# zbMATH — the first resource for mathematics

A justification of the linear Koiter and Naghdi models for totally clamped shells, subjected to ‘nonadmissible’ loads. (Justification des modèles linéaires de Koiter et de Naghdi pour des coques totalement encastrées soumises à des forces “non admissibles”.) (French) Zbl 1107.74323
Summary: We consider a shell, i.e., a three-dimensional body with a small thickness (denoted by $$2\varepsilon$$), which is clamped along its entire lateral boundary and subjected to regular loads. In the linear case, one can use the two-dimensional models of Koiter or Naghdi to calculate the displacement vector field of the shell. Some error estimates have been obtained in [V. Lods and C. Mardare, J. Elasticity 58, No. 2, 105–154 (2000; Zbl 0994.74046); Asymptotic Anal. 28, No. 1, 1–30 (2001; Zbl 1016.74038)] between these models and the three-dimensional displacement for flexural or membrane shells. Here, we do not make any assumptions on the geometry of the shell. In particular, the space of linearized inextensional displacements can be reduced to zero. The assumptions on the loads are weak: for instance, one can consider regular loads (in $$H^1$$) which do not depend on the transverse variable. We then establish a relative error estimate for the scaled linearized deformation tensor between Koiter’s model (or Naghdi’s model) and the three-dimensional model. These estimates hold though the limit model is not known. In addition, further assumptions on the data allow one to recover the error estimates concerning the displacements which have been proved in the papers cited above.”

##### MSC:
 74K25 Shells
Full Text: