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In this paper, a size-dependent Timoshenko beam is developed on the basis of the couple stress theory. The couple stress theory is a non-classic continuum theory capable of capturing the small-scale size effects on the mechanical behavior of structures, while the classical continuum theory is unable to predict the mechanical behavior accurately when the characteristic size of structures is close to the material length scale parameter. The governing differential equations of motion are derived for the couple-stress Timoshenko beam using the principles of linear and angular momentum. Then, the general form of boundary conditions and generally valid closed-form analytical solutions are obtained for the axial deformation, bending deflection, and the rotation angle of cross sections in the static cases. As an example, the closed-form analytical results are obtained for the response of a cantilever beam subjected to a static loading with a concentrated force at its free end. The results indicate that modeling on the basis of the couple stress theory causes more stiffness than modeling by the classical beam theory. In addition, the results indicate that the differences between the results of the proposed model and those based on the classical Euler-Bernoulli and classical Timoshenko beam theories are significant when the beam thickness is comparable to its material length scale parameter.Keywords Size-dependent behavior · Timoshenko beam · Couple stress theory · Material length scale parameter

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The purpose of this paper is the enhancement of the AFM sensitivity through the selection of an optimized FGM micro cantilever beam. In this paper, resonant frequencies and sensitivities of first two modes of micro cantilever which is made of functionally graded materials are investigated and a relationship is developed to evaluate the sensitivity of FGM micro cantilever. Effect of volume fraction of materials and surface contact stiffness on the resonant frequencies and sensitivities are studied. The rectangular FGM beam is modeled by an Euler-Bernoulli beam theory. It is assumed that beam is made of a mixture of metal and ceramic with properties varying through the thickness following a simple power law of n. This variation is a function of the volume fraction of the beam material constituents. The interaction between AFM tip and surface is modeled by a linear spring which expresses the contact stiffness. Results show that, increasing the ceramic volume fraction increases the resonant frequencies of both modes 1 and 2. When contact stiffness is small, for both modes, as ceramic volume fraction increases, sensitivities decreases, while for large contact stiffness, as ceramic volume fraction increases the sensitivities will be increased. Results also show that at each contact stiffness, there is a unique value of n at which the sensitivity is maximized. Using these values for n, the high quality and high contrast images can be obtained.

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