In the last two decades, the intense research activity at the micro- and nano-scale has highlighted the need to account for disparate levels of uncertainty from various sources and across scales. Αn integration of stochastic and multiscale methodologies is required to provide a rational framework for the analysis and design of heterogeneous materials and structures as even over-refined deterministic approaches are not able to account for this issue. The accurate stochastic modeling across multiple length scales becomes imperative especially due to the emergence of new engineered materials with complex microstructure.

Translation of deterministic multiscale methods into corresponding stochastic versions requires not only the development of highly efficient stochastic algorithms to deal with the “curse of dimension” problem, but also the knowledge of multiscale features of complex systems. Equally important is to ensure the adherence between stochastic models and reality by establishing a physical connection of probabilistic methods to fundamental materials science and experimental mechanics, as well as to understand the effect of uncertainties at each length scale on system response variability and reliability.

The aim of this Special Section is to present the most recent advances in the field of stochastic multiscale modeling of heterogeneous materials and structures. The Special Section comprises eight papers dealing with various related topics such as the homogenization of composite materials and in particular the effect of random microstructure on their apparent/effective stiffness and strength, the random field modeling of heterogeneous media, and the stochastic finite element analysis of composite structures with mechanical properties based on microstructure.

The computation of macroscopic properties through homogenization is a fundamental issue in the framework of multiscale modeling of random heterogeneous materials. This is the topic of two papers by Sakata and co-authors. The first paper by Sakata and Yamauchi entitled “Stochastic Elastic Property Evaluation With Stochastic Homogenization Analysis of a Resin Structure Made Using the FDM Method” deals with the evaluation of the stochastic elastic property of a resin structure, made using the fused deposition modeling (FDM) method, through experimental and numerical tests. The FDM method is an additive manufacturing method enabling the fabrication of complex shaped structures at a low cost. The appropriate numerical modeling for the computation of the apparent elastic property of the resin specimen is discussed, along with a comparison between the experimental and numerical results obtained using Monte Carlo simulation. In addition, the applicability of the perturbation-based hierarchical stochastic homogenization approach is examined for different fabrication densities.

The second paper by Sakata and Sakamoto entitled “A Local Sensitivity Based Multiscale Stochastic Stress Analysis of a Unidirectional Fiber Reinforced Composite Material Considering Random Location Variation of Multi Fibers” presents an efficient computational method (involving a local surrogate constructed by a successive sensitivity analysis) for estimating the probabilistic properties of the maximum microscopic stresses in a unidirectional fiber-reinforced composite with microscopic random variations of fiber locations. These microscopic random geometrical variations can cause a large variability of the microscopic stresses, which will have a significant influence on the apparent strength of the composite, although the effect on the homogenized elastic properties is small. Therefore, estimation of the random variation is important for the reliability-based design of a composite structure.

The following two papers investigate the effect of uncertainty in the microstructure on the homogenized properties of nanocomposites. In the article by Sanei et al. entitled “Effect of Nanocomposite Microstructure on Stochastic Elastic Properties: an FEA Study,” computer-simulated microstructures are used to investigate the effect of waviness, agglomeration, and orientation of carbon nanotubes on the elastic properties of nanocomposites. The generated microstructures are first converted to image-based 2D finite element models. A number of different realizations of the microstructures are generated to capture the stochastic response. It is shown that the elastic properties and the corresponding variability are driven by the morphological features of the microstructure.

The paper by Savvas and Stefanou entitled “The Effect of Material and Geometrical Uncertainty on the Homogenized Properties of Graphene Sheet-Reinforced Composites” focuses on the computational homogenization of graphene sheet-reinforced composites with randomly dispersed inclusions and uncertainty in the constituent materials. The uncertainty in the inclusion material is due to structural defects of the graphene lattice and is taken into account using random variables for each component of the elasticity tensor. It is shown that uncertainty in the matrix material and geometrical uncertainty have a significant effect on the statistical characteristics of the effective properties of the composite computed using Monte Carlo simulation, in contrast to uncertainty in the inclusion material which is of minor importance.

The paper by Pingaro et al. entitled “Homogenization of Random Porous Materials With Low Order Virtual Elements” presents a fast statistical homogenization procedure (FSHP) based on the virtual element method for computing the apparent properties of random porous microstructures. Porous media are described as bimaterial systems in which soft circular inclusions, with a very low value of material contrast, are randomly distributed in a continuous stiffer matrix. Several simulations are performed by varying the level of porosity, highlighting the effectiveness of FSHP in conjunction with virtual elements of degree one.

The determination of the macroscopic properties of composite materials with random microstructures is usually based on the concept of the representative volume element (RVE). This approach is useful for predicting a global response but smoothes the underlying spatial variability of the composite's properties resulting from the random microstructure. In the paper by Baxter and Acton entitled “Simulations of Non-Gaussian Property Fields Based on the Apparent Properties of Statistical Volume Elements,” Voronoi tessellation is used to partition RVEs into statistical volume elements (SVEs) and apparent properties are developed for each SVE. The resulting field of properties is characterized with respect to its spatial autocorrelation and probability distribution. These autocorrelation and distribution functions are used to simulate additional property fields with the same probabilistic characteristics. The paper presents an update to an existing simulation technique, which is illustrated for different random microstructures and SVE partitioning sizes.

In the paper by Acton et al. entitled “Effect of Volume Element Geometry on Convergence to a Representative Volume,” a Voronoi tessellation-based partitioning scheme is applied to generate SVE for mesoscale material strength characterization useful in fracture modeling. The resulting material property distributions are compared with those from SVE generated by square partitioning. The proportional limit stress of the SVE is used to approximate SVE strength. Superposition of elastic results is used to obtain failure strength distributions from boundary conditions at variable angles of loading. It is shown that strength properties converge more rapidly with increasing SVE size when Voronoi partitioning is used.

The constitutive parameters based on microstructure can be used in the framework of the stochastic finite element analysis of composite structures. The main aim of the paper by Strakowski and Kamiński entitled “Stochastic Finite Element Method Elasto-Plastic Analysis of the Necking Bar With Material Microdefects” is to study the significance of uncertain microdefects in structural steel subjected to tensile test on its elastoplastic large deformations with the use of the generalized stochastic perturbation method. The elastoplastic behavior of the macroscopically homogeneous material is defined by the Gurson–Tvergaard–Needleman constitutive model and the parameters involved are considered as random variables with Gaussian probability distribution. A comparison with Monte Carlo simulation as well as the semi-analytical integral technique based on the same polynomial bases confirms applicability of the proposed method for moderate input uncertainty.

In summary, the papers of this Special Section provide a glimpse of the current research trends in the field of stochastic multiscale modeling which aims at improving the safety and reliability of heterogeneous materials and structures. The Guest Editor would like to thank the authors for their contributions and the reviewers who carefully examined the submissions and provided valuable comments helping to improve the quality of the included papers.