Equivalent Representative Volume Element Research Articles

Crystal plasticity enhanced direct cyclic analysis of cyclic behaviour of LPBF-manufactured AISI 316L

The fatigue performance of Laser Powder Bed Fusion manufactured 316L stainless steel is critical for cyclic loading applications. Additive manufacturing of 316L produces a characteristic microstructure consisting of grains with a fine cellular structure within. This microstructure shows a comparatively high fatigue strength combined with a high defect tolerance. The purpose of this paper is to numerically analyse the local response of this microstructure to cyclic loading and, in particular, the joint action between lack of fusion defects and the surrounding grain structure. The traditional incremental analysis method for polycrystalline alloys is computationally expensive. Therefore, a direct cyclic analysis combined with a phenomenological crystal plasticity model is used in this work to predict the stable cyclic response of the material, resulting in a much lower computational cost. Three statistically equivalent representative volume elements are generated. The considered crystal plasticity parameters are calibrated using experimental tensile stress–strain curves. The comparison with incremental analysis demonstrates the accuracy and efficiency of the proposed method. The shakedown limit, based on the maximum load that allows the convergent cyclic response, significantly depends on the local grain orientation around lack of fusion defects, revealing a reason for the large scatter in the fatigue strength.

Macroscopic compressive strength study of historical grey bricks based on microscopic scale

Determining the macroscopic compressive strength of original grey bricks in historical structures has become a crucial topic due to rules and legislation about the preservation of cultural artifacts and illnesses like weathering. This work aims to provide a microscale approach for estimating the macroscopic compressive strength of historical grey bricks. This method, based on nanoindentation and mercury intrusion porosimetry, divides the historical grey bricks into two representative volume elements (RVEs) at the observation scale with continuous transition: A microscale Mori-Tanaka equivalent RVE encompassing microstructural features, which nanoindentation, X-ray diffraction, scanning electron microscopy, mercury intrusion porosimetry, and other tests obtain the features. A mesoscale RVE representing the two-phase model of a "porous inclusions-equivalent matrix." Then, the prediction of macroscopic compressive strength of the historical grey brick is realized by the simulation of the uniaxial compression of the mesoscale RVE. For the three types of historical grey bricks (A, B, and C), the errors in the predictions obtained by the multiscale micromechanical models, in comparison to the macroscopic experimental results, are 13.67%, 10.34%, and 5.22%, respectively. Compared to conventional mechanical experiments and non-destructive methods, this approach exhibits comprehensive information acquisition, minimal influence from human factors, and minimal subject disturbance. It is well-suited for obtaining performance parameters of intrinsic materials in historical buildings, grottoes, nuclear power plants, etc.

COUPLED CRYSTAL PLASTICITY PHASE-FIELD MODEL FOR DUCTILE FRACTURE IN POLYCRYSTALLINE MICROSTRUCTURES

A wavelet-enriched adaptive hierarchical, coupled crystal plasticity, phase-field finite element model is developed in this work to simulate crack initiation and propagation in complex polycrystalline microstructures. The model accommodates initial material anisotropy and crack tension-compression asymmetry through orthogonal decomposition of stored elastic strain energy into tensile and compressive counterparts. The crack evolution is driven by stored elastic and defect energies, resulting from slip and hardening of crystallographic slips systems. A finite element model is used to simulate the fracture process in a statistically equivalent representative volume element reconstructed from electron back-scattered diffraction scans of experimental microstructures. Multiple numerical simulations with the model exhibits microstructurally sensitive crack propagation characteristics.

