Statistical Volume Elements Research Articles

Size Dependency of Elastic and Plastic Properties of Metallic Polycrystals Using Statistical Volume Elements

We present an efficient approach to evaluate the size dependency of elastic and plastic properties of metallic polycrystalline materials. Specifically, we consider different volume fractions of ferrite and martensite phases for the construction of three macroscopic domains. Statistical Volume Elements (SVEs) of different sizes are extracted from these domains using the moving window method. Linear and Crystal Plasticity (CP) simulations provide elastic and plastic properties of the SVEs such as the bulk and shear moduli, yield strength, and hardening modulus. We use a variation-based criterion to determine the Representative Volume Element (RVE) size of these properties. This RVE size corresponds to a size beyond which the given property can be idealized as homogeneous. We also use anisotropy indices and an additional RVE size criterion to determine the size limits beyond which these properties can be idealized as isotropic. Numerical results show that the plastic properties often reach their homogeneity and isotropy limits at larger sizes compared to elastic properties. This effect is more pronounced for the hardening modulus compared to the yield strength.

A statistical volume element-based procedure for the prediction of the mechanical and electrical response of an epoxy-PZT self-sensing layer for application in composite laminates

Structural Health Monitoring (SHM) techniques are being developed to continuously oversee defects in composite structures. Within this context, research is focusing on the development of new types of sensors with high sensitivity and proper integration in the laminate.In this work, the mechanical and electrical properties of a recently developed piezoelectric composite material made of a Lead Zirconate Titanate (PZT) powder embedded in an epoxy matrix are evaluated with finite element simulations of plane strain Statistical Volume Elements (SVEs). The homogenized properties are then implemented in a second finite element model of a composite specimen with the embedded self-sensing material and loaded in compression. The electrical sensitivity is evaluated as a function of the distance between the signal electrodes.The results show that the finite element models with the homogenized properties have decreasing sensitivity with increasing electrodes distance, in agreement with the experimental results from another work, in which Glass Fiber Reinforced Polymer (GFRP) laminates with the embedded piezoelectric composite are loaded in compression and tested for output signal.

A 3D voxel-based mesostructure generator for finite element modelling of tow-based discontinuous composites

Tow-based discontinuous composites manufactured with ultra-thin tapes display high stiffness, strength, and in-plane isotropy, thus competing with composite laminates. Their complex 3D microstructure affects the mechanical response, in turn demanding 3D generators that capture the tape waviness, resin pockets, and thickness and fibre content variations. The present work proposes an automated numerical framework combining a 3D voxel-based mesostructure generator with finite element models. A modified 3D random sequential absorption technique is developed with bin-guided allocation, draping, and thickness control. A statistical study is used to size the statistical volume elements and predict the elastic properties of thick, thin, and ultra-thin tow-based discontinuous composites. The results are compared with the experimental values from the literature. Despite uncertainties in physical tape properties, the resulting stiffnesses are predicted with good accuracy.

Statistical Homogenization of Elastic and Fracture Properties of a Sample Selective Laser Melting Material

Selective laser melting (SLM) is an additive manufacturing technique commonly used in the rapid prototyping of components. The complexity of the SLM microstructure poses a unique challenge to deriving effective mechanical properties at different length scales. Representative volume elements (RVEs) are often used to homogenize the material properties of composites. Instead of RVEs, we use statistical volume elements (SVEs) to homogenize the elastic and fracture properties of the material. This relates the inherent variation of a material’s microstructure to the variation in its mechanical properties at different observation scales. The convergence to the RVE limit is examined from two perspectives: the stability of the mean value as the SVE size increases for the mean-based approach, and the tendency of the normalized variation in homogenized properties to zero as the SVE size increases for the variation-based approach. Fracture properties tend to make the RVE limit slower than do elastic properties from both perspectives. There are also differences between vertical (normal to printing plane) and horizontal (in-plane) properties. While the elastic properties tend to make the RVE limit faster for the horizontal direction, i.e., having a smaller variation and more stable mean value, the fracture properties exhibit the opposite effect. We attributed these differences to the geometry of the melt pools.

FE‐NN: Efficient‐scale transition for heterogeneous microstructures using neural networks

AbstractNumerical modeling and optimization of advanced composite materials can require huge computational effort when considering their heterogeneous mesostructure and interactions between different material phases within the framework of multiscale modeling. Employing machine learning methods for computational homogenization enables the reduction of computational effort for the evaluation of the mesostructural behavior while retaining high accuracy. Classically, one unit cell with representative characteristics of the material is chosen for the description of the heterogeneous structure, which presents a simplification of the actual composite. This contribution presents a neural network‐based approach for computational homogenization of composite materials with the ability to consider arbitrary compositions of the mesostructure. Therefore, various statistical volume elements and their respective constitutive responses are evaluated. Thereby, the naturally occurring fluctuation within the composition of the phases can be considered. Different approaches using distinct metrics to represent the arbitrary mesostructures are investigated in terms of required computational effort and accuracy.

