High-contrast Composites Research Articles

Upscaling of a diffusion problem with flux jump in high contrast composites

We study a diffusion problem in a periodic fibered composite consisting of two materials with highly contrasting diffusion coefficients. An important feature of our problem is the existence of an imperfect interface between the two phases of the composite medium, characterized by the continuity of the solution and the presence of a jump flux. By applying suitable homogenization techniques, we get the limit problems corresponding to two different choices for the flux jump.

Homogenization of high-contrast composites under differential constraints

Abstract We derive, by means of variational techniques, a limiting description for a class of integral functionals under linear differential constraints. The functionals are designed to encode the energy of a high-contrast composite, that is, a heterogeneous material which, at a microscopic level, consists of a periodically perforated matrix whose cavities are occupied by a filling with very different physical properties. Our main result provides a Γ-convergence analysis as the periodicity tends to zero, and shows that the variational limit of the functionals at stake is the sum of two contributions, one resulting from the energy stored in the matrix and the other from the energy stored in the inclusions. As a consequence of the underlying high-contrast structure, the study is faced with a lack of coercivity with respect to the standard topologies in L p {L^{p}} , which we tackle by means of two-scale convergence techniques. In order to handle the differential constraints, instead, we establish new results about the existence of potentials and of constraint-preserving extension operators for linear, k-th order, homogeneous differential operators with constant coefficients and constant rank.

Singularities of the stress concentration in the presence of 𝐶^{1,𝛼}-inclusions with core-shell geometry

In high-contrast composites, if an inclusion is in close proximity to the matrix boundary, then the stress, which is represented by the gradient of a solution to the Lamé systems of linear elasticity, may exhibit the singularities with respect to the distance ε \varepsilon between them. In this paper, we establish the asymptotic formulas of the stress concentration for core-shell geometry with C 1 , α C^{1,\alpha } boundaries in all dimensions by precisely capturing all the blow-up factor matrices, as the distance ε \varepsilon between interfacial boundaries of a core and a surrounding shell goes to zero. Further, a direct application of these blow-up factor matrices gives the optimal gradient estimates.

Using Deep Learning to Predict Fracture Patterns in Crystalline Solids

Summary Fracture is a catastrophic process whose understanding is critical for evaluating the integrity and sustainability of engineering materials. Here, we present a machine-learning approach to predict fracture processes connecting molecular simulation into a physics-based data-driven multiscale model. Based on atomistic modeling and a novel image-processing approach, we compile a comprehensive training dataset featuring fracture patterns and toughness values for different crystal orientations. Assessments of the predictive power of the machine-learning model shows excellent agreement not only regarding the computed fracture patterns but also the fracture toughness values and is examined for both mode I and mode II loading conditions. We further examine the ability of predicting fracture patterns in bicrystalline materials and material with gradients of microstructural crystal orientation. These results further underscore the excellent predictive power of our model. Potential applications of this model could be widely applied in material design.

Homogenization and multicontinuum models for high contrast composites

We consider the high contrast periodic composite materials having high ratio ω >>1 of physical constants of the components such as the heat conductivity or Young’s moduli. The ratio of the characteristic microscopic size and the characteristic macroscopic size is a small parameter e. If the component having high constants has the shape of small isolated particles then the homogenized (macroscopic, effective) model demonstrates a loss of the effective wave velocity. If the topology of the component having high constants corresponds to several (at least two) connected periodic sets disconnected one from other and if ε2ω>>1 or ε2ω = const then the macroscopic model is described by a multicontinuum, i.e. several continua in each point.

Establishing structure-property localization linkages for elastic deformation of three-dimensional high contrast composites using deep learning approaches

Data-driven methods are attracting growing attention in the field of materials science. In particular, it is now becoming clear that machine learning approaches offer a unique avenue for successfully mining practically useful process-structure-property (PSP) linkages from a variety of materials data. Most previous efforts in this direction have relied on feature design (i.e., the identification of the salient features of the material microstructure to be included in the PSP linkages). However due to the rich complexity of features in most heterogeneous materials systems, it has been difficult to identify a set of consistent features that are transferable from one material system to another. With flexible architecture and remarkable learning capability, the emergent deep learning approaches offer a new path forward that circumvents the feature design step. In this work, we demonstrate the implementation of a deep learning feature-engineering-free approach to the prediction of the microscale elastic strain field in a given three-dimensional voxel-based microstructure of a high-contrast two-phase composite. The results show that deep learning approaches can implicitly learn salient information about local neighborhood details, and significantly outperform state-of-the-art methods.

Deep learning approaches for mining structure-property linkages in high contrast composites from simulation datasets

Data-driven methods are emerging as an important toolset in the studies of multiscale, multiphysics, materials phenomena. More specifically, data mining and machine learning methods offer an efficient toolset for extracting and curating the important correlations controlling these multiscale materials phenomena in high-value reduced-order forms called process-structure-property (PSP) linkages. Traditional machine learning methods usually depend on intensive feature engineering, and have enjoyed some success in establishing the desired PSP linkages. In contrast, deep learning approaches provide a feature-engineering-free framework with high learning capability. In this work, a deep learning approach is designed and implemented to model an elastic homogenization structure-property linkage in a high contrast composite material system. More specifically, the proposed deep learning model is employed to capture the nonlinear mapping between the three-dimensional material microstructure and its macroscale (effective) stiffness. It is demonstrated that this end-to-end framework can predict the effective stiffness of high contrast elastic composites with a wide of range of microstructures, while exhibiting high accuracy and low computational cost for new evaluations.

Time domain finite element estimates of dynamic stiffness of viscoelastic composites with stiff spherical inclusions

A weighted residual time domain unstructured mesh finite element approach was used to obtain numerical estimates for the dynamic stiffness of medium and high stiffness contrast composites consisting of viscoelastic epoxy matrix filled with stiff spherical inclusions. Both random and regular microstructure composites were studied. It was shown that over the broad temperature and inclusion fraction ranges studied, the generalized self-consistent model ( Christensen and Lo, 1979 ) provided accurate predictions for the effective dynamic stiffness of random composites obeying classical Percus–Yevick hard sphere statistics. Assuming sphere flocculation, we have studied the role of microstructural effects and found that upon increasing stiffness contrast, they become progressively more important for both the effective storage and loss moduli. As a consequence, for the reliable predictions of effective stiffness of high contrast composites with rubber like matrices, one should necessarily use homogenization models accounting not only for the inclusion concentration but also for the finer microstructural details. However, it was also found that for such high stiffness contrast viscoelastic composites, the ratio of their effective storage and loss shear moduli (the damping factor) remains essentially unchanged as compared to that of pure unfilled matrix so it is the invariability of the damping factor that can be viewed as a special signature of underlying micromechanical mechanism of reinforcement.

Asymptotic Approximation of the Dirichlet to Neumann Map of High Contrast Conductive Media

We present an asymptotic study of the Dirichlet to Neumann map of high contrast composite media with perfectly conducting inclusions that are close to touching. The result is an explicit characterization of the map in the asymptotic limit of the distance between the particles tending to zero. Known homogenization results state that the effective conductivity of the high contrast composites is determined by resistor networks with topology defined by the geometric distribution of the inclusions. The result in this paper is stronger than homogenization because it applies to arbitrary boundary conditions. In particular, it accounts for the coupling of boundary current flow induced by oscillatory boundary potentials and the flow in the network.

On the quasistatic effective elastic moduli for elastic waves in three-dimensional phononic crystals

Effective elastic moduli for 3D solid–solid phononic crystals of arbitrary anisotropy and oblique lattice structure are formulated analytically using the plane-wave expansion (PWE) method and the recently proposed monodromy-matrix (MM) method. The latter approach employs Fourier series in two dimensions with direct numerical integration along the third direction. As a result, the MM method converges much quicker to the exact moduli in comparison with the PWE as the number of Fourier coefficients increases. The MM method yields a more explicit formula than previous results, enabling a closed-form upper bound on the effective Christoffel tensor. The MM approach significantly improves the efficiency and accuracy of evaluating effective wave speeds for high-contrast composites and for configurations of closely spaced inclusions, as demonstrated by three-dimensional examples.

Evaluation of the effective speed of sound in phononic crystals by the monodromy matrix method (L)

A scheme for evaluating the effective quasistatic speed of sound c in two- and three-dimensional periodic materials is reported. The approach uses a monodromy-matrix operator to enable direct integration in one of the coordinates and exponentially fast convergence in others. As a result, the solution for c has a more closed form than previous formulas. It significantly improves the efficiency and accuracy of evaluating c for high-contrast composites as demonstrated by a two-dimensional scalar-wave example with extreme behavior.

Tuneability amplification factor and loss of high-contrast composites

The problem of tuneability of nonlinear ferroelectric–dielectric composite materials is addressed. Attention is concentrated on the analysis of the tuneability amplification factor, K, of a composite material, which is introduced as the ratio of the effective tuneability of a composite material to the tuneability of its ferroelectric (tuneable) component. Previously, ferroelectric–dielectric composite materials have been designed with an effective tuneability amplification factor slightly greater than 1 (i.e. 1.1–1.4) [A.K. Tagantsev et al., J. Electroceramics 11 (2003) p.5; A.G. Kolpakov et al., J. Electroceramics 18 (2007) p.129]. It is demonstrated that the tuneability amplification factor can take values significantly greater than 1. Numerical prototypes (structural designs integrated with finite element method tools) of microstructures are presented with K in the range 3 to 30. The effective tuneability of nonlinear composite materials strongly depends on the microgeometry and microtopology of the material and, in general, cannot be described in terms of volume fraction of components of composite material. In the designs presented, the increased tuneability is due to concentration of the high electric field in narrow regions with carefully selected geometry. The problem of loss in ferroelectric–dielectric composite materials is discussed. In the general case, the loss tangent stays between that of the components of the composite. For high-contrast ferroelectric–dielectric composite materials, the effective loss tangent is practically equal to the loss tangent of the ferroelectric.

A discrete network approximation for effective conductivity of non-Ohmic high-contrast composites

A discrete network approximation for effective conductivity of non-Ohmic high-contrast composites