Determination of entropy for special classes of temperature-rate and strain-rate dependent inelastic materials

International Journal of Engineering Science 106 (2016) 220–228

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Determination of entropy for special classes of temperature-rate and strain-rate dependent inelastic materials A. Khoeini, A. Imam∗ Department of mechanical and aerospace engineering, Science and Research branch, Islamic Azad University, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 17 October 2015 Revised 1 May 2016 Accepted 28 June 2016 Available online 7 July 2016 Keywords: Continuum mechanics Thermomechanics Entropy Temperature-rate Strain-rate

a b s t r a c t With the use of a reduced energy equation and the Green–Naghdi dissipation inequality, the entropy function is determined in terms of the Helmholtz free energy and other functions that are known or can be determined for certain classes of materials with temperature-rate and strain-rate dependent thermomechanical constitutive response functions. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction The first law of thermodynamics, also known as the balance of energy, asserts the equivalence of the mechanical energy and heat and their conservation for every material volume of the body in every thermomechanical process. In the first law, it is assumed that mechanical energy can be transformed into heat and conversely, with no restrictions placed on such transformations. Experimentally it is known that transformation of mechanical energy into heat (for example, through friction) is not limited by any restrictions, however, the reverse transformation of heat into mechanical energy has certain limitations. This fact is presented in the form of a number of statements commonly referred to as “the second law of thermodynamics”. It is often stipulated that the various statements of the second law are equivalent (e.g., Pippard, 1966; Planck, 1945; Schmidt, 1949; ter Haar & Wergeland, 1966; Zemansky, 1996). However, efforts to place exact restrictions on the exchange of heat and mechanical energy in the context of continuum mechanics have been rather controversial, although certain measure of agreement related to specific materials has been achieved. Many of the differences appear to be due to the concept of entropy; none of the statements of the second law involve entropy or appear to lead to its existence, except in special cases. Many researchers in continuum mechanics have postulated the existence of a scalar field, called “entropy”, and so far a number of different approaches have been proposed which incorporate the concept of entropy and lead to a form of the second law of thermodynamics (Naghdi, 1980). Others have attempted to show the existence of the entropy for certain classes of materials (Casey, 2005, 2011). The role of the second law becomes particularly important with respect to the behavior of dissipative materials wherein restrictions are placed on the constitutive response functions of the material. One of the earliest such attempts is due to Coleman and Noll (1963) where they proposed the Clausius–Duhem inequality as a statement of the second law for ∗

Corresponding author. E-mail addresses: [email protected] (A. Khoeini), [email protected] (A. Imam).

http://dx.doi.org/10.1016/j.ijengsci.2016.06.007 0020-7225/© 2016 Elsevier Ltd. All rights reserved.

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221

every admissible thermomechanical processes. Subsequently, with the help of this inequality restrictions were placed on the constitutive relations for some classes of materials (e.g., Acharya & Shawki, 1996; Carlson, 1972; Green & Naghdi, 1965, 1966). Some restrictions of this kind do seem to embody concepts in various statements of the second law, making the Clausius–Duhem inequality widely regarded as representing the second law of thermodynamics. Although the results obtained by using this approach often receive widespread acceptance, the inequality has been the subject of some criticism (e.g., Day, 1972, 1977; Green & Naghdi, 1977, 1978a). Some writers in the field of thermomechanics assume the validity of the Clausius–Duhem inequality with the entropy assumed to be a primitive quantity despite the fact that no standard of entropy exists, nor does any procedure exist by which the entropy of a given system could be compared with that of an entropy standard, if such existed (Rivlin, 1986). Some other writers assume the entropy to be a functional of the process history up to the present instant (Rivlin, 1986). If the constitutive response functions are independent of temperature-rate, by employing the Clausius–Duhem inequality, entropy as a function can be determined in terms of the Helmholtz free energy. But if the constitutive response functions are dependent on the time derivative of temperature, entropy cannot be determined from the Clausius–Duhem inequality and remains undetermined. Green and Laws (1972) proposed a generalization of the Clausius–Duhem inequality wherein the absolute temperature was replaced by a function requiring a constitutive equation, which was later called thermodynamic temperature by some writers (Suhubi, 1975). They assumed that the thermodynamic temperature is positive and at states of equilibrium reduces to the absolute temperature. In some works through the use of this inequality, the entropy is determined in terms of the thermodynamic temperature and the Helmholtz free energy for materials whose constitutive response functions include dependence on the first and higher time derivatives of the temperature (Batra, 1976; Green & Lindsay, 1972; Suhubi, 1975, 1982). However, no prescription is given for determining the thermodynamic temperature itself much reminiscent of the case where the entropy was introduced as a primitive quantity. Green and Naghdi (1977, 1978a) proposed a dissipation inequality embodying certain aspects of the second law of thermodynamics which was not subject to the apparent limitations of the Clausius–Duhem inequality. Some researchers employed this inequality to place restrictions on the constitutive response functions (e.g., Green & Naghdi, 1978b, 1978c, 1984; Rubin, 1986, 2015; Johnson & Imam, 1999). The aim of the present paper is to determine the entropy for some classes of temperature-rate and strain-rate dependent inelastic materials. For this purpose the reduced energy equation and the Green–Naghdi inequality are employed to obtain an expression for the entropy in terms of the Helmholtz free energy and other response functions that are known or otherwise can be determined. 2. Basic equations 2.1. Conservation laws and energy equation Consider a finite body B with material points X, and identify the material point X in B with its position X in a fixed reference configuration. A motion of the point X at any time t is defined by a differentiable vector function χ in the form of x = χ(X, t ). In the present configuration at time t, the body B occupies a region of space R bounded by a closed surface ∂ R. Similarly, in the present configuration, an arbitrary subset p⊆B of the body B occupies the region p, bounded by a closed surface ∂ p. Then, the deformation gradient tensor F, the velocity vector at time t, the right Cauchy–Green deformation tensor C, the Lagrangian strain tensor E are given by