Self-consistent homogenization-based parametrically upscaled continuum damage mechanics model for composites subjected to high strain-rate loading

This paper develops a Parametrically Upscaled Continuum Damage Mechanics (PUCDM) model for carbon fiber/epoxy matrix composites subjected to high strain-rate loading. The PUCDM model for predicting high strain-rate induced deformation and damage evolution at the macroscopic scale, explicitly incorporates microstructural morphology and micro-inertia in its constitutive coefficients. It is developed using self-consistent homogenization that uses a concurrent multiscale model to account for the effect of micro-inertia and stress wave interaction with the microstructure. The concurrent model embeds a statistically equivalent representative volume element (SERVE) of the microstructure in an exterior domain, whose constitutive and damage response are described by the PUCDM model. Micromechanical modeling for the composite microstructure involves a unified cohesive zone enhanced phase-field damage model for fiber-matrix interface debonding, fiber breakage, and matrix cracking. The PUCDM model parameters for the upscaled domain are determined through micro-macro-scale energy equivalence, along with traction reciprocity and displacement continuity at the SERVE-exterior domain interface that are enforced using a Lagrange multiplier-based approach. Simulations using the concurrent model analyze stress wave propagation and damage evolution in composite microstructures with three different fiber volume fractions at multiple strain rates in the range of 102 − 105 s−1. The fiber volume fraction and spatial distributions affect the overall stiffness and damage evolution in the PUCDM model. Results of the analysis demonstrate the need for representation of micro-inertia, strain rates and microstructure morphology in the PUCDM-based stiffness and damage model parameters for high strain-rates above [Formula: see text].

Elastic stress transfer model and homogenized constitutive equation for the multi-phase geomaterials

Multi-phase geomaterials can be classified as the particulate composites composed of finer matrix and dispersed inclusions. With the extremely nonuniform stress and strain field, prediction of the overall mechanical response from the known mechanical properties and mixed proportion of constituents is a challenging task for multi-phase geomaterials. In this paper, using a concentric composite cylinder as the equivalent representative volume element (RVE), the analytical solutions of stress transfer model are derived from the modified shear-lag analysis. Through the mechanical homogenization of internal stress/strain field, a constitutive equation has been established in the elastic regime, containing a closed-form solution (S-L method) for prediction of effective Young's modulus (EHom). By the comparison with classical models, numerical simulations and experimental results, the homogenized constitutive equation (EHom) has been validated. Meanwhile, parametric studies of proposed solutions have been conducted to discuss the effect of meso-mechanical variables (e.g., mechanical contrast and volume proportion of constituents) on the stress transfer mechanism and the overall mechanical response, providing the useful insights and explanations to previous studies.

A Comparison of Statistically Equivalent and Realistic Microstructural Representative Volume Elements for Crystal Plasticity Models

Two methods used to construct a microstructural representative volume element (RVE) were evaluated for their accuracy when used in a crystal plasticity-based finite element (CP-FE) model. The RVE-based CP-FE model has been shown to accurately predict the complete tensile stress–strain response of a Ti–6Al–4V alloy manufactured by laser powder bed fusion. Each method utilized a different image-based technique to create a three-dimensional (3D) RVE from electron backscatter diffraction (EBSD) images. The first method, referred to as the realistic RVE (R-RVE), reconstructed a physical 3D microstructure of the alloy from a series of parallel EBSD images obtained using serial-sectioning (or slicing). The second method captures key information from three orthogonal EBSD images to create a statistically equivalent microstructural RVE (SERVE). Based on the R-RVEs and SERVEs, the CP-FE model was then used to predict the complete tensile stress–strain response of the alloy, including the post-necking damage progression. The accuracy of the predicted stress–strain responses using the R-RVEs and SERVEs was assessed, including the effects of each microstructure descriptor. The results show that the R-RVE and the SERVE offer comparable accuracy for the CP-FE purposes of this study.

