In the previous paper, the authors studied on the deformation of polycrystalline copper by X-ray diffraction, and it was suggested that the difference in work-hardening rate of each grain due to its orientation played an important role in the deformation behaviour of polycrystalline copper. That is, in such metals as copper, the plastic anisotropy of crystals must be considered in the problem of plasticity of polycrystals.
G.B. Greenough have introduced the relation between the microscopic residual stress among the grains resulting from the uniaxial tension (stress of the 2nd kind) and the residual lattice strains measured by the X-ray diffraction from several crystal planes perpendicular to stress axis. However, Greenough assumed, according to Taylor's theory, that the work-hardening of the grain depended only on the total shear strain on the slip planes, which does not seem to be adequate for such metals as copper. Furthermore, Greenough put assumptions that the total axial strain was equal in all grains under the loading condition and the crystal was elastically isotropic.
In this paper, the basic relation between the strains measured by X-rays and the crystal plasticity was investigated in more general case, namely, in arbitrary state of mutual interference between the grains and for elastically and plastically anisotropic metals.
The constraint ratio
n was defined at the beginning as,
Δε=n[Δε]Δσ=0
where Δεand Δσ were microscopic strain and stress in a grain respectively, and were given as, Δε=ε-ε, Δσ=σ-σ, where ε and σ were the mean values of strains and stresses of all grains, respectively. Then, the case of
n=0 corresponds to constant strain condition, while the case of
n=1 to constant stress.
Under the assumption that X-ray diffraction plane is at a right angle with tensile axis, the following two relations which give the X-ray measured strains Δε
ey associated with tensile loading have been found to hold (where Δε
ey is the mean value of Δε
ey for all the reflecting grains). First, in the loading condition,
Δε
eyL=s
12'·ΔF(ε)(1-n
L)
s12' shows the elastic compliance concerning the co-ordinate peculiar to the specimen, and ΔF shows the difference in work-hardening curve of a grain from its mean value of all grains, and
nL shows the constraint ratio at the loading time. Next, after unloading the residual strain is given as,
Δε
eyLU≅s
12'{ΔF(ε)(1-n
L)+Δs
11'/(s
11')
2ε(1-n
U)}
where
nU is the constraint ratio during the unloading process. From these results, it is considered that X-ray measurements have to be done in the loading condition as well as in the unloading one in order to get more information on the plastic property of polycrystals. It may be possible to know the value of constraint ratio
n from X-ray measurements, if the value of Δ
F (that is, the relation between crystal orientation and work-hardening curve) is known from the experiment on the single crystal.
Discussion was also made on elastic anisotropy, and it was suggested that its effect was prominent in such strongly anisotropic metals as copper.
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