Numerical simulation of fiber reinforced composite materials––two procedures

In this work, two methodologies for the analysis of unidirectional fiber reinforced composite materials are presented.The first methodology used is a generalized anisotropic large strains elasto-plastic constitutive model for the analysis of multiphase materials. It is based on the mixing theory of basic substance. It is the manager of the several constitutive laws of the different compounds and it allows to consider the interaction between the compounds of the composite materials. In fiber reinforced composite materials, the constitutive behavior of the matrix is isotropic, whereas the fiber is considered orthotropic. So, one of the constitutive model used in the mixing theory needs to consider this characteristic. The non-linear anisotropic theory showed in this work is a generalization of the classic isotropic plasticity theory (A Continuum Constitutive Model to Simulate the Mechanical Behavior of Composite Materials, PhD Thesis, Universidad Politécnica de Cataluña, 2000). It is based in a one-to-one transformation of the stress and strain spaces by means of a four rank tensor.The second methodology used is based on the homogenization theory. This theory divided the composite material problem into two scales: macroscopic and microscopic scale. In macroscopic level the composite material is assuming as a homogeneous material, whereas in microscopic level a unit volume called cell represents the composite (Tratamiento Numérico de Materiales Compuestos Mediante la teorı́ de Homogeneización, PhD Thesis, Universidad Politécnica, de Cataluña 2001). This formulation presents a new viewpoint of the homogenization theory in which can be found the equations that relate both scales. The solution is obtained using a coupled parallel code based on the finite elements method for each scale problem.     

A three-invariant cap model with isotropic–kinematic hardening rule and associated plasticity for granular materials

In this paper, a three-invariant cap model is developed for the isotropic–kinematic hardening and associated plasticity of granular materials. The model is based on the concepts of elasticity and plasticity theories together with an associated flow rule and a work hardening law for plastic deformations of granulars. The hardening rule is defined by its decomposition into the isotropic and kinematic material functions. The constitutive elasto-plastic matrix and its components are derived by using the definition of yield surface, material functions and non-linear elastic behavior, as function of hardening parameters. The model assessment and procedure for determination of material parameters are described. Finally, the applicability of proposed plasticity model is demonstrated in numerical simulation of several triaxial and confining pressure tests on different granular materials, including: wheat, rape, synthetic granulate and sand.     

Micromechanics of elasto-plastic materials reinforced with ellipsoidal inclusions

The effective mechanical behavior of an elasto-plastic matrix reinforced with a random and homogeneous distribution of aligned elastic ellipsoids was obtained by the finite element simulation of a representative volume element (RVE) of the microstructure and by homogenization methods. In the latter, the composite behavior was modeled by linearization of the local behavior through the use of the tangent or secant stiffness tensors of the phases. “Quasi-exact” results for the tensile deformation were attained by averaging of the stress-strain curves coming from the numerical simulation of RVEs containing a few dozens of ellipsoids. These results were used as benchmarks to assess the accuracy of the homogenization models. The best approximations to the reference numerical results were provided by the incremental and the second-order secant methods, while the classical or first-order secant approach overestimated the composite flow stress, particularly when the composite was deformed in the longitudinal direction. The discrepancies among the homogenization models and the numerical results were assessed from the analysis of the stress and strain microfields provided by the numerical simulations, which demonstrated the dominant effect of the localization of the plastic strain in the matrix on the accuracy of the homogenization models.     

The criterion for determining the direction of crack propagation in a random pattern composites

Presented paper contains results of fracture analysis of brittle composite materials with a random distribution of grains. The composite structure has been modelled as an isotropic matrix that surrounds circular grains with random diameters and space position. Analyses were preformed for the rectangular “numerical sample” by finite element method. FE mesh for the examples were generated using the authors’ computer program RandomGrain. Fracture analyses were accomplished with the authors’ computer program CrackPath3 executing the “fine mesh window” technique. Calculations were preformed in 2D space assuming the plane stress state. Current efforts focus on brittle materials such as rocks or concrete.     

