Elastic property prediction by finite element analysis with random distribution of materials for heterogeneous solids

In the present paper, finite element method is employed to predict the effective material properties of heterogeneous materials via random distributions of the constituent materials. With the random distributing strategy, massive parametric analysis via finite element becomes feasible for multi-phase heterogeneous solids. Using a two-phase bi-continuous material as an example, the effects of the specimen size with respect to the characteristic size of the micro-structural size and the element density on the predicted effective properties are considered. The numerical predictions of the effective properties are checked by two analytical bounds which were proposed by Hashin and Shtrikemn (1963) through the principle of variation and the matrix-fiber model. Some discussions on the finite element prediction are also made to clarify the status of the present work in the composite mechanics research.     

Effective Elastic Property Estimation for Bi-continuous Heterogeneous Solids

In the present study we performed finite element simulation for bi-continuous heterogeneous solids via a random distribution of materials to predict effective elastic properties. With a random distributing scheme, a statistical analysis via finite element becomes feasible for the multi-phase heterogeneous solids. Using a two-phase bi-continuous material as example, the numerical prediction of the effective properties is obtained in terms of a mean value and standard deviation with a sample size of 30 for each of given volume fraction. The finite element simulation results fall within the analytical bounds proposed by Hashin and Shtrikman (1963) based on the principle of variation. Comparison between the effective modulus based on the present bio-continuous morphology with the matrix-fiber configuration shows big difference.     

Multiscale stochastic finite element modeling of random elastic heterogeneous materials

In Xu et al. (Comput Struct 87:1416–1426, 2009) a novel Green-function-based multiscale stochastic finite element method (MSFEM) was proposed to model boundary value problems involving random heterogeneous materials. In this paper, we describe in detail computational aspects of the MSFEM explicitly across macro–meso–micro scales. Different numerical algorithms are introduced and compared in terms of numerical accuracy and convergence.     

A numerical method for homogenization in non-linear elasticity

In this work, homogenization of heterogeneous materials in the context of elasticity is addressed, where the effective constitutive behavior of a heterogeneous material is sought. Both linear and non-linear elastic regimes are considered. Central to the homogenization process is the identification of a statistically representative volume element (RVE) for the heterogeneous material. In the linear regime, aspects of this identification is investigated and a numerical scheme is introduced to determine the RVE size. The approach followed in the linear regime is extended to the non-linear regime by introducing stress–strain state characterization parameters. Next, the concept of a material map, where one identifies the constitutive behavior of a material in a discrete sense, is discussed together with its implementation in the finite element method. The homogenization of the non-linearly elastic heterogeneous material is then realized through the computation of its effective material map using a numerically identified RVE. It is shown that the use of material maps for the macroscopic analysis of heterogeneous structures leads to significant reductions in computation time.     

A numerical approach for determining the effective elastic symmetries of particulate-polymer composites

In this paper we deal with the problem of determining on the one hand the effective elastic properties of particulate-polymer composite materials and on the other hand the actual degree of symmetry of the resulting homogenised material. This twofold purpose has been accomplished by building a 2D as well as a 3D finite element model of the heterogeneous material and by using the strain-energy based numerical homogenisation technique. Both finite element models are able to reproduce with a good level of accuracy the real microstructure of the composite material by considering a random distribution of both particles and air bubbles (that are generated by the fabrication process). To assess the effectiveness of the proposed models, we present a numerical study to determine the effective elastic properties of the composite along with a comparison with the existing analytical and experimental results taken from literature and a sensitivity analysis in terms of the spatial distribution of the particles of the unit cell. Numerical results show that both models are able to provide the equivalent elastic properties with a very good level of accuracy when compared to experimental results and that the particulate-reinforced polymer composite could show, depending on the particles volume fraction and arrangement, an isotropic or a cubic elastic symmetry.     

