Computational Homogenization Procedure Research Articles

A multi-scale formulation for polycrystalline materials accounting for cohesive micro-cracks: Homogenisation of the traction-separation law

This contribution investigates the modelling of failure in polycrystalline materials due to the nucleation and propagation of cohesive cracks at the micro-scale. First, a two-scale formulation that enables the analysis of failure, even when the evolution and propagation of cohesive micro-cracks induce material instability, is proposed. The development of this macro-continuous/micro-discontinuous formulation is based on the Method of Multi-Scale Virtual Power, from which dynamic equilibrium and homogenisation equations are derived, ensuring a variationally consistent scale transition. The respective minimal kinematical constraint is retrieved and enforced with the Lagrange multiplier method. The formulation is also extended for periodic boundary conditions. The discretisation of the resulting equilibrium equations, with the finite element method and the generalised-α method, is described in detail.Second, a fracture-based computational homogenisation procedure is developed to obtain homogenised traction-separation laws and fracture properties from microstructural volume elements. It is derived from a crack-based Hill-Mandel principle and employs a novel energetic-based damage variable, within the Park-Paulino-Roesler cohesive model, proposed to define the crack domain and compute the crack homogenised quantities. A strategy for accurately computing the homogenised unit normal vector of the equivalent macroscopic crack is also suggested.Numerical examples demonstrate the formulation’s ability to analyse fracture problems, including a study on the impact of different microscopic boundary conditions. This work also reveals the convergence of the traction-separation law with increasing microstructure model size when using the fracture-based computational homogenisation. Furthermore, it investigates how advanced deformation mechanisms, like slip plasticity and martensitic transformation, affect intergranular crack propagation mechanisms.

Isotropic yield surfaces for porous ductile materials: complete geometric representation by a computational homogenisation procedure

PurposeThe present paper aims to explore a computational homogenisation procedure to investigate the full geometric representation of yield surfaces for isotropic porous ductile media. The effects of cell morphology and imposed boundary conditions are assessed. The sensitivity of the yield surfaces to the Lode angle is also investigated in detail.Design/methodology/approachThe microscale of the material is modelled by the concept of Representative Volume Element (RVE) or unit cell, which is numerically simulated through three-dimensional finite element analyses. Numerous loading conditions are considered to create complete yield surfaces encompassing high, intermediate and low triaxialities. The influence of cell morphology on the yield surfaces is assessed considering a spherical cell with spherical void and a cubic RVE with spherical void, both under uniform strain boundary condition. The use of spherical cell is interesting as preferential directions in the effective behaviour are avoided. The periodic boundary condition, which favours strain localization, is imposed on the cubic RVE to compare the results. Small strains are assumed and the cell matrix is considered as a perfect elasto-plastic material following the von Mises yield criterion.FindingsDifferent morphologies for the cell imply in different yield conditions for the same load situations. The yield surfaces in correspondence to periodic boundary condition show significant differences compared to those obtained by imposing uniform strain boundary condition. The stress Lode angle has a strong influence on the geometry of the yield surfaces considering low and intermediate triaxialities.Originality/valueThe exhaustive computational study of the effects of cell morphologies and imposed boundary conditions fills a gap in the full representation of the flow surfaces. The homogenisation-based strategy allows us to further investigate the influence of the Lode angle on the yield surfaces.

A Green’s function-based approach to the concentration tensor fields in arbitrary elastic microstructures

Computational homogenization based on FEM models is the gold standard when it comes to homogenization over a representative volume element (RVE), of so-called complex material microstructures, i.e., such which cannot be satisfactorily represented by an assemblage of homogeneous subdomains called phases. As a complement to the aforementioned models, which depend on the boundary conditions applied to the representative volume element and which, as a rule, do not give direct access to the macro-micro-relations in terms of concentration tensors, we here introduce a Green’s function-based homogenization method for arbitrary inhomogeneous microstructures: Inspired by the ideas underlying traditional phase-based homogenization schemes, such as the Mori-Tanaka or the self-consistent model, the new method rests on mapping, through the strain average rule, the microscopic strain fields associated with an auxiliary problem to the macroscopic strains subjected to the RVE. Thereby, the auxiliary problem is defined on a homogeneous infinite matrix subjected to homogeneous auxiliary strains and to inhomogeneous (fluctuating) polarization stresses representing the fluctuations of the microstiffness field, i.e., the complex microstructure within the RVE. The corresponding microscopic strains appear as the solution of a Fredholm integral equation, delivering a multilinear operator linking the homogeneous auxiliary strains to the microscopic strains. This operator, together with the aforementioned mapping, eventually allows for completing the model in terms of concentration tensor and homogenized stiffness quantification. This is illustrated by example of a sinusoidally fluctuating microstructure, whereby the corresponding singular convolution integrals are analytically evaluated from the solution of the Poisson’s equation, and this evaluation strategy is then analytically verified through a Cauchy principal value analysis, and numerically validated by a state-of-the-art FFT homogenization procedure. For the given example, the novel analytical method is several thousand times faster than an FTT-based computational homogenization procedure.

A mixed stress-strain driven computational homogenization of spiral strands

Spiral strands subjected to tensile force and bending loading display a nonlinear dissipative behavior due to frictional interactions between their elementary wires. This study aims to provide an efficient method, based on a computational homogenization procedure, to accurately characterize the nonlinear response of such strands. By using 1D beam elements in both micro- and macro-scale, homogenization is performed along the axial direction of a representative volume element (RVE), leading to expressing a boundary value problem on RVE, driven in a mixed manner by either strains or resulting forces or moments. The boundary value problem on the RVE is solved using an in–house implicit finite element solver for finite strain, considering all frictional contact interactions. A method is proposed to predict the bending moment’s evolution for any curvature variation from the simulation results of only one bending loading test on the RVE. The nonlinear behavior of the strand in the micro-scale identified through this offline technique can then be used in the macro-model to simulate various bending loading tests under constant tensile load. Results obtained with the multiscale model are compared to those provided by direct numerical simulation to demonstrate the validity of the proposed approach.

Stiffness and strength evaluation of lattice-based mechanical metamaterials by decoupled two-scale analysis

There is an emerging class of artificial materials, so-called mechanical metamaterials. By deliberately designing microscopic internal mechanisms, these materials exhibit exotic properties, which distinguish the mechanical metamaterials from natural materials. As additive manufacturing technology has been rapidly developed and its manufacturing quality has been improved, those conceptual artificial materials are ready to fabricate. This paper presents a decoupled two-scale analysis framework to efficiently evaluate the stiffness and strength characteristics of lattice-based mechanical metamaterials. Structural characteristics of lattice-based mechanical metamaterials are effectively captured by accommodating a computational homogenization method to lattice-based structures. In the computational homogenization procedure, a unit cell of the periodically distributed structure is modeled as representative finite elements. Equivalent stiffness properties of the unit cell are calculated based on the computational homogenization procedure. The equivalent structural properties are then used to perform macroscopic structural analysis of lattice-based structures in an effective manner. Similarly, a local/microscopic stress distribution within the periodic lattice unit cell is recovered by transforming macroscopic sectional loads, which are obtained from the macroscopic structural analysis, through a stress amplification matrix. The microscopic stress distribution obtained with the two-scale analysis can be used for strength evaluation. Subsequently, numerical results are validated by comparing them with experimental results providing actual behaviors of lattice-based mechanical metamaterials. This work presents the novel use of the multi-scale analysis scheme combining the computational homogenization method and the stress field transformation method enabling structural evaluations to design mechanical metamaterials with structural integrity. This study also demonstrates the feasibility and capability of the presented approach to effectively predict stiffness and strength characteristics of lattice-based structures.

