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Fiber orientation tensors represent averaged measures of fiber orientations inside a microstructure. Although, orientation-dependent material models are commonly used to describe the mechanical properties of representative microstructure, the influence of changing or differing microstructure on the material response is rarely investigated systematically for directional measures which are more precise than second-order fiber orientation tensors. For the special case of planar orientation distributions, a set of admissible fiber orientation tensors of fourth-order is known. Fiber orientation distributions reconstructed from given orientation tensors are of interest both for numerical averaging schemes in material models and visualization of the directional information itself. Focusing on the special case of planar orientations, this paper draws the geometric picture of fiber orientation distribution functions reconstructed from fourth-order fiber orientation tensors. The developed methodology can be adopted to study the dependence of material models on planar fourth-order fiber orientation tensors. Within the set of admissible fiber orientation tensors, a subset of distinct tensors is identified. Advantages and disadvantages of the description of planar orientation states in two- or three-dimensional tensor frameworks are demonstrated. Reconstruction of fiber orientation distributions is performed by truncated Fourier series and additionally by deploying a maximum entropy method. The combination of the set of admissible and distinct fiber orientation tensors and reconstruction methods leads to the variety of reconstructed fiber orientation distributions. This variety is visualized by arrangements of polar plots within the parameter space of fiber orientation tensors. This visualization shows the influence of averaged orientation measures on reconstructed orientation distributions and can be used to study any simulation method or quantity which is defined as a function of planar fourth-order fiber orientation tensors. [ABSTRACT FROM AUTHOR]

For material modeling of microstructured media, an accurate characterization of the underlying microstructure is indispensable. Mathematically speaking, the overall goal of microstructure characterization is to find simple functionals which describe the geometric shape as well as the composition of the microstructures under consideration and enable distinguishing microstructures with distinct effective material behavior. For this purpose, we propose using Minkowski tensors, in general, and the quadratic normal tensor, in particular, and introduce a computational algorithm applicable to voxel‐based microstructure representations. Rooted in the mathematical field of integral geometry, Minkowski tensors associate a tensor to rather general geometric shapes, which make them suitable for a wide range of microstructured material classes. Furthermore, they satisfy additivity and continuity properties, which makes them suitable and robust for large‐scale applications. We present a modular algorithm for computing the quadratic normal tensor of digital microstructures. We demonstrate multigrid convergence for selected numerical examples and apply our approach to a variety of microstructures. Strikingly, the presented algorithm remains unaffected by inaccurate computation of the interface area. The quadratic normal tensor may be used for engineering purposes, such as mean field homogenization or as target value for generating synthetic microstructures. [ABSTRACT FROM AUTHOR]

Fiber orientation tensors are established descriptors of fiber orientation states in (thermo-)mechanical material models for fiber-reinforced composites. In this paper, the variety of fourth-order orientation tensors is analyzed and specified by parameterizations and admissible parameter ranges. The combination of parameterizations and admissible parameter ranges allows for studies on the mechanical response of different fiber architectures. Linear invariant decomposition with focus on index symmetry leads to a novel compact hierarchical parameterization, which highlights the central role of the isotropic state. Deviation from the isotropic state is given by a triclinic harmonic tensor with simplified structure in the orientation coordinate system, which is spanned by the second-order orientation tensor. Material symmetries reduce the number of independent parameters. The requirement of positive-semi-definiteness defines admissible ranges of independent parameters. Admissible parameter ranges for transversely isotropic and planar cases are given in a compact closed form and the orthotropic variety is visualized and discussed in detail. Sets of discrete unit vectors, leading to selected orientation states, are given. [ABSTRACT FROM AUTHOR]

Bi-planar sliding is one of main instabilities for bedding rock slope and is mainly dominated by the geometrical and strength properties of the weak structural plane. Traditional stability evaluation combines limit equilibrium method (LEM) with shear strength reduction (SSR) to derive the deterministic safety factor against slope stability. However, considering the uncertainties inherent in the geometrical and strength properties of the weak structural plane, as well as the variations of rock mass properties, the safety factor of rock bedding slopes against bi-planar sliding cannot be deterministically calculated. This study proposes a framework for probabilistic assessment on rock bedding slope stability against bi-planar sliding. Surrogate model for factor of safety against bi-planar sliding from LEM and SSR is developed based on back-propagation neural network (BPNN). The BPNN model, together with the design variables is implemented into the Excel Spreadsheet First-Order Reliability Method for the reliability assessment. [ABSTRACT FROM AUTHOR]

The nonlinear mechanical behavior of viscoelastic materials is modeled based on the Schapery integral model containing internal variables. In this context, a new approach for the strain‐dependent material properties is introduced considering a one‐dimensional formulation with strain‐dependent nonlinear functions for an oscillatory load case. In addition to the viscoelastic storage and loss modulus, the higher order harmonic oscillations in the stress response are computed and compared to experimental data from Fourier transform rheology of Polyamide 6 (PA6). The comparison reveals a good agreement between the predictions of the nonlinear model and the experimental data for the higher harmonic intensity I2/1. [ABSTRACT FROM AUTHOR]