Anisotropy paramder evaluation in textured titanium alloy sheets

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1 Anisotropy paramder evaluation in textured titanium alloy sheets Dr. V.N. SEREBRYANY, Dr. R.G. KOKNAEV '.. All - Russia Institute of Light Alloys, Russia, Mose.ow, Gorbunova str., :-. ANNOTATION In this study, a technique.of the calculation 9f.a planar yield strength anisotropy and of a normal anisotropy coefficient on the basis normalized single crystal yield surface (SCYS) averaged with account. for the texture as a weight func~ioq. is presented. The SCYS was calculated according to Zacks and Taylor-Bishop-Hill models. in ternis of operated slip and twinning systems ensured the. p\astic deformation of the investigated titanium alloy; and also in terms of the critical-resolved shear stress..(crss) on an each system from the operated deformation ones. The texture weight function in the form of orientation distribution function (ODF).coefficients has been defined. from pole figures. The.technique for the titanium sheets had an equiaxed micn>structure with the different grain value has been tested. Key words: plastic anisotropy, texture, titanillm, sheet, model; computer calculation. 1. INTRODUCTION Most engineering materials are polycrystalline aggregates having a crystallographic texture, whose anisotropic mechanical properties can be predicted as an appropriate average over the microscopic behaviour of the constituent grains. In the case of cubic crystal, the concept of a single crystal yield surface (SCYS) has been successfully used to understand polycrystal plastic anisotropy development in terms of single crystal anisotropy [ 1,2 ]. Several approaches which relate single crystal to polycrystalline behavior have been suggested [ 2 ]. Among these is the isostress technique first employed by Zacks to relate the critical-resolved shear stress ( CRSS) of fee metals to the polycrystalline yield stress (PYS). This PYS was found by averaging the values of uniaxial stress necessary to reach the CRSS ( usually on a single system) in fee single crystals of selected orientation. Such an approach follows directly from a knowledge of the CRSS, the identity of the slip systems, and the orientation of the single crystals chosen to represent the polycrystalline aggregate. In this case a sterss continuity is satisfied only in an average sense while no provision is made for ensuring strain continuity. On the other hand, isostrain models ensure grain by grain strain continuity but make no provision for a stress continuity. However, in order to enforce an arbitrary strain state on a selected single crystal, it is necessary to have a procedure for selecting a sufficient number of independent deformation systems to accommodate the given strain. Two such isostrain procedures are the minimum shear procedure of Taylor and the maximum work procedure of Bishop and Hill. In order to use the last approach, it is necessary to find all possible stress states. which can simultaneously operate at least five independent deformation systems. Once all such polyslip stress states are known, the stress state which will operate to impose a particular strain state is selected by the maximum work procedure. For materials of hexagonal structure such as titanium and titanium alloys, much less work of this type has been done, though a better understanding of their anisotropic plastic behavior is of great technological importance. The reason is mainly to be found in the complexities of deformation modes presented in hep materials. They not only differ from one material to another, but within the same material the active deformation modes and their CRSS depend on the composition, the temperature, the strain rate, the stress state and the previous deformation history [ 3 ]. The isostress analysis was applied by several authors [ 3,4,5 ] to explain cold-rolling texture development in titanium and to predict the yield strength anisotropy in textured titanium alloys. The isostrain analysis with the use the maximum work procedure was applied by Thornburg and Piehler [ 6 ] to investigate texture development during cold-rolling in titanium and titanium-aluminium alloys and by Dervin [ 7] to predict the normal anisotropy coefficient in titanium sheet. The objective of this paper is to devise a technique for evaluation of the plastic anisotropy in titanium sheets on the basis SCYS calculated with the use Zacks and Taylor-Bishop-Hill models and averaged with account for the texture as a weight function. 698

