International Journal of Solids and Structures 41 (2004) 7377–7398 www.elsevier.com/locate/ijsolstr
On thermodynamic potentials in linear thermoelasticity V.A. Lubarda *
Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411, USA Received 13 March 2004; received in revised form 28 May 2004 Available online 13 August 2004
Abstract
The four thermodynamic potentials, the internal energy u ¼ uð ij; sÞ, the Helmholtz free energy f ¼ f ð ij; T Þ, the Gibbs energy g ¼ gðrij; T Þ and the enthalpy h ¼ hðrij; sÞ are derived, independently of each other, by using the Duh- amel–Neumann extension of Hooke’s law and an assumed linear dependence of the specific heat on temperature. A systematic procedure is then presented to express all thermodynamic potentials in terms of four possible pairs of independent state variables. This procedure circumvents a tedious transition from one potential to another, based on the formal change of variables, and inversions of the stress–strain and entropy–temperature relations. The general results are applied to uniaxial loading paths under isothermal, adiabatic, constant stress, and constant strain conditions. An interplay of adiabatic and isothermal elastic constants in the expressions for exchanged heat along certain ther- modynamic paths is indicated. Ó 2004 Elsevier Ltd. All rights reserved.
Keywords: Thermoelasticity; Thermodynamic potentials; Internal energy; Helmholtz free energy; Gibbs energy; Enthalpy; Entropy; Specific heats
1. Introduction
The structure of the constitutive equations relating the stress, strain, entropy, and temperature in linear thermoelasticity is well-known (e.g., Boley and Weiner, 1960; Sneddon, 1974). Most commonly, the deri- vation of these equations proceeds by assuming a quadratic representation of the Helmholtz free energy in terms of strain and temperature, with the coefficients specified in accord with the observed isothermal elastic behavior, the coefficient of thermal expansion, and the specific heat. This yields (e.g., Kovalenko, 1969) 0 1 2 cV 2 f ð ij; T Þ¼ kT kk þ l ij ij jT a0ðT T0Þ kk ðT T0Þ s0ðT T0Þþf0; ð1Þ 2 2T0
* Tel.: +1-858-534-3169; fax: +1-858-534-5698. E-mail address: [email protected] (V.A. Lubarda).
0020-7683/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2004.05.070 7378 V.A. Lubarda / International Journal of Solids and Structures 41 (2004) 7377–7398 where kT and l are the isothermal Lame elastic constants, jT ¼ kT þ 2l=3 is the isothermal bulk modulus, 0 and a0, cV , and s0 are, respectively, the coefficient of volumetric thermal expansion, the specific heat at constant strain, and the specific entropy, all in the reference state with temperature T0. The corresponding free energy (per unit reference volume) is f ð0; T0Þ¼f0. The stress and entropy in the deformed state are the gradients of f with respect to strain and temperature. The specific heat at constant strain, associated with Eq. (1) is os o2f T c ¼ T ¼ T ¼ c0 : ð2Þ V o o 2 V T T T0 Once the Helmholtz free energy is specified as a function of strain and temperature, the internal energy u ¼ f þ Ts can be expressed in terms of the same independent variables by simple substitution of Eq. (1) and the corresponding expression for the entropy. This yields (e.g., Ziegler, 1977)
0 1 2 cV 2 2 uð ij; T Þ¼ kT kk þ l ij ij þ jT a0T0 kk þ ðT T0 Þþu0: ð3Þ 2 2T0 In the sequel, it will be assumed that the internal energy vanishes in the reference state, so that
u0 ¼ 0; f0 ¼ T0s0: ð4Þ However, the internal energy is a thermodynamic potential whose natural independent state variables are strain and entropy, rather than strain and temperature, since the energy equation specifies the increment of internal energy as
