Analyses of Plane-strain Compression Using the Upper Bound Method

Traditional upper bound analyses for plane-strain compression leads to low load prediction values, for a large ratio range of contact Length (L) to thickness (h). These predicted load values are based on deformation fields consisted of rigid regions separated by planes upon which discrete shear occurs. In this study, the relatively simple deformation fields consisted of odd number of triangles such like, 1, 3 and 5, are used. A general minimum solution for this class of upper bounds is derived and found to occur when the base of the center triangle (w) (= 2L/ (n+1)), where n is an odd integer ≥3. Consequently, whether still lower values would occur with this class of field when the ratio(R), of the field triangles side lengths is varied, has been systematically investigated. Thus, the upper bound loads are calculated over a wide range of the ratio (R) and the deformation field parameters. These parameters include the thickness (h) the contact Length (L) and the base of the center triangle (w). It is found that, all the minimum load values occur at unity R ratio and the minimum values of h, L and w. The minimum values obtained for hopt, Lopt and wopt are 1.466, 3.266 and 2.0 units, respectively. Also, the corresponding overall minimum optimal upper bound value found is P/2k = 1.93.