Korn’s constant for a parallelepiped with a free face or pair of faces.

*(English)*Zbl 1001.74560Summary: Korn’s inequality is studied for vector fields (the ‘displacements’) on a rectangular parallelepiped under some special boundary conditions (abbreviated below to BCs). The BCs are posed on two pairs of faces or, additionally, on one more face. The remaining pair of faces/face is ‘free’ (by which is meant that the displacements are arbitrary on it). The BCs (where they are posed) are of one of the following three types: (1) zero displacements (‘clamping’), (2) tangential displacements (‘sliding’), or (3) normal displacements (‘bending’). The relative dimensions of the parallelepiped are arbitrary.

The exact values of Korn’s constant \(k\) are found in some of the cases considered, and the rigorous upper bounds for it are found in the others. In particular, it is established that \(k=4\) when all the faces but one are clamped (i.e., for the displacements on a half-space with a compact support), the same result being valid for some other problems considered, where the BCs are posed on all the faces but one. Additionally, in the case when a pair of faces is free, the explicit simple formula for Korn’s constant \(k\) versus ‘thickness’ (by which is meant the mutual distance of free faces) is found; in particular, this yields that \(k\) approaches exponentially from above the value of four as the thickness increases.

The results can be applied to the problem of determination of the bounds of elastic stability for the bodies of appropriate shape under specified BCs (e.g., for the plates of arbitrary thickness, clamped over the contour).

The exact values of Korn’s constant \(k\) are found in some of the cases considered, and the rigorous upper bounds for it are found in the others. In particular, it is established that \(k=4\) when all the faces but one are clamped (i.e., for the displacements on a half-space with a compact support), the same result being valid for some other problems considered, where the BCs are posed on all the faces but one. Additionally, in the case when a pair of faces is free, the explicit simple formula for Korn’s constant \(k\) versus ‘thickness’ (by which is meant the mutual distance of free faces) is found; in particular, this yields that \(k\) approaches exponentially from above the value of four as the thickness increases.

The results can be applied to the problem of determination of the bounds of elastic stability for the bodies of appropriate shape under specified BCs (e.g., for the plates of arbitrary thickness, clamped over the contour).

##### MSC:

74G45 | Bounds for solutions of equilibrium problems in solid mechanics |

74B05 | Classical linear elasticity |

35Q72 | Other PDE from mechanics (MSC2000) |

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\textit{E. I. Ryzhak}, Math. Mech. Solids 4, No. 1, 35--55 (1999; Zbl 1001.74560)

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