Accessing the elastic properties of cubic materials with diffraction

Transcription

1 Accessing the elastic properties of cubic materials with diffraction methods Thomas Gniiupel-HeroldlJ, Paul C. Brand and Henry J. Prask National Institute of Standards and Technology, Center for Neutron Research, Gaithersburg, MD 20899, U. S. A., 2University of Maryland, Department of Materials and Nuclear Engineering, College Park, MD , U S. A. INTRODUCTION Diffraction based methods for residual stress analysis probe the strain of a body rather than its stress state. Thus, the conversion into stress requires elastic constants of various kinds which themselves depend on the anisotropy of the crystal lattice over the plane hkl being probed, and the macroscopic state of anisotropy of the specimen. The latter point refers to the state of texture as well as effects arising from the size and the shape of the grains whose entirety constitutes the specimen. These diffraction elastic constants (DEC) can be calculated from the single crystal elastic constants (SCEC), which represent the most general description of the elastic properties of a material [ 1,2,3]. They can be expressed as fourth rank tensors either of the compliances cijkl or stiffnesses sjjkl. The importance of their knowledge originates from the fact that the whole variety of polycrystalline elasticity can be described by means of the SCEC. The SCEC can be measured either directly from single crystals or from polycrystals in load experiments in which a series of planes hkl is probed by diffraction [4]. The SCEC can be calculated due to the fixed relationship between the orientation of the grains contributing to the Bragg reflection and the direction of the applied load. In the following, a short outline of the theory and its results will be given. The determination of the SCEC from single crystals is to a large extent the domain of ultrasonic resonance. The limitations of that method reveal themselves on multiphase single crystals like r/r hardened superalloys. Ultrasonic resonance is not phase sensitive so that the measured signal represents an average over the phase constituents. Therefore the phase resolved SCEC cannot be measured directly on the composite material. Diffraction methods, on the other hand, can select each phase separately, thus providing independent data about the strain-load dependence of each phase. The theoretical background of the determination of the SCEC as well as first results from a single crystal superalloy will be discussed CALCULATION OF THE SINGLE CRYSTAL ELASTIC CONSTANTS FROM THE DIFFRACTION ELASTIC CONSTANTS The theory belonging to this subject has been outlined in detail elsewhere [4], so here only a short overview is given. The method can be described as a general inversion of the calculation of the DEC from the SCEC. Although a number of methods solving the calculation of the DEC allow the inversion of their equations under certain circumstances, a closer look at the physical meaning of the DEC shows that a more general approach is needed. The DEC describe the average strain response for a particular plane hkl in a large number of grains in a polycrystal. All these grains have in common that the direction [hkl] is parallel for all grains with the rotation about [hkl] as the one remaining degree of freedom. Due to this fixed

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3 relationship, the DEC still reflect the anisotropic nature of the single crystal. On the other hand, there is also an averaging which originates from the large variety of mechanical boundary conditions of this particular group of grains, each surrounded by other grains of unknown orientation. The effect of this averaging can be demonstrated by means of modulus bodies, which are three-dimensional visualizations of elastic constants and their hkl dependence (Fig. 1). For the single crystal, Young s modulus is calculated by means of Ehk, = 1 Sll,, -2h,, -SlI22 -~ lzi2f Poisson s ratio for the single crystal can be obtained by using r = h2k2 + h212 + k212 (h2 +k2 +12) V hkl = 2 - Ehkl (2(c,,,, + 2cH22 )> (2) (1) FIGURE 1: Dependence of Young s Modulus E (top) and of v (bottom) on the crystal direction hkl for nickel. The numerical values of the SCEC have been taken from [5]. The figures on the left hand side represent the case of a single crystal. On the right hand side, the Krijner model for grain-grain interaction was used as described in [ l-4,6]. Obviously, the hkl dependence of Young s modulus becomes smoother as a consequence of the averaging due to the grain-grain interaction in a polycrystal, but the principal features of anisotropy are still maintained. The general recipe for getting the SCEC from the DEC can therefore be formulated as follows: Calculate a set of DEC using a model as described in [4] and compare them to the measured DEC. Minimize x2 by refining the SCEC as free parameters in a least square loop using eqn. (3). X2=2 $ s, (hkl, cvk,),,, - t s2 (hkl, cv~l )c,c s, (hkl, ciikl ),,, - SI (hkz, cw )co,c * i=l o(+%) 41) 11 + mill (3)

