Waves in a 2-D Elastic Space

Abstract

An elastic layer can be understood as two parallel surfaces bounding a 3-D elastic space. In such circumstances the presence of the bounding surfaces limits potential directions of propagation of elastic waves, so it can be said that an elastic layer is a case of a 2-D elastic space.

5.1 SH-Waves in an Elastic Layer

An elastic layer can be understood as two parallel surfaces bounding a 3-D elastic space. In such circumstances the presence of the bounding surfaces limits potential directions of propagation of elastic waves, so it can be said that an elastic layer is a case of a 2-D elastic space. Although two parallel surfaces represent a every specific example of 2-D space, it significantly helps to simplify not only the analysis but also the mathematical description of the wave phenomenon, as well as to formulate some general conclusions.

Fig. 5.1

Formation of the pattern of elastic SH-waves propagating in an aluminium layer, in consecutive moments in time. Results of numerical computations by TD-SFEM

Propagation of SH-waves in a thin elastic layer was briefly discussed on the occasion of the analysis of Love surface waves in Sect. 4.2. In the current section this analysis is presented in more detail. First of all it should be observed that the presence of a bounding surface in a 3-D elastic space results in wave reflection and subsequent interaction of the reflected waves with the incident waves. At certain conditions these reflected and incident waves are able to form specific wave patterns, which are called wave modes. Such wave modes could be observed in the case of Love surface waves and because of their dispersive nature they were strongly frequency dependent, as presented in Fig. 4.8. However, such wave modes were not observed in the case of Rayleigh or Stoneley surface waves.

The presence of two surfaces parallel to the xy plane, bounding a 3-D elastic space, leads to the entrapment of the propagating waves within the elastic layer as well as the formation of various wave modes, as presented in Fig. 5.1. Following the considerations presented in Sect. 4.2 it can be repeated that the displacement field associated with the propagation of a plane harmonic wave $\textbf{u}$ within a layer of thickness h has the already known form. As before the nature of this wave requires that the displacement components $u_{x}$ and $u_{z}$ must vanish, i.e. $u_{x} = u_{z} = 0$, and the only non-zero displacement component is $u_{y}$, which remains a function of the spatial coordinates x, z and time t:

$$\begin{aligned} u_{y}(x,z,t) = \hat{u}_{y}(z)e^{ik(x-ct)} \end{aligned}$$

(5.1)

where now c denotes the phase velocity of the wave propagating in the direction of the x axis and resulting from a coupled interaction of the incident SH-waves and their reflections from the layer boundaries. This interaction leads to the creation of a standing wave in the direction perpendicular to the xy plane, which is the direction of the z axis.

As before a simple substitution of Eq. (5.1) into the governing equations of motion (3.5), or simply into the resulting equation from Eq. (3.24), leads to a well-known ordinary differential equation given by Eq. (4.25):

$$\begin{aligned} \frac{d^{2}\hat{u}_{y}(z)}{dz^{2}} + \beta ^{2}\hat{u}_{y}(z) = 0, \quad \beta = k\sqrt{\dfrac{c^{2}}{c^{2}_{S}} - 1} \end{aligned}$$

(5.2)

which is formally rewritten here for the sake of completeness, where $|z| \le a$, with $2a = h$ and h denoting the thickness of the elastic layer.

This equation has a solution composed of two independent harmonic functions: one being the symmetric cosine function and the other the antisymmetric sine function:

$$\begin{aligned} \hat{u}_{y}(z) = A_{1}\sin \beta z + A_{2}\cos \beta z \end{aligned}$$

(5.3)

Now, the application of the traction-free boundary conditions for the stress component $\tau _{yz}$ at the free surface of the elastic layer, i.e. for $z = \pm a$, allows one to obtain the dispersion relation for SH-waves propagating in the elastic layer, as illustrated in Figs. 5.2 and 5.3.

