Multiscale Nonlinear Constitutive Modeling of Carbon Nanostructures Based on Interatomic Potentials

Continuum-based modeling of nanostructures is an efficient and suitable method to study the behavior of these structures when the deformation can be considered homogeneous. This paper is concerned about multiscale nonlinear tensorial constitutive modeling of carbon nanostructures based on the interatomic potentials. The proposed constitutive model is a tensorial equation relating the second Piola-Kirchhoff stress tensor to Green-Lagrange strain tensor. For carbon nanotubes, some modifications are made on the planar representative volume element (RVE) to account for the curved atomic structure resulting a non-planar RVE. Using the proposed constitutive model, the elastic behavior of the graphene sheet and carbon nanotube are studied.     

A nonlocal computational homogenization of softening quasi-brittle materials

In this paper, a computational counterpart of the experimental investigation is presented based on a nonlocal computational homogenization technique for extracting damage model parameters in quasi-brittle materials with softening behavior. The technique is illustrated by introducing the macroscopic nonlocal strain to eliminate the mesh sensitivity in the macroscale level as well as the size dependence of the representative volume element (RVE) in the first-order continuous homogenization. The macroscopic nonlocal strains are computed at each direction, and both the local and nonlocal strains are transferred to the microscale level. Two RVEs with similar geometries and material properties are introduced for each macroscopic Gauss point, in which the microscopic damage variables and the macroscale consistent tangent modulus and its derivatives are obtained by imposing the macroscopic nonlocal strain on the first RVE, and the macroscopic stress is computed by employing the microscopic damage variables and imposing the macroscopic local strain over the second RVE. Finally, numerical examples are solved to illustrate the performance of the proposed nonlocal computational homogenization technique for softening quasi-brittle materials.     

A second-order two-scale homogenization procedure using 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 \(C^{1}\) two dimensional triangular finite elements, while the \(C^{0}\) 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 \(C^{1}-C^{0}\) 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.     

A two‐scale failure model for heterogeneous materials: numerical implementation based on the finite element method

In the first part of this contribution, a brief theoretical revision of the mechanical and variational foundations of a Failure‐Oriented Multiscale Formulation devised for modeling failure in heterogeneous materials is described. The proposed model considers two well separated physical length scales, namely: (i) the macroscale where nucleation and evolution of a cohesive surface is considered as a medium to characterize the degradation phenomenon occurring at the lower length scale, and (ii) the microscale where some mechanical processes that lead to the material failure are taking place, such as strain localization, damage, shear band formation, and so on. These processes are modeled using the concept of Representative Volume Element (RVE). On the macroscale, the traction separation response, characterizing the mechanical behavior of the cohesive interface, is a result of the failure processes simulated in the microscale. The traction separation response is obtained by a particular homogenization technique applied on specific RVE sub‐domains. Standard, as well as, Non‐Standard boundary conditions are consistently derived in order to preserve objectivity of the homogenized response with respect to the micro‐cell size. In the second part of the paper, and as an original contribution, the detailed numerical implementation of the two‐scale model based on the finite element method is presented. Special attention is devoted to the topics, which are distinctive of the Failure‐Oriented Multiscale Formulation, such as: (i) the finite element technologies adopted in each scale along with their corresponding algorithmic expressions, (ii) the generalized treatment given to the kinematical boundary conditions in the RVE, and (iii) how these kinematical restrictions affect the capturing of macroscopic material instability modes and the posterior evolution of failure at the RVE level. Finally, a set of numerical simulations is performed in order to show the potentialities of the proposed methodology, as well as, to compare and validate the numerical solutions furnished by the two‐scale model with respect to a direct numerical simulation approach. Copyright © 2013 John Wiley & Sons, Ltd.     

Two-scale analysis for deformation-induced anisotropy of polycrystalline metals

The anisotropic macroscopic mechanical behavior of polycrystalline metals is characterized by incorporating the microscopic constitutive model of single crystal plasticity into the two-scale modeling based on the mathematical homogenization theory, which enables us to derive both micro- and macro-scale governing equations. The two-scale simulations are conducted to evaluate the macroscopic anisotropy induced by microscopic plastic deformation histories of the polycrystalline aggregate. In the simulations, the representative volume element (RVE) composed of several crystal grains is uniformly loaded in one direction, unloaded to macroscopically zero stress in a certain stage of deformation and then re-loaded in the different directions. The last re-loading calculations provide different macroscopic responses of the RVE, which can be the appearance of material anisotropy. We then try to examine the effects of the intergranular and intragranular behaviors on the anisotropy by means of various illustrations of microscopic plastic deformation process without referring to the change of crystallographic orientations.     

