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A thermomechanical fracture modeling and simulation for functionally graded solids using a residual-strain formulation

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Abstract

This paper addresses mixed-mode crack growth in two-dimensional functionally graded solids under thermomechanical loads, and investigates the effect of mechanical and thermal loads as well as the T-stress on their crack growth behavior. A novel residual strain-based formulation in the interaction integral method is developed and used for the accurate evaluation of mixed-mode stress intensity factors and/or the T-stress. Simulation of mixed-mode crack propagation in functionally graded materials including solid oxide fuel cells under thermomechanical loads is performed by means of the finite element method and the generalized interaction integrals in conjunction with a remeshing algorithm. An iterative procedure is used for crack growth simulation including the calculation of mixed-mode stress intensity factors and/or the T-stress by means of the generalized interaction integral method, determination of crack growth direction and crack initiation condition based on selected fracture criteria, and local automatic remeshing along the crack path. The present approach employs a user-defined crack increment at the beginning of the simulation. Crack trajectories and fracture parameters obtained by the present simulation for thermomechanical loads are assessed for some numerical examples in comparison with those for mechanical loads.

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References

  • Abanto-Bueno J, Lambros J (2002) Investigation of crack growth in functionally graded materials using digital image correlation. Eng Fract Mech 69(14–16): 1695–1711

    Article  Google Scholar 

  • Anderson TL (1995) Fracture mechanics: fundamentals and applications. CRC Press LLC, Boca Raton

    MATH  Google Scholar 

  • Barthel K, Rambert C (1999) Thermal spraying and performance of graded composite cathodes as SOFC-component. Mater Sci Forum 308(311): 800–805

    Article  Google Scholar 

  • Bittencourt TN, Wawrzynek PA, Ingraffea AR (1996) Quasi-static simulation of crack propagation for 2D LEFM problems. Eng Fract Mech 55(2): 321–334

    Article  Google Scholar 

  • Cannillo V, Montorsi M, Siligardi C, Sola A, de Portu G, Micele L, Pezzotti G (2006) Microscale computational simulation and experimental measurement of thermal residual stresses in glass-alumina functionally graded materials. J Eur Ceram Soc 26: 1411–1419

    Article  CAS  Google Scholar 

  • Carpenter RD, Liang WW, Paulino GH, Gibeling JC, Munir ZA (1999) Fracture testing and analysis of a layered functionally graded Ti/TiB beam in 3-point bending. Mater Sci Forum 308–311: 837–842

    Article  Google Scholar 

  • Comi C, Mariani S (2007) Extended finite element simulation of quasi-brittle fracture in functionally graded materials. Comput Methods Appl Mech Eng 196(41–44): 4013–4026

    Article  MATH  Google Scholar 

  • Cotterell B, Rice JR (1980) Slightly curved or kinked cracks. Int J Fract 16(2): 155–169

    Article  Google Scholar 

  • Desplat JL (1996) Recent development on oxigenated thermionic energy converter—overview. In: Proceedings of the fourth international symposium on functionally gradient materials, Tsukuba City, Japan

  • Dolbow JE, Gosz M (2002) On the computation of mixed-mode stress intensity factors in functionally graded materials. Int J Solids Struct 39(9): 2557–2574

    Article  MATH  Google Scholar 

  • Eftis J, Subramonian N, Liebowitz H (1977) Crack border stress and displacement equations revisited. Eng Fract Mech 9(1): 189–210

    Article  Google Scholar 

  • Eischen JW (1987) Fracture of non-homogeneous materials. Int J Fract 34(1): 3–22

    Google Scholar 

  • Erdogan F, Sih GC (1963) On the crack extension in plates under plane loading and transverse shear. ASME J Basic Eng 85(4): 519–527

    Google Scholar 

  • Erdogan F, Wu BH (1996) Crack problems in FGM layers under thermal stresses. J Thermal Stresses 19(3): 237–265

    Article  Google Scholar 

  • Erdogan F, Wu BH (1997) The surface crack problem for a plate with functionally graded properties. J Appl Mech Trans ASME 64(3): 449–456

    MATH  Google Scholar 

  • Getto H, Ishihara SJ (1996) Development of the fire retardant door with functional gradient wood. In: Proceedings of the fourth international symposium on functionally gradient materials, Tsukuba City, Japan

  • Gu L-C, Noda N, Wu L (2008) Thermal fracture model for a functionally graded plate with a crack normal to the syrfaces and arbitrary thermomechanical properties. Compos Sci Technol 68: 1034–1041

