2.13 - Failure of Functionally Graded Materials

Comprehensive Structural Integrity: Nine Volume Set

Volumn 1-9, Issue , 2003, Pages 607-644

  Paulino, G H ^a   Jin, Z H ^a   Dodds, R H ^a  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85146769286     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1016/B0-08-043749-4/02101-7     Document Type: Chapter

References (143)

1. J. Aboudi, M.-J. Pindera and S.M. Arnold (1999) Higher-order theory for functionally graded materials. Composites Part B: Eng. 30B 777-832.
2. R. Alex and L. Schovanec (1996) An anti-plane crack in a nonhomogeneous viscoelastic body. Eng. Fract. Mech. 55 727-735.
3. American Society for Testing and Materials, 1996, “Standard E1820-96,” ASTM, Philadelphia, PA.
4. American Society for Testing and Materials, 1997, “PS 070-97,” ASTM, West Conshohocken, PA.
5. T.L. Anderson (1995) Fracture Mechanics: Fundamentals and Applications 2nd edn. Boca Raton: CRC Press
6. G. Anlas, M.H. Santare and J. Lambros (2000) Numerical calculation of stress intensity factors in functionally graded materials. Int. J. Fract. 104 131-143.
7. 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. Eng. Fract. Mech. 60 479-487.
8. G. Bao and H. Cai (1997) Delamination cracking in functionally graded coating/metal substrate systems. Acta Mater. 45 1055-1066.
9. G. Bao and L. Wang (1995) Multiple cracking in functionally graded ceramic/metal coatings. Int. J. Solids Struct. 32 2853-2871.
10. T.L. Becker Jr., R.M. Cannon and R.O. Ritchie (2001) Finite crack kinking and T-stresses in functionally graded materials. Int. J. Solids Struct. 38 5545-5563.
11. Y. Benveniste (1987) A new approach to the application of Mori–Tanaka's theory in composite materials. Mech. Mater. 6 147-157.
12. H. Cai and G. Bao (1998) Crack bridging in functionally graded coatings. Int. J. Solids Struct. 35 701-717.
13. G.T. Camacho and M. Ortiz (1996) Computational modelling of impact damage in brittle materials. Int. J. Solids Struct. 33 2899-2938.
14. 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. Mater. Sci. Forum 308–311 837-842.
15. R.D. Carpenter, G.H. Paulino, Z.A. Munir and J.C. Gibeling (2000) A novel technique to generate sharp cracks in metallic/ceramic functionally graded materials by reverse 4-point bending. Scr. Mater. 43 547-552.
16. E.M. Carrillo-Heian, R.D. Carpenter, G.H. Paulino, J.C. Gibeling and Z.A. Munir (2001) Dense layered MoSi2/SiC functionally graded composites formed by field-activated synthesis. J. Am. Ceram. Soc. 84 962-968.
17. E.M. Carrillo-Heian, C. Unuvar, J.C. Gibeling, G.H. Paulino and Z.A. Munir (2001) Simultaneous synthesis and densification of Niobium Silicide/Niobium composites. Scr. Mater. 45 405-412.
18. Y.-S. Chan, G.H. Paulino and A.C. Fannjiang (2001) The crack problem for nonhomogeneous materials under antiplane shear loading—a displacement based formulation. Int. J. Solids Struct. 38 2989-3005.
19. Y.-S. Chan, A. C. Fannjiang and G. H. Paulino, 2003, Integral equations with hypersingular kernels—theory and applications to fracture mechanics. Int. J. Eng. Sci., in press.
20. Y.F. Chen and F. Erdogan (1996) The interface crack problem for a nonhomogeneous coating bonded to a homogeneous substrate. J. Mech. Phys. Solids 44 771-787.
21. G.P. Cherepanov (1967) Crack propagation in continuous media. Prikl. Mat. Mekh. (Appl. Math. Mech.) 31 467-488.
22. G.P. Cherepanov (1968) Cracks in solids. Int. J. Solids Struct. 4 811-831.
23. H.J. Choi (1996) Bonded dissimilar strips with a crack perpendicular to the functionally graded interface. Int. J. Solids Struct. 33 4101-4117.
24. H.J. Choi (1996) An analysis of cracking in a layered medium with a functionally graded nonhomogeneous interface. ASME J. Appl. Mech. 63 479-486.
25. H.J. Choi (2001) The problem for bonded half-plane containing a crack at an arbitrary angle to the graded interfacial zone. Int. J. Solids Struct. 38 6559-6588.
26. H.J. Choi (2001) Effects of graded layering on the tip behavior of a vertical crack in a substrate under frictional Hertzian contact. Eng. Fract. Mech. 68 1033-1059.
27. H.J. Choi, K.Y. Lee and T.E. Jin (1998) Collinear cracks in a layered half-plane with a graded nonhomogeneous interfacial zone: Part I. Mechanical response. Int. J. Fract. 94 103-122.
28. H.J. Choi, T.E. Jin and K.Y. Lee (1998) Collinear cracks in a layered half-plane with a graded nonhomogeneous interfacial zone: Part II. Thermal shock response. Int. J. Fract. 94 123-135.
29. R.M. Christensen (1971) Theory of Viscoelasticity New York: Academic Press
30. R.D. Cook, D.S. Malkus and M.E. Plesha (1989) Concepts and Applications of Finite Element Analysis 3rd edn. New York: Wiley
31. B. Cotterell and J.R. Rice (1980) Slightly curved or kinked cracks. Int. J. Fract. 16 155-169.
32. S. Dag and F. Erdogan (2002) A surface crack in a graded medium loaded by a sliding rigid stamp. Eng. Fract. Mech. 69(14–16), 1729-1751.
33. F. Delale and F. Erdogan (1983) The crack problem for a nonhomogeneous plane. ASME J. Appl. Mech. 50 609-614.
34. F. Delale and F. Erdogan (1988) On the mechanical modeling of an interfacial region in bonded half-planes. ASME J. Appl. Mech. 55 533-545.
35. G.J. Dvorak and J. Zuiker (1994) The effective properties of functionally graded composites I. Extension of Mori–Tanaka method to linearly varying fields. Composite Eng. 4 19-35.
36. J.W. Eischen (1987) Fracture of nonhomogeneous materials. Int. J. Fract. 34 3-22.
37. J.W. Eischen (1987) An improved method for computing the J 2-integral. Eng. Fract. Mech. 26 691-700.
38. F. Erdogan (1978) Mixed boundary-value problems in mechanics. S. Nemat-Nasser (Eds) Mechanics Today vol. 4 New York: Pergamon 1-86.
39. F. Erdogan (1995) Fracture mechanics of functionally graded materials. Composite Eng. 5 753-770.
40. F. Erdogan, G.D. Gupta and T.S. Cook (1973) Numerical solution of singular integral equations. G.C. Sih (Eds) Mechanics of Fracture vol. 1 Leyden: Noordhoff 368-425.
41. F. Erdogan, A.C. Kaya and P.F. Joseph (1991) The crack problem in bonded nonhomogeneous materials. ASME J. Appl. Mech. 58 410-418.
42. F. Erdogan and M. Ozturk (1995) Periodic cracking of functionally graded coating. Int. J. Eng. Sci. 33 2179-2195.
43. F. Erdogan and B.H. Wu (1996) Crack problems in FGM layers under thermal stresses. J. Thermal Stresses 19 237-265.
44. F. Erdogan and B.H. Wu (1997) The surface crack problem for a plate with functionally graded properties. ASME J. Appl. Mech. 64 449-456.
45. J.D. Eshelby (1951) The force on an elastic singularity. Phil. Trans. Roy. Soc. A244 87-112.
46. L. Ewart and S. Suresh (1986) Dynamic fatigue crack growth in polycrystalline alumina under cyclic compression. J. Mater. Sci. Lett. 5 774-778.
47. A. Feng and Z.A. Munir (1994) Field-assisted self-propagating synthesis of β-SiC. J. Appl. Phys. 76 1927-1928.
48. A. Feng and Z.A. Munir (1995) The effect of an electric field on self-sustained combustion synthesis: Part I. Modeling studies. Metall. Trans. 26B 581-586.
