Two field multibody method for periodic homogenization in fracture mechanics of nonlinear heterogeneous materials

This paper presents a new computational approach dedicated to the fracture of nonlinear heterogeneous materials. This approach extends the standard periodic homogenization problem to a two field cohesive-volumetric finite element scheme. This two field finite element formulation is written as a generalization Non-Smooth Contact Dynamics framework involving Frictional Cohesive Zone Models. The associated numerical platform allows to simulate, at finite strain, the fracture of nonlinear composites from crack initiation to post-fracture behavior. The ability of this computational approach is illustrated by the fracture of the hydrided Zircaloy under transient loading.     

Boundary element method homogenization of the periodic linear elastic fiber composites

The paper presented is devoted to the Boundary Element Method based homogenization of the periodic transversely isotropic linear elastic fiber-reinforced composites. The composite material under consideration has deterministically defined elastic properties while its components are perfectly bonded. To have a good comparison with the FEM-based computational techniques used previously, the additional Finite Element discretization is presented and compared numerically against BEM homogenization implementation on the example of engineering glass–epoxy composite. The homogenization method proposed has rather general characteristics and, as it is shown, can be easily extended on n-component composites. On the contrary, we can consider and homogenize the heterogeneous media with randomly defined material properties using Monte-Carlo simulation technique or second order perturbation second probabilistic moment approach.     

On the homogenization of elastic-plastic periodic composites by the microlocal parameter approach

Summary In this paper it is shown how the nonstandard homogenization method of periodic unelastic composites leading to the microlocal parameter theories, [1], [2] can be applied to the Prandtl-Reuss elastic-plastic periodic materials with the kinematic work hardening. The problem is analysed within the theory of small elastic-plastic deformations. The cylindrical thick walled laminated tube under pressure is taken as an example of an application of the proposed approach.With 5 Figures     

Fourier series expansion in non-orthogonal coordinate system for the homogenization of linear viscoelastic periodic composites

The Eshelby method and the Fourier series are used in order to determine the linear elastic and viscoelastic properties of composites with periodically distributed inclusions in a non-orthogonal coordinate system. The relaxation moduli provided by the proposed method are compared with the moduli obtained via FEM for a unidirectional composite. This comparison gives good results. The procedure results very useful for periodic composites with hexagonal symmetry, such as some transversely isotropic composites.     

A homogenization technique for heat transfer in periodic granular materials

A homogenization technique is proposed to simulate the thermal conduction of periodic granular materials in vacuum. The effective thermal conductivity (ETC) and effective volumetric heat capacity (EVHC) can be obtained from the granular represent volume element (RVE) via average techniques: average heat flux and average temperature gradient can be formulated by the positions and heat flows of particles on the boundaries of the RVE as well as of the contact pairs within the RVE. With the thermal boundary condition imposed on the border region around the granular RVE, the ETC of the granular RVE can be computed from the average heat flux and average temperature gradient obtained from thermal discrete element method (DEM) simulations. The simulation results indicate that the ETC of the granular assembly consisting of simple-cubic arranged spheres coincides with the theoretical prediction. The homogenization technique is performed to obtain the ETC of the RVE consisting of random packed particles and the results exhibit the anisotropy of the thermal conduction properties of the RVE. Both the ETC and EVHC obtained are then employed to simulate the thermal conduction procedure in periodic granular materials with finite element analyses, which give the similar results of temperature profile and conduction properties as the DEM simulations.     

A second-order reduced asymptotic homogenization approach for nonlinear periodic heterogeneous materials

An efficient second-order reduced asymptotic homogenization approach is developed for nonlinear heterogeneous media with large periodic microstructure. The two salient features of the proposed approach are (i) an asymptotic higher-order nonlinear homogenization that does not require higher-order continuity of the coarse-scale solution and (ii) an efficient model reduction scheme for solving higher-order nonlinear unit cell problems at a fraction of computational cost in comparison to the direct computational homogenization. The former is a consequence of a sequential solution of increasing order solutions, which permits evaluation of higher-order coarse-scale derivatives by postprocessing from the zeroth-order solution. The efficiency and accuracy of the formulation in comparison to the classical zeroth-order homogenization and direct numerical simulations are assessed on hyperelastic and elastoplastic periodic structures.     

