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

• Use mesoscale elements to statistically characterize elastic and fracture behavior. • Establish convergence metrics appropriate to define isotropic and heterogeneous limits. • Compare representative volume sizes with respect to heterogeneity and anisotropy. • Develop methods to decrease inaccuracy introduced by mesoscale partitioning scheme. • Relate volume element homogenization to stochastic partial differential equations. In this paper, a method is presented to describe the statistical behavior of mesoscale material property fields, which are useful in macroscopic fracture simulation. The use of representative volume element-based properties can result in unsatisfactory fracture patterns because this approach eliminates random variations. In contrast, statistical volume elements can approximate random and inhomogeneous material properties. We compare two material property convergence metrics based on 1) vanishing variations of a property and 2) the convergence of property mean value versus volume element size. We examine the trends with which the properties of a two-phase composite tend toward homogeneous and isotropic limits. Compared to fracture properties, elastic properties reach their homogeneous and isotropic limit at smaller sizes. Accordingly, a volume element size can be chosen such that only the apparent fracture strength remains random and inhomogeneous. This reduces the computational cost and has proven useful in capturing realistic fracture results. In addition, the geometry of the mesoscale partitioning scheme has an important effect on homogenized properties. The intersection of straight edges with inclusions results in nonphysical size effects, increases anisotropy of properties, and results in 30 to 60 times larger representative volume elements compared to a partitioning scheme based on Voronoi homogenization. [ABSTRACT FROM AUTHOR]

We use square Statistical Volume Elements (SVEs) to homogenize elastic and fracture properties of a carbon-nanofiber reinforced polymer. The effect of SVE size and boundary condition (BC) on homogenized properties is investigated. Three different criteria are proposed for the convergence of properties and reaching the Representative Volume Element (RVE) limit: (1) Variation-based criterion examines the trend at which the variation of a property decreases versus SVE size; (2) Mean-based criterion examines the rate at which the SVE-mean of property converges to its terminal value; (3) BC-based criterion examines whether the homogenized properties can be independent of the choice of BC. These criteria show that the geometric property of fiber volume fraction/mass density converges to its homogeneous limit first followed by the elasticity tensor. Fracture strengths have the most complex response as they remain highly inhomogeneous, random, anisotropic, and size-dependent for the SVE sizes considered. This is further discussed in the context of stochastic multiscale material modeling and well-known size effect models for fracture strength. Furthermore, the square shape of the SVE is shown to induce nonphysical bias angles wherein elastic and fracture properties often take their maximum or minimum values at integer multiples of 45 degrees. The traction BC is shown to be superior to displacement BC by having a weaker form of this anisotropy and higher strength values. Moreover, for the anisotropic domain anisotropy indices converge from below and above for traction and displacement BCs, respectively; thus resembling Reuss and Voigt hierarchy of bounds for elastic properties. • Derive elastic and fracture properties using Statistical Volume Elements (SVEs). • Compare the angular bias in homogenized properties for displacement and traction BCs. • Determine the Representative Volume Element (RVE) size from three perspectives. • Discuss inhomogeneity/randomness, anisotropy, and size-dependency of properties. • Relate the use of SVE-homogenized properties for stochastic differential equations. [ABSTRACT FROM AUTHOR]

