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This work presents a framework to study the quasi-static brittle fracture in functionally graded materials with random material properties using a non-intrusive phase field method. The model accounts for variation in the material and fracture properties. Further, the effective Young’s modulus, the gradient index and the fracture properties of the constituent materials are considered as random input variables. The output randomness is represented by polynomial chaos expansion. Two-dimensional benchmark numerical examples are presented to demonstrate the benefits of the non-intrusive approach to study fracture characteristics. The influence of the randomness in the material properties on the peak load carrying capacity is considered as a primary quantity of interest in this study. A systematic parametric study is carried out to bring out the sensitivity of the input randomness on the stochastic output response using Sobol indices.

This paper aims to study the effect of stress-diffusion interactions and its effect on the localization of the plastic strain in an elastoplastic material using a fully coupled chemo-mechanical sys...

In this work, the fracture behavior of an epoxy coated hydrating cement paste system is investigated with the aid of a slant shear test. A finite element model based on the phase-field method (PFM) is used to simulate the crack propagation and understand the transitions from structural to interfacial fracture and vice versa. At construction sites, it is a usual practice to coat epoxy only after the concrete is completely cured, i.e. after 28 days. However, during hydration, the surface energy of the cement paste varies with time. Therefore, it is necessary to investigate the temporal dependency of the interfacial strength on the instant (time) of coating. With this in mind, we investigated fracture processes in the cement paste, coated with epoxy at various time instants, from an early age (2 days) to the later age (28 and 50 days) of cement paste. From the experimental and numerical results, the cohesive failure of cement paste (bulk or structural failure) is primarily observed for 2-days hydrated cement paste. In contrast, adhesive failure at the interface between cement paste and epoxy (interface fracture) is observed for 28 and 50-days hydrated cement paste. In our fracture analysis, we have incorporated the surface roughness and the bulk porosity of cement paste which evolves over time during the hydration process. From the results of numerical analysis, it is found that the fracture behavior of the cement-epoxy system can be controlled by tuning the material properties of the cement paste and the interface i.e., elastic modulus, E and fracture energy G c . Also, it is observed that the surface roughness and the bulk porosity of cement paste significantly influence the crack trajectory as well as the load–displacement response of the cement-epoxy system.

In this work, we propose an adaptive phase field method (PFM) to simulate quasi-static brittle fracture problems. The phase field equations are solved using the scaled boundary finite element method (SBFEM). The adaptive refinement strategy is based on an error indicator evaluated directly from the solutions of the SBFEM without any need for stress recovery techniques. Quadtree meshes are adapted to perform mesh refinement. The polygons with hanging nodes in the quadtree decomposition are treated as n − sided polygons within the framework of the SBFEM and do not require any special treatment in contrast to the conventional finite element method. Several benchmark problems are used to demonstrate the robustness and the efficacy of the proposed technique. The adaptive refinement strategy reduces the mesh burden when adopting the PFM to model fracture. Numerical results show an improvement in the computational efficiency in terms of the number of elements required in the standard PFM without compromising the accuracy of the solution.

The fracture processes of the constant/variable stiffness composite laminate (CSCL/VSCL) subjected to uni-axial tension is investigated using the phase-field method. The main objectives of this work are: (a) to study the crack propagation path and (b) to investigate the influence of the fiber-matrix interface on the crack path. In particular, the emphasis is to explore the role played by the orientation of the fiber and the inter-fiber spacing on the fracture pattern in the composite. From the numerical study, it is inferred that: (a) the cracking pattern of the composite is significantly influenced by the orientation of the fiber and (b) the load carrying capacity of the composite enhances with the increase in the orientation of the fiber. For a larger inter-fiber spacing, the load-displacement curve does not alter significantly for different orientations of the fiber. Furthermore, a novel approach to predict the load-displacement curve of the composite using an equivalent homogenized orthotropic material is presented, showing a very good agreement between them. It is seen that the location of the crack in the VSCL influences the failure load.

In this chapter, by employing a basis B-splines based finite element method, the natural frequency, critical buckling load and critical aerodynamic pressure of tow-steered composite laminate is numerically studied. The distinguishing feature of tow-steered composites when compared to conventional laminated composites is that in the former, a spatial variation of the fiber is considered. The plate kinematics accounts for the transverse shear deformations and an artificial shear correction factor is introduced to alleviate the shear locking problem. The effect of plate thickness, spatial variation of the fiber and the boundary conditions are systematically studied.

