On the form of free energy and specific heat in coupled thermo-elasticity with isotropic damage

This paper concerns the form of the free energy function in coupled thermo-mechanical problems with isotropic damage, particularly where damage accrues through both mechanical and thermal strains, and when the material is exposed to elevated temperatures. We show that with the normal assumption of a constant speci®c heat coecient, mechanical dissipation is negative and so the second law of thermodynamics is violated. This is true even for a general isotropic damage model that allows independent damage on the Youngʹs modulus and the bulk modulus. Our approach is to make speci®c heat damage dependent, and under these conditions, we show positive dissipation for a range of problems; for the case of concrete, at least, there is material evidence supporting this model. For elevated temperatures, we use the logarithmic form of thermal energy, again showing positive dissipation. Comparisons between the forms show a notable di€erence in the energy transformed to heat, which is signi®cant when reintroduced into the computation