Chapter 3 Thermodynamics In order to complete the formulation we need to express the stress tensor T and the heat-flux vector q in terms of other variables. These expressions are known as constitutive relations. As we will see in this Chapter, these relations are not completely arbitrary, as the second principle of Thermodynamics imposes certain restrictions on them. 3.1 econd Principle of Thermodynamics The state of a continuum is defined by certain state variables. Two state variables were already introduced, i.e., the internal energy, e, and the density, ρ. Another is the specific volume τ = 1 ρ (3.1) Other state variables include the temperature,, and the entropy,, which are defined below. The second principle of thermodynamics states that there exist two variables, the temperature,, and the entropy,, that satisfy the following condition d 1 ρ d dt r d 1 q n d (3.2) where q is the heat-flux vector introduced in the preceding chapter. The equal sign holds only for reversible phenomena. We will see that this equation yields, without any further major assumptions, that the heat flows towards regions of lower temperature, and other important facts, such as the structure of the stress tensors for certain specialized continua to be defined later (i.e., viscous fluids and elastic solids). Note that, using Gauss theorem and the second Reynolds transport theorem, Eq. 3.2, may be 31
32 rewritten as ρ D Dt d 1 ( q ) r d + div d 0 (3.3) In the limit, as the volume goes to zero, one obtains ρ D Dt 1 ( q ) r + div = ρ D Dt 1 r + 1 div q + q grad ( ) 1 0 (3.4) which is the differential form of the second principle of thermodynamics. Equation 3.4 may be combined with the conservation law for thermodynamic energy, to yield ρ De Dt ρ D Dt ρde Dt 1 = r div q + T : D (3.5) q grad + T : D 0 (3.6) 3.2 Thermodynamics of olids We define a solid to be a continuum for which the state is defined by the strain tensor, E, and by a thermodynamic variable, such as the entropy or the temperature. Note that the strain tensor, E, has the same role that the density has for fluid (recall that the trace of the strain rate tensor is div v, which is related to the density through the continuity equation). It is convenient to use as the primary variables, the entropy,, and the strain tensor, E. As the fundamental equation of state we will use the equation relating the internal energy, e, to the primary state variables, and E, e = e(, E) (3.7) Introducing the following notations e = e (3.8) E e E = e E (3.9) one obtains De Dt = e D Dt + e E : DE Dt (3.10)
33 Combining this with Eq. 3.6 and recalling that (see ection 1.3) one obtains 1 or, DE Dt = D (3.11) ρ ( e ) D Dt 1 q grad + T : D ρe E : D 0 (3.12) ρ ( e ) D Dt 1 q grad + (T ρe E) : D 0 (3.13) We will restrict ourselves to those solids in which the condition is satisfied individually by each term. Consider first the second term which yields q grad 0 (3.14) On the other hand, the third yields ( T ρ e ) E : D 0 (3.15) which indicates that T rev := ρ e E (3.16) is the portion of the stress tensor that corresponds to a reversible phenomenon. Finally, consider the first term. Noting that, both e and are state variables, and that, for a given state the sign of D/Dt is arbitrary, we deduce that = e (3.17) E 3.2.1 Entropy Evolution Equation Note that combining Eq. 3.10 and 3.17 with Eq. 3.11 one obtains De Dt = D Dt + e E : D (3.18) The entropy evolution equation is obtained by combining this equation with the differential form of the first law of thermodynamics, Eq. 3.5, ρ D Dt = r div q + ( T ρ e E ) : D (3.19) This indicates that the entropy rate is caused by the local heat generation, the local heat flux, and the work (always positive) done by the irreversible portion of the stress tensor T irr := T ρ e. 1 The following conclusions can be reached more generally considering a nonlinear kinematics, i.e., taking into account a more general formula instead of Eq. 3.11 and introducing the concept of Piola-Kirchhoff stress tensor, see ection 1.4. E
34 3.2.2 Elastic olids A solid is called elastic if there exists a function ê (elastic energy) such that T = ρ dê de (3.20) For instance, if q = 0, r = 0, and T irr = 0, then = constant (Eq. 3.19) and hence Eq. 3.16 is identified with Eq. 3.20, with ê e. 2 For this type of material the work done by the internal stresses, I T = T : Dd (3.22) may be expressed as the time derivative, at constant entropy, of the elastic energy, which is defined as E = For, combining this with Eqs. 3.20 and 1.56 one obtains I T = ρ e E : DE Dt d = ρ De d = d Dt dt where E := ρ e d (3.23) ρ e d = de dt (3.24) ρe d (3.25) is the internal energy associated to the continuum with material volume. Note that in this case, considering the Eqs. 3.22 and 3.24, the conservation of mechanical energy given by Eq. 2.30 becomes ( ) d v 2 ρ dt 2 + e d = ρf v d + n Tv d (3.26) Next, consider the Cauchy law in the static case multiplied by an arbitrary virtual displacement δu and then integrated in the material volume of the solid, i.e., (div T + ρf) δu d = 0 (3.27) 2 Also, if we can assume = 0 = constant (i.e., the case of infinitive termal conductivity in the solid, no presence of heat sources and boundary conditions = 0 = constant, if T irr = 0, one could obtain T = ρ A E (3.21) where A is the free energy previously introduced and the elastic energy ê coincides with A.
