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substrate system during indentation by a flat cylindrical punch
Scripta Materialia 55 (2006) 315–318 www.actamat-journals.com
An improved relation for the effective elastic compliance of a film/substrate system during indentation by a flat cylindrical punch Haitao Xua and G.M. Pharra,b,* a
University of Tennessee, Department of Materials Science and Engineering, Knoxville, TN 37996-2200, United States b Oak Ridge National Laboratory, Metals and Ceramics Division, Oak Ridge, TN 37831-6093, United States Received 19 January 2006; accepted 28 April 2006 Available online 26 May 2006
Measurement of the mechanical properties of thin films on substrates by load and depth sensing indentation methods such as nanoindentation often requires accurate descriptions for the effective elastic compliance of the film/substrate system. Here, a simple modification of the commonly used solution derived by Gao et al. [H. Gao, C.H. Chiu, J. Lee, Int. J. Solids Struct. (1992) 2471] is presented, that significantly improves its accuracy and range of applicability, as demonstrated by comparison with finite element simulations. Ó 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Nanoindentation; Elastic modulus; Thin films; Finite element analysis
The displacements of the indenter measured during nanoindentation of a thin film of one material deposited on a substrate of another clearly depend on the mechanical properties of both the film and the substrate [2,3]. As a consequence, accurate measurement of the film’s mechanical properties, like the elastic modulus from nanoindentation load–displacement data, can often be achieved only with good analytical solutions for the effective elastic compliance of the film/substrate system. Unfortunately, this is a complex problem for which no exact solution exists, even for the simplest indenter geometries, and one is thus forced to rely on approximations. Of the numerous approximate solutions that have been developed [1,4–9], those involving no adjustable parameters are generally the most valuable since there is usually no convenient way to determine the adjustable parameters experimentally without knowing the film properties a priori. In this regard, the approximate solutions for indentation by a flat cylindrical punch developed by Gao et al. [1], Bec et al. [8], and King [4] as implemented by Saha and Nix [9] have received considerable attention. Gao’s solution has proven especially useful because it can be written in a simple, closed algebraic form. A recent review by Mencik et al. [10] con* Corresponding author. Address: The University of Tennessee, Department of Materials Science and Engineering, Knoxville, TN 37996-2200, United States; e-mail: [email protected]
cluded that the mathematical form of nanoindentation data obtained with a Berkovich triangular pyramidal indenter is well described by Gao’s solution, although no estimate of the accuracy of the solution could be provided because the properties of the films examined were not known independently. Gao’s solution is based on a first order elastic perturbation method in which a known exact solution for a homogeneous half-space is modified to account for different elastic constants in a thin region near the surface representing the film. The formulation is given in terms of the shear modulus, l, and Poisson’s ratio, m. As shown in Figure 1, the solution applies to the indentation of a thin film of thickness, t, on a semi-infinite substrate indented by frictionless, rigid, flat cylindrical punch of radius, a, loaded to a force, F, to produce a displacement, h. For a homogeneous material, the exact solution is given by [11] 1m dh ; ð1Þ ¼ 4a l dF where the material parameter (1 m)/l is called the effective compliance [3]. It can be determined experimentally from the known radius of the punch, a, and the measured compliance dh/dF. In Gao’s analysis, a perturbation to the half-space to account for a difference in the film and substrate properties is accomplished in two separate ways: either (1) the initially homogeneous
1359-6462/$ - see front matter Ó 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2006.04.037
316
H. Xu, G. M. Pharr / Scripta Materialia 55 (2006) 315–318
F a
h
μf , υ f
r
t
μS , υ S
z
Figure 1. Geometry used to describe indentation of a film/substrate system by a flat cylindrical punch.
