay behaviour is governed by an ordinary differential equation. For more general situations, the compatibility relations, which underlie the method, lead to partial differential equations for the growth-decay behaviour. The resolution of this difficulty through transformation to a more suitable set of coordinates in which the desired equations are ordinary creates an analytical impasse as the choice of coordinates in the compatibility relations is fixed. In order to effect a more complete solution to the nonlinear wave propagation problem one needs information as to the dynamical state of affairs behind the wave front. It has been suggested [i], [2], that such information may be elicited via the singular surface technique by finding the jumps in the n-th order derivatives of the deformation, n ~ 2. These jumps could then be used as the coefficients in a series expansion at the wave front which would allow calculation into the region behind the wave front. In fact, such a process has been successfully effected in several recent papers [3], [4], [5], pertaining to the dynamics of plates and shells. As suggested above, one of the major roadblocks in executing such a scheme stems from the basic nature of the compatibility relations. Moreover, these relations when iterated for the higher order jumps are of such complexity as to render their application almost impossible, if in fact they are even available [6]. In this paper we present an extremely concise form of the iterated compatibility relations which are readily useful for finding the basic equations for the higher order waves. Moreover, their formulation involves an 37

38

H. COHEN and MARCELO EPSTEIN

arbitrary equations

choice of coordinates on the singular are derived in the following section.

surface

in space-time.

These

In the next section we apply these relations to the equations of motion of a nonlinear elastic solid and obtain the basic equations which govern deformation waves of all orders, n > 2. These equations are analyzed in the subsequent section, wherein both the growth-decay and induced wave equations are extracted from them. The first of these equations is in general a partial differential equation which is linear for waves of all orders, except n = 2. The latter equations are algebraic in nature and give the induced higher order wave associated with a primary wave of order one lower. In the final section we give an a l g o r i t h m for solving problems in nonlinear elastodynamics which utilizes the wave front expansion mentioned earlier. In particular, the procedure involves t r a n s f o r m i n g to coordinates obtained after solving for the characteristics of the growth-decay equations, which then may be employed to render these into ordinary differential equations. We surmise, that the ultimate success of the overall process will depend upon a suitable marriage of analytical with computer oriented numerical methods.

The Iterated

Let

S

Compatibility

Relations

be a moving surface

in space, which

represented

by ~(X I) - t = 0, where

system and

t

presents

denotes

the time.

transformation

In a space-time

which allows S to be represented Let ~ = ~(XI,t) formation

be a tensor

equation

Evaluation comprise equations

(i) results

a system of surface

to be

coordinate

m a n i f o l d this equation

Following

[7],

re-

[8], we introduce

defined by _ T ,

by th e equation

(i)

T = 0

of place and time.

Introducing

the trans-

in , ¢(yi)

of ~ at % = 0 yields

(2) results

, t = ~(yI)

function

: ~(yi

S.

in space-time

X I = yI

we assume

(XI), I = 1,2,3 are a spatial

the fixed surface S swept out by

the coordinate

for simplicity,

_ ~)

: ~(yi

, ~)

(2)

the value of ~ on S as a function of yI, which

coordinates

on S.

A simple

calculation

using

in •

2

~'i : ~'i + ~ ¢'i ' ~ : -

~

'

(3)

where

3X I

Let S be singular in ~ across S.

'

for ~, i.e.

~t

'

3yI

,

(4)

~T

[~] = [~] # 0 on S, where

Since these jumps are defined on S only,

[~] denotes

the jump

it follows that

[~]

N O N L I N E A R ELASTIC WAVE P R O P A G A T I O N

is a function of the surface hei~la, [6] - s e c t i o n

174,

arbitrary

on S.

direction

the surface equivalent arrive

coordinate to

coordinates

yl only.

