On separation of energies in viscoelasticity

M~es

Research Commtmieations, %'ol.24, No. 2, pp. 167-177, 1997

Pergamon

Copyright© 1997ElsevierSciencetad Printed in the USA. All rights reserved 0093-6413/97 $17.00 + .00

Pll S0093-6413(97)00009-8 O N SEPARATION OF ENERGIES IN VISCOELASTICITY S. HAZANOV H - Consultants av. V. Ruffy 24, CH-I012 Lausanne Switzerland (Received 2 April 1996; acceptedfor print 5 November 1996)

ABSTRACT Separation of the storaged and dissipated energies in viscoelastic deformation is considered. This is a key problem for the construction of viscoelastic minimum principles and for the micromechanics of heterogeneous materials with memory. The notion of the viscoelastic free energy functional is discussed, thermodynamic admissibility conditions are established. An engineering analysis is realized through the method of harmonic strain regimes, influence of the loss and the storage moduli on the dissipation rate is studied. For the Volterra-Frechet integral expansion approach, necessary conditions on the general form of a free energy viscoelastic functional are formulated. The obtained results are used to examine the thermodynamic validity of certain classic viscoelastic models, like that of Staverman-Schwarzl. Through the spectral method, this energy representation is shown to correspond to a generalized Maxwell model. © 1997 Elsevier Science Ltd

"

h INTRODUCTION Separation of the storaged and dissipated energies in viscoelastic deformation is a key problem for construction of viscoelastic minimum principles and thus for the micromechanics of materials with memory (Rabotnov (1980), Christensen (1982)). Consider classic constitutive equations of linear viscoelasticity: c=foo

;

~=roE

(1)

In (1) o and 8 are respectively the stress and the deformation tensors, f (t) and r (t) the creep and the relaxation functions tensors, and o the Stieltjes convolution operator. Consider the second law of thermodynamics in its uniaxial isothermal form : D= ere- F>0

(2)

In (2) F is the free energy density and D the dissipation rate density. Notation ~b means throughout the paper a derivative of the function ~0 with respect to time. The problem under study is to construct the functionals F and D basing on constitutive laws (1), that means to separate the storaged and the dissipated energies in viscoelastic deformation. Though trivial for purely elastic or purely dissipative materials, the situation in a viscoelastic case is more complicated. This is confirmed by the number of classical and recent works concerning this problem and arriving sometimes to contradicting conclusions (see Coleman (1964), Breuer and Onat (1964), Day (1972), Ferry (1980), Rabotnov (1980), Tschoegl (1980), Christensen (1982), Pipkin (1986)). The most 167

168

S. HAZANOV relevant theory on this subject is developed in Coleman and Owen (1970), Day (1972) and Fabrizio and Morro (1992). The formulated problem is fundamental for the construction of viscoelastic minimum principles and for micromechanics of heterogeneous materials with memory (Rabotnov (1980), Christensen (1982), Huet (1992), Hazanov (1995) and (1996)). Indeed, a standard viscoelastic energy principle is formulated as a problem of minimizing a functional H

d r - Ir, D

D

(3)

3D a

in the class of kinematically admissible strains/displacements ~, ~ . In (3) * is a linear operator defined by the form of the free energy functional F. Solutions evidently depend on the choice of the functional F. For homogeneous materials, alternative methods of solution can be found, but homogeneisation and estimation of the effective properties of heterogeneous viscoelastic materials inevitably bring us back to the energy separation problem. 2. FORMULATION OF THE PROBLEM An engineering analysis of the problem is provided. We work in terms of linear theory, because in the actual state-of-art of nonlinear viscoelasticity, it would be premature to attack the nonlinear separation problem. The only objective fact known about the free energy in viscoelasticity is that for an instantaneous modulus r0 not equal to zero, free energy is a stress potential in an isothermal deformation (see for example Day (1972), Rabotnov (1980)): ~F c = --

(4)

In (4) the notion of partial Frechet derivative is used, when only actual values of the deformation tensor are counted and not its history. Consider the most general free energy expression in linear viscoelasticity, that follows from the Frechet-Volterra expansion (see for example Christensen ( 1982)): F=~i

i K(t-u,t-v)de(v)de(u)

