Statistical Representative Volume Element Research Articles

A statistical RVE model for effective mechanical properties and contact forces in lithium-ion porous electrodes

A statistical representative volume element for a cathodic NMC active porous materials is developed as a periodic face-centered-cubic structure. Finite element analyses (FEA) are conducted to calculate effective elastic properties and contact force distributions for various numbers of particle–particle and binder-particle contacts. The statistics for the RVE is attained through simulations of stochastic distributions of contact conditions. The variations of effective properties with changing particle–particle and binder-particle contacts are compared to experimental results in the literature. The stiffening behavior at compressive loading is correlated with increasing interparticle connections in the RVE. The average values and the statistical distribution of particle–particle and binder-particle contact forces that result from particle swelling are as well investigated. The significance of contact force distributions, for both particle–particle and binder-particle contacts, on potential particle cracking and binder debonding is addressed. It is noted that particle–particle contact forces appear that are 4–5 times larger than their corresponding average values. Conclusions drawn from differences in average contact forces between particle–particle and particle-binder contacts as well as normalized standard deviations of contact forces are utilized to improve a previously developed analytical model for effective stiffness properties and contact forces. Excellent agreements are found in comparisons to the numerical simulations.

Generation and homogenization of foamed polymer RVEs: microstructure-mechanical properties relationship

The main purpose of this study is to better understand the relationship between the microstructure of foamed polymer and mechanical properties. X-Ray tomography was performed on polypropylene foam specimens machined from injected plates. These plates were obtained for different thicknesses and exhibit different microstructure morphologies. The tomography scans were first digitalized and meshed. Then, numerical simulations were performed on representative volume elements (RVEs) to get homogeneous mechanical property of the material using a parallel C++ library Cimlib developed at CEMEF. Numerical methods described in this study focused on immerged or body-fitted strategy (FE context - level set framework - meshing adaptation) for exact and statistical RVEs generation. The numerical results were compared to experimental testing performed under tension and compression at different strain rates using image correlation. Good agreement was observed between simulations performed at the mesoscale and experimental tests carried out at the macroscale for both real and statistical RVEs. This methodology opens a way for the development of digital materials designed for specific mechanical properties.

SRVE modeling of particulate polymer matrix composites with irregularly shaped inclusions: Application to a green stone composite

Particulate polymer matrix composites (PPMCs) play a significant role in a wide range of applications from tissue engineering to aero-structures. Modeling thermoelastic properties of PPMCs can save sizable experimental time and costs, as it provides the capability of predicting the composite’s response to different loading conditions with acceptable accuracies. Micromechanical modeling approach is employed in this investigation to predict the thermoelastic properties of a new particulate polymer matrix composite, made of granite powder as inclusion and the acrylonitrile butadiene styrene (ABS) as matrix—called green stone composite. The reinforcing particles are modeled according to their shape irregularities. Namely, a statistical representative volume element (SRVE) is developed using the concept of integral range, with randomly distributed particles with irregular shapes. Experiments have been conducted to examine the validity of the proposed modeling approach. Numerical and experimental results both show that adding granite powder to pure ABS up to 38% (volume fraction) can notably increase the composite’s thermoelastic properties (bulk and shear moduli as well as thermal conductivity), as high as 200%.

Investigation of the microstructure adjustment by velocity variations during the directional solidification of Al-Ag-Cu with the phase-field method

Directional solidification is a favored process to manufacture homogeneous microstructures which lead to macroscopically unique properties for a material. The dependence of the spacing and type of the arising microstructure from the solidification velocity for constant velocities is extensively investigated. However the effect of changes in the solidification velocity on the resulting microstructure adjustment processes is still unclear. Therefore large-scale (3D+t) simulations of the ternary eutectic system Ag-Al-Cu with changing solidification velocities are conducted with a phase-field model based on the grand potential approach. To study the spatially complex rearrangement process during velocity changes in statistical representative volume elements, simulations with different velocity profiles are calculated in large-scale domains. The results show, that the evolving microstructure continuously rearranges by splitting and merging of the rods despite constant growth conditions. By increasing the velocity, the microstructure refines by splitting of the Al2Cu phase. Whereas by decreasing the velocity, the microstructure coarsens by overgrowing events of both intermetallic phases.

