Some remarks on the growth behavior of one-dimensional shock waves in non-linear materials

Inc. J. Non-Lm~ar

Mechanics.

Vol. 9. pp 147 157.

Pergamon

Press 1974.

Prmfed in Great Brimn

SOME REMARKS ON THE GROWTH BEHAVIOR OF ONE-DIMENSIONAL SHOCK WAVES IN NON-LINEAR MATERIALS FREDERICK Lecturer, Department Technion-Israel Institute (Received

BLOOM

of Applied Mathematics, of Technology, Haifa, Israel 18 May 1973)

Abstract-The growth behavior of both compressive and expansive one-dimensional shock waves which propagate into an unstrained region of a non-linear material exhibiting anelastic response. in the sense of Eckart, is analyzed. In each case, a differential equation governing the growth of the amplitude of the shock is derived and it is shown that a critical strain gradient may be defined. The growth behavior of the waves closely resembles the growth behavior of compressive and expansive shock waves propagating in sufficiently smooth non-linear materials with fading memory, i.e., in materials which can be approximated by linear viscoelastic materials for small relative strains. 1. INTRODUCTION In this paper we study the growth behavior of one-dimensional shock waves which propagate into an unstrained region of an anelastic material of class zero. Such materials are defined by the property that there exists a non-empty class of continuous motions such that the Cauchy stress at a material particle, in any such motion, arises in response to deformations from a preferred time-dependent reference configuration associated with the motion. Anelastic materials of this kind were introduced by Eckart[l] and his theory was subsequently reviewed by Truesdell;[2] it is thought that such a material may serve as a, good model for an elastic body which possesses a continuous distribution of dislocations with time-dependent dislocation density. A study of the connection between dislocation motion and the concent of anelastic response has been carried out by Bloom and Wang in [3]; anelastic materials are shown here to be closely related to the materially uniform inhomogeneous elastic materials considered by No11[4] and Wang.[5] We consider, in the present work, the growth behavior of both compressive and expansive shock waves; for each case a differential equation governing the growth behavior of the wave is derived and it is shown that a critical strain gradient may be defined. Both of the derived equations very closely resemble the differential equations found by Chen and Gurtin[6] in their study of the growth behavior of compressive and expansive shock waves propagating in sufficiently smooth non-linear materials with memory, i.e., in materials which can be approximated by linear viscoelastic materials for small relative strains. Indeed, when the preferred time-dependent reference configuration of the anelastic material is defined by a homogeneous motion of a given fixed reference configuration and the initial value of the slope of the associated stress relaxation function depends in a simple and explicit way on the given strain history, the critical strain gradients for the anelastic material assume forms which are entirely similar to those for the materials with memory considered in [6]. For background material connected with the behavior of shock waves in a continuum as well as for a further discussion of some of the kinematical conditions and results that are utilized in 93, of the present work, the reader is referred to [7]. 147

148

FRLDFRICK BLOOM

2.

CONSTITUTIVE

ASSUMPTIONS

All our considerations in this paper shall be one-dimensional. A fixed reference configuration R of the body will be a closed interval of the real line and a motion of the body is a real-valued function 5 defined on Rx( - co, co), which gives the place .Y= <(X, t) at time te( - a, co) of that material point which occupies the place X in R. We let F(X, t) = Zx<(X, t), a(X, r) = ?,j”(X. t), and i’(X. t) = i:<(X, t) denote, respectively, the deformation gradient, velocity. and acceleration fields which correspond to 2. The constitutive assumption we are making in the present paper is then given via Dejirzition 2.1. A material body is said to be anelastic of class zero if there exists a non-empty class A? of Co motions defined on Rx( - x, co) and a C” real-valued Elastic Response Function 6 such that the following is true (a) In any rest history? at the deformation gradient F. the Cauchy stress T(X, t) is given by T(X, t) = &(F,). (b) For each 5 E A there exists a motion ,LQ defined on Rx( - co, a) such that the Cauchy stress at the particle X and the time t in the motion 5 is given by 7(X. t) = 8(&(X,

t)).

