minimum principles for linear elastodynamics

Journal of Elasticity, Vol. 8, No. 1, January 1978 Sijthoff & Noordhoff International Publishers Alphen aan den Rijn Printed in The Netherlands

Minimum principles got linear elastodynamics

R O B E R T REISS

Division of Materials Engineering, The University of Iowa, Iowa City, Iowa 52242, USA

(Received August 30, 1976; revised November 23, 1976)

ABSTRACT

Two minimum principles which take into account inhomogeneous initial conditions are presented within the context of the linear dynamic theory of elasticity. One principle, formulated in terms of displacements alone, is the dynamic counterpart to the static principle of minimum potential energy; the other principle is formulated in terms of stresses alone, but has no counterpart in elastostatics. Both principles are motivated by taking Laplace transforms of the field equations and boundary values and then using established minimum functionals in the "transform domain". The introduction of an appropriate "weight function" enables one to get back to the original time domain while preserving the minimum character of the transformed functionals.

RESUME

Deux principes minimum qui fiennent compte de conditions initiales non lin6aires sont pr6sent6s sans le contexte de la th6orie dynamique lin6alre de l'61asticit6. Un des principes, exprim6s en fonction seulement des d6placements, est la contrepartie dynamique du principe statique de l'energie potentieUe minimum; le second principe, exprim6 en fonction des efforts seulement, n'a pas de contrepartie en 61asticit6 statique. Les deux principes sont obtenus en prenant les transform6s de Laplace des 6quations differentielles et des conditions aux limites, puis en utilisant dans le "domaine transform6" des fonctionnels minimum connus. En introduisant une "weight function" appropri6e, il est possible de retourner au dornaine temporel d'origine tout en pr6servant le caractere minimum des fonctionnels transform6s.

1. Introduction

A m e t h o d of ob ta in ing var ia t ional solut ions to in i t ia l -value p rob lems was in t roduced

by G u r t i n [1] for l inear e las todynamics . Subsequent ly , G u r t i n appl ied his m e t h o d to

the in i t ia l -va lue wave p r o b l e m and the in i t ia l -value hea t conduc t ion p r o b l e m [2].

G u r t i n ' s concept has since b e e n appl ied to the l inear theory of viscoelasticity [3],

thermoviscoelas t ic i ty [4], coupled thermoelas t ic i ty [5, 6, 7], thermoelas t ic i ty with

micros t ruc ture [8], e lec t romagnetoe las t ic i ty [9], and coupled t he r mo-

e lec t romagnetoe las t ic i ty [10].

Journal of Elasticity 8 (1978) 35-45

36 R . R e i s s

The essential idea of Gurtin is the elimination of the initial conditions through the convolution integral. Thus a set of "reduced" integro-differential equations replaces the original field equations and initial conditions. The variational representation is then formulated directly through these reduced equations. It Should be emphasized that Gurtin's concept will produce variational principles which, in general, are not extremum principles. However, Benthien and Gurtin [11] recently showed that the Laplace transform of many of the Gurtin-type stationary functionals are in fact minimum functionals in the transform domain.

In this paper, two minimum functionals are established in the original space-time domain. They are derived from minimum principles in the transform domain. Inversion of these functionals merely recovers stationary principles of Gurtin-type. A special technique involving a "weight function" is used in order to get back to the original domain while preserving the minimum character of the transformed functional.

It should be remarked that the methods of this paper are applicable in areas other than elastodynamics. In fact, minimum principles for many other initial-value linear problems such as the wave equation and heat conduction equation may be routinely derived from minimum principles in their respective transform domains [11].

2. Notat ion and definitions

Let x~ (i = 1, 2, 3) denote the Cartesian components of a space vector x referred to an inertial rectangular Cartesian coordinate system. The closed, bounded and regular (in the sense of Kellogg) region of space occupied by an elastic medium is denoted by /~ for which the interior is R and boundary is OR. Furthermore, OR is the union of two complementary regular subsets OR, and 0R~ whose closures are denoted by 0/~, and 0/~, respectively.