We have by Hadamard's

that d[@]/ds = [@,l]dyI/ds, By choosing

ds

39

where

ds represents

along the directions

system on S, we see that the foregoing

[~],I = [~'I]"

By an iterated

application

an

defined by

equation

is

of this equation we

at the relations k

k A

,

] =

[~]

[~ ii...in_ k where

for present and later convenience k ^ ^ k k notations ~ = ~k~/~T and ~ =

~k~/~tk.

introduced

in [6] - section

174,

,

we have

one might

appropriately

compatibility

In order to obtain a convenient

and concise

fundamental

we utilize

relations

introduced

the shorthand

In conformity with the terminology

(5) as the iterated superficial

lity relations,

(5)

'I 1 ..-In_ k

refer to equations

relations.

form of the so-called

the t r a n s f o r m a t i o n

equations

compatibi-

(3) as well as the

(5), to obtain m

m

m+ 1

m

m

[~,i] = (-l)m{[~],i + [ ~ ]~,i } , [~,ij] = (-l)m{[~],ij m+

1

111+ 1

m+ 2

In+ 1

+ [ ~ ],i ~'a + [ ~ ]'J ~'I + [ ~ ]~'I ~'a + [ ~ ]~'la } Of course,

these equations

the compatibility the primary

relations

partial

second order,

these will

tions,

derivatives

attention

tangent

replacement worthwhile

geometric

which are in general

transversal

coordinates

remarking

dent of the metrical

to

that the d e r i v a t i o n properties

of space.

of motion

S

from

rela-

make use of a decomposition

to S, as well as a special Here, we used

only to S along the arbitrary yI.

as

- are of

compatibility

time derivative.

to S, in the direction

for the derivative n o r m a l

However,

for our purpose.

and k i n e m a t i c

and n o r m a l

tangent

derivatives.

to provide

(6) differ in appearance

Those relations

to S - the displacement

tions defined by the surface derivative

that equations

both tangent

iteration

- the equations

turn out to be sufficient

176 and 181.

in terms of derivatives derivative

equations

form of the iterated

[6] - sections

via further

for h i g h e r order spatial

differential

We call to the readers' the standard

may be generalized

(6)

direc-

In addition we have used a parallel

to the time axis,

in the usual procedure.

of equations

(6) is totally

as a

It is indepen-

40

H. COHEN and MARCELO EPSTEIN

Application

to Acceleration Waves

The equations

of motion of a homogeneous

nonlinear

elastic solid are

AKkQq xq,KQ + Po fk = p o ~k where

(XK) and (Xk) are rectangular Cartesian

ence configuration

and in space,

(7)

'

coordinate

respectively,

systems in the refer-

Po is the reference density,

fk is the body force per unit mass and the functions AKkQq = AKkQq(Xp,p) the elasticities

of the material.

The assumption

made for a desire to avoid notational limitation.

of material homogeneity

are is

awkwardness and not for any theoretical

For the same reason we shall assume the material to be undeformed

and initially at rest. The latter assumption will allow us to make subsequent

use of the simple

identity n ( l)n+ I [a~1 = ~a ] = _ where ~a, a = i, ..., n, are

n

but which vanish ahead of S.

In equation

n a =~l

quantities

[~a ]

(8)

,

discontinuous

at the wave front,

(8) ~ denotes the continuous product

symbol. Under the simplifying

assumptions made above the n-th material

of the equations of motion

n

m(n

I m=l P=I

(7) can be written as

n m m

Cmp

time derivative

ml

. . x. . . xq,KQ AKkQqlli I . . Ipip Ii 'If

+ ~kQq

n n+2 Xq,KQ = 0o Xk

x. Ip ip~

'

(9)

with m I >

m 2 >_

...

>_ m p

>_ 1

, m I +

m 2

+

...

+

mp

=

m

(i0)

,

~PAKkQq AKkQqIlil'''Ipip

In writing equations

(9) - (ii) we have generalized to several variables

formula of Fa~ di Bruno efficients

C

mp

.

(11) ~xi l,II'''~xip,Ip

the

[9], which provides a recipe for calculating the co-

The important

fact for our purpose is that C

ml

= I for all m.

