On the equivalence of slip-line fields and work principles for rigid–plastic body in plane strain

Extremum/work principles for a rigid–plastic body have been discussed in classical theory of plasticity to be of immense significance. Unfortunately, till now, these extremum theorems have been used only as a crude method of obtaining the limit load of a rigid–plastic body, using successive approximations by upper and lower bound estimates. On the other hand slip-line fields (SLF) have been extensively used not only for evaluation of limit load but also for obtaining sufficiently accurate estimates of stresses in the plastic region as well as in the vicinity of crack tip. Till now, these two methods of plastic analyses, that is, the work principles and SLF have remained more or less independent apart from the fact that both are upper bounds as they use kinematically admissible velocity fields. Recently, a new load bounding technique, modified upper bound (MUB) Approach, was proposed by Khan and Ghosh [Khan, I.A., Ghosh, A.K., 2007. A modified upper bound approach to limit analysis for plane strain deeply cracked specimens. International Journal of Solids and Structures 44 (10), 3114–3135]. In this article, a rigorous mathematical basis of this load bounding technique is presented and it is demonstrated that the method is actually a new form of the general extremum/work principles. The equivalence of this new form of work principle, that is, MUB with the classical SLF analysis, for a rigid–plastic material in plane strain, has been discussed in detail. Since plastic deformation fields depend on specimen geometry and type of loading specific cases have been considered. Both cracked and uncracked configurations have been analysed to establish this equivalence in general. Various simplifications resulting from the use of this new load bounding technique over SLF method has been demonstrated. Several standard problems of plane strain analysed by SLF method and validated by experiments in past have been considered in this article. As a novel application of the proposed method, single-edge-cracked plate under combined bending and tensile load has been analysed. For this specimen SLF solutions are available only for bending with small tensile load (defined in Section 3.2.4) while classical upper bound solutions are valid for bending with large tensile load. In this work a completely analytical formulation for yield locus for the entire range of tensile and bending load has been obtained. Apart from accurate evaluation of limit load, detailed evaluation of crack tip stresses and hence constraint near the crack tip has been performed using this new form of work principles.