Statistical Volume Elements Research Articles

Computational Determination of Macroscopic Mechanical and Thermal Material Properties for Different Morphological Variants of Cast Iron

The sensitivity of macroscopic mechanical and thermal properties of grey cast iron is computationally investigated for a variety of graphite morphologies over a wide temperature range. In order to represent common graphite morphologies according to EN ISO 945-1, a synthetic approach is used to algorithmically generate simulation domains. The developed mechanical and thermal model is applied in a large simulation study. The study includes statistical volume elements of the graphite morphology classes GJL-150 and IA2 to IA5, with 10, 11 and 12 v.−% of graphite precipitations, respectively, for a temperature range from 20 to 750 °C. Homogenised macroscopic quantities, such as the Young’s moduli, Poisson’s ratios, yield strengths and thermal conductivities, are predicted for different morphology classes by applying simulation and data analysis tools of the research data infrastructure Kadi4Mat. This is the first work to determine the mechanical and thermal properties of the morphology classes defined in EN ISO 945-1.

The effects of heterogeneous mechanical properties on the response of a ductile material

We investigate numerically the small-strain, elastic–plastic response of statistically isotropic materials with non-uniform spatial distributions of mechanical properties. The numerical predictions are compared to simple bounds derived analytically. We explore systematically the effects of heterogeneity on the macroscopic stiffness, strength, asymmetry, stability and size dependence. Monte Carlo analyses of the response of statistical volume elements are conducted at different strain triaxiality using computational homogenisation, and allow exploring the macroscopic yield behaviour of the heterogeneous material. We illustrate quantitatively how the pressure-sensitivity of the yield surface of the solid increases with heterogeneity in the elastic response. We use the simple analytical models developed here to derive an approximate scaling law linking the fatigue endurance threshold of metallic alloys to their stiffness, yield strength and tensile strength.

Accessing pore microstructure–property relationships for additively manufactured materials

AbstractUnderstanding structure–property (SP) relationships is essential for accelerating materials innovation. Still being in the state of ongoing research and development, this is especially true for additive manufacturing (AM) in which process‐induced imperfections like pores and microstructural variations significantly influence the material's properties. That is why, the present work aims at proposing an approach for accessing pore SP relationships for AM materials. For this purpose, crystal plasticity (CP) simulations on reconstructed domains based on experimental measurements are employed to allow for a microstructure‐sensitive investigation. For the considered Ti–6Al–4V specimen manufactured by laser powder bed fusion, the microstructure and pore characteristics are obtained by utilizing light microscopy and X‐ray computed tomography at the microscale. Employing suitable statistical analysis and reconstruction, statistical volume elements with reconstructed pore distributions are created. Using them, microscale CP simulations are performed to obtain fatigue indicating parameters. Employing a further statistical analysis, fatigue ranking parameters are derived for a comparison of different microstructures. Additionally, a comparison with the empirical Murakami's square root area concept is made. Results from first numerical studies underline the potential of the approach for understanding and improving AM materials.

Analysis of Fatigue Indicator Parameters for Ti-6Al-4V microstructures using extreme value statistics in the transition fatigue regime

Extreme value (EV) statistics have been previously utilized to model the tails of distributions of computed Fatigue Indicator Parameters (FIPs) to provide a method of ranking microstructures in terms of resistance to fatigue crack formation under identical loading conditions. FIPs that capture the two primary transgranular mechanisms for fatigue crack formation and early stages of growth in bimodal Ti-6Al-4V are computed for cyclic strain-controlled loading conditions corresponding to transition fatigue where there is perceptible macroscopic plastic cyclic strain, but the polycrystal cyclic elastic strain range dominates the cyclic plastic strain range. A statistical volume element (SVE) ensemble approach is employed, in which a large number of microstructure samples are constructed for simulation. The peaks-over-threshold method is used to identify extreme values within the FIP distributions which are then fit to the Generalized Pareto Distributions, enabling rank-ordering of various microstructures with regard to resistance to fatigue crack formation. Microstructure attributes including the grain size, orientation, and nearest neighbor arrangements are quantified in the neighborhood of FIP ‘hot spots’.

