Refined dimensional reduction for isotropic elastic Cosserat shells with initial curvature

Abstract

Using a geometrically motivated 8-parameter ansatz through the thickness, we reduce a three-dimensional shell-like geometrically nonlinear Cosserat material to a fully two-dimensional shell model. Curvature effects are fully taken into account. For elastic isotropic Cosserat materials, the integration through the thickness can be performed analytically and a generalized plane stress condition allows for a closed-form expression of the thickness stretch and the nonsymmetric shift of the midsurface in bending. We obtain an explicit form of the elastic strain energy density for Cosserat shells, including terms up to order O(h 5 ) in the shell thickness h. This energy density is expressed as a quadratic function of the nonlinear elastic shell strain tensor and the bending-curvature tensor, with coefficients depending on the initial curvature of the shell.

Derivation of a refined six-parameter shell model: descent from the three-dimensional Cosserat elasticity using a method of classical shell theory

“…To compare our results with other six-parameter shell models, we write the strain energy density in an alternative useful form (equation (113)). We pay special attention to the comparison with the Cosserat shell model of order O ( h 5 ) , which has been presented recently in Bîrsan et al [12]. Although the derivation methods are different, we obtain the same form of the strain energy density, except for the coefficients of the transverse shear energy, which are unequal.…”

“…where ε α β is the two-dimensional alternator ε 12 = − ε 21 = 1 , ε 11 = ε 22 = 0 ) and a ( x 1 , x 2 ) determines the elemental area of the surface ω ξ In view of equations (1) and (13), we can show that (see f. (46) in Bîrsan et al [12])…”

“…where W = W ( E ¯ , Γ ) is the elastically stored energy density. Using the Cosserat model for isotropic materials presented in Bîrsan et al [12] and Neff et al [17], we assume the following representation for the energy density:…”

“…First of all, the derivation method and starting point in Bîrsan et al [12] is different, since the deformation function φ is assumed to be quadratic in x 3 More precisely, the following ansatz is adopted (see f. (65) in Bîrsan et al [12])…”

A higher order geometrically nonlinear Cosserat‐shell model with initial curvature effects

“…where E e := Q T e Grad s m − a is the elastic shell strain tensor , K e := Q T e axl Q e,α Q T e ⊗ a α is the elastic shell bending-curvature tensor, and we denote a α = y 0,α , a = Grad s y 0 , b = −Grad s n 0 , c = −n 0 × a [2,3,1].…”