Analyzing effects of surface roughness, voids, and particle–matrix interfacial bonding on the failure response of a heterogeneous adhesive

An automated computational framework is introduced to link the microstructure to the micromechanical behavior of a heterogeneous adhesive composed of an epoxy matrix and embedded silica particles. A new reconstruction algorithm is employed to synthesize 3D periodic microstructural models of this materials system based on morphological and statistical data extracted from micro-computed tomography images. A non-iterative mesh generation algorithm is implemented in parallel to transform resulting virtual microstructures into finite element (FE) models. The continuum ductile and cohesive damage models used for simulating the adhesive’s failure response are calibrated with experimental data and the statistical representative volume element (SRVE) is identified. We have then carried out several high-fidelity FE simulations to investigate the effects of pre-existing voids in the adhesive layer, surface roughness of adherends, and the bonding strength along particles–matrix interfaces on the failure response of the adhesive subject to tensile, compressive, and shear loads.

The parallel multi-physics phase-field framework Pace3D

The phase-field method has been established for the numerical investigation of various microstructure evolution processes. The accurate description of these complex processes requires large domains and suitable models, allowing to couple several physical fields in statistical representative volume elements. To simplify the implementation of new models and to reduce the simulation run time, different frameworks have been developed. In this work, the parallel multi-physics phase-field framework Pace3D is introduced. The general structure of the solver, its modules and the parallelization are described. For increasing the performance of the implemented phase-field models, various optimization techniques are outlined. To efficiently store the simulation results, different data formats and parallel writing mechanisms are presented. The performance of an optimized implementation for a specific phase-field model is analyzed on a single core, showing a good peak performance. For a single node, the memory bandwidth is analyzed and ruled out as possible bottleneck. In addition, a proper weak scaling behavior is demonstrated on the three supercomputers ForHLR I, ForHLR II and Hazel Hen, for up to 96 100 cores.

A highly efficient 3D level-set grain growth algorithm tailored for ccNUMA architecture

A highly efficient simulation model for 2D and 3D grain growth was developed based on the level-set method. The model introduces modern computational concepts to achieve excellent performance on parallel computer architectures. Strong scalability was measured on cache-coherent non-uniform memory access (ccNUMA) architectures. To achieve this, the proposed approach considers the application of local level-set functions at the grain level. Ideal and non-ideal grain growth was simulated in 3D with the objective to study the evolution of statistical representative volume elements in polycrystals. In addition, microstructure evolution in an anisotropic magnetic material affected by an external magnetic field was simulated.

Determination of void statistics and statistical representative volume elements in carbon fiber-reinforced thermoplastic prepregs

Void consolidation of high-performance thermoplastic composites strongly depends not only on average void content but also on the distributions of void size, shape, and location within the prepreg materials. High-resolution 3-D X-ray microcomputed tomography shows that voids in carbon/poly(ether ether ketone) (AS4/APC2) prepreg are rodlike with major axis along the fiber axial direction. In order to accurately characterize the void microstructure, a detailed study was conducted to quantify the statistical distribution of void content, void length, equivalent void diameter, and void aspect ratio. Resolution of 1.48 μm/pixel provided a balance of measurement accuracy and inspection time. Suitable statistical distribution functions were found to describe the void length, diameter, and aspect ratio. For each void property, a sub-statistical representative volume element (SRVE) was determined. The SRVE of the overall void microstructure is defined as the maximum dimensions of the sub-SRVEs. In case of AS4/APC2 tape, the SRVE was found to be 6.1 mm in length (fiber direction), 27 mm in width (transverse to fiber direction), and 0.18 mm of full prepreg thickness.