Machine learning aided multiscale magnetostatics

Computational material modeling using advanced numerical techniques speeds up the design process and reduces the costs of developing new engineering products. In the field of multiscale modeling, huge computation efforts are expected for modeling heterogeneous materials while trying to reach high accuracy levels. In this work, a machine learning approach, namely the convolutional neural network (CNN), is developed as a solution providing a high level of accuracy while being computationally efficient. The input for the CNN model consists of two/three-dimensional images of artificial periodic and biphasic microstructures in the form of nonoverlapping and overlapping, mono- and polydisperse circular/spherical inclusion systems, which are generated by a random sequential inhibition process. These correspond to Statistical Volume Elements (SVE). Considering linear magnetostatics at the microscale, the output is the apparent permeability of the SVE. Training and testing data for the apparent properties is produced with finite element method-based two-scale asymptotic homogenization. The model efficiency is revealed by employing some representative examples in two and three-dimensional settings. In this regard, the performance of the CNN model is assessed with the applied computational homogenization method relating to the accuracy and computational efficiency. The results with the CNN model show high accuracy in predicting the homogenized permeability and a significant decrease in computation time.

Modeling the statistical distribution of fatigue crack formation lifetime in large volumes of polycrystalline microstructures

Microstructure-scale interactions involving crystallographic texture (orientation and disorientation distributions), distributions of grain shape and size, nearest neighbor grains/phases, etc. in polycrystals can be simulated using the Crystal Plasticity Finite Element Method (CPFEM). Digital statistical volume element (SVE) instantiations that comprise a significant number of grains are analyzed by CPFEM to compute fatigue indicator parameters (FIPs) which are used as surrogate measures of the driving force for fatigue crack formation within the first grain or nucleant phase. The computed maximum FIPs usually increase in magnitude with larger numbers of realistic microstructure instantiations or SVEs analyzed. This work predicts the extreme value distribution (EVD) of the maximum FIPs associated with large engineering components comprised of up to 108 SVEs using a recently developed upscaling scheme, based on statistical information identified from simulations involving only hundreds of SVEs, with each SVE containing nominally 264 grains. This scheme is numerically validated by extensive simulations for samples of duplex Ti-6Al-4V microstructure models with sharp transverse texture strained in two characteristic directions. The size of the training dataset for a reliable prediction is determined for different textures and loading directions based on uncertainty analysis. Finally, the statistical distribution of fatigue crack formation lifetime (FCFL) is correlated with the EVD of the maximum FIPs, facilitating quantitative exploration of the effect of crystallographic texture and sample size on the FCFL.

Elastic and strength properties of statistical volume elements: Determination of isotropic and homogeneous size limits

In this paper, a method is presented to describe the statistical behavior of mesoscale material property fields, which are useful in macroscopic fracture simulation. The use of representative volume element-based properties can result in unsatisfactory fracture patterns because this approach eliminates random variations. In contrast, statistical volume elements can approximate random and inhomogeneous material properties. We compare two material property convergence metrics based on 1) vanishing variations of a property and 2) the convergence of property mean value versus volume element size. We examine the trends with which the properties of a two-phase composite tend toward homogeneous and isotropic limits. Compared to fracture properties, elastic properties reach their homogeneous and isotropic limit at smaller sizes. Accordingly, a volume element size can be chosen such that only the apparent fracture strength remains random and inhomogeneous. This reduces the computational cost and has proven useful in capturing realistic fracture results. In addition, the geometry of the mesoscale partitioning scheme has an important effect on homogenized properties. The intersection of straight edges with inclusions results in nonphysical size effects, increases anisotropy of properties, and results in 30 to 60 times larger representative volume elements compared to a partitioning scheme based on Voronoi homogenization.

