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

Abstract

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.