General Theorems and Formulations

General Theo Chapter VIIand Formulat As the title of this chapter indicates, it is comprised of a collection of results which have general applicability and usefulness. For the most part these results are extensions to viscoelasticity of some of the well-known theorems of elasticity. For example the classical variational theorems and minimum theorems of elasticity will be shown to possess several different types of generalizations to viscoelasticity. The first item of study here is the establishment of the uniqueness of solution of the coupled thermoviscoelastic boundary value problem. 7.1. U N I Q U E N E S S O F S O L U T I O N O F C O U P L E D THERMOVISCOELASTIC BOUNDARY VALUE

PROBLEM

Now we give a general proof which establishes the uniqueness of solution for linear thermoviscoelastic problems. This proof of the uniqueness of solution is a generalization to the nonisothermal case of the isothermal uniqueness theorem given by Onat and Breuer [7.1]. The relevant linear equations from Chapter 3, which govern the dynamic, nonisothermal, anisotropic, coupled viscoelasticity theory, are given by (3.45)(3.52), where the equations of motion (3.45), with inertia terms included, are taken. The Laplace transform of these relations, with zero initial conditions for t < 0 are assumed to exist. These relations are given by (3.53)-(3.57), and they are repeated here for convenience. The equations of motion, the strain displacement relations, and the stress constitutive relations are respectively given by ^

+

^

-

^

_

*u = i( u + i,i) 9ifi ü

and

δα = sGijkihi —s

(7.1)

ü

(7.2 (7.3)

where s is the transform variable and θ(χ{ , t) is the infinitesimal temperature deviation from the base temperature T0 . The heat conduction equation is (kM

BM =

ΛΒ0 +

(7.4)

where ki} and the relaxation functions m(t) and ψα(ί) are mechanical properties of the material. The boundary conditions are given by δί3ηό = Sj üi = Ai 9=ô

on Β σ , on Bu, on B1

(7.5) (7.6) (7.7) 229

230

V I I . GENERAL THEOREMS A N D F O R M U L A T I O N S

and on

=0

(7.8)

B2

where Βσ , Bu , Βλ , and # 2 are the appropriate subregions of the boundary, assumed to be constant with time. The uniqueness theorem appropriate to thermoviscoelastic boundary value problems is now stated. The isotropic coupled thermoviscoelastic boundary value problem, governed by field equations and boundary conditions (3.45)-(3.52), possesses a unique solution, provided the Laplace transform of all field variables is assumed to exist with transformed field equations and boundary conditions (7.1 )—(7.8), and provided the initial values of the relaxation functions are positive definite and the thermal conductivity is positive semidefinite, as Gim(0)

yijYkl

> 0,

m(0) > 0,

kijYi7j

> 0

where yi5 is any symmetric second order tensor. To prove this theorem, we assume that two separate solutions of the field equations, boundary conditions, and initial conditions exist, with the field variables given by

ij IJ ij I) (1)

(2)

σ«>

σ«>

E

Q(2)

0(1)

where these are assumed to be continuous functions of time and of the spatial ά coordinates. We designate the difference solution, w/, ef^, σ?· , and θ , by

uftXi,

{

0 = u l\Xi

, t) - uf(Xi,

t),

> 0 = «S'fa > t) - 4F{ , t),

(7.9)

XI

4(*È , t) = σ§\Χί and

(2)

, t) - σ (χ{ , t)

e*( , t) ew( , o^( , /) Xi

=

Xi

Ο

Xi

-

ά

We note that the difference solution, w/(x 2 , *), €^(χ{, t), σ^(χ{, ί), and θ (χ{, t), satisfies homogeneous boundary conditions. Because of this, we can write the following relation: f &φβ*άα

J

Β

= 0

(7.10)

7.1.

UNIQUENESS OF THERMOVISCOELASTIC B O U N D A R Y VALUE PROBLEM

231

relative to the Laplace transformed variables. We apply the divergence theorem to (7.10) and use the form of (7.1) and (7.2) appropriate to the difference solution to get

Jy J*

[ps%%<

+ 4 4 ] dv

= 0.

(7.11)

We now use the difference form of (7.3) in (7.11) to obtain

Jy f

+

sGm*i&i - stpidfi*]

The last term in (7.12) will be eliminated by using the heat conduction equad tion (7.4) in difference form and multiplied by B : (kJT,)

m%

= s*m$*Y +

Steffi.

(7.13)

It is necessary to express the first term in (7.13) in an alternate form. To do this we note the following identity:

k S B% da = *Β d

f

d

i3

0

since 9 = 0 on B1 and kiß inj = 0 on B2 , where B1 and B2 are complementary parts of the boundary B. By applying the divergence theorem to (7.14), we find that d

d

d

Now we solve for e ß There results

j

v

[psHiW

from (7.13), using (7.15), and substitute it into (7.12).

+ sGmli%ilx

+^

θψ,ι

+

sm(e"f]

dv =

0.

We consider real values of s such that s > s0 , where s0 is the solution's furthest singularity to the right on the real axis. We recall that l i m sGim(s)

=

Gim(0) (7.17)

and lim sfh(s) = m(0).

Now it is required that the initial values of the relaxation functions be positive definite; thus, Gim(0) YijYki > 0 (7.18) m(0) > 0.

(7.16)

232

V I I . GENERAL THEOREMS A N D F O R M U L A T I O N S

It then follows that for s sufficiently large, sGijkl(s) y^y^i > 0 and sm(s) > 0. If it is also required that kiiyiyi > 0, and if s is sufficiently large, then Eq. (7.16) is composed of all nonnegative terms, which to be satisfied requires d

Uf = B = 0

for

s > s0

and sufficiently large.

(7.19)

A theorem from Doetsch [7.2] states that if the Laplace transform of a continuous function is zero on an infinite set of points on the real axis, which constitute an arithmetic progression, the function itself is identically zero. Applying this D to the present situation reveals that the difference functions uf and 6 must be identically zero, as O (7.20) UA = ßD = Since the difference solution is identically zero, the solution of the coupled thermoviscoelastic boundary value problem must be unique. This sufficiency proof follows from the assumptions k^y^yj ^ 0, m(0) > 0, and G z W(0)y^y f cj > 0. It is remarkable that only the initial values of the relaxation functions are involved in the proof. The use of the Laplace transformation to establish the uniqueness theorem here is, in a sense, artificial. Although integral transform techniques are a highly valuable tool in solving boundary value problems, their use in establishing general theorems is less appealing because of the implied restrictions which must accompany their use. The Laplace transform is used in this section only because of the simplicity it affords in this particular proof; it is not an essential part of a uniqueness proof. The isothermal uniqueness theorem given in Section 2.2 did not use an integral transform method. References were given in Section 2.2 to several different isothermal uniqueness theorems. Sternberg and Gurtin [7.3] have given a nonisothermal uniqueness theorem, applicable to different conditions from those considered here. 7.2. R E P R E S E N T A T I O N I N

TERMS

OF DISPLACEMENTFUNCTIONS

In the theory of elasticity, two particularly useful procedures for uncoupling the equations of equilibrium are those of the Galerkin vector and the PapkovitchNeuber stress function. We turn now to the extension to viscoelasticity of these elasticity uncoupling procedures. This extension depends upon the use of the Stieltjes convolution notation introduced in Section 1.2. We recall that the elasticity formulation with no body forces has the displacement form of the equations of equilibrium as V u + ΓΖΓΝ 2

W ·u = 0

(7.21)

7.2.

REPRESENTATION I N TERMS OF D I S P L A C E M E N T F U N C T I O N S

233

where U is the displacement vector, V is the Laplacian operator, and V · U is the scalar product. The Galerkin vector solution of these equations is given by U = (l/2/x)[2(l - v) V*g - VV · g]

(7.22)

where 4

V g = 0.

