Begell House Inc.
International Journal for Multiscale Computational Engineering
JMC
1543-1649
2
4
2004
Preface to Special Issue onMultiscale Computational Homogenization: From Microstructure to Properties
4
Marc
Geers
Eindhoven University of Technology
A Comparison Between an Embedded FE2 Approach and a TFA-Like Model
18
Pascale
Kanoute
ONERA, DMSE-LCME, 29, Avenue de la Division Leclerc, BP72, F-92322 Chatillon, France
Frederic
Feyel
Onera − The French Aerospace Lab, F-92322 Chatillon, France
Nicolas
Carrere
ONERA, DMSE-LCME, 29, Avenue de la Division Leclerc, BP72, F-92322 Chatillon, France
Two multiscale models are considered in this paper: one is based on an imbricated FE2 approach, while the second rests on a transformation field analysis (TFA) framework. Both models are presented and compared. They are similar regarding the computation cost for nonlinear problems. This conclusion is not obvious since a finite element computation of the representative volume element is usually considered to be more resource consuming than a simple phenomenological model. In fact, a nonlinear TFA model is not a simple model: it involves costly operations and may be even more expensive than a direct finite element computation. Special attention is paid to the microscale spatial discretization. A new method called "subvolumes reduction" is presented to reduce the number of subvolumes used in the TFA model, while preserving a good and controlled accuracy of the results. Various discretizations of the same problem are presented to discuss this method.
Analysis and Numerical Simulation of Discontinuous Displacements Modeling Fine Scale Damage in a Continuum Under Mixed-Mode Loading
27
Krishna
Garikipati
University of Michigan
A continuum damage degradation model capable of representing mixed-mode failure is analyzed. The damage criteria are represented by multiple surfaces that bound the elastic domain in stress space. The compliance tensor is treated as an internal variable and evolves with damage. The damage evolution law is associative and of a nonhardening nature. A distributional framework is adopted for the kinematics. In order to model fine scale features, such as microcracks and microvoids, it is assumed that the solution admits discontinuous displacements. This implies singular distributional strain fields. Necessary conditions are arrived at for the existence of such solutions. It is demonstrated that an interpretation consistent with the presence of strongly discontinuous solutions is possible for this damage model. Furthermore, the analysis leads to a law that dictates the evolution of the solution in the postbifurcation regime. This is combined with an unloading modulus that degrades as damage progresses. Computations are performed in the framework of the Enhanced Strain Finite Element Method. The strain field is enhanced with functions capable of representing singular distributions. Several numerical examples that demonstrate independence of element size and mesh alignment are presented.
Multiscale Computational Strategy With Time and Space Homogenization: A Radial-Type Approximation Technique for Solving Microproblems
18
Anthony
Nouy
Centrale Nantes
Pierre
Ladeveze
University Paris VI
A new multiscale computational strategy for the analysis of structures (such as composite structures) described in detail both in space and in time was introduced recently. This strategy is iterative and involves an automatic homogenization procedure in space as well as in time. At each iteration, this procedure requires the resolution of a large number of linear evolution equations, called the microproblems, on the microscale. In this paper, we present a robust approximate resolution technique for these microproblems based on the concept of radial approximation. This very general technique, which leads to the construction of a relevant reduced basis of space functions, is particularly suitable for the analysis of composite structures.
Size of a Representative Volume Element in a Second-Order Computational Homogenization Framework
24
Varvara G.
Kouznetsova
Department of Mechanical Engineering, Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven; and Netherlands Institute for Metals Research, Rotterdamseweg 137 2628 AL Delft, The Netherlands
Marc
Geers
Eindhoven University of Technology
W. A. M.
Brekelmans
Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
In this paper the intrinsic role of the size of the microstructural representative volume element (RVE) in a second-order computational homogenization is investigated. The presented second-order computational homogenization is an extension of the classical first-order computational homogenization scheme and is based on a proper incorporation of the macroscopic gradient of the deformation tensor and the associated higher-order stress measure into the multiscale framework. The macroscopic homogenized continuum obtained through this scheme is the full second gradient continuum. It is demonstrated with several examples that the size of the microstructural RVE used in a second-order computational homogenization scheme may be related to the length scale of the associated macroscopic homogenized higher-order continuum. It is shown that the analytical second-order homogenization of a microstructurally homogeneous linearly elastic material leads to the second gradient elastic Mindlin's continuum on the macroscale, where the resulting macroscopic length scale parameter is proportional to the RVE size. Several numerical microstructural and multiscale analyses reveal the significance of the contribution of the physical and geometrical nonlinearities in the relation between the RVE size and the calculated macroscopic response. Based on the obtained results, some conclusions are drawn with respect to the choice of the microstructural RVE in the second-order computational homogenization analysis.
