tailieunhanh - A cauchy like problem in plane elasticity

Let rl be a bounded domain in the plane, representing an elastic body. Let ro be a portion of the boundary r of rl, ro being assumed to be parallcd to the x - axis. It is proposed to determine the stress field in n from the displacements and surface stresses given on r 0 . Under the assumption of plane stress, it is shown that ux + uy is a harmonic function. An Airy stress function is introduced, from which the stress field is computed. | Vietnam Journal of Mechanics, VAST, Vol. 29, No. 3 (2007), pp. 215 - 248 Special Issue Dedicated to the Memory of Prof. Nguyen Van Dao A CAUCHY LIKE PROBLEM IN PLANE ELASTICITY* DANG DINH ANG, NGUYEN DUNG Institute of Applied Mechanics, VAST, Hochiminh City Abstract. Let rl be a bounded domain in the plane, representing an elastic body. Let ro be a portion of the boundary r of rl, ro being assumed to be parallcd to the x axis. It is proposed to determine the stress field in from the displacements and surface stresses given on r 0 . Under the assumption of plane stress, it is shown that ux + uy is a harmonic function. An Airy stress function is introduced, from which the stress field is computed. n Consider an elastic body represented by a bounded domain 0 in the plane. Let ro be a portion of the boundary r of 0 assumed to be paralled to the x - axis (cf. Fig. 1). \Ve propose to determine the stress field in 0 from the displacements and surface stresses given on ro. y 0 w x Fig. 1 Cauchy like problems in plane elasticity are treated in [1], [4] and others (cf. References). For a derivation of basic relations on stresses and displacements, we follow {TG]. Assume plane stress. \Ve denote the displacements in the x - and y -directions respectively *Supported by the Council for Natural Sciences of Vietnam. 246 Dang Dinh Ang, Nguyen Dung by u and v and the stress components by ax, ay and Txy· Now, we have Ex ou = -;:;;-; ux Ey ov = -;:;;-; uy 'Yxy 1 Du ov = -;:;;-+ -;:;;-, uy ux (1) 1 Ex= E(ax - Vay), Ey = E(ay - Vax), { . "( xy = 1 GTxy (2) 2(1+v) = E Txy In the absence of body forces, we have Oax + OTxy = O ox (3) oy Oay oy + OTxy _ O ox - (4) . These are t he equilibrium equations for our problem. We now derive the compatibility equations. \Ve have OU Ex = -;:;;-, uX Ey av = -;:;;-, uy OU 'Yxy av = -;:;;-+ -;:;;-uy ux (5) from which we get upon differentiating with recpect to y, then with respect to x o 2Ex o