FAILURE ANALYSIS OF BRITTLE MATERIALS USING MULTISCALE PHASE FIELD METHOD

Abstract

The prediction of fracture failure in engineering components and structures is of great importance as it finds critical applications in structural, mechanical, aerospace, robotics and biomedical fields. All engineering components and structures are inherently made of heterogeneous materials. Understanding the crack growth phenomena of these materials is much more complex as crack experiences multiple crack nucleation, crack arrest, crack deflection, crack branching and crack merging. This complex crack growth behavior drastically affects the overall fracture toughness and load carrying capacity of the heterogeneous materials. Thus, understanding the failure mechanisms in terms of crack initiation and propagation is a topic of great interest to the research community. Over the years, various numerical approaches (discrete and smeared/diffused) have been proposed to predict the failures based on crack initiation and propagation. In discrete approaches, a sharp crack is modeled by conformal mesh or by modifying displacement approximation to represent jump in the displacement. In addition to this, meandering crack topology is tracked and updated using different criteria. Alternatively, in a smeared approach such as phase field method (PFM), fracture failure is governed by two coupled partial differential equations. A crack in the phase field method is defined by a continuous scalar parameter known as phase field variable. The value of this parameter indicates the fractured and intact state of the material. The evolution of phase field variable is governed by the energy released during deformation. In the phase field model, the width of smeared crack is controlled by a length scale parameter for its practical feasibility. Thus, this approach has the capability to predict crack nucleation, crack branching, merging of cracks, crack growth on a complex path and accurate representation of crack front in 3-D crack growth simulations. The simulation of fracture failure using standard PFM requires highly dense (refined) mesh to obtain small length scale parameter for the representation of a sharp crack in the structure. In phase field modeling, the requirement of fine mesh in the entire domain or locally refined meshes (non-adaptive) leads to high computational efforts. Moreover, fracture analysis of heterogeneous material with a large number of small discontinuities (voids/inclusions) requires a large number of finite elements to satisfy the need of conformal meshing. This makes the computational model very bulky and inefficient for the purpose of simulation. To overcome these shortcomings of the standard phase field model, there is need to develop a multiscale framework based phase field method which not only provides adaptively refined mesh but also efficiently simulates the brittle fracture in the homogenous and heterogeneous materials. Initially, to enhance the numerical efficiency of the existing phase field model, the phase field method (PFM) is coupled with the multiscale finite element method (MsFEM). An Abstract iii adaptive mesh refinement strategy is developed to improve the computational efficiency of the phase field model. This coupled approach with adaptive mesh refinement strategy is named as adaptive phase field method (AMPFM). The developed AMPFM codes are used to simulate quasi-static brittle fracture in homogenous material. The obtained AMPFM results for some standard cases are validated through reference solutions available in the literature. AMPFM is further extended to model complex crack growth mechanism (i.e., crack arrest, crack deflection, crack coalescence, and multiple cracks initiation) observed in heterogeneous materials (bi-material, fibre reinforced composite and particulate composite). The interaction of a pre-existing crack with weak or strong interface between matrix-fibre in fiber reinforced composite is also studied. The developed multiscale AMPFM framework is found very useful for modeling the periodically distributed heterogeneities in the particulate composite. Moreover, it is demonstrated that the use of AMPFM significantly reduces the size of the computational model. Crack nucleation and growth starts in the vicinity of a pre-existing crack. Therefore, a local moving phase field method (LMPFM) has been proposed in which the phase field crack evolutions are solved in a small local region of the domain. In LMPFM, PFM, XFEM and MsFEM are combined in such a way that the MsFEM acts as a medium to couple coarse mesh with fine mesh, and XFEM models the crack surface outside the phase field region. Thus, LMPFM not only provides an adaptively refined mesh in a small region near the smeared discontinuity but also fulfils the prerequisite of refined mesh for solving phase field crack evolution equations. The results obtained by LMPFM for crack propagation in homogenous materials are compared with the standard PFM. It is also demonstrated that the LMPFM is computationally more efficient as compared to standard PFM and AMPFM. Finally, LMPFM is applied to simulate crack growth in periodically heterogeneous material containing three different types of inhomogeneities such as (a) periodically distributed voids, (b) periodically distributed stiff inclusions with perfect interface, and (c) periodically distributed stiff inclusions with weak interface (finite thickness). The results obtained by LMPFM for heterogeneous material are compared with results of homogenous material. The capability of AMPFM and LMPFM is demonstrated by simulating brittle fracture in homogenous and heterogeneous structures. For few standard problems, crack growth paths and load-displacement curves are compared with standard PFM and results available in the literature. A good agreement was observed. Both AMPFM and LMPFM provide a mesh refinement scheme, which automatically keep track of growing crack and refines the domain near the crack. This mesh refinement scheme significantly reduces the size of the computational model.