Elasticity: Thermodynamic Treatment André Zaoui, Claude Stolz

Home , Elastic energy, Rubber elasticity

To cite this version:

André Zaoui, Claude Stolz. Elasticity: Thermodynamic Treatment. Encyclopedia of Materials: Sci- ence and Technology, Elsevier Science, pp.2445-2449, 2000, 10.1016/B0-08-043152-6/00436-8. hal- 00112293

HAL Id: hal-00112293 https://hal.archives-ouvertes.fr/hal-00112293 Submitted on 31 Oct 2018

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Elasticity: Thermodynamic Treatment A similar treatment can be applied to the second principle that relates to the existence of an absolute The elastic behavior of materials is usually described temperature, T, and of a thermodynamic function, the by direct stress–strain relationships which use the fact entropy, obeying a constitutive inequality ensuring that there is a one-to-one connection between the the positivity of the dissipated energy. This leads to the strain and stress tensors ε and σ; some mechanical following local inequality: elastic potentials can then be defined so as to express E G ! the stress tensor as the derivative of the strain potential q r with respect to the strain tensor, or conversely the ρscjdiv k & 0 (2) T T latter as the derivative of the complementary (stress) F H potential with respect to the former. where s is the entropy per unit mass. Elimination of the Nevertheless, these potentials are restricted to thermal source, r, between Eqns. (1) and (2) and use of specific thermodynamic conditions. This purely mech- the free energy per unit mass, f l ekTs, lead to the anical treatment may be generalized and improved by so-called Clausius–Duhem inequality integrating its temperature dependence and by con- necting the elastic potentials with the classical thermo- ! dynamic functions within a consistent thermodynamic q ! Φ l σ : εckρ( fjc sTc )k :]T & 0 (3) framework. This leads to a more general definition of T thermoelasticity (Sect. 1), which can be used for a better understanding of the properties of the elastic which expresses the non-negativity of the dissipation, moduli (Sect. 2), for a natural definition of rubber Φ, per unit volume. Note that the total dissipation, Φ, elasticity (Sect. 3), and for the prediction of the includes both an ‘intrinsic’ part, which is directly effective thermal expansion coefficients of hetero- connected to the! mechanical behavior, and a thermal geneous materials (Sect. 4). part (k(!q\T):]T ), associated with conduction.

