The following article is a component of the August 1998 (vol. 50, no. 8)
JOM and is presented as JOM-e. Such articles appear exclusively on the web and do not have print equivalents. |
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CONTENTS |
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The experimental study of the creep deformation^{2-5} and rupture^{6-8} of these materials has been investigated by several researchers. The results of these efforts suggest that the fully lamellar microstructure is generally stronger than equiaxed microstructures for dual-phase titanium aluminides at elevated temperatures. In order to have a better understanding of the creep deformation and subsequent damage mechanisms in titanium aluminides, numerical simulation and analyses were performed using finite-element (FE) techniques.

Figure 1. A schematic showing the present unit cell with boundary conditions. |

Several models were considered with and without the grain-boundary sliding. Grain-boundary sliding under the action of externally applied stress was simulated using slide lines along the grain boundaries. In all of the analyses performed, it was assumed that the grain-boundary sliding takes place relatively freely (i.e., frictionless sliding), compared to the resistance against other creep-deformation mechanisms. This is a reasonable assumption for grain-boundary sliding as long as the temperature is sufficiently high, and the stress level, as well as the creep strain rate, is low.

The possibility of lath-boundary sliding in a fully lamellar model under creep conditions was also modeled in a previous study.^{10} The overall strain rates in such models were compared with the overall strain rate in fully lamellar models in the absence of any grain- or lath-boundary sliding. It was observed that the lath-boundary sliding gives rise to an overall strain-rate magnification of 100 times the overall strain rate obtained in the model without any boundary sliding. A comparison of these strain rates with the experimental data showed that the strain rate obtained in the model with lath-boundary sliding is unrealistically high, while the strain rate observed in the fully lamellar model in the absence of boundary sliding is similar to the experimental data. Moreover, transmission electron microscopy observations revealed that the density of superdislocations does not change along the lath boundaries during the course of creep deformation.^{11} Based on these findings, it is unlikely that the / phase boundary sliding occurs to any great degree during creep deformation.

Power-law creep is modeled according to

(1) |

where _{e} is the von Mises stress, A is a constant for a given material, and n is the stress exponent.

The effect of grain-boundary sliding on creep deformation and damage has been found to be important for many materials.^{12} Grain-boundary sliding increases the level of stress inside the grains, which results in an increase in the overall strain rate of the material. As a result of grain-boundary sliding, Equation 1 takes the following form^{13-15}

(2) |

where f is the stress-enhancement factor. Several researchers

Figure 4. A comparison of overall strain rate vs. strain for dual-phase equiaxed and fully lamellar TiAl microstructures. Models that include grain-boundary sliding are indicated by "GBS." |

The overall strain rate obtained in the duplex and fully lamellar microstructures are shown in Figure 4. Both microstructures exhibit nearly identical strain rate in the presence of grain-boundary sliding. However, the duplex microstructure exhibited a lower overall strain rate compared to that in the fully lamellar model in the absence of grain-boundary sliding.

(3) |

Where, _{n} is the average normal stress on the cavitated grain boundary. The sintering stress,
_{s}, is given as
_{s} = 2
_{s}sin (
/a, where _{s} is the surface free energy. The cavity tip angle,
, is obtained from the relation

(4) |

where _{b} is the grain-boundary free energy. The effective area fraction of the cavitated grain boundary, , is given in Reference 21.

Tvergaard^{22} has shown that the rate of void growth due to dislocation creep under low triaxial stress conditions is given by

(5) |

where _{n} = 3/2n, and _{n} = (n - 1)(n + 0.4310/n^{2}). The parameters _{m} and _{e} are the remote mean stress and Mises stress, respectively. The function h() accounts for the shape of the cavity, which is given as

(6) |

The total volumetric cavity growth rate can be determined by summing Equations 3 and 5; thus

(7) |

The corresponding rate of change in cavity radius is given as

(8) |

The growth of cavities on grain-boundary facets results in the separation of grains by the plating of atoms out onto the grain boundary from the surface of the cavities.^{23} The average separation can be expressed in terms of cavity volume V and cavity half-spacing b as = V/b^{2}. Therefore, the rate of separation of the grains, , is

(9) |

where is given by Equation 7. Van der Giessen and Tvergaard^{24} have shown that the rate of change of void spacing, , can be correlated with the true strain rate, _{p}, in the grain-boundary plane as

(10) |

It is assumed that the spatial distribution of the cavitating facets follows a periodic order so that we can confine our analysis to a unit cell. Only transverse grain boundaries with respect to the direction of externally applied stresses are assumed to cavitate. Figure 1 shows one quarter of the unit cell with boundary conditions. It can be seen from this figure that the dimensions of the unit cell are specified in terms of the number of grains in the y direction, m_{1}; the number of grains in the z direction, m_{2}; and the initial radius of the grain-boundary facets, R_{0}.

