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Numerical models of creep cavitation in single phase, dual phase and fully lamellar titanium aluminide
.kta marer. Vol. 45, No. 11, pp. 46154626, 1997 0 1997 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 135976454/97 $17.00 + 0.00 PII: S1359-6454(97)00131-6
Pergamon
NUMERICAL MODELS OF CREEP CAVITATION IN SINGLE PHASE, DUAL PHASE AND FULLY LAMELLAR TITANIUM ALUMINIDE A. CHAKRABORTY Materials
Science and Engineering,
and J. C. EARTHMAN
University
of California,
Irvine,
CA 92697-2700,
U.S.A.
(Received 7 January 1997; accepted 21 March 1997)
Abstract-Numerical simulations of the high temperature creep constrained cavitation in single phase y. equiaxed dual phase CO+ y and fully lamellar LO+ y TiAl intermetallic alloy microstructures have been performed. Nonlinear viscous secondary creep deformation is modeled in each phase using finite element techniques, Additional models of these alloys were developed that incorporate grain boundary sliding in addition to the dislocation creep flow within each phase. The cavitation in the models is based on the
modified equations of Needleman and Rice. It was found that grain boundary sliding strongly enhances the cavity growth in all of the models analyzed. The present results indicate that the relatively long creep life observed experimentally for fully lamellar TiAl is primarily due to the suppression of grain boundary sliding as a result of the serrated nature of the grain boundaries. 0 1997 Acta Metallurgica Inc.
1. INTRODUCTION
Titanium aluminide intermetallic alloys are finding growing application in power generation and the automobile industry owing to their attractive high temperature properties such as high specific strength, higher creep and oxidation resistance [l]. Creep rupture tests of titanium aluminide alloys with equiaxed duplex and fully lamellar LYE + y microstructures have been performed recently by many researchers [2-51. Mitao et al. [2] showed that the creep rupture time for a fully lamellar alloy having straight grain boundaries is lower by a factor of 3-20, depending on the test temperature and applied stress, than that for a fully lamellar alloy having serrated grain boundaries under identical test conditions. They observed extended wedge cracks in a fully lamellar microstructure with straight grain boundaries and cavities in a fully lamellar microstructure with serrated grain boundaries. Shih et al. [3] have studied the creep rupture behavior of equiaxed duplex and fully lamellar microstructures of TiAl. They showed that a fully lamellar microstructure with serrated grain boundaries requires 260 MPa of applied stress at 810°C to sustain a rupture time of 100 h while a duplex equiaxed microstructure requires only 127 MPa of applied stress to undergo the same rupture time at the same temperature. Tonnes et al. [4] have also studied the creep rupture behavior of duplex and fully lamellar microstructures. They found that the rupture time for a fully lamellar TiAl microstructure is six times higher than that for an equiaxed duplex TiAl microstructure under identical applied stress and temperature. Hayes and McQuay [5] have
performed a creep deformation and damage behavior study of fully lamellar TiAl microstructures. Metallographic examinations of tested specimens revealed that cavities are present at former c( grain boundaries which serve as lath colony boundaries throughout the microstructure. Thus it is apparent that boundary cavitation is a primary damage mechanism in fully lamellar TiAl microstructures. Creep cavitation is the dominant failure mechanism of many metals and alloys under creep conditions and is generally characterized by diffusive growth of cavities at grain boundaries [6]. Experimental observations have revealed that the cavities tend to nucleate at grain boundaries which are approximately transverse to the externally applied stress direction [7-91. These cavities grow at a steady pace during the overall creep deformation of a specimen and finally coalesce, leading to the formation of microcracks. These microcracks then grow to eventually cause rupture of the specimen. The effect of grain boundary sliding on creep deformation and damage has been found to be important for many materials [6]. Grain boundary sliding increases the level of stress inside the grains which results in an increase in the overall strain rate of the material. The effective creep strain rate for a power law material without sliding grain boundaries can be expressed as ;:=Aa:
(1)
where 6, is the von Mises stress, A is a constant for a given material and n is the stress exponent. As a
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result of grain boundary sliding, equation (1) takes the following form [IO-123: ;E = A(&)
(2)
where f is the stress enhancement factor. Several researchers [lO-151 in the past used finite element techniques to determine the stress enhancement factor f. These models indicate that f can range from 1.1 to 1.3 depending on the stress exponent n. Specifically, f represents an enhancement of the stress driving creep facilitated by dislocation motion. It is well established that dislocation creep processes often constrain the rate of cavity growth [8, 16, 171. Thus a higher dislocation creep rate resulting from stress redistribution by grain boundary sliding can enhance cavity growth which in turn reduces the rupture time. A model for grain boundary cavitation was first developed by Hull and Rimmer [7]. In their analysis, the cavitated grain boundary facet was considered to be a square array of voids lying in the grain boundary plane normal to the direction of the externally applied stress. Grains were considered rigid in this model. Beere and Speight [16] modified this model by considering the effect of plastic creep flow on diffusive void growth. They suggested that a coupling between creep flow of the grains and the diffusive growth of the cavities results in a shorter diffusion distance compared to that in the rigid grain model of Hull and Rimmer. Edward and Ashby [18] also gave a similar treatment. Chuang et al. [19] developed a model for non-equilibrium diffusive growth of grain boundary cavities and gave expressions for creep rupture time for different shapes of cavities. Subsequently, Needleman and Rice [17] presented a rigorous finite element (FE) model of the diffusive growth of grain boundary cavities with the plastic creep flow in the material surrounding the cavities. They introduced a characteristic diffusion length, Z’, which is given by Y = @&)“3.
(3)
Here, ge is the remote effective von Mises stress and 2 is the corresponding effective creep strain rate which are obtained from equation (1). 9 is defined by 9 = D&R/kT
(4)
where De is the grain boundary diffusivity, & is the grain boundary thickness, R is the atomic volume and kT is the energy per atom measure of temperature. _Y denotes the effective radius of the grain boundary diffusion zone around the cavity. Material outside this zone primarily deforms solely by dislocation creep while within this zone grain boundary diffusion from the cavity surface contributes to the deformation. Dyson [20] was the first to point out that when cavitating grain boundary facets are relatively isolated from one another, the rate at which the
cavities grow must be compatible with the dislocation creep flow in the surrounding matrix material. Following the work of Dyson, Rice [21] developed a model of the effect of constraints on the diffusive cavitation of isolated grain boundary facets. He considered a more realistic case of cavitation with grain boundary sliding. Grain boundary sliding was assumed to occur freely. A key point demonstrated by his analysis is that the rupture time for cavitating isolated grain boundary facets is controlled by the overall dislocation creep rate. Anderson and Rice [13] carried out a three dimensional analysis of constrained cavitation in nickel. They modeled the grains by Wigner-Seitz cells and used a stress based variational principle to perform the analysis. The grain boundaries were assumed to slide freely and the grains deform in a viscous manner under the action of externally applied stress. Their results indicate that the rupture time is affected strongly by the proximity of other cavitating facets. A number of numerical analyses have been carried out to study grain boundary cavitation in both the absence and presence of grain boundary sliding [22-271. All of these models were used to analyze grain boundary cavitation in single phase equiaxed microstructures. Accordingly, a detailed study of creep cavitation that is applicable to dual phase titanium aluminide microstructures is lacking. In particular, no numerical simulation studies of grain boundary cavitation in either equiaxed or lamellar titanium aluminide intermetallics have been reported. A model and analysis of the grain boundary cavitation in single phase y TiAl, dual phase (~(2+ y) equiaxed and fully lamellar (~1~+ y) titanium aluminide was performed in the present work. Cavitation is assumed to occur on the transverse grain boundaries of the microstructure. Some of the models were developed to include the effects of grain boundary sliding. Cavities are assumed to be of quasi-equilibrium shape and the nucleation of cavities is assumed to occur upon loading. Cavity growth equations used in the present study are based on Needleman and Rice’s equations [17] as modified subsequently by Tvergaard [22]. 2. MODEL FORMULATION
2.1. Overview At very low tensile stresses, grain boundary diffusion is the dominant mechanism and creep flow is negligible [17]. The volumetric growth rate of equilibrium shaped cavities in this case is given (after Rice and Needleman [ 171) as (in -
IJl = 4=g log(l/i)
(1 - 43,
- (3 - [)(l - o/2
(5)
where 0” is the average normal stress on the cavitated grain boundary. The sintering stress, cr,, is given as crs= 2y, sin($)/a, where ys is the surface free energy.
CHAKRABORTY
The cavity tip angle, following relation:
and EARTHMAN:
$,
cos( $) =
is obtained
NUMERICAL
from
the
$
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h(ll/) accounts is given as h($) =
s
CAVITATION
The growth of the cavities is dominated by grain boundary diffusion when a/2’ 6 0.1. By contrast, void growth is driven by dislocation creep flow when a/_!? is large. However, in a transition region, 0.2 < a/2? < 20, both mechanisms combine to yield a relatively high cavity growth rate [17]. Thus, both cavity growth mechanisms must be addressed for accurate predictions of cavity growth over a broad range of conditions. Equations for void growth owing to dislocation creep were developed by Budiansky et al. [29] for high triaxial stress conditions. Tvergaard [22] modified these equations to make them applicable for the low triaxiality range and have the form
(7) where c(, = 3/2n and /In = (n - l)(n + 0.4319)/n2. The parameters 6, and 6, are the remote mean stress and von Mises stress, respectively. The function
(1 + cos +)-I - $0,
Applied Stress Direction symbol
showing grain boundary cavitation. (c, denotes the cavity tip angle.
sin $.
(8)
1/
The above equation is based on the assumption that surface diffusion in the voids is sufficiently rapid to retain a spherical cap (equilibrium) shape. Cavities grow as a result of diffusion of surface atoms of the cavity to adjacent grain boundaries. If the void growth is facilitated by creep deformation, the cavities will tend to elongate in the tensile direction. On the other hand, the cavities tend to become lenticular when grain boundary diffusion is the only mechanism of void growth. It is rapid surface diffusion combined with grain boundary diffusion and dislocation creep that results in equilibrium shaped cavities [17]. The total volumetric cavity growth rate can be determined by summing equations (5) and (7); thus 3= The corresponding given as
vi, + ri,,
for 5 < 10.
(9)
rate of change of cavity radius is
P
a=Tia@j.
(10)
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 [21]. The average separation 6 can be expressed in terms of cavity volume T/and cavity half spacing b as 6 = - Vhb2. Therefore, the rate of separation of the grains, 8, is [22]
d 1. b=2tP.
1. Schematic
$
where 3 is given by equation (9). The contribution of the second term in equation (11) becomes important when finite strains in the zones close to the cavities result in significant changes in the void spacing. The rate of change of void spacing, 6, can be correlated with the true strain rate, &,, in the grain boundary plane under plane strain conditions as in Ref. [27]:
Applied Stress Direction
Fig.
4617
for the shape of the cavity which
[ where yb is the grain boundary free energy. The area fraction of the cavitated grain boundary, [, is given by c = (a/b)’ where b is the cavity half spacing as shown in Fig. 1. However, Sham and Needleman [28] have shown from FE analyses that c in equation (5) should take the following form:
IN TiAl
The
(12)
Van der Giessen and Tvergaard [27] argued that, since the cavity spacing is significantly smaller than the cavitating facet length, it is reasonable to assume that a continuous variation of 6 along the grain boundary is representative of the cavities there. For simplicity, we assume that all the cavities along a facet are of same size and are spaced equally. In this case, 8 does not vary along the facet length and a single representative value of d is determined for a given cavitating facet.
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2.2. Method of analysis 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 2 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, ml, the number of grains in the z direction, m2, and the initial radius of the grain boundary facets, R,,. Finite element techniques were used to perform the present modeling and analysis in the plane strain configuration. The FE mesh for single phase equiaxed hexagonal grain structures is shown in Fig. 3. Four noded quadrilateral and constant strain triangular
elements were used to model all of the meshes used in the present study. A general purpose nonlinear implicit two dimensional FE code for solid mechanics, NIKE2D [30], and its associated pre- and post-processors were used to analyze all of the cases presented here. Grain boundary sliding under the action of externally applied stresses was simulated using slide lines along the grain boundaries. Details on the use of slide lines in FE models can be found elsewhere [30,31]. In all of the analyses performed here, it is assumed that grain boundary sliding takes place freely. This assumption should be particularly valid for intermetallic materials, such as TiAl alloys, which have a relatively low dislocation mobility under the conditions considered, making boundary sliding more prevalent as a deformation mechanism [32, 331. Furthermore, Hayes and Martin [34] have
Z -d; =t) dy
F
m2 2
0
e
R,
H
. v =o
t
I
A _)
4 ml
Fig. 2. Schematic
showing
the present
unit cell with boundary
conditions.
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singularity at the grain boundary triple points is minimal there and thus the stresses and strains are more representative of the overall stress on the boundary. For fully lamellar models, this position is also within a y lath which is under a higher stress compared to neighboring ~(2laths since ~12is weaker than y under the present creep conditions [15]. The length of the time steps during the initial period of cavitation is kept very small so that numerical stability is maintained. Normal stresses, on, at the cavitating grain boundary elements are evaluated at each time step to assure that they are greater than zero. Whenever the normal stress reaches zero in any element at a cavitating boundary, that particular element edge is treated as a free surface from that time step onward. The problem formulation was verified by running a few case studies and comparing the results with those obtained by van der Giessen and Tvergaard [27] for creep cavitation in a homogeneous polycrystalline material with and without grain boundary sliding. Fig. 3. FE mesh of the single phase equiaxed hexagonal grain structure.
observed grain boundary triple point wedge cracks in single phase y TiAl which suggest that the friction
stress against grain boundary sliding in this material is relatively low under creep conditions. The boundary conditions of the present models, illustrated in Fig. 2, 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, u = 0 and ; = 0, 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, U, at all times. Stress. C, is applied on boundary CF. External stress is maintained constant all the time. The boundary conditions at the cavitating grain boundary are modeled after Tvergaard [22]. In each time increment, the facet OH was subjected to an incremental displacement 6 given as 6=$
(13)
where s is given by equation (11) and At is the time increment. Stresses u,, ge and 0” used in equations (5) and (7) and the true strain rate $ used in evaluating equation (11) are obtained directly from the elements at the center of the cavitating facet at each time step. The center of the cavitating facet is chosen as this sampling point because the influence of the
2.3. Modeling
TiAl microstructures
Single phase y TiAl is modeled using the unit cell and mesh described in the previous section. For comparison, both models where grain boundary sliding is operative and suppressed were performed. Although there is ample evidence to indicate that grain boundary sliding is operative in single phase TiAl [34], this comparison reveals the effect of grain boundary sliding on cavity growth and ultimately rupture time. Two forms of dual phase c(~+ y microstructures are analyzed here. The first, DX-1, is modeled by considering a unit cell which has equiaxed grains of x2 phase surrounded by six equiaxed grains of y phase. Thus the mesh used here is the same as that used in the single phase model except that the central grain, 4, is composed of clZ phase. The boundary conditions are identical to those in the previous case. The second model of the dual phase structure, DX-2, is configured as grains 3 and 5 of CI~phase and all other grains, including grain 4, of y phase. Other than this difference, the mesh used for the DX-2 model is the same as that used for the DX-1 model. The fully lamellar (FL) t(2+ y microstructure consists of lamellar grains with laths of y (TiAl) and c12(T&Al) phase [1] with the same volume fraction of x2 as that for the equiaxed dual phase model (33 vol.%). A representative unit cell of the FL microstructure consists of equiaxed grains containing both lamellae of c(~and y phases as shown in Fig. 4. It has been shown that the tl to y transformation temperature in processing the FL TiAl and Al content determines whether the grain boundaries will be straight or serrated [2]. Furthermore, it has been proposed that serrated grain boundaries inhibit grain boundary sliding during creep deformation [2, 3, 5, 1.51.Considering this hypothesis, models of
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Fig. 4. FE mesh for the fully lamellar TiAl models (FL-l and FL-2). The tl~ laths are indicated by the darker shading.
fully lamellar TiAl intermetallics are analyzed in the present work for both the case of cavitation with the presence of grain boundary sliding and the case of cavitation without the influence of grain boundary sliding. We refer to the mesh, shown in Fig. 4 as either FL-l or FL-2 depending on the loading and boundary conditions. For the FL-l models, tensile stress Z is applied at the face CF of the unit cell OACF and all other boundary conditions are the same as for the single phase and dual phase models. For the FL-2 models, tensile stre;js X is applied at the face OA and the boundaries OF and CF are constrained to have zero displacements along the y and z directions respectively. The nodes at boundaries OA are constrained to move together along the z direction and the nodes at boundaries AC are constrained to move together along the y direction at all times. As before, grain boundary triple points are pinned together to have the same y and z displacements at all times. The cavitating facet is
OH in all models except for the FL-2 models in which it is the facet EF. The last stages of cavity coalescence involve rapid tearing of the ligaments between the cavities and it is reasonable to assume that this process begins to occur once the cavitation damage, o, reaches about 0.6 [6]. Accordingly, a critical level of cavitation is assumed to occur when w = 0.6 in all of the present analyses. The mechanical properties of each constituent phase are given in Table 1 [35,36]. The angular orientations of laths in the FL models are chosen at random to represent realistic microstructures and those are listed in Table 2. In all of the analyses, it is assumed that the cavities nucleate upon loading and there is no nucleation of any new cavities during Table 1. Properties Phase name y TiAl m T&Al
of titanium
aluminides
E (GPa)
Y
150.0 102.0
0.241 0.28
at 1080 K [35, 361
A (MPa-“s-l) 2.51 x 1O-‘5 1.27 x lo-l9
n 7.8 5.5
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Table 2. Angular lath orientation in the grains of the FL models as defined in Fig. 2 Grain No.
I
Orientation, cp (degrees) 10.0 125.2 99.1 49.3 19.6 169.5 X.8
2 3 4 5 6 7
the rest of the simulation. The following initial values are assumed: (~/b)~ = 0.1 and (bo/Ro) = 0.4. It should be noted that the above assumptions are uniformly made for all of the present models. Riedel [6] observed that the initial size of the cavities in metals and alloys is typically 0.2-l pm. Accordingly, we assume a representative initial value of a to be 0.7 pm. We note that the model results are relatively insensitive to the initial cavity radius as long as it is within a range of reasonable values. For most metals and alloys, the grain boundary surface energy, yI is between 1 and 2 J/m2 [37]. We estimate y> = pLb/ 10 = 1.69 J/m’-, where b is the magnitude of the burgers vector and p is the shear modulus, following the work of Chu and Thompson [37]. The sintering stress, g\, is found to be 4.54 MPa for a cavity radius of a = 0.7 pm which is negligibly small compared to the externally applied stress of 120 MPa. Accordingly, 0, is taken to be zero in the present models. We also assume that the properties y, and $ are the same at all the grain boundaries of the single phase y, dual phase equiaxed (7 + a,) and fully lamellar (7 + CI?) TiAl models. Reference time, tE, is calculated for the fully lamellar alloy and used to normalize the time for all models. The values of (u/Y)~ are found to be 0.031, 0.045 and 0.047 for the single phase, dual phase equiaxed and the fully lamellar models, respectively. Therefore, grain boundary diffusion is the dominant mechanism of the growth of the cavities. Here, initial values of Y0 are obtained by evaluating equation (3) using 2 values obtained from finite element analyses of secondary creep deformation in the respective models assuming no cavitation or grain boundary sliding. In the present work. grain boundary diffusivity & is assumed to have the same value for the single phase models and the ~‘1)’grain boundaries in the dual phase equiaxed models. Since & of the y phase is higher than that for the z2 phase [38, 391, we have assumed that the De of the fully lamellar alloy is approximately equal to that for the 7 phase. Grain boundary properties of ;’ TiAl are given in Table 3. Although the above Table 3. Grain boundary properties of TiAl Property DH 6s i n
Value 7.5825 x 10 ‘m:/s [38] 0.001 /un [3X] 700 [6] 4.62 x IO-“- m’ [38]
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assumptions could result in significant errors predicting absolute rupture time, they appear to sufficient for an accurate comparison of susceptibility of different phase morphologies grain boundary cavitation. 3. RESULTS
in be the to
AND DISCUSSION
3.1. Cavitation in single phase y, dual phase equiaxed and ,fullJt lamellar models Figure S(a) illustrates the distribution of the axial stress or normalized by the externally applied stress C in the single phase 7 TiAl model with no grain boundary sliding (GBS) for w = 0.6. The cavitating facet is OH in all the models in this section except for FL-2 models in which it is the facet EF. The region adjacent to the grain boundary triple point H is heavily stressed as can be seen from this figure. The axial stress at the center of the cavitating facet approaches zero as eminent cavity coalescence leads to microcrack formation. The normalized axial stress distribution in the presence of GBS in single phase *J TiAl is shown in Fig. 5(b). In contrast to the previous figure, the stresses at triple point H are not concentrated in the presence of GBS. Rather, the GBS facilitates the redistribution of stresses from triple point H to other transverse boundaries. However, it should be noted that as a result of the cavitation, facets RS and KB are stressed to a higher level compared to other transverse facets. Thus, further cavity nucleation would be favored at these boundaries which, if facilitated, would substantially advance the creep rupture process. Figure 6(a) shows the fringe pattern of normalized axial stress in the dual phase equiaxed model (DX-1) in the absence of CBS. The stress pattern in this figure is very similar to that in Fig. 5(a) except that the presence of the Q~grain at the center of the unit cell leads to a stress concentration adjacent to triple point K. It is apparent that this stress concentration could aid in nucleating new cavities at grain boundary KB. The distribution of a,/C for the DX-1 model in the presence of GBS is illustrated in Fig. 6(b). The stress pattern is similar to that in Fig. 5(b) except that the regions along facets CD, KB and IA are found to be under a greater stress. These enhanced stresses are due to the load shedding of the central c(?grain to the adjacent y grains. Figure 7(a) shows the normalized axial stress distribution in the dual phase equiaxed model (DX-2) in the absence of GBS. It is observed here that the different location of c(2grains, compared to that in the DX-1 model, does not alter the character of the stress distribution other than to shift the stress concentration from triple point K, observed in Fig. 6(a), to I in the DX-2 model. Thus, a similar load shedding behavior of the CI?grains is observed here also. The normalized axial stress distribution in the
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DX-2 model in the presence of GBS is illustrated in Fig. 7(b). The stress pattern is also similar here compared to that for the DX-1 model as shown in Fig. 6(b). However, we note that the stresses are greater along KB and lower along IA for DX-2 compared to those for DX-1. Hence, it appears that GBS redistributes the stresses in a manner that reverses the trend noted in the comparison of Figs 6(a) and 7(a). It is also observed in Fig. 7(b) that the level of stress in grain 2 is higher than that for grain 2 in the DX-I model. This difference is due to the load shedding from grain 5 in the DX-2 model which is made up of u2 phase. Figure 8(a) shows the normalized axial stress distribution in the fully lamellar model (FL-l) in the absence of GBS. It is observed that a2 laths that are more closely aligned with the applied stress direction redistribute more stress to the adjacent y laths. This results in inhomogeneous stress distributions along the grain boundaries RS and PQ. However, the stress level discontinuities are more extreme at boundary RS than those at PQ. This is apparently due to the more isostress orientation of the lamellae in grain 5 as well as the proximity of RS to the cavitating facet. The distribution of the normalized axial stress in the FL-l model in the presence of GBS is illustrated in Fig. 8(b). It is evident from this figure that the transverse grain boundaries RS and KB are heavily stressed regions in the microstructure. Cavitation at grain boundary OH again redistributes stress from grain 1 to adjacent grains. However, a high level of stress at grain boundary RS ensues as a result of the load shedding of the a2 laths at this grain boundary and owing to the sliding of the inclined grain boundaries. It is interesting to note that the transverse grain boundary KB is stressed to a higher level than either boundary GJ or PQ, although these boundaries have lath orientations that are less characteristic of an isostress arrangement. It appears that the higher stress at boundary KB is due to the fact that it lies in the path of the highest stresses which arise from the growth of the cavities at grain boundary OH. This characteristic of the stress field is also evident in Figs S(b), 6(b) and 7(b). The normalized axial stress distribution in the FL-2 model is shown in Fig. 9(a). Here, the presence of isostress lamellae immediately ahead of the cavitating facet (in grain 5) facilitates the formation of a more uniform stress field in this region compared to those in the corresponding region of the FL- 1 model [grain 3 in Fig. 8(a)]. Figure 9(b) illustrates the normalized axial stress distribution in the FL-2 model in the presence of GBS. It is observed that the grain boundaries PQ and KB are in regions of high stress concentration due to grain boundary sliding coupled with creep constrained cavitation. These two boundaries are along the direction of the maximum stress, as are
MODELS
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boundaries RS and KB for the FL-l model, that results from the cavitation. The development of creep cavitation damage w = a/b as a function of normalized time t/tE is illustrated in Fig. 10, for the present models. It is evident from this figure that grain boundary sliding substantially reduces the cavity coalescence time for all the microstructures considered. In the absence of sliding, the single phase model exhibits the strongest resistance against creep cavitation. In the presence of grain boundary sliding, the dual phase equiaxed model, DX-2, is the strongest in terms of resistance to creep cavity growth while DX-1 exhibits the weakest behavior. This implies that the cavity growth will be slower if cavitation occurs at the y/y grain boundary adjacent to the tl2 grain compared to a nearby y/y grain boundary that is in an inclined direction from the c(~grain. We note that the overall stresses above the cavitating facet in Fig. 7(b) (DX-2) are higher than those in the same region in Fig. 6(b) (DX-1). As a result, less stress is distributed to facets IA and CD of the DX-2 model. Apparently, this gives rise to more constraint on the cavity growth in DX-2 as compared to that for DX-1 as indicated by the difference in cavity growth rates in Fig. 10. The two fully lamellar models FL-l and FL-2 yielded identical results for cavitation in the presence of GBS. This similarity indicates that the effect of lamellae orientation on creep cavitation is not as pronounced in the presence of grain boundary sliding. In the absence of GBS, the FL-2 model is found to be stronger than the FL-l model. The important point here is that a difference in cavity coalescence time for the two FL models is only evident when GBS is suppressed. This is likely the case for the FL microstructure which normally exhibits serrated grain boundaries. The present results indicate that cavities at a grain boundary with intersecting lath orientations which are isostress in
_..
