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Research Papers

Uniaxial Tension of a Class of Compressible Solids With Plastic Non-Normality

[+] Author and Article Information
Nisha Mohan

Division of Engineering and Applied Sciences,
California Institute of Technology,
Pasadena, CA 91125
e-mail: nmohan@caltech.edu

Justine Cheng

Polytechnic School,
Pasadena, CA 91106
e-mail: jcheng13@students.polytechnic.org

Julia R. Greer

Division of Engineering and Applied Sciences,
California Institute of Technology,
Pasadena, CA 91125
e-mail: jrgreer@caltech.edu

Alan Needleman

Department of Material Science and Engineering,
University of North Texas,
Denton, TX 76207
e-mail: needle@unt.edu

Manuscript received February 7, 2013; final manuscript received March 6, 2013; accepted manuscript posted April 10, 2013; published online May 31, 2013. Editor: Yonggang Huang.

J. Appl. Mech 80(4), 040912 (May 31, 2013) (8 pages) Paper No: JAM-13-1064; doi: 10.1115/1.4024179 History: Received February 07, 2013; Revised March 06, 2013; Accepted April 10, 2013

Motivated by a model that qualitatively captured the response of vertically aligned carbon nanotube (VACNT) pillars in uniaxial compression, we consider the uniaxial tensile response of a class of compressible elastic-viscoplastic solids. In Hutchens et al. [“Analysis of Uniaxial Compression of Vertically Aligned Carbon Nanotubes,” J. Mech. Phys. Solids, 59, pp. 2227–2237 (2011), Erratum 60, 1753–1756 (2012)] an elastic viscoplastic constitutive relation with plastic compressibility, plastic non-normality, and a hardening-softening-hardening hardness function was used to model experimentally obtained uniaxial compression data of cylindrical VACNT micropillars. Complex deformation modes were found in uniaxial compression, which include a sequential buckling-like collapse of the type seen in experiments. These complex deformation modes led to the overall stress-strain signature of the pillar not being of the same form as the input material hardness function. A fundamental question that motivates exploring the deformation of this class of materials—both experimentally and theoretically—is how to extract the intrinsic material response from simple tests. In this study we explore the relation between the input material response and the overall stress strain behavior in uniaxial tension using the constitutive framework of Hutchens et al. A simple one-dimensional analysis reveals the types of instability modes to be expected. Dynamic, finite deformation finite element calculations are carried out to explore the dependence of diffuse necking, localized necking, and propagating band deformation modes on characteristics of the hardness function. Attention is devoted to uncovering implications for obtaining intrinsic material properties of complex hierarchical structures; for example, vertically aligned carbon nanotubes (VACNTs), from uniaxial tension experiments.

