Stochastic Material Properties Research Articles

Uncertainty quantification of acoustic metamaterial bandgaps with stochastic material properties and geometric defects

Acoustic metamaterials are a subject of increasing study and utility. Through designed combinations of geometries with material properties, acoustic metamaterials can be built to arbitrarily manipulate acoustic waves for various applications. Despite the theoretical advances in this field, however, acoustic metamaterials have seen limited penetration into industry and commercial use. This is largely due to the difficulty of manufacturing the intricate geometries that are integral to their function and the sensitivity of metamaterial designs to material batch variability and manufacturing defects. Capturing the effects of stochastic material properties and geometric defects requires empirical testing of manufactured samples, but this can quickly become prohibitively expensive with higher precision requirements or with an increasing number of input variables. This paper demonstrates how uncertainty quantification techniques, and more specifically the use of polynomial chaos expansions and spectral projections, can be used to greatly reduce sampling needs for characterizing acoustic metamaterial dispersion curves. With a novel method of encoding geometric defects in a 1D, interpretable, resolution-independent way, our uncertainty quantification approach allows for both stochastic material properties and geometric defects to be considered simultaneously. Two to three orders of magnitude sampling reductions down to ∼100 and ∼101 were achieved in 1D and 7D input space scenarios, respectively. Remarkably, this reduction in sampling was possible while preserving accurate output probability distributions of the metamaterial performance characteristics (bandgap size and location).

Continuum versus micromechanical modeling of corneal biomechanics

Two alternative numerical models of the human cornea are used to simulate the mechanical response under the action of a physiological intraocular pressure (IOP). The first model is continuum or macromechanical, considering the stromal tissue as a bulk material with stochastic distribution of the spatial variability of reinforcing collagen fibers. The second model is discrete or micromechanical, considering the sole collagen-crosslink stiffening micro-structure. The geometry of the two models is reconstructed from corneal topographer images. Simulations consider the behavior of a healthy cornea and of a keratoconus cornea. For the keratoconus the material properties of a portion of the cornea are reduced to 1/8 of the values used for the healthy tissue. It is found that, for suitable choice of the material parameters for the discrete model, in the healthy case the mechanical responses of the two models are fully comparable. In the keratoconus case, both models capture with comparable accuracy the anterior shape of the conus; in addition, the discrete model is able to describe the tissue thinning typical of the pathology. Despite the inclusion of stochastic material properties, starting from a healthy condition, continuum models of the cornea are not able to predict the thinning of a keratoconus cornea, while the inclusion of the underlying collagen microstructure allows for a proper description of pathologic mechanical behaviors.

Evaluating a Stochastic Singularity Level in a V-Notch Isotropic Domain Due to Randomness in Material Properties

Abstract The singular level of a v-notched isotropic domain is crucial as it relates to the domain’s stress intensity factor (SIF), which is essential for failure prediction. Typically, the singular level is calculated using deterministic material properties (like mean values), but this can misrepresent the severity of the singularity. The singular level’s accuracy is influenced by the stochastic nature of material properties, particularly Poisson’s ratio. This paper presents a stochastic approximation of eigenvalues for an isotropic v-notched domain with stochastic material properties. While both Young’s modulus and Poisson’s ratio are independent stochastic variables, only the stochasticity of Poisson’s ratio affects the eigenvalues. The generalized polynomial chaos method is applied to these stochastic eigenvalues, resulting in a polynomial approximation based on the domain’s stochastic Poisson ratio. The polynomial’s deterministic constants are derived from eigenvalues computed using chosen deterministic material properties according to the stochastic properties of Young’s modulus and Poisson’s ratio. A numerical example illustrates the stochastic approximation of the first two eigenvalues, showing fast convergence as the number of polynomials increases. Results are validated against the Monte Carlo method, highlighting the importance of stochastic approximation in accurately predicting the singular level, which may be more severe than expected when using deterministic approximations. These findings have significant practical implications, as they underscore the need to consider stochastic material properties in structural design and failure prediction.

Modeling of Stochastic Material Properties with User Subroutines in SIMULIA Abaqus

This article describes a software module component integrated with the SIMULIA Abaqus engineering analysis software package and designed to simulate random values of material parameters in a finite element model based on specified statistical characteristics, with the possibility of taking into account the physical nonlinearity of material behavior under various combinations of loads and influences. The target group of materials under study is materials of load-bearing elements of building structures, such as concrete, stone, steel. This software module can be recommended for use by specialists, engineers and scientists engaged in probabilistic analysis of the reliability of structures of buildings and structures, apparatus, machines, devices, with the combined use of complexes of computer modeling and engineering analysis. Has a certificate of state registration of the computer program «AS for modeling stochastic properties of materials» No. 2019667439 dated 12.24.2019.

