The nonlocal theory of elasticity, the points undergo translational motion as in the classical case, but the stress at a point depends on the strain in a region near that point (Kroner, 1967, Eringen, 1972. The constitutive equation for stress is, in terms of the position vector x of points in the solid,
in which the Lame coefficients for an isotropic material become spatial functions of the distance from the point under consideration in the material. The nonlocal integral can be expanded in a series of spatial derivatives of strain to reveal sensitivity to gradients of strain and higher gradients. Nonlocal elasticity is a type of generalized continuum theory.
As for physical interpretation, the nonlocal theory incorporates long range interactions between points in a continuum model. Such long range interactions occur between charged atoms or molecules in a solid. Long range forces may also be considered to propagate along fibers or laminae in a composite material (Ilcewicz, et. al, 1981, 1981). Nonlocal elasticity has also been applied in composite analysis by Drugan (2003) and Monetto and Drugan (2004). Among the specific implications of these results, it is found that the minimum representative volume element (RVE) size estimate for composites containing spherical inclusions / voids using the Verlet-Weis improvement is significantly larger at higher inclusion / void volume fractions (0.3-0.64) than prior estimates.
A simplified form of nonlocality was presented in a differential form that entails an approximation to the nonlocal integral (Eringen 1983). Such a formulation introduces sensitivity to gradients of the stress or strain. Such gradient forms, termed nonlocal, have been presented (e. g. Peddieson 2003) in the context of nano-scale systems. Application of generalized continuum theory is not restricted to nano-scale. The essential aspect is that the size of the micro-structure not be negligible in comparison with the size scale of the experiment.
As for interpretation of experimental results for elastic or viscoelastic properties, we use the Cosserat theory of elasticity (or viscoelasticity) because
(i) there are six elastic constants for an isotropic Cosserat elastic solid in contrast to an infinite number of possible kernel functions for an isotropic nonlocal solid,
(ii) exact analytical solutions are known for the interpretation of bending and torsion experiments within Cosserat elasticity;
(iii) evidence of asymmetric stress, allowed in Cosserat elasticity, has been presented experimentally.
Cosserat solids are sensitive to gradients of strain; they incorporate a specific kind of nonlocality.
Nonlocal behavior is known in piezoelectric materials (Bursian et al., 1968, 1974; Ma and Cross, 2006).
References
Kroner, E., "Elasticity theory of materials with long range cohesive forces", Int. J. Solids and Structures, 3, 731-742, 1967.
Eringen, A. C., "Linear theory of nonlocal elasticity and dispersion of plane waves", Int. J. Engng Sci, 10, 425-435, 1972.
Ilcewicz, L., Narasimhan, and Wilson, J., "An experimental verification of nonlocal fracture criterion", Engineering Fracture Mechanics, 14, 801-808, 1981.
Ilcewicz, L., Kennedy, T. C., and Shaar, C., "Experimental application of a generalized continuum model to nondestructive testing", J. Materials Science Letters, 4, 434-438, 1985.
W. J. Drugan, "Two Exact Micromechanics-Based Nonlocal Constitutive Equations for Random Linear Elastic Composite Materials," Journal of the Mechanics and Physics of Solids, 51,1745-1772, (2003).
I. Monetto and W. J. Drugan, "A Micromechanics-Based Nonlocal Constitutive Equation for Elastic Composites Containing randomly oriented spheroidal Heterogeneities", Journal of the Mechanics and Physics of Solids, 52, 359-393, (2004)
Eringen, A. C. On differential equations of nonlocal elasticity and solutions of screw dislocations and surface waves, J. Appl. Phys. 54, 4703-4710, 1983
Bursian, E. V. and Trunov, N. N., "Nonlocal piezoelectric effect", Sov. Physics Solid State, 16 (4) 760-762, (1974).
Bursian, E. V. and Zaikovskii, O. I., "Changes in the curvature of a ferroelectric film due to polarization", Sov. Physics Solid State, 10 (5) 1121-1124, (1968).
Lakes, R. S., "The role of gradient effects in the piezoelectricity of bone", IEEE Trans. Biomed. Eng., BME-27 (5), 282-283, (1980). Stress gradient effects in piezoelectricity are obtained from general nonlocality considerations. A nonlocal elastic and nonlocal piezoelectric continuum representation of bone is appropriate in view of bone's structure. More recently, gradient effects in piezoelectricity have been called "flexoelectricity" of "flexoelectric" materials. Get pdf
Ma, W. and Cross, L. E., "Flexoelectricity of barium titanate", Appl. Phys. Lett. 88, 232902 (2006).