Bauschinger and size effects in thin-film plasticity due to defect-energy of geometrical necessary dislocations

The Bauschinger and size effects in the thinfilm plasticity theory arising from the defect-energy of geometrically necessary dislocations (GNDs) are analytically investigated in this paper. Firstly, this defect-energy is deduced based on the elastic interactions of coupling dislocations (or pile-ups) moving on the closed neighboring slip plane. This energy is a quadratic function of the GNDs density, and includes an elastic interaction coefficient and an energetic length scale L. By incorporating it into the work- conjugate strain gradient plasticity theory of Gurtin, an energetic stress associated with this defect energy is obtained, which just plays the role of back stress in the kinematic hardening model. Then this back-stress hardening model is used to investigate the Bauschinger and size effects in the tension problem of single crystal Al films with passivation layers. The tension stress in the film shows a reverse dependence on the film thickness h. By comparing it with discrete-dislocation simulation results, the length scale L is determined, which is just several slip plane spacing, and accords well with our physical interpretation for the defect- energy. The Bauschinger effect after unloading is analyzed by combining this back-stress hardening model with a friction model. The effects of film thickness and pre-strain on the reversed plastic strain after unloading are quantified and qualitatively compared with experiment results.     

Multiple interacting cracks in an orthotropic layer

The stress fields in an orthotropic layer containing climb and glide edge dislocations are obtained by means of the complex Fourier transform. Stress analysis in the intact layer under in-plane point loads is also carried out. These solutions are employed to derive integral equations for the layers weakened by several interacting cracks subject to in-plane deformation. The integral equations are of Cauchy singular type. These equations are solved numerically for the density of dislocations on a crack surface. The dislocation densities are utilized to derive stress intensity factor for cracks. Several examples are solved and the interaction between the two cracks is investigated.     

Study of size effects in thin films by means of a crystal plasticity theory based on DiFT

In a recent publication, we derived the mesoscale continuum theory of plasticity for multiple-slip systems of parallel edge dislocations, motivated by the statistical-based nonlocal continuum crystal plasticity theory for single-glide given by Yefimov et al. [2004b. A comparison of a statistical-mechanics based plasticity model with discrete dislocation plasticity simulations. J. Mech. Phys. Solids 52, 279-300]. In this dislocation field theory (DiFT) the transport equations for both the total dislocation density and geometrically necessary dislocation (GND) density on each slip system were obtained from the Peach-Koehler interactions through both single and pair dislocation correlations. The effect of pair correlation interactions manifested itself in the form of a back stress in addition to the external shear and the self-consistent internal stress. We here present the study of size effects in single crystalline thin films with symmetric double slip using the novel continuum theory. Two boundary value problems are analyzed: (1) stress relaxation in thin films on substrates subject to thermal loading, and (2) simple shear in constrained films. In these problems, earlier discrete dislocation simulations had shown that size effects are born out of layers of dislocations developing near constrained interfaces. These boundary layers depend on slip orientations and applied loading but are insensitive to the film thickness. We investigate the stress response to changes in controlled parameters in both problems. Comparisons with previous discrete dislocation simulations are discussed.     

Surface versus bulk nucleation of dislocations during contact

The indentation of single crystals by a periodic array of flat rigid contacts is analyzed using discrete dislocation plasticity. Plane strain analyses are carried out with the dislocations all of edge character and modeled as line singularities in a linear elastic solid. The limiting cases of frictionless and perfectly sticking contacts are considered. The effects of contact size, dislocation source density, and dislocation obstacle density and strength on the evolution of the mean indentation pressure are explored, but the main focus is on contrasting the response of crystals having dislocation sources on the surface with that of crystals having dislocation sources in the bulk. When there are only bulk sources, the mean contact pressure for sufficiently large contacts is independent of the friction condition, whereas for sufficiently small contact sizes, there is a significant dependence on the friction condition. When there are only surface dislocation sources the mean contact pressure increases much more rapidly with indentation depth than when bulk sources are present and the mean contact pressure is very sensitive to the strength of the obstacles to dislocation glide. Also, on unloading a layer of tensile residual stress develops when surface dislocation sources dominate.     