1. Thermodynamic Definition of Thermoelasticity 1.2 Thermoelasticity 1.1 General Case Thermoelastic behavior is defined by stating that the The thermodynamic framework relies on the two thermodynamic functions e and s as well as the stress σ fundamental (conservation) classical principles, for tensor, , are one-to-one functions of the independent ε which local expressions can be derived from the global state variables and T. In this! case, the intrinsic ] ones (Germain et al. 1983). Infinitesimal strains as well dissipation does not depend on T in Eqn. (3). This as quasi-static evolutions only are considered. For any means that both the intrinsic and the thermal dis- given body, V, subjected to purely thermodynamic sipation must be non-negative separately. From the processes, the first principle expresses the fact that, definition whatever the mechanical behavior, the variation of the cf cf total (i.e., internal plus kinetic) energy is equilibrated fcl εc j Tc cε ij c by the variation of the mechanical work of the applied ij T forces and by the variation of the heat received by the body. it follows for any evolution (ε0, TI ) from the equilibrium When thermal conduction only occurs through the state (ε, T ): c ! !n boundary V, with q the heat flux vector and the E G E G outward unit normal, and with r denoting the specific cf cf σ kρ εc kρ sj Tc & 0 (4) heat production rate, the global heat rate, Q, is given ij cε ij cT l ! k!c !:! l ! k ! F ij H F H by Q VrdV V q n dS V(r div(q))dV. Tak- ing into account the equilibrium equations, which are The independent variables ε0 and TI may have arbitrary used to connect the stress field with the exterior values, including zero; this results in the following loading, the local expression of the first principle relationships: finally reads cf cf cf ! σ l ρ (ε, T ), i.e., σ l ρ ; s lk (ε, T ) (5) ρ c l σ εcj k cε ij cε c e : r div(q) (1) ij T where ρ is the mass per unit volume and e the internal which are the constitutive equations of thermo- energy per unit mass. Here, variables with a dot above elasticity (Salenc: on 1995). them denote time derivatives and, for second-order The free energy, f(ε, T ), from which the mechanical tensors a and b, the notation a : b represents a : b l behavior can be derived completely for any thermo- Σ l ijaijbij aijbij. dynamic conditions, can be considered as a thermo- 1 dynamic potential, which is more general and powerful The isothermal compliances are given by Eqn. (7); the than the mechanical strain or stress potentials. The adiabatic ones result from Eqn. (6) and from the second expression of Eqn. (5) can be used, in associ- condition ds l 0. When the temperature dependence ation with the heat equation and boundary conditions, of ρ is neglected, the order of magnitude of the to derive the coupled thermomechanical evolution of a difference between Miso and Mad can be estimated by thermoelastic body. assimilating ε to the thermal strain, εth l α∆T, so that cε\c $ α c \c \ α An alternative treatment, relying on the state T ,and s TtoCp T,with beingthethermal σ ε description through the variables (instead of ) and expansion tensor and Cp the specific heat at constant T, can be developed by using the stability of any stress. deformed equilibrium state. It makes use of the This results in potential f * defined by f * l (1\ρ)σ : εkf. The associated constitutive equations are then T M iso kM ad $ α α (10) ijkl ijkl ρC ij kl cf * cf * p ε l ρ (σ, T ), s l (σ, T ) (6) ij cσ c ad ij T (note that M also exhibits diagonal symmetry). For α l αδ δ isotropic materials ( ij ij, where ij is the Kronecker symbol), and usual conditions, this proves 2. Elastic Moduli and Compliances that the relative difference between isotropic and adiabatic compliances hardly exceeds a few percent. 2.1 Symmetry However, this difference, which can only be under- When the potentials f(ε, T )orf *(σ, T ) are quadratic, stood from the thermodynamic treatment of elasticity, the usual case of linear thermoelasticity is recovered: could have significant consequences in some extreme the constitutive equations (Eqns. (5) and (6)) yield a cases (dynamics, very high pressure or temperature linear relationship between ε and σ through the variations, etc.). classical elastic moduli C and compliances S, respectively. When this is not the case, the same equations can be used to connect linearly infinitesimal 3. Crystal vs. Rubber Elasticity variations of stress and strain through ‘‘tangent’’ From Eqn. (5) and the relationship between f, e, and s, L M moduli and compliances . Their thermodynamic the constitutive equations of thermoelasticity can be definition ensures the symmetry of these fourth-order put in the form tensors. For example, for an isothermal evolution, L and M cf ce cs are simply given by the partial derivatives, at constant σ l ρ (ε,T ) l ρ kρT (11) temperature, cσ\cε and cε\cσ. Specification of the cε cε cε indices and use of Eqns (5) and (6) lead to This expression clearly shows that thermoelasticity cσ c#f cε c#f * results from two sources: the strain dependence of the L l ij l ρ , M l ij l ρ (7) ijkl cε cε cε ijkl cσ cσ cσ internal energy and that of the entropy. According to kl ij kl kl ij kl the materials under consideration, the first source or the second can be predominant: Classical properties of second-order derivatives ensure (i) some materials, especially crystals and poly- that the isothermal elastic tangent moduli and com- crystals, cannot deform very much in the elastic range pliances exhibit the diagonal symmetry and there are very small changes of the entropy under L l L , M l M (8) deformation, whereas the strain sensitivity of the ijkl klij ijkl klij internal energy is quite high, owing to the nature of the σ $ ρc \cε This result is, of course, also valid for C and S in the interatomic interactions (so that e ); case of linear elasticity. (ii) other materials, such as elastomers (Mark and Lal 1982), can suffer very large reversible distortions of the macromolecular chains in the rubber elasticity regime, which strongly alters the entropy, almost 2.2 Isothermal and Adiabatic Compliances without modification of the internal energy (so that The elastic moduli and compliances depend on the σ $kρT cs\cε). thermodynamic regime. This can be illustrated by For the first case, linear elasticity is generally an comparing isothermal (T constant) Miso and adiabatic appropriate model; the elastic moduli are quite high (s constant) Mad compliances (Franc: ois et al. 1998). and do not depend very much on temperature. On the They are defined by contrary, rubber elasticity is characterized by much lower moduli (by a factor of 10$–10% on average) and dε l Miso :dσ,dT l 0; dε l Mad :dσ,ds l 0 (9) a stronger temperature dependence. Unusual be-

2 F/S0 expansion coeﬃcients especially beneﬁts from a thermodynamic treatment of elasticity. (i) Since potentials are additive quantities, the overall total free energy, F, directly derives from the local one by integration

1 ρ` F(E, τ) l f ρf(ε, τ) g l & ρf(ε, τ)dΩ (12) Ω Ω –q∂s/∂k where ρ- is the overall mass per unit volume and q∂e/∂k f…g denotes a spatial average. Both at the local 0 and the global levels, the free energy is quadratic and the constitutive equations are similar in form. From T(K) Tg the local relationships

