Peridynamics for Local Interactions

Abstract

Within classical continuum mechanics, a material point can only interact with other material points in its nearest neighborhood. As depicted in Fig. 3.1, the material point $k$ at location ${{\mathbf{x}}_{(k) }}$ can only have interactions with the material points labeled as $(k-1)$, $(k+1)$, $(k-m)$, $(k+m)$, $(k-n)$, and $(k+n)$. These interactions are represented by “internal traction vectors.” For the material point $k$ that is located on a surface whose unit normal is ${{\mathbf{n}}^T}=({n_x},{n_y},{n_z})$, the components of a traction vector, ${{\mathbf{T}}^T}=({T_x},{T_y},{T_z})$, are related to the Cauchy stress components as

$$ \left\{ \begin{array}{lll} {{T_x}} \\{{T_y}} \\{{T_z}} \\\end{array} \right\}=\left[ \begin{array}{lll} {{\sigma_{xx(k) }}} & {{\sigma_{xy(k) }}} & {{\sigma_{xz(k) }}} \\{{\sigma_{xy(k) }}} & {{\sigma_{yy(k) }}} & {{\sigma_{yz(k) }}} \\{{\sigma_{xz(k) }}} & {{\sigma_{yz(k) }}} & {{\sigma_{zz(k) }}} \\\end{array} \right]\left\{ \begin{array}{lll} {{n_x}} \\{{n_y}} \\{{n_z}} \\\end{array} \right\}, $$

in which $({\sigma_{xx(k) }},{\sigma_{yy(k) }},{\sigma_{zz(k) }})$ and $({\sigma_{xy(k) }},{\sigma_{xz(k) }},{\sigma_{yz(k) }})$ are the normal and shear stress components, respectively.