A. Piva Assistant Professor, Department of Physics, University of Bologna, 40126 Bologna, Italy E. Radi Istituto di Scienza delle Construzioni, University of Bologna, 40136, Bologna, Italy Elastodynamic Local Fields for a Crack Running in an Orthotopic Medium The dynamic stress and displacement fields in the neighborhood of the tip of a crack propagating in an orthotropic medium are obtained. The approach deals with the methods of linear algebra to transform the equations of motion into a first-order elliptic system whose solution is sought under the assumption that the local dis- placement field may be represented under a scheme of separated variables. The analytical approach has enabled the distinction between two kinds of orthotropic materials for which explicit espressions of the near-tip stress fields are obtained. Some results are presented graphically also in order to compare them with the numerical solution given in a quoted reference. 1 Introduction A number of powerful analytical approaches for the solution of a propagating crack problem in an anisotropic elastic me- dium are available. Among the most notable of these there is the Stroh's for- mulation (1962) which provides an elegant method of treating steady-state problems in anisotropic elasticity. A modified version of the Stroh's method has been applied by Atkinson (1964) and by Atkinson and Head (1966) to de- velop the steady-state model of a propagating crack in an anisotropic medium. One of the most general features, common to plane elas- todynamic solutions for moving cracks through an elastic solid, is the representation of the spatial dependence of the elastic fields in the neighborhood of the crack tip, under a scheme of separated variables. This result has been well established by Freund and Clifton (1974) in the case of a crack moving nonuniformly in an iso- tropic medium. A work that directly relates to the present analysis is that of Achenbach and Bazant (1975) in which the above repre- sentation has been used to obtain elastodynamic near-tip fields for traction-free cracks running in isotropic and orthotropic materials. For the case of isotropic materials they found the closed-form solution for the spatial dependence of the elastic fields, whereas a numerical approach was used for the ortho- tropic case. In the present paper the problem of a crack propagating at a time-dependent velocity in an orthotropic medium is revisited to obtain closed-form expressions for the near-tip elastic fields. In solving the problem, use is made of an approach recently Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the JOURNAL OF APPLIED ME- CHANICS. Discussion on this paper should be addressed to the Technical Editor, Prof. Leon M. Keer, The Technological Institute, Northwestern University, Evanston, IL 60208, and will be accepted until two months after final publication of the paper itself in the JOURNAL OF APPLIED MECHANICS. Manuscript received by the ASME Applied Mechanics Division, June 12, 1989; final revision, July 30, 1990. proposed by Piva (1987) and by Piva and Viola (1988) to transform the equations of motion into a first-order elliptic system which is then solved following the initiative of the previously cited authors. In addition, the method of solution allows to distinguish between two kinds of orthotropic materials for which signif- icant differences on the near-tip stress field are pointed out. 2 Mathematical Preliminaries Consider an infinite orthotropic elastic body and a crack moving with a time-dependent speed c(t) along the x-axis of a Cartesian coordinate system 0(x, y, z) whose axes are of elastic symmetry. By referring to coordinates (x, y) attached to the moving crack tip and following the considerations re- ported by Achenbach and Bazant (1975), the system of equa- tions governing the elastodynamic displacement field in a deleted neighborhood of the crack tip may be written as d 2 u „„ d 2 v d 2 u n d 2 v d 2 u dx 2 dxdy + ai tf = °> (1«) (lb) where u = u(x, y, t) and v = v(x, y, f) are the displacement components and 2/3 = 20,- C\2 + C 66 c,i(l-M,)' c 12 + ^66 c 66 (l-M 2 2 ) c 66 'c u (l-MJ)' «r c 22 c 66 (i~M 2 2 Y The coefficients c,y are parameters related to the elastic cos- c u /p, v\ = tants, Mi = c z (t)/vi, Mi = c^(t)/i/ 2 , uf c 66 /p, and p is the mass density. The stress-strain equations for an orthotropic elastic body are 982 / Vol. 58, DECEMBER 1991 Transactions of the ASME Copyright © 1991 by ASME Downloaded 17 Sep 2009 to 155.185.228.41. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm