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Erratum to: “Curvature-dependent surface energy and implications for nanostructures” [J. Mech. Phys. Solids 59 (2011) 2103–2115]
Journal of the Mechanics and Physics of Solids 60 (2012) 1241–1242
Contents lists available at SciVerse ScienceDirect
Journal of the Mechanics and Physics of Solids journal homepage: www.elsevier.com/locate/jmps
Erratum
Erratum to: ‘‘Curvature-dependent surface energy and implications for nanostructures’’ [J. Mech. Phys. Solids 59 (2011) 2103–2115] P. Chhapadia a, P. Mohammadi a, P. Sharma a,b,n a b
Department of Mechanical Engineering, University of Houston, Houston, TX 77204, USA Department of Physics, University of Houston, Houston, TX 77204, USA
a r t i c l e i n f o Article history: Received 27 December 2011 Accepted 4 January 2012 Available online 28 February 2012
In this erratum, we correct some errors present in our paper (Chhpadia et al., 2011). Our conclusions are unaltered: 1. Let en be the outward unit normal to the surface then Is ¼ PIP and P ¼ Ien en :
ð1Þ
P is the projection operator to the subspace orthogonal to en, and Is is the identity mapped to the tangential surface. So the Gurtin–Murdoch surface energy function is expressed as
g ¼ g0 þ t0 Is ES þ 12C0 E2S
ð2Þ
In the original paper, we omitted the term Is from t0 Is ES : 2. In Section 5, we used the orthogonal least squares method to fit the atomistic simulation results to the theoretical model. The values for coefficients C0 and C1 should be corrected as follows: For Fig. 6 (i.e.) /100S axially oriented nanowire, the coefficients are: C0 ¼ 0.22265 eV/A˚ 2 and C1 ¼ 2.15062 eV. For Fig. 7 (i.e.) /110S axially oriented nanowire, the coefficients are: C0 ¼1.71629 eV/A˚ 2 and C1 ¼7.68365 eV. For the two orientations we have analyzed, we ˚ obtain: t/100S ¼ 3.108, and t/110S ¼2.1158 A. 3. In addition, Eq. (47) should be corrected to (Miller and Shenoy, 2000) Eef f 4C 0 ¼ 1þ E aE
DOI of original article: 10.1016/j.jmps.2011.06.007 n Corresponding author at: Department of Mechanical Engineering, University of Houston, Houston, TX 77204, USA. Tel.: þ 1 713 743 4256. E-mail address: [email protected] (P. Sharma). 0022-5096/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2012.01.004
ð3Þ
1242
P. Chhapadia et al. / J. Mech. Phys. Solids 60 (2012) 1241–1242
4. Finally, Eq. (61) should be corrected to 2 1 6ðrtÞ 12h ðtu bu tl bl Þ 72hrtbu bl =E 72rtðcu þ cl Þ 72rtðcu þ cl Þðbu þ bl Þ=Eh ¼ þ 2 2 3 2 R Eh Eh Eh þ2h ðbu þbl Þ þ 12hbu bl =E þ 12ðcu þ cl Þ þ 12ðcu þ cl Þðbu þ bl Þ=Eh
ð4Þ
And consequently (62) is converted to 0 1 2 2h ðtu bu tl bl Þ=rt 12hbu bl =E 12ðcu þ cl Þ 12ðcu þ cl Þðbu þ bl Þ=Eh 1 6ðrtÞ @ ¼ 1þ A 2 3 2 R Eh Eh þ 2h ðbu þ bl Þ þ 12hbu bl =E þ12ðcu þcl Þ þ 12ðcu þ cl Þðbu þbl Þ=Eh
ð5Þ
Therefore, for a 2 nm thick beam with /1 1 0S lateral surfaces, (5) will give us ! 1 6ðrtÞ 12ðcu þcl Þ ¼ 1 2 3 R Eh ½Eh þ 12ðcu þ cl Þ 6ðrtÞ ð0:957Þ: ¼ 2 Eh
ð6Þ
We see that the curvature of the nanowire changes by a value of 4.21% when we include the Steigmann–Ogden correction rather than 17.21% as indicated in the original paper (Chhapadia et al., 2011). As in the original paper our conclusions are the same. Steigmann–Ogden correction is small in general. In some cases this correction is however, essential, e.g. in explaining the atomistically observed discrepancy of the elastic modulus under bending and tension, stability of surfaces when compression is involved and in defining the so-called surface thickness. References Chhapadia, P., Mohammadi, P., Sharma, P., 2011. Curvature-dependent surface energy and implications for nanostructures. J. Mech. Phys. Solids 59, 2103–2115. Miller, R.E., Shenoy, V.B., 2000. Size dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139–147.