(5)

K(x,y) is a continuous, symmetric kernel and has first partial derivatives with respect to x and y. K is supposed to be such a kernel that functional F in relation (5) is nonnegative definite. The constitutive law of the material then will be:

ENERGY SEPARATIONIN VISCOELASTICITY

t~= J K(0,t-v)dE(v) ;

K(0, t ) = r (t)

169

(6)

Note that (4) and (6) are compatible. (4) induces that the free energy viscoelastic potential can be presented as: F=t~

-

r° 82 +Q 2

(7)

In (7) Q is the functional only of the strain history and not of the actual state. Q is defined by the following conditions obtained from (2), (4) and (7): t

OQ=o Oe

•

Q<-e ~ r'(t-u)de.(u)

(8)

Relationships (8) hold automatically for any spring-dashpot model and for hereditary models of the type (6), with the instantaneous modulus not equal to zero. Thus, the problem under study is to find a functional Q that satisfies conditions (8) in the class of thermodynamically admissible relaxation functions. Another important aim of the paper is to examine the thermodynamic validity of certain classic phenomenological free energy models. 3. FREE ENERGY REPRESENTATIONS Due to the difficulties pointed out above, the problem of viscoelastic free energy functional is avoided in many important papers on viscoelasticity, like that of Coleman (1964), Day (1972), Fabrizio&Morro (1992) and Kuiken (1994), where no explicit free energy expressions are given. In the actual state-of-art, the most well-known is the Staverman&Schwarzl (1952) representation, established for polymers on the base of the statistical physics concept: F = ~-i i r(2t-u-v)d~(v)dE(u) 0

(9)

0

(9) is a particular case of representation (5) in supposition that: K(x, y) = K(x + y)

(10)

Through the spectral method, (9) can be related with spring-dashpot assemblies. Indeed, consider the definition of the relaxation spectrum p(/J) :

r(t) = 5p([t)exp(- t //J)dp + r. 0

In (11) r.. is the long-term modulus. Then (9) becomes:

(11)

170

S. HAZANOV

F=5

p(~) exp(-(2t-u-v)/la)dll)de(v)de(u)+r--= oo o o 2 (12)

~!p(l~) exp(-(t-u)lll)de(u) dll+r--~ But (12) resembles the free energy expression for a generalized M~/xwell model with an infinite number of branches and constitutive law (13): t

o" = j'p(/z)(J'exp(-(t - u) / / a ) d e ( u ) ) d , u + r e 0

(13)

0

Thus, (9) is tightly related with the spring-dahspot approach. Reciprocally, because of its integral form, (I 1) can be interpreted as a relaxation function of a generalized Maxwell model with an infinite number of branches and a constitutive law (13). Corresponding free energy expression leads after simple transformations to (9):

F=~[O 0 t

t

exp(-(t-u)/ll)de(u

d~+r --=

LO

~

=

2

I

~f~ ~p(ll)(fexp(-(2t-u-v)/l.t)dll)de(v)de(u)+r~= O0

0

r (2t- u

(14)

0

v) d e ( v ) d e ( u )

oo

If (9) is written through the creep function, it will correspond to a generalized Kelvin model. As any spring-dashpot system is equivalent to a generalized Maxwell or Kelvin model, we concude that the spectral method demonstrates a fundamental relationship between the Staverman-Schwarzl energy expression and spring-dashpot assemblies. This confirms the viewpoint of Coussy (1987). Despite its interest, relationships (11)-(14) cannot pretend to give a general method of viscoelastic free energy functional construction, at least because in our opinion, not all viscoelastic systems can be reduced to spring-dashpot assemblies. Model (9) is often quoted in literature [Christensen (1982), Coussy (1987), Huet (1992)] and is known to describe pretty well the behavior of rheological systems, especially of spring-dashpot assemblies. But its thermodynamic admissibility still needs to be examined, especially because this model defines the free energy fimctional only on the base of the relaxation function, that in principle is not correct (Breuer and Onat (1964), Day (1972)). An alternative phenomenological model is a quadratic representation of Rabotnov (1980): F