Computational Evaluation of Strain Gradient Elasticity Constants
21
R. H. J.
Peerlings
Department of Mechanical Engineering, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands
N. A.
Fleck
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, CB2 1PZ, UK
Classical effective descriptions of heterogeneous materials fail to capture the influence of the spatial scale of the heterogeneity on the overall response of components. This influence may become important when the scale at which the effective continuum fields vary approaches that of the microstructure of the material and may then give rise to size effects and other deviations from the classical theory. These effects can be successfully captured by continuum theories that include a material length scale, such as strain gradient theories. However, the precise relation between the microstructure, on the one hand, and the length scale and other properties of the effective modeling, on the other, are usually unknown. A rigorous link between these two scales of observation is provided by an extension of the classical asymptotic homogenization theory, which was proposed by Smyshlyaev and Cherednichenko (J. Mech. Phys. Solids 48:1325−1358, 2000) for the scalar problem of antiplane shear. In the present contribution, this method is extended to three-dimensional linear elasticity. It requires the solution of a series of boundary value problems on the periodic cell that characterizes the microstructure. A finite element solution strategy is developed for this purpose. The resulting fields can be used to determine the effective higher-order elasticity constants required in the Toupin-Mindlin strain gradient theory. The method has been applied to a matrix-inclusion composite, showing that higher-order terms become more important as the stiffness contrast between inclusion and matrix increases.
Multiscale Model for Damage Analysis in Fiber-Reinforced Composites with Interfacial Debonding
24
Somnath
Ghosh
Department of Civil Engineering, Johns Hopkins University, Baltimore, Maryland 21218, USA
Prasanna
Raghavan
Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210
This paper presents an adaptive multilevel computational model for the multiscale analysis of composite structures with damage due to fiber/matrix interfacial debonding. The method combines continuum damage modeling with displacement based FEM with a microstructurally explicit modeling of interfacial debonding by the Voronoi cell FEM (VCFEM). Three computational levels of hierarchy with different resolutions are introduced to reduce modeling and discretization errors due to an inappropriate resolution. They are (a) level-0 of a pure macroscopic analysis, for which a continuum damage mechanics (CDM) model is developed from homogenization of micromechanical variables that evolve with interfacial debonding; (b) level-1 of a coupled macroscopic-microscopic modeling to implement adequate criteria for switching from macroscopic analyses to pure microscopic analyses; and (c) level-2 regions of a pure microscopic modeling with explicit interfacial debonding. The CDM model for a level-0 analysis is constructed from rigorous VCFEM-based micromechanical analysis of the representative volume element (RVE) followed by homogenization. A numerical example of a composite laminate with localized loading is solved to demonstrate the limitations of CDM models and to demonstrate the effectiveness of the multiscale approach in predicting failure due to interfacial debonding.
Nonlinear viscoelastic analysis of statistically homogeneous random composites
29
Michal
Sejnoha
Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Thakurova 7,166 29 Prague 6, Czech Republic
R.
Valenta
Faculty of Civil Engineering, Department of Structural Mechanics, Czech Technical University in Prague, Thakurova 7,166 29 Prague 6, Czech Republic
Jan
Zeman
Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Thakurova 7,166 29 Prague 6, Czech Republic; Centre of Excellence IT4Innovations, VSB-TU Ostrava, 17 listopadu 15/2172 708 33 Ostrava-Poruba, Czech Republic
Owing to the high computational cost in the analysis of large composite structures through a multiscale or hierarchical modeling, an efficient treatment of complex material systems at individual scales is of paramount importance. Limiting the attention to the level of constituents, the present paper offers a prosperous modeling strategy for the prediction of nonlinear viscoelastic response of fibrous graphite-epoxy composite systems with possibly random distribution of fibers within a transverse plane section of the composite aggregate. If such a material can be marked as statistically homogeneous and the mechanisms driving the material response fall within a category of the first-order homogenization scheme the variational principles of Hashin and Shtrikman emerge as an appealing option in the solution of uncoupled micro-macro computational homogenization. The material statistics up to two-point probability function that are used to describe the morphology of such a microstructure can be then directly incorporated into variational formulations to provide bounds on the effective material response of the assumed composite medium. In the present formulation the Hashin-Shtrikman variational principles are further extended to account for the presence of various transformation fields defined as local eigenstrain or eigenstress distributions in the phases. The evolution of such eigen-fields is examined here within a framework of the nonlinear viscoelastic behavior of a polymeric matrix conveniently described by the Leonov model. A fully implicit integration scheme is implemented to enhance the stability and efficiency of the underlying numerical analysis. A special choice of reference medium with a deformation-dependent shear modulus is proposed in order to improve the redistribution of averaged local fields due to local stress inhomogeneities associated with nonlinear viscoelastic response of the matrix phase. The present modeling strategy is further promoted by a good agreement of the results, including estimated effective thermoelastic properties, with the predictions of a direct microstructural computation.