A consistent nonlocal scheme based on filters for the homogenization of heterogeneous linear materials with non-separated scales

AbstractIn this work, the question of homogenizing linear elastic, heterogeneous materials with periodic microstructures in the case of non-separated scales is addressed. A framework if proposed, where the notion of mesoscopic strain and stress fields are defined by appropriate integral operators which act as low-pass filters on the fine scale fluctuations. The present theory extends the classical linear homogenization by substituting averaging operators by integral operators, and localization tensors by nonlocal operators involving appropriate Green functions. As a result, the obtained constitutive relationship at the mesoscale appears to be nonlocal. Compared to nonlocal elastic models introduced from a phenomenological point of view, the nonlocal behavior has been fully derived from the study of the microstructure. A discrete version of the theory is presented, where the mesoscopic strain field is approximated as a linear combination of basis functions. It allows computing the mesoscopic nonlocal operator by means of a finite number of transformation tensors, which can be computed numerically on the unit cell

Variational Foundations and Generalized Unified Theory of RVE-Based Multiscale Models

A unified variational theory is proposed for a general class of multiscale models based on the concept of Representative Volume Element. The entire theory lies on three fundamental principles: (1) kinematical admissibility, whereby the macro- and micro-scale kinematics are defined and linked in a physically meaningful way; (2) duality, through which the natures of the force- and stress-like quantities are uniquely identified as the duals (power-conjugates) of the adopted kinematical variables; and (3) the Principle of Multiscale Virtual Power, a generalization of the well-known Hill-Mandel Principle of Macrohomogeneity, from which equilibrium equations and homogenization relations for the force- and stress-like quantities are unequivocally obtained by straightforward variational arguments. The proposed theory provides a clear, logically-structured framework within which existing formulations can be rationally justified and new, more general multiscale models can be rigorously derived in well-defined steps. Its generality allows the treatment of problems involving phenomena as diverse as dynamics, higher order strain effects, material failure with kinematical discontinuities, fluid mechanics and coupled multi-physics. This is illustrated in a number of examples where a range of models is systematically derived by following the same steps. Due to the variational basis of the theory, the format in which derived models are presented is naturally well suited for discretization by finite element-based or related methods of numerical approximation. Numerical examples illustrate the use of resulting models, including a non-conventional failure-oriented model with discontinuous kinematics, in practical computations

Surface plasticity: theory and computation

Surfaces of solids behave differently from the bulk due to different atomic rearrangements and processes such as oxidation or aging. Such behavior can become markedly dominant at the nanoscale due to the large ratio of surface area to bulk volume. The surface elasticity theory (Gurtin and Murdoch in Arch Ration Mech Anal 57(4):291–323, 1975) has proven to be a powerful strategy to capture the size-dependent response of nano-materials. While the surface elasticity theory is well-established to date, surface plasticity still remains elusive and poorly understood. The objective of this contribution is to establish a thermodynamically consistent surface elastoplasticity theory for finite deformations. A phenomenological isotropic plasticity model for the surface is developed based on the postulated elastoplastic multiplicative decomposition of the surface superficial deformation gradient. The non-linear governing equations and the weak forms thereof are derived. The numerical implementation is carried out using the finite element method and the consistent elastoplastic tangent of the surface contribution is derived. Finally, a series of numerical examples provide further insight into the problem and elucidate the key features of the proposed theory. © 2017 Springer-Verlag GmbH Germany, part of Springer Natur

Auxetic orthotropic materials: Numerical determination of a phenomenological spline-based stored density energy and its implementation for finite element analysis

Abstract Auxetic materials, which have negative Poisson’s ratio, show potential to be used in many interesting applications. Finite element analysis (FEA) is an important phase in implementing auxetic materials, but may become computationally expensive because simulation often needs microscale details and a fine mesh. It is also necessary to check that topological aspects of the microscale reflects not only micro but macromechanical behavior. This work presents a phenomenological approach to the problem using data-driven spline-based techniques to properly characterize orthotropic auxetic material requiring neither analytical constraints nor micromechanics, expanding on previous methods for isotropic materials. Hyperelastic energies of auxetic orthotropic material are determined from experimental data by solving the equilibrium differential functional equations directly, so no fitting or analytical estimation is necessary. This offers two advantages; (i) it allows the FEA study of orthotropic auxetic materials without requiring micromechanics considerations, reducing modeling and computational time costs by two to three orders of magnitude; (ii) it adapts the hyperelastic energies to the nature of the material with precision, which could be critical in scenarios where accuracy is essential (e.g. robotic surgery)

Multiscale modeling of microstructure-property relations

The recent decades have seen significant progress in linking the mechanical performance of materials to their underlying microstructure. This article presents an overview of some of these achievements, trends, and challenges. Attention is given to methods initially developed for micromechanics and their gradual evolution toward powerful multiscale methods. Various methods have been proposed for bridging scales in mechanics of materials, all aiming for efficiency and accuracy. Computational homogenization is one of these powerful approaches, now used systematically for the assessment of structure–property relations. Novel solution methods and model reduction techniques provide tools to speed up the structure–property analysis, whereby large-scale computations have been made possible. Truly fast analyses of microstructures may be expected in the near future

Computational second-order homogenization of materials with effective anisotropic strain-gradient behavior

International audienceA computational homogenization method to determine the effective parameters of Mindlin's Strain Gradient Elasticity (SGE) model from a local heterogeneous Cauchy linear material is developed. The devised method, which is an extension of the classical one based on the use of Quadratic Boundary Conditions, intents to correct the well-known non-physical problem of persistent gradient effects when the Representative Volume Element (RVE) is homogeneous. Those spurious effects are eliminated by introducing a microstructure-dependent body force field in the homogenization scheme together with alternative definitions of the localization ten-sors. With these modifications, and by a simple application of the superposition principle, the higher-order stiffness tensors of SGE are computed from elementary numerical calculations on RVE. Within this new framework, the convergence of SGE effective properties is investigated with respect to the size of the RVE. Finally, a C 1-FEM procedure for simulating the behavior of the effective material at the macro scale is developed. We show that the proposed model is consistent with the solutions arising from asymptotic analysis and that the computed effective tensors verify the expected invariance properties for several classes of anisotropy. We also point out an issue that the present model shares with asymptotic-based solutions in the case of soft inclusions. Applications to anisotropic effective strain-gradient materials are provided, as well as comparisons between fully meshed structures and equivalent homogeneous models

Uncertainty quantification in computational stochastic multiscale analysis of nonlinear elastic materials

This paper is devoted to a computational stochastic multiscale analysis of nonlinear structures made up of heterogeneous hyperelastic materials. At the microscale level, the nonlinear constitutive equation of the material is characterized by a stochastic potential for which a polynomial chaos representation is used. The geometry of the microstructure is random and characterized by a high number of random parameters. The method is based on a deterministic non-concurrent multiscale approach devoted to micro-macro nonlinear mechanics which leads us to characterize the nonlinear constitutive equation with an explicit continuous form of the strain energy density function with respect to the large scale Cauchy Green strain states. To overcome the curse of dimensionality, due to the high number of involved random variables, the problem is transformed into another one consisting in identifying the potential on a polynomial chaos expansion. Several strategies, based on novel algorithms dedicated to high stochastic dimension, are used and adapted for the class of multi-modal random variables which may characterize the potential. Numerical examples, at both small and large scales, allow analyzing the efficiency of the approach through comparisons with classical methods