E L S E V I E R Wave Motion 19 (1994) 125-133 L

Shock waves in viscoelastic materials with voids

Antonio Scalia Dipartimento di Matematica, Universitit di Catania, Via le A. Doria 6, 95125 Catania, Italy

Received 18 January 1993, Revised 17 September 19.93

Abstract

The paper is concerned with the linear theory of viscoelastic materials with voids. The jump conditions and growth equations which govern the propagation of shock waves are derived and studied,

1. Introduction

The theory of elastic materials with voids has been a subject of intensive study in recent years. The linear theory of elastic materials with voids is one of those obtained by extending the classical theory of elasticity to treat materials with microstructure. The nonlinear version of the theory was established by Nunziato and Cowin [ l ] who later derived a linear version (cf. [2] ).

The basic feature of the theory of elastic materials with voids is the concept of a material for which the bulk density is written as the product of two fields, the matrix material density field and the volume fraction field. The above theory is expected to find applications in the treatment of the mechanics of granular materials like rock, soils and manufactured porous bodies.

Extensions of this theory to thermoelastic materials and viscoelastic materials exist (cf. [ 3-6] ). Wave propagation in elastic materials with voids has been studied in various papers (cf. [7-10] ).

The object of this paper is to study the propagation of shock waves in a linear viscoelastic material with voids. In the framework of classical viscoelasticity this subject was studied in various papers (cf. [ l 1-13] ). Firstly, we

derive propagation conditions for shock waves. Then, the coupling between the discontinuities is studied in detail.

2. Basic equations and general properties

We refer the motion of the continuum to a fixed system of rectangular Cartesian axes Oxi (i = 1, 2, 3). The conventions adopted with regard to tensor indices are as follows: Latin indices are understood to range over the integers ( 1, 2, 3) whereas Greek indices are confined to the range ( l, 2). The usual summation convention applies to all indices. Moreover, subscripts preceded by a comma denote partial differentiation with respect to the corre- sponding Cartesian coordinate and a superposed dot denotes the material derivative with respect to the time.

We consider an arbitrary open region V in the continuum, bounded by a surface aVat time t, and we suppose that P is the corresponding region in the domain B occupied by the undeformed continuum.

0165-2125/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0165-2125 ( 93 ) E0040-W

126 A. Scalia/Wave Motion 19 (1994) 125-133

A continuum with voids satisfies the global balance law of linear momentum (cf. [ 1,2] )

d

P P aP

and the law of balance of equilibrated force

d do+fh,n, , P P 8P

(2.1)

(2.2)

where p is the density in the undeformed state, u~ the displacement vector, v~ = ~, f, the body force, tji the stress tensor, k the equilibrated inertia, 9 the change in volume fraction field from the constant reference volume fraction, I the extrinsic equilibrated body force, g the intrinsic equilibrated body force and h~ the equilibrated stress.

Equations (2.1) and (2.2) must be supplemented by constitutive equations. In the case of a viscoelastic body with a center of symmetry the linear constitutive equations are (cf. [3,5 ] )

to(x, t) = i [Gij,,~(x, t-s)~s(x, s) +Bij(x, t-s)~(x, s)] ds,

(2.3)

hi(x, t) = i Air(x' t-s)(o.j(x, s) ds,

g(x , t )=- i [(B°(x't-s)eq(x's)+L(x't-s)(~(x's)] ds, (x , t )~BX(-~, +~),

where

e o = ½(u, a + uj.,) . (2.4)

The constitutive functions Go, s, Bij, A/j and L are assumed to be continuous and possess continuous partial derivatives of all orders with respect to their arguments. Moreover, the constitutive functions have the following symmetries:

Gijrs = Grsij -- Gjirs, Ao ~- Aji , BO = Bji.

We introduce the notations

,~l- I ~-Gqrs(X, O)~q~rs + 2Bo(x, 0)~o3,+Ao(x, 0) r, zj +L(x , 0)3, 2 ,

~ 2 = -G,~rdX, O)~u~rs- 2t~u(X, 0)~03,--A,~(X, 0 ) r , ~ - t~(X, 0)3, 2 ,

where s%, r~ and 3, are real numbers and ~:# = ~ . Throughout this paper we assume that 3 r t and ~-2 are positive definite quadratic forms, i.e.

