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Calculation of Autocatalytic Phase Transformation Zones in Cracked and Uncracked Zirconia Ceramics

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Abstract

Stabilized zirconia ceramics can undergo a stress-induced tetragonal-to-monoclinic phase transformation. A transformation zone with compressive stresses develops at crack tips, leading to an increased fracture toughness which depends on the size and geometry of the transformation zone. In this paper, transformation zones in cracked and uncracked bodies for the case of autocatalytic phase transformation are computed using the weight function method and Eshelby's method. The results are compared with experimentally observed transformation zones.

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References

  1. T. Liu, Y.-W. Mai, G. Grathwohl, Cyclic Fatigue Crack Propagation Behaviour of 9Ce-TZP Ceramics with Different Grain Size, J. Am. Ceram. Soc. 76, 2601–2606, 1993.

    Google Scholar 

  2. T. Liu, Y.-W. Mai, M.V. Swain, G. Grathwohl, JR-+a Curves of 9 mol % Ce-TZP Ceramics with Different Grain Size and Specimen Geometry, Science and Technology of Zirconia, 1993.

  3. T. Liu, M.V. Swain, G. Grathwohl, Effects of Grain Size and Specimen Geometry on the Transformation and R-Curve Behaviour of 9Ce-TZP Ceramics, J.Mat. Sci. 29, 835–843, 1994.

    Google Scholar 

  4. A.G. Evans, R.M. Cannon, Overview: Toughening of Brittle Solids by Martensitic Transformations, Acta Metall. 34, 761–800, 1986.

    Google Scholar 

  5. B.C. Muddle, R.J.H. Hannink, Crystallography of the Tetragonal to Monoclinic Transformation in MgOPartially-Stabilized Zirconia, J. Am Ceram. Soc. 69, 547–555, 1986.

    Google Scholar 

  6. T. Mori, K. Tanaka, Average Stress in Matrix and Average Elastic Energy of Materials with Misfitting Inclusions, Acta Metall. 21, 571–574, 1973.

    Google Scholar 

  7. R.M. McMeeking, A.G. Evans, Mechanics of Transformation-Toughening in Brittle Materials, J. Am. Ceram. Soc. 65, 242–246, 1982.

    Google Scholar 

  8. G.A. Gogotsi, V.P. Zavada, M.V. Swain, Mechanical Property Characterisation of a 9 mol% Ce-TZP Ceramic Material – I. Flexural Response, J. Europ. Ceram. Soc. 15, 1185–1192, 1995.

    Google Scholar 

  9. R.H.J. Hannink, M.V. Swain, Metastability of the Martensitic Transformation in a 12 mol% Ceria-Zirconia Alloy: I, Deformation and Fracture Observations, J. Am. Ceram. Soc. 72, 90–98, 1989.

    Google Scholar 

  10. R. Matt, Statisches und Zyklisches Ermüdungsverhalten Umwandlungsverstärkter ZrO2-Werkstoffe, Dissertation, Universität Karlsruhe, IKM 016, 1996.

  11. P.E. Reyes-Morel, J.-S. Cherng, I-W. Chen, Transformation Plasticity of CeO2-Stabilized Tetragonal Zirconia Polycrystals: I, Stress Assistance and Autocatalysis, J. Am. Ceram. Soc. 71, 343–353, 1988.

    Google Scholar 

  12. Q. Sun, Z. Zhao, W. Chen, X. Qing, X. Xu, F. Dai, Experimental Study of Stress-Induced Localized Transformation Plastic Zones in Tetragonal Zirconia Polycrystalline Ceramics, J. Am. Ceram. Soc. 77, 1352–1356, 1994.

    Google Scholar 

  13. G.A. Gogotsi, V.P. Zavada, M.V. Swain, Mechanical Property Characterization of 9 Mol% Ce-TZP Ceramic Material – II. Fracture Toughness, J. Europ. Ceram. Soc. 16, 545–551, 1996.

    Google Scholar 

  14. G. Grathwohl, T. Liu, Crack Resistance and Fatigue of Transforming Ceramics: II, CeO2-Stabilized Tetragonal ZrO2, J. Am. Ceram. Soc. 74, 3028–3034, 1991.

    Google Scholar 

  15. L.R.F. Rose, M.V. Swain, Transformation Zone Shape in Ceria-Partially-Stabilized Zirconia, Acta Metall. 36, 955–962, 1988.

    Google Scholar 

  16. I.W. Chen, P.E. Reyes-Morel, Implication of Transformation Plasticity in ZrO2-Containing Ceramics: I, Shear and Dilatation Effects, J. Am. Ceram. Soc. 69, 181–189, 1986.

