Variational Asymptotic Method For Unit Cell Homogenization Research Articles

Modeling of Cementitious Representative Volume Element with Various Water–Cement Ratios

This study uses the finite element method (FEM) to measure the mechanical properties of microstructure-based cementitious representative volume elements (RVEs) with various water–cement ratios (W/Cs) generated by CEMHYD3D. The finite element boundary condition effects that significantly and computationally change the elastic properties are studied and discussed. Various boundary conditions in ABAQUS are applied and compared with the results obtained using the variational asymptotic method for unit cell homogenization (VAMUCH). This comparison is conducted using ANSYS. This study aims to analyze and determine the effect of different boundary conditions in detail on the prediction of the elastic properties of cementitious RVE with various W/Cs and identify the best approach in this regard. Results show that Young’s, shear, and bulk moduli decrease with the increase in W/C and the boundary conditions in ABAQUS influence the outcomes, particularly on bulk modulus and Poisson’s ratio.

Crack formation within ceramics via coupled multiscale genome and XFEM predictions under various loading conditions

Because of the complex heterogeneous microstructure of ceramics, predicting crack formation within ceramics is still a challenge. The extended finite element method (XFEM) serves as a good tool for fracture prediction but is incapable of considering heterogeneous microstructure. In this paper, a numerical framework is developed to model cracks within ceramics by coupling a multiscale genome model with XFEM. XFEM is embedded in the formulation of the multiscale genome through the variational asymptotic method for unit cell homogenization (VAMUCH). The implementation of both multiscale genome model and XFEM retains the capabilities of XFEM in modeling fracture while providing accurate predictions by considering heterogeneous microstructure. The crack formation within SiC ceramics under different loading conditions is simulated in comparison with experiments in order to assess the validity of the proposed method. It is shown that the developed model captures the typical characteristics of crack formation within silicon carbide (SiC) ceramics under bending and indentation loadings. The predicted cutting forces and crack depth exhibit a good agreement with the experimental results during machining processes.

Multi-scale genome modeling for predicting fracture strength of silicon carbide ceramics

The variational asymptotic method for unit cell homogenization (VAMUCH) has emerged as a general purpose micromechanics approach capable of predicting the effective properties of heterogeneous materials and recovering the local fields. In this study, a novel micromechanics approach has been developed enabling VAMUCH to homogenize heterogeneous microstructure and predict its crack formation through a multi-scale materials genome model. A variational form for homogenization is formulated in combination with a cohesive zone model. The weak form of the problem is derived using an asymptotic method, discretized using finite element formulations, and implemented into VAMUCH. The advantages of the present approach are demonstrated through homogenizing silicon carbide ceramics and predicting its fracture strength. Both the elastic properties and fracture strength can be predicted in a computationally efficient manner using this approach compared with the multi-scale finite element model.

Multi-coating inhomogeneities approach for composite materials with temperature-dependent constituents under small strain and finite thermal perturbation assumptions

Abstract A multi-coated inhomogeneities mean field micromechanics approach is proposed to predict the overall thermoelastic properties, namely, elasticities, thermal expansions and heat capacity, of a thermoelastic composite material with temperature-dependent constituents under finite temperature changes and small strain assumptions. First, the Helmholtz potential density for small strain finite thermoelasticity has been presented. Then the fundamental solution based on Green's function of thermoelasticity problem has been used to derive the general expressions for the elastic and thermal strains concentration tensors with temperature-dependent material properties. Numerical examples based on the multi-coating Mori-Tanaka model are used to quantify the differences of the effective properties based on the small strain finite thermoelasticity in comparison to the linear thermoelasticity. The predictions of the present mean field micromechanics model are compared to those of the variational asymptotic method for unit cell homogenization (VAMUCH), a finite element based micromechanics model.

