Skip to main content
SpringerLink
Log in

Fracture Mechanics of Functionally Graded Materials

  • Chapter
Advances in Mechanics and Mathematics

Part of the book series: Advances in Mechanics and Mathematics ((AMMA,volume 4))

Abstract

Functionally graded materials (FGMs) represent a new concept of tailoring materials with microstructural and properties gradients to achieve optimized performance. FGMs were originally conceived as high temperature resistant materials for aircraft and aerospace applications. The FGM concept has since spread to other areas, for example, tribological coatings, diesel engines, energy conversion systems, optical fibers and filters, wireless communication devices, dental implants and posts, and so on. The interest in FGMs has been growing rapidly in recent years due to their unique characteristics and potential advantages.

Fracture mechanics concerns failure of materials and structures by crack initiation and propagation. Successful applications of FGMs require thorough understanding of fracture behavior of such materials. This article reviews the progress in fracture mechanics of FGMs; introduces fracture mechanics concepts in FGMs including crack tip elastic fields, K—dominance, fracture toughness and R—curve; describes typical methods to obtain fracture parameters and to simulate failure processes, such as integral transform/integral equation method and cohesive zone approach; and discusses some areas that need substantial future efforts. Special topics including thermal, viscoelastic and dynamic fracture are also discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • J. Aboudi, M.-J. Pindera and S. M. Arnold, 1999, Higher-order theory for functionally graded materials. Composites Part B-Engineering, 30B, 777–832.

    Google Scholar 

  • A. M. Afsar and H. Sekine, 2000, Crack spacing effect on the brittle fracture characteristics of semi-infinite functionally graded materials with periodic edge cracks. International Journal of Fracture, 102, L61–L66.

    Google Scholar 

  • T. L. Anderson, 1995, Fracture Mechanics: Fundamentals and Applications, second edition, CRC Press, Boca Raton.

    MATH  Google Scholar 

  • W. T. Ang and D. L. Clements, 1987, On some crack problems for in-homogeneous elastic materials. International Journal of Solids and Structures, 23, 1089–1104.

    MATH  Google Scholar 

  • G. Anlas, J. Lambros and M. H. Santare, 2002, Dominance of asymptotic crack tip fields in elastic functionally graded materials. International Journal of Fracture, 115, 193–204.

    Google Scholar 

  • G. Anlas, M. H. Santare and J. Lambros, 2000, Numerical calculation of stress intensity factors in functionally graded materials. International Journal of Fracture, 104, 131–143.

    Google Scholar 

  • N. D. Aparicio and C. Atkinson, 1994, Plane dynamic crack propagation in a nonhomogeneous viscoelastic strip. International Journal of Engineering Science, 32, 209–228.

    MathSciNet  MATH  Google Scholar 

  • C. Atkinson, 1974, Some aspects of dynamic crack propagation: a review and some generalisations. In: G. C. Sih (ed.), Prospects of Fracture Mechanics, Noordhoff International Publishing, Leyden, The Netherlands, pp. 337–350.

    Google Scholar 

  • C. Atkinson, 1975, Some results on crack propagation in media with spatially varying elastic moduli. International Journal of Fracture, 11, 619–628.

    Google Scholar 

  • C. Atkinson, 1977a, On stress singularities and interfaces in linear elastic fracture mechanics. International Journal of Fracture, 13, 807–820.

    MathSciNet  Google Scholar 

  • C. Atkinson, 1977b, Dynamic crack problems in dissimilar media. In: G. C. Sih (ed.), Mechanics of Fracture, Vol. 4: Elastodynamic Crack Problems, Noordhoff International Publishing, Leyden, The Netherlands, pp. 213–248.

    Google Scholar 

  • C. Atkinson and R. V. Craster, 1995, Theoretical aspects of fracture mechanics. Progress in Aerospace Sciences, 31, 1–83.

    ADS  Google Scholar 

  • C. Atkinson and R. D. List, 1978, Steady state crack propagation into media with spatially varying elastic properties. International Journal of Engineering Science, 16, 717–730.

    MATH  Google Scholar 

  • R. Babaei and S. A. Lukasiewicz, 1998, Dynamic response of a crack in a functionally graded material between two dissimilar half planes under anti-plane shear impact load. Engineering Fracture Mechanics, 60, 479–487.

    Google Scholar 

  • H. Balke, H.-A. Bahr, A. S. Semenov, G. Kirchhoff and H.-J. Weiss, 2001, Graded thermal barrier coatings: cracking due to laser irradiation and determining of interface toughness. In: K. Trumble, K. Bowman, I. Reimanis and S. Sampath (eds.), Ceramic Transactions, Vol. 114: Functionally Graded Materials 2000, American Ceramic Society, Westerville, Ohio, pp. 201–212.

    Google Scholar 

  • Y. Bansal and M.-J. Pindera, 2002, Efficient reformulation of the thermoelastic higher-order theory for FGMs. NASA Contractor Report: NASA/CR-2002–211909.

    Google Scholar 

  • Y. Bansal, Y. Zhong and M.-J. Pindera, 2003, Efficient reformulation of the higher-order theory for FGMs. Materials Science Forum, 423–4, 769–776.

    Google Scholar 

  • G. Bao and H. Cai, 1997, Delamination cracking in functionally graded coating/metal substrate systems. Acta Materialia, 45, 1055–1066.

    Google Scholar 

  • G. Bao and C. Y. Hui, 1990, Effects of interface debonding on the toughness of ductile particle reinforced ceramics. International Journal of Solids and Structures, 26, 631–642.

    Google Scholar 

  • G. Bao and L. Wang, 1995, Multiple cracking in functionally graded ceramic-metal coatings. International Journal of Solids and Structures, 32, 2853–2871.

    MATH  Google Scholar 

  • G. Bao and F. Zok, 1993, On the strength of ductile particle reinforced brittle matrix composites. Acta Metallurgica et Materialia, 41, 3515–3524.

    Google Scholar 

  • J. F. Bartolome, J. S. Moya, J. Requena, J. Llorca and M. Anglada, 1998, Fatigure crack growth behavior in mullite/alumina functionally graded ceramics. Journal of the American Ceramic Society, 81, 1502–1508.

    Google Scholar 

  • T. L. Becker, R. M. Cannon and R. O. Ritchie, 2001, Finite crack kinking and T-stresses in functionally graded materials. International Journal of Solids and Structures, 38, 5545–5563.

