Size effects and length scales in gradient plasticity and dislocation dynamics

Deformation in metals is size dependent over length scales ranging from a few nanometers to 100 μm. Two frameworks, dislocation dynamics and gradient plasticity, each having a length scale with different interpretation are discussed. It is suggested that various dislocation arrangements produce different physical behavior with different gradients and length scales.     

Dislocation dynamics simulations of plasticity in Fe laths at low temperature

Plastic deformation in 16MND5 steel made of Fe laths is investigated using three-dimensional dislocation dynamics (DD) simulations, adapted to treat the body-centred cubic crystalline structure, strained in the ductile to brittle transition temperature range. In that regime, the edge segment velocity is proportional to the local effective resolved shear stress, whereas the screw segments follow a thermal activation scheme. The adopted cross-slip rules are derived from atomistic simulations, implemented in the DD code using a kinetic Monte Carlo algorithm. Specific loading and boundary conditions are worked out, with a view to accounting for the bainitic microstructure of the steel and its specific deformation mode. In these conditions, the implemented cross-slip behaviour is shown to play an essential role in the development of specific dislocation arrangements forming at different temperatures, also observed in 16MND5 steel. The presented results also provide insights on dislocation-based deformation mechanisms possibly involved in damage initiation.     

Discrete dislocation plasticity modeling of short cracks in single crystals

The mode-I crack growth behavior of geometrically similar edge-cracked single crystal specimens of varying size subject to both monotonic and cyclic axial loading is analyzed using discrete dislocation dynamics. Plastic deformation is modeled through the motion of edge dislocations in an elastic solid with the lattice resistance to dislocation motion, dislocation nucleation, dislocation interaction with obstacles and dislocation annihilation incorporated through a set of constitutive rules. The fracture properties are specified through an irreversible cohesive relation. Under monotonic loading conditions, with the applied stress below the yield strength of the uncracked specimen, the initiation of crack growth is found to be governed by the mode-I stress intensity factor, calculated from the applied stress, with the value of K_init decreasing slightly with crack size due to the reduction in shielding associated with dislocations near a free surface. Under cyclic loading, the fatigue threshold is ΔK-governed for sufficiently long cracks. Below a critical crack size the value of ΔK_I at the fatigue threshold is found to decrease substantially with crack size and progressive cyclic crack growth occurs even when K_max is less than that required for the initiation of crack crack growth in an elastic solid. The reduction in the fatigue threshold with crack size is associated with a progressive increase in internal stress under cyclic loading. However, for sufficiently small cracks, the dislocation structure generated is sparse and the internal stresses and plastic dissipation associated with this structure alone are not sufficient to drive fatigue crack growth.     

GNDs in nonlocal plasticity theories: lessons from discrete dislocation simulations

Solutions to simple boundary value problems using discrete dislocation plasticity exhibit key features associated with an accumulation of geometrically necessary dislocations: size effects and long-range stresses. Comparisons of the predictions of nonlocal plasticity theories with the discrete dislocation results provide an assessment of the strengths and weaknesses of various phenomenological theories and also provide a means of identifying appropriate nonlocal boundary conditions.     

Initial dislocation structures in 3-D discrete dislocation dynamics and their influence on microscale plasticity

Realistic dislocation network topologies were generated by relaxing an initially pinning point free dislocation loop structure using three-dimensional discrete dislocation dynamics simulations. Traction-free finite-sized samples were used. Subsequently, these equilibrated structures were subjected to tensile loading and their mechanical behavior was investigated with respect to the initial configuration. A strong mechanical size effect was found. The flow stress at 0.2% plastic deformation scales with specimen size with an exponent between ?0.6 and ?0.9, depending on the initial structure and size regime. During relaxation, a mechanism, also favored by cross-slip, is identified which leads to rather stable pinning points. These pinning points are comparable to those of the isolated Frank–Read sources often used as a starting configuration in previous discrete dislocation dynamics simulations. These nodes act as quite stable dislocation sources, which can be activated multiple times. The influence of this source mechanism on the mechanical properties of small-scale specimens is discussed.     