du ¼ rij d ij þ T ds: ð5Þ
The desired representation u ¼ uð ij; sÞ can be obtained from u ¼ f þ Ts by eliminating the temperature in terms of strain and entropy via Eq. (1). The end result is
1 2 jT a0T0 T0 2 uð ij; sÞ¼ kS kk þ l ij ij 0 ðs s0Þ kk þ 0 ðs s0Þ þ T0ðs s0Þ: ð6Þ 2 cV 2cV
The adiabatic (isentropic) Lame constant kS is related to its isothermal counterpart kT by
2 a0T0 2 kS ¼ kT þ 0 jT : ð7Þ cV The purely algebraic transition from Eqs. (3)–(6) is simple, but little indicative of the underlying ther- modynamics connecting Eqs. (5) and (6). An independent derivation of (6), starting from the energy equation (5), and utilizing the experimental data embedded in the Duhamel–Neumann extension of Hooke’s law, and the assumed specific heat behavior, is therefore desirable. The systematic procedure to achieve this, and to derive the expressions for other thermodynamic potentials, the Helmholtz free energy f ¼ f ð ij; T Þ, the Gibbs energy g ¼ gðrij; T Þ, and the enthalpy h ¼ hðrij; sÞ, is presented. The four ther- modynamic potentials are then expressed in terms of four possible pairs of independent state variables: ð ij; T Þ, ð ij; sÞ, ðrij; T Þ,andðrij; sÞ. This furnishes a set of 16 alternative expressions, four for each ther- modynamic potential. Analogous results in the scalar setting, using pressure and volume as pertinent variables, is commonly utilized in materials science thermodynamics (Swalin, 1972; DeHoff, 1993; Ragone, 1995). The obtained general results are applied to uniaxial and spherical stress and strain states, which are of importance in high-pressure material testing (e.g., Lubarda, 1986; Lubarda et al., 2004). Particular attention is given to uniaxial loading under isothermal, adiabatic, constant stress, and constant strain condition. A simple interplay of adiabatic and isothermal elastic constants in the expressions for exchanged heat along certain thermodynamic paths is obtained. V.A. Lubarda / International Journal of Solids and Structures 41 (2004) 7377–7398 7379
2. Thermodynamics potentials in terms of their natural independent state variables
The four thermodynamic potentials are derived in this section in terms of their natural independent state variables. The derivation is in each case based only on the Duhamel–Neumann extension of Hooke’s law, and an assumed linear dependence of the specific heat on temperature.
2.1. Internal energy u ¼ u( ij; s)
The increment of internal energy is expressed in terms of the increments of strain and entropy by the energy equation (5). Since u is a state function, du is a perfect differential, and the Maxwell relation holds orij oT o ¼ o : ð8Þ s ij s
The thermodynamic potential u ¼ uð ij; sÞ is sought corresponding to the Duhamel–Neumann expression
rij ¼ kT kkdij þ 2l ij jT a0ðT T0Þdij; ð9Þ and an assumed linear dependence of the specific heat on temperature
0 T cV ¼ cV : ð10Þ T0 The Kronecker delta is denoted by d . By partial differentiation from Eq. (9) it follows that ij orij orij oT oT o ¼ o o ¼ jT a0 o dij; ð11Þ s T s s so that the Maxwell relation (8) gives oT oT o ¼ jT a0 o dij: ð12Þ ij s s The thermodynamic definition of the specific heat at constant strain is os cV ¼ T o ; ð13Þ T which, in conjunction with Eq. (10), specifies the temperature gradient oT T ¼ 0 : ð14Þ o 0 s cV The substitution into Eq. (12) yields oT j a T ¼ T 0 0 d : ð15Þ o 0 ij ij s cV The joint integration of Eqs. (14) and (15) provides