F=

∂χ , ∂X

v=

∂χ = x˙ , ∂t

C = FTF,

E=

1 (C − I ) 2

(2.1)

where FT denotes the transpose of F. The conservation laws of classical continuum mechanics for mass, momentum and moment of momentum which can be stated for every material part of B occupying a region p in the present configuration, under suitable continuity assumptions, have the local form

ρ˙ + ρ div v = 0

(2.2a)

div T + ρ b = ρ v˙

(2.2b)

T = TT

(2.2c)

where the following relationship holds

t = Tn

(2.2d)

In Eqs. (2.2a) to (2.2d), ρ is the mass density in the present configuration, b is the external body force per unit mass acting on the body B in the present configuration, t is the stress vector, n is the outward unit normal to the surface ∂ p, T is

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the stress tensor with TT as its transpose, and div stands for the divergence operator with respect to the place x, keeping t fixed. Next, the thermal properties of the body are defined: the absolute temperature at each material point is represented by a scalar field θ = (X, t ) > 0 and the existence of the following functions are admitted r = r (X, t ): external rate of supply of heat per unit mass, −h = −h(X, t; n ): heat flux into the body across the surface ∂ p measured per unit area of the surface ∂ p per unit time. Further, it is recalled that

∂ , ∂x

g=

go =

∂ = FTg ∂X

(2.3)

where g denotes the spatial temperature gradient vector and go the referential temperature gradient vector. Additionally, the quantities s=s(X,t) and k = k(X, t ) are defined by

r

s=

(2.4a)

θ

k=

h

(2.4b)

θ

In the above equations s is the external rate of supply of entropy per unit mass and k is the surface flux of entropy per unit area of ∂ p per unit time. Further, the existence of the following scalar fields is assumed (Naghdi, 1980; Green & Naghdi, 1977). η = ηˆ (X, t ) : entropy per unit mass, ξ = ξˆ (X, t ) : internal rate of production of entropy per unit mass, ε = εˆ (X, t ) : internal energy per unit mass. For a body susceptible to both mechanical and thermal effects, the balance of entropy is postulated in the following form (Green & Naghdi, 1977; Naghdi, 1980; Green & Naghdi, 1991).

d dt



p

ρη dv =



p

ρ (s + ξ )dv −



∂p

k da

(2.5)

By usual procedures, using Eq. (2.5), it can be shown that (Green & Naghdi, 1977) k is linear in n, i.e.

k = p·n

(2.6)

where p is called the entropy flux vector. Then, Eqs. (2.4b) and (2.6) imply that h = θ p · n wherein the spatial heat flux vector q is defined by

q = θp

(2.7)

The referential and spatial heat flux vectors are related by

F qo = Jq

(2.8)

where J = ρρo = det F > 0 and ρ o is the mass density in the reference configuration. with suitable continuity assumptions Eq. (2.5) yields

ρ η˙ = ρ (s + ξ ) − div p

(2.9)

