Resonance Responses of Geometrically Imperfect Functionally Graded Extensible Microbeams

This paper aims at analyzing the size-dependent nonlinear dynamical behavior of a geometrically imperfect microbeam made of a functionally graded (FG) material, taking into account the longitudinal, transverse, and rotational motions. The size-dependent property is modeled by means of the modified couple stress theory, the shear deformation and rotary inertia are modeled using the Timoshenko beam theory, and the graded material property in the beam thickness direction is modeled via the Mori–Tanaka homogenization technique. The kinetic and size-dependent potential energies of the system are developed as functions of the longitudinal, transverse, and rotational motions. On the basis of an energy method, the continuous models of the system motion are obtained. Upon application of a weighted-residual method, the reduced-order model is obtained. A continuation method along with an eigenvalue extraction technique is utilized for the nonlinear and linear analyses, respectively. A special attention is paid on the effects of the material gradient index, the imperfection amplitude, and the length-scale parameter on the system dynamical response.

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Modeling size-dependent behaviors of axially functionally graded Bernoulli-Euler micro-beam

Mechanical Engineering Science ◽

10.33142/mes.v2i1.2617 ◽

2020 ◽

Vol 2 (1) ◽

Author(s):

Shuai WANG ◽

Zetian KANG ◽

Shichen ZHOU ◽

Bo ZHOU ◽

Shifeng XUE

Keyword(s):

Couple Stress ◽

Couple Stress Theory ◽

Beam Theory ◽

Axial Direction ◽

Modified Couple Stress Theory ◽

Functionally Graded ◽

Mechanical Behaviors ◽

Euler Beam ◽

Poisson Effect ◽

Size Dependent

This work focus on the mechanical behaviors, which are related to the size effect, functionally graded (FG) effect and Poisson effect, of an axially functionally graded (AFG) micro-beam whose elastic modulus varies according to sinusoidal law along its axial direction. The displacement field of the AFG micro-beam is set according to the Bernoulli-Euler beam theory. Employing the modified couple stress theory (MCST), the components of strain, curvature, stress and couple stress are expressed by the second derivative of the deflection of the AFG micro-beam. A size-dependent model related to FG effect and Poisson effect, which includes the formulations of bending stiffness, deflection, normal stress and couple stress, is developed to predict the mechanical behaviors of the AFG micro-beam by employing the principle of minimum potential energy. The mechanical behaviors of a simply supported AFG micro-beam are numerically investigated using the developed model for demonstrating the size effects, FG effects and Poisson effects of the AFG micro-beam. Results show that the mechanical behaviors of AFG micro-beams are distinctly size-dependent only when the ratio of micro-beam height to material length-scale parameter is small enough. The FG parameter is an important factor that determines and regulates the size-dependent behaviors of AFG micro-beams. The influences of Poisson’s ratio on the mechanical behaviors of AFG micro-beams are not negligible, and should be also considered in the design and analysis of an AFG micro-beam. This work supplies a theoretical basis and a technical reference for the design and analysis of AFG micro-beams in the related regions.

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Asymmetric Oscillations of AFG Microscale Nonuniform Deformable Timoshenko Beams

Vibration ◽

10.3390/vibration2020013 ◽

2019 ◽

Vol 2 (2) ◽

pp. 201-221 ◽

Cited By ~ 1

Author(s):

Mergen H. Ghayesh ◽

Ali Farajpour ◽

Hamed Farokhi

Keyword(s):

Timoshenko Beam ◽

Couple Stress Theory ◽

Beam Theory ◽

Modified Couple Stress Theory ◽

Functionally Graded ◽

Timoshenko Beam Theory ◽

Cross Sectional ◽

Stress Theory ◽

Size Dependent ◽

Modified Couple Stress

A nonlinear vibration analysis is conducted on the mechanical behavior of axially functionally graded (AFG) microscale Timoshenko nonuniform beams. Asymmetry is due to both the nonuniform material mixture and geometric nonuniformity. Using the Timoshenko beam theory, the continuous models for translation/rotation are developed via an energy balance. Size-dependence is incorporated via the modified couple stress theory and the rotation via the Timoshenko beam theory. Galerkin’s method of discretization is applied and numerical simulations are conducted for a size-dependent vibration of the AFG microscale beam. Effects of material gradient index and axial change in the cross-sectional area on the force and frequency diagrams are investigated.

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