Errors in the r value when based upon total strain measurements

Abstract

The r value appears in many theoretical accounts of plastic anisotropy. In the interpretation of r as a formability parameter, three methods have been employed to establish errors that arise when r is found from using total strains rather than plastic strains. Inaccuracies are likely to arise where fully instrumented uninterrupted tensile testing is used for rapid quality control of sheet stock supply. Here, the calculations for r and those for other mechanical properties, including strength, ductility, and the n value, are usually fully automated, their values being displayed at the end of the test. This procedure foregoes the greater time consumed with the preferred procedure involving a repeated loading-unloading from within the plastic region. With unloading, the elastic strain recovers, allowing r to be found correctly as a ratio between its two components of plastic strain. Without unloading, the implication is that the r value has been based upon total strain. Certainly, it is traditional within the metal-forming industry to calculate the hardening exponent n from total strains. It is, of course, possible to calculate the elastic strain to be removed at a particular stress level, from the elastic constants, but this is unlikely to be incorporated into a procedure where elastic strains are regarded as negligibly small. The present analyses confirm that this would be true for the larger strains attained in forming a product from sheet steel, the error in r being less than -1 per cent beyond 10 per cent strain. However, it appears that the error accelerates with diminishing strain. For example, an error in r of -3 per cent applies to a 1 per cent total strain. Moreover, when elastic and plastic strains reduce further to a comparable magnitude in microstrain, the error in r rises to -15 per cent. The latter has a bearing upon other areas of plastic anisotropy, including the elastic-plastic analyses of structures and the determination of the subsequent yield surface.