Theoretical Investigation of Ψ-Splitting After PlasticDeformation of Two-Phase Materials

e0 is the average normal strain of the reflecting grains pointing in the N h direction.Nh is perpendicular to the reflecting planes. ij corresponds to the mean stress ob mined by averaging over the total volume of the reflecting phase. S and S2 are the X-ray elastic constants. The subscripts of (r and e refer to the coordinates of the speci men. The 3-direction p3 is defined to be perpendicular to the flat surface which is investigated with the X-rays. Nh then has the components Nhl (sin cosp), Nh2 (simg h h 3 sinq)) and N 3 (cos). 1/and q) are the angles between N and P and between P and the projection of N" on the 1-2 plane, respectively. The equation (1) is valid for macroscopic isotropic elasticity of the diffracting phase. 13 and/or 23 terms occur if the principal axes of are tilted with respect to p3. Because these terms are multiplied in equation (1) by sin 2g, eg will be different from ep_W. This effect is called "-splitting" and may be observed after machining (milling, turning, grinding...) or directional shot peening of metals. With few exceptions, wsplitting was observed only for multiphase materials. It is strongest in the plane which contains p3 and the direction of machining or shot peening and it is smaller or even disappears in the plane perpendicular to this direction. More details are given in the references [2] [3] [4]. In principle, wsplitting may be related to very different origins namely (i) special texture of the surface layer, (ii) surface roughness and/or (iii) complex stress fields which yield a tilting of the principal axes of the average stress and strain tensors with respect tO p3o


INTRODUCTION
X-ray stress analysis is based on the sin2g law which is described in more detail in a paper by Krier et al. [1] published in these proceedings. For the present discussions, we need the more complete equation relating the elastic strain e0 with all tensor components (ij of the stress, including the cr3j-terms 2 2 E -S 2 loll COS (P +(Y22 sin p + (Y12 sin20-Or33 ] sin2g + S [311+3] 1 +--2 $2 [3 + 13 cosp sin2g + (Y23 sinq sin2g] (1) e0 is the average normal strain of the reflecting grains pointing in the N h direction.N h is perpendicular to the reflecting planes. ij corresponds to the mean stress ob mined by averaging over the total volume of the reflecting phase. S and S 2 are the X-ray elastic constants. The subscripts of (r and e refer to the coordinates of the speci men. The 3-direction p3 is defined to be perpendicular to the flat surface which is investigated with the X-rays. N h then has the components Nhl (sin cosp), Nh2 (simg h h 3 sinq)) and N 3 (cos). 1/and q) are the angles between N and P and between P and the projection of N" on the 1-2 plane, respectively. The equation (1) is valid for macroscopic isotropic elasticity of the diffracting phase.
13 and/or 23 terms occur if the principal axes of are tilted with respect to p3. Because these terms are multiplied in equation (1) by sin 2g, eg will be different from ep_W. This effect is called "-splitting" and may be observed after machining (milling, turning, grinding...) or directional shot peening of metals. With few exceptions, wsplitting was observed only for multiphase materials. It is strongest in the plane which contains p3 and the direction of machining or shot peening and it is smaller or even disappears in the plane perpendicular to this direction. More details are given in the references [2] [3] [4]. In principle, wsplitting may be related to very different origins namely (i) special texture of the surface layer, (ii) surface roughness and/or (iii) complex stress fields which yield a tilting of the principal axes of the average stress and strain tensors with respect tO p3o We study only the last mentioned case for which eq. 1 applies and which seems to be the most frequent one. We start from a micromechanical model which is based on the plastic inclusion problem of a material containing a hard and a ductile phase. We neglect surface effects and absorption of the X-rays. THEORY We consider a material which is homogeneous on a macroscopic scale but heterogeneous on a microscopic level and composed of a ductile matrix and ellipsoidal nonductile inclusions of volume fraction f. The grains of both phases have the same isotropic elastic behaviour described by the shear modulus I.t and Poisson's ratio v. This material undergoes a global deformation E composed of an elastic deformation E e and a plastic part EP. Describing by e pM and e pE (= 0) the mean plastic strain in the matrix (M) and the inclusions (E) respectively, one can write (2) The solution of the problem of an inclusion in a matrix (Eshelby [5], Kr6ner [6]) allows us to write, using the Einstein convention -45 If E e 0 (no applied elastic strains) and since e pE 0, equation (3) can be written as eE ij (IiM-Sijka) F (4) where Iijkl is the identity tensor and Sijkl is the Eshelby tensor. The normal average strain of the inclusions becomes for the N h direction E The corresponding value of eM0g is obtained from the equilibrium conditions for E e 0, i.e. M f E g, _f g (6) Equation (5) indicates that there are two sources for wsplitting namely anisotropic S or EP tensors with tilted principal axes with respect to p3. We [7] recently studied the last mentioned case which is observed for spherical inclusions and which yields an isotropic S-tensor of the form [5]  In the special case of EP corresponding to oblique shot peening p FI==--F E=Ex otherEii=O we obtained the following equation for eMv (8)

APPLICATIONS AND DISCUSSION
We investigate penny-shape ellipsoidal inclusions with axes a b and c/a << 1.
If S is referred to these axes, it has only the following non-vanishing components [8] 13-8v n:c_ S m =1-l'2vrc, S 33 S rczz The orientation of the inclusions with respect to the P-coordinates is described analogous to N h by the angles c between p3 and c, corresponding to , and y corresponding to q. We f'trst discuss perpendicular shot peening, which yields the plastic strain The term which is proportional to C yields the -splitting seen in Fig. 2  It seems evident that phase-specific average 33 stresses will also occur for specimens consisting of randomly oriented penny packages. This is a quite realistic description of lamellar eutectics and means that, e.g., r33 stresses might occur in rolled steel plates.
As a last example, we studied the combination of both EP and S-induced splitting. EP, according to equations (5) and (6), yields the dashed drawn oij-curves show in fig.3. These look similar to the previous results with the exception of t13. It is now seen to be negative over the whole tx-mnge. Finally, we should point out that our calculations are based on the hypothesis of a purely elastic accomodation of the plastic incompatibilities which overestimates the internal stresses. Thus the effects shown in the figures are too strong. In reality, there will be partial plastic accomodation which yields stress gradients. A line profile analysis of the diffraction peaks might give informations about these gradients. This would be very helpful for understanding the interactions between a metallic matrix and the reinforcements of composite materials.