Author: Neff P.

Source: Proceedings Section A: Mathematics - Royal Society of Edinburgh, Volume 132, Number 1, 15 February 2002 , pp. 221-243(23)

Publisher: Royal Society of Edinburgh

Abstract:

In this paper we prove a Korn-type inequality with non-constant coefficients which arises from applications in elasto-plasticity at large deformations. More precisely, let R³ be a bounded Lipschitz domain and let be a smooth part of the boundary with non-vanishing two-dimensional Lebesgue measure. Define H_o^1,2(,) := { H^1,2() | _| = 0} and let F_p, F_p^-1 C¹ (bar, GL(3,R)) be given with det F_p(x) ⁺ > 0. Moreover, suppose that Rot F_p C¹(bar, M^3×3). Then

c⁺ > 0 H_o^1,2(, ) : || · F_p^-1(x) + F_p^-T(x) · ^T ||_L2()² c⁺ ||||_H1,2()².

Clearly, this result generalizes the classical Korn's first inequality

c⁺ > 0 H_o^1,2(, ) : || + ^T||_L2() c⁺ ||||_H1,2()²

which is just our result with F = 11. With slight modifications, we are also able to treat forms of the type

||F_p(x) · · G(x) + G(x)^T · ^T · F_p^T(x)||^p, 1 < p < .

Document Type: Research article

Publication date: 2002-02-15

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