Relationships between the material parameters of continuum-kinematics-inspired peridynamics and isotropic linear elasticity for two-dimensional problems

Continuum-kinematics-inspired Peridynamics (CPD) has been recently proposed as a geometrically exact formulation of peridynamics that is also thermodynamically and variationally consistent. CPD can capture the Poisson effect exactly, unlike the original formulation of peridynamics (PD). Due to its geometrically exact nature, CPD does not suffer from zero-energy modes and displacement oscillations that may be observed in state-based PD formulations. For a two-dimensional analysis, CPD builds upon one-neighbor and two-neighbor interactions. The one-neighbor interactions of CPD are equivalent to the bond-based interactions of the original PD formalism. Two-neighbor interactions, however, are key in CPD since the basic notions of classical continuum kinematics, namely length and area, are preserved exactly. The isotropic two-dimensional CPD formulation of non-local elasticity therefore involves two material constants, namely C1 and C2, associated with length and area, respectively. This manuscript aims to establish relationships between the material parameters of CPD and isotropic linear elasticity for an affine deformation in a two-dimensional setting. It is shown that each of the CPD material parameters can be expressed in terms of any pairs of isotropic linear elasticity constants, such as Lamé parameters. Finally, we establish the admissible ranges for CPD material parameters.