A Step-by-Step Equivalent Microprediction Method for the Mechanical Properties of Composite Solid Propellants considering Dewetting Damage

Reliable prediction of the macromechanical properties of composite solid propellants in the microscale can accelerate the development of propellants with high mechanical properties. According to the characteristics of the composition ratio of a four-component hydroxyl-terminated polybutadiene (HTPB) propellant with the component ammonium perchlorate (AP), hydroxyl-terminated polybutadiene, aluminum powder (AL), and cyclotrimethylenetrinitramine (or RDX for short), an improved random delivery algorithm was developed to separately model filler particles with the different sizes. A step-by-step equivalent representative volume element (RVE) model was generated to reflect the microstructures of the propellant. The isotropy and uniformity of the RVE model were also tested using a two-point probability function. The Park-Paulino-Roesler (PPR) cohesive model was introduced to simulate the particle debonding (or dewetting) in solid propellant. The stress-strain curves of the propellant were obtained by the macroscopic test with the extension rate 200 mm/min at different temperatures. Based on these experimental data, the 8 characteristic parameters suitable for the microinterface of the propellant were obtained by using an inversion optimization method. A microscale finite element prediction model of the propellant considering dewetting damage was constructed to study the evolution process of the microdamage of the propellant. The predicted stress-strain curves of the propellant under different loading conditions were in good agreement with the test results.

Statistical Analyses of the Strengths of Particulate Reinforced Metal Matrix Composites (PRMMCs) Subjected to Multiple Tensile and Shear Stresses

For design and application of particulate reinforced metal matrix composites (PRMMCs), it is essential to predict the material strengths and understand how do they relate to constituents and microstructural features. To this end, a computational approach consists of the direct methods, homogenization, and statistical analyses is introduced in our previous studies. Since failure of PRMMC materials are often caused by time-varied combinations of tensile and shear stresses, the established approach is extended in the present work to take into account of these situations. In this paper, ultimate strengths and endurance limits of an exemplary PRMMC material, WC-Co, are predicted under three independently varied tensile and shear stresses. In order to cover the entire load space with least amount of weight factors, a new method for generating optimally distributed weight factors in an n dimensional space is formulated. Employing weight factors determined by this algorithm, direct method calculations were performed on many statistically equivalent representative volume elements (SERVE) samples. Through analyzing statistical characteristics associated with results the study suggests a simplified approach to estimate the material strength under superposed stresses without solving the difficult high dimensional shakedown problem.

Analytical elastoplastic model for stress and strain of the equivalent representative volume element and material testing application

For unidirectional loaded specimens consisting of isotropic-homogeneous ductile materials, an analytical elastoplastic model for the equivalent stress and strain of the representative volume element at the median point were derived based on energy density equivalence and dimensional analysis. Then, a data processing method for various specimen types was proposed to obtain the equivalent stress–strain in real time without presetting the constitutive parameters. Finite element analyses of six specimen types and experiments with four selected specimen types were conducted to verify the method. The results show that the stress–strain curves obtained using this method were consistent with the preset stress–strain curves in the finite element analysis and with the standard tensile results. Using the proposed model and relevant data processing method to obtain the stress–strain curves is effective and, with the theoretical basis, could promote the application of non-tradition and small specimens for obtaining the material mechanical properties.

Predicting the complete tensile properties of additively manufactured Ti-6Al-4V by integrating three-dimensional microstructure statistics with a crystal plasticity model

A multiscale finite element model integrating microstructure statistics with an enhanced three-dimensional (3D) crystal plasticity model including damage has been developed to predict the complete tensile stress-strain response of Ti-6Al-4V manufactured by selective laser melting. A statistically equivalent representative volume element (SERVE) was constructed from an orthogonal set of electron backscatter diffraction (EBSD) images capturing key statistical information such as the spatial distribution of the grain size and the crystallographic and morphological orientation of the grains. Working from this 3D SERVE as a statistical model of the real microstructure, the crystal plasticity finite element (CP-FE) method was then used to predict the complete tensile stress-strain response of the material, including damage post necking. For the first time, crack-band theory (CBT) was incorporated into the CP-FF model to accurately predict ductility and to minimize mesh size sensitivity, which is a common issue in existing models. The effects of the 3D SERVE size on the stress-strain response was investigated through a rigorous sensitivity analysis. The model was calibrated using one sample and validated against a second sample with a different microstructure and properties. The new multiscale model provides a basis for a comprehensive Integrated Computational Materials Engineering (ICME) tool to enable the rational design of new high-performance materials.

