AN ELASTIC CONSTANTS DATABASE AND XEC CALCULATOR FOR USE IN XRD RESIDUAL STRESS ANALYSIS

Transcription

1 Copyright(c)JCPDS-International Centre for Diffraction Data 2001,Advances in X-ray Analysis,Vol AN ELASTIC CONSTANTS DATABASE AND XEC CALCULATOR FOR USE IN XRD RESIDUAL STRESS ANALYSIS A.C. Vermeulen Philips Analytical, Lelyweg 1, 7602 EA Almelo, The Netherlands AIBSTRACT The creation and set-up of an electronic database for elastic constants is described. The database combines different types of elastic constants found in several literature sources converted to a single unit system. Calculation models like Voigt, Reuss and Hill are included for materials with cu.bic, hexagonal and tetragonal crystal symmetry. Checks are performed on the data entries in order to prevent the introduction of new typing errors. Checks on consistency of redundant data are performed to detect printing errors and rounding errors as occurring in the original literature sources. Approximately 8% of the checked data entries needed to be corrected for printing errors and rounding errors. With this database, proper XRD residual stress analysis becomes available for a larger public. INTRODUCTION A well-known practical problem in evaluating XRD residual stress measurements is the poor availability of X-ray elastic constants (XECs) Si and Y&. The elastic constants found in various literature sources are presented in different types and different units. The types for elastic constants vary between: (i) isotropic -bulk- constants 1,2Y3, (ii) X-ray elastic constants for a specific { hkl} reflection 3,4,5,6, or (iii) single crystal constants 2*5*7s. The values are given in: (a) SI units, (b) CGS units, (c) imperial units, or (d) other units. The single crystal elastic constants can be used in model calculations to generate X-ray elastic constants. Various models are found in the literature for the calculation of X-ray elastic constants using single crystal elastic constants as compliances, stiffnesses or both. Unfortunately not all literature sources supply both compliances and stiffnesses. To overcome the practical problems mentioned above a database has been created dedicated to XjRD residual stress analysis lo. All three different types of elastic constants are included in the database and presented in SI units. The X-ray elastic constants (XECs) Si and Y2S2 are calculated with the built-in XEC calculator. Single crystal elastic constants are automatically converted when necessary. The current version of this database contains over 400 (factory) entries, which have been checked on consistency in order to exclude new typing errors and remove existing printing errors of the literature sources. The method used for this consistency check will be presented. Extending the database by entering new (user-defined) values is possible. The XEC calculator is available for the added data as well. With these properties this database is a useful addition to the evaluation software for XRD residual stress measurements lo.

2 This document was presented at the Denver X-ray Conference (DXC) on Applications of X-ray Analysis. Sponsored by the International Centre for Diffraction Data (ICDD). This document is provided by ICDD in cooperation with the authors and presenters of the DXC for the express purpose of educating the scientific community. All copyrights for the document are retained by ICDD. Usage is restricted for the purposes of education and scientific research. DXC Website ICDD Website -

3 Copyright(c)JCPDS-International Centre for Diffraction Data 2001,Advances in X-ray Analysis,Vol DATABASE ORGANIZATION The X-ray elastic constants Si and?4 SZ as required in sin2 ly analysis can be obtained from three different types of elastic constants in the literature. The database is organized in the same way (see Figure 1). The three types for elastic constants are: (i) isotropic elastic constants (i.e. Young s modulus E and Poisson s ratio V) as obtained on bulk (polycrystalline) materials 2,3. Also calculated values are present. (ii) X-ray elastic constants for a specific { hkl} reflection, Sihk2 and?4 Szhkl, as measured in an X- ray experiment with applied stresses 3,435,6. Also calculated values are present 3. (iii) single crystal elastic constants (i.e. compliances Sij and/or stiffnesses Cij), which are obtained on single crystals by other techniques 2, [ Isotropi;;;nstants 1 Single crystal constants Automatic conversion c <=> s Stiffnesses Cll, c12,... I I Compliances Sll, s12,... Pig. 1 Organization of the elastic constants database and built-in XEC calculator according to the three different types of elastic constants as available in the literature. UNITS CONVERSION The values of the various types of elastic constants are given in various units belonging to different unit systems. The unit systems as found in the literature are: (a) SI units (e.g. GPa, TPa-, N/m2), (b) CGS units (e.g. dyn/cm2), (c) imperial units (e.g. psi, ksi, lb/in2), or (d) other units (e.g. mm2/kp). To allow comparison, all the values of the database are converted into SI units (GPa and TPa- ). The factors used to convert the original units into the SI system are given in Table 1.