Residual stresses in thin film systems: Effects of lattice mismatch,thermal mismatch and interface dislocations

This paper explores the mechanisms of the residual stress generation in thin film systems with large lattice mismatch strain, aiming to underpin the key mechanism for the observed variation of residual stress with the film thickness. Thermal mismatch, lattice mismatch and interface misfit dislocations caused by the disparity of the material layers were investigated in detail. The study revealed that the thickness-dependence of the residual stresses found in experiments cannot be elucidated by thermal mismatch, lattice mismatch, or their coupled effect. Instead, the interface misfit dislocations play the key role, leading to the variation of residual stresses in the films of thickness ranging from 100 nm to 500 nm. The agreement between the theoretical analysis and experimental results indicates that the effect of misfit dislocation is far-reaching and that the elastic analysis of dislocation, resolved by the finite element method, is sensible in predicting the residual stress distribution. It was quantitatively confirmed that dislocation density has a significant effect on the overall film stresses, but dislocation distribution has a negligible influence. Since the lattice mismatch strain varies with temperature, it was finally confirmed that the critical dislocation density that leads to the measured residual stress variation with film thickness should be determined from the lattice mismatch strain at the deposition temperature.     

Thickness effects in polycrystalline thin films: Surface constraint versus interior constraint

The uniaxial tension behavior of polycrystalline thin films, in which all grain boundaries (GBs) are penetrable by dislocations, is investigated by two-dimensional discrete dislocation dynamics (DDD) method with a penetrable dislocation-GB interaction model. In order to study thickness effect on the tensile strength of thin films with and without surface treatment, three types of thin films are comparatively considered, including the thin films without surface treatment, with surface passivation layers (SPLs) of nanometer thickness and with surface grain refinement zones (SGRZs) consisting of nano-sized grains. Our results show that thickness effects and their underlying dislocation mechanisms are quite distinct among different types of thin films. The thicker thin films without surface treatment are stronger than the thinner ones; however, opposite thickness effects are captured in the thin films with SPLs or SGRZs. Moreover, the underlying dislocation mechanisms of the same thickness effects of thin films with SPLs and SGRZs are different. In the thin films with SPLs, the thickness effect is caused by the sharp increase of dislocation density near the film-passivation interface, while it is mainly due to the sharp decrease of dislocation density within the refined surface grains of the thin films with SGRZs. No matter in what type of thin films, thickness effect gradually disappears when the number of grains in the thickness direction is large enough. Our analysis reveals that general mechanism of those thickness effects lies in the competition between the exterior surface-constraint and interior GB-constraint on gliding dislocations.     

In-plane analysis of a cracked orthotropic half-plane

The stress fields in an orthotropic half-plane containing Volterra type climb and glide edge dislocations under plane stress condition are derived. The dislocation solutions are utilized to formulate integral equations for dislocation density functions on the surface of smooth cracks embedded in the half-plane under in-plane loads. The integral equations are of Cauchy singular type which are solved numerically. The dislocation density functions are employed to evaluate modes I and II stress intensity factors for multiple cracks with different configurations.     

Elastodynamic analysis of a plane weakened by several cracks

The stress fields in an infinite plane containing Volterra type climb and glide edge dislocations under time-harmonic excitation are derived. The dislocation solutions are utilized to formulate integral equations for dislocation density functions on the surfaces of smooth cracks. The integral equations are of Cauchy singular type which are solved numerically for several different cases of crack configurations and arrangements. The results are used to evaluate modes I and II stress intensity factors for multiple smooth cracks.     

Determination of interaction forces between parallel dislocations by the evaluation of J integrals of plane elasticity

The Peach–Koehler expressions for the glide and climb components of the force exerted on a straight dislocation in an infinite isotropic medium by another straight dislocation are derived by evaluating the plane and antiplane strain versions of J integrals around the center of the dislocation. After expressing the elastic fields as the sums of elastic fields of each dislocation, the energy momentum tensor is decomposed into three parts. It is shown that only one part, involving mixed products from the two dislocation fields, makes a nonvanishing contribution to J integrals and the corresponding dislocation forces. Three examples are considered, with dislocations on parallel or intersecting slip planes. For two edge dislocations on orthogonal slip planes, there are two equilibrium configurations in which the glide and climb components of the dislocation force simultaneously vanish. The interactions between two different types of screw dislocations and a nearby circular void, as well as between parallel line forces in an infinite or semi-infinite medium, are then evaluated.     

Plastic deformation of freestanding thin films: Experiments and modeling

Experimental measurements and computational results for the evolution of plastic deformation in freestanding thin films are compared. In the experiments, the stress-strain response of two sets of Cu films is determined in the plane-strain bulge test. One set of samples consists of electroplated Cu films, while the other set is sputter-deposited. Unpassivated films, films passivated on one side and films passivated on both sides are considered. The calculations are carried out within a two-dimensional plane strain framework with the dislocations modeled as line singularities in an isotropic elastic solid. The film is modeled by a unit cell consisting of eight grains, each of which has three slip systems. The film is initially free of dislocations which then nucleate from a specified distribution of Frank-Read sources. The grain boundaries and any film-passivation layer interfaces are taken to be impenetrable to dislocations. Both the experiments and the computations show: (i) a flow strength for the passivated films that is greater than for the unpassivated films and (ii) hysteresis and a Bauschinger effect that increases with increasing pre-strain for passivated films, while for unpassivated films hysteresis and a Bauschinger effect are small or absent. Furthermore, the experimental measurements and computational results for the 0.2% offset yield strength stress, and the evolution of hysteresis and of the Bauschinger effect are in good quantitative agreement.     