49. A. Feng and Z.A. Munir (1995) The effect of an electric field on self-sustained combustion synthesis: Part II. Field-assisted synthesis of β-SiC. Metall. Trans. 26B 587-593.
50. T. Fett, D. Munz and Y.Y. Yang (2000) Direct adjustment procedure for weight function analysis of graded properties. Fatigue Fract. Eng. Mater. Struct. 23 191-198.
51. L.B. Freund (1990) Dynamic Fracture Mechanics Cambridge: Cambridge University Press
52. Y.C. Fung (1965) Foundations of Solid Mechanics Englewood Cliffs, New Jersey: Prentice-Hall
53. A.E. Giannakopoulos, S. Suresh, M. Finot and M. Olsson (1995) Elastoplastic analysis of thermal cycling: layered materials with compositional gradients. Acta Metall. Mater. 43 1335-1354.
54. W.A. Gooch, B.H. Chen, M.S. Burkins, R. Palicka, J. Rubin and R. Ravichandran (1999) Development and ballistic testing of a functionally gradient ceramic/metal applique. Mater. Sci. Forum 308–311 614-621.
55. P. Gu and R.J. Asaro (1997) Cracks in functionally graded materials. Int. J. Solids Struct. 34 1-17.
56. P. Gu and R.J. Asaro (1997) Crack deflection in functionally graded materials. Int. J. Solids Struct. 34 3085-3098.
57. P. Gu, M. Dao and R.J. Asaro (1999) A simplified method for calculating the crack-tip fields of functionally graded materials using the domain integral. ASME J. Appl. Mech. 66 101-108.
58. Z. Hashin and S. Shtrikman (1963) A variational approach to the theory of the elastic behavior of multiphase materials. J. Mech. Phys. Solids 11 127-140.
59. J.M. Herrmann and L. Schovanec (1990) Quasi-static mode III fracture in a nonhomogeneous viscoelastic body. Acta Mech. 85 235-249.
60. J.M. Herrmann and L. Schovanec (1994) Dynamic steady-state mode III fracture in a nonhomogeneous viscoelastic body. Acta Mech. 106 41-54.
61. R. Hill (1965) A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 13 213-222.
62. T. Hirai (1996) Functionally gradient materials. R.J. Brook (Eds) Materials Science and Technology, Vol. 17B, Processing of Ceramics, Part 2 Weinheim, Germany: VCH Verlagsgesellschaft mbH 292-341.
63. T. Hirano, J. Teraki and T. Yamada, 1990, On the design of functionally gradient materials. In: “Proceedings of the 1st International Symposium on Functionally Gradient Materials, Sendai, Japan, 1990,” eds. M. Yamanouchi, M. Koizumi, T. Hirai and I. Shiota.
64. T. Honein and G. Herrmann (1997) Conservation laws in nonhomogeneous plane elastostatics. J. Mech. Phys. Solids 45 789-805.
65. G.R. Irwin (1957) Analysis of stresses and strains near the end of a crack traversing a plate. ASME J. Appl. Mech. 24 354-361.
66. B. Ilschner (1996) Processing-microstructure-property relationships in graded materials. J. Mech. Phys. Solids 44 647-656.
67. S. Itou (2001) Transient dynamic stress intensity factors around a crack in a nonhomogeneous interfacial layer between dissimilar elastic half-planes. Int. J. Solids Struct. 38 3631-3645.
68. Z.-H. Jin and R.C. Batra (1996) Some basic fracture mechanics concepts in functionally graded materials. J. Mech. Phys. Solids 44 1221-1235.
69. Z.-H. Jin and R.C. Batra (1996) Stress intensity relaxation at the tip of an edge crack in a functionally graded material subjected to a thermal shock. J. Thermal Stresses 19 317-339.
70. Z.-H. Jin and R.C. Batra (1996) Interface cracking between functionally graded coatings and a substrate under antiplane shear. Int. J. Eng. Sci. 34 1705-1716.
71. Z.-H. Jin and R.C. Batra (1998) R-curve and strength behavior of a functionally graded material. Mater. Sci. Eng. A242 70-76.
