Abstract
In this study, the feasibility of using Lamb waves in functionally graded (FG) nano copper layered wafers in nondestructive evaluation is evaluated. The elastic parameters and mass densities of these wafers vary with thickness due to the variation in grain size. The power series technique is used to solve the governing equations with variable coefficients. To analyze multilayered structures, of which the material parameters are continuous but underivable, a modified transfer matrix method is proposed and combined with the power series method. Results show that multiple modes of Lamb waves exist in FG nano copper wafers. Moreover, the gradient property leads to a decrease in phase velocity, and the absolute value of the phase velocity variation is positively correlated with the gradient coefficient. The phase velocity variation and variation rate in Mode 2 are smaller than those in other modes. The findings indicate that Mode 4 is recommended for nondestructive evaluation. However, if the number of layers is greater than four, the dispersion curves of the Lamb waves in the multilayer structures tend to coincide with those in the equivalent uniform structures. The results of this study provide theoretical guidance for the nondestructive evaluation of FG nanomaterial layered structures.
Highlights
Since the introduction of functionally graded (FG) nano copper in 2011 [1], this novel material with a graded grain size distribution has attracted increasing scientific interest due to its high strength and high ductility [2,3,4]
The findings suggest the difference in the Lamb wave propagation in a multilayered FG nano copper wafer weakens with the increase in the number of layers
The modified transfer matrix method based on the power series solution is proposed to solve the wave propagation problem in a multilayered wafer
Summary
Since the introduction of functionally graded (FG) nano copper in 2011 [1], this novel material with a graded grain size distribution has attracted increasing scientific interest due to its high strength and high ductility [2,3,4]. The WKB method is used to study the horizontal shear waves in different FG layered structures with one displacement component [22,23] This approach is only applicable to wave propagation problems with large wave numbers [20]. The transfer matrix method, which is based on the continuity of the stress and displacement in the interface, is used to solve the wave propagation problem in multilayered structures [15,34]. The Lamb wave propagation problem in a multilayered FG nano metal wafer with continuous and non-differentiable material parameters is solved through a modified transfer matrix method combined with the power series technique. If the grain size of the FG nano copper wafer varies with thickness and is a function of x3, the corresponding material parameters, including the elastic parameters and mass density, are not constants, but are functions of x3. The governing equations for the mechanical displacements are defined as:
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