Table of contents

This special issue constitutes the Proceedings of the IUTAM Symposium on Plasticity at the Micron Scale, held at the Technical University of Denmark, 21–25 May 2006. The purpose of this symposium was to gather a group of leading scientists working in areas of importance to length scale dependent plasticity. This includes work on phenomenological strain gradient plasticity models, studies making use of discrete dislocation models, and even atomic level models. Experimental investigations are central to all this, as all the models focus on developing an improved understanding of real observed phenomena.

The opening lecture by Professor N A Fleck, Cambridge University, discussed experimental as well as theoretical approaches. Also, recent results for the surface roughness at grain boundaries were presented based on experiments and crystal plasticity modelling.

A number of presentations focused on experiments for metals at a small length scale, e.g. using indenters or a small single crystal compression test. It was found that there are causes of the size effects other than the geometrically necessary dislocations related to strain gradients. Several lectures on scale dependent phenomenological plasticity theories discussed different methods of incorporating the characteristic material length. This included lower order plasticity theories as well as higher order theories, within standard plasticity models or crystal plasticity. Differences in the ways of incorporating higher order boundary conditions were the subject of much discussion.

Various methods for discrete dislocation modelling of plastic deformation were used in some of the presentations to obtain a more detailed understanding of length scale effects in metals. This included large scale computations for dislocation dynamics as well as new statistical mechanics approaches to averaging of dislocation plasticity. Furthermore, at a somewhat larger length scale, applications of scale dependent plasticity to granular media and to cellular solids were discussed.

The symposium consisted of thirty-six lectures, all of which were invited based on strong expertise in the area. Some of the lectures are not represented in this special issue, mainly because of prior commitments to publish elsewhere.

The international Scientific Committee responsible for the symposium comprised the following:

• Professor V Tvergaard (Chairman) Denmark

• Professor A Benallal France

• Professor N A Fleck UK

• Professor L B Freund (IUTAM Representative) USA

• Professor E van der Giessen The Netherlands

• Professor J W Hutchinson USA

• Professor A Needleman USA

• Professor B Svendsen Germany

The Committee gratefully acknowledges financial support for the symposium from the International Union of Theoretical and Applied Mechanics, from Novo Nordisk A/S and from the Villum Kann Rasmussen Foundation.

In the organization of all parts of the symposium the enthusiastic participation of Dr C F Niordson and Dr P Redanz was invaluable. The smooth running of the symposium also owes much to the efforts and organizational skills of Bente Andersen.

A thin aluminium sheet comprising of large polycrystals is pulled in uniaxial tension and the resulting surface profile is measured in a scanning electron microscope. The surface profile near the grain boundaries reveals a local deformation pattern of width of a few micrometres and is strong evidence for strain gradient effects. Numerical analyses of a bicrystal undergoing in-plane tensile deformation are also studied using a strain gradient crystal plasticity theory and also by using a strain gradient plasticity theory for an isotropic solid. Both theories include an internal material length scale. An interfacial potential that penalizes the dislocations in crossing the grain boundary is included in the analysis. The results indicate that the surface profile is strongly dependent upon the choice of this potential and on the material length scale.

In this paper, we perform crystal plasticity analyses of micro-bending of thin f.c.c. metal foils having thicknesses ranging from 10 to 50 µm. The scale dependent crystal plasticity model used here is a viscoplastic finite strain version of the model proposed by Ohashi (2005 Int. J. Plast.21 2071–88), in which the mean free path of moving dislocations is determined by a function of the densities of statistically stored dislocations (SSDs) and geometrically necessary dislocations (GNDs), while the slip resistance for each slip system is determined only by the density of SSDs through a Bailey–Hirsch type relation. The computational results are compared with the experimental results for Ni foils, reported in Stölken and Evans (1998 Acta Mater.46 5109–15). The validity of the current model and the direction of future development of the 'physically-based' scale dependent crystal plasticity models are discussed.

A gradient plasticity theory for small deformations is presented within the framework of nonlocal continuum thermodynamics. The second principle (Clausius–Duhem inequality), enriched by an additional term named energy residual, is employed in conjunction with the concepts of insulation condition and locality recovery condition, in order to derive all the pertinent restrictions upon the constitutive equations. These include the expressions of the energy residual and of the plastic dissipation density, as well as the PDEs governing the gradient kinematic and isotropic hardening of the material, together with the related higher-order boundary conditions for both the fixed and the moving boundaries. Other formulations, which apparently do not make use of an energy residual, are shown to contain a latent one.

A two-dimensional implementation of a continuum dislocation-based model is presented. This model describes the evolution of dislocation fields during plastic deformation. Since it accounts for the self-force of the curved dislocations carrying the plastic deformation, it is length-scale dependent. The model is embedded in a standard plane strain continuum crystal plasticity framework which is numerically implemented by the finite element method. As a test problem, the simple shearing of a constrained crystalline strip is considered. Then, the response of a representative cell of a two-dimensional composite structure to shear load is simulated. It is shown that the model is able to reproduce the source-shortening effect that is observed in deformed composite materials.

The intrinsic size-effect for porous metals is investigated. The analyses are carried out numerically using a finite strain generalization of a higher order strain gradient plasticity model. Results for plane strain growth of cylindrical voids are presented in terms of response curves and curves of relative void growth. The influence of void size compared with a constitutive length parameter is analysed and it is shown that strain gradient hardening suppresses void growth on the micron scale. This increased resistance to void growth is accompanied by an increase in the overall strength of the material. For porous materials with small void volume fractions under highly triaxial tension, void growth is analysed through cavitation instabilities using a finite element Rayleigh–Ritz procedure. Cavitation instabilities are found to be delayed for small voids, so that higher stress levels are needed in order to obtain unstable growth. Cavitation diagrams for cylindrical voids are compared with cavitation diagrams for axi-symmetric void growth of initially spherical voids.

In the present report, the competition between dissipative plastic strain gradient effects in the bulk and in an interface is investigated within a strain gradient plasticity framework. The model of the interface is analysed in terms of hardening behaviour and strength for the case of a thin film with an elastic–plastic interface. It is found that the yield strength of the film is increased by length scale effects both in the bulk material and the interface. The effect is governed by quite a simple rule, namely the weakest link of bulk and interface. In addition, if the interface is allowed to harden, three regions are observed for the bulk (interior) and interface of the film during an increasing load: (i) elastic bulk and rigid interface, (ii) both bulk and interface plastic and (iii) plastic bulk and rigid interface. The properties of the model are illustrated with numerical results from a parametric study.

The isotropic single-parameter strain-gradient plasticity model of Fleck and Hutchinson (2001 J. Mech. Phys. Solids49 2245–71) is generalized in order to account for plastic anisotropy in a finite strain elastic–viscoplastic formulation. Plastic anisotropy is introduced through a phenomenological yield surface, which is described by a homogeneous function of degree 1. The anisotropy affects the plastic work done by the conventional effective plastic strain while the plastic work done by the strain-gradients remains unchanged. A modified measure of the effective stress arises naturally in the formulation. Local plasticity (isotropic and anisotropic) as well as nonlocal isotropic plasticity are special cases of this anisotropic nonlocal plasticity model. Results for an anisotropic nonlocal aluminium containing voids are obtained by a finite element implementation of the model.

Classical constitutive models of phenomenological plasticity/viscoplasticity rely heavily on yield functions to distinguish plastic flow from reversible elastic deformation. Physically based yield functions are utilized here for body-centred cubic (bcc) and face-centred cubic (fcc) types of metal structures in investigating necking and dynamic shear localizations over a wide range of temperatures and strain rates. The consistency model is employed in determining the increment of the viscoplastic multiplier and consequently a proper definition for the continuum elasto-viscoplastic tangent modulus is derived. Mesh-independent results are obtained using the finite element analysis in investigating the localization behaviour for tantalum, vanadium and niobium for bcc metals and OFHC copper for fcc metals.

Two sets of indentation and sliding discrete dislocation plasticity analyses are carried out to investigate the initiation of frictional sliding between a rigid asperity and a single crystal film. Most calculations are carried out for sinusoidal asperities, but for comparison purposes some results are presented for wedge-shaped asperities. In one set of calculations the friction coefficient is evaluated from separate indentation and sliding calculations while in another set the indentation and sliding processes are carried out sequentially. Both sets of friction calculations predict a similar friction stress versus contact size relation with the friction stress dominated by adhesion at small contact sizes, being plasticity governed at large contact sizes and being strongly size dependent at intermediate values of the contact size. Remarkably, the predicted values of the friction coefficient are similar for both sets of calculations even though the predicted deformation fields and dislocation structures differ significantly.

