# Finite versus small strain discrete dislocation analysis of cantilever bending of single crystals

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## Abstract

Plastic size effects in single crystals are investigated by using finite strain and small strain discrete dislocation plasticity to analyse the response of cantilever beam specimens. Crystals with both one and two active slip systems are analysed, as well as specimens with different beam aspect ratios. Over the range of specimen sizes analysed here, the bending stress versus applied tip displacement response has a strong hardening plastic component. This hardening rate increases with decreasing specimen size. The hardening rates are slightly lower when the finite strain discrete dislocation plasticity (DDP) formulation is employed as curving of the slip planes is accounted for in the finite strain formulation. This relaxes the back-stresses in the dislocation pile-ups and thereby reduces the hardening rate. Our calculations show that in line with the pure bending case, the bending stress in cantilever bending displays a plastic size dependence. However, unlike pure bending, the bending flow strength of the larger aspect ratio cantilever beams is appreciably smaller. This is attributed to the fact that for the same applied bending stress, longer beams have lower shear forces acting upon them and this results in a lower density of statistically stored dislocations.

### Keywords

Dislocations Bending Finite strain Size effects## 1 Introduction

Over the past years, computational solid mechanics has become an integral part of theoretical materials science. Significant attention has been focused on mesoscale continuum mechanics where material properties and structural dimensions are important. Such formulations are intermediate between a direct atomistic simulation and an unstructured continuum description of the deformation processes. A variety of theoretical frameworks are emerging to describe inelastic deformation at the mesoscale: in this study we shall focus on one of these methods viz. discrete dislocation plasticity (DDP). In discrete dislocation plasticity, the dislocations are treated as line singularities in an elastic solid. A many-body interaction problem involving the discrete dislocations needs to be solved together with a complimentary more conventional elasticity boundary value problem.

- (1)
*Geometrically necessary dislocations*(GNDs). This has been the focus of phenomenological higher order plasticity theories [1, 2] and is used to explain size effects in bending, indentation, etc. - (2)
*Constrained dislocation glide by internal interfaces such as grain boundaries.*This is typically the main reason for size effects in phenomena such as the Hall-Petch effect [3, 4]. - (3)
*Source limited plasticity.*Continuum plasticity assumes that at any point where a flow condition is met, plastic deformation will take place. Nevertheless, this does not need to be the case at very small sizes such as frictional asperity contact [5]. - (4)
*Dislocation starvation.*The size effects in micro-pillar compression have been primarily associated with this effect [6, 7, 8] where dislocations exit the specimen at a rate faster than they are nucleated, and thus there is no build-up of dislocations within the specimen.

Bending of specimens has been extensively used to investigate size effects in crystalline plasticity since the early work of Stölken and Evans [27]. These experiments attempted to impose a pure bending stress field on the specimens, but did not directly measure the moment-curvature relation. More recently, small scale bending experiments were conducted by first using focussed ion beam (FIB) to cut cantilever beams from single crystals and then applying tip loads using nano-indentation instruments [28, 29]. These experiments enable the direct measurement of load versus displacement relations and are used to infer plasticity size effects. All bending experiments involve significant lattice rotations and changes in the specimen geometry. Discrete dislocation plasticity investigations of such experiments [11, 30, 31] have neglected these finite strain effects and usually attributed size effects in bending to GNDs.

The primary question addressed in this study is whether lattice rotations and the associated curving of the slip planes affect the influence of GNDs in governing plasticity size effects in bending. To address this question we shall employ the recently developed finite strain DDP formulation of Irani et al. [32]. The outline of the paper is as follows: First, the small strain and finite strain DDP formulations are briefly described. Next, the finite strain and small strain DDP predictions of the cantilever bending problem are presented for geometrically self-similar beams. These predictions are then used to rationalise size effects in cantilever bending experiments.