Fig. 5.2

Dispersion curves for: a the phase velocity, b the group velocity, for symmetric modes of SH-waves SH$_{n} (n=0,2,4,\ldots )$ propagating in a 10 mm thick aluminium layer

Fig. 5.3

Dispersion curves for: a the phase velocity, b the group velocity, for antisymmetric modes of SH-waves SH$_{n} (n=1,3,5,\ldots )$ propagating in a 10 mm thick aluminium layer

This condition can be formally presented in the following form:

$$\begin{aligned} \tau _{yz}|_{\pm a} = 0 \rightarrow A_{1}\cos \beta a \pm A_{2}\sin \beta a = 0 \end{aligned}$$

(5.4)

Indeed, it can be found that Eq. (5.4) is satisfied in two cases [1]:

• when $A_{1} = 0$ and $\hat{u}_{y}(z) = A_{2}\cos \beta z$ then:

  $$\begin{aligned} A_{2}\sin \beta a = 0 \rightarrow \beta _{n}a = n\pi , \quad n = 0,1,2,\ldots \end{aligned}$$

  or alternatively:

  $$\begin{aligned} \beta _{n}a = \dfrac{n\pi }{2} \rightarrow \beta _{n}h = n\pi , \quad n = 0,2,4,\ldots \end{aligned}$$

  (5.5)

• when $A_{2} = 0$ and $\hat{u}_{y}(z) = A_{1}\sin \beta z$ then:

  $$\begin{aligned} A_{1}\cos \beta a = 0 \rightarrow \beta _{n}a = \dfrac{(2n-1)\pi }{2}, \quad n = 1,2,3,\ldots \end{aligned}$$

  or alternatively:

  $$\begin{aligned} \beta _{n}a = \dfrac{n\pi }{2} \rightarrow \beta _{n}h = n\pi , \quad n = 1,3,5,\ldots \end{aligned}$$

  (5.6)

It is interesting to note that the case of $n = 0,2,4,\ldots$ corresponds to the infinite number of symmetric modes of SH-waves propagating within the elastic layer and standing waves described by the symmetric (even) cosine function, while the case of $n = 1,3,5,\ldots$ corresponds to the infinite number of antisymmetric modes of SH-waves propagating within the elastic layer and standing waves described by the antisymmetric (odd) sine function. In the literature [1,2,3] modes of SH-waves are very often noted SH$_{n}(n = 0,1,2,\ldots )$ and their associated displacement amplitude profiles are presented in Fig. 5.4.

Fig. 5.4

Relative displacement amplitudes of the transverse displacement $u_{y}$ for modes of SH-waves propagating in an elastic layer: symmetric SH$_{n} (n = 0,2,4,\ldots )$ and antisymmetric SH$_{n} (n=1,3,5,\ldots )$

It should be stressed that the displacement amplitude profiles of both symmetric and antisymmetric modes of SH-waves remain independent of the wave number k and the angular frequency $\omega$.

Finally, by using the definition of the parameter $\beta$ in Eq. (5.2) as well as the resulting relations from Eq. (5.5) and Eq. (5.6), the dispersion relation for SH-waves propagating within an elastic layer of thickness h, for symmetric and antisymmetric modes, can be expressed as a function of the wave number k in the following way:

$$\begin{aligned} c(k) = c_{S}\sqrt{1 + \left( \dfrac{n\pi }{kh}\right) ^{2}}, \quad n = 0,1,2,\ldots \end{aligned}$$

(5.7)

or alternatively as a function of the angular frequency $\omega$¹ as:

$$\begin{aligned} c(\omega ) = \frac{c_{S}}{\sqrt{1-\left( \dfrac{n\pi c_{S}}{\omega h}\right) ^{2}}}, \quad n = 0,1,2,\ldots \end{aligned}$$

(5.8)

It can be also noted from Figs. 5.2 and 5.3 that SH-waves propagating within an elastic layer are dispersive except for the fundamental mode, i.e. for $n = 0$, which is a non-dispersive fundamental symmetric mode SH$_{0}$. Moreover, it can be also seen that in the case of the fundamental mode the phase velocity of the SH-waves propagating in such a layer is equal to the phase velocity of SH-waves propagating in a 3-D elastic space.

5.2 Lamb Waves in an Elastic Layer

In a similar manner to SH-waves propagating in an elastic layer discussed previously in Sect. 5.1 the presence of two surfaces parallel to the xy plane bounding a 3-D elastic space leads to the entrapment of the propagating waves within the elastic layer, as presented in Fig. 5.5. As a result of this, it also leads to the formation of various wave modes, known as symmetric and antisymmetric modes of Lamb waves.

Lamb waves, very often referred to as Rayleigh-Lamb waves, are a different type of wave from SH-waves, which can also propagate in an elastic layer. Their propagation in an elastic layer is very similar to the propagation of Rayleigh waves discussed in Sect. 4.1. There a detailed discussion examined the formation of Rayleigh waves as a direct consequence of the presence of a free surface bounding a 3-D elastic space, where two types of waves can propagate freely as P-waves and S-waves. The presence of that free surface results in a coupled interaction between P-waves and S-waves, which required a complex mathematical description.

In the case Lamb waves the nature of the wave motion requires that the displacement component $u_{y}$ must vanish, i.e. $u_{y} = 0$, while the other two displacement components $u_{x}$ and $u_{z}$ are retained, remaining only a function of the spatial coordinates x and z as well as of time t. Thus the wave motion stays fully independent of the spatial coordinate y. As before it is assumed that $|z| < a$, with $2a = h$ where h denotes the thickness of the layer.

Fig. 5.5

Formation of the fundamental symmetric Lamb wave mode S$_{0}$ propagating in an aluminium layer in consecutive moments in time. Results of numerical computations by TD-SFEM

The investigation of Lamb waves benefits a lot from the decomposition of the displacement field, according to Eq. (3.25) from Sect. 3.2, into the scalar potential $\phi$ and the vector potential $\mathbf {\Psi }$. Bearing in mind the assumptions made about the nature of the wave motion it can be noted in the case of Lamb waves that Eq. (3.25) reduces to the following form:

$$\begin{aligned} \left\{ \!\! \begin{array}{l} u_{x}(x,z,t) = \dfrac{\partial \phi }{\partial x} - \dfrac{\partial \psi _{z}}{\partial z} \\ u_{z}(x,z,t) = \dfrac{\partial \phi }{\partial z} + \dfrac{\partial \psi _{z}}{\partial x} \end{array} \right. \end{aligned}$$

(5.9)

where now $\phi = \phi (x,z,t)$ and $\psi _{z} = \psi _{z}(x,z,t)$ represent two scalar potentials.

Both scalar potentials $\phi$ and $\psi _{z}$ must satisfy appropriate wave equations given by Eq. (3.21) and Eq. (3.22). For this reason it can be formally written that:

$$\begin{aligned} \left\{ \!\! \begin{array}{l} c^{2}_{P}\nabla ^{2}\phi = \ddot{\phi }\quad \text {or} \quad \dfrac{\partial ^{2}\phi }{\partial x^{2}} + \dfrac{\partial ^{2}\phi }{\partial z^{2}} = \dfrac{1}{c^{2}_{P}}\dfrac{\partial ^{2}\phi }{\partial t^{2}} \\ c^{2}_{S}\nabla ^{2}\psi _{z} = \ddot{\psi }_{z} \quad \text {or} \quad \dfrac{\partial ^{2}\psi _{y}}{\partial x^{2}} + \dfrac{\partial ^{2}\psi _{y}}{\partial z^{2}} = \dfrac{1}{c^{2}_{S}}\dfrac{\partial ^{2}\psi _{y}}{\partial t^{2}} \end{array} \right. \end{aligned}$$

(5.10)

Solutions to these wave equations (5.10) can be assumed to have well-known forms:

$$\begin{aligned} \left\{ \!\! \begin{array}{l} \phi (x,z,t) = \Phi (z) e^{ik(x - ct)} \\ \psi _{z}(x,z,t) = \Psi (z) e^{ik(x - ct)} \end{array} \right. \end{aligned}$$

(5.11)

where $\Phi$ and $\Psi$ represent unknown functions dependent only on the spatial coordinate z and where c is the phase velocity of Lamb waves propagating in the direction of the x axis.

A simple substitution of Eq. (5.11) into the appropriate wave equations, expressed by Eq. (5.10), leads to two ordinary differential equations:

$$\begin{aligned} \left\{ \!\! \begin{array}{l} \dfrac{d^{2}\Phi (z)}{dz^{2}} + p^{2} \Phi (z) = 0 \\ \dfrac{d^{2}\Psi (z)}{dz^{2}} + q^{2} \Psi (z) = 0 \end{array} \right. \end{aligned}$$

(5.12)

with:

$$\begin{aligned} \left\{ \!\! \begin{array}{l} p = k\sqrt{\dfrac{c^{2}}{c^{2}_{P}} - 1} = \sqrt{\dfrac{\omega ^{2}}{c^{2}_{P}} - k^{2}} \\ q = k\sqrt{\dfrac{c^{2}}{c^{2}_{S}} - 1} = \sqrt{\dfrac{\omega ^{2}}{c^{2}_{S}} - k^{2}} \end{array} \right. \end{aligned}$$

(5.13)

where p and q are certain constants dependent on the angular frequency $\omega$, the wave number k as well as the velocities of P-waves and SH-waves propagating in a 3-D elastic medium.

The displacement field associated with propagation of Lamb waves in an elastic layer, thanks to Eq. (5.11), can be expressed as:

$$\begin{aligned} \left\{ \!\! \begin{array}{l} u_{x}(x,z,t) = \left[ ik\Phi (z) - \dfrac{d\Psi (z)}{dz}\right] e^{ik(x - ct)} \\ u_{z}(x,z,t) = \left[ \dfrac{d\Phi (z)}{dz} + ik\Psi (z)\right] e^{ik(x - ct)} \end{array} \right. \end{aligned}$$

(5.14)

which helps to express the strain field in the elastic layer directly as dependent on the unknown functions $\Phi$ and $\Psi$:

$$\begin{aligned} \left\{ \!\! \begin{array}{l} \epsilon _{xx} = \dfrac{\partial u_{x}}{\partial x} = -\left[ k^{2}\Phi (z) + ik\dfrac{d\Psi (z)}{dz}\right] e^{ik(x - ct)} \\ \epsilon _{zz} = \dfrac{\partial u_{z}}{\partial z} = \left[ \dfrac{d^{2}\Phi (z)}{dz^{2}} + ik\dfrac{d\Psi (z)}{dz}\right] e^{ik(x - ct)} \\ \gamma _{xz} = \dfrac{\partial u_{x}}{\partial z} + \dfrac{\partial u_{z}}{\partial x} = \left[ 2ik\dfrac{d\Phi (z)}{dz} - k^{2}\Psi (z) - \dfrac{d^{2}\Psi (z)}{dz^{2}}\right] e^{ik(x - ct)} \\ \end{array} \right. \end{aligned}$$

(5.15)

Since solutions to the ordinary differential equations given by Eq. (5.12) are also well-known:

$$\begin{aligned} \left\{ \!\! \begin{array}{l} \Phi (z) = A_{1}\sin pz + A_{2}\cos pz \\ \Psi (z) = B_{1}\sin qz + B_{2}\cos qz \end{array} \right. \end{aligned}$$

(5.16)

the application of the traction-free boundary conditions for the stress components $\sigma _{zz}$ and $\tau _{xz}$ at the free surface of the elastic layer, i.e. for $z = \pm a$:

$$\begin{aligned} \sigma _{zz}(x,z,t) = \tau _{xz}(x,z,t) = 0, \quad \text {for} \quad z = \pm a \end{aligned}$$

(5.17)

allows one to obtain the dispersion relations for Lamb waves propagating in the elastic layer.

The stress components $\sigma _{zz}$ and $\tau _{xz}$ can be expressed by the use of Hooke’s law, as was presented in Eq. (4.9) in Sect. 4.1, which results in the following two equations:

$$\begin{aligned} \left\{ \!\! \begin{array}{l} \sigma _{zz}|_{\pm a} = 0 \rightarrow k^{2}\lambda \Phi (\pm a) - (\lambda + 2\mu )\dfrac{d^{2}\Phi (\pm a)}{dz^{2}} \\ \qquad \qquad \qquad - 2ik\mu \dfrac{d\Psi (\pm a)}{dz} = 0 \\ \tau _{xz}|_{\pm a} = 0 \rightarrow \mu \left[ 2ik\dfrac{d\Phi (\pm a)}{dz} - k^{2}\Psi (\pm a) - \dfrac{d^{2}\Psi (\pm a)}{dz^{2}}\right] = 0 \\ \end{array} \right. \end{aligned}$$

(5.18)

where as before the factor $e^{ik(x-ct)}$ is not present, since the traction-free boundary conditions must be satisfied for propagating Lamb waves independently of the spatial coordinate x and time t.