Extracting edge stress intensity function for isotropic 3D cracked domains having stochastic material properties

The Stress Intensity Factors (SIFs) are fundamental and valuable parameters in fracture mechanics since many failure criteria involve them. In the case of a 3D crack, the SIFs extraction, based on a post-processing solution using the Finite-Element Method (FEM), is performed repeatably at several points along the singular edge (as a point-wise method) to evaluate the functional representation of the SIF along the crack edge. The Quasi Dual Function Method (QDFM) provides a polynomial approximation of the SIFs along the edge and eliminates the need for point-wise repeated calculations along the edge. Herein, we extend the QDFM to provide a broader 3D extraction method that includes stochasticity in the material properties in the case of an isotropic cracked domain having a straight edge, using the generalized Polynomial chaos (gPC). We apply the 2D gPC SIF extraction method over the QDFM to provide a stochastic Edge Stress Intensity Function (ESIF) polynomial approximation for 3D cracked domains with stochastic material properties (stochastic variables). The obtained ESIF is expressed using three families of orthogonal polynomials: Jacobi polynomials which approximate the deterministic solution of the ESIF (as part of the QDFM) and two families of orthogonal polynomials, selected by the probability distribution function of the material properties. The proposed method provides a functional polynomial presentation of the stochastic ESIF along the crack edge. Numerical example problems are provided where the stochastic approximation of the ESIF is computed. The stochastic properties of the ESIF: Mean, Variance, Skewness and Kurtosis are calculated, and the convergence of these properties is examined. The stochastic approximation of the ESIF is compared with results obtained using the 2D point-wise gPC SIFs extraction method. The proposed method is very efficient compared with the known point-wise methods since it provides a continuous function along the singular edge, unlike the point methods that require a new calculation for each point separately. The numerical examples demonstrate the robustness and high accuracy of the proposed method.

Seismic fragility of lightly reinforced concrete school building typologies with different masonry infill configurations

This paper presents the outcome of a research study conducted to establish seismic fragilities of school building typologies in Sri Lanka. The school buildings in Sri Lanka can be characterised as lightly reinforced concrete (RC) buildings, infilled with masonry walls (IMW). However, they are categorised into two typologies based on the structural layouts used (1) Type 1 (T01) and (2) Type 2 (T02). Although, the school buildings can be grouped into two typologies, variabilities in terms IMW configurations and their arrangements are observed among those school buildings. These variabilities in terms of building typologies as well as IMW arrangements were taken to establish seismic fragility curves. The seismic performances of the school buildings were numerically assessed, where in total 640 building cases were analysed by varying typologies, IMW configurations, and stochastic material properties. Since, the RC school buildings are lightly reinforced, a simplified, yet a novel approach was followed to account the shear failure of RC columns under seismic actions. Then, four damage thresholds were established (slight, moderate, severe and collapse) and the corresponding fragility curves are presented in terms of school building typologies considered. Finally, based on the fragility curves, damage probability matrices of the building typologies were established.

A hybrid proper orthogonal decomposition-heteroscedastic sparse Gaussian process regression model for evaluating structural reliability with correlated stochastic material properties

Data-driven material models have shown advantages in recent studies since they can directly use the existing data and improve the model performances further when additional data is available. However, few data-driven material models in the previous studies considered the uncertainty and stochastic correlations of the material properties. In this work, the hybrid Proper Orthogonal Decomposition-Heteroscedastic Sparse Gaussian Process Regression (POD-HSGPR) framework is proposed for predicting stochastic material behaviors and their correlation. The two material behaviors case studies on the metal strength and the rock joint behavior have demonstrated that the proposed POD-HSGPR-based stochastic material model can effectively capture the material properties, material uncertainty, and stochastic correlation between the material behaviors directly from the experimental dataset. The proposed POD-HSGPR model is then applied to a rock slope structure problem to investigate the influence of the correlation of material properties on structural behavior. The results indicate that the proposed POD-HSGPR model could effectively quantify the correlation effect in a non-parametric format and avoid the overestimation of structural reliability.

