Variant of strain gradient elasticity with simplified formulation of traction boundary value problems

In this paper, we show that within the class of isotropic Mindlin-Toupin gradient theories there exist a particular variant of the theory, which allows a completely simplified form of traction boundary value problems. This theory can be obtained assuming that the high-grade part of the strain energy density depends only on the vector-type quantities, that are the gradient of dilatation and the curl of small rotation. Such incomplete gradient theory becomes positive semi-definite and at the same time it obeys the strong ellipticity conditions of the general Mindlin-Toupin first strain gradient elasticity. Based on the variational approach it is shown that the equilibrium equations of the developed theory and its definition for the surface traction can be given only in terms of the total stresses (like in classical elasticity). Such formulation can be useful for derivation of the closed form solutions for the problems with traction-type boundary conditions. Examples of the solutions for the problem of cylindrical bending and for the inplane crack tip fields are presented. It is shown that considered theory allows to obtain a regularized solution for the crack problems and at the same time it does not predict a non-physical infinite increase of the material's rigidity under bending.