Calculation of the yield locus of textured polycrystals using the taylor and the relaxed taylor theory

Several authors have proposed methods which use the Taylor–Bishop–Hill theory in order to calculate the yield surface of textured samples of which the O.D.F. is known.


INTRODUCTION
The plastic anisotropy of a material is often described by a yield surface. It is the geometrical representation of the stress states for which plastic yielding occurs. Such stress state is described by a tensor $, represented by a matrix S for a given reference system in 29 30 P. VAN HOUTFE physical space. The tensor S is symmetric, thus Hence there are only 6 independent components S 0. A six dimensional space is sufficient to represent a yield surface. An analytical representation would be: constant (2) For isotropic materials, plastic yielding only depends on the value of the principal stresses, not on the principal stress directions. This means that the yield locus can be represented in a three dimensional principal stress space. This simplification is however impossible for anisotropic materials. For materials, for which plastic yielding is independent of the hydrostatic component S" of the stress the number of dimensions can nevertheless be reduced to five: s"= (s.)/3 ( 3) This allows to replace the stress $ by the deviatoric stress $': s a os" for which the following property always holds (4) There are only five independent deviatoric stress components.
Materials for which plastic yieldings is independent frbm the hydrostatic component of the stress do not change their volume during plastic deformation. This is expressed by , 0 In which E are the components of the plastic strain rate tensor !.
We will consider Eq. (6) as valid throughout this paper.
The number of mechanical tests required to measure a five dimensional yield locus is tremendous. Therefore, it is often assumed that the anisotropy is only caused by the crystallographic anisotropy of the sample. It is then theoretically derived using crystal plasticity laws. This is not so difficult for a single crystal. For ii) a reliable model for the plastic yielding of a polycrystal. In this paper, we will consider the O.D.F. measured by means of the serial expansion method (Bunge, 1982). The Taylor-Bishop-Hill theory will be chosen as model for polycrystal plasticity (Gil Sevillano, Van Houtte and Aernoudt, 1980). This means, that the resulting yield curves are only valid for offsets that are sufficiently large to reach multiple slip in every grain (a few percent). The elasto-plastic transition range cannot be studied.
Various authors have already proposed such methods (Bunge, 1970;Parniere and Roesch, 1975;Grumbach, Parniere, Roesch and Sauzay, 1975;Bunge, 1979;Bunge, Schulze and Grzesik, 1980;Sowerby, Da Viana and Davies, 1980). They did not intend to construct a complete yield locus in five dimensional stress space, but limited the analysis to a plane section of it, showing a yield curve as a function of S1 and S2. These are the 11 and 22 components of the stress tensor in the [x, x, x] reference system of a tensile test sample, cut from a rolled sheet which itself has a [Xl, X2, X3] reference system (Figure 1). The analytical description of the section plane is given by" S3 S1 S2 0 (7) )t' FIGURE 1 Tensile test sample cut from a rolled sheet. Reference system of the sheet: xl rolling direction (RD); x2 transverse direction (TD); x3 sheet plane normal (SPN). Reference system of the test sample: x tensile axis (xt); x x3 and x transverse direction of the sample (perpendicular to x and x). 32 P. VAN HOUTTE The implementation of this condition is not easy, as will be demonstrated below. In previous work (Canova, Kocks, Tom6 and Jonas, 1985;Bunge et al., 1980;Sowerby et al., 1980), it was replaced by a much easier condition: 0 (8) xl, x2 and x3 usually are twofold symmetry axes of the texture of a rolled sheet. It can be demonstrated (see below) that in that case Eq. (7) and Eq. (8) are equivalent for tr 0 and c :r/2 but not for other values of c (the angle between the rolling direction and the tensile axisFigure 1). The physical meaning of this is, that for angles other than 0 or r/2, the plastic response of the anisotropic material to a uniaxial tensile stress may be a plastic strain for which xt is not a direction of principal strain, i.e. /2 may be different from zero. Models which neglect this fact by using Eqs. (8) instead of Eqs. (7) do not produce true yield loci but so-called "principal strain yield loci".
The method proposed by Mols, Van Praet and Van Houtte (1984) really uses Eqs. (7) and produces a polyhedron-shaped envelope of the five dimensional yield surface. It is stored in computer memory under the form of the equations of 51408 hyperplanes (the facets of the polyhedron). The computer can then at will produce any plane section, for example those that correspond to S1-S2 sections for any value of re, but also S2-S2 sections. The method is suited for cubic metals without Bauschinger effect. Monoclinic and orthorhombic sample symmetries can be treated. A serious drawback is the amount of calculations that must be performed, which prevents interactive exploitation of the yield surface using a graphical terminal.