    Article  Google Scholar 

  • Gu P, Asaro RJ (1997) Crack deflection in functionally graded materials. Int J Solids Struct 34(24): 3085–3098

    Article  MATH  Google Scholar 

  • Hart NT, Brandon NP, Day MJ, Shemilt JE (2001) Functionally graded cathodes for solid oxide fuel cells. J Mater Sci 36: 1077–1085

    Article  CAS  Google Scholar 

  • Hart NT, Brandon NP, Day MJ, Lapena-Rey N (2002) Functionally graded composite cathodes for solid oxide fuel cells. J Power Sources 106: 42–50

    Article  CAS  Google Scholar 

  • Hirano T, Teraki J, Yamada T (1990) On the design of functionally gradient materials. In: Yamanouochi M, Koizumi M, Hirai T, Shiota I (eds) Proceedings of the first international symposium on functionally gradient materials. Sendai, Japan, pp 5–10

    Google Scholar 

  • Holtappels P, Bagger C (2002) Fabrication and performance of advanced multi-layer SOFC cathodes. J Eur Ceram Soc 22: 41–48

    Article  CAS  Google Scholar 

  • Hussain MA, Pu SL, Underwood J (1974) Strain energy release rate for a crack under combined mode I and mode II. In: Paris PC, Irwin GR (eds) Fracture analysis ASTM STP 560. American Society for Testing and Materials, Philadelphia, pp 2–28

    Google Scholar 

  • Hutchinson JW, Suo Z (1992) Mixed mode cracking in layered materials. Adv Appl Mech 29: 63–191

    Article  MATH  Google Scholar 

  • Igari T, Notomi A, Tsunoda H, Hida K, Kotoh T, Kunishima S (1990) Material properties of functionally gradient material for fast breeder reactor. In: Yamanouochi M, Koizumi M, Hirai T, Shiota I (eds) Proceedings of the first international symposium on functionally gradient materials. Sendai, Japan, pp 209–214

    Google Scholar 

  • Ilschner B (1996) Processing-microstructure-property relationships in graded materials. J Mech Phys Solids 44(5): 647–656

    Article  CAS  ADS  Google Scholar 

  • Ivers-Tiffee E, Weber A, Herbstritt D (2001) Materials and technologies for sofc-components. J Eur Ceram Soc 21: 1805–1811

    Article  CAS  Google Scholar 

  • Jin Z-H, Feng YZ (2008) Thermal fracture resistance of a functionally graded coating with periodic edge cracks. Surf Coat Technol 202: 4189–4197

    Article  CAS  Google Scholar 

  • Jin Z-H, Paulino GH, Dodds RH Jr (2002) Finite element investigation of quasi-static crack growth in functionally graded materials using a novel cohesive zone fracture model. J Appl Mech Trans ASME 69(3): 370–379

    MATH  CAS  Google Scholar 

  • Kawasaki A, Watanabe R (2002) Thermal fracture behavior of metal/ceramic functionally graded materials. Eng Fract Mech 69: 1713–1728

    Article  Google Scholar 

  • KC A, Kim J-H (2008) Interaction integrals for thermal fracture of functionally graded materials. Eng Fract Mech 75: 2542–2565

    Article  Google Scholar 

  • Kim J-H (2003) Mixed-mode crack propagation in functionally graded materials. Ph.D. Thesis, University of Illinois at Urbana-Champaign

  • Kim J-H, Paulino GH (2002) Isoparametric graded finite elements for nonhomogeneous isotropic and orthotropic materials. J Appl Mech Trans ASME 69(4): 502–514

    Article  MATH  Google Scholar 

  • Kim J-H, Paulino GH (2003) 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(10): 1457–1497

    Article  MATH  Google Scholar 

  • Kim J-H, Paulino GH (2003) The interaction integral for fracture of orthotropic functionally graded materials: Evaluation of stress intensity factors. Int J Solids Struct 40(15): 3967–4001

    Article  MATH  Google Scholar 

  • Kim J-H, Paulino GH (2004) Simulation of crack propagation in functionally graded materials under mixed-mode and non-proportional loading. Int J Mech Mater Des 1: 63–94

    Article  Google Scholar 

  • Kim J-H, Paulino GH (2005) Consistent formulations of the interaction integral method for fracture of functionally graded materials. J Appl Mech Trans ASME 72: 351–364