material is taken to have the properties of the substrate with the surface layer perturbed to the properties of the film, or (2) the initially homogeneous material is given the properties of the film with the perturbation transforming the lower portion to the properties of the substrate. Letting lf and mf denote the properties of the film and ls and ms those of the substrate, the effective compliance when the perturbation occurs in the film is given by 1m 1 ms ðmf ms Þ ðl ls Þ I1 f ¼ 1 I0 ð2Þ l eff ð1 ms Þ ls ls where the weighting functions I0 and I1 are functions of the normalized film thickness n = t/a given by I 0 ðnÞ ¼
2 arctan n p 1 1 þ n2 n þ ð1 2ms Þn ln 2pð1 ms Þ 1 þ n2 n2 ð3Þ
and I 1 ðnÞ ¼
2 n 1 þ n2 arctan n þ ln : p p n2
ð4Þ
The mathematical form of Eq. (2) shows that the function I0 accounts for differences in shear modulus, whereas I1 accounts for Poisson’s ratio effects. Both weighting functions have simple limits, approaching zero as t/a ! 0 and increasing to unity as t/a ! 1. Although not given explicitly in Gao’s paper, the equivalent expression when the perturbation takes place in the substrate is given by 1m 1 mf ðms mf Þ ðl lf Þ ð1 I 1 Þ s ¼ 1 ð1 I 0 Þ : lf l eff ð1 mf Þ lf ð5Þ
Noting that Eqs. (2) and (5) must, to first order, give the same result, and that the solution must degenerate to the homogenous solution in the limiting cases t/a ! 0 (i.e., a homogeneous material with properties of the substrate) and t/a ! 1 (i.e., a homogeneous material with properties of the film), Gao suggested that the expressions be combined to the simple form: 1m 1 ms ðmf ms ÞI 1 ¼ : ð6Þ l eff ls þ ðlf ls ÞI 0
Comparison with finite element simulations showed that Eq. (6) is accurate to within 7% provided the film and substrate shear moduli vary by no more than a factor of 2 or 3. Larger variations were not considered, presumably due to the first order nature of the approximation. Although the limits when t/a ! 0 and t/a ! 1 are correctly described by Eq. (6), there are other physically important limits that must also be considered. One is the limit lf ! 1, for which Eq. (6) reduces to an effective compliance of zero, i.e., the film–substrate system is infinitely rigid. This is what is expected since the film as modeled is infinite in its lateral extent, implying that the force exerted by the indenter is distributed evenly through the rigid film over an infinite area, producing vanishingly small stresses in the substrate. Another important limit is ls ! 1, for which Eq. (6) reduces to 1m 1 ms ðmf ms ÞI 1 ð7Þ ¼ l eff ls ð1 I 0 Þ This limit, corresponding to a compliant film on a rigid substrate, is clearly incorrect since the effective compliance in this case should depend primarily on the shear modulus of the film, not the substrate. This leads to important inaccuracies in Gao’s solution when lf < ls. What we wish to show here is that the problem in Gao’s solution when lf < ls can largely be eliminated by a simple modification to the solution that maintains the spirit of the first order perturbation approximation. Finite element methods are used to show that the new relation is remarkably accurate, even when the substrate modulus is as much as 10 times greater than that of the film. On the other hand, the new relation does not give the correct limit when lf ! 1, but even so, its accuracy is still as good as or better than Eq. (6). In addition, it is shown that a simple weighted average of the two solutions can be used to further improve the accuracy. The new relation is derived by simple substitutions into Eqs. (2) and (5) and combination of the resulting equations using a slightly different method than that used by Gao. Without loss of first order accuracy, the term ls appearing in the denominator of the last term on the right hand side of Eq. (2) can be replaced by lf giving 1m 1 ms ðmf ms Þ ðlf ls Þ I1 ¼ 1 I 0 ; ð8Þ l eff ð1 ms Þ lf ls and the term lf appearing in the denominator of the last term on the right hand side of Eq. (5) can be replaced by ls to give: 1m 1 mf ðms mf Þ ðl lf Þ ð1 I 1 Þ s ¼ 1 ð1 I 0 Þ : lf l eff ð1 mf Þ ls ð9Þ
Making the substitution into Eq. (2) ensures that when lf < ls, the effective compliance depends primarily on the modulus of the film rather than the substrate. The substitution into Eq. (5) is needed to assure that the two equations can be combined in a self-consistent manner.