The customary indicial notation is employed throughout this paper. Thus sub- scripts are understood to range over the integers (1, 2, 3); a repeated subscript implies summation over that subscript, and a comma followed by a subscript denotes partial differentiation with respect to the indicated Cartesian coordinate. Differentia- tion of a function with respect to time t will be denoted by a dot above the function. Functions of x and t are said to be of class C M'N if they are continuously differentiable at least M times in x~ and at least N times in t. The domain of definition for such functions is the Cartesian product of the domain of x and the domain of t. Thus (x, t) ~ _ff × [0, oo) means x ~/~ and t ~ [0, ~). And finally, a function f ( x , t) wi l l be said to be bounded at ~ if l i m t ~ f ( x , t) exists for each x in the domain of definition of f.

We now formulate the mixed boundary value problem for linear elastodynamics. Let o-, i (x, t), u~ (x, t) and F~ (x, t) denote the Cartesian components of the stress tensor or, displacement vector u and specific body force vector F, respectively. Then, the equations of motion

o'~s. j + oF~ = oi~, o'~j = o-ji on R x (0, o~), (2.1)

Minimum principles for linear elastodynamics 37

and the stress-displacement relations,

o'~j = Ciik~Uk,l on R x (0, ~) (2.2)

form a complete system of field equations. In equation (2.1), O(x) denotes the mass density of the elastic medium, while in equation (2.2), Cijk~(x) are the Cartesian components of the elastic stiffness tensor C.

The boundary and initial conditions associated with the field equations (2.1) and (2.2) are:

T~ =- o-~in , = ~ on 0Ro- × [0, ~), (2.3)

u~ = ~ on OR, x [0, m), (2.4)

and

~(x, o)= uP(x),

on /~ (2.5)

¢(x, o)= a°(x).

In equation (2.3), n is the outward normal unit vector to OR, and T(x, t) is the traction vector. Moreover, ~(x, t), ~-(x, t), uP(x) and ri°(x) are considered as given functions.

The following assumptions on the material properties will be used throughout this paper:

(a) p(x) is strictly positive and continuous on /~ , (b) F(x, t) is continuous o n / ~ x [0, 00) and bounded at ~, (c) C (x) is (1) continuous on /~, (2) positive definite on /~, and (3) obey the

symmetry properties:

C, jkt = Gikl = Ckui on /~. (2.6)

As a result of the properties of C, the stress-displacement relations (2.2) may also be expressed by

u(,.,~ =--½(t~,, + u,a ) = Ki,~zcrkl on R x (0, oo). (2.7)

Here, Kijkt(x) are the Cartesian components of the positive definite compliance tensor K and they satisfy the same symmetry and regularity requirements as C. Also, the symbol u(i,j) denotes the symmetric part of the displacement gradient.

Finally, it is convenient to introduce the set of admissible weight functions ~. It will be said that g(t) ~ ~ if, and only if,

(a) g(t) exists for t6 [0, w) and is the Laplace transform of some non-negative continuous function G(s) which has a finite number of zeroes on its domain of definition s ~ [0, ~),

g(t) = ~o~G(s)e -~' ds, (2.8)

38 R. Reiss

(b) the improper double integrals

Io fo g(t+r) dtd% g'(t+ r) dtd't, g"(t + ~') dtd'r

exist. Here g' denotes the derivative of g with respect to its indicated argument. An example of g ( t ) ~ is provided by ( t+a)-" where n > 2 and a > 0 . In this case G(s) = [s "-1 exp (-as)]/(n - 1)!.

3. A minimum principle in terms of displacements

A vector-valued function u*(x, t) will be called a kinematically admissible displacement field if (a) u*e C ~'a on/~ x [0, ~), (b) u* and li* are bounded a t % and (c) u* meets the displacement boundary values (2.4).