=

(~e- r° e2 ( a - r 0 e): +p 2 2

(15)

ENERGY SEPARATIONIN VISCOELASTICITY

171

where p is a parameter corresponding to the type of the model under study and the third term of the right part of (11) contains in fact all the strain history. 4. HARMONIC OSCILLATIONS In principle, any free energy representation can be directly checked through conditions (8). But straightforw.ard verifications not always permit to construct new free energy functionals. Hence, alternatives should be found. As it was remarked above, objective conclusions on the free energy functional form demand some information additional to the knowledge of the constitutive law. In view of this, let us turn to the case of a harmonic strain regime that gives certain advantages for the energy analysis. Consider an arbitrary strain regime from the class of piecewise differentiable functions of bounded variation. Thus, it can be expanded in a Fourier series (Korn (1968)). Then, after an analysis of a harmonic strain law case, one can generalize the obtained results on an arbitrary regime from the class of the functions that can be expanded in Fourier series. That is why a harmonic regime can serve as a basic model for a general study. Consider a strain law: e = ~o sin mt

(16)

Then from (1) the energy rate will be written through the storage and the loss moduli G1 (co) and G2 (co) as: ~ = e~ co(G, sin cot cos cot + G 2 cos 2 cot)

(17)

Traditionally, in such a regime one is interested only in the energy variation over the total cycle. But the situation inside the cycle can be also of importance and this is where arises the problem of energy separation. Various authors resolve it differently. For example, Pipkin (1986) from the fact that the first term in the RHS of (13) is a full differential, supposes that this is the second term in the RHS that corresponds to the dissipated energy: D = eZo 03 G z cos ~ cot

(1 8)

On the other hand, Rabotnov (1980) considers the work over the total cycle and writes down the current dissipation rate as: D = - ~ 2 m G2 z

(19)

Let us focus on assumptions (18) and (19), important at least because they lie in the origin of the fundamental terms of the "loss" and the "storage" moduli. The only objective fact concerning the dissipation rate is that in an elastic material it is equal to zero and that in a viscous liquid on the contrary, all the work done over the body is dissipated. Certainly, in a linear viscoelastic case, the value of the rate of dissipation

172

S. HAZANOV over a total cycle always corresponds to expression (19). But because of the lack of information on the general form of the free energy functional, there is no reason to accept

a priori intuitional assumptions, such as dependence of the dissipation rate only on the loss modulus. To illustrate this conclusion, consider a generalized Maxwell model, with Ei and rli respectively the spring modulus and the dashpot viscosity in a branch N ° i. The instantaneous dissipation rate is: N co 1/x i z_~i=l--% ( ( 1 / x , ) 2+co2c°scot+ ( 1 / x ) 2 +0) 2 sin cot)2

D = e 2 c o 2 ~-, E;

o

(20)

In (20) ~i = 1]i ] Ei and the storage and the loss moduli are respectively :

G,=r

u E; +092Z;:1 (l/'t'i) 2+092

• G2 = c o ~ E; / r~ ' i=t (1/'t]) 2+092

(21)

(20) shows that even for a classic model the dissipation rate depends on both G2 and GI. This heretic conclusion concerns any spring-dashpot model, that means the only case where it is known for sure how to calculate the free energy and the dissipation rate, or in other words to separate the storaged and the dissipated energies. Thus, contrary to some authors' opinions, only the total dissipation rate (that means taken over a cycle) is proportional to the loss modulus, and not the instantaneous one. But the classic condition imposing the positivity of the loss modulus, still rests valid. Returning to the basic relationship (17), we admit that for harmonic deformations, the instantaneous dissipation rate as a positive definite function has the following form: D = ~ co (A 2 cos z cot + B 2 sin 2 cot + Csincotcoscot)

(22)

To assure the validity of (17) and the positivity of expression (22), coefficients A, B and C must be related as:

A2+B2=G2

;

tC]<<_2IABI

(23)