Dislocation dynamics simulations of dislocation interactions and stresses in thin films

The dislocation interactions that stop threading dislocations (threads) during relaxation at increasing applied strains in single-crystal thin films are investigated using large-scale three-dimensional dislocation dynamics simulations. Threads were observed to stop via interactions with both threads and misfit dislocations (misfits). Both types of interactions were shown to depend on stress inhomogeneity. Low-stress regions enabled threads to stop in weak thread–misfit interactions even at high average film stresses. Threads were also concentrated in low-stress regions, which facilitated their interaction with other threads. Threads accumulated in thread–thread interactions, and stopped only temporarily in thread–misfit interactions. The mean free path for dislocation motion is shown to be accurately predicted from details of the inhomogeneous stress state arising from the applied strain and the misfit structure. These behaviors are analyzed to present a more complete picture of film strength, strain hardening and relaxation.     

Origin of gradient terms in plasticity at different length scales

The origins of gradient terms in plasticity are discussed for three different cases related to atomic distance, dislocations, and finite grain size.     

Discrete dislocation plasticity analysis of single crystalline thin beam under combined cyclic tension and bending

The cyclic plastic response of a single crystalline thin beam subject to combined cyclic tension and bending is analyzed using two-dimensional discrete dislocation plasticity. In this contribution, special attention is paid to the difference in the inherent mechanism of the size effect for different cyclic loads. Results show that the cyclic plastic response has a strong size effect for both cyclic pure tension–compression and pure bending. However, the inherent mechanisms are different. The dislocation starvation mechanism dominates the cyclic tension–compression while the geometrically necessary dislocation dominates the cyclic pure bending. When the combined cyclic tension and bending are applied to the thin beam, the cyclic moment–rotation response shows strong size effect while the stress–strain response shows weak or even no size effect. In addition, it is also found that the cyclic loading paths have considerable influences on the shape of the cyclic stress–strain loops.     

Mechanical and energetical analysis of molecular dynamics simulations of dislocation–defect interactions

In this paper, we present a method of analysis on the continuum scale of molecular dynamics simulations of dislocation–defect interactions. It is shown how the applied work can be decomposed into its elementary components: the elastic strain energy, the dissipated work against the Peierls stress and the curvature energy needed to enable the dislocation, pinned by the defect, to bow out. While the dissipated work is entirely extracted from the system in molecular statics, the elastic and curvature energies contribute to a large increase in the internal energy. The curvature energy is evaluated and compared with predictions of the line tension model. This analysis allows the calculation of the dislocation–defect interaction energy and the determination of the predominant features in the unpinning process.     

Discrete dislocation dynamics modelling of microvoid growth and its intrinsic mechanism in single crystals

In the present paper, an infinite face-centered cubic single crystal containing an isolated cylindrical micron-sized void, which is subjected to proportional and monotonically uniform equal biaxial tension loading, is adopted to study the scale-dependent void growth and its intrinsic mechanism by employing a two-dimensional planar discrete dislocation dynamic framework. First, a typical dislocation distribution near the microvoid is presented and the void growth mechanism is revealed by dislocation shear loop expansion for each of three typical fcc slip systems. The effect of size on void growth is then investigated. The general conclusion that voids at the micron or submicron scale are less susceptible to growth than larger ones is drawn. Another result, which cannot be deduced from the continuum theories, is also achieved: at the micron or submicron scale, larger voids grow smoothly with remote strain, while smaller voids usually grow in a “leapfrog” manner. Specifically, when the void is even smaller, it grows in an approximately linear-elastic manner since only few dislocations are present around the void. Further analyses indicate that these size effects are closely related to the dislocation density on the void surface and the dislocation mobility around the void. Finally, the influences of the dislocation sources/obstacles density and their random distribution in materials on the void growth are studied briefly. Results show that there exists remarkable scatter in the microvoid growth due to random distribution of the dislocation sources or obstacles, especially for voids at the submicron scale. These results are helpful for us in understanding the size-dependent damage mechanism at the micron or submicron scale.     

Geometrically necessary dislocation and size-dependent plasticity

There has recently been a strong interest in modeling size-dependent plasticity in metals based on the concept of geometrically necessary dislocations. This article presents a brief summary of our viewpoints on geometrically necessary dislocations and their role in the development of continuum plasticity theories with an intrinsic material length scale.     