    Article  MATH  Google Scholar 

  • Koike Y (1992) Graded-index and single mode polymer optical fibers. In: Chiang LY, Garito AG, Sandman DJ (eds) Electrical, optical, and magnetic properties of organic solid state materials, vol. 247, Materials Research Society Proceedings, Pittsburgh, PA, p 817

  • Konda N, Erdogan F (1994) The mixed mode crack problem in a nonhomogeneous elastic medium. Eng Fract Mech 47(4): 533–545

    Article  Google Scholar 

  • Lambros J, Cheeseman BA, Santare MH (2000) Experimental investigation of the quasi-static fracture of functionally graded materials. Int J Solids Struct 37(27): 3715–3732

    Article  MATH  Google Scholar 

  • Lin CY, McShane HB, Rawlings RD (1994) Structure and properties of functionally gradient aluminium alloy 2124/SiC composites. Mater Sci Technol 10(7): 659–664

    CAS  Google Scholar 

  • Lui Y, Compson C, Lui M (2004) Nanostructured and functionally graded cathodes for intermediate temperature solid oxide fuel cells. J Power Sources 138: 194–198

    Article  CAS  Google Scholar 

  • Michell JH (1900) Elementary distributions of plane stress. Proc Lond Math Soc 32: 35–61

    Google Scholar 

  • Noda N (1999) Thermal stresses in functionally graded materials. J Thermal Stresses 22(4–5): 477–512

    Article  MathSciNet  Google Scholar 

  • Oonishi H, Noda T, Ito S, Kohda A, Ishimaru H, Yamamoto M, Tsuji E (1994) Effect of hydroxyapatite coating on bone growth into porous titanium alloy implants under loaded conditions. J Appl Biomater 5(1): 23–27

    Article  CAS  Google Scholar 

  • Oral A, Lambros J, Anlas G. (2008) Crack initiation in functionally graded materials under mixed mode loading: experiments and simulations. J Appl Mech Trans ASME 75(5):051100 (8 pages)

    Google Scholar 

  • Osaka T, Matsubara H, Homma T, Mitamura S, Noda K (1990) Microstructural study of electroless-plated CoNiReP/NiMoP double-layered media for perpendicular magnetic recording. Jap J Appl Phys 29(10): 1939–1943

    Article  CAS  ADS  Google Scholar 

  • Palaniswamy K, Knauss WG (1978) On the problem of crack extension in brittle solids under general loading. In: Nemat-Nasser S (eds) Mechanics today vol. 4. Pergamon Press, Oxford, pp 87–148

    Google Scholar 

  • Paulino GH, Dong Z A novel application of the singular integral equation approach to evaluate T-stress in functionally graded materials. Unknown, in preparation

  • Paulino GH, Kim J-H (2004) A new approach to compute T-stress in functionally graded materials using the interaction integral method. Eng Fract Mech 71(13–14): 1907–1950

    Article  Google Scholar 

  • Paulino GH, Kim J-H (2005) On the Poisson’s ratio effect on mixed-mode stress intensity factors and T-stress in functionally graded materials. Int J Comput Eng Sci 5(4): 833–861

    Article  Google Scholar 

  • Paulino GH, Kim J-H (2005) On accurate numerical evaluation of stress intensity factors and T-stress in functionally graded materials. Mater Sci Forum 492(493): 403–408

    Google Scholar 

  • Raju IS, Shivakumar KN (1990) An equivalent domain integral method in the two-dimensional analysis of mixed mode crack problems. Eng Fract Mech 37(4): 707–725

    Article  Google Scholar 

  • Rao BN, Rahman S (2003) Mesh-free analysis of cracks in isotropic functionally graded materials. Eng Fract Mech 70(1): 1–27

    Article  Google Scholar 

  • Rice JR (1968) A path-independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech Trans ASME 35(2): 379–386

    MathSciNet  Google Scholar 

  • Rousseau C-E, Tippur HV (2000) Compositionally graded materials with cracks normal to the elastic gradient. Acta Materialia 48(16): 4021–4033

    Article  CAS  Google Scholar 

  • Sih GC (1974) Strain energy density factor applied to mixed mode crack problems. Int J Fract 10(3): 305–321

    Article  Google Scholar 

  • Sladek J, Sladek V, Zhang Ch (2008) Evaluation of the stress intensity factors for cracks in continuously nonhomogeneous solids, part i: interaction integral for elastodynamics and thermoelasticity. Mech Adv Mater Struct 15: 438–443