Computational generation of multiphase asphalt nanostructures using random fields

AbstractThis study presents a novel methodology to generate computational replicates of nanostructures of multiphase materials, such as asphalt binders, by integrating image analysis techniques with stochastic random field (RF) modeling. Image analysis techniques are used to identify and segment nanostructure images obtained by atomic force microscopy, while RF is used to model the spatial distribution of their material properties. The results of this process are images showing probable arrangements of nanostructures with stochastic material properties that replicate the experimentally obtained images. The computationally generated nanostructures are then used as inputs in a finite element model to evaluate the effect of heterogeneity on their mechanical response. The efficacy of the developed approach is demonstrated through simulations of asphalt binders’ nanostructures, which reveal novel insights regarding their nanoscale mechanical behavior and response. The FE simulations provided the link between the distribution of nanoscale properties of asphalt binders and variations in their mechanical response. The application of this methodology expands the body of knowledge beyond the deterministic analysis of asphalt binders toward probabilistic analysis and uncertainty quantification that considers their heterogeneous, multiphase structures. Consequently, the methodology can be used to design multiphase materials, such as asphaltic blends, with tailored properties and enhanced performance.

Application of Variability Response Function to Cross Laminated Timber

Variation in material properties leads to variation in stiffness, or displacement response, of structural systems. This paper presents an efficient method to calculate the variability in stiffness of a three-layer Cross Laminated Timber (CLT) panel as a result of heterogeneous elastic properties in the constituent lumber. Typically, problems involving stochastic material properties are solved by Monte Carlo simulations, which is computationally expensive for complex systems. The Variability Response Function is a well-established concept for efficient evaluation of the variance of stochastic systems where properties are modeled by random fields. A new method to study the variation in CLT panel stiffness is developed by applying the Variability Response Function to CLT geometry. Using this method, novel CLT panel layups can be efficiently studied, increasing opportunities for traditionally low-value species to be used in CLT panels. Three case studies are provided to demonstrate the application and utility of the developed method.

A discrete-continuum coupled numerical method for fracturing behavior in concrete dams considering material heterogeneity

This study aims to propose a systematic discrete-continuum coupled computational approach for fracturing behavior in concrete dams, with consideration of material heterogeneity. Innovative techniques are developed to realize batch insertion of cohesive element and parameter randomization of material property, and their feasibility and efficiency are verified with fracturing modelling of concrete specimens. Based on numerical investigation of hydraulic and thermal loads in discrete-continuum mesh scheme, a complete approach of fracturing modelling in concrete dams with stochastic material property is established. A case study with Chencun arch dam is conducted, and the numerical results exhibit high consistency with observational information, which demonstrates that the proposed method is feasible to characterize the mechanical performance and fracturing behavior of concrete dams with material heterogeneity.

Estimation of High Structural Reliability Involving Nonlinear Dependencies Based on Linear Correlations

Stochastic nonlinear dependencies have been reported extensively between different uncertain parameters or in their time or spatial variance. However, the description of dependency is commonly not provided except a linear correlation. The structural reliability incorporating nonlinear dependencies thus needs to be addressed based on the linear correlations. This paper first demonstrates the capture of nonlinear dependency by fitting various bivariate non-Gaussian copulas to limited data samples of structural material properties. The vine copula model is used to enable a flexible modeling of multiple nonlinear dependencies by mapping the linear correlations into the non-Gaussian copula parameters. A sequential search strategy is applied to achieve the estimate of numerous copula parameters, and a simplified algorithm is further designed for reliability involving stationary stochastic processes. The subset simulation is then adopted to efficiently generate random variables from the corresponding distribution for high reliability evaluation. Two examples including a frame structure with different stochastic material properties and a cantilever beam with spatially variable stochastic modulus are investigated to discuss the possible effects of nonlinear dependency on structural reliability. Since the dependency can be determined qualitatively from limited data, the proposed method provides a feasible way for reliability evaluation with prescriptions on correlated stochastic parameters.

Strain response of fiber-reinforced ceramic-matrix composites subjected to stress-rupture with stochastic load at intermediate temperature

To ensure the reliability and safety of ceramic-matrix composites (CMCs) hot-section components used in the aero engines, it is necessary to perform the strain response of CMCs at intermediate temperatures (600 to 1000 °C) under stress-rupture with stochastic loading. In this paper, the strain response of SiC/SiC composite under stress-rupture with stochastic load at intermediate temperatures is investigated. Multiple damage mechanisms of matrix cracking, interface debonding and oxidation, and fiber’s oxidation and fracture are considered. Matrix crack spacing, interface oxidation and debonding length, fiber’s broken probability, and intact fiber’s stress are determined using micromechanical damage models. Experimental strain response and internal damage evolution of SiC/SiC composite under constant and stochastic stress are predicted. Effects of stochastic stress level and stochastic time, material properties, damage state and environment temperature on composite’s strain response and stress-rupture life are discussed. When stochastic stress level and environment temperature increase, the composite’s strain at the stochastic stress increases, the time for the interface complete debonding and oxidation decreases, and the stress-rupture life decreases.