Discrete dislocation modeling of fatigue crack propagation

Analyses of the growth of a plane strain crack subject to remote mode I cyclic loading under small-scale yielding are carried out using discrete dislocation dynamics. Cracks along a metal–rigid substrate interface and in a single crystal are studied. The formulation is the same as that used to analyze crack growth under monotonic loading conditions, differing only in the remote stress intensity factor being a cyclic function of time. Plastic deformation is modeled through the motion of edge dislocations in an elastic solid with the lattice resistance to dislocation motion, dislocation nucleation, dislocation interaction with obstacles and dislocation annihilation being incorporated through a set of constitutive rules. An irreversible relation is specified between the opening traction and the displacement jump across a cohesive surface ahead of the initial crack tip in order to simulate cyclic loading in an oxidizing environment. The cyclic crack growth rate log(da/dN) versus applied stress intensity factor range log(ΔK_I) curve that emerges naturally from the solution of the boundary value problem shows distinct threshold and Paris law regimes. Paris law exponents in the range 4 to 8 are obtained for the parameters employed here. Furthermore, rather uniformly spaced slip bands corresponding to surface striations develop in the wakes of the propagating cracks.     

Analytic treatment of metallic multilayer strength at all length scales: Influence of dislocation sources

Metallic multilayers exhibit a very pronounced size effect where the mechanical strength depends on the layer thickness. Traditionally the Hall–Petch relation is used to account for the size effect. However, rigorous application of dislocation pileup theory predicts significant deviation from the Hall–Petch relation due to elastic inhomogeneity, discreteness of dislocations and dislocation source operation. Elastic inhomogeneity leads to anomalous scaling where the scaling exponent deviates from 1/2 of the classical Hall–Petch relation. The discrete dislocation effect is properly accounted for by a piecewise approach that can be applied at all length scales. In this article, a key step in the formulation is taken: the dislocation source characteristics are taken into consideration. Thus, all the three effects are accounted for. Analytic formulas linking yield stress to microscopic interface strength, dislocation source activation stress and other easily obtainable parameters (the Burgers vector, the elastic constants of constituent materials, crystal structure and layer thickness) are provided for all length scales. The model is then applied to Cu/Ni multilayers and the predicted strength is compared with experimental data.     

Creep deformation behavior of Sn-3.5Ag solder at small length scales

To adequately characterize the behavior of solder spheres in electronic packaging, their mechanical behavior needs to be studied at small-length scales. The creep behavior of single Sn-3.5Ag solder spheres on a copper substrate was studied between 25°C and 130°C using a microforce testing system. In the low-stress regime, the creep stress exponent changed from 6 at lower temperatures to 4 at higher temperatures, indicating creep by dislocation climb. The activation energy for creep was also found to be temperature dependent, and correlated with values for dislocation core diffusion and lattice diffusion in pure tin. A change in the stress exponent with increasing stress was also observed and explained in terms of a threshold stress for dislocation motion, due to the presence of obstacles in the form of Ag₃Sn particles. For more information, contact N. Chawla, Arizona State University, Department of Chemical and Materials Engineering, Ira A. Fulton School of Engineering, Tempe, AZ 85287-6006; e-mail nchawla@asu.edu.     

Crystal plasticity simulations using discrete Fourier transforms

In this paper, we explore efficient representation of all of the functions central to crystal plasticity simulations in their complete respective domains using discrete Fourier transforms (DFTs). This new DFT approach allows for compact representation and fast retrieval of crystal plasticity solutions for a crystal of any orientation subjected to any deformation mode. The approach has been successfully applied to a rigid–viscoplastic Taylor-type model for face-centered cubic polycrystals. It is observed that the novel approach described herein is able to speed up the conventional crystal plasticity computations by two orders of magnitude. Details of this approach are described and validated in this paper through a few example case studies.     

聚晶金刚石压痕行为的离散位错动力学仿真分析

依据位错理论,建立了聚晶金刚石(polycrystalline diamond,PCD)离散位错动力学-有限元耦合的压痕仿真模型。通过离散位错动力学仿真,获得了在微压痕作用下PCD的位错云图,揭示了金刚石微粉颗粒尺寸及颗粒体积分数对聚晶金刚石压痕特性的影响规律,并分析了聚晶金刚石的位错演化机制及其失效行为。研究结果表明:采用离散位错动力学分析方法能够准确描述PCD的位错运动过程,PCD微粉颗粒尺寸增加会增大晶体位错滑移程度,金刚石颗粒体积分数增加则导致位错形核密度增加,最终将引起聚晶金刚石材料断裂强度下降。将离散位错理论引入到为分析聚晶金刚石位错演化的行为中,将为研究PCD在切削刀具等工程应用中材料的失效行为提供理论基础。