This paper aims at presenting a survey of the fractional derivative acoustic wave equations, which have been developed in recent decades to describe the observed frequency-dependent attenuation and scattering of acoustic wave propagating through complex media. The derivation of these models and their underlying elastoviscous constitutive relationships are reviewed, and the successful applications and numerical simulations are also highlighted. The different fractional derivative acoustic wave equations characterizing viscous dissipation are analyzed and compared with each other, along with the connections and differences between these models. These model equations are mainly classified into two categories: temporal and spatial fractional derivative models. The statistical interpretation for the range of power-law indices is presented with the help of Lévy stable distribution. In addition, the fractional derivative biharmonic wave equations governing scattering attenuation are introduced and can be viewed as a generalization of viscous dissipative attenuation models.

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The three-dimensional free vibration and time response of rotating functionally graded (FG) cantilevered beams are studied. Material properties of functionally graded beams are assumed to change gradually through both the width and the thickness in power-law form. The second-kind Lagrange’s equations are used in conjunction with the Ritz method to derive the comprehensive coupling dynamic equations for the axial, chordwise, and flapwise motions. The trial functions of deformations are taken as the products of the Chebyshev polynomials and the corresponding boundary functions. Nonlinear coupling deformations are considered to capture the dynamic stiffening effect due to the rotating motion. The influences of the material gradient index and rotational speed on modal characteristics are investigated by the state space method. The eigenvalue loci veering phenomena with modal conversions are exhibited. The time responses indicate that the deformations of rotating functionally graded beams are greatly affected by the material gradient index. It is shown that for large deformation problems, using Chebyshev polynomials is more efficient in computing precision and robustness than using other polynomials.

Shape memory polymers (SMPs) and shape memory polymer composites have drawn considerable attention in recent years for their shape memory effects. A unified modeling approach is proposed to describe thermomechanical behaviors and shape memory effects of thermally activated amorphous SMPs and SMP-based syntactic foam by using the generalized finite deformation multiple relaxation viscoelastic theory coupled with time-temperature superposition property. In this paper, the thermoviscoelastic parameters are determined from a single dynamic mechanical analysis temperature sweep at a constant frequency. The relaxation time strongly depends on the temperature and the variation follows the time-temperature superposition principle. The horizontal shift factor can be obtained by the Williams-Landel-Ferry equation at temperatures above or close to the reference temperature (T r ), and by the Arrhenius equation at temperatures below T r . As the Arruda-Boyce eight-chain model captures the hyperelastic behavior of the material up to large deformation, it is used here to describe partial material behaviors. The thermal expansion coefficient of the material is regarded as temperature dependent. Comparisons between the model results and the thermomechanical experiments presented in the literature show an acceptable agreement.

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The free vibration of rotating tapered Timoshenko beams (TBs) made of the axially functionally graded materials (FGMs) is studied. The Chebyshev polynomials multiplied by the boundary functions are selected as the admissible functions in the Ritz minimization procedure, which is called the Chebyshev–Ritz method. As such, the geometric boundary conditions are satisfied, while the numerical robustness is guaranteed through use of the Chebyshev polynomials. The Ritz approach provides an upper bound of the exact frequencies. The effectiveness of the method is confirmed in the convergence and comparison studies. The effects of hub radius ratio, rotational speed ratio, taper ratios, rotary inertia and material gradient on the natural frequencies of the TBs for six different sets of boundary conditions are studied in detail.

The free vibration of rotating functionally graded nanobeams under different boundary conditions is studied based on nonlocal elasticity theory within the framework of Euler-Bernoulli and Timoshenko beam theories. The thickness-wise material gradient variation of the nanobeam is considered. By introducing a second-order axial shortening term into the displacement field, the governing equations of motion of the present new nonlocal model of rotating nanobeams are derived by the Hamilton’s principle. The nonlocal differential equations are solved through the Galerkin method. The present nonlocal models are validated through the convergence and comparison studies. Numerical results are presented to investigate the influences of the nonlocal parameter, angular velocity, material gradient variation together with slenderness ratio on the vibration of rotating FG nanobeams with different boundary conditions. Totally different from stationary nanobeams, the rotating nanobeams with relatively high angular velocity could produce larger fundamental frequencies than local counterparts. Additionally, the axial stretching-transverse bending coupled vibration is perfectly shown through the frequency loci veering and modal conversion.

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