Overall thermo-magneto-electro-elastic properties of multiferroics composite materials with arbitrary heterogeneities spatial distributions

This study extends our recent work (Koutsawa, 2014) to multiferroics composite materials with temperature-independent properties. The key ingredient in this development is the computation of the so-called interaction tensors between the multiferroics inhomogeneities which are for the first time presented here for the magneto-electro-elastic coupling. The present mean field based micromechanics model is able to account for the morphological and topological textures of the multiferroics heterogeneities. Numerical examples based on the multiple site Mori–Tanaka model are used to quantify the prediction capabilities of the proposed micromechanics scheme. The predictions of the micromechanics model are compared to those of the variational asymptotic method for unit-cell homogenization (VAMUCH), a finite element based micromechanics approach. There is a good quantitative agreement between the mean field model predictions and numerical results obtained with VAMUCH. The model is then applied to three typical packing arrangements of particulate thermo-magneto-electro-elastic composites, namely, Body Centered Cubic (BCC), Face Centered Cubic (FCC) and Simple Cubic (SC) to demonstrate its capability to account for the topological texture of multiferroics composite materials microstructures.

Computational evaluation of effective stress relaxation behavior of polymer composites

This paper presents a micromechanics model to characterize the effective stress relaxation stiffness of polymer composites. The linear viscoelastic behavior of polymer material was modeled by hereditary integral. The proposed model was established based on the variational asymptotic method for unit cell homogenization (VAMUCH). All computations with this model were accomplished in the time domain, hence the Laplace transform and inversion commonly used for linear viscoelastic composites are not needed in this theory. The accuracy and efficiency of the proposed model were verified by comparing with the results and utilization of finite element models developed using ABAQUS.

A micromechanics approach to homogenizing elasto-viscoplastic heterogeneous materials

The variational asymptotic method for unit cell homogenization (VAMUCH) has emerged as a general-purpose micromechanics code capable of predicting the effective properties of heterogeneous materials and recovering the local fields. The objective of this paper is to propose a micromechanics approach enabling VAMUCH to homogenize elasto-viscoplastic heterogeneous materials. An affine formulation of the constitutive relations for an elasto-viscoplastic constituent, which exhibits viscoplastic anisotropy and combined isotropic–kinematic hardening, is derived. The weak form of the problem is derived using an asymptotic method, discretized using finite elements, and implemented into VAMUCH. The new features of VAMUCH are validated with examples such as homogenizing binary, fiber-reinforced, and particle-reinforced composites. VAMUCH is found to be capable of handling various microstructure, complex material models, complex loading conditions, and complex loading paths. More sophisticated material models can be implemented into it.

Effective electromechanical coupling coefficient of adaptive structures with integrated multi-functional piezoelectric structural fiber composites

This paper presents a linear computational homogenization framework to evaluate the effective (or generalized) electromechanical coupling coefficient (EMCC) of adaptive structures with piezoelectric structural fiber (PSF) composite elements. The PSF consists of a silicon carbide (SiC) or carbon core fiber as reinforcement to a fragile piezo-ceramic shell. For the micro-scale analysis, a micromechanics model based on the variational asymptotic method for unit cell homogenization (VAMUCH) is used to evaluate the overall electromechanical properties of the PSF composites. At the macro-scale, a finite element (FE) analysis with the commercial FE code ABAQUS is performed to evaluate the effective EMCC for structures with the PSF composite patches. The EMCC is postprocessed from free-vibrations analysis under short-circuit (SC) and open-circuit (OC) electrodes of the patches. This linear two-scale computational framework may be useful for the optimal design of active structure multi-functional composites which can be used for multi-functional applications such as structural health monitoring, power harvest, vibration sensing and control, damping, and shape control through anisotropic actuation.

Variational asymptotic micromechanics modeling of heterogeneous magnetostrictive composite materials

A new micromechanics model is developed to predict the effective properties as well as the local fields of heterogeneous magnetostrictive composite materials using the variational asymptotic method for unit cell homogenization (VAMUCH), a recently developed micromechanics modeling technique. Starting from the total magnetic enthalpy of the heterogenous continuum, we formulate the micromechanics model as a constrained minimization problem taking advantage of the fact that the size of the microstructure is small compared to the macroscopic size of the material. To handle realistic microstructures in engineering applications, we implement this new model using the finite element method. A few examples are used to demonstrate the application and accuracy of the proposed theory and the companion computer program-VAMUCH.