(7.23)

Thus, the solution of the elastic equations of equilibrium is reduced to the problem of finding a biharmonic vector function g. An alternative procedure is that of the Papkovitch-Neuber stress function, whereby the displacement vector is taken as U = (l/2/x)[V( 9 + R · Ψ ) - 4(1 - ν ) Ψ ]

(7.24)

where R is the position vector from the origin of the coordinate system and 2

V V = ν Ψ = 0.

(7.25)

In this case, the integration of the equations of equilibrium is reduced to the problem of finding four harmonic functions. For viscoelasticity, the equations of (quasi-static) equilibrium, in a form comparable to (7.21), are given by 2

V U * dGx + iVV · U * d(G1 + 2G2) = 0

(7.26)

where the Stieltjes convolution notation of Section 1.2 is used. This form is easily verified, using aijtj = 0 along with (2.3), G^t) = 2μ(ί), G2(t) — 3k(t)y and the strain displacement relations. The generalization of the elastic Galerkin's vector to viscoelasticity is given by 2

U = 2 V g * d(2G1 + G2) - VV - g * d(Gx + 2G2)

(7.27)

where V% = 0.

(7.28)

To prove that this representation satisfies the equations of equilibrium, (7.27) is directly substituted into (7.26). In order to do this, we use (7.27) to obtain the following two forms: 2

2

4

V U = 2(V g) * d(2G1 + G2) - V [VV · g * d(G1 + 2G 2)]

(7.29)

and 2

VV · U = VV · [2 V g * d(2G1 + G2) - VV · g * d(Gx + 2G 2)].

(7.30)

Now we note that the first term on the right-hand side of (7.29) vanishes, owing to (7.28). Then by substituting (7.29) and (7.30) into (7.26) we get the form 2

- V [ V V · g * d(Gx + 2G2)] * dGx 2 + iVV · [2 V g * d(2Gx + G2)] * d(Gx + 2G 2) - iVV · [VV · g * d(G1 + 2G 2)] * d{Gx + 2G 2) = 0.

(7.31)

234

V I I . GENERAL THEOREMS A N D F O R M U L A T I O N S 2

2

We use the vector analysis identity, V V div = V div V = V div V div, and the associativity and distributivity properties of the Stieltjes convolutions (see Section 1.2) in (7.31). This reduces (7.31) to the simple form 2

V V div g * [-d(G1

+ 2G2) * dG1 + d(Gx) * d(Gx + 2G 2)] = 0. (7.32)

Finally, we use the commutivity property of Stieltjes convolutions to show that (7.32) is satisfied, and the proof is complete. The generalization of the elastic Papkovich-Neuber stress function to viscoelasticity is given by u = ν(φ + r · ψ) * d(G1 + 2G2) - 4ψ * d(2G1 + G2)

(7.33)

where 2

V V = ν ψ = 0.

(7.34)

The proof of this representation follows, as with the Galerkin vector, by substituting (7.33) into (7.26), using the properties of the Stieltjes convolutions, and using (7.34). The comparable representation forms, when the body forces are nonzero, are given in Gurtin and Sternberg [7.4]. These two Stieltjes convolutions displacement representations in viscoelasticity reduce the integration problem to simple forms. An alternative procedure to these methods would be to take an integral transform of the governing viscoelastic field equations and boundary conditions and apply the elasticity-type Galerkin vector representation, or the Papkovich-Neuber representation, directly in the transform plane. The uncoupling procedure would be formally the same as in elasticity; and, indeed, an elasticity solution obtained by this means could directly be converted to the transformed viscoelastic solution. This latter procedure would, in general, be easier to apply than the Stieltjes convolutions forms of (7.27) and (7.33). But there is an important distinction which must be drawn between these two alternative methods of uncoupling the equations of equilibrium. The Stieltjes convolution forms could be applied in problems in which integral transform methods are not applicable. For example, the contact problems discussed in Section 5.1 are problems for which integral transform methods do not apply. The problem of the indentation of a viscoelastic half space by a curved indentor was discussed in Section 5.1. For this problem the complete and general analysis was given by Graham [7.5], using the Stieltjes convolution form of the Papkovich-Neuber stress function just discussed. 7.3.

RECIPROCAL THEOREM

A viscoelactic reciprocal theorem will be proved which is analogous to the one in elasticity theory. In doing this we find it convenient to follow the treatment

7.3.

RECIPROCAL T H E O R E M

235

of Gurtin and Sternberg [7.4] and utilize the formalized notation for Stieltjes convolutions, as given in Section 2.2. The viscoelastic reciprocal theorem is now stated, and the usual notation is used whereby , σ{, and ut refer to the components of body forces, boundary stress vector components, and displacements respectively. An isotropic viscoelastic body, when subjected to two different states of loading with corresponding body forces, surface stresses, and displacements, F{, σ{, u{ and F/, σ/, u/, respectively, has afield variable solution which satisfies the relationship [σ, *

du/] da -F-

[Fi * du/] dv =

[σ/ * du/\ da +

v

"

B

v

J

[F/ * du/\ dv. (7.35)

The proof of this theorem begins by applying the divergence theorem to and using the equations of equilibrium to write the left-hand side of (7.35), symbolized by L, as

(7.35)

L =

We write

Jβ

[σ, * du/]

f

(7.36)

da +

f

Jy

[Ft * du/]

f

dv =

Jy

[α,,· * & y dv.

(7.36)

in deviatoric and dilatational component form to get L =

\ [Sij J y

Now we use the stress strain relations

*

(1.22)

de'u

in

Jy

L = \

* den * àéiS + \G2

(7.37) *

+

%akk * dej3] dv.

(7.37)

to obtain

dekk

*

(7.38)

dc^] dv.

We use the associativity and commutivity properties of Stieltjes convolutions (Section 1.2) to obtain the forms

0 * de * de'a = G * d(e * de'y) = G * die'u * de ) de\i de . iS

Ύ

X = Gx

1

{j

if

*

*

(7.39)

ti

Relation (7.39) and a similar one for the second term in (7.38) then gives (7.38) as

Jy

L = f [Gi * de'u * deu + | G 2 * d€kk * d€„] dv or

L = f Κ , · *
Jy

The comparison of (7.36) and

(7.40)

completes the proof of

(7.40) (7.35).

236

V I I . GENERAL THEOREMS A N D F O R M U L A T I O N S

An interesting special case of this theorem results when the two loading systems have a form involving a separation of the spatial and time portions of the variable. In this case, the two systems of loading terms have the forms: Unprimed System Fi(*i,t)

=Fi{xt)g(t)

Ui{xiyt)

=Äi(xi)g{t)

> *H =

σ«(*ι

on

Bu

*i( i)g(t) x

(7.41)

on

Ba

and Primed System F^x^t)

=

F/(xi)g(t)

ui(xiyt)=Â/(xi)g(t)

°u{Xi

on Bu

> *K =

(7.42)

s/(Xi)g(t)

on BQ .

It must be noted that, even though these forms involve a separation of variables, the resulting solutions of the viscoelastic boundary value problems would not in general have a separation of variables form. As we argued in Section 2.3, for a separation of variables solution to exist, the displacement and stress boundary conditions must have a related but different time variation, which is in contrast to the situation here. The viscoelastic reciprocal theorem (7.35) appropriate to these two loading states asserts that

f [a * dg] Δ- da + ί Si[Ui * dg] da + ί P {u( * dg] dv t

t

=

Jf

B

[ σ / * dg] At da +

Jf

B

V

* dg] da + J f

F/[Ui

* dg] dv

(7.43)

u 0 where the commutivity of the Stieltjes convolutions has been used. We invert the order of space and time integration in (7.43) to obtain the form [ f

lJ

aj/

B -

u Jf

da +

f

J

B

Β

5,11/

da -

σ Jf

da +

f

J

Β

S{ut

FiU/ dv

V

da -

JV f

Ρ<\

dv] J* dg = 0.