Statistical Analysis of Mechanical Stressing in Short Fiber Reinforced Composites by Means of Statistical and Representative Volume Elements

Analyzing representative volume elements with the finite element method is one method to calculate the local stress at the microscale of short fiber reinforced plastics. It can be shown with Monte-Carlo simulations that the stress distribution depends on the local arrangement of the fibers and is therefore unique for each fiber constellation. In this contribution the stress distribution and the effective composite properties are examined as a function of the considered volume of the representative volume elements. Moreover, the influence of locally varying fiber volume fraction is examined, using statistical volume elements. The results show that the average stress probability distribution is independent of the number of fibers and independent of local fluctuation of the fiber volume fraction. Furthermore, it is derived from the stress distributions that the statistical deviation of the effective composite properties should not be neglected in the case of injection molded components. A finite element analysis indicates that the macroscopic stresses and strains on component level are significantly influenced by local, statistical fluctuation of the composite properties.

A combined experimental–numerical approach for permeability characterization of engineering textiles

AbstractExtensive experimental test programs are required for the characterization of textile permeability, which is essential for the design of liquid composite molding processes. In this study, an experimental–numerical approach is presented, aiming to partially substitute experiments by numerical simulations. This approach uses 3D microscale simulations to evaluate the permeability within rovings and attribute these values to statistical volume elements of the textile at the mesoscale. To further improve accuracy, a calibration method has been defined. In order to validate the functionality of this approach the permeability of a glass fiber woven and a non‐crimp fabric at fiber volume contents (FVCs) between 50 and 60% were predicted. For the non‐calibrated but virtually compacted models with FVCs of 55 and 60%, the deviation ranges from −27% to +42%. This seems acceptable considering the typical scatter in experimental tests, for example, a CV of 20–50% was measured in recent permeability benchmarks. The numerically determined permeability was then used as input for numerical filling simulations at part level (macroscale). The resulting filling times were then compared to results of simulations based on experimental input values.

Investigation of the forming behavior of carbon fiber yarns on microscopic scale with detailed statistical volume elements

Simulations at microscopic scale are subject of research in many fields of continuous fiber reinforced materials. They allow the virtual characterization with clearly defined boundary conditions and the investigation of effects that cannot easily be observed in experimental tests, such as crack initiation. While the influence of differently detailed modeling, such as variable fiber radii, is well investigated for consolidated composite materials, that influence is not well investigated for dry yarns. The effect of different modeling aspects on the shearing of statistical volume elements (SVE) is object of this study. Those aspects are in particular the SVE’s size and side ratio, the deformation velocity, the friction coefficient and the initial stress of the elements. The investigations have shown that the deformation of the SVEs is very sensitive to changes of the named parameters. Especially parameters that influence the number of fiber contacts or the friction forces have a noticeable effect on the shear stress. In addition, changes in the velocity have a significant impact on the deformation. Therefore, it is necessary to characterize the fiber and yarn parameters precisely and transfer them into the simulation model to obtain a realistic deformation behavior.

Application of statistical functions to the numerical modelling of ceramic foam: From characterisation of CT-data via generation of the virtual microstructure to estimation of effective elastic properties

This paper illustrates a numerical methodology to quantify and recreate the microstructure of a ceramic foam using statistical functions and physical descriptors (together referred to as microstructure descriptors). The microstructure descriptors were employed to characterize a real foam material obtained from X-ray tomography (CT) scans. The statistical functions were further used to determine an appropriate size of a statistical volume element (SVE) within the foam sample to be further used in numerical determination of effective elastic properties. A ranking method was also developed to reduce the number of realizations of SVEs used in this calculation. Further, a novel reconstruction procedure was developed to synthesize artificial microstructure of the foam material that had the same microstructure descriptors as the real foam sample. Effective elastic properties of these artificial microstructures were determined as well. Comparison of these properties with experimental results showed that the statistical functions can be used to minimize the computational effort required for determining effective elastic properties of real microstructures. Further, the reconstruction methodology generates microstructure that matches the real microstructure not only in terms of its microstructural descriptors but also in terms of the effective elastic material properties.

Random Material Property Fields of 3D Concrete Microstructures Based on CT Image Reconstruction.

The purpose of this paper is to determine the random spatially varying elastic properties of concrete at various scales taking into account its highly heterogeneous microstructure. The reconstruction of concrete microstructure is based on computed tomography (CT) images of a cubic concrete specimen. The variability of the local volume fraction of the constituents (pores, cement paste and aggregates) is quantified and mesoscale random fields of the elasticity tensor are computed from a number of statistical volume elements obtained by applying the moving window method on the specimen along with computational homogenization. Based on the statistical characteristics of the mesoscale random fields, it is possible to assess the effect of randomness in microstructure on the mechanical behavior of concrete.

