Microstructural Uncertainty Research Articles

Multiscale simulation of spatially correlated microstructure via a latent space representation

When deformation gradients act on the scale of the microstructure of a part due to geometry and loading, spatial correlations and finite-size effects in simulation cells cannot be neglected. We propose a multiscale method that accounts for these effects using a variational autoencoder to encode the structure–property map of the stochastic volume elements making up the statistical description of the part. In this paradigm the autoencoder can be used to directly encode the microstructure or, alternatively, its latent space can be sampled to provide likely realizations. We demonstrate the method on three examples using the common additively manufactured material AlSi10Mg in: (a) a comparison with direct numerical simulation of the part microstructure, (b) a push forward of microstructural uncertainty to performance quantities of interest, and (c) a simulation of functional gradation of a part with stochastic microstructure.

Uncertainty analysis for prediction of elastic properties of unidirectional bamboo fiber-reinforced composites

This paper presents a virtual framework to obtain the elastic response of unidirectional (UD) continuous bamboo fiber-reinforced composites based on the micromechanical modeling approach. The virtual framework involves the development of an algorithm for the generation of stochastic Representative Volume Elements (RVEs), taking into account the micro-structural uncertainty and variability of the bio-based bamboo fibers and their composites. The developed algorithm and model are validated with experimental characterizations and theoretical models. Different statistical micro-structural realizations with random fiber packing, fiber geometries, and variable RVE sizes are compared to investigate the effects of these parameters on the accuracy of the prediction of the elastic properties of bamboo fiber-reinforced composites. All models and loading scenarios are applied automatically through Python scripting in the Abaqus finite element (FE) solver. The uncertainty analysis of input parameters is carried out through parametric studies. This virtual micromechanical framework can be further linked in the future to macroscale models of bamboo or other natural fiber composites for multiscale prediction of the behavior of biocomposite structures under in-service loading.

Study on stochastic behavior of particle system in hot mix asphalt mixture from a meso-structural perspective

The variability and uncertainty bring great trouble to the application and in-depth research of asphalt mixture. This study aimed to detect the microstructural differences between the asphalt specimens after fixing the experimental and environmental inputs. To this end, the digital image correlation (DIC) method was utilized to propose reliable skeletal and disruptive meso-parameters. Besides, the close packing theory and digital sieving technique were combined to classify the internal particles of the mixture structure. Afterward, the stochastic behaviors of the particle systems containing the dominant aggregate size range (DASR) and interstitial component (IC) aggregates were thoroughly discussed. Finally, the disruption effect of IC on the skeletal structure was also investigated. The analysis results show that the microstructure of the asphalt mixture was a typical random system with a great deal of contingency. The aggregate contact index (ACI) was the most variable among the overall skeletal parameters, with a variation coefficient (COV) of 13.24 %. Besides, almost all the microstructural indexes indicated that about 4% of the specimens in parallel tests showed significant variability behavior, e.g., Na,13.2 > Na,16.0. Besides, the microstructural uncertainty of the minority class, e.g., the 16-mm aggregates with a 4% ∼7% COV higher than the other classes, was particularly significant. Finally, the interference effect was proven to focus on the contact behavior of DASR particles, thus damaging the DASR-based system. This study not only emphasizes the impact of variability but provides implications for controlling the performance of asphalt pavement.

Topology optimization under microscale uncertainty using stochastic gradients

This paper considers the design of structures made of engineered materials, accounting for uncertainty in material properties. We present a topology optimization approach that optimizes the structural shape and topology at the macroscale assuming design-independent uncertain microstructures. The structural geometry at the macroscale is described by an explicit level set approach, and the macroscopic structural response is predicted by the eXtended Finite Element Method (XFEM). We describe the microscopic layout by either an analytic geometric model with uncertain parameters or a level-cut from a Gaussian random field. The macroscale properties of the microstructured material are predicted by homogenization. Considering the large number of possible microscale configurations, one of the main challenges of solving such topology optimization problems is the computational cost of estimating the statistical moments of the cost and constraint functions and their gradients with respect to the design variables. Methods for predicting these moments, such as Monte Carlo sampling, and Taylor series and polynomial chaos expansions often require a large number of random samples resulting in an impractical computation. To reduce this cost, we propose an approach wherein, at every design iteration, we only use a small number of microstructure configurations to generate an independent, stochastic approximation of the gradients. These gradients are then used either with a gradient descent algorithm, namely Adaptive Moment (Adam), or the globally convergent method of moving asymptotes (GCMMA). Three numerical examples from structural mechanics are used to show that the proposed approach provides a computationally efficient way for macroscale topology optimization in the presence of microstructural uncertainty and enables the designers to consider a new class of problems that are out of reach today with conventional tools.

