Calculation of Wave Dispersion Curves in Multilayered Composite-Metal Plates

The major purpose of this paper is the development of wave dispersion curves calculation in multilayered composite-metal plates. At first, equations ofmotion and characteristic equations for the free waves on a single-layered orthotropic plate are presented. Since direction of wave propagation in composite materials is effective on equations of motion and dispersion curves, two different cases are considered: propagation of wave along an axis of material symmetry and along off-axes of material symmetry. Then, presented equations are extended for a multilayered orthotropic composite-metal plate using the transfer matrix method in which a global transfer matrix may be extracted which relates stresses and displacements on the top layer to those on the bottom one. By satisfying appropriate boundary conditions on the outer boundaries, wave characteristic equations and then dispersion curves are obtained. Moreover, presented equations may be applied to other materials such as monoclinic, transversely isotropic, cubic, and isotropic materials. To verify the solution procedure, a number of numerical illustrations for a single-layered orthotropic and double-layered orthotropic-metal are presented.


Introduction
Nondestructive inspection based on the propagation of elastic waves that relies generally on calculation of dispersion curves plays an important role in damage identification in multilayered structures.From this point of view, calculation of dispersion curves is one of the essential stages for inspection of structures.
Most of the researches have been done on the singlelayered isotropic or quasi-isotropic materials [1,2].However, Nayfeh [3] has presented a transfer matrix technique to obtain the dispersion curves of elastic waves propagating in multilayered anisotropic media, that is, composite laminates.Demcenko and Mazeika [4] have developed global matrix method to calculate dispersion curves of multilayered isotropic plate.Lowe [5] has presented a review of development of the matrix method and global matrix method to obtain dispersion curves in isotropic materials.Also, they have discussed the problems with large value of frequency thickness, material damping, and leaky wave.Verma [6] has investigated the harmonic thermoelastic waves and dispersion relations in layered anisotropic plate by utilizing the transfer matrix method.
Nowadays, composite-metal structures have developed significantly in the industry because of the great mechanical properties rather than other materials.Furthermore, damage detection using wave propagation is one of the most interesting topics that have attracted the attention of researchers.Thus, in this paper, we extend the relations of dispersion curves for multilayered composite-metal plates using the transfer matrix method presented in previous studies.Presented procedure can be used for damage detection in multilayered composite-metal structures using the wave propagation methods. 3 of the global coordinate system ( 1 ,  2 ,  3 ) and consider  1 −  2 plane coincides with the upper surface of the plate.Generally, the global coordinate system ( 1 ,  2 ,  3 ) does not coincide with the principle material coordinate system (  1 ,   2 ,   3 ).In this paper two different cases are considered: wave propagation along principle-axes,   1 or   2 , and wave propagation along off-principle-axes (see Figure 1).

Wave Propagation along
and stress-strain relations are given by where   ,   ,   are the components of stress, strain, and displacement, respectively, and  and   are the material density and stiffness coefficient, respectively.Furthermore, the linear strain-displacement relations are Substituting ( 2) and ( 3) into (1a), (1b), and (1c) concludes the three following equations: Two coupled equations, (4a) and (4c), are related to pressure wave (P wave) and shear vertical wave (SV wave).Also, the uncoupled equation (4b) is related to horizontal wave (SH wave).Formal solutions for these equations are assumed as follows [3]: where   and  are displacement amplitude along direction    and phase velocity in direction   1 , respectively, and (= /), , and  are   1 -component of wavenumber and ratio of   3 , circular frequency, and   1 -components of wavenumber, respectively.For clarification, the rest of the solution procedure for coupled and uncoupled equations is described next.
(a) Solution for SH wave: according to (5), it is assumed the solution of (4b) has the following form: substituting ( 6) into (4b) results in Also, stress-strain relation is thus Using ( 8) and (10) and after some simplifications, stress and displacement at   3 = 0 may be related to those at   3 = ℎ (ℎ is thickness of plate) by following the relation where []is transfer matrix and is given by By invoking free stress boundary conditions at lower and upper surfaces of plate, characteristic equation for singlelayered plate may be obtained as follows: Substituting  = / into (13) results in an equation with two unknown parameters  and  called "dispersion equation of SH wave." (b) Solution for P and SV waves: according to (5), suggested solutions of (4a) and (4c) are Substituting ( 14) into (4a) and (4c) results in To obtain nontrivial solution, determinant of coefficients matrix of displacement vector is vanished.Therefore, a fourdegree polynomial equation in  is obtained as follows: There are four solutions for  or in other words two pairs of solutions for  2 as follows: therefore Also, stress-strain relation is thus where Using ( 18), (20), and (21) and after some simplifications, stress and displacement at   3 = 0 may be related to those at   3 = ℎ (ℎ is thickness of plate) by following the relation where [] is transfer matrix and is given by By invoking free stress boundary conditions at lower and upper surfaces of plate, characteristic equation for singlelayered plate may be obtained as follows: Substituting  = / into (24) results in an equation with two unknown parameters  and  called "dispersion equation of P and SV waves."

Wave Propagation along
Off-Principle-Axes.Assume wave propagation along off-principle-axis  1 (or  2 ) with angle  with respect to principle-axis   1 (or   2 ) as shown in Figure 1(b).In this situation, (4a), (4b), and (4c) are coupled for all materials except isotropic one.The reason is that transformed stiffness matrix from system (  1 ,   2 ,   3 ) into system ( 1 ,  2 ,  3 ) has a monoclinic form as follows: ( By setting determinant [()] equal to zero, a six-degree polynomial equation is obtained as follows: There are four solutions for  as follows: Therefore, displacements and stress components are given by where  Using ( 29) and (30) and after some simplifications, stress and displacement at  3 = 0 may be related to those at  3 =   3 = ℎ (ℎ is thickness of plate) by following the relation where []is transfer matrix and is given by  be extended for a multilayered composite-metal plate (see Figure 2) by "transform matrix method." In this method by applying the continuity of the displacement and stress components at the layer interfaces of plate, displacements and stress components at lower surface can be related to those at upper one.For instance, in Figure 2, transfer matrix between  3 = 0 and  3 =  is given by where  and  refer to metal layer and anisotropic composite layer, respectively.By satisfying appropriate boundary condition on the outer boundaries, wave characteristic equations are obtained.It is noteworthy that a local coordinate system (  1 ,   2 ,   3 )  is considered for each layer  to obtain   using the presented equations in Section 2.1.

Dispersion Curves of Waves.
Presented dispersion equations of waves in previous section such as (13), (24), and (33) may be used for plotting of dispersion curves of phase velocity  as function of frequency (= /2).These curves are important tools in damage detection techniques based on the wave propagation.However, the dispersion equations of waves do not generally have analytical solution and can be solved by numerical methods.wave propagates along principle-axes of orthotropic layer.Dispersion curves of SH wave and P and SV waves are shown in Figure 5.

Conclusion
In this paper, wave dispersion relations of single anisotropic layer were presented.Two different cases were considered: propagation of wave along an axis of material symmetry and along off-axes of material symmetry.Since compositemetal structures have developed in last years, presented equations were extended for a multilayered composite-metal plate using the transfer matrix method.Suggested procedure is straightforward and applicable for damage detection based on wave propagation.However, the cost of calculations increases for large number of layers.

Figure 5 :
Figure 5: Wave propagation along principle-axes of a double-layered Al-orthotropic plate.