    MATH  Google Scholar 

  • T. L. Becker, R. M. Cannon and R. O. Ritchie, 2002, Statistical fracture modeling: crack path and fracture criteria with application to homogeneous and functionally graded materials. Engineering Fracture Mechanics, 69, 1521–1555.

    Google Scholar 

  • Y. Benveniste, 1987, A new approach to the application of Mori-Tanaka’s theory in composite materials. Mechanics of Materials, 6, 147–157.

    Google Scholar 

  • M. B. Bever and P. E. Duwez, 1972, Gradients in composite materials. Materials Science and Engineering, 10, 1–8.

    Google Scholar 

  • O. Bleeck, D. Munz, W. Schaller and Y. Y. Yang, 1998, Effect of a graded interlayer on the stress intensity factor of cracks in a joint under thermal loading. Engineering Fracture Mechanics, 60, 615–623.

    Google Scholar 

  • H. A. Bruck and A. L. Gershon, 2002, Three-dimensional effects near the interface in a functionally graded Ni-Al2O3 plate specimen. International Journal of Solids and Structures, 39, 547–557.

    MATH  Google Scholar 

  • B. Budiansky, 1965, On the elastic moduli of some heterogeneous materials. Journal of the Mechanics and Physics of Solids, 13, 223–227.

    ADS  Google Scholar 

  • B. Budiansky, J. C. Amazigo and A. G. Evans, 1988, Small-scale crack bridging and the fracture toughness of particulate-reinforced ceramics. Journal of the Mechanics and Physics of Solids, 36, 167–187.

    ADS  Google Scholar 

  • H. Cai and G. Bao, 1998, Crack bridging in functionally graded coatings. International Journal of Solids and Structures, 35, 701–717.

    MATH  Google Scholar 

  • R. D. Carpenter, 2000, Experimental fracture measurements of functionally graded materials. Ph.D. thesis, University of California, Davis, CA, USA (to be published).

    Google Scholar 

  • R. D. Carpenter, W. W. Liang, G. H. Paulino, J. C. Gibeling and Z. A. Munir, 1999, Fracture testing and analysis of a layered functionally graded Ti/TiB beam in 3-point bending. Materials Science Forum, 308–311, 837–842.

    Google Scholar 

  • H. S. Carslaw and J. C. Jaeger, 1959, Conduction of Heat in Solids, Clarendon Press, Oxford.

    Google Scholar 

  • Y.-S. Chan, G. H. Paulino and A. C. Fanjiang, 2001, Gradient elasticity theory for mode I crack in functionally graded materials. In: K. Trumble, K. Bowman, I. Reimanis and S. Sampath (eds.), Ceramic Transactions, Vol. 114: Functionally Graded Materials 2000, American Ceramic Society, Westerville, Ohio, pp. 731–738.

    Google Scholar 

  • J. Chapa, K. Rozenburg, I. Reimanis and E. D. Steffler, 2001, Fracture in ductile/brittle graded composites. In: K. Trumble, K. Bowman, I. Reimanis and S. Sampath (eds.), Ceramic Transactions, Vol. 114: Functionally Graded Materials 2000, American Ceramic Society, Westerville, Ohio, pp. 797–804.

    Google Scholar 

  • Y. F. Chen and F. Erdogan, 1996, The interface crack problem for a nonhomogeneous coating bonded to a homogeneous substrate. Journal of the Mechanics and Physics of Solids, 44, 771–787.

    ADS  Google Scholar 

  • S.-H. Chi and Y.-L. Chung, 2003, Cracking in coating-substrate composites with multi-layered and FGM coatings. Engineering Fracture Mechanics, 70, 1227–1355.

    Google Scholar 

  • H. J. Choi, 1996, Bonded dissimilar strips with a crack perpendicular to the functionally graded interface. International Journal of Solids and Structures, 33, 4101–4117.

    MATH  Google Scholar 

  • H. J. Choi, 2001a, The problem for bonded half-planes containing a crack at an arbitrary angle to the graded interfacial zone. International Journal of Solids and Structures, 38, 6559–6588.

    MATH  Google Scholar 

  • H. J. Choi, 2001b, Effects of graded layering on the tip behavior of a vertical crack in a substrate under factional Hertzian contact. Engineering Fracture Mechanics, 68, 1033–1059.

    Google Scholar 

  • H. J. Choi, 2002, Driving forces and kinking of an oblique crack in bonded nonhomogeneous materials. Archive of Applied Mechanics, 72, 342–362.

    ADS  MATH  Google Scholar 

  • H. J. Choi, K. Y. Lee and T. E. Jin, 1998a, Collinear cracks in a layered half-plane with a graded nonhomogeneous interfacial zone — Part I: mechanical response. International Journal of Fracture, 94, 103–122.

    Google Scholar 

  • H. J. Choi, T. E. Jin and K. Y. lee, 1998b, Collinear cracks in a layered half-plane with a graded nonhomogeneous interfacial zone — Part II: thermal shock response. International Journal of Fracture, 94, 123–135.

    Google Scholar 

  • R. M. Christensen, 1969, Viscoelastic properties of heterogeneous materials. Journal of the Mechanics and Physics of Solids, 17, 23–41.

    ADS  Google Scholar 

  • R. M. Christensen, 1971, Theory of Viscoelasticity, Academic Press, New York.

    Google Scholar 

  • R. M. Christensen, 1979, Mechanics of Composite Materials, John Wiley & Sons, New York.

    Google Scholar 

  • R. M. Christensen and K. H. Lo, 1979, Solutions for effective shear properties in three phase sphere and cylinder models. Journal of the Mechanics and Physics of Solids, 27, 315–330.

    ADS  MATH  Google Scholar 

  • T. J. Chung, A. Neubrand and J. Rodel, 2002, Effect of residual stress on the fracture toughness of Al2O3/Al gradient materials. Key Engineering Materials, 206–2, 965–968.

    Google Scholar 

  • T. J. Chung, A. Neubrand, J. Rodel and T. Fett, 2001, Fracture toughness and R-curve behavior of Al2O3/Al FGMs. In: K. Trumble, K. Bowman, I. Reimanis and S. Sampath (eds.), Ceramic Transactions, Vol. 114: Functionally Graded Materials 2000, American Ceramic Society, Westerville, Ohio, pp. 789–796.