A simplified version of this method is presented in this paper.
The amount of calculations has been reduced 30 times, which makes an interactive exploitation possible. One additional assumption was necessary: at least one principal direction of the stress should coincide with one twofold symmetry axis of the texture of the sample (sample symmetry, not crystal symmetry, as explained by Bunge (1982)). Nearly all practical applications known to the author can nevertheless be treated by this less-general method. It must be pointed out that there are other ways as well to improve the classical methods for yield locus prediction. Berveiller CALCULATION OF THE YIELD LOCUS 33 and Zaoui (1984) for example proposed to replace the Taylor-Bishop-Hill theory by an elasto-plastic self-consistent method, which would give more precise results for small offset yield loci.

Introduction
The simplification that is presented in this paper is an application of a theorem concerning the symmetry properties of anisotropic materials.
Suppose that an anisotropic sample possesses a twofold rotation axis of symmetry, at least with regard to those material properties that control plastic anisotropy. Then this axis will be a principal direction of strain when it is a principal direction of stress.
A few examples will illustrate the principle of the twofold rotation symmetry axis of the sample. The simplest assumption (also adopted in the present paper) is, that the anisotropy only depends on the crystallographic texture. The twofold rotation symmetry axis that is meant here belongs to the sample symmetry of the texture, not to the crystal symmetry (see e.g. Bunge (1982)).

34
P. VAN HOUTFE FIGURE 3 Choice of the reference system of the specimen used for pole figure measurements on a torsion test sample. X is the shear direction, x2 the shear plane normal and x3 the radial direction.
pole figures. It is seen that a rotation r around SPN transforms both pole figures into themselves. So the SPN is a twofold symmetry axis of these pole figures. In this example, the property would not only hold for these two pole figures, but for all pole figure of this sample. The SPN hence qualifies as a twofold symmetry axis to which the theorem applies. The rolling direction and the transverse direction would qualify as well. Figure 3 shows a schematic representation of a torsion test sample. After torsion, the properties of such sample are no longer homogeneous. Consider an infinitely small piece of material, at the surface of the sample. A (x, x2, x3) reference system is placed on it in Figure 3. CALCULATION OF THE YIELD LOCUS 35 these pole figures. It is a twofold rotation axis of them and also of all other pole figures that can be made of this texture. It qualifies as a twofold rotation axis to which the theorem applies, at least for that tiny part of material, at the surface of the sample, of which the texture was measured. Note that the radial direction is a direction of principal strain and stress during the torsional deformation.
Note concerning the implementation of the "sample symmetry" in a three-dimensional description of a crystallographic texture Pole figures such as those represented by Fig. 2 have "symmetry axes"" the two-dimensional figure is transformed into itself by a reflection with respect to these axes. The latter then are often thought to be the intersections of "mirror planes" of the threedimensional texture with the plane of the figure. This extrapolation is incorrect: a "mirror plane" cannot be used to define a symmetry relation of a three-dimensional texture, although the concept has often been used, even in the same context as here (Canova et al., 1985).
A description of a three-dimensional texture essentially is a set of crystallite orientations, together with a corresponding set of volume fractions. An orthogonal reference system is put on the crystal lattice of each crystallite. For cubic metals, it is possible to adopt a convention that requires these to be right-turning reference systems. This simplifying convention is not always possible (Esling et al., 1980). Three Euler angles then describe the orientation of such reference system with respect to the external reference system of the sample. Roughly speaking, an operation (defined within the set of crystallite orientations) which relates an arbitrary crystal orientation gA to a crystal orientation g is a "sample-symmetry operation" when the total volume of crystals with orientation gt is equal to the total volume of crystals with orientation gn. It is implicitly assumed, that gA and gn are both elements of the set of orientations i.e. they must both describe the orientation of a right-turning reference system with respect to the sample reference system. A separate definition of special Euler angles would be required if one wanted to describe the orientation of a left-turning crystal reference system. Symmetry operators in physical space are normally used to define 36 P. VAN HOUTTE symmetry operators in orientation space. A twofold rotation axis qualifies for this: one takes the reference system of crystal A (which is an object in physical space), rotates it over angle r around the twofold rotation axis and one gets another valid crystal reference system of which the orientation gB can be calculated. On the other hand, a reflection in a mirror plane (performed in physical space) does not qualify. It would transform the right-turning reference system of crystal A into a left-turning reference system that is put on the lattice of crystal B. No Euler angles exist for this left-turning reference system. Note that a rotation by an angle r around the normal of the mirror plane would also put the crystal reference system of crystal A on the lattice of crystal B, hence offering a valid alternative to the forbidden reflection.