0
4
8
12
16
20
2A
28
% Fig. 10. Plot of creep damage at facet # 1 of Single Phase (SP), Dual Phase Equiaxed (DX) and Fully Lamellar (FL) models vs normalized time.
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Fig
5. Normalized axial stress (u:/Z) distribution in the single phase y TiAl during creep cavitation central grain boundary OH (a) the absence and (b) in the presence of GBS.
Fig
6. Normalized axial stress (a:/Z) distribution in the dual phase co + y TiAl (DX-I) during creep cavitation at central grain boundary OH (a) without GBS and (b) in the presence of GBS.
Fig
7. Normalized axial stress (al/Z) distribution in the dual phase w + ;’ TiAl (DX-2) during creep cavitation at central grain boundary OH (a) without GBS and (b) in the presence of GBS.
at
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Fig. 8. Normalized axial stress (uz/X) distribution in the fully lamellar TiAl (FL-l) during creep cavitation at central grain boundary OH (a) without GBS and (b) in the presence of GBS. It should be noted tfiat the fringe levels in Fig. 8(a) are different from those in previous figures.
Fig. 9. Normalized axial stress (uz/X) distribution in the fully lamellar TiAl (FL-2) during creep cavitation at central grain boundary EF (a) without GBS and (b) in the presence of GBS. It should be noted that the fringe levels in Fig. 9(a) are different from those in previous figures.
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NUMERICAL
nature will tend to grow slower compared to the case where the intersecting lath orientations are closer to an isostrain orientation. Lath orientations that are nearly parallel to the applied stress direction in the zone adjacent to the cavitating grain boundary result in load shedding from a2 laths to y laths which then produces less constraint on the cavitation process. 4. CONCLUSIONS
The present model results indicate that the single phase TiAl microstructure exhibits maximum strength against grain boundary cavitation in the absence of grain boundary sliding. In the presence of grain boundary sliding, the dual phase equiaxed microstructure exhibits the greatest resistance against cavitation when cavity growth occurs at the r/r grain boundary adjacent to the CQgrain. Conversely, this microstructure is the most susceptible to cavitation damage when cavitation occurs at an inclined angle to a neighboring a2 grain. The effect of neighboring lath orientation on grain boundary cavitation for the fully lamellar microstructure is negligible in the presence of grain boundary sliding. However, in the absence of grain boundary sliding an isostress orientation of the laths in the grains adjacent to the cavitating grain boundary delays the cavity growth process. Based on the present results, the relatively long rupture times observed experimentally for fully lamellar TiAl may be primarily attributed to the suppression of grain boundary sliding in this microstructure. Acknorvledgements-Support for this work from the Air Force Office of Scientific Research under grant No. F49620-94-0 137 is gratefully acknowledged. The authors would like to thank the Lawrence Livermore National Laboratory for providing the FE codes used in the present work through their software collaborators program. REFERENCES Kim, Y. W., JOM, 1994, 46, 30. Mitao, S., Tsuyama, S. and Minakawa, K., in Microstructure/Property Relationships in Titanium Aluminides and Alloys, ed. Y. W. Kim and R. R. Boyer. The Minerals, Metals and Materials Society, Warrendale, 1991, p. 297. Shih, D. S., Huang, S. C., Starr, G. K., Jang, H. and Chestnut, J. C., in Microstructure/Property -Relationshins in Titanium Aluminides and Allovs. , ed. Y. W. Kim and R. R. Boyer. The Minerals, Metals and Materials Society, Warrendale, 1991, p. 135. Tonnes, C., Rosier, J., Baumann, R. and Thumann, M., in Structural Intermetallics, ed. R. Darolia et al. The Minerals, Metals and Materials Society, Warrendale, 1993, p. 241. Hayes, R. W. and McQuay, P. A., Scripta metall. mater., 1994, 30, 259. Reidel, H., Fracture at High Temperatures. SpringerVerlaa, New York, 1986. Hull,-D. and Rimmer, D. E., Phil. mag. A, 1959, 4, 613. Cocks, A. C. F. and Ashby, M. F., Prog. Mater. Sci., 1982, 27, 244.
MODELS
OF CREEP
CAVITATION
IN TiAl
4625
J. C., Eggeler, G. and Ilschner, 9. Nix, W. D., Earthman, B., Acta metall., 1989, 37, 1067. F., Int. J. Solids Struct., 1980, 16, 847. 10. Ghahremani, 11. Hsia, K. J., Parks, D. M. and Argon, A. S., Mech. mater., 1991, 11, 43. 12. Crossman, F. W. and Ashby, M. F., Acta metall.. 1975, 23, 425. P. M. and Rice, J. R., Acta metall., 1985, 13. Anderson, 33, 409. 14. Rodin, G. J. and Dib, M. W., in Advances in Fracture Research, Vol. 2, ed. K. Salama et al. Pergamon Press, Oxford, 1989, p. 1835. A. and Earthman, J. C., Metal/. Mater. 15. Chakraborty, Trans. A, 1997. 28A, 919. 16. Beere, W. and Speight, M. V., Metal. Sci., 1978, 12, 172. A. and Rice, J. R., Acta metall., 1980, 17. Needleman, 28, 1315. 18. Edward, G. H. and Ashby, M. F., Acta metall., 1979, 27, 1505. 19. Chuang, T., Kagawa, K. I., Rice, J. R. and Sills, L. B., Acta metall., 1979, 27, 265. 20. Dyson, B. F.. Metal Sri., 1976, 10, 349. 21. Rice, J. R., Acta metall., 1981, 29, 675. 22. Tvergaard, V., J. Mech. Phys. of Solids, 1984, 32, 373. 23. Tvergaard, V., J. Mech. Phys. of Solids, 1985, 33, 447. 24. Tvergaard, V., Int. J. Solids Struct., 1985, 21, 279. 25. van der Giessen, E. and Tvergaard, V., Int. J. Fracture, 1991,48, 153. 26. van der Giessen. E. and Tvergaard, V., Acta metall., 1994, 42, 959. 27. van der Giessen, E. and Tvergaard, V., Mech. mater., 1994, 17, 47. 28. Sham, T.-L. and Needleman, A., Acta metall., 1983, 31, 919. 29. Budiansky, B., Hutchinson, J. W. and Slutsky, S., in Mechanics of Solids, The Rodney Hill 60th Anniversary Volume, ed. H. G. Hopkins and M. J. Sewell. Pergamon Press, Oxford, 1982. p. 13. 30. Engelmann, B. and Hallquist, J. O., NIKE2D User Livermore National Laboratory, Manual. Lawrence Livermore, California, April 1991. 31. Hallquist, J. O., Goudreau, G. L. and Benson, D. J., Comp. Meth. Appl. Mech. Engng, 1985, 51, 107. 32. Hug, G., Loiseau, A. and Veyssiere, P., Phil. Mag. A, 1988, 57, 499. 33. Kim, H. K., Wolfenstine, J. and Earthman, J. C., Scripta metall. mater., 1992, 27, 1067. 34. Hayes. R. W. and Martin, P. L., Acta metall., 1995, 43, 2761. 35. Schafrik, R. E., Metall. Mater. Trans. A, 1977. 8, 1003. 36. Bartholomeusz, M. F., Yang, Q. and Wert, J. A., Scripta Metall. Mater., 1993, 29, 389. 37. Chu, W.-Y. and Thompson, A. W., Metall. Mater. Trans. A, 1992, 23, 1299. 38. Soboyejo. W. 0. and Ledrich, R. J., in Structural Intermetallics, ed. R. Darolia et al. The Minerals, Metals and Materials Society, Warrendale, 1993, p. 353. 39. Mehrer, H., Mater. Trans., JIM, 1996, 31, 1259.
APPENDIX Nomenclature
cavity creep grain cavity
radius stress coefficient area half spacing
4626
m2
n RO fE
At U u V ;
CHAKRABORTY
and EARTHMAN:
NUMERICAL MODELS OF CREEP CAVITATION IN TiAl
Burgers vector grain boundary diffusivity grain boundary diffusion parameter Young’s modulus stress enhancement factor Boltzmann’s constant characteristic of grain boundary diffusion number of grains in the y direction in the unit cell number of grains in the z direction in the unit cell creep stress exponent initial radius of grain boundary facet reference time for creep strains to equal elastic strains time increment displacement in the y direction displacement rate in the y direction displacement in the z direction displacement rate in the z direction
cavity volume grain boundary free energy surface free energy rate of grain boundary separation grain boundary thickness effective creep strain rate true strain rate in the grain boundary plane area fraction of cavitating grain boundary shear modulus density of cavitating facets in the unit cell von Mises stress mean stress average normal stress at grain boundary sintering stress axial stress externally applied stress externally applied von Mises stress cavity tip angle cavitation damage parameter atomic volume
Pergamon
NUMERICAL MODELS OF CREEP CAVITATION IN SINGLE PHASE, DUAL PHASE AND FULLY LAMELLAR TITANIUM ALUMINIDE A. CHAKRABORTY Materials
Science and Engineering,
and J. C. EARTHMAN
University
of California,
Irvine,
CA 92697-2700,
U.S.A.