4 Copyright(c)JCPDS-International Centre for Diffraction Data 2001,Advances in X-ray Analysis,Vol Table 1 Conversion factors used in the elastic constants database. Elastic Original Multiply Converted Ref. constant unit with unit E 10+ N/m2 = 10 GPa 1 E 1 GPa = 1 GPa 235 E 1O+3 N/mm2 = 1 GPa 3 (lv2sj1 1O+3 ksi = GPa 6 hs2 GPa- = 1000 TPa- 6 1hs2 lo * psi- = TPa- 5 s,, Y2& 1o 6 mm2/n = 1 TPa- s1, 1Y2s2 10e6 MPa- = 1 TPa 473 c, lo+ dyn/cm2 = 10 GPa 7 cij lo+ Pa = 10 GPa 5 Gj 1 GPa = 1 GPa 2 sij lo-l3 cm2/dyn = 1 TPa- 7 Sij 10-l Pa- = 10 TPa- 5 Sij 1O-5 mm2/kp = TPa- 8 Sif 1 TPa- = 1 TPa- 2 XEC CALCULATION MODELS To calculate residual stresses X-ray elastic constants, models are required. The most commonly known model is the isotropic model using the elastic constants Young s modulus E and the Poisson s ratio v 5: The resulting XECs are independent of the { hkl} reflection used to analyze residual stress. (lab) MIore advanced models describe the grain interactions and are based on single crystal elastic constants. The Voigt model assumes that all grains have the same state of strain, which violates the mechanical equilibrium. The calculated XECs are independent of {I&Z}. The Reuss model assumes that all grains have the same state of stress, which introduces discontinuities at the grain boundaries. The calculated XECs are { hkz}-dependent. A special case with the Reuss model is the Quasi-isotropic model based on an average { hkl} orientation. The Hill model (also indicated as Neerfeld-Hill) is based on the proof that the Voigt and Reuss models are the theoretical limits and is simply the arithmetic average of the two models. Examples are given in Figure 2 for the cubic material Fe and in Figure 3 for the hexagonal materials Zn and Zr. More complete descriptions of the Voigt and Reuss models can be found in the literature 7,8,9 for cubic, hexagonal and tetragonal symmetries. Lower symmetries are also described 9, but are not considered as being highly important from a technological point of view. Note that the ca.lculations for the Voigt model use only stiffnesses and for the Reuss model only compliances 7,8,9. This requires an automatic conversion for the situation that only one of the two sets (i.e. compliances or stiffnesses) of single crystal elastic constants is available (see also Figure 1).

5 Copyright(c)JCPDS-International Centre for Diffraction Data 2001,Advances in X-ray Analysis,Vol I 2 Reuss 4 - I I Quasi- I isotropic 2, I I I Fe Fig. 2 Example of model calculations for 1/2& with the Voigt, Reuss and Hill models for the cubic material Fe. The parameter on the horizontal axis is the orientation factor 3c which varies from 0 for the (100) reflection to 1 for the (111) reflection. All models show linear behavior. The quasi-isotropic orientation occurs at 3r=O.6. (100) 3r (111) The Eshelby-KrSner model describes the interaction of a grain with the surrounding matrix with the assumption of equilibrium in both stress and strain. The calculations of the Eshelby-Kr6ner model are more complex compared to the Hill model. Despite the simplicity of the Hill model the results are close to Eshelby-Kr6ner. Therefore Eshelby-KrGner is not offered as one of the models in the XEC calculator (see also Figure 1). A logical modification to the Hill model is the possibility to apply different weighing factors for the Voigt and Reuss components, which allows results even closer to the Eshelby-Krijner model. 35 :30 25 (D ii 20 cn,- 15 Zn ;tu 12 ik $ 10 I 10 8 IQuasi- / isotrodc (100) cos%jl (001) (100) co&g (001) Fig. 3 Example of model calculations for %S2 with the Voigt, Reuss and Hill models for the hexagonal materials Zn and Zr. The parameter on the horizontal axis is the orientation factor cos2p, which varies from 0 for the (100) reflection to 1 for the (001) reflection. The Reuss and Hill models show a parabolic behavior. The quasi-isotropic orientation occurs at cos2q =0.5. Remarkable is the Zr example where the quasi-isotropic model predicts an average value close to the maximum of the parabolic line.

