The cauchy problem for a one dimensional nonlinear elastic peridynamic model

This paper studies the Cauchy problem for a one-dimensional nonlinear peridynamic model describing the dynamic response of an infinitely long elastic bar. The issues of local well-posedness and smoothness of the solutions are discussed. The existence of a global solution is proved first in the sublinear case and then for nonlinearities of degree at most three. The conditions for finite-time blow-up of solutions are established.

[1]  Q. Du,et al.  MATHEMATICAL ANALYSIS FOR THE PERIDYNAMIC NONLOCAL CONTINUUM THEORY , 2011 .

[2]  Etienne Emmrich,et al.  The peridynamic equation of motion in non-local elasticity theory , 2006 .

[3]  I. Kunin,et al.  Elastic Media with Microstructure II , 1982 .

[4]  S. Silling,et al.  Deformation of a Peridynamic Bar , 2003 .

[5]  E. Emmrich,et al.  Analysis and Numerical Approximation of an Integro-differential Equation Modeling Non-local Effects in Linear Elasticity , 2007 .

[6]  O. Ladyzhenskaya,et al.  The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types , 1978 .

[7]  A. Erkip,et al.  Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity , 2009 .

[8]  E. Emmrich,et al.  On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity , 2007 .

[9]  Dominik Rogula,et al.  Nonlocal theory of material media , 1982 .

[10]  Elastic media with microstructure vols. I and II , 1986 .

[11]  A. Erkip,et al.  Blow-up and global existence for a general class of nonlocal nonlinear coupled wave equations , 2010, 1009.1263.

[12]  A. Eringen,et al.  Nonlocal Continuum Field Theories , 2002 .

[13]  S. Silling.  Reformulation of Elasticity Theory for Discontinuities and Long-Range Forces , 2000 .

[14]  Olaf Weckner,et al.  The effect of long-range forces on the dynamics of a bar , 2005 .

[15]  A. Erkip,et al.  The Cauchy problem for a class of two-dimensional nonlocal nonlinear wave equations governing anti-plane shear motions in elastic materials , 2010, 1009.1266.