(7.44)

u σ For (7.44) to be satisfied for all values of time, it is necessary that the term in the brackets vanish. When that form is multiplied by g(t)y it gives

Jβ

σ-u- da +

Jy

F-μΙ

dv =

Jβ da +

Jy

J P / w , dv.

(7.45)

V A R I A T I O N A L THEOREMS

7.4.

237

This special case of the viscoelastic reciprocal theorem, applicable only when (7.41) and (7.42) are appropriate, has exactly the same form as the elastic reciprocal theorem. A particularly interesting application of the viscoelastic reciprocal theorem is found in deriving a formula for the volume change of a body, analogous to the comparable elasticity formula, given for example, by Sokolnikoff [7.6]. This is given as a problem at the end of the chapter. Sternberg [7.7] has shown how the viscoelastic reciprocal theorem and several other general theorems can be adapted to nonisothermal conditions through the use of the body force analogy. 7.4. V A R I A T I O N A L T H E O R E M S

There have been several derivations of variational theorems in quasi-static viscoelasticity; for example, Onat [7.8], Schapery [7.9], Biot [7.10], Christensen [7.11], and Gurtin [7.12]. Probably the derivation by Gurtin is the most general treatment. In his approach, the Stieltjes convolution notation is used in a manner similar to that just employed in the previous two sections. The variational theorems derived by Christensen, however, are the only ones for which associated minimum theorems have been established. Accordingly, both of these types of theorems are discussed here, for quasi-static conditions. Also, undisturbed conditions are assumed for t < 0. The relations of Section 2.1 which define the quasi-static viscoelastic boundary value problem are rewritten here for convenience as

*a = i( ij + i,i)> α= dr, G u

u

(7-4

σ

J

ο

f

Gim(t

— τ)r ^C (

τ

)

{j = Gim

or €

a = j

T

Jmiif — ) —

= Sf

r

>

€ =

a

Jim * dakl,

(7.48) (7.49)

°iu +Fi=0, σφι

d

on

Βσ,

(7.50)

on

Bu

(7.51)

and

where S{ and At are the prescribed stresses and displacements on the boundary, and the general anisotropic case is being considered. In the variational theorems to be formulated here we will assume that the stresses, strains, and displace-

*

de

k

238

V I I . GENERAL THEOREMS A N D F O R M U L A T I O N S

merits are continuous and continuously differentiable. Under certain conditions this restriction can be relaxed somewhat, see for example Ref. [7.12]. The procedure now is to establish a functional F> for which Eqs. (7.46)-(7.51) imply that its variation 8F vanishes; that is, 8F = 0. Exactly what is meant by the variation of a functional is now defined. We take, for example, a functional F of a single history parameter e(t — s)> as oo

(7.52)

F = φ (φ - s), e(t)) s=0

with the usual dependence upon the current value e(t). We assume that this functional, for another history e(t — s) + [8e(t — s)], can be expanded about that for the history e(t — s) to give

φ (φ -S)+ δφ - s), φ)) = φ (φ - ί), φ + ±δ*φ(φ - s), φ) I δφ bo

oo

oo

s=0

+ - .

(7.53) 1

Such an expansion will be assumed to exist for the functionals of interest here. It may be noted that (7.53) is an expansion for functionals which is completely oo

analogous to the Taylor series expansion for functions. In (7.53), δψ ( ) symbolizes a functional of histories e(t — s) and Se(t — s) which is linear in Se(t — s) oo 2

and is called the first order Fréchet differential. Similarly, 8 φ ( ) is the second order Fréchet differential, and is quadratic in the history 8e(t — s). The higher order functionals are correspondingly defined. We define the first variation of the functional F, or simply 8F, as

8FUOi = / Γ ψl (e(t -s)+ 8e(t - S)OL, 6(0) + Φ «t - s), (t) + 8e{t) €

=0

s=0

Sa

(7.54) where α is a real constant. Now we perform the operation indicated in (7.54) upon the functional in (7.52), using (7.53), to show that 1 Sufficient

conditions

for the existence

of the

expansion

(7.53)

are discussed

in

Chapter 8.

8F = 8φ (φ - s), φ) I δφ - s)) + JL, φ (φ - s), φ)) δφ). s-0 \ ) s=0

l

V

€

(7.55)

=0

7.4.

VARIATIONAL THEOREMS

239

Thus, the first variation of the functional is nothing more than the first order Fréchet differential in the expansion of the functional of the history e(t — s) + 8e(t — s) about that for the history e(t — s), plus the partial derivative term. Functionals having more than one tensor valued history argument, as will be considered here, have variations defined similar to that in (7.54) and (7.55). We define the functional Fx by F

G

i=

\ [i mi

*

d€

Jy +

f

[at * dad

* de

if

kl

da +

f

—

[(σ, -

a

i$

* de

i3

—

{a + F ) * iitj

{

S{) * dut] da

dut] dv

(7.56)

where Fi, A{;, and St are given quantities, oi = σί}η3· on the boundary, and the coordinate dependence of all variables is understood. We are now in a position to state the first variational theorem as: Thefirstvariation

SF

F

,

(7.56),

vanishes if and on ± of the functional x defined by if thefieldequations and boundary conditions (7.46)—(7.51 ) are satisfied. To prove this theorem we begin by taking the variations in the histories

«i(T)» €«W»

a n d σ

τ

α( )

as b e i n

g g

i v en

b

y

« « ( τ ) + δ , 0( τ ) α ,

and

σ

τ

α( )

+

(7.57)

τδσα

«( )

where α is a real number, and δκ,^δ), δ € # ( τ ) , and δ σ ί ?· ( τ ) are the arbitrary, but sufficiently smooth, changes in the histories. The resulting first variation of the functional , from (7.56), is then given by

δ

^ι =Jί y[Gijki * d€ * dSe — a * dhc — (G + F^ * dhui — Sa * dut kl

i}

ijtj

+

f j

[ δ σ , * dAi] da + Bu

tj

ijtj

ί

[(*,· Ba

S{) *
(7.58)

J

de =

where, in the first term, the commutativity property G i m * d8ei3 * kl G mi * dekl * d&€i}- and the symmetry relation Gijkl = GkUj have both been used. The term $B [δσ, * du{] da is subtracted from the integral over Bu and added

240

V I I . GENERAL THEOREMS A N D F O R M U L A T I O N S

to the last term in the Ba integral. By using the divergence theorem, (7.58) can be rearranged into the form δ

^ ι = ί [{Gim * d€kl - σίό) * d8eiS - ( σ ^ · + Ft) * dSu{

Jy

dha ] dhUi]

— (€ij — \uitj — \ujti) + f

[fo - St) *

*

i0

dv

da + f

[(Ai -

The satisfaction of the field equations and boundary conditions (7.46)—(7.51) then requires that the first variation of Fx vanish, giving *FX = 0.

(7.60

It may also be argued that, for the variations 8u{ , 8c i :) , and δσ# taken as arbitrarily specified, the variational equation 8FX = 0 gives as Euler equations the field equations and boundary conditions (7.46)—(7.51). This completes the proof of the first variational theorem. The variational theorem just given is a generalization to viscoelasticity of the Hu-Washizu variational theorem in elasticity. It is consequently expected that the Hellinger-Reissner elasticity variational theorem could be generalized to account for viscoelastic effects. This is now done. We taken the functional F2 as defined by

2 J= y[hJmi * a * ki

F

d(7

d(J

— \Gij * d(uiti + ujti) + Fi * du^ dv + f J

[ai * d(Ui - A^] da + f

Bu

[Si* dui] da

Ba

J

where now the variations in ut and σίό are taken to be arbitrary. The second variational theorem is stated as: The first variation SF2 of the functional F2 , defined by (7.61), vanishes if and and if conditions (7.48)—(7.51) are satisfied, with € the field equations u and boundary iven

ij =

M

i,i) g Ui(r) + SUi(r)oc σ

τ

τδσα

α( ) + «( ) i( f.j + To effect the proof we take variations in the histories wt(r) and σ#(τ) as being given by

7.4.