Effect of microstructural variations on the failure response of a nano-enhanced polymer: a homogenization-based statistical analysis

Statistical Volume Elements (SVEs) are employed to evaluate homogenized mesoscopic ductile failure response of a carbon nanofiber reinforced composite under uniaxial tensile and compressive loadings. In the mesoscale analysis, after virtual reconstruction of the material microstructure, 2D finite element models are generated for each SVE using a non-iterative meshing algorithm named CISAMR, which fully automates the modeling process. The ductile damage response of each SVE is then simulated to derive its homogenized stress-strain curve. Corresponding initiation, maximum, and failure turning points, together with local fiber volume fraction and homogenized bulk modulus, are defined as mesoscopic Quantities of Interest (QoIs). While compressive loadings have slightly higher strength and nearly twice higher strains at failure compared to similar tensile cases, both loadings yield similar coefficients of variance for most QoIs. Further, a stochastic bulk damage model is calibrated from mesoscopic responses, which takes bilinear and linear forms versus strain for tensile and compressive loadings, respectively. Finally, cross-correlations are made between different QoIs, showing lower strain-based QoIs and higher strengths correspond to either higher local fiber volume fractions or more fibers being aligned with the loading direction.

Representative cell modeling strategy of 2.5D woven composites considering the randomness of weft cross-section for mechanical properties prediction

In view of the large difference between the traditional unit-cell model (UCM) and the actual microstructure of 2.5D woven composites, which will affect the mechanical properties and failure behavior, a representative volume element (RVE) modeling strategy considering the randomness of weft cross-section (WFCS) is developed, and the established model is called statistical volume element (SVE) in this work. Based on the microscopy observation of warp cross-section (WPCS) and WFCS, some geometric parameters are counted, and thereby the SVE can be established. Then, the mechanical properties of 2.5D woven composites, including the stiffness and strengths in the warp and weft directions, are predicted by the SVE and UCM, respectively. Compared with the experimental results, the maximum errors predicted by the SVE and UCM are 3.84% and 15.23%, respectively. Therefore, the mechanical properties predicted by the SVE are more accurate than those predicted by the UCM. Furthermore, considering the prediction accuracy and calculation cost comprehensively, the authors suggest that the SVE and UCM should be adopted for the warp-loaded and weft-loaded progressive damage process, respectively. This work provides a method for the RVE modeling of 2.5D woven composites considering randomness, which can be extended to the modeling of other textile composites.

Effects of algorithmic simulation parameters on the prediction of extreme value fatigue indicator parameters in duplex Ti-6Al-4V

Fatigue Indicator Parameters (FIPs) serve as surrogate measures of the driving force to form and grow transgranular fatigue cracks in metals and alloys. FIPs are volume averaged over sub-grain regions using a crystal plasticity finite element method (CPFEM) model of polycrystalline microstructures. Questions naturally arise regarding the ability to computationally simulate a sufficient volume of microstructure samples to objectively build a converged estimate of the distribution of maximum FIPs above some threshold, referred to as an Extreme Value Distribution (EVD). A reliable EVD can be estimated by considering FIPs computed from samples comprised of a sufficient number of grains. A recently developed statistical technique is exploited in this work to predict the maximum FIPs in a large volume of duplex Ti-6Al-4V. Effects of several crucial algorithmic parameters in the CPFEM simulations are explored, including the sub-grain FIP averaging volume, the statistical volume element (SVE) size of each microstructure instantiation, number of SVEs necessary to project extreme value FIPs for larger volumes, and finite element mesh density. The SVE size defines the number of grains in a single simulation and therefore controls the number of nearest and second nearest neighbor grain interactions that dominate FIP EVDs. This interplays closely with the number of SVEs needed to establish reliable estimates of FIP EVDs for larger volumes of microstructure. We demonstrate that the number of grains sampled is a critical consideration in these types of CPFEM simulations aimed at EVDs. The statistical technique for estimation of EVD FIPs is relatively insensitive to both the SVE size and mesh density of a single simulation, provided a sufficient number of individual grains are sampled to capture dominant effects of microstructure heterogeneity. Furthermore, averaging FIPs over different sub-grain volumes results in consistent predictions of the maximum FIPs.

