A new framework in nonlocal mechanics

Abstract

A new theoretical framework in nonlocal mechanics is defined, based on the concept of influence functions between material points within the continuum. The traditional idea of a fixed and isotropic representative volume is abandoned and the nonlocality is introduced via an influence function, which defines a nonlocal interaction between material points. The general framework developed is exemplified by the description of damage as a scalar internal variable: the local damage rate at a given point can be expressed as a path integral involving the influence functions and the values of the local rate of damage transported along each path. The properties satisfied by the influence function are first evidenced and the influence function is given an explicit expression, using a path integration technique. The concept of a representative volume is further defined as an outcome of the stationarity of the internal entropy production with respect to the path. An implicit equation which defines the representative volume is formulated. A numerical implementation of the proposed concepts is performed in the case of interfacial damage. The strength of the nonlocal interaction is further incorporated into the space geometry, so that a metric characteristic of a Riemanian space is coupled to the internal variable distribution. It appears that the curvature characterises the strength of the nonlocal interaction.