Recent developments in the theory of nonlinearly elastic plates and shells

Author(s):  
D Steigmann
2016 ◽  
Vol 124 (2) ◽  
pp. 243-278 ◽  
Author(s):  
Alexey B. Stepanov ◽  
Stuart S. Antman

2019 ◽  
Vol 968 ◽  
pp. 496-510
Author(s):  
Anatoly Grigorievich Zelensky

Classical and non-classical refined theories of plates and shells, based on various hypotheses [1-7], for a wide class of boundary problems, can not describe with sufficient accuracy the SSS of plates and shells. These are boundary problems in which the plates and shells undergo local and burst loads, have openings, sharp changes in mechanical and geometric parameters (MGP). The problem also applies to such elements of constructions that have a considerable thickness or large gradient of SSS variations. The above theories in such cases yield results that can differ significantly from those obtained in a three-dimensional formulation. According to the logic in such theories, the accuracy of solving boundary problems is limited by accepted hypotheses and it is impossible to improve the accuracy in principle. SSS components are usually depicted in the form of a small number of members. The systems of differential equations (DE) obtained here have basically a low order. On the other hand, the solution of boundary value problems for non-thin elastic plates and shells in a three-dimensional formulation [8] is associated with great mathematical difficulties. Only in limited cases, the three-dimensional problem of the theory of elasticity for plates and shells provides an opportunity to find an analytical solution. The complexity of the solution in the exact three-dimensional formulation is greatly enhanced if complex boundary conditions or physically nonlinear problems are considered. Theories in which hypotheses are not used, and SSS components are depicted in the form of infinite series in transverse coordinates, will be called mathematical. The approximation of the SSS component can be adopted in the form of various lines [9-16], and the construction of a three-dimensional problem to two-dimensional can be accomplished by various methods: projective [9, 14, 16], variational [12, 13, 15, 17]. The effectiveness and accuracy of one or another variant of mathematical theory (MT) depends on the complex methodology for obtaining the basic equations.


1989 ◽  
Vol 42 (11) ◽  
pp. 295-303 ◽  
Author(s):  
Cornelius O. Horgan

The simplifications arising in elasticity theory from consideration of resultant boundary conditions instead of mathematically exact pointwise conditions have been the key to widespread application of the subject. Thus, for example, theories for strength of materials, plates, and shells rely on such relaxed boundary conditions for their development. The justification of this approximation is usually based on some form of the celebrated Saint-Venant’s principle. A comprehensive survey of contemporary research concerning Saint-Venant’s principle (covering primarily the period 1965–1981) was given by Horgan and Knowles (1983). Since that time, several developments have taken place demonstrating continued interest in understanding the ramifications of Saint-Venant’s principle from both a physical and mathematical point of view. In this article we review these developments, thus providing an update on contributions to this fundamental engineering principle.


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