In this work, we extend the recently proposed adaptive phase field method to model fracture in orthotropic functionally graded materials (FGMs). A recovery type error indicator combined with quadtree decomposition is employed for adaptive mesh refinement. The proposed approach is capable of capturing the fracture process with a localized mesh refinement that provides notable gains in computational efficiency. The implementation is validated against experimental data and other numerical experiments on orthotropic materials with different material orientations. The results reveal an increase in the stiffness and the maximum force with increasing material orientation angle. The study is then extended to the analysis of orthotropic FGMs. It is observed that, if the gradation in fracture properties is neglected, the material gradient plays a secondary role, with the fracture behaviour being dominated by the orthotropy of the material. However, when the toughness increases along the crack propagation path, a substantial gain in fracture resistance is observed.

In the recent years, the phase field method for simulating fracture problems has received considerable attention. This is due to the salient features of the method: 1) it can be incorporated into any conventional finite element software; 2) has a scalar damage variable is used to represent the discontinuous surface implicitly and 3) the crack initiation and subsequent propagation and branching are treated with less complexity. Within this framework, the linear momentum equations are coupled with the diffusion type equation, which describes the evolution of the damage variable. The coupled nonlinear system of partial differential equations are solved in a ‘staggered’ approach. The present work discusses the implementation of the phase field method for brittle fracture within the open-source finite element software, FEniCS. The FEniCS provides a framework for the automated solutions of the partial differential equations. The details of the implementation which forms the core of the analysis are presented. The implementation is validated by solving a few benchmark problems and comparing the results with the open literature.

We present a phase field formulation for fracture in functionally graded materials (FGMs). The model builds upon homogenization theory and accounts for the spatial variation of elastic and fracture properties. Several paradigmatic case studies are addressed to demonstrate the potential of the proposed modelling framework. Specifically, we (i) gain insight into the crack growth resistance of FGMs by conducting numerical experiments over a wide range of material gradation profiles and orientations, (ii) accurately reproduce the crack trajectories observed in graded photodegradable copolymers and glass-filled epoxy FGMs, (iii) benchmark our predictions with results from alternative numerical methodologies, and (iv) model complex crack paths and failure in three dimensional functionally graded solids. The suitability of phase field fracture methods in capturing the crack deflections intrinsic to crack tip mode-mixity due to material gradients is demonstrated. Material gradient profiles that prevent unstable fracture and enhance crack growth resistance are identified: this provides the foundation for the design of fracture resistant FGMs. The finite element code developed can be downloaded from www.empaneda.com/codes., E. Martínez-Pañeda acknowledges financial support from the Royal Commission for the 1851 Exhibition through their Research Fellowship programme (RF496/2018).

Hydraulic fracture (or hydrofracture) can promote the propagation of meltwater-filled surface crevasses in glaciers and, in some cases, lead to full-depth penetration that can enhance basal sliding and iceberg calving. Here, we propose a novel poro-damage phase field model for hydrofracturing of glacier crevasses, wherein the crevasse is represented by a nonlocal damage zone and the effect of hydrostatic pressure due to surface meltwater is incorporated based on Biot’s poroelasticity theory. We find that the elastic strain energy decomposition scheme of Lo et al. (2019) with an appropriate fracture energy threshold can adequately represent the asymmetric tensile–compressive fracture behavior of glacier ice subjected to self-gravity loading. We assessed the performance of the model against analytical linear elastic fracture mechanics solutions by comparing their predictions of maximum crevasse penetration depth. The model simulates both surface crevasse propagation in the interior region of the glacier, as well as cliff failure in the terminus region. The excellent performance of the proposed model for air/water-filled surface crevasses in idealized land- and marine-terminating grounded glaciers illustrates its applicability to studying the dynamic response of glaciers to atmospheric warming.

In this work, we propose a framework for an adaptive contact analysis in deformable solids using the effective error indicator from the scaled boundary finite element method (SBFEM) with a quadtree decomposition. Further, the SBFEM is implemented with the commercial finite element software, Abaqus, to perform the contact analysis by employing the user element subroutine (UEL) feature. The SBFEM error indicator coupled with the quadtree decomposition is implemented in Matlab and allowed to interact with the Abaqus using .inp file for an adaptive refinement. The detailed implementation of the framework, input data format, and the UEL subroutine which is one of the key features of the proposed work are clearly explained. The effectiveness of the proposed framework is demonstrated by solving several contact problems of engineering significance. The developed SBFEM code can be downloaded from https://github.com/nsundar/sbfem .

In this paper, an adaptive phase-field scaled boundary finite element method for fracture in functionally graded material (FGM) is presented. The model accounts for spatial variation in the material and fracture properties. The quadtree decomposition is adopted for refinement, and the refinement is based on an error indicator evaluated directly from the solutions of the scaled boundary finite element method. This combination makes it a suitable choice to study fracture using the phase field method, as it reduces the mesh burden. A few standard benchmark numerical examples are solved to demonstrate the improvement in computational efficiency in terms of the number of degrees of freedom.