Computation of Mixed-Mode Stress Intensity Factors for Cracks in Three-Dimensional Functionally Graded Solids
Publication: Journal of Engineering Mechanics
Volume 132, Issue 1
Abstract
This work applies a two-state interaction integral to obtain stress intensity factors along cracks in three-dimensional functionally graded materials. The procedures are applicable to planar cracks with curved fronts under mechanical loading, including crack-face tractions. Interaction-integral terms necessary to capture the effects of material nonhomogeneity are identical in form to terms that arise due to crack-front curvature. A discussion reviews the origin and effects of these terms, and an approximate interaction-integral expression that omits terms arising due to curvature is used in this work to compute stress intensity factors. The selection of terms is driven by requirements imposed by material nonhomogeneity in conjunction with appropriate mesh discretization along the crack front. Aspects of the numerical implementation with (isoparametric) graded finite elements are addressed, and examples demonstrate the accuracy of the proposed method.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The writers thank the support of the NASA Graduate Student Researchers Program (NASANGT 2-52271) and the NASA-Ames Engineering for Complex Systems Program (NASANAG 2-1424). Dr. Tina Panontin at Ames serves as the technical monitor for these programs. The writers also gratefully acknowledge the National Science Foundation (NSF) Mechanics and Materials Program (NSFCMS-0115954). The first writer wishes to thank Professor J.-H. Kim of the University of Connecticut for many helpful conversations, and the anonymous reviewers for useful suggestions. Any opinions, findings, conclusions, or recommendations expressed in this publication do not necessarily reflect the views of the sponsors.
References
Anderson, T. L. (1995). Fracture mechanics, fundamentals and applications, 2nd Ed., CRC, New York.
Anlas, G., Santare, M. H., and Lambros, J. (2000). “Numerical calculation of stress intensity factors in functionally graded materials.” Int. J. Fract., 104, 103–143.
Anlas, G., Santare, M. H., and Lambros, J. (2002). “Dominance of asymptotic crack tip fields in elastic functionally graded materials.” Int. J. Fract., 115, 193–204.
Chan, Y.-S., Fannjiang, A. C., and Paulino, G. H. (2003). “Integral equations with hypersingular kernels—Theory and applications to fracture mechanics.” Int. J. Eng. Sci., 41, 683–720.
Chan, Y.-S., Paulino, G. H., and Fannjiang, A. C. (2001). “The crack problem for nonhomogeneous materials under antiplane shear loading—A displacement based formulation.” Int. J. Solids Struct., 38, 2989–3005.
Chen, F. H. K., and Shield, R. T. (1977). “Conservation laws in elasticity of the J-integral type.” Z. Angew. Math. Phys., 28, 1–22.
Choi, N. Y., and Earmme, Y. Y. (1992). “Evaluation of stress intensity factors in a circular arc-shaped interfacial crack using -integral.” Mech. Mater., 14, 141–153.
Choules, B. D., Kokini, K., and Taylor, T. A. (2001). “Thermal fracture of ceramic thermal barrier coatings under high heat flux with time-dependent behavior—Part 1—Experimental results.” Mater. Sci. Eng., A, 299, 296–304.
Cook, R. D., Malkus, D. S., Plesha, M. E., and Witt, R. J. (2002). Concepts and applications of finite element analysis, Wiley, New York.
Dag, S., and Erdogan, F. (2002). “A surface crack in a graded medium under general loading conditions.” ASME J. Appl. Mech., 69, 580–588.
Dhondt, G. (1998). “On corner point singularities along a quarter circular crack subject to shear loading.” Int. J. Fract., 89, L33–L38.
Dhondt, G. (2001). “3-D mixed-mode -calculations with the interaction integral method and the quarter point element stress method.” Commun. Numer. Methods Eng., 17, 303–307.
Dhondt, G. (2002). “Mixed-mode -calculations in anisotropic materials.” Eng. Fract. Mech., 69, 909–922.
Dolbow, J., and Gosz, M. (2002). “On the computation of mixed-mode stress intensity factors in functionally graded materials.” Int. J. Solids Struct., 39, 2557–2574.
Eischen, J. W. (1987). “Fracture of nonhomogeneous materials.” Int. J. Fract., 34, 3–22.
FEACrack 2.6; user’s manual. (2005). Structural Reliability Technology, 1898 South Flatiron Court, #235, Boulder, Colo. 80301.
Forth, S. C., Favrow, L. H., Keat, W. D., and Newman, J. A. (2003). “Three-dimensional mixed-mode fatigue crack growth in a functionally graded titanium alloy.” Eng. Fract. Mech., 70, 2175–2185.