2 TITANIUM"99: SCIENCE AND TECHNOLOGY 2. EXPERIMENT AL DETAILS 2.1. MATERIALS The commercially pure titanium sheets were obtained in the recrystallized condition with the variable concentrations of the interstitial impurities (oxygen and nitrogen) as given in Table 1. Table 1. Some interstitial impurity concentrations of the commercially pure titanium sheets Type N wt% wt% The different schedules of the rolling and annealing for the titanium sheets were used to provide the equiaxed grain structure with a variable recrystallized a -grain size MECHANICAL TESTING There are two main types of the anisotropy of mechanical properties in metal sheets: a nonnal and planar anisotropy. The first type is estimated by the nonnal anisotropy coefficient ( R ) equal a ratio of lateral to normal strains, and the second one is defined by the yield strength anisotropy in the sheet plane. These anisotropy parameters of the sheet material were evaluated using uniaxial tensile tests and tensile samples were machined from the titanium sheet at the different a - angles to rolling direction TEXTURE ANALYSIS Crystallographic textures were characterized by inverse and direct pole figures for titanium sheets of type I and type 2, respectively. Inverse pole figures ( IPF ) were obtained for three orthononnal directions of the sheet: rolling direction ( RD ), transverse direction ( TD ) and nonnal direction ( ND ) by the procedure presented in paper [ 8 ]. The direct pole figures were obtained by the reflection method using Cuka - radiation on a special texture goniometer. Intensity data were collected up to 7 from the center of the pole figure. A correction for geometric defocussing was invoked since the sample has been titled. The geometric defocussing factor is obtained by measuring the intensity of a powder. standart as a function of the tilt angle. Basal { 4}, prismatic { 1O1} and pyramidal { 111}, { 112} pole figures were obtained for the surface and core layers of the titanium sheets. 3. PREDICTION OF ANISOTROPY PARAMETERS 3.1. ISOSTRAIN TAYLOR-BISHOP-HILL ( TBH) APPROACH We assume a three-axial plastic deformation according to the defonnation tensor 1 -q -(1-q) (1) i.e. an elongation dr; in the x 1 - direction and shortening -dryq and -dr;(l- q) in the x 2 and x 3 directions, respectively. If this deformation is imposed on a single crystal (grain) a deformational work will be needed which may be written in the general form 699

3 da = T M(q g )d77, (2) where r is a certain stress factor. Mis a geometrical factor which depends on the contraction ratio q of the deformation tensor and on the orientation g' of the crystal axes with respect io the principal axes of the strain tensor. According to the isostrain TBH-model an each grain was assumed to undergo the same strain state as the polycrystalline material, i.e. the grain deformation tensor ( 1 ) is equal to the sample one dsi = dst. (3) We now assume a polycrystalline material having a certain texture described by.odf f (g), where g is the orientation of the crystal axes with respect to the sample coordinate axes, e.g. RD, TD, ND, respectively. The principal strain axes may deviate from the sample coordinate axes by the rotation g. Then the orientation g' m Eq. ( 2) is expressed in terms of the orientation g referred to the sample coordinate system by the rotation (4) which means that at first the rotation g and then the rotation g are to be carried out. The mean. value of the deformation work taken over all crystal orientations is then given by.c5) where M (q,g ) = J M(q,g g )f(g)d(g' ). (6) The integral can easily by calculated if both functions are developed into series ~f _generalized spherical harmonics of appropriate symmetry [ 9 ] and M is also expanded into a power series of q. "' f(g)= I (7)..l.=(2) µ=1(1) n=(2) L M(.1.).I. r M (q,g') = L I. I Imf; r:v (g' )qp. (8)..l.=(2) µ=!(!) v=(2) p=o{l) We futher assume that g be a rotation through the angle a about the x 3 axis, i.e. the sample ND. Then M(q,a) takes on the form [IO] M(.I.) M (q,a)= L L.l.=(2) µ=1(1) ). I v;.(2) r mf;c:v cosvaqp I~-- p=o(i) 2,1, +I (9) Ifa sample is elongated in a free tensile experiment then d77 is fixed by the experimental conditions but not the conraction ratio q. It will take on such a value that the deformational work is minimum which requires 8M(q,a) = O. aq (1) 7

