On the Validity of Saint Venant’s Principle in Finite Strain Plasticity

Diffusion of effects from a local disturbance in a homogeneous stress field is analyzed within the framework of continuum plasticity. An axial rate of decay for the variability of the effects of a nonuniform disturbance imposed on one end of a long circular cylinder is determined as a function of the axial stress (or stretch). The analysis considers constitutive equations corresponding to incompressible, finite strain versions of J₂ flow and deformation theories. Both theories result in effects that oscillate and decay exponentially with distance from the imposed disturbance; the rate of decay decreases as the uniaxial tension increases. Deformation theory predicts a larger rate of decay than flow theory except within a small range of stress near the ultimate (necking) load; at necking the rate of decay vanishes.