6 Copyright(c)JCPDS-International Centre for Diffraction Data 2001,Advances in X-ray Analysis,Vol CONSISTENCY CHECKS Creating an electronic database without checking the entries on errors does not make sense. Therefore checks on consistency have been performed in order to avoid the introduction of new typing errors and to remove existing printing errors of the literature sources. The X-ray elastic constants in the literature are sometimes presented with redundant data 3. The dalta is presented in two pairs: the X-ray elastic constants St and?&$ and the corresponding isotropic elastic constants E and v. This allows checking if the four constants are consistent w:ith respect to each other. For each of the constants a verification value is calculated and compared with the original literature entry. The verification formulas used are: S 1,check _ - lit + lit &heck = - El, 1 - sl,lit t2ab) E check = sl,lit + $ s2,1rt vchcxk = Sl,, + i SW Since each value can be calculated using only two other constants the consistency of triplets can be checked using the general formula: An entry requires closer inspection if # Ax I 21% (5) Clhecks on the X-ray elastic constants could be performed on 109 entries and resulted in the discovery of 8 errors in the original literature source 3. Four of the errors occurred in the used data (i.e. Sr and Wz) and are repaired in the database. Unfortunately it was not possible to perform a check on consistency on other entries with single pairs Sr and 1/2S24 or with only YiSz 5, 6. Obviously these entries are checked on the introduction of new typing errors only. The isotropic elastic constants in the literature are also sometimes presented with redundant data lr2. The data is presented with four values: Young s modulus E, Poisson s ratio v, shear (or rigidity) modulus G and compression (or bulk) modulus K. As above this allows checking if the four constants are consistent with respect to each other. For each of the constants a verification value is calculated and compared with the original literature entry. The verification formulas used are: 9KlitGlit 3&t - 2% E rheck = (3K, + G,,,) vcheck = 2(3K,, + G,,,) Elit K Elit Gcheck = 2(1 + lit 1 check = 3(1-224,) Clhecks on the isotropic elastic constants could be performed on 78 entries and resulted in the discovery of 3 errors in the original literature sources tw

7 Copyright(c)JCPDS-International Centre for Diffraction Data 2001,Advances in X-ray Analysis,Vol The single crystal elastic constants are frequently presented in double sets (i.e. compliances and stiffnesses) 1,2,5,7. The checks on consistency are performed by converting the compliances into stiffnesses and vice versa. Again the verification data is compared with the original literature va.lues. Checks on the single crystal elastic constants could be performed on 68 entries and resulted in the discovery of 10 errors in the original literature sources. Four values were a factor 10 too large 2 or too small 7, four other values were severely rounded 5 or contained a typing error 7 and for two values the minus sign was missing 7. CONCLUSIONS An elastic constants database has been created removing all practical problems in obtaining XlECs. The built-in XEC calculator offers various commonly used models for cubic, hexagonal and tetragonal crystal symmetries. Elastic constants from various literature sources and in various units systems are converted into SI units. A check for consistency on the literature data was necessary. Approximately 8% of the checked entries needed to be corrected because of serious printing errors. With this database, XRD residual stress analysis is made accessible to a larger public. RIEFERENCES [l] Wawra, H., 2. Metallkde, 1978, 69, [2] Smithells, C.J.; Brandes, E.A., Metal Reference Book, 5th ed., Butterworths: London, [J] Eigenmann, B.; Scholtes, B.; Macherauch, E., Mat.-wiss. u. Werkstofiech., 1989,20, 3 14-,325. [4] Hauk, V.M.; Macherauch, E., Advances In X-ray analysis, 1983,27, [5] Noyan, I.C.; Cohen, J.B., Residual Stress, 1987, Springer-Verlag: New York. [6] Cullity, B.D., Elements ofx-ray DifSraction, 2nd ed., 1978, Addison-Wesley: Reading. [7] Evenschor, P.D.; Frijlich, W.; Hauk, V., 2. MetaZZkde, 1971,62, [8] Bollenrath, F.; Hauk, V.; Mtiller, E.H., Z. MetaZZkde, 1967,58, [9] Behnken, H.; Hauk, V., Z. MetaZZkde, 1986,77, [IO] X Pert Stress PW3208, Software for Residual Stress Analysis, Philips Analytical, The Netherlands,