Then (17) becomes: Ok = e~ co [ (A 2 cos 2 cot + (G2 -AZ)s in2 cot + Csincotcoscot) + (G 2 - a 2 ) ( c o s 2 cot- sin 2 cot) + (G I - C)sincotcoscot ]

(24)

The main problem now is to find an expression relating coefficients A and C with the storage and loss moduli of the material. 5. ADMISSIBILITY CONDITIONS In a harmonic strain regime, equation (7) becomes: F = (Eo)2 (G2 cos cot sin cot + (GI - ro / 2) sin 2cot) + Q (~o sin cot )

(25)

ENERGY SEPARATION IN VISCOELASTICITY

173

Basing on (21) and (25), suppose that functional Q in this case can be similarly presented as: Q (Co sin tot ) = (eo) 2 (or cos 2 oat + 13 sin 2 tot - 7 sin oat cos oat) (26) Then the instantaneous free energy and dissipation rate correspondingly become: F = (eo) 2 (or cos 2oat + ([3 + GI - ro/2) sin 2tot + (G2 - )') cos oat sin oat )

(27)

D = (~_.o)2 oa (7 cos 2ot + (G2 - ~/) sin 2oat - (2or + ro - 213 - G1) sin mt cos rot)

(28)

Indispensable positive definiteness of bilinear forms (27-28) leads, according to relationships (23), to restrictions on coefficients 7 and [~:

Gt-Y<2~/o~(fl+G~-ro/2 ) ;12a+ro-2,O-G,l<_24y(G2-),i

(29)

(29) are general thermodynamic conditions on the form of the functional Q in a harmonic strain regime. 6. ANALYSIS OF PARTICULAR ENERGY MODELS a) For expression (6) resulting from the functional Volterra-Frechet expansion, we obtain: t

t

D = - I IK'(t-u,t-v)de(v)de(u)

; K'(x,y)=dK/dx=dK/dy

(30)

In a harmonic regime one has:

D

2 2 - c°s~ tot Eoto

ii

K'(x,y)costoxcosolydxdy+ sin 2 tot

oo

+ s i n tot c o s

ii

K'(x,y)sinoaxsintoydxdy

oo

tot ~ ~ K'(x, y)sin to(x + y) dx dy 00

(31) Calculating the quantity of the energy dissipated during one cycle and comparing it with expressions (17) and (27), we come to two conditions on the kernel K:

i i K'(x,y)costo(x - y)dxdy = -i(r(t) - r~)cosoatdt O0

(32)

0

-[~K'(x, ylsin to(x + yldxdy < O0

(33)

<_2

y

x,y)costoxcosootydxdy)(oo

K (x,y)sintoxsin OO

These are the necessary admissibility conditions on the general viscoelastic law (6).

174

S. HAZANOV b) For the Rabotnov quadratic expression (15), simple calculations give: D = c r e - P = - (~e + ro e k - p ( o -

ro e)((~ - ro k) . . . . . eo2 0 [ G2(r o - G Z)pcos 2 oat +

+ ( G 2 - p G z ( r o - G 1))sin 2 oat + ((r o - G] ) - p ( ( r o - G 1)2 _ G~)sinoatcosoat

]

(34)

Thus, one has: A 2 = p G2 (ro - G1), B 2= G2 (1+ p G1 - pro),

(35) 2 A B = ro - G1 + p (G2) 2 - p (ro - GI) 2 (35) gives that :

ro-G, P = (ro - G~)2 + G 2

(36)

(15) supposes that parameter p is a constant, but due to (36) this holds only for simplest mechanical spring-dashpot models. Indeed, already for a generalized Maxwell twobranch model one obtains:

Gl =Eo+El

G2=EI

092 0) 2 (1/,rl)2 +092 ÷E2 (1/r2)2+092

09/r] +E 2 09 / "t"2 (1 / ,t. )2 + 0~2 (1 / ,r2)2 + 092

(37)