With reference to the present configuration, the first law of thermodynamics can be written as

d dt



p

ρ

 1 2

   v · v + ε dv = ρ (r + b · v)dv + (t · v − h ) da p

∂p

(2.10)

Making use of Eqs. (2.2a) to (2.2d), (2.4b) and (2.6), with suitable continuity assumptions, the first law in Eq. (2.10) can be cast in the local form

ρ ε˙ = ρ r + T · D − div (θ p)

(2.11)

where D is the rate of deformation tensor1 . Using Eqs. (2.4a) and (2.9) to eliminate the external field r in Eq. (2.11) yields

−ρ (ε˙ − θ η˙ ) + T · D − ρξ θ − p · g = 0

(2.12)

Eq. (2.12) is referred to as the reduced energy equation (Naghdi, 1980; Green & Naghdi, 1977). Introducing the Helmholtz free energy ψ = ψˆ (X, t ) per unit mass by

ψ = ε − θη

(2.13)

The reduced energy Eq. (2.12) may be written in the alternative form

−ρ 1



 ˙ ψ˙ + θη + T · D − ρξ θ − p · g = 0

The inner product of two second order tensors is defined as A · B = tr (ABT ) where BT is the transpose of B.

(2.14)

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2.2. The second law of thermodynamics For elastic materials, the response functions ψ , η and ɛ depend only on the deformation gradient F and the temperature θ , and are independent of their rates and the temperature gradient g. Such materials are considered non-dissipative, while if the constitutive response functions ψ , η and ɛ also include dependence on any of the variables F˙ , θ˙ , g or their higher space and time derivatives, the material is said to be dissipative. It is now assumed that the constitutive response functions ψ , η and ɛ also include dependence on the list of variables F˙ , θ˙ , g and their higher space and time derivatives, which are referred to collectively as a set . Further, let ψ  , η , ɛ denote the respective values of ψ , η and ɛ when the set is put equal to zero in the response functions. Thus, for example,

ε = εˆ (θ , F , )

(2.15a)

ε  = ε (θ , F ) = εˆ (θ , F , 0 )

(2.15b)

= (θ˙ , F˙ , g, . . . )

(2.15c)

When the is set equal to zero, Eq. (2.13) becomes

ψ  = ε  − θ η

(2.16)

Green and Naghdi (1977, 1978a) proposed the following inequality embodying certain aspects of the second law of thermodynamics.





−ρ ε˙ − θ η˙  + T · D ≥ 0

(2.17)

Henceforth the above inequality is referred to as the Green–Naghdi inequality. 3. Entropy determination In the foregoing the existence of entropy as a scalar function in the thermomechanical response of the material has been assumed. In this Section methods for determining the entropy are developed by employing the reduced energy equation and the Green–Naghdi inequality. For a body it is convenient to introduce the symmetric Piola-Kirchhoff stress tensor S, and its relation to the Cauchy stress tensor as

J T · D = S · E˙

F SF T = J T ,

(3.1)

where the superposed dot represents material time derivative. By employing Eq. (3.1) and Jq · g = qo · go , the reduced energy Eq. (2.12) and the inequality (2.17) can be recast in the reference configuration as

−ρo (ε˙ − θ η˙ ) + S · E˙ − ρo ξ θ −





1

θ

qo · go = 0

−ρo ε˙ − θ η˙  + S · E˙ ≥ 0

(3.2) (3.3)

3.1. Classes of materials whose constitutive response functions include dependence on higher time derivatives of temperature, strain-rate, and rate of temperature gradient The constitutive response functions for the class of materials considered in this subsection may be written as

  ... ε = εˆ θ , θ˙ , θ¨ , θ , E, E˙ , go , g˙ o ,

ε  = εˆ (θ , E )

(3.4a)

  ... η = ηˆ θ , θ˙ , θ¨ , θ , E, E˙ , go , g˙ o ,

η = ηˆ (θ , E )

(3.4b)



S = Sˆ

...