A Numerical Method for Simulating the Microscopic Damage Evolution in Composites Under Uniaxial Transverse Tension

In this paper, a new numerical method that combines a surface-based cohesive model and extended finite element method (XFEM) without predefining the crack paths is presented to simulate the microscopic damage evolution in composites under uniaxial transverse tension. The proposed method is verified to accurately capture the crack kinking into the matrix after fiber/matrix debonding. A statistical representative volume element (SRVE) under periodic boundary conditions is used to approximate the microstructure of the composites. The interface parameters of the cohesive models are investigated, in which the initial interface stiffness has a great effect on the predictions of the fiber/matrix debonding. The detailed debonding states of SRVE with strong and weak interfaces are compared based on the surface-based and element-based cohesive models. The mechanism of damage in composites under transverse tension is described as the appearance of the interface cracks and their induced matrix micro-cracking, both of which coalesce into transversal macro-cracks. Good agreement is found between the predictions of the model and the in situ experimental observations, demonstrating the efficiency of the presented model for simulating the microscopic damage evolution in composites.

A multiscale approach and microstructure design of the elastic composite behavior reinforced with natural particles

The morphology characterization and computational methods favored numerical simulation and design of microstructures. Indeed, the multiscale approaches enable us to determine the elastic properties of materials. In this paper, the objective is to develop a three-dimensional microstructure of biocomposites containing natural particles. The biocomposite is made of polypropylene matrix mixed with natural fillers. The image is obtained using the microscope. We describe a serial sectioning process and finite element simulations to reproduce, visualize and model these microstructures. Statistical methods are introduced to study the representativity of specimen. The statistical representative volume element is introduced to determine the minimum volume which provides the representativeness. This statistical volume is compared with experimental and numerical ones.

Grain-level statistical plasticity analysis on strain cycle fatigue of a FCC metal

The micromechanical strain cycle fatigue-life is systematically investigated by the micro-level numerical simulation, compared with symmetrical strain cycle experiments of copper, focusing on the characteristics of polycrystalline aggregation and the mechanism of microscale plastic deformation. A methodology to predict the low-cycle fatigue life by micro-level simulation along with statistical analysis is proposed through the following steps: (1) A crystal plasticity model is developed based on the nonlinear kinematic hardening mechanism of crystal slipping system. This model is applied to the calculations of crystal grain interior stresses and plastic strains. (2) A statistical representative volume element (SRVE) is constructed for a pure copper as a material model which features a polycrystalline Voronoi aggregation consisting of a number of crystal grains. This SRVE can be used for statistical analysis of the material inhomogeneous stresses and strains during cycle loading. (3) The simulations are performed to model the experimental cycle evolution of strain fatigue by using the SRVE under the symmetrical tensile–compressive loading. (4) Statistical and micromechanical analyses are carried out for the inhomogeneous interior stresses and strains of the SRVE of the polycrystalline copper in the low cycle regime. The resulting analysis can render the microscale interpretation and numerical simulation for the low-cycle fatigue evolution accordingly.

3-D microstructure reconstruction of polymer nano-composite using FIB–SEM and statistical correlation function

Abstract 3-D reconstruction of Halloysite nanotube (HNT) polypropylene composite has been performed using two different methods. In the first method, several slices of the composite material were obtained using focused ion beam (FIB), and scanning electron microscopy (SEM). A representative volume element (RVE) of the real material’s micro/nanostructures was then constructed by stacking these morphological images using VCAT® software. In the second method, SEM images of the nano-composite were used to extract statistical two-point correlation function (TPCF), for reconstruction of an RVE of the nano-composite. The resulting RVEs obtained from both methods were meshed for finite element (FE) simulation of deformation under tension and shear loadings. The FE results were then used to compute the stiffness tensor of the nano-composite. In the statistical approach, the TPCF was obtained from a none-Eigen microstructure which can partially reflect statistical information of the microstructure. The mechanical constants obtained from statistical RVEs using FEM approach shows a 5.7% error compared with those obtained from real RVE, which could be attributed to the approximation using TPCF [1] . It is concluded that the statistical method using TPCF alone can produce an approximate microstructure that should be modified using other statistical descriptor such as two-point cluster function and lineal path function to have better reconstruction of heterogeneous nano-composites [2] .

