Research Papers

Fundamental Formulation for Transformation Toughening in Anisotropic Solids

[+] Author and Article Information
Lifeng Ma

S&V Lab,
Department of Engineering Mechanics,
Xi'an Jiaotong University,
710049, China

Alexander M. Korsunsky

Department of Engineering Science,
University of Oxford,
Oxford OX1 3PJ, UK

Robert M. McMeeking

Department of Mechanical Engineering and
Materials Department,
University of California,
Santa Barbara, CA 93106;
School of Engineering,
University of Aberdeen,
Aberdeen AB24 3UE, UK;
INM - Leibniz Institute for New Materials,
Campus D2 2,
66123 Saarbrücken, Germany

Manuscript received May 18, 2011; final manuscript received April 23, 2012; accepted manuscript posted January 22, 2013; published online July 12, 2013. Assoc. Editor: Kenneth M. Liechti.

J. Appl. Mech 80(5), 051001 (Jul 12, 2013) (9 pages) Paper No: JAM-11-1153; doi: 10.1115/1.4023476 History: Received May 18, 2011; Revised April 23, 2012; Accepted January 22, 2013

In this paper the problem of transformation toughening in anisotropic solids is addressed in the framework of Stroh formalism. The fundamental solutions for a transformed strain nucleus located in an infinite anisotropic elastic plane are derived first. Furthermore, the solution for the interaction of a crack tip with a residual strain nucleus is obtained. On the basis of these expressions, fundamental formulations are presented for the toughening arising from transformations using the Green's function method. Finally, a representative example is studied to demonstrate the relevance of the fundamental formulation.