jT a0T0 T0 T ¼ 0 kk þ 0 ðs s0ÞþT0: ð16Þ cV cV When this is inserted into Eq. (9), we obtain an expression for stress in terms of strain and entropy
jT a0T0 rij ¼ kS kkdij þ 2l ij 0 ðs s0Þdij: ð17Þ cV 7380 V.A. Lubarda / International Journal of Solids and Structures 41 (2004) 7377–7398
The adiabatic Lame constant kS is related to its isothermal counterpart kT by Eq. (7). By using Eqs. (16) and (17), the joint integration of ou ou rij ¼ o ; T ¼ o ; ð18Þ ij s s yields a desired expression for the internal energy in terms of its natural independent variables ij and s. This is
1 2 jT a0T0 T0 2 uð ij; sÞ¼ kS kk þ l ij ij 0 ðs s0Þ kk þ 0 ðs s0Þ þ T0ðs s0Þ: ð19Þ 2 cV 2cV
2.2. Helmholtz free energy f ¼ f ( ij; T )
An independent derivation of the Helmholtz free energy function f ¼ f ð ij; T Þ again begins with the pair of expressions (9) and (10). The increment of f is
df ¼ rij d ij sdT ; ð20Þ with the Maxwell relation orij os o ¼ o : ð21Þ T ij T
By evaluating the temperature gradient of stress from Eq. (9), and by substituting the result into Eq. (21), we find os o ¼ jT a0dij: ð22Þ ij T
The integration of above, in conjunction with o 0 s cV o ¼ ; ð23Þ T T0 provides the entropy expression 0 cV s ¼ jT a0 kk þ ðT T0Þþs0: ð24Þ T0 By using Eqs. (9) and (24), the joint integration of of of rij ¼ o ; s ¼ o ; ð25Þ ij T T yields a desired expression for the Helmholtz free energy in terms of its natural independent variables ij and T . This is 0 1 2 cV 2 f ð ij; T Þ¼ kT kk þ l ij ij jT a0ðT T0Þ kk ðT T0Þ s0T : ð26Þ 2 2T0 V.A. Lubarda / International Journal of Solids and Structures 41 (2004) 7377–7398 7381
2.3. Gibbs energy g ¼ g(rij; T )
The increment of the Gibbs energy is
dg ¼ ij drij sdT ; ð27Þ with the Maxwell relation o ij os o ¼ o : ð28Þ T r rij T
To derive the function gðrij; T Þ, independently of the connection g ¼ f rij ij and without tedious change of variables, we begin with the thermoelastic stress–strain relation and an expression for the specific heat, i.e., 1 mT a0 ij ¼ rij rkkdij þ ðT T0Þdij; ð29Þ 2l 1 þ mT 3
0 T cP ðT Þ¼cP : ð30Þ T0 The first one is a simple extension of Hooke’s law to include thermal strain, and the second one is the assumed linear dependence of the specific heat at constant stress on temperature. The thermodynamic definition of the specific heat cP is os cP ¼ T o : ð31Þ T r By differentiating Eq. (29) to evaluate the temperature gradient of strain, and by substituting the result into the Maxwell relation (28), we find os a0 o ¼ dij: ð32Þ rij T 3 The integration of this, in conjunction with o 0 s cP o ¼ ; ð33Þ T r T0 provides the entropy expression 0 a0 cP s ¼ rkk þ ðT T0Þþs0: ð34Þ 3 T0 Using Eqs. (29) and (34), the joint integration of og og ij ¼ o ; s ¼ o ; ð35Þ rij T T r yields a desired expression for the Gibbs energy in terms of its natural independent variables rij and T (e.g., Fung, 1965; Kovalenko, 1969). This is 0 1 mT 2 a0 cP 2 gðrij; T Þ¼ rijrij rkk ðT T0Þrkk ðT T0Þ s0T : ð36Þ 4l 1 þ mT 3 2T0 7382 V.A. Lubarda / International Journal of Solids and Structures 41 (2004) 7377–7398
0 0 The relationship between the specific heats cP and cV can be obtained in various ways. For example, by reconciling the entropy expressions (24) and (34), and by using the relationship 1 kk ¼ rkk þ a0ðT T0Þ; ð37Þ 3jT following from (27), it is found that 0 0 2 cP cV ¼ jT a0T0: ð38Þ
2.4. Enthalpy function h ¼ h(rij; s)
The increment of enthalpy is
dh ¼ ij drij þ T ds; ð39Þ with the Maxwell relation o ij oT o ¼ o : ð40Þ s r rij s
