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Characterization of indentation size effects in epoxy
Polymer Testing 40 (2014) 70e78
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Polymer Testing journal homepage: www.elsevier.com/locate/polytest
Material properties
Characterization of indentation size effects in epoxy F. Alisafaei*, Chung-Souk Han 1, Nishant Lakhera 2 University of Wyoming, Department of Mechanical Engineering, 1000 E. University Ave., Laramie, WY, 82071, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 13 July 2014 Accepted 25 August 2014 Available online 3 September 2014
Nanoindentation tests were conducted at different indentation depths, ranging from 3000 to 30 nm, to study the characteristics of indentation size effects in epoxy. Taking the effects of tip imperfection/bluntness into account, nanoindentation tests performed with a threesided Berkovich tip exhibited strong increases in the hardness with decreasing indentation depth. However, comparative testing with a spherical tip with a radius of 250 microns, showed no changes in the material properties with respect to the distance from the surface. Consequently, different deformation mechanisms must be present in the indentation tests with Berkovich and spherical tips, suggesting effects related to second order gradients of the displacement (such as strain or rotation gradients), which increase with a Berkovich tip but essentially stay constant with a spherical tip when indentation depth decreases. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Polymers Nanoindentation Length scale dependent deformation
1. Introduction Indentation probing of materials [1e3] on the microand nanometer length scales has revealed size dependent deformation of various materials [4e5]. In addition to micro and nanoindentation tests, size effects have been also observed in micro beam bending tests [6e7] (see [8] for corresponding finite element simulation), microtorsion of thin wires [9] and tension of geometrically similar perforated plates (plates with central holes) [10]. In metals, such size effects at micron and submicron length scales are usually associated with geometrically necessary dislocations arising from inhomogeneous plastic deformation [11]. Such a notion cannot be applied to size effects in polymers [12e16] as, in contrast to metals, length scale dependent deformation has been also observed in pure elastic deformation [7,17e21]. However, the literature on
the rationale for size effects in polymers is quite sparse [3,13e25]. In order to shed light on the characteristics of indentation size effects in polymers, polydimethylsiloxane (PDMS) had been previously studied in [21] where plastic deformation was essentially absent. Here, epoxy is chosen as it shows significant plastic deformation and has been studied before in indentation [13,15] and beam bending [7]. The purpose of this study is the corroboration of previous results in the literature on epoxy and an in-depth examination of indentation size effect characteristics that have not been published before. In addition to the indentation depth dependent hardness (already studied in [13,15]), here also the indentation depth dependent characteristics of dissipation, plastic deformation and stiffness are examined at the indentation depth range of 30 to 3000 nm. 2. Experiments
* Corresponding author. Tel.: þ1 307 343 3589. E-mail addresses: [email protected] (F. Alisafaei), [email protected] (C.-S. Han), [email protected] (N. Lakhera). 1 Tel.: þ1 307 766 2930. 2 Current address: Freescale Semiconductor Inc., 6501 W. William Cannon Drive, Austin, TX 78735, USA. Tel.: þ1 307 399 1796. http://dx.doi.org/10.1016/j.polymertesting.2014.08.012 0142-9418/© 2014 Elsevier Ltd. All rights reserved.
2.1. Sample preparation The epoxy samples were prepared by mixing the commercially available liquid epoxy resin, EPON 828 (bisphenol A-epichlorohydrin), as base agent with the
F. Alisafaei et al. / Polymer Testing 40 (2014) 70e78
curing agent EPIKURE 3223, diethylenetriamine (DETA). The required amount of epoxy resin (Momentive Specialty Chemicals, USA) was measured at room temperature in an aluminum dish (Fisherbrand, low form, weighing dish, 70 ml), to which DETA (Momentive Specialty Chemicals, USA) with 20 phr (parts per hundred resin) was added. The mixture was manually stirred with a rod for 2 minutes, after which the samples were kept in a vacuum chamber (Alcatel Drytel vacuum pump, 22 inches Hg pressure) for 20 minutes to remove trapped air. The samples were then cured in a conventional oven (Fisher-Scientific Isotemp force draft) at 100 C for 3 hours and slowly cooled down to room temperature. Finally, the samples were removed from the aluminum dish and annealed at 100 C for 2 hours to eliminate residual stresses. 2.2. Nanoindentation system
0.5
B
C
3. Experimental results and analysis 3.1. Universal hardness for an ideally sharp indenter tip The Martens hardness (also known as Universal hardness) HU is defined as [26]
HU ¼
0.3
Fmax ; As
(1)
where As is the nominal surface area of the indenter tip that has penetrated the original surface under the maximum applied force Fmax. Due to tip roundness/imperfections, according to [26], this definition should be only applied for indentation depths larger than 6 mm where the tip roundness is negligible. With a tip roundness radius of less than 20 nm, according to the manufacturer, our Berkovich tip is arguably very sharp where imperfections of the tip are negligible for indentation depths above about 250 nm, as will be seen below. Therefore, in a preliminary examination the uncorrected universal hardness
HUuncorrected ¼
Fmax ; Auncorrected s
(2)
is considered for simplicity where Auncorrected is the nominal s surface area of an ideally sharp indenter tip (without considering the effect of tip bluntness). The nominal surface area Auncorrected is obtained by s
Auncorrected ¼ Ch2 ; s
(3)
where h is the indention depth at Fmax (point B in Fig. 1) and C is a constant equal to 26.43 for a Berkovich indenter tip [26]. Fig. 2 shows HUuncorrected of epoxy at different h ranging
700 600
U
0.4
F (mN)
loading force was applied linearly in time and a loading time of 20 s was used for all spherical tests.
H uncorrected (MPa)
The nanoindentation tests were conducted using an Agilent G200 Nano Indenter system (Agilent Technologies, Oak Ridge, TN, USA) equipped with a Berkovich indenter tip which, according to the manufacturer's specifications, has a curvature radius of less than 20 nm. All Berkovich indentation tests were performed at 23 C using a dynamic contact module (DCM) in force controlled mode with linear loading and unloading force-time relations. A surface s s detection tolerance of Ktol ¼ 50 N/m was used, where Ktol is the stiffness threshold to recognize initial contact with the surface of the sample (see [21] for more details). The minimum distance between two individual indents was 50 mm. Fig. 1 depicts a typical load-displacement curve consisting of loading (section A-B with 20 s loading time), holding (section B-C with 50 s holding time) and unloading (section C-D with 20 s unloading time) sequences. Nanoindentation tests were carried out at different indentation depths ranging from 3000 to 30 nm associated with loading forces of about 32000 to 12 mN. In addition to the Berkovich indentation tests, a spherical tip with a curvature radius of 250 mm was also applied to probe the samples at indentation depths between 400 and 3700 nm. All spherical tests were conducted in force s controlled mode with an XP head and Ktol of 200 N/m. The
71
500 400 300
0.2
200
0.1
100 10
100
1,000
h (nm)
A
0 0
D 50
100
150
200
250
Indentation Depth (nm) Fig. 1. Typical load-displacement data.
300
350
Fig. 2. Uncorrected universal hardness of epoxy as a function of indentation depth h obtained with 5 s loading time. Each data point represents 3 indentations with an error bar being the standard error calculated by the standard deviations of averages divided by the square root of 3 (sample size).
F. Alisafaei et al. / Polymer Testing 40 (2014) 70e78
from 3000 to 30 nm obtained with 5 s loading time (note that h is measured at point B shown in Fig. 1). Each data point in Fig. 2 represents the average of 3 indentations, where an error bar has been added for each data point. As all indentation tests have been carried out in force controlled mode, the variation of h with respect to a specified Fmax could be shown. The related variation in h were however so small that the error bars for h have been omitted in Fig. 2. Strong increases in HUuncorrected are observed in Fig. 2 with decreasing h. As mentioned above, the results in Fig. 2 do not take the tip imperfection into account. Therefore, particularly for h < 200 nm, the experimental results should be adjusted to reflect the effect of tip bluntness. 3.2. Tip bluntness/imperfections
1
10
A
c
Auncorrected c
Acorrected c
1 0.1 0.01 0.001
10
100
1,000
h (nm) c Fig. 4. Different contact areas (AC ; Auncorrected ; Acorrected ) versus contact depth C C hc obtained for fused silica.