Theorem 1. Let th(x, t) and o-lj(x, t) be the solution to the field equations (2.1) and (2.2), boundary values (2.3) and (2.4) and initial conditions (2.5). Also let u~ e C x,2, o-ij c C 1,° on/~ x [0, ~) and assume that u~, ~- and crij are bounded at oo. Furthermore, let the given data satisfy the following regularity requirements:

0 . 0 (a) ui and ul are continuous on/~, (b) ~- is continuous on 0/~, × [0, w) and bounded at ~, (c) ~ is continuous on 0/~ x [0, ~), except possibly at points of discontinuities of

n where finite jump discontinuities are permissible, and is bounded at ~. Let U be the set of all kinematically admissible displacement fields. For each

u* e U and any given g e ~d, define the scalar-valued functional ~[u*; g] over U through

cb = ~ g(t + ~-) [pfi~(x, t)it.*, (x, ~') + C~jktuL(x , t)u*~.z(x, r)] dV dt d~"

- fo =/o =g(t+ ~'){fR pF~(x, t)u*(x, ~') dV + [ ~ ~(x, t)u*(x, ";)dA} dt dr

+ Io g(,)I. ,)Eu,.(x, 0)-uo(x)l-u,.(x, t) ti°(x)} dV dt

+ g(0)I, ou*(x, 0)[lu,*(x, 0)- u°(x)] dr. (3.1)

Then

dp[u*; g ] - ~ [ u ; g]--0 (3.2)

Moreover, the equality in (3.2) holds if, and only if, u*~ u.

Proof of Theorem 1. We first observe that each of the improper multiple integrals in (3.1) converges. Now let Au = u * - u. Then Au ~ C 1"1 on/~ x [0, ~) and satisfies the


displacement boundary condition

Au = 0 on OR~ x [0, ~). (3.3)

Let the left-hand side of (3.2) be denoted by A~. Then using (2.2), (2.5) and (3.1), we find

I/i L A~ = g(t+ ~') ON(x, t) A~(x, r) dVdtdr

+ g(t) p[~(x, t) au~(x, 0)-au~(x, t)tiO(x)] dVdt

+ fo~ i~g(t +'r){fR O',j(x, t) Ath,i(x, "r) dV

i ~ ~(x, t) AU4(X, "r) dA} dt d,

Io Io L - g(t+ 7) pF~(x, t) Au4(x, "r) dVdt dr

+½ Io~ i~g(t +'r) fR [o Aih.(x, t) Aih(x, ,)

+ Cijk~ Ath,j(x, t) Auk.t(x, 1-)] dVdtd'r

+ g(t) p A~(x, t) Au4(x, 0) dVdt

+ g(0)fR oA (x, o)Aui(x, 0, dV. (3.4)

The first integral in (3.4) is evaluated by reversing the order of space and time integrations, integrating by parts on ~', reversing the order of integration between t and T and then integrating by parts on t. 1 After once again changing the order of integration and combining this result with the second integral in (3.4), only

Io~ I~g( t+ ~')fR oil(x, t)Ath(x,,)dVdtd~ (3.5)

remains. The third integral in (3.4) is evaluated by applying the divergence theorem and equation (3.3), i.e.

- - i~ i~g( t+T)IR o',,,,(x, t)Au~(x,T)dVdtd'r. (3.6,

But the sum of the expressions (3.5), (3.6) and the fourth integral in (3.4) vanish

1 Reversal of the order of integration is justified since these integrals converge absolutely and have integrands that are continuous.

R. Reiss 40

identically by virtue of (2.!). Therefore, we have

A~ = A~[Au; g]

Io Io = g(t+~-) [pA~(x , t) A ri/- (x, -r)

+ C~jkt Au~,i(x, t) Auk, l(x, ~')] dV d td ' r

Iog(t)I. o A (x, 0) + d V dt

+~g(0) I R p Au~(x, 0) Au~(x, 0) dV. (3.7)

Now substitute equation (2.8) into (3.7) to explicitly eliminate g. After rearranging the orders of integration, one finally obtains

,x,~ = G(s ) {Os ~ zXv~(x, s) Avi(x, s)

+ C, jkl Av,,j(x, s) Avk,l(x, s)} dVds , (3.8)

where Av~(x, s) is the Laplace transform of Au~(x, t) which, by virtue of the hypothesis, exists for all s > 0. The right-hand side of (3.8) is dearly positive unless Ave(x, s) = 0 for all x ~/~ and all s c (0, oo), or equivalently, Au~ (x, t) =-- 0 on /~ × [0, 0o).

