Optimal exponentials of thickness in Korn’s inequalities for parabolic and elliptic shells. (English) Zbl 1460.74059
Summary: We consider the scaling of the optimal constant in Korn’s first inequality for elliptic and parabolic shells which was first given by Y. Grabovsky and D. Harutyunyan [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 35, No. 1, 267–282 (2018; Zbl 1395.74056)] with hints coming from the test functions constructed by P. E. Tovstik and A. L. Smirnov [Asymptotic methods in the buckling theory of elastic shells. River Edge, NJ: World Scientific (2001; Zbl 1066.74500)] on the level of formal asymptotic expansions. Here, we employ the Bochner technique in Remannian geometry to remove the assumption that the middle surface of the shell is given by one single principal coordinate, in particularly, including closed elliptic shells.
MSC:
| 74K25 | Shells |
| 74G10 | Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
| 53Z05 | Applications of differential geometry to physics |
Keywords:
asymptotic expansion; interpolation inequality; Bochner method; nonlinear elasticity; Riemannian geometryReferences:
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