Fig. 5.6

Relative displacement amplitudes for symmetric modes of Lamb waves S$_{n} (n=0,1,2,\ldots )$ propagating in a 10 mm thick aluminium layer, at the frequency $f = 750$ kHz

Fig. 5.7

Relative displacement amplitudes for antisymmetric modes of Lamb waves A$_{n} (n=0,1,2,\ldots )$ propagating in a 10 mm thick aluminium layer, at the frequency $f = 750$ kHz

It can be immediately seen that the system of two equations, resulting from the traction-free boundary conditions is dependent on four unknown constants: $A_{1}$, $A_{2}$, $B_{1}$ and $B_{2}$. However, its solution is possible thanks to the fact that the wave motion associated with propagation of Lamb waves can be split into two independent types of wave modes, i.e. symmetric modes and antisymmetric modes, in a very similar manner as it was in the case of SH-waves propagating in an elastic layer.

The modes of Lamb waves are noted in the literature [1,2,3] as S$_{n}(n = 0,1,2,\ldots )$ for symmetric modes and A$_{n}(n = 0,1,2,\ldots )$ for antisymmetric modes. In contrast to SH-waves propagating in an elastic layer, this time the associated displacement amplitudes of Lamb waves concern two components, i.e. the longitudinal displacement component $u_{x}$ and the transverse displacement component $u_{z}$, as presented Figs. 5.6 and 5.7.

Moreover, a closer examination of the displacement field expressed by Eq. (5.14) allows one to conclude that in the case of wave motion in the direction of the x axis (i.e. longitudinal displacement component $u_{x}$) symmetric modes must be associated with cosine functions in Eq. (5.16), while antisymmetric modes must be associated with sine functions, as clearly seen in Fig. 5.6. It is opposite in the case of wave motion in the direction of the z axis (i.e. transverse displacement component $u_{z}$), when symmetric modes must be associated with sine functions in Eq. (5.16), while antisymmetric modes must be associated with cosine functions, as presented in Fig. 5.7.

Following the notation used in Eq. (5.11) the displacement and stress fields can formally be expressed as plane harmonic waves [1]:

$$\begin{aligned} \left\{ \!\! \begin{array}{ll} u_{x}(x,z,t) = \hat{u}_{x}(z)e^{ik(x-ct)}, &{} \sigma _{zz}(x,z,t) = \hat{\sigma }_{zz}(z)e^{ik(x-ct)} \\ u_{z}(x,z,t) = \hat{u}_{z}(z)e^{ik(x-ct)}, &{} \tau _{xz}(x,z,t) = \hat{\tau }_{xz}(z)e^{ik(x-ct)} \end{array} \right. \end{aligned}$$

(5.19)

which together with all above considerations allows one to present the unknown functions of the spatial coordinate z in the case of symmetric modes of Lamb waves as:

$$\begin{aligned} \left\{ \!\! \begin{array}{l} \Phi (z) = A_{2} \cos pz \\ \Psi (z) = B_{1} \sin qz \\ \hat{u}_{x}(z) = ik A_{2} \cos pz - q B_{1} \cos qz \\ \hat{u}_{z}(z) = -p A_{2} \sin pz + ik B_{1} \sin qz \\ \hat{\sigma }_{zz}(z) = -[k^{2}+p^{2}(\lambda + 2\mu )] A_{2}\cos pz + 2\mu ikq B_{1}\cos qz \\ \hat{\tau }_{xz}(z) = -2\mu ikp A_{2}\sin pz + \mu (q^{2} - k^{2}) B_{1}\sin qz \end{array} \right. \end{aligned}$$

(5.20)

while in the case of antisymmetric modes of Lamb waves as:

$$\begin{aligned} \left\{ \!\! \begin{array}{l} \Phi (z) = A_{1} \sin pz \\ \Psi (z) = B_{2} \cos qz \\ \hat{u}_{x}(z) = ik A_{1} \sin pz + q B_{2} \sin qz \\ \hat{u}_{z}(z) = p A_{1} \cos pz + ik B_{2} \cos qz \\ \hat{\sigma }_{zz}(z) = -[k^{2} + p^{2}(\lambda + 2\mu )] A_{1}\sin pz - 2\mu ikq B_{2}\sin qz \\ \hat{\tau }_{xz}(z) = 2\mu ikp A_{1}\cos pz + \mu (q^{2} - k^{2}) B_{2}\cos qz \end{array} \right. \end{aligned}$$

(5.21)

In such a way the resulting the system of two homogeneous equations leading to the dispersion relations for Lamb waves propagating in an elastic layer are always dependent on two unknown constants: $A_{2}$ and $B_{1}$ in the case of symmetric modes of Lamb waves and $A_{1}$ and $B_{2}$ in the case of antisymmetric Lamb waves.

