Multiscale identification of apparent elastic properties at meso-scale for materials with complex microstructure using experimental imaging measurements (original) (raw)
Related papers
Materials, 2020
The aim of this work is to efficiently and robustly solve the statistical inverse problem related to the identification of the elastic properties at both macroscopic and mesoscopic scales of heterogeneous anisotropic materials with a complex microstructure that usually cannot be properly described in terms of their mechanical constituents at microscale. Within the context of linear elasticity theory, the apparent elasticity tensor field at a given mesoscale is modeled by a prior non-Gaussian tensor-valued random field. A general methodology using multiscale displacement field measurements simultaneously made at both macroscale and mesoscale has been recently proposed for the identification the hyperparameters of such a prior stochastic model by solving a multiscale statistical inverse problem using a stochastic computational model and some information from displacement fields at both macroscale and mesoscale. This paper contributes to the improvement of the computational efficiency,...
Validation of a Probabilistic Model for Mesoscale Elasticity Tensor of Random Polycrystals
2013
In this paper, we present validation of a probabilistic model for mesoscale elastic behavior of materials with microstructure. The linear elastic constitutive matrix of this model is described mathematically as a bounded random matrix. The bounds reflect theoretical constraints consistent with the theory of elasticity. We first introduce a statistical characterization of an experimental database on morphology and crystallography of polycrystalline microstructures. The resulting statistical model is used as a surrogate to further experimental data, required for calibration and validation. We then recall the construction of a probabilistic model for the random matrix characterizing the apparent elasticity tensor of a heterogeneous random medium. The calibration of this coarse scale probabilistic model using an experimental database of microstructural measurements and utilizing the developed microstructural simulation tool is briefly discussed. Before using the model as a predictive tool in a system level simulation for the purpose of detection and prognosis, the credibility of the model must be established through evaluating the degree of agreement between the predictions of the model and the observations. As such, a procedure is presented to validate the probabilistic model from simulated data resulting from subscale simulations. Suitable quantities of interest are introduced and predictive accuracy of the model is studied by comparing probability density functions of response quantities of interest. The validation task is exercised under both static and dynamic loading condition. The results indicate that the probabilistic model of mesoscale elasticity tensor is adequate to predict the response quantity of interest in the elastostatic regime. The scatter in the model predictions is found to be consistent with the fine scale response. In the case of elastodynamic, the model predicts the mean behavior for lower frequency for which we have a quasistatic regime.
On choosing effective elasticity tensors using a Monte-Carlo method
Acta Geophysica, 2014
A generally anisotropic elasticity tensor can be related to its closest counterparts in various symmetry classes. We refer to these counterparts as effective tensors in these classes. In finding effective tensors, we do not assume a priori orientations of their symmetry planes and axes. Knowledge of orientations of Hookean solids allows us to infer properties of materials represented by these solids. Obtaining orientations and parameter values of effective tensors is a highly nonlinear process involving finding absolute minima for orthogonal projections under all three-dimensional rotations. Given the standard deviations of the components of a generally anisotropic tensor, we examine the influence of measurement errors on the properties of effective tensors. We use a global optimization method to generate thousands of realizations of a generally anisotropic tensor, subject to errors. Using this optimization, we perform a Monte Carlo analysis of distances between that tensor and its ...
Stochastic modeling of mesoscopic elasticity random field
Mechanics of Materials, 2016
In the homogenization setting, the effective properties of a heterogeneous material can be retrieved from the solution of the so-called corrector problem. In some cases of practical interest, obtaining such a solution remains a challenging computational task requiring an extremely fine discretization of microstructural features. In this context, Bignonnet et al. recently proposed a framework where smooth mesoscopic elasticity random fields are defined through a filtering procedure. In this work, we investigate the capabilities of information-theoretic random field models to accurately represent such mesoscopic elasticity fields. The aim is to substantially reduce the homogenization cost through the use of coarser discretizations while solving mesoscale corrector problems. The analysis is performed on a simple but non-trivial model microstructure. First of all, we recall the theoretical background related to the filtering and multiscale frameworks, and subsequently characterize some statistical properties of the filtered stiffness field. Based on these properties, we further introduce a random field model and address its calibration through statistical estimators and the maximum likelihood principle. Finally, the validation of the model is discussed by comparing some quantities of interest that are obtained either from numerical experiments on the underlying random microstructure or from model-based simulations. It is shown that for the case under study, the information-theoretic model can be calibrated with a limited set of realizations and still allows for accurate predictions of the effective properties.
