On a micropolar theory of growing solids

E. V. Murashkin ; Yu. N. Radayev

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 24 (2020) no. 3, pp. 424-444

Résumé

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.

Détail

Keywords: micropolar hemitropic continuum, relative tensor, 3D printing, propagating growing surface, stress, differential constraint, complete system.
Mots-clés : microrotation, pseudoscalar, constitutive equation, rational relative invariant