V A R I A T I O N A L THEOREMS

241

where α is a real number and δ^(τ) and 8oi3(r) are the arbitrary changes in the histories. The functional F2 has the resulting first variation given by F

d(J

& 2 = ί [Jim * ki * dhai3 — \σί3 * d(8uitj + 8u3i)

Jy Bu

— \8σί3 * d(uu + ujti) + Ft * tfôifj dv + f [δσ< * - 4 ) + σ < * d8ui] da + f

J

J

Β

[Si * dhut] da

(7.62)

σ

where, in the first term, the commutativity property of Stieltjes convolutions has been used, along with ] i m = Jklij . The term \ B σχ• * d Su, is added and subtracted respectively from the last two terms 8F.in (7.62). By using the divergence theorem we can put (7.62) into the form 2 -

{[-i(Uij +

1 Jν B f

+ (aijtj

uSwi) + Jim + Fi) * dSui} dv

*

dakl] * d8oi3

ν

+ Jf

[(a, u

B

* «ίδσ,] ώζ - Jf

[(σ, - S{) * dSut] da. (7.63) a

We see that the satisfaction of the field equations (7.48) and (7.49) with € i3 = u u i( ij + j,i) along with the boundary conditions (7.50) and (7.51), gives the first variation of F2 as vanishing; thus, δ^ 2 - 0.

(7.64)

It may also be shown that δ ^ 2 = 0 gives (7.48)—(7.51 ) as Euler equations, presuming ei3 = \{ui3 + u3 t). This completes the proof of the second variational theorem. A variational theorem can be stated which is related to the first theorem given here but which has the simplified functional, given by ί

J

d€

[\G%m * ij * d€kl — Fi* dui] dv — |

V

B

* dut) da

J u a where. €i3 = \(uitj + In this case only the displacements are subject to the requirement that the displacement boundary conditions (7.51) are satisfied. Such a variational theorem is an extension to viscoelasticity of the theorem of stationary potential energy in elasticity. Similarly, a variational

j,i)-

242

V I I . GENERAL THEOREMS A N D F O R M U L A T I O N S

theorem can be stated which is related to the second theorem given here but which has a simplified functional given by

In this case, only the stresses are varied, subject only to the requirement that the equations of equilibrium (7.49) and the stress boundary conditions (7.50) are satisfied. This type of variational theorem is an extension to viscoelasticity, of the theorem of stationary complementary energy in elasticity. These restricted variational theorems are proved in Ref. [7.12]. The first and second variational theorems just proved provide the rigorous means of extending to viscoelasticity the approximate elasticity theories which are derived by elasticity variational theorems. For example, the Reissner elastic plate theory would have a viscoelastic counterpart which would be derived in the same manner as in the elastic case, but would use the second variational theorem given here. Actually, this type of formal use of these variational theorems is seldom made, since it is usually much easier to appeal to integral transform methods to directly obtain an approximate viscoelastic theory from the corresponding approximate elasticity theory. That is, elasticity variational theorems can be reinterpreted as viscoelastic variational theorems involving integral transformed field variables; and, thereby, the Euler equations obtained from a particular elasticity variational theorem can be reinterpreted as integral transformed viscoelastic Euler equations. It is important to note, however, that the direct use of the viscoelastic variational theorems is a rigorous and general means of establishing theories of mechanical behavior, while proceeding through the use of integral transform methods combined with the corresponding elasticity theories is necessarily a much less rigorous procedure. The variational theorems just given are from Gurtin [7.12]. Now we derive two different types of variational theorems, not using Stieltjes convolutions. These second two variational theorems are restricted to applications where separation of the time and spatial variables are allowed. Sufficient conditions to allow a separation of variables are outlined in Section 2.3. The corresponding restrictions on the mechanical properties are that the material be isotropic and that viscoelastic Poisson's ratio be a real constant. In the relations (7.46)—(7.51 ), governing the viscoelastic boundary value problems, the anisotropic constitutive relations (7.47) and (7.48) must be replaced by the isotropic forms as (7.65) and (7.66)

7.4.

and

V A R I A T I O N A L THEOREMS

e^^ifjit-rf^-dr.

243

(7.68)

We define the functional Fz by

where pA(t) is the isothermal form of the stored energy, and A(t) is the corresponding rate of dissipation of energy. At this point, it is worth noting that if the medium under consideration were elastic, the time integrations in (7.69) could be performed, and the resulting functional would be that of the Hu-Washizu variational theorem in elasticity, as was the case with Fx . The forms of pA(t) and A(t) are taken from the isothermal specialization of (3.22) and (3.25), from the thermodynamical derivation of Section 3.1. These are

and

where the two possible independent arguments of the relaxation function are taken in the additive form, and the viscoelastic Poisson's ratio is taken as a real constant.

244

V I I . GENERAL THEOREMS A N D F O R M U L A T I O N S

We substitute (7.70) and (7.71) into (7.69) to get the form of the functional F3 as

-2f f's, ^l*^ W

We take the first variation ofF 3 in (7.72) and use the divergence theorem to put it into the form

J γJ

ί { [ - € « + K«f j + *,·.<)] δσ^ ο

SF3 = 2 f

Π

_|_ V\

ΛΤ

2v) ù Η)Jί/τ < of Sût drda-2

kii F 3(1 ο^ δ T 7 7 + 4 (— ° JίW M( — — ?)^o( /) dr] + + ο +

- 2

ί

f

(S, -

T

δ

"~ ^ * " ^ ^ ί μ(τJ — η) 8έ„(η) άη ο

σ,)

ί

f

δσ,-ίώί - 4 )


where the time arguments, in addition to the coordinate dependence, are now understood, and the superimposed dot is differentiation with respect to the time variable. The conditions under which viscoelastic boundary value problems admit a separation of variables solution are discussed in Section 2.3. Such conditions will be assumed here, so that the field variables can be taken in the form

> 0 = ««•*(*•)(0> w

and

, ί) =

u(t)

y

{1.1A)

7.4.

V A R I A T I O N A L THEOREMS

245

Using (7.74) in the terms in (7.73) involving integrations with respect to τ, and taking variations only with respect to the spatial part of the field variables, we get

F

~ - 2

+ i) f

f'

dr dv

(St - ot) hùi drda-2

f

f

δσ,(ζ*, -

At) dr da.

(7.75)

We see that the separation of variables condition provides the key which allows the integrals in (7.73) to be combined into the simplified form in (7.75). Equations (7.46), ( 7 . 4 9 ) - ( 7 . 5 1 ) , (7.65), and (7.66) show that the first variation of F3 vanishes. That is, when these relations are satisfied, SF3 It follows from

= 0.

(7.76)

that the variational requirement, 8F3 = and (7.66) as Euler equations. These results are now stated as the third variational theorem: (7.69)

0,

gives

(7.46),

(7.49)-(7.51), (7.65),

For viscoelastic boundary value problems which admit a separation of variables form of solution^ the first variation 8F3 of the functional F3 , defined by (7.69), (7.70), and (7.71), vanishes if and only if thefieldequations and boundary conditions (7.46), (7.49)—(7.51), (7.65), and (7.66) are satisfied. Variations are allowed only in the spatial parts of thefieldvariables. A fourth type of variational theorem is derived, which, as with the second theorem, corresponds to the Hellinger-Reissner variational theorem of elasticity. In this case, we define the functional F 4 by

+

2

3W{T)

δτ

Γ(τ)

dr dv

(7.77)

246

V I I . GENERAL THEOREMS A N D F O R M U L A T I O N S

where

and

The thermodynamical significance of pW(t) and Γ(ί) has been established in Section 3.7. We substitute (7.78) and (7.79) into (7.77) to get

+ 2 ί

f' ^

[Ui(r)

-

(7.80)

At(r)] dr da.

