Continuum mesoscale theory inspired by plasticity

– We present a simple mesoscale field theory inspired by rate-independent plasticity that reflects the symmetry of the deformation process. We parameterize the plastic deformation by a scalar field which evolves with loading. The evolution equation for that field has the form of a Hamilton-Jacobi equation which gives rise to cusp-singularity formation. These cusps introduce irreversibilities analogous to those seen in plastic deformation of real materials: we observe a yield stress, work hardening, reversibility under unloading, and cell boundary formation. We call it plasticity when materials yield irreversibly at large external stresses. Macroscopically, plasticity is associated with three qualitative phenomena. To a good approximation, there is a threshold called the yield stress below which the deformation is reversible (see Fig. 1). A material pushed beyond its yield stress exhibits work hardening, through which the yield stress increases to match the maximum applied stress. Finally, the deformed crystal develops patterns, such as the cell structures observed in fcc metals [1, 2, 3]. While much is known about all three phenomena, a quantitative understanding based on mesoscopic continuum theory would be welcome—especially if it connects to microscopic properties of the atomic interactions in the material. In this paper, we will discuss a simple scalar field theory which naturally exhibits these three key features of plasticity. We do not claim to model in details plasticity in real materials but we believe that a continuum description of plasticity should share the key feature of our model equation: The transition between reversible and irreversible deformation is generated by singularities which occur at finite stress. (∗) E-mail: [email protected], Home page: http://www.lassp.cornell.edu/sethna/sethna.html (∗∗) E-mail: [email protected] (∗∗∗) E-mail: [email protected]