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.

SUMMARYIn this work we propose a method which combines the element-free Galerkin (EFG) with an extended partition of unity ÿnite element method (PUFEM), that is able to enforce, in some limiting sense, the essential boundary conditions as done in the ÿnite element method (FEM). The proposed extended PUFEM is based on the moving least square approximation (MLSA) and is capable of overcoming singularity problems, in the global shape functions, resulting from the consideration of linear and higher order base functions. With the objective of avoiding the presence of singular points, the extended PUFEM considers an extension of the support of the classical PUFE weight function. Since the extended PUFEM is closely related to the EFG method there is no need for special approximation functions with complex implementation procedures, and no use of the penalty and=or multiplier method is required in order to approximately impose the essential boundary condition. Thus, a relatively simple procedure is needed to combine both methods. In order to attest the performance of the method we consider the solution of an analytical elastic problem and also some coupled elastoplastic-damge problems.

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Assessing the overall instantaneous behavior and strength properties of jointed materials have been the subject of important investigations in the last decades, including phenomenological or micromechanics-based contributions. However, less attention has been dedicated to delayed component of deformation in such media. This issue is addressed in this paper, which is devoted to the formulation of a micromechanical approach to effective viscoelastic properties of jointed rocks with consideration of constituents aging. At the scale of representative elementary volume (REV), the joints are modeled as planar interfaces whose behavior is described by means of generalized viscoelastic state equations under normal and shear loading conditions. Closed-form expressions for the homogenized creep tensor are derived from solving an appropriate viscoelastic concentration problem stated on the REV. The local strain and displacement jump fields are analyzed by extending the concept of strain concentration to relate the components of joint displacement jump to macroscopic strain. Main features of the theoretical overall creep behavior, such as the anisotropy associated with the privileged joint orientations, are highlighted through explicit formulations in some particular configurations of the jointed medium. Finally, the ability of the approach to accurately reproduce the creep behavior of jointed media is assessed by comparison with experimental data as well as with finite element solutions derived in the context of multilayered stratified composite modeling.

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