Weak Form and the Virtual Work Equation

Let V be an arbitrary function, which is null on the displacement boundary  

We multiply (4.1a) by v(x), integrate it and introduce an integration by parts to the first term, then we can use the condition (4.4) and the Neumann boundary condition (4.1c), and we finally obtain the following weak form [WF], which is equivalent (see Note 4.1) to the strong form (4.1)  

We can regard the arbitrary function v as a virtual displacement 8u, then (4.4) corresponds to a null condition of the virtual displacement Su on x . We write a virtual strain as 

In this notation the weak form (4.5) is equivalent to the following virtual work equation [VW]  

Note 4.1 (Equivalence between the strong form and weak form Fundamental Lemma of the Variational Problem). Since the weak form (4.5) is derived from the strong form (4.1), the solution of the strong form is exactly the solution of the weak form. The converse is not always true. If the solution of the weak form can be regarded as sufficiently smooth, the converse is true, which is proved by the Fundamental Lemma of the Variational Problem. Note that the first term defined on the boundary of (4.5) can be considered separately from the rest of the terms defined in the domain, since v is arbitrary both on the boundary and in the 

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