Nonlinear Behavior of Ductile Quasi-homogeneous Solids

First M.T. Huber in 1904, and later Mises and Hencky suggested equivalent yield criteria for elastic-perfectly plastic solids in three-dimensional stress states. The H-M-H criterion is commonly used in structural design. But, the Huber-Hencky distortion energy formula and the Huber-Mises reduced stress formula do not give unique yielding measures for elastic-nonlinearly plastic solids. The yielding probability κ, which has been introduced by the author in 1954, serves the purpose for ductile elastic-nonlinearly plastic solids. This idea has been a part of a more general probability-based theory such that the yielding ratio κ and a cracking tensor λ are the damage measures for quasi-homogeneous continuous media. Structural concrete has been analyzed in earlier studies. In this study, nominally ductile materials are taken into consideration such as structural steel and aluminum alloys in normal temperatures. The log-normal probability distributions of plastic microstrength and microstress are accepted. Constitutive equations are derived with the yielding ratio κ as the coordinate of state. The Ramberg-Osgood σ-ε curve is taken as the empirical basis for evaluation of the probability distribution parameters. Two points of the curve are taken into account: the conventional yield strength f _y and the ultimate strength f _u . A numerical example indicates that both elastic and plastic compressible phases of the quasi-homogeneous solid is a likely model of behavior. A shear stress-strain curve is analytically derived. The conventional 0.2% permanent strain for the characteristic plastic strength f _y in a simple tension test applies approximately also to shear cases for the same yielding ratio κ _y at the characteristic strength level. The ultimate strength f _u will occur when the effective stress σ _eff (κ) attains its maximum level for a critical yielding ratio κ _cr ; however, it is not the maximum point σ _eff (ε) of the monotone Ramberg-Osgood curve. The characteristic κ _y and critical κ _cr values are verified in the case of shear.