Finite-element modelling of anisotropic single-crystal superalloy creep deformation based on dislocation densities of individual slip systems

Transcription

1 Johannes Preußner, Yegor Rudnik, Rainer Völkl, Uwe Glatzel Lehrstuhl Metallische Werkstoffe, Universität Bayreuth, Ludwig-Thoma-Straße 36b, Bayreuth, Germany Finite-element modelling of anisotropic single-crystal superalloy creep deformation based on dislocation densities of individual slip systems Dedicated to Professor Wolfgang Blum on the occasion of his 65th birthday A finite-element model is proposed which describes the creep behaviour of a two-phase single-crystal superalloy. For the matrix phase, a user-defined material model is used. It describes the creep behaviour based on evolution equations for dislocation densities on individual slip systems. An interaction matrix determines the influence of one glide system on the other. Due to the face-centred cubic (fcc) symmetry, 9 independent parameters of the interaction matrix can be distinguished, describing slip on octahedral and cube glide planes with a Burgers vector of the type a/2<110>. The finite-element method (FEM) was used to describe the stress strain behaviour as well as the dislocation evolution in matrix channels during creep deformation. Uniaxial stress in <100>, <110>, and <111> directions was considered. The evolution of the calculated dislocation densities and the resulting creep curve are compared with experimental observations. Keywords: Superalloy; Finite-element method; Dislocations; Creep; Anisotropic behaviour 1. Introduction Single-crystal nickel-base superalloys with high volume fraction of c 0 phase are materials with high performance at high temperatures. Turbine blades have to bear high centrifugal forces in hot gas streams. The excellent creep behaviour of Ni-base superalloys in [001] crystal direction makes them suitable for such highly strained components. Modern single-crystal superalloys such as CMSX-4 are able to resist creep stresses of 250 MPa at a temperature of 1255 K for up to 680 h [1]. This high performance can only be achieved by a gradually conducted solution heat treatment followed by a multi-step ageing process. The precipitation heat treatment yields L1 2 ordered c 0 cuboids with a mean edge length of about 500 nm, surrounded by narrow channels of the face-centred cubic (fcc) c matrix. The coherent c/c 0 interfaces are of {100} type. The hardening c 0 phase has several positive effects on the material properties, especially with high c 0 volume fractions of about 70 %. Due to the ordered L1 2 structure, the shear processes of the c 0 phase are much more complex, thus the resistance against shearing of the c 0 phase is higher than for the matrix. Since more energy is needed for dislocations to cut the c 0 precipitates, the deformation begins in the less resistant matrix phase, where they have to sharply bend due to the restrictions of the nearby c 0 phase. Another effect is the build-up of high internal stresses in the matrix phase caused by slightly different lattice parameters. The analysis of the c/c 0 microstructure with FEM has been initiated by Pollock and Argon [2], and Glatzel and Feller Kniepmeier [3]. There has been remarkable progress to model the microscopic and macroscopic high-temperature behaviour of these materials. An overview of the numerical analysis of single-crystal superalloys is given in [4]. For a review on the understanding of creep based on dislocation motion see [5]. The task of this work is to observe the stress strain-behaviour of the microstructure of a single-crystal superalloy and the dislocation formation during creep. A finite-element method is used to represent the c/c 0 microstructure and to introduce parameters like mechanical properties of both phases, c 0 volume fraction, coherency stresses, and external loads. A numeric creep model is implemented in the FEM program ANSYS [6]. It describes the creep behaviour through evolution equations for dislocation densities on individual slip systems. Octahedral and cube glide systems with Burgers vectors of the type a/2<110> are used. The creep behaviour and the hardening of the system are dependent on dislocation density and dislocation movement. An interaction matrix determines the influence of one glide system on the other. 12 octahedral and 6 cube glide systems are considered. This leads to an interaction matrix. Due to fcc symmetry, nine independent parameters of the interaction matrix can be distinguished. 2. Model description 2.1 Finite-element model Three finite-element models (FE models) were realised with ANSYS representing the three crystal directions <100>, <110>, and <111>. The first model was realised in [001] load direction. 1/8 of a c 0 cube with surrounding matrix phase on three sides has been selected as representative for the whole structure (see Fig. 1). The entire microstructure can be obtained by mirroring the faces of the chosen cubic section [7, 8]. A unit cell in context of this work is de- Z. Metallkd. 96 (2005) 6 Carl Hanser Verlag, München 595

