NASITE: Nagasaki Universit's Ac Title Author(s) Interference Analsis between Crack Plate b Bod Force Method Ino, Takuichiro; Ueno, Shohei; Saim Citation Ke Engineering Materials, 577-578, Issue Date 2014 UL http://hdl.handle.net/10069/34050 ight (2014) Trans Tech Publications. This document is downloaded http://naosite.lb.nagasaki-u.ac.jp
Interference Analsis Between Crack and General Inclusion in an Infinite Plate b Bod Force Method Takuichiro Ino 1,a, Shohei Ueno 1,b and Akihide Saimoto 2,c 1 Graduate Student, Mechanical Engineering Course, Graduate School of Engineering, Nagasaki Universit, 1-14 Bunko-machi, Nagasaki 852-8521, Japan 2 Corresponding Author: Graduate School of Engineering, Nagasaki Universit a taku20052006@gmail.com, b bali2 14@ahoo.co.jp, c s-aki@nagasaki-u.ac.jp Kewords: Inclusion, Graded Material, Force doublet, Principle of superposition Abstract. A continousl embedded force doublet over the particular region can be regarded as the distributing eigen strain. This fact implies that man sorts of inelastic strain can be replaced b the force doublet. In the present paper, the force doublet is used to alter the local constitutive relationship. As a result, a method for analzing the general inclusion problem in which the material properties of the inclusion are not onl different from those of the matri material but also can be even a function of spacial coordinate variables is proposed. The theoretical background of the present analsis is eplained followed b some representative numerical results. Introduction There are at least two advantages, if the Bod Force Method (BFM) is emploed rather than the conventional Finite Element Method (FEM), for the analsis of graded materials. The 1st advantage is that, there is no need to prepare the mesh division to the domain where the material composition is identical to the matri while the complete discretization of the total bod is indispensable in FEM. The 2nd advantage is that the location of crack is admitted onl along the boundar of each element in FEM while there is no such restriction in BFM. So far, the inclusion problems analzed b BFM were such a primitive one that the material composition or propert of the inclusion region is uniform though the value is different from that of surrounding matri material. M. Yatsuda, Y. Murakami and M. Isida[1] analzed the stress distribution in the vicinit of an elliptic inclusion embedded in the infinite plate under remote tension. The Young s modulus and the Poisson s ratio in the inclusion and the matri regions were uniform at (E I, ν I ) and (E M, ν M ) respectivel. The most characteristic point of their approach was the use of the elastic fields due to a point force acting in a uniform infinite medium that corresponds the matri and the inclusion regions independentl. To the contrast, in the present analsis, the force doublet is emploed in order to epress the presence of general inclusion whose composition can be even a function of the spacial coordinate variables as in graded materials. Based on the present strateg, the interference problem between a general inclusion and a line crack and so on were computed and the results were shown graphicall. Theoretical background The strateg for the treatment of general inclusion problem is illustrated in Fig.1. In the present analsis, the stress component at an arbitrar point P in the matri region (σij M (P )) and that of in the inclusion region (σij(p I )) is epressed b a superposition of the influence due
to remote stresses σij (P ) and the influence of force doublet of magnitude T ij (Q) embedded at a point Q[2,3]. σij M (P ) = σij (P ) + σij kl (P, Q)T kl (Q)d(Q) (1) σij(p I ) = σij (P ) + σij kl (P, Q)T kl (Q)d(Q) T ij (P ) (2) As in a same manner, the strain components corresponding to the matri (ε M ij (P )) and the inclusion (ε I ij(p )) region can be epressed as ε M ij (P ) = ε I ij(p ) = ε ij (P ) + ε kl ij (P, Q)T kl (Q)d(Q) (3) In Eq.1 3, i, j, k and l are indices that epress either of and. T (Q), T (Q) and T (Q) are unknown densit of force doublet at point Q to be embedded in the inclusion region (Q ). σij (P, Q), σ ij (P, Q) and σ ij (P, Q) are stress components at the reference point P due to a unit magnitude