Printed in the United Kingdom Model Calculations of Elastic Moduli in Highly Deformed Composites

Young's modulus of heavily deformed two-phase composites shows an unusually high 
increase after plastic deformation. It is assumed that this is due to two reasons, i.e. 
texture changes and changes of the moduli of the constitutive phases on the basis of 
non-linear elasticity theory and internal stresses of opposite sign in the phases. 
Expressions of the two contributions are given on the basis of simple model 
assumptions. It is estimated that the changes of shape and arrangement of the phases 
and shape and arrangement of the crystallites in the phases are only of minor 
importance.


INTRODUCTION
Composite materials can be produced by powder metallurgical methods.Thereby fine powders of the component materials are mixed, compacted and extruded at an appropriate temperature with deformation degrees in the range of 70-90% (see e.g.Wassermann  1981).This latter step is essential since it is the reason for the formation of a compact composite.The so obtained materials can be further deformed by rolling or wire drawing.Materials of this type consist of two (or more) phases the particles of which are elongated in one direction (fibres) or in two directions (bands).The 153 particles themselves are polycrystalline with, usually, strong textures.
Composites of this type exhibit a number of "anomalous" properties which means that these properties cannot be understood on the basis of the properties of the components, alone.Rather, the anomalies are due to phase interactions of various kinds.One of these anomalous properties is an unexpectedly strong increase of Young's modulus with the degree of deformation (Welch, Ratke,  Wassermann 1984).
The elastic properties of a multiphase, polycrystalline material may, in principle, depend on the following structural parameters" 1.The macroscopic properties are the weighted mean values of the properties of the phases.
1.1 These mean values depend on the shape and arrangement of the phase particles.2. The elastic properties of the particles themselves will, in general, be anisotropic.They are the orientation mean values of the crystallographic anisotropies of the corresponding crystallites.
2.1 These mean values, too, depend on the shape and arrangement of the crystallites within the particles (phases).
3. Finally, the elastic moduli of the constitutive crystallites themselves may deviate from their ideal values measured in big single crystals.This latter effect may be due, for instance, to: 3.1 lattice defects, e.g.dislocation substructures 3.2 internal stresses This latter influence will be particularly discussed in the following.The structural variable of point 1 is the volume fractions x of the phases (a) by which the mean values are to be weighted.Usually the linear mean value is considered but also the higher order terms of x may be included.If we consider only the variation of the elastic moduli with the degree of deformation, this term remains constant.
The shape and arrangement of the particles can be described by particle shape functions and phase correlation functions which have, however, not yet uniquely been defined nor are they taken quantitatively into account in a rigorous theory.The influence of shape and arrangement of the phases is generally smaller then the influence of their volume fractions.It expresses itself in the higher order terms of x in the mean value formula.Furthermore, if only the variation of the elastic moduli is considered then only the variation of this higher order effect may enter.Hence, the contribu- tion of this effect is also considered to be small.The controlling variable in point 2 is the textures of the phases.The textures are changed by deformation.This holds also for high deformation degrees were the type of texture is generally well developed.Nevertheless, the degree of the texture--expressed for instance in the texture indexmay still increase.Hence, the contribution of texture changes to the variation of the elastic moduli cannot be neglected.
It has also been found that the textures of the phases in multiphase composites may be strongly different from textures in single phase materials with the same degree of deformation (Wassermann, Bergmann, Frommeyer 1978).Furthermore, texture measurements by x-ray diffraction in multiphase materials may suffer from the effect of anisotropic absorption (Bunge, Liu,   Hanneforth, 1987).Hence, quantitative texture measurements in deformed composites must be carried out by neutron diffraction (Brokmeier, B6cker, Bunge in print) which has not yet been done for the above mentioned Fe-Ag composites.
The elastic moduli C of a textured single phase polycrystalline material can be calculated in a good approximation using the Hill average of the Voigt and the Reuss mean values (Bunge 1968, 1974,   Morris, 1969, 1974).For the general case of non-equiaxed grains, Hill's assumption must be formulated (1) whereby rn 0.5 corresponds to the original Hill approximation which is assumed to be valid in equiaxed grain structures (Hill 1952).In highly deformed composites this assumption cannot be made.Nevertheless, the value of rn can be assumed to be near to 0.5 and its variation with the degree of deformation will also not be very strong.Furthermore, the difference between (" and itself is only in the order of 10-20%.Hence, a change of m may result in a variation of/ of a few percent only.
The influence of lattice defects on the elastic properties of crystals is in principle well known.The quantitative dependence on the details of e.g.dislocation substructures is, however, not very well known.The development of dislocation substructures in highly deformed composites has been found different from corresponding single phase materials.Smaller dislocation cells were found in composites.On the other hand, very small particles in composites were found essentially dislocation free.Hence, it is difficult to estimate a contribution of changing substructure on the variation of the elastic moduli upon deformation.
Finally, the influence of the last one of the parameters, listed above, i.e. internal stresses, can be quantified more easily.When a composite is being deformed, the necessary stresses in the phases will be different according to the different yield strengths (including effects like dislocation hardeneing, texture hardeneing or softening, boundary influences expressed by a Hall-Petch relation and others).After deformation the applied stresses will relax when no more external stress is present.In this state the phases will develop internal stresses of opposite sign and rather high absolute values (see e.g.Cohen 1986).This requires to calculate the average elastic moduli of the stressed particles on the basis of non-linear elasticity.The same effect has been assumed by Janowski and Tsakalakos   (1985) for layered two-component materials.

