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Journal of the Mechanics and Physics of Solids

Journal of the Mechanics and Physics of Solids 60 (2012) 1753–1756

0022-50

http://d

DOI of on Corr

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journal homepage: www.elsevier.com/locate/jmps

Erratum

Erratum to ‘‘Analysis of uniaxial compression of vertically alignedcarbon nanotubes’’ [Journal of the Mechanics and Physics of Solids59 (2011) 2227–2237]

Shelby B. Hutchens a,n, Alan Needleman c, Julia R. Greer b

a Division of Chemistry and Chemical Engineering, California Institute of Technology, 1200 E. California Blvd, MC 210-41, Pasadena, CA 91106, USAb Division of Engineering and Applied Science, California Institute of Technology, 1200 E. California Blvd, MC 309-81, Pasadena, CA 91106, USAc Department of Materials Science and Engineering, University of North Texas, 1155 Union Circle #305310, Denton, TX 76203, USA

a r t i c l e i n f o

Article history:

Received 31 May 2012

Accepted 4 June 2012Available online 23 June 2012

We found an error in the code that implemented the constitutive relation in our recent paper, Hutchens et al. (2011).The viscoplastic relation implemented was of the form

dp¼

3

2

_ep

se½s�bp trðsÞI� ð1Þ

instead of Eq. (2) of Hutchens et al. (2011) and with se given by

s2e ¼

3

2½s : s�apðtrðsÞÞ

2�: ð2Þ

The constitutive relation reported in Hutchens et al. (2011) was equivalent to ap ¼ bp ¼ 0:20 which satisfies plasticnormality whereas the constitutive relation actually implemented corresponded to ap ¼ 0:20 and bp ¼ 0:28, which givesrise to plastic non-normality. Another consequence of the programming error was an incorrect constitutive update. Theterm containing the error was of the order Dt and hence was small. Nevertheless, there was a significant effect on thecomputed overall stress–strain response in that using the corrected code gives rise to high frequency oscillations. This is amesh artifact, as discussed by Ballarin et al. (2009). The kinetic energy is only a small fraction of the total work input,typically 0.1% or less over the strain range of interest, yet has a major effect on the overall stress–strain response. Since theresponse of interest is mainly quasi-static, the mesh induced oscillation effect can be minimized by computing the overallstress using the quasi-static expression

sn ¼1

UA0

ZVs : ðF�T

� ruÞ dV

�������� ð3Þ

where ð Þ�T denotes the transpose inverse, r is the gradient, U is the applied displacement and A0 is the original cross-sectional area at the loaded end.

96/$ - see front matter & 2012 Elsevier Ltd. All rights reserved.

x.doi.org/10.1016/j.jmps.2012.06.002

riginal article: http://dx.doi.org/10.1016/j.jmps.2011.05.002

esponding author.

ail addresses: hutch@alumni.caltech.edu, hutch@mail.pse.umass.edu (S.B. Hutchens).

0 0.05 0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

1

1.2

Fig. 1. A comparison of the overall nominal stress, sn, normalized by s0 versus true strain, Et , for a pillar as calculated with the quasi-static expression,

Eq. (3), (black) and using total nodal forces (gray). The material properties are those for Hutchens et al. (2011) Fig. 2 but with no property gradient.

Fig. 2. Comparison of the uniaxial response for the corrected calculation corresponding to that in Fig. 3 of Hutchens et al. (2011). (a) Normalized nominal

stress versus true strain responses for the corrected (black) and Fig. 3(b) of Hutchens et al. (2011) calculations (gray). The nominal stress for the corrected

results is computed using Eq. (3). (b) Outer displacement profiles for the corrected (black) and original (gray) calculations at Et ¼ 0:1 and 0.2.

S.B. Hutchens et al. / J. Mech. Phys. Solids 60 (2012) 1753–17561754

Fig. 1 compares values of the overall nominal stress, sn, calculated using the values of the nodal forces on thesurface where displacements are prescribed (pillar top) and for the quasi-static expression Eq. (3). As seen in Fig. 1, theonly noticeable difference in these two stress values occurs in the range of strains during which an instability is present. Amore complete discussion of the programming error and its effect on overall response is given in Needleman et al.(forthcoming).