Homogenization of elasto-plastic composites coupled with a nonlinear finite element analysis of the equivalent inclusion problem

This paper deals with the micromechanical modeling of particle reinforced elasto-plastic composites under general non-monotonic loading histories. Incremental mean-field (MF) homogenization models offer an excellent cost-effective solution, however there are cases where their predictions are inaccurate. Here, we assess the applicability of the equivalent inclusion representation, which sustains many homogenization schemes. To this end, MF models are fully coupled with a finite element (FE) solution of the equivalent inclusion problem (EIP). Consequently, Eshelby’s tensor is not used and most (but not all) approximations involved in the generalization of MF models from linear elasticity to the nonlinear regime are avoided. The proposal is implemented for Mori-Tanaka (M-T) and dilute inclusion models and applied to several composite systems with elasto-plastic matrix and spherical or ellipsoidal particles, subjected to various loadings (tension, plane strain, cyclic tension/compression). The predictions are verified against reference full-field FE simulations of multiparticle cells. Results show that the M-T model coupled with the nonlinear FE solution of the EIP is very accurate at the macro level up to 25% volume fraction of reinforcement, while the phase averages remain accurate as long as the volume fraction does not exceed 15%. The strain concentration tensor computed almost exactly from single inclusion FE analysis is compared against approximate expressions assumed by classical MF models. Implications for the development of advanced MF homogenization models are discussed.     

Micromechanical modeling and computation of elasto-plastic materials reinforced with distributed-orientation fibers

This paper deals with the mean-field homogenization of multiphase elasto-plastic materials reinforced with non-spherical and non-aligned inclusions. Most of the literature on the micro–macro modeling of elasto-plastic composites deals with fixed-orientation fibers but this paper is concerned with cases where the inclusions have a non-uniform orientation defined by an orientation distribution function (ODF). We propose a general two-step incremental formulation and the corresponding numerical algorithms which are able to deal with any rate-independent model for any phase as well as cyclic or otherwise non-proportional loadings. The formulation was implemented in the DIGIMAT (2003) software and the numerical predictions were validated against experimental data for several composite systems.     

本构模型对复合材料中X射线热击波数值模拟结果的影响

以碳酚醛靶中的一维应变热击波为例,讨论本构模型对热击波数值模拟结果的影响.数值模拟采用3种本构模型,即各向同性理想弹塑性本构模型、各向异性理想弹塑性本构模型及各向异性动态弹塑性本构模型.结果表明:与各向同性理想弹塑性、各向异性理想弹塑性本构模型相比,利用各向异性动态弹塑性本构模型获得的热击波应力峰值较小、应力峰值衰减较...     

A homogenization-based constitutive model for isotropic viscoplastic porous media

An approximate model based on the “second-order” nonlinear homogenization method is proposed to estimate the effective behavior of isotropic, viscoplastic, porous materials. The model is constructed in such a way that it reproduces exactly the behavior of a “composite-sphere assemblage” in the limit of hydrostatic loadings, and therefore coincides with the hydrostatic limit of Gurson’s criterion in the special case of ideal plasticity. As a consequence, the new model improves on earlier homogenization estimates, which have been found to be quite accurate for low triaxialities but overly stiff for sufficiently high triaxialities and nonlinearities. Additionally, the estimates delivered by the model exhibit a dependence on the third invariant of the macroscopic stress tensor, which has a nontrivial effect on the effective response of the material at moderate triaxialities. The proposed model is compared with exact results obtained for a special class of porous materials with sequentially laminated microstructures. The agreement is found to be quite good for the entire range of stress triaxialities, and all values of the porosity and nonlinearity considered.     

