Minimum Energy Characterizations for the Solution of Saint-Venant's Problem in the Theory of Shells

Abstract

The linear theory of Cosserat surfaces is employed to study Saint-Venant's problem for cylindrical shells of arbitrary cross-section. We prove minimum energy characterizations for the solution of the relaxed Saint-Venant's problem previously obtained. Then, we determine the global measures of strain appropriate to extension, bending, torsion and flexure for certain classes of solutions to the relaxed problem. (2000): 74K25, 74G05. Mathematics Subject Classifications

On a Problem of Truesdell for Anisotropic Elastic Shells

“…In [4], we have determined a solution of the extension-bending-torsion problem (P 1 ) expressed as a linear combination of four displacement fields v (1) , v (2) , v (3) and v (4) . These displacement fields v (k) are known and their expressions are recorded in the Appendix.…”

“…Returning to the extension-bending-torsion problem (P 1 ) for cylindrical Cosserat shells, we mention that in the isotropic case the constants a 3 , a α and a 4 which satisfy the system (20) can be interpreted as the global measures of axial strain, axial curvature and twist, respectively (see [3]). Suggested by (20), we consider the following problem: to define the functionals τ k (•), k = 1, ..., 4, on the set of solutions…”

On the theory of loaded general cylindrical Cosserat elastic shells

“…We note that the constants a 3 , a a and k can be interpreted as the global measures of axial strain, axial curvature and twist, respectively, for the extension-bending-torsion problem, while the coefficient D stands for the torsional rigidity of the cylindrical shell (see Bîrsan, 2005a). In the subsequent developments, we shall employ the field v[a i , k] to express the solution of our problem.…”

“…We mention that the solution of the relaxed Saint-Venant's problem for Cosserat shells has been presented in Bîrsan (2004Bîrsan ( , 2005a. Interesting results related to this problem have been obtained by different approaches in several studies, such as Berdichevsky et al (1992), Ladevèze et al (2004), Librescu and Song (2005).…”

Inequalities of Korn’s Type and Existence Results in the Theory of Cosserat Elastic Shells

“…been successfully applied to the study of Saint-Venant's problem for cylindrical thinwalled tubes in [5,6].…”