Fig. 2 shows a comparison of the results obtained using the corrected code and calculating sn from Eq. (3) with those ofFigs. 3(b) and (c) from Hutchens et al. (2011). Little qualitative difference between the two sets of results is observed, andthe stresses are of the same magnitude.

The deformation mode parameter space computed with the corrected code is shown in Fig. 3. The main change in theparameter space is that some points on the boundary between regions were shifted from one deformation mode toanother. In particular, the regime giving rise to a buckling-like deformation mode is slightly smaller than originallyreported. The deformation modes in all regimes except the regime marked by blue circles in Fig. 3 were qualitatively, andoften quantitatively, similar to those shown in Fig. 5 of Hutchens et al. (2011). In the regime marked by blue circles in Fig. 3the amplitude of the displacement was significantly reduced.

Fig. 4 shows the effect of a flow strength gradient for the same sets of parameters used in Fig. 7 of Hutchens et al.(2011). The hardening slope during the instability portion of the stress–strain response and its dependence on the inputproperty gradient is very similar to that originally reported as shown in Fig. 4(b). The buckling-like deformation pattern isless regular for the extreme gradients (200%, 10% reverse), especially for the case of the reverse gradient shown in Fig. 4(d).

Fig. 3. (Color online) Influence of the position and depth of the ‘well’ minimum of the hardening function, g, on the deformation mode for parameters

identical to Fig. 5 of Hutchens et al. (2011). The points computed are denoted by the same symbols as used in Fig. 5 of Hutchens et al. (2011).

Fig. 4. (Color online) The effect of a property gradient on the pillar’s true stress–true strain response. (a) Input axial gradient, Q(z), applied to s0 to give

the overall responses in (b). (b) Overall true stress–true strain responses (symbols/color) compared with results from Hutchens et al. (2011), Fig. 7(b)

(gray). The true stress for the corrected results is computed as A=A0 times the nominal stress value obtained from Eq. (3), with A the current cross

sectional area at the loaded end. (c) and (d) Displacement profiles et ¼ 0:05, 0.10, 0.15, and 0.20 for 20% and 10%-reverse gradients, respectively.

S.B. Hutchens et al. / J. Mech. Phys. Solids 60 (2012) 1753–1756 1755

In Hutchens et al. (2012) a microstructurally motivated basis was postulated for varying the parameters of thehardening function g. Dependence of the wavelength and amplitude of the buckling-like deformation mode on thosevariations was presented. Results obtained using the corrected code for the effects of material parameter variations

S.B. Hutchens et al. / J. Mech. Phys. Solids 60 (2012) 1753–17561756

on the wavelength and amplitude of the buckling-like deformation mode will be presented in the errata for thatpaper.

It is interesting to note that calculations in Needleman et al. (forthcoming) with plastic normality, either withap ¼ bp ¼ 0:20 or with ap ¼ bp ¼ 0:28 (all other material parameters fixed), did not give rise to an instability with abuckling-like deformation pattern. Thus, at least for the range of material input parameters considered, plastic non-normality is needed to obtain the type of deformation mode seen in the experiments. Plastic non-normality is typicallyassociated with frictional type behavior and this may be a clue as to the physical mechanisms responsible for the inelasticresponse of VACNTs.

References

Ballarin, V., Soler, M., Perlade, A., Lemoine, S., Forest, S., 2009. Mechanisms and modeling of bake-hardening steels: Part I uniaxial tension. Metal. Mater.Trans. A 40, 1367–1374.

Hutchens, S.B., Needleman, A., Greer, J.R., 2011. Analysis of uniaxial compression of vertically aligned carbon nanotubes. J. Mach. Phys. Solids 59,2227–2237.

Hutchens, S.B., Needleman, A., Greer, J.R., 2012. A microstructurally motivated description of the deformation of vertically aligned carbon nanotubestructures. Appl. Phys. Lett. 100, 121910.

Needleman, A., Hutchens, S.B., Mohan, N., Greer, J.R. Deformation of plastically compressible hardening-softening-hardening solids. Acta Mech. Sin.,forthcoming.