## LSU Doctoral Dissertations

#### Title

Non-local Methods in Fracture Dynamics

#### Degree

Doctor of Philosophy (PhD)

Mathematics

Dissertation

#### Abstract

We first introduce a regularized model for free fracture propagation based on non-local potentials. We work within the small deformation setting and the model is developed within a state based peridynamic formulation. At each instant of the evolution we identify the softening zone where strains lie above the strength of the material. We show that deformation discontinuities associated with flaws larger than the length scale of non-locality $\delta$ can become unstable and grow. An explicit inequality is found that shows that the volume of the softening zone goes to zero linearly with the length scale of non-local interaction. This scaling is consistent with the notion that a softening zone of width proportional to $\delta$ converges to a sharp fracture set as the length scale of nonlocal interaction goes to zero. Here the softening zone is interpreted as a regularization of the crack network. Inside quiescent regions with no cracks or softening the nonlocal operator converges to the local elastic operator at a rate proportional to the radius of nonlocal interaction. This model is designed to be calibrated to measured values of critical energy release rate, shear modulus, and bulk modulus of material samples. For this model one is not restricted to Poission ratios of $1/4$ and can choose the potentials so that small strain behavior is specified by the isotropic elasticity tensor for any material with prescribed shear and Lam\'e moduli.

Then a model for dynamic damage propagation is developed using non-local potentials.

The model is posed using a state based peridynamic formulation.

The resulting evolution is seen to be well posed. At each instant of the evolution we identify a damage set. On this set the local strain has exceeded critical values either for tensile or hydrostatic strain and damage has occurred. The damage set is nondecreasing with time and is associated with damage state variables defined at each point in the body. We show that a rate form of energy balance holds at each time during the evolution. Away from the damage set we show that the nonlocal model converges to the linear elastic model in the limit of vanishing nonlocal interaction.

6-29-2018

Lipton, Robert

COinS