Gradient plasticity theories which are used to describe size effects require nonstandard boundary conditions. We believe that the role of these boundary conditions in combination with a moving elastic–plastic boundary has so far not been well understood. In order to clarify the conditions which are required at such an internal boundary, semi-analytical solutions are derived for a foil in pure bending. Two variants of gradient theories are considered, for which the necessary conditions are significantly different.

Two-dimensional discrete dislocation simulations of indentation in the sub-micron range are presented for wedge indenters with a sharp tip and for indenters with a circular tip. Plane strain calculations are carried out for single crystals that are initially free of mobile dislocations and with all dislocations nucleating from a specified distribution of internal sources. The hardness is expressed in terms of the indentation force divided by the actual contact area accounting for roughness of the surface in contact with the indenter. For wedge indenters the hardness is found to decrease with increasing indentation depth, while for indenters with a circular tip the hardness increases somewhat with increasing indentation depth. However, at a given indentation depth, the indentation hardness of circular indenters increases with decreasing tip radius. The difference in hardness evolution for the two tip shapes is mainly due to the manner in which the evolution of the contact area depends on indenter tip shape. The nominal hardness, i.e. that based on the geometric contact area neglecting material sink-in or pile-up and surface roughness, is found to follow the inverse square root size dependence predicted by Nix and Gao [1] and by Swadener et al [2], even though the plastic zone found in the simulations differs significantly in shape and size from that assumed in deriving the scaling laws.

This paper studies dislocation induced internal stresses as introduced in a strain gradient crystal plasticity approach. The need for a second-order crystal plasticity approach is motivated in the context of the analysis of free-standing metal films used in radio-frequency micro-electro-mechanical systems (RF-MEMS). The presented second-order framework incorporates the influence of inter- and intragranular deformation inhomogeneities in the constitutive description, generally assumed to be the basic origin of scale dependent behaviour and associated size effects. Focusing on the dynamic loading of thin RF-MEMS films, attention is given to dislocation induced kinematical hardening effects. The corresponding internal stresses are discussed with respect to their contribution to the solution of the underlying elasto-plastic framework which implicitly accounts for elastic incompatibilities. Within the loading regime of interest, the response in loading–unloading–reloading and cyclic loading is assessed.

In multiphase carbon steels, irreversible martensitic transformations from a parent austenitic phase are often accompanied by plastic deformations in the surrounding phases and damage growth in the martensitic product phase. In the present communication the interactions between these complex processes are studied through three-dimensional numerical analyses, where the response of the austenitic phase is simulated using a phase-changing damage model and the response of the surrounding matrix is mimicked with a plasticity model. A numerical update algorithm is provided for the phase-changing damage model, which is based on a fully implicit backward Euler scheme formulated within the framework of finite deformations. The consistent tangent operator for the model is computed using a numerical differentiation technique. Special attention is given to the robust and accurate treatment of the various constraints related to the volume fractions and damaged volume fractions of the transformation parent and product phases. The numerical analyses of a multiphase carbon steel microstructure illustrate the effect of the transformation, plasticity and damage processes on the overall stress–strain response and on the transformation and damage evolutions of the sample for various austenite crystal orientations. The computed responses are in qualitative agreement with experimental results reported in the literature. In terms of the numerical model, a variation study of the mesh density shows that the numerical results tend to converge upon mesh refinement.

Three practically useful reductions of phenomenological mesoscopic field dislocation mechanics (PMFDM) are explored to evaluate some of their capabilities and limitations. Lower-order gradient plasticity arises as one of these reduced frameworks, and a boundary condition appropriate for the modelling of constrained plastic flow in 3D is demonstrated through FEM simulation.

To elaborate and fix ideas in a simplified context, we motivate the averaging procedure giving rise to PMFDM in a 1D context and show the possibility of obtaining a diffusion term in coarse evolution through a perturbative approach. Formally, the diffusion appears as a small, first-order correction to mean transport of excess dislocation density. Physically, it arises from a more refined modelling of the slip distortion rate and not as an additional energetic 'back' stress contribution. The approximate nature of the perturbative approach is assessed to highlight the importance of the transport term in the equation. We also comment on the emergence of stochasticity and memory effects in coarse response.

