Generalized solutions in the theory of plasticity

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Conditions prevailing on the surfaces of the strong velocity discontinuities in rigid-plastic media were studied by many workers, e.g. /1, 2/. However, in all cases known to the author the conditions were obtained by utilizing a passage to the limit, when the surface of the discontinuity was considered as a limit to which a layer tends, the layer undergoing an intense deformation and its thickness tending to zero. Meanwhile, it is desirable to obtain the conditions at the discontinuities by intrinsic means from the system of equations itself, without bringing in the irrelevant concepts on what represents the surface of the discontinuity. To this end the equations must be given in divergent form. In the theory of plasticity the main difficulties in this respect are encountered in connection with the law of flow and the law controlling the hardening. The present paper shows that certain generalization of the Mises principle makes it possible to impart to the inequality expressing it a divergent form and enables us to write it in integral form. From this it follows that in the incompressible plastic medium the surface of discontinuity in the tangential velocity component serves as the surface of maximum tangential stresses, with tangential stress directed along the velocity jump vector. In a compressible plastic medium the stress discontinuity is determined from the condition that the direction of the six-dimensional deformation velocity “vector” is continuous. We note that the integral form of the Mises inequality was used in /3/ to prove the existence and uniqueness of the solution. It was not, however, given in divergent form, and the conditions at the discontinuities were not considered. With regard to the equation describing the hardening law, it can be reduced to divergent form when the specific plastic work is used as the hardening parameter. The problem considered here is that of steady motion of a strip of finite thickness undergoing pure shear, in a rigid-plastic hardening medium. The emission of heat caused by plastic deformation and its effect on the functions of state and the forces of inertia are all taken into account. It is shown that under certain specified conditions the strip thickness may tend to zero, i.e. the appearance of isothermal velocity discontinuities is possible. The adiabatic and quasistatic cases are discussed. It is noted that the insufficiency of the continuous solutions in the connected perfect thermorigid-plastic medium was discussed in /4/. In important practical problems the thickness of the layer undergoing pure shear is small compared with the characteristic dimension and can be neglected. Therefore the discontinuous solutions can also be brought in in the general case when heat conduction is taken into account. In this case the condition of temperature continuity should be omitted. Some of the results of this paper were given in /5/.