Relation (3) for the nominal surface area As is valid for an ideally sharp tip or for sufficiently large h where the tip bluntness can be neglected. To assess the influence of tip bluntness, the pyramidal Berkovich indenter tip is modeled as an equivalent conical indenter tip so that the cone with an ideally sharp tip would have the same volume (and area) to depth relation as that of the ideally sharp Berkovich indenter tip [27e28]. Fig. 3 illustrates the equivalent conical indenter tip with a semi apex angle, q, equal to 70.3 [29]. The blunt indenter tip is modeled by a cone smoothly fitted with a spherical cap of radius R as shown in Fig. 3 [27]. The radius of the spherical cap has been determined by two different approaches. In the first approach [30], the contact area Ac (the sample area with which the tip is in actual contact) is determined as a function of contact depth hc (see [1] for its definition) from nanoindentation tests on fused silica (known not to exhibit length scale dependent deformation). The projected contact area Ac was obtained in the following form 1
100
Contact area ( µ m2)
72
1
Ac ¼ C0 h2c þ C1 hc þ C2 h2c þ C3 h4c þ C4 h8c ;
(4)
where C0, C1,…,C4 are constants determined by curve fitting [1,29]. In Fig. 4, the projected contact area Ac with respect to contact depth hc is illustrated. It can be seen that at small hc there is a significant difference between
the projected contact area Ac and the contact area of a which can be perfect Berkovich indenter tip Auncorrected c obtained by Auncorrected ¼ 24:56h2c : c
The difference between Ac and Auncorrected observed in c Fig. 4 illustrates the significance of the tip bluntness at small indentation depths. To compensate for the tip bluntness and to obtain the tip curvature radius R, the projected contact area Ac was fitted with the contact area of a rounded tip, Acorrected , which can be obtained as [30]. c 2
Acorrected ¼ 24:56ðhc þ hb Þ ; c
(6)
where hb is the difference in depth between the sharp and blunt tips, as shown in Fig. 3. Fitting of the projected contact area Ac with Eq. (6), see Fig. 4, hb was found to be 2.5 nm. For hb ¼ 2.5 nm, the tip curvature radius R was determined as 40.2 nm by geometrical considerations of Fig. 3 as hb : R¼ 1 (7) 1 sin q The tip curvature radius R was also determined by second approach applying Hertzian contact theory on fused silica at a very low loading force of F ¼ 1 mN [31] with its relation to h given as
4 E F ¼ R1=2 h3=2 ; 3 s 1 v2
Fig. 3. Schematic illustration of a conical indenter with a blunt tip.
(5)
(8)
where E and v are, respectively, elastic modulus and Poisson's ratio of the sample. Assuming a spherically shaped tip, applying such a small load and measuring the displacement h, the tip curvature radius R ¼ Rs can be obtained from Eq. (8) with the known elastic properties of fused silica (E and v) yielding R ¼ 24.4 nm, which is closer to the curvature radius of less than 20 nm specified by the manufacturer. However, in both cases, R is higher than the specified 20 nm which may indicate blunting of the tip due to use on a hard material like fused silica. Using the larger hb ¼ 2.5 nm and R ¼ 40.2 nm of the first approach, to be conservative, the universal hardness is
F. Alisafaei et al. / Polymer Testing 40 (2014) 70e78
2
Acorrected ¼ 26:43ðh þ hb Þ s
2
26:43R2 sin ð90 qÞtan2 ð90 qÞ
(9)
þ 2pR ð1 cosqÞ: 2
Fig. 5 illustrates the corrected universal hardness Fmax =Acorrected of epoxy over h. Compared with s
HUcorrected ¼ HUuncorrected
of Fig. 2, it can be seen in Fig. 5 that the tip roundness has considerable effect on the experimental results at small h. However, there is still a significant size effect in Fig. 5, illustrating that the tip imperfections cannot be the only source of depth dependent hardness in epoxy. To assess at what h the effect of tip bluntness becomes important, the corrected universal hardness HUcorrected is
plotted in Fig. 6 along with HUuncorrected (using Auncorrected ) s obtained with 5 s loading time. It can be seen that
700
H uncorrected 600
Universal Hardness (MPa)
recalculated to assess the effect of tip roundness on the size effect in epoxy. Therefore,Auncorrected in Eq. (2) is replaced s with the nominal surface area of a blunted tip, Acorrected , s which can be obtained by [20].
H
U corrected U
500 400 300 200 100 0 10
100
1,000
h (nm) Fig. 6. Effect of tip roundness on the indentation size effect in epoxy.
HUcorrected
starts to deviate from HUuncorrected below about h z 250 nm, which is almost 10 times higher than hb, while the effect of tip bluntness can be neglected for h > 250 nm. It may be worth noting that, particularly at small h, the data in Fig. 6 and the following experimental results is in parts less smooth, which may be related to surface roughness or possible local sample heterogeneity. 3.3. Corrected universal hardness and hardness model To assess the time dependence of the indentation size effects in epoxy, three different loading times of 5, 20, and 80 s were applied. Depth dependent hardness is observed for all loading times in Fig. 7 where HUcorrected increases with decreasing h. Fig. 7 also depicts the effect of loading time on
the hardness of epoxy. Consistent with the results reported in [32], the hardness decreases at large h with increasing loading time from 5 to 80 s. However, time dependence of the hardness markedly decreases with decreasing h (see [33e34] and references therein for time dependent behavior of polymeric materials during nanoindentation). The time-independent hardness model of Han and Nikolov [23] is considered here to analyze the size effect observed in our experiments. This model has been successfully applied to analyze the depth dependent deformation of various polymers [12,21]. The variation of this rotation gradient based hardness model with respect to h is given by
c[ ; H ¼ H0 1 þ h
(10)
where H0 (macroscopic hardness) is considered as the lower limit of H in H-h curve (see Fig. 7) and c[ is the
500 450
450
400
5 s , Experimental Results 20 s , Experimental Results 80 s , Experimental Results 5 s , Hardness Model (10) 20 s , Hardness Model (10) 80 s , Hardness Model (10)
400
H corrected (MPa)
350 300
U
H corrected (MPa)
73
300 250
U
250
350
200 150 10
100
1,000
200 150
h (nm) Fig. 5. Corrected universal hardness of epoxy as a function of indentation depth h obtained with 5 s loading time. Each data point represents 3 indentations with an error bar being the standard error calculated by the standard deviations of averages divided by the square root of 3 (sample size).
100 0
500
1000
1500
2000
2500
3000
h (nm) Fig. 7. Corrected universal hardness of epoxy together with the theoretical hardness model (10) for different loading times.
F. Alisafaei et al. / Polymer Testing 40 (2014) 70e78
material length scale parameter micromechanically motivated in [23]. Finding the parameters c[ and H0 by fitting Eq. (10) to the experimental data,3 Fig. 7 shows the resulting data obtained from the theoretical model together with the experimental results with reasonably good agreement between the experiments and the hardness model. 3.4. Change in stiffness with h In addition to tip bluntness, the accuracy of hardness values is also strongly dependent on a precise detection of the surface. To characterize the deformation behavior, the stiffness, i.e. Km ¼ vF=vh, at the maximum applied force is experimentally determined by calculating the derivative of the loading curve at the last data point (point B in Fig. 1) for each indentation test [21]. Therefore, Km is only dependent on the slope of force-depth curve (at h associated with the maximum applied force) and not on the absolute value of h at Fmax and, consequently, is not dependent on accurate surface detection. In Fig. 8, Km of epoxy experimentally obtained at different indentation depths is depicted for loading times of 5, 20 and 80 s (see [21] for more details of Km determination). Consistent with the hardness results of Fig. 7, it can be seen in Fig. 8 that at large depths (above 250 nm) the stiffness increases with decreasing loading time. Note that the same behavior is observed in Fig. 7 where at indentation depths above 250 nm the hardness increases with decreasing loading time. Also, similar to the hardness trend observed in Fig. 7, the stiffness of epoxy becomes time independent at small depths (nearly below 250 nm). It is noteworthy that in Fig. 8, the Km h relation becomes nonlinear at small indentation depths and Km does not seem to approach zero when h / 0. However, based on the classical local continuum theory (which does not account for length scale dependent deformation) Km is expected to approach zero when h / 0. This trend can be observed in the stiffness of fused silica (which does not exhibit size effects) shown in Fig. 9 where Km / 0 when h / 0. This behavior of epoxy can be also described with the hardness model (10) as set forth in the following.
Indentation Stiffness (kN / m)
74
5 s , Experimental Results 20 s , Experimental Results 80 s , Experimental Results 5 s , Stiffness Model (12) 20 s , Stiffness Model (12) 80 s , Stiffness Model (12) 5 s , Linear line K=2HoC h
10
1
10
100
Fig. 8. Indentation stiffness at Fmax versus h obtained for epoxy with different loading times.
of Fig. 7, the stiffness, K ¼ vF=vh, is obtained through the loading part as K ¼ H0 Cðc[ þ 2hÞ:
As depicted in Fig. 8 and discussed above, Km for epoxy appears to not approach zero when h / 0, illustrating the effect of size dependence on the indentation stiffness. A similar but much stronger trend has been also observed in PDMS where Km approximately reaches a plateau for small indentation depths [21]. Starting from a hardness model (10) that can be related to the universal hardness (1) as
120
where the parameters of H0 and c[ can be obtained by fitting the hardness model (10) to the experimental data
Km ( kN / m )
140
(11)
100 80 60 40 20 0 0
3
Fitted values are H0 ¼ 163 MPa and c[ ¼ 130 nm for 5 s loading time, H0 ¼ 158 MPa and c[ ¼ 135 nm for 20 s loading time, and H0 ¼ 145 MPa and c[ ¼ 150 nm for 80 s loading time.
(12)
The stiffness model (12), with the previously fitted parameters of H0 and c[, is depicted in Fig. 8 together with the experimentally determined indentation stiffness (as described in the previous section). As can be seen in Fig. 8, there is good agreement between the experiments and the theoretical stiffness model (12), corroborating the validity of the hardness model (10) to describe size effects in epoxy. Eq. (12) is also consistent with the trend observed in Fig. 8 where Km does not approach zero when h / 0. In Eq. (12), c[ becomes dominant over 2h at very small h. Therefore, for very small h, further decreases in h will result in negligible changes in the stiffness, which is in agreement with the experimental results obtained for Km in Fig. 8. For materials which do not exhibit size effects, i.e. c[ ¼ 0, Eq. (12) predicts
3.5. Theoretical analysis of Km
F c[ ; ¼ H0 1 þ Ch2 h
1,000
h (nm)
100
200
300
400
500
h (nm) Fig. 9. Indentation stiffness at Fmax versus h (fused silica).
600
F. Alisafaei et al. / Polymer Testing 40 (2014) 70e78
1
h /h f
a linear relation between stiffness and indentation depth, K ¼ 2H0Ch, with K / 0 for h / 0, which is consistent with the experimental results obtained for fused silica (see Fig. 9).
75
3.6. Dissipation and inelastic deformation
he ¼
WIe ; WI
5s 20 s 80 s
0.1 1
Welastic / Wtotal
To characterize the changes in indentation recovery with respect to h, the ratio of the final deformation hf (see Fig. 10) to the indentation depth at the maximum applied force, hf/h, is considered. The changes of hf/h with h are shown in Fig. 11 where again 5, 20, and 80 s are applied as loading and unloading times along with a holding time of 50 s. A decrease in the normalized parameter hf/h with decreasing h (particularly, for the range of 3000 to about 100 nm) is observed for all different loading times where hf/h is lower for higher loading times at all indentation depths. Similarly, the change in the ratio of the elastic indentation work WIe to the total indentation work WI ¼ WIe þ WIine (see Fig. 10),
0.1 10
100
1,000
h (nm)
Fig. 11. hf/h and he versus h (all tests were carried out with 50 s holding time together with the same times for loading and unloading).
(13) hardly any loading time dependence at large depths h > 2000 nm in PDMS). Thus, 20 phr epoxy (as a polymer in the glassy state) and PDMS (as a polymer in the rubbery state) have quite opposite characteristics with respect to loading time dependence and corresponding trends in dissipation. 3.7. Indentation hardness and elastic modulus Indentation hardness [1] can be also used to study the depth dependent plastic deformation as epoxy exhibits significant plastic deformation during indentation tests. Fig. 12 shows the increase of indentation hardness with decreasing h obtained for the same indentation depth range of 3000 to 30 nm with 20 s loading, 20 s holding, and 20 s unloading times (Poisson's ratio v ¼ 0.35). Comparing the indentation hardness of Fig. 12 with the universal
700
Indentation Hardness (MPa)
with h can be considered to illustrate changes in the deformation mechanism. This ratio has been also plotted in Fig. 11 where he increases with decreasing h for all loading/ unloading times. Furthermore, he decreases with decreasing loading time at different indentation depths, indicating that the energy dissipation (relative to the whole deformation) increases when the material is loaded/ unloaded faster. The changes in both normalized parameters hf/h and he with respect to h, as shown in Fig. 11, would indicate that the elastic part of the total deformation work increases with decreasing h. This behavior in epoxy is exactly opposite to that observed in PDMS and filled silicone rubber (there is a decrease in the elastic indentation recovery in both PDMS and filled silicone rubber with decreasing indentation depth [21,19]). In this respect, it should be also noted that the hardness and stiffness of epoxy become time independent at small depths (Figs. 7 and 8), while for PMDS (see [21]) the hardness is more loading time dependent at small h than at large h (there is
Force
500 400 300 200 100 0
Depth Fig. 10. Schematic loading-unloading curve.
Present Results Results in [15]
600
500
1000
1500
2000
2500
3000
h (nm) Fig. 12. Indentation hardness of epoxy (according to [1]) versus h with 20 s loading time, 20 s holding time and 20 s unloading time (Present Results).
76
F. Alisafaei et al. / Polymer Testing 40 (2014) 70e78
hardness of Fig. 7, the indentation hardness is slightly higher than the universal hardness as the contact area Ac (used for determining the indentation hardness) is usually smaller than the nominal area As. As shown in Fig. 12, the indentation hardness obtained from our experiments is in good agreement with an earlier study in [15] on an epoxy with 20% hardener. In addition to the indentation hardness shown in Fig. 12, the elastic modulus E has been also determined with the Berkovich tip following [1] for various h shown in Fig. 13. As can be seen therein, E (obtained with the Berkovich tip) increases with decreasing h which would indicate that the elastic modulus changes through depth, which is quite misleading. Performing experiments with a spherical indenter tip, it will be shown below that E does not change with h, as illustrated in Fig. 13.
4. Discussion 4.1. Influence of rotation gradients on the depth dependent deformation of polymers To illustrate the influence of rotation gradients on the indentation size effects observed in our experiments, comparative indentation tests were conducted using a spherical tip with a curvature radius of 250 mm. Rotation gradients are defined as cij ¼ 12 einm um;jn where um is the displacement field and einm is the permutation symbol. Corresponding to a geometrical analysis [35e36], indentations with such a spherical indenter tip will result in negligible changes in the rotation gradients cij with h (the rotation gradient cij is proportional to the radius of the spherical tip and essentially independent of h). Therefore, strain/rotation gradient based models [22e23] would not predict h dependence in the hardness. It has been also experimentally observed that applying such a spherical tip, rotation gradients essentially do not change with h [37]. In
Fig. 13, the elastic modulus of epoxy obtained with such a spherical tip (curvature radius of 250 mm) is shown where E has been determined according to the Hertzian theory (8) by assuming that the diamond spherical tip is rigid as the elastic modulus of diamond is much larger than that of epoxy. The lower limit of h has been restricted to 400 nm due to technical difficulties of applying a spherical tip at small h (such as effects of surface roughness and the presence of adhesion for which the Hertzian theory does not apply). Hardly any changes in E with h are present in Fig. 13 in the whole range of h ¼ 3700 to 400 nm, which is in agreement with the rotation gradient based models [22e23] as rotation gradients do not change with h when a spherical tip is applied. Therefore, results obtained for the elastic modulus shown in Fig. 13 illustrate that the elastic modulus of epoxy should be determined with caution at smaller h when a sharp pyramidal or conical tip is used as gradient effects may be present and, if possible, indentation tests should be performed at higher h to obtain reliable results. Consistent with the experimental results observed above, the size dependent deformation of polymers is here viewed as a result of an increase in rotation gradients cij (with decreasing h), which can be interpreted in terms of ~ F arising from the rotation gradients deformation work DW finite bending stiffness of polymer chains and their interactions (see [22e23] for more details). The total inden~ F yielding tation work WI can be augmented by DW I
~ F; WI ¼ WεIe þ WεIine þ DW I WεIe
(14) WεIine
where the local terms and are respectively the elastic and inelastic indentation works (the subscript ε indicates the local nature of these components). Also, note ~ F ¼ W e (because DW ~ F is reversible in nathat W e þ DW εI
I
I
I
ture) and WεIine ¼ WIine as represented in Fig. 10. According ~ F is expressed as a to [22e23], the nonlocal component DW I
function of rotation gradients by
Elastic Modulus (GPa)
10
Spherical indenter tip Berkovich indenter tip
8
6
As shown in [35e36], the higher order gradients cij increase with decreasing indentation depth h for sharp conical or pyramidal indenter tips. Based on the model developed in [22–23] and according to Eqs. (14) to (16), the ~ F will increase with decreasing nonlocal component DW I indentation depth h resulting in the higher hardness values observed in the experiments.
4
2
0 0
~ ~ F ¼ K cS cS ; (15) DW I 3 ij ij ~ is a constant parameter micromechanically where K motivated in [22e23] and cSij is the symmetric part of the rotation gradient tensor cij which is obtained as 1 (16) cSij ¼ cij þ cji ; 2
1000
2000
3000
4000
h (nm) Fig. 13. Elastic modulus of epoxy determined with a Berkovich indenter tip (according to Oliver & Pharr's approach [1] with 20 s loading, 20 s holding, and 20 s unloading times) and with a spherical indenter tip (via Hertzian theory with 20 s loading time).
4.2. Strain/rotation gradient based hardness models As discussed earlier, based on the rotation gradient theory [22e23], the indentation size effects observed in the experiments can be explained by increases in rotational
F. Alisafaei et al. / Polymer Testing 40 (2014) 70e78
gradients with decreasing h. However, in addition to the above mentioned rationale, a different strain gradient theory is suggested to explain indentation size effects in polymers. To predict strain gradient dependent deformation in polymers, Lam and Chong [13] developed a strain gradient plasticity model on the basis of molecular theory of yield for glassy polymers. In this model, the depth dependent deformation of polymers is motivated by a molecular kinking mechanism [14] which, similar to metals, is related to the plastic deformation. In this model, the proposed hardness varies with h as
H ¼ H
qffiffiffiffiffiffiffiffiffiffi 1 þ h =h ;
(17)
where, similar to the macroscopic hardness H0 in Eq. (10), H* is the hardness at infinite depth and h* is a constant dependent on material behavior and temperature [14]. Both parameters H* and h* are obtained by fitting the hardness model (17) to the experimental hardness data of Fig. 7. Based on the theoretical hardness model (17), similarly to obtaining Eq. (12), a stiffness model can be deduced as
KL&C ¼ H C
pffiffiffiffiffi 3 h h1=2 þ 2h ; 2
(18)
Indentation Stiffness (kN / m)
where KL&C is the indentation stiffness obtained from the hardness model of Lam and Chong [13]. It should be noted that, in contrast to the stiffness model (12), KL&C approaches zero when h / 0, which is not in agreement with the experimental results shown in Fig. 8. As mentioned in Subsection 3.4 and shown in Fig. 8, the material stiffness Km does not approach zero at small indentation depths. This trend would not be predicted by the stiffness model (18) as KL&C / 0 when h / 0. In Fig. 14, the theoretical stiffness models (12) and (18) are compared with the experimental results obtained with 80 s loading time. It can be seen that the stiffness model (12) is in better agreement with the
10
80 s , Experimental Results 80 s , Stiffness Model (12) 80 s , Stiffness Model (18)
100
experimental data, particularly at small h where Km does not seem to approach zero. The hardness models (17) and (10) are also based on different physical explanations of size effect in polymers. In the model developed in [13], size effect in polymers is explained by molecular kinking mechanisms associated with plastic deformation. Therefore, this model is not capable of predicting size effects in pure elastic deformation mechanisms observed by the authors [19e21] and others [7,17e18]. 5. Conclusions Indentation depth dependent deformation behavior of epoxy was experimentally observed by nanoindentation testing conducted at indentation depths ranging from 3000 to 30 nm with tip bluntness being carefully accounted for at small indentation depths. The indentation size effects and the change in the stiffness characteristics of epoxy observed in these experiments were analyzed by the indentation depth dependent based hardness and stiffness models, and good agreement was found between models and experimental results. To detect any changes of the material properties through depth and to demonstrate the influence of rotation gradients on the indentation size effect in epoxy, comparative tests with a spherical indenter tip were performed to obtain the elastic modulus at different probing depths. No changes in the elastic modulus were determined with probing depth which is in agreement with the rotation gradient based model [22e23] illustrating the significance of rotation gradients in the indentation size effects of epoxy.
Acknowledgments The support of this work through the National Science Foundation, Grants CMMI 1102764 and CMMI 1126860, is highly appreciated. The spherical indentation tests of Fig. 13 were conducted by Mr. S.H.R. Sanei which should be also acknowledged here.
References
1
10
77
1,000
h (nm) Fig. 14. Comparison of the indentation stiffness models (12) and (18) with experimental data.
[1] W.C. Oliver, G.M. Pharr, Improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments, J. Mater. Res. 7 (1992) 1564e1583. [2] Q. Liao, J. Huang, T. Zhu, C. Xiong, J. Fang, A hybrid model to determine mechanical properties of soft polymers by nanoindentation, Mech. Mater. 42 (2010) 1043e1047. [3] C.A. Tweedie, G. Constantinides, K.E. Lehman, D.J. Brill, G.S. Blackman, K.J. Van Vliet, Enhanced stiffness of amorphous polymer surfaces under confinement of localized contact loads, Advanced Mater 19 (2007) 2540e2546. [4] J.G. Swadener, E.P. George, G.M. Pharr, The correlation of the indentation size effect measured with indenters of various shapes, J. Mech. Phys. Solids 50 (2002) 681e694. [5] A.A. Elmustafa, D.S. Stone, Indentation size effect in polycrystalline F.C.C. metals, Acta Mater. 50 (2002) 3641e3650. [6] C. Motz, T. Schoberl, R. Pippan, Mechanical properties of micro-sized copper bending beams machined by the focused ion beam technique, Acta Mater. 53 (2005) 4269e4279. [7] D.C.C. Lam, F. Yang, A.C.M. Chong, J. Wang, P. Tong, Experiments and theory in strain gradient elasticity, J. Mech. Phys. Solids 51 (2003) 1477e1508. [8] N. Garg, C.-S. Han, A penalty finite element approach for couple stress elasticity, Comput. Mech. 52 (2013) 709e720.
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[9] N.A. Fleck, G.M. Muller, M.F. Ashby, J.W. Hutchinson, Strain gradient plasticity: theory and experiment, Acta Metall. Mater. 42 (1994) 475e487. [10] I. Tsagrakis, E.C. Aifantis, Recent developments in gradient plasticity e part I: formulation and size effects, ASME J. Eng. Mat. Technol. 124 (2002) 352e357. [11] Y. Huang, S. Qu, K.C. Hwang, M. Li, H. Gao, A conventional theory of mechanism-based strain gradient plasticity, Int. J. Plasticity 20 (2004) 753e782. [12] C.-S. Han, Influence of the molecular structure on indentation size effect in polymers, Mater. Sci. Eng. A 527 (2010) 619e624. [13] D.C.C. Lam, A.C.M. Chong, Indentation model and strain gradient plasticity law for glassy polymers, J. Mater. Res. 14 (1999) 3784e3788. [14] A.C.M. Chong, D.C.C. Lam, Strain gradient plasticity effect in indentation hardness of polymers, J. Mater. Res. 14 (1999) 4103e4110. [15] D.C.C. Lam, A.C.M. Chong, Effect of cross-link density on strain gradient plasticity in epoxy, Mater. Sci. Eng. A 281 (2000) 156e161. [16] B.J. Briscoe, L. Fiori, E. Pelillo, Nanoindentation of polymeric surfaces, J. Phys. D 31 (1998) 2395e2405. [17] Y.Y. Lim, M.M. Chaudhri, Indentation of elastic solids with a rigid Vickers pyramidal indenter, Mech. Mater. 38 (2006) 1213e1228. [18] C. Charitidis, Nanoscale deformation and nanomechanical properties of polydimethylsiloxane (PDMS), Ind. Eng. Chem. Res. 50 (2011) 565e570. [19] R.V.S. Tatiraju, C.-S. Han, Rate dependence of indentation size effects in filled silicone rubber, J. Mech. Mater. Struc. 5 (2010) 277e288. [20] A.J. Wrucke, C.-S. Han, P. Majumdar, Indentation size effect of multiple orders of magnitude in polydimethylsiloxane, J. Appl. Polym. 128 (2013) 258e264. [21] F. Alisafaei, C.-S. Han, S.H.R. Sanei, On the time and indentation depth dependence of hardness, dissipation and stiffness in polydimethylsiloxane, Polym. Test. 32 (2013) 1220e1228. [22] S. Nikolov, C.-S. Han, D. Raabe, On the origin of size effects in smallstrain elasticity of solid polymers, Int. J. Solids Struc. 44 (2007) 1582e1592. Corrigendum in Int. J. Sol. Struc., 44(2007), 7713e7713. [23] C.-S. Han, S. Nikolov, Indentation size effects of polymers and related rotation gradients, J. Mater. Res. 22 (2007) 1662e1672. [24] T.-Y. Zhang, W.-H. Xu, Surface effects on nanoindentation, J. Mater. Res. 17 (2002) 1715e1720.