Experimental and numerical investigations on the micro-damage behaviour of glass fibre-reinforced plastics

The formation and influence of micro-damage in glass fibre-reinforced plastics (GFRP) is carried out by numerical and experimental investigations. A quasi-static material characterisation is performed and the stiffness and strength of unidirectional, undamaged material in longitudinal and transverse direction as well as in the 23-plane are determined. Since the desired micro-damage could not be generated in unidirectional laminates due to abrupt failure, a [0∘/90∘]s layup has been identified to be most suitable to generate micro-damage in a reproducible way in the 90° layers, which is done using a tension–tension fatigue test program. During these experiments, micro-damage is detected optically and by means of ultrasonic birefringence, where the latter method shows a reduction in shear modulus due to fibre–matrix debonding prior to final inter-fibre failure. The shear modulus degradation is indirectly measured by the change in the velocity of sound. Since this method is limited to measurements in the 13- and 23-plane, all experiments are accompanied with microstructural simulations with detailed statistical volume elements (SVE), to study the influence of micro-damage under various loading conditions and directions. For loading in fibre direction, no influence on the mechanical properties is observed for the investigated fibre–matrix debonding. However, this is different for transverse loading, where a transverse shear stiffness loss of around 8% results in a decrease of transversal stiffness of around 18%, if the load direction is perpendicular to the damage plane. A similar behaviour is observed for the strength, which is reduced by a factor of 2 for the same configuration.

Effect of boundary condition and statistical volume element size on inhomogeneity and anisotropy of apparent properties

We use square Statistical Volume Elements (SVEs) to homogenize elastic and fracture properties of a carbon-nanofiber reinforced polymer. The effect of SVE size and boundary condition (BC) on homogenized properties is investigated. Three different criteria are proposed for the convergence of properties and reaching the Representative Volume Element (RVE) limit: (1) Variation-based criterion examines the trend at which the variation of a property decreases versus SVE size; (2) Mean-based criterion examines the rate at which the SVE-mean of property converges to its terminal value; (3) BC-based criterion examines whether the homogenized properties can be independent of the choice of BC. These criteria show that the geometric property of fiber volume fraction/mass density converges to its homogeneous limit first followed by the elasticity tensor. Fracture strengths have the most complex response as they remain highly inhomogeneous, random, anisotropic, and size-dependent for the SVE sizes considered. This is further discussed in the context of stochastic multiscale material modeling and well-known size effect models for fracture strength. Furthermore, the square shape of the SVE is shown to induce nonphysical bias angles wherein elastic and fracture properties often take their maximum or minimum values at integer multiples of 45 degrees. The traction BC is shown to be superior to displacement BC by having a weaker form of this anisotropy and higher strength values. Moreover, for the anisotropic domain anisotropy indices converge from below and above for traction and displacement BCs, respectively; thus resembling Reuss and Voigt hierarchy of bounds for elastic properties.

On generalized boundary conditions for mesoscopic volumes in computational homogenization

This paper investigates generalized boundary conditions for mesoscopic statistical volume elements (SVEs) in computational homogenization methods. Lagrange multipliers are used to enforce the average values of the macroscopic quantity over the SVE. Spring-like constraints are simultaneously applied at the boundary in order to take into account microscopically perturbed interaction between the mesoscopic volume and its surroundings. This approach provides apparent homogenized parameters, which are located between the results generated by classical uniform kinematic constraining and uniform static loading of the SVE. Since the method combines both these solutions, the term generalized boundary conditions is used. A single parameter with the dimension of length is recognized and adjusted to obtain good homogenization results, even for small SVEs. The method was implemented within the finite element framework and the formula for calculating apparent stiffness tensor, in the convenient form of a single matrix equation, was given. 2D and 3D reference homogenization examples are also provided. The presented approach is especially suitable for random composites, as no assumptions concerning the SVE geometry, such as periodicity, are necessary.

Uncertainty Quantification for Microstructure-Sensitive Fatigue Nucleation and Application to Titanium Alloy, Ti6242

Microstructure of polycrystalline materials has profound effects on fatigue crack initiation, and the inherent randomness in the material microstructure results in significant variability in fatigue life. This study investigates the effect of microstructural features on fatigue nucleation life of a polycrystalline material using an uncertainty quantification framework. Statistical volume elements (SVE) are constructed, where features are described as probability distributions and sampled using the Monte Carlo method. The concept of SVE serves as the tool for capturing the variability of microstructural features and consequent uncertainty in fatigue behavior. The response of each SVE under fatigue loading is predicted by the sparse dislocation density informed eigenstrain based reduced order homogenization model with high computational efficiency, and is further linked to the fatigue nucleation life through a fatigue indicator parameter (FIP). The aggregated FIP and its evolution are captured using a probabilistic description, and evolve as a function of time. The probability of fatigue nucleation is measured as the probability that the predicted FIP exceeds the local critical value which represents the ability of material to resist the fatigue load. The proposed framework is implemented and validated using the fatigue response of titanium alloy, Ti-6Al-2Sn-4Zr-2Mo (Ti-6242).