A micromechanical model of hardening, rate sensitivity and thermal softening in BCC single crystals

The present paper is concerned with the development of a micromechanical model of the hardening, rate-sensitivity and thermal softening of bcc crystals. In formulating the model, we specifically consider the following unit processes: double-kink formation and thermally activated motion of kinks; the close-range interactions between primary and forest dislocations, leading to the formation of jogs; the percolation motion of dislocations through a random array of forest dislocations introducing short-range obstacles of different strengths; dislocation multiplication due to breeding by double cross-slip; and dislocation pair annihilation. The model is found to capture salient features of the behavior of Ta crystals such as: the dependence of the initial yield point on temperature and strain rate; the presence of a marked stage I of easy glide, specially at low temperatures and high strain rates; the sharp onset of stage II hardening and its tendency to shift towards lower strains, and eventually disappear, as the temperature increases or the strain rate decreases; the parabolic stage II hardening at low strain rates or high temperatures; the stage II softening at high strain rates or low temperatures; the trend towards saturation at high strains; the temperature and strain-rate dependence of the saturation stress; and the orientation dependence of the hardening rate.     

Coupled DDD–FEM modeling on the mechanical behavior of microlayered metallic multilayer film at elevated temperature

To investigate the mechanical behavior of the microlayered metallic thin films (MMMFs) at elevated temperature, an enhanced discrete-continuous model (DCM), which couples rather than superposes the two-dimensional climb/glide-enabled discrete dislocation dynamics (2D-DDD) with the linearly elastic finite element method (FEM), is developed in this study. In the present coupling scheme, two especial treatments are made. One is to solve how the plastic strain captured by the DDD module is transferred properly to the FEM module as an eigen-strain; the other is to answer how the stress field computationally obtained by the FEM module is transferred accurately to the DDD module to drive those discrete dislocations moving correctly. With these two especial treatments, the interactions between adjacent dislocations and between dislocation pile-ups and inter-phase boundaries (IBs), which are crucial to the strengthening effect in MMMFs, are carefully taken into account. After verified by comparing the computationally predicted results with the theoretical solutions for a dislocation residing in a homogeneous material and nearby a bi-material interface, this 2D-DDD/FEM coupling scheme is used to model the tensile mechanical behaviors of MMMFs at elevated temperature. The strengthening mechanism of MMMFs and the layer thickness effect are studied in detail, with special attentions to the influence of dislocation climb on them.     

Non-steady state non-conformal contacts: friction and film thickness studies

Many machine elements work under non-steady state conditions. Velocity, load and geometry can be time-dependent, and this can influence in a significant way two important quantities of a lubricated contact: friction and film thickness. This work is devoted to non-conformal lubricated contacts under transient conditions and consists of two parts. In the first part a review is made of studies on transient conditions related to the systems themselves (as occurs in gears, valve train, piston assembly and rolling bearings), to particular operating conditions (as starting and stopping of the motion) and to variations localized in the contact (as the ones related to surface defects and roughness). The second part is specifically addressed to investigations on friction and film thickness under conditions of periodically variable velocity. Loops of film thickness and friction coefficient appear when these quantities are plotted as a function of the velocity, mainly due to squeeze effects. The friction is higher and the film thickness smaller for increasing speed than for decreasing speed. A time shift between the film thickness and the speed variations occurs. The loops amplitude increases by increasing the frequency of the speed variation.     

Discrete dislocation plasticity analysis of the grain size dependence of the flow strength of polycrystals

The grain size dependence of the flow strength of polycrystals is analyzed using plane strain, discrete dislocation plasticity. Dislocations are modeled as line singularities in a linear elastic solid and plasticity occurs through the collective motion of large numbers of dislocations. Constitutive rules are used to model lattice resistance to dislocation motion, as well as dislocation nucleation, dislocation annihilation and the interaction with obstacles. The materials analyzed consist of micron scale grains having either one or three slip systems and two types of grain arrangements: either a checker-board pattern or randomly dispersed with a specified volume fraction. Calculations are carried out for materials with either a high density of dislocation sources or a low density of dislocation sources. In all cases, the grain boundaries are taken to be impenetrable to dislocations. A Hall–Petch type relation is predicted with Hall–Petch exponents ranging from ≈0.3 to ≈1.6 depending on the number of slip systems, the grain arrangement, the dislocation source density and the range of grain sizes to which a Hall–Petch expression is fit. The grain size dependence of the flow strength is obtained even when no slip incompatibility exists between grains suggesting that slip blocking/transmission governs the Hall–Petch effect in the simulations.     