Uncertainty quantification of metallic microstructures using principal image moments

The present work uses Markov Random Field (MRF) algorithm to construct large-scale and statistically-equivalent samples from small-scale experimental data of metallic microstructures. While the MRF method can build such digital material representations in large computational domains, its algorithmic stochasticity (epistemic uncertainty) causes variations in the resulting microstructural features, such as the texture and grain topology. This work addresses the effects of the epistemic uncertainty on homogenized mechanical properties by characterizing the variations in the microstructural features using a shape descriptor based on the concept of moment invariants. In particular, 2D and 3D synthetic microstructure data for Titanium-7wt%Aluminum (Ti-7Al) alloy is generated with the MRF method using smaller-scale 2D experimental data. To quantify the uncertainty of the reconstructed synthetic samples, a graphical method and five different metrics of statistical variability are developed. Next, the propagation of the microstructural uncertainty on homogenized properties is studied using an analytical uncertainty quantification (UQ) algorithm and Gaussian Process Regression (GPR).

Multiscale Modeling of Metal-Ceramic Spatially Tailored Materials via Gaussian Process Regression and Peridynamics

In this paper, a micro-to-macro multiscale approach with peridynamics is proposed to study metal-ceramic composites. Since the volume fraction varies in the spatial domain, these composites are called spatially tailored materials (STMs). Microstructure uncertainties, including porosity, are considered at the microscale when conducting peridynamic modeling and simulation. The collected dataset is used to train probabilistic machine learning models via Gaussian process regression, which can stochastically predict material properties. The machine learning models play a role in passing the information from the microscale to the macroscale. Then, at the macroscale, peridynamics is employed to study the mechanics of STM structures with various volume fraction distributions.

Microstructure-guided deep material network for rapid nonlinear material modeling and uncertainty quantification

Modeling nonlinear materials with arbitrary microstructures and loading paths is crucial in structural analyses with heterogeneous materials with uncertainty. However, it is computationally prohibitive because (1) fine meshes are required to resolve the microstructures for nonlinear simulations, (2) enormous model evaluations are expected to simulate stochastic microstructures and the loading variabilities, and (3) nonlinear material models are generally expensive to solve. In this paper, we present the microstructure-guided deep material network (MgDMN), a deep-learning-based approach to model multi-phase materials and efficiently predict their mechanical response under virtually any loading paths and microstructure definitions considering plasticity. This work is built upon the earlier development of the deep material network (DMN) method for fast predictions of nonlinear homogenized responses of multi-phase materials using linear elastic training data for a predefined microstructure. We extend DMN to MgDMN for any microstructure definition by parameterizing the microstructure space, mapping a small set of DMNs to their respective microstructure parameters, and interpolating the DMNs to generate new models in the microstructure space. The proposed method achieves superb efficiency and accuracy since the interpolated DMNs do not need additional training to predict nonlinear behavior of any new microstructures. We demonstrate our approach via modeling and uncertainty quantification of short fiber reinforced polymer (SFRP) composites. We show that for this dual-phase material, only four base DMNs are needed to predict the nonlinear stress responses for any loading path given any fiber volume fraction, orientation states, and constituent properties as long as the combination is within the microstructure space enclosed by the base models. The accelerated modeling process enables complex structural simulations of heterogeneous materials and nonlinear stochastic analyses with uncertain microstructures, which are essential in computational materials engineering and multiscale modeling. Our research also draws insights into the impact of microstructure uncertainties on linear and nonlinear material behaviors.

Effects of the microstructural uncertainties on the poroelastic and the diffusive properties of mortar

The assessment of the durability of civil engineering structures subjected to several chemical attacks requires the development of chemo-poromechanical models. The mechanical and chemical degradations depend on several factors such as the initial composition of the porous medium. A multi-scale model is used to incorporate the multi-level microstructural properties of the mortar material. The present paper aims to study the effect of morphological and local material properties uncertainties on the poroelastic and diffusive properties of mortar estimated with the help of analytical homogenization. At first, the proposed model is validated for different cement paste and mortar by comparison to experimental results and micromechanical models. Secondly, based on a literature study, sensitivity and uncertainty analysis have been developed to assess the stochastic predictions of the multi-scale model. The main result highlights the predominant impact of the cement matrix phases (C-S-H) and interfacial transition area at the mortar scale. Furthermore, the sensitive analysis underlines that the material properties induce more variability than the volume fraction.