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.

Geometric partitioning schemes to reduce modeling bias in statistical volume elements smaller than the scale of isotropic and homogeneous size limits

In predicting failure of a material, small scale material properties are critical, and yet small scale random heterogeneities are difficult to characterize accurately and efficiently. In building fracture models, work has focused on statistical characterization of local random microstructure at small to intermediate scales (termed micro- or mesoscale). Many materials can be characterized as effectively isotropic and homogeneous at a large scale. In contrast, at a micro- or mesoscale, Statistical Volume Elements (SVE) often show significant anisotropy and heterogeneity. Characterizing the properties of SVE presents challenges because modeling choices, such as the length scale, boundary conditions applied, and boundary shape chosen, may introduce modeling bias.Modeling bias must be separated from the actual prediction of anisotropic and heterogeneous characteristics at the SVE level, in order to provide an accurate basis for fracture simulation. Much literature has been devoted to characterizing the size of an RVE in order to effectively homogenize a material, however, there is less study of the effective isotropic limit. The representative size may be different when approaching the homogeneous limit or the isotropic limit.In this work, elastic and strength properties will be evaluated at multiple scales using SVE with square and circular (so-called “regular”) geometry. A Voronoi tessellation based geometry, where aggregation of Voronoi cells is achieved using square or circular grids, is studied and compared with regular partitioning methods. Models will be tested on two types of microstructures, one that is isotropic at the macroscale, and one which contains a slight directional bias. Results for different SVE geometries will be compared to demonstrate how SVE modeling choices affect the characterization of anisotropy and heterogeneity at the micro- and mesoscale.

Automatic generation of statistical volume elements using multibody dynamics and an erosion-based homogenization method

Modeling particle based heterogeneous materials using statistical volume elements (SVE) for predicting its mechanical behavior can be tedious when the particles are densely packed or elongated. Positioning particles without creating overlaps and avoiding meshing problems are two obstacles frequently mentioned. To counter these obstacles, a new modeling methodology based on multibody dynamics (MBD) and on an erosion-based homogenization method is proposed. The CAD model of a SVE is first generated and particles causing meshing problems are excluded. Meshing and finite element analysis are automatically carried out and a subsequent erosion-based homogenization method is used to retrieve the macroscopic behavior of the SVE. To illustrate the potential of this new method, results obtained with a random sequential adsorption algorithm on non-eroded SVEs are compared with results obtained from the same SVEs submitted to our erosion method. These results are then compared with results obtained using the new MBD based approach.

Experimental-numerical analysis of microstructure-property linkages for additively manufactured materials

The innovation of new or improved products fabricated from additive manufacturing processes with desired properties depends on a multitude of trials as stated by the National Science and Technology Council (2011). Therefore, a systematic approach is essential to accelerate materials development. This can be realised by developing systematic materials knowledge in the form of process-structure-property relationships. In this envisioned framework, the present work aims to derive the structure-property linkages of additively manufactured Ti-6Al-4V alloy. The main focus is to investigate the influence of potential defects, in the form of pores, inherited from the fabrication process on the fatigue properties. The complicated polycrystalline microstructure, including porosity at a microscale, is obtained by processing light microscopy and x-ray computed tomography measurements. A detailed statistical analysis is performed to obtain a low-dimensional representation of the structure. Based on these statistical measures, a suitable reconstruction algorithm is developed to create pore distributions that are incorporated into synthetic statistical volume elements (SVEs) generated from DREAM.3D by Groeber and Jackson (2014). Using these SVEs, microscale crystal plasticity simulations in DAMASK, see Roters et al. (2019), are performed to obtain the material properties such as yield strength and fatigue indicator parameters (FIPs). A detailed numerical analysis is carried out to study the influence of pore statistics such as size distribution or porosity fraction. Data analysis is carried out to rank-order the SVEs based on FIPs. Furthermore, a comparison with Murakami’s empirical square root area concept is made.

Micro-mechanical analysis on random RVE size and shape in multiscale finite element modelling of unidirectional FRP composites

Developing a high-fidelity stochastic multi-scale modelling framework that can capture the structure–property relationship in a computationally efficient manner is of interest to the structural engineers working with composite structures. This work revisits the problem of defining representative volume elements (RVEs) for performing a combined computational homogenisation-dehomogenisation procedure to estimate critical material allowables of unidirectional fibre reinforced polymer (FRP) composites. Particularly, the size and shape effects of 3D RVEs are investigated by performing Monte Carlo simulation of RVE arrangements to analyse the statistical convergence of their geometric features and the predicted structural properties in comparison with commonly used unit-cell models. For the microstructural features adopted in this work, it has been demonstrated that appropriately sized RVE models in two different aspect ratios could be considered as 3D statistical volume elements (SVEs) to efficiently capture the properties of interest. Attention has also been drawn to the implications of domain shapes and aspect ratios when small SVE models are used in the modelling framework for regulating computational power.

