Abstract

Understanding the failure of brittle heterogeneous materials is essential in many applications. Heterogeneities in material properties are frequently modeled through random fields, which typically induces the need to solve finite element problems for a large number of realizations. In this context, we make use of reduced order modeling to solve these problems at an affordable computational cost. This paper proposes a reduced order modeling framework to predict crack propagation in brittle materials with random heterogeneities. The framework is based on a combination of the Proper Generalized Decomposition (PGD) method with Griffith’s global energy criterion. The PGD framework provides an explicit parametric solution for the physical response of the system. We illustrate that a non-intrusive sampling-based technique can be applied as a postprocessing operation on the explicit solution provided by PGD.We first validate the framework using a global energy approach on a deterministic two-dimensional linear elastic fracture mechanics benchmark. Subsequently, we apply the reduced order modeling approach to a stochastic fracture propagation problem.

Abstract

We present a phase field model to simulate brittle fracture in an initially straight Euler–Bernoulli beam, with generalization to curved beams. We start from formulating the problem with the principle of minimum potential energy in a 3D solid, with the displacement field and the phase field as primary arguments. We then select, for each cross section, representative fields that characterize the said cross section, including the beam deflection and rotation, and two independent ansatz variables within the cross section to represent the phase field. The problem then reduces to a minimization with only one-dimensional field variables. A feature of the proposed method is, without discretizing the phase field within the cross section, it can represent its variation within the cross section, allowing to simulate cracks partially going through the thickness due to bending as well as axial loads.

Abstract

The phase-field modeling and its adaptive moving mesh solution for studying initiation and propagation of brittle fracture will be presented. Challenges such as non-smoothness of the energy functional, violation of fracture boundary conditions, and the need for mesh adaptation, and possible remedies for these challenges will be discussed. In particular, a moving mesh finite element method will be presented for the numerical solution of the phase-field model for brittle fracture.