NON-LINEAR MEAN VALUES OF THE ELASTIC MODULI
In a simple model, we assume a two-phase material consisting of the phases a and fl with the volume fractions x and x a x + x a 1 (2) The composite may be deformed plastically whereby a stress-strain relation #() holds.Similarly, the stress-strain relations of the two phases be tr(e ) and trt(et).If the yield strengths of the phases are different, the phases will undergo different strains e and e t which are somehow related to the average strain (3) The latter one may be assumed to be the linear mean value of the strains and phases =X .e+Xtl, etl These strains contain elastic and plastic parts which need not to be distinguished here.In the present considerations, the specific forms of the functions in Eq. ( 3) do not matter.They allow, however, to express o and o t as functions of , i.e. o() and ot().The mean stress of the composite is assumed to be the linear mean value x".

+
(5) After deformation the macroscopic stress O is relaxed to zero.Then Eq. ( 5) refers to the residual stresses o and o in the two phases.It We assume, for the sake of simplicity, that this relaxation process is completely elastic as is shown in Figure 1.Then each phase undergoes the same elastic redeformation Ae which is also equal to the macroscopic redeformation.Within the linear elasticity theory it Where C and C a are the linear elastic stiffness tensors of the phases c and ft.With Eq. ( 5) and ( 6) one obtains o (?. zxe (8) with x '.C + x t.C o (9) Eq. ( 8) may be inverted Ae .0(10) where q is the elastic compliance tensor corresponding to (.This allows the residual stresses to be expressed in terms of the average stress (x C .S. (11) Or O= a -C o. " 0 (12)   Since the stresses a and o t reached during deformation are not very well known, anyway, it seems justified to use linear elasticity theory in Eq. ( 7) although we shall use later on.the non-linear theory for the smaller residual stresses.
The stress derivatives ( of the elastic constants used in Eq. ( 14) and ( 15) can be expressed by the third order elastic stiffnesses in the following way.With the definition O-" C, e -1--C (3)  where C 3 are the third order stiffnesses and S are the second order compliances.
It is seen (Eq.16, 17) that in the case of internal stresses and non-linear elasticity theory the mean value of the elastic moduli contains a term which depends on the internal stresses and which may thus increase with deformation due to increasing stresses.The third order elastic constants are numerically not very well known.
Hence, only the order of magnitude of the expected effect can be estimated.In the Landolt-B6rnstein data collection compiled by Hearmon (1979), values of Clll for iron are given which range from -4820 to -2830 GPa, the reported value for silver is -843 GPa.
Using these values and the values $11 for S in Eq. ( 22) (i.e. 7.67 (TPa) -and 22.9 (TPa)respectively) one obtains values of (Fe__ (Ag) ranging from 4.8 to 53, which seams quite reasonable.The increase of Young's modulus ('-t)/( found by Welch, Ratke and Wassermann (1984) was in the order of magnitude of 0.4 at 94% rolling reduction.According to Eq. ( 17) this requires Fe residual compressive stresses cr in the iron phase in the order of magnitude of 0.8% to 8% of the average Young's modulus (.Internal stresses in the order of magnitude of 1% of Young's modulus were frequently found in single phase materials (see e.g.Macherauch and Hauk 1983).Internal stresses in two-phase mate- rials must be assumed to be even higher (e.g.Cohen 1986).Hence, the estimated order of magnitude seems to be quite reasonable.
It is, thus, possible to understand an increase of 40% of Young's modulus upon rolling on the basis of non-linear elasticity.On the other hand, also texture changes must still be assumed in this range although quantitative texture measurements and calculations of Young's modulus on this basis were not yet carried out in the above-mentioned case.The texture effect and the internal stress effect may be additive or substractive in certain cases which may account for different results of the magnitude of increase of Young's modulus in two-phase compositers (e.g.Bevk et al. 1978, Bevk  1983).

