Applied Elasto-plasticity of Solids by T. Z. Blazynski

By T. Z. Blazynski

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1c, the externally applied system of loading results in the presence of direct stresses a, and shear stresses T, acting on all the faces of our elementary cube . The notation used there indicates the interchangeability of the subscripts i and j, which denote the surface on which the stress is present and the direction in which it acts. Direct stresses are thus denoted by i = j and shear stresses by i =1= j. 1c that this condition is satisfied if the nine com- ponent forces of the system are considered to be mutually balanced in the three perpendicular directions.

YdA. C 1 0.. xdA. 4 By projecting the component forces in directions normal and tangential to CD, we obtain the following equations SNdA N = axxdA N sin6 0 + ayydAN coslO - 2a x ydA N sin 0 cos 0 SgdA N = axxdA N sin 0 cos 0 --- ayydA N cos 0 sin 0 + axydA N (sin20 - cos20) or SN = 1(ax x + ayy) + 1(ayy -- axx)cos 20 -- a x y sin 20 Sg = 1(ayy -- a x x) sin 20 + ax y cos 20 The above equations give the variation of stress components with the angle O. To locate the principal planes , we consider the maximum and minimum values of sN respectively .

7 These depend for practical and theoretical reasons on (a) retaining in the same indentation, the incremental nature of loading, with the increments not exceeding 2 per cent in the plastic range, (b) maintaining the ratio b/h between 2 and 4 by changing the dies when required and (c) ensuring that w/b > 6. The basic stress-strain curve for the material thus obtained will be that of true stress and natural strain . 4 Torsion test The distortion of the specimen experienced in tensile and compressive testing, on the one hand, and the relatively low rates of straining obtainable, on the other , have led to the increasing use of the torsion test.

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