Extracting stress intensity factors for isotropic cracked domains having stochastic material properties

Material properties are inevitably stochastic due to the manufacturing process and the measurement procedure. In case of a cracked domain, the stochasticity of material properties (as stochastic variables) may manifest in the stress intensity factors (SIFs). Having the stochastic representation of the material properties, Young modulus and Poisson ratio (isotropic material), we approximate the SIFs for 2D cracked domains using the generalized polynomial chaos (gPC). The approximated SIF consists of two families of orthogonal polynomials, selected by the probability distribution function of the material properties. The polynomials are multiplied by a deterministic coefficient, consisting of deterministic SIFs, extracted from a finite element model according to the stochastic properties of both Young modulus and Poisson ratio. Numerical example problems are provided where the stochastic approximation of the SIF is computed. The obtained approximation of the SIF is compared with results obtained using the Monte Carlo method. The results demonstrate the efficiency and accuracy of the proposed method.

Time‐separated stochastic mechanics for simulating the visco‐elastic response of construction parts

AbstractIn a series of papers, we investigated the problem of efficiently analyzing visco‐elastic materials with stochastic material properties. The related evolution equation could be solved analytically which allowed for a stochastic series expansion around the mean for the internal variable. This, in turn, gave access to analytical expressions for the stochastic stress and reaction force in a finite element setting such that expectation and variation could be expressed in terms of simple formulas. Consequently, our analysis separated the stochastic properties of the material from any boundary problem and any loads, i.e., the stochastic behavior was separated from all time‐dependent conditions such that we referred our method as time‐separated stochastic mechanics. Numerical results of our formulas were in excellent agreement to representative Monte‐Carlo (MC) simulations. This holds true for both material point computations as well as finite element simulations. Particularly for finite element simulations, the time consumption for our approach is smaller by orders of magnitude.In this contribution, we recall the basic aspects of our works and present a brief overview. After presenting the key idea of the TSM, we recall the numerical treatment and present several examples which we compare to Monte‐Carlo results. Our results show a similar quality as the reference solution but at a computational effort that is orders of magnitude smaller.

On generative models as the basis for digital twins

AbstractA framework is proposed for generative models as a basis fordigital twinsormirrorsof structures. The proposal is based on the premise that deterministic models cannot account for the uncertainty present in most structural modeling applications. Two different types of generative models are considered here. The first is a physics-based model based on the stochastic finite element (SFE) method, which is widely used when modeling structures that have material and loading uncertainties imposed. Such models can be calibrated according to data from the structure and would be expected to outperform any other model if the modeling accurately captures the true underlying physics of the structure. The potential use of SFE models as digital mirrors is illustrated via application to a linear structure with stochastic material properties. For situations where the physical formulation of such models does not suffice, a data-driven framework is proposed, using machine learning and conditional generative adversarial networks (cGANs). The latter algorithm is used to learn the distribution of the quantity of interest in a structure with material nonlinearities and uncertainties. For the examples considered in this work, the data-driven cGANs model outperforms the physics-based approach. Finally, an example is shown where the two methods are coupled such that a hybrid model approach is demonstrated.

Ultra-efficient reconstruction of 3D microstructure and distribution of properties of random heterogeneous materials containing multiple phases

Ultra-fast 3D material microstructure reconstruction and quantitative structure-property mapping are crucial components of integrated computational material engineering (ICME). It is particularly challenging for modeling random heterogeneous materials such as alloys, composites, polymers, porous media, and granular matters, which exhibit strong randomness and variations of their material properties due to the hierarchical uncertainties associated with their complex microstructure at different length scales. An explicit mixture random field (MRF) model is proposed to characterize and reconstruct multi-phase stochastic material property and microstructure simultaneously. The proposed method is shown to have ultra-high computational efficiency and only requires minimal imaging and property input data. The material property field is modeled by this strongly non-Gaussian random field, which is generated by a nonlinear mapping from the underlying Gaussian random field explicitly. The corresponding microstructure is represented by discrete phase indicators obtained from the material property field. A decomposed Karhunen–Loève (K-L) expansion method is used to dramatically reduce the computational costs and memory requirement for high-dimensional and high-resolution generations. The feasibility and superior efficiency are demonstrated by reconstructing various materials, from 2D to 3D, bi-phase to multi-phase, isotropic to anisotropic materials. The results show that it only takes approximately seconds to reconstruct a 3D material with the resolution of 500×500×500 pixels. The elegant expression for explicit microstructure-property mapping with uncertainty quantification capability can be directly incorporated into the ICME framework for material design and optimization.