A stress-driven computational homogenization method based on complementary potential energy variational principle for elastic composites

Based on the complementary potential energy variational principle, in this work, we proposed a stress-driven homogenization procedure to compute overall effective material properties for elastic composites with locally heterogenous micro-structures. We have developed a novel incremental variational formulation for homogenization problems of both infinitesimal and finite deformations where the macro-stress-based complementary potential energy is obtained for hyperelastic materials for a global minimization problem with respect to fine-scale displacement fluctuation field. The point of departure of our approach is a general complementary variational principle formulation that can determine material responses of elastic composites with heterogeneous micro structures. We have implemented the proposed stress-driven computational homogenization procedure with the finite element method. By comparing the numerical results with the analytical method and the strain-driven homogenization method, we find that the stress-driven homogenization offers the lower bound estimate of materials properties for elastic composites.

An XFEM-based computational homogenization framework for thermal conductivity evaluation of composites with imperfectly bonded inclusions

In many engineering applications, such as thermal coating and structural insulation, 2D modelling is deemed sufficient to identify design parameters. For the effective thermal conductivity evaluation of heterogeneous composites, a 2D computational homogenization procedure is developed in this study. The inclusions are distributed randomly within the plane. The boundary conditions tested for homogenization procedure introduced include the periodic temperature, uniform temperature, and uniform flux boundary conditions. The bond between the inclusions and the surrounding matrix is assumed imperfect allowing partial heat transfer at the interface. Randomly distributed inclusions can be introduced without altering the underlying regular mesh of the matrix by using the XFEM, providing an efficient way of introducing the inclusions. Implementation details of the proposed computational homogenization scheme based on the XFEM are provided. The results are validated by comparisons with available analytical solutions. Effect of assumed boundary conditions on the results are shown. Parametric studies illustrate the influence of the interface properties, volume ratio of inclusions as well as the distributions of the inclusions on the effective thermal conductivity of composites.

FE2 simulations of magnetorheological elastomers: influence of microscopic boundary conditions, microstructures and free space on the macroscopic responses of MREs

In the current work, the response of heterogeneous magnetorheological elastomers (MREs) which are loaded by external magnetic fields in the absence and also in the presence of free space is studied. A fully-coupled two-scale finite element computational homogenization procedure is used to derive the material response at the macro-scale from the averaged response of the underlying micro-scale problem. Different combinations of boundary conditions, that satisfy the Hill-Mandel condition and are based on the primary variables of the magneto-elastic enthalpy and energy functionals are applied to solve the micro-scale boundary value problem. Furthermore, the influences of various microstructures on the macroscopic response of the MREs are investigated. The results indicate that the choice of microscopic boundary conditions and microstructure types can significantly affect the macroscopic responses of MREs.

Fast statistical homogenization procedure (FSHP) for particle random composites using virtual element method

Mechanical behaviour of particle composite materials is growing of interest to engineering applications. A computational homogenization procedure in conjunction with a statistical approach have been successfully adopted for the definition of the representative volume element (RVE) size, that in random media is an unknown of the problem, and of the related equivalent elastic moduli. Drawback of such a statistical approach to homogenization is the high computational cost, which prevents the possibility to perform series of parametric analyses. In this work, we propose a so-called fast statistical homogenization procedure (FSHP) developed within an integrated framework that automates all the steps to perform. Furthermore within the FSHP, we adopt the numerical framework of the virtual element method for numerical simulations to reduce the computational burden. The computational strategies and the discretization adopted allow us to efficiently solve the series (hundreds) of simulations and to rapidly converge to the RVE size detection.