For an acceleration wave we have by definition [xi] = [xi ] = 0 , [xi ] # 0

(12)

NONLINEAR ELASTIC WAVE PROPAGATION

Rewriting equations

41

(6) for the motion x i = xi(Xl,t) results in

n+ 1 n n n I] = (-l)n{[xi],l n + [ xi ]~'I } • [ x i , i J ] = (-I) [xi, n+l

n+l

n+l

n

{[~i],ij

n+2

(13)

+ [ xi ]'I *'J + [ xi ]'J 9'I + [ xi ]~'IJ + [ xi ]~'I ~•J} " By virtue of equations

(12) we get, in particular

•

A

[xi, I] = -

A

(14)

[xi]~, I , [xi,iJ] = [xi]~, I $,j

We now take the jump of equation (9) noting that by virtue of the above equations the highest order jump appearing therein will be n + 2.

Lumping to-

gether all of the terms not containing jumps of order higher than

n

under the

n

notation Rk, and using equations

(3)2, (8), (13), (14), and Cml = i, we obtain

for the jump of equation (9) n+2 A

(AQq

A

~'K @'Q - 0o ~kq )[ Xq ] + (AKkQqli ~'K ~'Q @'I[Xi ] n+l

n+l

n

+ AKkQq ,,KQ )[ Xq ] + (AKkQq + ~ k K q l , , K ~xq ],Q + Rk = 0

(15)

Note the essential feature of the equation, namely• that the coefficient of the highest order jump is exactly the one used in the propagation condition. Indeed, for n = 0, equation (15) reduces to the propagation condition.

In

general, we refer to equation (15) as the decay-induction equation since for any n k 1 it determines the growth-decay characteristics

of all primary waves

of order n + i• as well as the secondary or induced waves of order n + 2. Note n+l also that, except for n = i, the equation is linear in [ Xq ].

Analysis of Decay and Induction

As remarked above, for n = 0 in equation (15) we recover the propagation condition for acceleration waves, namely X (Qkp -Qo ~kp)[Xp ] = 0

where the acoustic tensor Qkp is defined by Qkp = AKkPp ¢'K @'P" trivial solutions of equation

(16)

(16)

, The non

can be obtained by first solving the

characteristic equation det ._(0kp- 0o ~k_)i ~ = 0

(17)

42

H. COHEN and MARCELO EPSTEIN

which is a highly non linear first order partial differential equaiton in ~. To each simple solution ~(~), ~ = i, 2, 3, of equation initial conditions,

(17), subject to given

there will correspond a wave surface S (~) whose speed of

propagation U (~) is given by U (~) = {~!~)- #!~)}-½.-

By a simple solution of

equation (17) we mean that the determinantal equation factors into three distinct terms, each of which gives rise to a unique partial differential equation for ~.

Corresponding to each ~(~) there will be an eigenvector field

](~)

of acceleration jumps

[xi

constant by equation

(16).

determined up to an arbitrary multiplicative

Choosing one of the ~(~) we write [~q](~) = no

sum on

a (~)

eq(~) ,

(18)

for the associated acceleration jump.

a (~) are a normalized solution of equation tude, respectively.

In equation

(18) e~ ~) and

(16) and its indeterminate magni-

We assume conditions are such that on the initial surface

So at t = 0, the acceleration jump is in fact a solution of equation ~(~)-, so as to avoid giving rise to the other wave types.

(16) for

Moreover, we assume

also that the initial conditions are such as to be compatible with whatever higher order induced waves are required by equation (15). We shall now extract the decay and induced wave equations from equation (15) (~) for the solution ~ From this point on, we shall drop the superscript for notational convenience. more use equations

If we multiply equation

(15) by ek, and further-

(16) and (18), we obtain the first order partial differen-

tial equations n+l n+l ^ n n n BQq[ Xq ],Q + Cq[ Xq] + D = 0 ,BQq = BkQ q ek , Cq = Ckq e k , m = K k ek ,

(19) BkQq : (AKkQq +AQkKq)*'K ' Ckq = AKkQq *'KQ + AKkQqli *'K *'Q *'I a e i

Equations

(19) I will be used to determine for all order waves the growth-decay

behaviour parallel to the primary wave

[~].

To extract the induced waves from equations

(15) we note that the operator

(Qkq - Po 6kq)' when restricted to the two-dimensional subspace ek orthogonal to ek, is non-singular and hence invertible.

