The peridynamic correspondence theory is based on the nonlocal peridynamic momentum equation. However, it can be applied to constitutive models from the local theory. Therefore, the local kinematic description of the continuum can be used (see e.g. Wriggers [
36] and Holzapfel [
16]). For an elastic body
\( \Omega \) the current position of a material point
\( \mathbf {x} \) is given by
$$\begin{aligned} \mathbf {x} = \mathbf {X} + \mathbf {u} \end{aligned}$$
(1)
with the initial position
\( \mathbf {X} \) and the displacement
\( \mathbf {u} \). The deformation gradient is defined as
$$\begin{aligned} \mathbf {F} = \frac{\partial \mathbf {x}}{\partial \mathbf {X}} = {\text {Grad}}\,\mathbf {x} \ \ \ \ \ \text {with}\ \ \ \ \ J = {\text {det}}\,\mathbf {F}\, \end{aligned}$$
(2)
where the Jacobian
J describes the volumetric part. We also define the right Cauchy-Green tensor
\( \mathbf {C} \) by
$$\begin{aligned} \mathbf {C} = {\mathbf {F}}^T \cdot {\mathbf {F}} \end{aligned}$$
(3)
and its isochoric part as
$$\begin{aligned} \mathbf {C}^{iso} = J^{-\frac{2}{3}} \mathbf {C}. \end{aligned}$$
(4)
The body
\( \Omega \) is subject to the momentum equation at each particle
\( \mathbf {X} \). It states in the non-local peridynamic form
$$\begin{aligned} \int _{H} \left( \mathbf {t} - \mathbf {t}' \right) \ d H + \rho _{0} \overline{\mathbf {b}} = \mathbf {0} \end{aligned}$$
(5)
as an integral over momentum transfers with neighboring material points. Herein,
\( \mathbf {t} \) and
\( \mathbf {t}' \) are called the pairwise force densities. While
\( \mathbf {t} \) stands for the force acting on
\( \mathbf {X} \) exerted by a neighboring material point
\( \mathbf {X}' \) inside the family
H (see Fig.
1), the pairwise part
\( \mathbf {t}' \) arises from the collective deformation of family
\( H' \) of
\( \mathbf {X}' \) and is considered by means of Newton’s third law.
\( \rho _{0}\overline{\mathbf {b}} \) is a body force acting on
\( \mathbf {X} \). The local counterpart is defined with respect to the initial configuration as
$$\begin{aligned} {\text {Div}}\,\mathbf {P} + \rho _{0} \overline{\mathbf {b}} = \mathbf {0} \end{aligned}$$
(6)
with the first Piola–Kirchhoff stress tensor
\( \mathbf {P} \). A correspondence formulation
\( \mathbf {t} = \mathbf {t} \left( \mathbf {P}\right) \) links the first Piola–Kirchhoff stress with a state of pairwise force densities in the discretized form (see for instance Silling et al. [
31], Madenci and Oterkus [
26] and Bode et al. [
4]). By applying the principle of virtual displacements on equations (
5) and (
6) the peridynamic virtual strain energy yields
$$\begin{aligned} \int _{H} \left( \mathbf {t} - \mathbf {t}' \right) \ d H \cdot \delta \mathbf {u} + \rho _{0} \overline{\mathbf {b}}\cdot \delta \mathbf {u} = 0 \end{aligned}$$
(7)
and the local counterpart excluding surface forces
$$\begin{aligned} -\mathbf {P}:\delta \mathbf {F} + \rho _{0} \overline{\mathbf {b}}\cdot \delta \mathbf {u} = 0. \end{aligned}$$
(8)
Using the correspondence formulation, the constitutive laws can be based on the local theory for elastic finite deformations. In case of incompressible material behavior the Neo-Hookean strain energy function
$$\begin{aligned} \Psi ^{i} = \frac{\mu }{2}\left( {\text {tr}}\left( \mathbf {C}^{iso}\right) -3\right) \end{aligned}$$
(9)
is used and in case of compressible material behavior extended to
$$\begin{aligned} \Psi = \frac{\mu }{2}\left( {\text {tr}}\left( \mathbf {C}^{iso}\right) -3\right) + \frac{K}{4} \left( J-1-2\ln {J}\right) . \end{aligned}$$
(10)
The bulk modulus
K and the shear modulus
\( \mu \) can be calculated from the Young’s modulus
E and the Poisson’s ratio
\( \nu \) or lame constants
\( \lambda \) and
\( \mu \). The derivation of the strain energy function with respect to the deformation gradient yields the first Piola–Kirchhoff stress
$$\begin{aligned} \mathbf {P} = \frac{\partial \Psi }{\partial \mathbf {F}}. \end{aligned}$$
(11)