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).

Multiscale Modeling for Texture and Grain Topology of Polycrystalline Microstructures Under Uncertainty

The present work addresses multiscale modeling for grain topology of polycrystalline microstructures under the effects of the microstructural uncertainties. The special focus is on the titanium-7wt%-aluminum alloy (Ti-7Al), which is a candidate material for many aerospace systems owing to its outstanding mechanical performance in elevated temperatures. The electron backscatter diffraction samples of Ti-7Al in small-scales are used to reconstruct the alloy microstructures in larger domains. The reconstruction provides a statistically equivalent synthetic representation to the small-scale test samples while introducing epistemic uncertainty on microstructural features. Here, the grain topology of the polycrystalline microstructures is quantified using shape moment invariants. These invariant parameters can capture the shapes of physical objects with a numerical representation, and they are invariant to shape transformations. To quantify the effects of the uncertainties on the homogenized properties of the microstructures, a surrogate model is developed as a function of shape moment invariants with Gaussian process regression by utilizing the experiments and crystal plasticity simulations for training data.

Uncertainty Quantification of Metallic Microstructures with Analytical and Machine Learning Based Approaches

Uncertainty in the microstructures has a significant influence on the material properties. The microstructural uncertainty arises from the fluctuations that occur during thermomechanical processing and can alter the expected material properties and performance by propagating over multiple length scales. It can even lead to the material failure if the deviations in the critical properties exceed a certain limit. We introduce a linear programming (LP) based method to quantify the effects of the microstructure uncertainty on the desired material properties of the titanium–7 wt % aluminum alloy, which is a candidate material for aerospace applications. The microstructure is represented using the orientation distribution function (ODF) approach. The LP problem solves for the mean values and covariance of the ODFs that maximize a volume-averaged linear material property. However, the analytical procedure is not applicable for maximizing nonlinear material properties where microstructural uncertainties are present. Therefore, an artificial neural network based sampling method is developed to estimate the mean values and covariance of the ODFs that satisfy design constraints and maximize the volume-averaged nonlinear material properties. A couple of other design problems are also illustrated to clarify the applications of the proposed models for both linear and nonlinear properties.

On the sensitivity of the design of composite sound absorbing structures

The acoustic properties of composite structures made of a perforated panel, an air gap, and a porous layer, can be studied numerically by a combined use of ad hoc optimization and sensitivity analysis methods. The methodology is briefly described and is systematically applied to a series of multi-layer configurations under manufacturing constraints. We specifically consider a foam layer of constant thickness presenting three different degrees of reticulations (pore opening). For each foam layer, the optimal geometrical parameters of the perforated panel and the cavity depth maximizing sound absorption under normal incidence are determined, together with the corresponding sensitivity indices. The simulation results are found in good agreement with interpretations and experimental data provided elsewhere. From a general perspective, the framework can be used to identify the most influential parameters within multiscale settings for acoustics studies, hence enabling robust design under material and microstructural uncertainties for instance.

Transverse mechanical properties of unidirectional FRP including resin-rich areas

Fiber reinforced plastic (FRP) is known to possess advantages of high strength and low density, and it is extensively used in engineering structures. However, most FRP have a certain level of microstructural uncertainty, and resin-rich areas in FRP can exist as inter-lamina resin layers or intra-lamina resin pockets. This work proposes a new algorithm to generate non-uniform random fiber distribution which contains resin-rich areas. The transverse tensile and compressive mechanical behavior of unidirectional FRP including resin-rich areas are studied using numerical simulation. The representative volume element (RVE) of the FRP with resin-rich areas is constructed. Matrix plastic deformation and interfacial debonding are described by extended Drucker-Prager model and cohesive zone model. Detrimental effect of resin-rich areas on the mean values and uncertainty of transversely ultimate strength and fracture strain is quantified. A new observation is that the resin-rich areas would result in large uncertainty on the ultimate strength of FRP, while its influence on the mean values of ultimate strength and fracture strain is limited. The large uncertainty on the tensile strength is attributed to that different morphology of resin-rich areas would lead to different cracks. The size of the resin-rich areas is also found to have an effect on transverse tensile and compressive mechanical properties.

