Compatibility of strains and the three-fold differentiability of the displacement field

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Abstract

The problem of the necessary class of smoothness of solutions to quasi-static problems of deformable solid mechanics in terms of displacements was discussed. It is shown that in order for the equations of compatibility of deformations to become identities when displacements are substituted in them, the existence of some third mixed derivatives of displacements is required. A counterexample for a linearly elastic compressible isotropic elastic medium was given. In this counterexample, the displacement field, being a doubly differentiable solution to the boundary value problem for the system of Lame equations in the entire domain, is not a solution to the displacement problem at all points in this domain.

About the authors

D. V. Georgievskii

Lomonosov Moscow State University; Ishlinsky Institute for Problems in Mechanics RAS; Moscow Center of Fundamental and Applied Mathematics

Author for correspondence.
Email: georgiev@mech.math.msu.su
Russian Federation, Moscow; Moscow; Moscow

References

  1. Nowacki W. Teoria sprezystosci. Warszawa: PWN, 1973.
  2. Georgievskii D.V. High-rank deformators and the Kroener incompatibility tensors with two-domensional structure of indices // Doklady Physics. 2019. V. 64. № 6. P. 256–257.
  3. Pobedria B.E. Numerical Methods in Theory of Elasticity and Plasticity. Moscow: Moscow Univ., 1995. [in Russian]

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