- Email: [email protected]
Continuum modeling of dislocation starvation and subsequent nucleation in nano-pillar compressions
Available online at www.sciencedirect.com
Scripta Materialia 66 (2012) 93–96 www.elsevier.com/locate/scriptamat
Continuum modeling of dislocation starvation and subsequent nucleation in nano-pillar compressions Antoine Je´rusalem,a,⇑ Ana Ferna´ndez,a Allison Kunzb and Julia R. Greerb a
IMDEA Materials Institute, Calle Profesor Aranguren, s/n, 28040 Madrid, Spain California Institute of Technology, MC 309-81, Pasadena, CA 91125-8100, USA
b
Received 24 September 2011; revised 4 October 2011; accepted 5 October 2011 Available online 11 October 2011
The mechanical behavior of single crystalline aluminum nano-pillars under uniaxial compression differs from bulk Al in that the former is characterized by a smoother transition from elasticity to plasticity. We propose an extension of the phenomenological model of dislocation starvation originally proposed in [Greer and Nix, Phys. Rev. B 73 (2006) 245410] additionally accounting for dislocation nucleation. The calibrated and validated continuum model successfully captures the intrinsic mechanisms leading to the transition from dislocation starvation to dislocation nucleation in fcc nano-pillars. Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Continuum model; Nanopillar; Crystal plasticity; Dislocation starvation
The enormous advances in fabrication processes, computational modeling, and experimental characterization methods at the submicron scale have catalyzed the emergence of a new era in materials science, specifically in the area of structural materials. Namely, the development of novel material systems with radically superior properties will be achieved through architectural control at the appropriate microstructural scales. This will result in the ability to develop structural materials with vastly superior properties [1]. In order to utilize these principles towards insertion into relevant structural applications, it is essential to assess mechanical properties and deformation mechanisms in surface-dominated structures with reduced dimensions (i.e. not via nanoindentation), as they will likely comprise the aforementioned architectural constituents. Recently, it has been shown that at the micron- and submicron scales, the presence of free surfaces in small-scale samples dramatically affects crystalline strength [2–5]. In these studies, cylindrical nano-pillars were fabricated mainly by the use of the focused ion beam (FIB) and, remarkably, the results of all of these reports for fcc metals show the same power-law dependence between the flow stress and sample size, implying that this scaling might be universal [6]. To date, a fundamental understanding of nano-scale mechanical response in surface dominated structures
⇑ Corresponding
author; E-mail imdea.org; [email protected]
addresses:
antoine.jerusalem@
with sub-micron dimensions is still elusive [7–10]. Because of the intrinsic nature of the continuum approach a priori contradicting the stochastic and highly size dependent nature of nanoscale mechanics, continuum finite element methods have been scarcely used to model nanopillar compressions. Among these efforts, Zhang et al. used an isotropic plasticity finite element model to provide a set of guidelines to minimize the experimental artifacts [11]. They mostly emphasize proper choice of pillar diameters and aspect ratios while minimizing both the vertical taper angle and indenter head-pillar top misalignment. Raabe et al. followed up on this work by making use of a crystal plasticity model focusing on studying the effect of crystal orientation and its stability on the deformation mechanisms [12]. Schuster et al. also built up on Zhang’s efforts by applying their model to metallic glass and evaluating the effect of specimen taper on the compressive strength [13]. This was further described by Wu et al. by specifically studying the geometrical constraint dependent stress distribution [14]. Shade et al. emphasized the non-constitutive law related hardening caused by the lateral stiffness of the indenter by means of crystal plasticity [15], and Chen et al. recently focused on local stress concentration in metallic glass pillar bending, using an isotropic plasticity model [16]. Finally, Hurtado and Ortiz proposed a non-local model accounting for surface effects during pillar compression [17], and Zhang and Aifantis developed a strain gradient model capturing the strain bursts in micropillars [18].
