Figure 4

Deformation of an infinitesimal fiber \(\boldsymbol{e_{R}}\) connecting two points in local and nonlocal kinematics. A material curve \(\Gamma _{0} \) passing trough x and its time evolution \(\Gamma _{t} \) are represented to highlight the differences between local and nonlocal kinematics. (a) In classical continuum mechanics the motion of fiber is described by an affine mapping \(\boldsymbol{e}\approx [(\boldsymbol{I}+\nabla _{\mathbf{x}}\mathbf{u}(\mathbf{x})] \boldsymbol{e_{R}} \) from material to geometric space; the continuous curve \(\Gamma _{0} \) can never break up. (b) In peridynamics (nonlocal theory ) the fiber displacement is finite and determined by the generic mapping \(\boldsymbol{e}=\boldsymbol{e_{R}}+\boldsymbol{\eta }\). No smoothness is required; then a continuous material curve \(\Gamma _{0} \) can break up