(2.1)

where P&X, I)~~~,(go~t’)(,;,?(.,!. (c) If R,(X, t)s&& ‘(X, t) then R;(X, t) --f 1 as t + -or; (and we assume, that L?,p<(X, t) # 0 on Rx-( - CG,“c) as (1,~~: 1 = l/?,pL:).

of course,

Remark. The definition above is the one-dimensional counterpart of the definition used in [l] with the additional restriction that fi consist of Co motions; the same definition, with I%?restricted to C’ motions, has been used in [8] to study the growth behavior of acceleration waves in materials exhibiting anelastic response. As in [S] we shall require that each motion ,LL~,associated with a motion 5 EM, be smooth enough so that all derivatives of PCwhich appear are jointly continuous in X and t for all (X, t) in Rx( - x. m); for the purposes of this work we require that each ,LL< be of class C* jointly in X and t. It should be clear that the chain-rule and the various definitions given above imply that &(X, t) = F(X, t)RE(X, t); @X, t) is thus the deformation gradient of that motion which takes us from the configuration of X in the motion ~1: at time t to the configuration of X in the motion < at time t. Thus for each motion ~EQ there corresponds a motion p; which, in effect, gives rise to a time dependent and, possibly, inhomogeneous reference configuration for the body; the stress in any such motion arises in response to deformations from this time dependent reference configuration defined by p< and these deformations are precisely the functions @X, t). Requirement (c) in the definition is imposed so as to insure consistency with the condition stated in part (a), i.e., that the body behaves elastically in all rest histories. If we now define an Instantaneous Anelastic Response Function .Fr by P< (F(X, t); X; t)%Y (F(X, t)R<(X, t)) ;
mean

that if F(X. r) = ;,;(X.

t) then F(X. Z) = F,(X).

~ x < 7i

(2.2) t. where t represents

One-dimensional

then by our smoothness assumptions

shock waves in non-linear materials

149

on d and p, we clearly have

“\ &(X, t) &@-
(2.3,)

@B,(F;

(2.3,)

x; t) = (c7i.@)} R,2(X, t)

and we note that Sg, a,P<, and d:9< are of class Co in X and t at all points (X, t) E Rx( - 00, co) where F(X, t) exists and is continuous. 3.

GENERAL

PROPERTIES

OF

SHOCK

WAVES

Let < be a motion in A. We assume that a shock wave C, relative to { exists in the body during some time interval T. Let Y(t) denote the material point at which the wave is to be found at time t. The trajectory of the wave is then given as C = {(X, tf / X = Y(t), t E 7’) and the intrinsic velocity of the wave is defined to be U, = (d/dt) Y(r). In saying that C is a shock wave relative to the motion i_r~_@we mean, explicitly, that < is continuous on a neighborhood _,,I*of EC,of class C2 on J,‘- C and that the functions -2, F, 2, p-. and ?F/dX suffer jump discontinuities across E. BY thejump I71 in a function f we understand the following: (3.1)

[f](t) = xJ*y,_ f(X, t) - x$y,+ f(X> 4

The two limits here are usually denoted by ,f-(t) and f’(t), respectively, and represent, when U, > 0 the values of f immediately behind and just in front of the wave. At the wave we have the following kinematical conditions: U,[F] = -[k]

(3.2,) (3.2,) (3.2,)

From the law of balance of momentum we easily find that, across C

P-1= -Pqxl

(3.3)

and away from C g

=. pjt,

(3.4)

In (3.3) and (3.4). p denotes the density (which is assumed to be constant) in R. Combining (3.2,) and (3.3) yields the following well known result for the velocity of the shock: (3.5) The quantity [F] is called the amplitude of the shock. If we take the jump of (3.4) and use the result in con.junction with (3.2,) and (3.2,) we find as the equation governing the growth behavior of the amplitude of the shock 7

(3.6)

a result which is independent of the constitutiv~ assumption for the stress. In order to exploit this relation we shall next calculate [c’r/?X] for an anelastic material of class 0.