(Received 7 January 1997; accepted 21 March 1997)
Abstract-Numerical simulations of the high temperature creep constrained cavitation in single phase y. equiaxed dual phase CO+ y and fully lamellar LO+ y TiAl intermetallic alloy microstructures have been performed. Nonlinear viscous secondary creep deformation is modeled in each phase using finite element techniques, Additional models of these alloys were developed that incorporate grain boundary sliding in addition to the dislocation creep flow within each phase. The cavitation in the models is based on the
modified equations of Needleman and Rice. It was found that grain boundary sliding strongly enhances the cavity growth in all of the models analyzed. The present results indicate that the relatively long creep life observed experimentally for fully lamellar TiAl is primarily due to the suppression of grain boundary sliding as a result of the serrated nature of the grain boundaries. 0 1997 Acta Metallurgica Inc.
1. INTRODUCTION
Titanium aluminide intermetallic alloys are finding growing application in power generation and the automobile industry owing to their attractive high temperature properties such as high specific strength, higher creep and oxidation resistance [l]. Creep rupture tests of titanium aluminide alloys with equiaxed duplex and fully lamellar LYE + y microstructures have been performed recently by many researchers [2-51. Mitao et al. [2] showed that the creep rupture time for a fully lamellar alloy having straight grain boundaries is lower by a factor of 3-20, depending on the test temperature and applied stress, than that for a fully lamellar alloy having serrated grain boundaries under identical test conditions. They observed extended wedge cracks in a fully lamellar microstructure with straight grain boundaries and cavities in a fully lamellar microstructure with serrated grain boundaries. Shih et al. [3] have studied the creep rupture behavior of equiaxed duplex and fully lamellar microstructures of TiAl. They showed that a fully lamellar microstructure with serrated grain boundaries requires 260 MPa of applied stress at 810°C to sustain a rupture time of 100 h while a duplex equiaxed microstructure requires only 127 MPa of applied stress to undergo the same rupture time at the same temperature. Tonnes et al. [4] have also studied the creep rupture behavior of duplex and fully lamellar microstructures. They found that the rupture time for a fully lamellar TiAl microstructure is six times higher than that for an equiaxed duplex TiAl microstructure under identical applied stress and temperature. Hayes and McQuay [5] have
performed a creep deformation and damage behavior study of fully lamellar TiAl microstructures. Metallographic examinations of tested specimens revealed that cavities are present at former c( grain boundaries which serve as lath colony boundaries throughout the microstructure. Thus it is apparent that boundary cavitation is a primary damage mechanism in fully lamellar TiAl microstructures. Creep cavitation is the dominant failure mechanism of many metals and alloys under creep conditions and is generally characterized by diffusive growth of cavities at grain boundaries [6]. Experimental observations have revealed that the cavities tend to nucleate at grain boundaries which are approximately transverse to the externally applied stress direction [7-91. These cavities grow at a steady pace during the overall creep deformation of a specimen and finally coalesce, leading to the formation of microcracks. These microcracks then grow to eventually cause rupture of the specimen. The effect of grain boundary sliding on creep deformation and damage has been found to be important for many materials [6]. Grain boundary sliding increases the level of stress inside the grains which results in an increase in the overall strain rate of the material. The effective creep strain rate for a power law material without sliding grain boundaries can be expressed as ;:=Aa:
(1)
where 6, is the von Mises stress, A is a constant for a given material and n is the stress exponent. As a
4615
4616
CHAKRABORTY
and EARTHMAN:
NUMERICAL MODELS OF CREEP CAVITATION IN TiAl
result of grain boundary sliding, equation (1) takes the following form [IO-123: ;E = A(&)
(2)
where f is the stress enhancement factor. Several researchers [lO-151 in the past used finite element techniques to determine the stress enhancement factor f. These models indicate that f can range from 1.1 to 1.3 depending on the stress exponent n. Specifically, f represents an enhancement of the stress driving creep facilitated by dislocation motion. It is well established that dislocation creep processes often constrain the rate of cavity growth [8, 16, 171. Thus a higher dislocation creep rate resulting from stress redistribution by grain boundary sliding can enhance cavity growth which in turn reduces the rupture time. A model for grain boundary cavitation was first developed by Hull and Rimmer [7]. In their analysis, the cavitated grain boundary facet was considered to be a square array of voids lying in the grain boundary plane normal to the direction of the externally applied stress. Grains were considered rigid in this model. Beere and Speight [16] modified this model by considering the effect of plastic creep flow on diffusive void growth. They suggested that a coupling between creep flow of the grains and the diffusive growth of the cavities results in a shorter diffusion distance compared to that in the rigid grain model of Hull and Rimmer. Edward and Ashby [18] also gave a similar treatment. Chuang et al. [19] developed a model for non-equilibrium diffusive growth of grain boundary cavities and gave expressions for creep rupture time for different shapes of cavities. Subsequently, Needleman and Rice [17] presented a rigorous finite element (FE) model of the diffusive growth of grain boundary cavities with the plastic creep flow in the material surrounding the cavities. They introduced a characteristic diffusion length, Z’, which is given by Y = @&)“3.
(3)
Here, ge is the remote effective von Mises stress and 2 is the corresponding effective creep strain rate which are obtained from equation (1). 9 is defined by 9 = D&R/kT
(4)
where De is the grain boundary diffusivity, & is the grain boundary thickness, R is the atomic volume and kT is the energy per atom measure of temperature. _Y denotes the effective radius of the grain boundary diffusion zone around the cavity. Material outside this zone primarily deforms solely by dislocation creep while within this zone grain boundary diffusion from the cavity surface contributes to the deformation. Dyson [20] was the first to point out that when cavitating grain boundary facets are relatively isolated from one another, the rate at which the
cavities grow must be compatible with the dislocation creep flow in the surrounding matrix material. Following the work of Dyson, Rice [21] developed a model of the effect of constraints on the diffusive cavitation of isolated grain boundary facets. He considered a more realistic case of cavitation with grain boundary sliding. Grain boundary sliding was assumed to occur freely. A key point demonstrated by his analysis is that the rupture time for cavitating isolated grain boundary facets is controlled by the overall dislocation creep rate. Anderson and Rice [13] carried out a three dimensional analysis of constrained cavitation in nickel. They modeled the grains by Wigner-Seitz cells and used a stress based variational principle to perform the analysis. The grain boundaries were assumed to slide freely and the grains deform in a viscous manner under the action of externally applied stress. Their results indicate that the rupture time is affected strongly by the proximity of other cavitating facets. A number of numerical analyses have been carried out to study grain boundary cavitation in both the absence and presence of grain boundary sliding [22-271. All of these models were used to analyze grain boundary cavitation in single phase equiaxed microstructures. Accordingly, a detailed study of creep cavitation that is applicable to dual phase titanium aluminide microstructures is lacking. In particular, no numerical simulation studies of grain boundary cavitation in either equiaxed or lamellar titanium aluminide intermetallics have been reported. A model and analysis of the grain boundary cavitation in single phase y TiAl, dual phase (~(2+ y) equiaxed and fully lamellar (~1~+ y) titanium aluminide was performed in the present work. Cavitation is assumed to occur on the transverse grain boundaries of the microstructure. Some of the models were developed to include the effects of grain boundary sliding. Cavities are assumed to be of quasi-equilibrium shape and the nucleation of cavities is assumed to occur upon loading. Cavity growth equations used in the present study are based on Needleman and Rice’s equations [17] as modified subsequently by Tvergaard [22]. 2. MODEL FORMULATION
2.1. Overview At very low tensile stresses, grain boundary diffusion is the dominant mechanism and creep flow is negligible [17]. The volumetric growth rate of equilibrium shaped cavities in this case is given (after Rice and Needleman [ 171) as (in -
IJl = 4=g log(l/i)
(1 - 43,
- (3 - [)(l - o/2
(5)
where 0” is the average normal stress on the cavitated grain boundary. The sintering stress, cr,, is given as crs= 2y, sin($)/a, where ys is the surface free energy.
CHAKRABORTY
The cavity tip angle, following relation:
and EARTHMAN:
$,
cos( $) =
is obtained
NUMERICAL
from
the
$
MODELS
OF CREEP
h(ll/) accounts is given as h($) =
s
CAVITATION
The growth of the cavities is dominated by grain boundary diffusion when a/2’ 6 0.1. By contrast, void growth is driven by dislocation creep flow when a/_!? is large. However, in a transition region, 0.2 < a/2? < 20, both mechanisms combine to yield a relatively high cavity growth rate [17]. Thus, both cavity growth mechanisms must be addressed for accurate predictions of cavity growth over a broad range of conditions. Equations for void growth owing to dislocation creep were developed by Budiansky et al. [29] for high triaxial stress conditions. Tvergaard [22] modified these equations to make them applicable for the low triaxiality range and have the form
(7) where c(, = 3/2n and /In = (n - l)(n + 0.4319)/n2. The parameters 6, and 6, are the remote mean stress and von Mises stress, respectively. The function
(1 + cos +)-I - $0,
Applied Stress Direction symbol
showing grain boundary cavitation. (c, denotes the cavity tip angle.
sin $.