Let Pkq denote the composition

of this inverse preceded by projection onto e k. results in

Applying Pkq to equation (15)

NONLINEAR ELASTIC WAVE PROPAGATION

43

n+2]~ Pkt{Ckq[n+l n21 n [ xt = Xq ] + BkQq[ Xq ],Q + ~ } which is an algebraic equation for the determination

,

(20)

of the induced higher

order waves for all n k 0. We note that the induced wave determines only the n+2 n+2 part [ xt ]~ of [ xt ] which is orthogonal to e k.

A Solution Algorithm

We now present an algorithm for utilizing the preceding analysis in solving problems in nonlinear wave propagation. (i)

The procedure envisaged is as follows:

Solve the partial differential

which defines the wave front S.

equation

(17) for a simple solution ¢

Use this solution in equation

((16) to find

the corresponding eigenvector ek as a function of YI" (ii)

For n = 1 in equation

(19) 1 use equation

(18) to obtain the first

decay equation as BQq eq a,Q + (BQq e q , Q + C q This is a first order nonlinear ential equation

and they coincide (7).

(by virtue of equation

for the amplitude

istics of equation

a

d Y2

d Y3

B lq e q

B 2q e q

B ~q e q

[8] with the bicharacteristics

of the equations of motion

(18), provides

Still with n = i, use equation

and substitute into equation

on S.

subspace of e k.

] = [~] (19)1

+ a ek

(23)

for n = 2 to obtain a linear partial diffare still given by equations

Use these to construct and solve the corresponding ordinary differential I

equation for a, which when combined with equations S.

The sol-

(20) to calculate the projection

erential equation for ~, whose characteristics

on

[~]

(21) into

Write [

(22).

(22)

,

for 'a' along these characteristics.

of [~k] on the complementary (iv)

The character-

and use these to transform equation

ution of this equation, with equation

[Xk ]

(19)6) partial differ-

of the acceleration wave.

d YI

ordinary differential equations

(iii)

(21)

(21) are given by

Find the characteristics

"4"

eq)a = 0 .

(18) and (23) yields

"'°

[%]

44

H. COHEN and MARCELO EPSTEIN

(v)

Increase

n

by one and repeat steps (iii) and (iv) recursively to

obtain the jumps for increasingly higher (iv)

n.

To obtain the solution into the region behind the wave front we

formally compute the Taylor's expansion N n ~ (-i) n [xi] Tn/n! , (24) n=2 n where T ~ 0, and the coefficients [xi] have been found by the preceding proA

A

xi(Yi,T ) =

cess and

N

is sufficiently large to ensure satisfactory convergence.

References

i.

F.G. Friedlander,

"Sound Pulses", Cambridge University Press, 1958.

2.

J.D. Achenbach, "Wave Propagation in Elastic Solids", North Holland, Amsterdam, 1973.

3.

H. Cohen, A.H. Shah, and D.P. Thambiratnam, "Transient and Time Harmonic Waves in Elastic Plates", Int. J. Solids Structures, 15, pp. 395-404,1979.

4.

D.P. Thambiratnam, A.H. Shah, and H. Cohen, "Axisymmetric Transients in Shells of Revolution", Earthqu. Engng. & Struct. Mech., 7, pp. 369-382, 1979.

5.

D.P. Thambiratnam, A.H. Shah, and H. Cohen, "Transients in Cylindrical Shells", Earthqu. Engng. & Struct. Mech., 8, pp. 17-30, 1980.

6.

C. Truesdell and R. Toupin, "The Classical Field Theories", Encyclopedia of Physics, Edited by S. FiHgge, Vol. III/l, Springer-Verlag, Berlin, 1960.

7.

E. Varley and E. Cumberbatch, "Nonlinear Theory of Wave Front Propagation", J. Inst. Maths. Applics., i, pp. 101-112, 1965.

8.

H. Cohen and M. Epstein, "Wave Fronts in Elastic Membranes", to appear in J. Elasticity, 1982.

9.

S. Roman, "The Formula of Faa di Bruno", American Math. Monthly, 87, pp. 805-809, 1980.

Acknowledgement

H. Cohen wishes to acknowledge and thank The University of Calgary and the Killam Foundation for naming him a Killam Visiting Scholar, during which time this paper was written. Both authors wish to acknowledge the support in part through their respective operating grants from the National Sciences & Engineering Research Council of Canada.