TAILIEUCHUNG - Uniqueness of elastic continuation in a semilinear elastic body
The authors prove a theorem on uniqueness of elastic continuation in a nonhomogeneous elastic solid with a displacement-dependent tension modulus, generalizing an earlier result by Ang, Ikehata, Trong and Yamamoto for a nonhomogeneous linear elastic solid. | Vietnam Journal of Mechanics, NCST of Vietnam Vol. 25 , 2003, No 1 (1 - 8) UNIQUENESS OF ELASTIC CONTINUATION IN A SEMILINEAR ELASTIC BODY DANG DINH ANG 1 NGUYEN DUNG 2 NGUYEN Vu HuY 1 AND DANG Due TRONG 1 1 2 University of Natural Sciences HoChiMinh City Institute of Applied Mechanics, HoChiMinh City ABSTRACT. The authors prove a theorem on uniqueness of elastic continuation in a nonhomogeneous elastic solid with a displacement-dependent tension modulus, generalizing an earlier result by Ang, Ikehata, Trong and Yamamoto for a nonhomogeneous linear elastic solid. Let D be a bounded domain in R 3 representing an elastic body. We consider the problem of uniqueness for the determination of the stress field in n from the displacements and surface stresses given on an open portion r of the boundary an of n, a problem referred to as one of elastic continuation. In [AITY], uniqueness of elastic continuation is proved for a nonhomogeneous linear elastic solid. In the present paper, we address the problem of uniqueness of elastic continuation in the case of a nonhomogeneous elastic solid with a tension modulus depending not only on x = (x 1 , x 2, x 3) but also on the displacement u = (u 1 , u 2 , u 3). More precisely, we shall assume >. = >.(x, u), (1) where >. is a multiple of the tension modulus (cf. [TG]). Let x = (x 1,x2,X3) be in DC R 3 . For (i,j,k) = (1,2,3),(2,3,1),(3,1,2), we denote by CJi, Tjk the components of the normal stress and of the shear stress corresponding to the xi-direction. We shall consider the following system (cf. [TG]) OCJi OTij OTik ++- = - xi oxi &xj &xk (2) subject to the boundary conditions (3) and (4) 1 where u0 = (u~, ug, ug) and n = (n1, n2, n3) is the outer unit normal vector to an. The displacement u = (u 1 , u2, u 3) and the stresses ai , Tjk satisfy the following relations (cf. [TG]) (5) (6) where (7) (8) From now on, we shall assume that and that ,\(x,u) > 0 for all x E 0, u E R 3 . Following is the main result
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