35 Integrating by part and using that yields ( T : δe + ρf δu) d + grad δu = δe (3.28) t δud = 0 (3.29) If one consider an elastic solid, in the same conditions presented at the beginning of this ection, one obtains ρ e E : δed + ρf δud + t δud = 0 (3.30) Moreover, as the energy e depends only on the strain tensor E one has also ρ e E : δed = ρδe d = δ ρe d = δe (3.31) Finally, as t and f are external loads and if they are, for instance, dependent by the position only, then a potential function for the external loads W e will exists such that δw e := ρf δud t δud (3.32) Therefore, Eq. 3.30 can be finally rewritten as δ (E + W e ) = 0 (3.33) which states that, for static problems, the total energy of an elastic solid defined as E + W e is stationary with respect to any arbitrary δu compatible with the prescribed constraints of the elastic solid. 3.2.3 Alternative Formulation Free Energy * The thermodynamic formulation for solids is often expressed in terms of the Helmholtz free energy A = e (3.34) (see Green and Naghdy, 1964). In this subsection, we outline this formulation and show the equivalence to the internal energy formulation. Combining Eqs. 3.34 and 3.6, one obtains ρ D Dt ρda Dt 1 q grad + T : D 0 (3.35)
36 The temperature,, and the strain tensor, E, are used as the primary variables. The fundamental equation of state is that relating the Helmholtz free energy, A, to the primary state variables, and E, A = A(, E) (3.36) Introducing the following notations A = A (3.37) E A E = A E (3.38) one obtains DA Dt = A D Dt + A E : DE Dt (3.39) Combining this with Eq. 3.35 and using Eqs. 3.11, one obtains ρ ( + A ) DT Dt 1 T q grad + ( ˆT + ρae ) : ˆD 0 (3.40) We will restrict ourselves to those solids in which the condition is satisfied individually by each term. Consider the first term. Note that, both A/ and are state variables, and that for a given state of A/ and we may choose D/Dt arbitrarily. Therefore we may deduce that = A (3.41) E The second term yields q grad 0 (3.42) and the third yields ( ˆT ρ A ) E : ˆD 0 (3.43) which indicates that A/ E is the portion of the stress tensor that corresponds to a reversible phenomenon. If the process is isothermal we have that the equal sign holds in Eq. 3.43. This implies ˆT = ρ A E (3.44)
37 For this type of process the work done by the internal stress I T = ρt : D d = ρ ˆT : ˆD d (3.45) may be expressed as the time derivative, at constant temperature, of the elastic energy, which is defined as E = For, combining this with Eqs. 3.44 and 3.11 one obtains I T = ρ A DE E Dt d = ρ DA d = D Dt Dt 3.3 Constitutive equations for isotropic solids ρ A d (3.46) ρ A d = DE Dt (3.47) In this section some essential topics on the constitutive relations for solids will be discussed: these relations will be supposed to be T = T(E) (3.48) and, by definition, these laws denote intrinsic proprieties of the continua. 3.3.1 Linearly-elastic and Hookean solids If a solid is elastic (see ec. 3.2.2) the internal energy e and, indeed, also the density ρ are function of the strain state given by E. This implies that in a neighborhood of a zero strain-state one has ρ = ρ 0 + ρ ϵ lm ϵ lm +... and e = e 0 + ˆτ ij ϵ ij + 1 0 2 ˆb ijkl ϵ ij ϵ kl +... (3.49) Therefore, in the same hypothesis of elastic solid, the stress tensor is given by 3 τ ij = ρ e ( = ρ 0 + ρ ϵ ij ϵ lm +...) (ˆτ ij + ˆb ) ijkl ϵ kl +... 0 ϵ lm (3.50) The previous expansion has been obtained considering the following symmetry property for ˆb ijlm 3 Note that, in general, ˆbijlm = ˆb lmij (3.51) e ε rs = 1 2 ˆb ijkl δ riδ sj ε kl + 1 2 ˆb ijkl ε ij δ rk δ sj = 1 2ˆb rskl ε kl + 1 2ˆb ijrsε ij
38 This essential property arises from the existence of the internal energy e: indeed, this implies for the second derivatives with respect to the state-space variables ϵ ij the chwarz identity which, using Eq. 3.49, gives Eq. 3.51. 2 e ϵ ij ϵ lm = 2 e ϵ lm ϵ ij (3.52) If the solid is linearly elastic, then one can generally consider the truncated expansion ( ) ρ τ ij = ρ 0ˆτ ij + ϵ lm ˆτ ij + ρ 0ˆbijkl ϵ kl = ρ 0ˆτ ij + ρ 0 b ijkl ϵ kl (3.53) 0 ϵ lm with b ijkl := ˆb ijkl + 1 ρ ρ 0 ˆτ ij (3.54) 0 ϵ kl where it is apparent that the contribute associated to the term ˆτ ij is due to a pre-stress condition, i.e., a stress condition that exists also when E = 0. If no pre-stress condition is present in the linearly elastic solid, then Eq. 3.53 becomes τ ij = C ijkl ϵ kl (3.55) where C ijkl := ρ 0ˆbijkl. Note that the symmetry property given by Eq. 3.51 also implies in this case for the elastic operator the internal constraint C ijlm = C lmij (3.56) Furthermore, combining in the same hypotheses Eq. 3.55 with Eq. 3.49 one has, avoiding any unessential constant (i.e., ρ 0 and e 0 ) ρe = 1 2 C ijklϵ ij ϵ kl = 1 2 τ ij ϵ ij (3.57) which means that for linearly elastic solid with no pre-stressed condition at zero strain, the elastic energy is given by E := ρed = 1 T : E d (3.58) 2 As further comment, it is worth to point out that if there is no pre-stress condition (i.e., ˆτ ij = 0) and if the term of higher order than the second of the elastic energy are neglected, then for a linearly elastic solid it can be assumed in the constitutive prescription ρ = ρ 0 = constant.
39 Note that the number of constants characterizing the elastic solid is generally equal to 81. However, the number can be reduced considering the some symmetry properties related to the tensor C ijkm. First, as previously mentioned, if the solid is linearly elastic, Eq. 3.20 becomes ρ 0 e ε ij C ijkm ε km (3.59) We have already pointed that the chwarz identity given by Eqs. 3.51-3.52, implies also the property given by Eq. 3.56. Furthermore, another symmetry property is a consequence of the symmetry of the stress tensor T: indeed, if C jikm ε km = τ ji = τ ij = C ijkm ε km, then C ijkm = C jikm (3.60) Finally, combining Eqs. 3.56 and 3.60 one has C ijkm = C kmij = C mkij = C ijmk (3.61) If one consider the relationships 3.60 and 3.61, the number of the essential constant relating the six essential components of the strain tensor and the six essential components of the stress tensor should reduce from 81 to 21. This have suggested the engineering notation representing the tensor E with the six component vector v ε and the tensor T with the six component vector v τ linearly related by the 6 6 simmetric matrix C such that v τ = Cv ε (3.62) Note that the symmetry property for the matrix C implies again that the number of the essential elastic constants defining a linearly elastic solid reduces to 21. A Hookean, namely, isotropic solid is a special case of a linearly-elastic solid and may be defined in a similar way as the tokesian fluid: i.e., on the base of a supposed isotropy, one may assume that a proportionality relationship between the corresponding spheric and deviatoric portions of the tensors T and E exists. This implies that 4 T = c 1 E (3.63) T D = c 2 E D (3.64) where c 1 and c 2 are two constants depending upon the solid type. Note that in this case no thermodynamic consequences can be done a priori on these constants. 4 Note that the constitutive relation for solids should be prescribed for ˆT and not for T: however, as a linearized relationship between stresses and strains has been obtained, the distinction between stress tensor and Piola-Kirchhoff stress vanishes.