((1/ 2"2)2 + ¢-02)2E1 ] 't'~ +((1/"t',) 2 + 092)2E 2 / "t'~

P= [((1/"r2) 2 +0) 2) E l / "t, +((1/z']) 2 +092) E2 / ,r2]((1 / v,): +092)((1 / ,t.2)2 +0)2) Thus, the quadratic free energy model (15) leads to a contradiction, at least in a supposition of a constant parameter p. c) In the case of the Staverman-Schwarzl model (9), one obtains from (18) :

A2(09) = G2(09)- c02~ J ( r ( x + y ) - r ~ ) c o s 0 9 x s i n 0 9 y d x d y ; 0 o B2(09) =092~ ~(r(x+y)-r.)cos09xsin09ydxdy; 0 o C

(38)

G~ (09) - r + 09=~ ~ ( r ( x + y) - r~) cos 09(x + y) dx dy ;

o o In order to correspond to equation (22), conditions (23) must hold. To verify the second of them, one has to simplify the expressions for A, B and C in (38). This can be done through the technique of the two-variables Carson transform R**(p, q) (see for example Ditkin & Prudnikov (1967)): R**(p, q) = (pR*(q) - qR*(p)) / (p - q) where R* is a standard 1D Carson transform.

(39)

ENERGY SEPARATIONIN VISCOELASTICITY

175

Posing in formule (39), p = -i to, q = -i to, we obtain additional formulae: ~

I I q)(x + y)cos03(x + y)dxdy= o

o

d

-7-[g,(z)sin03zdz a03~o

(40)

I I tp(x+y)sin03(x+y)dxdy= - 7[~(z)cos03zdz d ~ o o a03~o Then (38) give: A2

1 =7(G2+toG~). , B2 = 7 ( G 2 - t o G;); C=toG:

(41)

Thus, the basic thermodynamic condition (23) will be written for this model as:

to2((G02+((G~)2) _ 6~

(42)

Let us check whether Staverman-Schwarz theory satisfies the basic inequality (42). Simple calculations show that for a generalized Maxwell model it holds. But models based on spring-dashpot representation are not indeed hereditary, because their constitutive laws can be rewritten in a differential form, without stress-strain history. That is why it is preferable to consider more sophisticated models, for example with relaxation function :

r(t)

fJ 2 - t , Ol

(43)

This relaxation function, smooth, positive and monotone decreasing, satisfies the fading memory condition and is thermodynamically admissible. The storage and the loss moduli in this case are: G1

=

2-

sin to+ tocosto co(1 + toz )

"

, G2

=

I tosin to - costo -t to co(1 + to2 )

--

(44)

But then thermodynamic condition (42) is violated in several points. Take for example 03 equal to 1. Then: G( (1) = -1. 263 ; G ~ ( 1 ) = - 0 . 1 9 ;

G2(1)=1.151;

(45)

Substitution into (42) shows that LHS of (42) is equal to 1.6311, while RHS of (42) is equal to 1.3248, so that the thermodynamic admissibility condition (42) is violated. We have come to a contradiction: the dissipation rate calculated through (8), can become negative, though the chosen constitutive law is thermodynamically admissible. This shows that Staverman-Schwarzl free energy representation should be better treated not as a universal relationship, but as a phenomenological model irreplaceable in applications, especially for exponential and power-type creep-relaxation functions. One can also mention another important result following from (41). These relationships state that in the framework of the Staverman-Schwarzl theory, the

