Boundary Value Problems
In this chapter, the field equations derived in the previous chapter are summarized and supplemented by boundary conditions. This results in the boundary value problems of compressible and (nearly) incompressible finite hyperelasticity within both Newtonian and Eshelbian mechanics. The derivations are performed in terms of their strong and weak forms and supplemented by appropriate linearizations that are used within the iterative Newton-Raphson scheme. In the special case of the first iteration, this yields the linearized elasticity problem. It will also be demonstrated how further simplifications result in the Poisson (membrane) and uniaxial problems. Attention is focused in this chapter on the derivations of the variational forms of the boundary value problems. These forms are also known as the weak forms because the strong forms only need to be satisfied in an integral rather than in a pointwise sense. Moreover, integration by parts reduces the differentiability requirements for the functions involved in the weak forms. These are the key ingredients needed to develop numerical methods of the Galerkin type, as will be presented in the subsequent chapter.
- Hellinger, E.: Die allgemeinen Ansätze der Mechanik der Kontinua. In: Klein, F., Müller, C. (eds.) Enzyklopädie der Mathematischen Wissenschaften, pp. 601–694. Teubner, Leipzig (1914)Google Scholar
- Lax, P.D., Milgram, A.N.: Parabolic equations. In: Annals of Mathematics Studies, vol. 33, pp. 167–190. Princeton University Press, Princeton (1954)Google Scholar
- Prange, G.: Das Extremum der Formänderungsarbeit—Habilitation thesis, Hannover, Germany. In: Knothe, K. (ed.) Institut für Geschichte der Naturwissenschaften, München, 1999. Erwin Rauner, München (1916)Google Scholar
- Raviart, P.-A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Mathematical Aspects of Finite Element Methods (Proceedings of the Conference, Consiglio Naz. delle Ricerche (C. N. R.), Rome, 1975). Lecture Notes in Mathematics, vol. 606, pp. 292–315. Springer, Berlin (1977)Google Scholar
- Rüter, M., Stein, E.: On the duality of global finite element discretization error-control in small strain Newtonian and Eshelbian mechanics. Technische Mechanik 23, 265–282 (2003)Google Scholar