Application of the variational asymptotic method for unit cell homogenization in the prediction of mechanical properties for microcellular plastics

This study presents the application of the micromechanical variational asymptotic method for unit cell homogenization (VAMUCH) with a three-dimensional unit cell structure and a coupled, macroscaled finite element analysis for analyzing and predicting the effective elastic properties of microcellular injection-molded plastics. A series of injection-molded plastic samples, which include polylactic acid, low-density polyethylene, polypropylene, polystyrene, and thermoplastic polyurethane, with microcellular-foamed structures were produced and their mechanical properties were compared with predicted values. The results showed that, for most material samples, the numerical prediction was in fairly good agreement with experimental results, which suggests the applicability and reliability of VAMUCH in analyzing the mechanical properties of porous materials. Other findings were that the material characteristics (e.g. brittleness and ductility) influenced the predicted results and that VAMUCH prediction could be improved when the unit cell structure was more representative of the real composition.

X-FEM implementation of VAMUCH: Application to active structural fiber multi-functional composite materials

In multiscale analysis of composite materials, there is usually a need to solve microstructures problems with complex geometries. The variational asymptotic method for unit cell homogenization (VAMUCH) is a recently developed variant of the asymptotic homogenization approach. In contrast to conventional asymptotic methods, VAMUCH carries out an asymptotic analysis of the variational statement, synthesizing the merits of both variational methods and asymptotic methods. This work gives an outline of the Extended Finite Element Method (X-FEM) implementation of VAMUCH for complex multi-material structures. The X-FEM allows one to use meshes not necessarily matching the physical surface of the problem while retaining the accuracy of the classical finite element approach. For material interfaces, this is achieved by introducing an enrichment strategy. The X-FEM/VAMUCH approach is applied successfully to many examples reported in the VAMUCH literature. Numerical experiments on the periodic homogenization of complex unit cells demonstrate the accuracy and simplicity of the X-FEM/VAMUCH approach.

Analysis and optimization of heterogeneous materials using the variational asymptotic method for unit cell homogenization

The variational asymptotic method for unit cell homogenization is used to find the sensitivity of the effective properties of periodically heterogeneous materials, within a periodic base-cell. The sensitivities are found by the direct differentiation of the variational asymptotic method for unit cell homogenization (VAMUCH) and by the method of adjoint variables. This sensitivity theory is implemented using the finite element method and the engineering program VAMUCH. The methodology is used to design the periodic microstructure of a material that allows obtaining prescribed constitutive properties. The microstructure is modeled as a 2D periodic structure, but a complete set of 3D material properties are obtained. Furthermore, the present methodology can be used to perform the micromechanical analysis and related sensitivity analysis of heterogeneous materials that have 3D periodic structures. The effective material properties of the artificially mixed materials of the microstructure are obtained by the density approach, in which the solid material and void are mixed artificially.

Variational asymptotic micromechanics modeling of heterogeneous piezoelectric materials

In this paper, a new micromechanics model is developed to predict the effective properties and local fields of heterogeneous piezoelectric materials using the variational asymptotic method for unit cell homogenization (VAMUCH), a recently developed micromechanics modeling technique. Starting from the total electric enthalpy of the heterogenous continuum, we formulate the micromechanics model as a constrained minimization problem using the variational asymptotic method. To handle realistic microstructures in engineering applications, we implement this new model using the finite element method. For validation, a few examples are used to demonstrate the application and accuracy of this theory and the companion computer program – VAMUCH.

Variational asymptotic method for unit cell homogenization of periodically heterogeneous materials

A new micromechanics model, namely, the variational asymptotic method for unit cell homogenization (VAMUCH), is developed to predict the effective properties of periodically heterogeneous materials and recover the local fields. Considering the periodicity as a small parameter, we can formulate a variational statement of the unit cell through an asymptotic expansion of the energy functional. It is shown that the governing differential equations and periodic boundary conditions of mathematical homogenization theories (MHT) can be reproduced from this variational statement. In comparison to other approaches, VAMUCH does not rely on ad hoc assumptions, has the same rigor as MHT, has a straightforward numerical implementation, and can calculate the complete set of properties simultaneously without using multiple loadings. This theory is implemented using the finite element method and an engineering program, VAMUCH, is developed for micromechanical analysis of unit cells. Many examples of binary composites, fiber reinforced composites, and particle reinforced composites are used to demonstrate the application, power, and accuracy of the theory and the code of VAMUCH.