    Google Scholar 

  • B. Cotterell, 1965, On brittle fracture paths. International Journal of Fracture Mechanics, 1, 96–103.

    Google Scholar 

  • B. Cotterell and J. R. Rice, 1980, Slightly curved or kinked cracks. International Journal of Fracture, 16, 155–169.

    Google Scholar 

  • R. V. Craster and C. Atkinson, 1994, Mixed boundary-value-problems in nonhomogeneous elastic-materials. Quarterly Journal of Mechanics and Applied Mathematics, 47, 183–206.

    MathSciNet  MATH  Google Scholar 

  • S. Dag and F. Erdogan, 2002, A surface crack in a graded medium under general loading conditions. ASME Journal of Applied Mechanics, 69, 580–588.

    ADS  MATH  Google Scholar 

  • F. Delale and F. Erdogan, 1983, The crack problem for a nonhomogeneous plane. ASME Journal of Applied Mechanics, 50, 609–614.

    ADS  MATH  Google Scholar 

  • F. Delale and F. Erdogan, 1988, On the mechanical modeling of an interfacial region in bonded half-planes. ASME Journal of Applied Mechanics, 55, 317–324.

    ADS  Google Scholar 

  • R. S. Dhaliwal and B. M. Singh, 1978, On the theory of elasticity of a non-homogeneous medium. Journal of Elasticity, 8, 211–219.

    MATH  Google Scholar 

  • R. H. Dodds Jr, C. F. Shih and T. L. Anderson, 1993, Continuum and micro-mechanics treatment of constraint in fracture. International Journal Fracture, 64, 101–133.

    ADS  Google Scholar 

  • J. E. Dolbow and M. Gosz, 2002, On the computation of mixed-mode stress intensity factors in functionally graded materials. International Journal of Solids and Structures, 39, 2557–2574.

    MATH  Google Scholar 

  • J. W. Eischen, 1987a, Fracture of nonhomogeneous materials. International Journal of Fracture, 34, 3–22.

    Google Scholar 

  • J. W. Eischen, 1987b, An improved method for computing the J 2 integral. Engineering Fracture Mechanics, 26, 691–700.

    Google Scholar 

  • S. El-Borgi, L. Hidri and F. Erdogan, 2001, Stress intensity factors for a crack arbitrarily oriented in a functionally graded layer. In: K. Trumble, K. Bowman, I. Reimanis and S. Sampath (eds.), Ceramic Transactions, Vol. 114: Functionally Graded Materials 2000, American Ceramic Society, Westerville, Ohio, pp. 723–730.

    Google Scholar 

  • F. Erdogan, 1978, Mixed boundary-value problems in mechanics. In: S. Nemat-Nasser (ed.), Mechanics Today, Vol. 4, Pergamon Press, New York, pp. 1–86.

    Google Scholar 

  • F. Erdogan, 1985, The crack problem for bonded nonhomogeneous materials under antiplane shear loading. ASME Journal of Applied Mechanics, 52, 823–828.

    ADS  MathSciNet  MATH  Google Scholar 

  • F. Erdogan, 1995, Fracture mechanics of functionally graded materials. Composite Engineering, 5, 753–770.

    Google Scholar 

  • F. Erdogan, G. D. Gupta and T. S. Cook, 1973, Numerical solution of singular integral equations. In: G. C. Sih (ed.), Mechanics of Fracture, Vol. 1: Methods of Analysis and Solutions of Crack Problems, Noordhoff International Publishing, Leyden, The Netherlands, pp. 368–425.

    Google Scholar 

  • F. Erdogan and M. Ozturk, 1995, Periodic cracking of functionally graded coating. International Journal of Engineering Science, 33, 2179–2195.

    MathSciNet  MATH  Google Scholar 

  • F. Erdogan and G. C. Sih, 1963, On the crack extension in plates under plane loading and transverse shear. Journal of Basic Engineering, 85, 519–527.

    Google Scholar 

  • F. Erdogan and B. H. Wu, 1996, Crack problems in FGM layers under thermal stresses. Journal of Thermal Stresses, 19, 237–265.

    Google Scholar 

  • F. Erdogan and B. H. Wu, 1997, The surface crack problem for a plate with functionally graded properties. ASME Journal of Applied Mechanics, 64, 449–456.

    ADS  MATH  Google Scholar 

  • A. G. Evans and R. M. McMeeking, 1986, On the toughening of ceramics by strong reinforcements. Acta Metallurgica, 34, 2435–2441.

    Google Scholar 

  • T. Y. Fan, 1990, Introduction to Fracture Dynamics, Beijing Institute of Technology Press, Beijing.

    Google Scholar 

  • T. Fett, D. Munz and Y. Y. Yang, 2000, Direct adjustment procedure for weight functions of graded properties. Fatigue and Fracture of Engineering Materials and Structures, 23, 191–198.

    Google Scholar 

  • L. B. Freund, 1990, Dynamic Fracture Mechanics, Cambridge Unversity Press, Cambridge.

    MATH  Google Scholar 

  • T. Fujimoto and N. Noda, 2001a, Two crack growth in a functionally graded plate under thermal shock. Journal of Thermal Stresses, 24, 847–862.

    Google Scholar 

  • T. Fujimoto and N. Noda, 2001b, Influence of the compositional profile of functionally graded material on the crack path under thermal shock. Journal of the American Ceramic Society, 84, 1480–1486.

    Google Scholar 

  • A. Gerasoulis and R. P. Srivastav, 1980, A Griffith crack problem for a nonhomogeneous medium. International Journal of Engineering Science, 18, 239–247.

    MATH  Google Scholar 

  • A. E. Giannakopoulos, S. Suresh, M. Finot and M. Olsson, 1995, Elastoplastic analysis of thermal cycling: layered materials with compositional gradients. Acta Metallurgica et Materialia, 43, 1335–1354.

    Google Scholar 

  • P. Gu and R. J. Asaro, 1997a, Cracks in functionally graded materials. International Journal of Solids and Structures, 34, 1–17.

    MATH  Google Scholar 

  • P. Gu and R. J. Asaro, 1997b, Crack deflection in functionally graded materials. International Journal of Solids and Structures, 34, 3085–3098.

    MATH  Google Scholar 

  • P. Gu, M. Dao and R. J. Asaro, 1999, A simplified method for calculating the crack-tip field of functionally graded materials using the domain integral. ASME Journal of Applied Mechanics, 66, 101–108.