By taking the first variation of F4 from theorem, we get

(7.80),

and using the divergence

+ ï^ii f 7( - *?) Sii άη + Ji ί /(τ - 7?) δί„
J

if

ο

*Ό

δ> ί

+

/(τ

12(Γ+?) "· J0 Γ

(1 — 2ι>)

+ 2

ά

ί Γ* (^ - σ,·)

JΒ J0 σ

άη

~ ^»

)

δκ,

- J) J Bu * 0

rfr^+2Î Ç 8ai(Ui

f

A-

da

(7.8

7.4.

VARIATIONAL

247

THEOREMS

where, as before, the time arguments are understood and the superimposed dot is differentiation with respect to the time variable. As with the variational theorem for F3 , conditions are assumed such that a separation of variables solution is admitted in the form of (7.74). By using this in the terms in (7.81), which involve integrations with respect to r, and admitting variations only in the spatial part of the field variables, we obtain

+ +

σ(τ) [- ' + ' + f U + i f J(r - η) S {Ui j

Uj i]

8

kik

&?„(*,) σ(τ) [ -

+ 2

f

f

!ψ

+

^

(

1~ +

2

^

(St - ai) SUi drda+2

J Ba J 0

J* / ( τ -

ί

η) σ „ ( * , , η) άη] j dr

Ç 8ai{Ui

-

dv

dr da.

(7.82)

^ Bu J 0

Equations (7.46), (7.49)-(7.51), (7.67), and (7.68) show that the first variation of F\ must vanish. Thus, 8F4 = 0.

(7.83)

It follows from (7.77) that the variational equation 8F4 = 0 gives as Euler equations (7.49)-(7.51) and \{uu

8

+ uu) - f uKk = i

J(t - r) ^

dr

(7.84)

and

These results are now combined in the statement of the fourth variational theorem: For viscoelastic boundary value problems which admit a separation of variables form of solution, thefirstvariation hF± of the functionalF\ , defined by (7.77), (7.78), and (7.79), vanishes if and only if the field equations and boundary conditions (7.49)(7.51), (7.84), and (7.85) are satisfied. Variations are allowed only in the spatial parts of thefieldvariables. Although these last two variational theorems are more restrictive than the first two, the conditions for their applicability are revealing. That is, through the employment of the separation of variables technique these variational theorems are found as straightforward extensions of the corresponding elasticity theorems, but the revealing aspect of this is that it can only occur under the

248

V I I . GENERAL THEOREMS A N D F O R M U L A T I O N S

conditions which permit a separation of variables solution. Furthermore, this close relationship between elastic and viscoelastic variational theorems under separation of variables conditions suggests that under the same conditions elastic and viscoelastic minimum theorems might be similarly related. This possibility is next investigated, and it is indeed found that viscoelastic minimum theorems can be established. However, as will be seen, these viscoelastic minimum theorems are not the simple generalizations of the corresponding elastic theorems as occurred in the case of variational theorems.

7.5.

MINIMUM

THEOREMS

The variational theorems just considered establish the stationary value of certain functionals. Under certain restrictive conditions, the much stronger result will be obtained whereby particular functionals can be shown to not only have a stationary character, but also a minimum character. The minimum theorems to be proved here correspond to the last two variational theorems of the preceding section. The viscoelastic boundary value problem is again posed by the relations of the preceding section, namely (7.46), (7.49)-(7.51), and (7.65)-(7.68). As in the previous variational theorems, it is assumed that the stresses, strains, and displacements are continuous and continuously differentiable. The first minimum theorem is associated with the theorem of minimum potential energy in elasticity and in fact reduces to it in the case of elasticity. We define the functional Mx as dr

dv (7.86)

where pA(t) and A(t) are defined by (7.70) and (7.71), respectively. Using these, as in the third variational theorem derivation, Mx is written as

;(η)

dr] — 2 F f ( r ) W z( r ) j dr

dv

0

(7.87) Conditions are assumed, such that a separation of variables solution applies, in the form of (7.74). See Section 2.3 for the statement of such conditions.

7.5.

M I N I M U M THEOREMS

249

Furthermore, we take variations from the solution of the boundary value problem by varying only the spatial parts of the field variables in (7.74). Under these conditions the first minimum theorem can be stated as: For a viscoelastic boundary value problem, which admits a separation of variables form of solution, define as admissible states only those states for which (a) the displacements satisfy the displacement boundary conditions, (7.51), and (b) the displacements differ from the state of the solution only in the spatial rather than in the time part of the solution. Of these admissible states, the one which is the solution of the boundary value problem posed by (7.46), (7.49)-(7.51), (7.65), and (7.66) renders the functional Μλ , defined by (7.86), (7.70), and (7.71), a minimum, compared with Mx evaluated for any other admissible state, if

for the process under consideration. The proof begins by letting the admissible displacement in accordance with (7.74), have the form Ufa,

field

, t),

(7.88)

t) = Ufa) u(t) + Sûfa) u(t)

where ü^x^ u(t) is the exact solution displacement field, and the varied displacement is seen to involve variations only in the spatial portions of the displacement field. Under these conditions, the minimum theorem to be proved is the statement that ißTj - Μλ > 0 (7.89) where Μλ is defined by (7.87) for the field variables of the solution of the viscoelastic boundary value problem, while M1 represents the value of (7.87) for any admissible displacement field. We expand (7.89), using (7.87) and (7.88). Then,

j

- Μλ =

[2iii8ijj 8iiiSëji]Ü{r) 2 ( 3 (

+

+ liieren + Sëifièij] Ù(r) ί μ(τ — η j

/_^2v)

ίο

J

- IFfa

δΜ,·ά(τ)|

dr dv - 2

μ

Ύ {

~

V)

ύ

( ) η άη

0

f

f*

S fa

δβ,·ώ(τ)

dr da.

(7.90)

The last integral is written as an integral over the entire boundary, since δ ^ = 0

250

V I I . GENERAL THEOREMS A N D F O R M U L A T I O N S

over Bu , and then it is converted to a volume integral through the divergence theorem. This results in

^

-

Mx

=

J" J *

j [ 3f f

j

^ 2 v ) Jo ^ é

μ(τ -

+

[4

-

2[aijtj

+

2δ4·Ζ/(τ)

î?)

ii(v)

" ^ ^

dt] -

(

7 )?

^

"

§ σ ( τ )

"

]

&

"

ώ

( )τ

2%(τ)] δ ^ ώ ( τ )

+ i%] δκ,·ώ(τ)

η) ^ΰ{η)

/x(r -

άη\
(7.91)

The first three bracketed terms vanish since the solution must satisfy the stress strain relations ( 7 . 6 5 ) and ( 7 . 6 6 ) , and the equations of equilibrium, ( 7 . 4 9 ) . Equation ( 7 . 9 1 ) is left as -

Mx

=

f

J

V

where δσ#(ί) = δ,·,·

μ(* 3(1 — 2v) J

0

[δέ„(τ) δσ,,(τ)] dr dv

f

τ)

(7.92)

J 0

dr + 2\

^ tfr

μ(ί ~ r) J

0

or

rfr. (7 93)

Replacing the symbol δβ^ by eio and δσ^ by σ ί5 in (7.93) it has the form of the viscoelastic stress strain relation. It then follows from (7.92) and (7.93) that (7.89) is satisfied if

J

Γ^ω^#-^>0 0

T

(7.94)

C

for the process or processes examined. Thus, the minimum theorem is true for all materials for which the work done is nonnegative. The implications of a nonnegative work requirement have been discussed in Section 3.3. It was found there that if the stored energy and the rate of dissipation of energy are taken as nonnegative then the nonnegative work requirement (7.94) is always satisfied. Furthermore, for relaxation functions represented by a positive constant plus a series of positive decaying exponentials, as in (2.68), these conditions are always satisfied and this minimum theorem is applicable to all processes. It is reasonable to expect the existence of a second minimum theorem which corresponds to the theorem of minimum complementary energy in elasticity and

7.5.