2 Fig. 1. FE-model simulating creep in [001] load direction. The matrix channels can be divided into one horizontal and two vertical channels that are symmetric to the load direction. The figure shows the c 0 cube in the microstructure with the representative cut-out (left) and the realised FE model (right). fined as a c 0 cube with the surrounding matrix channels (half the thickness of the c 0 separation spacing). In the case of [001] load direction, the unit cell contains eight representative FE models. Horizontal and vertical matrix channels, which are perpendicular and parallel to the load direction, respectively, can be distinguished. The <110> model represents 1/4 of the unit cell. It includes the gable channel, which is parallel to the load direction and the two roof channels, one of them completely, and the second partly (see Fig. 2). The roof channels are inclined by 45 to the load direction. The load axis is chosen along [101]. Every face of the cube cut-out is a mirror plane. The <111> model is more complex. The polyhedron consists of six <110> side surfaces and two <111> bases. The load axis is realised in [111] direction. In the middle of the structure one c 0 cube is situated. The FE-model comprises three unit cells (Fig. 3). The matrix channels are aligned symmetrically to the load axis. The boundary conditions are chosen such that each surface remains flat during the creep calculation, except the base surfaces of the <111> model. These planes are not real mirror planes, but the microstructure can be obtained by putting the polyhedra on top of each other. Thus, in the model these surface planes need to have the same surface distortion. Due to cubic crystal symmetry, the anisotropic elastic properties can be specified by three independent variables, Fig. 2. FE model simulating creep in [101] load direction. The surfaces of the FE structure need to be planar as these are the mirror planes in the microstructure with reference to the load direction. Fig. 3. Cut-out of the microstructure to simulate creep in <111> load direction. The figure shows the microstructure with the chosen cut-out (upper left), the truncated c 0 cubes of the chosen section with one entire cube in the middle (lower) and the realised finite-element model with six planar surface planes (mirror planes) and two planes with equal boundary conditions (upper right). for example, Young s modulus E, shear modulus G, and Poisson s ratio m, all of them in <001> direction. The internal misfit stresses were introduced by using different thermal expansion coefficients for c and c 0 and virtually heating the structure such that the misfit between the two phases is simulated. This procedure has been described earlier [7 9] Material model Two different material laws were used for the two phases. The c 0 phase can deform elastically until the von Mises stress exceeds the yield stress of 750 MPa. For higher stresses, the material deforms ideally plastic. The material model used here is called bilinear isotropic hardening in [6]. For the matrix phase, a user-defined material model is applied. It considers dislocation densities on individual slip systems and is described in detail in [10, 11]. The proposed model is based on the work of Alexander and Haasen [12]. Despite its phenomenological aspect, the model of Alexander and Haasen has been widely accepted and has been followed up in numerous works [13, 14]. Starting from the Orowan equation, the plastic shear strain increments can be calculated: _eðtþ ¼ qðtþ j~bj vðtþ ð1þ where j~bj is the absolute value of a Burgers vector of type a/2 <110>, q(t) is the time-dependent dislocation density, and v(t) is the dislocation velocity. 596 Z. Metallkd. 96 (2005) 6