Recognition and Quantification of Dual Phase Titanium Alloy Microstructures Using Convolutional Neural Networks

Recent advances in machine learning and image recognition tools/methods are being used to address fundamental challenges in materials engineering, such as the automated extraction of statistical information from dual phase titanium alloy microstructure images to support rapid engineering decision making. Initially, this work was performed by extracting dense layer outputs from a pretrained convolutional neural network (CNN), running the high dimensional image vectors through a principal component analysis, and fitting a logistic regression model for image classification. Kfold cross validation results reported a mean validation accuracy of 83% over 19 different material pedigrees. Furthermore, it was shown that fine-tuning the pre-trained network was able to improve image classification accuracy by nearly 10% over the baseline. These image classification models were then used to determine and justify statistically equivalent representative volume elements (SERVE). Lastly, a convolutional neural network was trained and validated to make quantitative predictions from a synthetic and real, two-phase image datasets. This paper explores the application of convolutional neural networks for microstructure analysis in the context of aerospace engineering and material quality.

A dispersive homogenization model for composites and its RVE existence

An asymptotic homogenization model considering wave dispersion in composites is investigated. In this approach, the effect of the microstructure through heterogeneity-induced wave dispersion is characterised by an acceleration gradient term scaled by a “dispersion tensor”. This dispersion tensor is computed within a statistically equivalent representative volume element (RVE). One-dimensional and two-dimensional elastic wave propagation problems are studied. It is found that the dispersive multiscale model shows a considerable improvement over the non-dispersive model in capturing the dynamic response of heterogeneous materials. To test the existence of an RVE for a realistic microstructure for unidirectional fiber-reinforced composites, a statistics study is performed to calculate the homogenized properties with increasing microstructure size. It is found that the convergence of the dispersion tensor is sensitive to the spatial distribution pattern. A calibration study on a composite microstructure with realistic spatial distribution shows that convergence is found although only with a relatively large micromodel.

Computationally fast morphological descriptor-based microstructure reconstruction algorithms for particulate composites

In this paper, we propose a series of novel algorithms for microstructure reconstruction based on constituent morphologies. The proposed algorithm can generate statistically equivalent representative volume elements (RVE) from shape repository consisting of the actual particles from micro-computed tomography (CT) images. The reconstruction algorithm is also general in that it can adopt any morphological descriptors. CDFs errors of the morphological descriptors between the targeted and reconstructed RVEs are minimized within the proposed algorithmic framework. High reliability of the microstructure reconstructions stems from unique and efficient design of the algorithmic framework. One of the novelties of the proposed algorithm is that it does not rely on time-consuming two-point correlation function (TPCF) calculations while it can replicate distributions keeping realistic morphologies of particles. To verify the proposed reconstruction algorithm, the TPCF and homogenized effective properties were compared between targeted and reconstructed RVEs. Moreover, mechanical responses of the targeted and reconstructed RVEs are evaluated and compared by 3D finite element analyses.

A Bayesian statistics based investigation of binder hardening’s influence on the effective strength of particulate reinforced metal matrix composites (PRMMC)

In order to understand how hardening of the binder phase in particulate reinforced metal matrix composites (PRMMC) influences the effective strength, we present in this work a numerical framework consisting of the direct method (DM) and statistical models. Using this approach we created a large number of statistically equivalent representative volume element (SERVE) models to represent an exemplary PRMMC material WC-20 Wt.% Co and predicted its effective strengths using DM. After the global strength was calculated from each SERVE sample all derived data are interpreted by Bayesian network and diagnostic testing. By doing so the relationship between material strength and few selected characteristics have been clarified. The study shows the formulated approach as a novel means for investigating how the overall mechanical properties of random heterogeneous materials react to a certain constituent. Meanwhile, the study also demonstrates how statistical models, in particular the Bayesian network, can be used as a powerful supplement to micromechanical models for result analysis and knowledge discovery.