(8)
1/
The above equation is based on the assumption that surface diffusion in the voids is sufficiently rapid to retain a spherical cap (equilibrium) shape. Cavities grow as a result of diffusion of surface atoms of the cavity to adjacent grain boundaries. If the void growth is facilitated by creep deformation, the cavities will tend to elongate in the tensile direction. On the other hand, the cavities tend to become lenticular when grain boundary diffusion is the only mechanism of void growth. It is rapid surface diffusion combined with grain boundary diffusion and dislocation creep that results in equilibrium shaped cavities [17]. The total volumetric cavity growth rate can be determined by summing equations (5) and (7); thus 3= The corresponding given as
vi, + ri,,
for 5 < 10.
(9)
rate of change of cavity radius is
P
a=Tia@j.
(10)
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 [21]. The average separation 6 can be expressed in terms of cavity volume T/and cavity half spacing b as 6 = - Vhb2. Therefore, the rate of separation of the grains, 8, is [22]
d 1. b=2tP.
1. Schematic
$
where 3 is given by equation (9). The contribution of the second term in equation (11) becomes important when finite strains in the zones close to the cavities result in significant changes in the void spacing. The rate of change of void spacing, 6, can be correlated with the true strain rate, &,, in the grain boundary plane under plane strain conditions as in Ref. [27]:
Applied Stress Direction
Fig.
4617
for the shape of the cavity which
[ where yb is the grain boundary free energy. The area fraction of the cavitated grain boundary, [, is given by c = (a/b)’ where b is the cavity half spacing as shown in Fig. 1. However, Sham and Needleman [28] have shown from FE analyses that c in equation (5) should take the following form:
IN TiAl
The
(12)
Van der Giessen and Tvergaard [27] argued that, since the cavity spacing is significantly smaller than the cavitating facet length, it is reasonable to assume that a continuous variation of 6 along the grain boundary is representative of the cavities there. For simplicity, we assume that all the cavities along a facet are of same size and are spaced equally. In this case, 8 does not vary along the facet length and a single representative value of d is determined for a given cavitating facet.
4618
and EARTHMAN:
CHAKRABORTY
NUMERICAL MODELS OF CREEP CAVITATION IN TiAl
2.2. Method of analysis 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 2 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, ml, the number of grains in the z direction, m2, and the initial radius of the grain boundary facets, R,,. Finite element techniques were used to perform the present modeling and analysis in the plane strain configuration. The FE mesh for single phase equiaxed hexagonal grain structures is shown in Fig. 3. Four noded quadrilateral and constant strain triangular
elements were used to model all of the meshes used in the present study. A general purpose nonlinear implicit two dimensional FE code for solid mechanics, NIKE2D [30], and its associated pre- and post-processors were used to analyze all of the cases presented here. Grain boundary sliding under the action of externally applied stresses was simulated using slide lines along the grain boundaries. Details on the use of slide lines in FE models can be found elsewhere [30,31]. In all of the analyses performed here, it is assumed that grain boundary sliding takes place freely. This assumption should be particularly valid for intermetallic materials, such as TiAl alloys, which have a relatively low dislocation mobility under the conditions considered, making boundary sliding more prevalent as a deformation mechanism [32, 331. Furthermore, Hayes and Martin [34] have
Z -d; =t) dy
F
m2 2
0
e
R,
H
. v =o
t
I
A _)
4 ml
Fig. 2. Schematic
showing
the present
unit cell with boundary
conditions.
CHAKRABORTY
and EARTHMAN:
NUMERICAL
MODELS OF CREEP CAVITATION IN TiAl
4619
singularity at the grain boundary triple points is minimal there and thus the stresses and strains are more representative of the overall stress on the boundary. For fully lamellar models, this position is also within a y lath which is under a higher stress compared to neighboring ~(2laths since ~12is weaker than y under the present creep conditions [15]. The length of the time steps during the initial period of cavitation is kept very small so that numerical stability is maintained. Normal stresses, on, at the cavitating grain boundary elements are evaluated at each time step to assure that they are greater than zero. Whenever the normal stress reaches zero in any element at a cavitating boundary, that particular element edge is treated as a free surface from that time step onward. The problem formulation was verified by running a few case studies and comparing the results with those obtained by van der Giessen and Tvergaard [27] for creep cavitation in a homogeneous polycrystalline material with and without grain boundary sliding. Fig. 3. FE mesh of the single phase equiaxed hexagonal grain structure.
observed grain boundary triple point wedge cracks in single phase y TiAl which suggest that the friction
stress against grain boundary sliding in this material is relatively low under creep conditions. The boundary conditions of the present models, illustrated in Fig. 2, 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, u = 0 and ; = 0, 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, U, at all times. Stress. C, is applied on boundary CF. External stress is maintained constant all the time. The boundary conditions at the cavitating grain boundary are modeled after Tvergaard [22]. In each time increment, the facet OH was subjected to an incremental displacement 6 given as 6=$
(13)
where s is given by equation (11) and At is the time increment. Stresses u,, ge and 0” used in equations (5) and (7) and the true strain rate $ used in evaluating equation (11) are obtained directly from the elements at the center of the cavitating facet at each time step. The center of the cavitating facet is chosen as this sampling point because the influence of the
2.3. Modeling
TiAl microstructures
Single phase y TiAl is modeled using the unit cell and mesh described in the previous section. For comparison, both models where grain boundary sliding is operative and suppressed were performed. Although there is ample evidence to indicate that grain boundary sliding is operative in single phase TiAl [34], this comparison reveals the effect of grain boundary sliding on cavity growth and ultimately rupture time. Two forms of dual phase c(~+ y microstructures are analyzed here. The first, DX-1, is modeled by considering a unit cell which has equiaxed grains of x2 phase surrounded by six equiaxed grains of y phase. Thus the mesh used here is the same as that used in the single phase model except that the central grain, 4, is composed of clZ phase. The boundary conditions are identical to those in the previous case. The second model of the dual phase structure, DX-2, is configured as grains 3 and 5 of CI~phase and all other grains, including grain 4, of y phase. Other than this difference, the mesh used for the DX-2 model is the same as that used for the DX-1 model. The fully lamellar (FL) t(2+ y microstructure consists of lamellar grains with laths of y (TiAl) and c12(T&Al) phase [1] with the same volume fraction of x2 as that for the equiaxed dual phase model (33 vol.%). A representative unit cell of the FL microstructure consists of equiaxed grains containing both lamellae of c(~and y phases as shown in Fig. 4. It has been shown that the tl to y transformation temperature in processing the FL TiAl and Al content determines whether the grain boundaries will be straight or serrated [2]. Furthermore, it has been proposed that serrated grain boundaries inhibit grain boundary sliding during creep deformation [2, 3, 5, 1.51.Considering this hypothesis, models of
4620
CHAKRABORTY
and EARTHMAN:
NUMERICAL
MODELS OF CREEP CAVITATION IN TiAl
Fig. 4. FE mesh for the fully lamellar TiAl models (FL-l and FL-2). The tl~ laths are indicated by the darker shading.
fully lamellar TiAl intermetallics are analyzed in the present work for both the case of cavitation with the presence of grain boundary sliding and the case of cavitation without the influence of grain boundary sliding. We refer to the mesh, shown in Fig. 4 as either FL-l or FL-2 depending on the loading and boundary conditions. For the FL-l models, tensile stress Z is applied at the face CF of the unit cell OACF and all other boundary conditions are the same as for the single phase and dual phase models. For the FL-2 models, tensile stre;js X is applied at the face OA and the boundaries OF and CF are constrained to have zero displacements along the y and z directions respectively. The nodes at boundaries OA are constrained to move together along the z direction and the nodes at boundaries AC are constrained to move together along the y direction at all times. As before, grain boundary triple points are pinned together to have the same y and z displacements at all times. The cavitating facet is
OH in all models except for the FL-2 models in which it is the facet EF. The last stages of cavity coalescence involve rapid tearing of the ligaments between the cavities and it is reasonable to assume that this process begins to occur once the cavitation damage, o, reaches about 0.6 [6]. Accordingly, a critical level of cavitation is assumed to occur when w = 0.6 in all of the present analyses. The mechanical properties of each constituent phase are given in Table 1 [35,36]. The angular orientations of laths in the FL models are chosen at random to represent realistic microstructures and those are listed in Table 2. In all of the analyses, it is assumed that the cavities nucleate upon loading and there is no nucleation of any new cavities during Table 1. Properties Phase name y TiAl m T&Al
of titanium
aluminides
E (GPa)
Y
150.0 102.0
0.241 0.28
at 1080 K [35, 361
A (MPa-“s-l) 2.51 x 1O-‘5 1.27 x lo-l9
n 7.8 5.5
CHAKRABORTY
and EARTHMAN:
NUMERICAL
Table 2. Angular lath orientation in the grains of the FL models as defined in Fig. 2 Grain No.
