*Mathematical Analysis of the Nonlocal State Based Peridynamic Models*

**Kun Zhou**

Penn State University

Department of Mathematics

`zhou@math.psu.edu`

**Abstract**: We develop a calculus for nonlocal operator of
vector-valued functions. It allows us to reduce the regularity requirements
associated with the classical identities. We then use the nonlocal calculus
to reformulate the linear peridynamic state models, and to show that under
certain stability condition and some mild requirements on the influence
functions, the state model is well-posed, and its energy space is equivalent
to the L2 space. We also relate the local limit of the linear peridynamic
state model to the Navier equation of linear elasticity with arbitrary Poisson
ratio.