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A reduced form of shakedown kinematic theorem

A reduced form of the shakedown kinematic theorem without time integrals is deduced for Tresca material, which is equivalent to the original one when the principal directions of plastic deformations everywhere in a structure remain unchanged during loading cycles. | Vietnam Journal of Mechanics, NCST of Vietnam Vol. 24, 2002, No 1 (25 - 34) A REDUCED FORM OF SHAKEDOWN KINEMATIC THEOREM Due CHINH Institute of Mechanics, NCST of Vietnam PHAM ABSTRACT. A reduced form of the shakedown kinematic theorem without time integrals is deduced for Tresca material, which is equivalent to the original one when the principal directions of plastic deformations everywhere in a_structure remain unchanged during loading cycles. 1. Introduction An elastio-perfectly plastic body subjected to loading cycles, though not undergoing instantaneous plastic collapse, may fail because plastic deformations accumulated during cycles increase indefinitely (leading to incremental collapse) or change signs endlessly (leading to low cycle fatigue). On the other hand, it may happen that no further plastic deformation occurs after one or a few cycles - that is, all subsequent cycles are elastic. In that case the body is said to shake down. The shakedown (quasistatic) theory, which contains the plastic limit theory ([1], [2]) as its limit case, has been comprehensively presented in the classical work of Koiter [3) and is naturally extended for general dynamic processes ([4], [5], [6]). The primary kinematic theorem formulated by Koiter is difficult to be used because of its complexity caused by the presence of time integrals. To eliminate them, Gokhfeld [7) and Sawczuk [8] (see also [9]) have restricted admissible plastic deformation cycles to a special proportional monotonous deformation mode (called the perfect incremental collapse one) , and obtained subsequently an upper bound on the shakedown safety factor. With a broader admissible set of proportional (but need not to be monotonous) plastic deformations at every point x E V, we [10] succeeded in deducing a better bound on the shakedown factor . Other reduced forms (without time integrals) of the kinematic theorem for certain problems are obtained in [11], [12). Let ue(x, t) denote the fictitious stress .