    Google Scholar 

  17. I.W. Chen, P.E. Reyes-Morel, Transformation Plasticity and Transformation Toughening in Mg-PSZ and Ce-TZP, Mat. Res. Soc. Symp. Proc. 78, 1987.

  18. I.W. Chen, Model of Transformation Toughening in Brittle Material, J. Am. Ceram. Soc. 74, 2564–2572, 1991.

    Google Scholar 

  19. P.E. Reyes-Morel, J.-S. Cherng, I-W. Chen, Transformation Plasticity of CeO2-Stabilized Tetragonal Zirconia Polycrystals: II, Pseudoelasticity and Shape Memory Effect, J. Am. Ceram. Soc. 71, 648–657, 1988.

    Google Scholar 

  20. Q. Sun, K.C. Hwang, S.W. Yu, A Micromechanics Constitutive Model of Transformation Plasticity with Shear and Dilatation Effect, J. Mech. Phys. Sol. 39, 507–524, 1991.

    Google Scholar 

  21. Y. Gillet, M.-A. Meunier, V. Brailovski, F. Trochu, E. Patoor, M. Berveiller, Comparison of Thermomechanical Models for Shape Memory Alloy Springs, Journal de Physique IV C8, 1165–1170, 1995.

    Google Scholar 

  22. E. Patoor, M. El Amrani, A. Eberhardt, M. Berveiller, Determination of the Origin for the Dissymmetry Observed between Tensile and Compression Tests on Shape Memory Alloys, Journal de Physique IV C2, 495–500, 1995.

    Google Scholar 

  23. E. Patoor, A. Eberhardt, M. Berveiller, Micromechanical Modelling of Superelasticity in Shape Memory Alloys, Journal de Physique IV C1, 277–292, 1996.

    Google Scholar 

  24. J.-C. Videau, G. Cailletaud, A. Pineau, Modélisation des Effets Mécaniques des Transformations de Phases pour le Calcul de Structures, Journal de Physique IV C3, 227–233, 1994.

    Google Scholar 

  25. J.-C. Videau, G. Cailletaud, A. Pineau, Experimental Study of the Transformation-Induced Plasticity in a Cr-Ni-Mo-Al-Ti Steel, Journal de Physique IV C1, 465–474, 1996.

    Google Scholar 

  26. D.M. Stump, The Role Of Shear Stresses and Shear Strains in Transformation Toughening, Phil. Mag. A64, 879–902, 1991.

    Google Scholar 

  27. D.M. Stump, Autocatalysis: the Self-Induced Growth of Martensitic Phase Transformations in Ceramics, Acta Metall. Mat. 42, 3027–3033, 1994.

    Google Scholar 

  28. G. Rauchs, T. Fett, D. Munz, R. Oberacker, Tetragonal-to-monoclinic Phase Transformation in CeO2-stabilized Zirconia under Multiaxial Loading, J. Europ. Ceram. Soc. 22, 841–849, 2002.

    Google Scholar 

  29. B. Budiansky, J.W. Hutchinson, J.C. Lambropoulos, Continuum Theory of Dilatant Transformation Toughening in Ceramics, Int. J. Sol. Struct. 19, 337–356, 1983.

    Google Scholar 

  30. R.H. Dauskardt, D.K. Veirs, R.O. Ritchie, Spatially Resolved Raman Spectroscopy Study of Transformed Zones in Magnesia-Partially-Stabilized Zirconia, J. Am. Ceram. Soc. 72, 1124–1130, 1989.

    Google Scholar 

  31. A.G. Evans, A.H. Heuer, Review – Transformation Toughening in Ceramics: Martensitic Transformations in Crack-Tip Stress Fields, J. Am. Ceram. Soc. 63, 241–248, 1980.

    Google Scholar 

  32. L. Wenfang, M. Jilong, D. Shanyi, Transformation Criterion of Phase Transformation Toughening Ceramics with Misoriented Microcracks, J. Mat. Sci. 32, 4149–4152, 1997.

    Google Scholar 

  33. Y.P. Chiu, On the Stress Field Due to Initial Strains in a Cuboid Surrounded by an Infinite Elastic Space, J. Appl. Mech., 587–590, 1977.

  34. T. Mura, P.C. Cheng, The Elastic Field Outside an Ellipsoidal Inclusion, J. Appl. Mech., 591–594, 1977.

  35. S. Schmauder, H. Schubert, Significance of Internal Stresses for the Martensitic Transformation in Yttria-Stabilized Tetragonal Zirconia Polycrystals during Degradation, J. Am. Ceram. Soc. 69, 534–540, 1986.

    Google Scholar 

  36. J.C. Lambropoulos, Shear, Shape and Orientation Effects in Transformation Toughening, Int. J. Sol. Struct. 22, 1083–1106, 1986.