~-~ > 0 , ~ r 2 > 0 for any ~ij:~0, r ~ 0 , 3,~0, ~o=~ i . (2.5)

The positive definiteness of the quadratic form 9-~ is equivalent to the positive definiteness of the internal energy density of an elastic material with voids with the constitutive coefficients Gqrs(X, 0 ) , Bq(x, 0) , Aij(x, O) and

A. Scalia / Wave Motion 19 (1994) 125-133 127

L ( X , 0). This condition has been studied in detail in [ 1,2]. The restriction upon 3-2 is compatible with the second law of thermodynamics and has been extensively used in classical viscoelasticity (cf. [ 13,14,17 ] ).

We shall study the case of homogeneous and isotropic bodies, for which

Gij~.~(x, t) = a(t)6ii6~ + lz( t ) [ 6i~Sj~ + 6i~6/,] ,

a o ( x , t) --- or(t) 6q, B o ( x , t ) = [3(t)8o., L ( x , t) = y ( t ) .

In this case the conditions (2.5) imply that (el. [2] )

A ( 0 ) + 2 / x ( 0 ) > 0 , / z ( 0 ) > 0 , o r (0 )>0 ; ,~(0) +2/2(0) < 0 , / 2 ( 0 ) < 0 , & ( 0 ) < 0 . (2.6)

To the above equations, we adjoin the initial history condition

u~(x, t ) = 0 , ~(x, t) = 0 V(x, t) ~ B × ( -o% 0 ) . (2.7)

Let ,~ be a moving surface defined by the equations

x~ =x~(O ~, 0 2, t) ,

where/9 ~, O 2 are curvilinear coordinates on the surface. We suppose that the above functions are continuously differentiable with respect to their arguments and that ,~ is smooth in the sense that the matrix (i~x~/OO") has rank two. The metric tensor of the surface is given by

a~a -~- xi.axi, a . (2.8)

In what follows we denote by n~ the unit normal to ~. We note that (cf. [ 14] )

nini = 1 , nixi: a = 0 , x i : ~ = b ~ a n i , ni; a = - a a P b ~ x ~ : a , (2.9)

where indices followed by a semicolon represent covariant partial differentiation based on the metric of ~, b,,o is the second fundamental form of the surface and a "~ are the elements of the inverse of matrix (a,~o). We have

a~'ax~:~xj:a = 6ij - n in j , t a - i ~ a i . • " - ~" ",~a, (2.10)

where H is the mean curvature of the surface. L e t f be a continuously differentiable function ofx~ and t on each side of the moving surface ,~. We assume that

fsuffers a jump discontinuity across ,~. In what follows we denote by If] the jump of the functionfacross ,~. The discontinuities in the first and second derivatives o f f satisfy the following relations (cf. [ 15] ):

&t

[f0] = a "a(B:,, + a aPb,~aA w) (nix~:a + njxi:a) + a '~Ba UP(A:,, ~ - b~,,,B)xi;ax/: o + C n i n / ,

- VB x~;~ + a ~ A : ~ V:~ - C n, ,

where

(2.11)

128 A. Scalia/Wave Motion 19 (1994) 125-133

is the convected derivative for an observer moving with the surface, V the speed of propagation of the surface, and

A = [f] , B = [f.,n~] , C= [f.on, nj] . (2.12)

We shall use the following lemma.

Lenuna 2.1 ( Fisher and Gurtin [ 11 ] ). Let,~ and ~ be functions on B × ( - oo, + oo ) with the following properties:

( i) g is continuous,

(ii) .,( is continuous everywhere except for a possible jump discontinuity across ~;

(iii) ~ is bounded on every compact subset o f B × ( - oo, + o~ ) .

Then the function ,,n defined by

/rz(x, t) = i ~ ( x , t - s ) / f ( x , s) ds

o

is continuous on B× ( -oo, +oo).