Meso-macro modelling of fibre-reinforced rubber-like composites exhibiting large elastoplastic deformation

Many composites consist of a fabric structure embedded in a matrix material. As an example, in the present paper, the case of pneumatic membranes is considered. Fibres are often made of material which shows noticeable plastic deformation. The stiffness of the fibres determines the overall stiffness of the material such that the correct modelling of the orthotropy of the composite is very important. In addition, the structure experiences large deformations which must be accounted for. Suitable models for this type of materials are therefore derived in the framework of finite anisotropic plasticity. A main problem is, however, the lack of experimental data in the literature. For this reason, a computer model of the composite is set up for numerical experiments. In this way, sufficient data can be generated. The present continuum mechanical model based on these “artificial” test data can be efficiently implemented into a finite element formulation. Using a special integration algorithm, the non-linear equation system consisting initially of 10 equations reduces to two non-linear scalar equations.     

General mean-field homogenization schemes for viscoelastic composites containing multiple phases of coated inclusions

We consider matrix materials reinforced with multiple phases of coated inclusions. All materials are linear viscoelastic. We present general schemes for the prediction of the effective properties based on mean-field homogenization. There are four contributions in this work. First, we present a two-step homogenization procedure in a general setting which besides the usual assumptions of Eshelby-based models, does not suffer any restriction in terms of material properties, aspect ratio or orientation. Second, for a matrix reinforced with coated inclusions, we propose two general homogenization schemes, a two-step method and a two-level recursive scheme. We develop and compare the mathematical expressions obtained by the two schemes and a generalized Mori–Tanaka (M–T) model. Third, for a two-phase composite, either standalone or stemming from two-step or two-level schemes, we use a double-inclusion model based on a closed-form but non-trivial interpolation between M–T and inverse M–T estimates. Fourth, we conduct an extensive validation of the proposed schemes as well as others against experimental data and unit cell finite element simulations for a variety of viscoelastic composite materials. Under severe conditions, the proposed schemes perform much better than other existing homogenization methods.     

Determination of the effective elastic–plastic response of metal–ceramic composites

The mechanical response of metal–ceramic composites is analysed through a homogenization model accounting for the mechanical behaviour of the constituent materials. In order to achieve this purpose a nonlinear homogenization method based on the phase field approach has been suitably implemented into a numerical code. A prescribed homogenized strain state is applied to a unit volume element of a metal–ceramic composite with proportional loading in which all components of the strain tensor are proportional to one scalar parameter. The mechanical response of the material has been modeled by considering a von Mises plasticity model for the metal phase and a Drucker–Prager associative elastic–plastic material model for the ceramic phase. A two stages plasticity has been obtained in which inelastic strain develops in the metal phase followed by a fully plastic response. A comparison with a finite element model of the stress–strain response of an axisymmetric unit cell has been carried out with the purpose to validate the homogenization based modeling presented in the paper. Plastic parameters of a Drucker–Prager yield surface for the homogenized composite have been calculated at different materials compositions. Associative Drucker–Prager plasticity has been found to be accurate for high ceramic content.     

A simple meso-macro model based on SQP for the non-linear analysis of masonry double curvature structures

A 3D model for the evaluation of the non-linear behavior of masonry double curvature structures is presented. In the model, the heterogeneous assemblage of blocks is substituted with a macroscopically equivalent homogeneous non-linear material. At the meso-scale, a curved running bond representative element of volume (REV) constituted by a central block interconnected with its six neighbors is discretized through of a few six-noded rigid wedge elements and rectangular interfaces. Non linearity is concentrated exclusively on joints reduced to interface, exhibiting a frictional behavior with limited tensile and compressive strength with softening. The macroscopic homogenous masonry behavior is then evaluated on the REV imposing separately increasing internal actions (in-plane membrane actions, meridian and parallel bending, torsion and out-of-plane shear). This simplified approach allows to estimate heuristically the macroscopic stress–strain behavior of masonry at the meso-scale. The non-linear behavior so obtained is then implemented at a structural level in a novel FE non-linear code, relying on an assemblage of rigid infinitely resistant six-noded wedge elements and non-linear interfaces, exhibiting deterioration of the mechanical properties.Several numerical examples are analyzed, consisting of two different typologies of masonry arches (a parabolic vault and an arch in a so-called “skew” disposition), a ribbed cross vault, a hemispherical dome and a cloister vault. To fully assess numerical results, additional non-linear FE analyses are presented. In particular, a simplified model is proposed, which relies in performing at a structural level a preliminary limit analysis – which allows to identify the failure mechanism – and subsequently in modeling masonry through elastic elements and non-linear interfaces placed only in correspondence or near the failure mechanism provided by limit analysis. Simulations performed through an equivalent macroscopic material with orthotropic behavior and possible softening are also presented, along with existing experimental evidences (where available), in order to have a full insight into the capabilities and limitations of the approach proposed.     