In a similar manner as before the two system of homogeneous equations resulting from the traction-free boundary conditions at the free surface of the elastic layer, i.e. for $z = \pm a$, for either symmetric or antisymmetric mode of Lamb waves, have non-trivial solutions only if their determinant vanishes. These conditions lead to the two characteristic equations.

In the case of symmetric modes of Lamb waves this characteristic equation can be expressed as:

$$\begin{aligned} (k^{2} - q^{2})[\lambda k^{2} + (\lambda + 2\mu )p^{2}]\sin qa\cos pa = 4\mu k^{2}pq\sin pa\cos qa \end{aligned}$$

(5.22)

which can be simplified after necessary mathematical manipulations and rearrangements of terms² to its well-know form:

$$\begin{aligned} \frac{\tan qa}{\tan pa} = - \frac{4k^{2} pq}{(k^{2} - q^{2})^{2}}, \quad \text {for}\,symmetric\,modes \end{aligned}$$

(5.23)

In the case of antisymmetric modes of Lamb waves the corresponding characteristic equation can be expressed as:

$$\begin{aligned} (k^{2} - q^{2})[\lambda k^{2} + (\lambda + 2\mu )p^{2}]\sin pa\cos qa = 4\mu k^{2}pq\sin qa\cos pa \end{aligned}$$

(5.24)

which can be simplified after the same mathematical manipulations and rearrangements to its well-know form:

$$\begin{aligned} \frac{\tan qa}{\tan pa} = - \frac{(k^{2} - q^{2})^{2}}{4k^{2} pq}, \quad \text {for}\,antisymmetric\,modes \end{aligned}$$

(5.25)

Fig. 5.8

Dispersion curves for: a the phase velocity, b the group velocity, for symmetric modes of Lamb waves S$_{n} (n=0,1,2,\ldots )$ propagating in a 10 mm thick aluminium layer

Fig. 5.9

Dispersion curves for: a the phase velocity, b the group velocity, for antisymmetric modes of Lamb waves A$_{n} (n=0,1,2,\ldots )$ propagating in a 10 mm thick aluminium layer

Fig. 5.10

Relative displacement amplitudes as a function of the frequency f for the fundamental symmetric mode of Lamb waves S$_{0}$ propagating in a 10 mm thick aluminium layer

Fig. 5.11

Relative displacement amplitudes as a function of the frequency f for the fundamental antisymmetric mode of Lamb waves A$_{0}$ propagating in a 10 mm thick aluminium layer

The characteristic equations for symmetric and antisymmetric modes of Lamb waves, given by Eqs. (5.23) and (5.25), represent transcendental equations, which cannot be solved analytically. Instead, they can be effectively solved numerically leading to desired dispersion curves for Lamb waves propagating in an elastic layer—see Appendix B. Such dispersion curves obtained numerically are presented in Figs. 5.8 and 5.9 for symmetric and antisymmetric modes of Lamb waves.

It is interesting to note that the dispersion curves obtained for the fundamental symmetric S$_{0}$ and antisymmetric A$_{0}$ modes of Lamb waves are strongly correlated with Rayleigh surface waves, as the speed of Rayleigh waves $c_{R}$ is their high frequency limit.

Moreover, it can also be seen from Eqs. (5.20) and (5.20) that displacement amplitude profiles of symmetric and antisymmetric modes of Lamb waves are directly dependent on the wave number k, hence they are dispersive and as such must be also dependent on the angular frequency $\omega$. For this reason these displacement amplitude profiles evolve with changes in the wave number k or the angular frequency $\omega$, which is not observed in the case of SH-waves discussed in Sect. 5.1. This variation is also clearly observed in Figs. 5.10 and 5.11 as a function of the frequency f, due to the well-known relationship between the angular and cyclic frequencies $\omega = 2\pi f$.