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Interpreting strain bursts and size effects in micropillars using gradient plasticity
Materials Science and Engineering A 528 (2011) 5036–5043
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Interpreting strain bursts and size effects in micropillars using gradient plasticity X. Zhang a,b , K.E. Aifantis b,c,∗ a
School of Civil Engineering and Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China Lab of Mechanics and Materials, Aristotle University of Thessaloniki, Greece c Physics, Michigan Technological University, USA b
a r t i c l e
i n f o
Article history: Received 10 December 2010 Received in revised form 15 February 2011 Accepted 16 February 2011 Available online 22 February 2011 Keywords: Micropillar compression Strain gradient Size effects Strain bursts
a b s t r a c t Size effects and strain bursts that are observed in compression experiments of single crystalline micropillars are interpreted using a gradient plasticity model that can capture the process of sequential slip and heterogeneous yielding of thin material layers. According to in situ experiments during compression subgrains and significant strain gradients develop, while deformation occurs through slip layers in the gauge region. In the multilayer strain gradient model, the higher order stress is discontinuous across the interface between a plastic layer and an elastic layer, but it becomes continuous across the interface between two plastic layers. Strain bursts occur when two neighboring layers yield. Based on this hypothesis the experimental stress–strain curves with strain bursts observed in micropillars can be fitted by properly selecting the number of layers that yield and the ratio of the internal length over the specimen size; the modulus and the yield stress are obtained from the experimental curves while the hardening modulus evolves during deformation based on the dislocation mechanisms. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Numerous studies have shown that the plastic deformation at the micron and sub-micron scales is dramatically different from that of the bulk material. Corresponding size effects that have been observed in torsion [1], indentation [2], and bending [3] are attributed to the presence of strain gradients developed during microscopically heterogeneous plastic deformation. In [4–6] it has been shown how this original gradient plasticity model [7] can conveniently describe such effects through the extra, internal length and specimen size dependent terms entering the relevant “load-deformation” relations. Conventional theories are incapable to explain size effects due to the lack of a length scale in the constitutive relations and the absence of spatial gradients and/or volume integrals accounting for non-local interactions. Size effects were not originally expected in tension/compression experiments as no macroscopic strain gradients were present, they have been experimentally observed, however, and gradient plasticity [4–6] was successfully used to interpret them by introducing “microscopic” rather than “macroscopic” strain gradients, which are obviously absent from tension/compression configurations. Recently, strong and rather peculiar size effects have also been observed in compression experiments of micro- and nano-pillars
∗ Corresponding author at: Aristotle University of Thessaloniki, Lab of Mechanics and Materials, Box 468, GR 54124 Thessaloniki, Greece. Fax: +30 2310 995921. E-mail address: [email protected] (K.E. Aifantis). 0921-5093/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2011.02.049
fabricated by the focused ion beam (FIB) method, where the flow stress increases when the diameter decreases [8–16]. Several researchers attributed the size effect to the FIB-induced precipitate strengthening and surface amorphization [15,16], however, pillars fabricated without the FIB method [17] also give the same size effect as the FIB-fabricated FCC micropillars, demonstrating that the size effects are not related to the fabrication method but to the underlying microstructure. In contrast to the smooth macroscopic stress–strain curve obtained during bulk compression, the stress–strain curves of micropillars contain several discrete slip bursts where the stress remains almost constant, while the strain “jumps” discontinuously to increasing values. The situation is reminiscent to “displacement bursts” observed during nanoindentation tests near grain boundaries; a phenomenon which was also interpreted by using gradient plasticity with an interfacial energy [18]. In [8] it is argued that such kind of micropillar related deformation phenomena occur in the stochastic multiplication-limited exhaustion regime, and both size effects and strain bursts were attributed to the truncation of dislocation sources and varying dislocation mechanisms at small volumes; for example it was argued that dislocations at such volumes are induced at large stresses and hence the dislocation velocity is very high and they accumulate at the pillar surface. But experimental evidence documenting these interpretations for micropillars is lacking. Some researchers have argued that the uniaxial compression behavior of micropillars can be attributed to the damage layer during the FIB method, but the experiments of [9] and the TEM
X. Zhang, K.E. Aifantis / Materials Science and Engineering A 528 (2011) 5036–5043
observations for nanosized pillars [14] indicate that the thin outer surface is not strong enough to impede the dislocation motion and has little effect on strength. Particularly, for nanosized pillars, according to [14] the bursts arise because the dislocations generated are completely squeezed out of the deforming crystal. Hence, the most accepted explanation of strain bursts is attributed to the occurrence of a change in the deformation mechanism in the micropillar specimens due to the limited space for dislocation activity. Based on this experimental evidence, the works of [9,19], introduced the so called “dislocation starvation” and “re-nucleation” mechanism to explain the observed hardening phenomena, and they developed a numerical iterative algorithm to calculate the evolution of the dislocation density and the corresponding stresses and strains step by step at a constant strain rate. The stress–strain curve generated by this procedure can predict the strain burst phenomenon by allowing initial elastic deformation followed by a stage of declined stress and then elastic loading again, therefore, only one strain burst could be captured in [19], as opposed to the multiple bursts that are observed experimentally [8]. Furthermore, in [13] the jerky and stochastic deformation nature of the micropillars was predicted using a Monte Carlo approach; it was possible to obtain lower and upper bounds of the stress–strain curve that enveloped the dispersed experimental curves. More detailed recent experiments, however, by Mass and co-workers [20] have documented through synchrotron microdiffraction on micropillars that significant dislocation activity takes place inside the pillars, forming dislocation structures during deformation. Significant geometrically necessary dislocations are generated during compression giving rise to strain gradients [20]. The assumed mechanism of dislocation starvation is therefore not the case for micropillars, and it is only within gradient plasticity, that their stress-strain response can be understood. Furthermore, in [21] it is argued that the deformation is macroscopically homogeneous, but microscopically heterogeneous, and although the geometrically necessary dislocation/GND density vanishes macroscopically, it is present locally. The simulations of [22] also show that the deformation is heterogeneous from the onset of deformation and is confined to the discrete slip bands. The SEM images taken after deforming the pillars [8,13] suggest that although the deformation is applied homogeneously it develops in an inhomogeneous manner, since the end regions of the pillars appear non-deformed, while the center of the sample is defined by multiple plastic slip zones. In the present paper, therefore, a different point of view and interpretation for micropillar deformation is proposed, by focusing on the effect of “microscopic strain gradients” that develop as a result of the heterogeneous deformation that occurs in the observed “slip” layers adjacent to the “non-slip” or elastic regions that the specimen is divided to during compression. In order to explicitly capture the strain gradient effects, the strain gradient needs to be considered as an independent variable within a strain gradient plasticity model. The third order stress that arises in gradient theories accounts for the significant changes in the internal stresses as the plastic layers yield. In this connection it should be noted that this criterion for the occurance of a strain burst is contradictory to that of [14], which suggests that bursts occur when all dislocations are “squeezed” from the pillar surface. However, it should be noted that the experimental evidence of [14] is for nanosized pillars, whereas the present study models deformation for micro scale pillars, in which it is impossible to squeeze all dislocations out of the pillar, yet strain bursts are still observed, due to the localization of GNDs as documented in [20]. In Section 2 the basic structure of gradient plasticity for a multilayer specimen is outlined. In Section 3 a simplified 1D case is considered to describe deformation of compressed micropillars
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which yield by slip zones that are activated at different stress levels. When the adjacent slip zones are activated sequentially, a corresponding strain burst occurs in the stress–strain plot; it is therefore possible for this model to predict strain bursts observed experimentally. In particular, the gradient plasticity model is shown to be in agreement with the experimental stress–strain curves obtained for 1 m, 2.4 m and 5.2 m diameter micropillars [8]. This is the first continuum mechanics model to predict the rather complex deformation behavior of micropillars with a minimal computational effort. 2. Gradient plasticity model The gradient plasticity theory used in the following is similar to that in [23–25]. In three dimensions, the basic constitutive assumptions for the stress ij , the internal stress sij and the 3rd order stress ijk are ∂V
p
ij = Lijkl (εkl − εkl ), sij = −ij +
p
∂εij
, ijk =
∂V p
∂εij,k
,
(1) p
where Lijkl is the stiffness tensor, εij is the total strain, εij is the plastic strain and V a plastic potential. The corresponding field equations are ij,j = 0;
sij = ijk,k
(2)
which are solved along with the external boundary conditions on the outer surface ij nj = ti0
or ui = u¯ i
and
ijk nk = m0ij
p
p
εij = ε¯ ij
or
(3)
and the internal boundary conditions across the interface of adjacent layers
冀ij nj 冁 = 0 or ui = u¯ i and 冀ijk nk 冁 = 0 or εpij = ε¯ pij
(4)
Here the stress ij and higher order stress ijk are assumed to be continuous across internal boundaries when the two adjacent layers both yield. But this continuity of ij and ijk does not imply that their conjugate variables are also continuous across the interface. p In fact, plastic strain gradient εij,k is discontinuous across such an internal boundary due to the different properties of the neighboring layers (e.g. elastic stiffness, hardening modulus, etc.) after yielding. In accordance with the above, the present multilayer strain gradient plasticity scheme assumes that the upper and lower portions of the pillar specimens deform elastically, whereas severe plastic deformation occurs in the gauge region through sequential yielding of adjacent discrete slip zones. The pillar is therefore divided into elastic and plastic thin layers that are characterized by different elastic moduli, yield stresses, hardening moduli and internal lengths, which is reminiscent to the sub-grain microstructure observed in the micro-pillars of [20]. In particular, yielding is governed by a one-dimensional gradient-dependent plastic potential of the form [26] p
p
ys
Vi (εi , εi,x ) = i
p 1 p 2 2 p 2 ε + ˇi (ε ) + (ε ) , i i i i,x
(5)
2
ys
where the subscript i denotes the ith layer, i is the yield stress, ˇi is the hardening modulus (accounting for statistically stored dislocations/SSDs) and i is an internal length (accounting for geometrically necessary dislocation/GNDs). With this definition of the plastic potential, Eq. (1) becomes p
i = Ei (εi − εi ),
ys
p
si = −i + i + ˇi εi ,
p
i = ˇi 2i εi,x .