To derive the function hðrij; sÞ, we shall again begin with the expressions (29) and (30). By partial dif- ferentiation from Eq. (29) it follows that o ij o ij oT a0 oT o ¼ o o ¼ o dij: ð41Þ s r T r s r 3 s r The substitution into the Maxwell relation (40) gives oT a oT a T ¼ 0 d ¼ 0 0 d : ð42Þ o o ij 0 ij rij s 3 s r 3cP The definition (31), in conjunction with (30), was used in the last step. The joint integration of Eq. (42) and oT T ¼ 0 ; ð43Þ o 0 s r cP provides the temperature expression
a0T0 T0 T ¼ 0 rkk þ 0 ðs s0ÞþT0: ð44Þ 3cP cP When this is substituted into Eq. (29), there follows 1 mS a0T0 ij ¼ rij rkkdij þ 0 ðs s0Þdij: ð45Þ 2l 1 þ mS 3cP
The adiabatic Poisson’s ratio mS is related to its isothermal counterpart mT by 2 mT þ 2lð1 þ mT Þa mS 2lð1 þ mSÞa a0T0 mS ¼ ; mT ¼ ; a ¼ 0 : ð46Þ 1 2lð1 þ mT Þa 1 þ 2lð1 þ mS Þa 9cP Note that the adiabatic and isothermal Young’s moduli are related by 2 1 1 a0T0 ¼ 0 ; ð47Þ ET ES 9cP V.A. Lubarda / International Journal of Solids and Structures 41 (2004) 7377–7398 7383 in variance with an expression given by Fung (1965, p. 389), where the specific heat at constant strain appears in the denominator on the right-hand side. A simple relationship is also recorded (e.g., Chadwick, 1960)
0 cP jS 0 ¼ : ð48Þ cV jT
This easily follows by noting, from Eqs. (16) and (44), that for adiabatic loading
jT a0T0 a0T0 T0 T ¼ 0 kk ¼ 0 rkk: ð49Þ cV 3cP
Since for adiabatic loading rkk ¼ 3jS kk, the substitution into (49) yields (48). Returning to the enthalpy function, by using Eqs. (44) and (45), the joint integration of oh oh ij ¼ o ; T ¼ o ; ð50Þ rij s s r yields the expression for the enthalpy in terms of its natural independent variables r and s. This is ij 1 mS 2 a0T0 T0 2 hðrij; sÞ¼ rijrij rkk 0 ðs s0Þrkk þ 0 ðs s0Þ þ T0ðs s0Þ: ð51Þ 4l 1 þ mS 3cP 2cP
2.5. List of thermodynamic potentials in terms of their natural independent state variables
1 2 jT a0T0 T0 2 uð ij; sÞ¼ kS kk þ l ij ij 0 ðs s0Þ kk þ 0 ðs s0Þ þ T0ðs s0Þð52Þ 2 cV 2cV
0 1 2 cV 2 f ð ij; T Þ¼ kT kk þ l ij ij jT a0ðT T0Þ kk ðT T0Þ s0T ð53Þ 2 2T0 0 1 mT 2 a0 cP 2 gðrij; T Þ¼ rijrij rkk ðT T0Þrkk ðT T0Þ s0T ð54Þ 4l 1 þ mT 3 2T0 1 mS 2 a0T0 T0 2 hðrij; sÞ¼ rijrij rkk 0 ðs s0Þrkk þ 0 ðs s0Þ þ T0ðs s0Þð55Þ 4l 1 þ mS 3cP 2cP
3. Internal energy in terms of four pairs of independent state variables
The internal energy was expressed in Section 2.1 in terms of its natural independent state variables as u ¼ uð ij; sÞ. We now derive its representation in terms of other three pairs of independent state variables.
3.1. Function u ¼ u( ij; T )
We start with the relationship
uð ij; T Þ¼f ð ij; T ÞþTs: ð56Þ 7384 V.A. Lubarda / International Journal of Solids and Structures 41 (2004) 7377–7398
The partial derivative with respect to strain at constant temperature is ou of os o ¼ o þ T o : ð57Þ ij T ij T ij T Since of os orij rij ¼ o ; o ¼ o ; ð58Þ ij T ij T T we obtain ou orij o ¼ rij T o : ð59Þ ij T T By using the Duhamel–Neumann expression (9), this gives ou o ¼ kT kkdij þ 2l ij þ jT a0T0dij: ð60Þ ij T On the other hand, by taking a partial derivative of Eq. (56) with respect to temperature at constant strain, there follows ou os T T c0 ; 61 o ¼ o ¼ V ð Þ T T T0 o o having regard to s ¼ ð f = T Þ . Therefore, upon joint integration of Eqs. (60) and (61), there follows 0 1 2 cV 2 2 uð ij; T Þ¼ kT kk þ l ij ij þ jT a0T0 kk þ ðT T0 Þ: ð62Þ 2 2T0
This could have been also obtained directly from the relationship u ¼ f ð ij; T ÞþTs, by using (24) and (26) (e.g., Ziegler, 1977, p. 118; Haddow and Ogden, 1990, p. 165, 167). The corresponding expression for 0 internal energy in the case when it is assumed that cV ¼ cV can be found in Chadwick (1960, p. 275), or Noda et al. (2003, p. 445).