3 . S h o c k w a v e s

The balance laws (2.1) and (2.2) have the form

f f dt pF dv = Ginl da + Q d r , ( 3. l ) p (3p p

which, at singular surface Z, is equivalent to the condition (cf. [ 15] )

- V[pF] = [G~]n~. (3.2)

By a wave of order N, where N>~ 1 is an integer, we mean a solution (ui, ~p) of Eqs. (2 .1 ) - (2 .4 ) , with these properties:

( a ) the functions u~ and ~0, together with their first N - I derivatives with respect to their arguments, are continuous on B × ( -oo, +oo);

(/3) the Nth order derivatives of the four-dimensional vector U = (u~, q~) have jump discontinuities across a propagating smooth surface ,~, but are continuous functions of x and t elsewhere;

(",/) the body forces f~ and I are continuous and, if N>~ 2, N - 1 times continuously differentiable on B × ( - o~ +oo).

When N = 1 the surface Z is called a shock wave. If we apply (3.2) to the Eqs. ( 2.1 ) and (2.2), we obtain the conditions (cf. [ 15 ] )

pV[f~i] + [tj~lnj = 0 , tskV[ ~b] + [h,]n~ = 0 . (3.3)

In the case of shock waves, from (2.11 ) we obtain

[ u, .j] = ~ in~ , [ ~o.A = ~'n~, [Lii] = - V~:,,

where

F .il

[~bl = - V~', (3.4)

(3.5)

A. Scalia/Wave Motion 19 (1994) 125-133 129

It follows from (2.3) and (2.7) that (cf. [ 13,16] )

tij(x, t) =Gi/rs(X, O)er.,(x, t) + Bij(x, 0)(p(x, t) + i {Gij,s(x, t -S)ers(X, s) +/~i/(x, t - s )~o(x , s) } ds , 0

hi(x, t) =Aij(x, O)(p4(x, t) + i Aij(x, t-s)~o.j(x, s) ds , 0

g(x, t) = -Bi j (x , O)eij(x, t) - L ( x , O)~0(x, t) - i { (/~°(x' t - s )e i j ( x , s) +L(x, t - s )~o(x , s)} ds. (3.6) 0

If we take the jump in (3.6), in view of Lemma 2.1, (2.4) and (2.5) we obtain

[tij] -~Gijrs(O)~rns, [hi] =Aij(O)~nj, [g] = -Bi/(O)~inj , (3.7)

where Gijrs( O ) -~ Gijrs( X, 0 ) , A q( O ) = Aij( x , 0 ) , Bij( O ) = Bij( x , 0 ) . The assumptions about the constitutive functions made in Section 2 allow the integration by parts used in deriving of (3.6).

Taking into account the relations (3.4) and (3.7), the conditions (3.3) become

(Gu,s(O) nin, - pV 28 u) ~.~ = O, (3.8)

( Aij( O )ninj - pkV 2) ~ = O . (3.9)

If ~ 0 , we say that the wave is a macro shock wave. Eq. (3.8) admits a nontrivial solution for ~., if and only if

det( Gii~( O )n/nr - pV 2 8i.,) = 0 .

In the case of isotropic materials, this equation reduces to

(c~-V~)~(c~-V~)-~o,

where

c 2 _ - 1 ( ) t ( 0 ) + 2 1 z ( 0 ) ) P

c~ -- 1 p , (0) . (3.10) P

It is an immediate consequence of Eq. (3.8) that the longitudinal macro shock waves (for which s¢~=~i) propagate with the speed V = c~. The speed of propagation of transverse macro shock waves (for which ~:in i = 0) is V = c 2.

If ~'~ 0, the wave is a shock wave of compaction or distension. In the case of homogeneous and isotropic bodies the possible speed of propagation of this wave is V = c3 where

1 c~-- ~, o,(0). (3.11)

Let us study now the growth of the waves. The local form of the balance laws (2. i ) and (2.2) are

lji.j "f- Pfi = Pid'i ( 3 . 1 2 )

and

hci + p / + g = p k ¢ , (3.13)

130 A. Scalia / Wave Motion 19 (1994) 125-133

respectively. Taking into account that V is constant for all waves, from (2.11 ) and (3.5) we obtain

[ i i i ] = V 2 t . t i - 2 V ~ , [ t i ; . j ]=( -V l x i+ ~ ) n j - V a " a ~ ; : j c j : / j ,