Proof of the theorem
The theorem is proved by negative demonstration. Let (xl, X2, X3 be a fixed reference system. (x], x, x) is a reference system fixed to the sample. Initially, they coincide. Let x3=xt3 then be the twofold symmetry axis of the sample. Consider now that the sample is loaded and plastically deformed in such way, that x3 x is a principal direction of stress. This means that: $23 $13 0 (9) This can for example be achieved by a uniaxial tensile test in the direction x3. Two different thought experiments are now carried out on the same sample: 1) A. the sample is loaded; plastic deformation takes place B. the deformed sample is rotated by an angle r around x3.
2) A. the sample is rotated by an angle az around x3, and so is the loading mechanism.
B. the sample is loaded; plastic deformation takes place.
The net result of both experiments must be identical. Now look what happens when during plastic deformation, x3 x would not be a principal direction of strain. Assume for example that there would be a non-zero shear/31, caused by a simple shear along the Ox2x3 plane in the Ox direction.  Figure 5a shows the initial shape of the sample. Figure 5b shows the sample after experiment 1A. Figure 5c shows the result after thought experiment lB. Figure 5d gives the sample after thought experiment 2A. Now note that the configuration of Figure 5d is physically exactly equivalent to the configuration of Figure 5a, because x3 is a twofold symmetry axis of the sample, and at the same time a principal direction of stress. The operation (2B) is physically totally equivalent to the thought experiment (1A) since, because of Eq. (9), the stress tensor (expressed in the fixed reference system (x, x, x3) remains unchanged after a rotation of the loading mechanism by an angle :r around x3.
So the loading operations (1A) and (2B) are physically equivalent and act upon physically equivalent samples (5a and 5d). Hence the result must also be the same. For this reason, Figure 5e is a copy of Figure 5b. identical, as they should be. Hence the assumption that x3 x is not a principal direction of the strain must be incorrect. The same idea can also be expressed as follows: when x3 simultaneously is a twofold rotation symmetry axes of the sample and of the loading mode, then it must remain so during and after plastic deformation.

DESCRIPTION OF THE METHOD
The classical method of calculating a yield surface of a textured polycrystal consists of finding a set of hyperplanes which are tangent to it Sowerby et al., 1980;Mols et al., 1984;Canova et al., 1985). Each of these hyperplanes corre.spond to a certain strain mode, described by a strain rate tensor Eij. Let g be short-hand rotation for a crystal orientation (Bunge, 1982). The Taylor-Bishop-Hill theory (Gil Sevillano et al., 1980) or a relaxed version of it (Van Houtte, 1984), can calculate the Taylor factor M for a crystal with orientation g (with respect to the specimen reference system Xl, x2, x3). Let 's be the rate of slip on slip system s. The Taylor factor then means: M= Is /eq (10) --eq is the equivalent strain rate (of the macroscopic strain) as defined by some convention. The von Mises convention would be: q ((2/3)ijY.,;,j)1/2 (11) It can then be shown (Gil Sevillano et al., 1980) that for that crystal, the plastic work rate per unit volume is given by: M eq in which is the critical resolved shear stress of the metal crystal (assumed to be equal on all slip systems). Equation (12) (14) The average Taylor factor/(/no longer is a function of the crystal orientation g, but still is one of the strain mode /ij. f(g) is the crystal orientation distribution function of the textured metal, which must be measured (Bunge, 1982). The averaging procedure (Eq. (14)) is made easier when the series expansion method is used.
Assume that, for a given strain mode Eij, a Taylor-Bishop-Hill computer code or a relaxed version of it was used to obtain the M values for all possible crystal orientations g, and that these data were used to obtain the series expansion of M itself (it will be explained below how this work can be organized): . , mtat'tat(g) (15) in which T"(g) are symmetrical generalized spherical harmonics of order l, , v (see Bunge, 1982). m are the corresponding coefficients. A set of such coefficients describes a function M(g) with a certain precision, which depends on the degree L of the series expansion. The coefficients m ' do not depend on the texture. So several sets of them (for several deformation modes ) can be stored on disk or on tape once and for all. For a given strain mode/ij, /17/can be obtained using Eq. (16).