Comparison of lamellar nanostructured cubical and spherical statistical volume elements in terms of crystallographic texture and deformation behaviour

This study presents the numerical homogenization of spherical and cubical statistical volume element (SVE). Numerical homogenization with periodic boundary conditions for spherical and cubical SVEs has been employed. The lamellar structure of eutectic Copper (Cu)-Silver (Ag) has been focused for the investigation of deformation behaviour and texture evolution. The initial texture for cold-drawn eutectic Cu-Ag composite has been approximated by a set of 200 discrete orientations. The comparison of deformation behaviour and the crystallographic texture has been investigated for two different SVEs. It is demonstrated that the SVEs with cubical shape induce an anisotropy when compared with the spherical SVEs. The numerical results for cubical and spherical SVEs in terms of deformation behaviour and crystallographic texture are in good agreement with the experimental observations.

A probabilistic upscaling of microstructural randomness in modeling mesoscale elastic properties of concrete

We present a modeling framework for propagating the uncertainty, stemming from random heterogeneity of concrete microstructure, to its mesoscale elastic properties at the level of statistical volume element. The microstructure of concrete is modeled as a three-phase composite, consisting of aggregates (inclusions) in mortar (matrix), and interfacial transition zone (ITZ), a thin high porosity zone surrounding aggregates. There is inherent randomness in the composition and mechanical properties of these constituents, which in turn causes spatial heterogeneity in the local material behavior. We present a statistical characterization process to quantify the uncertainty associated with local aggregate volume fraction and aggregate size distribution. We then adopt and test a probabilistic model based on the random matrix theory and maximum entropy principle to construct a stochastic description for bounded apparent elasticity tensors at the mesoscale level, in which the bounds are obtained through a computational calibration process based on the theory of micromechanics and numerical homogenization. The framework provides an efficient way to model and simulate the local statistical fluctuations of material properties that are linked to phenomena occurring at the subscale level of heterogeneity.

Effects of microstructure on high cycle fatigue properties of dual-phase Ti alloy: combined nonlocal CPFE simulations and extreme value statistics

The nonlocal crystal plasticity finite element (CPFE) simulations and extreme value statistics were combined to study the effects of microstructure on the high cycle fatigue (HCF) behavior of dual-phase Ti alloy. A modified Armstrong-Frederick nonlinear kinematic hardening equation accounting for cyclic softening effect was employed in the crystal plasticity constitutive model. Three-dimensional equiaxed microstructure models and two-dimensional duplex microstructure models with real lamellar structure were generated based on Voronoi method, serving as statistical volume elements (SVEs). The effects of morphological and crystallographic features, including grain size, grain orientation, phase volume fraction and lamellae width, on the fatigue performance were investigated. By simulating multiple SVEs, extreme value distributions of the driving force for fatigue crack formation were predicted using Fatemi-Socie (FS) parameter as a fatigue indicator parameter (FIP). Meanwhile, whether the extreme values of geometrically necessary dislocation (GND) density can be taken as a FIP was discussed with compared to FS FIP. The GND density, related to local stress and strain gradient, has considerable potential as a FIP to estimate the fatigue performance of titanium alloys. Based on simulated results, it is suggested that microstructure with small grain size, low volume fraction of primary α grains, and thinner α lamellae width have the lowest probability of crack formation during HCF.

Uncertainty propagation in reduced order models based on crystal plasticity

In this work, an uncertainty propagation study is performed based on simulated ensembles of statistical volume elements (SVE) used to inform a reduced order internal state variable model, focusing on model form and uncertainties in numerical simulations. A rate-dependent polycrystal plasticity finite element model (CP-FEM) of cartridge brass is calibrated to room-temperature uniaxial tension testing data of an annealed sample. Model forms are considered with and without the inclusion of back stress in the CP model. Three sizes of SVE are explored for simulation uncertainty effects. Effects are explored with regard to the parameters of a Bammann–Chiesa–Johnson (BCJ) macroscopic viscoplasticity model, calibrated element-wise to SVEs from an ensemble modeled using CP-FEM simulations exploring quasi-static uniaxial tension. Variability in stress–strain response at multiple defined length-scales is measured. The selection of model form is discussed. The numerical simulation uncertainty of the model reduction process is quantified.

