Journal of Samara State Technical University, Ser. Physical and Mathematical SciencesJournal of Samara State Technical University, Ser. Physical and Mathematical SciencesВестник Самарского государственного технического университета. Серия «Физико-математические науки»1991-86152310-7081Samara State Technical University6086710.14498/vsgtu1792ArticlesСтатьиResearch ArticleOn a micropolar theory of growing solidsО микрополярной 3D-теории растущих телMurashkinEugenii ValeryevichМурашкинЕвгений Валерьевич

Candidate of physico-mathematical sciences, no status

кандидат физико-математических наук, без звания

murashkin@dvo.ruRadayevYuri NikolaevichРадаевЮрий Николаевич

Doctor of physico-mathematical sciences, Professor

доктор физико-математических наук, профессор

y.radayev@gmail.comIshlinsky Institute for Problems in Mechanics of the Russian Academy of SciencesИнститут проблем механики им. А. Ю. Ишлинского Российской академии наук01102020243VOL 24, NO3 (2020)ТОМ 24, №3 (2020)42444414022021Copyright ©; 2020, Samara State Technical UniversityCopyright ©; 2020, Авторы, Самарский государственный технический университет2020Samara State Technical UniversityАвторы, Самарский государственный технический университетhttps://creativecommons.org/licenses/by/4.0https://journals.eco-vector.com/1991-8615/article/view/60867

The present paper is devoted to the problem of boundary conditions formulation in the growing micropolar solid mechanics. The static equations of the micropolar continuum in terms of relative tensors (pseudotensors) are derived due to virtual work principle for a solid of constant staff. The constitutive quadratic form of the elastic potential (treated as an absolute scalar) for a linear hemitropic micropolar solid is presented and discussed. The constitutive equations for symmetric and antisymmetric parts of force and couple stress tensors are given. The final forms of the static equations for the hemitropic micropolar continuum in terms of displacements and microrotations rates are obtained including the case of growing processes. A transformation of the equilibrium equations is proposed to obtain boundary conditions on the propagating growing surface in terms of relative tensors in the form of differential constraints. Those are valid for a wide range of materials and metamaterials. The algebra of rational relative invariants is intensively used for deriving the constitutive relations on the growing surface. Systems of joint algebraic rational relative invariants for force, couple stress tensors and also unit normal and tangent vectors to propagating growing surface are obtained, including systems of invariants sensitive to mirror reflections and 3D-space inversions.

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.

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