θ , θ˙ , θ¨ , θ , E, E˙ , go , g˙ o



  ... ξ = ξˆ θ , θ˙ , θ¨ , θ , E, E˙ , go , g˙ o

(3.4c) (3.4d)

Using Eq. (3.4a), the inequality (3.3) is expanded to yield



−ρo

∂ εˆ ∂ εˆ ˙ ∂ εˆ ¨ ∂ εˆ ... ∂ εˆ ∂ εˆ ˙ ∂ εˆ ¨ ∂ εˆ ... θ+ θ+ + ·E+ ·E+ · g˙ + · g¨ + ρo θ η˙  + S · E˙ ≥ 0 θ+ ∂θ ∂ go o ∂ g˙ o o ∂ θ ∂E ∂ E˙ ∂ θ¨ ∂ θ˙

(3.5)

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.... ¨ ¨ For given values of the variables which .... occur in the argument of εˆ in Eq. (3.4a), the rates θ , E and go and may be chosen ¨ arbitrarily; and since the coefficients of θ , E and g¨o in inequality (3.5) are independent of these rates it is concluded that

∂ εˆ ... = 0, ∂θ

∂ εˆ = 0, ∂ E˙

∂ εˆ = 0 ⇒ ε = εˆ (θ , θ˙ , θ¨ , E, go ) ∂ g˙ o

(3.6)

The reduced energy Eq. (3.2) is assumed to hold for all thermomechanical processes. Using Eqs. (3.4a) to (3.4d) and (3.6) and the same argument as above, it follows that

∂ ηˆ ... = 0, ∂θ

∂ ηˆ = 0, ∂ E˙

∂ ηˆ = 0 ⇒ η = ηˆ (θ , θ˙ , θ¨ , E, go ) ∂ g˙ o

(3.7)

Eq. (3.7) indicates that the entropy is independent of E˙ , thereby setting E˙ = 0 in the inequality (3.5) yields

∂ εˆ = 0, ∂ θ¨

  ∂ εˆ = 0 ⇒ ε = εˆ θ , θ˙ , E ∂ go

if

E˙ = 0

(3.8)

The inequality (3.5) obtains for E˙ = 0



−ρo

∂ εˆ ˙ ∂ εˆ ¨ θ+ θ + ρo θ η˙  ≥ 0 ∂θ ∂ θ˙

(3.9)

The above inequality must holds for all thermomechanical processes, therefore

∂ εˆ = 0 ⇒ ε = εˆ (θ , E ) = ε  ∂ θ˙

if

E˙ = 0

(3.10)

Using Eqs. (2.13), (2.16) and (3.10) it follows that

˙  ε˙ = ε˙  = ψ˙  + θ η˙  + θη

(3.11)

Substituting Eq. (3.11) into the inequality (3.3), recalling that ψ  = ψˆ  (θ , E ) and setting E˙ = 0 it is found that



−ρo

∂ ψˆ  + η θ˙ ≥ 0 ∂θ

(3.12)

The inequality (3.12) holds for all thermomechanical processes, thus

η = −

∂ ψˆ  ∂θ

(3.13)

Also using Eqs. (2.13), (2.16) and (3.10), it may be concluded that

η=

 1  ψ − ψ + θ η

(3.14)

θ

which if combined with Eq. (3.13) yields

 ∂ ψˆ   η= ψ −ψ −θ θ ∂θ 1

(3.15)

The above expression for the entropy was obtained under the condition that E˙ = 0 but since entropy is independent of E˙ , as deduced in Eq. (3.7) , the expression obtained in Eq. (3.15) is in general valid and is not limited to the processes with vanishing strain-rate. Further, if the constitutive response functions in Eqs. (3.4a) to (3.4d) depend only on the variables (θ , θ˙ , E, E˙ , go ), by

ψ employing the method presented herein it can be shown that ε = ε  , ψ = ψ  , η = η and the entropy is given by η = − ∂∂θ . It does not appear that for classes of materials considered in this subsection the Clausius–Duhem inequality alone could be used to find an expression for the entropy. In addition, it should be noted that Eq. (3.15) remains valid even if the response functions in Eqs. (3.4a) to (3.4d) includethe fourth and higher order time derivatives of temperature and/or the second and higher order time derivatives of temperature gradient. However, for classes of materials whose constitutive response functions are independent of the time derivative of temperature, starting with inequality ( 3.3 ), by a similar reasoning as ˆ

ψ above, it can be concluded that the entropy is η = − ∂∂θ ; a result that can also be obtained by employing the Clausius– Duhem inequality2 . From Eqs. (2.13), (3.6) and (3.7) it is found that ψ = ψˆ (θ , θ˙ , θ¨ , E, go ), and making use of Eqs. (2.13), (3.2) and (3.15) the internal rate of production of entropy is determined to be ˆ



