Journal ArticleDOI
Computational micro-to-macro transitions of discretized microstructures undergoing small strains
Christian Miehe,A. Koch +1 more
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In this paper, the Lagrangian multiplier method is used for the computation of equilibrium states and the overall properties of discretized microstructures, where the overall macroscopic deformation is controlled by three boundary conditions: linear displacements, constant tractions and periodic displacements.Abstract:
The paper investigates algorithms for the computation of homogenized stresses and overall tangent moduli of microstructures undergoing small strains. Typically, these microstructures define representative volumes of nonlinear heterogeneous materials such as inelastic composites, polycrystalline aggregates or particle assemblies. We consider a priori given discretized microstructures, without focusing on details of specific discretization techniques in space and time. The key contribution of the paper is the construction of a family of algorithms and matrix representations of the overall properties of discretized microstructures. It is shown that the overall stresses and tangent moduli of a typical microstructure may exclusively be defined in terms of discrete forces and stiffness properties on the boundary. We focus on deformation-driven microstructures, where the overall macroscopic deformation is controlled. In this context, three classical types of boundary conditions are investigated: (i) linear displacements, (ii) constant tractions and (iii) periodic displacements and antiperiodic tractions. Incorporated by the Lagrangian multiplier method, these constraints generate three classes of algorithms for the computation of equilibrium states and the overall properties of microstructures. The proposed algorithms and matrix representations of the overall properties are formally independent of the interior spatial structure and the local constitutive response of the microstructure and are therefore applicable to a broad class of model problems. We demonstrate their performance for some representative model problems including elastic–plastic deformations of composite materials.read more
Citations
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Multi-scale computational homogenization: Trends and challenges
TL;DR: This paper reviews the state of the art of a particular, yet powerful, method, i.e. computational homogenization, and discusses the main trends since the early developments up to the ongoing contributions and upcoming challenges in the field.
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Multi-scale second-order computational homogenization of multi-phase materials : a nested finite element solution strategy
TL;DR: In this paper, a second-order computational homogenization approach is applied for the multi-scale analysis of simple shear of a constrained heterogeneous strip, where a pronounced boundary size effect appears.
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A multilevel finite element method (FE2) to describe the response of highly non-linear structures using generalized continua
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A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials
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Computational homogenization for the multi-scale analysis of multi-phase materials
TL;DR: A submitted manuscript is the version of the article upon submission and before peer-review as mentioned in this paper, while a published version is the final layout of the paper including the volume, issue and page numbers.
References
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Book
Non-Homogeneous Media and Vibration Theory
TL;DR: In this article, a spectral perturbation of spectral families and applications to self-adjoint eigenvalue problems are discussed, as well as the Trotter-Kato theorem and related topics.
Book
Micromechanics: Overall Properties of Heterogeneous Materials
TL;DR: In this paper, the authors introduce basic elements of elasticity theory: foundations geometric foundations, kinematic foundations, dynamic foundations, constitutive relations elastostatic problems of linear elasticity boundary value problems and extremum principles three-dimensional problems solution of singular problems.
Journal ArticleDOI
Analysis of Composite Materials—A Survey
TL;DR: In this paper, the authors review the analysis of composite materials from the applied mechanics and engineering science point of view, including elasticity, thermal expansion, moisture swelling, viscoelasticity, conductivity, static strength, and fatigue failure.
Journal ArticleDOI
A numerical method for computing the overall response of nonlinear composites with complex microstructure
Hervé Moulinec,Pierre Suquet +1 more
TL;DR: An alternate method based on Fourier series which avoids meshing and which makes direct use of microstructure images is proposed, based on the exact expression of the Green function of a linear elastic and homogeneous comparison material.
Journal ArticleDOI
Effective properties of composite materials with periodic microstructure : a computational approach
TL;DR: In this paper, two different families of numerical methods are considered to solve the problem of a homogeneous linear reference material undergoing a nonhomogeneous periodic eigenstrain, and the relative merits of the two methods are compared and several examples are discussed.