Bounds on the effective response for gradient crystal inelasticity based on homogenization and virtual testing

This paper presents the application of variationally consistent selective homogenization applied to a polycrystal with a subscale model of gradient-enhanced crystal inelasticity. Although the full gradient problem is solved on Statistical Volume Elements (SVEs), the resulting macroscale problem has the formal character of a standard local continuum. A semi-dual format of gradient inelasticity is exploited, whereby the unknown global variables are the displacements and the energetic micro-stresses on each slip-system. The corresponding time-discrete variational formulation of the SVE-problem defines a saddle point of an associated incremental potential. Focus is placed on the computation of statistical bounds on the effective energy, based on virtual testing on SVEs and an argument of ergodicity. As it turns out, suitable combinations of Dirichlet and Neumann conditions pertinent to the standard equilibrium and the micro-force balance, respectively, will have to be imposed. Whereas arguments leading to the upper bound are quite straightforward, those leading to the lower bound are significantly more involved; hence, a viable approximation of the lower bound is computed in this paper. Numerical evaluations of the effective strain energy confirm the theoretical predictions. Furthermore, heuristic arguments for the resulting macroscale stress-strain relations are numerically confirmed.     

Simulations of deformation and damage processes of SiCp/Al composites during tension

The deformation, damage and failure behaviors of 17 vol.% SiCp/2009Al composite were studied by microscopic finite element (FE) models based on a representative volume element (RVE) and a unit cell. The RVE having a 3D realistic microstructure was constructed via computational modeling technique, in which an interface phase with an average thickness of 50 nm was generated for assessing the effects of interfacial properties. Modeling results showed that the RVE based FE model was more accurate than the unit cell based one. Based on the RVE, the predicted stress-strain curve and the fracture morphology agreed well with the experimental results. Furthermore, lower interface strength resulted in lower flow stress and ductile damage of interface phase, thereby leading to decreased elongation. It was revealed that the stress concentration factor of SiC was ～2.0: the average stress in SiC particles reached ～1200 MPa, while that of the composite reached ～600 MPa.     

Computational micromechanical analysis of the representative volume element of bituminous composite materials

Micromechanical computational modeling is used in this study to determine the smallest domain, or Representative Volume Element (RVE), that can be used to characterize the effective properties of composite materials such as Asphalt Concrete (AC). Computational Finite Element (FE) micromechanical modeling was coupled with digital image analysis of surface scans of AC specimens. Three mixtures with varying Nominal Maximum Aggregate Size (NMAS) of 4.75 mm, 12.5 mm, and 25 mm, were prepared for digital image analysis and computational micromechanical modeling. The effects of window size and phase modulus mismatch on the apparent viscoelastic response of the composite were numerically examined. A good agreement was observed in the RVE size predictions based on micromechanical computational modeling and image analysis. Micromechanical results indicated that a degradation in the matrix stiffness increases the corresponding RVE size. Statistical homogeneity was observed for window sizes equal to two to three times the NMAS. A model was presented for relating the degree of statistical homogeneity associated with each window size for materials with varying inclusion dimensions.     

On the contribution of carbon nanotube deformation to piezoresistivity of carbon nanotube/polymer composites

The change in electrical resistance due to mechanical deformation of carbon nanotube (CNT)/polymer composites can be rationalized in terms of two effects: (i) changes in the composite electrical resistivity due to changes in the CNT network configuration and (ii) deformation of the CNTs themselves. The contribution of CNT dimensional changes (ii) to the piezoresistivity of CNT/polymer composites is investigated here. An analytical model based exclusively on dimensional changes which describes the CNT change of electrical resistance in terms of its mechanical deformation is proposed. A micromechanics approach and finite element analysis are performed to correlate the macroscale composite strain to the individual CNT strain. The CNT change of electrical resistance is quantified for different matrix elastic moduli and CNT weight fractions. The CNT/polymer composite is also modeled as an effective continuum material in terms of both its electrical and mechanical responses so that the effect of dimensional changes on the global piezoresistivity can be investigated. Based on the modeling predictions and previous experimental results, it is estimated that the CNT change of resistance due to the macroscale composite strain is marginal (∼5%) compared to the total composite change of resistance commonly measured in the laboratory, suggesting that the dominant effect in the piezoresistivity of CNT/polymer composites is the change in the CNT network configuration.     

