This paper presents a reformulation of elasticity theory for materials with discontinuities and long-range forces, known as the peridynamic model. The peridynamic formulation avoids the use of spatial derivatives, which are problematic at discontinuities, and instead uses integration to compute forces between particles. This allows the same equations to be applied both on and off discontinuities, such as cracks. The model is based on a pairwise force function that depends on the relative displacement and position of particles. The formulation is isotropic, meaning it is independent of orientation, and can be microelastic, meaning it conserves energy. The model is shown to be equivalent to conventional elasticity theory in certain cases, but it allows for the modeling of discontinuities and long-range forces that are not possible in conventional elasticity. The paper also discusses the linearization of the model and the conditions for material stability. The peridynamic model is shown to be more general than conventional elasticity theory, but it is also less general in some respects. The paper concludes that the peridynamic model is a useful tool for modeling materials with discontinuities and long-range forces.This paper presents a reformulation of elasticity theory for materials with discontinuities and long-range forces, known as the peridynamic model. The peridynamic formulation avoids the use of spatial derivatives, which are problematic at discontinuities, and instead uses integration to compute forces between particles. This allows the same equations to be applied both on and off discontinuities, such as cracks. The model is based on a pairwise force function that depends on the relative displacement and position of particles. The formulation is isotropic, meaning it is independent of orientation, and can be microelastic, meaning it conserves energy. The model is shown to be equivalent to conventional elasticity theory in certain cases, but it allows for the modeling of discontinuities and long-range forces that are not possible in conventional elasticity. The paper also discusses the linearization of the model and the conditions for material stability. The peridynamic model is shown to be more general than conventional elasticity theory, but it is also less general in some respects. The paper concludes that the peridynamic model is a useful tool for modeling materials with discontinuities and long-range forces.