M I N I M U M THEOREMS

251

reduces to it under the proper conditions. To establish such a theorem we define the functional M2 as

where />ΐ^(τ) and Γ'(τ) are defined by (7.78) and (7.79) respectively. As with the first minimum theorem, conditions are assumed such that a separation of variables solution applies in the form of (7.74). The variations are permitted only in the spatial parts of the field variables. With these agreements the second minimum theorem can be stated as: For a viscoelastic boundary value problem, which admits a separation of variables form ofsolution, define as admissible states only those states for which (a) the stresses satisfy the equilibrium equations (7.49), (b) the stresses satisfy the stress boundary conditions (7.50), and (c) the stresses differ from the state of the solution only in the spatial rather than in the time part of the solution. Of these admissible states, the one which is the solution of the boundary value problem posed by (7.46), (7.49)-(7.51), (7.67), and (7.68), renders the functional M2, defined by (7.95), (7.78), and(7J9), a minimum, compared with M2 evaluated for any admissible state, if

for the process under consideration. Under these conditions the minimum theorem to be proved is the statement that M2 - M2 > 0 (7.96) where M2 is defined by (7.95) for the field variables of the solution of the viscoelastic boundary value problem, while m2 represents the value of (7.95) for any admissible stress field. We let the admissible stress field ô^fa , i) have the form where σ^(χ{) a(t) is the exact solution result and the varied stress field is seen to involve only a spatial variation from that of the solution. To obtain the conditions under which this minimum theorem is true, we start by expanding (7.96), using (7.95). This gives

= \ ^ %y^v) SSifiSij] f

[2ôii8âjj+SôiiSâjj]ά{τ)

M2-M2

v

+ £[2ί«δί« + - 2 f

δσ^σ(τ)

ZI,(T)

dr da.

f ~
η)ά{

0

σ(τ)

J(r - η) σ(η)

(7.98)

dr d

V I I . GENERAL THEOREMS A N D F O R M U L A T I O N S

252

Sd =

Β

The last integral is written as an integral over the entire boundary, since 0 over {j σ . Using the divergence theorem and ijtj = 0, (7.98) is w as

Sâ

Ü - M = IJ^j [\~^ J(r - V) àM d - f„(r)] δ^,σ(τ) 2

(

t

3 (

2

e

V

+ [J* Κ* - V) hi(v) V - 2β«(τ)] δΐσ(τ) d

+ φ δσ(τ) /(τ - η) |δ4σ(τ) J* 3ί

+

Κτ-η)

}ί

8σ{ίσ(η) άη (7.99)

δ^σ(η) άη \ dr dv.

The first two bracketed terms in (7.99) vanish since the solution satisfies the stress strain relations (7.67) and (7.68). Relation (7.99) becomes ^ 2

-

f

M2 = f

[δσ„(τ)

δε,,(τ)]

(7.100)

dr dv

where Ä

M

m o

f*

1

- j J ο

/ K t

ï

-

T )

ë 8

T

^^( ) j

, s

t

Π

he^

- ô t -

dT

+

δ

* ^n^y

-

)

2V

C

J ο

F T (

R t

"

χ ^δσ^(τ)

j

σ-

*^

T )

1

0

1)

We replace the symbols by and δ σ ΐ7 by in (7.101) to give (7.101 ί3 the form of the viscoelastic stress strain relation. It follows from (7.100) and (7.101) that (7.96) is satisfied if

^^-e^dr^O

(7.102)

for the process being studied. The requirement (7.102) is not as easily interpreted as the nonnegative work requirement (7.94) of the first minimum theorem. Certainly, in the limiting case of an elastic material, (7.102) implies / > 0 where / is the elastic shear compliance. But, in the case of viscoelastic materials, it is easy to show that (7.102) can be violated. For example, using nothing more complicated than a one-dimensional Maxwell-model stress strain relation, for two consecutive step function applications of stress, it can be shown that (7.102) is violated. On the other hand, (7.102) is readily seen to be satisfied under the conditions of a creep test, if /(0) ^ 0, and Poisson's ratio is suitably restricted. Although the first minimum theorem is applicable to typical viscoelastic materials, the conditions under which the second minimum theorem is applicable for realistic mechanical properties characterization imposes restrictions upon the processes for which (7.102) can be satisfied. Thus, the usefulness of this

7.5.

MINIMUM

THEOREMS

253

second minimum theorem is somewhat limited. Nevertheless, there are interesting special conditions under which it is applicable. The conditions of the creep test have been mentioned as one such case. Another is that of the condition of steady state harmonic oscillation. Although it is, in principle, possible to show this latter application by directly restricting the processes in the two theorems just given, in practice it is easier to derive two new minimum theorems, directly appropriate to steady state harmonic oscillation conditions. We now do this to arrive at the third and fourth minimum theorems. The proof of the next theorem is only briefly outlined, and that of the last theorem is left as an exercise, since they have many elements in common with the two previous theorems. The viscoelastic boundary value problem now posed is given by the adaptation to steady state harmonic conditions of the relations (7.46), (7.49)-(7.51), and (7.65)-(7.68). As with the first minimum theorem, the one fo be given now is associated with the theorem of minimum potential energy in elasticity. We define the functional M 3 as M

3

= ^LF \μ*(* ) ω

j

v

[

^

2v)

€ii€jj

6ij6ij +

~ ] ~ί F{Ut dv

SiUida

Β

(7.103) where ω is the frequency of oscillation, μ.*(/ω) is the complex modulus in shear, and viscoelastic Poisson's ratio is again a real constant. The field variables are all harmonic functions of time, as must be the specified boundary displacements and tractions and the body forces. Variations from the solution of the boundary value problem are taken only in the spatial parts of the field variables and not in the harmonic function of time part. The third minimum theorem is now stated as:

\

For a harmonic oscillation viscoelastic boundary value problem, define as admissible states only those states for which (a) the displacements satisfy the displacement boundary conditions and (b) the displacements differ from the state of the solution only in the spatial rather than the time part of the solution. With Re μ*(ΐω) > 0,

Im μ*(ΐώ) > 0,

and

— 1 < ν < 1/2

then Re(M 3 — M 3 ) > 0

and

Im(7#3 - M 3 ) > 0

where Mz is the functional (7.103) evaluated for the solution of the boundary value problem and M 3 is (7.103) evaluated for any admissible displacement state. We begin the proof of this theorem by defining an admissible displacement field u^Xi, t) in the form (7.104)

ui(Xi, t) = [üiiXi) + 8üi(Xi)] e

u

254

V I I . GENERAL THEOREMS A N D F O R M U L A T I O N S iajt

where Ufa) e designates the solution of the boundary value problem and necessarily Süfa) = 0 on Bu . By following a lengthy procedure similar to that of the first minimum theorem, (7.103) and (7.104) are combined to give - M 3 = μ*(ίω) j

(

[3(j

+ ^ ν ) δβ,,δ^, + δ ^ , ] dv

(7.105)