    Google Scholar 

  37. C. Collins, Computation of Twinning, The IMA Volumes in Mathematics and its Applications, Vol. 54: Microstructure and Phase Transition, D. Kinderlehrer, R. James, M. Luskin, J.L. Ericksen, Springer-Verlag, 1994.

  38. K.Y. Lam, J.M. Zhang, Transformation Twinning in Monoclinic Zirconia Particles, Acta Metall. 40, 1395–1401, 1992.

    Google Scholar 

  39. K.Y. Lam, J.M. Zhang, P.P. Ong, A Micromechanical Model for ZrO2-Toughened Ceramics, Mech. Mat. 19, 227–238, 1995.

    Google Scholar 

  40. J.M. Zhang, K.Y. Lam, On Transformation Shear of Precipitated Zirconia Particles, Acta Metall. Mat. 41, 1773–1782, 1993.

    Google Scholar 

  41. J.M. Zhang, K.Y. Lam, Transformation Shear of Precipitated ZrO2 Particles in the Presence of Multi-Mode Twinning, Int. J. Sol. Struct. 31, 517–532, 1994.

    Google Scholar 

  42. L. Wenfang, M. Jilong, D. Shanyi, The Stiffness and Strength of Transformation Toughening Ceramics with Misoriented Microcracks, J. Mat. Sci. 29, 4252–4255, 1994.

    Google Scholar 

  43. T. Fett, Application of the Weight Function and Boundary Collocation Method to the Calculation of Initial Phase Transformation Zones, Eng. Fract. Mech. 52, 853–863, 1995.

    Google Scholar 

  44. D.M. Stump, B. Budiansky, Finite Cracks in Transformation-Toughened Ceramics, Acta Metall. 37, 3297–3304, 1989.

    Google Scholar 

  45. T. Fett, Contributions to the R-Curve Behaviour of Ceramic Materials, KfK-Bericht 5291, 1994.

  46. T. Fett, Residual Crack Profiles under Weak Phase-Transformation Conditions, Eng. Fract. Mech. 56, 275–284, 1997.

    Google Scholar 

  47. T. Fett, Phase Transformation Zones and Shielding Stress Intensity Factors for Cracks in Finite Bodies, Eng. Fract. Mech. 59, 47–55, 1998.

    Google Scholar 

  48. T. Fett, Computation of Residual Stress Intensity Factors for Non-Propagated Cracks with Phase-Transformation Zones in Finite Bodies, Eng. Fract. Mech. 56, 397–408, 1997.

    Google Scholar 

  49. J.C. Lambropoulos, Constitutive Laws for Ceramics Exhibiting Stress-Induced Martensitic Transformation, Mar. Res. Soc. Symp. 78, 35–41, 1987.

    Google Scholar 

  50. G.T.M. Stam, E. van der Giessen, Analysis of Supercritical Transformation with Dilatation and hear Effects during Crack Growth in Ceramics, Topics in Applied Mechanics, eds J.F. Dijksman, F.T.M. Nieuwstadt, 137–146, 1993.

  51. G.T.M. Stam, E. van der Giessen, P. Meijers, Effect of Transformation-Induced Shear Strains on Crack Growth in Zirconia-Containing Ceramics, Int. J. Sol. Struct. 31, 1923–1948, 1994.

    Google Scholar 

  52. G.T.M. Stam, Analysis of Crack Growth in Materials with Transformation Plasticity, Report No. 978, Delft University of Technology.

  53. S. Suresh, J.R. Brockenbrough, Theory and Experiments of Fracture in Cyclic Compression: Single Phase Ceramics, Transforming Ceramics and Ceramic Composites, Acta Metall. 36, 1455–1470, 1988.

    Google Scholar 

  54. S.A. Silling, Numerical Studies of Loss of Ellipticity Near Singularities in an Elastic Material, J. Elasticity 19, 213–239, 1988.

    Google Scholar 

  55. J.K. Knowles, E. Sternberg, On the Failure of Ellipticity and the Emergence of Discontinuous Deformation Gradients in Plane Finite Elastostatics, J. Elasticity 8, 329–379, 1978.

    Google Scholar 

  56. R. Abeyaratne, G.-H. Jiang, Dilatationally Nonlinear Elastic Materials – I. Some Theory, Int. J. Sol. Struct. 25, 1201–1219, 1989.

    Google Scholar 

  57. R. Abeyaratne, G.-H. Jiang, Dilatationally Nonlinear Elastic Materials – II. An Example Illustrating Stress Concentration Reduction, Int. J. Sol. Struct. 25, 1221–1233, 1989.

