^{1}

^{2}

^{*}

^{1}

^{3}

^{1}

Analyzed the data: PM BC M-CM. Wrote the paper: PM M-CM. Conceived and designed the study: M-CM PM. Performed the simulations: BC M-CM. Performed the elasticity theory calculations: PM.

The authors have declared that no competing interests exist.

Nanoindentation techniques recently developed to measure the mechanical response of crystals under external loading conditions reveal new phenomena upon decreasing sample size below the microscale. At small length scales, material resistance to irreversible deformation depends on sample morphology. Here we study the mechanisms of yield and plastic flow in inherently

Over the past years, experimental investigations have gathered increasing evidence that plastic deformation of crystalline materials proceeds through intermittent bursts of activity

While several nanoindentation techniques have been developed to measure material resistance to irreversible deformation and plastic flow, the experimental observation of plasticity at the nanoscale still represents an enormous challenge

We first consider the compression of perfect crystals of various sizes and aspect ratios. Crystals are simulated as two-dimensional aggregates of short-range interacting monodisperse particles in their lowest energy configuration, that is a triangular lattice in the

where

(Left) Schematic representation of a uniaxial compression test on a perfect crystal monolayer. (Right) We obtain i) a serrated-flow stress response under displacement (or strain) control conditions, i.e. ruling the position of the bounding walls (inside the grey boxes), or ii) a staircase shaped curve under force or stress control conditions, i.e. regulating the applied force on the driving walls. The dashed line signals the location of the yield point.

The coupled Eqs. (1) for

In both protocols, the response is initially elastic. In a perfect crystal, the elastic limit is reached as soon as the motion of a pair of opposite sign edge dislocations is activated, as in

(Left) Perfect crystal, a dislocation pair is nucleated. Dislocations are represented as pairs of 5- and 7-coordinated particles, in blue and red respectively. (Center) Instantaneous velocity field of the particles. (Right) Surface-disordered crystal, dislocations may be individually nucleated at surface steps.

By moving, dislocations allow the system to slip plastically and emit/dissipate part of the stored elastic energy. The value of both the yield stress

Smaller systems are stronger, but also different aspect ratios

According to the literature, in larger systems the dependence of the yield stress on the system size follows a power law

The connection between the yield stress and the boundary effects can be easily visualized in our simple model system. Up to values of the stress very close to

where

Equation (3) establishes a connection between the yield stress and dislocation strain distribution. It also bears implicitly the information about the dependence of the yield stress on the system size and geometry.

The essence of our problem then lies in the stress distribution that accounts for

By means of elasticity theory, we can demonstrate that the stress fields produced inside the sample by an edge dislocation close to a rigid boundary are long ranged and decay as

with boundary conditions such that the full

where

(Left) Shear stress distribution around an edge dislocation in an infinite medium. The right side of the picture is shaded, indicating that we are interested in how the stress field changes upon imposing a rigid boundary. (Center) Shear stress distribution around an edge dislocation located near a rigid boundary. (Right) Shear stress distribution due to a pair of opposite sign dislocations, confined within two rigid walls. Stresses rapidly vanish in the

From the above result, the elastic strain tensor

In the case of our simulations, however, Eq. (6) may seem of limited help in calculating the strain energy

We can conclude that the configuration in

As soon as the yield point is reached, the response of the system to further loading differentiates depending on the deformation protocol. Under conditions of displacement control, stress-strain curves are characterized by serrated yielding, while they assume a staircase shape under conditions of stress control. We emulate realistic realizations of compressed samples by introducing randomness at free boundaries as follows. The initial state of each realization is obtained from the perfect crystal by extracting a random number of particles from one free surface and relocating them at random positions on the opposing surface. In this way both the number of particles and the linear size of all simulated specimens are kept constant, while the morphologies of their free surfaces are allowed to vary stochastically. Strain plateaus

Left: (a) Distribution of platen displacements

Due to the limited system size, moving dislocations easily leave the sample through free boundaries. Pioneering studies have shown that in sub-micrometer Ni samples, pure mechanical loading can induce dislocation depletion within the sample

Plastic event sizes are commonly quantified in experiments by looking at the amount of energy

Here the dissipated energy

Let us first consider the case of force control. If focusing on a single platen displacement event, we have

(a) Temporal evolution of the nominal strain

Under displacement control conditions, instead, the statistical analysis of plastic flow and dissipated energy can be performed by looking at the distribution of stress drops (

Compared to numerical studies of size effects in dislocation dynamics at the microscale

As for the origin of the anomalous avalanche exponents, we should remark that the novel behavior is related to the inability of the system to store large numbers of dislocations. In the absence of collective behavior, plastic flow departs from the traditional picture of cooperative dislocation organization. In fact our simulations show a behavior which approximately recalls a sequence of load-unload events, much in the spirit of stick-slip dynamics or fracture/failure mechanics. We notice that an energy release exponent very close to

In conclusion, we have shown that the onset of plasticity at small scales is mediated by few dislocations. The number and arrangement of nucleated dislocations must account for the distribution of stress stored inside the crystal during the elastic-loading regime, allowing one to estimate the dependence of the yield stress on sample size and geometry. Our results confirm that both size and shape are crucial factors in determining the strength of materials at these scales. We find that plastic flow occurs in an intermittent manner reminiscent of irreversible deformation at larger length scales. Plastic avalanches of broadly distributed sizes are still observed, however, the absence of dislocation storage has important effects on the scaling characteristics of viscoplastic dynamics, which ultimately violate the

The first appearance of a dislocation pair signals the onset of yield for the perfect system. As soon as each dislocation reaches the opposite rigid boundary, plastic activity stops and the the first event is over. The animation consists of 7 snapshots of the dynamics. Top: dislocations are represented as pairs of 5- and 7-coordinated particles, in blue and red respectively. Bottom: velocity field corresponding to the dislocation configuration above. The modulus of the velocity vector is represented. Lower velocities are in red, higher in violet, according to the color scheme of the visible spectrum.

(MOV)

Time evolution of the velocity field at the time-steps shown in Video 1. Moving dislocations trigger particle motion and thus elastic energy dissipation.

(MOV)

Dislocation dynamics in the flow regime. Plastic events correspond to the activation of few dislocations at a time. Dislocation storage is not observed for such small systems. The animation consists of 20 snapshots of the dynamics. Conventions are as in Video 1.

(MOV)

The authors are grateful to M. Zaiser, L. Laurson and I. Groma for stimulating discussions.