A coupled-mode theory for periodic piezoelectric composites

A coupled oscillator model to calculate the resonance spectrum of a one-dimensional piezoelectric composite plate, used in ultrasonic transducers, is proposed. Two resonant modes, one produced by the elastic wave reflection on the plate boundaries (thickness resonance) and the other by the reflection on the periodic discontinuities (lateral resonance) are considered. A Kronig-Penney model is used to calculate the lateral resonances. The thickness resonance is obtained with an effective medium model. The coupling of these two modes is described by a biquadratic equation whose solutions are the resonant frequencies of the piezoelectric composite plate. A criterion for a distribution of phases to keep the spurious lateral resonances away from the thickness resonance vicinity is obtained.     

A homogenization theory for the overall creep of isotropic viscoplastic composites

Summary A field-fluctuation method is introduced into the secant-viscosity framework to evaluate the homogenized effective stress of the heterogeneously deformed elastic-viscoplastic matrix in an isotropic composite. Two microgeometries are considered here: one is reinforced with spherical particles and the other with randomly oriented thin discs. The time-dependent creep strains of the elastic-viscoplastic composites are then calculated as a function of inclusion volume concentration. As these two microgeometries are known to provide the lower and upper bounds of the effective moduli in elasticity, the creep strains associated with these two inclusion shapes are believed to set the upper and lower ranges of the overall creep strain for all inclusion shapes. Detailed comparison with a previously developed direct work-rate method is also made. While the effective stress of the ductile matrix-and therefore the overall creep strain of the composite-are higher by the direct work-rate method, the difference between the two is found to be small. However, when the Laplace inversion of the effective secant viscosities of the viscoelastic comparison composite can be cast in an explicit form, the field-fluctuation method will provide a complete evaluation of the effective stress, and is also substantially simpler than the direct work-rate method.Dedicated to Prof. Dr. Dr. h. c. Franz Ziegler on the occasion of his 60th birthday     

A BEM formulation for homogenization of composites with randomly distributed fibers

In recent papers by the authors, deterministic models of distribution of fibers in composite structures have been studied. For example, problems related to optimization, homogenization, localization, etc., have been solved. The extended Hashin–Shtrikman (H–S) variational principles served as a starting point (eigenparameters were involved in the formulations), and the comparative medium was introduced. The BEM formulations were then admissible and efficient. The formulations of the above-mentioned problems require the restriction of geometry of the fibers to certain ‘locally reasonable’ structures, e.g. to periodic or pseudo-periodic cells.Since the condition of regular distribution of fibers is violated in applications, and a random distribution is more probable, another extension of the H–S principles is needed. In this paper, the problem is extended to the case of statistically distributed fibers. H–S variational principles are formulated in terms of statistical characteristics in the domain and the eigenparameters are also involved, affected by the statistical values. Following the H–S principles, an integral formulation is stated (again, thanks to the use of the comparative medium such a formulation is admissible) in a representative volume, which contains no longer regular geometry of the fibers. The boundary element method has then a special form, which is advantageous particularly for two-phase media.The above-mentioned formulation of H–S variational principles with randomly distributed fields of fibers can be extended to non-linear problems (plasticity, debonding) by introducing transformation fields (eigenstresses or eigenstrains, which are involved in the formulations for completeness).The results form the research presented in this paper basically apply to homogenization of diaphysal implants. But, there is a wide range of applications of the theory introduced in this paper. Due to results from tests on the bearing composite frame of a bicycle, which has a similar structure for certain types of composites of the diaphysal implants, a typical cross-section of the bearing frame of a bicycle is studied as an example. The frame is built of a graphite-epoxy composite.     