of force doublet acting at the source point Q into, and directions, respectivel. In a same manner, ε ij (P, Q), ε ij (P, Q) and ε ij (P, Q) are strain components at the reference point P due to a unit magnitude of force doublet acting at the source point Q into, and directions, respectivel. The constitutive equation between components of stress and strain follows σ M ij (P ) = D M ijkl(p )ε M kl(p ), ε M ij (P ) = C M ijkl(p )σ M kl (P ) (4) σ I ij(p ) = D I ijkl(p )ε I kl(p ), ε I ij(p ) = C I ijkl(p )σ I kl(p ) (5) where D M ijkl and CM ijkl are the stiffness and compliance tensor for matri material and DI ijkl and C I ijkl are those for inclusion material. As seen in Eq.3, εm ij (P ) and ε I ij(p ) has the identical epression. Therefore, the following relation is derived. C I ijkl(p )σ I kl(p ) = C M ijkl(p )σ M kl (P ) (6) Eq.6 defines the condition through which the unknown magnitude of bod force doublet T ij (P ) at point P is determined as T ij (P ) = [ E ijαβ Dijkl(P I )Cklαβ(P M ) ] [ ] σαβ(p ) + σαβ(p, st Q)T st (Q)d(Q) (7) where E ijαβ is a unit matri of order 3 3. σ σ (a) (b) γ (Q) γ (Q) γ (Q) tension of plate with inclusion simple tension of matri material Γ matri material with force doublets Figure 1: Analsis of inclusion problem b force doublets
Numerical eample In order to verif the applicabilit of the present method, stress distribution in an infinite plate with a circular hole subjected to a uniform tensile stress at infinit was analzed. A circular hole can be considered a kind of inclusion whose modulus of elasticit is zero. Fig.2 shows the analzed σ distribution along the ais. In this analsis, the circular area corresponding to a circular hole was divided with regularl distributed number of triangles NT. As seen, the numerical solution ehibited a good agreement with theoretical value. Figure 3,4 and 5 shows the σ distribution along the ais for the cases of single circular inclusion (Fig.3), interference between a circular inclusion and a circular hole (Fig.4) and interference between a circular inclusion and a crack (Fig.5). The used number of triangles for the inclusion part was fied at NT=1024. Conclusion The bod force doublet approach for solving the general inclusion problem was proposed. As the present method does not require an of special fundamental solution, an arbitrar inclusion problem can be solved in a same manner. For simplicit, the magnitude of force doublet for each triangular area was assumed at constant in the present stud, however, higher order element could be easil introduced and epected to bring a further accurate solution. eferences [1] Yatsuda, A., Murakami, Y. and Isida, M., Stress field due to an interference of two elliptic inclusions, Journal of the Japan Societ of Mechanical Engineers, Ser. A, 51-464, pp.1057-1065 (1985), in Japanese. [2] Chen, D. H. and Nisitani, H., Etension of bod force method to elastic-plastic problems, Journal of the Japan Societ of Mechanical Engineers, Ser. A, 51-462, pp.571-578 (1985), in Japanese. [3] Chen, D. H. and Nisitani, H., Stress field in a composite based on bod force method, Journal of the Japan Societ of Mechanical Engineers, Ser. A, 57-540, pp.1815-1821 (1991), in Japanese. 4.0 NT=16 NT=256 NT=64 NT=1024 σ/σ 2 eact NT = 1024 NT = 256 NT = 64 NT = 16 1.1 1.2 1.3 1.4 / σ Figure 2: Triangular division of the circular area corresponding to a circular hole and the calculated σ distribution along the ais (NT: number of triangles)
σ σ/σ E I /E M = E I /E M = E I /E M =(/+3)/2 E I /E M =(/+2) 2 - -4.0 - - - 4.0 Figure 3: σ distribution along the ais for the problem of tension of an infinite plate with single circular inclusion of various characteristics (ν I = ν M = ) / σ d/=4.0 d σ/σ hole - E I /E M = E I /E M = E I /E M = /+3 E I /E M =(/+2) 2-4.0 - - - 4.0 / Figure 4: σ distribution along the ais for the problem of tension of an infinite plate with circular inclusion of various characteristics (ν I = ν M = ) and a circular hole 4.0 σ dc 2a σ/σ dc/=4.0 a/= E I /E M = E I /E M = E I /E M = /+3 E I /E M =(/+2) 2-4.0 - - - 4.0 / Figure 5: σ distribution along the ais for the problem of tension of an infinite plate with circular inclusion of various characteristics (ν I = ν M = ) and a crack