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《Acta Scicentiarum Naturalum Universitis Pekinesis》 1956-04
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PLASTIC FLOW BETWEEN PARALLEL RIGID PLATES UNDER COMPRESSION

Wang Jen  
The present paper treats the compression of a rectangular block between two parallel rough plates as a problem in the theory of plane strain for perfectly plastic-rigid materials.At first, the plastic-rigid theory of plane strain was outlined, then, the solution to the present problem is briefly surveyed. In section 4, the case that is left out in the present literature, viz. when the width-height ratio lies between 1 and 3.64 for partially rough plates is solved. In this treatment, the coefficient of friction ν is considered as constant along the contact surfaces. For eachμ, a critical value of the ratio w0/h is given. When w/hw0/h, the plastic region extends across the entire block, i. e. the unrestricted plastic flow is possible, before the frictional force is put to full use. In this case, there is no slip along the contact surfaces and the solution is the same as if the contact were perfectly rough.When w0/hw/hwo/h + 2 tan μo=w1/h, where μo is the frictional angle shown in Fig. 3, the slip line field can be constructed in a similar manner as initiated by A. F. Green (ref. 5). In the present case, however, a new parameter connected with μ should be added and the semi-inverse type of construction becomes very complicated.For w/h, the construction is again straightforward. The graphical construction used is described in detail in Appendix Ⅱ.The results of the solution are expressed in terms of Figs. 6 and 7, which show respectively: the average pressure on the plates vs. the width-height ratio for different values of μ and the pressure distributions along the plates. Although the construction for the intermediate case, w0/hw/hw1/h, was not canied out, it is clearly seen in Fig. 6, that the straight line interpolation cannot be far from that of the true solution.Analytic expresions in terms of a rapidly convergent series for the nodal points in a slip line field defined by equal circular arcs are given in Appendix I. The computation was compared to the results obtained by R. Hill (ref. 1) using graphical construction with fairly rough meshes. The comparison shows that the graphical construction used is accurate for all practical purposes. From these expressions we obtain the analytic expression for wo/h in terms of the frictional angle connected with μ(Eq. 11).Finally, a short discussion on the graphical construction used for the case of constant μ is given in Appendix Ⅱ.
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