150

FREDERICKBLEW 4. SHOCK

WAVES

IN

ANELASTIC

MATERIALS

I

We shall consider shock waves which propagate into an unstrained region of an anelastic body. Since the strain at the particle X at time c is defined by E(X, 1) = F(X. t) - 1 = &4(X, t) - 1

(4.1)

we shall assume that, for the motion 5 E A, F(X, t) = 1 for all X 3 Y(t) and t E T. Thus [&F] = (&F)),

[P] = P-.

(4.2)

Now, from (2.1) we have

at all (X, t) # (Y(t), t). From our smoothness assumptions

on the preferred motion p, it

fOllOWS that

g

=

I

mw*m

1 (Y(t),t) f

CF‘w~l&~5

(4.4)

I (y(w).

I

If we make use of (2.3,) we may rewrite this as & i

= [(&P&&F]

+ [F(i?FFC)]C?xIn R, l(~(~),t).

1

(4.5)

Then when we take account of (4.2) the result above is Let us set R,, = &In R, l(~(~).~). easily seen to reduce to

2T =(&,F<)((&F)+R,,,F-), i-l (4.6)

i;X

where

(a,&-%?,@-#;

x; t)l17=F-(X.t)=IY,f),t).

We now make the following definitions: E;- = d,F
F-(x.t)=(~(r),t)

=

M(F-R<(Y(d,

t))&(J%),

t)

(4.7)

will be called the instantaneous tangent modulus for the material, in the motion corresponding to the value of the deformation gradient behind the wave. Also E- = (a~~~)i,=,-,.~,,,=(,,,,,)

= ~~~~~_~~(Y(~), ~)~~~(Y(t),t)

<, (4-S)

is the instantaneous second-order modulus corresponding to F-. Similarly E: and &! will denote, respectively, the instantaneous tangent modulus and instantaneous second-order modulus corresponding to the value (F = 1) of the deformation gradient just ahead of the wave. Thus (4.6) may be rewritten in the form

I-I= ?T

dX

Et- {(c&F)_

-tR<,,bq

(4.9)

and if we substitute this result into (3.6) we derive the following equation for the growth behavior of the wave: (4.10)

One-dimensional

5.

SHOCK

shock waves in non-linear

WAVES

IN

ANELASTIC

151

materials

MATERIALS

II

We shall consider the growth behavior of compressive and expansive shock waves in anelastic materials. To begin with let us notice that, by (3.5)

(5.1)

F--l

However, our smoothness assumption on &
which, in turn, means that we may rewrite (5.1) as

pt..‘: = E: -t-@:(F- - I)fO((F-

- 1)).

(5.3)

In a similar fashion we easily find that E;

= E,! +&(F-

- l)+O((F-

- 1))

(5.4)

and thus E; -pU;?

= +I?#-

- l)+O((F-

- 1)).

(5.5)

We now make the two following assumptions concerning the instantaneous stress-strain law. (i) In compression, i.e. for E = F- 1 < 0 the instantaneous stress-strain law is concave from below, i.e. 8iLF, < 0 for F d 1. (ii) In expansion, i.e. for E = F - 1 > 0, the instantaneous stress-strain law is concave from above, i.e. a;@-, > 0 for F 2 1. For a compressive shock wave [a] < O(F- <: 1) and for an expansive shock wave [E] > O(F- > 1). From (5.1) or (5.5) it is clear that E; >

pu,”

(5.6)

for both compressive and expansive shocks. Now, if we differentiate [T] with respect to t by making use of (3.5) we find that

(5.7) On the other hand, [T] = F<(F-, to t yields

4Tl _I_ dt

=

E-

Y(t), t) - P<(l, Y(t), t) and so differentiation

dF-t drfiG;-G:)U~+(H,--H:)

with respect

(5.8)

152

FREDERICK BLOOM

where G, , G: , H,-, and H,’ are defined

as follows :

G; = (S,.Y,)

= F=F_IX.t)=(Y(t),r)

G;

lim

X-Y(t)

(5.9,)

=

(&F-;)

= F=l(X,I)=(Y(t),t)

lim

X-Y(r)’

{~FB)F(X.I)~

(5.9,)

H,-= (&,F,)

From

(5.9,) and (5.9,) we note the simple relation (5.10)

If the motion homogeneous

/.q, which defines our preferred motion of R, i.e. &(X.r)

other simplifications R,(X, t) = l/&(t)(=

time-dependent

XsR,

= &(t)X+c(t);

result. We assume that R<(t)) and so (5.9,))(5.9,)

reference

configuration,

teT

is a

(5.11)

there are no zeros of P<(t) for TV T. Thus reduce to

G,- =0

(5.12,)

G; = 0

(5.12,)

H,- = E;(F-d,lnRi}

(5.124

H: = E:{Ztln&}.

