A model-reduction approach to the micromechanical analysis of polycrystalline materials

Michel, Jean‐Claude; Suquet, Pierre

doi:10.1007/s00466-015-1248-9

Cited by 33 publications

(33 citation statements)

References 42 publications

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“…However, without a proper physical insight in the problem to be reduced, it is likely that these general procedures will not deliver satisfactory answers. The model initially proposed by the authors ( [3]) and further developed in [7][8][9] and by other authors ( [4, 5]) is limited in scope to materials with a microstructure, comprised of constituents with a certain type of constitutive relations involving internal variables. In turn, the approximations on which the model is based are physically sound in this context.…”

Section: Reduced-order Modelsmentioning

confidence: 99%

“…The first simplification is that the free-energy function w of the individual constituents is assumed to be quadratic with respect to ε and α (when this is not the case a work-around has been proposed in [7] and [8]). Then the thermoelastic problem (2) becomes linear, its solution can be expressed by superposition as…”

Section: Approximate Effective Potentials and Nonlinear Homogenizationmentioning

confidence: 99%

“…Using the decomposition (7), the effective free-energy w can be expressed explicitly in terms of ε and ξ (see [7,8]). …”

Section: (K)mentioning

confidence: 99%

“…The first approximation consists in replacing ψ by its tangent second-order expansion (TSO) in each individual phase ( [7,8]):…”

Section: (K)mentioning

confidence: 99%

“…The first data set is that identified in [12]. The other two data sets correspond to different values of the latent hardening parameters and have been chosen to highlight the sensitivity of the response to the hardening parameters ( [8]). The predictions of the two NTFA models for the overall response of the polycrystalline aggregate with these three different data sets are compared in Figs 1a and 1b with full-field simulations.…”

Section: (K)mentioning

confidence: 99%

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Model‐reduction in micromechanics of polycrystalline materials

Michel

Suquet

2017

Proc Appl Math and Mech

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The present study is devoted to the generalization of the Nonuniform Transformation Field Analysis (NTFA), a modelreduction approach introduced by the authors. First, the local fields of internal variables are decomposed on a reduced basis of modes. Second, the effective (average) dissipation potential of the phases is replaced by accurate approximations. The reduced evolution equations of the models, in other words the homogenized constitutive relations, can be entirely expressed explicitly in terms of quantities which are pre-computed "off-line". The example of creep of polycrystalline ice is used to assess the accuracy of the models. Their predictions, both the overall response and the local response, are shown to be in good agreement with full-field simulations with a significant speed-up. Reduced-order modelsA common engineering practice in the analysis of composite (or polycrystalline) structures is to use effective or homogenized material properties instead of taking into account all details of the individual phase properties. Unfortunately when the individual constituents are nonlinear, the exact description of the effective constitutive relations requires the determination of the local fields (at the microscopic scale). For structural computations, the consequence of this theoretical result is that the two levels of computation, the level of the structure and the level of the unit-cell, remain intimately coupled. The nested resolution of these coupled problems (known as FE 2 analysis) is so far limited by their formidable size. It is therefore quite natural to resort to model-reduction techniques achieving a compromise between analytical approaches, which are costless but often very limited by nonlinearity, and full-field simulations which resolve all complex details of the exact solutions, but come at a very high cost. This is the aim of reduced-order modelling (ROM) in general ( [1, 2]). However, ROM is often understood as a way to reduce the cost of a numerical simulation. In homogenization problems, the objective is not only to reduce the computational cost, but also to arrive at an explicit constitutive model, where internal variables are identified and evolution equation for these internal variables are explicitly derived. The model which is reduced is not only the computational model, but also and primarily, the constitutive model.General procedures to achieve the model reduction do exist, at least for the computational model. However, without a proper physical insight in the problem to be reduced, it is likely that these general procedures will not deliver satisfactory answers. The model initially proposed by the authors ( [3]) and further developed in [7][8][9] and by other authors ( [4, 5]) is limited in scope to materials with a microstructure, comprised of constituents with a certain type of constitutive relations involving internal variables. In turn, the approximations on which the model is based are physically sound in this context. The three main ingredients of the model are as follows....

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Section: Reduced-order Modelsmentioning

confidence: 99%

Section: Approximate Effective Potentials and Nonlinear Homogenizationmentioning

confidence: 99%

“…Using the decomposition (7), the effective free-energy w can be expressed explicitly in terms of ε and ξ (see [7,8]). …”

Section: (K)mentioning

confidence: 99%

“…The first approximation consists in replacing ψ by its tangent second-order expansion (TSO) in each individual phase ( [7,8]):…”

Section: (K)mentioning

confidence: 99%

Section: (K)mentioning

confidence: 99%

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Model‐reduction in micromechanics of polycrystalline materials

Michel

Suquet

2017

Proc Appl Math and Mech

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Numerical and Experimental Characterization of Elastic Properties of a Novel, Highly Homogeneous Interpenetrating Metal Ceramic Composite

et al. 2020

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An interpenetrating aluminum–alumina composite is presented, based on a ceramic foam manufactured via a novel slurry‐based route resulting in a highly homogeneous preform microstructure in contrast to other preform techniques. The metal matrix composite (MMC) is produced by infiltrating the open‐porous ceramic preform with molten aluminum at 700 °C using a Argon‐driven gas pressure infiltration process. The resulting MMC and the primary ceramic foam are investigated both numerically and experimentally in terms of microstructural characteristics. In addition, the mechanical behavior of the material as well as the structural and material interactions on the microscale are investigated. To characterize the MMC regarding mechanical isotropy, elastic properties are determined experimentally via ultrasonic phase spectroscopy (UPS). A fast Fourier transform (FFT) formulation is used to simulate the complex 3D microstructure with reasonable effort based on image‐data gathered from high‐resolution X‐ray computed tomography (CT) scans of the ceramic foam as computational grid. Simulations prove that the material properties are, indeed, considered as highly homogeneous with respect to the material microstructure. A comparison with effective experimental investigations confirms these findings.

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An algorithmically consistent macroscopic tangent operator for FFT‐based computational homogenization

Göküzüm

Keip

2017

Numerical Meth Engineering

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Summary The present work addresses a multiscale framework for fast‐Fourier‐transform–based computational homogenization. The framework considers the scale bridging between microscopic and macroscopic scales. While the macroscopic problem is discretized with finite elements, the microscopic problems are solved by means of fast‐Fourier‐transforms (FFTs) on periodic representative volume elements (RVEs). In such multiscale scenario, the computation of the effective properties of the microstructure is crucial. While effective quantities in terms of stresses and deformations can be computed from surface integrals along the boundary of the RVE, the computation of the associated moduli is not straightforward. The key contribution of the present paper is the derivation and implementation of an algorithmically consistent macroscopic tangent operator which directly resembles the effective moduli of the microstructure. The macroscopic tangent is derived by means of the classical Lippmann‐Schwinger equation and can be computed from a simple system of linear equations. This is performed through an efficient FFT‐based approach along with a conjugate gradient solver. The viability and efficiency of the method is demonstrated for a number of two‐ and three‐dimensional boundary value problems incorporating linear and nonlinear elasticity as well as viscoelastic material response.

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A model-reduction approach to the micromechanical analysis of polycrystalline materials

Cited by 33 publications

References 42 publications

Model‐reduction in micromechanics of polycrystalline materials

Model‐reduction in micromechanics of polycrystalline materials

Numerical and Experimental Characterization of Elastic Properties of a Novel, Highly Homogeneous Interpenetrating Metal Ceramic Composite

An algorithmically consistent macroscopic tangent operator for FFT‐based computational homogenization

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