ρo ξ θ = −ρo

θ˙ ∂ ψˆ ˙ ∂ ψˆ ¨ ∂ ψˆ ... ∂ ψˆ ˙ ∂ ψˆ θ+ θ+ + ·E+ · g˙ o + θ ¨ ∂θ ∂E ∂ go θ ∂θ ∂ θ˙



ψ − ψ − θ

∂ ψˆ  ∂θ



+ S · E˙ −

1

θ

qo · go

(3.16)

2 By employing the reduced energy equation (3.2) alone it can be shown that the entropy is η = − ∂∂θψ for classes of materials whose constitutive response functions are independent of the time derivative of temperature. ˆ

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225

From (2.13) and (3.15) the internal energy is given as

ε = ψ − θ

∂ ψˆ  ∂θ

(3.17)

Eq. (3.17) shows that the internal energy for the materials considered above is ε = εˆ (θ , E ) = ε  . 3.2. Classes of materials whose constitutive response functions include dependence on temperature-rate, rate of strain-rate, and temperature gradient The constitutive response functions considered in this subsection3 are of the form:

  ε = εˆ θ , θ˙ , E, E˙ , E¨ , go ,

ε  = εˆ (θ , E )

(3.18a)

  η = ηˆ θ , θ˙ , E, E˙ , E¨ , go ,

η’ = η’ (θ , E )

(3.18b)



S= S θ , θ˙ , E, E˙ , E¨ , go



(3.18c)

  ξ = ξˆ θ , θ˙ , E, E˙ , E¨ , go

(3.18d)

Employing (3.18a), inequality (3.3) is expanded to yield



−ρo

∂ εˆ ˙ ∂ εˆ ¨ ∂ εˆ ˙ ∂ εˆ ¨ ∂ εˆ ... ∂ εˆ θ+ θ+ ·E+ ·E+ ·E+ · g˙ + ρo θ η˙  + S · E˙ ≥ 0 ∂θ ∂E ∂ go o ∂ E¨ ∂ E˙ ∂ θ˙

(3.19)

From the above inequality it follows that

∂ εˆ = 0, ∂ θ˙

∂ εˆ = 0, ∂ E¨

  ∂ εˆ = 0 ⇒ ε = εˆ θ , E, E˙ ∂ go

(3.20)

The reduced energy Eq. (3.2) is assumed to hold for all thermomechanical processes which, considering Eq. (3.20), implies

∂ ηˆ = 0, ∂ θ˙

∂ ηˆ = 0, ∂ E¨

  ∂ ηˆ = 0 ⇒ η = ηˆ θ , E, E˙ ∂ go

(3.21)

In general, it is assumed that the constitutive response functions for ψ , η and ɛ also include dependence on the list of variables E˙ , θ˙ , go and their higher space and time derivatives in the set . In the case above it is further assumed that when is set equal to zero, i.e., when the response functions depend only on θ and E, the thermomechanical processes take place on a surface where there is no dissipation and for dissipation to occur the thermomechanical processes must lie outside of that surface where the set is no longer zero. This surface is shown in Fig. 1, where processes take place reversibly such as those associated with dissipation-less media, i.e., elastic materials, whose response functions depend only on θ and E. Using Eqs. (2.13) and (2.16) it follows that

η=

1

θ

ε − ε  + ψ  − ψ + θ η



(3.22)

where the superposed prime represents the value of the response function on the surface without dissipation as shown in Fig. 1. The process shown in Fig. 1 by the solid line starts at point m in the dissipation-less surface and ends at the point n. Thus, by determining the quantity (ε − ε  ), entropy can be obtained from Eq. (3.22) in general since it is assumed that ˆ

ψ 4 the function ψ is known and it was shown in Eq. (3.13) that η = − ∂∂θ . Further it is assumed that at time to , the material is in a state indicated by the point m on the dissipation-less surface where = 0 and at time t1 the material is in a state indicated by the point n where = 0. The process5 between the two points is shown schematically in Fig. 1 by a solid line.

3 It can be shown that for inelastic materials whose response functions are independent of the rate of strain rate, entropy is obtained from Eq. (3.15) and the internal energy for all thermomechanical processes, ε = εˆ (θ , E ) = ε  , obtained from Eq. (3.17). 4 It can be shown that for a large class of materials ε = ε  , hence Eq. (3.22) reduces to Eq. (3.15). Also for classes of materials whose constitutive response

functions are independent of temperature rate, from Eq. (3.22) it can be shown that η = − ∂∂θψ which is in agreement with the result obtained by using the Clausius-Duhem inequality for this class of materials. 5 θ and E at the end points n and m are equal. ˆ

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Fig. 1. surface without dissipation and a dissipative process originating from point m on this surface and ending at point n.