基于参数变分原理的含夹杂Voronoi单元法及非均质材料弹塑性计算

含夹杂Voronoi单元通过在基体单元中引入一任意夹杂, 可以更好地反映非均质材料中微结构特性。基于参数势能和余能原理, 推导了无夹杂和含夹杂Voronoi单元有限元列式, 并在此基础上形成二次规划求解模型。将含夹杂Voronoi单元应用于非均质材料宏观弹塑性性能预测计算中, 分析了非均质材料中夹杂对其宏观等效弹塑性力学性能的影响。数值结果与其它方法所得结果的比较证明了本文中所给出模型的正确性和工程可适用性。      

Extended multiscale finite element method: its basis and applications for mechanical analysis of heterogeneous materials

Extended multiscale finite element method (EMsFEM) has been proved to be an efficient method for the mechanical analysis of heterogeneous materials. The key factor for efficiency and accuracy of EMsFEM is the numerical base functions (NBFs). The paper summarizes the general method for constructing NBFs and proposes a generalized isoparametric interpolation based on the rigid displacement properties (RDPs) of NBFs. We prove that the NBFs constructed by linear, periodic and rotational angle boundary conditions satisfy the RDPs, which is independent with the shape and material properties of unit cells. The properties of NBFs for oversampling technique are also comprehensively discussed. The algorithm complexity is discussed in theoretical and numerical aspects, which concludes that the computation quantity of EMsFEM is much smaller than the direct solutions. The algorithm is validated by linear analysis of the materials with random impurities and holes and the efficiency is improved further by parallel computing.     

Nanoindentation analysis for local properties of ultrafine grained copper processed by high pressure torsion

Severe plastic deformation (SPD) techniques have recently been developed for producing bulk ultrafine grained metallic materials. High pressure torsion (HPT) produces finer microstructures than those achieved by other SPD processes because of the higher imposed strain and hydrostatic pressure. It is known that HPT-processed metals show a highly heterogeneous microstructure not only along the radius due to the nature of torsional deformation but also through the thickness. Since the sample size for HPT is small, the local properties of HPT-processed specimens have not been investigated yet. In this paper, we propose a method to obtain stress–strain curves from nanoindenting curves by combining the finite element method and the recursion method. The nanoindentation technique was employed to elucidate the local mechanical properties, especially the stress–strain behavior. The method to extract the stress–strain curves from the load–displacement curves obtained by nanoindentation tests was applied to the edge region of the HPT-processed sample. The extracted properties correlated well with experimental results qualitatively.     

Iterative mesh partitioning optimization for parallel nonlinear dynamic finite element analysis with direct substructuring

 This work presents a novel iterative approach for mesh partitioning optimization to promote the efficiency of parallel nonlinear dynamic finite element analysis with the direct substructure method, which involves static condensation of substructures' internal degrees of freedom. The proposed approach includes four major phases – initial partitioning, substructure workload prediction, element weights tuning, and partitioning results adjustment. The final three phases are performed iteratively until the workloads among the substructures are balanced reasonably. A substructure workload predictor that considers the sparsity and ordering of the substructure matrix is used in the proposed approach. Several numerical experiments conducted herein reveal that the proposed iterative mesh partitioning optimization often results in a superior workload balance among substructures and reduces the total elapsed time of the corresponding parallel nonlinear dynamic finite element analysis. Received 22 August 2001 / Accepted 20 January 2002     

The effect of loading time on flexible pavement dynamic response: a finite element analysis

Dynamic response of asphalt concrete (AC) pavements under moving load is a key component for accurate prediction of flexible pavement performance. The time and temperature dependency of AC materials calls for utilizing advanced material characterization and mechanistic theories, such as viscoelasticity and stress/strain analysis. In layered elastic analysis, as implemented in the new Mechanistic-Empirical Pavement Design Guide (MEPDG), the time dependency is accounted for by calculating the loading times at different AC layer depths. In this study, the time effect on pavement response was evaluated by means of the concept of “pseudo temperature.” With the pavement temperature measured from instrumented thermocouples, the time and temperature dependency of AC materials was integrated into one single factor, termed “effective temperature.” Via this effective temperature, pavement responses under a transient load were predicted through finite element analysis. In the finite element model, viscoelastic behavior of AC materials was characterized through relaxation moduli, while the layers with unbound granular material were assumed to be in an elastic mode. The analysis was conducted for two different AC mixtures in a simplified flexible pavement structure at two different seasons. Finite element analysis results reveal that the loading time has a more pronounced impact on pavement response in the summer for both asphalt types. The results indicate that for reasonable prediction of dynamic response in flexible pavements, the effect of the depth-dependent loading time on pavement temperature should be considered.     

