A Fully Nonlinear Thin Shell Model of Kirchhoff-Love Type
This work presents a fully nonlinear Kirchhoff-Love shell model. In contrast with shear flexible models, our approach is based on the Kirchhoff-Love theory for thin shells, so that transversal shear deformation is not accounted for.
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1 Introduction The main objective of this work is to present a fully nonlinear finite deformation thin shell formulation, which can be employed by numerical methods. To this end, the geometrically exact shell formulation developed in Pimenta (1993) and Campello et al. (2003), which represents an alternative to the work initiated by Simo et al. (1990), is now constrained to obey the Kirchhoff-Love assumption. As in Pimenta (1993) and Campello et al. (2003), our approach defines energetically conjugated generalized cross-sectional generalized stress and strains based on the concept of a shell director. Besides their practical importance, cross section quantities
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make easy the derivation of equilibrium equations in weak and strong sense, as well as the achievement of the tangent weak form, which is always symmetric for hyper-elastic materials and conservative loadings, even far from an equilibrium state. On the other hand, the concept of director allows the introduction of a parameter that describes thickness variation, as done in Campello et al. (2003). This is useful for the derivation of shell constitutive equations from 3-D finite strain constitutive equations by applying a simple consistent plane stress condition, which does not destroy the symmetry of the tangent operator. A plane reference configuration was assumed for the shell. Initially curved shells can then be regarded as a stress-free deformation from this configuration. This approach was already employed for rods and shells in Pimenta (1996) and Pimenta et al. (2009). It precludes the use of convective non-Cartesian coordinate systems and other complicate entities like Christoffel symbols and the second fundamental form. It simplifies the comprehension of tensor quantities as well, since only components on orthogonal systems are employed. Throughout the text, italic Greek or Latin lowercase letters a , b, !, B, C, ! denote scalars, bold italic Greek or Latin lowercase letters a, b, !, B, C, ! denote vectors and bold italic Greek or Latin capital letters A, B, ! denote second-order tensors, as well as linear operators built with them. Summation convention over repeated indices (subscripts) is adopted in the entire text, whereby Greek indices range from 1 to 2, while Latin indices range from 1 to 3.
2 Nonlinear Kirchhoff-Love shell theory 2.1
Kinematics
It is assumed at the outset that the shell is plane at the initial configuration, which is used as reference. This formulation can be straightforwardly used for plane finite elements. Let E 8 \e1r , e2r , e3r ^ be an orthogonal system. The vectors r eB , B 1,2 , are placed on the reference middle plane of the shell, as shown in Figure 1. Thus, e 3r is orthogonal to this plane. The position of the shell material points in the reference configuration can be described by Y [ ar ,
(1)
A Fully Nonlinear Thin Shell Model of Kirchhoff-Love Type
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where [ YBeBr
and
a r [e3r .
(2)
[ is contained in the middle plane and a r is the director.
e3
e2 a
e1
x z
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