Differential Hardening in Sheet Metal Under Biaxial Loading: A Theoretical Framework

A simple constitutive framework is proposed to take account of progressive changes in a stress-based criterion of yielding as plastic deformation accumulates. The analysis is relevant to a sheet of isotropic or orthotropic material which is subsequently loaded in its plane by biaxial tensions (σ₁, σ₂) in arbitrary fixed ratio. Under these circumstances there is evidence that texture development causes significant changes in the shapes of successive yield loci in (σ₁, σ₂) space. Here such departures from geometric similarity are explicitly linked to a presumed dependence on σ₁/σ₂ of the exponent m in a standard power-law representation of the stress-strain characteristic under each constant value of σ₁/σ₂. The linkage is formulated in detail for several types of representation, and it is shown how a family of yield loci associated with any conjectural dependence m(σ₁/σ₂) can be generated readily by computer graphics. Some typical examples are given, in particular when m (σ₁/σ₂) reflects the distinctive trends reported for certain kinds of brass and aluminum sheet (e.g., Wagoner, 1980; Stout and Hecker 1983).