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

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

Analysis on the variances of material and structural properties based on random field theory

Based on the random field theory (RFT) and the stochastic finite element method (SFEM), the variances of the mechanical properties of materials and structures are studied. Manufacturing processes can easily lead to the spatial variations of the load and the material properties such as moduli and density. Characterizing the elastic moduli, load and density with one-dimensional random fields, the analytical solutions for the coefficient of variations (COVs) of effective material moduli, displacement and natural frequencies of beams are obtained. Then, with the fiber and matrix properties, volume fraction modeled by two-dimensional random fields and the fiber angle as a single random variable, a Monte Carlo simulation (MCS) is performed to generate the variances of effective modulus of fiber-reinforced composite laminar plate. Compared with the previous numerical conclusions, the present results reveal that the variances of effective material properties and structural displacement are greatly dependent on both the random fields and the sizes of structures in theory.     

Characterization of the damage in nanocomposite materials by a.c. electrical properties: experiment and simulation

Evolution of a.c. electrical properties under large strain of random nanocomposite materials made of a soft thermoplastic insulating matrix and hard conductive fillers is investigated. The transport properties are directly linked with the macroscopic mechanical strain on the composites during uniaxial tensile test or to the time under relaxation, meaning that the method is suitable for monitoring microstructural evolution of such composites. The real part of the conductivity indicated the breaking of the percolating network, while the imaginary part gave information on the possible “spatial correlation” of the damage events. Two different filler shapes were used, i.e. spherical and stick-like (aspect ratio about 15), leading to quantitatively different results. The microstructural evolution was simulated with the help of a resistance–capacitance (RC) model for the electrical properties and with finite element analysis for the mechanical properties. This revised version was published online in September 2006 with corrections to the Cover Date.     

Exploring the relationship between ferrite fraction and morphology and the electromagnetic properties of steel

The link between the electromagnetic properties of steel and its microstructure is a complex one, depending on both phase fractions and morphology. In this paper, both analytical and three-dimensional finite element (3D FEM) modelling techniques were applied to the prediction of permeability for steel with a given ferrite fraction for random ferrite/austenite distributions. Experimental measurements from a multi-frequency electromagnetic sensor on samples generated by hot isostatic pressing (HIPping) of powder mixtures were used to evaluate the analytical and FEM predictions. Theoretical treatment of the relationship between the sensor output and the effective permeability is also given; in particular, it was found that the zero crossing frequency of the real part of the inductance is approximately linearly related to the permeability for high (> 40%) ferrite percentages. The EM sensor can therefore be used to identify the samples across the full range (0–100%) of ferrite percentages using both the zero crossing frequency (> 40%) and trans-impedance (0–40%). The effect of banded (non-random) microstructures on sensor output and the prediction of the upper and lower bounds of permeability are also discussed.     

Uncertainty analysis of thermo-hydro-mechanical coupled processes in heterogeneous porous media

In this paper we present an uncertainty analysis of thermo-hydro-mechanical (THM) coupled processes in a typical geothermal reservoir in crystalline rock. Fracture and matrix are treated conceptually as an equivalent porous medium, and the model is applied to available data from the Urach Spa and Falkenberg sites (Germany). The finite element method (FEM) is used for the numerical analysis of fully coupled THM processes, including thermal water flow, advective–diffusive heat transport, and thermoelasticity. Non-linearity in system behavior is introduced via temperature and pressure dependent fluid properties. Reservoir parameters are considered as spatially random variables and their realizations are generated using conditional Gaussian simulation. The related Monte-Carlo analysis of the coupled THM problem is computationally very expensive. To enhance computational efficiency, the parallel FEM based on domain decomposition technology using message passing interface (MPI) is utilized to conduct the numerous simulations. In the numerical analysis we considered two reservoir modes: undisturbed and stimulated. The uncertainty analysis we apply captures both the effects of heterogeneity and hydraulic stimulation near the injection borehole. The results show the influence of parameter ranges on reservoir evolution during long-term heat extraction, taking into account fully coupled thermo-hydro-mechanical processes. We found that the most significant factors in the analysis are permeability and heat capacity. The study demonstrates the importance of taking parameter uncertainties into account for geothermal reservoir evaluation in order to assess the viability of numerical modeling.     

