Representative Volume Element Simulations Research Articles

Simulation-free determination of microstructure representative volume element size via Fisher scores

A representative volume element (RVE) is a reasonably small unit of microstructure that can be simulated to obtain the same effective properties as the entire microstructure sample. Finite element (FE) simulation of RVEs, as opposed to much larger samples, saves computational expenses, especially in multiscale modeling. Therefore, it is desirable to have a framework that determines the RVE size prior to FE simulations. Existing methods select the RVE size based on when the FE-simulated properties of samples of increasing sizes converge with insignificant statistical variations, with the drawback being that many samples must be simulated. We propose a simulation-free alternative that determines the RVE size based only on a micrograph. The approach utilizes a machine learning model trained to implicitly characterize the stochastic nature of the input micrograph. The underlying rationale is to view RVE size as the smallest moving window size for which the stochastic nature of the microstructure within the window is stationary as the window moves across a large micrograph. For this purpose, we adapt a recently developed Fisher score-based framework for microstructure nonstationarity monitoring. Because the resulting RVE size is based solely on the micrograph and does not involve any FE simulation of specific properties, it constitutes an RVE for any property of interest that solely depends on the microstructure characteristics. Through numerical experiments of simple and complex microstructures, we validate our approach and show that our selected RVE sizes are consistent with when the chosen FE-simulated properties converge.

Micromechanical prediction of the elastic and plastic properties of sintered steels

One of the characteristic features of sintered steels is the porosity in their microstructure resulting from the compaction and sintering process. This porosity strongly influences the mechanical properties. To enhance the understanding for the structure–property relationship of sintered Astaloy®85Mo with 0.4 wt.%C, a micromechanical modelling approach based on face-centred cubic (fcc) representative volume elements (RVE) is proposed. The fcc-like periodic arrangement of the sintered particles in the RVE enables the consideration of a realistic non-spherical pore morphology. To compare the predictions with experimental results, accompanying uniaxial tensile tests are considered at different pore volume fractions after initial microstructure characterisation. In addition to the effect of pore volume fraction, the influence of sinter necks on the predicted overall strength is also systematically investigated. Despite the fairly simple nature of the underlying fcc structure, the RVE simulations are perfectly capable of reproducing the experimental trend, showing that the elasto-plastic properties decrease with increasing porosity. This is in contrast to analytical predictions, which underestimate the decrease in properties due to spherical pore assumptions. Moreover, the finite element-based simulations reveal a less pronounced influence of the sinter neck shape on the macroscopic behaviour, even though substantial differences in plastic strain localisation are discernible at the microscopic scale.

A homogenization-based model of the Gurson type for porous metals comprising randomly oriented spheroidal voids

In this work, we propose a new fully explicit isotropic, rate-independent, elasto-plastic model for porous materials comprising a random with uniform probability, isotropic distribution of randomly oriented spheroidal voids of the same shape. The proposed model is based on earlier homogenization estimates that use a linear comparison composite theory. The resulting expressions exhibit the simplicity of the well known Gurson model and, thus, its numerical implementation in a finite element code is straightforward. To assess the accuracy of the analytical model, we carry out detailed finite-strain, three-dimensional finite element (FE) simulations of representative volume elements (RVEs) with the corresponding microstructures. Proper minimal parameter calibration of the model leads to fairly accurate agreement of the analytical predictions with the corresponding FE average stresses and porosity evolution. We show, both analytically and numerically, that the initial aspect ratio of the voids has a significant effect on the homogenized yield surface of the porous material leading to extremely soft responses for flat oblate voids (e.g. aspect ratio less than 0.5) especially at large triaxialities. Finally, after numerical implementation of the model in a commercial finite element code (ABAQUS), we solve the industrially important problem of hole expansion and comment on the role of porous compressible plasticity versus classical incompressible plasticity.