Contents lists available at SciVerse ScienceDirect
Journal of the Mechanics and Physics of Solids journal homepage: www.elsevier.com/locate/jmps
Erratum
Erratum to: ‘‘Curvature-dependent surface energy and implications for nanostructures’’ [J. Mech. Phys. Solids 59 (2011) 2103–2115] P. Chhapadia a, P. Mohammadi a, P. Sharma a,b,n a b
Department of Mechanical Engineering, University of Houston, Houston, TX 77204, USA Department of Physics, University of Houston, Houston, TX 77204, USA
a r t i c l e i n f o Article history: Received 27 December 2011 Accepted 4 January 2012 Available online 28 February 2012
In this erratum, we correct some errors present in our paper (Chhpadia et al., 2011). Our conclusions are unaltered: 1. Let en be the outward unit normal to the surface then Is ¼ PIP and P ¼ Ien en :
ð1Þ
P is the projection operator to the subspace orthogonal to en, and Is is the identity mapped to the tangential surface. So the Gurtin–Murdoch surface energy function is expressed as
g ¼ g0 þ t0 Is ES þ 12C0 E2S
ð2Þ
In the original paper, we omitted the term Is from t0 Is ES : 2. In Section 5, we used the orthogonal least squares method to fit the atomistic simulation results to the theoretical model. The values for coefficients C0 and C1 should be corrected as follows: For Fig. 6 (i.e.) /100S axially oriented nanowire, the coefficients are: C0 ¼ 0.22265 eV/A˚ 2 and C1 ¼ 2.15062 eV. For Fig. 7 (i.e.) /110S axially oriented nanowire, the coefficients are: C0 ¼1.71629 eV/A˚ 2 and C1 ¼7.68365 eV. For the two orientations we have analyzed, we ˚ obtain: t/100S ¼ 3.108, and t/110S ¼2.1158 A. 3. In addition, Eq. (47) should be corrected to (Miller and Shenoy, 2000) Eef f 4C 0 ¼ 1þ E aE
DOI of original article: 10.1016/j.jmps.2011.06.007 n Corresponding author at: Department of Mechanical Engineering, University of Houston, Houston, TX 77204, USA. Tel.: þ 1 713 743 4256. E-mail address: [email protected] (P. Sharma). 0022-5096/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2012.01.004
ð3Þ
1242
P. Chhapadia et al. / J. Mech. Phys. Solids 60 (2012) 1241–1242
4. Finally, Eq. (61) should be corrected to 2 1 6ðrtÞ 12h ðtu bu tl bl Þ 72hrtbu bl =E 72rtðcu þ cl Þ 72rtðcu þ cl Þðbu þ bl Þ=Eh ¼ þ 2 2 3 2 R Eh Eh Eh þ2h ðbu þbl Þ þ 12hbu bl =E þ 12ðcu þ cl Þ þ 12ðcu þ cl Þðbu þ bl Þ=Eh
ð4Þ
And consequently (62) is converted to 0 1 2 2h ðtu bu tl bl Þ=rt 12hbu bl =E 12ðcu þ cl Þ 12ðcu þ cl Þðbu þ bl Þ=Eh 1 6ðrtÞ @ ¼ 1þ A 2 3 2 R Eh Eh þ 2h ðbu þ bl Þ þ 12hbu bl =E þ12ðcu þcl Þ þ 12ðcu þ cl Þðbu þbl Þ=Eh
ð5Þ
Therefore, for a 2 nm thick beam with /1 1 0S lateral surfaces, (5) will give us ! 1 6ðrtÞ 12ðcu þcl Þ ¼ 1 2 3 R Eh ½Eh þ 12ðcu þ cl Þ 6ðrtÞ ð0:957Þ: ¼ 2 Eh
ð6Þ
We see that the curvature of the nanowire changes by a value of 4.21% when we include the Steigmann–Ogden correction rather than 17.21% as indicated in the original paper (Chhapadia et al., 2011). As in the original paper our conclusions are the same. Steigmann–Ogden correction is small in general. In some cases this correction is however, essential, e.g. in explaining the atomistically observed discrepancy of the elastic modulus under bending and tension, stability of surfaces when compression is involved and in defining the so-called surface thickness. References Chhapadia, P., Mohammadi, P., Sharma, P., 2011. Curvature-dependent surface energy and implications for nanostructures. J. Mech. Phys. Solids 59, 2103–2115. Miller, R.E., Shenoy, V.B., 2000. Size dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139–147.