Influence of nonlinear spatial distribution of stress and strain on solving problems of solid mechanics

The stress and the strain should be defined as statistical variables averaged over the representative volume elements for any real continuum system. It is shown that their nonlinear spatial distributions undermine the classical framework of solid mechanics and may cause non-ignorable errors to the solutions. With considering the high-order gradients of the stress and the strain, a two-step solution scheme is proposed to compensate for the influence. Through a revisit to three simple but typical problems, i.e., the hole size-dependence of the fracture strength of perforated plates, the indentation depth-dependence of the measured elastic modulus by micro-indentation tests, and the tensile necking of metallic materials as well as hyperelastic materials, the effect of the nonlinear spatial distribution of stress and strain on solving these problems is illustrated. The observed size effect and the instability of deformation can be quantitatively explained if the effect is properly considered by the proposed method.