Harnessing deep learning for physics-informed prediction of composite strength with microstructural uncertainties

Representative volume elements (RVEs) are commonly utilized to analyze the effective properties of fiber reinforced composites based on their repetitive microstructures and the constituent fiber and matrix properties. Intrinsically, the randomness of fiber distribution exists in composites even though the manufacturing process is strictly controlled. Such microstructural uncertainties essentially render the composite strength stochastic and difficult to characterize. In this research, a physics-informed deep learning framework is developed to analyze the variation of the strength of composite material with microstructural uncertainties. A random fiber packing algorithm is employed to sample the RVE images that are subsequently subjected to composite progressive damage analysis using the finite element method. The input–output relations acquired from this first-principle analysis are used as training data to facilitate deep learning that is capable of directly predicting the composite strength based on the RVE image. Two neural network architectures, a customized convolutional neural network (CNN) and a VGG16 transfer learning neural network, are established, with the view to unleashing the power of deep learning with small data size. This new framework significantly expedites the uncertainty analysis. It can directly take the spatial uncertainties in RVEs into account, outperforming other uncertainty quantification approaches. Systematic case investigations are conducted, in which the statistical cross-validation confirms the validity of the method. Owing to the highly efficient emulation, one can further carry out convergence analysis for uncertainty quantification. The results clearly demonstrate the effectiveness and capability of the proposed new framework for composite strength prediction. This framework is generic, which can be potentially extended into uncertainty quantification of other composite properties.

Digital Twin-driven framework for fatigue life prediction of steel bridges using a probabilistic multiscale model: Application to segmental orthotropic steel deck specimen

Accurate fatigue life prediction facilitates the fatigue maintenance of steel bridges. Since Digital Twin can simulate the lifecycle for physical objects at various scales, this study aims to provide a Digital Twin-driven framework for non-deterministic fatigue life prediction of steel bridges. A probabilistic multiscale model was developed to depict the fatigue evolution throughout the bridge lifecycle. The small crack initiation period was well described by the modified Fine and Bhat model considering microstructure uncertainties. After obtaining the critical model parameter via crystal plastic finite element simulation, the modified model was further calibrated using the assumed historical fatigue data in Digital Twin database. Based on the initiated half-penny-shaped small crack, the small crack initiation period was connected to the macrocrack extension period. Given the uncertainties of macrocrack propagation, the Paris’ law with random growth parameters was adopted. The Bayesian inference of the growth parameters realized the real-time calibration of the macrocrack growth model using Markov chain Monte Carlo simulation. The feasibility of the proposed framework was demonstrated through fatigue tests on a segmental steel deck specimen with mixed-mode deformed U-rib to diaphragm welded joints. The results show that the predicted fatigue initiation life and residual fatigue life are in good agreement with the experimentally observed life results. In summary, the proposed framework enhances our understanding of the fatigue evolution mechanism throughout the bridge lifecycle and provides an entirely new approach to accurately predict the fatigue life of steel bridges under various sources of uncertainties.

Reliability-based topology optimization with stochastic heterogeneous microstructure properties

With advanced manufacturing processes, many modern structural designs need to include microscale uncertainties of material properties and microstructures that can affect overall performances, such as microelectromechanical systems, micro–opto–electro–mechanical systems, and micro-optical electronics systems. Topology design optimization at this scale must consider uncertainties from material microstructures as the scale of the structure and material microstructure is comparable. Very few studies consider microstructure uncertainties in topology design due to their scales are much different in classical mechanical/civil engineering. A novel framework of reliability-based topology optimization is proposed to specifically address this gap for the design. The microscale uncertainties of heterogeneous materials are quantified by the explicit mixture random field model. Then, the material distribution is optimized by a heuristic updating scheme, and sensitivities of reliability constraints are calculated using the adjoint design-point-based importance sampling method. The proposed methodology simultaneously considers microscale hierarchical uncertainties in microstructures and material property variations for each phase. The feasibility of the framework is demonstrated by several numerical examples with different multi-phase materials. Compared with deterministic topology optimization and classical topology optimization with first-order reliability method approximations, the optimal design obtained from the proposed method can achieve accurate target reliability with minimized limit state function evaluations.

