27. Strain Gradient Plasticity: Deformation Patterning, Localization, and Fracture

In this chapter, two different strain gradient plasticity models based on non-convex plastic energies are presented and compared through analytical estimates and numerical experiments. The models are formulated in the simple one-dimensional setting, and their ability to reproduce heterogeneous plastic strain processes is analyzed, focusing on strain localization phenomena observed in metallic materials at different length scales. In a geometrically linear context, both models are based on the additive decomposition of the strain into elastic and plastic parts. Moreover, they share the same non-convex plastic energy, and they are both characterized by the same nonlocal plastic energy as well, i.e., a quadratic form of the plastic strain gradient. In the first model, proposed in Yalçinkaya et al. (Int J Solids Struct 49:2625–2636, 2012) and Yalcinkaya (Microstructure evolution in crystal plasticity: strain path effects and dislocation slip patterning. Ph.D. thesis, Eindhoven University of Technology, 2011), the plastic energy is assumed to be conservative, and plastic dissipation is introduced through a viscous term, which makes the formulation rate-dependent. In the second model, developed in Del Piero et al. (J Mech Mater Struct 8(2–4):109–151, 2013), the plastic term is supposed to be totally dissipative. As a result, plastic deformations are not recoverable, and the resulting framework is rate-independent, contrary to the first model. First, the evolution problems resulting from the two theories are analytically solved in a special simplified case, and correlations between the shape of the plastic potential and the modeling predictions are established. Then, the models are numerically implemented by finite elements, and numerical solutions of two different one-dimensional problems, associated with different plastic energies, are determined. In the first problem, a double-well plastic energy is considered, and the evolution of plastic slip patterning observed in crystals at the mesoscale is reproduced. In the second problem, a convex-concave plastic energy is used to simulate the macroscopic response of a tensile steel bar, which experiences the so-called necking process, with plastic strains localization and final coalescing into fracture. Numerical results provided by the two models are analyzed and compared.