This theorem is the counterpart to the principle of minimum potential energy for elastostatics. It should also be remarked that the theorem remains valid under a variety of different sets of regularity conditions on the data. These regularity conditions are chosen so that the integrals in (3.1) converge and that interchanging of the order of integration, where necessary, is justified.

4. A minimum principle in terms of stresses

We shall say that o-~(x, t) are the Cartesian components of a dynamically admissible stress tensor o'*(x, t) if: (a) * C 1'1 * and "* O'ij E o n / ~ × [0, oo), (b) o-ij o'ii are bounded at 0% (c) t r l j - o-j~ o n / ~ × [0, o0) and (d) t r * n j - T~ meets the traction boundary condition (2.3).

Theorem 2. Let u~(x, t) and o-ii(x, t) be the solution to the field equations (2.l) and (2.7), boundary values (2.3) and (2.4) and initial conditions (2.5). Also let u~ E C 1'z, trlj E C m on /~.×[0, oo) and assume that u~, ~, o-ij and 6-~j are all bounded at oo. Suppose that the given data satisfy the following regularity requirements:

(a) u ° is continuously differentiable o n / ~ and ri o is continuous on /~, (b) ~ is continuous for all x c 0/~,, continuously differentiable for all t e [0, oo) and

bounded at % (c) ~/- is bounded at %


(d) ~ and Ti are continuous on ~/~ x [0, o0), except possibly at points of discontinuities of n where finite jump discontinuities are permissible, and are bounded at O0

Let E be the set of all dynamically admissible stress tensors. For each o-* ~ E and for any given g c ~, define the scalar-valued functional O[o-*; g] on ~ through

L°L ° L 0=½ g(t + ~-) -1 , [o ,~,.(x, t)~*k(x, ~)

. * + Kijk~ij (x, t) 6-*t(x, ~')] d V dt dr

+ I f fog( t + ~')IR Fi(x,t)o'~a(x,~')dVdtd';

+Iog(t)IR * [o-,ja(x, t)6°(x) + 6"*i(x, t)u °(x) ] d V dt

+ Iog(t)IR * K, jklo-ii(x, O)d'*l(x, t) d V dt

#[ f - g(t+ r) ~.(x, t)7"~(x, r) dA dta'r J O R u

-f :g(t) i [~-(x, t)T~i (x, O)+ ~.(x, O)~(x , t)] dA dt R~

g(0) I ~(x, 0)7~ (x, 0) dA. 2 R~

Then

OEcr*; g] -0[o ' ; g]-->0,

where the equality holds if, and only if, tr*---tr.

(4.1)

(4.2)

Proof of theorem 2. The proof of this theorem follows along the same lines as the first theorem. We first note that each of the integrals in (4.1) exist. The orders of integration will be changed only for those integrals whose integrands are continuous. Define Ao-~j by

A o-ij =-- or~ -- o-ij.

Then Ao-~j c C 1'1 and satisfies

ATi------Ao-,nj=0 on OR ×[0, oo).

Denote the left-hand side of (4.2) by AO

(4.3)

(4.4)

and use (2.1), (2.5), (2.7), (4.1) and

42 R. Reiss

(4.4) to obtain

I0 fo AO = g( t+ ~-) /~.(x, t)Atrii,j(x, "r) dVdtdz

+ g( t+ z) ~,j(x, t) A6"ij(x, ~') dVdtd~

fog(t)fR [~,j(X, t)Atr, j(x, 0)+ u°j(x)Ad'i,(x, + t)] dV dt

+ Io - ~ g ( t + ' r ) f R ~(x,t)A~(x,':)dAdtd~"

- ~ g ( t ) f [~.(x,t) ATi(x,O)+~.(x,O)A~(x,t)]dAdt R=

g(0)f R [u°j(x) Ao-~j(x, O) + Zo-,i,j(x , 0) u°(x)] + dV

g(O)[ ~(x, o) AT,(x, o) dA Jo R=

+ Kijk, Ad-,j(x, t)Ad'kt(x, r) d V dt dr

+ g(t) K~jk~ Acr~j(x, O) Ad'k~(x, t) dVdt

+½g(0)IR K, jkt Atr, j(x, O) Ao'kt(x, 0) dV. (4.5)

The sum of the first six integrals in (4.5) vanishes identically. This is easily shown by integrating the first integral by parts twice, once in t and once in ~-, and then applying the divergence theorem to the next two integrals. The sum of the seventh and eighth integrals in (4.5) vanishes by virtue of the divergence theorem. The remaining integrals depend solely on Ao-i i. After substituting equation (2.8) into the remaining terms in (4.5) and changing the order of integration, we obtain

AO = ½ f~ G(S) IR [P-l A1"q,i Al"ik, k + s2Kiikt A~'ii Arkt] dV ds,

where A~-~j(x, s), the Laplace transform of Ao-~(x, t), exists for all s > 0 by virtue of the hypothesis. The integrand is clearly positive definite for all x c /~, and all s ~ (0, oo). Therefore AO = 0 if, and only if, Ao-lj ~- 0 on /~ x [0, o0).

This theorem has no counterpart in elastostatics. Like Theorem 1, however, this theorem may also be established under a variety of regularity conditions on the data.


5. Derivation of the minimum iunctionals

We shall now take up the derivation of the functionals dp and @. In the interest of clarity, all regularity assumptions will be omitted.

As Gurtin [2] pointed out, the principal difficulty in establishing variational principles for initial-value problems centres around these initial conditions. Thus Gurtin [1] eliminated the initial conditions through the use of convolution integrals. The field equations therefore become a set of linear integro-differential equations which form the basis for variational principles. An alternative procedure involves the use of Laplace transforms.

We proceed formally and take the Laplace transform of the field equations (2.1) and (2.2) or (2.7), and boundary values (2.3) and (2.4). Thus we obtain a transformed boundary value problem, i.e.

r~,,, + o [ , = o [ s % ~ - s u ° - a°], (5.1)

r,s = C, sk,v(k,O or v(i,s~= Kisklrkl, (5.2)

subject to

~,jr~---~(x, s )= ~(x, s) x e OR~, (5.3)

v,(x, s)= ¢J~(x, s) x ~ ORu. (5.4)

Here, ris(x, s), vi(x, s), [~(x, s) and ~(x, s) are, respectively, the Laplace transforms of o'~s(x, t), u~(x, t), F~(x, t) and T~(x, t).

With the exception of the first term on the right-hand side of (5.1), equations (5.1) through (5.4) are formally the same as those governing the mixed problem in elastostatics. Thus, analogous to the potential energy functional of elastostatics, we introduce the functional ~b[v*(., s); s]

6[v*(', s); s]------~ C, sk,v*,,N'~,t d V + - f p , , d V

+ (5.5)

We shall now quote a minimum principle given previously by Benthien and Gurtin [111.

Theorem 3. Let v*(x, s) be any vector-valued function meeting the transformed displacement values (5.4), and let v(x, s) be the exact solution to (5.1) through (5.4). Then

~b[v*(-, s); s] ~ ~b[v(-, s); s], (5.6)

where the equality sign in (5.6) holds if, and only if,

v*(., s) = v(., s).

44 R. Reiss

Equation (5.6) may be regarded as the principle of minimum transformed potential energy.

