Frequency analysis of SMA composite beam resting on Pasternak elastic foundation using four engineering beam theories

In the last decades, the production of smart materials have led to modern structures with superior properties. Among these materials one may points to the Shape Memory Alloys (SMAs) which show the capability of retaining the large plastic strains when exposed to outer temperature or traction loadings. The developement of SMA acuators in the forms of wire and stent have attracted many attentions in the fields of engineering and smart structures. In this regard, an analytical model is represented for the composite beam with shape memory alloy wires resting on the Pasternak elastic foundation. The composite beam is simply-supported in both sides and pre-strained SMA wires are embedded in the middle of the cross section. The governing equations of Euler-Bernoulli, Rayleigh, Shear and Timoshenko beams are extracted using the Hamilton's principle. By heating the beam, strain recovery operation will produce a tensile force along the beam. This tensile force in turn will produce a compressive force against the beam supports. The resulted force is modeled by martensite transformation. By normalizing the governing equations, analytical relations are provided to evaluate the exact solution of natural frequency. The validity of results are established by comparing the typical solution with similar solution in the literature. Based on the analyses, the effects of Pasternak foundation coefficients, number of SMA wires, thickness to span ratio, recoverable strain limit and span to width ratio on the natural frequency in temperature above the austenite finish temperature are found and represented by using the engineering beam theories