Non-classical aspects of Kirchhoff type shells
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ORIGINAL ARTICLE
Non‑classical aspects of Kirchhoff type shells Bensingh Dhas1 · Debasish Roy1 Received: 15 April 2020 / Accepted: 15 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Kirchhoff type shells are continuum models used to study the mechanics of thin elastic bodies; these are largely based on the theory of surfaces. Here, we report a reformulation of Kirchhoff shells using the theory of moving frames. This reformulation permits us to treat the deformation and the geometry of the shell as two separate entities. The structure equations which represent the familiar torsion and curvature free conditions (of the ambient space) are used to combine deformation and geometry in a compatible way. From such a perspective, Kirchhoff type theories have non-classical features which are similar to the equations of defect mechanics (theory of dislocations and disclinations). Using the proposed framework, we solve a boundary value problem and thus demonstrate, to an extent, the importance of moving frames. Keywords Kirchhoff shells · Moving frame · Structure equations · Defect mechanics
1 Introduction Kirchhoff type shell theories are known for a long time. They have been successfully applied to a wide range of problems in structural mechanics. Applications vary from the dynamics of cell membranes [5, 10] to stress analyses of an aircraft fuselage. A continuum model typically predicts the deformed state of a body using tools from geometry and thermodynamics. In classical continuum mechanics, the geometry of the body remains frozen; three dimensional elasticity is an example of one such theory. Conventionally, Kirchhoff type shells are considered to be within the realms of classical continuum mechanics. This perspective stems from a purely displacement based formalism of these shells. An alternate approach to Kirchhoff shells is to consider the geometry of the mid-surface and its deformation as separate entities. From such a viewpoint, this shell theory contains non-classical aspects which cannot be found in classical continuum mechanics. Srinivasa and Reddy [16] have classified non-local continuum models into two major groups; the first involves displacement as the primal field. In this class of models, non-locality is brought in by considering the energy * Debasish Roy [email protected] 1
Centre of Excellence in Advanced Mechanics of Materials, Indian Institute of Science, Bangalore, India
contribution due to higher gradients or by averaging the conventional strains over a neighbourhood. Examples include higher gradient models and integral type non-local models of Eringen [7]. The second class of non-local models involves the introduction of additional variables other than displacement. These additional variables are in general tensor fields. Micropolar [17] and micromorphic theories [8] are representative examples from this class. In this class of models, the coupling between non-locality and deformation is through energy considerations (first and second laws of thermodynamics).
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