Shear, principal, and equivalent strains in equal-channel angular deformation
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TRUE normal and shear strains are the only two types of essential strains in a deformed body. The mathematically rigorous and unique expressions for them, however, are complex to use except for very small deformations where simplified forms can be used. In engineering applications involving large plastic deformations, parameters, such as the engineering strain, elongation, reduction of area, ratio of compression, ratio of extension, and ratio of extrusion, are routinely used to measure the deformation intensity in tension, compression, drawing, or extrusion processes in order to avoid mathematical complexity. This practice is carried out at the expense of rigour and accuracy, although, in the previously mentioned processes, the major strain is longitudinal with the principal directions unchanged during deformation, and the parameters can generally be related to the true normal strain. Equal channel angular deformation (ECAD), on the other hand, subjects material to severe plastic strain in the form of simple shear.[1,2] The deformation process has been successfully used to refine grain sizes to below 1 m in many alloys including aluminum,[3,4] magnesium,[5] steel,[6] titanium,[7] and titanium aluminide.[8] In the ideal situation, where the material being deformed is a rigid ideal plastic body (i.e., with an elastic modulus of infinity and a constant flow stress) and there is no friction with the wall of the channels, the deformation is of simple shear along the plane of intersection of the two channels. It is desirable to know the strain and strain path in such a deformation. For simple shear, the engineering shear strain is defined as ␥ ⫽ tan
[1]
K. XIA, Senior Lecturer, is with the Department of Mechanical and Manufacturing Engineering, University of Melbourne, Victoria 3010, Australia. J. WANG, Professor and Deputy Dean, is with the School of Metallurgical Engineering, Xi’an University of Architecture and Technology, Xi’an 710055, People’s Republic of China. Manuscript submitted January 11, 2001. METALLURGICAL AND MATERIALS TRANSACTIONS A
where is the angular distortion caused by shear, as shown in Figure 1. Such a definition has been used to calculate the total shear strain in ECAD,[9] which, for the ideal situation, is
␥ ⫽ 2 cot
冢2冣 ⌽
[2]
where ⌽ is the angle between the two channels. The same expression for the shear strain can be reached based on slipline field theory[10] or an upper-bound solution.[11] Further, the total equivalent strain[9] or strain intensity[2] after one pass has been calculated to be e ⫽
␥ 冪3
⫽
2 ⌽ cot 2 冪3
[3]
This equivalent strain has been predominantly used to describe the degree of plastic deformation in ECAD. Such a simplistic approach, however, has not considered the principal strains and ignores the fact that the principal strain axes are known to rotate, whereas, the axes of the principal stresses remain fixed during simple shear, and the strain path is noncoaxial.[12,13] These considerations may have a significant effect on the understanding of the m
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