C/C-SiC缎纹编织复合材料孔隙缺陷的建模及其拉伸性能仿真

主要研究了随机孔隙缺陷在C/C-SiC缎纹编织复合材料中的有限元建模方法及其对拉伸性能的影响。基于C/C-SiC缎纹编织复合材料的细观结构和实验观察所得的微观形貌,得出孔隙缺陷具有随机分布特征,提出了一种三维随机碰撞算法模拟孔隙在复合材料中的分布,建立了含随机孔隙缺陷的C/C-SiC缎纹编织复合材料的有限元模型。采用有限元软件ABAQUS模拟了其在拉伸载荷下的力学行为,讨论了孔隙缺陷的尺寸和分布形式对材料拉伸性能的影响,并对试样进行了单轴拉伸实验测试,验证了数值模拟的有效性。结果表明,用本文方法建立的有限元模型符合含孔隙缺陷C/C-SiC缎纹编织复合材料的真实细观结构,相应的数值模拟结果也与试验数据吻合较好。本文的研究结果为含孔隙缺陷的缎纹编织复合材料及具有相似结构特征的复合材料的力学分析与优化设计提供了一种有效的方法。      

Predicting Effective Elastic Moduli and Strength of Ternary Blends with Core–Shell Structure by Second–Order Two–Scale Method

Core–shell particle–filled PA6/EPDM–g–MA/HDPE ternary blend has excellent mechanical properties. In this paper, effective elastic properties and tensile yield strength of the ternary blend are predicted by the second–order two– scale method, to investigate the relationship between morphology and mechanical properties. The method and the limit analysis for predicting mechanical properties of random heterogeneous materials are briefly introduced. Realistic morphology of the ternary blend including both core–shell particles and pure particles is simulated, and finite element mesh is generated. The unified strength theory is embedded in the method for the convenience of selecting a suitable yield criterion. The effective elastic moduli and tensile yield strength predicted by the method in this paper are compared with analytical and experimental results. Finally, effect of shell thickness in the core–shell particles on the effective elastic moduli and tensile yield strength is investigated.     

Voronoi cell finite element model based on micropolar theory of thermoelasticity for heterogeneous materials

In this paper, a new ‘Voronoi cell finite element model’ is developed for solving steady-state heat conduction and micropolar thermoelastic stress analysis problems in arbitrary heterogeneous materials. The method is based on the natural discretization of a multiple phase domain into basic structural elements by Dirichlet Tessellation. Tessellation process results in a network of polygons called Voronoi polygons. In this paper, formulations are developed for treating these polygons as elements in a finite element mesh. Furthermore, a composite Voronoi cell finite element model is developed to account for the presence of a second phase inclusion within a polygonal element. Various numerical examples are executed for validating the effectiveness of this model in the analysis of the temperature and stress fields for micropolar elastic materials. Effective material properties are derived for microstructures containing different distributions of second phase.     

Efficient parallel algorithms for elastic–plastic finite element analysis

This paper presents our new development of parallel finite element algorithms for elastic–plastic problems. The proposed method is based on dividing the original structure under consideration into a number of substructures which are treated as isolated finite element models via the interface conditions. Throughout the analysis, each processor stores only the information relevant to its substructure and generates the local stiffness matrix. A parallel substructure oriented preconditioned conjugate gradient method, which is combined with MR smoothing and diagonal storage scheme are employed to solve linear systems of equations. After having obtained the displacements of the problem under consideration, a substepping scheme is used to integrate elastic–plastic stress–strain relations. The procedure outlined controls the error of the computed stress by choosing each substep size automatically according to a prescribed tolerance. The combination of these algorithms shows a good speedup when increasing the number of processors and the effective solution of 3D elastic–plastic problems whose size is much too large for a single workstation becomes possible.