In predicting failure of a material, small scale material properties are critical, and yet small scale random heterogeneities are difficult to characterize accurately and efficiently. In building fracture models, work has focused on statistical characterization of local random microstructure at small to intermediate scales (termed micro- or mesoscale). Many materials can be characterized as effectively isotropic and homogeneous at a large scale. In contrast, at a micro- or mesoscale, Statistical Volume Elements (SVE) often show significant anisotropy and heterogeneity. Characterizing the properties of SVE presents challenges because modeling choices, such as the length scale, boundary conditions applied, and boundary shape chosen, may introduce modeling bias. Modeling bias must be separated from the actual prediction of anisotropic and heterogeneous characteristics at the SVE level, in order to provide an accurate basis for fracture simulation. Much literature has been devoted to characterizing the size of an RVE in order to effectively homogenize a material, however, there is less study of the effective isotropic limit. The representative size may be different when approaching the homogeneous limit or the isotropic limit. In this work, elastic and strength properties will be evaluated at multiple scales using SVE with square and circular (so-called "regular") geometry. A Voronoi tessellation based geometry, where aggregation of Voronoi cells is achieved using square or circular grids, is studied and compared with regular partitioning methods. Models will be tested on two types of microstructures, one that is isotropic at the macroscale, and one which contains a slight directional bias. Results for different SVE geometries will be compared to demonstrate how SVE modeling choices affect the characterization of anisotropy and heterogeneity at the micro- and mesoscale. • Statistical Volume Elements (SVE) are generated using different edge geometries • SVE elastic and material strength properties are evaluated at multiple scales • SVE edge geometry affects characterization of anisotropy and heterogeneity • Circular SVE edge geometry is preferable to square geometry for capturing anisotropy • SVE boundaries that avoid phase intersections improve modeling accuracy. [ABSTRACT FROM AUTHOR]

• Homogenized a mesoscopic fracture strength field using Statistical Volume Elements (SVEs). • Used Yuen-Long marble sample micocrack statistics for homogenization. • Studied the effect of the distribution shape of microcracks by using various Weibull models. • Studies the effect of the size of SVE and crack density on the homogenized strength field. • Microcrack distributions with narrower variation result in a stronger, tougher, and more brittle rock. To capture the randomness and inhomogeneity of rock at microscale, a statistical volume element (SVE) averaging approach is proposed. The microcrack statistics of a real-world Yuen-Long marble sample is used to realize 2D microcracked domains. The size effect, i. e. the decrease of the mean and variation of homogenized strength field by increasing SVE size, is analyzed. Increasing the crack density is shown to have a similar effect. While smaller SVEs maintain a greater level of inhomogeneity and are preferred for fracture analysis, it is shown that low density of microcracks pose a lower limit on the SVE size. Beside the actual power-law distribution of microcrack length, by varying the Weibull model shape parameter m other domains are created with different microcrack distribution shapes. Macroscopic fracture simulations, by the asynchronous Spacetime Discontinuous Galerkin (aSDG) method, study the effect of m for a uniaxial tensile problem. By increasing m from 0.5 to 4, the length distribution of microcracks become more uniform; this corresponds to a more uniform and stronger mesoscopic strength field, which results to about 3 and 6 times increase to macroscopic tensile strength and toughness, respectively. However, the more uniform length distribution of microcracks is shown to reduce rock brittleness. [ABSTRACT FROM AUTHOR]

• Space-angle Discontinuous Galerkin method is employed to solve radiative transfer equation. • DGM validated against exact solution. • Carried out convergence rate study. • Demonstrated the superior convergence of DGM simultaneously applied to space and angle. The radiative transfer equation (RTE) for a plane-parallel problem involving scattering, absorption and radiation is solved using the discontinuous Galerkin (DG) finite element method (FEM). Both space and angle directions are discretized by the DG method. Thus, while the method has a higher accuracy in angle direction than hybrid FEM-Discrete Ordinate (S N) and FEM-Spherical Harmonic (P N) methods, it removes the continuity constraint implied by the form of basis function for continuous FEMs in space and angle. The discrete formulation of the problem is presented for nonzero phase function and a variety of boundary conditions. The numerical results demonstrate a p + 1 convergence rate when the intensity is interpolated by an order p polynomial in both space and angle. The method is validated against exact solutions, and compared with other space-angle and hybrid FEMs for a few benchmark problems. The appropriateness of the DG formulation for problems with discontinuous solution is demonstrated by solving a problem with delta source term, where an in-element averaging of the source term eliminates negative intensity values for high order elements. Finally, a problem with an angular-line source term and a convergence study where the solution order is zero in angle are used to further demonstrate the advantages of the high order space-angle DG formulation; for the convergence study problem, the error was reduced by about 13 binary orders of magnitude, by increasing the order in both space and angle, rather than in space only. [ABSTRACT FROM AUTHOR]