    ADS  Google Scholar 

  • G. V. Guinea, M. Elices and J. Planas, 1997, On the initial shape of the softening function of cohesive materials. International Journal of Fracture, 87, 139–149.

    Google Scholar 

  • A. S. Gullerud, K. C. Koppenhoefer, Y. A. Roy and R. H. Dodds Jr, 2002, WARP3D — Release 14.2 Manual, Civil Engineering, Report No. UILU-ENG-95–2012, University of Illinois, Urbana, IL 61801, USA.

    Google Scholar 

  • G. D. Gupta and F. Erdogan, 1974, The problem of edge cracks in an infinite strip. ASME Journal of Applied Mechanics, 41, 1001–1006.

    ADS  MATH  Google Scholar 

  • Z. Hashin, 1965, Viscoelastic behavior of heterogeneous media. ASME Journal of Applied Mechanics, 32, 630–636.

    Google Scholar 

  • Z. Hashin, 1968, Assessment of the self consistent scheme approximation: conductivity of particulate composites. Journal of Composite Materials, 2, 284–300.

    Google Scholar 

  • D. P. H. Hasselman and G. E. Youngblood, 1978, Enhanced thermal stress resistance of structural ceramics with thermal conductivity gradient. Journal of the American Ceramic Society, 61, 49–52.

    Google Scholar 

  • H. Hatta and M. Taya, 1986, Equivalent inclusion method for steady state heat conduction in composites. International Journal of Engineering Science, 24, 1159–1172.

    ADS  MATH  Google Scholar 

  • K. Hayashi and S. Nemat-Nasser, 1981, Energy release rate and crack kinking under combined loading. ASME Journal of Applied Mechanics, 48, 520–524.

    ADS  MATH  Google Scholar 

  • J. M. Herrmann and L. Schovanec, 1990, Quasi-static mode III fracture in a nonhomogeneous viscoelastic body. Acta Mechanica, 85, 235–249.

    MathSciNet  MATH  Google Scholar 

  • J. M. Herrmann and L. Schovanec, 1994, Dynamic steady-state mode III fracture in a nonhomogeneous viscoelastic body. Acta Mechanica, 106, 41–54.

    MathSciNet  MATH  Google Scholar 

  • R. Hill, 1965, A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids, 13, 213–222.

    ADS  Google Scholar 

  • T. Hirai, 1996, Functionally gradient materials. In: R. J. Brook (ed.), Materials Science and Technology, Vol. 17B Processing of Ceramics, Part 2, VCH Verlagsgesellschaft mbH, Weinheim, Germany, pp. 292–341.

    Google Scholar 

  • T. Honein and G. Herrmann, 1997, Conservation laws in nonhomogeneous plane elastostatics. Journal of the Mechanics and Physics of Solids, 45, 789–805.

    ADS  MathSciNet  MATH  Google Scholar 

  • B. Ilschner, 1996, Processing-microstructure-property relationships in graded materials. Journal of the Mechanics and Physics of Solids, 44, 647–656.

    ADS  Google Scholar 

  • T. Ishiguro, A. Makino, N. Araki and N. Noda, 1993, Transient temperature response in functionally gradient materials. International Journal of Thermophysics, 14, 101–121.

    ADS  Google Scholar 

  • S. Itou, 2001, Transient dynamic stress intensity factors around a crack in a nonhomogeneous interfacial layer between dissimilar elastic half-planes. International Journal of Solids and Structures, 38, 3631–3645.

    MATH  Google Scholar 

  • L. Y. Jiang and X. D. Wang, 2002, On the dynamic crack propagation in an interphase with spatially varying elastic properties under inplane loading. International Journal of Fracture, 114, 225–244.

    Google Scholar 

  • Z.-H. Jin, 2002, An asymptotic solution of temperature field in a strip of a functionally graded material. International Communications in Heat and Mass Transfer, 29, 887–895.

    Google Scholar 

  • Z.-H. Jin, 2003, Some notes on the linear viscoelasticity of functionally graded materials. Mathematics and Mechanics of Solids (in press).

    Google Scholar 

  • Z.-H. Jin and R. C. Batra, 1996a, Some basic fracture mechanics concepts in functionally graded materials. J. Mech. Phys. Solids, 44, 1221–1235.

    ADS  Google Scholar 

  • Z.-H. Jin and R. C. Batra, 1996b, Interface cracking between functionally graded coatings and a substrate under antiplane shear. International Journal of Engineering Science, 34, 1705–1716.

    MATH  Google Scholar 

  • Z.-H. Jin and R. C. Batra, 1996c, Stress intensity relaxation at the tip of an edge crack in a functionally graded material subjected to a thermal shock. Journal of thermal stresses, 19, 317–339.

    Google Scholar 

  • Z.-H. Jin and R. C. Batra, 1998, R-curve and strength behavior of a functionally graded material. Materials Sci. and Engn., A242, 70–76.

    Google Scholar 

  • Z.-H. Jin and N. Noda, 1993, An internal crack parallel to the boundary of a nonhomogeneous half plane under thermal loading. International Journal of Engineering Science, 31, 793–806.

    ADS  MATH  Google Scholar 

  • Z.-H. Jin and N. Noda, 1994a, Crack-tip singular fields in nonhomogeneous materials. ASME J. Appl. Mech., 61, 738–740.

    ADS  MATH  Google Scholar 

  • Z.-H. Jin and N. Noda, 1994b, An edge crack in a nonhomogeneous half-plane under thermal loading. J. of Thermal Stresses, 17, 591–599.

    Google Scholar 

  • Z.-H. Jin and N. Noda, 1994c, Transient thermal stress intensity factors for a crack in a semi-infinite plane of a functionally graded material. International Journal of Solids and Structures, 31, 203–218.

    MATH  Google Scholar 

  • Z.-H. Jin and G. H. Paulino, 2001, Transient thermal stress analysis of an edge crack in a functionally graded material. International Journal of Fracture, 107, 73–98.

    Google Scholar 

  • Z.-H. Jin and G. H. Paulino, 2002, A viscoelastic functionally graded strip containing a crack subjected to in-plane loading. Engineering Fracture Mechanics, 69, 1769–1790.

