On the evaluation of the J-integral by a plastic crack tip element
Tóm tắt
Two crack tip elements are formulated for a stationary, mode I plastic crack in planar structures using hybrid assumed stress approach, based on the secant modulus and the Newton-Raphson schemes, respectively. The stress distribution in the crack tip element is assumed to be the HRR field superimposed by the regular polynomial terms. The formulated (hybrid) crack tip elements are compatible with the isoparametric element so that they can be used conveniently along with the conventional displacement-based finite elements. The intensity of the HRR stress field, the J-integral, is determined directly from the finite element equations together with the nodal displacements. The dominance of the HRR stress field at the crack tip is pertinent to the present approach, which depends on geometry and loading conditions. Since the J-integral is globally path-independent for nonlinear elastic materials (deformation plasticity model), in order to assess the accuracy and efficiency of the methodology as compared to the contour integration approach, numerical studies of common plane-stress cracked configurations are performed for these materials. The results indicate that for a sufficiently small crack tip element size, J from the present approach correlates well, within 6 percent difference, with that from the contour integration for a wide range of material hardening coefficients if the HRR zone exists at the crack tip. These highly accurate results for J from the crack tip stresses could not be achieved without using (newly) modified variational principles and a refined numerical technique. It should be emphasized that the present methodology also can be applied to cracks in J
2 flow materials under HRR dominance. In such case, the J integral may not be globally path independent, and hence it now must be determined from the stress and strain fields near the crack tip.
Tài liệu tham khảo
J.W. Hutchinson, Singular behavior at the end of a tensile crack in a hardening material. Journal of the Mechanics and Physics of Solids 16 (1968) 13–31.
J.R. Rice and G.F. Rosengren, Plane strain deformation near a crack tip in a power law hardening material. Journal of the Mechanics and Physics of Solids 16 (1968) 1–12.
J.R. Rice, A path independent integral and the approximate analysis of notches and cracks. Journal of Applied Mechanics 35 (1968) 379–386.
J.R. Rice, Mathematical analysis in the mechanics of fracture. In H. Liebowitz (ed.) Fracture: An Advanced Treatise 2 (1986) 191–311.
B. Moran and C.F. Shih, Crack tip and associated domain integrals from momentum and energy balance. Engineering Fracture Mechanics 27 (1987) 615–642.
C.F. Shih and M.D. Germain, Requirements for a one parameter characterization of crack tip fields by the HRR singularity. International Journal of Fracture 17 (1981) 27–43.
R.M. McMeeking and D.M. Parks, On criteria for J-dominance of crack tip fields in large scale yielding. In J.D. Landes, J.A. Begley, and G.A. Clarke (eds) Elastic-Plastic Fracture, ASTM STP 668 (1979) 175–194.
A. Needleman and V. Tvergaard, Crack tip stress and deformation fields in a solid with a vertex on its yield surface. In C.F. Shih and J.P. Gudas (eds) Elastic-Plastic Fracture, ASTM STP 803 (1983) 80–115.
S.N. Atluri and M. Nakagaki, J-integral estimates for strain-hardening materials in ductile fracture problems. AIAA Journal 15 (1977) 923–931.
P.D. Hilton and J.W. Hutchinson, Plastic intensity factors for cracked plates. Engineering Fracture Mechanics 3 (1971) 435–451.
P.D. Hilton, Finite Element Fracture Mechanics Analysis of Two-Dimensional and Axisymmetric Elastic and Elastic-Plastic Cracked Structure. Report AD/A-005 880 (1974).
D.M. Tracey, Finite elements for determination of crack tip elastic stress intensity factors. Engineering Fracture Mechanics 3 (1971) 255–265.
R.S. Barsorum, Triangular quarter-point elements as elastic and perfectly plastic crack tip element. International Journal Numerical Methods in Engineering 11 (1977) 85–98.
T.H.H. Pian, Derivation of element stiffness matrices by assumed stress distribution. AIAA Journal 2 (1964) 1333–1336.
T.H.H. Pian and P. Tong, Basis of finite element methods for solid continua. International Journal of Numerical Methods in Engineering 1 (1969) 3–28.
C.H. Luk, Assumed Stress Hybrid Finite Element Method for Fracture Mechanics and Elastic-Plastic Analysis. Technical Report AFOSR TR 73-0493 (1972).
Y. Yamada, S. Nakagiri and K. Takatsuka, Elastic-plastic analysis of Saint-Venant torsion problem by a hybrid stress model. International Journal of Numerical Methods in Engineering 15 (1972) 193–203.
A.J. Banard and P.W. Sharman, The elasto-plastic analysis of plates using hybrid finite elements. International Journal of Numerical Method in Engineering 10 (1976) 1343–1356.
R.L. Spilker, Elastic-Plastic Analysis by the Hybrid-Stress Model and the Initial Stress Approach. Technical Report AMMRC CRT-73-38 (1973).
N.P. O'Dowd and C.F. Shih, Family of crack tip fields characterized by a triaxiality parameter — I. Structure of fields. Journal of the Mechanics and Physics of Solids 39 (1991) 989–1015.
S.M. Sharma and N. Aravas, Determination of high-order terms in asymptotic elastoplastic crack tip solution. Journal of the Mechanics and Physics of Solids 39 (1991) 1043–1072.
A. Mendelson, Plasticity Theory and Application, MacMillan, New York (1968).
P.C.T. Chen, Constitutive Matrix and Finite Element Formulation for a Deformation Theory of Plasticity. Technical Report AMCMS 501A-11-84400-02 (1971).
C.F. Shih and A. Needleman, Fully plastic crack problems, Part 1: Solutions by a penalty method. Journal of Applied Mechanics 51 (1984) 48–56.
K. Washizu, Variational Method in Elasticity and Plasticity, Pergamon Press, Oxford (1968).
C.F. Shih, Table of Hutchinson-Rice-Rosengren Singular Field Quantities. Report MRL E-147, Brown University, Providence (1983).
C.N. Duong and D. Blair, Singular Stress Field Near a Plastic Crack Tip. Report MDC-K4518, McDonnel Douglas Corporation, Long Beach (1989).
G.C. Nayak and O.C. Zienkiewicz, Elasto-plastic stress analysis. A generalization for various constitutive relations including strain softening. International Journal of Numerical Methods in Engineering 5 (1972) 113–135.
Bath, Finite Element Procedure in Engineering Analysis, Prentice Hall (1989).
P.J. Davis and P. Rabinowitz, Numerical Integration, Blaisdell Publishing Company (1967).
V. Kumar, H.G.de Lorenzi, W.R. Andrew, C.F. Shih, M.D. German and D.F. Mowbray, Estimation Technique For The Prediction Of Elastic-Plastic Fracture Of Structural Components of Nuclear Systems. Fourth Semi Annual Report to EPRI, Contract No. RP1237–1, General Electric Company, Schenectady, New York (1981).
S. Jun, Z.H. Li, Z.J. Deng, and M.J. Tu, Crack tip constraint and J-controlling stable growth of crack in plane-stress case. Engineering Fracture Mechanics 39 (1991) 1045–1049.
S. Yang, Y.J. Chao, and M.A. Sutton, Complete theoretical-analysis for higher-order terms and the HRR zone at a crack tip for mode-I and mode-II loading of a hardening material. Acta Mechanica 98 (1993) 79–98.
T.J.R. Hughes, Generalization of selective integration procedures to anisotropic and nonlinear media. International Journal for Numerical Methods in Engineering 15 (1980) 1413–1418.