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1. Introduction We move now from the easy ride of Poisson problems and Bernoulli-Euler beams to the tougher road of elasticity in three dimensions. This Chapter summarizes the governing equations of linear elastostatics. Various notational systems are covered in sufficient detail to help readers with the literature of the subject, which is enormous and spans over two centuries. The governing equations are displayed in a Strong Form Tonti diagram.

For example if w B E = w F BC = 1 one obtains the so-called subdomain method, one of whose variants (in fluid applications) is the Fluid Volume Method. Other possibility is to take weights to be the same as the corresponding residuals, which leads to the two-century-old method of Least Squares. But for the Poisson equation a Variational Form exists. Why not try for the best? 3. 5) as the point of departure. 2 Also rename residual R as δ to emphasize that this will be hopefully the variation of a functional as yet unknown: δ BE = − ∇ · ρ∇u + s δu d V, δ V 2 F BC = ρ∇u · n − qˆ δu d S.

4 may be removed because they are strongly connected (2) The data boxes q and M, to the varied field M. 5 displays the five quantities (M, V M , κ M , wˆ A , θˆA ) that survive in the TCPE functional. 3. One can easily show that for the actual solution of the beam problem, U ∗ = U , a property valid for any linear elastic continuum. Furthermore U ∗ = 12 W ∗ . 5. The Hellinger-Reissner Functional The TPE and TCPE functionals are single-field, because there is only one master field that is varied: displacements in the former and moments in the latter.

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