3.2. Function u ¼ u(rij; s)
We conveniently start with the relationship
uðrij; sÞ¼hðrij; sÞþrij ij: ð63Þ The partial differentiation with respect to stress at constant entropy is ou o kl o ¼ rkl o ; ð64Þ rij s rij s since ¼ ðoh=or Þ . Incorporating Eq. (45), the above becomes ij ij s ou 1 mS o ¼ rij rkkdij : ð65Þ rij s 2l 1 þ mS On the other hand, by taking a partial derivative of Eq. (63) with respect to entropy at constant stress, there follows ou oh o a T ¼ þ r ij ¼ T þ 0 0 r ; ð66Þ o o ij o 0 kk s r s r s r 3cP V.A. Lubarda / International Journal of Solids and Structures 41 (2004) 7377–7398 7385 having regard to T ¼ðoh=osÞ . Since T is given by Eq. (44), this reduces to r ou T ¼ T þ 0 ðs s Þ: ð67Þ o 0 0 0 s r cP The joint integration of Eqs. (65) and (67) gives 1 mS 2 T0 2 uðrij; sÞ¼ rijrij rkk þ 0 ðs s0Þ þ T0ðs s0Þ: ð68Þ 4l 1 þ mS 2cP
3.3. Function u ¼ u(rij; T )
The derivation in this case proceeds by starting with the relationship
uðrij; T Þ¼gðrij; T Þþrij ij þ Ts: ð69Þ The partial differentiation with respect to stress at constant temperature is ou o kl o ij o ¼ rkl o þ T o : ð70Þ rij T rij T T r The connections were used og os o ij ij ¼ o ; o ¼ o : ð71Þ rij T rij T T r Incorporating Eq. (29) into (70) yields ou 1 mT a0 o ¼ rij rkkdij þ T dij: ð72Þ rij T 2l 1 þ mT 3 On the other hand, the partial derivative of Eq. (69) with respect to temperature at constant stress is ou o os a T r ij T 0 r c0 ; 73 o ¼ ij o þ o ¼ kk þ P ð Þ T r T r T r 3 T0 o o having regard to s ¼ ð g= T Þr. The joint integration of Eqs. (72) and (73) yields a desired expression 0 1 mT 2 a0 cP 2 2 uðrij; T Þ¼ rijrij rkk þ T rkk þ ðT T0 Þ: ð74Þ 4l 1 þ mT 3 2T0
3.4. List of internal energy functions in terms of four pairs of independent state variables
1 2 jT a0T0 T0 2 uð ij; sÞ¼ kS kk þ l ij ij 0 ðs s0Þ kk þ 0 ðs s0Þ þ T0ðs s0Þð75Þ 2 cV 2cV
0 1 2 cV 2 2 uð ij; T Þ¼ kT kk þ l ij ij þ jT a0T0 kk þ ðT T0 Þð76Þ 2 2T0 1 mS 2 T0 2 uðrij; sÞ¼ rijrij rkk þ 0 ðs s0Þ þ T0ðs s0Þð77Þ 4l 1 þ mS 2cP 7386 V.A. Lubarda / International Journal of Solids and Structures 41 (2004) 7377–7398 0 1 mT 2 a0 cP 2 2 uðrij; T Þ¼ rijrij rkk þ T rkk þ ðT T0 Þð78Þ 4l 1 þ mT 3 2T0
4. Helmholtz free energy in terms of four pairs of independent state variables
The Helmholtz free energy was expressed in Section 2.2 in terms of its natural independent state vari- ables as f ¼ f ð ij; T Þ. We now derive its representation in terms of other three pairs of independent state variables. The derivation is in many aspects analogous to that presented for the internal energy in Section 3. We accordingly record only its essential steps.
4.1. Function f ¼ f ( ij; s)
We start with the relationship
f ð ij; sÞ¼uð ij; sÞ Ts; ð79Þ to obtain of orij o ¼ rij s o ; ð80Þ ij s s and of j a T ¼ k d þ 2l þ T 0 0 s d : ð81Þ o S kk ij ij 0 0 ij ij s cV
On the other hand, of oT T ¼ s ¼ 0 s: ð82Þ o o 0 s s cV The joint integration of Eqs. (81) and (82) gives
1 2 jT a0T0 T0 2 2 f ð ij; sÞ¼ kS kk þ l ij ij þ 0 s0 kk 0 ðs s0Þ T0s0: ð83Þ 2 cV 2cV
4.2. Function f ¼ f (rij; T )
We start with the relationship