Fung, Y. C. (1965). Foundations of solid mechanics, Prentice-Hall, Englewood Cliffs, N.J.
Gosz, M., Dolbow, J., and Moran, B. (1998). “Domain integral formulation for stress intensity factor computation along curved three-dimensional interface cracks.” Int. J. Solids Struct., 35, 1763–1783.
Gosz, M., and Moran, B. (2002). “An interaction energy integral method for computation of mixed-mode stress intensity factors along non-planar crack fronts in three dimensions.” Eng. Fract. Mech., 69, 299–319.
Gu, P., Dao, M., and Asaro, R. J. (1999). “A simplified method for calculating the crack-tip field of functionally-graded materials using the domain integral.” ASME J. Appl. Mech., 66, 101–108.
Gullerud, A., et al. (2004). WARP3D Release 15 manual, Civil Engineering, Rep. No. UILU-ENG-95-2012, Univ. of Illinois at Urbana-Champaign, Urbana, Ill.
Huang, G.-Y., and Wang, Y.-S. (2004). “A new model for fracture analysis of a functionally graded interfacial zone under harmonic anti-plane loading.” Eng. Fract. Mech., 71, 1841–1851.
Jin, Z.-H., and Dodds, R. H., Jr. (2004). “Crack growth resistance behavior of a functionally graded material: Computational studies.” Eng. Fract. Mech., 71, 1651–1672.
Jin, Z.-H., Paulino, G. H., and Dodds, R. H., Jr. (2002). “Finite element investigation of quasistatic crack growth in functionally graded materials using a novel cohesive zone fracture model.” ASME J. Appl. Mech., 69, 370–379.
Jin, Z.-H., Paulino, G. H., and Dodds, R. H., Jr. (2003). “Cohesive fracture modeling of elastic-plastic crack growth in functionally graded materials.” Eng. Fract. Mech., 70, 1885–1912.
Kim, J.-H. (2003). “Mixed-mode crack propagation in functionally graded materials.” PhD thesis, Univ. of Illinois at Urbana-Champaign, Urbana, Ill.
Kim, J.-H., and Paulino, G. H. (2002a). “Finite element evaluation of mixed mode stress intensity factors in functionally graded materials.” Int. J. Numer. Methods Eng., 53, 1903–1935.
Kim, J.-H., and Paulino, G. H. (2002b). “Isoparametric graded finite elements for nonhomogeneous isotropic and orthotropic materials.” ASME J. Appl. Mech., 69, 502–514.
Kim, J.-H., and Paulino, G. H. (2003a). “Mixed-mode J-integral formulation and implementation using graded finite elements for fracture analysis of nonhomogeneous orthotropic materials.” Mech. Mater., 35, 107–128.
Kim, J.-H., and Paulino, G. H. (2003b). “T-stress, mixed-mode stress intensity factors, and crack initiation angles in functionally graded materials: A unified approach using the interaction integral method.” Comput. Methods Appl. Mech. Eng., 192, 1463–1494.
Kim, J.-H., and Paulino, G. H. (2003c). “The interaction integral for fracture of orthotropic functionally graded materials: Evaluation of stress intensity factors.” Int. J. Solids Struct., 40, 3967–4001.
Kim, J.-H., and Paulino, G. H. (2003d). “An accurate scheme for mixed-mode fracture analysis of functionally graded materials using the interaction integral and micromechanics models.” Int. J. Numer. Methods Eng., 58, 1457–1497.
Kim, J.-H., and Paulino, G. H. (2004). “T-stress in orthotropic functionally graded materials: Lekhnitskii and Stroh formalism.” Int. J. Fract., 126, 345–389.
Kim, Y. J., Kim, H.-G., and Im, S. (2001). “Mode decomposition of three-dimensional mixed-mode cracks via two-state integrals.” Int. J. Solids Struct., 38, 6405–6426.
Knowles, J. K., and Sternberg, E. (1972). “On a class of conservation laws in linearized and finite elastostatics.” Arch. Ration. Mech. Anal., 44, 187–211.
Konda, N., and Erdogan, F. (1994). “The mixed-mode crack problem in a nonhomogeneous elastic medium.” Eng. Fract. Mech., 47, 533–545.
Krysl, P., and Belytschko, T. (1999). “The element free Galerkin method for dynamic propagation of arbitrary 3-D cracks.” Int. J. Numer. Methods Eng., 44, 767–800.
Li, C., and Zou, Z. (1998a). “Internally circumferentially cracked cylinders with functionally graded material properties.” Int. J. Pressure Vessels Piping, 75, 499–507.