二维类桁架材料结构弹塑性分析

类桁架材料所构成结构的弹塑性行为的精确建模分析保证非常耗时, 为了 在保证精度的前提下提高此类问题的求解效率, 本文利用类桁架材料基本构件长细比 较大的特点,将材料单胞简化为桁架模型. 考虑到微单胞空间分布的周期性,基于 数值均匀化理论提出了类桁架材料结构的宏微观两 级弹塑性求解格式. 原问题转化为宏观上一个非线性弹性连续体计算问题和微观上多个小规 模桁架系统的弹塑性计算问题. 两个数值算例分别考虑了简单加载,非单调加载,规则宏观 结构和具有非完整单胞的较复杂宏观结构等问题. 与实际结构计算结果在精度和时间等方面 的比较验证了求解格式的有效性. 最后还探讨了算法的适用范围.     

A homogenization theory of strain gradient single crystal plasticity and its finite element discretization

In this study, a homogenization theory based on the Gurtin strain gradient formulation and its finite element discretization are developed for investigating the size effects on macroscopic responses of periodic materials. To derive the homogenization equations consisting of the relation of macroscopic stress, the weak form of stress balance, and the weak form of microforce balance, the Y-periodicity is used as additional, as well as standard, boundary conditions at the boundary of a unit cell. Then, by applying a tangent modulus method, a set of finite element equations is obtained from the homogenization equations. The computational stability and efficiency of this finite element discretization are verified by analyzing a model composite. Furthermore, a model polycrystal is analyzed for investigating the grain size dependence of polycrystal plasticity. In this analysis, the micro-clamped, micro-free, and defect-free conditions are considered as the additional boundary conditions at grain boundaries, and their effects are discussed.     

Kinematic hardening in large strain plasticity

A finite strain hyper elasto-plastic constitutive model capable to describe non-linear kinematic hardening as well as non-linear isotropic hardening is presented. In addition to the intermediate configuration and in order to model kinematic hardening, an additional configuration is introduced – the center configuration; both configurations are chosen to be isoclinic. The yield condition is formulated in terms of the Mandel stress and a back-stress with a structure similar to the Mandel stress.It is shown that the non-dissipative part of the plastic velocity gradient not governed by the thermodynamical framework and the corresponding quantity associated with the kinematic hardening influence the material behaviour to a large extent when kinematic hardening is present. However, for isotropic elasticity and isotropic hardening plasticity it is shown that the non-dissipative quantities have no influence upon the stress–strain relation.As an example, kinematic hardening von Mises plasticity is considered, which fulfils the plastic incompressibility condition and is independent of the hydrostatic pressure. To evaluate the response and to examine the influence of the non-dissipative quantities, simple shear is considered; no stress oscillations occur.     

Anisotropic strain hardening behavior in simple shear for cube textured aluminum alloy sheets

Finite element (FE) simulations of the simple shear test were conducted for 1050-O and 6022-T4 aluminum alloy sheet samples. Simulations were conducted with two different constitutive equations to account for plastic anisotropy: Either a recently proposed anisotropic yield function combined with an isotropic strain hardening law or a crystal plasticity model. The FE computed shear stress–shear strain curves were compared to the experimental curves measured for the two materials in previous works. Both phenomenological and polycrystal approaches led to results consistent with the experiments. These comparisons lead to a discussion concerning the assessment of anisotropic hardening in the simple shear test.