TEXTURE MEAN VALUES
The quantities C , C a and t , t t are texture mean values of the corresponding single crystal quantities taken with the two textures fO(g), ft,(g) of the two phases.If we assume that grain shape in the two phases does not change very much during deformation, i.e. m const in Eq. ( 1), then the texture mean values of C and t in Eq. (14, 15) can be calculated with either of the two approximations according to Voigt or Reuss.
In order to calculate the orientation mean values, the tensors C and t must be referred to a common sample coordinate system Kt.
The crystal coordinate system KB of a considered crystal may be related to KA by the rotation g gB g gm', g [gij] (23) The components of the tensors in the crystal coordinate system be Ckl o Cijklmn.The components in the system KA are then given by the transformation relations Tijkl (24) where T and T' are orientation factors and the Einstein summation is being used.The orientation mean values of these quantities are given by 'ijkl (26) abcdel "0 r,ijklmn C ijklmn abcdef where 2 and 2' are texture factors defined by "t'ijkl [,.abcd abcd,15 ]" f (g) dg (28) ijklmn t, ijklmn[ abcdef Zabcdef 1" f (g) de (29) The texture may be given in terms of the series expansion (see e.g.Bunge 1982) .=0 /=1 v=l Also the orientation factors T and T' of Eq. (24,25) can be expressed in terms of these functions (31) (32) In the case of cubic crystal symmetry and axial sample symmetry, the function qg(fl) is to be replaced by another function tp'(te) where a is the angle to the axis direction qg'(tr) a'-C cos 2a (42) In this case, only one texture coefficient is involved.

CONCLUSIONS
The observed, rather high increase of Young's modulus of two- phase composites after plastic deformation may have several reasons.Estimations show that the changes of the geometrical structural parameters such as the shape and arrangement of the particles of the phases and the shape and arrangement of the crystallites in the phases is only a second order effect, the contribution of which is assumed to be small.Internal stresses in the phases after deformation will have opposite sign in the two phases.If it is assumed that they reach an order of magnitude of 1% of Young's modulus then the stress dependence of the elastic constants must be taken into account.A variation of Young's modulus in the order of 40% as was observed, may thus be, in principle, understood.
A second contribution to the variation of Young's modulus with deformation will certainly be due to texture changes.In this respect, the influence of anisotropic absorption of x-rays in anisotropic two-phase materials must be taken into account.This effect may strongly falsify texture measurements by x-rays.Quantitative tex- ture measurements in such materials must thus be carried out by neutron diffraction where this effect does not occur.
The contribution of textures changes and that of residual stresses to the macroscopic Young's modulus may in principle be additive or substractive which may account for the different results found in different composites.

Figure 1
Figure 1 Plastic stress-strain curves of the phases of a two-phase composite (schematically) and the elastic relaxation.