Because it makes use of partial differential equations, the classical theory of continuum mechanics encounters immediate difficulties in applications involving fracture and other discontinuities. In spite of the development of special techniques for fracture mechanics, it perhaps remains a weakness in the classical theory that the fundamental equations cannot be applied directly when cracks are present. For this reason, an alternative formulation of solid mechanics called the peridynamic theory has been proposed. Because it is based on integral rather than differential equations, the mathematics of the peridynamic theory can be applied with equal validity to smooth deformations and to deformations containing singularities such as cracks.
When applied to brittle elastic membranes, the peridynamic model yields direct predictions of material deformation and failure, particularly by fracture and phase transformation. This approach avoids the need for supplemental equations that control the kinetics of crack growth; fracture occurs spontaneously in the model according to the equations of motion and constitutive model. Any number of cracks can appear and interact with each other. Another property of the theory is that due to its nonlocal nature, it avoids the unbounded strains at crack tips that are predicted by the classical theory of finite elasticity. Predicted phenomenology of dynamic fracture, including crack trajectories and velocities, also appear reasonable.
This talk will present the peridynamic theory in the context of large deformation of elastic membranes. Numerical examples will include dynamic fracture, oscillating crack paths, and peeling of thin sheets.
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