Nonlocal Models in Continuum Mechanics

The recent appearance of nonlocal methods is examined in the light of traditional continuum mechanics. A comparison of nonlocal approaches in the fields of solid and fluid mechanics reveals that no consistent definition of a nonlocal theory has been used. We suggest a definition based on the violation of the principle of local action in continuum mechanics. From the consideration of the implications of a nonlocal theory based on this definition, we conclude that constitutive relations with nonlocal terms can confuse the traditional separation of the roles between conservation laws and constitutive relations and bring into question the identification of constants in the constitutive equation as material parameters. The diversity of motivations for the nonlocal approaches are presented, resulting primarily from deficiencies in numerical solutions to practical problems. To illustrate these concepts, the history of nonlocal terms in the field of viscoelastic fluids is reviewed. A specific example of a viscoelastic constitutive relation that contains a stress diffusion term is applied to a simple shear flow and found not to be a physical description of any known fluid. We conclude by listing questions that should be asked of nonlocal approaches. [