-- trfl lu,.;j] - a ~,:,(n;xj:a+njx;:,)-a'~aa~Pb,,~,x;:axj:,+ tz, n~n;, (3.14)

where/ ,q= [ ui.,nrn,]. From (3.12) we obtain

[t,,.,l = p [ u ' ; ] . (3.15)

Clearly, by (2.11 ) we have

8 Vltj;jl = V[trcjnj]nr + Va~[t;i] :,,x;; a = - [tj;lnj +n j fit [tj;l + Va'~attj;];,,xj:a. (3.16)

If we differentiate the constitutive relation (3.6) ~ with respect to t and take the jump across ,~, in view of Lemma 2. ! and (3.4) 2 we get

8¢, = . - - +Gj~rs(O)(;,ns (3.17) It',;] -VGj;r~(O)g,n, VBo(O)f+Gj;,.,(O)n,.-~t VGj;,,(O)a"~¢~:~xs:a (') ,

where we have used the notation q ( ~ ) = ~. By (3.7), (3.14), (3.16) and (3.17), Eq. (3.15) reduces to

VlC,,,,(O)n, + + (¢,n,)+ 2pV 8t

+VGj;~,(O)a~a(£,n~).~,x~a+VGj;,s(O)nja~a£,:~s:a- (~) 0 . . . G)i~.,( )~:,nsnj=0 (3.18)

Now we restrict our attention to homogeneous and isotropic solids. If we use (2.6), ( 2.9 ), ( 2.1 O) and the relations

8n; a'~a~;:"x':a=a'~a(x~:a~;):"-2Hn;~" (5-T = 0 , (3.19)

then Eq. (3.18) becomes

V{ (A(O) + / x ( 0 ) )n;n, + p,(O)8i~ -pV28; , } Iz, + 2pV 2 ~ + Vfl(O)n~(,+ V(A(O)

+ p,(O) ) { a "an~(Xr:a~,),~ + a'~a( ~:,nr ):,~xi.a - 2H~,n~n~ } - 2HVlx(O) ~;

- ( A ( I ) ( O ) +/x(l)(O))sC~n~n;- p,(l)(O)~i = 0 . (3.20)

If we multiply (3.20) by n; and sum on i, in view of (2.9) and (3.19), we get

V{,~(0) + 2/~(0) - p V 2 } tz~n~ + V~(O)~ + 2pV 2 ~----~ • . (~t

+ V(,~(0) + / L ( 0 ) )a'~a(x,:a~,):,~ - 2V(M0) + 2/~(0) ) H E - ( ;~ (t) (0) + 2 /~J ) (0 ) ) ~= 0 , (3.21)

where ~:= ~:;n;. First we consider longitudinal macro shock waves. Then so; = ~ , V = c~ and x~:,~s~-- O. Now Eq. (3.21 ) for c~ ~: c3

becomes

A. Scalia/Wave Motion 19 (1994) 125-133 131

l 6~: d~ ¢ = ~ ( H - M ) , (3.22)

c~ 8t dn

where n is the distance measured along the normal to the wave surface and

A(')(O) + 2 ~ ( 1 ) ( 0 ) M = - (3.23)

2pc~

The inequalities (2.6) imply that M is a positive constant. It is known [ 14] that if at some instant to the mean

and Gaussian curvatures of surface are Ho and Ko, respectively, then at a subsequent time t the following relation:

Ho - n Ko H = (3.24)

1 - 2nHo + n 21(o

holds. It follows from (3.22) and (3.24) that

s¢=~( 1 - 2nHo + n : K o ) -1 /2 e x p ( - r i M ) , (3.25)

where ~ is the strength of the wave at time t-- to. In the case of a transverse macro shock waves V= c2 and if we assume that c2 4: c3, then we have ~r= 0 and Eq.

(3.21) reduces to

a ~a(x,:.agr):~, + n~l~ = O.

