Alternative derivation of the higher-order constitutive model for six-parameter elastic shells

Abstract

In this paper, we present a general method to derive the explicit constitutive relations for isotropic elastic 6-parameter shells made from a Cosserat material. The dimensional reduction procedure extends the methods of the classical shell theory to the case of Cosserat shells. Starting from the three-dimensional Cosserat parent model, we perform the integration over the thickness and obtain a consistent shell model of order $$ O(h^5) $$ O ( h 5 ) with respect to the shell thickness h. We derive the explicit form of the strain energy density for 6-parameter (Cosserat) shells, in which the constitutive coefficients are expressed in terms of the three-dimensional elasticity constants and depend on the initial curvature of the shell. The obtained form of the shell strain energy density is compared with other previous variants from the literature, and the advantages of our constitutive model are discussed.

On the Equilibrium Equations of Linear 6-Parameter Elastic Shells

“…The nonlinear strain measures commonly used in the 6-parameter elastic shell models (see, e.g., [7,8,10,11,18]) are the shell strain tensor…”

“…According to the virtual power statement (18), we have to equate this with the virtual power of external loads P, which is given by…”

“…In [18] we have derived a refined Cosserat shell model of order O(h 5 ) by a dimensional descent from the three-dimensional nonlinear Cosserat elasticity. This 6-parameter model is able to describe isotropic shells made of Cosserat materials.…”

Derivation of an elastic shell model of order hn as n tends to infinity

“…For instance, in the case of plates, a potential energy of order h 5 is considered in the work by Pruchnicki [10]. For geometrically nonlinear Cosserat elastic shells (or six-parameter shells), the models of order h 3 and h 5 have been presented in the work by Bîrsan and colleagues [11, 12, 13] and have been analyzed in matrix notation in the work by Ghiba et al [14, 15], where the existence of minimizers has been proved. Other approaches to higher-order shell models, including also numerical treatment, have been presented in the previous works [16, 17, 18], among others.…”

On solvability of initial boundary-value problems of micropolar elastic shells with rigid inclusions

“…The counterparts of translations and rotations are stress resultants and surface couple stresses including so-called drilling moment, that is, a moment related to the rotation about the normal to the base surface. The equations of the six-parameter shell theory could be derived using the through-the-thickness integration procedure [3–5] or within the direct approach as in Eremeyev and his colleagues [6,7]; see also [9–11] and the references therein. As a result, on the boundary of a micropolar shell, we have six load boundary conditions that give a possibility to describe the kinematics of multifolded shells or interaction of a shell with rigid bodies.…”