I
Orientation, cp (degrees) 10.0 125.2 99.1 49.3 19.6 169.5 X.8
2 3 4 5 6 7
the rest of the simulation. The following initial values are assumed: (~/b)~ = 0.1 and (bo/Ro) = 0.4. It should be noted that the above assumptions are uniformly made for all of the present models. Riedel [6] observed that the initial size of the cavities in metals and alloys is typically 0.2-l pm. Accordingly, we assume a representative initial value of a to be 0.7 pm. We note that the model results are relatively insensitive to the initial cavity radius as long as it is within a range of reasonable values. For most metals and alloys, the grain boundary surface energy, yI is between 1 and 2 J/m2 [37]. We estimate y> = pLb/ 10 = 1.69 J/m’-, where b is the magnitude of the burgers vector and p is the shear modulus, following the work of Chu and Thompson [37]. The sintering stress, g\, is found to be 4.54 MPa for a cavity radius of a = 0.7 pm which is negligibly small compared to the externally applied stress of 120 MPa. Accordingly, 0, is taken to be zero in the present models. We also assume that the properties y, and $ are the same at all the grain boundaries of the single phase y, dual phase equiaxed (7 + a,) and fully lamellar (7 + CI?) TiAl models. Reference time, tE, is calculated for the fully lamellar alloy and used to normalize the time for all models. The values of (u/Y)~ are found to be 0.031, 0.045 and 0.047 for the single phase, dual phase equiaxed and the fully lamellar models, respectively. Therefore, grain boundary diffusion is the dominant mechanism of the growth of the cavities. Here, initial values of Y0 are obtained by evaluating equation (3) using 2 values obtained from finite element analyses of secondary creep deformation in the respective models assuming no cavitation or grain boundary sliding. In the present work. grain boundary diffusivity & is assumed to have the same value for the single phase models and the ~‘1)’grain boundaries in the dual phase equiaxed models. Since & of the y phase is higher than that for the z2 phase [38, 391, we have assumed that the De of the fully lamellar alloy is approximately equal to that for the 7 phase. Grain boundary properties of ;’ TiAl are given in Table 3. Although the above Table 3. Grain boundary properties of TiAl Property DH 6s i n
Value 7.5825 x 10 ‘m:/s [38] 0.001 /un [3X] 700 [6] 4.62 x IO-“- m’ [38]
MODELS
OF CREEP
CAVITATION
IN TiAl
4621
assumptions could result in significant errors predicting absolute rupture time, they appear to sufficient for an accurate comparison of susceptibility of different phase morphologies grain boundary cavitation. 3. RESULTS
in be the to
AND DISCUSSION
3.1. Cavitation in single phase y, dual phase equiaxed and ,fullJt lamellar models Figure S(a) illustrates the distribution of the axial stress or normalized by the externally applied stress C in the single phase 7 TiAl model with no grain boundary sliding (GBS) for w = 0.6. The cavitating facet is OH in all the models in this section except for FL-2 models in which it is the facet EF. The region adjacent to the grain boundary triple point H is heavily stressed as can be seen from this figure. The axial stress at the center of the cavitating facet approaches zero as eminent cavity coalescence leads to microcrack formation. The normalized axial stress distribution in the presence of GBS in single phase *J TiAl is shown in Fig. 5(b). In contrast to the previous figure, the stresses at triple point H are not concentrated in the presence of GBS. Rather, the GBS facilitates the redistribution of stresses from triple point H to other transverse boundaries. However, it should be noted that as a result of the cavitation, facets RS and KB are stressed to a higher level compared to other transverse facets. Thus, further cavity nucleation would be favored at these boundaries which, if facilitated, would substantially advance the creep rupture process. Figure 6(a) shows the fringe pattern of normalized axial stress in the dual phase equiaxed model (DX-1) in the absence of CBS. The stress pattern in this figure is very similar to that in Fig. 5(a) except that the presence of the Q~grain at the center of the unit cell leads to a stress concentration adjacent to triple point K. It is apparent that this stress concentration could aid in nucleating new cavities at grain boundary KB. The distribution of a,/C for the DX-1 model in the presence of GBS is illustrated in Fig. 6(b). The stress pattern is similar to that in Fig. 5(b) except that the regions along facets CD, KB and IA are found to be under a greater stress. These enhanced stresses are due to the load shedding of the central c(?grain to the adjacent y grains. Figure 7(a) shows the normalized axial stress distribution in the dual phase equiaxed model (DX-2) in the absence of GBS. It is observed here that the different location of c(2grains, compared to that in the DX-1 model, does not alter the character of the stress distribution other than to shift the stress concentration from triple point K, observed in Fig. 6(a), to I in the DX-2 model. Thus, a similar load shedding behavior of the CI?grains is observed here also. The normalized axial stress distribution in the
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DX-2 model in the presence of GBS is illustrated in Fig. 7(b). The stress pattern is also similar here compared to that for the DX-1 model as shown in Fig. 6(b). However, we note that the stresses are greater along KB and lower along IA for DX-2 compared to those for DX-1. Hence, it appears that GBS redistributes the stresses in a manner that reverses the trend noted in the comparison of Figs 6(a) and 7(a). It is also observed in Fig. 7(b) that the level of stress in grain 2 is higher than that for grain 2 in the DX-I model. This difference is due to the load shedding from grain 5 in the DX-2 model which is made up of u2 phase. Figure 8(a) shows the normalized axial stress distribution in the fully lamellar model (FL-l) in the absence of GBS. It is observed that a2 laths that are more closely aligned with the applied stress direction redistribute more stress to the adjacent y laths. This results in inhomogeneous stress distributions along the grain boundaries RS and PQ. However, the stress level discontinuities are more extreme at boundary RS than those at PQ. This is apparently due to the more isostress orientation of the lamellae in grain 5 as well as the proximity of RS to the cavitating facet. The distribution of the normalized axial stress in the FL-l model in the presence of GBS is illustrated in Fig. 8(b). It is evident from this figure that the transverse grain boundaries RS and KB are heavily stressed regions in the microstructure. Cavitation at grain boundary OH again redistributes stress from grain 1 to adjacent grains. However, a high level of stress at grain boundary RS ensues as a result of the load shedding of the a2 laths at this grain boundary and owing to the sliding of the inclined grain boundaries. It is interesting to note that the transverse grain boundary KB is stressed to a higher level than either boundary GJ or PQ, although these boundaries have lath orientations that are less characteristic of an isostress arrangement. It appears that the higher stress at boundary KB is due to the fact that it lies in the path of the highest stresses which arise from the growth of the cavities at grain boundary OH. This characteristic of the stress field is also evident in Figs S(b), 6(b) and 7(b). The normalized axial stress distribution in the FL-2 model is shown in Fig. 9(a). Here, the presence of isostress lamellae immediately ahead of the cavitating facet (in grain 5) facilitates the formation of a more uniform stress field in this region compared to those in the corresponding region of the FL- 1 model [grain 3 in Fig. 8(a)]. Figure 9(b) illustrates the normalized axial stress distribution in the FL-2 model in the presence of GBS. It is observed that the grain boundaries PQ and KB are in regions of high stress concentration due to grain boundary sliding coupled with creep constrained cavitation. These two boundaries are along the direction of the maximum stress, as are
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boundaries RS and KB for the FL-l model, that results from the cavitation. The development of creep cavitation damage w = a/b as a function of normalized time t/tE is illustrated in Fig. 10, for the present models. It is evident from this figure that grain boundary sliding substantially reduces the cavity coalescence time for all the microstructures considered. In the absence of sliding, the single phase model exhibits the strongest resistance against creep cavitation. In the presence of grain boundary sliding, the dual phase equiaxed model, DX-2, is the strongest in terms of resistance to creep cavity growth while DX-1 exhibits the weakest behavior. This implies that the cavity growth will be slower if cavitation occurs at the y/y grain boundary adjacent to the tl2 grain compared to a nearby y/y grain boundary that is in an inclined direction from the c(~grain. We note that the overall stresses above the cavitating facet in Fig. 7(b) (DX-2) are higher than those in the same region in Fig. 6(b) (DX-1). As a result, less stress is distributed to facets IA and CD of the DX-2 model. Apparently, this gives rise to more constraint on the cavity growth in DX-2 as compared to that for DX-1 as indicated by the difference in cavity growth rates in Fig. 10. The two fully lamellar models FL-l and FL-2 yielded identical results for cavitation in the presence of GBS. This similarity indicates that the effect of lamellae orientation on creep cavitation is not as pronounced in the presence of grain boundary sliding. In the absence of GBS, the FL-2 model is found to be stronger than the FL-l model. The important point here is that a difference in cavity coalescence time for the two FL models is only evident when GBS is suppressed. This is likely the case for the FL microstructure which normally exhibits serrated grain boundaries. The present results indicate that cavities at a grain boundary with intersecting lath orientations which are isostress in
_..