    Google Scholar 

  58. R. Abeyaratne, J.K. Knowles, On the Driving Traction Acting on a Surface of Strain Discontinuity in a Continuum, J. Mech. Phys. Sol. 38, 345–360, 1990.

    Google Scholar 

  59. R. Abeyaratne, J.K. Knowles, On the Propagation of Maximally Dissipative Phase Boundaries in Solids, Quart. App. Math. L, 149–172, 1992.

    Google Scholar 

  60. Y.A. Chu, B. Moran, A Computational Model for Nucleation of Solid-Solid Phase Transformation, Modelling Simul. Mat. Sci. Eng. 3, 455–471, 1995.

    Google Scholar 

  61. J.K. Knowles, On the Dissipation Associated with Equilibrium Shocks in Finite Elasticity, J. Elasticity 9, 1979.

  62. E.N. Mamiya, J.C. Simo, Stress-Induced Phase Transformations in Elastic Solids under Anti-Plane Deformations: Dissipative Constitutive Models and Numerical Simulations, Comp. Meth. Appl. Mech. Eng. 133, 47–77, 1996.

    Google Scholar 

  63. A. Needleman, M. Ortiz, Effect of Boundaries and Interfaces on Shear-Band Localization, Int. J. Sol. Struct. 28, 859–877, 1991.

    Google Scholar 

  64. S.A. Silling, Consequences of the Maxwell Relation for Anti-Plane Shear Deformations of an Elastic Solid, J. Elasticity 19, 241–284, 1988.

    Google Scholar 

  65. A.G. Evans, N. Burlingame, M. Drory, W.M. Kriven, Martensitic Transformations in Zirconia – Particle Size Effects and Toughening, Acta Metall. 29, 447–456, 1980.

    Google Scholar 

  66. C.-H. Hsueh, P.F. Becher, Some Considerations of Nonideal Transformation-Zone Profile, J. Am. Ceram. Soc. 71, 494–497, 1988.

    Google Scholar 

  67. G. Rauchs, T. Fett, D. Munz, R. Oberacker, Time-independent and time-dependent deformation in Ce-TZP, J. Mat. Sci. Lett. 19, 1481–1484, 2000.

    Google Scholar 

  68. J.D. Eshelby, The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems, Proc. R. Soc. London Ser. A241, 376–396, 1957.

    Google Scholar 

  69. H.F. Bueckner, A Novel Principle for the Computation of Stress Intensity Factors, ZAMM 50, 529–546, 1970.

    Google Scholar 

  70. T. Fett, Stress Intensity Factors and Weight Functions for Cracks in Front of Notches, KfK-Report 5254, Kernforschungszentrum Karlsruhe, 1993.

  71. Fett, T., Munz, D., Stress intensity factors and weight functions, Computational Mechanics Publications, 1997, Southampton, UK.

    Google Scholar 

  72. J.R. Rice, Some Remarks on Elastic Crack-tip Stress Fields, Int. J. Sol. Struct. 8, 751–758, 1972.

    Google Scholar 

  73. T. Fett, E. Diegele, G. Rizzi, Calculation of Stress Fields near Inclusions by Use of the Fracture Mechanics Weight Function, Eng. Frac. Mech. 53, 17–22, 1996.

    Google Scholar 

  74. N.H. Aliabadi, D.P. Rooke, Numerical Fracture Mechanics, Computational Mechanics Publications, 1991.

  75. S.P. Timoshenko, J.N. Goodier, Theory of Elasticity, McGraw-Hill Publications, 1970.

  76. J.D. Eshelby, The Elastic Energy-Momentum Tensor, J. Elast. 5, 321–335, 1975.

    Google Scholar 

  77. H. Tada, P. Paris, G. Irwin, The Stress Analysis of Cracks Handbook, Del Research Corporation, 1973.

  78. G. Rauchs, T. Fett, D. Munz, R. Oberacker, Tetragonal-to-Monoclinic Phase Transformation in CeO2-Stabilised Zirconia under Uniaxial Loading, J. Europ. Ceram. Soc. 21, 2229–2241, 2001.

    Google Scholar 

  79. G. Rauchs, T. Fett, D. Munz, R-Curve Behaviour of 9Ce-TZP Zirconia Ceramics, Engng. Fract. Mech. 69, 389–401, 2002.

    Google Scholar 

  80. J.C. Amazigo, B. Budiansky, Steady-State Crack Growth in Supercritically Transforming Materials, Int. J. Sol. Struct. 24, 751–755, 1988.

    Google Scholar 

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Rauchs, G., Fett, T. & Munz, D. Calculation of Autocatalytic Phase Transformation Zones in Cracked and Uncracked Zirconia Ceramics. International Journal of Fracture 116, 121–140 (2002). https://doi.org/10.1023/A:1020134428040

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