Optimal plastic shape design via the boundary perturbation method

1 I n t r o d u c t i o n The boundary perturbation method (BPM) is now widely developed both in solid and fluid mechanics (although not always under this name). Older results are presented in the monograph by Morse and Feshbach (1953), and a recent survey by Guz and Nemysh (1987) gives 310 references (mostly Ukrainian and Russian). The monograph by Guz and Nemysh (1989) presents in detail two variants of this method. Recently, much attention to BPM applied to elastic problems has been paid by Parnes (1987, 1989) (eccentric loadings) and Gao (1991) (inclusions). Fewer papers are devoted to BPM in the problems of plasticity; they were initiated by Ilyushin (1940), Ivlev (1957) and Spencer (1962). Applications of BPM to optimal shape design are rather seldom; we mention here the paper by Schnack and Iancu (1989) who used numerically realized local perturbations in the elastic range. A series of papers by Kordas and her collaborators, started in 1970, used BPM in the investigation of fully plastic states at the stage of collapse of various perfectly plastic structural elements. Kordas and Zyczkowski (1970) considered noncircular shapes of cylinders under pressure; Kordas (1973) discussed pipe-lines of variable diameter; Kordas and Skraba (1977) analysed cylinders under pressure with bending; Kordas (1977) presented a general approach to the problem under consideration; Kordas (1979) considered noncircular shapes of disks under pressure; Dollar and Kordas (1980) discussed frame corners under bending, tension and shear; Kordas and Postrach (1990) analysed rotating disks. Full plastification at the stage of collapse is the first step towards optimization (in most cases the necessary condition), since the material in rigid or elastic zones at the stage of collapse is not properly utilized. In many cases, however, the above condition is not sufficient and then additional optimization is necessary. Examples of such additional optimization are given in the papers by Bochenek et al. (1983) and Egner et al. (1993). The first paper discussed plastic optimization of a doublyconnected cross-section of a bar under torsion with small bending; in this problem only one boundary condition along each contour holds, thus it is always possible to find a class of solutions satisfying that condition and then to perform the subsequent optimization. The second paper is devoted to the optimal design of yoke elements (ends of connecting rods, bolt joints, chain links), i.e. to plane problems of plasticity with two boundary conditions along each contour; it turns out that they may all be satisfied simultaneously and the optimal shape ensuring the maximal limit load-carrying capacity may be found. In both problems the circular (annular) shape was subject to perturbations. If the thickness (of beams, plates or shells) is assumed as the design variable, then the Prager-Shield theorem on optimal plastic design is particularly useful (Rozvany 1976; Lellep 1991). On the other hand, if the shape of the boundary is subject to variations, its applications are less effective, although possible (Mr6z 1963). In such cases, BPM may successfully be employed proVided that the optimal shape is not too different from a certain classical shape with a simple perfectly plastic solution. In the present paper we first consider general three-dimensional perturbations of a circular cylinder under internal pressure and then give two examples of the application to optimal plastic shape design. 2 Genera l p e r t u r b a t i o n s for a plas t ic c i rcular cylinder u n d e r in te rna l p ressure 2.1 Assumpt ions , basic solulion We consider bodies with a shape close to a circular cylinder under the following assumptions: the material is perfectly plastic, isotropic, incompressible, subject to the HuberMises-Hencky (HMIt) yield condition. In the basic, unperturbed state the cylinder with internal radius a 0 and external radius b 0 is in a plane-strain condition. Small strains are assumed throughout the paper. Hencky-Ilyushin or Levy-Mises constitutive equations are employed; they lead to the same results only with strains replaced by strain rates in the second case. The cylindrical coordinates r, 6, z are used. Under the assumption of uniform internal pressure Pa =