    Google Scholar 

  • Z.-H. Jin, G. H. Paulino and R. H. Dodds Jr., 2002, Finite element investigation of quasi-static crack growth in functionally graded materials using a novel cohesive zone fracture model. ASME Journal of Applied Mechanics, 69, 370–379.

    ADS  MATH  Google Scholar 

  • Z.-H. Jin, G. H. Paulino and R. H. Dodds Jr., 2003, Cohesive fracture modeling of elastic-plastic crack growth in functionally graded materials. Engineering Fracture Mechanics, 70, 1885–1912.

    Google Scholar 

  • J. Jitcharoen, N. P. Padture, A. E. Giannakopoulos and S. Suresh, 1998, Hertzian-crack suppression in ceramics with elastic-modulus-graded surfaces. Journal of the American Ceramic Society, 81, 2301–2308.

    Google Scholar 

  • S. Kadioglu, S. Dag and S. Yahsi, 1998, Crack problem for a functionally graded layer on an elastic foundation. International Journal of Fracture, 94, 63–77.

    Google Scholar 

  • M. K. Kassir, 1972, Note on the twisting deformation of a nonhomoge-neous shaft containing a circular crack. International Journal of Fracture Mechanics, 8, 325–334.

    Google Scholar 

  • M. K. Kassir and G. C. Sih, 1975, Mechanics of Fracture, Vol. 2: Three-Dimensional Crack Problems, Noordhoff International Publishing, Leyden, The Netherlands.

    Google Scholar 

  • A. Kawasaki and R. Watanabe, 1987, Finite element analysis of thermal stress of the metal ceramic multilayer composites with controlled compositional gradients. Journal of the Japan Institute of Metals, 51, 525–529.

    Google Scholar 

  • A. Kawasaki and R. Watanabe, 1993, Fabrication of disk-shaped functionally gradient materials by hot pressing and their thermomechanical performance. In: J. B. Holt, M. Koizumi, T. Hirai and Z. A. Munir (eds.), Ceramic Transactions, Vol. 34: Functionally Gradient Materials, American Ceramic Society, Westerville, Ohio, pp. 157–164.

    Google Scholar 

  • A. S. Kim, S. Suresh and C. F. Shih, 1997, Plasticity effects on fracture normal to interfaces with homogeneous and graded compositions. International Journal of Solids and Structures, 34, 3415–3432.

    MATH  Google Scholar 

  • J.-H. Kim, and G. H. Paulino, 2002, Finite element evaluation of mixed mode stress intensity factors in functionally graded materials. International Journal for Numerical Methods in Engineering, 52, 1903–1935.

    Google Scholar 

  • M. Koizumi, 1993, The concept of FGM. In: J. B. Holt, M. Koizumi, T. Hirai and Z. A. Munir (eds.), Ceramic Transactions, Vol. 34: Functionally Gradient Materials, American Ceramic Society, Westerville, Ohio, pp. 3–10.

    Google Scholar 

  • M. Koizumi, 1997, FGM activities in Japan. Composites Part B-Engineering, 28, 1–4.

    Google Scholar 

  • K. Kokini and M. Case, 1997, Initiation of surface and interface edge cracks in functionally graded ceramic thermal barrier coatings. ASME Journal of Engineering Materials and Technology, 119, 148–152.

    Google Scholar 

  • K. Kokini, J. DeJonge, S. Rangaraj and B. Beardsley, 2001, Thermal shock of functionally graded thermal barrier coatings. In: K. Trumble, K. Bowman, I. Reimanis and S. Sampath (eds.), Ceramic Transactions, Vol. 114: Functionally Graded Materials 2000, American Ceramic Society, Westerville, Ohio, pp. 213–221.

    Google Scholar 

  • N. Konda and F. Erdogan, 1994, The mixed mode crack problem in a nonhomogeneous elastic medium. Engineering Fracture Mechanics, 47, 533–545.

    Google Scholar 

  • V. D. Krstic, 1983, On the fracture of brittle-matrix/ductile-particle composites. Philosophical Magazine, A48, 695–708.

    ADS  Google Scholar 

  • Y. Kuroda, K Kusaka, A. Moro and M. Togawa, 1993, Evaluation tests of ZrO2/Ni functionally gradient materials for regeneratively cooled thrust engine applications. In: J. B. Holt, M. Koizumi, T. Hirai and Z. A. Munir (eds.), Ceramic Transactions, Vol. 34: Functionally Gradient Materials, American Ceramic Society, Westerville, Ohio, pp. 289–296.

    Google Scholar 

  • J. Lambros, M. H. Santare, H. Li and G. H. Sapna, 1999, A novel technique for the fabrication of laboratory scale model functionally graded materials. Experimental Mechanics, 39, 184–190.

    Google Scholar 

  • V. M. Levin, 1967, Thermal expansion coefficients of heterogeneous material. Mekh. Tver. Tela, 2, 88–94.

    Google Scholar 

  • C. Li, Z. Zou and Z. Duang, 2000, Multiple isoparametric finite element method for nonhomogeneous media. Mechanics Research Communications, 27, 137–142.

    MATH  Google Scholar 

  • C. Y. Li, G. J. Weng and Z. P. Duan, 2001, Dynamic behavior of a cylindrical crack in a functionally graded interlayer under torsional loading. International Journal of Solids and Structures, 38, 7473–7485.

    MATH  Google Scholar 

  • H. Li, J. Lambros, B. A. Cheeseman, M. H. Santare, 2000, Experimental investigation of the quasi-static fracture of functionally graded materials. International Journal of Solids and Structures, 37, 3715–3732.

    MATH  Google Scholar 

  • P. R. Marur and H. V. Tippur, 2000a, Numerical analysis of crack-tip fields in functionally graded materials with a crack normal to the elastic gradient. International Journal of Solids and Structures, 37, 5353–5370.

    MATH  Google Scholar 

  • P. R. Marur and H. V. Tippur, 2000b, Dynamic response of bimaterial and graded interface cracks under impact loading. International Journal of Fracture, 103, 95–109.

    Google Scholar 

  • P. A. Mataga, 1989, Deformation of crack-bridging ductile reinforcements in toughened brittle materials. Acta Metallurgica, 37, 3349–3359.

    Google Scholar 

  • S. A. Meguid, X. D. Wang and L. Y. Jiang, 2002, On the dynamic propagation of a finite crack in functionally graded materials. Engineering Fracture Mechanics, 69, 1753–1763.