The results of two sets of experiments to measure the elastic–plastic behaviour of gold at the nanometre length scale are reported. One set of experiments was on free-standing nanoscale single crystals of gold, and the other was on free-standing nanoscale specimens of open-celled nanoporous gold. Both types of specimens were fabricated from commercially available leaf which was either pure Au or a Au/Ag alloy following by dealloying of the Ag. Mechanical testing specimens of a 'dog-bone' shape were fabricated from the leaf using standard lithographic procedures after the leaf had been glued onto a silicon wafer. The thickness of the gauge portion of the specimens was about 100 nm, the width between 250 nm and 300 nm and the length 7 µm. The specimens were mechanically loaded with a nanoindenter (MTS) at the approximate midpoint of the gauge length. The resulting force–displacement curve of the single crystal gold was serrated and it was evident that slip localization occurred on individual slip systems; however, the early stages of the plastic deformation occurred in a non-localized manner. The results of detailed finite element analyses of the specimen suggest that the critical resolved shear stress of the gold single crystal was as high as 135 MPa which would lead to a maximum uniaxial stress of about 500 MPa after several per cent strain. The behaviour of the nanoporous gold was substantially different. It exhibited an apparent elastic behaviour until the point where it failed in an apparently brittle manner, although it is assumed that plastic deformation occurred in the ligaments prior to failure. The average elastic stiffness of three specimens was measured to be E_np = 8.8 GPa and the stress at ultimate failure averaged 190 MPa for the three specimens tested. Scaling arguments suggest that the stress in the individual ligaments could approach the theoretical shear strength.

Dislocation and grain-boundary processes contribute significantly to plastic behaviour in polycrystalline metals, but a full understanding of the interaction between these processes and their influence on plastic response has yet to be achieved. The coupled atomistic discrete-dislocation method is used to study edge dislocation pile-ups interacting with a Σ11-⟨1 1 3⟩ symmetric tilt boundary in Al at zero temperature under various loading conditions. Nucleation of grain-boundary dislocations (GBDs) at the dislocation/grain-boundary intersection is the dominant mechanism of deformation. Dislocation pile-ups modify both the stress state and the residual defects at the intersection, the latter due to multiple dislocation absorption into the boundary, and so change the local grain-boundary/dislocation interaction phenomena as compared with cases with a single dislocation. The deformation is irreversible upon unloading and reverse loading if multiple lattice dislocations absorb into the boundary and damage in the form of microvoids and loss of crystalline structure accumulates around the intersection. Based on these results, the criteria for dislocation transmission formulated by Lee, Robertson and Birnbaum are extended to include the influences of grain-boundary normal stress, shear stress on the leading pile-up dislocation and minimization of step height at the intersection. Two possible yield loci for the onset of GBD nucleation versus compressive stress and relevant shear stresses are derived from the simulations. These results, and similar studies on other boundaries and dislocation characters, guide the formulation of continuum constitutive behaviours for use in discrete-dislocation or strain-gradient plasticity modelling.

A large-scale computational and statistical strategy is presented to investigate the development of plastic strain heterogeneities and plasticity induced roughness at the free surface in multicrystalline films subjected to cyclic loading conditions, based on continuum crystal plasticity theory. The distribution of plastic strain in the grains and its evolution during cyclic straining are computed using the finite element method in films with different ratios of in-plane grain size and thickness, and as a function of grain orientation (grains with a {1 1 1} or a {0 0 1} plane parallel to the free surface and random orientations). Computations are made for 10 different realizations of aggregates containing 50 grains and one large aggregate with 225 grains. It is shown that overall cyclic hardening is accompanied by a significant increase in strain dispersion. The case of free-standing films is also addressed for comparison. The overall surface roughness is shown to saturate within 10 to 15 cycles. Plasticity induced roughness is due to the higher deformation of {0 0 1} and random grains and due to the sinking or rising at some grain boundaries.

A two-dimensional (2D) discrete dislocation plasticity framework, which incorporates some three-dimensional mechanisms through constitutive additions, is used to analyse the response to uniaxial tension of nanoscale bilayer thin films. Frank–Read sources, modelled as junction dipoles in 2D, act as sources of dislocations. Infinite, homogeneous medium fields of the discrete dislocations are superposed with a non-singular complementary field that enforces the boundary conditions and accounts for image stresses arising from the difference in elastic properties between the layers. The resulting boundary value problem is solved using the finite element method. Analysis has been carried out for fully coherent bilayer Al/Cu and Cu/Ni films oriented for double slip. The analysis accounts for the effects of three key mechanisms: resistance to dislocation nucleation and motion due to elastic modulus mismatch (e.g. Koehler barrier); single-dislocation bow-out within layers (Orowan process) and slip blocking at interfaces (Hall–Petch mechanism). The relative importance of each mechanism is studied as a function of the bilayer thickness. The results indicate a significant strengthening with decreasing bilayer thickness. Conclusions are drawn regarding the possible causes of the observed strengthening.