4 s1 and $s2 can be written in terms of the familiar quantities E and v s,(hkl)=+ +s2(hkl)= 17 hkl A successful fit of the SCEC requires the measurement both of Young s modulus E and the ratio E/v where v is Poisson s ratio. Additionally, the limitations and simplifications of the underlying model for the DEC have direct consequences on the accuracy of the SCEC. Any deviation from the conditions implied in the calculation of the DEC such as anisotropy due to texture, causes an increasing inaccuracy for the calculated SCEC. This is demonstrated in Table 1, whose results have been taken from [4]. The value of x,l, for copper is much worse than those for aluminum and steel. Table 1: Results of the refinement of the SCEC from the experimental data in [4] using the various models as described in [4]. The last column of the table contains the values of XLdas a measure of the quality of the refinement. Except for copper, the samples were texture free. The bracketed values are the uncertainties. material model used in refinement WI I [GPal CI 122 [GPal ~1212 PPal xi, Bollerath et al [I] (7.9) 65.3 (7.6) 28.5 (0.8) 0.65 aluminum Reuss-Voigt Average [7,8] (7.9) 65.2 (7.6) 28.3 (0.7) 0.65 hkl de Wit [2] (7.9) 65.3 (7.6) 28.5 (0.8) 0.65 Bollerath et al [l] (13.2) (13.4) 74.8 (3.7) 2.57 copper Reuss-Voigt Average [7,8] (13.5) (13.1) 70.6 (2.9) 2.57 de Wit [2] (13.7) (13.8) 74.2 (3.2) 1.70 Bollerath et al [l] (10.5) (10.5) (4.5) 0.79 ferritic steel Reuss-Voigt Average [7,8] (10.8) (10.2) (3.7) 0.79 de Wit [2] (10.5) (10.3) (3.7) 0.99 As long as there is no texture, the models used in [4] produce almost no difference in the SCEC. Larger differences as well as a decreased quality of the refinement is obtained for copper due to some residual texture. In this case, the anisotropy effect needs to be included in order to get better results. (4) MEASUREMENT OF THE SCEC ON SINGLE CRYSTALS As long as single crystals of sufficient size can be produced ultrasonic methods are favored, especially if the elastic limit of the material is so low that loading of the material produces only very little strain. There are exceptions, though, which result mainly from the averaging effect of ultrasonic waves traveling through a crystal. Inhomogeneities as second phase precipitations change the speed of sound but don t give a separate signal. Diffraction, by its phase selectivity, can be used to detect the response on external load for crystalline constituents. The experimental data are strains obtained from the shifts of the Bragg reflections hkl due to applied load.