Thus, with the help of (3.10), from (3.20) we obtain

1 6~ d~ c - 5 6-T = - ~ = ( H - P ) ~ , (3.26)

where

~(~)(0) p_- 2Oc~

It follows from (2.5) that P is a positive constant. From (3.26) we get

5 = ~ ( 1 - 2 n H o + n 2Ko) - l/2 exp( - nP) , ( 3.27 )

where ~ -- 5(to). Let us assume that the wave under consideration is a shock wave of compaction or distension. In this case V-- c3,

and we assume that c 3 ~ c l and c 3 ~ c 2 . Then ~¢~--0 and Eqs. (3.20) reduce to

{ (h(O) + /~ (0 ) )n , nr +/z (O) 8,r - p c ] 8,, ]/zr = - ~(O)n,~'. (3.28)

Clearly, from (3.28) we get

()t(0) + 2 # ( 0 ) - p c ~ ) / z , n , = - / 3 ( 0 ) ~'.

In view of the above relation, (3.28) becomes

O(0)n,~ I t , = p ( c ~ - c ~ ) " (3.29)

Thus, a macro-acceleration discontinuity is induced by a shock wave of compaction or distension. Let us consider now Eq. (2.2). From (3.13) we obtain

132 A. Scalia/Wave Motion 19 (1994) 125-133

[ h;.~] + [g ] = pk (V 2 r - 2V ~tt~) ,

where

r = [ (p.on;n;] .

We have

V[hci] = - [hilni +n; ~t [hi] + Va"a[hk]:"xk:a'

[ l~i] = a( O ) { ( - Vr + ~tt~) n; -- Va "a~;,,xi:a} + ott ' > ( O ) ,ni ,

[hi] = a(0)~'n,, [gl = - /3(0)~' ;n; ,

so that Eq. (3.30) becomes

V(a(0) - p k V 2 ) r - 2HVa(O)~-o t ( I ) (O)~- V~(O)(d+ 2pkV 2 3tt = 0 .

If V = c 3, c3 4= c~, c3 4= c2, then ~:, = 0 and from (3.32) we obtain the growth equation

1 6.__~ = ~ ' (H- A ) , c 3 (~t

where

A = a(I)(O) 2c~

Clearly, (2.6) implies that A is a positive constant. It follows from (3.33) that

~.= ~.o( 1 -2nHo + nZKo) -i/2 e x p ( - n A ) ,

where ~)= ~'(to). Moreover, if V--cl, cl ~ c3, then ~'= 0, and (3.32) becomes

(a(o) -~c~)r= /3(0)~.

Thus, a longitudinal macro shock wave induces an acceleration discontinuity in the wave of compaction.

(3.30)

(3.31)

(3.32)

(3.33)

4 . C o n c l u s i o n s

In this paper we have studied the propagation of shock waves in a linear viscoelastic material with voids. We assume that the body is homogeneous and isotropic and that the relaxation functions satisfy the conditions (2.6). The restrictions (2.6) are universally accepted and they have been used in various papers (cf. [ 1,2,13,15,17] ) in order to obtain uniqueness and asymptotic stability results. Under these hypotheses we studied the propagation of various types of singular surfaces and their attenuations in viscoelastic materials with voids. The mathematical techniques employed for the study of the surfaces of discontinuity are those developed by Hadamard and later extended by Thomas (cf. [ 14,15] ).

First, we derived the speeds of propagation of shock waves. It is found that the longitudinal macro shock waves

A. Scalia / Wave Motion 19 (1994) 125-133 133

(for which ~:i = ~-~i) propagate with the speed c l, and the transverse macro shock waves (for which ~:ani = 0) with the speed of propagation c2 (see (3.10) ). The speed of propagation of a shock wave of compaction or distension c3. The possible speeds of propagation of macro shock waves are determined solely by the macro relaxation functions, while the speed which governs the propagation of shock waves of compaction or distension is independent of these functions. In general, the speeds of propagation of shock waves will be different. Thus, in general, a shock wave is either a pure macro shock wave and has no discontinuity in the first derivatives of the volume fraction field associated with it, or it is a pure shock wave of compaction or distension with no associated velocity discontinuity. It is shown that a longitudinal macro shock wave induces an acceleration discontinuity in the wave of compaction or distension. Moreover, a macro acceleration discontinuity is induced by a shock wave of compaction of distension.

Finally, the growth equations which govern the propagation of shock waves are derived and studied.

Acknowledgement

I express my gratitude to the referees for their detailed criticism of the manuscript. This research has been performed under the auspices of the G.N.F.M. of the Italian C.N.R. and has been partially supported by the

M.U.R.S.T. under 40% and 60% contracts.

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