Se^fa)

where has the obvious identification Se^ = \(§üitj + Süjti) with th spatial parts of the displacement expression in (7.104), and has been decomposed into dilatational and deviatoric parts. The proof of the theorem follows directly from (7.105) for Be^ being real. Since represents deviations from the spatial part of the solution for Mfa) and the spatial part of the admissable displacement field to be real ü^Xi) must be real. This is true under the conditions of the theorem. It can be reasoned that, if viscoelastic Poisson's ratio were not a real constant or if the medium were not homogeneous, « z(# ?) would be complex, and the minimum theorem would be invalid under these conditions. It has been implicitly assumed that in boundary value problems having both prescribed stresses and displacements on the surface and having prescribed body forces, the phase angle between the prescribed surface tractions and body forces on the one hand, and the prescribed surface displacements on the other hand, must have the particular value which allows the spatial part of the displacement field to be real. This phase _1 angle is given by tan (Im /x*/Re μ*). The restriction here to homogeneous materials is in sharp contrast to the corresponding minimum theorem in elasticity, where the application to nonhomogeneous conditions is allowed. The last minimum theorem is the steady state harmonic counterpart of the second minimum theorem and the theorem of minimum complementary energy in elasticity. We define the functional M 4 through

ü^x^

Sü^Xi)

=^ 1 ^

S [^ttt v

**J

+

dv

- L„ \ · σΛda

(7 106)

where ω is the frequency of oscillation, /*(ιω) is the complex compliance in shear, and again viscoelastic Poisson's ratio is taken to be a real constant. The field variables are all harmonic functions of time, and variations from the solution of the boundary value problem are taken only in the spatial parts of the field variables and not in the harmonic function of time part. The fourth minimum theorem is now stated as: For a harmonic oscillation viscoelastic boundary value problem, define as admissible states only those states for which (a) the stresses satisfy the stress boundary conditions, (b) the stresses satisfy the equations of equilibrium aijfj + 7% = 0, and (c) the stresses

7.6.

O P T I M A L STRAIN HISTORY

255

differ from the state of the solution only in the spatial rather than the time part of the solution. With Re J*(iœ) > 0,

Im J*(iw) < 0,

and

-1 < ν < \

then Re(7#4 - M 4 ) > 0

a/z
Im(ifir4

- M 4) < 0

where M 4 w i/re functional (7.106) evaluated for the solution of the boundary value problem and M 4 is (7.106) evaluated for any admissible stress state. The proof of this theorem follows the lines of reasoning given in proving the previous minimum theorems. As an application of these minimum theorems, Christensen [7.13] has used them to derive bounds upon the effective complex moduli for two special types of composite viscoelastic materials. That is, in some cases where it is not possible to obtain exact solutions for the effective complex moduli of composite viscoelastic materials, these minimum theorems may be used to obtain bounds upon the real and imaginary parts of these moduli through the use of admissible displacement and stress fields. The two special types of composite materials in this application are those of a homogeneous material containing voids or perfectly rigid inclusions, both cases of which qualify under the dual categories of homogeneous materials and composite materials. This resolves the seeming inconsistency of the application of homogeneous material minimum theorems to composite media. There have been attempts to formulate minimum theorems using integral transformed field variables. Typically, these procedures err in failing to take into account the complex variable nature of the formulation.

7.6.

OPTIMAL STRAIN HISTORY

There are many types of problems in viscoelasticity that can be formulated as an optimal path problem. Such optimality problems have a close resemblance to the standard formulations of the calculus of variations. The minimum theorems of Section 7.5 were of such a type; another type will be given here. The complication over and above that of the calculus of variations is in accounting for the effect of the strain history of viscoelastic behavior. The specific problem to be studied here is that of finding the optimal strain history path in going from one strain state to another while minimizing the work done during a given time interval. This problem was solved by Breuer [7.14]. A different approach was given later to the same problem by Gurtin et al. [7.15]. We follow the latter procedure. We pose the problem as follows. In a particular viscoelastic material, the strain state is taken to be zero up until time t = 0.

256

V I I . GENERAL THEOREMS A N D F O R M U L A T I O N S

At time Τ the strain state has a value e0 that is constant thereafter. Find the strain history e(t) with

,(T) =

c(0) = 0, c0 in order to minimize the work done in deforming the material. By definition the work is W

=

a dj

Î

dt

)( 7 , 1 0 7

^ a^t~ '

Write the one dimensional stress strain relation as σ(ί) •= Ç

(7.108)

G(t - τ) ^ d r .

Combining (7.107) and (7.108) gives W(e) = C Ç G(t-r)

(7.109)

è(r) è(t) dr dt.

In seeking to minimize (7.109) we do not know if e(t) is continuous on the closed interval [0, T]. Certainly we cannot assume continuity at the outset of the derivation. Begin by integrating (7.108) by parts to obtain the alternative form

σ(ί) = G(0) e(t) + Ç G(t - τ) e(r) dr where G(t — τ) denotes the time derivative

(7.110)

G ( t - r ) = ^ G ( t - r ) .

Substitute (7.110) into (7.107), with the result W{e) = 1G(0) el + fè(t)

f Ù(t - τ) e(r) dr dt.

(7.111)

Integrate (7.111) by parts using e(0) = 0 and Leibnitz's rule to obtain

W{e) = where

- 6(0)

1G(0)

el f Ô(T + e0 T

j e (t) dt -•'o j e(t) [[' T

-

2

2

d

.

G(t -

r)

t)

e(t) dt

e{r) dr]

dt

(7.112)

7.6.

O P T I M A L STRAIN H I S T O R Y

257

The last term in (7.112) can be put into alternative form. To do this, begin with the following form and break the second integration into the intervals shown: & =

- * Γ

= - \ f

T

G(\t-r\)e(r)e(t)drdt

Γ

F

G(t -

r) e(r) e(t) dr dt

• Τ /» Τ

2 f

f

G(r-t)

(7.113)

e(r) e(t) dr dt.

Invert the orders of the limits in the last term and change the variables in it to obtain Θ =

- \

-

ί

T

e(t) \

1 \\{t')

Ç G(t

-

τ) e(r) dr] dt

F f ' Ö(t' -

τ') e(r') rfr'] dt'

(7.114)

where r = T - r \

t =

T-t'.

With a further change of dummy variables in the last term in (7.114) it is seen that the two terms are identical and can be combined to eliminate the factor of I . Furthermore, the ensuing term is identical to the last term in (7.112) and thus can be replaced by the term in (7.113). There results 2

W(e) = JG(0) e 0 + e0 f

- i

e(t) dt -

G(T-t)

2

ί

0(0)

e (t) dt

•τ

Γ

*Ό

F G(\t-T\)e(T)e(t)drdt.

(7.115)

Λ )

Write (7.115) as where

W(e)=A+L(e)+Q(e,e)

A=\G(0)el L(e) = e0f Q{e, ß) =

-0(0)

G(T-t) fe(t)

(7.117)

e(t) dt ß(t) dt -

\\ C T

G(\ t -

τ

I)

e(r) ß(t) dr dt.

The second form in (7.117) is linear, the last form bilinear and symmetric in the strain variables. Now form W{e

+ β) -

W(e) = Q(ß, β) + 2Q(e, β) + L(ß)

(7.118)

V I I . GENERAL THEOREMS A N D F O R M U L A T I O N S

258

Since Q(ß, β) ^ 0, for W(e) to be a minimum we must have 2Q(e,ß)+L(ß)=0

(7.119)

because these terms can depend upon the sign of β. Writing out ( 7 . 1 1 9 ) , using ( 7 . 1 1 7 ) , gives

fß(t)

[ - 2 6 ( 0 ) e(t) - f

G ( | t - Τ I ) € ( R ) dr + e0Ô(T

- F ) ] dt = 0 .

The Euler equation for this problem is thus given by setting the term in brackets equal to zero, yielding T

2 ( 5 ( 0 ) € ( 0 + F G ( | t - r I ) E ( R ) dr = e0Ô(T

- t).

(7.120)

We have made considerable progress. The minimum problem has now been reduced to the solution of an Euler equation in the strain history e(t). To solve the Euler equation, begin by examining the general characteristics of the solution. Let /(/)=€„-«(Γ-*)·

(7.121)

It can be verified that/(i) satisfies the same Euler equation, ( 7 . 1 2 0 ) , as E ( £ ) ; thus,

<τ -1)

<*) = Ό -

and the strain path is antisymmetric about Tj2. To proceed further requires specialization to a particular model of viscoelastic stress strain behavior. As an example, take the standard linear solid described in Problem 1.1. The relaxation function is given by G(t)

= ( G

t/T

0

- G J e- > + G

Œ

.