Using Eq. (3.20), dɛ is expanded as

∂ εˆ ∂ εˆ ∂ εˆ dθ + · dE + · dE˙ ∂θ ∂E ∂ E˙

dε =

(3.23)

The above equation can be integrated over the process shown in Fig. 1 to yield

ε − ε =



n m

dε =



n m

∂ εˆ dθ + ∂θ



n m

∂ εˆ · dE + ∂E



n m

∂ εˆ · dE˙ ∂ E˙

(3.24)

The quantity (ε − ε  ) being independent of path of integration, may be evaluated using the vertical path from m to n ∂ εˆ = 0, ∂ εˆ = 0 and Eq. (3.24) is reduced to along which changes of θ and E do not exist; therefore ∂θ ∂E

ε − ε =



∂ εˆ · dE˙ ∂ E˙

n m

(3.25)

To evaluate the integrand ∂ε˙ inequality (3.19) can be used. However in order to determine the entropy, in what follows ∂E attention is limited to the case where S is linear in E¨ , i.e.,



S=λ

     θ , θ˙ , E, E˙ , go trE¨ I + 2μ θ , θ˙ , E, E˙ , go E¨ + S θ , θ˙ , E, E˙ , go

(3.26)

Where S¯ is a tensor function of (θ , θ˙ , E, E˙ , go ), μ and λ are in general scalar functions of (θ , θ˙ , E, E˙ , go ). Using Eq. (3.26), from the inequality (3.19) and Eq. (3.20) it follows that



−ρo

   ∂ εˆ ∂ εˆ ˙ ∂ εˆ ˙ θ+ · E − θ η˙  − ρo − λ I · E˙ I − 2μ E˙ · E¨ + S¯ · E˙ ≥ 0 ∂θ ∂E ∂ E˙

(3.27)

Since inequality (3.27) holds for all thermomechanical processes, we may choose E¨ arbitrarily resulting in



∂ εˆ 1  = λ I · E˙ I + 2μE˙ ∂ E˙ ρo

(3.28)

Substituting Eq. (3.28) in Eq. (3.25) yields

ε − ε =

1

ρo



n m

 n   2 λ tr E˙ I · dE˙ + μE˙ · dE˙ ρo m

(3.29)

Substituting Eq. (3.29) in Eq. (3.22), the entropy at point n is found to be6

η=

1

θ



1

ρo



n

m

λ





tr E˙ I · dE˙ +

2

ρo



n m

μ

E˙ · dE˙ + ψ  − ψ − θ

∂ ψˆ  ∂θ



(3.30)

λ and μ are scalar function of θ , θ˙ , E, E˙ , go , but it was shown that η = ηˆ (θ , E, E˙ ), indicating that the sum of two integral in Eq. (3.30) must be independent of θ˙ and go . 6

ˆ

ψ . This is because θ and E at points n and m are equal and it was shown in Eq. (3.13) that η = − ∂∂θ

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227

Recalling that for this class of materials ψ = ψˆ (θ , E, E˙ ), and using Eqs. (2.13), (3.2) and (3.30), the internal rate of production of entropy is determined to be



∂ ψˆ ˙ ∂ ψˆ ˙ ∂ ψˆ ¨ ρo ξ θ = −ρo θ+ ·E+ ·E ∂θ ∂E ∂ E˙   n n   θ˙ ˙ ˙ − λ trE I · dE + 2 μ E˙ · dE˙ θ m m   ˆ 1 ψ ∂ + ρo ψ  − ψ − θ + S · E˙ − qo · go ∂θ θ

(3.31)

The Lagrangian strain tensor E and its rate can be decomposed into their so-called spherical and deviatoric parts as

E = E∗ +

1 (trE ) I 3

(3.32a)

E˙ = E˙ ∗ +

1 (trE˙ ) I 3

(3.32b)

where E∗ denotesthe deviatoricpart of the strain tensor. For the special case where λ and μ in Eq. (3.26) are independent of E˙ , i.e.,

