The Considère Diagram A Review of Uniaxial Stress-Strain Relationships for Finite Deformations

For metals it is widely accepted that the stress-strain diagram is symmetric with regard to tension and compression when plotted in terms of true rather than nominal quantities. Kinematic considerations state that the engineering definition of nominal strain exhibits fundamental deficiencies under rigid body rotations and deformations which are not truly infinitesimal. For these two reasons the direct tension behavior of metals is normally plotted in science-oriented publications in terms of true rather than nominal measures of stress and strain. In contrast, the structural engineering community prefers to plot stress-strain diagrams in terms of nominal rather than true measures of stress and strain which clearly depicts the ultimate strength of the material. There is a one-to one relationship between true and nominal measures, hence in principle it should make little difference which representation is used to display the main features of the direct tension test. Unfortunately, the true-stress-strain diagram does not depict the point of ultimate load resistance which distinguishes hardening from softening. It is this point in the deformation history where the the tangential stiffness of elastoplastic contitutive models looses positive definite properties and where the tangential localization operator indicates the formation of discontinuities e.g. in the form of shear bands (loss of ellipticity and positive wave speeds). Further, traditional finite element failure computations exhibit strong mesh sensitivity, one speaks of ‘loss of mesh objectivity’ if no special provisions are made to capture or regularize these discontinuities in the softening regime. In short, for failure analysis, it is the ultimate strength and the ductility which are of main interest. For this reason it is important to locate the ultimate strength which corresponds to the point of failure initiation at the material level. Since the true stress-strain diagram exhibits hardening and does not indicate the drastic difference between hardening and softening it is proposed to combine the two representations of the uniaxial tension experiment using the ‘Considère Diagram’ which superimposes the two measures of stress and strain in a single plot, see e.g. Considère [1], Nadai [2], or Meyers and Chawla [4].