In this work, we propose a homogenization formulation to model transient heat conduction in heterogeneous media that takes into account thermal inertia contributions, which arise from a finite description of the microscale. Rewriting the variational form of the transient heat conduction problem and making use of key assumptions, we arrive at a mathematical formulation that suggests an extension of the Hill-Mandel principle when considering non-null heat flux divergence in the representative volume element (RVE). Along the manuscript, we highlight that the main results of the proposed formulation are in agreement with recent advances in the field of computational homogenization applied to transient mechanical and heat flow problems. The proposed extension of the Hill-Mandel principle contributes to the understanding of the microscale thermal inertia effects incorporation into the multiscale framework. We also present the calculations needed for implementing the model and numerical results, which give support to the theoretical model developed. The numerical results highlight the importance of considering full transient aspects when dealing with multiscale heat conduction in heterogeneous media which are subjected to high thermal gradients. P m .x; t/ D c P Â.x; t/;yielding P D c P Â , where c is the specific heat.In this subsection, we study a transient heat conduction problem with heat generation in a homogeneous material. We compare the macroscopic multiscale solution with the analytical solution and a Quad8: Incomplete quadratic quadrilateral element. b Tri3: Linear triangular element. † † We use the expression 'volume elements' when referring to the one-scale (conventional) solution.

This paper investigates both theoretically and using finite elements the elastoplastic field induced by a pressurized spherical cavity expanding dynamically in an infinite medium modelled using the Gurson-Tvergaard-Needleman porous plasticity approach. The theoretical model, which assumes that the porosity is uniformly distributed in the material and the cavitation fields are self-similar, incorporates artificial viscous stresses into the original formulation of Cohen and Durban (2013b) to capture the shock waves that emerge at high cavitation velocities. The finite element calculations, performed in ABAQUS/Explicit (2013) using the Arbitrary Lagrangian Eulerian adaptive meshing available in the code, simulate the cavity expansion process in materials with uniform and non-uniform distributions of porosity. The finite element results show that the distribution of porosity has small influence on the cavitation velocity, as well as on the location of the shock wave, which are primarily determined by the cavity pressure and the average material properties. In contrast, it is shown that the intensity of the shock wave, evaluated based on the maximum value of the plastic strain rate within the shock, depends on the local material porosity. The ability of the theoretical model to reproduce the numerical results obtained for the various distributions of porosity used in this work is exposed and discussed.

This paper investigates the steady-state elastoplastic fields induced by a pressurized cylindrical cavity expanding dynamically in an anisotropic porous medium. For that task, we have developed a theoretical model which: (i) incorporates into the formalism developed by Cohen and Durban (2013b) the effect of plastic anisotropy using the constitutive framework developed by Benzerga and Besson ( 2001) and (ii) uses the artifical viscosity approach developed by Lew et al. (2001) to capture the shock waves that emerge at high cavity expansion velocities. We have shown that while the development of the shock waves is hardly affected by the material anisotropy, the directionality of the plastic properties does have an effect on the elastoplastic fields that evolve near the cavity. The importance of this effect is strongly dependent on the cavity expansion velocity, the initial porosity and the strain hardening of the material. In addition, the theoretical model has been used in conjunction with the Recht and Ipson (1963) formulas to assess the ballistic performance of porous anisotropic targets against high velocity perforation.

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