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References

Cao, A. Y., Dickrell, P. L., Sawyer, W. G., Ghasemi-Nejhad, M. N., and Ajayan, P. M., 2005, “Supercompressible Foamlike Carbon Nanotube Films,” Science, 310, pp. 1307–1310. [CrossRef] [PubMed]
Pathak, S., Cambaz, Z. G., Kalidindi, S. R., Swadener, J. G., and Gogotsi, Y., 2009, “Viscoelasticity and High Buckling Stress of Dense Carbon Nanotube Brushes,” Carbon, 47, pp. 1969–1976. [CrossRef]
Gogotsi, Y., 2010, “High-Temperature Rubber Made From Carbon Nanotubes,” Science, 330, pp. 1332–1333. [CrossRef] [PubMed]
Xu, M., Futaba, D. N., Yamada, T., Yumura, M., and Hata, K., 2010, “Carbon Nanotubes With Temperature-Invariant Viscoelasticity From −196 to 1000 °C,” Science, 330, pp. 1364–1368. [CrossRef] [PubMed]
Xu, M., Futaba, D. N., Yumura, T., and Hata, K., 2011, “Carbon Nanotubes With Temperature-Invariant Creep and Creep-Recovery From −190 to 970 °C,” Adv. Mater., 23, pp. 3686–3690. [CrossRef] [PubMed]
Zhang, Q., Lu, Y. C., Du, F., Dai, L., Baur, J., and Foster, D. C., 2010, “Viscoelastic Creep of Vertically Aligned Carbon Nanotubes,” J. Phys. D: Appl. Phys., 43, p. 315401. [CrossRef]
Mesarovic, S. D., McCarter, C. M., Bahr, D. F., Radhakrishnan, H., Richards, R. F., Richards, C. D., McClain, D., and Jiao, J., 2007, “Mechanical Behavior of a Carbon Nanotube Turf,” Scr. Mater., 56, pp. 157–160. [CrossRef]
McCarter, C. M., Richards, R. F., Mesarovic, S. D., Richards, C. D., Bahr, D. F., McClain, D., and Jiao, J., 2006, “Mechanical Compliance of Photolithographically Defined Vertically Aligned Carbon Nanotube Turf,” J. Mater. Sci., 41, pp. 7872–7878. [CrossRef]
Pathak, S., Lim, E., Pour Shahid Saeed Abadi, P., Graham, S., Cola, B., and Greer, J. R., 2012, “Higher Recovery and Better Energy Dissipation at Faster Strain Rates in Carbon Nanotube Bundles: An In-Situ Study,” ACS Nano, 26, pp. 2189–2197. [CrossRef]
Suhr, J., Victor, P., Sreekala, L. C. S., Zhang, X., Nalamasu, O., and Ajayan, P. M., 2007, “Fatigue Resistance of Aligned Carbon Nanotube Arrays Under Cyclic Compression,” Nat. Nanotech., 2, pp. 417–421. [CrossRef]
Yaglioglu, O., Cao, A., Hart, A. J., Martens, R., and Slocum, A. H., 2012, “Wide Range Control of Microstructure and Mechanical Properties of Carbon Nanotube Forests: A Comparison Between Fixed and Floating Catalyst CVD Techniques,” Adv. Func. Mats., 22, pp. 5028–5037. [CrossRef]
Zbib, A. A., Mesarovic, S. D., Lilleodden, E. T., McClain, D., Jiao, J., and Bahr, D. F., 2008, “The Coordinated Buckling of Carbon Nanotube Turfs Under Uniform Compression,” Nanotechnology, 19, p. 175704. [CrossRef] [PubMed]
Hutchens, S. B., Hall, L. J., and Greer, J. R., 2010, “In Situ Mechanical Testing Reveals Periodic Buckle Nucleation and Propagation in Carbon Nanotube Bundles,” Adv. Func. Mats., 20, pp. 2338–2346. [CrossRef]
Yaglioglu, O., 2007, “Carbon Nanotube Based Electromechanical Probes,” Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA.
Pathak, S., Mohan, N., Pour Shahid Saeed Abadia, P., Graham, S., Cola, B. A., and Greer, J. R., 2013, “Compressive Response of Vertically Aligned Carbon Nanotube Films Gleaned From In Situ Flat-Punch Indentations,” J. Mater. Res., 28(7), pp. 984–997. [CrossRef]
Cao, C., Reiner, A., Chung, C., Chang, S.-H., Kao, I., Kukta, R. V., and Korach, C. S., 2011, “Buckling Initiation and Displacement Dependence in Compression of Vertically Aligned Carbon Nanotube Arrays,” Carbon, 49, pp. 3190–3195. [CrossRef]
Maschmann, M. R., Qiuhong, Z., Feng, D., Liming, D., and BaurJ., 2011, “Length Dependent Foam-Like Mechanical Response of Axially Indented Vertically Oriented Carbon Nanotube Arrays,” Carbon, 49, pp. 386–397. [CrossRef]
Raney, J. R., Fraternali, F., Amendola, A., and Daraio, C., 2011, “Modeling and In Situ Identification of Material Parameters for Layered Structures Based on Carbon Nanotube Arrays,” Compos. Struct., 93, pp. 3013–3018. [CrossRef]
Tong, T., Zhao, Y., Delzeit, L., Kashani, A., Meyyappan, M., and Majumdar, A., 2008, “Height Independent Compressive Modulus of Vertically Aligned Carbon Nanotube Arrays,” Nano Letts., 8, pp. 511–515. [CrossRef]
Qiu, A., Bahr, D. F., Zbib, A. A., Bellou, A., Mesarovic, S. D., McClain, D., Hudson, W., Jiao, J., Kiener, D., and Cordill, M. D., 2011, “Local and Non-Local Behavior and Coordinated Buckling of CNT Turfs,” Carbon, 49, pp. 1430–1438. [CrossRef]
Bradford, P. D., Wang, X., Zhao, H., and Zhu, Y. T., 2011, “Tuning the Compressive Mechanical Properties of Carbon Nanotube Foam,” Carbon, 49, pp. 2834–2841. [CrossRef]
Hutchens, S. B., Needleman, A., and Greer, J. R., 2011, “Analysis of Uniaxial Compression of Vertically Aligned Carbon Nanotubes,” J. Mech. Phys. Solids, 59, pp. 2227–2237, [CrossRef]Errata60, pp. 1753–1756 (2012).
Needleman, A., Hutchens, S. B., Mohan, N., and Greer, J. R., 2012 “Deformation of Plastically Compressible Hardening-Softening-Hardening Solids,” Acta Mech. Sinica, 28, pp. 1115–1124. [CrossRef]
Pathak, S., Mohan, N., Decolvenaere, E., Needleman, A., Bedewy, M., Hart, A. J., and Greer, J. R., 2013, “Influence of Density Gradients on the Stress-Strain Response of Carbon Nanotube Micropillars,” (submitted).
Cola, B. A., Xu, J., and Fisher, T. S., 2009, “Contact Mechanics and Thermal Conductance of Carbon Nanotube Array Interfaces,” Int. J. Heat Mass Transfer, 52, pp. 3490–3503. [CrossRef]
Cho, C., Richards, D., Bahr, Jiao, J., and Richards, R., 2008, “Evaluation of Contacts for a MEMS Thermal Switch,” J. Micromech. Microeng., 18, p. 105012. [CrossRef]
Considére, A., 1885, “L'Emploi du fer et de l'acier,” Ann. Ponts Chaussées, 9, Ser. 6, pp. 574–775.
Ericksen, J. L., 1975, “Equilibrium of Bars,” J. Elast., 5, pp. 191–201. [CrossRef]
Hutchinson, J. W., and Neale, K. W., 1977, “Influence of Strain Rate Sensitivity on Necking Under Uniaxial Tension,” Acta Metall.25, pp. 839–846. [CrossRef]
James, R. D., 1979, “Co-Existent Phases in the One Dimensional Static Theory of Elastic Bars,” Arch. Rat. Mech. Anal., 72, pp. 99–140. [CrossRef]
Hutchinson, J. W., and Neale, K. W., 1983, “Neck Propagation,” J. Mech. Phys. Solids, 31, pp. 405–426. [CrossRef]
Abeyaratene, R., and Knowles, J. K., 1993, “A Continuum Model of a Thermoelastic Solid Capable of Undergoing Phase Transitions,” J. Mech. Phys. Solids, 41, pp. 541–571. [CrossRef]
Needleman, A., 1988, “Material Rate Dependence and Mesh Sensitivity in Localization Problems,” Comp. Meth. Appl. Mech. Eng., 67, pp. 69–85. [CrossRef]
Needleman, A., 1999 “Plastic Strain Localization in Metals,” The Integration of Material, Process and Product Design, L. Lalli, N. Zabaras, R. Becker, and S. Ghosh, eds., A. A. Balkema, Rotterdam, pp. 59–70.
Chater, E., and Hutchinson, J. W., 1984, “On the Propagation of Bulges and Buckles,” ASME J. Appl. Mech., 51, pp. 269–277. [CrossRef]
Graff, S., Forest, S., Strudel, J.-L., Prioul, C., Pilvin, P., Béchade, J.-L., 2004, “Strain Localization Phenomena Associated With Static and Dynamic Strain Ageing in Notched Specimens: Experiments and Finite Element Simulations,” Mater. Sci. Eng. A, 387–389, pp. 181–185. [CrossRef]
Hutchinson, J. W., and Miles, J. P., 1974, “Bifurcation Analysis of the Onset of Necking in an Elastic/Plastic Cylinder Under Uniaxial Tension,” J. Mech. Phys. Solids, 22, pp. 61–71. [CrossRef]
Rudnicki, J. W., and Rice, J. R., 1975, “Conditions for the Localization of Deformation in Pressure-Sensitive Dilatant Materials,” J. Mech. Phys. Solids, 23, pp. 371–394. [CrossRef]
Ballarin, V., Soler, M., Perlade, A., Lemoine, X., and Forest, S., 2009, “Mechanisms and Modeling of Bake-Hardening Steels—Part I: Uniaxial Tension,” Metall. Mater. Trans. A, 40, pp. 1367–1374. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