Multiscale modeling of the thermomechanical behavior in heterogeneous media embedding Phase Change Materials particles

In this work, a multiscale model for thermomechanical properties of composite structures containing phase change particles is developed. For the mechanical part, a classical linear computational homogenization procedure is employed. For the thermal part, due to the strong nonlinear, history-dependent thermal effects, a concurrent multiscale (FE2) method is extended to take into account the presence of Phase Change Materials particles (PCM) at the microscale. The PCM inclusions change from liquid to solid state in the range of room temperature. This phase change induces a modified macroscopic thermal behavior, which can be used e.g. to design materials with enhanced thermal inertia and reduce energy consumption in civil engineering constructions. The technique allows taking into account accurately the fully nonlinear, history-dependent thermal behavior through numerical calculations at the microscale based on a Representative Volume Element (RVE) and its effect at the macroscale. The method is applied to concrete material including paraffin wax PCM. The results show the benefits of the PCM on the thermal behavior, including shifted and smoothed temperature response as compared to materials without PCM particles.

Computational homogenization of tensile deformation behaviors of a third generation Al-Li alloy 2060-T8 using crystal plasticity finite element method

A computational homogenization procedure based on the crystal plasticity model was proposed herein to predict the tensile deformation behavior and investigate the anisotropic response of a novel third generation Al-Li alloy (AA2060-T8) at room temperature and different deformation conditions. To elucidate the in-grain deformation features, a representative volume element was constructed to reveal the microstructure of the real AA2060-T8 (polycrystalline material) in which each grain was discretized by many finite elements. Afterward, numerical results were assigned to each grain to explore the pre-texture formed by previous thermomechanical processes. A dislocation density-based crystal plasticity model was developed to infer the constitutive equation of each grain and simulate the plastic deformation of AA2060-T8 alloy. The material parameters used in the dislocation density-based crystal plasticity model were calibrated against a tensile stress-strain curve deformed at 30° with respect to rolling direction. The results obtained from the proposed computational homogenization method are in line with those obtained from experimentation. This indicates that the proposed computational homogenization method can definitely predict the tensile deformation behavior and capture the anisotropic responses of AA2060-T8 (polycrystalline materials) originating from deformation induced texture as well as the initially anisotropic texture.

Skin friction under pressure. The role of micromechanics

The role of contact pressure on skin friction has been documented in multiple experimental studies. Skin friction significantly raises in the low-pressure regime as load increases while, after a critical pressure value is reached, the coefficient of friction of skin against an external surface becomes mostly insensitive to contact pressure. However, up to now, no study has elucidated the qualitative and quantitative nature of the interplay between contact pressure, the material and microstructural properties of the skin, the size of an indenting slider and the resulting measured macroscopic coefficient of friction. A mechanistic understanding of these aspects is essential for guiding the rational design of products intended to interact with the skin through optimally-tuned surface and/or microstructural properties.Here, an anatomically-realistic 2D multi-layer finite element model of the skin was embedded within a computational contact homogenisation procedure. The main objective was to investigate the sensitivity of macroscopic skin friction to the parameters discussed above, in addition to the local (i.e. microscopic) coefficient of friction defined at skin asperity level. This was accomplished via the design of a large-scale computational experiment featuring 312 analyses. Results confirmed the potentially major role of finite deformations of skin asperities on the resulting macroscopic friction. This effect was shown to be modulated by the level of contact pressure and relative size of skin surface asperities compared to those of a rigid slider. The numerical study also corroborated experimental observations concerning the existence of two contact pressure regimes where macroscopic friction steeply and non-linearly increases up to a critical value, and then remains approximately constant as pressure increases further.The proposed computational modelling platform offers attractive features which are beyond the reach of current analytical models of skin friction, namely, the ability to accommodate arbitrary kinematics, non-linear constitutive properties and the complex skin microstructure.