The appropriate minimum functional in terms of stresses alone is

I IR - 1 . , saIR * * 0[~'*(', S); S]--~ O Tij.j~,k, kdV +-~ g, jdc~j'rkz dV

IR * f t~fhdA, (5.7) + ~-,j,j~ + su ° + a °] d V - s ~ d ~ , R u

Theorem 4. 2 Let ~'*(x, s) be any tensor valued function meeting the transformed traction values (5.3), and let ~" be the exact stress solution to (5.1) through (5.4). Then

0[T*(., s); s]--- 0[~(., s); s], (5.8)

where the equality sign in (5.8) holds if, and only if,

• *(., s )= ~(., s).

We have two minimum principles in the transformed domain. Our object now is to get back to the original domain/~ x [0, oo).

One approach is to take the inverse Laplace transform. Thus, by taking the inverse transforms of s a4~ and s-20 we are led to two of the variational principles by Gurtin (see equations (5.1) and (6.1) in Ref. [1]). In this approach, however, the minimum character of the transformed functionals ~b and 0 is lost; only their stationarity is preserved.

Therefore, we proceed by noting that multiplication of (5.6) and (5.8) by a non-negative function G(s) and then integration of the result from s = 0 to s = clearly preserves the inequalities.

Thus we form two new functionals

q~[u*; g]-- G(s)4[v*(', s); s] ds, (5.9)

and

I? O[r*; g]--- G(s)O[~r*(', s); s] ds. (5.10)

The integration is necessary to eliminate explicit dependence on s; the function G(s), with its associated properties, is introduced so that tile resulting improper integrals exist. Next, recall that

v*(., s) = u*(., t)e -~t dt (5.11)

and substitute this expression into (5.9), rearrange the orders of integration, use

2 This result was also obtained by Benthien and Gurtin [11].

M i n i m u m principles for l inear e las todynamics 45

(2.8) and integrate by parts; one will eventual ly arrive at (3.1). In a similar manner ,

equa t ion (5.10) is t r ans formed into equa t ion (4.1).

Acknowledgment

The au thor gratefully acknowledges Professor M. E. Gur t in ' s suggestions for the

p repa ra t ion of this manuscript .

REFERENCES

[1] Gurtin, M. E., Variational principles for linear elastodynamics. Archive for Rational Mechanics and Analysis 16 (1964) 34-50.

[2] Gurtin, M. E., Variational principles for linear initial-value problems. Quarterly of Applied Mathematics 22 (1964) 252-256.

[3] Leitman, M. J., Variational principles in the linear dynamic theory of viscoelasticity. Quarterly of Applied Mathematics 24 (1966) 37-46.

[4] Rafalski, P., Variational principle for the problem of dynamic thermal stresses in a linear visco- elastic body. Bulletin De L'Acadamie Polonaise Des Sciences, Serie des sciences techniques 16 (1968) 159-164 (see also pp. 205-210).

[5] Iesan, D., Principles variationnels dans la th6orie de la thermorlastieite couplre. Analele St. Ale Univ. Al. Cuza Din Iasi 12 (1966) 439-456.

[6] Rafalski, P., A variational principle for the coupled thermoelastie problem. International Journal of Engineering Science 6 (1968) 465-471.

[7] Nickell, R. E. and Sackman, J. L., Variational principles for linear coupled thermoelasticity. Quarterly of Applied Mathematics 26 (1968) 11-26-.

[8] Kline, K. A. and De Silva, C. N., Variational principles for linear coupled thermoelasticity with microstructure. International Journal of Solids and Structures 7 (1971) 129-142.

[9] Bytner, S., Variational principle for linear coupled electromagnetoelasticity of conductors. Bulletin De L'Academie Polonaise Des Sciences, Serie de sciences techniques 21 (1973) 491-497.

[10] Bytner, S., Variational principle for linear coupled thermo-electromagnetoelastieity of conductors. Bulletin De L'Academie Polonaise Des Sciences, Serie des sciences techniques 23 (1974) 207-212.

[11] Benthien, G. and Gttrtin, M. E., A principle of minimum transformed energy in linear elastodynamics. Journal of Applied Mechanics 37 (1970) 1147-1149.

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