5 Additional relations are needed between the directions [hkl], the crystal reference frame and the direction of the external load. A two phase nickelbased superalloy single crystal was used as sample material. The reference frames used in the subsequent calculations are shown in Fig. 2. t 0 ext FIGURE 2: Single Crystal in a four circle geometry with Euler angles o,x and $. The transformations between the laboratory frame, the <D frame and the crystal reference frame are described in detail in [9,10]. The CD frame is fixed to the (I axis and coincides with the laboratory frame if all instrument angles are zero. The crystal reference frame is established by the [ 1001, [OlO] and [OOl] lattice vectors. Examples for load-strain curves in various crystal directions are given in Figure 3. in [Ol I] orientation, sample in [Ol I] orientation, sample in [OOI] orientation, sample 2 t*! in [OOI] orientation sample, n 5k ij 001 in [OOI] orientation, sample 3 c ; O.OOl- * Qe n *Cyc- T$ 002 in [OOI] orientation, sample 3 it* % n n * %; in jool] arier1tr?tian. sample 2 T 2-20 in [OOI] orientation, sample applied load [MPa] FIGURE 3: Load-strain curves for different crystal directions and sample orietations. All results have been obtained by neutron diffraction. See the legend for the reflection measured and the respective sample orientation in which the stress was imposed. Note that the orientations of sample 2 and 3 were close to [OOl] but the difference between them was about 10. The results are different slopes of the stress-strain curves due to different orientation matrices. The error bars do not appear because they are smaller than the symbol size. The hkl dependent strain for a particular reflection hkl is related to the strain tensor as follows

6 h,h, C Ehki = h; + h; + h3 % (5) The superscript C denotes that s,f is determined in the crystal reference frame and the hi are the Miller indices. Now, Hooke s law is written as c cc Eij = ijklc kl (6) The stress tensor in the crystal reference frame can be expressed in terms of the laboratory frame as 0 ij c = Uik up k, 4 (7) The matrix us is the transformation between the crystal reference frame and the Q frame. It becomes the identity matrix if the [OOl] sample direction is parallel to zl& at x=$=0. Using (6) and (7), eqn. (5) now reads hihj E hkl = q + h2 + g s& k3u,3(t t3 It is preferrable to use the slope of the stress strain curves rather than the single stress-strain data pairs. Eqn. (8) becomes d& hkl _ hihj h; + q s;k, k3u13 (9) 84 This result is the basic equation for the least square refinement of the &jkl. Note that Uij is valid for one sample orientation. If different sample orientations are used it has to be recalculated. For example, a [OOl] oriented sample allows only the determination of sill1 and ~1122. At least one differently oriented sample is necessary to obtain also the shear modulus ~1212. In general, a sufficient large deviation from the [OOl] sample orientation enables the computation of all three independent &jr+ APPLICATION TO y/y NICKELBASED SUPERALLOY SINGLE CRYSTALS A rare but important example for two phase single crystals are the y/y superalloy single crystals the microstructure features usually cuboidal to spherical y particles surrounded by a fee y matrix. The basic features of the microstructure can be seen in Fig 4. The y precipitates themselves exhibit an ordered Ll2 structure whose lattice parameter is nonetheless only slightly different from the one of the matrix. This lattice mismatch is of the order of 0.1%. As long as particle sizes of about 600 nm are not exceeded the bonding of the y precipitates and the y lattice is coherent, i.e. each lattice plane in y has its continuation through the interface into the y matrix. The basic property for the separate determination of the SCEC for both phases is the ordered structure of y. Its main consequence is that the usual extinction rules for a fee lattice do not apply. In general, there is a non zero structure factor for every combination of hkl. This way the strain response of the y phase can be obtained independently. The fee lattice of the y matrix prohibits an undisturbed measurement of its reflections. For almost every superalloy the lattice mismatch is so small that the strongly overlapping y/y reflections cannot be separated due to the limited resolution power of most diffractometers. The numerical deconvolution of such overlapped profiles has been done [l l] in the past but is a very challenging computational task. Experimentally a separation can be achieved by diffractometers