Simplified spectral stochastic finite element formulations for uncertainty quantification of engineering structures

In general, the structure inputs including loading and physical specifications can be deterministic or stochastic and uncertainties can be modeled using a random variable, random field, or a stochastic process. A well-known method developed to analyze structures with uncertainties is the spectral stochastic finite element method (SSFEM). To improve the efficiency of SSFEM, this paper has developed three simplified formulations to separately solve problems involving stochastic loading and constant material properties, stochastic loading and random material properties and stochastic loading and stochastic material properties the main feature of which is that they need fewer finite element (FE) analyses to assess the statistical specifications of the response. To examine their accuracy and efficiency, three practical engineering problems: 1) Euler-Bernoulli beam, 2) square solid and 3) modeling the corrosion effect in a concrete structure were solved and the results were compared with those of the Monte Carlo simulation (MCS). To see if the proposed formulations can be used in structural reliability analyses, a limit state function (LSF) was defined for each problem and the results were compared with those of the corresponding MCS; their significant efficiency and enough potential was, thus, confirmed.

Assessment of extreme stresses and deflections on wind turbine blades with stochastic material properties using statistical extrapolation methods

Since the introduction of the 3rd edition of the IEC Standard 61400-1, designers of wind turbines are required to apply statistical extrapolation techniques, to estimate the extreme (ultimate) load values corresponding to fifty-year return period. In the present paper, the certification procedure is assessed under the uncertainty of the material properties using simulated load time series of the NREL 5MW Reference Wind Turbine. The uncertainty of the material properties is introduced in the elastic properties of the composite blades by using input data from the OptiDAT composite material database. Comparison of the estimated extreme loads and deflections of the blades as well as maximum stresses, also in connection to the Tsai Wu failure criterion, is performed for different material sets. It is found that the variability of the material properties does not affect the estimated ultimate flapwise moments (difference <1.5%) but affects the maximum flapwise deflection (differences ∼8%). It is concluded that for the levels of variation considered in the composite material properties, the coefficient of variation of the extreme stresses and the Tsai Wu failure criterion are in the order of 2% and 8% respectively.

Statistical extrapolation methods for estimating extreme loads on wind turbine blades under turbulent wind conditions and stochastic material properties

According to the IEC Standard 61400-1, designers of wind turbines are required to apply statistical extrapolation techniques to estimate the extreme (ultimate) load. In the present article, the certification procedure is assessed under the uncertainty of the material properties using a simulated load time series of the National Renewable Energy Laboratory 5MW reference wind turbine rotor. The uncertainty of the material properties is introduced in the elastic properties of the composite materials based on the OptiDAT composite material database. The assessment relies on the comparison of the estimated blade extreme loads and deflections, obtained for the reference and the stochastically varied material properties. It is found that the variability of the material properties does not affect the estimated ultimate moments (differences &lt; 1.5%) but affects the maximum flapwise deflection (differences ~8%). It is concluded that the peak over threshold peak extraction technique and the three-parameter Weibull fitting functions outperform among those considered in the article.

Modeling of viscoelastic structures with random material properties using time‐separated stochastic mechanics

SummaryModeling and simulation of materials with stochastic properties is an emerging field in both mathematics and mechanics. The most important goal is to compute the stochastic characteristics of the random stress, such as the expectation value and the standard deviation. An accurate approach are Monte Carlo simulations; however, they consume drastic computational power due to the large number of stochastic realizations that have to be simulated before convergence is achieved.In this paper, we show that a recently published approach for accurate modeling of viscoelastic materials with stochastic material properties at the material point level in the work of Junker and Nagel is also valid for macroscopic bodies. The method is based on a separation of random but time‐invariant variables and time‐dependent but deterministic variables for the strain response at the material point (time‐separated stochastic mechanics [TSM]). We recall the governing equations, derive a simplified form, and discuss the numerical implementation into a finite element routine. To validate our approach, we compare the TSM simulations with Monte Carlo simulations, which provide the “true” answer but at unaffordable computational costs. In contrast, the numerical effort of our approach is in the same range as for deterministic viscoelastic simulations.