3 Table 1. Classification of the interaction parameters. Coefficient Plane plane relation Plane type Burgers vector relation When considering individual slip systems, the effective stress acting on the particular system i needs to be calculated: s i ¼j^~b~b i r $ ^~n~n i j c 1 Same Octahedral- octahedral Same c 2 Same Octahedral- octahedral Different c 3 Different Octahedral- octahedral Same c 4 Different Octahedral- octahedral Different c 5 Same Cube- cube Same c 6 Same Cube- cube Different c 7 Different Cube- cube Different c 8 Different Octahedral- cube Same c 9 Different Octahedral-cube Different ð¼ m s r for uniaxial stress stateþ ð2þ where ^~b~b j is the direction of the normalised Burgers vector, r $ the stress tensor, ^~n~n j the normalised vector of the glide plane normal, and m s the Schmid factor (sf). Starting with an initial dislocation density, the dislocation velocity, v i, for each system can be determined:! X 18 pffiffiffiffiffiffiffiffiffi v i ¼ v 0 s i k 3 c ik q k ð3þ k¼1 v 0 is a constant factor which affects the value of the dislocation velocity at a given stress state and dislocation density, k 3 is a constant parameter, q k is the dislocation density, and c ik are the interaction constants between the slip systems. The evolution of the dislocation density, _q i, is given by: vffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi ux 18 X _q i ¼ v i k 2 q t 18 i q k s i k 1 k¼1 k¼1! pffiffiffiffiffiffiffiffiffi c ik q k ð4þ with the constant parameter k 2. k 1 equals agb, with the dislocation interaction constant a ranging between 0.1 and 1, the shear modulus G and the length of the Burgers vector b. Equations (3) and (4) contain scalar parameters. This is a slightly modified form of Eq. (10) in [11]. The model of Alexander and Haasen [12] is not valid for large creep times, because the dislocation velocity converges to zero. In reality, a constant, finite creep rate is observed. To describe this behaviour, the value of k 3 is less than the value of k 1 with the result that the dislocations are still moving even with a zero multiplication rate. In the fcc crystal structure, 18 slip systems (12 octahedral and 6 cube systems) can be distinguished. The components of the 1818 interaction matrix determine how strong the dislocation density in system j slows down the multiplication rate of the dislocation density in system i. A big value of the constant c ik signifies a strong interaction of both slip systems and thus a strong decrease in the multiplication of dislocation j caused by dislocation i. Due to crystal symmetry, the 1818 interaction matrix can be reduced to nine independent parameters, see Table 1. The macroscopic creep strain can be calculated by adding up the shear-strain increments of all glide systems, see [10]. Solving the system of 18 coupled differential equations allows us to identify the dislocation density and velocity for each glide system at every moment in a given volume by using FEM. 3. Simulation of creep tests A creep test on CMSX-4 with external stress of 500 MPa at a temperature of 1123 K was simulated with mechanical properties taken from [15]. The volume fraction of c 0 was 70 % with a misfit d of 0.2 %. The c 0 phase has been approximated with the material properties from [15] and a yield stress of 750 MPa. The parameters of the material model were taken from [10], whereas k 2 does not correspond exactly to k 2 in [10] and has been chosen 10m/N in this work. The nine independent parameters of the interaction matrix were chosen as shown in Table 2. Strong interaction between glide planes of the same type is assumed (c 1 and c 2 ). The interaction between glide planes with the same Burgers vector (c 3 and c 8 ) is presumed higher than the interaction between two systems with the same slip plane (c 2 and c 6 ). No interaction is assumed between two slip systems with different Burgers vectors and different slip planes (c 4, c 7 and c 9 ). Table 2. Interaction parameters applied to matrix phase for calculating creep. The classification of the parameters is given in Table 1. Parameter c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 c 9 Value The 18 slip systems for the fcc matrix phase are taken from Table 3. To compare the creep behaviour in different load directions the three orientations [001], [101], and [111], representing the corner orientations of the standard orientation triangle, have been chosen. The calculations were performed with ANSYS including a FORTRAN user-defined creep model for the matrix phase. The ANSYS x-direction equals [100], y = [010] and z = [001]. The creep tests were simulated in several time steps. The stress state is calculated by FEM, hence the dislocation velocity and density are determined for each element and, thus, the creep strain is computed which determines the stresses and elastic strains for the next time step, called explicit creep method [6]. Z. Metallkd. 96 (2005) 6 597

4 Table 3. Individual fcc slip systems with respective Schmid factors for the simulated load directions. No. b ~ ~n Schmid factor for [001] load direction Schmid factor for [101] load direction Schmid factor for [111] load direction octahedral 1 [110] (111) [101] (111) [011] (111) [110] (111) [101] (111) [011] (111) [110] (111) [101] (111) [011] (111) [110] (111) [101] (111) [011] (111) cube 13 [110] (001) [110] (001) [101] (010) [101] (010) [011] (100) In the following graphs, the time equals the actual simulated time steps. For comparison with real creep tests these values need to be scaled up. The parameters concerning the matrix, plotted in the graphs below, give the results for the elements in the middle of the respective matrix channels [001] creep 18 [011] (100) oct high sf 4 oct high sf 4 cube high sf The stresses in the horizontal and vertical channels during creep deformation are plotted in Fig. 4. In the horizontal channel (Fig. 4a), the von Mises stress is at a very high level at the beginning (800 MPa) but decreases to a value of approx. 100 MPa, because the stresses perpendicular to the load axis increase during creep. Thus, in spite of high stresses in load direction the shear stresses on the free dislocations in the matrix are decreasing. In the vertical channel (Fig. 4b) parallel to the (100) interface, the von Mises stresses are also decreasing as the stress in load [001] direction is transferred to the c 0 phase and the compressive stress in [010] direction decreases. The [100] compressive stress decreases for vertical channels parallel to the (010) channel, respectively. The stress progression affects the evolution of the dislocation density and the dislocation velocity. As the stresses are high in the horizontal channel at the beginning of the creep test, the dislocation densities increase quickly for each glide system with high Schmid factor with respect to [001] (Fig. 5a). These are the eight octahedral glide systems with Schmid factor 0.41 in a uniaxial stress state. The four slip systems with low Schmid factor do not multiply. Some cube glide systems multiply up to a density of about m 2. In the vertical channel, the stress state is more complex. The glide systems with Schmid factor zero with 3 cube high sf 6 oct medium sf Fig. 4. (a) Stresses in horizontal channel: [001] is the direction of external stress (z-direction). The stresses in [100] (XX) and [010] (YY) direction are equal. All shear stresses equal zero. (b) Stresses in vertical channel: all shear stresses equal zero. Although the stress state is much higher in the horizontal channel, the von Mises stresses in both channels have the same magnitude for large times. The abscissa represent the actual calculated time steps. 598 Z. Metallkd. 96 (2005) 6

5 Fig. 5. Dislocation evolution in (a) horizontal and (b) vertical channel. While every glide system with high Schmid factor reaches a high dislocation density in the horizontal channel, only every other glide system with high Schmid factor reaches a high density in the vertical channel. respect to [001] multiply clearly, see Fig. 5b. Highest densities are reached for the four octahedral systems with ~b ¼ ½011Š and ~b ¼ ½011Š. Systems with ~b ¼ ½101Š and ~b ¼ ½101Š reach a density of about m 2 in (010) vertical channels. Comparing the local creep rate for both channels (see Fig. 6a), it can be recognised that after reaching the maximum creep rate the horizontal channel creeps slower than the vertical channel. The von Mises stress decreases drastically and the generated dislocations build up high back stresses. Both effects slow down the dislocation velocity and, hence, the local creep rate [101] creep Considering creep in [101] direction, two roof and one gable channel can be distinguished. The stress progressions are shown in Fig. 7. The high equivalent stress in the roof channel is remarkable. The high r xy shear stresses and the big differences between the tensile stresses are responsible for the high von Mises stress and, thus, for high shear stresses on several dislocation systems. The gable channel holds a completely different stress state, where the misfit reduces the influence of the external stress. The effect on the dislocation densities is illustrated in Fig. 8. In the (001) roof channel, the slip systems No. 3 and 9 dominate the deformation process. The other two slip systems with high Schmid factor have the highest dislocation densities in the (100) roof channel, respectively. As the stress state changes during creep deformation (e. g., in [010] direction the compressive stresses change to tensile stresses, see Fig. 7a), specific slip systems multiply at larger times (see systems 4 and 10 in Fig. 8a). At the (010) gable channel, the dislocation network is more complex. Both octahedral and cube systems with high Schmid factor dominate the deformation process there. The overall dislocation density in the roof channel is about three times higher than in the gable channel. As the effective stress on the dislocations is big even for high deformations, the deformation rate of the roof channel is much higher than in the gable channel (Fig. 6b). The plastic deformation of the c 0 phase is lowest in <110> load direction. The c 0 deformation begins only at approx time units. Fig. 6. Local creep rates for the characteristic channels over time compared with the plastic deformation rates of the c 0 phase. The <100> model demonstrates the highest hardening effect; especially in the horizontal channel, the diminishing creep rate is remarkable (a). In <110>-direction, the roof channel dominates the creep deformation. The plastic deformation of c 0 begins at much larger times (b). In the <111> model, the three symmetric channels all hold a high creep rate (c). Except for <100> load direction, the plastic deformation of the c 0 cubes is negligible. Z. Metallkd. 96 (2005) 6 599

6 Fig. 7. Stress progress in (a) the (001) roof and (b) gable channel: The von Mises stresses in the roof channel do not decline. In the gable channel, the misfit stresses reduce the external stresses in [101] direction and the von Mises stress reaches a very low value. Fig. 8. Dislocation density evolution for (a) the roof channel and (b) the gable channel. The legends give the glide system number. While at the other diagrams symmetric glide systems are merged to one line, some glide systems in diagram (b) are combined to one line for clarity reasons. Fig. 9. (a) Stress evolution in the (010) channel at [111] load direction (a). The von Mises stress remains at a high level during creep deformation. (b) Dislocation evolution in (010) channel. One cube glide system reaches the highest dislocation density [111] creep The three matrix channels parallel to (100), (010), and (001) are all symmetric to the ½111Š load axis. The stress progression for the (010) channel is shown in Fig. 9a. At the (010) channel, one cube glide system (No. 15) reaches the highest dislocation density (Fig. 9b). At the (100) and (001) channels, one of the two other cube glide systems with high Schmid factor gains the highest density. Regarding only one channel, not every glide system with high Schmid factor multiplies clearly. But considering all channel the systems with high Schmid factors reach the highest dislocation densities. As both the dislocation density of several slip systems and the resolved shear stress on the systems are high, the matrix channels hold a high creep rate, see Fig. 6c. The plastic deformation of the c 0 cube is negligible because it decreases during creep deformation Creep curves The overall creep results can be obtained by analysing the displacements of the base surfaces of the FEM models for the three crystal directions. The results are displayed in Fig. 10. An influence of several parameters on the creep curves has been observed: The c 0 yield stress has the strongest impact on the <100> direction. As calculated in further computations, the steady-state creep rate rises with decreasing c 0 yield stress especially for the <100> direction. If the external stress exceeds the factor of (c 0 yield stress vol- 600 Z. Metallkd. 96 (2005) 6

