On one class of applied gradient models with simplified boundary problems


We consider the generic gradient elasticity theory of Mindlin-Tupin and try to establish a class of applied models of gradient elasticity, for which the boundary value problems of the gradient theory with static boundary conditions are divided into a sequence of two subtasks, one of which is classical. Such applied models are very effective in applications, because their solutions reduce exactly to a consistent solution of boundary value problems of the second and not of the fourth order. We consider gradient theories with a general structure of tensors of gradient modules that satisfy potentiality conditions and additional symmetry conditions, which is considered as a criterion of correctness. It is shown that their gradient tensors of the elastic modules are represented in the form of an expansion with respect to the tensor basis of five sixth-rank tensors, three of which satisfy a special property. Each of these basis tensors is represented as a convolution of fourth-rank tensors, and the corresponding quadratic form is a convolution of vectors. It is shown that for the traditional gradient Mindlin-Tupin theory, the “classical” static conditions on the body surface are not satisfied locally. However, if the gradient modules are represented as a convolution of the “classical” tensors of elastic moduli, then the set of the boundary value problems of such gradient theory admits a full fractionation of the initial boundary value problem into two: the “classical” boundary value problem and the “cohesive” boundary value problem. It is established the structure of the applied gradient models with such property of separating boundary value problems. They are particular cases of gradient elasticity theories with gradient modulus tensors, representable in the form of an expansion in three basis tensors of the sixth rank, satisfying the properties of the representation in the form of convolution via fourth-rank tensors.We formulated “vector” gradient Mindlin-Tupin model that preserves the classical form of static boundary conditions. Such a model leads to a specific variant of the gradient theory with a single non-classical modulus, or one-parametrical model. It is shown that the obtained gradient model can be considered as some generalization of the well-known applied theoryGradEla providing for it the separation of boundary value problems.