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

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Abstract
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).

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