A note on gradient thermoelasticity

This is a brief note to show how Aifantis’ simple model of gradient elasticity can be extended to account for temperature variations and heat transfer effects. This is done by assuming that both higher-order strain gradients and higher-order temperature gradients should be included in the constitutive equations for the stress tensor and the heat flux vector. A rigorous continuum thermomechanical theory can then be developed by adopting the generalized thermodynamic framework of Forest and Aifantis [1] which, among other ideas, also introduces the concepts of hypertemperature and hyperentropy. However, our approach here would be much simpler. This is because from an engineering point of view, we are interested in the form of governing differential equations to be used for the formulation and solution of boundary value problems, thus avoiding questions regarding the existence of a hyperentropy function and a corresponding suitable form of the second law of thermodynamics for continuous media. For simplicity, we also consider first equilibrium conditions when time variations have settled down.

Then, by following the review article of Carlson [2–Chapter C], the basic equations of stress equilibrium and equilibrium energy equation have the forms

where S is the stress tensor, and q is the heat flux vector. The body force vector and the scalar heat supply terms that generally appear in Eqs. (1)₁ and (1)₂, respectively, have been assumed to vanish.

The constitutive equations of classical linear thermoelasticity have the forms [2-chapter C]

where E is the strain tensor given in terms of the displacement u by the usual strain-displacement relation E=½ (∇u+∇uT), and Θ is the temperature difference field Θ=T-T₀ with T denoting its current and T₀ a reference uniform value. The fourth order tensor C˜ and the second order tensors (M, K) represent the elasticity, the stress-temperature and the conductivity (material property) fields, respectively. For the case of homogeneous and isotropic bodies, we respectively have

where (μ, λ) are the Lame’ moduli, m is the stress-temperature modulus, and k represents the conductivity. Then from Eqs. (1) to (3), it follows that the governing differential equations can be expressed as

A gradient generalization of the uncoupled equations of the classical equations of equilibrium thermoelasticity can be derived using the process described here. We retain the basic balance laws given by Eqs. (1)₁ and (1)₂ but express the strain and temperature fields (E, Θ) in the constitutive expressions given by Eqs. (2)₁ and (2)₂ in terms of its non-local counterparts. These are given by

where υ is the elementary volume of a sphere of radius R surrounding the point x under consideration (∣r∣≤R), and (f_E, f_Θ) are influence or weight functions. Then, by following an argument similar to that of Muhlhaus and Aifantis [3], we obtain the expressions below through a Taylor expansion around x and retention of terms up to the second order in the spatial derivatives. The expressions are given by

where (l_ε, l_θ) denote strain and temperature internal lengths depending on (R, f_E) and (R, f_Θ), respectively.

It follows that the classical constitutive equations of equilibrium thermoelasticity given by Eqs. (2)₁ and (2)₂ are replaced by their gradient counterparts

If we define a displacement field u in terms of the strain ε through the relationship ε=½ (∇u+∇uT), the governing differential equations of classical equilibrium thermoelasticity – for the case of homogeneous and isotropic bodies, i.e., when Eqs. (3)₁ and (3)₂ hold – are replaced by their gradient counterparts below

It is noted that the system of Eqs. (8)₁ and (8)₂ is uncoupled. This means that Eq. (8)₂ can be solved for θ first when appropriate boundary conditions are prescribed for the equilibrium higher-order heat conduction problem, after which Eq. (8)₁ can be solved for u when appropriate boundary conditions are prescribed for the equilibrium gradient elasticity problem.

On defining new quantities u^c and θ^c through the differential expressions

we can rewrite Eqs. (8)₁ and (8)₂ in the classical form

This suggests that the observations of Ru and Aifantis [4] may also apply to the case of gradient thermoelasticity. In other words, solutions of boundary value problems of the equilibrium theory of gradient thermoelasticity can be obtained in terms of solutions of corresponding boundary value problems of classical thermoelasticity, which can be achieved by using them as source functions in the inhomogeneous Helmholtz equations, given by Eqs. (9)₁ and (9)₂.

An extension of the preceding arguments for the uncoupled equilibrium theory of thermoelasticity to the coupled quasi-static case may formally be accomplished as described here. We first write the differential equations for the stress equilibrium and the energy balance in the form [2–chapter D]

which replace Eq. (1) of the equilibrium theory with c denoting the specific heat. It is noted that Eq. (11)₂ is a differential statement of the energy balance, with the term cΘ˙-T0M⋅E˙ resulting from the rate term Τη˙. Here, η is the entropy density, which in the classical theory of thermoelasticity, is a function of the temperature Θ and the strain E˙. By inserting Eqs. (2) and (3) into Eq. (11) we obtain the well-known equations of coupled quasi-static theory of linear thermoelasticity given by

The gradient generalizations of Eqs. (12)₁ and (12)₂ can be formally obtained by following the steps earlier introduced between Eqs. (4) and (8). In this way Eq. (12)₁ is generalized, as before, to Eq. (8)₁, while Eq. (12)₂ is generalized to the higher-order temperature-displacement equation given by

In the case that the solid is rigid (u≡0), Eq. (13) reduces to a generalized heat conduction equation of the form

where k*=clθ2,k**=klθ2. An equation of this form has been derived by Aifantis [5] for mass transport in polycrystals. For this reason, it should be pointed out that the arguments given here also apply for diffusion in stressed linear elastic solids and flow through deformable porous media. In fact, the second term of the r.h.s of Eq. (14) is similar to that introduced by Barenblatt for seepage in fissured rocks, whereas the third term of the r.h.s. of Eq. (14) is similar to that introduced by Cahn-Hilliard in the theory of spinodal decomposition. These are all discussed by Aifantis [5] in his work on the mechanics of diffusion in solids.