Some specific aspects of linear homogenization shell theory

In this paper we propose a multiscale linear shell theory for simulating the mechanical response of a highly heterogeneous shell of varying thickness. To resolve this issue, a higher-order stress-resultant shell formulation based on multiscale homogenization is considered. At the macroscopic scale level, we approximate the displacement field by a fourth-order Taylor–Young expansion in thickness. The transition between both the microscopic and the macroscopic scales is obtained through the introduction of a specific Hill–Mandel condition. Since we adopt the standard assumption of small strain which is used in linear elasticity, we can present a variant of the homogenization scheme which is valid for small strain. The nonlinearity of the previous model occurs from the assumption of large rotation of the transverse normal.

A NEW APPROACH TO LINEAR SHELL THEORY

We propose a new approach to the existence theory for quadratic minimization problems that arise in linear shell theory. The novelty consists in considering the linearized change of metric and change of curvature tensors as the new unknowns, instead of the displacement vector field as is customary. Such an approach naturally yields a constrained minimization problem, the constraints being ad hoc compatibility relations that these new unknowns must satisfy in order that they indeed correspond to a displacement vector field. Our major objective is thus to specify and justify such compatibility relations in appropriate function spaces. Interestingly, this result provides as a corollary a new proof of Korn's inequality on a surface. While the classical proof of this fundamental inequality essentially relies on a basic lemma of J. L. Lions, the keystone in the proposed approach is instead an appropriate weak version of a classical theorem of Poincaré. The existence of a solution to the above constrained minimization problem is then established, also providing as a simple corollary a new existence proof for the original quadratic minimization problem.

Blending Functions for Vibration Analysis of a Cylindrical Shell with an Oblique End

The free vibration problem of a cylindrical shell with an oblique end is considered. A theoretical solution based on the Sanders–Budiansky linear shell theory, and the differential quadrature method, is presented. The surface of the shell is first developed onto a plane, and the resulting irregular domain is then mapped, using blending functions, onto a square parent domain. The analysis is finally carried out in the parent domain. Two solutions are derived, using either trigonometric or polynomial trial functions in the circumferential direction of the domain. Convergence, validation and parametric studies are carried out. Results from the two solutions are compared with each other and with finite element results. The paper ends with an appropriate set of conclusions.