Plastic size effect analysis of lamellar composites using a discrete dislocation plasticity approach

Plastic size effect analysis of lamellar composites consisting of elastic and elastic-plastic layers is performed using a discrete dislocation plasticity approach, which is based on applying periodic homogenization to the superposition method for discrete dislocation plasticity. In this approach, the decomposition of displacements into macro and perturbed components circumvents the calculation of superposing displacement fields induced by dislocations in an infinitely homogeneous medium, resulting in two periodic boundary value problems specialized for the analysis of representative volume elements. The present approach is verified by analyzing a model lamellar composite that includes edge dislocations fixed at interfaces. The plastic size effects due to dislocation pile-ups at interfaces are also analyzed. The analysis shows that, strain hardening in elastic-plastic layers arises depending on two factors, namely the thickness and stiffness of elastic layers; and the gap between slip planes in adjacent elastic-plastic layers. In the case where the thickness of elastic layers is several dozen nm, strain hardening in elastic-plastic layers is restrained as the gap of the slip planes decreases. This particular effect is attributed to the long range stress due to pile-ups in adjacent elastic-plastic layers.     

Analysis of multiple axisymmetric annular cracks in a piezoelectric medium

An axisymmetric annular electric dislocation is defined. The solution of axisymmetric electric and Volterra climb and glide dislocations in an infinite transversely isotropic piezoelectric domain is obtained by means of Hankel transforms. The distributed dislocation technique is used to construct integral equations for a system of co-axial annular cracks with so-called permeable and impermeable electric boundary conditions on the crack faces where the domain is under axisymmetric electromechanical loading. These equations are solved numerically to obtain dislocation densities on the crack surfaces. The dislocation densities are employed to determine field intensity factors for a system of interacting annular and/or penny-shaped cracks.     

A multi-scale algorithm for dislocation creep at elevated temperatures

Dislocation creep at elevated temperatures plays an important role for plastic deformation in crystalline metals. When using traditional discrete dislocation dynamics(DDD) to capture this process, we often need to update the forces on N dislocations involving ～N~2 interactions. In this letter, we introduce a multi-scale algorithm to speed up the calculations by dividing a sample of interest into sub-domain grids:dislocations within a characteristic area interact following the conventional way, but their interaction with dislocations in other grids are simplified by lumping all dislocations in another grid as a super one. Such a multi-scale algorithm lowers the computational load to ～N 1.5. We employed this algorithm to model dislocation creep in Al-Mg alloy. The simulation leads to a power-law creep rate in consistent with experimental observations. The stress exponent of the power-law creep is a resultant of dislocations climb for ～5 and viscous dislocations glide for ～3.     

Elastic interaction between edge dislocation,concentrated force,point heat source and edge crack in a semi-infinite plane

We investigate the interaction between edge crack and edge dislocation as well as concentrated force and point heat source. The stress intensity factors at the edge crack tip and the image forces acting on the edge dislocation are calculated. The influence of the concentrated force, point heat source and edge dislocation on the shielding and anti-shielding effects to edge crack as well as the glide and climb forces acting on the edge dislocation is examined in detail. The results indicate that the shielding and anti-shielding effects increase acutely with the increment of concentrated force and point heat source. In addition, the glide and climb forces increase acutely with the decrement of the distance between dislocation and crack tip.     

Mixed mode axisymmetric annular cracks in a finite layer: Off angle crack initiation

The solutions of axisymmetric Volterra type climb and glide edge dislocations are obtained in a layer by means of the Hankel transforms. Utilizing the same procedure, Green’s function solution is obtained for a layer under self-equilibration normal ring traction. The distributed dislocation technique is used to construct integral equations for a system of co-axial annular cracks where the layer is under axisymmetric normal loads. These equations are solved numerically to obtain dislocation density on the cracks surfaces. The results are employed to determine stress intensity factors for annular and penny-shaped cracks and the interaction between two co-axial penny-shaped cracks is studied. Moreover, the stress intensity factors of the interacting cracks are determined such that they can be further used in conjunction with strain energy density (SED) failure criterion to obtain the possible direction of crack initiation that may not be apparent under mixed mode conditions.