A homogenization procedure for the numerical analysis of woven fabric composites

The paper presents a procedure for the numerical evaluation of the mechanical properties of woven fabric laminates. Woven fabrics usually present orthogonal interlaced yarns (warp and weft) and distribution of the fibers in the yarns and of the yarns in the composite may be considered regular. This allows us to apply the homogenization theory for periodic media both to the yarn and to the fabric. Three-dimensional finite element models are used in two steps to predict both the stiffness and the strength of woven fabric laminates. The model includes all the important parameters that influence the mechanical behavior: the lamina thickness, the yarn orientation, the fiber volume fraction and the mechanical characteristics of the components. The capabilities of the numerical model were verified studying the elastic behavior of a woven fabric laminate available in the literature and the ultimate strength of a glass fabric laminate experimentally investigated. The procedure, that can be implemented into commercial finite element codes, appears to be an efficient tool for the design of textile composites.     

A dual homogenization and finite element approach for material characterization of textile composites

A novel procedure for predicting the effective nonlinear elastic moduli of textile composites through a combined approach of the homogenization method and the finite element formulation is presented. The homogenization method is first applied to investigate the meso-microscopic material behavior of a single fiber yarn based on the properties of the constituent phases. The obtained results are compared to existing analytical and experimental results to validate the homogenization method. Very good agreements have been obtained. A unit cell is then built to enclose the characteristic periodic pattern in the textile composites. Various numerical tests such as uni-axial and bi-axial extension and trellising tests are performed by 3D finite element analysis on the unit cell. Characteristic behaviors of force versus displacement are obtained. Meanwhile, trial mechanical elastic constants are imposed on a four-node shell element with the same outer size as the unit cell to match the force–displacement curves. The effective nonlinear mechanical stiffness tensor is thus obtained numerically as functions of elemental strains. The procedure is exemplified on a plain weave glass composite and is validated by comparing to experimental data. Using the proposed approach, the nonlinear behavior of textile composites can be anticipated accurately and efficiently.     

A two-scale homogenization framework for nonlinear effective thermal conductivity of laminated composites

In this study, we formulate the effective temperature-dependent thermal conductivity of laminated composites. The studied laminated composites consist of laminas (plies) made of unidirectional fiber-reinforced matrix with various fiber orientations. The effective thermal conductivity is obtained through a two-scale homogenization scheme. A simplified micromechanical model of a unidirectional fiber-reinforced lamina is formulated at the lower scale. Thermal conductivities of fiber and matrix constituents are allowed to change with temperature. The upper scale uses a sublaminate model to homogenize temperature-dependent thermal conductivities of only a representative lamina stacking sequence in laminated composites. The effective thermal conductivity of each lamina, in the sublaminate model, is obtained using the simplified micromechanical model. The thermal conductivities from the micromechanical and sublaminate models represent average nonlinear properties of fictitiously homogeneous composite media. Interface conditions between fiber and matrix constituents and within laminas are assumed to be perfect. Experimental data available in the literature are used to verify the proposed multi-scale framework. We then analyze transient heat conduction in the homogenized composites. Temperature profiles, during transient heat conduction, in the homogenized composites are compared to the ones in heterogeneous composites. The heterogeneous composites, having different fiber arrangements and sizes, are modeled using finite element (FE) method.     