72. Z.-H. Jin and N. Noda (1994) Crack-tip singular fields in nonhomogeneous materials. ASME J. Appl. Mech. 61 738-740.
73. Z.-H. Jin and N. Noda (1994) An edge crack in a nonhomogeneous material under thermal loading. J. Thermal Stresses 17 591-599.
74. Z.-H. Jin and N. Noda (1994) Transient thermal stress intensity factors for a crack in a semi-infinite plane of a functionally graded material. Int. J. Solids Struct. 31 203-218.
75. Z.-H. Jin and G.H. Paulino (2001) Transient thermal stress analysis of an edge crack in a functionally graded material. Int. J. Fract. 107 73-98.
76. Z.-H. Jin and G.H. Paulino (2002) A viscoelastic functionally graded strip containing a crack subjected to in-plane loading. Eng. Fract. Mech. 69(14–16), 1769-1790.
77. Z.-H. Jin, G.H. Paulino and R.H. Dodds 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.
78. S. Kadioglu, S. Dag and S. Yahsi (1998) Crack problem for a functionally graded layer on an elastic foundation. Int. J. Fract. 94 63-77.
79. R. Kaczmarek, W. Robert, J. Jurewicz, M. I. Boulos and S. Dallaire, 1984, Microprocessor control of the spraying of graded coatings. In: “Plasma Processing and Synthesis of Materials Symposium,” eds. J. Szekely and D. Apelian. pp. 207--215, New York.
80. B.L. Karihaloo (1982) On crack kinking and curving. Mech. Mater. 1 189-201.
81. J.-H. Kim and G.H. Paulino (2002) Finite element evaluation of mixed mode stress intensity factors in functionally graded materials. Int. J. Numer. Methods Eng. 52 1903-1935.
82. K. Kokini and M. Case (1997) Initiation of surface and interface edge cracks in functionally graded ceramic thermal barrier coatings. ASME J. Eng. Mater. Technol. 119 148-152.
83. N. Konda and F. Erdogan (1994) The mixed mode crack problem in a nonhomogeneous elastic medium. Eng. Fract. Mech. 47 533-545.
84. J. Lambros, M.H. Santare, H. Li and G.H. Sapna III (1999) A novel technique for the fabrication of laboratory scale model functionally graded materials. Exp. Mech. 39 184-190.
85. Y.-D. Lee and F. Erdogan (1995) Residual/thermal stresses in FGM and laminated thermal barrier coatings. Int. J. Fract. 69 145-165.
86. 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. Int. J. Solids Struct. 38 7473-7485.
87. X. Ma, K. Tanihata, K. Miyamoto, Y. Kumakawa, S. Nagata, T. Yamada and T. Hirano (1992) Fabrication of TiC/Ni functionally graded materials and their mechanical and thermal properties. Ceram. Eng. Sci., Proc. 13 356-364.
88. A.J. Markworth, K.S. Ramesh and W.P. Parks (1995) Modelling studies applied to functionally graded materials. J. Mater. Sci. 30 2183-2193.
89. P.R. Marur and H.V. Tippur (1998) Evaluation of mechanical properties of functionally graded materials. J. Test. Eval. 26 539-545.
90. P.R. Marur and H.V. Tippur (2000) Dynamic response of bimaterial and graded interface cracks under impact loading. Int. J. Fract 103 95-109.
91. P.R. Marur and H.V. Tippur (2000) Numerical analysis of crack-tip fields functionally graded materials with a crack normal to the elastic gradient. Int. J. Solids Struct. 37 5353-5370.
92. S.A. Meguid, X.D. Wang and L.Y. Jiang (2002) On the dynamic propagation of a finite crack in functionally graded materials. Eng. Fract. Mech. 69(14–16), 1753-1768.
93. Y. Miyamoto, K. Tanihara, Y. Matsuzaki and X. Ma (1992) HIP in SHS technology. Int. J. Self-Propagation High-Temp. Synth. 1 147-154.
94. T. Mori and K. Tanaka (1973) Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Mater. 21 571-574.
95. Z.A. Munir (1996) The use of an electric field as a processing parameter in the combustion synthesis of ceramics and composite. Metall. Trans. 27A 2080-2085.