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Garvie, R. C., Hannink, R. H. J., and Pascoe, R. T., 1975, “Ceramic Steel?,” Nature, 258, pp. 703–704. [CrossRef]
Claussen, N., 1976, “Fracture Toughness of Al2O3 With an Unstabilized ZrO2 Dispersed Phase,” J. Am. Ceram. Soc., 59, pp. 49–51. [CrossRef]
Gupta, T. K., Bechtold, J. H., Kuznicki, R. C., Cadoff, L. H., and Rossing, B. R., 1977, “Stabilization of Tetragonal Phase in Polycrystalline Zirconia,” J. Mater. Sci., 12, pp. 2421–2426. [CrossRef]
Hannink, R. H. J., 1978, “Growth Morphology of the Tetragonal Phase in Partially Stabilized Zirconia,” J. Mater. Sci., 13, pp. 2487–2496. [CrossRef]
Evans, A. G., and Heuer, A. H., 1980, “Review—Transformation Toughening in Ceramics: Martensitic Transformations in Crack-Tip Stress Fields,” J. Am. Ceram. Soc., 63, pp. 241–248. [CrossRef]
Lange, F. F., 1982, “Transformation Toughening—Part 2, Contribution to Fracture Toughness,” J. Mater. Sci., 17, pp. 235–239. [CrossRef]
Munz, D., and Fett, T., 1998, Ceramics—Mechanical Properties, Failure Behaviour, Material Selection (Springer Series in Material Sciences), Springer, Berlin.
Hannink, R. H. J., Kelly, P. M., and Muddle, B. C., 2000, “Transformation Toughening in Zirconia-Containing Ceramics,” J. Am. Ceram. Soc., 83, pp. 461–487. [CrossRef]
Rauchs, G., Fett, T., Munz, D., and Oberacker, R., 2001, “Tetragonal-to-Monoclinic Phase Transformation in CeO2-Stabilized Zirconia Under Uniaxial Loading,” J. Eur. Ceram. Soc., 21, pp. 2229–2241. [CrossRef]
Rauchs, G., Fett, T., Munz, D., and Oberacker, R., 2002, “Tetragonal-to-Monoclinic Phase Transformation in CeO2-Stabilized Zirconia Under Multiaxial Loading,” J. Eur. Ceram. Soc., 22, pp. 841–849. [CrossRef]
Kelly, P. M., and Rose, L. R. F., 2002, “The Martensitic Transformation in Ceramics—Its Role in Transformation Toughening,” Prog. Mater. Sci., 47, pp. 463–557. [CrossRef]
Magnani, G., and Brillante, A., 2005, “Effect of the Composition and Sintering Process on Mechanical Properties and Residual Stresses in Zirconia-Alumina Composites,” J. Eur. Ceram. Soc., 25, pp. 3383–3392. [CrossRef]
Claussen, N., Ruehle, M., and Heuer, A. H., 1984, “Science and Technology of Zirconia II,” Advances in Ceramics, American Ceramic Society, Columbus, OH, Vol. 12.
Yang, W., and Zhu, T., 1998, “Switch-Toughening of Ferroelectrics Subjected to Electric Fields,” J. Mech. Phys. Solids, 46, pp. 291–311. [CrossRef]
Wang, J., Shi, S. Q., Chen, L., Q.Li, Y., and Zhang, T. Y., 2004, “Phase Field Simulations of Ferroelectric/Ferroelastic Polarization Switching,” Acta Mater., 52, pp. 749–764. [CrossRef]
Jones, J. L., Salz, C. R. J., and Hoffman, M., 2005, “Ferroelastic Fatigue of a Soft PZT Ceramic,” J. Am. Ceram. Soc., 88, pp. 2788–2792. [CrossRef]
Jones, J. L., and Hoffman, M., 2006, “R-Curve and Stress-Strain Behavior of Ferroelastic Ceramics,” J. Am. Ceram. Soc., 89, pp. 3721–3727. [CrossRef]
Jones, J. L., Motahari, S. M., Varlioglu, M., Lienert, U., Bernier, J. V., Hoffman, M., and Uestuendag, E., 2007, “Crack Tip Process Zone Domain Switching in a Soft Lead Zirconate Titanate Ceramic,” Acta Mater., 55, pp. 5538–5548. [CrossRef]
Pojprapai, S., Jones, J. L., Studer, A. J., Russell, J., Valanoor, N., and Hoffman, M., 2008, “Ferroelastic Domain Switching Fatigue in Lead Zirconate Titanate Ceramics,” Acta Mater., 56, pp. 1577–1587. [CrossRef]
McMeeking, R. M., and Evans, A. G., 1982, “Mechanics of Transformation-Toughening in Brittle Materials,” J. Am. Ceram. Soc., 65, pp. 242–246. [CrossRef]
Yi, S., and Gao, S., 2000, “Fracture Toughening Mechanism of Shape Memory Alloys Due to Martensite Transformation,” Int. J. Solids Struct., 37, pp. 5315–5327. [CrossRef]
Yi, S., Gao, S., and Shen, L. X., 2001, “Fracture Toughening Mechanism of Shape Memory Alloys Under Mixed-Mode Loading Due to Martensite Transformation,” Int. J. Solids Struct., 38, pp. 4463–4476. [CrossRef]
Li, Z. H., and Yang, L. H., 2002, “The Application of the Eshelby Equivalent Inclusion Method for Unifying Modulus and Transformation Toughening,” Int. J. Solids Struct., 39, pp. 5225–5240. [CrossRef]
Fischer, F. D., and Boehm, H. J., 2005, “On the Role of the Transformation Eigenstrain in the Growth or Shrinkage of Spheroidal Isotropic Precipitates,” Acta Mater., 53, pp. 367–374. [CrossRef]
Hom, C. L., and McMeeking, R. M., 1990, “Numerical Results for Transformation Toughening in Ceramics,” Int. J. Solids Struct., 26, pp. 1211–1223. [CrossRef]
Zeng, D., Katsube, N., and Soboyejo, W. O., 1999, “Simulation of Transformation Toughening of Heterogeneous Materials With Randomly Distributed Transforming Particles by Hybrid FEM,” Comput. Mech., 23, pp. 457–465. [CrossRef]
Zeng, D., Katsube, N., and Soboyejo, W. O., 2004, “Discrete Modeling of Transformation Toughening in Heterogeneous Materials,” Mech. Mater., 36, pp. 1057–1071. [CrossRef]
Vena, P., Gastaldi, D., Contro, R., and Petrini, L., 2006, “Finite Element Analysis of the Fatigue Crack Growth Rate in Transformation Toughening Ceramics,” Int. J. Plast., 22, pp. 895–920. [CrossRef]
Hutchinson, J. W., 1974, “On Steady Quasi-Static Crack Growth,” Harvard University Report DEAP S-8 (AFOSR-TR-74-1042).
Budiansky, B., Hutchinson, J. W., and Lambropoulos, J. C., 1983, “Continuum Theory of Dilatant Transformation Toughness in Ceramics,” Int. J. Solids Struct., 19, pp. 337–355. [CrossRef]
Lambropoulos, J. C., 1986, “Effect of Nucleation on Transformation Toughening,” J. Am. Ceram. Soc., 69, pp. 218–222. [CrossRef]
Rose, L. R. F., 1987, “The Mechanics of Transformation Toughening,” Proc. R. Soc. London Ser. A, 412, pp. 169–197. [CrossRef]
Tsukamoto, H., and Kotousov, A., 2006, “Transformation Toughening in Zirconia-Enriched Composites: Micromechanical Modeling,” Int. J. Fract., 139, pp. 161–168. [CrossRef]
Mura, T., 1987, Micromechanics of Defects in Solids, Martinus Nijhoff, Dordrecht, The Netherlands.
Love, A. E. H., 1927, Mathematical Theory of Elasticity, Cambridge University Press, Cambridge, UK.
Karihaloo, B. L., and Andreasen, J. H., 1996, Mechanics of Transformation Toughening and Related Topics, Elsevier, Amsterdam.
Li, Q., and Anderson, P. M., 2001, “A Compact Solution for the Stress Field From a Cuboidal Region With a Uniform Transformation Strain,” J. Elast., 64, pp. 237–245. [CrossRef]
Ting, T. C. T., 1996, Anisotropic Elasticity: Theory and Application, Oxford University Press, Oxford, UK.
Hirth, J. P., and Wagoner, R. H., 1976, “Elastic Fields of Line Defects in a Cracked Body,” Int. J. Solids Struct., 12, pp. 117–123. [CrossRef]
Ma, L., Lu, T. J., and Korsunsky, A. M., 2006, “Vector J-Integral Analysis of Crack Interaction With Pre-Existing Singularities,” ASME J. Appl. Mech., 73, pp. 876–883. [CrossRef]
Kisi, E. H., and Howard, C. J., 1998, “Elastic Constants of Tetragonal Zirconia Measured by a New Powder Diffraction Technique,” J. Am. Ceram. Soc., 81, pp. 1682–1684. [CrossRef]


Grahic Jump Location
Fig. 1

An infinitesimal element with transformation strain located in an infinite plane solid

Grahic Jump Location
Fig. 2

A transformation strain nucleus interacting with a semi-infinite crack in an infinite body

Grahic Jump Location
Fig. 3

Two subproblems forming the decomposition of the original problem in Fig. 2. (a) Transformation strain nucleus in an infinite plane without a crack. (b) Application on the semi-infinite crack surfaces of the negative of the tractions due to the transformation strain nucleus.

Grahic Jump Location
Fig. 4

A circular transformation strain zone in an anisotropic solid with a semi-infinite crack

Grahic Jump Location
Fig. 6

A crack is enclosed by a transformed wake

Grahic Jump Location
Fig. 5

Normalized stress intensity factor contributions due to a cylindrical transformation strain zone at angle θ




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