Plane asymptotic crack-tip solutions in gradient elasticity

INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES

卷:46

Article

Aravas, N.^1,3  Giannakopoulos, A. E.^2,3 

[1] Univ Thessaly, Dept Mech Engn, Volos 38334, Greece

[2] Univ Thessaly, Dept Civil Engn, Volos 38334, Greece

[3] Ctr Res & Technol Thessaly, Mechatron Inst, Volos 38500, Greece

关键词: Fracture mechanics;    Gradient elasticity;    Asymptotic solutions;   

DOI  :  10.1016/j.ijsolstr.2009.09.009

来源: Elsevier

【 摘 要 】

An asymptotic crack-tip solution under conditions of plane strain is developed for a material that obeys a special form of linear isotropic strain gradient elasticity. In particular, an elastic constitutive equation of the form tau = tau((0)) - l(2)del(2)tau((0)) is considered, where (tau, epsilon) are the stress and strain tensors, T-(0) = lambda epsilon(kk)delta + 2 mu epsilon, (lambda, mu) are the Lame constants, and R is a material length. Both symmetric (mode-I) and antisymmetric (mode-II) solutions are developed. The asymptotic solution predicts finite strains at the crack-tip. The mode-I crack-tip displacement field u is of the form u(1) = Ax(1) + l(r/l)(3/2)[A(1)(u) over tilde (11)(theta, nu) + A(2)(u) over tilde (12)(theta, nu)] + O(r(2)), u(2) = Bx(2) + l(r/l)(3/2)[A(1)(u) over tilde (21)(theta, nu) + A(2)(u) over tilde (22)(theta, nu)] + O(r(2)), where (X-1, X-2) and (r, 0) are crack-tip Cartesian and polar coordinates, respectively, v is Poisson's ratio, and (A, B, A(1), A(2)) are dimensionless constants determined by the complete solution of a boundary value problem. The A- and B-terms above correspond to uniform normal strains parallel (epsilon(11)) and normal (epsilon(22)) to the crack line, which do not contribute to the crack-tip energy release rate (J-integral). Detailed finite element calculations are carried out for an edge-cracked-panel (ECP) loaded by point forces and the asymptotic solution is verified. The region of dominance of the asymptotic solution for the ECP geometry analyzed is found to be of order l/10. The energy release rate is found to decrease with increasing e. (C) 2009 Elsevier Ltd. All rights reserved.

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