Interpretation of the size effects in micropillar compression by a strain gradient crystal plasticity theory

International Journal of Plasticity 116 (2019) 280–296

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Interpretation of the size effects in micropillar compression by a strain gradient crystal plasticity theory

T

Eduardo Bittencourt Center of Applied and Computational Mechanics, Department of Civil Engineering, Universidade Federal do Rio Grande do Sul, Av. Osvaldo Aranha, 99, 90035-190, Porto Alegre, RS, Brazil

ARTICLE INFO

ABSTRACT

Keywords: Microstructures Dislocations Crystal plasticity Finite elements Micropillar

This work presents a study of the behavior of micropillars compressed with a flat punch, considering a continuum model based on a higher-order strain gradient crystal plasticity theory. Hardening models are associated with the density of geometrically necessary dislocations. Size effects observed in the present calculations are governed by the development of slip-bands in the micropillar and the geometrically necessary dislocations that build up around them. Two critical factors are identified as defining the slip-band behavior. One is the aspect ratio considered for the micropillar. Shorter micropillars introduce geometrical restrictions to the free path for dislocations in the slip-band, which may result in the development of a substantial amount of geometrically necessary dislocations and size dependent hardening (strain hardening and strengthening). In longer micropillars the effect is not observed and then the proposed model is incapable of capturing size effects in these cases. The other factor is the hardening model used. Two possible models are considered: a dissipative model directly related to slip rate gradients and an energetic model derived from a defect energy. Qualitative comparisons with experiments indicate that dissipative hardening is a more adequate representation of micropillar compression than the energetic model used. Finally, it is shown that a large friction between punch and crystal may inhibit strengthening converting it into strain hardening. Low friction coefficients also tend to promote the concentration of plastic strains in one slip system.

1. Introduction Compression experiments of pillar-shaped single crystals in the micron and sub-micron range show size-dependent response; see e.g. Dimiduk et al. (2005) and Greer et al. (2005). Two types of size effects were observed by the authors: First, in terms of stressstrain response, with smaller being harder. This size effect is observed in the yield strength ( Y ) and in the hardening rate. Despite a considerable variability, experiments show that Y w n , where w is the pillar width or diameter and n is a positive value in general smaller than 1, see e.g. Dimiduk et al. (2005), Liu et al. (2016) and Papanikolaou et al. (2017). The second manifestation of the size effects is in terms of deformation mode. Deformation shifts from homogeneous to shear bands with size decrease. Experiments also show that the lateral stiffness of the experimental set up may have an influence in this type of size effect (Shade et al., 2009). If rotation of the crystal axis is allowed, a single shear zone is observed in small specimens. Micro and nanopillar compression is also characterized by the observation of strain bursts, Dimiduk et al. (2006). Different models have been proposed to understand the phenomenon such as in Zhang et al. (2013), Zhang et al. (2016), Papanikolaou et al. (2017), etc. The origin of size effects in micro and nanopillars is frequently attributed to dislocation starvation, exhaustion or changes in

E-mail address: [email protected]. https://doi.org/10.1016/j.ijplas.2019.01.011 Received 18 September 2018; Received in revised form 24 January 2019; Accepted 25 January 2019 Available online 30 January 2019 0749-6419/ © 2019 Elsevier Ltd. All rights reserved.

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dislocation nucleation mechanisms. Lower-scale models such as molecular dynamics or discrete dislocation dynamics (DDD) are in general necessary to capture these changes. A variety of such mechanisms have been proposed, see e.g. Dimiduk et al. (2005), Greer et al. (2005), Parthasarathy et al. (2007), Rao et al. (2007), Shao et al. (2012), Cui et al. (2014), Pan et al. (2015), Yaghoobi and Voyiadjis (2016), Papanikolaou et al. (2017), etc. Ledges and stress concentrations at the pillar surface are also considered as hardening factors in Akarapu et al. (2010). Attempts to incorporate phenomenologically the effects of dislocation nucleation mechanism into continuum model have also been done lately, see Liu et al. (2016), Lu et al. (2019), among others. A different approach is followed by Guruprasad and Benzerga (2008), Kiener et al. (2011) and Kondori et al. (2017) that attribute size-dependency to the development of geometrically necessary dislocation (GND) densities. In the simulations done by these authors, based on DDD models, the development of GNDs does not depend on the application of a macroscopic strain gradient and it is the result of evolving dislocation substructures inside the crystal. The authors consider that the approach is valid for at least w equal to 0.4 μm. Continuum models based on strain gradients or GND densities have also been applied to the micropillar compression problem such as in Kuroda (2011, 2013, 2017), Hurtado and Ortiz (2012), Husser et al. (2014), Lin et al. (2016) and Wang et al. (2016). Transition from homogeneous deformation to slip-band has been successfully captured (e.g. Husser et al., 2014; Lin et al., 2016), but a trend for the models to underestimate size effects is observed (Hurtado and Ortiz, 2012; Kuroda, 2013). Some distinct aspects remain also still relatively unexplored or unclear regarding the application of such models. Maybe the most important of the unexplored features is the influence of the micropillar geometrical relations on size effects. In Zhang et al. (2017) is shown that the ratio of the dimensions of the cross-section of the pillar may have an influence on these effects. It has been also shown, see for instance Volkert and Lilleodden (2006), Kiener et al. (2008), Senger et al. (2011), that shorter micropillars are related to greater size effects. However a comprehensive study regarding this issue from the perspective of higher-order crystal plasticity formulations was not done yet. Another aspect that should be more deeply investigated is the difficulty of GND based continuum models to capture strengthening,1 being the work by Hurtado and Ortiz (2012) maybe the only exception. In Demiral et al. (2017) strengthening is also captured, but a polar dislocation based model (Mayeur and McDowell, 2011, 2014) is used. Therefore it seems important to expand the study of the effects of alternative GND based hardening models to micropillar applications. As shown in Gurtin et al. (2007), Ohno and Okomura (2007), Bardella and Giacomini (2008), Garroni et al. (2010), Klusemann and Yalcinkaya (2013), Bardella et al. (2013), Wulfinghoff et al. (2015), El-Naaman et al. (2016), among others, these models may have a substantial impact on the apparent hardening. In the present work, some of the aspects described above are being tackled so a better understanding of limitations and possible applications of GND based continuum theories to the problem can be achieved. In the crystal plasticity theory used in this work, higher-order stresses work conjugate to the density of GNDs, following the model developed by Gurtin (2002, 2008). These stresses can be obtained from a defect energy and the hardening is then considered energetic. Alternatively, higher-order stresses can be also related directly to the rate of the density of GNDs. This hardening is called dissipative. Only these two models are tested and analyzed. This work aims to explore the behavior of these two particular models in problems of micropillar compression and it does not intend to make a thoroughly comparison with other continuum hardening formulations available nor to claim any superiority regarding them. Comparisons are made with different experiments, but the main goal is to observe how the two hardening models qualitatively represent such experiments. For this reason, material data used are not related to a specific material. The combination of different friction conditions between punch and crystal coupled with different higher-order boundary conditions is also being studied. Only plane-strain cases and edge dislocations are taken into account. The remaining of the paper is organized as follows: in section 2, continuum crystal plasticity formulation based on Gurtin's theory is briefly described. In section 3, the applications to micropillar compression considering different geometries and boundary conditions are presented. Discussion and comparisons of the obtained size effects with expected results are in section 4. In the Appendix, a simplified numerical model of the slip-band is formulated. 2. Continuum formulation The present formulation is based on the higher order strain gradient crystal plasticity developed by Gurtin (2002, 2008). A brief description is given below. Details of the formulation used in this work can be found in Bittencourt (2018). The deformation gradient F, is decomposed as,

Fij = Fik* Fkjp

(1)

where represents a transformation consisting only of crystallographic slipping ( ) on slip system β. For this system, and m( ) are unit vectors specifying the slip direction and the slip plane normal, respectively. The slip systems are fixed in this transformation. F * represents elastic stretching, elastic rotation and rigid body rotation. As elastic stretching is considered small in this work, the slip system definition is updated by the transformation

Fp

s( )

si ( ) = Rij*s j( where

R*

)

is the rotation tensor from the polar decomposition of

Cauchy stresses

1

and m ¯ i( ) = Rij*mj( ) ,

(2)

F*

.

can be obtained after integration of the lattice Jaumann rate stress

defined as follows:

The term “strengthening” in this work means the increase in the apparent yield strength due to effects related to higher-order stresses. 281

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ij

= Lijkl Dkl*

(3) *

where L is the Hooke tensor and D is the elastic part of the total rate of stretching D,

Dij = Dij* + Dijp .

(4)

From Cauchy stresses, the resolved stresses on the lattice are calculated as: ( )

= Pij(

)

(5)

ij

and,

Pij( ) =

1 () () (¯si m ¯j + m ¯ i( ) s¯j( ) ). 2

Finally, slip resistance ( ( )

=

|

( )

( ))

m 1

|( )

(6)

is defined by the rate-dependent relation below: ( )

0

,

(7)

0

is the slip rate on system β, 0 is a reference slip rate and m is a viscosity parameter. is defined later, in the In the equation, next subsection. The virtual-power can be written in two parts. Consider B the deformed volume of the body, Bt the associated surface where tractions ti = ij nj are applied (n is the outward normal to the surface) and u is the displacement, then: ( )

B

ij

( )

Dij dV =

Bt

ti ui dA

(8)

δ denotes a virtual field and (·) = ()/ t , where t is time. Considering also the gradients of the slip rates and their power conjugated, the microstress i( ) , then:

(

B

( )

( ))

( )

+

( ) i

( ) ,i

dV =

Bq

q(

)

( ) dA .

(9)

Bq is a surface in the deformed body where microtractions q( ) = are applied. If q( ) = 0 , dislocations can escape from the surface freely, which is known as the micro-free boundary condition. If dislocations cannot escape the surface, a condition in terms of plastic slips must be prescribed on the boundary, see Gurtin and Needleman (2005). This condition is known as the micro-hard boundary condition. In this work i( ) is a function of the density of GNDs ( ( ) ), which evolves according to the following equation: ( ) i ni

( )

=

s¯i(

)

( ) ,i .

(10)

= 0 in all cases. Initial value of densities is Hardening is defined through constitutive equations for ( ) and i( ) . ( ) will be only related to statistically stored dislocations (SSD). It could be associated also to GNDs, but this situation is not regarded in this work. i( ) is associated to GNDs considering two different mechanisms. These constitutive equations are described below. ( ) t=0

2.1. Hardening due to SSDs Hardening may be due to plastic slips, which then represents the effect of SSDs. A simple linear hardening, as described for instance in Peirce et al. (1983), is considered in this work: ( )

=

h |

( )|

(11)

and

h

= QH0 + (1

Q ) H0

(12)

.

The initial value of ( ) is 0 , which is, as Q and H0 , a material constant. The modulus H0 plays an important role in the present simulations because, if null, the hardening effect of SSDs is eliminated from calculations. 2.2. Hardening due to GNDs The microstress i( ) , and by consequence the hardening associated to GNDs, can have an energetic or dissipative nature. In the energetic hardening, i( ) is obtained from an augmented free energy by a defect energy. In this work the simplest possible case defined in Gurtin (2008) is considered, where the defect energy is taken as a quadratic function of the density of dislocations on the slip system under consideration, resulting an uncoupled energy. Then, the corresponding energetic microstress i( ) en has the following form: 282

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Fig. 1. Illustration of the problem formulation for the compression of a single crystal with two active slip systems. ( ) en i

=

J

1 2

0

( ) s¯ ( ) i

(13)

where is a characteristic material size and J = det F. The resulting microstress is dependent on accumulated density of GNDs and then it can be associated to the hardening due to pile up of dislocations on obstacles. In Bittencourt (2018), such hardening was associated to the presence of obstacles in discrete dislocation simulations in indentation problems. In the dissipative hardening, the microstress is defined in analogy to eq. (7) as: ( ) dis i

d( ) d0

( )

=

m

L2

( ) ,i , ( d )

(14)

where,

d( ) =

(

( ) )2

+ L2 (

( ) 2 ,i )

(15)

and L is a characteristic material size. Gurtin originally assumed m = m . Herein it will be considered m = 0 , which corresponds to ignore viscous effects on the dissipative microstress. Therefore, dissipative part of the microstresses is dependent on the instantaneous rate of slip gradient, while the energetic part depends on accumulated slip gradient. Finally, the total microstress can be calculated as: ( ) i

=

( ) en i

+

( ) dis . i

(16)

3. Numerical experimentation The geometry of the model problem is depicted in Fig. 1. The pillar is a single crystal in plane-strain and has height h and width w. Aspect ratios h/ w considered are 1.67, 2.5 and 3.75. Crystal is free to move and dislocations are free to escape on the sides of the pillar (x1 = 0, w ). On the bottom surface of the pillar ( x2 = 0 ) displacements u2 are blocked and displacements u1 are free. Also on the 283

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bottom, dislocations cannot escape the surface, or the micro-hard condition prevails. This condition is imposed considering that both plastic slips (1) and (2) are blocked. On the top surface ( x2 = h ) a rigid flat punch compresses the pillar at a rate U / h = 0.0027 s 1. Different friction coefficients f and higher-order boundary conditions are assumed between punch and crystal. Shear modulus is µ = 26.3 GPa, Poisson's ratio is taken to be = 0.33 and the flow strength is specified by 0 = 40 MPa. These data can be associated to aluminum. Viscosity parameter is m = 0.05, reference slip rate 0 = 0.0027 s 1 and H0 = 0 , unless stated differently, so the hardening due to SSD is not considered in most cases. Two symmetric slip systems are adopted,2 where = ± 60o . Inertial contributions are also considered and the solution of resulting equations is obtained by the explicit conditionally stable central difference method. On the left side of eq. (8) the volumetric integral of the term ˜ u¨i ui on B is added and on the left of eq. (9) on B is added. As the goal is to model quasi-static problems and not dynamic problems, the volumetric integral of the term c ˜ inertial properties ˜ and c are arbitrated in order that the resulting kinematic energy is low when compared to deformation energy associated to the problems under study, while still resulting in a reasonable stable time-step for the solution (see Bittencourt, 2014, for further details). The values adopted in this work are, respectively, 2700 Kgf/mm3 and 0.0064 mm2. As a result of the description above, fundamental aspects of real micropillar experiments are retained. Calculations are carried out using 20 × 50 4-node finite elements with uniform size. Standard bilinear interpolation functions and 2 × 2 integration points are used. Stress-strain relations are presented as the average axial stress ave per unit thickness versus axial strain ε. ave is obtained dividing contact force F by w and ε is obtained dividing punch displacement U by h. Accumulated plastic slip in system β is defined as:

|

( )|

=

t 0

|

( ) |dt .

(17)

Apparent yield strength Y is defined as the average axial stress related to a residual axial strain equal to 0.002. Experimentation is separated in two groups, depending on the hardening model considered, as follows: 3.1. Energetic hardening cases In the cases analyzed in this subsection, only the hardening associated to a defect energy is taken into consideration, eq. (13). Fig. 2 shows stress-strain results for this case. Depending on the boundary conditions, a general trend for smaller to be harder is observed. For cases in Fig. 2a, the micro-free condition is considered between punch and crystal. Hardening is only possible if a high friction coefficient3 is taken ( f >0.3). When friction is not considered, f =0, hardening is not observed as seen also in Fig. 2a. In Fig. 2b–c the micro-hard condition is taken into account between punch and crystal. Differently from cases in Fig. 2a, hardening practically does not depend on friction in these cases, as seen in Fig. 2b. Cases considering micro-hard condition also tend to present a slightly larger hardening than micro-free cases for the same size. Strengthening was not observed in cases where energetic hardening was considered alone, regardless the boundary conditions. Aspect ratio h/ w has a significant effect on results, as seen in Fig. 2c for w/ =0.5. In the case of the largest ratio, h/ w = 3.75, hardening is minimal. For the smallest ratio, case h/ w = 1.67, the effect is the opposite, with a substantial increase in hardening. This effect was also observed in experiments by Kiener et al. (2008) and Senger et al. (2011). The abrupt drops in stresses observed in Fig. 2 will be discussed in subsection 3.2. In order to understand the behavior observed in Fig. 2, distributions of accumulated plastic slips and GND densities are shown in Fig. 3. Only micro-hard cases and h/ w = 2.5 are analyzed, but the effects are similar for micro-free condition, if a high friction coefficient is considered. Aspect ratio influence on the microstructure is discussed in details in subsection 3.2. Fig. 3a–c depict slips right after the yield stress is reached ( =0.00184). For w/ = 0.5 (a), slips in system 1 occur in multiple planes and the result is a nearly homogeneous deformation, concentrated at the center of the pillar. The pillar then tends to deform in a barrel shape. When size decreases to w/ = 0.125 and 0.0625, Fig. 3b–c, slip distributions change with a trend to concentrate in fewer planes, forming a band. The transition is a well-known effect captured experimentally, see e.g. Dimiduk et al. (2005). The change in this distribution is caused by the evolution of the GND microstructure with size, as seen in Fig. 3d–f. In the smaller pillar, Fig. 3f, GND densities are approximately uniform in the band direction and present a gradient perpendicularly to it, being maximal around the boundary of the band and tending to be null inside, where plastic slip are concentrated. This description is coincident with typical band microstructures observed experimentally (Bay et al., 1989). The GND distribution is the opposite in the larger size, Fig. 3d, with densities presenting gradients mainly in the slip system direction. The intermediate size, Fig. 3e, presents a transition between the two cases. Inside the slip-band, dislocations are free to move. They can escape from the surface of the micropillar due to micro-free condition on the sides. Dashed lines drawn in Fig. 3c have the same initial direction of system 1 and define the limits for the dislocation free path for w/ h = 2.5. Then a maximal thickness bmax for the band can be defined. As the micropillar is reduced, bmax must follow the reduction. The dashed lines in Fig. 3c also show a misorientation between the slip-band and the slip system direction. The misorientation plays an important factor because it promotes the development of GNDs at the boundary of the band, as seen in eq. (10). The slip-bands seen in Fig. 3 have a similar configuration of a constrained strip under compression and shear. In order to confirm that 2 In order to have a more realistic representation of crystals a third horizontal slip system could be introduced, however it would not be activated by the applied loading in the present cases. 3 The high friction value used in the simulations, f = 10, physically represents sticking contact.

284

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Fig. 2. Average stress ave versus axial strain ε for energetic hardening. (a) Micro-free boundary condition for h/ w=2.5 and different friction coefficients. (b) Micro-hard boundary condition for h/ w = 2.5 and different friction coefficients. (c) Micro-hard boundary condition for different values of h/ w ; f =0 and w/ =0.5. (In all cases boundary conditions are related to the interface punch-crystal.)

such configuration is associated to size dependent hardening, a simplified numerical model for the slip-band is developed in the Appendix. The model confirms that a smaller thickness for the band, as well as large misorientations, create a greater strain hardening. Theses effects are also shown in the band model developed by Kratochvil and Kruzik (2016). The trend observed in early plastic strains continue to be observed throughout the entire deformation process, as seen in Fig. 3g–i, where accumulated slips for a larger strain, = 0.00536 , are shown. Slips in the larger crystal w/ = 0.5, remain diffusely distributed at the center of the pillar. For sizes w/ = 0.125 and 0.0625 the slip-band tends to be more sharply developed, when compared to distributions for = 0.00184 . A substantial increase of GNDs with deformation is also seen. However, densities of GNDs do not clearly increase with size decrease, see Fig. 3d–f and j-l, despite the much greater hardening. Therefore, in the cases studied in this 285

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Figure 3. (a–c) Accumulated slips in system 1 for = 0.00184 . (d–f) Density of GNDs in system 1 for = 0.00184 . (g–i) Accumulated slips in system 1 for = 0.00536 . (j–l) Density of GNDs in system 1 for = 0.00536 . Energetic hardening, micro-hard condition, f = 0 and h/ w = 2.5 are considered in all cases.

subsection, the distributions of GNDs play a more significant effect than their amount in the determination of size dependent hardening. These results indicate a break down of the Taylor rule, or a relation of the size dependent hardening to the square root of the density of GNDs is not observed.4 A similar conclusion was observed for instance in Dimiduk et al. (2005) and Guruprasad and Benzerga (2008). Distributions of slips and GNDs do not change substantially for larger strains and for this reason are not shown.

4 The Taylor rule relates hardening effects to the square root of total densities of dislocations. However, in the present simulations only GNDs are related to hardening, so the density of SSDs is in fact irrelevant.

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Fig. 4. Average stress ave versus axial strain ε for dissipative hardening. (a) Micro-free boundary condition for h/ w=2.5. (b) Micro-hard condition for h/ w = 2.5 and (c) for h/ w = 1.67 and 3.5. (In the latter case, sizes studied are w/ L=1, 0.5 and 0.25. Smaller values of w/ L correspond to larger values of stresses.)

3.2. Dissipative hardening cases In this subsection, only dissipative hardening is assumed, according to eq. (14). The trend for the smaller to be harder is again observed, as seen in Fig. 4, but some important differences are noticed when compared to energetic cases. First, size effects appear for relatively larger pillars. Second, under specific conditions, dissipative hardening can be associated to strengthening. For a micro-free boundary condition between punch and pillar, Fig. 4a, hardening is only possible for a very high friction condition, as also observed in energetic cases. Only strain hardening is observed. If the micro-hard condition replaces micro-free, then strengthening is observed if low friction coefficients ( f < 0.1) are considered, as shown in Fig. 4b–c. For larger friction coefficients, a trend for the strengthening to be eliminated or “converted” into strain hardening is noticed. This transformation is seen in Fig. 4b for f = 1. A reduction in Y is 287

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Fig. 5. Accumulated plastic slips in system 1 for = 0.00136 . (a–d) h/ w = 2.5; (e,f) h/ w = 3.75; (g,h) h/ w = 1.67. Dissipative hardening, micro-hard and frictionless cases are considered. (Contour level in all figures is the same, as seen in the case (a).)

also observed in experiments in Shade et al. (2009), when the lateral movement of the pillar is blocked. In DDD simulations in Akarapu et al. (2010), hardening in general tends to decrease for a higher friction as observed in Fig. 4b. Another difference in comparison with energetic cases is a tendency for an exhaustion or stagnation of the hardening produced by GNDs for sizes w/ L 0.25, approximately, regardless the ratio h/ w . This exhaustion can be observed in Fig. 4a where the rate of increase in hardening from w/ L = 0.5 to 0.25 seems to fade or decrease. The effect is also visible in Fig. 4c for h/ w = 1.67: in the case w/ L = 0.25 the increase in Y occurs for much larger plastic strains than in larger sizes. If relative sizes smaller than 0.25 are considered (not shown), strengthening effect tends to disappear and only strain hardening is then observed. Aspect ratio h/ w has again a significant effect on results, as seen in Fig. 4c. It is observed that, in cases h/ w = 3.75, only a very small increase in Y is observed with size decrease and, in cases h/ w = 1.67, the increase is substantial. In the latter cases, for sizes smaller than w/ L=2, at certain strain, numerical solution seems to reach a bifurcation point and then a significant drop of the stress is observed. The strain where bifurcation takes place increases with size decrease. In order to understand this and other effects observed in Fig. 4, the distribution of plastic slips and GNDs are analyzed in Figs. 5–7. Fig. 5 shows accumulated plastic slips for system 1 at the onset of plastic slips ( =0.00136). Only frictionless cases are shown. Before the onset of plasticity, deformations and stresses are uniform in the micropillar, which is under pure compression. Therefore no macroscopic strain gradient is imposed to it. Despite that, when critical resolved stress is reached, plastic slips occur in bands. The distribution is related to the GND structure developed, see Fig. 6. Therefore, results show that the present model is able to capture the development of GNDs even in the absence of macroscopic imposed strain gradients, if microscopic hard boundary conditions are considered between crystal and punch. Thickness of the bands decreases with size for h/ w = 2.5, see Fig. 5a–d. The effect explains the greater strengthening associated to smaller sizes, Fig. 4. However, comparing the evolution of the band thicknesses, it is possible to conclude that they start to reach a limit size for w/ L 0.5, Fig. 5d. This may be a possible explanation for the fade of the strengthening effect for sizes below or around w/ L=0.25 in general. In Fig. 5e–h, other aspect ratios are also depicted. Dashed lines drawn in Fig. 5f have the same initial direction of system 1 and define bmax for the dislocation free path for h/ w = 3.75. It is seen that the misorientation between slip-band and slip system direction is visually minimal. It is apparent from the figure that aspect ratio h/ w changes maximal slip-band thicknesses too. The equation below permits to calculate this change for the slip systems assumed in this work: 288

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Fig. 6. Distribution of GND densities in system 1 for = 0.00136 . (a–c) h/ w=2.5; (d,e) h/ w=3.75; (f,g) h/ w=1.67. Dissipative hardening and microhard cases are considered. Except for case (c), where f =10, in all other cases f =0. (Contour level in all figures is the same, as seen in the case (a).)

Fig. 7. Slip rates in system 1 at three different deformation levels (see Fig. 4c). Frictionless dissipative cases with h/ w = 1.67 and w/ L = 1 are considered.

bmax h = cos w w

sin

(18)

Then, the greater h/ w , the greater bmax , for a fixed w. So, pillars with larger aspect ratios are unconstrained to develop larger slipband thicknesses. Both effects (smaller misorientation and larger bands) are associated to a small size effect as shown by the model in the Appendix and confirmed in Fig. 4c for h/ w = 3.75. From eq. (18) it is possible to conclude also that for h/ w 1.73 a free path for dislocations is impossible. This fact is observed in 289

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Fig. 5g, where dashed lines indicate system 1 directions for case h/ w = 1.67. All dislocation paths are blocked by the micro-hard boundary conditions used at the top or bottom of the crystal. However, even in such circumstances, slips still find a way to rearrange in the band shape from the sides of the micropillar, see Fig. 5g–h, but with a cost of a considerable misorientation of the slip-band with the slip direction toward the center of the micropillar. The result is a considerable slip gradient build up and by consequence a large amount of GNDs. The effect is also reproduced in the slip-band model of the Appendix. Fig. 6 shows GND distributions for system 1 at the onset of plastic slips. Except for the case in Fig. 6c, all others are frictionless. Smaller pillars develop a greater density of GNDs. This size dependency is more evident for ratios h/ w = 2.5 and 1.67, Fig. 6a–b and f-g respectively. This effect was observed experimentally in Maass and Uchic (2012). Densities tend also to increase substantially with h/ w decrease, which is a consequence of the misorientation effect. The case in Fig. 6c, where a high friction is taken into account, shows a different distribution of GNDs at the plastic onset. This change is expected because friction introduces an external deformation gradient. The slip-band morphology does not take shape initially, restraining strengthening. However, the morphology ends up being formed with the continuation of the deformation 0.02 , hardening is the same regardless friction (size w/ L=0.5). Changes process. This is confirmed by Fig. 4b considering that, for observed in Fig. 6c also explain why strengthening is not observed in high friction micro-free cases, Fig. 4a. Finally, Fig. 7 shows a transition in the configuration of the plastic slip rates (system 1) that explains the sudden drops in average stresses seen in the stress-strain relation of Fig. 4c in the case of the ratio h/ w = 1.67. Configurations shown in Fig. 7a, b and c, correspond to points a, b and c in Fig. 4c, respectively. Initially plastic slips tend to arrange in the band shape from the sides of the micropillar, Fig. 7a. Observe that these bands have approximately the same angle of the slip system. With the course of the deformation process, plastic slips then tend to follow the GND-free path, Fig. 7b, which does not have the same orientation of the slip system. As already mentioned, such configuration is associated to a greater hardening. However, it turns out that the configuration is unstable and eventually slips suddenly return to their original shape, Fig. 7c. As a consequence, the level of stresses drops or returns to the same level it had at the onset of the plasticity. The effect described can be considered then a bifurcation in the solution caused by GNDs. The sudden drops of the average stresses observed in Fig. 2, for energetic cases in smaller pillars, have a similar explanation, or a GND configuration associated with a greater hardening becomes unstable from time to time. Due to double symmetric slip systems assumed and the low level of lattice rotations, distribution of slips and GNDs for system 2 (not shown) is a mirror of the results seen in Figs. 3, 5 and 6. The discussion about aspect ratio effects was not extended to energetic hardening and/or micro-free condition because, once slip-bands are formed, the effects are similar to cases discussed in the present subsection, with smaller h/ w ratios having the tendency to develop larger size effects due to larger misorientations. 4. Comparisons and discussion In this section, observed size effects are compared to results presented in the literature. Comparisons focus on the strain hardening modulus, apparent yield strength and the distributions of plastic slips, GNDs and lattice rotations. In order to define absolute sizes for the micropillars, characteristic material sizes L and must be attributed. A value equal to 1 μm was used in micropillar studies by Husser et al. (2014), based on an indentation study by Begley and Hutchinson (1998). Lin et al. (2016) used a value approximately equal to 0.4 μm. The value was based on a suggestion done by Gurtin (2010) for the OhnoOkumura-Shibata's GND free energy. Kuroda (2013), on the other side, used a much larger value in the order of 10–20 μm. However, due to the phenomenological nature of the theory used in the present work, a rule to define characteristic material sizes is inexistent. Due to this difficulty, it is common to make comparisons with DDD solutions to define characteristic sizes, such as in Bardella et al. (2013) and Bittencourt (2018). So, in order to have a guide or a clue to define such dimension in the present cases, the DDD simulations done by Kiener et al. (2011) and Kondori et al. (2017) are considered. They limited their applications to the study of GND effects to w 0.4 μm, where a sufficient number of dislocations were still present. In the present model, at least for dissipative hardening, the effectiveness to create size effects is lost below w/ L 0.25. In order to have a coincidence of the two limit sizes, this work will assume that L = 1.6 μm. This dimension was also used by Begley and Hutchinson (1998) to characterize annealed metals. The same value is taken for . It must be emphasized that, if a different value for L and were attributed, then a different range of sizes for the micropillars would emerge. 4.1. Strain hardening In this work, the secant strain hardening modulus (hsec ) is calculated according to definition seen in eq. (19). The modulus is calculated between = 0.01 and = 0.07 (or the maximal strain achieved by calculations).

hsec =

ave = 0.1

ave = 0.07

(19)

0.06

Results are depicted in Fig. 8a. Comparisons focus on DDD simulation results done by Guruprasad and Benzerga (2008), Kiener et al. (2011) and Kondori et al. (2017). These works were selected because hardening effects are associated to GNDs and not other features beyond the resolution of the present continuum model. Results in Kondori et al. (2017) and Kiener et al. (2011) are also confirmed by experiments. hsec for Guruprasad and Benzerga (2008) is taken for stage II hardening. As in the references h/ w ranges from 2 to 3, they are compared to results obtained in this work for h/ w = 2.5. The two curves related to Kiener et al. (2011) and to Guruprasad and Benzerga (2008), in Fig. 8a, represent the superior and inferior limits of the results presented by the authors. Both hardening models studied in the present work tend to underestimate absolute values of hsec . If some hardening due to SSD is 290

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Fig. 8. (a) The secant strain hardening modulus versus width w for h/ w = 2.5. (b) Apparent yield strength versus w for micro-hard, dissipative hardening and frictionless cases. In all cases, L = = 1.6 μm.

added to the model, a better fit is obtained overall as seen in Fig. 8a, where cases with µ / H0 = 131.5 are also included (Q = 1.4, eq. (12)). As expected, the SSD effect is size independent and it corresponds to displace the hsec curves vertically in the figure. In these cases, energetic hardening with micro-hard condition tends to present a good fit with Guruprasad and Benzerga (2008). A substantial size effect is then observed for sizes smaller than 0.8 μm. Dissipative hardening with micro-hard condition presents results that are an average of results seen in Kiener et al. (2011). 4.2. Strengthening Experiments show that Y is proportional to w n and also to (h/ w ) m . According to review in Papanikolaou et al. (2017), n is expected to be in the range 0.4–0.8 while smaller values are likely for m. Fig. 8b shows a logarithmic plot comparing Y to w in the present calculations. Besides the dimension w of the pillars, different aspect ratios are also taken into account. For h/ w = 3.75, 2.5 and 1.67, n values are 0.04, 0.14 and 0.42, respectively. These values were obtained by least-squares fit. Deviation from a linear relation seen in Fig. 8b is not visually larger than normally observed. Only the relation h/ w = 1.67 fits the expected experimental range. Also, the behavior observed in this case does not depart substantially from experiments done with aluminum, see Senger et al. (2011), where n ranges from 0.5 to 0.6 for h/ w = 1.5. However, even the results for h/ w = 2.5 can be considered acceptable because an excellent correlation of the present results with DDD simulations presented in Kiener et al. (2011) is possible. In the reference, the average value for n is equal to ≈0.13 for an aspect ratio h/ w = 2. Then, at least for the smaller h/ w values tested in this work (1.67 and 2.5), it can be concluded that the present model predicts correctly size effects and, by consequence, that such effects are mostly related to the density and the distribution of GNDs. The predominance of GND effects over other effects is also a general conclusion in Kondori et al. (2017), Kiener et al. (2011) and Guruprasad and Benzerga (2008) for w 0.4 μm. It is important to underscore that the strengthening observed in the present calculations is related to a greater amount of GNDs at the onset of plastic slips, see Fig. 6, and not only due to a redistribution of GNDs as in energetic cases, Fig. 3. In order to quantify this effect, the absolute density of GNDs per unit area, GND , is calculated for frictionless cases dividing average values of (1) in Fig. 6 by the Burgers vector size (see Gurtin, 2010). A value corresponding to aluminum, 0.285 nm, was taken. The result is Y ( GND ) p with p 0.2. In experiments done in Maass and Uchic (2012) it was also found that a greater strengthening was associated to a greater density of GNDs in early deformation stages of the compression. After processing the experimental results presented by these authors, it was possible to conclude that p 0.3. Therefore, despite the differences in p values, present simulations and experiments show a similar correlation between GND and Y . However, again the Taylor relation is not satisfied in either cases. This work reveals that the predominance of GNDs on size effects depends on the aspect ratio h/ w . In cases where h/ w is large (greater than 2.5), GNDs do not evolve in a way or quantity to develop size effects. The discrepancies are mainly related to strengthening and not strain hardening. It was shown by experiments in Kiener et al. (2008) and Senger et al. (2011) that the latter is in fact minimal or nonexistent in longer micropillars. However, strengthening observed in these experiments and also in multiple DDD simulations (see e.g. Akarapu et al., 2010; Papanikolaous et al., 2017) are much higher than the effects observed in the present work. So strengthening associated to longer micropillars should be related to events not considered in the present formulation, such as surface effects. These effects may introduce additional size dependent hardening terms, see Peng et al. (2018) for a general review on the subject. In the particular case of micropillar compression, the effect of surface steps caused by dislocation exiting free surfaces was incorporated in a continuum model studied by Hurtado and Ortiz (2012). The authors showed that the surface effects become an important parcel of the hardening when size decreases because the surface to volume ratio increases. The slip-band discussions 291

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developed in this work show that not only the size decrease is important to define the relative importance of the surfaces, but also the ratio h/ w . The surface length smax of the slip-band associated to bmax , see Fig. 3c, can be calculated for the slip systems assumed in this work as smax / w = h/ w tanθ. So, for a fixed size w, smax increases with h and then the relative importance of any surface event increases too. For the system orientations considered in this work, for h/ w equal or smaller than 1.73, smax = 0 and then surface effects tend to be irrelevant, regardless the size of the micropillar.5 Table 1 below shows the relations between h/ w , smax / w and the hardening exponent n. Then, clearly, any surface effect added would modify more significantly the h/ w = 3.75 case. It remains to be seen if the modifications would be enough to adjust n values to experimental values. 4.3. Rotations, accumulated slips and GND densities In Fig. 9, in-plane lattice rotations (a), accumulated plastic slips (b,c) and total GND density (d) are shown. The total GND density is calculated as the sum of the absolute values of ( ) , divided by the Burgers vector of the aluminum. In all cases friction is null, boundary condition is micro-hard, w=0.8 μm, h/ w = 2.5 and hardening is dissipative. Axial strain considered is 0.0752. As friction is null, the top surface can move horizontally. In the simulations it moves to the left. The result is a predominant positive in-plane lattice rotation (anti-clockwise direction) near the center of the pillar. The effect is much greater for larger micropillars (not shown). Slips in both systems are no longer symmetric, with plastic slips tending to concentrate in system 1, Fig. 9b. Therefore, present simulations are able to capture the tendency for micropillars to concentrate slips in only one system if rotation of the micropillar axis is allowed, as shown in Shade et al. (2009) and in DDD simulations in Akarapu et al. (2010). Such effect will only occur under the micro-hard boundary condition in the present model. Results presented in Fig. 9 are specially similar to results presented in the DDD simulations done by Kiener et al. (2011) (see Figs. 11 and 15 in the reference). For instance, in both simulations, GNDs tend to concentrate in the slip-bands, Fig. 9d. However, in the reference, GNDs tend to accumulate also at the center of the micropillar, where the present model does not detect such dislocations. The discrepancy can be related only to small geometrical differences between both cases. However, the explanation may be more complex. As discussed in Benzerga et al. (2004), GNDs in DDD formulations are formed under the shape of cell substructures in the crystal, regardless of strain gradients. The present model does not have resolution to “see” such cell structures. Then, the differences observed at the center of the pillar, considering the present calculations, can be also the result of an inherent lack of resolution of the model. In general, a better fit with experiments and DDD simulations is possible if the dissipative hardening model is assumed. This conclusion does not rule out effects related to energetic hardening. Depending on the crystal characteristics, such hardening may be important, as it was observed in the comparison with Guruprasad and Benzerga (2008), in Fig. 8a. However, it is important to stress that the energetic model considered alone is unable to explain most of the effects observed experimentally in micropillar compression. This conclusion cannot be extended to other energetic models, such as the models where less than quadratic forms of the defect energy are used (e.g. Ohno and Okomura, 2007). T GND

5. Summary In this work, a continuum model based on the Gurtin's strain gradient higher-order crystal plasticity theory was used to numerically simulate the behavior of micropillars in compression. The hardening models used, energetic and dissipative, are linked to the densities of GNDs. Therefore the only size dependent events captured by the present formulation are due to the presence of GNDs. This work shows that size effects are noteworthy on stress-strain relations when plastic slips occur in few slip planes developing then a slip-band morphology. A simplified model for the slip-band shows that its thickness and the misorientation of the slip systems regarding the direction of the band are fundamental variables that govern the behavior of the band. Both thickness and misorientation are related to the geometry of the pillar, i.e., width w and ratio h/ w . The results obtained by the present continuum model are comparable to experiments and DDD simulations if h/ w is smaller than certain value (2.5 for the properties assumed). As a consequence, distribution and densities of GNDs can be considered a controlling feature in defining size effects in these cases. For larger h/ w , the model seems to fail to capture correctly size effects, specially strengthening. One possible flaw of the model is the fact that hardening mechanisms associated to the free surface of the pillar are not being taken into consideration. The chances that such mechanisms are important grow because this work shows that the surface effects may be more important specifically for larger h/ w relations. Therefore, different hardening mechanisms should dominate the behavior of the micropillars depending on h/ w . It is also possible that the model used is unable to capture all the GND activity in micropillar compression. For instance, in Fig. 9d the GND activity at the center is smaller than the GND activity observed in DDD simulations. Definitions of the hardening model are essential to represent size effects. For the definition of the defect energy used in this work, it was possible to capture only strain hardening effects, while the dissipative model was able to represent strengthening or strain hardening, depending on the boundary conditions. Hardening associated to dissipative model is possible only above a certain limit size w. This limit is actually arbitrary because it depends on the attributed material characteristic size for the theory. In this work, a size equal to 1.6 μm was proposed resulting a limit dimension w 0.4 μm. Finally, combination of higher-order boundary conditions and friction conditions between punch and crystal can change 5 This formula considers the surface associated only to the free-path for dislocations. In practical terms it was shown, see Fig. 6, that slips end up being formed on the sides of the micropillar for aspect ratios smaller than 1.73.

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Table 1 Decrease in hardening n with increase of slip-band surface smax . h/w

smax /w

n

1.67 2.5 3.75

0 0.77 2.02

0.42 0.14 0.04

Fig. 9. (a) In-plane lattice rotations (anti-clockwise direction is positive); (b) accumulated plastic slips in system 1; (c) accumulated plastic slips in system 2; (d) total GND densities. In all cases friction is null, boundary condition is micro-hard, w=0.8 μm, h/ w = 2.5 and hardening is dissipative. = 0.0752 .

considerably results as well. If micro-free condition is used, size effects can be obtained only for high friction coefficients. If microhard condition is considered, friction effects tend to eliminate strengthening and convert it into strain hardening. It is possible that the difficulty to distinguish strengthening from strain hardening in some experiments is then caused by friction effects. In the case of low friction, a tendency for slips to concentrate in only one slip system is observed. Further analysis with the present models are being considered in cases where the pillar is composed by different layers of materials (see for instance Mu et al., 2014) as well as the effects of different orientations of the slip systems. Acknowledgment The author is pleased to acknowledge support from the Brazilian Government through a CNPq Fellowship. Appendix

Fig. A.1. Layer of the crystal under shear and compression. 293

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Fig. A.1 shows a simplified model for the slip-band developed in the micropillar compression. In this model a crystalline strip of height b in plane-strain is sheared along the x1-direction and compressed in the x2 -direction. b in the strip model represents the thickness of the slip-band in real micropillars. Slip system orientations in the model have approximately the same orientations as in the micropillar band that is formed in the direction of system 1, as seen in Figs. 3 and 5 (see more details regarding slip orientation below). Resolved stresses in both slip systems are initially the same in the micropillar. Displacements Us and Uc of the strip were then chosen in order to comply with this condition. The following boundary conditions are considered for the model:

u1 = 0,

u2 = 0 along x2 = 0;

u1 = Us = b ,

u2 = Uc

(A.1)

along x2 = b,

where is the prescribed shear. All field quantities are required to be periodic in x1 with period d. To reproduce approximately the slip distribution seen in the band, the following micro-hard boundary condition is taken on the top and bottom of the strip: (1)

(A.2)

= 0 along x2 = 0, b.

As a result, fields are not uniform in x2 . The misorientation observed between slip system and the slip-band directions, see e.g. Fig. 3c, is represented by the angle δ in Fig. A.1. Three possible values for δ are studied: 0, 5 and 10°. Values of Uc taken for these angles are,6 respectively, 0.51Us , 0.365Us and 0.178Us . In all cases, =0.0003 s−1. With regard to the finite element mesh, 1 × 100 elements with uniform size are used in the simulations. ave Fig. A.2 shows results for both hardening considered in this work.7 Results are presented as function of the overall shear stress 12 and , where the former is calculated as: ave 12

=

1 d

In Fig. A.2, nonlocal effect.

d 12 (x1,

b) dx1.

(A.3)

0 ave 12

is also parameterized by

Y 0,

which corresponds to the apparent yield shear stress for each δ used, without any

Fig. A.2. Overall shear stress versus shear for different sizes and misorientation angles (δ). When not indicated, curves refer to case where =5°.

In both cases in Fig. A.2, a size effect is observed only for δ = 5° and 10°. For δ = 0°, size effects are not noticed. This is an expected result because gradients of slips regarding system 1 are absent. As shown in Fig. 5f, misorientation between slip system 1 and the slip-band is practically absent for cases h/ w = 3.75. Then the model is able to explain why these cases do not present substantial hardening nor a significant presence of GNDs. For >0°, size effects then take place. Larger values for δ give rise to greater hardening effects due to greater slip gradients and GNDs being created, see Fig. A.3a. High values of δ may represent situations where a free-path for dislocations are not possible, such as the case h/ w = 1.67. In the cases were >0°, the following behavior is obtained by the strip model: a) For dissipative hardening, Fig. A.2b, the dominant size effect is strengthening. For smaller values of b, an increase in the apparent yield strength is observed. As it was shown that b decreases with the decrease of the micropillar size, the strip model then permits to understand size dependent strengthening observed in Fig. 4b–c. b) For energetic hardening, Fig. A.2a, only strain hardening is observed. For smaller values of b, there is an increase in the hardening.

6 Using these values for Uc , approximately the same resolved stresses are obtained for both slip systems. For Us in the opposite direction as shown in Fig. A.1, the relation between Uc and Us should be different to guarantee the same resolved stresses. However, even in this case, general conclusions of this Appendix do not change. 7 Small oscillations in Fig. A.2 are caused by inertial terms used in the solution method.

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Therefore the strip model is also able to explain size effects observed in micropillars in the case of energetic hardening, Fig. 2. The range of sizes where the energetic hardening is more effective is much smaller than the corresponding sizes where dissipative hardening is operating. This is also a characteristic observed in the micropillars. Finally, in Fig. A.3, distribution of GNDs (a) and accumulated slips (b) in system 1 for =0.016 are depicted. Only energetic hardening is considered. An increase in δ from 5 to 10° substantially increases the amount of GNDs, Fig. A.3a. However, a decrease in the thickness of the band not necessarily leads to a substantial increase in the amount of GNDs, at least for the deformation considered, as can be seen in the figure for b/ = 0.0313 and 0.0417 (cases where =5°). This characteristic helps to understand the small dependence of GND densities in Fig. 3 on the size. However, the accumulated slip in system 1 shows a visible difference between the two sizes, as seen in Fig. A.3b. The changes in accumulated slips are caused by the history of changes of GNDs throughout the course of deformation.

Fig. A.3. (a) Distribution of the densities of GNDs in system 1, in mm−1. (b) Distribution of accumulated slip in system 1. In both cases

= 0.016.

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