Quantitative study on the statistical properties of fibre architecture of genuine and numerical composite microstructures

A quantitative study is carried out regarding the statistical properties of the fibre architecture found in composite laminates and that generated numerically using Statistical Representative Volume Elements (SRVE’s). The aim is to determine the reliability and consistency of SRVE’s for representation of the composite microstructure as well as investigate the effect of a varying fibre radii distribution on the fibre architecture. Based on digital image analysis, the fibre architecture of unidirectional glass fibre composites with varying fibre content is recognised. The fibre architecture found is compared to a numerical microstructure generator using Monte Carlo simulations. It is shown that the numerical microstructure generator produces fibre arrangements that are statistically similar to the observed, which indicates a reliable and consistent SRVE. The microstructural effects of a parametric variation of the parameters for fibre radii distribution are simulated, and the influence on the fibre architecture is investigated.

Determination of the minimum size of a statistical representative volume element from a fibre-reinforced composite based on point pattern statistics

In a previous study, Trias et al. [1] determined the minimum size of a statistical representative volume element (SRVE) of a unidirectional fibre-reinforced composite primarily based on numerical analyses of the stress/strain field. In continuation of this, the present study determines the minimum size of an SRVE based on a statistical analysis on the spatial statistics of the fibre packing patterns found in genuine laminates, and those generated numerically using a microstructure generator.

Micromechanical Modelling for Bauschinger Effect of Dual Phase Steel

The aim of this study is to simulate the micromechanics of dual phase (DP) 600 steel during tension-compression deformation and apply the calculated stress-strain curves to identify material parameters of phenomenological constitutive models with focus on Bauschinger effects. Image-based and statistical representative volume element (RVE) models were built to perform virtual tension-compression tests. The effects of mesh size, dimension, martensite volume fraction and martensitic particle size on stress-strain response were also studied. The results showed that the RVE models gave good overall prediction compared with experiment data. Material parameters of Chaboche nonlinear kinematic model were identified by fitting the resulted cyclic strain-stress curve from virtual tests using optimization software LS-OPT.

Precise generation of complex statistical Representative Volume Elements (RVEs) in a finite element context

A precise procedure to generate complex 2D and 3D statistical virtual microstructures in a finite element (FE) context is described. This technique, based on Laguerre tessellations and advancing front method, level-set description of interfaces and anisotropic meshing adaptation, is detailed for equiaxial polycrystal and powder RVEs defined by a size distribution law. The capability of the procedure to respect statistical data could be insured by the advancing front method coupled with an optimization procedure depending on the nature of the considered RVE. Moreover, by applying a graph coloration technique, the number of level-set functions used to represent the cells and also in anisotropic mesh adaptation is greatly reduced.

Determination of the critical size of a statistical representative volume element (SRVE) for carbon reinforced polymers

Representative volume elements (RVEs) have been extensively used to estimate the elastic properties of fibre-reinforced composites. Most of them rely on the assumption of a periodic distribution of fibres which is not realistic. In order to reproduce damage phenomena, such as matrix cracking, it is necessary for volume element to represent properly the random distribution of fibres (distance to first neighbors, occurrence of clusters of fibres, etc.). Therefore, a statistical RVE (SRVE) should satisfy both mechanical and point pattern criteria. The present work establishes the size of a SRVE for a typical carbon fibre reinforced polymer. It is concluded that the minimum size is δ = L/ R = 50 ( L the side of the element and R the fibre radius).

A two-scale method for matrix cracking probability in fibre-reinforced composites based on a statistical representative volume element

This paper presents a two-scale methodology for evaluating the probability of matrix cracking in any point of the transverse section of a fibre-reinforced composite material. At the microscale, the random distribution of fibres has been analyzed by means of optical microscopy and the corresponding images have been used to generate realizations of statistical representative volume elements (SRVE). The size of these SRVEs was chosen in order to guarantee that they are representative of the material in both the mechanical and statistical senses and consequently, allow the statistical simulation of transverse random composites. Finite Element models of these real-microstructure SRVEs were solved with arbitrary values of the boundary conditions. From these results, probability density functions of the stress, strain and dilatational energy density were found. These results can be related to any value of the stress tensor at any point in the macroscale by means of a two-scale methodology. The presented methodology has been applied to three different carbon-fibre reinforced polymers (CFRP), and the results related to experimental data from tensile tests. This approach can be used in the engineering design process as a procedure to define critical strains, stresses or combinations of both to obtain a matrix cracking probability below a desired level.