On a differential constraint in the continuum theory of growing solids

Abstract

Предлагается один общий принцип постановки граничных условий в краевых задачах механики растущих тел. При выводе определяющих соотношений на поверхности наращивания используется аппарат алгебры рациональных инвариантов. Проведен вывод различных вариантов физически непротиворечивых дифференциальных ограничений на поверхности наращивания. Полученные условия справедливы для весьма широкого круга материалов и метаматериалов. Для использования сформулированных дифференциальных ограничений в конкретных приложениях необходима их экспериментальная идентификация. По этой причине полученные результаты могут служить общей основой в прикладных исследованиях по механике растущих тел.

On a system of independent arguments for constitutive tensor functions on the growing surface in micropolar continuum

“…Throughout the article, the description in Euler coordinates is used. Terminology and notations mainly correspond to the studies [22][23][24][25]. A propagating growing surface Σ in three-dimensional Euclidean space can be given by the implicit equation…”

On a micropolar theory of growing solids

“…288-292] by G. I. Bykovtsev. Here we use notation and terminology introducing in the studies [41][42][43], where the differential constraints for the force stress tensor are derived and discussed. Let's analytically define a propagating growing surface Σ in three-dimensional space by the equation…”

“…As previously shown (see for example, [41][42][43]), the transformation of equations of equilibrium (7) using a formula for the actual components of the force stress tensor…”

“…In equations (16)- (18) we use the notation adopted in [41][42][43]: is the stress jump related integral, Formulas for the components of the couple stress tensor, by analogy with the equations (16) can be assume in the following form…”

Application of surface growth model for a pathological process in a blood vessel's wall

“…The derivations of some boundary conditions using thermodynamics principles or conservation laws can be found in a number of studies. [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50] The principal variables of the boundary value problem for theory of surface growth for solids are the stress tensor and strain rate tensors and velocity vector. On the surface of growth, one needs to set a specific boundary condition depending on the curvature tensor of the growth surface as well as on tension and inflow rates of the incremented elements.…”