This report, authored by S. A. Silling and published by Sandia National Laboratories, proposes a new framework for continuum mechanics called the "peridynamic" formulation. This framework is designed to better model materials that naturally form discontinuities, such as cracks, by avoiding the need to specify the location of these discontinuities. The peridynamic equations of motion and equilibrium are derived, and the concept of material stability is investigated. The paper discusses the propagation of linear stress waves and wave dispersion relations. An example is provided to demonstrate how the reformulated approach can solve fracture problems using the same equations on or off the crack surface or tip. The report also explores isotropy, microelasticity, structureless materials, harmonic materials, and the relationship between the peridynamic and conventional theories of elasticity. It concludes with a discussion on linearization and the conditions for material stability.This report, authored by S. A. Silling and published by Sandia National Laboratories, proposes a new framework for continuum mechanics called the "peridynamic" formulation. This framework is designed to better model materials that naturally form discontinuities, such as cracks, by avoiding the need to specify the location of these discontinuities. The peridynamic equations of motion and equilibrium are derived, and the concept of material stability is investigated. The paper discusses the propagation of linear stress waves and wave dispersion relations. An example is provided to demonstrate how the reformulated approach can solve fracture problems using the same equations on or off the crack surface or tip. The report also explores isotropy, microelasticity, structureless materials, harmonic materials, and the relationship between the peridynamic and conventional theories of elasticity. It concludes with a discussion on linearization and the conditions for material stability.