Shakedown, Integrity and Design of Elastic Plastic Damaged Structures as Subjected to Cyclic Loading

Abstract

A method for direct design (i.e. without detailed investigation of the deformation process) of structures of elastic plastic isotropically damaged materials as subjected to cyclic loading is developed. The material is assumed to be stable. Shakedown and integrity conditions provide the possibility of obtaining the interval of safe values of a parameter 112 in design for which the structure preserves its safety. The damage process is assumed to be coupled with the plastic deformation development. Features of the post-adaptation stage lead to a min-max problem of mathematical programming, whose solution provides a relation between the limit values of the damage and hardening parameters. As the hardening parameter is assumed to be bounded, this relation issues in upper and lower bounds for the local actual limit values of the damage parameter admitted by the yield condition for the given loading program. The bounds result in a priori necessary and sufficient conditions of integrity. As the bounds depend on 1, the integrity conditions impose constraints to its safe values. A method for computing the purely elastic damaged response of the structure to the prescribed loading program is proposed. An example of the application of the developed method is given.Thus, fulfillment of the shakedown conditions is only necessary for the safety of solids subjected to cyclic loading. To guarantee its safety, the condition of local integrity at every point of the structure has to be satisfied, i.e. the parameters of the structure should be specified in such a way to assure the fulfillment both the shakedown and integrity conditions.Maier [1], Weichart and GrossWeege [2], Polizzotto et al. [3], Karadeniz and Ponter [4] and Zarka et al. [5] considered various aspects of applying the shakedown theory to problems of design. Giambanco et al. [6,7] employed the shakedown theory for the optimal design of certain structures, and investigated some related problems.The problem of application of the shakedown theory to the design of creeping elastic plastic mechanical systems has been the subject of the papers by Rose and White [8], Buckthorpe and White [9-11] and Toulios et al. [12].It is well known that the shakedown theory was originally developed for perfect elastic plastic materials [13][14][15]. Presently the question of extension of the shakedown theory to damaged elastic plastic solids is topical.Hachemi and Weichert [16] extended the static shakedown (Melan) theorem to elastic plastic isotropically damaged solids with unlimited linear kinematic strain hardening.Siemashko [17] developed a method of step-by-step inadaptation analysis for elasticplastic structures subjected to repeated loading, which accounts for nonlinear isotropic strain hardening, development of damage, and nonlinear geometrical effects.Polizzotto et al.[18] included the damage variable into a set of internal variables, and developed an elastic plastic damaged material model with associative constitutive relations and nonlinear elasticity. Employing the D-stability ...