Research Papers

Comprehensive Solutions for the Response of Freestanding Beams With Tensile Residual Stress Subject to Point-Loading

[+] Author and Article Information
John Gaskins

Graduate Research Assistant
Mechanical and Aerospace Engineering,
University of Virginia,
Charlottesville, VA 22904
e-mail: jtg2e@virginia.edu

N. Scott Barker

Electrical and Computer Engineering,
University of Virginia,
Charlottesville, Virginia 22904
e-mail: nsb6t@virginia.edu

Matthew R. Begley

Mechanical Engineering Department,
Materials Department,
University of California,
Santa Barbara, CA 93106
e-mail: begley@engr.ucsb.edu

The results are identical for plane strain, provided one substitutes (1 + ν)εR for εR and E/(1 − ν2) for E.

1Corresponding author.

Manuscript received April 2, 2013; final manuscript received June 6, 2013; accepted manuscript posted June 11, 2013; published online September 18, 2013. Assoc. Editor: Chad M. Landis.

J. Appl. Mech 81(3), 031008 (Sep 18, 2013) (7 pages) Paper No: JAM-13-1144; doi: 10.1115/1.4024785 History: Received April 02, 2013; Revised June 06, 2013; Accepted June 11, 2013

This paper provides comprehensive solutions for the load-deflection response of an elastic beam with tensile residual stresses subjected to point-loading. A highly accurate explicit approximation is derived from the exact implicit solution for moderate rotations, which greatly facilitates property extraction and the design of devices for materials characterization, actuation, and sensing. The approximation has less than 6% error across the entire range of loads, displacements, geometry, and residual stress levels. An illustration of the application of the theory is provided for microfabricated nickel beams. The explicit form provides straightforward estimates for the critical loads and deflection defining the limits where classical asymptotic limits (e.g., pretensioned membrane, plate, and nonlinear membrane) will be accurate. Regimes maps are presented that identify critical loads, displacements, and properties correspond to these behaviors. Finally, the explicit form also enables straightforward estimations of bending strains relative to stretching, which is useful in the design of materials experiments that can be approximated as uniform straining of the beams.

Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.


Zhang, T.-Y., Su, Y.-J., Qian, C.-F., Zhao, M.-H., and Chen, L.-Q., 2000, “Microbridge Testing of Silicon Nitride Thin Films Deposited on Silicon Wafers,” Acta Mater., 48(11), pp. 2843–2857. [CrossRef]
Espinosa, H., Prorok, B., and Fischer, M., 2003, “A Methodology for Determining Mechanical Properties of Freestanding Thin Films and MEMS Materials,” J. Mech. Phys. Solid., 51(1), pp. 47–67. [CrossRef]
Wang, L., and Prorok, B., 2007, “Characterization of the Strain Rate Dependent Behavior of Nanocrystalline Gold Films,” J. Mater. Res., 23, pp. 55–65. [CrossRef]
Herbert, E., Oliver, W., De Boer, M., and Pharr, G., 2009, “Measuring the Elastic Modulus and Residual Stress of Freestanding Thin Films Using Nanoindentation Techniques,” J. Mater. Res., 24(9), pp. 2974–2985. [CrossRef]
Denhoff, M. W., 2003, “A Measurement of Young's Modulus and Residual Stress in MEMS Bridges Using a Surface Profiler,” J. Micromech. Microeng., 13(5), pp. 686–692. [CrossRef]
Mulloni, V., Colpo, S., Faes, A., and Margesin, B., 2013, “A Simple Analytical Method for Residual Stress Measurement on Suspended MEM Structures Using Surface Profilometry,” J. Micromech. Microeng., 23(2), p. 025025. [CrossRef]
Heidelberg, A., Ngo, L. T., Wu, B., Phillips, M. A., Sharma, S., Kamins, T. I., Sader, J. E., and Boland, J. J., 2006, “A Generalized Description of the Elastic Properties of Nanowires,” Nano Lett., 6(6), pp. 1101–1106. [CrossRef] [PubMed]
Ngo, L. T., Almécija, D., Sader, J. E., Daly, B., Petkov, N., Holmes, J. D., Erts, D., and Boland, J. J., 2006, “Ultimate Strength Germanium Nanowires,” Nano Lett., 6(12), pp. 2964–2968. [CrossRef] [PubMed]
Wang, Z.-J., Liu, C., Li, Z., and Zhang, T.-Y., 2010, “Size Dependent Elastic Properties of Au Nanowires Under Bending and Tension—Surfaces Versus Core Nonlinearity,” J. Appl. Phys., 108(8), p. 083506. [CrossRef]
Zeng, D., and Zheng, Q., 2010, “Large Deflection Theory of Nanobeams,” Acta Mech. Solid. Sin., 23(5), pp. 394–399. [CrossRef]
Celik, E., Guven, I., and Madenci, E., 2011, “Mechanical Characterization of Nickel Nanowires by Using a Customized Atomic Force Microscope,” Nanotechnology, 22(15), p. 155702. [CrossRef] [PubMed]
Wang, X., Najem, J. F., Wong, S.-C., and Tak Wan, K., 2012, “A Nano-Cheese-Cutter to Directly Measure Interfacial Adhesion of Freestanding Nano-Fibers,” J. Appl. Phys., 111(2), p. 024315. [CrossRef]
Zhan, H., and Gu, Y., 2012, “Modified Beam Theories for Nanowires Considering Surface/Intrinsic Effects and Axial Extension Effect,” J. Appl. Phys., 111(8), p. 084305. [CrossRef]
Seker, E., Gaskins, J. T., Bart-Smith, H., Zhu, J., Reed, M. L., Zangari, G., Kelly, R., and Begley, M. R., 2007, “The Effects of Post-Fabrication Annealing on the Mechanical Properties of Freestanding Nanoporous Gold Structures,” Acta Mater., 55(14), pp. 4593–4602. [CrossRef]
Seker, E., Gaskins, J. T., Bart-Smith, H., Zhu, J., Reed, M. L., Zangari, G., Kelly, R., and Begley, M. R., 2008, “The Effects of Annealing Prior to Dealloying on the Mechanical Properties of Nanoporous Gold Microbeams,” Acta Mater., 56(3), pp. 324–332. [CrossRef]
Djalali, R., Chen, Y.-F., and Matsui, H., 2002, “Au Nanowire Fabrication From Sequenced Histidine-Rich Peptide,” J. Amer. Chem. Soc., 124(46), pp. 13660–13661. [CrossRef]
Xiao, Z., Han, C. Y., Welp, U., Wang, H., Kwok, W., Willing, G., Hiller, J., Cook, R., Miller, D., and Crabtree, G., 2002, “Fabrication of Alumina Nanotubes and Nanowires by Etching Porous Alumina Membranes,” Nano Lett., 2(11), pp. 1293–1297. [CrossRef]
Simmonds, J., Begley, M., and Komaragiri, U., 2005, “The Mechanical Response of Freestanding Circular Elastic Films Under Point and Pressure Loads,” ASME J. Appl. Mech., 72(2), pp. 203–212. [CrossRef]
Landau, L. D., and Lifshitz, E., 1970, Theory of Elasticity, Pergamon, Oxford.
Senturia, S., D., 2001, Microsystem Design, Kluwer Academic Publishers, Norwell, MA.
Espinosa, H., Zhu, Y., Fischer, M., and Hutchinson, J., 2003, “An Experimental/Computational Approach to Identify Moduli and Residual Stress in MEMS Radio-Frequency Switches,” Exper. Mech., 43(3), pp. 309–316. [CrossRef]
Oliver, W. C., and Pharr, G., 1992, “An Improved Technique for Determining Hardness and Elastic Modulus Using Load and Displacement Sensing Indentation Experiments,” J. Mater. Res., 7(6), pp. 1564–1583. [CrossRef]
Maner, K. C., Begley, M. R., and Oliver, W. C., 2004, “Nanomechanical Testing of Circular Freestanding Polymer Films With Sub-Micron Thickness,” Acta Mater., 52(19), pp. 5451–5460. [CrossRef]
Luo, J., Flewitt, A., Spearing, S., Fleck, N., and Milne, W., 2004, “Young's Modulus of Electroplated Ni Thin Film for MEMS Applications,” Mater. Lett., 58(17), pp. 2306–2309. [CrossRef]
He, S., Chang, J. S., Li, L., and Ho, H., 2009, “Characterization of Young's Modulus and Residual Stress Gradient of MetalMUMPs Electroplated Nickel Film,” Sensor. Actuat. A: Physical, 154(1), pp. 149–156. [CrossRef]
Majjad, H., Basrour, S., Delobelle, P., and Schmidt, M., 1999, “Dynamic Determination of Young's Modulus of Electroplated Nickel Used in LIGA Technique,” Sensor. Actuat. A: Physical, 74(1), pp. 148–151. [CrossRef]
Namazu, T., and Inoue, S., 2007, “Characterization of Single Crystal Silicon and Electroplated Nickel Films by Uniaxial Tensile Test With In Situ X-Ray Diffraction Measurement,” Fatique Fract. Eng. Mat. Struct., 30(1), pp. 13–20. [CrossRef]


Grahic Jump Location
Fig. 1

Exact and approximate load-deflection relationships for a broad range of prestretch

Grahic Jump Location
Fig. 2

(a) The full range of F(Λ) and (b) error in the predicted load as a function of applied deflection for values of ɛ¯R from 0–106 when c = 2.12

Grahic Jump Location
Fig. 3

Illustration of combinations of (a) normalized critical loads, (b) normalized critical displacements and normalized prestretch for which asymptotic solutions are accurate: the shaded region represents the transition from linear regimes to the membrane regime where the analytical solution can be used to extract material properties. Labeled vertical lines correspond to the range over which experimental data are fit in Sec. 5.

Grahic Jump Location
Fig. 4

Contours showing combinations of prestrain and deflections where the contribution of bending strain to the total strain in the beam is 1, 5, and 10%. For applied displacements greater than approximately five times the film thickness bending strains are negligible regardless the level of prestrain.

Grahic Jump Location
Fig. 5

Illustration of fabrication method for freestanding nickel beams. (a) Side and (b) top view of beam and photoresist mask prior to silicon etch. (c) Side and (d) top view postetch and lift off resist removal.

Grahic Jump Location
Fig. 6

(a) SEM of representative MEMS beam used in point-load test. (b) Side view of coordinates and deformation variables used in the analysis.

Grahic Jump Location
Fig. 7

Raw (a) and normalized (b) load-displacement curves. The inset graph in (a) shows the load and unload curve for the average of six tests on a single 81 nm thick beam showing repeatable results and negligible thermal drift over the duration of the test. Error bars in (a) and (b) are the average of tests on three different beams for each film thickness. Data shown is below 0.2% to ensure elastic behavior. In (b) data below 500 nm are truncated to ensure the beam and indenter are in full contact.




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In