Random field models of heterogeneous materials
International Journal of Solids and Structures, 1998
One of the main challenges in solid mechanics lies in the passage from a heterogeneous microstructure to an approximating continuum model. In many cases (e.g. stochastic finite elements, statistical fracture mechanics), the interest lies in resolution of stress and other dependent fields over scales not infinitely larger than the typical microscale. This may be accomplished with the help of a meso-scale window which becomes the classical representative volume element (RVE) in the infinite limit. It turns out that the material properties at such a mesoscale cannot be uniquely approximated by a random field of stiffness/compliance with locally isotropic realizations, but rather two random continuum fields with locally anisotropic realizations, corresponding, respectively, to essential and natural boundary conditions on the meso-scale, need to be introduced to bound the material response from above and from below. We study the first-and second-order characteristics of these two meso-scale random fields for anti-plane elastic response of random matrix-inclusion composites over a wide range of contrasts and aspect ratios. Special attention is given to the convergence of effective responses obtained from the essential and natural boundary conditions, which sheds light on the minimum size of an RVE. Additionally, the spatial correlation structure of the crack density tensor with the meso-scale moduli is studied.
The use of strain gradient theory for analysis of random media
International Journal of Solids and Structures
In this paper we consider the equations governing the response to a forcing field of an infinite statistically homogeneous medium with small fluctuations in the constants of elasticity. These equations were previously derived by Beran and McCoy [l]. The solution is obtained for the problem of a point force in an infinite medium and an analysis is presented to determine the ability of first strain gradient theory to approximate this solution. It is shown that a valid approximation can be obtained for the "slowfy" varying (in space) portion of the solution only if the square of the lengths introduced in gradient theory for an isotropic centrosymmetric material are negative. This requirement violates the requirement that the strain energy density of the gradient theory be positive definite. Further consequences of choosing these lengths to be imaginary are considered. The value of going to higher order gradient theories is also discussed.
Prediction of elastic properties of heterogeneous materials with complex microstructures
Journal of the Mechanics and Physics of Solids, 2007
The phase-field microelasticity (PFM) is adapted into a homogenization process to predict all the effective elastic constants of three-dimensional heterogeneous materials with complex microstructures. Comparison between the PFM approach and the Hashin-Shtrikman variational approach is also given. Using 3D images of two-phase heterogeneous media with regular and irregular microstructures, results indicate that the PFM approach can accurately take into account the effects of both elastic anisotropy and inhomogeneity of materials with arbitrary microstructure geometry, such as complex porous media with suspended inclusions. Published by Elsevier Ltd.
Computer-Aided Design, 2013
Building sensible processing-structure-property (PSP) links to gain fundamental insights and understanding of materials behavior has been the focus of many works in computational materials science. Microstructure characterization and reconstruction (MCR), coupled with machine learning techniques and materials modeling and simulation, is an important component of discovering PSP relations and inverse material design in the era of high-throughput computational materials science. In this article, we provide a comprehensive review of representative approaches for MCR and elaborate on their algorithmic details, computational costs, and how they fit into the PSP mapping problems. Multiple categories of MCR methods relying on statistical functions (such as n-point correlation functions), physical descriptors, spectral density function, texture synthesis, and supervised/unsupervised learning are reviewed. As no MCR method is applicable to the analysis and (inverse) design of all material systems, our
Effective Elasticity Tensors in Context of Random Errors
Journal of Elasticity, 2015
We introduce the effective elasticity tensor of a chosen material-symmetry class to represent a measured generally anisotropic elasticity tensor by minimizing the weighted Frobenius distance from the given tensor to its symmetric counterpart, where the weights are determined by the experimental errors. The resulting effective tensor is the highestlikelihood estimate within the specified symmetry class. Given two material-symmetry classes, with one included in the other, the weighted Frobenius distance from the given tensor to the two effective tensors can be used to decide between the two models-one with higher and one with lower symmetry-by means of the likelihood ratio test.