A model is developed for nano-indentation with spherical indenters. It is based on the Taylor dislocation model and is an extension of Qu et al's (2006 Int. J. Plast.22 1265–86) micro-indentation model for spherical indenters to nano-indentation. For relatively large indenters (e.g. radii on the order of 100 µm), the present model degenerates to Qu et al (2006 Int. J. Plast.22 1265–86). For small indenters (e.g. radii on the order of 10 µm), the maximum allowable geometrically necessary dislocation (GND) density is introduced to cap the GND density such that the latter does not become unrealistically high for small indenters. The present nano-indentation model agrees well with the indentation hardness data of iridium for both nano-and micro-indentation.

This study develops a continuum theory of small-deformation single crystal plasticity that includes 'strain gradients' and 'strain rate gradients' in the spirit of our recent papers (Gurtin and Anand 2005a, 2005b) for isotropic plasticity. Specifically, we develop a constitutive theory that allows for

• energetic microscopic stresses that arise as a consequence of an assumed dependence of the free energy on edge and screw dislocations densities induced by slip gradients; and

• dissipative microscopic stresses that depend on slip-rate gradients.

Our theory is based on a system of microscopic force balances—one balance for each slip system—with derivation based on the principle of virtual power. When augmented by constitutive relations for the microscopic stresses these force balances result in non-local flow rules in the form of second-order partial differential equations for the slips.

The Cauchy–Born rule is widely used as a standard method in continuum mechanics in order to construct descriptions of material behaviour using atomistic information. The main objective of the present work is to investigate the validity of this kinematic assumption, i.e. to determine the state when a transition to non-affine deformations is possible due to instabilities of the underlying atomic system. To this end, the results of the Cauchy–Born rule are compared with the results of direct molecular dynamics calculations on the one hand and with the results obtained by using the acoustic tensor of continuum mechanics on the other hand.

A bifurcation and stability analysis is carried out here for a bar made of a material obeying a gradient damage model with softening. We show that the associated initial boundary-value problem is ill posed and one should expect mesh sensitivity in numerical solutions. However, in contrast to what happens for the underlying local damage model, the damage localization zone has a finite thickness and stability arguments can help in the selection of solutions.

In non-close-packed crystalline lattices, e.g. bcc metals and intermetallics, the stress-state dependence of the Peierls barrier for the motion of a screw dislocation violates Schmid's law and leads to non-associated plastic flow in single crystals and polycrystalline aggregates (Bassani 1994 Adv. Appl. Mech.30 191–258, Bassani et al 2001 Mater. Sci. Eng. A 319–321 97–101). Plasticity models based upon distinct yield and flow functions are proposed to describe polycrystalline behaviour and specialized to the case of isotropic response. Studies of sheet necking using both a Marciniak–Kuczynski analysis and finite elements predict that non-associated flow has a significant effect on the evolution of inhomogenieties in the sheet. For nearly rate-insensitive response, intermittent bursts of strain arise as a consequence of non-associated flow, particularly for deformations near the plane-strain state. These strain bursts, which are due to instantaneously unstable deformation behaviour, are observed to be sensitive to mesh refinement in the case of a purely local constitutive description. Strain-gradient effects are introduced, which significantly improves convergence.

This short review for the special conference issue presents briefly experimental results on the interaction of cracks and dislocations with grain boundaries obtained from local measurements. Artificial notches parallel to slip bands were made with a focused ion beam (FIB), which could be directly positioned near special interfaces and grain boundaries. The growth rate of cracks initiated from these notches was measured as a function of the distance to defined boundaries using a replica technique. A microcrack passing through a grain boundary was reconstructed from FIB sections (nanotomography). The experiments clearly showed the strong effect of special boundaries on crack growth. Using a nanoindenter, the effect of grain boundaries on the load–displacement curve was then studied. In the experiments, the grain size was always chosen to be a little larger than the indenter size in order to guarantee that the dislocations nucleated below the tip could interact with the surrounding interfaces. The effect of grain boundaries on the indentation size effect and on the strain rate sensitivity was examined. The results clearly showed that below a grain size of 900 nm lateral boundaries override the indentation size effect and are responsible for the rate sensitivity of Ni below this grain size.