7 of very high resolution [12], which, on the other hand, usually means a strong drop in intensity. Because of these reasons and in order to assess the feasibility of the method, a more simplified approach was chosen. FIGURE 4: Micro structure of a r/y single crystal superalloy. The y cube s are aligned along crystallographic [ 1001, [Ol 0] and [OOl] directions. The main constituent of the material investigated here is nickel with about 70%. At this time, more information about the material cannot be disclosed. Based on the availability of the y reflections as well as the combined r+y reflections, we now make the assumption that the SCEC of the composite material are a linear mixture of the SCEC of their constituents. From this point, the following equations can be derived ijk! - s& = a sijkl sqk, = (1-A&)s;, +Aj& (10) The quantity AfYV denotes the volume fraction of y. For the samples investigated here the y volume fraction was estimated from SEM micrographs to 70%. The parameter a is a mixture factor which has to be refined. This way a connection is created between the weighting of the volume fractions and the assumption of the linear mixture. We obtain for the SCEC of the matrix a+(l-afy,ll-a) $jk, = ijkr (11) l-a&, These equations allow the use of the differences in the slopes of composite and superlattice reflections, which both belong to the same crystal direction. Examples are pairs of 002/001 and 022/011 reflections where the latter are superlattice reflections. A sufficiently large database for these differences can be created by using samples of different orientations. They are expressed as ---= a& hti a ;:,*,* hihj a0 a0 h2 -i-k2 +I2 (1-A&, Is;%, - s;;, )uk3z+ (12) The h*k l* denote the corresponding superlattice reflection. The results of the least square refinement are summarized in Tab. 2. The results for the composite resemble those of nickel with clll, =250 GPa, c1,22 =160 GPa and c,~,~ =118.5 GPa [5] which is caused by nickel as the main constituent in these samples. Therefore It can be concluded that the determination of the SCEC by measuring the lattice strain on loaded single crystals is feasible and it can be a valuable tool for multiphase materials

8 TABLE 2: Single crystal elastic constants for the composite material, the y matrix and the y precipitates. The last row of the table contains the uncertainties which have been obtained from the experimental uncertainties of the slopes in Fig.3. The result for the mixture parameter is a=o.o870~ CONCLUSIONS It has been demonstrated how diffraction methods can be used for the determination of the single crystal elastic constants (SCEC) both from polycrystalline materials and single crystals. For polycrystals a method was used which refines the SCEC with the diffraction elastic constants as input data. Single Crystals, by the clear relationship between crystal lattice, stress direction and lattice vector hkl, allow a more direct approach. The elastic deformation of the crystal lattice is probed in different crystal directions hkl from which the SCEC can be obtained by a least sqare refinement. A useful application of that method are y/y superalloy single crystals. ACKNOWLEDGEMENT The authors wish to thank Dr. Richard Fields for his valuable help and his support during the work on the superalloy single crystals. REFERENCES El1 [51 I31 II91 WI WI WI Bollenrath, F.,Hauk, V. & Mtiller, E. H. (1967), Z. MetaUk.58, 1,76-82 de Wit, R. (1997). J. Appl. Cryst. 30,5 lo-51 1 Behnken, H. & Hauk, V. (1986). Z. MetaUk. 77, Gnaupel-Herold, T., Brand, P. C., Prask, H. J., (1998). J. Appl. Cryst., in press Simmons, G. & Wang, H. (1971). Single Crystal Elastic Constants and Calculated Aggregate Properties: A Handbook, The M.I.T. Press, Cambridge, Massachusetts Kriiner, E. (1958). Zeitschr. J: Physik 151, Reuss, A. (1928). Zeitschr. Angew. Math. Mech. 9,49-58 Voigt, W. (1928). Lehrbuch der Kristallphysik, Teubner Verlag Berlin-Leipzig Busing, W. R., Levy, H. A. (1967), Angle Calculations for 3- and 4-Circle X-Ray and Neutron Diffractometers, Acta Cryst. 22, Reimers, W. (1992), Investigations of Large Grained Samples - Principles, in : H. T. Hutchings, A. D. Kravitz (Hrsg.), Measurement of Residual and Applied Stress Using Neutron Diffraction, Kluwer Academic Publishers, Dordrecht 1992, Mtiller, A., Reimers, W. (1996), phys. stat. sol. (a) 156, 1,47-58 Gnaupel-Herold, T., Reimers, W. Scripta Met. 33,4,