Computational homogenization for heterogeneous thin sheets

In this paper, a computational homogenization technique for thin-structured sheets is proposed, based on the computational homogenization concepts for first- and second-order continua. The actual three-dimensional (3D) heterogeneous sheet is represented by a homogenized shell continuum for which the constitutive response is obtained from the nested analysis of a microstructural representative volume element (RVE), incorporating the full thickness of the sheet and an in-plane representative cell of the macroscopic structure. At an in-plane integration point of the macroscopic shell, the generalized strains, i.e. the membrane deformation and the curvature, are used to formulate the boundary conditions for the microscale RVE problem. At the RVE scale, all microstructural constituents are modeled as an ordinary 3D continuum, described by the standard equilibrium and the constitutive equations. Upon proper averaging of the RVE response, the macroscopic generalized stress and the moment resultants are obtained. In this way, an in-plane homogenization is directly combined with a through thickness stress integration. From a macroscopic point of view, a (numerical) generalized stress–strain constitutive response at every macroscopic in-plane integration point is obtained. Additionally, the simultaneously resolved microscale RVE local deformation and stress fields provide valuable information for assessing the reliability of a particular microstructural design

Novel boundary conditions for strain localization analyses in microstructural volume elements

Multi-scale modeling frequently relies on microstructural representative volume elements (RVEs) on which macroscopic deformation is imposed through kinematical boundary conditions. A particular choice of these boundary conditions may influence the obtained effective properties. For strain localization and damage analyses, the RVE is pushed beyond the limits of its representative character, and the applied boundary conditions have a significant impact on the onset and the type of macroscopic material instability to be predicted. In this article, we propose a new type of boundary conditions for microstructural volume elements, called percolation-path-aligned boundary conditions. Intrinsically, these boundary conditions capture the constraining effect of the material surrounding the RVE upon developing localization bands. The alignment with evolving localization bands allows the highly strained band to cross the RVE and fully develop with minimal interference of the applied boundary conditions. For an illustration of the performance of the newly proposed boundary conditions, macroscopic deformation has been imposed on a voided elasto-plastic RVE using different types of boundary conditions. It is observed that the new RVE boundary conditions provide a good estimate for the effective stiffness, are not susceptible to spurious localization, and permit the development of a full strain localization band up to failure

A multiscale framework for localizing microstructures towards the onset of macroscopic discontinuity

This paper presents a multiscale computational homogenization model for the post localization behavior of a macroscale domain crossed by a cohesive discontinuity emanating from microstructural damage. The stress–strain and the cohesive macroscopic responses are obtained incorporating the underlying microstructure, in which the damage evolution results in the formation of a strain localization band. The macro structural kinematics entails a discontinuous displacement field and a non-uniform deformation field across the discontinuity. Novel scale transitions are formulated to provide a consistent coupling to the continuous microscale kinematics. From the solution of the micromechanical boundary value problem, the macroscale stress responses at both sides of the discontinuity are recovered, providing automatically the cohesive tractions at the interface. The effective displacement jump and deformation field discontinuity are derived from the same microscale analysis. This contribution focusses on scale transition relations and on the solution procedure at the microlevel; the highlights of the approach are demonstrated on microscale numerical examples. Coupled two-scale solution strategy will be presented in a subsequent paper

Energy consistency in homogenisation-based upscaling scheme for localisation in masonry shells

This paper presents an enhanced multi-scale framework for the failure of quasi-brittle thin shells as an improvement of the one proposed in Mercatoris and Massart (Int J Numer Methods Eng 85:1177-1206, 2011). The computational homogenisation-based multi-scale methodology is an attractive solution for heterogeneous materials when their characterisation becomes difficult because of complex evolving behaviour such as damage-induced anisotropy and localisation of degradation. An enhanced upscaling scheme for damage localisation in shell structures is proposed using a periodic computational homogenisation procedure and an energy equivalence between mesostructural material instabilities and aggregate macroscopic cracks. The structural cracking is treated by using embedded strong discontinuities incorporated in the shell formulation, the behaviour of which is deduced by an energetically consistent upscaling scheme. The effects of this energy equivalence are discussed based on results of multi-scale simulations of out-of-plane loaded masonry walls including flexural stair-case failure and compared to the results of direct numerical simulations. A good agreement is observed in terms of the load-bearing capacity and of associated energy dissipation. Based on the homogenisation procedure, the orientation of the structural-scale cracking is detected by means of an acoustic tensor-based failure detection adapted to shell kinematics. A multi-scale bifurcation analysis on a simple loading case is performed in order to discuss the selection of the cracking orientation based on energetic considerations. © 2012 Springer Science+Business Media B.V

Viscoplastic regularization of local damage models: revisited

Local damage models are known to produce pathological mesh dependent results. Regularization techniques are therefore mandatory if local damage models are used for academic research or industrial applications. The viscoplastic framework can be used for regularization of local damage models. Despite of the easy implementation of viscoplasticity, thismethod of regularization did not gain much popularity in comparison to the non-local or gradient damage models. This work is an effort to further explore viscoplastic regularization for quasi-static problems. The focus of this work is on ductile materials. Two different types of strain rate hardening\ud models i.e. the Power law (with a multiplicative strain rate part) and the simplified Bergström van Liempt (with an additive strain rate part) models are used in this study. The modified Lemaitre’s anisotropic damage model with a strain\ud rate dependency was used in this study. It was found that the primary viscoplastic length scale is a function of the hardening and softening (damage) parameters and does not depend upon the prescribed strain rate whereas the secondary length scale is a function of the strain rate. As damage grows, the effective regularization length gradually decreases.When the effective regularization length gets shorter than the element length numerical results become mesh dependent again. This loss of objectivity can not be solved but the effect can be minimized by selecting a very fine mesh or by prescribing high deformation velocities