Dependence of total elongations of superplastic materials on m
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I.
INTRODUCTION
SUPERPLASTIC flow is a rate-controlling process The study of the 0.-k relation is the fundamental topic of superplastic materials. Here the o" is the flow stress, and k is the strain rate. Successful application of the following equation to the study of the 0.-k relation curves of superplastic materials by Backofen established the foundation of the mechanics of superplasticity: l
cr = kk"
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where m is the strain-rate sensitivity index of flow stress, and k is a material constant. The log o'-log k curves of superplastic materials obtained by tensile test are sometimes straight lines; therefore the values of both m and k can be constant under certain conditionsfl However, the curves are usually S-shaped; therefore, the values of both m and k vary with the strain rate k under certain conditions. 345 For S-shaped curves, assuming that the difference between o1 and 0"2 corresponding to kj and e2 is very small, we can define the m value as follows: d log 0" log 0"1/0"2 m -- - -d log k log gt/#2
[2]
which was the base on which the strain-rate change method for measuring m values was developed by Backofen. Generally, it has been considered that the uniform stretching and the high total elongation of a material occurring during a tensile test are due to the fact that it has a high m value. 6 Therefore, m values are often used for expressing total elongations of materials. The value of m always varies with increase in strain (6); thus we can have the m0 (~v0), mi (mll, ml2, mr3 . . . . ), m F values corresponding to the initial strain, 60 (-- 0.00 pct), the instantaneous strains of each period during stretching, 61 (6t~, 6z2, tSt3. . . . ), and the total elongation, 6F, at fracture for every material under certain conditions, respectively. 7-1~ On being used without any symbols (i.e., as m and k), they can represent any of these values which is required for the condition studied. The dependence of m values on 6 values may be simply called the m-6 relationships including the curves and the
equations. For the mutual comparisons of relative superplasticity of different materials, the m value alone can not do so, because different materials of the same m value may have entirely different strains or total elongations, although the total elongation value alone can do well for this purpose. It seems that the best way is to use the 6/m ratio, which is either the strain or the total elongation caused by unit value of mt or mF, respectively, and could be called the "strength of superplasticity" of the m value of a material. We can have 6o/mo, 6~/m~, and 6F/mF ratios, although the first one is meaningless. The purposes of this paper are to review all the relationships including the curves and the equations published by the superplastic researchers and analyze their inadequate treatments in establishing them so that the study of the m-6 relationships may be made on a new base.
II.
T H E m-6 CURVES
The m-6 curves of some materials published by a few researchers are shown in Figures 1 through 9.~7-~9'~1'22 All
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