1359-6462/$ - see front matter Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2011.10.009
94
A. Je´rusalem et al. / Scripta Materialia 66 (2012) 93–96
Setting the burst-ridden stochastic nature of nanodeformation aside, we propose a model capable of capturing the macroscopic stress–strain behavior of single crystalline Al nano-pillar compressions. This involves the consideration of the phenomenological ‘dislocation starvation’ model originally proposed by Greer et al. [19,20], which rationalizes the initial stage of yielding by a starvation of the initially present mobile dislocations. A crystal plasticity model accounting for size-dependent dislocation starvation and subsequent nucleation is thus proposed, calibrated and validated against experiments. It has now been ubiquitously shown that even in the absence of strong strain gradients, small-scale uniaxial compression and tension experiments reveal a so-called size effect, or size-induced strengthening of single crystals in micron- and sub-micron structures. Deeply in the sub-micron regime, this size dependence has been at least in part rationalized by dislocation starvation followed by nucleation mechanism [2–4,6]. Based on the classical work of Johnston and Gilman [21,22], Greer and Nix proposed a phenomenological model [19,20] in which mobile dislocations initially present in the sub-micron pillar annihilate in the vicinity of a free surface. A follow-up on this approach accounting for the kinetics of nucleation was recently proposed by Nix and Lee [23]. Accounting for the resistance stress necessary to overcome the elastic dislocation interactions, and lengthening and bowing out of dislocations in their slip planes, the average dislocation resistance shear stress associated to the dislocation starvation is given by [20]: aa lb pffiffiffi ln ð1Þ sstarv ¼ 0:5lb q þ 1:4 4pað1 mÞ b where l, m, b and a are the shear modulus, the Poisson’s ratio, the Burgers vector and a constant of order unity, and where the instantaneous pillar diameter a and density q are given by: 8 0:5 > > < a ¼ a0 ð1 Þ ðd1Þ ð2Þ q ¼ q0 þ b a Mp > > : where q0 and a0 are the initial dislocation density and pillar diameter, and p are the engineering overall strain and plastic strain, M is the Schmid factor, and d is the breeding coefficient (representing the inverse of the distance a dislocation travels before replicating itself). We postulate here that the starvation of pre-existing dislocations and nucleation of new ones should be independently characterized by two reference critical resolved shear stresses (CRSSs). We thus propose another model based on Eq. (1) taking into account a reference (initial) starvation CRSS s0,starv and a reference nucleation CRSS s0,nucl such that, for each slip system i: ! ! p p i 1 starv s0;starv þ starv s0;nucl ; s0;nucl s0 ¼ Min ð3Þ p p where pffiffiffiffiffi s0;starv ¼ 0:5lb q0 þ 1:4
aa lb 0 ln 4pa0 ð1 mÞ b
ð4Þ
is a model parameter corresponding to and where starv p the plastic strain for which nucleation dislocation is more favorable than dislocation starvation, and at which the nucleation CRSS s0,nucl is reached. In this model, the CRSS is equal to the starvation CRSS at initial yielding, and then linearly increases as a function of p as the mobile dislocations are moving towards the free surface, while nucleation processes are becoming increasingly more prevalent. Once all mobile dislocations have been annihilated, plasticity is fully nucleation driven. The linear dependence has been chosen as a simple first approximation and is justified below. The rate-independent constitutive model adopted here is following Ref. [24]. The reader is invited to consult this reference for a complete description. In the following, we make the hypothesis that the crystal is small enough so that dislocations exit the crystal before interacting significantly (i.e. no hardening). As a consequence si ¼ si0 at all time (see Eq. (3)). For both the calibration and the validation, experimental results of mono-crystalline pillar compressions are chosen as reference [5]. These pillars are made of high purity aluminum (5N purity, ESPI Metals), annealed under vacuum at 350 °C overnight and electropolished by Able Electropolishing. Grain orientation maps were obtained by automatic indexing of electron backscatter diffraction (EBSD) patterns using a Zeiss1550 VP field emission scanning electron microscope equipped with an EBSD system from Oxford Instruments. Once a suitable grain was chosen, nanopillar samples were fabricated using a FEI Nova 200 scanning electron microscope with a focused ion beam (FIB), see Ref. [5] for more details. For each considered nano-pillar, the top diameter and tapering angle were extracted by means of a scanning electron microscopy (SEM). The height being the most ambiguous geometric parameter due to FIB undercutting, it was determined by a stiffness-matching method [25]. The average diameter calculated along with this height was also calculated. The geometrical parameters of the two chosen pillars, as well as their finite element characteristics are gathered in Table 1. Pillar A (see Table 1) was chosen for calibration. The elastic constants for aluminum at room temperature, the Burgers vector, the initial dislocation density for nanopillar (approximating its value for aluminum by the one for gold), and a were taken from the literature [20,26,27]. The two remaining parameters s0,nucl and were then independently adjusted to fit the experistarv p mental stress strain curve of the nanopillar indentation. Note that, by definition, the two calibrated parameters modify the final strength and the plastic strain at which this final strength is reached; as a consequence, their unicity is guaranteed. The simulations were done using Abaqus/Explicit [28] under quasi-static loading, clamping the base of the nanopillar, while applying a free slip boundary condition on the top surface. Both temporal and spatial convergence were checked. The parameters are given in Table 2, the stress–strain curve in Figure 1, and the resulting Von Mises stress field in Figure 2. As previously observed in other efforts [5,13], the experimental stress–strain curve appears to be abnormally compliant during the initial part of loading. This
A. Je´rusalem et al. / Scripta Materialia 66 (2012) 93–96
95
Table 1. Nanopillar finite element characteristics. Pillar
Top diameter
Tapering angle
Height
Average diameter
Euler angles
Schmid factor
Number of linear tetrahedra
A B
622.9 nm 655 nm
88.1° 88.7°
2.35 lm 2.15 lm
696 nm 700 nm
19.5°, 18.4°, 28.1° 269.1°, 15.5°, 85.5°
0.4879 0.4803
41,840 37,683
Table 2. Model parameters.
