Reviewed by Dr. John ChristodoulidesREVIEW -The methods of elastic-plastic analysis of continua were developed in response to increased efficiency and reliability requirements on modern structural components. The fundamentals of the theory of plasticity have already been implemented in a number of available computer codes. Among the methods of plastic analysis of structures, the theory of limit analysis has already become an engineering tool. The method employs the rigid-plastic model of material behavior and it is capable fo evaluating upper and lower bounds to the ultimate load intensity that can be sustained by a structure. If variable repeated loading is present, a suitable generalization of the limit analysis is introduced, known as the shakedown theory. Initial development of the shakedown theory dates back to the thirties and the shakedown analysis of structures under variable temperature fields were initiated by Prager in the fifties. The present text is among the rare comprehensive treatments of elastic-plastic structures under cyclic mechnaical and thermal loading.The concept of elastic-plastic response under variable repeated loading and temperature is first introduced herein for a simple bar system. The possible ensuing regimes of a stabilized response or an incremental collapse process are depicted in order to enhance reader understanding of the subject matter.All the concepts of applied mechanics pertinent to the contents are illustrated before the theorems of shakedown are stated and proved. Analogies existing between shakedown analysis and limit analysis are presented. The generalized stresses and strains used traditionally in plate and shell analysis are used and the deviation of limiting interaction surfaces for variable cyclic stresses is illustrated.The solution of shakedown problems can be obtained with the aid of mathematical programming techniques. The mathematical background in the involved process is outlined and detailed examples are given for simple structural elements under cyclic mechanical and thermal loading.The book devotes considerable attention to the analysis of rotating disks at cyclic variations of angular velocity and temperature. Shakedown diagrams are derived for a number of plate and shell problems, and thus their capacity under cyclic pressure and temperature profiles is identified. Progressive thermal cycling deformation (ratchetting) and contact problems are included. Finally theoretical proof of the existence and uniqueness of the elastic stabilization (shakedown) state is given.A wealth of information concerning elastic-plastic analysis of structural components under thermal and mechanical cycling is given in this monograph. It is concise, complete and it is considered extremely valuable in its field for both research and design.

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It is noted that calculations of the rated strength for bearing elements of extremely loaded structures, including nuclear power plants allow inelastic deformation of the materials of these elements. At the same time calculations of the low cycle fatigue require taking into account factors that are not observed under single loading including, i.e., kinetics of cyclic strains, cyclic creep, change of the mode of inelastic cyclic deformation upon normal operation. Moreover, the materials can be of different types: cyclically hardening, softened or stable. For the first type of materials at a soft loading with constant amplitude of stresses in cycles, the range of strains decreases with an increase in the number of cycles, but increases for the second one. Under a hard mode of loading with constant amplitude of strains the maximum stresses in a cycle for the hardening material increase, and, on the contrary, decrease for softened material. Moreover, the soft loading of softened material results in one-sided accumulation of plastic strains as the number of loading cycles increases. These circumstances must be taken into account both in the analytical description of the kinetics of deformation diagrams and in the corresponding calculation equations used in the strength standards. It is noted that at early stages of forming computation methods developed for these conditions, calculation of stresses was carried out in the assumption of ideal elasticity of the material. The use of such approach was attributed to the lack of available methods for addressing the problem of an inelastic cyclic deformation, complicated on the statement. The subsequent evolution of the theory of cyclic elastoplastic deformation, analytical and numerical solutions of cyclic boundary-value problems, developing of numerical methods of computation and powerful computer packages fundamentally changed the situation providing the possibility of analysis and modeling of physically and geometrically nonlinear deformation processes. It is shown that transition from the elastic adaptability (with an elastic deformation of the structure in a stable cycle) to a sign-variable flow is smooth and continuous, similar to the transition from the elastic to plastic deformation under a single loading. Such a mechanism is similar to conditional boundary of the transition from low cycle to high-cycle fatigue under a cyclic strain. At the same time, we offer to use in calculations the existing rather simple models and experimentally determined parameters of cyclic deformation diagrams of materials. In the modern statement of the problems under consideration, taking into account both the kinetics of cyclic and unilaterally accumulated deformations, with allowance for the manifestation of creep effects in cycles is of fundamental importance. This approach also makes it possible to take into account the acceleration of unsteady cyclic creep due to previous plastic deformation of a different sign, which can be rather significant.

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