Eq. (18) then becomes a linear equation for the unknown stress components Sij. In stress space, it represents a hyperplane that somewhere contains one point of the yield locus. So an envelope to the yield surface is obtained if a lot of these hyperplanes are constructed ( Figure 6). Note that Eq. (18) is independent of the convention for eq. Indeed: by using Eq. (10), it can be written as: ijSij "ic [s[ (19) in which/ is completely absent.  (6), it has only five independent components. This means that a tremendous number of distinct strain rate tensors must be considered when trying to cover all possible deformation modes within a certain angular resolution. Such a method has nevertheless been developed (Mols et al., 1984). After using it for some time, it was concluded that i) the textures of the considered samples nearly all had at least a twofold symmetry axis;

GENERATION OF DEFORMATION MODES AND CALCULATION OF THE CORRESPONDING M-VALUES
ii) nearly all stress states that were of some practical interest, and for which plane sections of the yield locus had to be made, were such that this twofold symmetry axis also was a principal direction of stress.
These two condiserations led to the idea, that the number of deformation modes that must be considered could perhaps be reduced strongly by limiting the method to stress states for which a twofold symmetry axis of the sample would be a principal direction of stress. The theorem that has been explained in section 2 permits to see how this stress restriction also restricts the strain modes that must be considered.
According to the theorem the x3 axis also is a principal direction of the plastic strain mode that corresponds to such stress state. Hence: for the plastic strain rate tensor. Let (xf, x, x) be a reference system that coincides with the principal strain axes.
Since x3=x, only one angle p suffices to characterize the orientation of this reference system with respect to the macroscopic reference system of the sample (Xl, xz, x3) (Figure 7). In the reference system (xfxx) the strain rate tensor can be 42 P. VAN The three components /v1, /2v2 and /3v3 and the angle qJ are sufficient to characterize a deformation mode that satisfies condition Eq. (21). This means that all possible deformation modes can be generated by selecting appropriate values for these parameters. The number of deformation modes to be considered must of course be finite in practical calculation. It depends on the desired angular resolution in tlJ and in principal strain space.

Generation of the deformation modes
Range for . We will let the angle W ( Figure 7) vary from 0 to ar in steps of at/18 (10). Note that the range from sr to 2at would produce identical strain rate tensors. In case of orthorhombic sample symmetry, this range can be reduced even more, at least for computational purposes (see Appendix I). /'3 -(1 ql)/ (26) q corresponds to the "contraction ratio q" defined by Bunge (1970). b) for 0 < q2 < 0.5" c) for 0 < q3 < 0.5: The value of M is not affected, unless there is asymmetry in the crystallographic deformation mechanisms. This finally leads to 798 different hyperplanes. However, for samples with orthorhombic sample symmetry and with symmetric slip systems, the /9/-values only need to be calculated for 210 of these hyperplanes" 21 deformations modes for 10 values of W between 0 and /2. The 588 other hyperplanes are then obtained from these by inversion of the strain rate tensor (see above) or by symmetry operations (see appendix I).
Expression for P-ij. Let the tensor Rij express the transformation law from the (xi) system to the (x) system (Fig. 7 (24)(25)(26)(27)(28)(29)(30)(31)(32) give the strain rate tensor as expressed in the (xP) system. Its expression in the reference system of the sample (x) is given by ii) The average ./-values of a polycrystal with a crystallographic texture are calculated using Eq. (14).
The practical implementation will now be discussed. For reasons of simplicity, only the case of orthorhombic sample symmetry and symmetric slip systems is explained. That means that the M-values need only be calculated for 210 deformation modes (see the previous sections). The method can very easily be adapted to treat the monoclinic case (e.g. torsion texture) or the case that the deformation mechanisms are asymmetric, e.g. when mechanical twinning is present.
The M-values which are needed for the data bank can be constructed by means of a Taylor-Bishop-Hill model or by means of the relaxed Taylor-Bishop-Hill theory, for metals with flat elongated grains (Van Houtte, 1984b). Figure 8a shows a schematic representation of such fiat grain. In the classical Taylor-Bishop-Hill theory, thereafter called fullconstraint theory (FC), it is assumed that the strain rate tensors are identical for all grains. This requires the simultaneous activation of at least five slip systems.