    Article  Google Scholar 

  • Sladek J, Sladek V, Zhang Ch, Tan CL (2006) Evaluation of fracture parameters for crack problems in fgm by a meshless method. J Theor Appl Mech 44: 603–636

    Google Scholar 

  • Smith DJ, Ayatollahi MR, Pavier MJ (2001) The role of T-stress in brittle fracture for linear elastic materials under mixed-mode loading. Fatigue Fract Eng Mater Struct 24(2): 137–150

    Article  Google Scholar 

  • Tilbrook MT, Moon RJ, Hoffman M (2005) Crack propagation in graded composites. Compos Sci Technol 65: 201–220

    Article  Google Scholar 

  • Tilbrook MT, Moon RJ, Hoffman M (2005) Finite element simulations of crack propagation in functionally graded materials under flexural loading. Eng Fract Mech 72: 2444–2467

    Article  Google Scholar 

  • Ueda Y, Ikeda K, Yao T, Aoki M (1983) Characteristics of brittle failure under general combined modes including those under bi-axial tensile loads. Eng Fract Mech 18(6): 1131–1158

    Article  Google Scholar 

  • Walters MC, Dodds RH Jr, Paulino GH (2005) Interaction integral procedures for 3-D curved cracks including surface tractions. Eng Fract Mech 60: 577–581

    Google Scholar 

  • Walters MC, Paulino GH, Dodds RH Jr (2004) Stress intensity factors for surface cracks in functionally graded materials under mode I thermomechanical loading. Int J Solids Struct 41: 1081–1118

    Article  MATH  Google Scholar 

  • Walters MC, Paulino GH, Dodds RH Jr (2006) Computation of mixed-mode stress intensity factors for cracks in three-dimensional functionally graded solids. J Eng Mech 132(1): 1–15

    Article  Google Scholar 

  • Wang Z, Nakamura T (2004) Simulations of crack propagation in elastic-plastic graded materials. Mech Mater 36: 601–622

    Article  Google Scholar 

  • Watanabe Y, Nakamura Y, Fukui Y, Nakanishi K (1993) A magnetic-functionally graded material manufactured with deformation-induced martensitic transformation. J Mater Sci Lett 12(5): 326–328

    Article  CAS  Google Scholar 

  • Watari F, Yokoyama A, Saso F, Uo M, Ohkawa S, Kawasaki T (1996) EPMA elemental mapping of functionally graded dental implant in biocompatibility test. In: Proceedings of the fourth international symposium on functionally gradient materials. Tsukuba City, Japan

  • Wawrzynek PA (1987) Interactive finite element analysis of fracture processes: an integrated approach. M.S. Thesis, Cornell University

  • Wawrzynek PA, Ingraffea AR (1991) Discrete modeling of crack propagation: theoretical aspects and implementation issues in two and three dimensions. Report 91-5, School of Civil Engineering and Environmental Engineering, Cornell University

  • Williams JG, Ewing PD (1972) Fracture under complex stress—the angled crack problem. Int J Fract 8(4): 416–441

    Google Scholar 

  • Williams ML (1957) On the stress distribution at the base of a stationary crack. J Appl Mech Trans ASME 24(1): 109–114

    MATH  Google Scholar 

  • Yau JF, Wang SS, Corten HT (1980) A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity. J Appl Mech Trans ASME 47(2): 335–341

    Article  MATH  Google Scholar 

  • Yildirim B (2006) An equivalent domain integral method for fracture analysis of functionally graded materials under thermal stresses. J Therm Stress 9: 371–397

    Article  Google Scholar 

  • Yildirim B, Dag S, Erdogan F (2005) Three dimensional fracture analysis of FGM coatings under thermomechanical loading. Int J Fract 132: 369–395

    Article  CAS  Google Scholar 

  • Zhang Z, Paulino GH (2005) Cohesive zone modeling of dynamic failure in homogeneous and functionally graded materials. Int J Plast 21: 1154–1195

    Google Scholar 

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  1. Department of Civil and Environmental Engineering, University of Connecticut, 261 Glenbrook Rd. U-2037, Storrs, CT, 06269, USA

    Ganesh Anandakumar & Jeong-Ho Kim

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  1. Ganesh Anandakumar
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  2. Jeong-Ho Kim
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Correspondence to Jeong-Ho Kim.

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Anandakumar, G., Kim, JH. A thermomechanical fracture modeling and simulation for functionally graded solids using a residual-strain formulation. Int J Fract 164, 31–55 (2010). https://doi.org/10.1007/s10704-010-9454-0

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