7 Fig. 10. Creep curves for the three orientations. The large hardening rate of the <100> orientation is striking. The <111> orientation creeps fastest. ume fraction c 0 ), the steady-state creep rate of the <100>-direction raises drastically, called overloading case in Reference [9]. Calculations with a c 0 volume fraction of 60 %, for example, result in steady-state creep rates of the same order of magnitude for the three crystal directions. The interaction parameters incorporating cube glide systems have a high influence on the <110> and <111> load directions. The stress progression is relatively independent of the interaction coefficients and is alike the results obtained by additional calculations with the Norton creep law. Therefore, the c 0 morphology has dominating influence on the anisotropy of the creep behaviour. The influence of crystallography and individual slip systems is negligible in the case of high c 0 volume fractions. 4. Discussion and outlook Extensive observations on the creep behaviour of the superalloy CMSX-4 have been made by Saß [16], who recorded creep curves and analysed the dislocation distribution by transmission electron microscopy (TEM) within particular channels. Considering the work of Link et al. [17], who observed dislocations with <100> Burgers vectors in c 0 and assumed that those are often misinterpreted as <110>, the dislocation evolution is in full agreement with this work. Only the very low primary creep rate of the <100> creep direction cannot be confirmed in the simulation yet. To save calculation time, the multiplication of the dislocations has been increased. This could have an effect on the hardening progression, especially on the maximum creep rate. In [001] loading, c 0 deformation is absolutely necessary to achieve a total creep strain of more than 0.5 %. This is due to the build, up of high stress levels in this orientation. To obtain better results and investigate further influences, the model can be extended. Dislocation annihilation can be included in the material model. Not only the matrix phase, but also the deformation of c 0 should be simulated with the presented model. The influence of the geometry of the c 0 precipitates can also be observed. In summary it can be ascertained: In [001] load case, we do need deformation of the c 0 phase (e. g., cutting by dislocations) to obtain overall strains higher than 0.5 %. Coherency stresses in the vertical channels and creep-induced tension stresses in the horizontal channel perpendicular to the load axis lead to low creep rates. The c 0 phase morphology (cuboidal particles with high volume fraction) determines the anisotropic creep behaviour. The influence of crystallographic slip on the anisotropy is negligible in the case of the here calculated two-phase material, but plays an important role in single-phase materials [11]. References [1] U. Glatzel: Microstructure and Internal Strains of Undeformed and Creep Deformed Samples of a Nickel-Base-Superalloy, Köster Verlag, Berlin (1994), ISBN [2] T. Pollock, A. Argon, in: S. Reichman, D. Duhl, G. Maurer, S. Antolovich, C. Lund (Eds.), Superalloys 1988, The Metallurgical Society (1988) 285. [3] U. Glatzel, M. Feller-Kniepmeier: Scripta Metall. 23 (1989) [4] E. Kirchner: Int. J. Plasticity 17 (2001) 907. [5] W. Blum, P. Eisenlohr, F. Breutinger: Metall. Mater. Trans. A 33 (2002) 291. [6] ANSYS Procedures Manual, Swanson Analysis Systems, Inc. (1994). [7] M. Probst-Hein, A. Dlouhy, G. Eggeler: Acta Mater. 47 (1999) [8] R. Völkl, U. Glatzel, M. Feller-Kniepmeier: Acta Mater. 46 (1998) [9] L. Müller, M. Feller-Kniepmeier: Scripta Metall. Mat. 29 (1993) 81. [10] H. Brehm: Modellierung des orientierungsabhängigen Kriechverhaltens einkristalliner Strukturen, Shaker Verlag, Aachen (2000) (ISBN ). [11] H. Brehm, U. Glatzel: Int. J. Plasticity 15 (1998) 285. [12] H. Alexander, P. Haasen: Solid State Physics 22 (1968) 27. [13] W. Blum, in: H. Mughrabi (Ed.), Materials Science and Technology, Vol.6, VHC Verlag, Weinheim (1993). [14] D. Caillard, J.L. Martin: Thermally Activated Mechanisms in Crystal Plasticity, Pergamon Materials Series, Elsevier Science Ltd, Cambridge, UK (2003). [15] D. Siebörger, H. Knake, U. Glatzel: Mater. Sci. Eng. A 298 (2001) 26. [16] V. Saß: Untersuchung der Anisotropie im Kriechverhalten der einkristallinen Nickelbasis-Superlegierung CMSX-4, Köster Verlag, Berlin (1997) (ISBN ). [17] T. Link, C. Knobloch, U.Glatzel: Scripta Mater. 40 (1999) 85. (Received December 17, 2005; accepted March 17, 2005) Correspondence address Johannes Preußner Universität Bayreuth Lehrstuhl Metallische Werkstoffe Ludwig-Thoma-Str. 36b, D Bayreuth Tel.: Fax: johannes.preussner@stud.uni-bayreuth.de Z. Metallkd. 96 (2005) 6 601