( 7 . 1 2 2 )

The other terms in ( 7 . 1 2 0 ) , 0 ( 0 ) , G(T - t\ G(t - Τ ) , and G(r - t\ can readily be formed from ( 7 . 1 2 2 ) . With ( 7 . 1 2 2 ) , the Euler equation ( 7 . 1 2 0 ) becomes -

— e(t) + T

L

rh

L

ί Te ^(r)

^0

dr + ^

L

\

J

t

h

e^ U(r)

T

dr = - T ^ -

L

V

/ T L

.

(7.123)

As a trial solution, try the linear function €(t)=c0 + Clt. (7.124) Substituting ( 7 . 1 2 4 ) in ( 7 . 1 2 3 ) gives terms that are independent of time, - I T T expressed as a coefficient of (i)°, and coefficients of t, e / I , and e*/ I. Setting

O P T I M A L STRAIN H I S T O R Y

7.6.

259

these coefficients equal to zero gives four equations, only the last two of which are not identically satisfied; i.e., c

- f

+ c

1

=

0

T l

(7.125)

The solution of (7.125) is e

c and

~

° T/tj + 2 (7.126)

0

c

°

€

r^r/x! + 2) ·

1

The complete solution for the optimal strain history is then

1
(7.127)

and the work done is given by G

7 128

^ = T( » + Î S ^ ) -

<· >

The important thing to observe from the solution (7.127) is that the optimal strain path involves jump discontinuities at t = 0 and t = T. The discontinuous nature of the optimal strain path appears to be inherent in typical problems of 1 viscoelastic optimization. For 7 —^0, W — \GQe\ , whereas for Τ —> oo, W = \Gœe\ , in accordance with physical expectations. Having succeeded in finding the optimal strain history for the standard linear solid, we proceed to the generalized Maxwell model. Let G(t) = X Gie~

t,Ti

(7.129)

+ Gx .

i=l The Euler equation (7.120) with the form (7.129) gives 2
i=l i r

T

ii= N

i e-

f ^€(T)

0

J

T l T i t , Ti

e

Tlri

άτ + Σ ^ \

= "€οΣTi - —»

T

i=i i

e- ^)

*r

J

t

(·

If one tries the linear • -ι form of solution (7.124) in (7.130), and then sets the

7130

260

VII. GENERAL THEOREMS AND FORMULATIONS

coefficients of the time terms equal to zero, there results more equations than there are unknowns. The linear form (7.124) is not correct, and it can be seen that exponential terms must occur in the solution. The general solution is given by Breuer [7.14]. The example given here shows the practicality of formulating optimal work and energy problems in viscoelasticity. Bounds on the work done in viscoelastic deformation have been given by Breuer [7.16], Day [7.17], and Spector [7.18]. A solution of maximum recoverable energy has been given by Breuer and Onat [7.19]. Other references to optimization problems, in relationship to residual stresses, are given in Section 3.8. PROBLEMS L L A uniqueness theorem for isothermal, anisotropic, dynamic, linear viscoelasticity has been established which does not require the existence of a Laplace transform, as was required in Section 7.1. This uniqueness theorem, using Volterra's method in the proof, is given in Ref. [7.20]. Formulate completely the proof of this theorem, following that given in Ref. [7.20]. 7*2* Using the reciprocal theorem given in Section 7.3, find the total volume change of a viscoelastic body subjected to surface tractions and body forces. T h e resulting formula involves only these surface tractions, the body forces, and the volumetric creep function. 7 3 * Derive viscoelastic variational theorems, using the Stieltjes convolution notation, which correspond to the theorems of stationary potential and complementary energy in elasticity. 7A. Obtain the complete proofs for the third and fourth minimum theorems of Section 7.5. 7*5* Verify relation (7.122) in the optimal strain history problem of Section 7.6. Discuss t h e possible m e a n s of minimizing t h e m a x i m u m stored energy in going from one strain state to another in a given t i m e interval.

REFERENCES 7.1.

7.2. 7.3.

Onat, E. T., and S. Breuer, "On Uniqueness in Linear Viscoelasticity," in Progress in Applied Mechanics (D. C. Drucker, ed.) (The Prager Anniversary Volume), p. 349. Macmillan, New York, 1963. Doetsch, G., Handbuch der Laplace-Transformation, Vol. 1, p. 74. Birkhäuser, Basel, 1950. Sternberg, E., and M. E. Gurtin, "Uniqueness in the Theory of Thermorheologically Simple Ablating Viscoelastic Solids," in Progress in Applied Mechanics (D. C. Drucker, ed.) (The Prager Anniversary Volume), p. 373. Macmillan, New York, 1963.

REFERENCES 7.4. 7.5. 7.6. 7.7. 7.8. 7.9. 7.10. 7.11. 7.12. 7.13. 7.14.

7.15. 7.16. 7.17. 7.18. 7.19. 7.20.

261

Gurtin, M. E., and E. Sternberg, "On the Linear Theory of Viscoelasticity," Arch. Ration. Mech. Anal. 11, 343 (1962). Graham, G. A. C , " T h e Contact Problem in the Linear Theory of Viscoelasticity," Int. J. Eng. Sei. 3, 27 (1965). SokolnikofT, I. S., Mathematical Theory of Elasticity, 2nd ed. McGraw-Hill, New York, 1956. Sternberg, E., "On the Analysis of Thermal Stresses in Viscoelastic Solids," Proc. 3rd Symp. Nav. Structural Mech. 348. Macmillan, New York, 1964. Onat, E. T., "On a Variational Principle in Linear Viscoelasticity," / . Mech. 1, 135 (1962). Schapery, R. A., "On the Time Dependence of Viscoelastic Variational Solutions," Quart. Appl. Math. 22, 207 (1964). Biot, M. A., "Linear Thermodynamics and the Mechanics of Solids," Proc. 3rd U.S. Nat. Cong. Appl. Mech. 1 (1958). Christensen, R. M., "Variational and Minimum Theorems for the Linear Theory of Viscoelasticity," Z. Angew. Math. Phys. 19, 233 (1968). Gurtin, M. E., "Variational Principles in the Linear Theory of Viscoelasticity," Arch. Ration. Mech. Anal. 13, 179 (1963). Christensen, R. M., "Viscoelastic Properties of Heterogeneous Media," / . Mech. Phys. Solids 17, 23 (1969). Breuer, S., " T h e Minimizing Strain-Rate History and the Resulting Greatest Lower Bound on Work in Linear Viscoalasticity," Z. Angew. Math. Mech. 49, 209 (1969). Gurtin, M. E., R. C. MacCamy, and L. F. Murphy, " O n Optimal Strain Paths in Linear Viscoelasticity," Quart. Appl. Math. 37, 151 (1979). Breuer, S., "Lower Bounds on Work in Linear Viscoelasticity," Quart. Appl. Math. 27, 139(1969). Day, W. A., "Improved Estimates for Least Work in Linear Viscoelasticity," Quart. J. Mech. Appl. Math. 32, 17 (1979). Spector, S. J., "On Monotonicity of the Optimal Strain Path in Linear Viscoelasticity," Quart. Appl. Math. 38, 369 (1980). Breuer, S., and E. T . Onat, " O n Recoverable Work in Linear Viscoelasticity," Z. Angew. Math. Phys. 15, 12 (1964). Edelstein, W. S., and M. E. Gurtin, "Uniqueness Theorems in the Linear Dynamic Theory of Anisotropie Viscoelastic Solids," Arch. Ration. Mech. Anal. 17, 47 (1964).