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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

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

Matthew R. Begley

Professor
Mem. ASME
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.

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Figures

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Fig. 1

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

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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

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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.

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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.

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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.

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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.

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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.

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