107.3 Gpa
C44
60.8 Gpa
l
28.3 Gpa
m
26 Gpa
0.35
b 2.8 10
350 300
Stress (MPa)
250 200 Experimental Simulation − reference Simulation − constrained top Simulation − with substrate
150 100 50 0
0
0.02
0.04
0.06 0.08 Strain
0.1
q0 10
0.12
Figure 1. Stress–strain curves of pillar A compression as extracted from (a) experiment, (b) simulation, (c) simulation with constrained top and (d) simulation with substrate.
Figure 2. (a) SEM image of the compressed nanopillar A, and Von Mises field (Pa) for a longitudinal cut of the simulation: (b) reference, (c) with constrained top and (d) with substrate.
apparent discrepancy with the simulation can be rationalized by the misalignment between the indenter flat punch tip and the pillar top (absent in the simulation), thus affecting the test accuracy and leading to an under-estimation of the elastic modulus [13]. Experimental and numerical studies have suggested that the effect of the nanoindenter lateral stiffness could lead to important changes in the deformation mechanism [3,15]; for instance, the stochastic behavior of bursts could be significantly decreased and/or a significant fictitious hardening (independent of the constitutive behavior) could be observed. Figures 1 and 2 also show the results for the same reference simulation but with the top surface fully constrained laterally. In agreement with Ref. [15], the stress–strain curve presents a significantly more important hardening and the pillar is more constrained, albeit with comparable maximum Von Mises stress values. As the lateral stiffness of nanoindenters is generally estimated at best, one could assume, as can be seen on Figure 1, that the simulated curves for an indenter with realistic lateral stiffness should lie in between these two curves, with and without lateral clamping. The influence of substrate was studied by additionally supporting the pillar with a thick layer of material; see
m
2
12
5 10 m
a
s0,nucl
starv p
1
160 MPa
5% (Pillar A only)
the results on Figures 1 and 2. These investigations showed a minimal variation in the elastic slope and starvation region, and more pronounced hardening in the nucleation, but less important than the ones arising from constraining the top surface. It should also be emphasized that this simulation assumed the same starvation behavior in the substrate as in the crystal; the substrate should theoretically have an even higher starvation CRSS, thus a priori alleviating the overall discrepancy. The curve given here consequently corresponds to the case of maximum discrepancy. Additionally, Figure 2 exhibits very similar Von Mises stress fields, thus confirming the relatively good approximation done by neglecting the substrate. The phenomenological model proposed in Ref. [20] is used alternatively here to study the evolution of dislocation density and flow stress assuming that no dislocations are nucleated. Noting that the deformation is mostly plastic ( ’ p), the remaining parameter, i.e. the breeding coefficient d, is calibrated so as to obtain . This condition is reached for q = 0 for p ¼ starv p d = 1.46 106 m1. Note that this value is slightly larger than the one used for gold in Ref. [20] and a factor 4 larger than the one given for aluminum in Ref. [29]. The dislocation density and the associated flow stress evolutions are given in Figure 3. As can be seen, an increase of the dislocation density is observed up to 15% deformation, as the gliding dislocations encounter (“breed”) others, pushing them out of the crystal. Eventually, these obstacles are overcome and dislocation starvation takes over the deformation process. This succession leads to a relatively stable flow stress at 20 MPa. The assumption of the model of linear transition between starvation and nucleation CRSS (see Eq. (3)), seems to be confirmed as a reasonable representation of a change from a constant starvation CRSS to a constant nucleation CRSS, where local 12
7
x 10
30
6 5 20 4 3 10
2
Flow stress (MPa)
C12
Dislocation density (m−2)
C11
1 0
0
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Strain
Figure 3. Dislocation density evolution (full line) and associated flow stress (dashed line) as calculated by the phenomenological model.
A. Je´rusalem et al. / Scripta Materialia 66 (2012) 93–96
96
Figure 4. Stress–strain curves of pillar B compression as extracted from experiment and simulations (“lower bound” being with free top surface and “upper bound” with laterally constrained top surface).
The Abaqus subroutine VUMAT used here is a modified version of the one used for HCP metals in Ref. [31], itself an extension of the one originally written (and kindly provided) by Dr. Alexander Staroselsky and Prof. Lallit Anand at MIT [24]. A.J. acknowledges support from the Juan de la Cierva grant from the Spanish Ministry of Science and Innovation, from the Amarout grant from the European Union and from the ESTRUMAT-S2009/MAT-1585 grant (Madrid Regional Government). A.F. acknowledges funding from the Caltech SURF program. J.R.G. gratefully acknowledges the financial support of the National Science Foundation through the CAREER award (DMR-0748267) and Office of Naval Research (Grant No. N000140910883).
