Deformation vs. flow and wavelet-based models of gradient plasticity: Examples of axial symmetry

The modest goal of this article – the first in a series of three dedicated to Kirk Valanis – is to discuss and compare (in terms of their constitutive and variational formulation, boundary conditions and size-dependent solutions to axially symmetric problems) a class of deformation and flow models of gradient plasticity derived from the same basic assumption of a gradient-dependent flow stress: That is, the dependence of the flow or effective stress image on the Laplacian image of the effective strain image. This assumption is also used to derive, through wavelet analysis, a scale-dependent constitutive model involving a scale or gage length parameter instead of a gradient term. All models exhibit size effects which may differ from one model to another according to the nature of the boundary conditions and the values of the gradient coefficients or the scale parameter used. A weak formulation of stress equilibrium is employed to derive corresponding extra boundary conditions for the deformation and flow models which are then examined in view of their solutions for problems of axial symmetry, and also in connection with the corresponding solutions of the scale-dependent model for which extra boundary conditions are not required. In particular, results for the stress/strain distributions and the description of the associated size effects for internally pressurized thick-walled cylinders are provided. A comparison with some available experimental results for the yielding behaviour of internally pressurized hollow cylinders is also made.