(6)
which upon combination with the one-dimensional counterpart of Eq. (2), defines the plastic strain, displacement and higher-order
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X. Zhang, K.E. Aifantis / Materials Science and Engineering A 528 (2011) 5036–5043
stress as p d2 εi dx2
−
p εi 2i
+
ys ¯ − i ˇi 2i
p
= 0 ⇒ εi =
¯
ys − i
ˇi
+ Ai ex/i + Bi e−x/i ,
(7)
ys
¯ − i dui ¯ ¯ p − εi − = 0 ⇒ ui = x+ x Ei Ei dx ˇi + i Ai ex/i − i Bi e−x/i + Ci .
= ˇi 2i εi,x = ˇi i (Ai ex/i − Bi e−x/i ). p
(8)
(9)
where ¯ is the applied stress and the constants (Ai , Bi , Ci ) are to be determined by the corresponding boundary condition [for elastically deforming layers it easily turns out that the displacement is given by the first and last term of Eq. (8)]. 3. Single crystalline micropillar compression As mentioned in the introduction the stress–strain response of micropillars during compression is dominated by size effects and strain bursts [8,9,27]. The slip zones observed in the SEM images and the corresponding serrations in the stress–strain curves, suggests that there is a relation between the strain burst and the dispersed shear layers in the deformed pillars. Particularly, in situ compression experiments of metallic micro-pillars [20] show that not only GNDs, but also a sub-grain structure forms in single crystal specimen, while for metallic glass pillars [28] it was illustrated that the jerky events in the load-displacement curves are associated with the initiation and propagation of the shear bands resulting in the jumps in the compression response. In modeling, therefore, the experimental data of [8], the following qualitative response is considered: (i) the stress–strain curve is characterized by initial elastic deformation, followed by the usual “knee” indicating yielding of the material; (ii) with continuous loading, however, individual slip zones are activated and discrete strain bursts occur. After each strain burst the micropillar undergoes work hardening until the next slip layer is activated and another strain burst occurs. Based on the experimental images of [8], it can be assumed that the upper most and lower most regions of the pillars do not deform plastically, whereas severe deformation, through slip, accumulates within the gauge region. The pillar is therefore divided into elastic and plastic zones. Within the gradient model, each slip zone is identified with a thin layer that is characterized by a different yield stress, elastic modulus and hardening modulus from the adjacent layers. The boundary value problem of micropillar compression can be solved in the 1D version of the model presented in Section 2. Therefore, there will be just one relevant component of the stress, strain, and displacement and it is appropriate to drop all suffixes, and to let x denote the coordinate in which there is variation. It should be noted that the strain gradient model presented in section 2 does not account for slip systems, and the one dimension simplification eliminates the possibility to consider anisotropy. The combination of the multilayer model proposed here with a strain gradient single crystal plasticity model [29], so as to capture both the anisotropic character and strain bursts in deformed micro-pillars is a future task. 3.1. Modeling stress–strain curves with one burst As documented in [8], slip bands are contained within the uniform gauge section of the pillar. The plastic deformation is heterogeneous within this region [20], and as the dislocation sources are randomly distributed [17], each “slip layer” is activated at a different stress level. The strain gradients that develop within the
slip layers dictate the strain bursts, since according to the theoretical model when the higher order stress, , is continuous across the slip layers (i.e. they both deform plastically) a strain burst occurs. Hence, the consecutive yielding of two layers gives one burst. The simplest possible case is to model a micropillar stress–strain curve which exhibits only one strain burst. In this case the pillar is divided into four layers, as shown in Fig. 1. Based on the SEM images [4] the upper and lower layers (1 and 4) are always elastic, whereas the plastically deformed region is assumed to consist of two “slip layers”. Here we assume, without loss of generality, that the yield stress of layer 2 is smaller than that of layer 3 ys ys (2 < 3 ).] A strain burst occurs when slip layer 3 yields after slip layer 2, and this is associated with the continuity of the 3rd order stress across the layer boundary. Before layer 3 yields, a higher order stress component exists in layer 2, while there is no such 3rd order stress in layer 3, thus, 2 (L1 + L2 ) = / 3 (L1 + L2 ). After yielding of layer 3, however, the 3rd order stress is required to be continuous across the boundary between the two adjacent plastic layers as dictated by the internal boundary conditions of Eq. (4); hence, 2 (L1 + L2 ) = 3 (L1 + L2 ). More details on a similar mathematical analysis for such type of strain burst behavior can be found in [25]. Initially (stage 1), all layers deform elastically with the same elastic modulus (E) and, therefore, the portion O–A of the stress–strain curve of Fig. 1 is given by ¯ = E ε¯ .
(10) ys 2 ,
layer 2 begins to yield and the At a critical stress, ¯ = above linear relationship gives the strain where stage 1 ends, ys ys i.e. ε¯ 2 = 2 /E. After layer 2 yields, the plastic strain and displacement expressions for layer 2 are given by Eqs. (7) and (8), while the other layers still deform elastically. The constants (A2 , B2 ) are determined from the plastic strain continuity conditions, p p i.e. ε2 (L1 ) = 0, ε2 (L1 + L2 ) = 0, while the constants (C1 -C4 ) are determined by assuming the following continuity and boundary conditions for the displacement field u: u1 (0) = 0, u1 (L1 ) = u2 (L1 ), u2 (L1 + L2 ) = u3 (L1 + L2 ), u3 (L1 + L2 + L3 ) = u4 (L1 + L2 + L3 ). Furthermore, the displacement of the top layer is related to the average strain as u4 (L) = ε¯ L. Hence solving this expression for ¯ gives the overall stress–strain relationship in stage 2 (segment AB of Fig. 1) as ys
ys
¯ = 2 + H2 (¯ε − ε¯ 2 ),
(11)
ys
ys
ys
with ε¯ 2 denoting the strain at which layer 2 yields (¯ε2 = 2 /E), ¯ ε¯ ) during and the quantity H2 being the strain hardening rate (d/d stage 2
H2 =
1 22 f2 1 − + tan h E ˇ2 ˇ2 L
∗2 2
−1 ,
where f2 = L2 /L and ∗2 = L2 /2 . [The notation fi = Li /L and ∗i = Li /i is adopted in the sequel for the ith layer.] It follows from the above discussion that the strain at which stage 1 ends is the same as the strain where stage 2 begins, therefore, the stress–strain curve from stage 1 to stage 2 is continuous, as shown in Fig. 1. Stage 2 ends when the stress triggers yielding ys of layer 3. The strain ε¯ 3 at the end of stage 2 can then be computed through the stress–strain relationship during stage 2, i.e. Eq. ys ys (11), by setting ¯ = 3 with 3 denoting the stress at which stage 3 begins. The result is ys ε¯ 3
ys
=
2 E
ys + (3
ys − 2 )
1 22 f2 1 − + tan h E ˇ2 ˇ2 L
∗2 2
.
(12)
X. Zhang, K.E. Aifantis / Materials Science and Engineering A 528 (2011) 5036–5043
5039
Fig. 1. Configuration used to fit stress–strain curve of micropillar with one strain burst.
During continuous deformation layer 3 also begins to yield at a ys critical stress = 3 . Now there are two plastic layers in the middle of the sample and two elastic zones at the ends. The plastic strain and displacement expressions for layers 2 and 3 are given by Eqs. (7) and (8), while layers 1 and 4 remain elastic. The constants (A2 , B2 , A3 , B3 ) are determined by allowing the plastic strain to be continuous throughout the whole pillar (εp = 0 on the interface between elastic and plastic layers) and by also ensuring continuity of both εp and across the boundary of the two plastically deforming slip layers 2 and 3 according to Eq. (4). p p p In other words, ε2 (L1 ) = 0, ε3 (L1 + L2 + L3 ) = 0, ε2 (L1 + L2 ) = p ε3 (L1 + L2 ), 2 (L1 + L2 ) = 3 (L1 + L2 ).The other four constants of integration (C1 –C4 ) are still determined by the displacement boundary and continuity conditions as before, and the overall stress–strain relationship (portion CD in Fig. 1) can be obtained through the boundary condition at u4 (L) as ys
ys
¯ = 3 + H3 (¯ε − ε¯ 3 ),
(13)
ys
ys
where ε¯ 3 is the strain corresponding to the yield stress, 3 , of layer 3 and H3 is the corresponding strain hardening rate
(d/d ¯ ε¯ ) given by H3 = E3 + P3 + G 3
−1
, where E3 = 1/E and
P3
= (f2 /ˇ2 + f3 /ˇ3 ) are the elastic and plastic contributions to is an hardening, whereas the strain gradient contribution G 3 explicit function of (ˇ’s, L’s, ’s and * ’ s) and is defined by
G3
∗ ∗ ∗ 2 ˇ2 2 + 3 (2ˇ3 − ˇ2 ) cot h(3 ) + ˇ2 csc h(3 ) tan h(2 /2) = ∗ ∗ ˇ2 2 cot h(2 ) + ˇ3 3 cot h(3 ) ˇ2 L
+
∗ ∗ ∗ 3 ˇ3 3 + 2 (2ˇ2 − ˇ3 ) cot h(2 ) + ˇ3 csc h(2 ) tan h(3 /2) . ˇ3 L ˇ2 2 cot h(∗2 ) + ˇ3 3 cot h(∗3 )
be analytically determined as
2 sin h(∗2 /2) cos h(∗3 /2) + 3 sin h(∗3 /2) cos h(∗2 /2)
ε3 =
L ˇ2 2 cos h(∗2 ) sin h(∗3 ) + ˇ3 3 cos h(∗3 ) sin h(∗2 )
×42 tan h
∗2 2
sin h
∗2 2
sin h
∗3 2
ys
ys
(3 − 2 ). (15)
The stress–strain curve depicted with the light green curve for the 1.0 m diameter pillar in Fig. 3(d) of [8] exhibited one strain burst, as the aforementioned 4-layer model predicts. This experimental curve (reproduced here with dots in Fig. 2) can, therefore, be divided into four segments similar to those of Fig. 1, for which the various stress–strain portions have been constructed. Since the stress–strain curve in [8] corresponds to shear deformation, the shear modulus, G, instead of the elastic modulus, E, should be used. The slope of the initial elastic region, however, in Fig. 3(d) of [8] is 6.1 GPa; hence it will be assumed that E = 6.1 GPa. The yield stress of the layys ys ys ers (2 , 3 ) are taken, according to the experiment, as 2 = ys 143 MPa and 3 = 175 MPa (Fig. 2). As the height-to-diameter ratio of the micropillars ranged from 2:1 to 3:1 in the experiments of [8], this ratio will be taken here as 2.5:1, L = 2.5 m. For simplicity, the pillar is divided into 4 equal-length layers L1 = L2 = L3 = L4 = 0.625 m, and the internal lengths of the plastically deforming layers are also set equal and smaller than Li : 2 = 3 = 0.5 m. The only remaining parameters to be determined are the hardening moduli ˇ2 and ˇ3 , the values of which may vary throughout the deformation process for each layer, according to the pertinent dislocation mechanisms that dominate at each stage. When the
ys
It then turns out that the strain, ε¯ 3 , at which stage 3 begins can be expressed by the following straight forward relation
ys
=
2 E
ys + (3
ys − 2 )
+
f2 1 1 22 + tan h − E ˇ2 ˇ2 L
∗2
150
2
2 sin h(∗2 /2) cos h(∗3 /2) + 3 sin h(∗3 /2) cos h(∗2 /2)
L ˇ2 2 cos h(∗2 ) sin h(∗3 ) + ˇ3 3 cos h(∗3 ) sin h(∗2 )
×42 tan h
∗2 2
sin h
∗2 2
sin h
∗3 2
100
50
ys (3
Stress MPa
ys ε¯ 3
ys − 2 ).
(14) Comparison of Eq. (12) with Eq. (14) indicates that the strain where stage 2 ends is not the same as the strain where stage 3 begins. The length of the corresponding discontinuity/strain burst ys ys (ε3 = ε¯ 3 − ε¯ 3 ) that occurs upon yielding of layer 3 can then
0
0.00
0.01
0.02
Strain
0.03
0.04
0.05
Fig. 2. Fit of the analytical model (solid line), Eqs. (10), (11), (13) and (15) to the experimental (dots) stress–strain curve of a 1 m diameter micropillar under compression (light green curve of Fig. 3d in [8]).
5040
X. Zhang, K.E. Aifantis / Materials Science and Engineering A 528 (2011) 5036–5043
Fig. 3. Configuration used to predict the stress–strain curve of micropillars with two strain bursts.
3.2. Modeling stress–strain curves with two strain bursts In Fig. 2 it was illustrated that this gradient plasticity multilayer model can capture very well the response of the particular micropil-
100 80
Stress MPa
second layer yields, a “knee” occurs in the stress–strain curve, indicating work hardening and, therefore, ˇ2 controls the deformation in the AB portion of Fig. 1; fitting Eq. (11) to the experimental data of Fig. 2 gives ˇ2 = 0.138 GPa. In continuing with the portion BC, it is noted that we have already an expression for the strain burst length ys when the stress is kept constant at 3 . Thus, Eq. (15) is used to fit the strain burst portion BC (or ε3 ) giving the same ˇ2 as before and predicting that ˇ3 = 0.117 GPa. Upon insertion of these values in Eq. (14) the strain at which the strain burst ends is predicted as ys ε¯ 3 = 0.48, which is precisely the experimentally observed value in Fig. 2. Finally, Eq. (13) is used to fit the portion of the stress–strain curve that follows the strain burst, giving ˇ2 = ˇ3 = 5.966 GPa. It should be noted here that some previous theoretical models [9] assume that the stress–strain response between strain bursts is fully elastic. This, however, has not been experimentally documented since through compression tests it is not possible to obtain a perfect elastic response, as even the slope of the initial stress–strain curve gives a much different modulus than tensile tests. This is due to the contact between the sample and the indenter, which allows for slip and elasto-plastic response is observed from the initial deformation stage. In the present model it is seen that re-hardening after the burst (as observed in [20]) can be modelled quite effectively.
60 40 20 0 0.000
0.005
0.010
0.015
Strain
0.020
0.025
Fig. 5. Fit of Eqs. (10), (11), (13), (15), (18) and (23) to experimental stress–strain curve of a 2.4 m diameter Ni micropillar under compression (dark purple curve in Fig. 3(d) in [8]).
lar. In most cases in [8], however, more than one strain burst was observed. Such behavior can also be captured within this multilayer model by inserting another slip layer as illustrated in Fig. 3. Initially all the layers deform elastically, then the middle layers (2, 3, and 4) yield one by one in the sequence shown in Fig. 4, while layers 1 and 5 remain always elastic. The dotted curves in Figs. 5 and 6 depict the experimental data from [8] that will be used to verify the multilayer model with two strain bursts. Before the fourth layer yields, the boundary conditions, and hence stress–strain relations in the five layer configuration are the
ys
ys
ys
Fig. 4. Dark layers from left to right indicate the sequential slip in micropillars that exhibit two strain bursts in their stress–strain response; it is assumed that 2 < 3 < 4 .
X. Zhang, K.E. Aifantis / Materials Science and Engineering A 528 (2011) 5036–5043
p
(16)
In continuing, the plastic strain (Eq. (7)) and displacement (Eq. (8)) expressions in the layers need to be evaluated. The constants (C1 –C5 ) are found through the boundary and continuity conditions:
50
0
0.00
0.01
0.02
Finally, the stress–strain expression for segment EF can be found, through the boundary condition at the upper elastic layer u5 (L) = ε¯ L. Hence, ys 4
ys + H4 (¯ε − ε¯ 4 ),
(18)
ys
ys
where ε¯ 4 is the strain corresponding to the stress, 4 and H4 is the strain hardening rate (d/d ¯ ε¯ ) after layer 4 yields H4 =
1 E4 + P4 + G 4
p 4
,
(19)
G 4
and represent the work hardening contributions that come from the elastic strain, plastic strain and the plastic strain gradient, respectively, E4 =
1 , E
f2 + f3 + f4 , ˇ
p
4 =
G 4 =−
2 tan h((L2 + L3 + L4 )/2) . ˇL (20)
0.03
0.04
0.05
Strain Fig. 7. Fitting gradient plasticity model (solid line) to experimental data (dots) allows for interpreting both size effects and strain bursts. Experimental data are taken from Fig. 3d of [8].
+
u1 (0) = 0, u1 (L1 ) = u2 (L1 ), u2 (L1 + L2 ) = u3 (L1 + L2 ), u3 (L1 + L2 + L3 ) = u4 (L1 + L2 + L3 ), (17) u4 (L1 + L2 + L3 + L4 ) = u5 (L1 + L2 + L3 + L4 ).
E4 ,
2.4 μm
100
p
ε2 (L1 ) = 0, ε4 (L1 + L2 + L3 + L4 ) = 0, p p ε2 (L1 + L2 ) = ε3 (L1 + L2 ), 2 (L1 + L2 ) = 3 (L1 + L2 ). p p ε3 (L1 + L2 + L3 ) = ε4 (L1 + L2 + L3 ), 3 (L1 + L2 + L3 ) = 4 (L1 + L2 + L3 ).
¯ =
1 μm
150 Stress MPa
same as that of the former four layer configuration. Hence, all the expressions required to generate the experimental plot, up to point D, of Fig. 3 are known (replacing L4 with L4 + L5 ). A stress–strain relation, however, needs to be obtained for segment EF of Fig. 3, along with an analytical expression for the length of the second strain burst. If the hardening modulus is kept different in each layer only numerical solutions can be obtained, hence in order to obtain analytical expressions it must be assumed that ˇ2 = ˇ3 = ˇ4 = ˇ and 2 = 3 = 4 = . The plastic strain and higher-order stressexpressions are the same as in Section 3.2, since V is kept the same for layer 4 of Fig. 4. Hence, the constants of integration (A2 –A4 , B2 –B4 ) in Eqs. (7) and (8) once slip occurs in layers 2, 3 and 4 can be determined through the following continuity conditions for the plastic strain and higher-order stress
5041
2 sec h ˇL
+ cos h
L + L + L 2 3 4 2
L − L 2 4 2
L + L 2 3
−sin h
sin h
cos h
2
cos h
L 3
2
L 4
2
L + L 3 4
sin h
2
L 2
2
ys
3
ys
4
.
(21)
Now that the stress–strain portions OA, AB, BC, CD and EF of Fig. 3 have been analytically defined by Eqs. (10), (11), (13), (15) and (18), respectively, the length of the second strain burst (portion DE of Fig. 3) observed in micropillar deformation must be calculated. ys First, the strain, ε¯ 4 , at the end of portion CD must be computed. This is done by setting the stress–strain relationship Eq. (13), equal ys to the stress, 4 , at which layer 4 begins to yield and solve for ε¯ , ys
ε¯ 4
ys
=
4 E −
ys
ys
+
4 (f2 + f3 + f4 )
ˇL
ˇ ys
ys
(3 − 2 ) sec h
ys
ys
+(4 − 2 ) tan h
ys
ys
f2 2 + f3 3 + f4 4
−
ˇ
L + L 2 3
sin h
2
L + L 2 3 2
ys
L − L 2 3 2 ys
+ (4 − 3 ) tan h
L + L 2 3 2
ys
ε¯ 4
ys
=
4 E
ys
ys
+
4 (f2 + f3 + f4 ) ˇ
−
ys
ys
f2 2 + f3 3 + f4 4 ˇ
Subtracting Eq. (22) from Eq. (21) gives the length of the discontiys ys nuity/strain burst, ε4 = ε¯ 4 − ε¯ 4 , that occurs upon yielding of layer 4 as ys
Stress MPa
30
20
10
0
ys
ys
ys
L
ys
ys
2 (2 − 4 ) + (3 − 2 ) cos h + (4 − 3 ) cos h ε4 =
L L +L L +L +L ˇL cos h 22 3 cos h 2 23 4 csc h 4
40
0.00
0.01
0.02
0.03
Strain
0.04
0.05
Fig. 6. Fit of Eqs. (11), (13), (15), (18) and (23) to experimental stress–strain curve of a 5.2 m diameter Ni micropillar under compression (dark green curve in Fig. 3(c) in [8]).
.
(22)
ys
The strain where the second strain burst ends, ε¯ 4 , is expressed as
ys
2
L2 +L3 2
.
(23)
Following the same fitting procedure as in Section 3.2, it is possible to fit the stress–strain curves for the 2.4 m and 5.2 m diameter micropillars that exhibited two strain bursts; curves colored dark purple in Fig. 3(d) and dark green in Fig. 3(c) of [8]. It should be noted that in performing the fit, the layers were taken to have equal thickness L1 = L2 = L3 = L4 = L5 = L/5. The parameters used for the fits are listed in Tables 1 and 2, and the comparison between the theoretical prediction and the experimental data are shown in Figs. 5 and 6, respectively. In order to better observe the size effects that the experiments document and the theory captures, Figs. 2 and 5 are combined (since they are taken from the same figure of [8]) to produce Fig. 7. Fig. 7 clearly shows that the present theoretical model can account for the size effects observed for pillars of varying diameters. It should be noted that the values for most of the parameters reported in the tables, were obtained directly from the observation of the stress–strain curves.
5042
X. Zhang, K.E. Aifantis / Materials Science and Engineering A 528 (2011) 5036–5043
Table 1 Parameters used to generate stress–strain plot for 2.4 m diameter micropillar. Shear modulus G(GPa) 5.2
ˇ2 (GPa) ˇ3 (GPa) ˇ4 (GPa)
Yield stress (MPa) ys ys 3 2
4
72
95
81
ys
Layer thickness L1 = L2 = L3 = L4 = L5 (m)
Inner length scale 2 = 3 (m)
1.2
0.9
AB (Layer 2 yield)
BC (1st strain burst)
CD (Layer 3 yield)
DE (2nd strain burst)
EF (Layer 4 yield)
0.161
0.161 0.348
3.893 3.893
0.555 0.555 0.555
6.402 6.402 6.402
Table 2 Parameters used to generate stress–strain plot for 5.2 m diameter micropillar. Shear modulus G(GPa) 2.485
ˇ2 (GPa) ˇ3 (GPa) ˇ4 (GPa)
Yield stress (MPa) ys ys 2 3
4
28
39
34
ys
Layer thickness L1 = L2 = L3 = L4 = L5 (m)
Inner length scale 2 = 3 (m)
2.6
1.95
AB (Layer 2 yield)
BC (1st strain burst)
CD (Layer 3 yield)
DE (2nd strain burst)
EF (Layer 4 yield)
0.034
0.034 0.101
0.191 0.191
0.021 0.021 0.021
0.679 0.679 0.679
The yield stress for each layer differs according to the experimenys tal data, since i was taken directly from the experimental data (Fig. 3d of [8]) as being the stress at which the “knee” or strain bursts occurred in the stress–strain plot, whereas the modulus was the slope of the initial linear portion of the experimental curves. Hence, the hardening moduli and internal lengths were the only fit parameters. Although only up to two strain bursts were captured here, it is possible to capture more by increasing the number of slip layers; more slip layers in the gradient model correspond to more strain bursts. 4. Summary and conclusions In the present study strain gradients were used for the first time to capture the size effects and strain bursts that are experimentally observed in single crystal Ni micropillars under compression [8]. According to [20] significant strain gradients develop during compression and a sub-grain structure forms, while the gauge region is deformed through slip. In using, therefore, strain gradient plasticity the pillar is sectioned into equal length slip zones, each of which has its own yield stress. The zones are allowed to yield sequentially and a strain burst arises when two adjacent zones deform plastically, capturing therefore the experimentally observed strain bursts. The higher-order stress is the underlying mechanism for the strain burst, as it is discontinuous at the interface between a plastic layer and its neighboring elastic layer, but it becomes continuous when two neighboring layers begin deforming plastically. The transition from discontinuity to continuity of the higher-order stress across the internal boundary between two neighboring layers results in the strain burst. The analytical stress–strain expressions that arise from gradient plasticity were in precise agreement with the experimental stress–strain curves for 1 m, 2.4 m and 5.2 m diameter micron pillars that exhibited 1 or 2 strain bursts during deformation [8]. In addition to capturing the experimental data of [8] the present model was also able to provide upper and lower bounds [30] for the stress–strain curves observed in compression of Al micropillars [31]. It is possible to modify the gradient model so as to capture additional strain bursts as some experimental curves indicate, but then the solutions become numerical and it is beyond the scope of the present study, whose main purpose was to illus-
trate a computationally lean continuum mechanics framework that can predict the stress–strain curves during micropillar compression. In concluding it can be mentioned that stochasticity in the present model enters through uncertainty in the boundary conditions (fuzzy sets) and/or uncertainty on the yield stress. These are features to be taken into account later through a combined “deterministic gradient vs. random microstructure” model as advanced, for example, in [32]. Acknowledgement This work would not be possible without KEA’s European Research Council Starting Grant MINATRAN 211166. References [1] N.A. Fleck, G.M. Muller, M.F. Ashby, J.W. Hutchinson, Acta Metall. Mater. 42 (1994) 475–487. [2] W.D. Nix, H. Gao, J. Mech. Phys. Solids 46 (1998) 411–425. [3] J.S. Stölken, A.G. Evans, Acta Mater. 46 (1998) 5109–5115. [4] E.C. Aifantis, Int. J. Fracture 95 (1999) 299–314. [5] E.C. Aifantis, Mech. Mater. 35 (2003) 259–280. [6] A.A. Konstantinidis, E.C. Aifantis, J. Eng. Mater-t. Asme 124 (2002) 358–364. [7] E.C. Aifantis, J. Eng. Mater-t. Asme 106 (1984) 326–330. [8] D.M. Dimiduk, M.D. Uchic, T.A. Parthasarathy, Acta Mater. 53 (2005) 4065–4077. [9] J.R. Greer, W.D. Nix, Phys. Rev. B 73 (2006). [10] M.D. Uchic, D.M. Dimiduk, J.N. Florando, W.D. Nix, Science 305 (2004) 986–989. [11] J.R. Greer, W.C. Oliver, W.D. Nix, Acta Mater. 53 (2005) 1821–1830. [12] C.A. Volkert, E.T. Lilleodden, Philos. Mag. 86 (2006) 5567–5579. [13] K.S. Ng, A.H.W. Ngan, Acta Mater. 56 (2008) 1712–1720. [14] Z.W. Shan, R.K. Mishra, S.A. Syed Asif, O.L. Warren, A.M. Minor, Nat. Mater. 7 (2008) 115–119. [15] D. Kiener, C. Motz, M. Rester, M. Jenko, G. Dehm, Mater. Sci. Eng. A 459 (2007) 262–272. [16] H. Bei, S. Shim, M.K. Miller, G.M. Pharr, E.P. George, Appl. Phys. Lett. 91 (2007) 111915–111917. [17] A.T. Jennings, M.J. Burek, J.R. Greer, Phys. Rev. Lett. 104 (2010) 135503. [18] (a) K. Aifantis, A. Ngan, Mater. Sci. Eng. A 459 (2007) 251–261; (a) K. Aifantis, W. Soer, J. Dehosson, J. Willis, Acta Mater. 54 (2006) 5077–5085. [19] J.R. Greer, Rev. Adv. Mater. Sci. 13 (2006) 59–70. [20] (a) R. Maß, S. Van Petegem, C.N. Borca, H. Van Swygenhoven, Mater. Sci. Eng. A 524 (2009) 40–45; (b) R. Maß, S. Van Petegem, J. Zimmermann, C.N. Borca, H. Van Swygenhoven, Scripta Mater 59 (2008) 471–474; (c) R. Maaß, S. Van Petegem, H. Van Swygenhoven, P.M. Derlet, C.A. Volkert, D. Grolimund, Phys. Rev. Lett 99 (2007) 145505. [21] P.J. Guruprasad, A.A. Benzerga, J. Mech. Phys. Solids 56 (2008) 132–156.
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Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Interpreting strain bursts and size effects in micropillars using gradient plasticity X. Zhang a,b , K.E. Aifantis b,c,∗ a
School of Civil Engineering and Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China Lab of Mechanics and Materials, Aristotle University of Thessaloniki, Greece c Physics, Michigan Technological University, USA b
a r t i c l e
i n f o
Article history: Received 10 December 2010 Received in revised form 15 February 2011 Accepted 16 February 2011 Available online 22 February 2011 Keywords: Micropillar compression Strain gradient Size effects Strain bursts
a b s t r a c t Size effects and strain bursts that are observed in compression experiments of single crystalline micropillars are interpreted using a gradient plasticity model that can capture the process of sequential slip and heterogeneous yielding of thin material layers. According to in situ experiments during compression subgrains and significant strain gradients develop, while deformation occurs through slip layers in the gauge region. In the multilayer strain gradient model, the higher order stress is discontinuous across the interface between a plastic layer and an elastic layer, but it becomes continuous across the interface between two plastic layers. Strain bursts occur when two neighboring layers yield. Based on this hypothesis the experimental stress–strain curves with strain bursts observed in micropillars can be fitted by properly selecting the number of layers that yield and the ratio of the internal length over the specimen size; the modulus and the yield stress are obtained from the experimental curves while the hardening modulus evolves during deformation based on the dislocation mechanisms. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Numerous studies have shown that the plastic deformation at the micron and sub-micron scales is dramatically different from that of the bulk material. Corresponding size effects that have been observed in torsion [1], indentation [2], and bending [3] are attributed to the presence of strain gradients developed during microscopically heterogeneous plastic deformation. In [4–6] it has been shown how this original gradient plasticity model [7] can conveniently describe such effects through the extra, internal length and specimen size dependent terms entering the relevant “load-deformation” relations. Conventional theories are incapable to explain size effects due to the lack of a length scale in the constitutive relations and the absence of spatial gradients and/or volume integrals accounting for non-local interactions. Size effects were not originally expected in tension/compression experiments as no macroscopic strain gradients were present, they have been experimentally observed, however, and gradient plasticity [4–6] was successfully used to interpret them by introducing “microscopic” rather than “macroscopic” strain gradients, which are obviously absent from tension/compression configurations. Recently, strong and rather peculiar size effects have also been observed in compression experiments of micro- and nano-pillars
∗ Corresponding author at: Aristotle University of Thessaloniki, Lab of Mechanics and Materials, Box 468, GR 54124 Thessaloniki, Greece. Fax: +30 2310 995921. E-mail address: [email protected] (K.E. Aifantis). 0921-5093/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2011.02.049
fabricated by the focused ion beam (FIB) method, where the flow stress increases when the diameter decreases [8–16]. Several researchers attributed the size effect to the FIB-induced precipitate strengthening and surface amorphization [15,16], however, pillars fabricated without the FIB method [17] also give the same size effect as the FIB-fabricated FCC micropillars, demonstrating that the size effects are not related to the fabrication method but to the underlying microstructure. In contrast to the smooth macroscopic stress–strain curve obtained during bulk compression, the stress–strain curves of micropillars contain several discrete slip bursts where the stress remains almost constant, while the strain “jumps” discontinuously to increasing values. The situation is reminiscent to “displacement bursts” observed during nanoindentation tests near grain boundaries; a phenomenon which was also interpreted by using gradient plasticity with an interfacial energy [18]. In [8] it is argued that such kind of micropillar related deformation phenomena occur in the stochastic multiplication-limited exhaustion regime, and both size effects and strain bursts were attributed to the truncation of dislocation sources and varying dislocation mechanisms at small volumes; for example it was argued that dislocations at such volumes are induced at large stresses and hence the dislocation velocity is very high and they accumulate at the pillar surface. But experimental evidence documenting these interpretations for micropillars is lacking. Some researchers have argued that the uniaxial compression behavior of micropillars can be attributed to the damage layer during the FIB method, but the experiments of [9] and the TEM
X. Zhang, K.E. Aifantis / Materials Science and Engineering A 528 (2011) 5036–5043
observations for nanosized pillars [14] indicate that the thin outer surface is not strong enough to impede the dislocation motion and has little effect on strength. Particularly, for nanosized pillars, according to [14] the bursts arise because the dislocations generated are completely squeezed out of the deforming crystal. Hence, the most accepted explanation of strain bursts is attributed to the occurrence of a change in the deformation mechanism in the micropillar specimens due to the limited space for dislocation activity. Based on this experimental evidence, the works of [9,19], introduced the so called “dislocation starvation” and “re-nucleation” mechanism to explain the observed hardening phenomena, and they developed a numerical iterative algorithm to calculate the evolution of the dislocation density and the corresponding stresses and strains step by step at a constant strain rate. The stress–strain curve generated by this procedure can predict the strain burst phenomenon by allowing initial elastic deformation followed by a stage of declined stress and then elastic loading again, therefore, only one strain burst could be captured in [19], as opposed to the multiple bursts that are observed experimentally [8]. Furthermore, in [13] the jerky and stochastic deformation nature of the micropillars was predicted using a Monte Carlo approach; it was possible to obtain lower and upper bounds of the stress–strain curve that enveloped the dispersed experimental curves. More detailed recent experiments, however, by Mass and co-workers [20] have documented through synchrotron microdiffraction on micropillars that significant dislocation activity takes place inside the pillars, forming dislocation structures during deformation. Significant geometrically necessary dislocations are generated during compression giving rise to strain gradients [20]. The assumed mechanism of dislocation starvation is therefore not the case for micropillars, and it is only within gradient plasticity, that their stress-strain response can be understood. Furthermore, in [21] it is argued that the deformation is macroscopically homogeneous, but microscopically heterogeneous, and although the geometrically necessary dislocation/GND density vanishes macroscopically, it is present locally. The simulations of [22] also show that the deformation is heterogeneous from the onset of deformation and is confined to the discrete slip bands. The SEM images taken after deforming the pillars [8,13] suggest that although the deformation is applied homogeneously it develops in an inhomogeneous manner, since the end regions of the pillars appear non-deformed, while the center of the sample is defined by multiple plastic slip zones. In the present paper, therefore, a different point of view and interpretation for micropillar deformation is proposed, by focusing on the effect of “microscopic strain gradients” that develop as a result of the heterogeneous deformation that occurs in the observed “slip” layers adjacent to the “non-slip” or elastic regions that the specimen is divided to during compression. In order to explicitly capture the strain gradient effects, the strain gradient needs to be considered as an independent variable within a strain gradient plasticity model. The third order stress that arises in gradient theories accounts for the significant changes in the internal stresses as the plastic layers yield. In this connection it should be noted that this criterion for the occurance of a strain burst is contradictory to that of [14], which suggests that bursts occur when all dislocations are “squeezed” from the pillar surface. However, it should be noted that the experimental evidence of [14] is for nanosized pillars, whereas the present study models deformation for micro scale pillars, in which it is impossible to squeeze all dislocations out of the pillar, yet strain bursts are still observed, due to the localization of GNDs as documented in [20]. In Section 2 the basic structure of gradient plasticity for a multilayer specimen is outlined. In Section 3 a simplified 1D case is considered to describe deformation of compressed micropillars
5037
which yield by slip zones that are activated at different stress levels. When the adjacent slip zones are activated sequentially, a corresponding strain burst occurs in the stress–strain plot; it is therefore possible for this model to predict strain bursts observed experimentally. In particular, the gradient plasticity model is shown to be in agreement with the experimental stress–strain curves obtained for 1 m, 2.4 m and 5.2 m diameter micropillars [8]. This is the first continuum mechanics model to predict the rather complex deformation behavior of micropillars with a minimal computational effort. 2. Gradient plasticity model The gradient plasticity theory used in the following is similar to that in [23–25]. In three dimensions, the basic constitutive assumptions for the stress ij , the internal stress sij and the 3rd order stress ijk are ∂V
p
ij = Lijkl (εkl − εkl ), sij = −ij +
p
∂εij
, ijk =
∂V p
∂εij,k
,
(1) p
where Lijkl is the stiffness tensor, εij is the total strain, εij is the plastic strain and V a plastic potential. The corresponding field equations are ij,j = 0;
sij = ijk,k
(2)
which are solved along with the external boundary conditions on the outer surface ij nj = ti0
or ui = u¯ i
and
ijk nk = m0ij
p
p
εij = ε¯ ij
or
(3)
and the internal boundary conditions across the interface of adjacent layers
冀ij nj 冁 = 0 or ui = u¯ i and 冀ijk nk 冁 = 0 or εpij = ε¯ pij
(4)
Here the stress ij and higher order stress ijk are assumed to be continuous across internal boundaries when the two adjacent layers both yield. But this continuity of ij and ijk does not imply that their conjugate variables are also continuous across the interface. p In fact, plastic strain gradient εij,k is discontinuous across such an internal boundary due to the different properties of the neighboring layers (e.g. elastic stiffness, hardening modulus, etc.) after yielding. In accordance with the above, the present multilayer strain gradient plasticity scheme assumes that the upper and lower portions of the pillar specimens deform elastically, whereas severe plastic deformation occurs in the gauge region through sequential yielding of adjacent discrete slip zones. The pillar is therefore divided into elastic and plastic thin layers that are characterized by different elastic moduli, yield stresses, hardening moduli and internal lengths, which is reminiscent to the sub-grain microstructure observed in the micro-pillars of [20]. In particular, yielding is governed by a one-dimensional gradient-dependent plastic potential of the form [26] p
p
ys
Vi (εi , εi,x ) = i
p 1 p 2 2 p 2 ε + ˇi (ε ) + (ε ) , i i i i,x
(5)
2
ys
where the subscript i denotes the ith layer, i is the yield stress, ˇi is the hardening modulus (accounting for statistically stored dislocations/SSDs) and i is an internal length (accounting for geometrically necessary dislocation/GNDs). With this definition of the plastic potential, Eq. (1) becomes p
i = Ei (εi − εi ),
ys
p
si = −i + i + ˇi εi ,
p
i = ˇi 2i εi,x .
(6)
which upon combination with the one-dimensional counterpart of Eq. (2), defines the plastic strain, displacement and higher-order
5038
X. Zhang, K.E. Aifantis / Materials Science and Engineering A 528 (2011) 5036–5043
stress as p d2 εi dx2
−
p εi 2i
+
ys ¯ − i ˇi 2i
p
= 0 ⇒ εi =
¯
ys − i
ˇi
+ Ai ex/i + Bi e−x/i ,
(7)
ys
¯ − i dui ¯ ¯ p − εi − = 0 ⇒ ui = x+ x Ei Ei dx ˇi + i Ai ex/i − i Bi e−x/i + Ci .
= ˇi 2i εi,x = ˇi i (Ai ex/i − Bi e−x/i ). p
(8)
(9)
where ¯ is the applied stress and the constants (Ai , Bi , Ci ) are to be determined by the corresponding boundary condition [for elastically deforming layers it easily turns out that the displacement is given by the first and last term of Eq. (8)]. 3. Single crystalline micropillar compression As mentioned in the introduction the stress–strain response of micropillars during compression is dominated by size effects and strain bursts [8,9,27]. The slip zones observed in the SEM images and the corresponding serrations in the stress–strain curves, suggests that there is a relation between the strain burst and the dispersed shear layers in the deformed pillars. Particularly, in situ compression experiments of metallic micro-pillars [20] show that not only GNDs, but also a sub-grain structure forms in single crystal specimen, while for metallic glass pillars [28] it was illustrated that the jerky events in the load-displacement curves are associated with the initiation and propagation of the shear bands resulting in the jumps in the compression response. In modeling, therefore, the experimental data of [8], the following qualitative response is considered: (i) the stress–strain curve is characterized by initial elastic deformation, followed by the usual “knee” indicating yielding of the material; (ii) with continuous loading, however, individual slip zones are activated and discrete strain bursts occur. After each strain burst the micropillar undergoes work hardening until the next slip layer is activated and another strain burst occurs. Based on the experimental images of [8], it can be assumed that the upper most and lower most regions of the pillars do not deform plastically, whereas severe deformation, through slip, accumulates within the gauge region. The pillar is therefore divided into elastic and plastic zones. Within the gradient model, each slip zone is identified with a thin layer that is characterized by a different yield stress, elastic modulus and hardening modulus from the adjacent layers. The boundary value problem of micropillar compression can be solved in the 1D version of the model presented in Section 2. Therefore, there will be just one relevant component of the stress, strain, and displacement and it is appropriate to drop all suffixes, and to let x denote the coordinate in which there is variation. It should be noted that the strain gradient model presented in section 2 does not account for slip systems, and the one dimension simplification eliminates the possibility to consider anisotropy. The combination of the multilayer model proposed here with a strain gradient single crystal plasticity model [29], so as to capture both the anisotropic character and strain bursts in deformed micro-pillars is a future task. 3.1. Modeling stress–strain curves with one burst As documented in [8], slip bands are contained within the uniform gauge section of the pillar. The plastic deformation is heterogeneous within this region [20], and as the dislocation sources are randomly distributed [17], each “slip layer” is activated at a different stress level. The strain gradients that develop within the
slip layers dictate the strain bursts, since according to the theoretical model when the higher order stress, , is continuous across the slip layers (i.e. they both deform plastically) a strain burst occurs. Hence, the consecutive yielding of two layers gives one burst. The simplest possible case is to model a micropillar stress–strain curve which exhibits only one strain burst. In this case the pillar is divided into four layers, as shown in Fig. 1. Based on the SEM images [4] the upper and lower layers (1 and 4) are always elastic, whereas the plastically deformed region is assumed to consist of two “slip layers”. Here we assume, without loss of generality, that the yield stress of layer 2 is smaller than that of layer 3 ys ys (2 < 3 ).] A strain burst occurs when slip layer 3 yields after slip layer 2, and this is associated with the continuity of the 3rd order stress across the layer boundary. Before layer 3 yields, a higher order stress component exists in layer 2, while there is no such 3rd order stress in layer 3, thus, 2 (L1 + L2 ) = / 3 (L1 + L2 ). After yielding of layer 3, however, the 3rd order stress is required to be continuous across the boundary between the two adjacent plastic layers as dictated by the internal boundary conditions of Eq. (4); hence, 2 (L1 + L2 ) = 3 (L1 + L2 ). More details on a similar mathematical analysis for such type of strain burst behavior can be found in [25]. Initially (stage 1), all layers deform elastically with the same elastic modulus (E) and, therefore, the portion O–A of the stress–strain curve of Fig. 1 is given by ¯ = E ε¯ .
(10) ys 2 ,
layer 2 begins to yield and the At a critical stress, ¯ = above linear relationship gives the strain where stage 1 ends, ys ys i.e. ε¯ 2 = 2 /E. After layer 2 yields, the plastic strain and displacement expressions for layer 2 are given by Eqs. (7) and (8), while the other layers still deform elastically. The constants (A2 , B2 ) are determined from the plastic strain continuity conditions, p p i.e. ε2 (L1 ) = 0, ε2 (L1 + L2 ) = 0, while the constants (C1 -C4 ) are determined by assuming the following continuity and boundary conditions for the displacement field u: u1 (0) = 0, u1 (L1 ) = u2 (L1 ), u2 (L1 + L2 ) = u3 (L1 + L2 ), u3 (L1 + L2 + L3 ) = u4 (L1 + L2 + L3 ). Furthermore, the displacement of the top layer is related to the average strain as u4 (L) = ε¯ L. Hence solving this expression for ¯ gives the overall stress–strain relationship in stage 2 (segment AB of Fig. 1) as ys
ys
¯ = 2 + H2 (¯ε − ε¯ 2 ),
(11)
ys
ys
ys
with ε¯ 2 denoting the strain at which layer 2 yields (¯ε2 = 2 /E), ¯ ε¯ ) during and the quantity H2 being the strain hardening rate (d/d stage 2
H2 =
1 22 f2 1 − + tan h E ˇ2 ˇ2 L
∗2 2
−1 ,
where f2 = L2 /L and ∗2 = L2 /2 . [The notation fi = Li /L and ∗i = Li /i is adopted in the sequel for the ith layer.] It follows from the above discussion that the strain at which stage 1 ends is the same as the strain where stage 2 begins, therefore, the stress–strain curve from stage 1 to stage 2 is continuous, as shown in Fig. 1. Stage 2 ends when the stress triggers yielding ys of layer 3. The strain ε¯ 3 at the end of stage 2 can then be computed through the stress–strain relationship during stage 2, i.e. Eq. ys ys (11), by setting ¯ = 3 with 3 denoting the stress at which stage 3 begins. The result is ys ε¯ 3
ys
=
2 E
ys + (3
ys − 2 )
1 22 f2 1 − + tan h E ˇ2 ˇ2 L
∗2 2
.
(12)
X. Zhang, K.E. Aifantis / Materials Science and Engineering A 528 (2011) 5036–5043
5039
Fig. 1. Configuration used to fit stress–strain curve of micropillar with one strain burst.
During continuous deformation layer 3 also begins to yield at a ys critical stress = 3 . Now there are two plastic layers in the middle of the sample and two elastic zones at the ends. The plastic strain and displacement expressions for layers 2 and 3 are given by Eqs. (7) and (8), while layers 1 and 4 remain elastic. The constants (A2 , B2 , A3 , B3 ) are determined by allowing the plastic strain to be continuous throughout the whole pillar (εp = 0 on the interface between elastic and plastic layers) and by also ensuring continuity of both εp and across the boundary of the two plastically deforming slip layers 2 and 3 according to Eq. (4). p p p In other words, ε2 (L1 ) = 0, ε3 (L1 + L2 + L3 ) = 0, ε2 (L1 + L2 ) = p ε3 (L1 + L2 ), 2 (L1 + L2 ) = 3 (L1 + L2 ).The other four constants of integration (C1 –C4 ) are still determined by the displacement boundary and continuity conditions as before, and the overall stress–strain relationship (portion CD in Fig. 1) can be obtained through the boundary condition at u4 (L) as ys
ys
¯ = 3 + H3 (¯ε − ε¯ 3 ),
(13)
ys
ys
where ε¯ 3 is the strain corresponding to the yield stress, 3 , of layer 3 and H3 is the corresponding strain hardening rate
(d/d ¯ ε¯ ) given by H3 = E3 + P3 + G 3
−1
, where E3 = 1/E and
P3
= (f2 /ˇ2 + f3 /ˇ3 ) are the elastic and plastic contributions to is an hardening, whereas the strain gradient contribution G 3 explicit function of (ˇ’s, L’s, ’s and * ’ s) and is defined by
G3
∗ ∗ ∗ 2 ˇ2 2 + 3 (2ˇ3 − ˇ2 ) cot h(3 ) + ˇ2 csc h(3 ) tan h(2 /2) = ∗ ∗ ˇ2 2 cot h(2 ) + ˇ3 3 cot h(3 ) ˇ2 L
+
∗ ∗ ∗ 3 ˇ3 3 + 2 (2ˇ2 − ˇ3 ) cot h(2 ) + ˇ3 csc h(2 ) tan h(3 /2) . ˇ3 L ˇ2 2 cot h(∗2 ) + ˇ3 3 cot h(∗3 )
be analytically determined as
2 sin h(∗2 /2) cos h(∗3 /2) + 3 sin h(∗3 /2) cos h(∗2 /2)
ε3 =
L ˇ2 2 cos h(∗2 ) sin h(∗3 ) + ˇ3 3 cos h(∗3 ) sin h(∗2 )
×42 tan h
∗2 2
sin h
∗2 2
sin h
∗3 2
ys
ys
(3 − 2 ). (15)
The stress–strain curve depicted with the light green curve for the 1.0 m diameter pillar in Fig. 3(d) of [8] exhibited one strain burst, as the aforementioned 4-layer model predicts. This experimental curve (reproduced here with dots in Fig. 2) can, therefore, be divided into four segments similar to those of Fig. 1, for which the various stress–strain portions have been constructed. Since the stress–strain curve in [8] corresponds to shear deformation, the shear modulus, G, instead of the elastic modulus, E, should be used. The slope of the initial elastic region, however, in Fig. 3(d) of [8] is 6.1 GPa; hence it will be assumed that E = 6.1 GPa. The yield stress of the layys ys ys ers (2 , 3 ) are taken, according to the experiment, as 2 = ys 143 MPa and 3 = 175 MPa (Fig. 2). As the height-to-diameter ratio of the micropillars ranged from 2:1 to 3:1 in the experiments of [8], this ratio will be taken here as 2.5:1, L = 2.5 m. For simplicity, the pillar is divided into 4 equal-length layers L1 = L2 = L3 = L4 = 0.625 m, and the internal lengths of the plastically deforming layers are also set equal and smaller than Li : 2 = 3 = 0.5 m. The only remaining parameters to be determined are the hardening moduli ˇ2 and ˇ3 , the values of which may vary throughout the deformation process for each layer, according to the pertinent dislocation mechanisms that dominate at each stage. When the
ys
It then turns out that the strain, ε¯ 3 , at which stage 3 begins can be expressed by the following straight forward relation
ys
=
2 E
ys + (3
ys − 2 )
+
f2 1 1 22 + tan h − E ˇ2 ˇ2 L
∗2
150
2
2 sin h(∗2 /2) cos h(∗3 /2) + 3 sin h(∗3 /2) cos h(∗2 /2)
L ˇ2 2 cos h(∗2 ) sin h(∗3 ) + ˇ3 3 cos h(∗3 ) sin h(∗2 )
×42 tan h
∗2 2
sin h
∗2 2
sin h
∗3 2
100
50
ys (3
Stress MPa
ys ε¯ 3
ys − 2 ).
(14) Comparison of Eq. (12) with Eq. (14) indicates that the strain where stage 2 ends is not the same as the strain where stage 3 begins. The length of the corresponding discontinuity/strain burst ys ys (ε3 = ε¯ 3 − ε¯ 3 ) that occurs upon yielding of layer 3 can then
0
0.00
0.01
0.02
Strain
0.03
0.04
0.05
Fig. 2. Fit of the analytical model (solid line), Eqs. (10), (11), (13) and (15) to the experimental (dots) stress–strain curve of a 1 m diameter micropillar under compression (light green curve of Fig. 3d in [8]).
5040
X. Zhang, K.E. Aifantis / Materials Science and Engineering A 528 (2011) 5036–5043
Fig. 3. Configuration used to predict the stress–strain curve of micropillars with two strain bursts.
3.2. Modeling stress–strain curves with two strain bursts In Fig. 2 it was illustrated that this gradient plasticity multilayer model can capture very well the response of the particular micropil-
100 80
Stress MPa
second layer yields, a “knee” occurs in the stress–strain curve, indicating work hardening and, therefore, ˇ2 controls the deformation in the AB portion of Fig. 1; fitting Eq. (11) to the experimental data of Fig. 2 gives ˇ2 = 0.138 GPa. In continuing with the portion BC, it is noted that we have already an expression for the strain burst length ys when the stress is kept constant at 3 . Thus, Eq. (15) is used to fit the strain burst portion BC (or ε3 ) giving the same ˇ2 as before and predicting that ˇ3 = 0.117 GPa. Upon insertion of these values in Eq. (14) the strain at which the strain burst ends is predicted as ys ε¯ 3 = 0.48, which is precisely the experimentally observed value in Fig. 2. Finally, Eq. (13) is used to fit the portion of the stress–strain curve that follows the strain burst, giving ˇ2 = ˇ3 = 5.966 GPa. It should be noted here that some previous theoretical models [9] assume that the stress–strain response between strain bursts is fully elastic. This, however, has not been experimentally documented since through compression tests it is not possible to obtain a perfect elastic response, as even the slope of the initial stress–strain curve gives a much different modulus than tensile tests. This is due to the contact between the sample and the indenter, which allows for slip and elasto-plastic response is observed from the initial deformation stage. In the present model it is seen that re-hardening after the burst (as observed in [20]) can be modelled quite effectively.
60 40 20 0 0.000
0.005
0.010
0.015
Strain
0.020
0.025
Fig. 5. Fit of Eqs. (10), (11), (13), (15), (18) and (23) to experimental stress–strain curve of a 2.4 m diameter Ni micropillar under compression (dark purple curve in Fig. 3(d) in [8]).
lar. In most cases in [8], however, more than one strain burst was observed. Such behavior can also be captured within this multilayer model by inserting another slip layer as illustrated in Fig. 3. Initially all the layers deform elastically, then the middle layers (2, 3, and 4) yield one by one in the sequence shown in Fig. 4, while layers 1 and 5 remain always elastic. The dotted curves in Figs. 5 and 6 depict the experimental data from [8] that will be used to verify the multilayer model with two strain bursts. Before the fourth layer yields, the boundary conditions, and hence stress–strain relations in the five layer configuration are the
ys
ys
ys
Fig. 4. Dark layers from left to right indicate the sequential slip in micropillars that exhibit two strain bursts in their stress–strain response; it is assumed that 2 < 3 < 4 .
X. Zhang, K.E. Aifantis / Materials Science and Engineering A 528 (2011) 5036–5043
p
(16)
In continuing, the plastic strain (Eq. (7)) and displacement (Eq. (8)) expressions in the layers need to be evaluated. The constants (C1 –C5 ) are found through the boundary and continuity conditions:
50
0
0.00
0.01
0.02
Finally, the stress–strain expression for segment EF can be found, through the boundary condition at the upper elastic layer u5 (L) = ε¯ L. Hence, ys 4
ys + H4 (¯ε − ε¯ 4 ),
(18)
ys
ys
where ε¯ 4 is the strain corresponding to the stress, 4 and H4 is the strain hardening rate (d/d ¯ ε¯ ) after layer 4 yields H4 =
1 E4 + P4 + G 4
p 4
,
(19)
G 4
and represent the work hardening contributions that come from the elastic strain, plastic strain and the plastic strain gradient, respectively, E4 =
1 , E
f2 + f3 + f4 , ˇ
p
4 =
G 4 =−
2 tan h((L2 + L3 + L4 )/2) . ˇL (20)
0.03
0.04
0.05
Strain Fig. 7. Fitting gradient plasticity model (solid line) to experimental data (dots) allows for interpreting both size effects and strain bursts. Experimental data are taken from Fig. 3d of [8].
+
u1 (0) = 0, u1 (L1 ) = u2 (L1 ), u2 (L1 + L2 ) = u3 (L1 + L2 ), u3 (L1 + L2 + L3 ) = u4 (L1 + L2 + L3 ), (17) u4 (L1 + L2 + L3 + L4 ) = u5 (L1 + L2 + L3 + L4 ).
E4 ,
2.4 μm
100
p
ε2 (L1 ) = 0, ε4 (L1 + L2 + L3 + L4 ) = 0, p p ε2 (L1 + L2 ) = ε3 (L1 + L2 ), 2 (L1 + L2 ) = 3 (L1 + L2 ). p p ε3 (L1 + L2 + L3 ) = ε4 (L1 + L2 + L3 ), 3 (L1 + L2 + L3 ) = 4 (L1 + L2 + L3 ).
¯ =
1 μm
150 Stress MPa
same as that of the former four layer configuration. Hence, all the expressions required to generate the experimental plot, up to point D, of Fig. 3 are known (replacing L4 with L4 + L5 ). A stress–strain relation, however, needs to be obtained for segment EF of Fig. 3, along with an analytical expression for the length of the second strain burst. If the hardening modulus is kept different in each layer only numerical solutions can be obtained, hence in order to obtain analytical expressions it must be assumed that ˇ2 = ˇ3 = ˇ4 = ˇ and 2 = 3 = 4 = . The plastic strain and higher-order stressexpressions are the same as in Section 3.2, since V is kept the same for layer 4 of Fig. 4. Hence, the constants of integration (A2 –A4 , B2 –B4 ) in Eqs. (7) and (8) once slip occurs in layers 2, 3 and 4 can be determined through the following continuity conditions for the plastic strain and higher-order stress
5041
2 sec h ˇL
+ cos h
L + L + L 2 3 4 2
L − L 2 4 2
L + L 2 3
−sin h
sin h
cos h
2
cos h
L 3
2
L 4
2
L + L 3 4
sin h
2
L 2
2
ys
3
ys
4
.
(21)
Now that the stress–strain portions OA, AB, BC, CD and EF of Fig. 3 have been analytically defined by Eqs. (10), (11), (13), (15) and (18), respectively, the length of the second strain burst (portion DE of Fig. 3) observed in micropillar deformation must be calculated. ys First, the strain, ε¯ 4 , at the end of portion CD must be computed. This is done by setting the stress–strain relationship Eq. (13), equal ys to the stress, 4 , at which layer 4 begins to yield and solve for ε¯ , ys
ε¯ 4
ys
=
4 E −
ys
ys
+
4 (f2 + f3 + f4 )
ˇL
ˇ ys
ys
(3 − 2 ) sec h
ys
ys
+(4 − 2 ) tan h
ys
ys
f2 2 + f3 3 + f4 4
−
ˇ
L + L 2 3
sin h
2
L + L 2 3 2
ys
L − L 2 3 2 ys
+ (4 − 3 ) tan h
L + L 2 3 2
ys
ε¯ 4
ys
=
4 E
ys
ys
+
4 (f2 + f3 + f4 ) ˇ
−
ys
ys
f2 2 + f3 3 + f4 4 ˇ
Subtracting Eq. (22) from Eq. (21) gives the length of the discontiys ys nuity/strain burst, ε4 = ε¯ 4 − ε¯ 4 , that occurs upon yielding of layer 4 as ys
Stress MPa
30
20
10
0
ys
ys
ys
L
ys
ys
2 (2 − 4 ) + (3 − 2 ) cos h + (4 − 3 ) cos h ε4 =
L L +L L +L +L ˇL cos h 22 3 cos h 2 23 4 csc h 4
40
0.00
0.01
0.02
0.03
Strain
0.04
0.05
Fig. 6. Fit of Eqs. (11), (13), (15), (18) and (23) to experimental stress–strain curve of a 5.2 m diameter Ni micropillar under compression (dark green curve in Fig. 3(c) in [8]).
.
(22)
ys
The strain where the second strain burst ends, ε¯ 4 , is expressed as
ys
2
L2 +L3 2
.
(23)
Following the same fitting procedure as in Section 3.2, it is possible to fit the stress–strain curves for the 2.4 m and 5.2 m diameter micropillars that exhibited two strain bursts; curves colored dark purple in Fig. 3(d) and dark green in Fig. 3(c) of [8]. It should be noted that in performing the fit, the layers were taken to have equal thickness L1 = L2 = L3 = L4 = L5 = L/5. The parameters used for the fits are listed in Tables 1 and 2, and the comparison between the theoretical prediction and the experimental data are shown in Figs. 5 and 6, respectively. In order to better observe the size effects that the experiments document and the theory captures, Figs. 2 and 5 are combined (since they are taken from the same figure of [8]) to produce Fig. 7. Fig. 7 clearly shows that the present theoretical model can account for the size effects observed for pillars of varying diameters. It should be noted that the values for most of the parameters reported in the tables, were obtained directly from the observation of the stress–strain curves.
5042
X. Zhang, K.E. Aifantis / Materials Science and Engineering A 528 (2011) 5036–5043
Table 1 Parameters used to generate stress–strain plot for 2.4 m diameter micropillar. Shear modulus G(GPa) 5.2
ˇ2 (GPa) ˇ3 (GPa) ˇ4 (GPa)
Yield stress (MPa) ys ys 3 2
4
72
95
81
ys
Layer thickness L1 = L2 = L3 = L4 = L5 (m)
Inner length scale 2 = 3 (m)
1.2
0.9
AB (Layer 2 yield)
BC (1st strain burst)
CD (Layer 3 yield)
DE (2nd strain burst)
EF (Layer 4 yield)
0.161
0.161 0.348
3.893 3.893
0.555 0.555 0.555
6.402 6.402 6.402
Table 2 Parameters used to generate stress–strain plot for 5.2 m diameter micropillar. Shear modulus G(GPa) 2.485
ˇ2 (GPa) ˇ3 (GPa) ˇ4 (GPa)
Yield stress (MPa) ys ys 2 3
4
28
39
34
ys
Layer thickness L1 = L2 = L3 = L4 = L5 (m)
Inner length scale 2 = 3 (m)
2.6
1.95
AB (Layer 2 yield)
BC (1st strain burst)
CD (Layer 3 yield)
DE (2nd strain burst)
EF (Layer 4 yield)
0.034
0.034 0.101
0.191 0.191
0.021 0.021 0.021
0.679 0.679 0.679
The yield stress for each layer differs according to the experimenys tal data, since i was taken directly from the experimental data (Fig. 3d of [8]) as being the stress at which the “knee” or strain bursts occurred in the stress–strain plot, whereas the modulus was the slope of the initial linear portion of the experimental curves. Hence, the hardening moduli and internal lengths were the only fit parameters. Although only up to two strain bursts were captured here, it is possible to capture more by increasing the number of slip layers; more slip layers in the gradient model correspond to more strain bursts. 4. Summary and conclusions In the present study strain gradients were used for the first time to capture the size effects and strain bursts that are experimentally observed in single crystal Ni micropillars under compression [8]. According to [20] significant strain gradients develop during compression and a sub-grain structure forms, while the gauge region is deformed through slip. In using, therefore, strain gradient plasticity the pillar is sectioned into equal length slip zones, each of which has its own yield stress. The zones are allowed to yield sequentially and a strain burst arises when two adjacent zones deform plastically, capturing therefore the experimentally observed strain bursts. The higher-order stress is the underlying mechanism for the strain burst, as it is discontinuous at the interface between a plastic layer and its neighboring elastic layer, but it becomes continuous when two neighboring layers begin deforming plastically. The transition from discontinuity to continuity of the higher-order stress across the internal boundary between two neighboring layers results in the strain burst. The analytical stress–strain expressions that arise from gradient plasticity were in precise agreement with the experimental stress–strain curves for 1 m, 2.4 m and 5.2 m diameter micron pillars that exhibited 1 or 2 strain bursts during deformation [8]. In addition to capturing the experimental data of [8] the present model was also able to provide upper and lower bounds [30] for the stress–strain curves observed in compression of Al micropillars [31]. It is possible to modify the gradient model so as to capture additional strain bursts as some experimental curves indicate, but then the solutions become numerical and it is beyond the scope of the present study, whose main purpose was to illus-
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