obstacles such as Lomer–Cotrell locks leading to the creation of single-arm sources eventually lead to new dislocation nucleations [30]. The breeding coefficient can now be used directly to calculate the starvation strain of other pillars by solving ¼ 5:8%. q = 0 in Eq. (2). For pillar B (see Table 1), starv p The model is finally validated against the experimental results of this new pillar by using the same set of calibrated parameters (Table 2) but with this new starvation strain, see Figure 4. Here, again, the level of strength and the changes of curvature of the stress–strain curves are very well captured, thus validating the model. Whereas the initial starvation CRSS and starvation strain are directly defined by means of Eqs. (4) and (2), respectively, the nucleation CRSS remains a free parameter. In other words, for a different crystal size, this parameter either needs to be calibrated or evaluated from other non-local methods accounting for example for surface energy, see Ref. [17]. With an increase of diameter, the initial starvation CRSS and the starvation strain, respectively decreases and increases. However, the subsequent hardening after initial starvation yield is expected to be much more important, reaching the nucleation CRSS (a priori much smaller at larger diameters) at very small strains [20]. In other words, for large crystals, full starvation is never reached, as nucleation becomes more favorable earlier; the starvation strain and the strain at which nucleation occurs are different. The model can then be adapted . by considering the minimum of both for starv p Note finally that the number of initial dislocations in the nano-pillars considered here is of the order of 10, meaning that the prominent burst behavior observed at such scales needs to be considered in future work. To this end, their incorporation in a nanoscale continuum model will require taking into account the stochastic nature of this phenomenon. In this letter, we proposed a crystal plasticity continuum model accounting for dislocation starvation and nucleation, based on the phenomenological model of Greer et al. [19,20]. The model has been found to capture very closely the smoother transition to plasticity observed in nanopillar compressions, characterized by two plastic zones representative of dislocation starvation and nucleation, respectively. Remarkably, only one free parameter remains and still needs to be characterized by other non-local means.
[1] L. Valdevit, A.J. Jacobsen, J.R. Greer, W.B. Carter, J. Am. Ceram. Soc. 94 (2011) 15–34. [2] G. Dehm, Prog. Mater. Sci. 54 (2009) 664–688. [3] M. Uchic, P. Shade, D. Dimiduk, Annu. Rev. Mater. Res. 39 (2009) 361–386. [4] O. Kraft, P. Gruber, R. Monig, D. Weygand, Annu. Rev. Mater. Res. 40 (2010) 293–317. [5] A. Kunz, S. Pathak, J. Greer, Acta Mater. 59 (2011) 4416–4424. [6] J. Greer, J. De Hosson, Prog. Mater. Sci. 56 (2011) 654–724. [7] L. Zepeda-Ruiz, B. Sadigh, J. Biener, A. Hodge, A. Hamza, Appl. Phys. Lett. 91 (2007) 101907. [8] J. Marian, J. Knap, Int. J. Multiscale Comput. Eng. 5 (2007) 287–294. [9] C. Weinberger, W. Cai, PNAS 105 (2008). [10] C. Weinberger, A. Jennings, K. Kang, J. Greer, J. Mech. Phys. Solids, in press, doi:10.1016/j.jmps.2011.09.010. [11] H. Zhang, B. Schuster, Q. Wei, K. Ramesh, Scripta Mater. 54 (2006) 181–186. [12] D. Raabe, D. Ma, F. Roters, Acta Mater. 55 (2007) 4567– 4583. [13] B. Schuster, Q. Wei, T. Hufnagel, K. Ramesh, Acta Mater. 56 (2008) 5091–5100. [14] X. Wu, Y. Guo, Q. Wei, W. Wang, Acta Mater. 57 (2009) 3562–3571. [15] P. Shade, R. Wheeler, Y. Choi, M. Uchic, D. Dimiduk, H. Fraser, Acta Mater. 57 (2009) 4580–4587. [16] C. Chen, Y. Pei, J.D. Hosson, Acta Mater. 58 (2010) 189–200. [17] D. Hurtado, Multiscale Modeling of Microcrystalline Materials, Ph.D. Thesis, Caltech, 2011. [18] X. Zhang, K. Aifantis, Mater. Sci. Eng. A 528 (2011) 5036–5043. [19] J. Greer, W. Nix, Phys. Rev. B 73 (2006) 1–6. [20] J. Greer, Rev. Adv. Mater. Sci. 13 (2006) 59–70. [21] W. Johnston, J. Gilman, J. Appl. Phys. 30 (1959) 129–144. [22] J. Gilman, Micromechanics of Flow in Solids, McGrawHill, New York, NY, 1969. [23] W. Nix, S. Lee, Philos. Mag. 91 (2010) 1084–1096. [24] A. Staroselsky, Crystal Plasticity Due to Slip and Twinning, Ph.D. Thesis, Massachusetts Institute of Technology, 1998. [25] J. Greer, W. Oliver, W. Nix, Acta Mater. 53 (2005) 1821–1830. [26] J. Vallin, M. Mongy, K. Salama, O. Beckman, J. Appl. Phys. 35 (1964) 1825–1826. [27] S. Crandall, N. Dahl, T. Lardner, An Introduction to the Mechanics of Solids, McGraw-Hill, 1999. [28] Dassault Syste`mes, Abaqus/Explicit
350 300
Stress (MPa)
250 200
Experimental Simulation − lower bound Simulation − upper bound
150 100 50 0
0
0.02
0.04
0.06 0.08 Strain
0.1
0.12
Scripta Materialia 66 (2012) 93–96 www.elsevier.com/locate/scriptamat
Continuum modeling of dislocation starvation and subsequent nucleation in nano-pillar compressions Antoine Je´rusalem,a,⇑ Ana Ferna´ndez,a Allison Kunzb and Julia R. Greerb a
IMDEA Materials Institute, Calle Profesor Aranguren, s/n, 28040 Madrid, Spain California Institute of Technology, MC 309-81, Pasadena, CA 91125-8100, USA
b
Received 24 September 2011; revised 4 October 2011; accepted 5 October 2011 Available online 11 October 2011
The mechanical behavior of single crystalline aluminum nano-pillars under uniaxial compression differs from bulk Al in that the former is characterized by a smoother transition from elasticity to plasticity. We propose an extension of the phenomenological model of dislocation starvation originally proposed in [Greer and Nix, Phys. Rev. B 73 (2006) 245410] additionally accounting for dislocation nucleation. The calibrated and validated continuum model successfully captures the intrinsic mechanisms leading to the transition from dislocation starvation to dislocation nucleation in fcc nano-pillars. Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Continuum model; Nanopillar; Crystal plasticity; Dislocation starvation
The enormous advances in fabrication processes, computational modeling, and experimental characterization methods at the submicron scale have catalyzed the emergence of a new era in materials science, specifically in the area of structural materials. Namely, the development of novel material systems with radically superior properties will be achieved through architectural control at the appropriate microstructural scales. This will result in the ability to develop structural materials with vastly superior properties [1]. In order to utilize these principles towards insertion into relevant structural applications, it is essential to assess mechanical properties and deformation mechanisms in surface-dominated structures with reduced dimensions (i.e. not via nanoindentation), as they will likely comprise the aforementioned architectural constituents. Recently, it has been shown that at the micron- and submicron scales, the presence of free surfaces in small-scale samples dramatically affects crystalline strength [2–5]. In these studies, cylindrical nano-pillars were fabricated mainly by the use of the focused ion beam (FIB) and, remarkably, the results of all of these reports for fcc metals show the same power-law dependence between the flow stress and sample size, implying that this scaling might be universal [6]. To date, a fundamental understanding of nano-scale mechanical response in surface dominated structures
⇑ Corresponding
author; E-mail imdea.org; [email protected]
addresses:
antoine.jerusalem@
with sub-micron dimensions is still elusive [7–10]. Because of the intrinsic nature of the continuum approach a priori contradicting the stochastic and highly size dependent nature of nanoscale mechanics, continuum finite element methods have been scarcely used to model nanopillar compressions. Among these efforts, Zhang et al. used an isotropic plasticity finite element model to provide a set of guidelines to minimize the experimental artifacts [11]. They mostly emphasize proper choice of pillar diameters and aspect ratios while minimizing both the vertical taper angle and indenter head-pillar top misalignment. Raabe et al. followed up on this work by making use of a crystal plasticity model focusing on studying the effect of crystal orientation and its stability on the deformation mechanisms [12]. Schuster et al. also built up on Zhang’s efforts by applying their model to metallic glass and evaluating the effect of specimen taper on the compressive strength [13]. This was further described by Wu et al. by specifically studying the geometrical constraint dependent stress distribution [14]. Shade et al. emphasized the non-constitutive law related hardening caused by the lateral stiffness of the indenter by means of crystal plasticity [15], and Chen et al. recently focused on local stress concentration in metallic glass pillar bending, using an isotropic plasticity model [16]. Finally, Hurtado and Ortiz proposed a non-local model accounting for surface effects during pillar compression [17], and Zhang and Aifantis developed a strain gradient model capturing the strain bursts in micropillars [18].
1359-6462/$ - see front matter Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2011.10.009
94
A. Je´rusalem et al. / Scripta Materialia 66 (2012) 93–96
Setting the burst-ridden stochastic nature of nanodeformation aside, we propose a model capable of capturing the macroscopic stress–strain behavior of single crystalline Al nano-pillar compressions. This involves the consideration of the phenomenological ‘dislocation starvation’ model originally proposed by Greer et al. [19,20], which rationalizes the initial stage of yielding by a starvation of the initially present mobile dislocations. A crystal plasticity model accounting for size-dependent dislocation starvation and subsequent nucleation is thus proposed, calibrated and validated against experiments. It has now been ubiquitously shown that even in the absence of strong strain gradients, small-scale uniaxial compression and tension experiments reveal a so-called size effect, or size-induced strengthening of single crystals in micron- and sub-micron structures. Deeply in the sub-micron regime, this size dependence has been at least in part rationalized by dislocation starvation followed by nucleation mechanism [2–4,6]. Based on the classical work of Johnston and Gilman [21,22], Greer and Nix proposed a phenomenological model [19,20] in which mobile dislocations initially present in the sub-micron pillar annihilate in the vicinity of a free surface. A follow-up on this approach accounting for the kinetics of nucleation was recently proposed by Nix and Lee [23]. Accounting for the resistance stress necessary to overcome the elastic dislocation interactions, and lengthening and bowing out of dislocations in their slip planes, the average dislocation resistance shear stress associated to the dislocation starvation is given by [20]: aa lb pffiffiffi ln ð1Þ sstarv ¼ 0:5lb q þ 1:4 4pað1 mÞ b where l, m, b and a are the shear modulus, the Poisson’s ratio, the Burgers vector and a constant of order unity, and where the instantaneous pillar diameter a and density q are given by: 8 0:5 > > < a ¼ a0 ð1 Þ ðd1Þ ð2Þ q ¼ q0 þ b a Mp > > : where q0 and a0 are the initial dislocation density and pillar diameter, and p are the engineering overall strain and plastic strain, M is the Schmid factor, and d is the breeding coefficient (representing the inverse of the distance a dislocation travels before replicating itself). We postulate here that the starvation of pre-existing dislocations and nucleation of new ones should be independently characterized by two reference critical resolved shear stresses (CRSSs). We thus propose another model based on Eq. (1) taking into account a reference (initial) starvation CRSS s0,starv and a reference nucleation CRSS s0,nucl such that, for each slip system i: ! ! p p i 1 starv s0;starv þ starv s0;nucl ; s0;nucl s0 ¼ Min ð3Þ p p where pffiffiffiffiffi s0;starv ¼ 0:5lb q0 þ 1:4
aa lb 0 ln 4pa0 ð1 mÞ b
ð4Þ
is a model parameter corresponding to and where starv p the plastic strain for which nucleation dislocation is more favorable than dislocation starvation, and at which the nucleation CRSS s0,nucl is reached. In this model, the CRSS is equal to the starvation CRSS at initial yielding, and then linearly increases as a function of p as the mobile dislocations are moving towards the free surface, while nucleation processes are becoming increasingly more prevalent. Once all mobile dislocations have been annihilated, plasticity is fully nucleation driven. The linear dependence has been chosen as a simple first approximation and is justified below. The rate-independent constitutive model adopted here is following Ref. [24]. The reader is invited to consult this reference for a complete description. In the following, we make the hypothesis that the crystal is small enough so that dislocations exit the crystal before interacting significantly (i.e. no hardening). As a consequence si ¼ si0 at all time (see Eq. (3)). For both the calibration and the validation, experimental results of mono-crystalline pillar compressions are chosen as reference [5]. These pillars are made of high purity aluminum (5N purity, ESPI Metals), annealed under vacuum at 350 °C overnight and electropolished by Able Electropolishing. Grain orientation maps were obtained by automatic indexing of electron backscatter diffraction (EBSD) patterns using a Zeiss1550 VP field emission scanning electron microscope equipped with an EBSD system from Oxford Instruments. Once a suitable grain was chosen, nanopillar samples were fabricated using a FEI Nova 200 scanning electron microscope with a focused ion beam (FIB), see Ref. [5] for more details. For each considered nano-pillar, the top diameter and tapering angle were extracted by means of a scanning electron microscopy (SEM). The height being the most ambiguous geometric parameter due to FIB undercutting, it was determined by a stiffness-matching method [25]. The average diameter calculated along with this height was also calculated. The geometrical parameters of the two chosen pillars, as well as their finite element characteristics are gathered in Table 1. Pillar A (see Table 1) was chosen for calibration. The elastic constants for aluminum at room temperature, the Burgers vector, the initial dislocation density for nanopillar (approximating its value for aluminum by the one for gold), and a were taken from the literature [20,26,27]. The two remaining parameters s0,nucl and were then independently adjusted to fit the experistarv p mental stress strain curve of the nanopillar indentation. Note that, by definition, the two calibrated parameters modify the final strength and the plastic strain at which this final strength is reached; as a consequence, their unicity is guaranteed. The simulations were done using Abaqus/Explicit [28] under quasi-static loading, clamping the base of the nanopillar, while applying a free slip boundary condition on the top surface. Both temporal and spatial convergence were checked. The parameters are given in Table 2, the stress–strain curve in Figure 1, and the resulting Von Mises stress field in Figure 2. As previously observed in other efforts [5,13], the experimental stress–strain curve appears to be abnormally compliant during the initial part of loading. This
A. Je´rusalem et al. / Scripta Materialia 66 (2012) 93–96
95
Table 1. Nanopillar finite element characteristics. Pillar
Top diameter
Tapering angle
Height
Average diameter
Euler angles
Schmid factor
Number of linear tetrahedra
A B
622.9 nm 655 nm
88.1° 88.7°
2.35 lm 2.15 lm
696 nm 700 nm
19.5°, 18.4°, 28.1° 269.1°, 15.5°, 85.5°
0.4879 0.4803
41,840 37,683
Table 2. Model parameters.
107.3 Gpa
C44
60.8 Gpa
l
28.3 Gpa
m
26 Gpa
0.35
b 2.8 10
350 300
Stress (MPa)
250 200 Experimental Simulation − reference Simulation − constrained top Simulation − with substrate
150 100 50 0
0
0.02
0.04
0.06 0.08 Strain
0.1
q0 10
0.12
Figure 1. Stress–strain curves of pillar A compression as extracted from (a) experiment, (b) simulation, (c) simulation with constrained top and (d) simulation with substrate.
Figure 2. (a) SEM image of the compressed nanopillar A, and Von Mises field (Pa) for a longitudinal cut of the simulation: (b) reference, (c) with constrained top and (d) with substrate.
apparent discrepancy with the simulation can be rationalized by the misalignment between the indenter flat punch tip and the pillar top (absent in the simulation), thus affecting the test accuracy and leading to an under-estimation of the elastic modulus [13]. Experimental and numerical studies have suggested that the effect of the nanoindenter lateral stiffness could lead to important changes in the deformation mechanism [3,15]; for instance, the stochastic behavior of bursts could be significantly decreased and/or a significant fictitious hardening (independent of the constitutive behavior) could be observed. Figures 1 and 2 also show the results for the same reference simulation but with the top surface fully constrained laterally. In agreement with Ref. [15], the stress–strain curve presents a significantly more important hardening and the pillar is more constrained, albeit with comparable maximum Von Mises stress values. As the lateral stiffness of nanoindenters is generally estimated at best, one could assume, as can be seen on Figure 1, that the simulated curves for an indenter with realistic lateral stiffness should lie in between these two curves, with and without lateral clamping. The influence of substrate was studied by additionally supporting the pillar with a thick layer of material; see
m
2
12
5 10 m
a
s0,nucl
starv p
1
160 MPa
5% (Pillar A only)
the results on Figures 1 and 2. These investigations showed a minimal variation in the elastic slope and starvation region, and more pronounced hardening in the nucleation, but less important than the ones arising from constraining the top surface. It should also be emphasized that this simulation assumed the same starvation behavior in the substrate as in the crystal; the substrate should theoretically have an even higher starvation CRSS, thus a priori alleviating the overall discrepancy. The curve given here consequently corresponds to the case of maximum discrepancy. Additionally, Figure 2 exhibits very similar Von Mises stress fields, thus confirming the relatively good approximation done by neglecting the substrate. The phenomenological model proposed in Ref. [20] is used alternatively here to study the evolution of dislocation density and flow stress assuming that no dislocations are nucleated. Noting that the deformation is mostly plastic ( ’ p), the remaining parameter, i.e. the breeding coefficient d, is calibrated so as to obtain . This condition is reached for q = 0 for p ¼ starv p d = 1.46 106 m1. Note that this value is slightly larger than the one used for gold in Ref. [20] and a factor 4 larger than the one given for aluminum in Ref. [29]. The dislocation density and the associated flow stress evolutions are given in Figure 3. As can be seen, an increase of the dislocation density is observed up to 15% deformation, as the gliding dislocations encounter (“breed”) others, pushing them out of the crystal. Eventually, these obstacles are overcome and dislocation starvation takes over the deformation process. This succession leads to a relatively stable flow stress at 20 MPa. The assumption of the model of linear transition between starvation and nucleation CRSS (see Eq. (3)), seems to be confirmed as a reasonable representation of a change from a constant starvation CRSS to a constant nucleation CRSS, where local 12
7
x 10
30
6 5 20 4 3 10
2
Flow stress (MPa)
C12
Dislocation density (m−2)
C11
1 0
0
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Strain
Figure 3. Dislocation density evolution (full line) and associated flow stress (dashed line) as calculated by the phenomenological model.
A. Je´rusalem et al. / Scripta Materialia 66 (2012) 93–96
96
Figure 4. Stress–strain curves of pillar B compression as extracted from experiment and simulations (“lower bound” being with free top surface and “upper bound” with laterally constrained top surface).
The Abaqus subroutine VUMAT used here is a modified version of the one used for HCP metals in Ref. [31], itself an extension of the one originally written (and kindly provided) by Dr. Alexander Staroselsky and Prof. Lallit Anand at MIT [24]. A.J. acknowledges support from the Juan de la Cierva grant from the Spanish Ministry of Science and Innovation, from the Amarout grant from the European Union and from the ESTRUMAT-S2009/MAT-1585 grant (Madrid Regional Government). A.F. acknowledges funding from the Caltech SURF program. J.R.G. gratefully acknowledges the financial support of the National Science Foundation through the CAREER award (DMR-0748267) and Office of Naval Research (Grant No. N000140910883).
obstacles such as Lomer–Cotrell locks leading to the creation of single-arm sources eventually lead to new dislocation nucleations [30]. The breeding coefficient can now be used directly to calculate the starvation strain of other pillars by solving ¼ 5:8%. q = 0 in Eq. (2). For pillar B (see Table 1), starv p The model is finally validated against the experimental results of this new pillar by using the same set of calibrated parameters (Table 2) but with this new starvation strain, see Figure 4. Here, again, the level of strength and the changes of curvature of the stress–strain curves are very well captured, thus validating the model. Whereas the initial starvation CRSS and starvation strain are directly defined by means of Eqs. (4) and (2), respectively, the nucleation CRSS remains a free parameter. In other words, for a different crystal size, this parameter either needs to be calibrated or evaluated from other non-local methods accounting for example for surface energy, see Ref. [17]. With an increase of diameter, the initial starvation CRSS and the starvation strain, respectively decreases and increases. However, the subsequent hardening after initial starvation yield is expected to be much more important, reaching the nucleation CRSS (a priori much smaller at larger diameters) at very small strains [20]. In other words, for large crystals, full starvation is never reached, as nucleation becomes more favorable earlier; the starvation strain and the strain at which nucleation occurs are different. The model can then be adapted . by considering the minimum of both for starv p Note finally that the number of initial dislocations in the nano-pillars considered here is of the order of 10, meaning that the prominent burst behavior observed at such scales needs to be considered in future work. To this end, their incorporation in a nanoscale continuum model will require taking into account the stochastic nature of this phenomenon. In this letter, we proposed a crystal plasticity continuum model accounting for dislocation starvation and nucleation, based on the phenomenological model of Greer et al. [19,20]. The model has been found to capture very closely the smoother transition to plasticity observed in nanopillar compressions, characterized by two plastic zones representative of dislocation starvation and nucleation, respectively. Remarkably, only one free parameter remains and still needs to be characterized by other non-local means.
[1] L. Valdevit, A.J. Jacobsen, J.R. Greer, W.B. Carter, J. Am. Ceram. Soc. 94 (2011) 15–34. [2] G. Dehm, Prog. Mater. Sci. 54 (2009) 664–688. [3] M. Uchic, P. Shade, D. Dimiduk, Annu. Rev. Mater. Res. 39 (2009) 361–386. [4] O. Kraft, P. Gruber, R. Monig, D. Weygand, Annu. Rev. Mater. Res. 40 (2010) 293–317. [5] A. Kunz, S. Pathak, J. Greer, Acta Mater. 59 (2011) 4416–4424. [6] J. Greer, J. De Hosson, Prog. Mater. Sci. 56 (2011) 654–724. [7] L. Zepeda-Ruiz, B. Sadigh, J. Biener, A. Hodge, A. Hamza, Appl. Phys. Lett. 91 (2007) 101907. [8] J. Marian, J. Knap, Int. J. Multiscale Comput. Eng. 5 (2007) 287–294. [9] C. Weinberger, W. Cai, PNAS 105 (2008). [10] C. Weinberger, A. Jennings, K. Kang, J. Greer, J. Mech. Phys. Solids, in press, doi:10.1016/j.jmps.2011.09.010. [11] H. Zhang, B. Schuster, Q. Wei, K. Ramesh, Scripta Mater. 54 (2006) 181–186. [12] D. Raabe, D. Ma, F. Roters, Acta Mater. 55 (2007) 4567– 4583. [13] B. Schuster, Q. Wei, T. Hufnagel, K. Ramesh, Acta Mater. 56 (2008) 5091–5100. [14] X. Wu, Y. Guo, Q. Wei, W. Wang, Acta Mater. 57 (2009) 3562–3571. [15] P. Shade, R. Wheeler, Y. Choi, M. Uchic, D. Dimiduk, H. Fraser, Acta Mater. 57 (2009) 4580–4587. [16] C. Chen, Y. Pei, J.D. Hosson, Acta Mater. 58 (2010) 189–200. [17] D. Hurtado, Multiscale Modeling of Microcrystalline Materials, Ph.D. Thesis, Caltech, 2011. [18] X. Zhang, K. Aifantis, Mater. Sci. Eng. A 528 (2011) 5036–5043. [19] J. Greer, W. Nix, Phys. Rev. B 73 (2006) 1–6. [20] J. Greer, Rev. Adv. Mater. Sci. 13 (2006) 59–70. [21] W. Johnston, J. Gilman, J. Appl. Phys. 30 (1959) 129–144. [22] J. Gilman, Micromechanics of Flow in Solids, McGrawHill, New York, NY, 1969. [23] W. Nix, S. Lee, Philos. Mag. 91 (2010) 1084–1096. [24] A. Staroselsky, Crystal Plasticity Due to Slip and Twinning, Ph.D. Thesis, Massachusetts Institute of Technology, 1998. [25] J. Greer, W. Oliver, W. Nix, Acta Mater. 53 (2005) 1821–1830. [26] J. Vallin, M. Mongy, K. Salama, O. Beckman, J. Appl. Phys. 35 (1964) 1825–1826. [27] S. Crandall, N. Dahl, T. Lardner, An Introduction to the Mechanics of Solids, McGraw-Hill, 1999. [28] Dassault Syste`mes, Abaqus/Explicit
350 300
Stress (MPa)
250 200
Experimental Simulation − lower bound Simulation − upper bound
150 100 50 0
0
0.02
0.04
0.06 0.08 Strain
0.1
0.12