The boundary conditions of the present models, illustrated in Figure 1, are implemented in such a way that the grain boundaries can slide freely without separation. Nodes at the grain-boundary triple points are pinned together to meet strain compatibility conditions. Symmetry boundary conditions, and , are applied on sides OF and OA of the cell OACF, respectively. This unit cell forms a larger cell when reflected about OA and OF, and repetition of this process builds a representative bulk material sample. Nodes on cell boundary AC are constrained to have the same y displacement, u, while nodes on boundary CF are constrained to have the same z displacement, v, at all times. Stress, , is applied on boundary CF. External stress is maintained constant all the time. A detailed discussion of the problem formulation and method of analysis is given in Reference 25.

Figure 6a shows the normalized axial stress distribution in the fully lamellar model in the absence of grain-boundary sliding. As observed in the case of creep deformation, it is found that the

Figure 6b illustrates the normalized axial stress distribution in the same model in the presence of grain-boundary sliding. Grain-boundary sliding resulted in higher stress concentration compared to that in the absence of grain-boundary sliding.

The development of creep cavitation damage, = a/b, as a function of normalized time, t/t

The effect of interaction between the cavitating facets was also investigated in the fully lamellar models.^{26} It was observed that the effect of interaction is more pronounced when cavitating facets are on the neighboring transverse grain boundaries. Conversely, when cavitating facets are not the nearest-neighbor grain boundaries, a negative interaction (in that the cavity growth is lower that that in an isolated facet) can be produced between the two facets. However, a negative interaction is only predicted in the presence of grain-boundary sliding.

Figure 7. A plot of creep damage at facet one of duplex and fully lamellar models vs. normalized time. |

The effect of lamellae orientation on creep cavitation damage in the fully lamellar microstructures is found to be insignificant in the presence of grain-boundary sliding. However, the isostress orientation of laths in the grains adjacent to the cavitating grain boundary is found to delay the cavity growth and, thereby, increases the rupture time. However, this beneficial effect may not be seen for microstructures with random lath orientation.

2. R.E. Schafrik,

3. M.F. Bartholomeusz, Q Yang, and J.A. Wert,

4. Helmet Mehrer,

5. D.S. Shih et al.,

6. S. Mitao, S. Tsuyama, and K. Minakawa,

7. W.O. Soboyejo and R.J. Lederich,

8. R.W. Hayes and P.L. Martin,

9. B. Engelmann and J.O. Hallquist,

10. A. Chakraborty and J.C. Earthman,

11. M. Es-souni, A. Bartels, and R. Wagner,

12. H. Riedel,

13. F. Ghahremani,

14. K.J. Hsia, D.M. Parks, and A.S. Argon,

15. F.W. Crossman and M.F. Ashby,

16. P.M. Anderson and J.R. Rice,

17. G.J. Rodin and M.W. Dib,

18. W. Beere and M.V. Speight,

19. A. Needleman and J.R. Rice,

20. A.C.F. Cocks and M.F. Ahsby,

21. T.-L. Sham and A. Needleman,

22. V. Tvergaard,

23. J.R. Rice,

24. E. van der Giessen and V. Tvergaard,

25. A. Chakraborty and J.C. Earthman,

26. A. Chakraborty and J.C. Earthman,

**ABOUT THE AUTHORS**

**A. Chakraborty** earned his Ph.D. in engineering in 1997 from the University of California at Irvine. He is currently a scientist with Tanner Research.

**J.C. Earthman** earned his Ph.D. in materials science and engineering from Stanford University in 1985. He is currently an associate professor in the Department of Chemical and Biochemical Engineering and Materials Science at the University of California at Irvine.

**For more information, contact A. Chakraborty, Tanner Research, 180 North Vindeo Avenue, Pasadena, California 91107; (626) 432-5766; fax (626) 432-5705; e-mail anirban@tanner.com.**

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