Machine learning in multiscale modeling of spatially tailored materials with microstructure uncertainties

In this paper, a novel hierarchical micro–macro multiscale modeling enhanced via machine learning is proposed. Machine learning plays an important role in this multiscale framework to pass the information from the microscale to the macroscale. This multiscale method provides a new approach to study the mechanics of a metal-ceramic (Ti-TiB2) spatially tailored material in which the volume fractions vary in space at the macroscale. Data sets, collected from microscale simulations, are used to train machine learning regression and classification models. Those predictive models are then implemented in the macroscale simulations to study dynamical responses of spatially tailored Ti-TiB2 structures under various loading conditions. As a difference from other reported works, microstructure uncertainties are considered in this paper so that an artificial neural network is trained as the machine learning classification model to predict the failure probability at the macroscale, which depends on the volume fraction and the deformation (i.e., the strain).

Stochastic nonlinear analysis of unidirectional fiber composites using image-based microstructural uncertainty quantification

We present a data-driven nonlinear uncertainty quantification and propagation framework to study the microstructure-induced stochastic performance of unidirectional (UD) carbon fiber reinforced polymer (CFRP) composites. The proposed approach integrates (1) microscopic image characterization, (2) stochastic microstructure reconstruction, and (3) efficient multiscale finite element simulations enabled by self-consistent clustering (SCA) analysis. To model the complex microstructural variability, the proposed UQ methods take the non-Gaussian uncertainty sources into account through a distribution-free sampling approach leveraging nonparametric and asymptotic statistical tools. A hierarchical conditional sampling strategy enables the simultaneous sampling of multiple sources of uncertainties. Our approach provides insights into the impact of microstructural variabilities, which are shown to have an increasing impact on the nonlinear responses of UD CFRP parts under progressive compression loading and ultimately on the failure rate over time. We discover that before CFRP parts start to fail, a characteristic time period emerges with distinctive uncertainty distributions specific to the microstructure variability and the probability of failure. Identifying the failure time period is crucial to the reliability prediction, which is an essential component of CFRP design.

Computational Design of Microstructures With Stochastic Property Closures

Abstract The present work addresses a stochastic computational solution to define the property closures of polycrystalline materials under uncertainty. The uncertainty in material systems arises from the natural stochasticity of the microstructures as a result of the fluctuations in deformation processes. The microstructural uncertainty impacts the performance of engineering components by causing unanticipated anisotropy in properties. We utilize an analytical uncertainty quantification algorithm to describe the microstructural stochasticity and model its propagation on the volume-averaged material properties. The stochastic solution will be integrated into linear programming to generate the property closure that shows all possible values of the volume-averaged material properties under the uncertainty. We demonstrate example applications for stiffness parameters of α-Titanium, and multi-physics parameters (stiffness, yield strength, magnetostrictive strain) of Galfenol. Significant differences observed between stochastic and deterministic closures imply the importance of considering the microstructural uncertainty when modeling and designing materials.

Uncertainty-quantified parametrically homogenized constitutive models (UQ-PHCMs) for dual-phase \u03b1/\u03b2 titanium alloys

This paper develops an uncertainty-quantified parametrically homogenized constitutive model (UQ-PHCM) for dual-phase α/β titanium alloys such as Ti6242S. Their microstructures are characterized by primary α-grains consisting of hcp crystals and transformed β-grains consisting of alternating laths of α (hcp) and β (bcc) phases. The PHCMs bridge length-scales through explicit microstructural representation in structure-scale constitutive models. The forms of equations are chosen to reflect fundamental deformation characteristics such as anisotropy, length-scale dependent flow stresses, tension-compression asymmetry, strain-rate dependency, and cyclic hardening under reversed loading conditions. Constitutive coefficients are functions of representative aggregated microstructural parameters or RAMPs that represent distributions of crystallographic orientation and morphology. The functional forms are determined by machine learning tools operating on a data-set generated by crystal plasticity FE analysis. For the dual phase alloys, an equivalent PHCM is developed from a weighted averaging rule to obtain the equivalent material response from individual PHCM responses of primary α and transformed β phases. The PHCMs are readily incorporated in FE codes like ABAQUS through user-defined material modeling windows such as UMAT. Significantly reduced number of solution variables in the PHCM simulations compared to micromechanical models, make them several orders of magnitude more efficient, but with comparable accuracy. Bayesian inference along with a Taylor-expansion based uncertainty propagation method is employed to quantify and propagate different uncertainties in PHCM such as model reduction error, data sparsity error and microstructural uncertainty. Numerical examples demonstrate the accuracy of PHCM and the relative importance of different sources of uncertainty.