4 The solution ofeq. ( 1) with Eq. ( 9) yields q = qmin (a) which is assumed to be the actual contraction ratio related to the normal anisotropy coefficient R by R(a) = qmin (a) 1- qmin (a) (11) Substituting q = q min (a) in Eq. ( 9 ) may be obtained M ( q min, a) values suited to the conditions of the tensile experiment. Then, the planar yield strength anisotropy may be defined from the relation M(q,a)/ M(q,O) ISOSTRESS ZACKS APPROACH Whereas the TBH assumption is expressed entirely in terms of strain uniformity, the Zacks model refers to the relation between the stress state of an aggregate and its constituent grains. The tensile test is assumed to be represented by the following uniaxial stress tensor "11" ap = lj (12) In this representation the stress state is assumed to be uniform across the various grains c - p au-au (13) In this case the deformational work of the given crystal under uniaxial loading may be written in the form (14) Using the analogous procedures presented by the Eq. ( 5 )-( 9) we receive M(a) in the form - L M(A.) ;. mµvcµv cosva M (a) = L L L ---'"..i.'---"-.i =(2) µ=l(i) v=(2) 2A + 1 (15) Then the planar yield strength anisotropy may be defined from the relation M(a)/ M(O) CALCULATED PROCEDURES The plastic deformation under the axial loading of the polycrystalline titanium was supposed to caused by a basal, prism and pyramidal slip of dislocations along <112> direction and by { 112}<11 I> and { 1122} < 1123> twinning to provide for c - axis extension and compression strain. The deformation system CRSS values were defined with according to the temperature, the strain rate ap.d the oxygen and nitrogen contents of the titanium [ 4, 5, 7 ]. In isostrain approach we applied the maximum work procedure of Bishop and Hill, being all possible stress states which can simultaneously operate at least five independent deformation systems we calculated using the Thornburg and Piehler technique [ 6 ]. For a given q-value M(q,g) magnitude is defined by a ratio the maximum for given grain orientation deformational work normalized to an unit strain to CRSS-value of the prism slip system. In isostress approach the grain deformation with the g' - orientation performed by the slip or twinning in the crystallographic plane system with the maximum shearing stress [ 4 ]. Then for the given grain M (g' )-value is defined by the ratio of the axial tensile stress ( for 71

5 example, flow stress ) to this shearing stress. Then the analogous procedures were repeated for the different orientations g = {<pp <I>, <p 2 } (where <pp <I>, <p 2 are the Euler angles) varied through ~ <p 1 ~ 9, ~ <I> ~ 9, ~ <p 2 ~ 6 ranges with steps 5 for an each angle and.5 for q -value ( O< q < I ). As a result, we apply the normalized M(q,g:) and M(g') orientation functions. Then, using the abovedecribed procedure we defined the angle dependence of the yield strength and the nonnal anisotropy coefficient in titanium sheets, being the ODF coefficients er have been calculated by the series expansion method from pole figures [ 5, 8 ]. Relying on the above-decribed algorithm we developed the special computer programs. 4. RESULTS AND DISCUSSION The CRSS values for the slip and twinning systems of the investigated titanium sheets are listed in Table 2 [ 5,7 ]. Table 2. The CRSS values for the slip and twinning systems of the investigated titanium sheets Type r,mpa { 1 I }<112> {2}<112> {111}<112> {112}<11 l> {l 122}<1123> The ODF coefficients and the mean diameters of the recrystallized a -grain of the investigated titanium sheets are listed in Table 3. The ODF coefficients and microstructure parameters of the investigated titanium sheets Table 3 Type D,µm ODF coefficients C2uo C2uz. C4uu C/2 C44 I 15,952 -, ,429 1,248 I 52 2,125 -,27,98 -,41 1, '' In the case of. the plastic defonnation considered here the order L =4 seems to be sufficient [ 9 ], i.e. we used only the five er and mf; coefficients. The nonnalized calculated according to Zacks and TBH-models M(a)/ M(O) values.for the titanium sheets (Type 1) with the mean diameter of the recrystallized grain equal 15 µm are presented in Fig. I. The calculated values are compared to the relative experimental values obtained from mechanical tests. The last values also are presented in Fig. I. The analogous planar calculated and experimental distributions for the titanium sheets ( Type 1 ) with the mean size of the grain equal 52 µm are presented in Fig.2. 72