Data-driven multiscale modelling of granular materials via knowledge transfer and sharing

Machine learning approaches have found immense potential to revolutionise the constitutive modelling of granular materials. However, data scarcity poses a significant challenge to this emerging paradigm. This study aims to tackle this issue by presenting two transfer learning-based strategies that harness well-established constitutive knowledge and similar material data to reduce data demands for data-driven material modelling. The first approach utilises phenomenological constitutive models to generate massive synthetic data which reflect the targeted material behaviour to train a base model. This base model is then repurposed for a new task based on numerical simulation data via transfer learning. The other approach involves using available material data to train a base model, which is then applied to other new materials that are similar but with limited data. The proposed transfer learning methods are tested on both particle-scale simulations of representative volume elements (RVEs) and hierarchical multiscale modelling of boundary value problems (BVPs) of granular materials. The trained data-driven material model is embedded in numerical simulations with the finite element method (FEM) to validate its accuracy, efficiency, and stability. The results demonstrate that transfer learning can effectively achieve high-quality machine learning predictions with limited data. The transfer learning strategy presented in this study is expected to be widely applicable to small data-driven material modelling.

Local defect formation in short glass fibre reinforced polymers – micro-mechanical simulations and interrupted in-situ experiments

Discontinuous fibre reinforced polymers are widely used in various industry sectors and often replace conventional materials, due to lower production costs and their lightweight structure. For improvement of the component design, detailed knowledge of the failure mechanisms are necessary. To better understand the defect formation and thus the micro-mechanics, the strain behaviour in single fibres was analysed by micro-mechanical simulations of Representative Volume Elements (RVE). Therefore, selected fibres – similar orientated as in the real structure – were chosen for detailed analysis. Additionally, the defect formation next to selected fibres was investigated by X-ray computed tomography (CT). Furthermore, the critical fibre length was estimated based on the protruding fibre length of the fracture surface. Overall the simulation results correspond to theory. However, the detailed local inspection of the experimental volume data showed a rather strong influence of neighbouring fibres.

Discrete element models for elasto-plastic wave propagation in nonlinear architected materials

Nonlinear architected materials are known to exhibit a plethora of dynamic phenomena that enhance our capacity to manipulate elastic waves. Since these properties stem from complex, subwavelength geometry, dynamic simulations at high resolutions are often intractable at scales of interest. Therefore, prior studies have turned to effective medium models that capture essential properties in the long-wavelength limit. One class of such models is a periodic structure composed of discrete mass, spring, and damper elements, whose constitutive relationships are computed to match behaviors observed in experiments or fine-scale simulations of representative volume elements. While models of this type have been implemented successfully in many cases, the majority of literature has been focused on recoverable deformation, possibly including linear damping. However, history-dependent effects, such as friction and plasticity, have been much less explored. In this work, we develop a discrete element modeling framework for nonlinear architected materials undergoing plastic deformation. Due to the history- and rate-dependent characteristics of plasticity, the framework generally yields a system of differential-algebraic equations whose computational cost is significantly greater than an elastic system of comparable size. We demonstrate the method using several examples from the literature and explore means to obtain phenomenological plasticity models for general geometries.

Effect of raster and layer characteristics on tensile behavior and failure of FFF printed PLA samples by representative volume element model

In the fused filament fabrication (FFF) based additive manufacturing process, finding optimum printing parameters for achieving the required mechanical properties of the FFF-built part is challenging. In this study, a representative volume element (RVE) based mesoscale approach was developed to describe the influences of printing parameters on the mechanical behaviors of the 3D printed parts. It was shown that the stress-strain curves up to failure obtained from RVE simulations were well verified by experimental tensile test data of printed PLA samples. Then, effective tensile properties of samples manufactured using different raster angles (0°, 45°/−45°, and 90°) and a wide range of layer heights and widths were predicted and correlated with their respective local damage occurrences. The raster angle strongly affected the elastic modulus and tensile strength. The orientation between interlayer voids and loading direction governed local stress distribution, interface failure, and total deformation of FFF samples. An increased layer height and decreased layer width resulted in a more significant fraction of voids between layers and thus lowered stiffness and tensile strength. The introduced RVE model can serve as a simple tool for determining homogenized responses and studying local stress-strain developments and failure of complex printed parts according to the used printing parameters.