Parametrically homogenized constitutive models (PHCMs) from micromechanical crystal plasticity FE simulations, part I: Sensitivity analysis and parameter identification for Titanium alloys

This is the first of a two part paper that systematically develops a finite deformation elasto-plastic, parametrically homogenized constitutive model (PHCM) for structural-scale macroscopic simulations of Titanium alloy Ti6242S. The PHCMs are thermodynamically consistent, reduced-order continuum models that incorporate functional forms of representative aggregated microstructural parameters (RAMPs) in their constitutive coefficients. These models are designed to represent all characteristics of macroscopic response such as anisotropy, tension-compression asymmetry, strain-rate and temperature dependence, and are consistent with microscopic crystal plasticity relations. An image-based size and rate-dependent crystal plasticity FE (CPFE) model is developed for creating a data-base needed for deriving functional forms of PHCM coefficients using machine learning methods. The first part of this sequence summarizes the generation of microstructure-based statistically equivalent representative volume elements (M-SERVEs) and the image-based CPFE model. Extensive sensitivity analysis is conducted to unravel the effect of specific RAMPs on the overall material response, and hence the PHCM constitutive coefficients. Sobol sensitivity analysis is performed to evaluate the sensitivities of constitutive coefficients in PHCM with respect to RAMPs to examine their suitability for representation. In the second part (Kotha et al., 2019), machine learning will be used with the database to create functional forms of the constitutive coefficients in the PHCMs.

Generating 3D virtual microstructures and statistically equivalent RVEs for subgranular gamma-gamma’ microstructures of nickel-based superalloys

Nickel-based superalloys are characterized by γ phase (FCC) nickel solid solution and γ′ phase (L12) intermetallics in their subgrain microstructures. The γ′ precipitates act as obstacles to the motion of dislocations and affect the mechanical behavior. It is important to capture the statistics of these microstructures in effective micromechanical models of these materials at the sub-grain scales. This paper develops a comprehensive suite of computational methods for generating optimally designed representative volume elements from experimental data on the material microstructure. Starting from 3D input datasets generated by a dual-beam focused ion beam (FIB-SEM) system, computational algorithms are developed to generate statistically equivalent virtual microstructures (SEVMs) and microstructure-based statistically equivalent representative volume elements (M-SERVEs) of γ-γ′ microstructures of Ni-based superalloys. Image processing, microstructural characterization, feature parametrization and representation of statistical distributions are conducted, along with novel interpolation schemes for statistically accurate precipitate packing. Subsequently, methods are developed for identifying the M-SERVE as an optimal subset of the SEVM, meeting requirements of statistical equivalence with respect to specific microstructural descriptors. Parametric representation of geometric features enable the incorporation of microstructural descriptors in parametrically homogenized constitutive models.

Microstructure and property based statistically equivalent RVEs for polycrystalline-polyphase aluminum alloys

This paper develops necessary preprocessors for image-based micromechanical analysis of polycrystalline-polyphase microstructures of Al alloys such as Al7075-T651. Starting from input data in the form of electron back scatter diffraction (EBSD) and scanning electron microscopy (SEM) maps of orthogonal surfaces of experimental specimens, the paper creates a robust methodology for generating 3D statistically equivalent virtual microstructures (3D-SEVMs) by 3D stereological projection of 2D statistical distribution and correlation functions using a genetic algorithm (GA)-based numerical algorithm. Validation studies of the SEVM reconstruction process by comparing with morphological and crystallographic distributions of grains and precipitates from the experiments. Next, microstructure-based statistically equivalent representative volume element (M-SERVE) that corresponds to the minimum sized SERVE for convergence of morphological or crystallographic distributions, are established using the Kolmogorov–Smirnov (KS) tests. Finally, property-based statistically equivalent RVE (P-SERVE), defined as the smallest SERVE for predicting response functions (both effective and local), are estimated by conducting crystal plasticity finite-element simulations. Convergence plots of material response functions are used to assess the P-SERVE. Comparison of M-SERVE and P-SERVE sizes are made to comprehend microstructure-property relations.