On semi‐analytical probabilistic finite element method for homogenization of the periodic fiber‐reinforced composites

The main aim of this paper is a development of the semi‐analytical probabilistic version of the finite element method (FEM) related to the homogenization problem. This approach is based on the global version of the response function method and symbolic integral calculation of basic probabilistic moments of the homogenized tensor and is applied in conjunction with the effective modules method. It originates from the generalized stochastic perturbation‐based FEM, where Taylor expansion with random parameters is not necessary now and is simply replaced with the integration of the response functions. The hybrid computational implementation of the system MAPLE with homogenization‐oriented FEM code MCCEFF is invented to provide probabilistic analysis of the homogenized elasticity tensor for the periodic fiber‐reinforced composites. Although numerical illustration deals with a homogenization of a composite with material properties defined as Gaussian random variables, other composite parameters as well as other probabilistic distributions may be taken into account. The methodology is independent of the boundary value problem considered and may be useful for general numerical solutions using finite or boundary elements, finite differences or volumes as well as for meshless numerical strategies. Copyright © 2011 John Wiley & Sons, Ltd.     

Dissipation inequality-based periodic homogenization of wavy materials

In this paper we present an internal variable-based homogenization of a composite made of wavy elastic-perfectly plastic layers. In the context of a strain-driven process, the macrostress and the effective yield surface are expressed in terms of the residual stresses, which act as hardening parameters in the effective behavior of the composite. Moreover, an approximate two-steps homogenization scheme useful for composites made of matrix with wavy inclusions is proposed and a comparison with one computational and one semi-analytical homogenization method is presented.     

XFEM modeling and homogenization of magnetoactive composites

This paper addresses the application of the extended finite element method (XFEM) to the modeling of two-dimensional coupled magneto-mechanical field problems. Continuum formulations of the stationary magnetic and the coupled magneto-mechanical boundary problem are outlined, and the corresponding weak forms are derived. The XFEM is applied to generate numerical models of a representative volume element, characterizing a magnetoactive composite material. Weak discontinuities occurring at material interfaces are modeled numerically by an enriched approximation of the primary field variables. In order to reduce the complexity of the representation of curved interfaces, an element local approach is proposed which allows an automated computation of the level set values. The composite’s effective behavior and its coupled magneto-mechanical response are computed numerically by a homogenization procedure. The scale transition process is based on the energy equivalence condition, which is satisfied by using periodic boundary conditions.     

A general model for the mechanics of saturated-unsaturated porous materials

A general predictive model for the mechanical analysis of isothermal and non-isothermal saturated-unsaturated porous materials is presented. The model is developed along the lines of Biot's theory and applies both for high water content and for low to medium water content in the pore space. Due to the similarity of the matrices in both situations, even if the transfer mechanisms are different, a single computer program can handle all of them. Examples belonging to both domains in the isothermal case as well as to heat and mass transfer in deforming porous media are shown.     

Asymptotic homogenization algorithm for reinforced metal-matrix elasto-plastic composites

The theory of the two-scale convergence was applied to homogenization of initial flow stresses and hardening constants in some exponential hardening laws for elasto-plastic composites with a periodic microstructure. The theory is based on the fact that both the elastic and the plastic part of the stress field two-scale converge to a limit, which can be factorized by parts, one of which depends only on the macroscopic, and the other one – only on the microscopic characteristics. The first factor is represented in terms of the homogenized stress tensor and the second factor – in terms of stress concentration tensor, that relates to the micro-geometry and elastic or plastic micro-properties of composite components. The theory was applied to a composite, that consists of the metallic elasto-plastic matrix with Ludwik and Hocket–Sherby hardening law and pure elastic silica inclusions. Results were compared with those of averaging based on the self-consistent methods.     

Wavelet-based homogenization of unidirectional multiscale composites

The main aim is to present a homogenization algorithm for the multiscale heterogeneous (composite) materials, which is based on the wavelet representation of material properties and the relevant multiscale reduction. It is shown that classical homogenization method used before for two-scale composites (with micro and macro scales) is a special case of general multiresolutional strategy, where a single scale parameter tends to 0. The approach presented is applied to unidirectional wavelet-based homogenization of linear elasticity heterogeneous problem and to wave propagation, which may be applied in conjunction with various discrete numerical methods for efficient modeling of heterogeneous solids, fluids and multiphase media.