Abstract
In this paper the influence of microstructure on the free vibration of geometrically similar heterogeneous beams with free-free boundary conditions was numerically investigated by detailed finite element analysis (FEA) to identify and quantify any effect of beam size on transverse modal frequencies when the microstructural scale is comparable to the overall size. ANSYS Mechanical APDL was used to generate specific unit cells at the microstructural scale comprised of two isotropic materials with different material properties. Unit cell variants containing voids and inclusions were considered. At the macroscopic scale, four beam sizes consisting of one, two, three or four layers of defined unit cells were represented by repeatedly regenerating the unit cell as necessary. In all four beam sizes the aspect ratio was kept constant. Changes to the volume fractions of each material were introduced while keeping the homogenized properties of the beam fixed. The influence of the beam surface morphology on the results was also investigated. The ANSYS results were compared with the analytical results from solution to Timoshenko beam and nonlocal Timoshenko beam as well as numerical results for a Micropolar beam. In nonlocal Timoshenko beams the Eringen’s small length scale coefficients were estimated for some of the studied models. Numerical analyses based on Micropolar theory were carried out to study the modal frequencies and a method was suggested to estimate characteristic length in bending and coupling number via transverse vibration which verifies the use of Micropolar elasticity theory in dynamic analysis.
Highlights
In recent years the progress in technologies such as aerospace, biomedical, nanotechnology etc. have demanded the need for the application of small scale structures and that has created a whole new era for researchers to investigate the dynamic behaviour of structures where the classical theories of elasticity become increasingly invalid to use in cases such as small scale heterogeneous beams
Characteristic length does not vary with beam size and only depends on volume fraction
And 8, the finite element results from ANSYS and Micropolar control volume finite element method (CVFEM) codes for mode 1 and 2 are compared after convergence of the iteration process; mω2 is obtained within two limits of N=0 and N=0.9 (N value at higher bound must not equal 1 due to numerical errors it causes.)
Summary
In recent years the progress in technologies such as aerospace, biomedical, nanotechnology etc. have demanded the need for the application of small scale structures and that has created a whole new era for researchers to investigate the dynamic behaviour of structures where the classical theories of elasticity become increasingly invalid to use in cases such as small scale heterogeneous beams. Available results on the influence of size effect on the behaviour of heterogeneous materials reported by researches show deviation from elastic theories in static loading cases when the beam or plate L/h ratio reduces [9]–[15]. A 2D Micropolar strip loaded at one end was investigated by Nakamura & Lake [16] and the influences of elastic constants especially coupling number and characteristic length are investigated. They concluded that for a very small characteristic length (in comparison with the strip’s width), the rate of stress/or strain energy decreases as the characteristic length increases. Beveridge et al investigated the Micropolar behaviour of perforated beams in 3 point bending and by using a control volume finite element method and iteration to model static 3 point bending test results determined the coupling number for the models[10]
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