Generalizing J 2 flow theory: Fundamental issues in strain gradient plasticity

Abstract

It has not been a simple matter to obtain a sound extension of the classical J 2 flow theory of plasticity that incorporates a dependence on plastic strain gradients and that is capable of capturing size-dependent behaviour of metals at the micron scale. Two classes of basic extensions of classical J 2 theory have been proposed: one with increments in higher order stresses related to increments of strain gradients and the other characterized by the higher order stresses themselves expressed in terms of increments of strain gradients. The theories proposed by Muhlhaus and Aifantis in 1991 and Fleck and Hutchinson in 2001 are in the first class, and, as formulated, these do not always satisfy thermodynamic requirements on plastic dissipation. On the other hand, theories of the second class proposed by Gudmundson in 2004 and Gurtin and Anand in 2009 have the physical deficiency that the higher order stress quantities can change discontinuously for bodies subject to arbitrarily small load changes. The present paper lays out this background to the quest for a sound phenomenological extension of the rateindependent J 2 flow theory of plasticity to include a dependence on gradients of plastic strain. A modification of the Fleck-Hutchinson formulation that ensures its thermodynamic integrity is presented and contrasted with a comparable formulation of the second class where in the higher order stresses are expressed in terms of the plastic strain rate. Both versions are constructed to reduce to the classical J 2 flow theory of plasticity when the gradients can be neglected and to coincide with the simpler and more readily formulated J 2 deformation theory of gradient plasticity for deformation histories characterized by proportional straining.

Strain Gradient Plasticity-Based Modeling of Damage and Fracture

“…This has been numerically corroborated by Bardella and Panteghini [56] and by Martínez-Pañeda et al [57]. This physically uncertain response may favor the incremental modeling approach suggested by Hutchinson [22] where increments of all stress magnitudes are expressed in terms of increments of strain. However, a thermodynamically-consistent formulation with incremental constitutive relations for dissipative gradient contributions has not been proposed yet.…”

“…However, as discussed by Hutchinson [22], this interpretation requires the input tensile curve to have a constant tangent modulus, which is not a realistic restriction for a general plasticity model. Moreover, from a physical standpoint it seems likely that some of the work associated with τ ijk should be dissipative.…”

“…However, a number of open research issues remain to be addressed. Very recently Fleck, Hutchinson and Willis [18,22,23] noted that non-incremental dissipative higher order terms may lead to a delay in plastic flow under certain non-proportional loading conditions. This has been numerically corroborated by Bardella and Panteghini [56] and by Martínez-Pañeda et al [57].…”

“…Positive plastic work was later ensured by employing dissipative higher order stresses constitutively related to increments of strain [6,15,16]. But it has been very recently noticed that this non-incremental formulation may lead to a delay in plastic flow under certain non-proportional loading conditions [18,22,23]. It is therefore manifest that continuum modeling of size effects in metal plasticity is under continuous development.…”

On modeling micro-structural evolution using a higher order strain gradient continuum theory

“…While there is an ongoing discussion on whether size-effects result in strain gradient strengthening (delayed yield point) or strain gradient hardening, non-trivial size-effects are well recognized. Since the first gradient dependent theory appeared some 30 years ago, micromechanically based crystal plasticity models have been a highly active research subject and recent contributions include Hutchinson (2012); Öztop et al (2012); van Beers et al (2013); Klusemann and Yalçinkaya (2013); Niordson and Kysar (2014). Different families of higher order strain gradient plasticity theories have evolved, but a common feature for all is that they incorporate micro-structural boundary conditions, enabling constraints on plastic flow or associated field quantities, as opposed to the so-called lower order theories (e.g Bassani, 2001; Huang et al, 2004) for which unexpected strain localization behavior was pointed out by Niordson and Hutchinson (2003).…”