Analysis of 3D Planar Crack Problems by Body Force Method

Derivation of the integral equation for general 3D crack problems was examined based on the theory of body force method. In the present analysis, stress intensity factors (SIFs) along a front of arbitrary shaped 3D planar crack are obtained directly only by solving simultaneous equations expressing a boundary condition. The crack surface is discretized using number of triangular elements and the variation of the force doublet embedded in each triangle is assumed at constant. The derived boundary integral equation was transformed into a set of simultaneous equations and was solved computationally. In order to improve the accuracy of the numerically examined boundary integral, a polar transformation scheme combined with Tayler expansion of the fundamental solutions is introduced. Not only a single crack problem but also an interference among coplanar cracks can be calculated using the unique program developed in this research. It was verified that as the number of triangular elements increases, the evaluated SIF converges to the reference solution. Introduction SIF determination plays a central role in linear elastic fracture mechanics. Since SIF was proposed by Irwin [1] to express displacements and stresses in the vicinity of crack tip, several analytical techniques have been developed for a variety of common crack configurations; however, these analytical solutions are limited to simple crack geometries and loading conditions. Advances in numerical modeling procedures have opened new doors for fracture mechanics analysis. Advanced researches have been carried out based on FEM, BEM and other numerical techniques but may be undesirable due to excessive modeling and computational time. Nisitani [2] first applied the body force method (BFM) to crack problems. The BFM has been provided highly accurate solutions of stress concentration factors and SIFs of practically important problems, but the treated 3D problems are considerably limited than 2D problems till now. Isida and Tsuru [3] proposed a new technique for 3D crack analysis based on BFM. Noda [4] discussed numerical solutions using singular integral equation of the BFM for 2D and 3D cracks problems. However, these solutions are limited to crack geometries and also need to solve equations for each problem. In the present research, a numerical program based on BFM for versatile purpose has been developed. And is used to analyze a various 3D planar crack problems. Crack to crack interaction can change the stress distribution near the macro crack tip. Thus the study of interacting cracks subjected to a given set of external load is extremely important for the purpose of design and life prediction of mechanical structures. Till now interaction between cavity problems was solved based on BFM [5]. In the present paper, the interference effect between different-shaped planner cracks is also presented. Theoretical analysis In BFM, the solution of any elastic problem is transformed into a problem of a complete infinite domain without any crack nor notch. That is, a boundary of a given problem is replaced by an equivalent imaginary boundary along which body forces and body force doublets are embedded [6]. Consider an arbitrary oriented planar cracks of general shape in an infinite solid. Take the global coordinate system O -XYZ and local coordinate system O′-xyz of the crack as shown in Fig. 1a. For the analysis of the crack, the global coordinate is transformed to local coordinate system where z-axis is normal to the crack surface. On the idea of the body force method, the problem is reduced to sets of integral equations in which the density of force doublets are unknowns to be determined. Let σzz, τzx and τyz be the stress component in local coordinate system due to the fundamental force doublets distributed over ther crack surface. The stress component in local coordinate system are as follows. σzz(P) = σzz (P) + ∫∫[σzz (P, Q)γzz(Q) + σzz (P, Q)γzx(Q)+σzz (P, Q)γyz(Q)] Ωc dΩc(Q) (1) τzx(P) = τzx ∞ (P) + ∫∫[τzx (P, Q)γzz(Q) + τzx (P, Q)γzx(Q) + τzx (P, Q)γyz(Q)]dΩc(Q) Ωc (2) τyz(P) = τyz ∞ (P) + ∫∫[τyz (P, Q)γzz(Q) + τyz (P, Q)γzx(Q)+τyz (P, Q)γyz(Q)] Ωc dΩc(Q) (3) In these equations, P(x,y,z) is a reference point, Q(ξ, η, ζ) is a source point, Ωc is an imaginary crack surface, γzz , γzx and γyz are the unknown functions called density of standard force doublets. These equation includes nine fundamental solutions which can be derived from the Kelvin solution (a stress field due to a point force acting in an infinite solid). Among the fundamental solutions of body force doublet, three of them are listed bellow as an example and similarly it is possible to derive the others solutions. σzz (P, Q) = 1 − 2ν 8π(1 − ν)2 [ 1 r3 + 6 (z − ζ) r5 − 15 (z − ζ) r7 ] (4) τzx (P, Q) = 3(1 − 2ν) 8π(1 − ν)2 (x − ξ)(z − ζ) [ 1 r5 − 5 (z − ζ) r7 ] (5) τyz (P, Q) = 3(1 − 2ν) 8π(1 − ν)2 (y − η)(z − ζ) [ 1 r5 − 5 (z − ζ) r7 ] (6) Where r = (x − ξ) + (y − η) + (z − ζ) and ν is the Poisson’s ratio. Fig. 1: a) Global and local coordinates systems; b) Planar triangle surface element x,ξ y,η