Li, C., and Zou, Z. (1998b). “Stress intensity factors for a functionally graded material cylinder with an external circumferential crack.” Fatigue Fract. Eng. Mater. Struct., 21, 1447–1457.
Li, C., Zou, Z., and Duan, Z. (1999). “Stress intensity factors for functionally graded solid cylinders.” Eng. Fract. Mech., 63, 735–749.
Li, C., Zou, Z., and Duan, Z. (2000). “Multiple isoparametric finite element method for nonhomogeneous media.” Mech. Res. Commun., 27, 137–142.
Nahta, R., and Moran, B. (1993). “Domain integrals for axisymmetric interface problems.” Int. J. Solids Struct., 30, 2027–2040.
Nakamura, T. (1991). “Three-dimensional stress fields of elastic interface cracks.” ASME J. Appl. Mech., 58, 939–946.
Nakamura, T., and Parks, D. M. (1989). “Antisymmetrical 3-D stress field near the crack front of a thin elastic plate.” Int. J. Solids Struct., 25, 1411–1426.
Ozturk, M., and Erdogan, F. (1995). “An axisymmetric crack in bonded materials with a nonhomogeneous interfacial zone under torsion.” ASME J. Appl. Mech., 62, 116–125.
Ozturk, M., and Erdogan, F. (1996). “Axisymmetric crack problem in bonded materials with a graded interfacial region.” Int. J. Solids Struct., 33, 193–219.
Paulino, G. H., Jin, Z.-H., and Dodds, R. H., Jr. (2002). “Failure of functionally graded materials.” Comprehensive structural integrity, Vol. 2, B. Karihaloo and W. G. Knauss, eds., Elsevier, New York.
Paulino, G. H., and Kim, J.-H. (2004). “A new approach to compute T-stress in functionally graded materials using the interaction integral method.” Eng. Fract. Mech., 71, 1907–1950.
Pook, L. P. (1994). “Some implications of corner point singularities.” Eng. Fract. Mech., 48, 367–378.
Rangaraj, S., and Kokini, K. (2003). “Interface thermal fracture in functionally graded zirconia-mullite-bond coat alloy thermal barrier coatings.” Acta Mater., 51, 251–267.
Rao, B. N., and Rahman, S. (2003). “Mesh-free analysis of cracks in isotropic functionally graded materials.” Eng. Fract. Mech., 70, 1–27.
Rice, J. R. (1968). “A path independent integral and the approximate analysis of strain concentration by notches and cracks.” J. Appl. Mech., 35, 379–386.
Shih, C. F., Moran, B., and Nakamura, T. (1986). “Energy release rate along a three-dimensional crack front in a thermally stressed body.” Int. J. Fract., 30, 79–102.
Stern, M., Becker, E. B., and Dunham, R. S. (1976). “A contour integral computation of mixed-mode stress intensity factors.” Int. J. Fract., 12, 359–368.
Suresh, S., and Mortensen, A. (1998). Fundamentals of functionally graded materials, Institute of Materials, London.
Sutradhar, S., and Paulino, G. H. (2004). “Symmetric Galerkin boundary element computation of -stress and stress intensity factors for mixed-mode cracks by the interaction integral method.” Eng. Anal. Boundary Elem., 28, 1335–1350.
Walters, M. C., Paulino, G. H., and Dodds, R. H., Jr. (2004). “Stress intensity factors for surface cracks in functionally graded materials under mode-I thermomechanical loading.” Int. J. Solids Struct., 41, 1081–1118.
Walters, M. C., Paulino, G. H., and Dodds, R. H., Jr. (2005). “Interaction integral procedures for 3-D curved cracks including surface tractions.” Eng. Fract. Mech., 72, 1635–1663.
Wang, X. (2004). “Elastic -stress solutions for penny-shaped cracks under tension and bending.” Eng. Fract. Mech., 71, 2283–2298.
Wei, L. W., Edwards, L., and Fitzpatrick, M. E. (2002). “FE analysis of stresses and stress intensity factors of interfacial cracks in a CTS specimen.” Eng. Fract. Mech., 69, 85–90.
Williams, M. L. (1957). “On the stress distribution at the base of a stationary crack.” ASME J. Appl. Mech., 24, 109–114.
Yau, J. F., Wang, S. S., and Corten, H. T. (1980). “A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity.” ASME J. Appl. Mech., 47, 335–341.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 132 • Issue 1 • January 2006
Pages: 1 - 15
Copyright
© 2006 ASCE.
History
Received: Aug 16, 2004
Accepted: Mar 14, 2005
Published online: Jan 1, 2006
Published in print: Jan 2006
Notes
Note. Associate Editor: Yunping Xi
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.