H. Xu, G. M. Pharr / Scripta Materialia 55 (2006) 315–318
This new relation for the effective compliance has a particularly simple form since the first term in square brackets on the right hand side incorporates all the Poisson ratio effects while the second term accounts for the effects of the shear modulus. The equation maintains the correct limits when t/a ! 0 and t/a ! 1, but more importantly, the rigid substrate limit ls ! 1 is given by 1m I0 ¼ ½1 ms þ ðms mf ÞI 1 ; ð11Þ l eff lf which depends only on the shear modulus of the film and not the substrate. On the other hand, the rigid film limit lf ! 1 is given by 1m 1 I0 ¼ ½1 ms þ ðms mf ÞI 1 ; ð12Þ l eff ls is not correct since the effective compliance should be zero in this case. In this regard, Eq. (10) might be expected to work better when lf < ls and Gao’s relation (Eq. (6)) when lf > ls. How well these relations perform is now assessed by finite element simulation. Indentation with a cylindrical punch was simulated using the ABAQUSÒ finite element code. The punch was modeled as a rigid cylinder with a radius of 1 lm and the film/substrate system as a cylinder 100 lm high and 100 lm in radius with the film bonded to the substrate. To determine the effects of different indenter radius to film thickness ratios a/t, the film thickness was varied from 0.1 lm to 10 lm. The punch and specimen were modeled as an axisymmetric body with a small element size of 0.040 lm in the contact area. Further away, a coarser mesh was employed with fixed boundary conditions at the bottom of the specimen and along its centerline. The sample was modeled as an elastically isotropic material with Young’s modulus E and Poisson’s ratio m. The modulus of the substrate was fixed at Es = 10 GPa while the film moduli were varied in the range Ef = 1–100 GPa. Poisson’s ratio for the film and substrate were fixed at 0.3 in all calculations, i.e., Poisson ratio effects were not considered. The indenter was driven in to a depth of 0.050 lm, with the specimen compliance determined from the load–displacement data generated in the simulations. To verify the mesh and simulation procedures, indentation of a homogeneous material (i.e., film and substrate properties the same) was examined for comparison to the well-known theoretical predictions [11]. The mesh was refined until the differences in the simulated and theoretical compliances were less than 1%. Results of the finite element simulations are compared to predictions of the new relation (Eq. (10)) in Figure 2 and Gao’s relation (Eq. (6)) in Figure 3. In both figures, the range of film to substrate shear moduli
[(1-ν)/μ]eff /[(1- νs )/μs ]
10.0
New Relation
1.0
ξ =t/a
0.1 0.1
FEA Eq.10
10.0 3.33 1.00 0.33 0.10
1.0
10.0
μf / μs Figure 2. Comparison of finite element analysis results (FEA) to predictions of the new relation for the effective compliance, Eq. (10).
10.0
[(1-ν)/μ]eff /[(1-νs )/μs ]
To combine the relations, we propose a slightly different combination scheme. Gao combined Eqs. (2) and (5) to the form of Eq. (6) using the first order approximation 1 a b ffi (1 a)/(1 + b) (a 1 and b 1). Here, we use 1 a b ffi (1 a)(1 b), which when combined with Eqs. (8) and (9) gives: 1m ð1 I 0 Þ I 0 ¼ ½1 ms þ ðms mf ÞI 1 þ ð10Þ l eff ls lf
317
Gao's Relation
1.0
ξ =t/a
0.1 0.1
FEA Eq.6
10.0 3.33 1.00 0.33 0.10
1.0
10.0
μf / μs Figure 3. Comparison of finite element analysis results (FEA) to predictions of the Gao’s relation for the effective compliance, Eq. (6).