In the relaxed constraints (RC) Taylor-Bishop-Hill theory, this condition is somewhat "relaxed." In the "lath" version of the theory, the/13 component is left free: it need not be equal for all crystallites . (Figure 8b)..Only four slip systems need to be active. Finally, the E13 and the E23 components are both free in the "pancake" version of the theory (Figure 8c) (Van Houtte, 1984b).
The use of these models for the construction of a data bank of M-values will be discussed in the following order: (i) "lath" model, (ii) "pancake" model; (iii) FC model.

Using the "lath" model
The M-values must be calculated for all crystal orientation with respect of the sample reference systems (xi). In practice, the M-values are only calculated for the nodal points of a grid in Euler space. The following grid could for example be used for cubic The reason for this is, that the M(g) functions that are calculated for the 21 deformation modes associated to the case W 0 can be used to construct the M(g) function associated to other angles W as well.
Let g be the crystal orientation with respect to the reference system of the sample (xi) and let gP represent the orientation of the same crystal with respect to the system (xP) (Figure 7).
Theorem. The M-values of individual crystallites, as calculated by the pancake version of the relaxed Taylor theory, depend on gP and -p on Eq, but not on tlJ.
Proof. Let us construct two distinct deformation modes using the same values for/ff but different values for W 0 (A) W4:0 (B) (Fig. 7) In case (A), we would write the equations of the pancake version of the relaxed Taylor theory (Van Houtte, 1982) in the reference system (xi), in case (B) we would write them in the reference system (x/P) as shown in Figure 7.
The formal expressions that these equations have in a reference system (xP) depend on (Van Houtte, 1982)" the crystal orientation gP; the nature of the slip and/or twinning systems, which is the same in case A and B; the expression for the strain rate tensor that describes the imposed strain. It is the same in case (A) and (B) because we have deliberately chosen to use the same values for/ff in both cases.
the mathematical expression of the components of the strain rate tensor that have been relaxed. In the pancake version of the relaxed Taylor model, /13 and /23, expressed in the reference system of the sample, are relaxed. This means, that the strain rate of an individual crystallite is allowed to differ from the imposed strain rate by a "freedom strain rate tension" given by:   In a first step, the symmetric (rnP)'-coefficients must be desymmetrisized with respect to the "sample symmetry" (Bunge, 1982 (mP) (me)", (me) ,-" This is done by means of the following formulas: The coefficients of M(g) must now be made symmetric again: In order to be able to treat monoclinic textures (such as torsion textures), a type of m TM coefficients that has a monoclinic sample symmetry must be used. A method of constructing an orthogonal set of monoclinic coefficients and functions has been described by Van Houtte, Aernoudt and Sekine (1981) The definition of C' and C" for monoclinic sample symmetrices has already been given elsewhere (Van Houtte et al., 1981). In case of orthorhombic sample symmetry (e.g. rolling textures), C" are the usual C-coefficients and the C7 " are zero. ii) the latter are used to compute analytically the intersections with the 210 hyperplanes.
The plane section is defined in a user-friendly form. Figure 9 shows the general form of such plane section.
It is a curve plotted in a (x, y) reference system. The user wants the x-axis to represent a certain stress component and the y-axis another. He merely needs to define the unit vectors S,, and S in terms of stress components. Let the six dimensions of stress space 54 P. VAN HOUTTE Y Sx x FIGURE 9 (x, y) is the orthogonal coordinate system of a two dimensional section of stress space. It is defined by specifying the unit vectors S,, and Sy in terms of stress components. respectively correspond to S, $22, $33, $23, $3 and S2. Suppose that one wants a S, S2-section.
It is clear that in view of condition (21), the $23 and S31 components must remain zero. It is also possible to ask for two predetermined unit vectors $ and Sy that represent a section by the :r-plane. The vectors $ and Sy can also be specified in terms of stresses that are expressed in a reference system fixed to a tensile test specimen cut from the sheet at an angle c from the rolling direction (Figure 1). Eq. (18) can be written as follows (regarding Eq. (21)): PllSll -b P22S22 + P33S33 -1-P2S2 :/17/ The P values that correspond to the 798 hyperplanes are stored once and for all.