    Google Scholar 

  • Y. Miyamoto, Y. Kang, K. Tanihara, J. Lin, M. Niino, S. Hirai and T. Umakoshi, 1997, Application of graded MoSi2/Al2O3/Ni/Al2O3/MoSi2 to high temperature electrode of SiGe thermoelectric device. In: A. Ghosh, Y. Miyamoto, I. Reimanis and J. J. Lannutti (eds.), Ceramic Transactions, Vol. 76: Functionally Graded Materials, American Ceramic Society, Westerville, Ohio, pp. 135–148.

    Google Scholar 

  • R. J. Moon, K. J. Bowman, K. P. Trumble and J. Rodel, 2001, Fracture resistance curve behavior of multilayered alumina-zirconia composites produced by centrifugation. Acta Materialia, 49, 995–1003.

    Google Scholar 

  • R. J. Moon, M. Hoffman, J. Hilden, K. J. Bowman, K. P. Trumble and J. Rodel, 2002, R—curve behavior in alumina-zirconia composites with repeating graded layers. Engineering Fracture Mechanics, 69, 1647–1665.

    Google Scholar 

  • T. Mori and K. Tanaka, 1973, Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Materialia, 21, 571–574.

    Google Scholar 

  • N. I. Muskhelishvili, 1953, Singular Integral Equation, Noordhoff, Groningen.

    Google Scholar 

  • Needleman A, 1987, A continum model for void nucleation by inclusion debonding. ASME Journal of Applied Mechanics, 54, 525–531.

    ADS  MATH  Google Scholar 

  • G. Nelson and A. Ezis, 1996, Functionally graded material (FGM) armor in the TiB/Ti system (U). CERCOM Report, Vista, CA 92083, USA.

    Google Scholar 

  • M. Nemat-Alla and N. Noda, 2000, Edge crack problem in a semi-infinity FGM plate with a bidirectional coefficient of thermal expansion under two dimensional thermal loading. Acta Mechanica, 144, 211–229.

    MATH  Google Scholar 

  • M. Nevalainen and R. H. Dodds Jr, 1995, Numerical investigation of 3-D constraint effects on brittle fracture in SE(B) and C(T) specimens. International Journal of Fracture, 74, 131–161.

    Google Scholar 

  • N. Noda, 1986, Thermal stresses in materials with temperature-dependent properties. In: R. B. Hetnarski (ed.), Thermal Stresses I, Elsevier Science, North-Holland, pp. 391–438.

    Google Scholar 

  • N. Noda, 1991, Thermal stresses in materials with temperature-dependent properties. ASME Applied Mechanics Review, 44, 383–397.

    ADS  Google Scholar 

  • N. Noda, 1997, Thermal stress intensity factor for functionally gradient plate with an edge crack. Journal of Thermal Stresses, 20, 373–387.

    Google Scholar 

  • N. Noda, 1999, Thermal stresses in functionally graded materials. Journal of Thermal Stresses, 22, 477–512.

    MathSciNet  Google Scholar 

  • N. Noda and Z.-H. Jin, 1993, Thermal stress intensity factors for a crack in a strip of a functionally graded material. International Journal of Solids and Structures, 30, 1039–1056.

    MATH  Google Scholar 

  • N. Noda and Z.-H. Jin, 1995, Crack tip singularity fields in nonhomoge-neous body under thermal stress fields. JSME International Journal, Series A, 38, 364–369.

    Google Scholar 

  • N. Noda and T. Tsuji, 1991a, Steady thermal stresses in a plate of a functionally gradient material. Trans. Japan Soc. Mech. Eng., A57, 625–631.

    Google Scholar 

  • N. Noda and T. Tsuji, 1991b, Steady thermal stresses in a plate of a functionally gradient material with temperature-dependent properties. Trans. Japan Soc. Mech. Eng., A57, 98–103.

    Google Scholar 

  • Y. Obata and N. Noda, 1993a, Transient thermal stresses in a plate of functionally gradient material. In: J. B. Holt, M. Koizumi, T. Hirai and Z. A. Munir (eds.), Ceramic Transactions, Vol. 34: Functionally Gradient Materials, American Ceramic Society, Westerville, Ohio, pp. 403–410.

    Google Scholar 

  • Y. Obata and N. Noda, 1993b, Unsteady thermal stresses in a functionally gradient material plate. Trans. Japan Soc. Mech. Eng., A59, 1090–1103.

    ADS  Google Scholar 

  • M. Ortiz and A. Pandolfi, 1999, Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis. International Journal for Numerical Methods in Engineering, 44, 1267–1282.

    MATH  Google Scholar 

  • M. N. Ozisik, 1980, Heat Conduction, John Wiley & Sons, New York.

    Google Scholar 

  • M. Ozturk and F. Erdogan, 1996, Axisymmetric crack problem in bonded materials with a graded interfacial region. International Journal of Solids and Structures, 33, 193–219.

    MATH  Google Scholar 

  • M. Ozturk and F. Erdogan, 1997, Mode I crack problem in an inhomogeneous orthotropic medium. International Journal of Engineering Science, 35, 869–883.

    MATH  Google Scholar 

  • M. Ozturk and F. Erdogan, 1999, The mixed mode crack problem in an inhomogeneous orthotropic medium. International Journal of Fracture, 98, 243–261.

    Google Scholar 

  • V. Parameswaran and A. Shukla, 1999, Crack tip stress fields for dynamic fracture in functionally gradient materials. Mechanics of Materials, 31, 579–596.

    Google Scholar 

  • V. Parameswaran and A. Shukla, 2002, Near-tip out of plane displacement fields for dynamic crack propagation in functionally graded materials. Mechanics Research Communications, 29, 397–405.

    MATH  Google Scholar 

  • G. H. Paulino, R. D. Carpenter, W. W. Liang, Z. A. Munir and J. C. Gibeling, 2001, Fracture testing and finite element modeling of pure Titanium. Engineering Fracture Mechanics, 68, 1417–1432.

    Google Scholar 

  • G. H. Paulino and Z.-H. Jin, 2001, Correspondence principle in viscoelas-tic functionally graded materials. ASME Journal of Applied Mechanics, 68, 129–132.

    ADS  MathSciNet  MATH  Google Scholar 

  • M.-J. Pindera, 1991, Local/global stiffness matrix formulation for composite materials and structures. Composites Engineering, 1, 69–83.