(5.124

Where 8, In Ri = 8,ln &l (yct),~).We shall return to the special case expressed later on in this paper. If we now compare (5.8) with (5.7) we see that

2p~~(F~-l)~=(Ei~-p~~)~+(G,-oC’,+(H,-H,1).

by (5.11)

(5.13)

One-dimensional

shock waves in non-linear

153

materials

Let us consider first the ease of a compressive shock wave, i.e., F- < 1. We may set /3*= (6; - G:) U, + (H; - H,l) in (5.13) and solve for dCi,/dt to get dt’, -_= dt

K d/F--l/ -__l.---4pU, dt

PI 2pz@- - 11

(5.14)

where

&(Et-

-vu:).

(F--1(

(5.15)

Clearly k: = K(F-) and as F- -P 1 it follows, from (5.5), that ti -+ Ej. Also by (5,6), x: > 0. Now for the case of a compressive shock (4.10) reads (5.16) where we have set ~1,= E-&/p.

Substituting for dU,Jdt in (5.16) from (5.14) we have

which we may rewrite as

By ~~corporatillg the first twa terms an the right hand side of this expression into the last term we arrive at the final result (5.19)

where (5.20) is called the critical strain gradient. If the preferred motion p5 is given by a homogeneous motion of R such as (5.11) then ct, = 0 and J_*reduces to X = ,&/k-U,jFP - IIf. ffwe assume that U, > 0 then (1X19),together with the fact that x’> 0, yield the following T~z~ore~. Consider a compressive shock wave ~rapagating in an anelastic material of class 0 and assume that the region ahead of the wave is unstrained. Assume also that the instantaneous stress-strain law in compression is concave from below. Then

where a* is the critical strain gradient defined by (5.20). iBy

(S.lZ,)-j5.12.4,

ff, = d,lnl?~jE;F-

-E:).

154

FREDERICK BLOOM

Remark. Chen and Gurtin[6] have considered the growth of one-dimensional shock waves in non-linear materials with memory, i.e., they take as their constitutive equation (5.21)

T(t) = F_(&‘) where

2x(X, t-s)

Et(S)=

?X

---l,O
(5.22)

The domain DF of B is a cone of strain histories E’ in the normed space D consisting of all integrable functions f on [0, co) which vanish on some sub-interval (so, cc) and have norm 11fll = I,f(O)I + 1: If’(s)I ds; the admissible strain histories satisfy the natural restriction: - 1 < .$(S) < m, 0 < s < co. Chen and Gurtin also assume that, for small relative strains, the material can be approximated by a linear viscoelastic material, i.e., given any history F’E D.9 there exists a smooth (relaxation) function G on [0, co) with the property that if EYEDB is a strain history, with y’ = E^--e’ the corresponding relative strain history. then F(a)

3oG’(s)y’(~)d~+0(l/y’Il)

= F(E’) + G(O)y’(O) + s

as d + E’ in the norm of D. In general is made explicit by writing

both G(s) and G’(s) depend

G(s) = F&s’; s),

where EE( - 1, co). Let material and define

to be the instantaneous

CT~(E)

=

tangent

F(E*)

on

E’;

this dependence

G’(s) = Y(F’; s)

and the functionals g(. , .) and G’(. , .) are assumed Chen and Gurtin proceed to show the following: let E*(S)=

(5.23)

0

E*

(5.24)

to be continuous on D.ix[O, co). be the history in D.,- defined by

L ;r===, ;

o.

be the instantaneous

modulus

corresponding

(5.25) response

to the strain

function

s-(t)

for the

behind

the

wave. Then the growth behavior of a compressive shock wave is governed precisely by an equation of the form (5.19) whenever the region ahead of the shock has been unstrained at all past times, where

ti,2c&,L:Z) (F--1(

(5.26)

‘Ot

and

.&W’W KU*

(5.27)

.