The method of finding their intersections with the plane section CALCULATION OF THE YIELD LOCUS 55 defined by S and S consists in expressing Pq in a different reference system (in stress space) for which $ or $ belong to the set of basis vectors. It then becomes very easy to find the intersection. The method has been described in detail elsewhere (Mols, 1982).
The computer code that draws the yield surface section also makes a printout of the strain tensors normal to the yield curve, including those strain components that are invisible in the plane section itself.

EXAMPLES OF RESULTS
A few examples of results both for f.c.c, metals and b.c.c, metals will be given here.
The  (111)), with equal critical resolved shear stresses on all systems. The resulting Mvalues are somewhat smaller than those obtained for f.c.c, metals. Figure 10a shows the :r-plane section for an isotropic f.c.c, metal, according to the full constraints model. The result of the relaxed constraints (pancake) model is given by Figure 10b.
The yield curve of Figure 10a looks like a Tresca yield curve (hexagon) with rounded corners. Compared to this, the stress levels of the "pancake" curve ( Figure 10b) are ca. 10% smaller. It is interesting to note, that for $22 $33 0, the radius of curvature is very small, suggesting a vertex. Figure lla shows the :r-plane section for an isotropic b.c.c, metal (F.C. Method). This curve is much more rounded off than the f.c.c. curve of Figure 10a; it is closer to the von Mises circle than to the Tresca hexagon. Figure llb shows the same according to the relaxed constraints (pancake) model. Again, the stress levels are reduced; a region of small radius of curvature is observed near the points $22 $33 0, but not to the extent of suggesting a vertex. Figure 12a shows the S1-$22 section of a b.c.c, metal, namely a  (Van Houtte, 1983 and1984a). The fullconstraints method was used for the curves of Figure 12a, while the pancake method was used for Figure 12b. In both cases the tr 0 and the tr 45 curves are given (see Figure 1).
According to these calculations, there should be in-plane shearing when performing uniaxial tensile tests at 45. The values of the /12//11 ratios were taken from the printouts that accompany the curves of Figure 12a-b. At $22 0, a value of -0.143 was found for the full-constraints case, and a value of 0.143 for the pancake case, suggesting a quite different strain anisotropy. Exactly which model applies best is the object of an experimental investigation under course. However, the curves of Figure 12a and Figure 12b predict the same R-values (2.45 for tr 0 and 1.67 for tr 45) for the full constraint and the relaxed cases.
The method can of course also be applied to a theoretically predicted deformation texture. Figure 13a-b show an example of this. A b.c.c, rolling texture was simulated using the pancake version of the relaxed Taylor theory, extended with Wintenberger's second-order selection criterion for those cases where the solution for the slip are not unique (Renouard and Wintenberger, 1981). A uniform texture consisting of 147 separate crystallites was chosen as starting texture. Then 50 increments of 1% thickness reduction (plane strain) were simulated, followed by 30 increments of 5% strain each. A total true strain e 2 was so obtained, representing a thickness reduction of 86%. A continuous O.D.F. was derived from the 147 discrete orientations in the following way: For each orientation, a Gaussian distribution was considered: is the minimal rotation angle, required to rotate the considered crystallite to the orientation g. 0 is the spread of the Gauss distribution. Its value was 7. So is chosen in order to normalize the texture function: The O.D.F. of the texture as a whole is the weighted average of the 147 texture functions given by Eq. (69). It is quite easy to CALCULATION OF THE YIELD LOCUS 61 calculate both the even and the odd C'V-coefficients of such texture (Bunge, 1982). These coefficients were used to calculate the yield locus. Figure 13a shows a $11-$22 section (both for a=0 and a=45), using the "full constraint" M-values" (although the simulated texture was produced by the relaxed constraints method).
The printout of the strain tensors provided the following results: R-values: 2.45 for 0 and 2.31 for c 45 /1//1 -0.15 for a 45 Figure 13b shows the S1-$1 section of the yield loci calculated using the "pancake" M-values.
R-values: 1.7 for a 0 and 2.4 for a 45 /// 0.15 for a 45 Figure 14a shows the S2a-Sa section of the yield locus of the surface of a solid copper bar after torsion testing. The total shear at the surface was equal to 5. The {100} and {111} pole figures are given by Figure 4; the O.D.F. has been reported earlier (Van Houtte et al., 1981). Figure 3 gives the conventions for the reference axes. The component $1 is the shear stress induced by the torsional deformation. Saa is the normal stress in axial direction.
