Strain-Gradient Plasticity
Motivation
Classical plasticity theory predicts that the yield stress and hardening response of a metal are independent of the specimen size. Experiments at the micron scale — micro-torsion tests on thin wires, micro-indentation, micro-bending — consistently show the opposite: smaller is stronger. This size effect cannot be captured by a local plastic flow rule because the flow rule involves no length scale. Strain-gradient plasticity (SGP) remedies this by incorporating the gradient of the plastic strain, or equivalently the geometrically necessary dislocation density, into the free energy and dissipation potential. The gradient introduces a material length scale $\ell$ that controls the size effect, and the central challenge becomes: how do the energetic and dissipative contributions of the gradient compete, and how are they reflected in macroscopic observables?
This research thread addressed two canonical SGP problems — torsion of a cylindrical bar and simple shear of a slab — in close collaboration with L. Giacomelli and M. Chiricotto (Roma La Sapienza).
Details
Mathematical formulation: a nonlocal degenerate parabolic system (2011)
Before studying specific boundary-value problems, the general mathematical structure of SGP was analyzed at the PDE level:
- Bertsch, Dal Passo, Giacomelli & Tomassetti (2011) — A nonlocal and fully nonlinear degenerate parabolic system from strain-gradient plasticity. The incremental constitutive relations of SGP, when written in rate form, yield a system of PDEs that is simultaneously nonlocal (due to the gradient terms), fully nonlinear (due to the yield condition), and degenerate parabolic (due to the presence of a free boundary separating elastic and plastic zones). This paper established existence and uniqueness of weak solutions for this system, and identified the regularity of the elastic–plastic interface. The result provided the functional-analytic foundation for the subsequent mechanical analyses.
Energetic scale effects in torsion (2012)
The first benchmark is the torsion of a thin wire. Classical theory predicts a linear torque–twist relation for a given cross-section; experiments show that the slope increases as the wire radius decreases. This is the energetic size effect, attributable to the storage of gradient energy.
- Chiricotto, Giacomelli & Tomassetti (2012) — Torsion in strain-gradient plasticity: energetic scale effects. The SGP problem for a circular cross-section under anti-plane loading was formulated as a variational problem with the Gurtin–Anand free energy, which includes a term quadratic in the gradient of the plastic strain. Exact solutions were derived in the fully plastic regime, showing that the torque scales as $R^3 + c \ell R$ for a wire of radius $R$, where $c$ is a non-dimensional constant that depends on the ratio of the gradient modulus to the classical modulus. The $\ell R$ term — the size effect — dominates for small radii and disappears as $\ell \to 0$, recovering the classical result.
Dissipative scale effects in simple shear (2016)
The second benchmark is the shear of a slab constrained between two rigid plates. In this geometry the gradient effects are purely dissipative (arising from the dissipation potential, not the free energy), and their influence on the response is qualitatively different from the energetic case.
- Chiricotto, Giacomelli & Tomassetti (2016) — Dissipative scale effects in strain-gradient plasticity: the case of simple shear. The dissipative SGP model for simple shear was analyzed. In contrast to the energetic torsion problem, here the plastic strain is constrained to satisfy a flow rule with gradient (Fleck–Willis-type model). The analysis revealed a sharp transition: for $\ell < \ell_c$ (a critical length determined by the layer thickness and the applied shear rate) the plastic zone fills the entire slab, while for $\ell > \ell_c$ a boundary layer of thickness $\ell$ forms near each plate, and the bulk remains elastic. This elastic core phenomenon is a purely dissipative size effect with no counterpart in classical plasticity.
Summary and connections
The three papers form a coherent unit: the PDE analysis (2011) established well-posedness; the torsion paper (2012) characterized the energetic size effect; the simple-shear paper (2016) characterized the dissipative size effect. Together they demonstrate that the two mechanisms produce qualitatively different phenomenology — the energetic effect modifies the hardening slope continuously, while the dissipative effect produces a threshold transition between two distinct regimes.
The connection to the diffusion thread is not coincidental: both invoke nonlinear PDE analysis for problems with free boundaries and involve the same collaboration with L. Giacomelli. The mathematical techniques developed for SGP (monotone operator theory, Moreau–Yosida regularization) were later reused for the Cahn–Hilliard analysis described in that thread.