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ABSTRACT A nonlinear variational auto‐encoder (NLVAE) is developed to reconstruct the plane strain stress field in a solid with embedded cracks subjected to uniaxial tension, uniaxial compression, and shear loading paths. Latent features are sampled from a skew‐normal distribution, which allows encoding marked variations of the features of the stress field across the load steps. The NLVAE is trained and tested based upon stress maps generated with the finite element method (FEM) with cohesive zone elements (CZEs). The NLVAE successfully captures stress concentrations that develop across the loading steps as a result of crack propagation, especially when enhanced disentanglement is emphasized during training. Some latent variables consistently emerge as significant across various microstructure descriptors and loading paths. Correlations observed between the evolution of fabric descriptors and that of their significant stress latent features indicate that the NLVAE can capture important microstructure transitions during the loading process. Crack connectivity, crack eccentricity, and the distribution of zones of highly connected opened cracks versus zones with no cracks are the fabric descriptors that best explain the sequences of latent features that are the most important for the reconstruction of the stress field. Notably, the distributional shape, tail behavior, and symmetry of microstructure descriptor distributions have more influence on the stress field than basic measures of central tendency and spread. 

Abstract A pruned VGG19 model subjected to Axial Coronal Sagittal (ACS) convolutions and a custom VGG16 model are benchmarked to predict 3D fabric descriptors from a set of 2D images. The data used for training and testing are extracted from a set of 600 3D biphase microstructures created numerically. Fabric descriptors calculated from the 3D microstructures constitute the ground truth, while the input data are obtained by slicing the 3D microstructures in each direction of space at regular intervals. The computational cost to train the custom ACS-VGG19 model increases linearly withp(the number of images extracted in each direction of space), and increasingpdoes not improve the performance of the model - or only does so marginally. The best performing ACS-VGG19 model provides a MAPE of 2 to 5% for the means of aggregate size, aspect ratios and solidity, but cannot be used to estimate orientations. The custom VGG16 yields a MAPE of 2% or less for the means of aggregate size, distance to nearest neighbor, aspect ratios and solidity. The MAPE is less than 3% for the mean roundness, and in the range of 5-7% for the aggregate volume fraction and the mean diagonal components of the orientation matrix. Increasingpimproves the performance of the custom VGG16 model, but becomes cost ineffective beyond 3 images per direction. For both models, the aggregate volume fraction is predicted with less accuracy than higher order descriptors, which is attributed to the bias given by the loss function towards highly-correlated descriptors. Both models perform better to predict means than standard deviations, which are noisy quantities. The custom VGG16 model performs better than the pruned version of the ACS-VGG19 model, likely because it contains 3 times (p= 1) to 28 times (p= 10) less parameters than the ACS-VGG19 model, allowing better and faster cnvergence, with less data. The custom VGG16 model predicts the second and third invariants of the orientation matrix with a MAPE of 2.8% and 8.9%, respectively, which suggests that the model can predict orientation descriptors regardless of the orientation of the input images. 

This paper investigates the validity of two different analytical homogenization methods: the Mori–Tanaka mean-field theory and Milton’s correlation function-dependent bounds. We focus on biphase linearly elastic transversely isotropic composites. The composites consist of a matrix reinforced with long fibers of either circular or irregular cross section shapes formed by overlapping circles, with different degrees of radius polydispersity. The Mori–Tanaka effective stiffness depends on the phase moduli, volume fractions, and on a few geometric descriptors of the fibers that can be readily evaluated. In contrast, the computation of Milton’s bounds requires finer knowledge of the microstructure, in terms of two and three-point spatial correlation functions, which are not always analytically tractable. We thus consider very specific random microstructure geometries with known correlation functions. The effective moduli estimates of the two methods are validated against the results of numerical homogenization using the finite element method. It is shown that the Mori– Tanaka predictions of the effective transverse bulk modulus are significantly more accurate than those of the transverse and axial shear moduli. In addition, the predictions of the scheme generally deteriorate with an increasing fiber volume fraction. By contrast, the average of Milton’s upper and lower bounds provides a highly accurate estimate for all three independent effective moduli, without any limitation on the fiber concentration. This study highlights the indisputable effect of the spatial correlation functions on the effective properties of composites, and aspires to pave the way towards the development of more predictive, correlation function-dependent homogenization methods. 

This study aims to detect in which microstructure conditions the Mori–Tanaka scheme is inappropriate to calculate the effective stiffness of a two-phase matrix-inclusion system.We analyze the discrepancy between numerical and Mori–Tanaka stiffness estimates in two-dimensional (2D) solids with crack-like flat cavities. The maximum transfer entropy that occurs between a microstructure feature and a stiffness component discrepancy cannot only detect the phase change between a Mori–Tanaka-like cracked solid to a non-Mori–Tanaka-like cracked solid, but also reveal at which load step that phase change first occurs and which microstructure features most affect that phase change. Further analysis with a binary classifier based on a support vector machine (SVM) algorithm shows that the systematic calculation of nine microstructure features based on six statistical crack network descriptors at each step of a loading path can inform the detection of a microstructure transition. The microstructure features identified here could thus be used to trigger the transition from one homogenization scheme to another during incremental stiffness updates, for example, during the simulation of a load path. 

The representation of material microstructure in most existing analytical homogenization models is condensed into a set of well-known ‘‘classical’’ microstructure descriptors, such as the volume fraction and morphology of the individual phases of a composite. This study considers an enriched set of micro descriptors, containing both those ‘‘classical’’ descriptors as well as ‘‘non-classical’’ descriptors which quantify the spatial correlations of any two given micro material points inside a random heterogeneous material. We focus on 2D composites consisting of a matrix with embedded inhomogeneities (or inclusions) of random spatial arrangement. Both phases are treated as homogeneous, linearly elastic and isotropic. Starting from a rich database of reference microstructures, new datasets of perturbed microstructures are created, by inducing changes emulating the physical processes of inclusion nucleation and growth. All microstructures are characterized using the enriched set of micro descriptors, while their apparent stiffness tensor is computed numerically with the finite element (FE) method. A sensitivity analysis between the changes of the micro descriptors and corresponding changes of the apparent stiffness tensor reveals that the ‘‘non-classical’’ descriptors are consistently highly important to the macroscopic behavior. This suggests that enhanced homogenization models, made dependent on the identified pertinent ‘‘non-classical’’ micro descriptors, could be of higher predictive capability than existing approaches.