    Google Scholar 

  • M.-J. Pindera, S. M. Arnold, J. Aboudi and D. Hui (eds.), 1994, Use of composites in functionally graded materials. Composites Engineering, 4, No. 1.

    Google Scholar 

    Google Scholar 

  • M.-J. Pindera, J. Aboudi, S. M. Arnold and W. F. Jones (eds.), 1995, Use of composites in multi-phased and functionally graded materials. Composites Engineering, 5, No. 7.

    Google Scholar 

    Google Scholar 

  • M.-J. Pindera, J. Aboudi, A. M. Glaeser and S. M. Arnold (eds.), 1997, Use of composites in multi-phased and functionally graded materials. Composites Part B-Engineering, 28, Nos. 1/2.

    Google Scholar 

    Google Scholar 

  • M.-J. Pindera, J. Aboudi, and S. M. Arnold, 2002, Analysis of spallation mechanism in thermal barrier coatings with graded bond coats using the higher-order theory for FGMs. Engineering Fracture Mechanics, 69, 1587–1606.

    Google Scholar 

  • B. H. Rabin and R. J. Heaps, 1993, Powder processing of Ni-Al2O3 FGM. In: J. B. Holt, M. Koizumi, T. Hirai and Z. A. Munir (eds.), Ceramic Transactions, Vol. 34: Functionally graded Materials, American Ceramic Society, Westerville, OH, pp. 173–180.

    Google Scholar 

  • B. N. Rao and S. Rahman, 2003, Mesh-free analysis of cracks in isotropic functionally graded materials. Engineering Fracture Mechanics, 70, 1–27.

    Google Scholar 

  • I. Reimanis and J. Chapa, 2003, The strength of functionally graded joints: crack paths and residual stress for cracks perpendicular to the gradient. Materials Science Forum, 423–4, 593–598.

    Google Scholar 

  • T. Reiter, G. J. Dvorak and V. Tvergaard, 1997, Micromechanical models for graded composite materials. Journal of the Mechanics and Physics of Solids, 45, 1281–1302.

    ADS  Google Scholar 

  • T. Reiter and G. J. Dvorak, 1998, Micromechanical models for graded composite materials: II. Thermomechanical loading. Journal of the Mechanics and Physics of Solids, 45, 1655–1673.

    ADS  Google Scholar 

  • J. R. Rice, 1968, A path independent integral and the approximate analysis of strain concentration by notches and cracks. ASME Journal of Applied Mechanics, 35, 379–386.

    Google Scholar 

  • C. Rogers and D. L. Clements, 1978, Bergman integral operator in inhomogeneous elasticity. Quarterly of Applied Mathematics, 36, 315–321.

    MathSciNet  MATH  Google Scholar 

  • C.-E. Rousseau and H. V. Tippur, 2000, Compositionally graded materials with cracks normal to the elastic gradient. Acta Materialia, 48, 4021–4033.

    Google Scholar 

  • C.-E. Rousseau and H. V. Tippur, 2001, Influence of elastic gradient profiles on dynamically loaded functionally graded materials with cracks along the gradient. International Journal of Solids and Structures, 38, 7839–7856.

    MATH  Google Scholar 

  • C.-E. Rousseau and H. V. Tippur, 2002, Evaluation of crack tip fields and stress intensity factors in functionally graded elastic materials: cracks parallel to elastic gradient. International Journal of Fracture, 114, 87–111.

    Google Scholar 

  • Y. A. Roy and R. H. Dodds Jr., 2001, Simulation of ductile crack growth in thin aluminum panels using 3-d surface cohesive elements. International Journal of Fracture, 110, 21–45.

    Google Scholar 

  • E. F. Rybicki and M. F. Kanninen, 1977, A finite element calculation of stress intensity factors by a modified crack closure integral. Engineering Fracture Mechanics, 9, 931–938.

    Google Scholar 

  • Ch. Sand, J. Adle and R. Lenk, 1999, A new concept for manufacturing sintered materials with a three dimensional composition gradient using a silicon carbide — titanium carbide composite. Materials Science Forum, 308–311, 65–70.

    Google Scholar 

  • L. Schovanec and J. R. Walton, 1987a, The quasi-static propagation of a plane strain crack in a power-law inhomogeneous linearly viscoelastic body. Acta Mechanica, 67, 61–77.

    MathSciNet  MATH  Google Scholar 

  • L. Schovanec and J. R. Walton, 1987b, The energy release rate for a quasi-static mode I crack in a nonhomogeneous linearly viscoelastic body. Engineering Fracture Mechanics, 28, 445–454.

    Google Scholar 

  • L. Schovanec and J. R. Walton, 1988, On the order of stress singularity for an antiplane shear crack at the interface of two bonded inhomogeneous elastic materials. ASME Journal of Applied Mechanics, 55, 234–236.

    ADS  Google Scholar 

  • R. A. Shapery, 1978, A method for predicting crack growth in nonhomogeneous viscoelastic media. International Journal of Fracture, 14, 293–309.

    Google Scholar 

  • N. I. Shbeeb, W. K. Binienda and K. L. Kreider, 1999a, Analysis of the driving forces for multiple cracks in an infinite nonhomogeneous plate, part I: theoretical analysis. ASME Journal of Applied Mechanics, 66, 492–500.

    ADS  Google Scholar 

  • N. I. Shbeeb, W. K. Binienda and K. L. Kreider, 1999b, Analysis of the driving forces for multiple cracks in an infinite nonhomogeneous plate, part II: numerical solutions. ASME Journal of Applied Mechanics, 66, 501–506.

    ADS  Google Scholar 

  • N. I. Shbeeb and W. K. Binienda, 2001, Analysis of an interface crack for a functionally graded strip sandwiched between two homogeneous layers of finite thickness. Engineering Fracture Mechanics, 64, 693–720.

    Google Scholar 

  • L. X. Shen and S. W. Yu, 1997, An asymptotic method for analyzing the stress in a functionally gradient material layer on a surface of a structural component. Acta Mechanica Solida Sinica, 10, 36–47.

    Google Scholar 

  • C. F. Shih, H. G. deLorenzi and M. D. German, 1976, Crack extension modeling with singular quadratic isoparametric elements. International Journal of Fracture, 12, 647–651.

