Dissipative scale effects in strain-gradient plasticity: the case of simple shear

We analyze dissipative scale effects within a one-dimensional theory, developed in [L. Anand et al. (2005) J. Mech. Phys. Solids 53], which describes plastic flow in a thin strip undergoing simple shear. We give a variational characterization of the {\emph{ yield (shear) stress}} --- the threshold for the inset of plastic flow --- and we use this characterization, together with results from [M. Amar et al. (2011) J. Math. Anal. Appl. 397], to obtain an explicit relation between the yield stress and the height of the strip. The relation we obtain confirms that thinner specimens are stronger, in the sense that they display higher yield stress

Weak solutions to thin-film equations with contact-line friction

We consider the thin-film equation with a prototypical contact-line condition modeling the effect of frictional forces at the contact line where liquid, solid, and air meet. We show that such condition, relating flux with contact angle, naturally emerges from applying a thermodynamic argument due to Weiqing Ren and Weinan E [Commun. Math. Sci. 9 (2011), 597–606] directly into the framework of lubrication approximation. For the resulting free boundary problem, we prove global existence of weak solutions, as well as global existence and uniqueness of approximating solutions which satisfy the contact line condition pointwise. The analysis crucially relies on new contractivity estimates for the location of the free boundary

Droplets spreading with contact-line friction: lubrication approximation and traveling wave solutions

We consider the spreading, driven by surface tension, of a thin liquid droplet on a plane solid surface. In lubrication approximation, this phenomenon may be modeled by a class of free boundary problems for fourth order nonlinear degenerate parabolic equations, the free boundary being defined as the contact line where liquid, solid and vapor meet. Our interest is on an effective free boundary condition which has been recently proposed by Ren and E: it includes into the model the effect of frictional forces at the contact line, which arises from the deviation of the contact angle from its equilibrium value. In this note we outline the lubrication approximation of this condition, we describe the dissipative structure and the traveling wave profiles of the resulting free boundary problem, and we prove existence and uniqueness of a class of traveling wave solutions which naturally emerges from the formal asymptotic analysis

Scaling laws for droplets spreading under contact-line friction

This manuscript is concerned with the spreading of a liquid droplet on a plane solid surface. The focus is on effective conditions which relate the speed of the contact line (where liquid, solid, and vapor meet) to the microscopic contact angle. One such condition has been recently proposed by Weiqing Ren and Weinan E [Phys. Fluids 19, 022101, 2007]: it includes into the model the effect of frictional forces at the contact line, which arise from unbalanced components of the Young's stress. In lubrication approximation, the spreading of thin droplets may be modeled by a class of free boundary problems for fourth order nonlinear degenerate parabolic equations. For speed-dependent contact angle conditions of rather general form, a matched asymptotic study of these problems is worked out, relating the macroscopic contact angle to the speed of the contact line. For the specific model of Hen and E, ODE arguments are then applied to infer the intermediate scaling laws and their timescales of validity: in complete wetting, they depend crucially on the relative strength of surface friction (at the liquid-solid interface) versus contact-line friction; in partial wetting, they also depend on the magnitude of the static contact-angle

Torsion in strain-gradient plasticity : energetic scale effects

We study elasto-plastic torsion in a thin wire within the framework of the strain-gradient plasticity theory elaborated by Gurtin and Anand in 2005. The theory in question envisages two material scales: an energetic length-scale, which takes into account the so-called "geometrically-necessary dislocations" through a dependence of the free energy on the Burgers tensor, and a dissipative length-scale. For the rate-independent case with null dissipative length-scale, we construct and characterize a special class of solutions to the evolution problem. With the aid of such characterization, we estimate the dependence on the energetic scale of the ratio between the torque and the twist. Our analysis confirms that the energetic scale is responsible for size-dependent strain-hardening, with thinner wires being stronger. We also detect, and quantify in terms of the energetic length-scale, both a critical twist, after which the wire becomes fully plastified, and two boundary layers near the external boundary of the wire and near the boundary of the plastified region, respectively

Mass-constrained minimization of a one-homogeneous functional arising in strain-gradient plasticity

We consider a minimization problem for a one-homogeneous functional, under the constraint that functions have a prescribed mean value. The problem originates from a one-dimensional strain-gradient theory of plasticity developed in [L. Anand, M.E. Gurtin, S.P. Lele, C. Gething, A one-dimensional theory of strain-gradient plasticity: formulation, analysis, numerical results, J. Mech. Phys. Solids 53 (2005) 1789-1826]: the minimizer represents the (normalized) time-derivative of the plastic strain in a strip-shaped sample undergoing simple shear with a given shear stress. Of particular interest in this theory is the dependence of solutions on a "dissipative length-scale" which quantifies the rate of free energy dissipation due to the plastic-strain flow. In arbitrary space dimension we identify the relaxation of the functional, we characterize its sub-differential, and we prove the existence of a minimizer. In addition, we identify a relation between the value of the minimum, the shear stress, and the Lagrange multiplier of the problem, and we use it to infer a monotonicity property of the shear stress with respect to the dissipative length-scale. Such property confirms that the strain-gradient theory under consideration is able to model the experimental evidence that smaller samples have higher relative strength. In one space dimension, where the model is proposed, we also prove uniqueness, regularity, and qualitative properties of the minimizer in the space SBV. (C) 2012 Elsevier Inc. All rights reserved