The hardness function g(ɛp) as a function of plastic strain ɛp for material A (h2=5.0, h3=15.0, and ɛ2=0.6) and for material B (h2=h3=5.0)

Grahic Jump Location
Fig. 2

Overall nominal stress sn and true stress σt versus ɛt=ln(H/H0) for material A (h2=5.0, h3=15.0, and ɛ2=0.6) and for material B (h2=h3=5.0)

Grahic Jump Location
Fig. 3

Distributions of normalized plastic strain rate e·=ɛp/(W/H0) for material A: (a) ɛt=0.50 and (b) ɛt=0.76

Grahic Jump Location
Fig. 4

Distributions of normalized plastic strain rate e·=ɛp/(W/H0) for material B: (a) ɛt=0.50 and (b) ɛt=0.76

Grahic Jump Location
Fig. 5

The hardness function g(ɛp) as a function of plastic strain ɛp for material B (h2=h3=5.0), material C (h2=h3=1.0), and material D (h2=h3=0.5)

Grahic Jump Location
Fig. 6

Overall nominal stress sn and true stress σt versus ɛt=ln(H/H0) for material B (h2=h3=5.0), material C (h2=h3=1.0), and material D (h2=h3=0.5)

Grahic Jump Location
Fig. 7

Distributions of normalized plastic strain rate e·=ɛ·p/(W/H0): (a) material C at ɛt=0.30 and (b) material D at ɛt=0.14

Grahic Jump Location
Fig. 8

The hardness function g(ɛp) as a function of plastic strain ɛp for material E, h2=-3.90, h3=15.0, ɛ2=0.6, material F, h2=0.5, h3=15.0, ɛ2=0.6, and material G, h2=-3.90, h3=15.0, ɛ2=5.0

Grahic Jump Location
Fig. 9

Overall nominal stress sn and true stress σt versus ɛt=ln(H/H0) for material E, h2=-3.90,h3=15.0, ɛ2=0.6

Grahic Jump Location
Fig. 10

Distributions of normalized plastic strain rate e·=ɛ·p/(W/H0) for material E, h2=-3.90,h3=15.0, ɛ2=0.6: (a) ɛt=0.20, (b) ɛt=0.40, and (c) ɛt=0.50

Grahic Jump Location
Fig. 11

Overall nominal stress sn and true stress σt versus ɛt=ln(H/H0): material F, h2=0.5, h3=15.0, ɛ2=0.6 and material G, h2=-3.90, h3=15.0, ɛ2=5.0

Grahic Jump Location
Fig. 12

Distributions of normalized plastic strain rate e·=ɛ·p/(W/H0) for material F, h2=0.5, h3=15.0, ɛ2=0.6: (a) ɛt=0.20 and (b) ɛt=0.50

Grahic Jump Location
Fig. 13

Distributions of normalized plastic strain rate e·=ɛp/(W/H0) for material G, h2=-3.90, h3=15.0, ɛ2=5.0: (a) ɛt=0.20 and (b) ɛt=0.50

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