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 post-processing 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.
Highlights
One of the important goals in engineering design is to avoid catastrophic failure
The simulation of fracture evolution in the linear elastic fracture mechanics (LEFM) framework typically involves a stepwise incrementation of the crack path based on the evaluation of the fracture criterion, which implies that a linear elasticity problem must be solved at each step in the propagation process
When we compute the mean of the error epgd(lc) over the complete parameter domain, i.e., error measure (35), we observe from Fig. 5b that this mean energy error is essentially independent of the mesh size for the finer meshes (h 0.25)
Summary
One of the important goals in engineering design is to avoid catastrophic failure. Besides, in many applications, it is often crucial to understand the failure processes. The simulation of fracture evolution in the LEFM framework typically involves a stepwise incrementation of the crack path based on the evaluation of the fracture criterion, which implies that a linear elasticity problem (with a tip singularity) must be solved at each step in the propagation process. This finite element procedure is typically computationally expensive because, on account of accuracy and stability requirements, the crack length increments must generally be small, and because some form of mesh adaptation is required to accommodate changes in fracture geometry.
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