Micromechanical analysis of age‐induced strength reduction in a multiaxially loaded short‐fiber reinforced thermoplastic

AbstractIn this study, we demonstrate that the strength of a short‐fiber reinforced polymer as measured in uniaxial tension is not a good indicator for its strength under multiaxial loading conditions. To illustrate this fact, the influence of physical aging on the strength of a 20 wt% short‐fiber reinforced polycarbonate is studied for a uniaxial and biaxial loading condition. Results demonstrate that aging strongly reduces the strength in multiaxial loading, whereas, it increases in uniaxial loading. To rationalize these observations, a micromechanical analysis of the local stress state is performed using three‐dimensional (3D) representative volume elements (RVE) in combination with a constitutive model that adequately describes the intrinsic deformation response of the matrix. RVE simulations demonstrate that biaxial loading results in higher local hydrostatic stresses and triaxiality than the uniaxial loadcase for the composite system considered, and aging results in a shift towards higher values. The high triaxiality, the magnitude of the hydrostatic stress, and the shift in hydrostatic stress upon aging, presents a clear rationalization why physical aging induces a strength reduction in biaxial loading and not in uniaxial loading. These results highlight that care should be taken when predicting strength under complex loading conditions with only uniaxial data available.

A multiscale viscoelastic constitutive model of unidirectional carbon fiber reinforced PEEK over a wide temperature range

Due to super toughness and good repairability, the fiber reinforced polyether ether ketone (PEEK) composites are promising to replace metals in the primary-load bearing structures for the next generation aircraft. However, at the high service temperature close to or even above the glass transition temperature of PEEK, composite properties strongly depend on the loading temperature and time, bringing great challenges to their applications. How to evaluate their temperature/time dependent properties over a wide temperature range is the key issue in both the structure design and the process design. This study proposes a multiscale viscoelastic constitutive model for the composites to separately consider the individual contribution of the intrinsic matrix properties and the fiber induced strain concentration. The generalized Maxwell model for the PEEK viscoelastic properties was extended to a three-dimensional case by the genetic integration method. Based on the bottom-up approach, the isotropic viscoelastic matrix and the anisotropic linear elastic reinforcement were considered to construct a multiscale model of the fiber reinforced PEEK composites. To verify the developed model, the unidirectional carbon fiber reinforced PEEK composite samples were fabricated and tested. By the PEEK matrix relaxation tests, the composite transverse tensile tests and the representative volume element (RVE) simulations, the model parameters were calibrated step by step. Good agreements among the experimental results, the RVE simulations, and the model predictions under different tensile states over a wide temperature range (25–200 °C), proved that this model could be used to predict and to design the relaxation behaviors of the fiber reinforced PEEK composites.

Data‐based prediction of the viscoelastic behavior of short fiber reinforced composites

AbstractThe viscoelastic behavior of short fiber reinforced polymers (SFRPs) partly depends on different microstructural parameters such as the local fiber orientation distribution. To account for this by simulation on component level, two‐scale methods couple simulations on the micro‐ and macroscale, which involve considerable computational costs. To circumvent this problem, the generation of a viscoelastic surrogate model is presented here. For that purpose, an adaptive sampling technique is investigated and data are obtained by creep simulations of representative volume elements (RVEs) using a fast Fourier transform (FFT) based homogenization method. Numerical tests confirm the high accuracy of the surrogate model. The possibility of using that model for efficient material optimization is shown.

Revisiting the empirical particle‐fluid coupling model used in DEM‐CFD by high‐resolution DEM‐LBM‐IMB simulations: A 2D perspective

AbstractThe work investigates the applicability of the unresolved Computational Fluid Dynamics and Discrete Element Method (CFDDEM) technique based on empirical equations for fluid‐particle coupling. We first carry out a series of representative volume element simulations using the high‐resolution particle‐resolved Lattice Boltzmann method and Discrete Element Method (LBMDEM) coupled by an Immersed Moving Boundary (IMB) scheme. Then, we compare the results obtained by both LBMDEM and empirical equations used in unresolved CFDDEM with analytical solutions. It is found that the existing empirical equations used in solving fluid‐particle interactions in 2D CFDDEM fail to accurately calculate the hydrodynamic force applied to solid particles. The underlying reason is that the existing empirical models are obtained based on 3D experimental results and thus are not applicable to 2D problems. Based on the simulation results, a new drag coefficient model is then proposed. The estimated drag forces using the new model are compared favourably with the simulated ones, indicating the good performance of the proposed model.