96. Z.A. Munir (1997) Field effects in self-propagating solid state synthesis reactions. Solid State Ionics 101 991-1001.
97. Z. A. Munir, W. Lai and K. Ewald, 1995, US Patent no. 5380409, January 10.
98. G. Nelson and A. Ezis, 1996, “Functionally graded material (FGM) armor in the TiB/Ti system (U),” CERCOM Report, Vista, CA 92083, USA.
99. 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 Mech. 144 211-229.
100. S. Nemat-Nasser and M. Hori (1993) Micromechanics: Overall Properties of Heterogeneous Materials Amsterdam, The Netherlands: North-Holland
101. N. Noda (1999) Thermal stresses in functionally graded materials. J. Thermal Stresses 22 477-512.
102. N. Noda and Z.-H. Jin (1993) Thermal stress intensity for a crack in a strip of a functionally graded material. Int. J. Solids Struct. 30 1039-1056.
103. R.M. Ogorkiewicz (1970) Engineering Properties of Thermoplastics London: Wiley Interscience
104. M. Ozturk and F. Erdogan (1997) Mode I crack problem in an inhomogeneous orthotropic medium. Int. J. Eng. Sci. 35 869-883.
105. M. Ozturk and F. Erdogan (1999) The mixed mode crack problem in an inhomogeneous orthotropic medium. Int. J. Fract. 98 243-261.
106. V. Parameswaran and A. Shukla (1998) Dynamic fracture of functionally gradient material having discrete property variation. J. Mater. Sci. 33 3303-3311.
107. V. Parameswaran and A. Shukla (1999) Crack tip stress fields for dynamic fracture in functionally gradient materials. Mech. Mater. 31 579-596.
108. V. Parameswaran and A. Shukla (2002) Asymptotic stress fields for stationary cracks along the gradient in functionally gradient materials. ASME J. Appl. Mech. 69 240-243.
109. G.H. Paulino and Z.-H. Jin (2001) Correspondence principle in viscoelastic functionally graded materials. ASME J. Appl. Mech. 68 129-132.
110. G.H. Paulino and Z.-H. Jin (2001) Viscoelastic functionally graded materials subjected to antiplane shear fracture. ASME J. Appl. Mech. 68 284-293.
111. G.H. Paulino and Z.-H. Jin (2001) A crack in a viscoelastic functionally graded material layer embedded between two dissimilar homogeneous viscoelastic layers – antiplane shear analysis. Int. J. Fract. 111 283-303.
112. G. H. Paulino and J. H. Kim, 2003, The weak patch test for nonhomogeneous materials modeled with graded finite elements, submitted for journal publication.
113. M.-J. Pindera, J. Aboudi and S.M. Arnold (1998) Thermomechanical analysis of functionally graded thermal barrier coatings with different microstructural scales. J. Am. Ceramic Soc. 81 1525-1536.
114. I.S. Raju (1987) Calculation of strain-energy release rates with high order and singular finite elements. Eng. Fract. Mech. 28 251-274.
115. I.S. Raju and K.N. Shivakumar (1990) An equivalent domain integral method in the two-dimensional analysis of mixed mode crack problems. Eng. Fract. Mech. 37 707-725.
116. T.S. Ramamurthy, T. Krishmamurthy, K.B. Narayana, K. Vijayakumar and B. Dataguru (1986) Modified crack closure integral method with quarter point elements. Mech. Res. Commun. 13(4), 179-186.
117. T. Reiter and G.J. Dvorak (1998) Micromechanical models for graded composite materials: II. Thermomechanical loading. J. Mech. Phys. Solids 45 1655-1673.
118. T. Reiter, G.J. Dvorak and V. Tvergaard (1997) Micromechanical models for graded composite materials. J. Mech. Phys. Solids 45 1281-1302.
119. J.R. Rice (1968) A path independent integral and approximate analysis of strain concentration by notches and cracks. ASME J. Appl. Mech. 35 379-386.
120. C.-E. Rousseau and H.V. Tippur (2000) Compositionally graded materials with cracks normal to the elastic gradient. Acta Mater. 48 4021-4033.