176

S. HAZANOV dissipation rate in general depends not only on the loss modulus, but also on the storage modulus, that is confirmed by the given above calculations on the spring-dashpot assemblies. Nevertheless, the classic conception that the energy amount dissipated over a cycle is determined only by the loss modulus, certainly rests valid. For the Staverman-Schwarzl theory, condition (32) is satisfied, but not condition (33), as it was shown above. Hence, this free energy representation cannot be the general solution of the problem (8). It is well known (Day (1972), Rabotnov (1980), Christensen (1982)) that to construct a free energy functional, one needs some information additional to the constitutive law (the principle of non-uniqueness of the free energy). What information is it and where can it be taken from? This question still is to be studied, that can be possibly done through the analysis of relationships (29), (32), (33). 7. CONCLUSIONS 1) We provide an engineering analysis of the sense and structure of the free energy notion in viscoelasticity. Thermodynamic admissibility conditions (8) on the form of viscoelastic free energy functional are established. 2) Because of the lack of additional experimental evidence, thermodynamic analysis is done for harmonic strain regimes. This method gives an opportunity to translate afterwards the obtained results on arbitrary loadings, at least for piecewise differentiable functions of bounded variation. 3) It is shown that, contrary to some authors' opinions, not instantaneous dissipation rate but only the total one (that means taken over a cycle) is proportional to the loss modulus. This is illustrated on the example of a traditional spring-dashpot assembly. In general, both the loss and the storage moduli should be considered responsible for dissipation. 4) For the Volterra-Frechet presentation, necessary admissibility conditions on the form of a viscoelastic free energy functional are formulated, relationships (29). 5) Analysis is given for the most well-known free energy representations, StavermanSchwarzl and Rabotnov models. The relationship of the Staverman-Schwarzl representation with spring-dashpot assemblies is demonstrated through the spectral method. Both studied models are shown to be sometimes inconsistent with the admissibility criteria established in the paper, and therefore cannot have universal validity. The Staverman-Schwarzl free energy expression should be considered as a powerful phenomenological model, useful in numerous applications. 6) Separation of the storaged and dissipated energies in viscoelasticity rests a key problem in the establishment of viscoelastic minimum principles and in micromechanics of heterogeneous materials with memory. To resolve it, we must precise the source and quantity of information (additional to the constitutive law), that uniquely defines the free

ENERGY SEPARATIONIN VISCOELASTICITY energy functional of a linear viscoelastic hereditary model. Harmonic regime method is shown to be a powerful tool for such analysis. ACKNOWLEDGEMENT The author would like to acknowledge prof. C. Huet for numerous useful discussions. REFERENCES Breuer, S. and Onat, E.T. (1964) On the determination of free energy in linear viscoelastic solids, J. Appl. Math. Phys., 15, 184-191. Christensen, R. (1982) Theory of viscoelasticity. An Introduction, Academ. Press, N. Y. Coleman, B.D. (1964) Thermodynamics of materials with memory, Arch. Rational

Mech. Anal., 17, 1-46. Coleman, B.D., Owen, D. R. (1970) On the thermodynamics of materials with memory. Arch. Rational Mech. Anal., 36, 245-269. Coussy, O. (1987) Materials with memory and generalized potentials, C. R. Acad. Sci. Paris, t. 305, s6rie II,,765-768. Day, A. (1972) The Thermodynamics of Simple Materials with Fadng Memory, Springer-Verlag, New York. Ditkin, V., Prudnikov, A.(1967) Formulaire pour le calcul op6rationnel, Masson., Paris. Fabrizio, M., Morro, A. (1992) Mathematical problems in linear viscoelasticity, SIAM, Philadelphia. Ferry, John D. (1980) Viscoelastic properties of polymers, John Wiley & Sons, New York. Hazanov, S. (1995) New class of creep-relaxation functions", Intern. J. of Solids and Structures, 32, 2, 165-172. Hazanov, S. (1996) Viscoelastic minimum principles revisited, Transactions ASME, Journal of Applied Mechanics, v. 63, 551-554. Huet, C. (1992) Minimum principles for viscoelasticity, Europ. J. of Mechanics, A/Solids, 1l, 653-684. Kuiken, Gerard D.C. (1994) Thermodynamics of irreversible processes, John Wiley&Sons, N.Y. Korn, G.A., Korn, T.M. (1968) Mathematical Handbook, Acad.Press, New York. Pipkin, A.C. (1986) Lectures on viscoelasticity theory, Springer-Verlag, New York. Rabotnov, Y. (1980) Elements of Hereditary Solid Mechanics, Mir, Moscow. Staverman, A.J., Schwarzl, F. (1952) Thermodynamics of Viscoelastic Behaviour, Proceedings of the Academy of Sciences, The Netherlands, 55,474. Tschoegl, N. W. (1980) The theory of linear viscoelastic behaviour, Acad.Press, N.-Y.

177