Abstract
To investigate Lamb waves in thin films made of functionally graded viscoelastic material, we deduce the governing equation with respect to the displacement component and solve these partial differential equations with complex variable coefficients based on a power series method. To solve the transcendental equations in the form of a series with complex coefficients, we propose and optimize the minimum module approximation (MMA) method. The power series solution agrees well with the exact analytical solution when the material varies along its thickness following the same exponential function. When material parameters vary with thickness with the same function, the effect of the gradient properties on the wave velocity is limited and that on the wave structure is obvious. The influence of the gradient parameter on the dispersion property and the damping coefficient are discussed. The results should provide nondestructive evaluation for viscoelastic material and the MMA method is suggested for obtaining numerical results of the asymptotic solution for attenuated waves, including waves in viscoelastic structures, piezoelectric semiconductor structures, and so on.
Highlights
Lamb waves, which are a type of plain strain wave in a thin film or a plate with a traction-free boundary, are widely used in nondestructive evaluation
Scientists have directed more attention to Lamb waves in plates made of various materials, including viscoelastic materials [2], functionally graded materials (FGMs) [3], piezoelectric materials [4], and piezoelectric–piezomagnetic materials [5]
When the power series method is employed to solve the wave propagation problem in a functionally graded viscoelastic material (FGVM) structure, the dispersion equation, which is a transcendental equation with complex numbers in series form, is difficult to solve based on the above numerical simulation method
Summary
Lamb waves, which are a type of plain strain wave in a thin film or a plate with a traction-free boundary, are widely used in nondestructive evaluation. Researchers suggested that these equations can be solved by using a power series method [11,24] and a Legendre polynomial method [25,26], which are fit for solving the wave propagation problem in heterogeneous structures in arbitrary cases in which material parameters vary continuously and slowly. When the power series method is employed to solve the wave propagation problem in a functionally graded viscoelastic material (FGVM) structure, the dispersion equation, which is a transcendental equation with complex numbers in series form, is difficult to solve based on the above numerical simulation method. The dispersion and attenuation characteristics of Lamb wave propagation under different gradient parameters are discussed, and the damping coefficients are analyzed Conclusions based on these results can provide a theoretical basis for nonhomogeneous viscoelastic structure nondestructive testing
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