Generalised Hooke's law for an element of a shell

Abstract

Introduction We are now ready to establish the mechanical properties of a typical small element of a thin uniform elastic shell. We have already decided to replace the shell itself by a model consisting of a surface, and we must now furnish this surface with appropriate mechanical properties. This task is equivalent to the well-known piece of work in the classical theory of beams in which the beam is shown to be equivalent to a ‘line’ endowed with a flexural stiffness EI , where E is Young's modulus of elasticity of the material and I is a geometrical property of the cross-section. But the present task is more complex than the corresponding one for the beam, in two distinct ways. First, an element of a shell is two dimensional , whereas an element of a beam is one dimensional. Second, an element of shell is in general curved rather than flat. A basic idea, which was proposed in the early days of shell theory, is that in relation to the specification of the mechanical properties of an element of a shell it is legitimate to proceed as if the element were flat, and not curved. Legitimate, that is, as a ‘first approximation’. Much work has been done by many authors on the question of the degree of inaccuracy which is introduced by this idea: see, for example, Novozhilov (1964), Naghdi (1963). We shall not attempt to justify this idea formally.