## 2 Formulation of the problem

The bending/tensile response of single crystals is investigated here using both small and finite strain discrete dislocation plasticity in a two-dimensional (2D) setting. The crystals are taken to be elastically isotropic with Young’s modulus *E* and Poisson’s ratio \(\nu \). Plane strain conditions are assumed with deformations restricted to the \(X_1-X_2\) plane. Here, we briefly summarise both the small and finite strain formulations. For more details see the references cited.

### 2.1 Small strain discrete dislocation plasticity

*t*, the body contains

*N*edge dislocations, with the dislocations being represented as line singularities in an elastic medium, with Burgers vector

*b*. In this method, in order to obtain the solution, the problem is decomposed in two additive parts and the field quantities are computed using superposition. First, the singular \(({\tilde{\;}})\) field associated with the

*N*dislocations are calculated analytically. Typically, solutions of dislocations fields in an infinite medium are used to represent the \(({\tilde{\;}})\) field, but it is equally possible to use other solutions such as those for dislocations in a half-space. Next, the complete solution is obtained by adding an image \(({\hat{\;}})\) field that ensures the boundary conditions are satisfied. Thus, the displacements, strains, and stresses are expressed as

*I*exceeds a critical value \(\tau _\mathrm{nuc}b\) during a time period \(t_\mathrm{nuc}\). The sign of the dipole is determined by the sign of the resolved shear stress along the slip plane. Furthermore, the distance between the two nucleated dislocations, \(L_\mathrm{nuc}\), is taken such that the attractive stress field, which the dislocations exert on each other is equilibrated by a shear stress of magnitude \(\tau _\mathrm{nuc}\). Annihilation of two opposite signed dislocations on a slip plane occurs when they are within a material-dependent critical annihilation distance, \(L_e\). The magnitude of the glide velocity \(v^{(I)}\) along the slip direction of dislocation

*I*is taken to be linearly related to the Peach-Koehler force \(f^{(I)}\) through the drag relation

*B*is the drag coefficient. Obstacles to dislocation motion are modelled as points associated with a slip plane. The dislocations gliding on the slip plane of an obstacle, get pinned as they try to pass through this obstacle. Moreover, obstacles release the pinned dislocations when the Peach-Koehler force on the obstacle exceeds \(\tau _\mathrm{obs}b\).

### 2.2 Finite strain discrete dislocation plasticity

#### 2.2.1 Finite strain discrete dislocation plasticity constitutive rules

*t*there are

*N*dislocations in the body and the Peach-Koehler force \(f^{(I)}\) on dislocation

*I*is given by

*J*at the position of dislocation

*I*while \(m^{*(\alpha )}\) is the unit vector normal to slip plane \(\alpha \) and \(b^{*(I)}_j\) the Burgers vector of dislocation

*I*in the current configuration. The Peach-Koehler force includes the long-range interactions with all other dislocations in the material. This force determines the evolution of the dislocation structure, accounting for glide, generation, annihilation, pinning at and releasing from the obstacles. Similar to the small strain case, the magnitude of the glide velocity along the current slip direction of dislocation

*I*is taken to be linearly related to the Peach-Koehler force such that the velocity \(v_i^{(I)}\) of dislocation

*I*is given as

*B*is the drag coefficient and \(s_i^{*(\alpha )}\) is the unit vector along the slip direction of slip system \(\alpha \) in the current configuration. Here, it is assumed that the drag coefficient

*B*is constant throughout the body. We also do not account for any changes in the resistance to dislocation motion near a free surface associated with the energy required to create a new free surface when the dislocation exits.

New dislocation pairs are generated by simulating Frank-Read sources. In two dimensions, this is mimicked by discrete point sources on a slip system which generate a dislocation dipole with their Burgers vectors aligned with \(s_i^{*(\alpha )}\). This occurs when the magnitude of the Peach-Koehler force at that source exceeds a critical value \(\tau _\text {nuc}b\) during a time period \(t_\text {nuc}\). The sign of the dipole is determined by the sign of the resolved shear stress along the current slip direction. The distance between the two dislocations at nucleation, \(L_\text {nuc}\), is taken such that the attractive stress field that the dislocations exert on each other is equilibrated by a shear stress of magnitude \(\tau _\text {nuc}\). Annihilation of two opposite signed dislocations on slip system \(\alpha \) occurs when they are sufficiently close together. This is modelled by eliminating the two dislocations when they are within a material-dependent critical annihilation distance \(L_{e}\). Unlike in the small strain formulation where only opposite signed dislocations on a given slip plane can annihilate each other, in the finite strain context opposite signed dislocations on a given *slip system* can annihilate each other. Thus, annihilation of two opposite signed dislocations on a particular slip system occurs when they are within a distance equal to \(L_{e}\), irrespective of their current slip planes. Obstacles to dislocation motion are modelled as points associated with a slip system. Dislocations on the slip system of an obstacle, get pinned as they try to pass through that point. Again, unlike the small strain case, dislocations and obstacles are associated with a slip system rather than a slip plane. Thus, dislocations that glide on the slip system of an obstacle and are within a distance equal to \(L_{e}\) from it, get pinned to that obstacle. Pinned dislocations can only pass through an obstacle when their Peach-Koehler force exceeds an obstacle dependent value \(\tau _\text {obs}b\).

### 2.3 Specification of the cantilever beam bending boundary value problem

*a*, width

*W*, and a root of width \(W_R\). We fix the origin of the coordinate system at the cantilever root such that \(X_1\)-axis coincides with the axis of symmetry of the specimen and the loading is applied at the cantilever tip at \((X_1,X_2)=(L_R+a,W/2)\). The boundary conditions imposed to simulate cantilever bending are

*P*to the applied displacement imposes a bending moment \(M\equiv Pa\) at the root of the gauge section of the cantilever beam and a loading rate \(|{\dot{\delta }}|/a = 500 /\mathrm{s}\) is employed.

- (1)
*Short beams.*These beams have dimensions with ratios \(a/W=3\), \(W_R/W=3\), and \(L_R/W=5.6\). The taper angle as defined in Fig. 2 is then \(\alpha =20^\mathrm{\circ }\). The three geometrically self-similar specimens that are analysed are specified by the widths \(W=0.5\), 1.0, \(1.5\;\upmu \hbox {m}\). - (2)
*Long beams.*The only difference in this case is that the gauge section aspect ratio is increased to \(a/W=6\) with the beam root dimension ratios kept fixed at the same values, i.e. \(W_R/W=3\) and \(L_R/W=5.6\), such that again \(\alpha =20^\mathrm{\circ }\). For this case we also analyse three geometrically self-similar specimens with \(W=0.5\), 1.0, \(1.5\;\upmu \hbox {m}\).

### 2.4 Reference properties

The following set of material properties is used in both the small strain and finite strain DDP simulations. The crystal is assumed to be elastically isotropic with Young’s modulus \(E = 70\) GPa and Poisson’s ratio \(\nu = 0.33\), which are representative values for aluminium. In the undeformed configuration, the slip planes are spaced 100*b* apart, where \(b = 0.25\) nm is the magnitude of the Burgers vector. Simulations are reported for two crystallographic configurations of the crystal: (1) in the crystal oriented for single slip there is a single slip system oriented at \(\phi _1=30^{\circ }\) with respect to the \(X_1\)-axis in the undeformed configuration and (2) in the crystal orientated for symmetric double slip, there are two slip systems at \(\phi _1=30^{\circ }\) and \(\phi _2=-30^{\circ }\) with respect to the \(X_1\)-axis in the undeformed configuration.

## 3 Comparison of finite strain and small strain DDP predictions

Pure bending of crystals using discrete dislocation plasticity has been investigated by a number of authors initiated by the pioneering work of Cleveringa et al. [11]. However, most small-scale experiments are unable to generate pure bending loading with experiments typically carried out in a cantilever beam setting [28]. In cantilever beam bending, stress gradients exist along both the beam axis and width. Moreover, in this setting the imposed loads also generate shear stresses. In their study, Tarleton et al. [31] performed small strain DDP calculations of cantilever beam bending and argued that the GND structure formed in cantilever beam bending differed from that observed in pure bending. Finite strain effects such as lattice rotations can have a significant influence on the GND structures in bending and in this section, we aim to quantify these effects.

### 3.1 Uniaxial tensile/compressive response of the crystal

- (1)
The small strain DDP results are nearly identical to the finite strain predictions, except for the single slip \(W=0.5\;\upmu \hbox {m}\) specimen where the small strain results show a larger hardening regime.

- (2)
There is negligible tension/compression asymmetry due to any finite strain effect for both the single and double slip cases.

- (3)
The crystals oriented for single slip display an initial elastic response followed by a mildly hardening plastic behaviour. By contrast, in the double slip configuration the crystals display nearly no plastic hardening.

- (4)
The crystals oriented for single slip have a small plastic size effect with the strength of the \(W=1.0\;\upmu \hbox {m}\) specimens slightly lower compared to the \(W=0.5\;\upmu \hbox {m}\) specimens. No discernible size effect is observed for the crystals oriented in double slip.

### 3.2 Cantilever beam bending

*bending stress*measure

*P*. In the following, we present the results by describing the evolution of the bending stress \(\sigma _b\) as a function of a measure of beam rotation \(\theta \equiv |\delta |/a\). In this section, we will focus on understanding the impact of finite deformations on the bending response of the material, and this aim will be achieved by comparing the predictions of finite strain and small strain DDP formulations for \(\sigma _b\).

Predictions of \(\sigma _b\) versus \(\theta \) are presented in Fig. 7a, b for the single and double slip configurations, respectively, of the short beams. These figures show that after an initially elastic response, there is a knee in the \(\sigma _b\) versus \(\theta \) curve corresponding to the onset of dislocation activity and the initiation of plastic deformation. However, unlike the uniaxial loading case, the bending stress displays a hardening plastic response. Moreover, the bending stress in the plastic regime is size dependent with the smaller specimens exhibiting a larger bending stress. This is consistent with a wide body of both experimental [27, 28] and computational [11, 30, 31] investigations that suggest that the bending stress is size dependent in the micron size regime. We shall explore the origins of this size effect in more details in Sect. 4. Very briefly, the size effect is primarily a manifestation of the so-called GNDs. As will be seen subsequently, in these simulations the GNDs are primarily dislocation pile-ups that cause back-stresses and thereby plastic hardening. These dislocation pile-ups and their associated back-stresses lead to the size effects seen here. Moreover, the plastic hardening is more pronounced in the crystals in the single slip configuration (Fig. 7a) compared to the double slip case (Fig. 7b). In order to rationalise this observation we plot the dislocation structures in the \(W=0.5\) and \(1.5\;\upmu \hbox {m}\) specimens at \(\theta =0.045\) in Fig. 8a, b, respectively, for the crystals oriented in the single slip configuration. The corresponding plots for crystals oriented in the double slip configuration are included in Fig. 8c, d, respectively. In Fig. 8 contours of the lattice rotation \(\varphi \) are also included. Longer dislocation pile-ups are observed in the specimens with only one active slip system and we attribute the stronger plastic hardening to these longer pile-ups. This higher hardening rate is also consistent with the fact that plasticity is more constrained when only one active slip system is present compared to the double slip case and also that under uniaxial loading the single slip configuration exhibited a higher plastic hardening rate compared to the double slip case (Fig. 4).

## 4 Size effects in cantilever beam bending

Over the range of the cantilever tip displacements investigated here, finite strain effects were shown not to be significant. We thus proceed to investigate the underlying plasticity mechanisms for cantilever beam bending using the small strain DDP formulation. We first restrict our attention to the *short beam* specimens and then continue with exploring the effect of the beam aspect ratio *a* / *W*.

Predictions of \(\sigma _b\) versus \(\theta \) are included in Fig. 7a, b, respectively, for specimen sizes \(W=0.5\), 1.0, \(1.5\;\upmu \hbox {m}\). As was mentioned in Sect. 3, these results indicate that the bending stress is strongly hardening in the plastic regime, and the single slip configuration is stronger compared to the double slip case. Moreover, a plastic size effect exists with the bending stress in the plastic regime being larger for the smaller specimens.

*W*for the geometrically self-similar short beams is plotted in Fig. 10. The error bars in Fig. 10 indicate the scatter in the results based on the different realisations of the source and obstacle distributions computed here. The variation of the bending flow stress with

*W*is reasonably well described by a relation of the form

*W*. This curve is included in Fig. 10 and with \(W_0=1.0\;\upmu \hbox {m}\) and \(n\approx 0.3\), it is seen to fit the data with reasonable accuracy for both the slip configurations of the short beam specimens.

The distributions of dislocations and contours of the stress component \(\sigma _{11}\) are plotted in Fig. 11 at an applied rotation \(\theta =0.045\). The results shown in Fig. 11a, b are for the \(W=0.5\) and \(1.5\;\upmu \hbox {m}\) short beam specimens, respectively, with the crystals oriented for single slip. These figures demonstrate that in the cantilever beams the dislocation activity is concentrated around the root of the gauge section of the specimen. The vast majority of these dislocations have the same sign as their oppositely signed counterparts have exited the specimen through the free surfaces. This implies that in these specimens there is a significant net Burgers vector due to the dislocations and hence a large GND density in Nye’s terminology [36]. These GNDs are dislocation pile-up structures and result in back-stresses on the dislocation sources counteracting their further operation. The size effect is largely associated with these GNDs as argued extensively in Ref. [1, 27].

### 4.1 Effect of cantilever aspect ratio

*a*/

*W*increases, the shear collapse load increases with respect to the bending collapse load and hence, the deformation of the beam will become increasingly bending dominated. This basic notion holds irrespective of the details of the assumed plastic constitutive properties of the beam material. Thus, in order to investigate the contribution of the shear force on the plastic properties reported above, we proceed to investigate the bending response of the

*long beams*with a gauge section aspect ratio \(a/W=6\) (as detailed in Sect. 2.3).

Predictions of \(\sigma _{f}\) for the long beam specimens are included in Fig. 10 along with the short beam results. It can be observed that the overall trends are very similar with \(\sigma _{f}\) decreasing with increasing *W* and the double slip case being slightly weaker compared to the single slip configuration. However, as it is shown in Fig. 10 the bending flow strength of the long beam specimens is appreciably smaller compared to the equivalent short beam samples. This is rationalised by noting that the long beams also have a lower dislocation density compared to the short beam case. We attribute this to the fact that for the same applied bending stress, the longer beams have lower shear forces (and stresses) acting upon them and this results in a lower density of statistically stored dislocations (SSDs). To clarify this, we plot in Fig. 11c, d the dislocation structure and the stress distributions in the \(W=0.5\) and \(1.5\;\upmu \hbox {m}\) long beam specimens, respectively, at \(\theta =0.045\). The dislocation structures in the long beam specimens are almost exclusively comprised of same-signed dislocations while there are relatively more opposite signed dislocations in the short beam specimens; see Fig. 11a, b. This confirms that the long beam specimens have a lower density of SSDs compared to the short beam case. Consequently, the lower dislocation density accounts for the lower bending flow strength in the long beam specimens, as dislocation motion is less inhibited by a dense dislocation structure.

The results for the long beam specimens demonstrate that the reduction in the SSD density does not change the trend of the variation of \(\sigma _{f}\) with *W*. Moreover, the bending flow stress is reasonably well described by Eq. (24) with \(W_0=1.0\;\upmu \hbox {m}\) and \(n\approx 0.3\). Thus, these results suggest that although the value of \(\sigma _0\) in Eq. (24) depends on the specimen aspect ratio, the value of \(\sigma _{f}/\sigma _0\) only depends on specimen width *W*.

## 5 Discussion

Initiated by the pioneering work of Stölken and Evans [27], numerous studies have confirmed that the bending strength of crystalline materials increases with decreasing specimen size; see for example Ref. [28]. This so-called size effect has traditionally been attributed to the fact that the GND density, which is associated with the imposed curvature in bending, increases with decreasing specimen size. The vast majority of models employed to capture this size effect are the so-called strain gradient plasticity models (see for example Ref. [37]), which explicitly include the contribution of GNDs to the strength and thereby capture the observed size effect. These continuum models thus have an in-built length scale as an additional parameter, which is calibrated either via experiments or lower length scale models. Another class of closely related models uses the concept of the Nye [36] dislocation density tensor to estimate the GND density, which enhances the physical basis of these higher order plasticity theories [38].

The drive to improve the mechanistic basis of the higher order plasticity theories has led to the development of what is now known as continuum dislocation plasticity (CDP) with the aim of describing the behaviour of the ensembles of dislocations via continuum field equations. For example, Sandfeld et al. [39] presented a numerical implementation of such a theory and applied it to the plane-strain micro-bending problem. Later, Le and Nguyen [40] (also see Refs. [41, 42]) proposed analytical descriptions for bending of single crystal beams within the CDP setting. In their work, in addition to the usual GND size effects, they also included the effect of specimen size dependent dislocation nucleation.

In the current study we employ discrete dislocation plasticity wherein the dislocation structures and the associated stresses are generated as a natural outcome of the boundary value problem. The main difference between the discrete dislocation plasticity and continuum models discussed above, is that in the discrete setting, GNDs and SSDs cannot be differentiated without including an artificial averaging length scale. Moreover, in the cantilever bending problem analysed here the lattice curvature varies over the length of the beam. Thus, it is not possible to quantify the GND density in a sensible manner from these discrete dislocation plasticity calculations. Therefore, making direct comparisons between the discrete dislocation plasticity and continuum plasticity descriptions is not instructive. Nevertheless, there is an excellent agreement between the discrete dislocation plasticity predictions and measurements of the size effects besides a good qualitative agreement with the above mentioned continuum plasticity predictions.

## 6 Concluding remarks

Cantilever beam configurations are commonly used to analyse the bending response of micro-sized specimens. Here, we have analysed the bending of tip loaded single crystal cantilever beam specimens with one and two active slip systems using DDP. Both finite strain and small strain DDP formulations are employed to investigate the influence of lattice rotations and changes in specimen geometry on the bending response of the specimens.

Over the range of specimen sizes analysed here, the bending stress versus applied tip displacement response has a strong plastic hardening component. This hardening rate increases with decreasing specimen size, and thus the bending flow strength is size dependent. The hardening rates are slightly lower when the finite strain DDP formulation is employed as curving of the slip planes is accounted for in the finite strain formulation. This curving relaxes the back-stresses in the dislocation pile-ups that result due to the bending stress field and thus reduces the hardening rate. However, over the range of applied tip displacements analysed here the small strain DDP formulation is shown to capture with excellent accuracy all the key features of the bending response of end loaded cantilever beams.

The small strain DDP formulation is then used to analyse size effects in these cantilever beam specimens. In line with the well-known pure bending case, the bending stress is shown to display a plastic size dependence. Moreover, the bending flow strength of beams with a larger aspect ratio is shown to be appreciably smaller compared to the equivalent short beam samples. We attribute this to the fact that for the same applied bending stress, the longer beams have lower shear forces (and stresses) acting upon them, and this results in a lower density of statistically stored dislocations: this reduction in the dislocation density lowers the bending strength.

## Notes

### Acknowledgements

The support from Eindhoven University of Technology is gratefully acknowledged.

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