If we restrict our attention to the important case of (2.1) where ,u< is given by a homogeneous motion of the form (5.11) then, by virtue of the remark following (5.20). we have the following result: Theorem. The growth behavior of a compressive shock wave propagating in an anelastic t

Provided

the instantaneous

stress- strain law in compression

is concave

from below

One-dimensional

shock waves in non-linear

155

materials

material of class 0, whose preferred time-dependent reference configuration is defined by a homogeneous motion of R, and the growth behavior of a compressive shock wave propagating in a sufficiently smooth? non-linear material with memory are both governed by the differential equation

where ti is of the form K = 2(E,-pU;)/lF-

- 11.

(5.29)

Here E, represents the instantaneous tangent modulus, for either material. corresponding to the strain E-(t) behind the wave at time t, U, is the corresponding speed of propagation and for the anelastic material f(&_, t)@qJ,/I&- I while for the material

with memory

and a given strain history

f(&-, t)=2IG’(O)I where G(s) is the stress relaxation For expansive for compressive expansive shock that the region governed by the

(5.30) E’E D.9

= 2(9’(s’; 0)l

(5.31)

function.

shock waves, in anelastic materials, results entirely analogous to those shock waves are obtained and we shall only state the results. For an wave (F- > 1) propagating in an anelastic material of class 0 (and such ahead of the wave is unstrained) the growth behavior of the wave is differential equation dF-

ti(F_ - 1) {&

(5.32)

(W-j,

where K is defined by (5.15) and where 2, the critical strain gradient

for this case, is given by (5.33)

If the preferred motion ,ur is given CC,= 0 and 1 reduces to -QJKU&Fthe following conditions apply:

by a homogeneous motion of R such as (5.11) then - 1). From (5.32) it is easily seen that, in this case,

(i;,F)_
>I+$<0

(S,F)_=fi+o, tie. the approximation (5.23) is shown, by Coleman and Gurtin.[7] smoothness assumptions which may be imposed on the response functional.

to be a consequence

of certain

1%

FREDERKK

BLOOM

Rema&. For expansive shocks propagating in a sufficiently smooth non-linear material with memory, which is such that the region abead of the wave has been unstrained at all past times, Chen and Gurtin[Sf nave shown that the growth behavior of the shock is governed by an equation of precisely the same form as (5.32) where their f is given by

fi = 2G’(O)//
(5.34)

It should be obvious that the previous theorem may be modified so as to apply to expansive shock waves in both anelastic materials of class 0 and sufficiently smooth non-linear materials with memory. In fact the common governing differential equation is of the form

where for the anelastic materiaf g(&-, t)!E - p&E-

(5.36)

g(e-, t)gg2G’(O) = 2%‘(e”;0).

(5.37)

and for the material with memory

Ackrmwledgment-The

research

reported

here was carried

out while the author

held Tcchnion

grant

110-504.

I. C. Eckart, The thermodynamics of irreversibfc processus.--IV. The theory of elasticity and anelasticity. Pfi_tJs.RPIL 7X373- 382 11948). 7 ^. C, A. Trues&II. The mechanicat foundations ofeiasticity and Ruid mdanics. 3. Rat. :tfd~. Anal. I, 125 300 (1952). in anelastic hodies. Ar&. I\‘trr. ilfcc%. 3. F. Bloom and C. C. Wang, Material uniformity and inhomogeneity Annl. (in press). 4. W. Noll. Materially uniform simple bodies with inhomogeneities. Arc/t. Rut. M&t. Arrnl. 27, l-132 (1967). structures of simple bodies, a mathematical foundation for the theory of 5. C. C. Wang, On the geometric continuous distributions of dislocations, Arch. Rat. Mcch. .4nal. 27. 33 92 (1967). shock waves in materials with memory. 6. P. .I. Chen and M. E. Gurtin. On the growth of one-dimensional Arch. Rat. Mrch. And. 36. 33-46 (1970). I. B. D. Coleman and M. E. Gurtin, Waves in materials with memory. II. 4rch. Rat. Mcch. ,4~0f. 19. 219 265( 19h5). acceleration waves in non-linear materials with anclastic 8. F. Bloom. Growth and decay of one-dimensional response, Accepted for Publication -to appear in the Inr. .I. engna Sci.

