Nonlinear Analysis of Instability in Tension

A nonlinear theory of tensile instability (necking) of a bar is formulated. The material of this bar is assumed to be homogeneous, isotropic; obeys the Rambert-Osgood law and has a permissible Poisson's ratio. Numerical solutions are obtained, for the case of Poisson's ratio equals to 0.5, by using the modified Euler's method. Nondimensional load-length diagrams of the bar are presented graphically for various material parameters. If the tensile load is monotonically increasing all the time, tensile instability will occur. From this nonlinear analysis, it is found that the ultimate nominal stress of tensile instability, for a bar with finite strength, can be much greater than that predicted by the classical theory. The discrepancy between these two theories is explained. The validity of the classical theory is analyzed.