      θ , θ˙ , E, go trE¨ I + 2μ θ , θ˙ , E, go E¨ + S θ , θ˙ , E, E˙ , go

S=λ

it follows that

1



ρo

n m

   λ n  ˙  ˙∗ 1  ˙  λ trE˙ I · dE˙ = tr E I · d E + tr E I ρo m 3  n    n    1    λ λ = tr E˙ I · d E˙ ∗ + tr E˙ I · d tr E˙ I ρo m ρo m 3 λ  ˙  ˙ ∗ 1  ˙ 2 n λ  ˙ 2 = tr E I · E + tr E = tr E ρo 2 2 ρo m

(3.33)

(3.34)

Finally, substituting Eqs. (3.34) in Eq. (3.29), the entropy at the arbitrary point n for the special case where λ and μ are independent of E˙ is found to be

η=

1



θ



1

ρo

 ˆ ψ ∂ λ  ˙ 2  tr E + μE˙ · E˙ + ψ − ψ − θ 2 ∂θ

(3.35)

And the internal rate of production of entropy per unit mass is given as



∂ ψˆ ˙ ∂ ψˆ ˙ ∂ ψˆ ¨ θ+ ·E+ ·E ∂θ ∂E ∂ E˙ 

 2 θ˙ λ ∂ ψˆ   ˙ ˙ ˙ − (tr E ) + μE · E + ρo ψ − ψ − θ θ 2 ∂θ

ρo ξ θ = −ρo

+ S · E˙ −

1

θ

qo · go

(3.36)

As an example, we consider a material whose stress response function is given by Eq. (3.33) and the Helmholtz free energy function is given as (Green & Naghdi, 1993)

1 2

1 2

ρo ψ = λ∗ (tr E )2 + μ∗ tr (E )2 − c

Eβ ∗ θ2 − θ tr E θo 3 ( 1 − 2 ν )

(3.37)

where λ∗ and μ∗ are Lame constants, E is the Young’s modulus of elasticity, ν is the Poisson’s ratio, θ o is the reference temperature,c is the specific heat and β ∗ is the coefficient of volume expansion. Then the entropy and the internal rate of production of entropy are determined, according to Eqs. (3.35) and (3.36), to be

ρo η =

1



θ

ρo ξ θ = −λ

θ Eβ ∗ λ  ˙ 2 tr E + μE˙ · E˙ + c + tr E 2 θo 3 ( 1 − 2 ν )

∗

(tr E ) I · E˙ − 2μ∗ E · E˙ +

Eβ ∗θ θ˙ I · E˙ − 3 ( 1 − 2ν ) θ

(3.38)



1 λ  ˙ 2 ˙ ˙ trE + μE · E + S · E˙ − qo · go 2 θ

(3.39)