Scaling in Anti-Plane Elasticity on Random Shear Modulus Fields with Fractal and Hurst Effects

The scale dependence of the effective anti-plane shear modulus response in microstructures with statistical ergodicity and spatial wide-sense stationarity is investigated. In particular, Cauchy and Dagum autocorrelation functions which can decouple the fractal and the Hurst effects are used to describe the random shear modulus fields. The resulting stochastic boundary value problems (BVPs) are set up in line with the Hill–Mandel condition of elastostatics for different sizes of statistical volume elements (SVEs). These BVPs are solved using a physics-based cellular automaton (CA) method that is applicable for anti-plane elasticity to study the scaling of SVEs towards a representative volume element (RVE). This progression from SVE to RVE is described through a scaling function, which is best approximated by the same form as the Cauchy and Dagum autocorrelation functions. The scaling function is obtained by fitting the scaling data from simulations conducted over a large number of random field realizations. The numerical simulation results show that the scaling function is strongly dependent on the fractal dimension D, the Hurst parameter H, and the mesoscale δ, and is weakly dependent on the autocorrelation function. Specifically, it is found that a larger D and a smaller H results in a higher rate of convergence towards an RVE with respect to δ.

Simulated effects of sample size and grain neighborhood on the modeling of extreme value fatigue response

Assessing the size of representative volume elements (RVEs) for fatigue-related applications is challenging. A RVE relevant to random microstructure requires a volume of material that is sufficiently large to capture the grain/phase heterogeneity that captures all statistical moments of the distribution of the driving force for fatigue crack formation at “hot spot” grains. Consequently, the large size of a microstructure RVE required to study fatigue phenomena is largely computationally intractable and difficult to explore. A more realistic objective in this work is to systematically study, as a function of the size of a statistical sample of microstructure, trends towards convergence of the simulated distribution of driving force for fatigue crack formation. The present work accordingly leverages the recently developed open-source PRISMS-Fatigue framework [Yaghoobi et al., npj Comput. Mater., 7, 38 (2021)] to examine the trends in convergence of extreme value distributions (EVD) of Fatigue Indicator Parameters (FIPs) in progressively larger polycrystalline microstructure realizations of FCC Al alloy 7075-T6 using crystal plasticity finite element method simulations. The results are compared to the traditional method in which ensembles of statistical volume elements (SVEs) are simulated to build up statistics intended to approximate those associated with a larger volume of material. The convergence of EVDs with increase of size of a SVE of microstructure is closely related to the extent of grain nearest neighbor (NN) interactions. Accordingly, the sensitivity of the local micromechanical response at hot spot grains is quantitatively investigated by systematically varying the orientations of NN grains. Results indicate that SVEs with cubic crystallographic texture tend towards convergence of the EVD of FIPs with tens of thousands of grains while the random and rolled textures require larger volumes. Simple relationships based on microstructure parameters (e.g., Schmid Factor, grain size, NN misorientation) do not completely correlate to fatigue hot spot grains. Finally, the sensitivity of the extreme value fatigue response at hot spot grains extends to the 3rd NN when a single neighborhood grain orientation is altered.

Computational homogenisation based extraction of transverse tensile cohesive responses of cortical bone tissue.

The numerical assessment of fracture properties of cortical bone is important in providing suggestions on patient-specific clinical treatments. We present a generic finite element modelling framework incorporating computational fracture approaches and computational homogenisation techniques. Finite element computations for statistical volume elements (SVEs) at the microscale are performed for different sizes with random osteon packing with a fixed volume fraction. These SVEs are loaded in the transverse direction under tension. The minimal SVE size in terms of ensuring a representative effective cohesive law is suggested to be 0.6 mm. Since cement lines as weak interfaces play a key role in bone fracture, the effects of their fracture properties on the effective fracture strength and toughness are investigated. The extracted effective fracture properties can be used as homogenised inputs to a discrete crack simulation at macroscopic or structural scale. The extrinsic toughening mechanisms observed in the SVE models are discussed with a comparison against experimental observations from the literature, giving beneficial insights to cortical bone failure.