f ðrij; T Þ¼gðrij; T Þþrij ij; ð84Þ to obtain of o kl o ¼ rkl o ; ð85Þ rij T rij T V.A. Lubarda / International Journal of Solids and Structures 41 (2004) 7377–7398 7387 and of 1 mT o ¼ rij rkkdij : ð86Þ rij T 2l 1 þ mT
On the other hand, o 0 f a0 cP o ¼ s þ rkk ¼ ðT T0Þ s0: ð87Þ T r 3 T0
The joint integration of Eqs. (86) and (87) gives 0 1 mT 2 cP 2 f ðrij; T Þ¼ rijrij rkk ðT T0Þ s0T : ð88Þ 4l 1 þ mT 2T0
4.3. Function f ¼ f (rij; s)
We start with the relationship
f ðrij; sÞ¼hðrij; sÞþrij ij Ts; ð89Þ to obtain of o kl o ij o ¼ rkl o þ s o ; ð90Þ rij s rij s s r and of 1 m a T ¼ r S r d þ 0 0 sd : ð91Þ o ij kk ij 0 ij rij s 2l 1 þ mS 3cP On the other hand, of o oT T a ¼ r ij s ¼ 0 0 r s : ð92Þ o ij o o 0 kk s r s r s r cP 3 The joint integration of Eqs. (91) and (92) gives 1 mS 2 a0T0 T0 2 2 f ðrij; sÞ¼ rijrij rkk þ 0 srkk 0 ðs s0Þ T0s0: ð93Þ 4l 1 þ mS 3cP 2cP
4.4. List of Helmholtz free energy functions in terms of four pairs of independent state variables
0 1 2 cV 2 f ð ij; T Þ¼ kT kk þ l ij ij jT a0ðT T0Þ kk ðT T0Þ s0T ð94Þ 2 2T0
1 2 jT a0T0 T0 2 2 f ð ij; sÞ¼ kS kk þ l ij ij þ 0 s0 kk 0 ðs s0Þ T0s0 ð95Þ 2 cV 2cV 7388 V.A. Lubarda / International Journal of Solids and Structures 41 (2004) 7377–7398 0 1 mT 2 cP 2 f ðrij; T Þ¼ rijrij rkk ðT T0Þ s0T ð96Þ 4l 1 þ mT 2T0 1 mS 2 a0T0 T0 2 2 f ðrij; sÞ¼ rijrij rkk þ 0 srkk 0 ðs s0Þ T0s0 ð97Þ 4l 1 þ mS 3cP 2cP
5. Gibbs energy in terms of four pairs of independent state variables
The Gibbs energy was expressed in Section 2.3 in terms of its natural independent state variables g ¼ gðrij; T Þ. We now derive its representation in terms of other three pairs of independent state variables, recording only essential steps.
5.1. Function g ¼ g(rij; s)
We start with the relationship
gðrij; sÞ¼hðrij; sÞ Ts; ð98Þ to obtain og o ij o ¼ ij þ s o ; ð99Þ rij s s r and og 1 m a T ¼ r S r d þ 0 0 s d : ð100Þ o ij kk ij 0 0 ij rij s 2l 1 þ mS 3cP On the other hand, og oT T ¼ s ¼ 0 s: ð101Þ o o 0 s r s r cP The joint integration of Eqs. (100) and (101) gives 1 mS 2 a0T0 T0 2 2 gðrij; sÞ¼ rijrij rkk þ 0 s0rkk 0 ðs s0Þ T0s0: ð102Þ 4l 1 þ mS 3cP 2cP
5.2. Function g ¼ g( ij; T )
We start with the relationship
gð ij; T Þ¼f ð ij; T Þ rij ij; ð103Þ to obtain og orkl o ¼ kl o ; ð104Þ rij T ij T V.A. Lubarda / International Journal of Solids and Structures 41 (2004) 7377–7398 7389 and og o ¼ ðkT kkdij þ 2l ijÞ: ð105Þ ij T On the other hand, o o 0 g rij cV o ¼ s ij o ¼ s þ jT a0 kk ¼ ðT T0Þ s0: ð106Þ T T T0
The joint integration of Eqs. (105) and (106) gives 0 1 2 cV 2 gð ij; T Þ¼ kT kk þ l ij ij ðT T0Þ s0T : ð107Þ 2 2T0
5.3. Function g ¼ g( ij; s)
We start with the relationship
gð ij; sÞ¼uð ij; sÞ rij ij Ts; ð108Þ to obtain og orkl orij o ¼ kl o s o ; ð109Þ ij s ij s s and og j a T ¼ ðk d þ 2l Þþ T 0 0 sd : ð110Þ o S kk ij ij 0 ij ij s cV On the other hand, og or oT T ¼ ij s ¼ 0 ðj a sÞ: ð111Þ o ij o o 0 T 0 kk s s s cV The joint integration of Eqs. (110) and (111) gives 1 2 jT a0T0 T0 2 2 gð ij; sÞ¼ kS kk þ l ij ij þ 0 s kk 0 ðs s0Þ T0s0: ð112Þ 2 cV 2cV
5.4. List of Gibbs energy functions in terms of four pairs of independent state variables