228

A. Khoeini, A. Imam / International Journal of Engineering Science 106 (2016) 220–228

4. Conclusion Many researchers in continuum mechanics assume the existence of entropy as a functional of the process history up to the present instant and attempt to determine it for many materials in terms of other known functions. However, if the constitutive response functions depend on the time derivative of temperature and its higher time derivatives, employing the Clausius–Duhem inequality, as an embodiment of the second law of thermodynamics, does not enable one to determine the entropy explicitly and hence it remains an unknown function. By employing the generalized Clausius-Duhem inequality proposed by Green and Laws, the entropy can be determined in terms of a thermodynamic temperature and the Helmholtz free energy function for such materials. However, there does not appear to be any way of determining the thermodynamic temperature either a priori or through a prescribed procedure. In the present paper it has been shown that by employing the Green–Naghdi inequality, embodying certain aspects of the second law of thermodynamics, and the reduced energy equation the entropy function can be determined explicitly in terms of the Helmholtz free energy and other known functions for some classes of inelastic materials whose response functions depend on the temperature-rate and the rate of strain. References Acharya, A., & Shawki, T. G. (1996). The Clausius–Duhem inequality and the structure of rate-independent plasticity. International Journal of Plasticity, 12(2), 229–238. Batra, R. C. (1976). Thermodynamics of non-simple elastic materials. Journal of Elasticity, 6, 451–456. Carlson, D. E. (1972). Linear thermoelasticity. In S. Flügge (Ed.). In Handbuch der physik: vol. VIa/2 (pp. 297–345). Berlin: Springer. Coleman, B. D., & Noll, W. (1963). The thermodynamics of elastic materials with heat conduction and viscosity. Archive for Rational Mechanics and Analysis, 13, 167–177. Casey, J. (2005). A remark on Cauchy-elasticity. International Journal of Non-Linear Mechanics, 40, 331–339. Casey, J. (2011). Nonlinear Thermoelastic materials with viscosity, and subject to internal constraints: A classical continuum thermodynamics approach. Journal of Elasticity, 104, 91–104. Day, W. A. (1972). The thermodynamics of simple materials with fading memory. Springer tracts in natural philosophy: Volume 22. Berlin: Springer-Verlag Inc. Day, W. A. (1977). An objection to using entropy as a primitive concept in continuum thermodynamics. Acta Mechanica, 27, 251–255. Green, A. E., & Naghdi, P. M. (1965). A general theory of an elastic-plastic continuum. Archive for Rational Mechanics and Analysis, 18, 251–281. Green, A. E., & Naghdi, P. M. (1966). A thermodynamic development of elastic-plastic continua. In L. I. Sedov, & H. Parkus (Eds.), Irreversible aspects of continuum mechanics and transfer of physical characteristics in moving fluids (pp. 117–131). Springer: Wien-New\sYork. Green, A. E., & Laws, N. (1972). On entropy production inequality. Archive for Rational Mechanics and Analysis, 45, 47–53. Green, A. E., & Lindsay, K. A. (1972). Thermoelasticity. Journal of Elasticity, 2, 1–7. Green, A. E., & Naghdi, P. M. (1977). On thermodynamics and the nature of the second law. In Proceedings of the Royal Society of London: A357 (pp. 253–270). Green, A. E., & Naghdi, P. M. (1978a). The Second law of thermodynamics and cyclic processes. Journal of Applied Mechanics, 45, 487–492. Green, A. E., & Naghdi, P. M. (1978b). On thermodynamics and the nature of the second law for mixtures of interacting continua. Quarterly Journal of Mechanics and Applied Mathematics, 31, 265–293. Green, A. E., & Naghdi, P. M. (1978c). On thermodynamic restrictions in the theory of elastic-plastic materials. Acta Mechanica, 30, 157–162. Green, A. E., & Naghdi, P. M. (1984). Aspects of the second law of thermodynamics in the presence of electromagnetic effects. Quarterly Journal of Mechanics and Applied Mathematics, 37, 179–193. Green, A. E., & Naghdi, P. M. (1991). A demonstration of consistency of an entropy balance with balance of energy. Journal of Applied Mathematics and Physics (ZAMP), 42, 159–168. Green, A. E., & Naghdi, P. M. (1993). Thermoelasticity without energy dissipation. Journal of Elasticity, 31, 189–208. Johnson, G. C., & Imam, A. (1999). On the determination of entropy from a scalar potential for special inelastic materials. In J. Casey, & R. Abeyaratne (Eds.), Finite thermoelasticity (AMD vol. 236) (pp. 97–104). New York: ASME Press. Pippard, A. B. (1966). The elements of classical thermodynamics. Cambridge: Cambridge University Press. Naghdi, P. M. (1980). On the role of the second law of thermodynamics in mechanics of materials. Energy, 5, 771–781. Planck, M. (1945). Treatise on thermodynamics (3th ed.). New York: Dover Publications. Rivlin, R. S. (1986). Reflections on certains aspects of thermomechanics. In G. Grioli (Ed.), Finite thermoelasticity, contributi del centro linceo interdisciplinare di scienze matemitiche e loro applicazione (pp. 11–43). Rome: Accademia Nazionaledei Lincei. Reprinted: Rivlin, R. S. (1997). Reflections on certains aspects of thermomechanics. In: G.L., Barenblatt, & D. Joseph (Eds.), collected papers of R. S. Rivlin (pp. 430-463). New York: Springer-Verlag Inc. Rubin, M. B. (1986). An elastic-viscoplastic model for large deformation. International Journal of Engineering Science, 24(7), 1083–1095. Rubin, M. B. (2015). An Eulerian formulation of nonlinear thermomechanics and electrodynamics of moving anisotropic elastic solids. International Journal of Engineering Science, 97, 69–83. Schmidt, E. (1949). Thermodynamics. Oxford: Clarendon Press. Suhubi, E. S. (1975). Thermoelastic solids. In A. C. Eringen (Ed.). In Continuum physics: Vol.2 (pp. 191–265). New York: Academic Press. Suhubi, E. S. (1982). A generalized theory of simple thermomechanical materials. International Journal of Engineering Science, 20(2), 365–371. ter Haar, D., & Wergeland, H. (1966). Elements of thermodynamics (7th ed.). New York: Addison-Wesey. Zemansky, M. W. (1996). Heat and thermodynamics. New York: McGraw-Hill.