基于RVE单元的石砌体有效模量计算方法EI北大核心CSCD

石砌体结构的力学性能研究已经取得一定的成果,但很少有研究考虑构造的随机性对石砌体结构性能产生的影响。该文基于代表性体积单元(RVE单元),针对几何构造具有一定随机性的石砌体,提出了一种有效模量计算方法,并与文献的试验和数值模拟结果对比,证明了所得有效模量的准确性。分别根据该方法和有限尺度测试窗法,对比分析了藏式古建石砌体的有效模量,并探讨了RVE单元尺寸大小对有效模量的影响。最后提出了一个基于所得有效模量建立砌体整体式模型的建模方法,并与传统分离式建模和基于有限尺度窗口法的整体式建模进行对比分析。研究结果表明:该方法与有限尺度测试窗法所得有效模量的数值大小和变化趋势相近,但该方法弥补了有限尺度测试窗法不能获得泊松比的缺陷;RVE单元尺寸的增大,使得其组元分布逐渐向完整结构的组元分布接近,导致大部分的Voigt、Reuss有效模量分量呈现先快后慢的收敛趋势;单元尺寸变化对厚度方向的轴向模量几乎没有影响,对其他面外的模量分量影响较小,对面内的模量分量影响较大;基于该方法所建整体式模型,可以在节约计算成本的同时,较为精确复制分离式模型的宏观变形。     

Multi-scale analysis of the early damage mechanics of ferritized ductile iron

A multiscale computational homogenization method for the modeling of hydro-mechanical coupling problem for quasi-brittle materials is developed. The present method is based on an asymptotic expansion homogenization combined with the semi-concurrent finite element modelling approach. Modified periodic boundary conditions and a molecular dynamics (MD) based inclusion or filler generation procedure are devised for the hydro-mechanical coupling problem. A modified elastic damage constitutive model and a damage induced permeability law have been developed for the hydraulic fracturing. The statistical convergence of the microscale representative volume element (RVE) model regarding the RVE characteristic size is studied. It was found that the RVE characteristic size is determined by both the mechanical and hydraulic properties of the RVE simultaneously. The present method is validated by the experimental results for brittle material. The damage zone and crack propagation path captured by the present method is compared with the experimental results (Chitrala et al. in J Pet Sci Eng 108:151–161, 2013). The results show that the present method is an effective for the modelling of hydro-mechanical coupling for brittle materials.     

Modeling the Response of 3D Textile Composites under Compressive Loads to Predict Compressive Strength

The compression response of 3D woven textile composites (3DWC) that consist of glass fiber tows and a polymer matrix material is studied using a combination of experiments and finite element based analyses. A previous study reported by the authors consisted of an experimental investigation of 3DWC under high strain rate loading, Pankow, Salvi, Waas, Yen, and Ghiorse (2011). Those experimental results were explained by using the finite element method to analyze the high rate deformation response of representative volume elements (RVEs) of the 3DWC, Pankow, Waas, Yen, and Ghiorse (2012). In this paper, the same modeling strategy is used to examine the quasi-static, compressive deformation response of 3DWC. The effect of using different numbers of the textile repeat unit architecture in the RVE, on the predicted compression strength, is examined. The transitions in failure modes that are seen in experiments are seen to be captured by the model that is presented here.     

Material spatial randomness: From statistical to representative volume element

The material spatial randomness forces one to re-examine various basic concepts of continuum solid mechanics. In this paper we focus on the Representative Volume Element (RVE) that is commonly taken for granted in most of deterministic as well as in stochastic solid mechanics, although in the latter case it is called the Statistical Volume Element (SVE). The key issue is the scale over which homogenization is being carried out—it is called the mesoscale, separating the microscale (level of microheterogeneities) from the macroscale (level of RVE). As the mesoscale grows, the SVE tends to become the RVE. This occurs in terms of two hierarchies of bounds stemming from Dirichlet and Neumann boundary value problems on the mesoscale, respectively. Since generally there is no periodicity in real random media, the RVE can only be approached approximately on finite scales. We review results on this subject in the settings of linear elasticity, finite elasticity, plasticity, viscoelasticity, thermoelasticity, and permeability.     

Prediction of damage-growth based fatigue life of polycrystalline materials using a microstructural modeling approach

A new finite element-based mesoscale model is developed to simulate the localization of deformation and the growth of microstructurally short fatigue cracks in crystalline materials by considering the anisotropic behavior of the individual grains. The inelastic hysteresis energy is used as a criterion to predict the fatigue crack initiation and propagation. This criterion in conjunction with continuum damage modeling provides a strong tool for studying the behavior of materials under cyclic loading at the level of the microstructure. The model predictions are validated against an austenitic stainless steel alloy experimental data. The results show that a combined microstructural and continuum damage modeling approach is able to express the overall fatigue behavior of crystalline materials at the macroscale based on the microstructural features. It correctly predicts the crack initiation on slip bands and at inclusions in low-cycle and high-cycle fatigue, respectively, in agreement with experimental observations reported in the literature.     

Multi‐scale constitutive modelling of heterogeneous materials with a gradient‐enhanced computational homogenization scheme

A gradient‐enhanced computational homogenization procedure, that allows for the modelling of microstructural size effects, is proposed within a general non‐linear framework. In this approach the macroscopic deformation gradient tensor and its gradient are imposed on a microstructural representative volume element (RVE). This enables us to incorporate the microstructural size and to account for non‐uniform macroscopic deformation fields within the microstructural cell. Every microstructural constituent is modelled as a classical continuum and the RVE problem is formulated in terms of standard equilibrium and boundary conditions. From the solution of the microstructural boundary value problem, the macroscopic stress tensor and the higher‐order stress tensor are derived based on an extension of the Hill–Mandel condition. This automatically delivers the microstructurally based constitutive response of the higher‐order macro continuum and deals with the microstructural size in a natural way. Several examples illustrate the approach, particularly the microstructural size effects. Copyright © 2002 John Wiley & Sons, Ltd.     

Woven fabric permeability: From textile deformation to fluid flow mesoscale simulations

A two-step methodology is proposed in order to estimate from numerical simulations the permeability of deformed woven fabrics. Firstly, the shear deformation of a glass plain weave until the shear locking is studied from a mesoscale analysis achieved with a representative volume element (RVE) of the periodic plain weave. Simulations have been carried out within the scope of large transformations, accounting for yarn–yarn contacts, and assuming that yarns behave as hypoelastic materials with transverse isotropy. From the simulated deformed solid RVE, a complementary periodic fluid RVE is then built and the slow flow of an incompressible Newtonian fluid within it is investigated. This allows to compute, in a second step, the permeability of the deformed plain weave. The role of the shear deformation on the permeability of multi-layers or single layer preforms is discussed.     

基于RVE单元的藏式古建石砌体均质化研究

在普通砖石砌体结构方面的均质化研究已经较为完善,而在构造、材料存在随机性的古建筑砌体结构方面的均质化研究相对欠缺。该文以有限尺度测试窗法为基础,提出了一种选择砌体结构代表性体积单元(RVE单元)的方法,并与试验和传统有限元模拟结果对比,验证了所提方法的可行性。在此基础上,该文进行了藏式古建石砌体结构RVE单元的选择,探讨了RVE单元的尺寸大小和所包含的组元分布对等效模量的影响,并基于所选RVE单元建立了藏式石砌体结构的均质化模型和整体式模型。结果表明:该选择方法适用于周期性和准周期性砌体结构,能选出与完整结构力学性能接近的RVE单元。随着RVE单元尺寸变大,其Voigt、Reuss等效模量会逐渐向完整结构的模量收敛,呈现先快后慢的变化趋势;组元分布的不同会改变等效模量的收敛程度,但在较大尺寸的RVE单元上,组元分布的影响将被体积造成的影响抵消。该文所建均质化模型能代替传统有限元模型进行局部结构的分析,并给出藏式古建石砌体结构的应力分布规律;所建整体模型能代替传统有限元模型,较为精确模拟结构整体的宏观变形。     

Tsallis entropy in dual homogenization of random composites using the stochastic finite element method

This work concerns an application of the Tsallis entropy to homogenization problem of the fiber‐reinforced and also of the particle‐filled composites with random material and geometrical characteristics. Calculation of the effective material parameters is done with two alternative homogenization methods—the first is based upon the deformation energy of the Representative Volume Element (RVE) subjected to the few specific deformations, while the second uses explicitly the so‐called homogenization functions determined under periodic boundary conditions imposed on this RVE. Probabilistic homogenization is made with the use of three concurrent non‐deterministic methods, namely Monte‐Carlo simulation, iterative generalized stochastic perturbation technique as well as the semi‐analytical approach. The last two approaches are based on the Least Squares Method with polynomial basis of the statistically optimized order— this basis serves for further differentiation in the 10th‐order stochastic perturbation technique, while semi‐analytical method uses it in probabilistic integrals. These three approaches are implemented all as the extensions of the traditional Finite Element Method (FEM) with contrastively different mesh sizes, and they serve in computations of Tsallis entropies of the homogenized tensor components as the functions of input coefficient of variation.     

Automatic generation of 2D micromechanical finite element model of silicon–carbide/aluminum metal matrix composites: Effects of the boundary conditions

Two-dimensional finite element (FE) simulations of the deformation and damage evolution of Silicon–Carbide (SiC) particle reinforced aluminum alloy composite including interphase are carried out for different microstructures and particle volume fractions of the composites. A program is developed for the automatic generation of 2D micromechanical FE-models with randomly distributed SiC particles. In order to simulate the damage process in aluminum alloy matrix and SiC particles, a damage parameter based on the stress triaxial indicator and the maximum principal stress criterion based elastic brittle damage model are developed within Abaqus/Standard Subroutine USDFLD, respectively. An Abaqus/Standard Subroutine MPC, which allows defining multi-point constraints, is developed to realize the symmetric boundary condition (SBC) and periodic boundary condition (PBC). A series of computational experiments are performed to study the influence of boundary condition, particle number and volume fraction of the representative volume element (RVE) on composite stiffness and strength properties.