A Kinematic Method for Shakedown and Limit Analysis of Periodic Heterogeneous Media

In this Chapter the kinematic (second, Koiter’s) shakedown theorem is applied to the representative volume of periodic heterogeneous media with Huber-Mises local plastic behavior. The adopted formulation of shakedown analysis is based on periodicity bound

  • PDF / 1,881,131 Bytes
  • 18 Pages / 481.89 x 691.654 pts Page_size
  • 34 Downloads / 195 Views

DOWNLOAD

REPORT


Abstract. In this Chapter the kinematic (second, Koiter's) shakedown theorem is applied to the representative volume of periodic heterogeneous media with Huber-Mises local plastic behavior. The adopted formulation of shakedown analysis is based on periodicity boundary conditions, conventional finite element modeling and penalization enforcement of plastic incompressibility. A cost-effective iterative solution procedure is discussed and computationally tested. Numerical tests and engineering applications are presented with reference to perforated plates and metal-matrix unidirectional fiber-reinforced composites.

1 Introduction In several engineering situations and advanced technologies, structural analysis and design concern heterogeneous systems which exhibit ductility of material behaviour and periodicity of texture as main basic features. Two typical categories of such systems are considered in what follows: perforated steel plates, often used in power plants; metal-matrix unidirectional fiberreinforced composites, employed particularly by aerospace industries. In view of the expected ductile inelastic behaviour, lack of shakedown (SD) under variablerepeated loads or, as a special case, plastic collapse under monotonically increasing loads can reasonably be regarded as the main critical events with respect to which the safety margins must be assessed. Periodic space distribution of geometric and physical properties makes it possible and, clearly, computationally very advantageous, to select as the space domain of the limit state analysis problem the "representative volume" (RV). This is defined as the minimum volume which contains all geometrical and physical information about the heterogeneous medium and can be conceived as generating the whole solid by repeated translations, see e.g. Suquet ( 1982). In the above circumstances a practically meaningful shakedown analysis (SDA) problem can be formulated in the spirit of homogenization theory as follows, (Carvelli et al., 1999b, 2000; Dvorak et al., 1994; Pouter and Leckie, 1998a). A "load domain" is assigned in the space of average (or "macroscopic") external actions, i.e. a region of that space within which they slowly (without inertia forces) fluctuate in time a supposedly unbounded number of times according to unknown time histories. The structural analysis problem is to achieve "directly" (i.e. not by stepby-step inelastic computations along loading histories, which are unknown) practically essential information on the "safety factor", i.e. the number s such that SD occurs if the "load multiplier" J..l is below s (J..ts). Collapse for J..L>s means here that either incremental collapse or alternating plasticity occurs because yielding (dissipative) processes never cease locally (at the microscopic level).

D. Weichert et al. (eds.), Inelastic Behaviour of Structures under Variable Repeated Loads © Springer-Verlag Wien 2002

116

G. Maier, V. Carvelli

The objectives and limitations of what follows in this Chapter are clarified by the following remarks. (a) T