Growth of random polyhedral void in structural steel based on micromechanical RVE simulations

This paper investigates growth of random polyhedral voids (RPVs) in structural steel, which is a critical stage for ductile fracture of steel. An in-situ high-resolution μXCT experiment shows that most micro-voids in structural steels can be characterized by RPVs. In this study, a micromechanical representative volume element (RVE) model containing an RPV was established with periodic boundary conditions. Comparison of RPV growths with spherical and elliptical void growths was conducted for model validation. On this basis, effects of geometric parameters of the RPV on void growth were analyzed through Python-based parametric modeling in ABAQUS with respect to the aspect ratio, orientation, sphericity, initial void volume fraction, respectively. The results found that growth of the RPV decreases with an increasing initial aspect ratio of the RPV. Growth of the RPV increases as the initial sphericity of RPV decreases. When the initial orientation of RPV tends to be perpendicular to the main tensile direction, a greater void growth will be obtained. The initial volume fraction of RPV has a minor effect on void growth compared with the other geometric parameters. For engineering application, accurate and simplified formulae were proposed to evaluate growth of microscopic RPVs, respectively.

Nano-vascularized polymers: how nanochannels impact the mechanical behaviour at the macroscale

In recent decades, nature has inspired the engineering of many structural and smart materials. Nano-vascularization has been stimulating research on advanced materials for novel biomedical, orthopaedic, industrial, and aeronautical applications. The continuous development of nano-vascularized materials requires a more accurate understanding of their mechanical behaviour. This work provides a multiscale methodology to predict the macroscale properties of nano-vascularized materials. The methodology is experimentally validated for the case of a nano-vascularized epoxy resin manufactured using sacrificial electrospun nanofibers. It is based on the development of representative volume elements (RVEs) that encompass information about both the nanochannels distribution and on the mechanical properties of the material at different dimensional scales. The RVEs simulations allowed obtaining a homogenized model describing the nano-vascularized material properties and studying the most intimate failure mechanisms. A virtual stress tomographic investigation on the RVEs was adopted as a digital twin to reveal the damage evolution and the actual failure mechanisms of the nano-vascularized material: damage occurred mainly at the nanochannels intersections, particularly where the intersections become dense. Interestingly, the simulations revealed a correlation between the stress state and the formation of feather markings as well as local failures on the nanochannels linking directions, as evidenced by SEM analysis.

Damage Evolution of Hot Stamped Boron Steels Subjected to Various Stress States: Macro/Micro-Scale Experiments and Simulations.

Hot stamping components with tailored mechanical properties have excellent safety-related performance in the field of lightweight manufacturing. In this paper, the constitutive relation and damage evolution of bainite, martensite, and mixed bainite/martensite (B/M) phase were studied. Two-dimensional representative volume element (RVE) models were constructed according to microstructure characteristics. The constitutive relations of individual phases were defined based on the dislocation strengthening theory. Results showed that the damage initiation and evolution of martensite and bainite phases can well described by the Lou-Huh damage criterion (DF2015) determined by the hybrid experimental–numerical method. The calibrated damage parameters of each phase were applied to the numerical simulation, followed by the 2D RVE simulations of B/M phase under different stress states. To study the influence of martensite volume fraction (Vm) and distribution of damage evolution, the void nucleation and growth were evaluated by RVEs and verified by scanning electron microscope (SEM). Three types of void nucleation modes under different stress states were experimentally and numerically studied. The results showed that with the increase of Vm and varying martensite distribution, the nucleation location of voids move from bainite to martensite.

A ductile fracture criterion under warm-working conditions based on the multiscale model combining molecular dynamics with finite element methods

The ductile fracture (DF) criterion has an important guiding significance for process simulation in predicting material safety and formability. However, knowledge regarding DF micro-mechanisms, especially temperature impact, is limited. Temperature impacts micro-void-nucleation via molecular dynamics simulations, and temperature impacts mesoscale-void-growth via representative volume element simulations of periodic boundary conditions are studied. Results of these simulations show that elevated temperature promotes micro-void-nucleation but has no apparent effect on mesoscale-void-growth. This paper develops an uncoupled DF criterion considering the effects of micro-void-nucleation, mesoscale-void-growth, and void-coalescence on damage from the multiscale viewpoint. The range of application of the developed criterion is from room temperature to below recrystallization temperature. The fracture locus of AA 2024-T351 alloy and 316LN stainless steel is constructed using the criterion and compared with the Lou criterion (Lou et al., 2012), the B&W criterion (Bai and Wierzbicki, 2008), and the Shang criterion (Shang et al., 2018). The results show that the New criterion provides better predictability. The DF criterion is also calibrated and validated using seven types of specimens processed from additive manufacturing AlSi10Mg. The Force-stroke curve and fracture morphology indicate that the criterion satisfactorily predicts DF onset in warm-working conditions and at various stress states. Therefore, the present study provides an in-depth understanding of the micro-mechanism of temperature on DF.