6 TITAN!UM'99: SCIENCE AND TECHNOLOGY Z-model 'I'BH-model ~ ~ ~ - --n n - ~---n a-~ ~ a, degree Figure I. The planar calculated and experimental () distributions of the yield strength for the titanium sheet with the mean size of the grain equal 15 µm Z-model -- 'I'BH-model ~ ~~ --~ ~~ a, degree Figure 2. The planar calculated and experimental () distributions of the yield strength for the titanium sheet with the mean size of the grain equal 52 µm. The planar R - value distribution calculated according to TBH-model for titanium sheet ( Type 2 ) is presented in Fig.3. As is seen in Fig. I and Fig.2, the calculated data agree rather well with the experimental results, being the coincidence these results is better for the data obtained with use Zacks model in the coarser grained titanium sheet. As is seen in Fig. 3, the calculated planar R - value distribution correlate within the limits of experimental errors with values obtained from mechanical tests of speciments cut at the different a - angles to RD. 73

7 6 ~ a, degree Figure 3. The planar calculated and experimental (D) R -value distributions for the titanium sheet (Type 2 ). 5. CONCLUSIONS The anisotropy parameters ( yield strength anisotropy and normal anisotropy coefficient) predictions based on the ODF coefficients up to L =4 and the SCYS calculated in the context of the isostress and isostrain models, obtained in this study, lead to rather well agreement with tensile test results. 6. REFERENCES I. H.J. BUNGE: "Technological Applications oftexture Analysis", Z. Meta/lkde,1985,16, p P. LEQUEV and J.J. JONAS: "Modelling of the Plastic Anisotropy of Textured Sheet'',Meta/lurgical Transactions, l 988,19A, p. l F. LARSON and A. ZARKADES: "Properties of Textured Titanium Alloys'', MCJC Report, 1974, 76pp. 4. V.N. SEREBRYANY and R.G. KOKNAEV:[in Russian] "Relation between the crystallographic texture and the yield strength anisotropy of titanium alloy sheets", Color metals, 1984, 2, p V.N. SEREBRY ANY and R.G. KOKNAEV:[in Russian] "On the yield strength anisotropy in the sheets from VTI- and VT6ch alloys", Jzv. AN USSR, Metals, 199, 6, p D.R. THORNBURG and H.R. PIEHLER: "An Analysis of Constrained Deformation by Slip and Twinning in Hexagonal Close Packed Metals and Alloys", Metallurgical Transactions, 1975, 6A, p P. DERVIN: " Analyse quantitative des textures cristallographiques de materiaux de systeme cubique on hexagonal. Applications a!'aluminium et au titane. Relation avec l'anisotropie de deformation plastique dans le cas de tolles minces de titane'', These Docteur lngenier, Paris, 1978, l 95pp. 8. A.A. RUSAKOV and V.N. SEREBRY ANY:[in Russian] " A contribution to the question of the inverse pole figures development of the hexagonal metals and alloys", Zavodskaya Laboratory., 1984, 5,.p H.J. BUNGE: "Three Dimensional Texture Analysis", International Materials Reviews, 1987, 32, p I. H.J. BUNGE and others: "The Relation between Preferred Orientation and the Lankford Parameter r of Plastic Anisotropy'', Arch. Eisenhuttenwes, 1981, 52,p l. 74

Influence of Texture on the Plastic Anisotropy of Mg Alloy Determined by Experimental and Numerical Investigation

International Journal of Innovations in Materials Science and Engineering (IMSE), VOL. 1, NO. 3 108 Influence of Texture on the Plastic Anisotropy of Mg Alloy Determined by Experimental and Numerical Investigation