ИСТИНА

Structural inhomogeneity in materially uniform solids may be caused by a variety of physical and technological reasons. Layer by layer (LbL) film synthesizing, surface growth, remodeling, etc are among them. Structural inhomogeneity is closely related to laminar or fibrillar distribution of defects and specific spatial structure of deformation incompatibility. The latter results in unwanted features, particularly, in residual stresses and distortion of geometrical shape of desired structure. These factors are associated with critical parameters in modern high-precision technologies, particularly, in additive manufacturing, and considered to be thereby essential constituents in corresponding mathematical models. In present communication the methods of geometrical mechanics and their application in modeling of stress-strain state in structurally inhomogeneous solids are developed. It is known for simple materials that theory of affine connections gives an elegant mathematical formalization of the concept of a materially uniform (in particular, stress-free) non-Euclidean reference form. One can obtain affine connection on material manifold by defining parallel transport as the transformation of the tangent vector, in which its inverse image with respect to locally uniform embeddings does not change. This leads to Weitzenbock space (the space of absolute parallelism, or teleparallelism) and gives a clear interpretation of the material connection in terms of the local non-degenerate linear transformations which return an elementary volume of simple material into uniform state. The methods in question are based on the representation of a body and physical space in terms of differentiable manifolds. These manifolds are endowed with specific Riemannian metrics and affine connections, which are non-Euclidian in general. Affine connection on the physical manifold is defined a priori by considerations which are independent from the properties of the solid, while the connection on material manifold is defined by intrinsic properties of the body. Deformation, stress and power measures are formalized by smooth mappings defined on smooth manifolds which represent the body and physical space or vector bundles over them. Tangent map, which locally represents embedding of body into physical space, plays role of deformation gradient. The measures of local deformation are introduced as pull-back of spatial metrics which defines generalized right Cauchy-Green tensor and push-forward of the material metrics, which defines generalized left Cauchy-Green tensor. Forces are interpreted as covectors, i.e. as a linear functionals, which act on the velocity vectors of material points and results power. Accordingly, the stress fields are interpreted as covector-valued exterior two-forms and body forces as scalar-valued three-forms. Tensor fields of different types are considered as elements of the uniformly graded structure over pair: the material manifold and the physical one. The abstract theory of integration based on exterior forms formalism can be adapted to the elements of this structure which allows one to formulate the power balance equation on the material manifold (similarly to reference description in the classical mechanics of compatible deformation) and on the physical one (similarly to the spatial description). The balance equations in terms of Cartan's exterior covariant derivative can be obtained from the general principle of covariance.