is 0.1 6 lf/ls 6 10, that is, the modulus ratios span a range much greater than the factor of 2 considered in Gao’s work. Close inspection of the data shows that the new relation gives more accurate predictions over the entire range, but particularly so when lf < ls. In fact, for lf < ls, Eq. (10) is usually well within 10% of all the finite element results whereas Gao’s solution may deviate by as much as a factor of 2. The new relation thus provides a much better approximation, especially when the film is more compliant than the substrate. Further inspection of the results in Figures 2 and 3 reveals that the finite element results generally fall somewhere between the predictions of Eqs. (6) and (10), suggesting that a weighted average of the two solutions might be used to further improve the accuracy. Letting a be the weighting factor (0 6 a 6 1), the weighted effective compliance can be written as: 1m ð1 I 0 Þ I 0 ¼ a½1 ms þ ðms mf ÞI 1 þ l eff ls lf þ ð1 aÞ
1 ms ðmf ms ÞI 1 ; ls þ ðlf ls ÞI 0
ð13Þ
where a = 1 corresponds to the compliance given by the new relation and a = 0 to Gao’s relation. A numerical examination of the data revealed that the best value of
318
H. Xu, G. M. Pharr / Scripta Materialia 55 (2006) 315–318
Table 1. Weighting factors a that minimize the maximum difference between finite element analysis results and Eq. (13) for different values of n = t/a n = t/a
10.00 3.33 1.00 0.33 0.10
lf < ls
lf > ls
Weighting factor a
Maximum difference (%)
Weighting factor a
Maximum difference (%)
0.99 0.97 0.93 0.82 0.72
<1 <1 1.00 1.16 1.46
0.62 0.65 0.67 0.74 0.79
1.19 2.54 1.97 <1 <1
a depends on n = t/a in the manner prescribed in Table 1. Thus, if one knows the contact radius and the film thickness, the appropriate weighting factor can be estimated. Unfortunately, there may often be instances in experimental work when the contact radius is not known well, due, for instance, to pile-up or sink-in. In this case, it would be useful to have a single value that can be used to provide a reasonable approximation. By examining the data we have found that a = 0.89 works well for lf < ls (accuracy of better than 9% over the entire range), and a = 0.69 is a good choice for lf > ls (accuracy of better than 6% over the entire range). It should be noted that the accuracies listed here
are those for the extreme cases; for 95% of the finite element results, the accuracy is better than 5%. This work was sponsored by the Division of Materials Science and Engineering, Office of Basic Energy Sciences, US Department of Energy, and the SHaRE User Center at Oak Ridge National Laboratory, under contract DE-AC05-00OR22725 with UT-Battelle, LLC (GMP). Many of the original ideas for this work were developed in the Ph.D. dissertation of Dr. Haitao Song, Rice University, Houston, TX. [1] H. Gao, C.H. Chiu, J. Lee, Int. J. Solids Struct. (1992) 2471. [2] W.D. Nix, Metall. Trans. (1989) 2217. [3] G.M. Pharr, W.C. Oliver, MRS Bull. (1992) 28. [4] R.B. King, Int. J. Solids Struct. (1987) 1657. [5] H.Y. Yu, S.C. Sanday, B.B. Rath, J. Mech. Phys. Solids (1990) 745. [6] A.K. Bhattacharya, W.D. Nix, Int. J. Solids Struct. (1988) 1287. [7] X. Chen, J.J. Vlassak, J. Mater. Res. (2001) 2974. [8] S. Bec, A. Tonck, J.M. Georges, E. Georges, J.L. Loubet, Philos. Mag. (1996) 1061. [9] R. Saha, W.D. Nix, Acta Mater. (2002) 23. [10] J. Mencik, D. Munz, E. Quandt, E.R. Weppelmann, M.V. Swain, J. Mater. Res. (1997) 2475. [11] I.N. Sneddon, Int. J. Eng. Sci. (1965) 47.
An improved relation for the effective elastic compliance of a film/substrate system during indentation by a flat cylindrical punch Haitao Xua and G.M. Pharra,b,* a
University of Tennessee, Department of Materials Science and Engineering, Knoxville, TN 37996-2200, United States b Oak Ridge National Laboratory, Metals and Ceramics Division, Oak Ridge, TN 37831-6093, United States Received 19 January 2006; accepted 28 April 2006 Available online 26 May 2006
Measurement of the mechanical properties of thin films on substrates by load and depth sensing indentation methods such as nanoindentation often requires accurate descriptions for the effective elastic compliance of the film/substrate system. Here, a simple modification of the commonly used solution derived by Gao et al. [H. Gao, C.H. Chiu, J. Lee, Int. J. Solids Struct. (1992) 2471] is presented, that significantly improves its accuracy and range of applicability, as demonstrated by comparison with finite element simulations. Ó 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Nanoindentation; Elastic modulus; Thin films; Finite element analysis
The displacements of the indenter measured during nanoindentation of a thin film of one material deposited on a substrate of another clearly depend on the mechanical properties of both the film and the substrate [2,3]. As a consequence, accurate measurement of the film’s mechanical properties, like the elastic modulus from nanoindentation load–displacement data, can often be achieved only with good analytical solutions for the effective elastic compliance of the film/substrate system. Unfortunately, this is a complex problem for which no exact solution exists, even for the simplest indenter geometries, and one is thus forced to rely on approximations. Of the numerous approximate solutions that have been developed [1,4–9], those involving no adjustable parameters are generally the most valuable since there is usually no convenient way to determine the adjustable parameters experimentally without knowing the film properties a priori. In this regard, the approximate solutions for indentation by a flat cylindrical punch developed by Gao et al. [1], Bec et al. [8], and King [4] as implemented by Saha and Nix [9] have received considerable attention. Gao’s solution has proven especially useful because it can be written in a simple, closed algebraic form. A recent review by Mencik et al. [10] con* Corresponding author. Address: The University of Tennessee, Department of Materials Science and Engineering, Knoxville, TN 37996-2200, United States; e-mail: [email protected]
cluded that the mathematical form of nanoindentation data obtained with a Berkovich triangular pyramidal indenter is well described by Gao’s solution, although no estimate of the accuracy of the solution could be provided because the properties of the films examined were not known independently. Gao’s solution is based on a first order elastic perturbation method in which a known exact solution for a homogeneous half-space is modified to account for different elastic constants in a thin region near the surface representing the film. The formulation is given in terms of the shear modulus, l, and Poisson’s ratio, m. As shown in Figure 1, the solution applies to the indentation of a thin film of thickness, t, on a semi-infinite substrate indented by frictionless, rigid, flat cylindrical punch of radius, a, loaded to a force, F, to produce a displacement, h. For a homogeneous material, the exact solution is given by [11] 1m dh ; ð1Þ ¼ 4a l dF where the material parameter (1 m)/l is called the effective compliance [3]. It can be determined experimentally from the known radius of the punch, a, and the measured compliance dh/dF. In Gao’s analysis, a perturbation to the half-space to account for a difference in the film and substrate properties is accomplished in two separate ways: either (1) the initially homogeneous
1359-6462/$ - see front matter Ó 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2006.04.037
316
H. Xu, G. M. Pharr / Scripta Materialia 55 (2006) 315–318
F a
h
μf , υ f
r
t
μS , υ S
z
Figure 1. Geometry used to describe indentation of a film/substrate system by a flat cylindrical punch.
material is taken to have the properties of the substrate with the surface layer perturbed to the properties of the film, or (2) the initially homogeneous material is given the properties of the film with the perturbation transforming the lower portion to the properties of the substrate. Letting lf and mf denote the properties of the film and ls and ms those of the substrate, the effective compliance when the perturbation occurs in the film is given by 1m 1 ms ðmf ms Þ ðl ls Þ I1 f ¼ 1 I0 ð2Þ l eff ð1 ms Þ ls ls where the weighting functions I0 and I1 are functions of the normalized film thickness n = t/a given by I 0 ðnÞ ¼
2 arctan n p 1 1 þ n2 n þ ð1 2ms Þn ln 2pð1 ms Þ 1 þ n2 n2 ð3Þ
and I 1 ðnÞ ¼
2 n 1 þ n2 arctan n þ ln : p p n2
ð4Þ
The mathematical form of Eq. (2) shows that the function I0 accounts for differences in shear modulus, whereas I1 accounts for Poisson’s ratio effects. Both weighting functions have simple limits, approaching zero as t/a ! 0 and increasing to unity as t/a ! 1. Although not given explicitly in Gao’s paper, the equivalent expression when the perturbation takes place in the substrate is given by 1m 1 mf ðms mf Þ ðl lf Þ ð1 I 1 Þ s ¼ 1 ð1 I 0 Þ : lf l eff ð1 mf Þ lf ð5Þ
Noting that Eqs. (2) and (5) must, to first order, give the same result, and that the solution must degenerate to the homogenous solution in the limiting cases t/a ! 0 (i.e., a homogeneous material with properties of the substrate) and t/a ! 1 (i.e., a homogeneous material with properties of the film), Gao suggested that the expressions be combined to the simple form: 1m 1 ms ðmf ms ÞI 1 ¼ : ð6Þ l eff ls þ ðlf ls ÞI 0
Comparison with finite element simulations showed that Eq. (6) is accurate to within 7% provided the film and substrate shear moduli vary by no more than a factor of 2 or 3. Larger variations were not considered, presumably due to the first order nature of the approximation. Although the limits when t/a ! 0 and t/a ! 1 are correctly described by Eq. (6), there are other physically important limits that must also be considered. One is the limit lf ! 1, for which Eq. (6) reduces to an effective compliance of zero, i.e., the film–substrate system is infinitely rigid. This is what is expected since the film as modeled is infinite in its lateral extent, implying that the force exerted by the indenter is distributed evenly through the rigid film over an infinite area, producing vanishingly small stresses in the substrate. Another important limit is ls ! 1, for which Eq. (6) reduces to 1m 1 ms ðmf ms ÞI 1 ð7Þ ¼ l eff ls ð1 I 0 Þ This limit, corresponding to a compliant film on a rigid substrate, is clearly incorrect since the effective compliance in this case should depend primarily on the shear modulus of the film, not the substrate. This leads to important inaccuracies in Gao’s solution when lf < ls. What we wish to show here is that the problem in Gao’s solution when lf < ls can largely be eliminated by a simple modification to the solution that maintains the spirit of the first order perturbation approximation. Finite element methods are used to show that the new relation is remarkably accurate, even when the substrate modulus is as much as 10 times greater than that of the film. On the other hand, the new relation does not give the correct limit when lf ! 1, but even so, its accuracy is still as good as or better than Eq. (6). In addition, it is shown that a simple weighted average of the two solutions can be used to further improve the accuracy. The new relation is derived by simple substitutions into Eqs. (2) and (5) and combination of the resulting equations using a slightly different method than that used by Gao. Without loss of first order accuracy, the term ls appearing in the denominator of the last term on the right hand side of Eq. (2) can be replaced by lf giving 1m 1 ms ðmf ms Þ ðlf ls Þ I1 ¼ 1 I 0 ; ð8Þ l eff ð1 ms Þ lf ls and the term lf appearing in the denominator of the last term on the right hand side of Eq. (5) can be replaced by ls to give: 1m 1 mf ðms mf Þ ðl lf Þ ð1 I 1 Þ s ¼ 1 ð1 I 0 Þ : lf l eff ð1 mf Þ ls ð9Þ
Making the substitution into Eq. (2) ensures that when lf < ls, the effective compliance depends primarily on the modulus of the film rather than the substrate. The substitution into Eq. (5) is needed to assure that the two equations can be combined in a self-consistent manner.
H. Xu, G. M. Pharr / Scripta Materialia 55 (2006) 315–318
This new relation for the effective compliance has a particularly simple form since the first term in square brackets on the right hand side incorporates all the Poisson ratio effects while the second term accounts for the effects of the shear modulus. The equation maintains the correct limits when t/a ! 0 and t/a ! 1, but more importantly, the rigid substrate limit ls ! 1 is given by 1m I0 ¼ ½1 ms þ ðms mf ÞI 1 ; ð11Þ l eff lf which depends only on the shear modulus of the film and not the substrate. On the other hand, the rigid film limit lf ! 1 is given by 1m 1 I0 ¼ ½1 ms þ ðms mf ÞI 1 ; ð12Þ l eff ls is not correct since the effective compliance should be zero in this case. In this regard, Eq. (10) might be expected to work better when lf < ls and Gao’s relation (Eq. (6)) when lf > ls. How well these relations perform is now assessed by finite element simulation. Indentation with a cylindrical punch was simulated using the ABAQUSÒ finite element code. The punch was modeled as a rigid cylinder with a radius of 1 lm and the film/substrate system as a cylinder 100 lm high and 100 lm in radius with the film bonded to the substrate. To determine the effects of different indenter radius to film thickness ratios a/t, the film thickness was varied from 0.1 lm to 10 lm. The punch and specimen were modeled as an axisymmetric body with a small element size of 0.040 lm in the contact area. Further away, a coarser mesh was employed with fixed boundary conditions at the bottom of the specimen and along its centerline. The sample was modeled as an elastically isotropic material with Young’s modulus E and Poisson’s ratio m. The modulus of the substrate was fixed at Es = 10 GPa while the film moduli were varied in the range Ef = 1–100 GPa. Poisson’s ratio for the film and substrate were fixed at 0.3 in all calculations, i.e., Poisson ratio effects were not considered. The indenter was driven in to a depth of 0.050 lm, with the specimen compliance determined from the load–displacement data generated in the simulations. To verify the mesh and simulation procedures, indentation of a homogeneous material (i.e., film and substrate properties the same) was examined for comparison to the well-known theoretical predictions [11]. The mesh was refined until the differences in the simulated and theoretical compliances were less than 1%. Results of the finite element simulations are compared to predictions of the new relation (Eq. (10)) in Figure 2 and Gao’s relation (Eq. (6)) in Figure 3. In both figures, the range of film to substrate shear moduli
[(1-ν)/μ]eff /[(1- νs )/μs ]
10.0
New Relation
1.0
ξ =t/a
0.1 0.1
FEA Eq.10
10.0 3.33 1.00 0.33 0.10
1.0
10.0
μf / μs Figure 2. Comparison of finite element analysis results (FEA) to predictions of the new relation for the effective compliance, Eq. (10).
10.0
[(1-ν)/μ]eff /[(1-νs )/μs ]
To combine the relations, we propose a slightly different combination scheme. Gao combined Eqs. (2) and (5) to the form of Eq. (6) using the first order approximation 1 a b ffi (1 a)/(1 + b) (a 1 and b 1). Here, we use 1 a b ffi (1 a)(1 b), which when combined with Eqs. (8) and (9) gives: 1m ð1 I 0 Þ I 0 ¼ ½1 ms þ ðms mf ÞI 1 þ ð10Þ l eff ls lf
317
Gao's Relation
1.0
ξ =t/a
0.1 0.1
FEA Eq.6
10.0 3.33 1.00 0.33 0.10
1.0
10.0
μf / μs Figure 3. Comparison of finite element analysis results (FEA) to predictions of the Gao’s relation for the effective compliance, Eq. (6).
