# zbMATH — the first resource for mathematics

Korn’s constant for a parallelepiped with a free face or pair of faces. (English) Zbl 1001.74560
Summary: 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).

##### MSC:
 74G45 Bounds for solutions of equilibrium problems in solid mechanics 74B05 Classical linear elasticity 35Q72 Other PDE from mechanics (MSC2000)
Full Text:
##### References:
 [1] Friedrichs, K.O., Ann. of Math. 48 pp 441– (1947) · Zbl 0029.17002 · doi:10.2307/1969180 [2] Horgan, C.O., SIAM Review 37 pp 491– (1995) · Zbl 0840.73010 · doi:10.1137/1037123 [3] Holden, J.T, Arch. Rat. Mech. Anal. 17 pp 171– (1964) · Zbl 0125.13005 · doi:10.1007/BF00282436 [4] Payne, L.E., Arch. Rat. Mech. Anal. 8 pp 85– (1961) · doi:10.1007/BF00277432 [5] Bernstein, B., Arch. Rat. Mech. Anal. 6 pp 51– (1960) · Zbl 0094.30001 · doi:10.1007/BF00276153 [6] Dafermos, C.M., ZAMP 19 pp 913– (1968) · Zbl 0169.55904 · doi:10.1007/BF01602271 [7] Horgan, C.O., Arch. Rat. Mech. Anal. 40 pp 384– (1971) · Zbl 0223.73011 · doi:10.1007/BF00251798 [8] Ryzhak, E.I., Quart. J. Mech. Appl. Math. 47 pp 663– (1994) · Zbl 0820.73036 · doi:10.1093/qjmam/47.4.663 [9] Zygmund, A., Trigonometric series (1959) · Zbl 0085.05601 [10] Kaczmarz, S., Theorie der orthogonalreihen (1935) · JFM 61.1119.05 [11] Ito, H., J. Elasticity 24 pp 43– (1990) · Zbl 0772.35007 · doi:10.1007/BF00115553
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.