A stochastic multi-scale approach for the modeling of thermo-elastic damping in micro-resonators

The aim of this work is to study the thermo-elastic quality factor (Q) of micro-resonators with a stochastic multi-scale approach. In the design of high-Q micro-resonators, thermo-elastic damping is one of the major dissipation mechanisms, which may have detrimental effects on the quality factor, and has to be predicted accurately. Since material uncertainties are inherent to and unavoidable in micro-electromechanical systems (MEMS), the effects of those variations have to be considered in the modeling in order to ensure the required MEMS performance. To this end, a coupled thermo-mechanical stochastic multi-scale approach is developed in this paper. Thermo-mechanical micro-models of polycrystalline materials are used to represent micro-structure realizations. A computational homogenization procedure is then applied on these statistical volume elements to obtain the stochastic characterizations of the elasticity, thermal expansion, and conductivity tensors at the meso-scale. Spatially correlated meso-scale random fields can thus be generated to represent the stochastic behavior of the homogenized material properties. Finally, the distribution of the thermo-elastic quality factor of MEMS resonators is studied through a stochastic finite element method using as input the generated stochastic random field.

An intuitive computational multi-scale methodology and tool for the dynamic modelling of viscoelastic composites and structures

This paper proposes an intuitive computational multi-scale homogenisation procedure and tool for the estimation of the effective static and mechanical properties of complex viscoelastic composite material and structures. The proposed solution consists in computing numerically the complex effective properties (storage and loss moduli) as function of frequency. The developed numerical tool is coupled with ABAQUS FEA in order to ease the uptake of the technology code and allows for an accurate and automatic simulation of composite engineering structures with substantially less human intervention and a rational control of the error. Details concerning the numerical implementation and a selection of representative numerical examples are provided.

Algorithmic two-scale transition for magneto-electro-mechanically coupled problems: FE2-scheme: Localization and homogenization

In this paper we present a computational homogenization procedure for the simulation of magneto-electro-mechanically coupled boundary value problems (bvp)s on two scales. We derive the basic equations for the localization and the homogenization of the individual field variables and give an algorithmic expression for the effective tangent moduli. The resulting algorithmic two-scale transition procedure is implemented into an FE2-method, which allows us to compute macroscopic boundary value problems in consideration of attached microscopic representative volume elements. The challenge in the simulation of magneto-electro-mechanically coupled materials is the modeling of the complicated interactions between magnetical, electrical and mechanical quantities on both scales. A primer example for such interactions is given by magneto-electric composites. Thus, we apply the presented method to the two-scale simulation of a variety of versions of magneto-electric composites. In detail, we consider two-phase composites composed of (i) piezomagnetic and piezoelectric, (ii) piezomagnetic and non-linear electrostrictive, as well as (iii) piezomagnetic and non-linear dissipative ferroelectric phase. Based on numerical examples, we discuss aspects of the applicability, the numerical stability as well as the predictive capability of the proposed method.

Grain‐scale‐based simulation of granular material

AbstractIn recent years, new developments in experimental mechanics, e. g., particle‐tracking methods using X‐ray micro‐tomography, have provided a new insight into the material behaviour of granular media. The possibility to capture the individual grain displacement and rotation of all particles within a sample provides a new fundament for a reliable numerical simulation of granulates. This is, in particular, the case for deformation localisation in shear zones as they are observed in bi‐ and triaxial compression tests, where the exact thickness of the localised zone as well as the amount of accompanying grain displacements and rotations are measurable in a precise and direct way. In this context, the Discrete‐Element (DE) Method has become a promising and frequently used modelling technique for granular materials. In contrast to macroscopically based continuum‐mechanical approaches, the DE approach treats each particle as a rigid and uncrushably body and is, therefore, defined on a microscopic level of description. This consequently allows a direct comparison with experimental data from grain‐scale tracking methods. In the present contribution, the DE method is applied in order to solve problems of localisation in granular media. Therefore, constitutive inter‐particle contact laws are formulated for forces and torques which are transferred between interacting particles. With a given solution of initial‐boundary‐value problems on the microscale, a computational homogenisation procedure leads to macroscopic quantities purely stemming from microscopic information. Herein, a particle‐centre‐based volume averaging technique leads to macroscopic quantities which correspond to quantities known from micropolar continuum theories. The contribution presents a framework for the homogenisation in all time steps in a fully 3‐dimensional context and discusses the activation of micropolar effects in the localised zone. Furthermore, the admissible size of the chosen Representative Elementary Volume (REV) is addressed. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Yield surfaces of heterogeneous media with debonded inclusions

Purpose – The purpose of this paper is to present knowledge in estimating yield surfaces of heterogeneous media by use of homogenization, especially where the macroscopic behaviour is driven by weak interfaces between phase constituents. Design/methodology/approach – A computational homogenization procedure is used to determine the yield surface of a Representative Volume Element (RVE) that contains a fully debonded inclusion embedded within ideally plastic matrix, whereby the interface is modelled by a Coulomb type friction law. Findings – The macroscopic behaviour of the RVE is shown to coincide an RVE with a hole for expanding loads, whereas for compressive loads, it was shown to approach an RVE with a fully bonded inclusion. Originality/value – The present paper builds on Gurson’s work in estimating macroscopic yield surfaces of heterogeneous materials. The work is novel in the sense that there had been no previous publications discussing influence of weak interfaces on yield surfaces.

A stochastic computational multiscale approach; Application to MEMS resonators

Abstract The aim of this work is to develop a stochastic multiscale model for polycrystalline materials, which accounts for the uncertainties in the micro-structure. At the finest scale, we model the micro-structure using a random Voronoi tessellation, each grain being assigned a random orientation. Then, we apply a computational homogenization procedure on statistical volume elements to obtain a stochastic characterization of the elasticity tensor at the meso-scale. A random field of the meso-scale elasticity tensor can thus be generated based on the information obtained from the SVE simulations. Finally, using a stochastic finite element method, these meso-scale uncertainties are propagated to the coarser scale. As an illustration we study the resonance frequencies of MEMS micro-beams made of poly-silicon materials, and we show that the stochastic multiscale approach predicts results in agreement with a Monte Carlo analysis applied directly on the fine finite-element model, i.e. with an explicit discretization of the grains.

Second-Order Computational Homogenization Scheme Preserving Microlevel <i>C</i><sup>1</sup> Continuity

The paper deals with a new second-order computational homogenization procedure for modeling of heterogeneous materials at small strains, where C1 continuity is preserved at the microlevel. The multiscale model is based on the Aifantis theory of gradient elasticity. The C1 two dimensional triangular finite element used for the discretization of macro-and microlevel is described. Contrary to the C1 - C0 transition, here besides the displacements, the displacement gradients are included into the boundary conditions on the representative volume element (RVE). According to the second order continuum at microlevel, the relevant homogenization relations are derived. Finally, the performance of the algorithms derived is investigated. Dependency of homogenized stresses on mesh density and microstructural parameter l are examined in simple loading cases.

A second-order two-scale homogenization procedure using $$C^{1}$$ C 1 macrolevel discretization

The present study deals with a second-order two-scale computational homogenization procedure for modeling deformation responses of heterogeneous materials at small strains. The macro to micro transition and the application of generalized periodic boundary conditions on the representative volume element (RVE) at the microlevel are investigated. The structure at macroscale level is discretized by the $$C^{1}$$ C 1 two dimensional triangular finite elements, while the $$C^{0}$$ C 0 quadrilateral finite element is used for the discretization of the RVE. The finite element formulations and the new proposed multiscale scheme have been implemented into the finite element software ABAQUS using user subroutines derived. Due to the $$C^{1}-C^{0}$$ C 1 - C 0 continuity transition, an additional integral condition on microlevel fluctuation field has to be imposed, as expected. The integration has been performed using various numerical integration techniques and the results obtained are compared in a few examples. It is concluded that only trapezoidal rule gives a physically based deformed shape of the RVE. Finally, the efficiency and accuracy of the proposed multiscale homogenization approach are demonstrated by the modeling of a shear layer problem, usually used as a benchmark in multiscale analyses.