40 However, one obtains T = c 1 E + c 2 E D = (c 1 c 2 )E + c 2 E = (c 1 c 2 ) 1 3 I + c 2E (3.65) where is the trace of the tensor E. If one defines the following Lamé constants c 2 2 = µ c 1 c 2 3 the following final relationship for Hookean solid can be obtained: = λ (3.66) T = λi + 2µE (3.67) Note that two essential constants are necessary to define constitutively a Hookean solid. Also in this case, similar constitutive relationship can be obtained developing by the linear algebra four hypothesis for the Hookean solid similar to those done on the tokesian fluid in the previous ection. Note that, considering Eq. 3.58, one has 5 E := ρed = 1 T : E = 1 [λ (ε 11 + ε 22 + ε 33 ) δ ij + 2µε ij ] ε ij M 2 M 2 ( ) M λ = 2 2 + µ E : E d (3.71) M that represents the internal (elastic) energy of an elastic hookean body of material volume M in term deformation tensor. 6 Note that for an Hookean solid the energy E is positive if (and only if) λ > 0 and µ > 0: this implies that (see Eq. 3.72 and µ G) E > 0 G > 0 0 < ν < 1 2 (3.73) 5 Another way to demonstrate the Eq. XX is that shown in the following. Equations 3.20 and Eq. 3.67 becomes, in terms of Carthesian co-ordinates ρ e ε ij = λ(ε 11 + ε 22 + ε 33)δ ij + 2µε ij (3.68) As ε ij are the state-space variable of the solid, assuming that ρ does not practically depend on the ε ij, there exists the exact differential 3 e de = dε ij (3.69) ε ij i,j Then, assuming that e εij =0 = 0, one obtains (e.g., integrating on a Carthesian path from the origin to a generic point in the state-space ε ij, or alternatively, using Eq. 3.58) ρe = 1 2 τijεij = λ 2 (ε11 + ε22 + ε33)2 + µ 3 ε 2 ij (3.70) i.e., integrating on a generic material volume M and using absolute notation one obtain the Eq. 3.71. 6 For the case of a purely axial rod with rod axis, for instance, in direction 1, rod length l rod, and rod section area A, the strain state is described by the strain entries ε 11, ε 22, and ε 33 only. Then, Eq. 3.71 yields (ε 22 = νε 11 i,j
41 where E, G, and ν are the Young modulus, the shear modulus and the Poisson coefficient respectively. and ε 33 = νε 11): E rod := ρed = M M ε 2 11 [ λ 2 (1 2ν)2 + µ ( 1 + 2ν 2) ] d = 1 2 lrod 0 EA ε 2 11dx 1 where ν and E are the Poisson coefficient and the Young Modulus respectively, and the relationships have been used. λ = νe (1 + ν)(1 2ν) µ = E 2(1 + ν) (3.72)
42
Bibliography [1] errin, J., Mathematical Principles of Classical Fluid Mechanics, in Ed.:. Fluegge, Encyclopedia of Physics, ol. III/1, pp.125 263, 1959. [2] Green, A. E., and Naghdy, P. M., A General Theory of an Elastic Plastic Continuum, Archives of Rational Mechanics, ol. 18, pp. 251-281, 1964. [3] Green and Rivlin, Archives of Rational Mechanics, ol. 17, 113 1964 [4] Morino, L. Mastroddi, F., Introduction to Theoretical Aeroelasticity for Aircraft Design, in preparation. 43
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