    Google Scholar 

  • L. S. Sigl, P. A. Mataga, B. J. Dalgleish, R. M. McMeeking and A. G. Evans, 1988, On the toughness of brittle materials reinforced with a ductile phase. Acta Metallurgica, 36, 945–953.

    Google Scholar 

  • Z. Suo, S. Ho, X. Gong, 1993, Notch ductile-to-brittle transition due to localized inelastic band. ASME Journal of Engineering Materials and Technology, 115, 319–326.

    Google Scholar 

  • S. Suresh and A. Mortensen, 1998, Functionally Graded Materials, The Institute of Materials, IOM Communications Ltd, London.

    Google Scholar 

  • I. Tamura, Y. Tomota and H. Ozawa, 1973, Strength and ductility of Fe-Ni-C alloys composed of austenite and martensite with various strength. In: Proceedings of the Third International Conference on Strength of Metals and Alloys, Vol. 1, Institute of Metals, Cambridge, pp. 611–615.

    Google Scholar 

  • Y. Tanigawa, 1995, Some basic thermoelastic problems for nonhomogeneous materials. ASME Applied Mechanics Review, 48, 287–300.

    ADS  Google Scholar 

  • Y. Tanigawa, T. Akai, R. Kawamura and N. Oka, 1996, Transient heat conduction and thermal stress problems of a nonhomogeneous plate with temperature-dependent material properties. Journal of Thermal Stresses, 19, 77–102.

    Google Scholar 

  • Y. Tanigawa, T. Muraki and R. Kawamura, 1996, Evaluation of axisymmetric steady thermal stress and thermal stress intensity factor in Kassier’s nonhomogeneous infinite body with a penny-shaped crack. JSME International Journal, Series A, 39, 540–547.

    Google Scholar 

  • K. Tohgo, M. Sakaguchi and H. Ishii, 1996, Applicability of fracture mechanics in strength evaluation of functionally graded materials. JSME International Journal, Series A, 39, 479–488.

    Google Scholar 

  • V. Tvergaard, 2002, Theoretical investigation of the effect of plasticity on crack growth along a functionally graded region between dissimilar elastic-plastic solids. Engineering Fracture Mechanics, 69, 1635–1645.

    Google Scholar 

  • V. Tvergaard and J. W. Hutchinson, 1992, The relation between crack growth resistance and fracture process parameters in elastic-plastic solids. Journal of the Mechanics and Physics of Solids, 40, 1377–1392.

    ADS  MATH  Google Scholar 

  • S. Ueda, 2001, Thermal shock fracture in a W-Cu divertor plate with a functionally graded nonhomogeneous interface. Journal of Thermal Stresses, 24, 1021–1041.

    Google Scholar 

  • S. Ueda, 2002, Transient thermal singular stresses of multiple cracking in a W-Cu functionally graded divertor plate. Journal of Thermal Stresses, 25, 83–95.

    Google Scholar 

  • H. Uzun, T. C. Lindley, H. B. McShane and R. D. Rawlings, 2001, Fatigue crack growth behavior of 2124/SiC/10p functionally graded materials. Metallurgical and Materials Transactions, A32, 1831–1849.

    Google Scholar 

  • B. L. Wang, J. C. Han and S. Y. Du, 1999, Dynamic response for functionally graded materials with penny-shaped cracks. Acta Mechanica Solida Sinica, 12, 106–113.

    MATH  Google Scholar 

  • B. L. Wang, J. C. Han and S. Y. Du, 2000, Thermoelastic fracture mechanics for nonhomogeneous material subjected to unsteady thermal load. ASME Journal of Applied Mechanics, 67, 87–95.

    ADS  MATH  Google Scholar 

  • B. L. Wang and N. Noda, 2003, Thermally loaded functionally graded materials with embedded defects. Journal of Thermal Stresses, 26, 25–39.

    Google Scholar 

  • X. Wang, Z. Zou and D. Wang, 1997, On the penny-shaped crack in a nonhomogeneous interlayer of adjoining two different elastic materials. International Journal of Solids and Structures, 34, 3911–3921.

    MATH  Google Scholar 

  • G. J. Weng, 1984, Some elastic properties of reinforced solids with special reference to isotropic ones containging spherical inclusions. International Journal of Engineering Science, 22, 845–856.

    MATH  Google Scholar 

  • M. L. Williams, 1957, On the stress distribution at the base of a stationary crack. ASME Journal of Applied Mechanics, 24, 109–114.

    MATH  Google Scholar 

  • R. L. Williamson, B. H. Rabin and J. T. Drake, 1993, Finite element analysis of thermal residual stresses at graded ceramic-metal interfaces, part I: model description and geometrical effects. Journal of Applied Physics, 74, 1310–1320.

    ADS  Google Scholar 

  • C.-C. Wu, P. He and Z. Li, 2002, Extension of J integral to dynamic fracture of functional graded material and numerical analysis. Computers and Structures, 80, 411–416.

    Google Scholar 

  • S. W. Yu and R. Li, 2003, Cohesive model for thin graded film/substrate interfacial cleavage fracture. Materials Science Forum, 423–4, 651–658.

    Google Scholar 

  • Y. Zhong and M.-J. Pindera, 2002, Efficient reformulation of HOTFGM: heat conduction with variable thermal conductivity. NASA Contractor Report: NASA/CR-2002–211910.

    Google Scholar 

  • Z. Z. Zou, S. X. Wu and C. Y. Li, 2000, On the multiple isoparametric finite element method and computation of stress intensity factor for cracks in FGMs. Key Engineering Materials, 183–187, 511–516.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN, 47907-1282, USA

    Zhi-He Jin

Authors
  1. Zhi-He Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, Virginia Polytechnic Institute & State University, Blacksburg, VA, 24061, USA

    David Y. Gao

  2. Department of Mathematics, University of Glasgow, Glasgow, G12 8QW, Scotland, UK

    Ray W. Ogden

  3. Department of Mathematics, University Gardens, Glasgow, G12 8QW, Scotland, UK

    Ray W. Ogden

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Kluwer Academic Publishers

About this chapter

Cite this chapter

Jin, ZH. (2003). Fracture Mechanics of Functionally Graded Materials. In: Gao, D.Y., Ogden, R.W. (eds) Advances in Mechanics and Mathematics. Advances in Mechanics and Mathematics, vol 4. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0247-6_1

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-0247-6_1

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7959-1

  • Online ISBN: 978-1-4613-0247-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation