Effects of characteristic element length on fracture energy dissipation in continuum damage mechanics models

Progressive damage finite element (FE) analysis methods based on continuum damage mechanics (CDM) use mesh regularization algorithms to ensure that fracture energy dissipation predictions are independent of problem discretization. Mesh regularization algorithms require some geometric input related to the discretization. When using crack band theory for mesh regularization, a characteristic element length is used to approximate the width of the region affected by the continuum crack, i.e., the crack band width. Inaccuracy in representing the crack band width significantly affects predictions in terms of fracture energy dissipation. For square elements misaligned by 45°, using a typical line length across an element rather than the crack band width overestimates dissipated fracture energy by 41%. Not accounting for element aspect ratio underestimates dissipated fracture energy by 29% and 50% for ratios of two and four, respectively. Herein, methods for calculating characteristic element lengths in fiber-reinforced materials are presented that account for meshes being misaligned with respect to material directions, element aspect ratio, and element skew. The limits of applicability of different crack band width approximations are explored through numerical crack growth studies and center notch tension FE analyses for different discretizations. Results are compared in terms of fracture energy dissipation to linear elastic fracture mechanics. Analyses with the proposed characteristic element lengths predict consistent fracture energy dissipation with various meshes. The proposed methods and the included studies on potential error in fracture energy dissipation provide analysts the basis to better understand error in CDM model predictions associated with simplified FE model preprocessing.

Fracture Models and Effect of Fibers on Fracture Properties of Cementitious Composites—A Review

Cementitious composites have good ductility and pseudo-crack control. However, in practical applications of these composites, the external load and environmental erosion eventually form a large crack in the matrix, resulting in matrix fracture. The fracture of cementitious composite materials causes not only structural insufficiency, but also economic losses associated with the maintenance and reinforcement of cementitious composite components. Therefore, it is necessary to study the fracture properties of cementitious composites for preventing the fracture of the matrix. In this paper, a multi-crack cracking model, fictitious crack model, crack band model, pseudo-strain hardening model, and double-K fracture model for cementitious composites are presented, and their advantages and disadvantages are analyzed. The multi-crack cracking model can determine the optimal mixing amount of fibers in the matrix. The fictitious crack model and crack band model are stress softening models describing the cohesion in the fracture process area. The pseudo-strain hardening model is mainly applied to ductile materials. The double-K fracture model mainly describes the fracture process of concrete. Additionally, the effects of polyvinyl alcohol (PVA) fibers and steel fibers (SFs) on the fracture properties of the matrix are analyzed. The fracture properties of cementitious composite can be greatly improved by adding 1.5–2% PVA fiber or 4% steel fiber (SF). The fracture property of cementitious composite can also be improved by adding 1.5% steel fiber and 1% PVA fiber. However, there are many problems to be solved for the application of cementitious composites in actual engineering. Therefore, further research is needed to solve the fracture problems frequently encountered in engineering.

Development and Evaluation of Crack Band Model Implemented Progressive Failure Analysis Method for Notched Composite Laminate

Progressive failure analysis (PFA) is widely used to predict the failure behavior of composite materials. As a structure becomes more complex with discontinuities, prediction of failure becomes more difficult and mesh dependence must be taken into account. In this study, a PFA model was developed using the Hashin failure criterion and crack band model. The failure initiation was evaluated using the Hashin failure criterion. If failure initiation occurred, the damage variables at each failure mode (fiber tension and compression; matrix tension and compression) were calculated according to linear softening degradation and they were then used to derive the damaged stiffness matrix. This matrix reflected a degraded material, and PFA was continued until the damage variables became “1,” implying complete material failure. A series of processes were performed using the finite element method program ABAQUS with a user-defined material subroutine. To evaluate the proposed PFA model, experimental results of open-hole composite laminate tests were compared with the obtained numerical results. The strain behaviors were compared using a digital image correlation system. The obtained numerical results were in good agreement with the experimental ones.

Strain localization in reduced-order asymptotic homogenization

A reduced-order asymptotic homogenization-based multiscale technique that can capture damage and inelastic effects in composite materials is proposed. This technique is based on a two-scale homogenization procedure where eigenstrain representation accounts for the inelastic response and the computational efforts are alleviated by a reduction-of-order technique. Macroscale stress is derived by calculating the influence tensors from analysis of a representative volume element. At microscale, the damage in the material is modeled using a framework based on continuum damage mechanics. To solve the problem of strain localization, a method of alteration of the stress–strain relation of microconstituents based on the dissipated fracture energy in a crack band is implemented. The issue of spurious postfailure artificial stiffness at macroscale is discussed and the effect of increasing the order to alleviate this problem is checked. Verification studies demonstrated that the proposed formulation predicts the macroscale response and also captures the damage- and plasticity-induced inelastic strains.

Branching of hydraulic cracks enabling permeability of gas or oil shale with closed natural fractures

While hydraulic fracturing technology, aka fracking (or fraccing, frac), has become highly developed and astonishingly successful, a consistent formulation of the associated fracture mechanics that would not conflict with some observations is still unavailable. It is attempted here. Classical fracture mechanics, as well as current commercial software, predict vertical cracks to propagate without branching from the perforations of the horizontal well casing, which are typically spaced at 10 m or more. However, to explain the gas production rate at the wellhead, the crack spacing would have to be only about 0.1 m, which would increase the overall gas permeability of shale mass about 10,000×. This permeability increase has generally been attributed to a preexisting system of orthogonal natural cracks, whose spacing is about 0.1 m. However, their average age is about 100 million years, and a recent analysis indicated that these cracks must have been completely closed by secondary creep of shale in less than a million years. Here it is considered that the tectonic events that produced the natural cracks in shale must have also created weak layers with nanocracking or microcracking damage. It is numerically demonstrated that seepage forces and a greatly enhanced permeability along the weak layers, with a greatly increased transverse Biot coefficient, must cause the fracking to engender lateral branching and the opening of hydraulic cracks along the weak layers, even if these cracks are initially almost closed. A finite element crack band model, based on a recently developed anisotropic spherocylindrical microplane constitutive law, demonstrates these findings [Rahimi-Aghdam S, et al. (2018) arXiv:1212.11023].