Integropartial differential equations of the linear theory of nonlocal elasticity are reduced to singular partial differential equations for a special class of physically admissible kernels. Solutions are obtained for the screw dislocation and surface waves. Experimental observations and atomic lattice dynamics appear to support the theoretical results very nicely.

On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves

Some materials may naturally form discontinuities such as cracks as a result of deformation. As an aid to the modeling of such materials, a new framework for the basic equations of continuum mechanics, called the "peridynamic" formulation, is proposed. The propagation of linear stress waves in the new theory is discussed, and wave dispersion relations are derived. Material stability and its connection with wave propagation is investigated. It is demonstrated by an example that the reformulated approach permits the solution of fracture problems using the same equations either on or off the crack surface or crack tip. This is an advantage for modeling problems in which the location of a crack is not known in advance.

https://digital.library.unt.edu/ark:/67531/metadc664843/m2/1/high_res_d/1895.pdf

Reformulation of Elasticity Theory for Discontinuities and Long-Range Forces

Couple stress based strain gradient theory for elasticity

Matter is commonly found in the form of materials. Analytical mechanics turned its back upon this fact, creating the centrally useful but abstract concepts of the mass point and the rigid body, in which matter manifests itself only through its inertia, independent of its constitution; “modern” physics likewise turns its back, since it concerns solely the small particles of matter, declining to face the problem of how a specimen made up of such particles will behave in the typical circumstances in which we meet it. Materials, however, continue to furnish the masses of matter we see and use from day to day: air, water, earth, flesh, wood, stone, steel, concrete, glass, rubber, ... All are deformable. A theory aiming to describe their mechanical behavior must take heed of their deformability and represent the definite principles it obeys.

The non-linear field theories of mechanics

A generalization of Einstein's gravitational theory is discussed in which the spin of matter as well as its mass plays a dynamical role. The spin of matter couples to a non-Riemannian structure in space-time, Cartan's torsion tensor. The theory which emerges from taking this coupling into account, the ${U}_{4}$ theory of gravitation, predicts, in addition to the usual infinite-range gravitational interaction medicated by the metric field, a new, very weak, spin contact interaction of gravitational origin. We summarize here all the available theoretical evidence that argues for admitting spin and torsion into a relativistic gravitational theory. Not least among this evidence is the demonstration that the ${U}_{4}$ theory arises as a local gauge theory for the Poincar\'e group in space-time. The deviations of the ${U}_{4}$ theory from standard general relativity are estimated, and the prospects for further theoretical development are assessed.

A. Cemal Eringen

Papers

On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves

On nonlocal elasticity

Nonlocal polar elastic continua

Microcontinuum field theories

Nonlinear theory of simple micro-elastic solids-i