[25] T.-Y. Zhang, W.-H. Xu, M.-H. Zhao, The role of plastic deformation of rough surfaces in the size-dependent hardness, Acta Mater. 52 (2004) 57e68. [26] ISO 14577-1 Metallic Materials e Instrumented Indentation Test for Hardness and Materials Parameters e Part 1: Test Method, International Organization for Standardization, 2002. [27] W. Chen, M. Li, T. Zhang, Y.-T. Cheng, C.-M. Cheng, Influence of indenter tip roundness on hardness behavior in nanoindentation, Mater. Sci. Eng. A 445e446 (2007) 323e327. [28] Y.-T. Cheng, C.-M. Cheng, Further analysis of indentation loading curves: effects of tip rounding on mechanical property measurements, J. Mater. Res. 13 (1998) 1059e1064. [29] N. Yu, A.A. Polycarpou, T.F. Conry, Tip-radius effect in finite element modeling of sub-50 nm shallow nanoindentation, Thin Solid Films 450 (2004) 295e303. [30] J.-Y. Kim, B.-W. Lee, D.T. Read, D. Kwon, Influence of tip bluntness on the size-dependent nanoindentation hardness, Scripta Mater. 52 (2005) 353e358. [31] M. Troyon, L. Huang, Correction factor for contact area in nanoindentation measurements, J. Mater. Res. 20 (2005) 610e617. [32] X.-G. Tang, M. Hou, R. Truss, J. Zou, W. Yang, Z.-G. Dong, H. Huang, An unexpected plasticization phenomenon and a constant of the change rate of viscoelastic properties for polymers during nanoindentation test, J. Appl. Polym. 122 (2011) 885e890. [33] T.F. Juliano, M.R. VanLandingham, C.A. Tweedie, K.J. Van Vliet, Multiscale creep compliance of epoxy networks at elevated temperatures, Exp. Mech. 47 (2007) 99e105. [34] M.L. Oyen, Analytical techniques for indentation of viscoelastic materials, Phil. Mag 86 (2006) 5626e5641. [35] W.D. Nix, H. Gao, Indentation size effects in crystalline materials: a law for strain gradient plasticity, J. Mech. Phys. Solids 46 (1998) 411e425. [36] W.W. Gerberich, N.I. Tymiak, J.C. Grunlan, M.F. Horstemeyer, M.I. Baskes, Interpretations of indentation size effects, ASME J. Appl. Mech. 69 (2002) 433e442. [37] C.-S. Han, S.H.R. Sanei, F. Alisafaei, On the Origin of Indentation Size Effects and Depth Dependent Elastic Moduli in Polymers, Submitted, 2014.
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Polymer Testing journal homepage: www.elsevier.com/locate/polytest
Material properties
Characterization of indentation size effects in epoxy F. Alisafaei*, Chung-Souk Han 1, Nishant Lakhera 2 University of Wyoming, Department of Mechanical Engineering, 1000 E. University Ave., Laramie, WY, 82071, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 13 July 2014 Accepted 25 August 2014 Available online 3 September 2014
Nanoindentation tests were conducted at different indentation depths, ranging from 3000 to 30 nm, to study the characteristics of indentation size effects in epoxy. Taking the effects of tip imperfection/bluntness into account, nanoindentation tests performed with a threesided Berkovich tip exhibited strong increases in the hardness with decreasing indentation depth. However, comparative testing with a spherical tip with a radius of 250 microns, showed no changes in the material properties with respect to the distance from the surface. Consequently, different deformation mechanisms must be present in the indentation tests with Berkovich and spherical tips, suggesting effects related to second order gradients of the displacement (such as strain or rotation gradients), which increase with a Berkovich tip but essentially stay constant with a spherical tip when indentation depth decreases. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Polymers Nanoindentation Length scale dependent deformation
1. Introduction Indentation probing of materials [1e3] on the microand nanometer length scales has revealed size dependent deformation of various materials [4e5]. In addition to micro and nanoindentation tests, size effects have been also observed in micro beam bending tests [6e7] (see [8] for corresponding finite element simulation), microtorsion of thin wires [9] and tension of geometrically similar perforated plates (plates with central holes) [10]. In metals, such size effects at micron and submicron length scales are usually associated with geometrically necessary dislocations arising from inhomogeneous plastic deformation [11]. Such a notion cannot be applied to size effects in polymers [12e16] as, in contrast to metals, length scale dependent deformation has been also observed in pure elastic deformation [7,17e21]. However, the literature on
the rationale for size effects in polymers is quite sparse [3,13e25]. In order to shed light on the characteristics of indentation size effects in polymers, polydimethylsiloxane (PDMS) had been previously studied in [21] where plastic deformation was essentially absent. Here, epoxy is chosen as it shows significant plastic deformation and has been studied before in indentation [13,15] and beam bending [7]. The purpose of this study is the corroboration of previous results in the literature on epoxy and an in-depth examination of indentation size effect characteristics that have not been published before. In addition to the indentation depth dependent hardness (already studied in [13,15]), here also the indentation depth dependent characteristics of dissipation, plastic deformation and stiffness are examined at the indentation depth range of 30 to 3000 nm. 2. Experiments
* Corresponding author. Tel.: þ1 307 343 3589. E-mail addresses: [email protected] (F. Alisafaei), [email protected] (C.-S. Han), [email protected] (N. Lakhera). 1 Tel.: þ1 307 766 2930. 2 Current address: Freescale Semiconductor Inc., 6501 W. William Cannon Drive, Austin, TX 78735, USA. Tel.: þ1 307 399 1796. http://dx.doi.org/10.1016/j.polymertesting.2014.08.012 0142-9418/© 2014 Elsevier Ltd. All rights reserved.
2.1. Sample preparation The epoxy samples were prepared by mixing the commercially available liquid epoxy resin, EPON 828 (bisphenol A-epichlorohydrin), as base agent with the
F. Alisafaei et al. / Polymer Testing 40 (2014) 70e78
curing agent EPIKURE 3223, diethylenetriamine (DETA). The required amount of epoxy resin (Momentive Specialty Chemicals, USA) was measured at room temperature in an aluminum dish (Fisherbrand, low form, weighing dish, 70 ml), to which DETA (Momentive Specialty Chemicals, USA) with 20 phr (parts per hundred resin) was added. The mixture was manually stirred with a rod for 2 minutes, after which the samples were kept in a vacuum chamber (Alcatel Drytel vacuum pump, 22 inches Hg pressure) for 20 minutes to remove trapped air. The samples were then cured in a conventional oven (Fisher-Scientific Isotemp force draft) at 100 C for 3 hours and slowly cooled down to room temperature. Finally, the samples were removed from the aluminum dish and annealed at 100 C for 2 hours to eliminate residual stresses. 2.2. Nanoindentation system
0.5
B
C
3. Experimental results and analysis 3.1. Universal hardness for an ideally sharp indenter tip The Martens hardness (also known as Universal hardness) HU is defined as [26]
HU ¼
0.3
Fmax ; As
(1)
where As is the nominal surface area of the indenter tip that has penetrated the original surface under the maximum applied force Fmax. Due to tip roundness/imperfections, according to [26], this definition should be only applied for indentation depths larger than 6 mm where the tip roundness is negligible. With a tip roundness radius of less than 20 nm, according to the manufacturer, our Berkovich tip is arguably very sharp where imperfections of the tip are negligible for indentation depths above about 250 nm, as will be seen below. Therefore, in a preliminary examination the uncorrected universal hardness
HUuncorrected ¼
Fmax ; Auncorrected s
(2)
is considered for simplicity where Auncorrected is the nominal s surface area of an ideally sharp indenter tip (without considering the effect of tip bluntness). The nominal surface area Auncorrected is obtained by s
Auncorrected ¼ Ch2 ; s
(3)
where h is the indention depth at Fmax (point B in Fig. 1) and C is a constant equal to 26.43 for a Berkovich indenter tip [26]. Fig. 2 shows HUuncorrected of epoxy at different h ranging
700 600
U
0.4
F (mN)
loading force was applied linearly in time and a loading time of 20 s was used for all spherical tests.
H uncorrected (MPa)
The nanoindentation tests were conducted using an Agilent G200 Nano Indenter system (Agilent Technologies, Oak Ridge, TN, USA) equipped with a Berkovich indenter tip which, according to the manufacturer's specifications, has a curvature radius of less than 20 nm. All Berkovich indentation tests were performed at 23 C using a dynamic contact module (DCM) in force controlled mode with linear loading and unloading force-time relations. A surface s s detection tolerance of Ktol ¼ 50 N/m was used, where Ktol is the stiffness threshold to recognize initial contact with the surface of the sample (see [21] for more details). The minimum distance between two individual indents was 50 mm. Fig. 1 depicts a typical load-displacement curve consisting of loading (section A-B with 20 s loading time), holding (section B-C with 50 s holding time) and unloading (section C-D with 20 s unloading time) sequences. Nanoindentation tests were carried out at different indentation depths ranging from 3000 to 30 nm associated with loading forces of about 32000 to 12 mN. In addition to the Berkovich indentation tests, a spherical tip with a curvature radius of 250 mm was also applied to probe the samples at indentation depths between 400 and 3700 nm. All spherical tests were conducted in force s controlled mode with an XP head and Ktol of 200 N/m. The
71
500 400 300
0.2
200
0.1
100 10
100
1,000
h (nm)
A
0 0
D 50
100
150
200
250
Indentation Depth (nm) Fig. 1. Typical load-displacement data.
300
350
Fig. 2. Uncorrected universal hardness of epoxy as a function of indentation depth h obtained with 5 s loading time. Each data point represents 3 indentations with an error bar being the standard error calculated by the standard deviations of averages divided by the square root of 3 (sample size).
F. Alisafaei et al. / Polymer Testing 40 (2014) 70e78
from 3000 to 30 nm obtained with 5 s loading time (note that h is measured at point B shown in Fig. 1). Each data point in Fig. 2 represents the average of 3 indentations, where an error bar has been added for each data point. As all indentation tests have been carried out in force controlled mode, the variation of h with respect to a specified Fmax could be shown. The related variation in h were however so small that the error bars for h have been omitted in Fig. 2. Strong increases in HUuncorrected are observed in Fig. 2 with decreasing h. As mentioned above, the results in Fig. 2 do not take the tip imperfection into account. Therefore, particularly for h < 200 nm, the experimental results should be adjusted to reflect the effect of tip bluntness. 3.2. Tip bluntness/imperfections
1
10
A
c
Auncorrected c
Acorrected c
1 0.1 0.01 0.001
10
100
1,000
h (nm) c Fig. 4. Different contact areas (AC ; Auncorrected ; Acorrected ) versus contact depth C C hc obtained for fused silica.
Relation (3) for the nominal surface area As is valid for an ideally sharp tip or for sufficiently large h where the tip bluntness can be neglected. To assess the influence of tip bluntness, the pyramidal Berkovich indenter tip is modeled as an equivalent conical indenter tip so that the cone with an ideally sharp tip would have the same volume (and area) to depth relation as that of the ideally sharp Berkovich indenter tip [27e28]. Fig. 3 illustrates the equivalent conical indenter tip with a semi apex angle, q, equal to 70.3 [29]. The blunt indenter tip is modeled by a cone smoothly fitted with a spherical cap of radius R as shown in Fig. 3 [27]. The radius of the spherical cap has been determined by two different approaches. In the first approach [30], the contact area Ac (the sample area with which the tip is in actual contact) is determined as a function of contact depth hc (see [1] for its definition) from nanoindentation tests on fused silica (known not to exhibit length scale dependent deformation). The projected contact area Ac was obtained in the following form 1
100
Contact area ( µ m2)
72
1
Ac ¼ C0 h2c þ C1 hc þ C2 h2c þ C3 h4c þ C4 h8c ;
(4)
where C0, C1,…,C4 are constants determined by curve fitting [1,29]. In Fig. 4, the projected contact area Ac with respect to contact depth hc is illustrated. It can be seen that at small hc there is a significant difference between
the projected contact area Ac and the contact area of a which can be perfect Berkovich indenter tip Auncorrected c obtained by Auncorrected ¼ 24:56h2c : c
The difference between Ac and Auncorrected observed in c Fig. 4 illustrates the significance of the tip bluntness at small indentation depths. To compensate for the tip bluntness and to obtain the tip curvature radius R, the projected contact area Ac was fitted with the contact area of a rounded tip, Acorrected , which can be obtained as [30]. c 2
Acorrected ¼ 24:56ðhc þ hb Þ ; c
(6)
where hb is the difference in depth between the sharp and blunt tips, as shown in Fig. 3. Fitting of the projected contact area Ac with Eq. (6), see Fig. 4, hb was found to be 2.5 nm. For hb ¼ 2.5 nm, the tip curvature radius R was determined as 40.2 nm by geometrical considerations of Fig. 3 as hb : R¼ 1 (7) 1 sin q The tip curvature radius R was also determined by second approach applying Hertzian contact theory on fused silica at a very low loading force of F ¼ 1 mN [31] with its relation to h given as
4 E F ¼ R1=2 h3=2 ; 3 s 1 v2
Fig. 3. Schematic illustration of a conical indenter with a blunt tip.
(5)
(8)
where E and v are, respectively, elastic modulus and Poisson's ratio of the sample. Assuming a spherically shaped tip, applying such a small load and measuring the displacement h, the tip curvature radius R ¼ Rs can be obtained from Eq. (8) with the known elastic properties of fused silica (E and v) yielding R ¼ 24.4 nm, which is closer to the curvature radius of less than 20 nm specified by the manufacturer. However, in both cases, R is higher than the specified 20 nm which may indicate blunting of the tip due to use on a hard material like fused silica. Using the larger hb ¼ 2.5 nm and R ¼ 40.2 nm of the first approach, to be conservative, the universal hardness is
F. Alisafaei et al. / Polymer Testing 40 (2014) 70e78
2
Acorrected ¼ 26:43ðh þ hb Þ s
2
26:43R2 sin ð90 qÞtan2 ð90 qÞ
(9)
þ 2pR ð1 cosqÞ: 2
Fig. 5 illustrates the corrected universal hardness Fmax =Acorrected of epoxy over h. Compared with s
HUcorrected ¼ HUuncorrected
of Fig. 2, it can be seen in Fig. 5 that the tip roundness has considerable effect on the experimental results at small h. However, there is still a significant size effect in Fig. 5, illustrating that the tip imperfections cannot be the only source of depth dependent hardness in epoxy. To assess at what h the effect of tip bluntness becomes important, the corrected universal hardness HUcorrected is
plotted in Fig. 6 along with HUuncorrected (using Auncorrected ) s obtained with 5 s loading time. It can be seen that
700
H uncorrected 600
Universal Hardness (MPa)
recalculated to assess the effect of tip roundness on the size effect in epoxy. Therefore,Auncorrected in Eq. (2) is replaced s with the nominal surface area of a blunted tip, Acorrected , s which can be obtained by [20].
H
U corrected U
500 400 300 200 100 0 10
100
1,000
h (nm) Fig. 6. Effect of tip roundness on the indentation size effect in epoxy.
HUcorrected
starts to deviate from HUuncorrected below about h z 250 nm, which is almost 10 times higher than hb, while the effect of tip bluntness can be neglected for h > 250 nm. It may be worth noting that, particularly at small h, the data in Fig. 6 and the following experimental results is in parts less smooth, which may be related to surface roughness or possible local sample heterogeneity. 3.3. Corrected universal hardness and hardness model To assess the time dependence of the indentation size effects in epoxy, three different loading times of 5, 20, and 80 s were applied. Depth dependent hardness is observed for all loading times in Fig. 7 where HUcorrected increases with decreasing h. Fig. 7 also depicts the effect of loading time on
the hardness of epoxy. Consistent with the results reported in [32], the hardness decreases at large h with increasing loading time from 5 to 80 s. However, time dependence of the hardness markedly decreases with decreasing h (see [33e34] and references therein for time dependent behavior of polymeric materials during nanoindentation). The time-independent hardness model of Han and Nikolov [23] is considered here to analyze the size effect observed in our experiments. This model has been successfully applied to analyze the depth dependent deformation of various polymers [12,21]. The variation of this rotation gradient based hardness model with respect to h is given by
c[ ; H ¼ H0 1 þ h
(10)
where H0 (macroscopic hardness) is considered as the lower limit of H in H-h curve (see Fig. 7) and c[ is the
500 450
450
400
5 s , Experimental Results 20 s , Experimental Results 80 s , Experimental Results 5 s , Hardness Model (10) 20 s , Hardness Model (10) 80 s , Hardness Model (10)
400
H corrected (MPa)
350 300
U
H corrected (MPa)
73
300 250
U
250
350
200 150 10
100
1,000
200 150
h (nm) Fig. 5. Corrected universal hardness of epoxy as a function of indentation depth h obtained with 5 s loading time. Each data point represents 3 indentations with an error bar being the standard error calculated by the standard deviations of averages divided by the square root of 3 (sample size).
100 0
500
1000
1500
2000
2500
3000
h (nm) Fig. 7. Corrected universal hardness of epoxy together with the theoretical hardness model (10) for different loading times.
F. Alisafaei et al. / Polymer Testing 40 (2014) 70e78
material length scale parameter micromechanically motivated in [23]. Finding the parameters c[ and H0 by fitting Eq. (10) to the experimental data,3 Fig. 7 shows the resulting data obtained from the theoretical model together with the experimental results with reasonably good agreement between the experiments and the hardness model. 3.4. Change in stiffness with h In addition to tip bluntness, the accuracy of hardness values is also strongly dependent on a precise detection of the surface. To characterize the deformation behavior, the stiffness, i.e. Km ¼ vF=vh, at the maximum applied force is experimentally determined by calculating the derivative of the loading curve at the last data point (point B in Fig. 1) for each indentation test [21]. Therefore, Km is only dependent on the slope of force-depth curve (at h associated with the maximum applied force) and not on the absolute value of h at Fmax and, consequently, is not dependent on accurate surface detection. In Fig. 8, Km of epoxy experimentally obtained at different indentation depths is depicted for loading times of 5, 20 and 80 s (see [21] for more details of Km determination). Consistent with the hardness results of Fig. 7, it can be seen in Fig. 8 that at large depths (above 250 nm) the stiffness increases with decreasing loading time. Note that the same behavior is observed in Fig. 7 where at indentation depths above 250 nm the hardness increases with decreasing loading time. Also, similar to the hardness trend observed in Fig. 7, the stiffness of epoxy becomes time independent at small depths (nearly below 250 nm). It is noteworthy that in Fig. 8, the Km h relation becomes nonlinear at small indentation depths and Km does not seem to approach zero when h / 0. However, based on the classical local continuum theory (which does not account for length scale dependent deformation) Km is expected to approach zero when h / 0. This trend can be observed in the stiffness of fused silica (which does not exhibit size effects) shown in Fig. 9 where Km / 0 when h / 0. This behavior of epoxy can be also described with the hardness model (10) as set forth in the following.
Indentation Stiffness (kN / m)
74
5 s , Experimental Results 20 s , Experimental Results 80 s , Experimental Results 5 s , Stiffness Model (12) 20 s , Stiffness Model (12) 80 s , Stiffness Model (12) 5 s , Linear line K=2HoC h
10
1
10
100
Fig. 8. Indentation stiffness at Fmax versus h obtained for epoxy with different loading times.
of Fig. 7, the stiffness, K ¼ vF=vh, is obtained through the loading part as K ¼ H0 Cðc[ þ 2hÞ:
As depicted in Fig. 8 and discussed above, Km for epoxy appears to not approach zero when h / 0, illustrating the effect of size dependence on the indentation stiffness. A similar but much stronger trend has been also observed in PDMS where Km approximately reaches a plateau for small indentation depths [21]. Starting from a hardness model (10) that can be related to the universal hardness (1) as
120
where the parameters of H0 and c[ can be obtained by fitting the hardness model (10) to the experimental data
Km ( kN / m )
140
(11)
100 80 60 40 20 0 0
3
Fitted values are H0 ¼ 163 MPa and c[ ¼ 130 nm for 5 s loading time, H0 ¼ 158 MPa and c[ ¼ 135 nm for 20 s loading time, and H0 ¼ 145 MPa and c[ ¼ 150 nm for 80 s loading time.
(12)
The stiffness model (12), with the previously fitted parameters of H0 and c[, is depicted in Fig. 8 together with the experimentally determined indentation stiffness (as described in the previous section). As can be seen in Fig. 8, there is good agreement between the experiments and the theoretical stiffness model (12), corroborating the validity of the hardness model (10) to describe size effects in epoxy. Eq. (12) is also consistent with the trend observed in Fig. 8 where Km does not approach zero when h / 0. In Eq. (12), c[ becomes dominant over 2h at very small h. Therefore, for very small h, further decreases in h will result in negligible changes in the stiffness, which is in agreement with the experimental results obtained for Km in Fig. 8. For materials which do not exhibit size effects, i.e. c[ ¼ 0, Eq. (12) predicts
3.5. Theoretical analysis of Km
F c[ ; ¼ H0 1 þ Ch2 h
1,000
h (nm)
100
200
300
400
500
h (nm) Fig. 9. Indentation stiffness at Fmax versus h (fused silica).
600
F. Alisafaei et al. / Polymer Testing 40 (2014) 70e78
1
h /h f
a linear relation between stiffness and indentation depth, K ¼ 2H0Ch, with K / 0 for h / 0, which is consistent with the experimental results obtained for fused silica (see Fig. 9).
75
3.6. Dissipation and inelastic deformation
he ¼
WIe ; WI
5s 20 s 80 s
0.1 1
Welastic / Wtotal
To characterize the changes in indentation recovery with respect to h, the ratio of the final deformation hf (see Fig. 10) to the indentation depth at the maximum applied force, hf/h, is considered. The changes of hf/h with h are shown in Fig. 11 where again 5, 20, and 80 s are applied as loading and unloading times along with a holding time of 50 s. A decrease in the normalized parameter hf/h with decreasing h (particularly, for the range of 3000 to about 100 nm) is observed for all different loading times where hf/h is lower for higher loading times at all indentation depths. Similarly, the change in the ratio of the elastic indentation work WIe to the total indentation work WI ¼ WIe þ WIine (see Fig. 10),
0.1 10
100
1,000
h (nm)
Fig. 11. hf/h and he versus h (all tests were carried out with 50 s holding time together with the same times for loading and unloading).
(13) hardly any loading time dependence at large depths h > 2000 nm in PDMS). Thus, 20 phr epoxy (as a polymer in the glassy state) and PDMS (as a polymer in the rubbery state) have quite opposite characteristics with respect to loading time dependence and corresponding trends in dissipation. 3.7. Indentation hardness and elastic modulus Indentation hardness [1] can be also used to study the depth dependent plastic deformation as epoxy exhibits significant plastic deformation during indentation tests. Fig. 12 shows the increase of indentation hardness with decreasing h obtained for the same indentation depth range of 3000 to 30 nm with 20 s loading, 20 s holding, and 20 s unloading times (Poisson's ratio v ¼ 0.35). Comparing the indentation hardness of Fig. 12 with the universal
700
Indentation Hardness (MPa)
with h can be considered to illustrate changes in the deformation mechanism. This ratio has been also plotted in Fig. 11 where he increases with decreasing h for all loading/ unloading times. Furthermore, he decreases with decreasing loading time at different indentation depths, indicating that the energy dissipation (relative to the whole deformation) increases when the material is loaded/ unloaded faster. The changes in both normalized parameters hf/h and he with respect to h, as shown in Fig. 11, would indicate that the elastic part of the total deformation work increases with decreasing h. This behavior in epoxy is exactly opposite to that observed in PDMS and filled silicone rubber (there is a decrease in the elastic indentation recovery in both PDMS and filled silicone rubber with decreasing indentation depth [21,19]). In this respect, it should be also noted that the hardness and stiffness of epoxy become time independent at small depths (Figs. 7 and 8), while for PMDS (see [21]) the hardness is more loading time dependent at small h than at large h (there is
Force
500 400 300 200 100 0
Depth Fig. 10. Schematic loading-unloading curve.
Present Results Results in [15]
600
500
1000
1500
2000
2500
3000
h (nm) Fig. 12. Indentation hardness of epoxy (according to [1]) versus h with 20 s loading time, 20 s holding time and 20 s unloading time (Present Results).
76
F. Alisafaei et al. / Polymer Testing 40 (2014) 70e78
hardness of Fig. 7, the indentation hardness is slightly higher than the universal hardness as the contact area Ac (used for determining the indentation hardness) is usually smaller than the nominal area As. As shown in Fig. 12, the indentation hardness obtained from our experiments is in good agreement with an earlier study in [15] on an epoxy with 20% hardener. In addition to the indentation hardness shown in Fig. 12, the elastic modulus E has been also determined with the Berkovich tip following [1] for various h shown in Fig. 13. As can be seen therein, E (obtained with the Berkovich tip) increases with decreasing h which would indicate that the elastic modulus changes through depth, which is quite misleading. Performing experiments with a spherical indenter tip, it will be shown below that E does not change with h, as illustrated in Fig. 13.
4. Discussion 4.1. Influence of rotation gradients on the depth dependent deformation of polymers To illustrate the influence of rotation gradients on the indentation size effects observed in our experiments, comparative indentation tests were conducted using a spherical tip with a curvature radius of 250 mm. Rotation gradients are defined as cij ¼ 12 einm um;jn where um is the displacement field and einm is the permutation symbol. Corresponding to a geometrical analysis [35e36], indentations with such a spherical indenter tip will result in negligible changes in the rotation gradients cij with h (the rotation gradient cij is proportional to the radius of the spherical tip and essentially independent of h). Therefore, strain/rotation gradient based models [22e23] would not predict h dependence in the hardness. It has been also experimentally observed that applying such a spherical tip, rotation gradients essentially do not change with h [37]. In
Fig. 13, the elastic modulus of epoxy obtained with such a spherical tip (curvature radius of 250 mm) is shown where E has been determined according to the Hertzian theory (8) by assuming that the diamond spherical tip is rigid as the elastic modulus of diamond is much larger than that of epoxy. The lower limit of h has been restricted to 400 nm due to technical difficulties of applying a spherical tip at small h (such as effects of surface roughness and the presence of adhesion for which the Hertzian theory does not apply). Hardly any changes in E with h are present in Fig. 13 in the whole range of h ¼ 3700 to 400 nm, which is in agreement with the rotation gradient based models [22e23] as rotation gradients do not change with h when a spherical tip is applied. Therefore, results obtained for the elastic modulus shown in Fig. 13 illustrate that the elastic modulus of epoxy should be determined with caution at smaller h when a sharp pyramidal or conical tip is used as gradient effects may be present and, if possible, indentation tests should be performed at higher h to obtain reliable results. Consistent with the experimental results observed above, the size dependent deformation of polymers is here viewed as a result of an increase in rotation gradients cij (with decreasing h), which can be interpreted in terms of ~ F arising from the rotation gradients deformation work DW finite bending stiffness of polymer chains and their interactions (see [22e23] for more details). The total inden~ F yielding tation work WI can be augmented by DW I
~ F; WI ¼ WεIe þ WεIine þ DW I WεIe
(14) WεIine
where the local terms and are respectively the elastic and inelastic indentation works (the subscript ε indicates the local nature of these components). Also, note ~ F ¼ W e (because DW ~ F is reversible in nathat W e þ DW εI
I
I
I
ture) and WεIine ¼ WIine as represented in Fig. 10. According ~ F is expressed as a to [22e23], the nonlocal component DW I
function of rotation gradients by
Elastic Modulus (GPa)
10
Spherical indenter tip Berkovich indenter tip
8
6
As shown in [35e36], the higher order gradients cij increase with decreasing indentation depth h for sharp conical or pyramidal indenter tips. Based on the model developed in [22–23] and according to Eqs. (14) to (16), the ~ F will increase with decreasing nonlocal component DW I indentation depth h resulting in the higher hardness values observed in the experiments.
4
2
0 0
~ ~ F ¼ K cS cS ; (15) DW I 3 ij ij ~ is a constant parameter micromechanically where K motivated in [22e23] and cSij is the symmetric part of the rotation gradient tensor cij which is obtained as 1 (16) cSij ¼ cij þ cji ; 2
1000
2000
3000
4000
h (nm) Fig. 13. Elastic modulus of epoxy determined with a Berkovich indenter tip (according to Oliver & Pharr's approach [1] with 20 s loading, 20 s holding, and 20 s unloading times) and with a spherical indenter tip (via Hertzian theory with 20 s loading time).
4.2. Strain/rotation gradient based hardness models As discussed earlier, based on the rotation gradient theory [22e23], the indentation size effects observed in the experiments can be explained by increases in rotational
F. Alisafaei et al. / Polymer Testing 40 (2014) 70e78
gradients with decreasing h. However, in addition to the above mentioned rationale, a different strain gradient theory is suggested to explain indentation size effects in polymers. To predict strain gradient dependent deformation in polymers, Lam and Chong [13] developed a strain gradient plasticity model on the basis of molecular theory of yield for glassy polymers. In this model, the depth dependent deformation of polymers is motivated by a molecular kinking mechanism [14] which, similar to metals, is related to the plastic deformation. In this model, the proposed hardness varies with h as
H ¼ H
qffiffiffiffiffiffiffiffiffiffi 1 þ h =h ;
(17)
where, similar to the macroscopic hardness H0 in Eq. (10), H* is the hardness at infinite depth and h* is a constant dependent on material behavior and temperature [14]. Both parameters H* and h* are obtained by fitting the hardness model (17) to the experimental hardness data of Fig. 7. Based on the theoretical hardness model (17), similarly to obtaining Eq. (12), a stiffness model can be deduced as
KL&C ¼ H C
pffiffiffiffiffi 3 h h1=2 þ 2h ; 2
(18)
Indentation Stiffness (kN / m)
where KL&C is the indentation stiffness obtained from the hardness model of Lam and Chong [13]. It should be noted that, in contrast to the stiffness model (12), KL&C approaches zero when h / 0, which is not in agreement with the experimental results shown in Fig. 8. As mentioned in Subsection 3.4 and shown in Fig. 8, the material stiffness Km does not approach zero at small indentation depths. This trend would not be predicted by the stiffness model (18) as KL&C / 0 when h / 0. In Fig. 14, the theoretical stiffness models (12) and (18) are compared with the experimental results obtained with 80 s loading time. It can be seen that the stiffness model (12) is in better agreement with the
10
80 s , Experimental Results 80 s , Stiffness Model (12) 80 s , Stiffness Model (18)
100
experimental data, particularly at small h where Km does not seem to approach zero. The hardness models (17) and (10) are also based on different physical explanations of size effect in polymers. In the model developed in [13], size effect in polymers is explained by molecular kinking mechanisms associated with plastic deformation. Therefore, this model is not capable of predicting size effects in pure elastic deformation mechanisms observed by the authors [19e21] and others [7,17e18]. 5. Conclusions Indentation depth dependent deformation behavior of epoxy was experimentally observed by nanoindentation testing conducted at indentation depths ranging from 3000 to 30 nm with tip bluntness being carefully accounted for at small indentation depths. The indentation size effects and the change in the stiffness characteristics of epoxy observed in these experiments were analyzed by the indentation depth dependent based hardness and stiffness models, and good agreement was found between models and experimental results. To detect any changes of the material properties through depth and to demonstrate the influence of rotation gradients on the indentation size effect in epoxy, comparative tests with a spherical indenter tip were performed to obtain the elastic modulus at different probing depths. No changes in the elastic modulus were determined with probing depth which is in agreement with the rotation gradient based model [22e23] illustrating the significance of rotation gradients in the indentation size effects of epoxy.
Acknowledgments The support of this work through the National Science Foundation, Grants CMMI 1102764 and CMMI 1126860, is highly appreciated. The spherical indentation tests of Fig. 13 were conducted by Mr. S.H.R. Sanei which should be also acknowledged here.
References
1
10
77
1,000
h (nm) Fig. 14. Comparison of the indentation stiffness models (12) and (18) with experimental data.
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