0
4
8
12
16
20
2A
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% Fig. 10. Plot of creep damage at facet # 1 of Single Phase (SP), Dual Phase Equiaxed (DX) and Fully Lamellar (FL) models vs normalized time.
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Fig
5. Normalized axial stress (u:/Z) distribution in the single phase y TiAl during creep cavitation central grain boundary OH (a) the absence and (b) in the presence of GBS.
Fig
6. Normalized axial stress (a:/Z) distribution in the dual phase co + y TiAl (DX-I) during creep cavitation at central grain boundary OH (a) without GBS and (b) in the presence of GBS.
Fig
7. Normalized axial stress (al/Z) distribution in the dual phase w + ;’ TiAl (DX-2) during creep cavitation at central grain boundary OH (a) without GBS and (b) in the presence of GBS.
at
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Fig. 8. Normalized axial stress (uz/X) distribution in the fully lamellar TiAl (FL-l) during creep cavitation at central grain boundary OH (a) without GBS and (b) in the presence of GBS. It should be noted tfiat the fringe levels in Fig. 8(a) are different from those in previous figures.
Fig. 9. Normalized axial stress (uz/X) distribution in the fully lamellar TiAl (FL-2) during creep cavitation at central grain boundary EF (a) without GBS and (b) in the presence of GBS. It should be noted that the fringe levels in Fig. 9(a) are different from those in previous figures.
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nature will tend to grow slower compared to the case where the intersecting lath orientations are closer to an isostrain orientation. Lath orientations that are nearly parallel to the applied stress direction in the zone adjacent to the cavitating grain boundary result in load shedding from a2 laths to y laths which then produces less constraint on the cavitation process. 4. CONCLUSIONS
The present model results indicate that the single phase TiAl microstructure exhibits maximum strength against grain boundary cavitation in the absence of grain boundary sliding. In the presence of grain boundary sliding, the dual phase equiaxed microstructure exhibits the greatest resistance against cavitation when cavity growth occurs at the r/r grain boundary adjacent to the CQgrain. Conversely, this microstructure is the most susceptible to cavitation damage when cavitation occurs at an inclined angle to a neighboring a2 grain. The effect of neighboring lath orientation on grain boundary cavitation for the fully lamellar microstructure is negligible in the presence of grain boundary sliding. However, in the absence of grain boundary sliding an isostress orientation of the laths in the grains adjacent to the cavitating grain boundary delays the cavity growth process. Based on the present results, the relatively long rupture times observed experimentally for fully lamellar TiAl may be primarily attributed to the suppression of grain boundary sliding in this microstructure. Acknorvledgements-Support for this work from the Air Force Office of Scientific Research under grant No. F49620-94-0 137 is gratefully acknowledged. The authors would like to thank the Lawrence Livermore National Laboratory for providing the FE codes used in the present work through their software collaborators program. REFERENCES Kim, Y. W., JOM, 1994, 46, 30. Mitao, S., Tsuyama, S. and Minakawa, K., in Microstructure/Property Relationships in Titanium Aluminides and Alloys, ed. Y. W. Kim and R. R. Boyer. The Minerals, Metals and Materials Society, Warrendale, 1991, p. 297. Shih, D. S., Huang, S. C., Starr, G. K., Jang, H. and Chestnut, J. C., in Microstructure/Property -Relationshins in Titanium Aluminides and Allovs. , ed. Y. W. Kim and R. R. Boyer. The Minerals, Metals and Materials Society, Warrendale, 1991, p. 135. Tonnes, C., Rosier, J., Baumann, R. and Thumann, M., in Structural Intermetallics, ed. R. Darolia et al. The Minerals, Metals and Materials Society, Warrendale, 1993, p. 241. Hayes, R. W. and McQuay, P. A., Scripta metall. mater., 1994, 30, 259. Reidel, H., Fracture at High Temperatures. SpringerVerlaa, New York, 1986. Hull,-D. and Rimmer, D. E., Phil. mag. A, 1959, 4, 613. Cocks, A. C. F. and Ashby, M. F., Prog. Mater. Sci., 1982, 27, 244.
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J. C., Eggeler, G. and Ilschner, 9. Nix, W. D., Earthman, B., Acta metall., 1989, 37, 1067. F., Int. J. Solids Struct., 1980, 16, 847. 10. Ghahremani, 11. Hsia, K. J., Parks, D. M. and Argon, A. S., Mech. mater., 1991, 11, 43. 12. Crossman, F. W. and Ashby, M. F., Acta metall.. 1975, 23, 425. P. M. and Rice, J. R., Acta metall., 1985, 13. Anderson, 33, 409. 14. Rodin, G. J. and Dib, M. W., in Advances in Fracture Research, Vol. 2, ed. K. Salama et al. Pergamon Press, Oxford, 1989, p. 1835. A. and Earthman, J. C., Metal/. Mater. 15. Chakraborty, Trans. A, 1997. 28A, 919. 16. Beere, W. and Speight, M. V., Metal. Sci., 1978, 12, 172. A. and Rice, J. R., Acta metall., 1980, 17. Needleman, 28, 1315. 18. Edward, G. H. and Ashby, M. F., Acta metall., 1979, 27, 1505. 19. Chuang, T., Kagawa, K. I., Rice, J. R. and Sills, L. B., Acta metall., 1979, 27, 265. 20. Dyson, B. F.. Metal Sri., 1976, 10, 349. 21. Rice, J. R., Acta metall., 1981, 29, 675. 22. Tvergaard, V., J. Mech. Phys. of Solids, 1984, 32, 373. 23. Tvergaard, V., J. Mech. Phys. of Solids, 1985, 33, 447. 24. Tvergaard, V., Int. J. Solids Struct., 1985, 21, 279. 25. van der Giessen, E. and Tvergaard, V., Int. J. Fracture, 1991,48, 153. 26. van der Giessen. E. and Tvergaard, V., Acta metall., 1994, 42, 959. 27. van der Giessen, E. and Tvergaard, V., Mech. mater., 1994, 17, 47. 28. Sham, T.-L. and Needleman, A., Acta metall., 1983, 31, 919. 29. Budiansky, B., Hutchinson, J. W. and Slutsky, S., in Mechanics of Solids, The Rodney Hill 60th Anniversary Volume, ed. H. G. Hopkins and M. J. Sewell. Pergamon Press, Oxford, 1982. p. 13. 30. Engelmann, B. and Hallquist, J. O., NIKE2D User Livermore National Laboratory, Manual. Lawrence Livermore, California, April 1991. 31. Hallquist, J. O., Goudreau, G. L. and Benson, D. J., Comp. Meth. Appl. Mech. Engng, 1985, 51, 107. 32. Hug, G., Loiseau, A. and Veyssiere, P., Phil. Mag. A, 1988, 57, 499. 33. Kim, H. K., Wolfenstine, J. and Earthman, J. C., Scripta metall. mater., 1992, 27, 1067. 34. Hayes. R. W. and Martin, P. L., Acta metall., 1995, 43, 2761. 35. Schafrik, R. E., Metall. Mater. Trans. A, 1977. 8, 1003. 36. Bartholomeusz, M. F., Yang, Q. and Wert, J. A., Scripta Metall. Mater., 1993, 29, 389. 37. Chu, W.-Y. and Thompson, A. W., Metall. Mater. Trans. A, 1992, 23, 1299. 38. Soboyejo. W. 0. and Ledrich, R. J., in Structural Intermetallics, ed. R. Darolia et al. The Minerals, Metals and Materials Society, Warrendale, 1993, p. 353. 39. Mehrer, H., Mater. Trans., JIM, 1996, 31, 1259.
APPENDIX Nomenclature
cavity creep grain cavity
radius stress coefficient area half spacing
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At U u V ;
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Burgers vector grain boundary diffusivity grain boundary diffusion parameter Young’s modulus stress enhancement factor Boltzmann’s constant characteristic of grain boundary diffusion number of grains in the y direction in the unit cell number of grains in the z direction in the unit cell creep stress exponent initial radius of grain boundary facet reference time for creep strains to equal elastic strains time increment displacement in the y direction displacement rate in the y direction displacement in the z direction displacement rate in the z direction
cavity volume grain boundary free energy surface free energy rate of grain boundary separation grain boundary thickness effective creep strain rate true strain rate in the grain boundary plane area fraction of cavitating grain boundary shear modulus density of cavitating facets in the unit cell von Mises stress mean stress average normal stress at grain boundary sintering stress axial stress externally applied stress externally applied von Mises stress cavity tip angle cavitation damage parameter atomic volume