Suppose a free end torsion test. At first sight, one would think that all stress components are zero except S. This stress is represented by A in Fig. 14a. The normal to the yield locus is not perpendicular to the Sa axis. This means that in addition to the shearing, there should be some extension of the sample. However, the printout of the strain tensors reveals that this is not an axisymmetric elongation, but a plane strain elongation, the strain in radial direction being zero. Such deformation would be possible if the sample was a thinwalled tube. In addition to the torsional shearing, the tube would become longer; its radius would become smaller, while the wall thickness would remain constant. The rate of change of the radius would equal minus the rate of change of the length. This has also been found by Canova et al. (1985), who used a theoretically predicted torsion texture in order to derive the yield surface.
The elongation of a solid bar must however be axisymmetric. This means that the rate of change of its radius must be equal to minus half the rate of length change. This would require the building up of 62 I. _t CALCULATION OF THE YIELD LOCUS 63 a positive Sll (tangential) normal stress, while probably also inducing axial stresses $22. The free end indeed does not require that $22 is zero at the surface, only that the average of the $22 component over the entire cross section must be zero.
An exact solution of the problem of predicting the ratio of shearing/elongation of a solid bar during a torsion test would be quite complex. The stress components Sxx, $22, $33 and Sx2 are all expected to vary strongly from the surface to the centre. The stress equilibrium equations should be taken into account. Moreover, the texture and the degree of work hardening also vary from the surface to the centre.
An approximate solution that only accounts for the outside layer is possible. One way of doing it would be to transform the measured O.D.F. into a set of discrete crystal orientations. Then a procedure which has been proposed by Gil Sevillano, Van Houtte and Aernoudt (1975) can be carried out. A Taylor The most probable value of r/ is the one that corresponds to the minimum of M.
The macroscopic stress tensor would then be obtained as the average of the Bishop-Hill stress tensor of the individual crystallites, at least when the theorem holds that the macroscopic stress of a polycrystalline aggregate equals the average of the microscopic stresses of the individual crystallites, as has been suggested by Canova et al. (1985).
Another method, which avoids the transformation of the Ccoefficients of the O.D.F. into a discrete set of crystallite orientations, will be explained below. It is based on a thorough exploration of the yield surface. The basic idea is to look for stress states for which the strain rate tensors (given by the normals to the yield locus) are of the type given by Eqs. (71-72).
Because of the theorem described in section 2, Eq. (72) Let x and y be the coordinates of the points of the resulting yield curve (Figure 9). It is seen that, besides Eqs. ( This plastic work must be produced by the external torque that is applied to the test sample. The twist angle that is realized is proportional to the shear E2. So the "resistance" of the material against twisting is proportional to the quantity ff'//2. Of all stress states for which the strain rate satisfies Eqs. (71-72) (for example, the stress state B in Figure 14b), the one that minimizes this quantity is supposed to be the most probable. Solutions were found in the sections p 0 , 15 , 30 and 45. For more information, an additional section at/9 37.5 was produced.
The minimum for //12 was found for p 30 and 37.5 and for a positive value of r/, the ratio of the axial elongation to the macroscopic shear.f Figure 15 shows the evolution of I///2 as a function of/9. For reasons of clarity, only the solutions for which r/ was positive are shown. The figure also shows the associated values of r/, $12 $11 and $22 ($33 being zero).
In the vicinity of the minimum, the values of 1///a2 and r/were found to be quite intensitive to p. The ratio between $11 and $22 is of course sensitive to 19, much less however their average. According to these calculations, the rate of elongation r/should be no less than 11.5%. The stress state consists of a shear stress (as expected in a torsion test) combined with a far from negligible biaxial tensile stress. The elongation rate r/ is probably overestimated when compared to experimental results, the main reason being the neglection of the role of the inner layers of the sample, which probably have a smaller tendency for elongation than the surface layer. It can also be expected that the hydrostatic component of the stress would decrease from the surface to the centre of the sample, thus reducing the stresses $11 and $22 and provoking a compressive stress in radial direction. Figure 16 shows true yield curves (obtained by the present method) and some "principal strain yield loci" obtained for a sheet with a very sharp texture: a Gaussian distribution with a narrow spread (0 7 , Eq. (69)). Its centre of gravity lies exactly on the (110) [001] (Goss) orientation. The F.C. M-factors for b.c.c, metals were used. The yield curves have been calculated for tr 22.5 and te 45 (Figure 1). It is seen that the differences between the true yield curve and the principal strain yield curve are quite spectacular.
