**Spandan Maiti**, Michigan Technological University

**ABSTRACT**

An improvement in the fundamental understanding of crack propagation in geological formations will contribute to advancement of knowledge and help in hazard mitigation and extraction of natural resources. Current approaches to modeling crack propagation consider mostly phenomena occurring ahead of the crack tip. However, the significant effect of microstructures present at the crack wake remains unaddressed, resulting in incomplete understanding. We pursued a research program to advance the fundamental understanding of crack propagation in geological materials by following a central hypothesis that the effective fracture toughness of geological material is significantly altered by the presence of microstructural features at the crack level, operative at the wake of the propagating crack. Novel numerical algorithms have been developed to simulate arbitrary fast crack propagation in a domain, as well as the sensitivity analysis of various input parameters on the output.

**INTRODUCTION**

Our central hypothesis for the proposed research is that the microstructural features at the crack level are operative at the wake of the propagating crack and it significantly alters the effective fracture toughness of the material. Unlike in homogeneous materials, a crack cannot propagate in its own plane in face of these heterogeneities thus giving rise to the crack surface features mentioned above. We proposed to test the central hypothesis by evaluating the effect of both grain bridging and presence of pre-existing flaws on crack propagation, and quantify their effect on the fracture toughness. This endeavor resulted in two computational challenges: (1) A computational technique for arbitrary fast crack propagation in a domain, and (2) development of sensitivity analysis technique to study the effect of microstructural features on fracture toughness. To address the first challenge, we developed a new technique called Generalized ‚ohesive Element (GCE) method within the framework of finite element analysis that can nucleate and propagate a fast crack in an arbitrary direction. For the second challenge, we have developed an ANOVA decomposition based algorithm that can compute the sensitivity indices for input parameters for a given output. These two directions will be discussed briefly in subsequent sections.

**RESEARCH ACHIEVEMENTS**

In the presence of heterogeneities, a crack in a geological material can deflect out of its plane that can significantly affect the apparent fracture toughness. We have developed a novel computational technique to simulate arbitrary crack motion in a domain. Salient features are discussed below:

**Generalized Cohesive Element (GCE)
modeling:** An alternative approach to the classical theory of fracture
mechanics is the cohesive zone modeling. However, typical cohesive element
algorithms are restricted to the edges of the finite elements, thus making the
potential crack path mesh dependent. We have developed a new technique that can
pass a crack through a finite element, thus approximating actual crack path
closely. The original element is split (Figure 1) into two child elements to
allow for the crack path within the element. Cracks are nucleated whenever
maximum principal value reaches the failure strength of the material. Once
nucleated, it can propagate through adjacent elements
subjected to the satisfaction of the failure criterion. The direction of

Figure 2. A single tetrahedral finite element and associated graph. |

**Mesh topology: **Most challenging feature
of the said technique is the enumeration of mesh topology. As new child
elements are created, the topology of the finite element mesh changes
continually, and element connectivity table need to be updated continuously. We
have devised an efficient way to handle the resulting dynamic data structure.
The mesh is represented as a graph as shown in Figure 2, and all the cutting
and duplication operations, that give rise to the changing topology, are
performed on the graph itself. The connectivity table required for finite
element computations can be extracted from the graph at any point.

**Sensitivity analysis: **The computational
model is represented as ANOVA decomposition with the coefficients of the
expansion as the variances of different input parameters:

In the above
equation, the term *f*_{0 }gives
the mean effect which is a zeroth-order constant. The first order term *f _{i }(x_{i}) *gives the
individual contribution of an input variable

**PROJECT OUTCOME**

Figure 3. Simulations of arbitrary crack path with generalized cohesive elements. |

Moving Mountains; Dr. Surpless

Desert Sea Fossils; Dr. Olszewski

Lighting Up Metals; Dr. Assefa

Ecological Polymers; Dr. Miller

Copyright © 2011 American Chemical Society