A computational and experimental comparison on the nucleation of fatigue cracks in statistical volume elements

The failure of micron-scale metallic components presents significant variability as a result of their size being comparable to microstructural length scales. Indeed, these components do not represent the bulk of the material but correspond to statistical volume elements (SVEs). This work investigates the role of SVEs on fatigue crack nucleation with a novel comparison between microbeam experiments and microstructure-sensitive simulations. We recreate multiple microstructural computational realizations to estimate fatigue crack nucleation lives and orientations by means of physics-based crystal plasticity models. We demonstrate a unique approach to validate microstructure sensitive models and quantify the fatigue crack stochasticity associated with small volumes.

Prediction of maximum fatigue indicator parameters for duplex Ti–6Al–4V using extreme value theory

Fatigue Indicator Parameters (FIPs) based on the cyclic plastic strain are used as surrogate measures of the driving force for fatigue crack formation. For a given microstructure, the Extreme Value Distribution (EVD) of FIPs can be populated using results of a number of digital Statistical Volume Element (SVE) instantiations analyzed by the crystal plasticity finite element method. The number of microstructure instantiations affects the maximum FIPs computed. To predict the maximum FIPs in a large volume of material using simulation results from a limited number of SVEs, we proposed a statistical approach based on extreme value theory. The predicted maximum FIP values are compared directly to simulation results of 1000 SVEs to validate the proposed method for duplex Ti–6Al–4V. It is shown that simulations of only 100 SVEs suffice to identify the statistical information for a reliable prediction of the maximum FIPs in polycrystalline duplex Ti–6Al–4V with initial random texture.

Reliability increase of masonry characteristics estimation by a sampling algorithm applied to thermographic digital images

In this paper the estimation of masonry characteristics by means of thermographic images, enhanced by sampling Kantorovich algorithm, is taken into account. In particular, the convergence of the Statistical Volume Element (SVE) to the Representative Volume Element (RVE) is analyzed. It is found that the enhancement, obtained by the proposed procedure, allows a faster convergence of SVE to RVE and a reduced coefficient of variation of the estimates obtained using a partition of the image with windows of smaller dimensions. Moreover, the effects of the uncertainties in the parameters involved in the reconstruction of texture from thermographic image, such as those used in morphological operators employed in digital image processing, involving the environmental conditions of the samples and the properties of the kernel utilized in the image enhancement algorithm, have been studied in order to assess their influence on the estimated mechanical characteristics. The performed analyses contribute to increase the reliability of the thermography as tool for identifying of masonries covered with plaster or frescoes, which is a very frequent case in vulnerability analysis of historical buildings.

A Three-Dimensional Statistical Volume Element for Histology Informed Micromechanical Modeling of Brain White Matter.

This study presents a novel statistical volume element (SVE) for micromechanical modeling of the white matter structures, with histology-informed randomized distribution of axonal tracts within the extracellular matrix. The model was constructed based on the probability distribution functions obtained from the results of diffusion tensor imaging as well as the histological observations of scanning electron micrograph, at two structures of white matter susceptible to traumatic brain injury, i.e. corpus callosum and corona radiata. A simplistic representative volume element (RVE) with symmetrical arrangement of fully alligned axonal fibers was also created as a reference for comparison. A parametric study was conducted to find the optimum grid and edge size which ensured the periodicity and ergodicity of the SVE and RVE models. A multi-objective evolutionary optimization procedure was used to find the hyperelastic and viscoelastic material constants of the constituents, based on the experimentally reported responses of corpus callosum to axonal and transverse loadings. The optimal material properties were then used to predict the homogenized and localized responses of corpus callosum and corona radiata. The results indicated similar homogenized responses of the SVE and RVE under quasi-static extension, which were in good agreement with the experimental data. Under shear strain, however, the models exhibited different behaviors, with the SVE model showing much closer response to the experimental observations. Moreover, the SVE model displayed a significantly better agreement with the reports of the experiments at high strain rates. The results suggest that the randomized fiber architecture has a great influence on the validity of the micromechanical models of white matter, with a distinguished impact on the model's response to shear deformation and high strain rates. Moreover, it can provide a more detailed presentation of the localized responses of the tissue substructures, including the stress concentrations around the low caliber axonal tracts, which is critical for studying the axonal injury mechanisms.