R&m+-On analpe le phenomknc dc ia croissaucc d’ondcs de choc unid~i~ens~on~cj~cs dc compression ou de dilatation qui se propagent dans unc region non deform6c dun mate&u non Iineaire montrant une ri-ponse int-lastique au sens d‘Ec.ckart. Dans chaque cas on Ctahlit une kquation didrenticlle regissant la croissance et I’amphtude du chat et on montrc qu’on peut definir un gradient de deformation critiyuc. 1.e comportement de la croissance des ondes resscmhlc de pres au comportement de la croissance d’ondes de choc. de compression et de dilatation. se propageant dans dcs materiaux non lineaires suffisamment tcndrcs avcc une m&moire cvanouissantc, c’est a dire des materiaux qui peuvent &tre consider&s comme des materiaux viscoelastiqucs lineaires pour des deformations relativement pctites.

Zusammenfassung-Das Wachstums~er~iten von eindimensionalen, Druck- und Zugspanaungen verursacbendcn Stosswellen, die sich in ein dehnuagsfreics G&et eines nichtlintaren Materials mit anelatischem Verbalten im Sinne Eckarts ausbreiten, wird untersucht. Fur j&en FatI wird einc ~~~erentia~gleichun~ hergefeitet, die das Wachstum der ~to~amplitude bestimmt. und es wird

One-dimensional

shock waves in non-linear

157

materials

gereigt, dass ein kritischer Dehnungsgradient definiert werden kann. Das Wachstumsverhalten der Wellen ist sehr ahnlich dem Wachstumsverhalten von Druck- und Zugspannung verursachenden Stosswellen. die sich in ein hinreichend gleichmlssiges nichtlineares Material mit verschwindendem Erinnerungsvermogen ausbreiten. d.h. in ein Material. das fir kleine relative Dehnungen durch ein lineares viskoelastisches Material angenahert werden kann.

AHHoTaUHR IUlipeHkUi. pIEin)‘,

PaccMarpesaeTcfl

OAHOMePHbIC

06na&SOI&My

PaCCMaTpHBaeMbIX Ha,,aCTaHWZ

Mat@.

H&lie HapaCTEUO4HX MaTCpkWIZtX

CJIy’iaCB

aIIII,,OKCNMH,,OBEiTb

HeynpyrMMM

BbIBOAllTCII

CKBYK8.

nOKa3aH0,

HapacTaromnx

ynapebtx

YAapHbIX

C 3aT,‘XtUO”@

uapacraroqnx CBOkTBaMH

YTO CyIWCTByeT B

nanfroM

BO,‘IH GKBTHII II paCIUH,,eHRE HaCJIeACTBeHHOCTbH),

,SfHefiHbIMH

II0

BR3KOyII,,jTHMH

cmarnfl

ypaBHCHW?, KpHTHYCCKllii

bfarepriane

B AOCTtlTO’iHO hlWIbIX

MaTC&Is

BCeX

O~HCblBEUOlLVie Q,kU,HeHT

nanobfmiaer

Ae@O,,-

noseHe-

FJIaAK&iX HeJIMH’StiHbIX

TO CCTb B MaTCpPfaJIaX, MOACJVIMH jW%

H pac-

HeBO3M,‘“@HHOM,’

B CMbIC,Ie 3KKapTa.

iW~~ep’2HLWUIbHbIC B0n~

~0nH

ynaonbtx

BOJIHbI paCtIpOCTpaHSILOTC~

Ht2ILiHefiHbIMA

aMIIJIkiTy~b1

Iloseaeiuie

nonenerine

YAapHbIe

KOTOPbIe

&,OpMa,,,&.

MO)1(HO