# The First Shell Theory: The Love-Kirchhoff-Love Model

Several theories have been proposed and each has its application areas and limitations. In general, there are two groups of basic assumptions on which the theories of the shells are based. The first group includes the assumptions for the theories of

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[Tas1965], Martin and Drew [Mar1989], whose theory was based on Donnell’s equations, and the work done by Chao [Cha1960] whose analysis was based on Timoshenko’s buckling equations. All the theories discussed above are based on the classical shell theory where the Love-Kirchhoff assumptions are used. The Love-Kirchhoff assumptions amount to treating shells as infinitely rigid in the transverse direction by neglecting transverse strains. The theory underestimates deflections and stresses and overestimates natural frequencies and buckling loads. Since the transverse shear moduli of advanced composite materials are usually very low compared to the in-plane moduli, the transverse shearing strains must be taken into account for an accurate representation of the response of laminated plates and shells. Numerous plate and shell theories which account for transverse shear deformations are documented in the

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Introducing a polynomial description of the displacements in thickness direction, he opened the door for shell models that match the three-dimensional theory with arbitrary exactness. Nevertheless, these shear-deformable shell models are commonly called shells with Reissner-Mindlin kinematics due to the origin of the assumption of shear-deformable cross-sections. Following the naming of Bischoff [Bis1999], the pure displacement-based shell formulations with shear deformation are called five-parameter formulation. This name relates to the use of five kinematic degrees of freedom (3 displacements and 2 rotational degrees of freedom).

The fundamental idea of shell models is based on the dimensional reduction of the 3D continuum: Taking advantage of the disparity in length scale, the semi-discretization of the continuum is performed to end up in a two-dimensional problem description. This discretization in thickness direction is completely independent from the (later) discretization within the shell mid-surface. Starting from the mechanical model, two general approaches can be