Modelling nonlocality in continuum physics via constitutive functions depending on spatial gradients

What is it about?

The paper is devoted to the modelling of nonlocality in continuum physics through constitutive functions that depend on suitable gradients. For definiteness the attention is addressed to elastic solids, heat conductors, and magnetic solids. Models are developed where both the requirements of the second law of thermodynamics and the balance equations are satisfied for the constitutive functions that involve gradients of strain, temperature, heat flux, and magnetization. Concerning elastic and magnetic solids, it is shown that, depending on the chosen variables, the standard symmetry property of the stress holds identically. The models so developed are free from any hyperstress tensor frequently considered in the literature.

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Why is it important?

The purpose of this paper is to establish thermodynamically-consistent nonlocal models of elasticity, heat conduction, and magnetism without any appeal to hyperstresses and corresponding powers. The consistency is determined by deriving the restrictions placed by the Clausius-Duhem inequality as the statement of the second law. The novelty of our application of the Clausius-Duhem inequality is to regard the entropy production as a constitutive function as is the case for the entropy flux. Furthermore our procedure avoids the introduction of ad-hoc energy fluxes and merely assumes (for non-polar solids) the stress power as the scalar product between the Cauchy stress and the stretching tensor, as is the case for classical local theories.

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