On a Differential Constraint in Asymmetric Theories of the Mechanics of Growing Solids

2022

J. Phys.: Conf. Ser.

The present study is devoted to the boundary value problems statements for the growing materials with microstructural features. The general form of tensor relations on the propagating growing surface is derived as a consequence of the conservation laws of momentum and angular momentum. The necessary system of independent arguments of constitutive differential constraints on the growing surface in micropolar continuum is determined. A complete set of joint rational invariants of the system of tensors and vectors that determine the thermodynamics of the production process of a woven 3D material is given and discussed. An invariant-complete formulation of the constitutive relations on the growing surface is obtained.

Section: Boundary Conditions On the Propagating Growing Surface In Ma...mentioning

confidence: 99%

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

2022

J. Phys.: Conf. Ser.

“…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…”

Section: Differential Constraints On a Micropolar Growing Surfacementioning

confidence: 99%

“…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…”

Section: Differential Constraints On a Micropolar Growing Surfacementioning

confidence: 99%

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On a micropolar theory of growing solids

Вестник Самарского государственного технического университета.

Radayev

2020

Обсуждается принцип вывода граничных условий в краевых задачах механики растущих микрополярных тел. Приводится вывод уравнений динамики микрополярного континуума в терминах относительных тензоров для тел постоянного состава. Указана определяющая квадратичная форма упругого потенцила (абсолютного скаляра) для линейного гемитропного микрополярного тела. Выведены определяющие соотношения для симметричных и антисимметричных частей тензоров силовых и моментных напряжений. Получены конечные формы уравнений динамики гемитропного микрополярного континуума в терминах скоростей перемещений и микровращений. Полученные динамические уравнения для тел постоянного состава остаются справедливыми и в теориях растущих тел. Предложена процедура преобразования уравнений равновесия для получения граничных условий на поверхности наращивания в терминах относительных тензоров в форме дифференциальных ограничений. Полученные условия справедливы для весьма широкого круга материалов и метаматериалов. При выводе определяющих соотношений на поверхности наращивания активно используется аппарат алгебры рациональных относительных инвариантов. Получены полные системы совместных относительных инвариантов для тензоров силовых, моментных напряжений и единичного вектора нормали, в том числе системы инвариантов, не выдерживающие зеркальных отражений.

“…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.…”

Section: Mechanical Background Of Growth Modelsmentioning

confidence: 99%

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

Math Methods in App Sciences

Dats

Стадник

2020

The present study deals with a problem of modeling a surface growth process.Governing principles for mechanics of growing solids have been formulated.Within the framework of the surface growth theory, different variants of boundary value problems are discussed. A human blood vessel under pathological growth processes has been considered as an example of a surface growth process.Vessels are simulated by a long thick-walled circular cylinder. The boundary value problem of a surface growth for elastic thick-walled vessels is solved. The analytical solution in terms of velocities for parameters of stress-strain state has been obtained. Condition of thickness has allowed us to study strain-stress state of cylinder surfaces using approximation of infinitesimal deformations. The stress-strain state characteristics are numerically computed and graphically analyzed using various mechanical parameters of the surface growth processes. Two cases of boundary value problems, namely, deformation of two-and multilayer cylinder have been considered. General analytical solution for thick multilayer cylinder deformation has been obtained. Analytical solution is based on the Lamé problem for elastic material. Comparison of multilayer and single-layer solutions is performed.