A void closure model based on hydrostatic integration and the Lode parameter for additive manufacturing AlSi10Mg

The void closure model is urgently needed for the subsequent hot forming process of additive manufacturing (AM) products. The influence of temperature, Lode parameter, and hydrostatic integration (Gm) on void closure was studied by representative volume element (RVE) simulations. The results show that Gm and the Lode parameter significantly influence void closure, and the temperature effect is related to material properties. A new model accounting for Gm and Lode parameter in evaluating void closure was developed using linear regression with an average error of 2.1% in comparison to RVE simulations. X-ray computed tomography (CT) was carried out at different high reductions for the AM cylinder's hot forging experiments. The new and stress-triaxiality-based (STB) models were integrated into the ABAQUS and compared with the CT measurements. Results indicate that the model is more precise than the STB model, suggesting that the model can be appropriately applied to forecast void closure during the hot forging process for AM products.

Data-driven discovery of interpretable causal relations for deep learning material laws with uncertainty propagation

This paper presents a computational framework that generates ensemble predictive mechanics models with uncertainty quantification (UQ). We first develop a causal discovery algorithm to infer causal relations among time-history data measured during each representative volume element (RVE) simulation through a directed acyclic graph. With multiple plausible sets of causal relationships estimated from multiple RVE simulations, the predictions are propagated in the derived causal graph while using a deep neural network equipped with dropout layers as a Bayesian approximation for UQ. We select two representative numerical examples (traction-separation laws for frictional interfaces, elastoplasticity models for granular assembles) to examine the accuracy and robustness of the proposed causal discovery method for the common material law predictions in civil engineering applications.

A Multiscale Approach for the Numerical Simulation of Turbulent Flows with Droplets.

A multiscale approach for the detailed simulation of water droplets dispersed in a turbulent airflow is presented. The multiscale procedure combines a novel representative volume element (RVE) with the Pseudo Direct Numerical Simulation (P-DNS) method. The solution at the coarse-scale relies on a synthetic model, constructed using precomputed offline RVE simulations and an alternating digital tree, to characterize the non-linear dynamic response at the fine-scale. A set of numerical experiments for a wide range of volume fractions, particle distribution sizes, and external shear forces in the RVE are carried out. Quantitative results of the statistically stationary turbulent state are obtained, and the turbulence modulation phenomenon due to the presence of droplets is discussed. The developed synthetic model is then employed to solve global scale simulations of flows with airborne droplets via the P-DNS method. Improved predictions are obtained for flow conditions where turbulence modulation is noticeable.

A Novel Approach to Predict the Process-Induced Mechanical Behavior of Additively Manufactured Materials

The grain structure and texture of additively manufactured materials depend strongly on the local temperature gradients during the solidification of the material. These grain structures and textures influence the mechanical behavior, ranging from isotropy to transversal and orthotropic symmetry. In the present contribution, a cellular automaton is used to model the grain growth during selective electron beam melting. The resulting grain structures and textures serve as input for a mesoscopic mechanical model. The mechanical behavior on the mesoscale is modeled by means of gradient-enhanced crystal plasticity, applying the finite element method. Computational homogenization is applied to determine the resulting macroscopic elastic and plastic properties of the additively manufactured metals. A general orthotropic yield criterion is identified by means of the initial yield loci computed with mesoscopic simulations of representative volume elements. The numerical results are partly validated with experimental data.

Evolution of the stiffness tetrad in fiber-reinforced materials under large plastic strain

The main aim of this work is to track the evolution of the stiffness tetrad during large plastic strain. Therefore, the framework of a general finite plasticity theory is developed. Some special cases are examined, and the case of a material plasticity theory is considered more closely. Its main feature is that the elasticity law changes during plastic deformations, for which we develop an approach. As sample materials, we use three types of fiber-reinforced composites. For numerical experiments and verification of the model’s predictions, finite element simulations of representative volume elements for uni-, bi- and tri-directional reinforced materials with periodic boundary conditions are used. From these, we extract the stiffness tetrads before and after large deformations of the material. We quantify the change of the stiffness tetrads due to the fiber reorientation. Finally, we propose an analytical evolution with three parameters that account reasonably well for the evolution of the stiffness tetrad.