The same holds for the R-values: for example for o =45 , the "principal strain" yield curve predicts a value of 0.786 and the "true" yield curve a value of 0.287.
A similar comparison carried out for the experimental O.D.F. of Figure 12a however only led to a very small difference between the "true" yield curve and the principal strain yield curve. At present it " This means, that the sample should elongate when a torsion test would be carried out in the same sense as the prestraining. The minimum for W/E12 gives the most probable solution for p. Due to lack of resolution of the method, it was impossible to differentiate between p-30 and p--37.5. The corresponding values of the rate of elongation r/, the torsional stress $12, the axial stress $22 and the normal stress in tangential direction S are also shown.
is not known whether this holds for all textures of polycrystalline samples that are of industrial interest. More experience must be gained before a definitive conclusion can be made.
An additional computer program permitted to produce at wish plane sections through it. This method used 51408 hyperplanes. Although the calculations were not unreasonably lengthy, an interactive exploitation proved to be too expensive for a university laboratory. A truly low-cost procedure is proposed in the present paper. It can be applied to textures for which the x3-axis is a two-fold "sample" symmetry axis. The complete yield locus is no longer obtained, but only its section by the ($11-S2-$33-$12) subspace. It however is complete in that subspace. Although the method is less general than its predecessor, it nearly covers all practical applications.
The average M-factors needed for the hyperplanes are first calculated. In a next step, the user defines the plane sections that he wants. "True" sections are obtained: when e.g. a ($11-$20 section is p.roduced, then the shear stress $12 is truly zero, and the shear rate Ela is computed together with the shear rates /11 and 22. The values of such predicted shears (produced by the older method) have been found to correlate positively with experimentally observed shearing of tensile samples cut from textured sheets (Mols et al., 1984).
The M-value data bases used in the present model for the first time not only include the usual M-values of f.c.c, metals ({111}(110) slip) but also those of b.c.c, metals ({110} and {112} slip planes, (111) slip directions). In this regard, there is no need for approximate solutions such as those offered by the "pencil glide" model. Not only the full constraints Taylor-Bishop-Hill model was used, but also the "pancake" version of the relaxed Taylor model (Van Houtte, 1982). This model is thought to be suited for rolled metal sheets which have very fiat grains. Work to verify this experimentally is in progress. The examples given above show, that the method can be used to study the plastic response of anisotropic materials. The experience gained so far suggests, that i) the yield curves themselves are found with a reasonable precision.
ii) the accuracy of the directions of the normals to the yield curves, i.e. the strain modes, suffer from the relatively limited CALCULATION OF THE YIELD LOCUS 69 number of hyperplanes used for the analysis (angular resolution ca. 10). It is useless to apply smoothing procedures to the yield curves in the plane sections since we want to know all components of the vectors normal to them, i.e. not only the 2 components that are seen in the plane section, but the 2 other "invisible" components that also exist in the ($11-$22-$33-$12) subspace as well. A sound multidimensional interpolation procedure is needed. Such procedure has in fact been developed and will be reported in the near future.
When comparing this method to older "principal strain" methods based on series expansions, it can be concluded that: the new method perhaps needs more computer time, but this is not a decisive factor since the required time is quite small anyway (except for the calculation of the M-value data bank, which has to be performed only once); For very sharp (artificial) textures, large differences were found between "true" yield curves produced by the new method and "principal strain yield loci" for tr angles other than 0 or 90 ( Figure 1). Until now, an important difference of this kind was not found for polycrystalline samples of industrial interest.
Nevertheless, even in such cases, in-plane shears E12 of the order of 10% of the elongation strain were quite often found by the new method. Needless to say that the "principal strain" methods are unable to calculate these shears.
It is finally repeated here that in this paper, no attempt was undertaken to replace the (FC or RC) Taylor-Bishop-Hill model by an elasto-plastic model. This means that only large offset yield loci (a few percent plastic strain) can be calculated, since it is implicitly assumed that the multislip stage has been reached in all grains.
In future it could be attempted to combine the method presented here and which essentially provides in a means for exploring the whole S11-$22-$33-S12 subspace, with a more sophisticated model for polycrystal plasticity (such as an elasto-plastic self-consistent model). It would then be possible to predict small offset yield loci (within the elasto-plastic transition range) as well.