121. C.-E. Rousseau and H.V. Tippur (2001) Influence of elastic gradient profiles on dynamically loaded functionally graded materials with cracks along the gradient. Int. J. Solids Struct. 38 7839-7856.
122. Y.A. Roy and R.H. Dodds Jr. (2001) Simulation of ductile crack growth in thin aluminum panels using 3D surface cohesive elements. Int. J. Fract. 110 21-45.
123. E.F. Rybicki and M.F. Kanninen (1977) A finite element calculation of stress intensity factors by a modified crack closure integral. Eng. Fract. Mech. 9 931-938.
124. M. Sakai and R.C. Bradt (1993) Fracture toughness testing of brittle materials. Int. Mater. Rev. 38 53-78.
125. M.H. Santare and J. Lambros (2000) Use of graded finite elements to model the behavior of nonhomogeneous materials. ASME J. Appl. Mech. 67 819-822.
126. N. Sata (1993) Characteristics of SiC-TiB2 composites as the surface layer of SiC–TiB2–Cu functionally gradient material produced by self-propagating high-temperature synthesis. Ceram. Trans. 34 109-116.
127. L. Schovanec and J.R. Walton (1987) The quasi-static propagation of a plane strain crack in a power-law inhomogeneous linearly viscoelastic body. Acta Mech. 67 61-77.
128. L. Schovanec and J.R. Walton (1987) The energy release rate for a quasi-static mode I crack in a nonhomogeneous linearly viscoelastic body. Eng. Fract. Mech. 28 445-454.
129. N.I. Shbeeb, W.K. Binienda and K.L. Kreider (1999) Analysis of the driving forces for multiple cracks in an infinite nonhomogeneous plate: Part I. Theoretical analysis. ASME J. Appl. Mech. 66 492-500.
130. N.I. Shbeeb, W.K. Binienda and K.L. Kreider (1999) Analysis of the driving forces for multiple cracks in an infinite nonhomogeneous plate: Part II. Numerical solutions. ASME J. Appl. Mech. 66 501-506.
131. C.F. Shih, H.G. de Lorenzi and M.D. German (1976) Crack extension modeling with singular quadratic isoparametric elements. Int. J. Fract. 12 647-651.
132. K. Suganuma, T. Okamoto, M. Koizumi and M. Shimada (1984) Effect of interlayers in ceramic-metal joints with thermal expansion mismatches. J. Am. Ceram. Soc. 67 C256-C257.
133. S. Suresh and A. Mortensen (1998) Functionally Graded Materials London: The Institute of Materials, IOM Communications
134. I. Tamura, Y. Tomota and H. Ozawa, 1973, “Proceedings of the 3rd International Conference on Strength of Metals and Alloys,” Institute of Metals, Cambridge, vol. 1, p. 611.
135. S.P. Timoshenko and J.N. Goodier (1973) Theory of Elasticity New York: McGraw-Hill
136. B.L. Wang, J.C. Han and S.Y. Du (2000) Crack problems in nonhomogeneous composite subjected to dynamic loading. Int. J. Solids Struct. 37 1251-1274.
137. X. Wang, Z. Zou and D. Wang (1997) On the penny-shaded crack in a nonhomogeneous interlayer of adjoining two different elastic materials. Int. J. Solids Struct. 34 3911-3921.
138. M.L. Williams (1957) On the stress distribution at the base of a stationary crack. ASME J. Appl. Mech. 24 109-114.
139. 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. J. Appl. Phy. 74 1310-1320.
140. H. Xue and Z.A. Munir (1995) Extending the compositional limit of combustion synthesized B4–TiB2 composites by field activation. Metall. Trans. 27 475-480.
141. H. Xue and Z.A. Munir (1997) Synthesis of AlN-SiC composites and solid solution by field-activated self-propagating combustion. J. Eur. Ceram. Soc. 17 1787-1792.
142. W. Yang and C.F. Shih (1994) Fracture along an interlayer. Int. J. Solids Struct. 31 985-1002.
143. J.R. Zuiker (1995) Functionally graded materials: choice of micromechanics model and limitations in property variation. Composite Eng. 5 807-819.