is 0.1 6 lf/ls 6 10, that is, the modulus ratios span a range much greater than the factor of 2 considered in Gao’s work. Close inspection of the data shows that the new relation gives more accurate predictions over the entire range, but particularly so when lf < ls. In fact, for lf < ls, Eq. (10) is usually well within 10% of all the finite element results whereas Gao’s solution may deviate by as much as a factor of 2. The new relation thus provides a much better approximation, especially when the film is more compliant than the substrate. Further inspection of the results in Figures 2 and 3 reveals that the finite element results generally fall somewhere between the predictions of Eqs. (6) and (10), suggesting that a weighted average of the two solutions might be used to further improve the accuracy. Letting a be the weighting factor (0 6 a 6 1), the weighted effective compliance can be written as: 1m ð1 I 0 Þ I 0 ¼ a½1 ms þ ðms mf ÞI 1 þ l eff ls lf þ ð1 aÞ
1 ms ðmf ms ÞI 1 ; ls þ ðlf ls ÞI 0
ð13Þ
where a = 1 corresponds to the compliance given by the new relation and a = 0 to Gao’s relation. A numerical examination of the data revealed that the best value of
318
H. Xu, G. M. Pharr / Scripta Materialia 55 (2006) 315–318
Table 1. Weighting factors a that minimize the maximum difference between finite element analysis results and Eq. (13) for different values of n = t/a n = t/a
10.00 3.33 1.00 0.33 0.10
lf < ls
lf > ls
Weighting factor a
Maximum difference (%)
Weighting factor a
Maximum difference (%)
0.99 0.97 0.93 0.82 0.72
<1 <1 1.00 1.16 1.46
0.62 0.65 0.67 0.74 0.79
1.19 2.54 1.97 <1 <1
a depends on n = t/a in the manner prescribed in Table 1. Thus, if one knows the contact radius and the film thickness, the appropriate weighting factor can be estimated. Unfortunately, there may often be instances in experimental work when the contact radius is not known well, due, for instance, to pile-up or sink-in. In this case, it would be useful to have a single value that can be used to provide a reasonable approximation. By examining the data we have found that a = 0.89 works well for lf < ls (accuracy of better than 9% over the entire range), and a = 0.69 is a good choice for lf > ls (accuracy of better than 6% over the entire range). It should be noted that the accuracies listed here
are those for the extreme cases; for 95% of the finite element results, the accuracy is better than 5%. This work was sponsored by the Division of Materials Science and Engineering, Office of Basic Energy Sciences, US Department of Energy, and the SHaRE User Center at Oak Ridge National Laboratory, under contract DE-AC05-00OR22725 with UT-Battelle, LLC (GMP). Many of the original ideas for this work were developed in the Ph.D. dissertation of Dr. Haitao Song, Rice University, Houston, TX. [1] H. Gao, C.H. Chiu, J. Lee, Int. J. Solids Struct. (1992) 2471. [2] W.D. Nix, Metall. Trans. (1989) 2217. [3] G.M. Pharr, W.C. Oliver, MRS Bull. (1992) 28. [4] R.B. King, Int. J. Solids Struct. (1987) 1657. [5] H.Y. Yu, S.C. Sanday, B.B. Rath, J. Mech. Phys. Solids (1990) 745. [6] A.K. Bhattacharya, W.D. Nix, Int. J. Solids Struct. (1988) 1287. [7] X. Chen, J.J. Vlassak, J. Mater. Res. (2001) 2974. [8] S. Bec, A. Tonck, J.M. Georges, E. Georges, J.L. Loubet, Philos. Mag. (1996) 1061. [9] R. Saha, W.D. Nix, Acta Mater. (2002) 23. [10] J. Mencik, D. Munz, E. Quandt, E.R. Weppelmann, M.V. Swain, J. Mater. Res. (1997) 2475. [11] I.N. Sneddon, Int. J. Eng. Sci. (1965) 47.