Microstructure and Property-Based Statistically Equivalent Representative Volume Elements for Polycrystalline Ni-Based Superalloys Containing Annealing Twins

This paper has three major objectives related to the development of computational micromechanics models of Ni-based superalloys, containing a large number of annealing twins. The first is the development of a robust methodology for generating 3D statistically equivalent virtual polycrystalline microstructures (3D-SEVPM) of Ni-based superalloys. Starting from electron backscattered diffraction (EBSD) images of sections, the method develops distributions and correlation functions of various morphological and crystallographic parameters. To incorporate twins in the parent grain microstructure, the joint probability of the number of twins and parent grain size, and the conditional probability distributions of twin thickness and twin distance are determined. Subsequently, a method is devised for inserting twins following the distribution functions. The overall methodology is validated by successfully comparing various statistics of the virtual microstructures with 3D EBSD data. The second objective is to establish the microstructure-based statistically equivalent representative volume element or M-SERVE that corresponds to the minimum SERVE size at which the statistics of any morphological or crystallographic feature converge to that of the experimental data. The Kolmogorov–Smirnov (KS) test is conducted to assess the convergence of the M-SERVE size. The final objective is to estimate the property-based statistically equivalent RVE or P-SERVE, defined as the smallest SERVE, which should be analyzed to predict effective material properties. The crystal plasticity finite-element model is used to simulate SERVEs, from which the overall material response is computed. Convergence plots of material properties including the yield strength and hardening rate are used to assess the P-SERVE. A smaller P-SERVE compared to the M-SERVE indicates that the characteristic features of twins implemented in determining the M-SERVE are more stringent than those for determining material properties.

A method to numerically predict the loading ratio dependency of long crack propagation rates under cyclic loading

The investigation of cyclic crack propagation rates (CPR) for different loading ratios is very elaborate. A multiscale numerical approach to predict high cycle fatigue (HCF) strength for a ferritic-pearlitic steel has been modified to be applicable for the numerical calculation of its cyclic CPR. The original approach involves a series of numerical and mechanism-based analytical models. Microstructural features of the material are statistically characterized for the generation of synthetically representative volume element (RVE) models of the microstructure. A combined isotropic and kinematic hardening model for crystal plasticity is used to include the microdeformation behavior under cyclic loading in the model. The simulation of cyclic loading with numerous of these statistically equivalent RVEs results in fatigue indicator parameter (FIP) fields which differ from one RVE to the other. With regard to the weakest link theory, only the highest FIP in each RVE is extracted for further fatigue calculations. Theses FIPs are distributed by an extreme value distribution function which can be uniquely described by two parameters. These parameters are used to determine the crack propagation rate. For the comparison of results from experiment and simulations, a concept is proposed which links the globally applied stress amplitude on the RVEs to the cyclic stress intensity factor. The model needs to be calibrated by cyclic CPR results conducted with arbitrarily chosen loading ratios. The validation of the calibrated model is achieved by the prediction and comparison of cyclic CPR at a different loading ratio. The obtained numerical values are in good agreement with the experimental results.

Stochastic multiscale modeling and simulation framework for concrete

In this paper we present a computational framework for generating a realistic representative volume element of concrete, which reflects its inherent structural randomness. Computed tomography (CT) images are employed to provide the necessary information for the geometric statistical characterization of aggregate and defect (voids, pores, and micro-cracks) distributions. A Monte-Carlo simulation is used to generate 1000 realizations of statistically equivalent representative volume element (SERVE) and finite element predictions of SERVEs elastic and inelastic response are compared with experiments.