Figure 1 cf cf Σ l ρ l C : εkτφ, s lkρ l φ : εjC τ, Temperature dependence of the force per unit area of the cε cτ ν initial cross-section needed to maintain a given elongation for an elastomer sample. φ l C : α (13) havior, which can be understood easily from a uniaxial where Cν is the specific heat at constant strain, the version of Eqn. (11) (i.e., F $kρT cs\cλ), is observed, following overall relationships can be derived (Stolz such as shortening under load on heating or heat 1986): production by elongation. The entropic origin of rubber elasticity is easily proved by the temperature cF dependence of the force per unit area of the initial l f σ g l ρ` l Ceff : EkτΦeff, \ cE cross-section, F S!, needed to provoke a given elongation λ (new length over initial length). c Typical experimental data obey a two-slope scheme F eff eff ρ` s l f ρs g lkρ` l Φ : EjCν τ, (Fig. 1). Below the glass transition temperature, Tg, the cτ force abruptly decreases with increasing temperature, Φeff l eff αeff as observed for crystalline materials; rubber elasticity C : (14) & is dominant for T Tg, as proved by the increasing (almost linear) variation of the stress with temperature where the superscript ‘‘eff’’ refers to ‘‘effective’’ macro- (the slope, given by kρcs\cλ, is positive since the scopic quantities. entropy decreases with an increasing elongation) and (ii) Owing to the linear character of the whole ε ! by the very low value of the back-extrapolated stress at problem, the strain field, (x), depends linearly on the τ 0K from the rubber-type response (kρce\cλ $ 0), loading parameters (E, ), i.e., which shows that the strain dependence of the internal energy is negligible. ! ! ! ε(x) l A(x):Ejτη(x) (15)

and a decomposition of the problem into two el- 4. Heterogeneous Materials (Linear ementary ones, with E l 0 and τ l 0 respectively, can Thermoelasticity) be carried out. This results in the following relationships: This section is concerned with the prediction of the effective thermoelastic properties of heterogeneous Ceff l f C : A g, Φeff l f φ : A g, materials, such as polycrystals or composite materials, eff l f gjf φ η g from those of the constituents (grains or phases). Since Cν Cv : (16) the exact internal geometry of any representative volume element, Ω, of such materials is generally not These expressions show especially that the effective known, these properties can only be estimated or thermal expansion tensor, αeff, given by Ceff : αeff l bounded. For the sake of simplicity, restriction is f (C : α):A g, as well as the effective heat capacity at eff made to linear behavior, isothermal conditions (with constant strain, Cν , do not reduce to the spatial τ l k T T!, a uniform change of temperature from the averages of the corresponding quantities. This results reference temperature, T!, in the initial natural state), basically from the fact that the thermal strain field is and homogeneous strain conditions, E. The derivation generally not compatible, which is responsible for the of bounds for the overall elastic moduli and thermal presence of a thermal residual (self-equilibrated) stress

3 ﬁeld and its associated stored elastic energy. The comprehensive framework for the derivation of foregoing treatment can be developed in order to thermoelastic properties of heterogeneous materials, derive bounds for αeff and to perform a reﬁned analysis including the analysis of thermal stresses at the local of the thermal stresses, which cannot be obtained from level. a purely mechanical treatment of thermoelasticity. See also: Mechanics of Materials

5. Conclusions The foregoing derivations have provided some illus- Bibliography trations of the benefits that can be gained from a Franc: ois D, Pineau A, Zaoui A 1998 Mechanical Behaiour of thermodynamic treatment of elasticity. The main Materials, Kluwer, Dordrecht, The Netherlands, Vol. 1 advantage lies in the fact that the temperature de- Germain P, Nguyen Q S, Suquet P 1983 Continuum thermo- pendence of the elastic behavior is naturally taken into dynamics. ASME J. Appl. Mech. 50, 1010–20 account so that the associated constitutive equations Mark J E, Lal J 1982 Elastomers and Rubber Elasticity. American Chemical Society, Washington, DC are valid for any thermodynamic conditions instead of Salenc: on J 1995 MeT canique du Continu. Ellipses, Paris, Vol. 2 being specified for particular ones (e.g., isothermal, Stolz C 1986 General relationships between micro and macro- adiabatic, etc.). Moreover, definite relationships can scales for nonlinear behaviour of heterogeneous media. In: be derived between the elastic characteristics exhibited Gittus J, Zarka J (eds.) Modelling Small Deformations of for different such conditions. This treatment yields a Polycrystals. Elsevier, London simple and meaningful interpretation of the entropic nature of rubber elasticity; in addition, it offers a A. Zaoui and C. Stolz