A new formulation of the second law in continuum physics and new methods to exploit it

What is it about?

The paper revisits the formulation of the second law in continuum physics and investigates new methods of exploitation. Both the entropy flux and the entropy production are taken to be expressed by constitutive equations. In three-dimensional settings, vectors and tensors are in order and they occur through inner products in the inequality representing the second law; a representation formula, which is quite uncommon in the literature, produces the general solution whenever the sought equations are considered in rate-type forms. Next, the occurrence of the entropy production as a constitutive function is shown to produce a wider set of physically admissible models. Furthermore the constitutive property of the entropy production results in an additional, essential term in the evolution equation of rate-type materials, as is the case for Duhem-like hysteretic models. This feature of thermodynamically consistent hysteretic materials is exemplified for elastic--plastic materials. The representation formula is shown to allow more general non-local properties while the constitutive entropy production proves essential for the modeling of hysteresis.

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Why is it important?

Three main points have to emerge from this paper. First, the occurrence of a nonzero extra entropy flux proves essential whenever we look for non-local terms involving higher-order gradients of temperature and deformation. Second, in three dimensional settings, vectors and tensors are in order and they occur through inner products in the inequality representing the second law. A representation formula, quite uncommon in the literature, produces the general solution whenever the sought equations are expressed in rate-type forms. Third, the occurrence of the entropy production as a constitutive function is essential in the thermodynamically consistent modeling of hysteretic~materials. The entropy production allows the completion of rate-type hysteretic equations, as with Duhem-like models. This feature is exemplified in this paper for elastic--plastic materials, though the analogue can be performed for magnetic or electric hysteresis. As is shown in this paper, both the use of the representation formula and the entropy production as a constitutive function turn out to be decisive improvements in the elaboration of material modeling.

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