Limit analysis of a cylindrical tube of a random inhomogeneous plastic material under internal pressure

We examine the effects of random spatial fluctuations in the yield limit on the strength of a cylindrical tube subjected to internal pressure in plane strain conditions. The analysis is conducted along the lines laid in [1], where a generalization of the method of slip-lines to random media was outlined. The basic role in this formulation is played by a Representative Volume Element (RVE) of scale δ = L/d, where L is the scale of observation while d is the size of a single grain in a polycrystalline medium. Since the microstructure is random, the yield condition of an RVE at a given δ is random and, hence, the approximating continuum is a random medium. Due to the presence of material inhomogeneities in random media, the deterministic characteristics (slip-lines) of the homogeneous medium problems are replaced by forward evolution cones representing the ranges of scatter in stochastic characteristics. Consequently, the logarithmic spirals of the deterministic problem are randomly distorted lines and extend the external radius of the plastic zone. The tube has thus to be made thicker in order to withstand the same limit internal pressure as the one made of a homogeneous material described by the average 〈k〉. The analysis may easily be extended to an arbitrary axisymmetric traction on the hole.