On infinite group velocity in composite plates

Abstract

The analytical and geometrical conditions for the possible appearance of the infinite group velocity (IGV) points belonging to the dispersion curves of Lamb waves propagating in traction-free plates are studied by a combined method comprising Cauchy sextic formalism and the exponential fundamental matrix method. According to the obtained geometrical condition, the IGV corresponds to the coincidence of a tangent line to any of the dispersion curves with a straight line passing through the origin. The developed technique is demonstrated by applying to the analysis of Lamb wave dispersion in three-layered plates with a soft inner core.

Keywords

Acoustic wave traction-free plate phase velocity group velocity infinite group velocity

Introduction

The current research is concerned with the analysis of group velocity; deriving a relation between phase velocity and circular frequency for infinite group velocity (IGV); and, finding zones with large group velocity in three-layered plates with a soft inner core layer.

An overview

Group velocity

According to^1,2 and some later works^3–5 the scalar-valued group velocity $c_{g}$ can be defined as

c_{g} = \frac{\partial ω}{\partial r},

(1.1)

where

ω

is the circular frequency, and

r

is the wavenumber

r = \frac{ω}{c},

(1.2)

herein,

c

is the phase velocity.

It has also been found that thus defined group velocity of travelling harmonic waves in plates may be take positive, zero, and negative values.^6–12 The zero group velocity (ZGV) plays an important role in the acoustic non-destructive testing of materials^13–16 and geophysical formations.^17,18 It is also known that the zone in the vicinity of ZGV is prone to large oscillations, which may cause the plate “ringing;” see.^19–24

Group slowness, group velocity, and zero group velocity

Following,²⁵ introduce slowness associated with phase and group velocities

s = c^{- 1}; s_{g} = c_{g}^{- 1} .

(1.3)

With equation (1.3), it is possible to find a convenient expression for

c_{g}

.^26,27

s_{g} \equiv \frac{\partial r}{\partial ω} = \frac{c (ω) - ω \partial_{ω} c (ω)}{c {(ω)}^{2}},

(1.4)

from where

c_{g} = \frac{c^{2} (ω)}{c (ω) - ω \partial_{ω} c (ω)} = \frac{c (ω)}{1 - ω \partial_{ω} (\ln c (ω))} .

(1.5)

The last equation yields the following conditions for the ZGV²⁰

\begin{array}{l} Z G V_{I} : c (ω) = 0 & ω \partial_{ω} (\ln c (ω)) \neq 1 \\ Z G V_{I I} : c (ω) \neq 0 & ω \partial_{ω} (\ln c (ω)) \to \pm \infty \end{array} .

(1.6)

While the first condition contains no valuable solutions, the last condition breaks down into two more conditions

\begin{array}{l} Z G V_{I I_{A}} : c (ω) \to \infty & | ω \partial_{ω} (\ln c (ω)) | \to \infty & | ω \partial_{ω} (\ln c (ω)) | = O (c (ω)) \\ Z G V_{I I_{B}} : c (ω) < \infty & ω \partial_{ω} (\ln c (ω)) \to \pm \infty . \end{array}

(1.7)

Condition

Z G V_{I I_{A}}

corresponds to the real resonances associated with the appearance of standing waves in a plate;^6,28 and condition

Z G V_{I I_{B}}

gives the ZGV associated with the transverse quasi resonance phenomenon,^19,20,29,30 occurring at a finite phase velocity. Note also, that condition

Z G V_{I I_{B}}

can be rewritten in terms of

ω (c)

, yielding³¹

Z G V_{I I_{B}} : c (ω) < \infty & \partial_{c} ω (c) = 0 .

(1.8)

Infinite group velocity

Formally, the infinite group velocity stems from equation (1.5), yielding

\begin{array}{l} I G V_{I} : c (ω) \to \infty & | ω \partial_{ω} (\ln c (ω)) | < \infty \\ I G V_{I I} : 0 < c (ω) < \infty & ω \partial_{ω} (\ln c (ω)) = 1 \end{array} .

(1.9)

The first condition corresponds to the appearance of stationary waves, when

c (ω) \to \infty

⁶; while, the condition

I G V_{I I}

means the appearance of the IGV at a finite phase velocity. This condition can also be rewritten in terms of

c

, which is considered as an independent variable

I G V_{I I} : 0 < c < \infty & \partial_{c} ω (c) = \frac{ω (c)}{c} .

(1.10)

Geometrically, equation (1.10) corresponds to the coincidence of the tangent

\partial_{c} ω (c)

with a straight line

c - ω (c)

in the corresponding plane; see Figure 1.

The situation shown in Figure 1 is at least rare, much more often the dispersion curves have either a negative tangent, or a very small positive; see Figure 2.

Group velocity and flux of energy

Assuming that a medium has no change in entropy, which means the absence of the mechanical energy dissipation,³² has found that the group velocity defined by equation (1.1) coincides with the velocity of energy propagation ( $c_{e}$ )

c_{g} = c_{e} .

(1.11)

Later on, condition (1.11) was extrapolated to inhomogeneous or discontinuous media.³³ According to³³ the velocity of energy propagation is defined, as

c_{e} \equiv \frac{J \cdot n}{W},

(1.12)

where

J \cdot n

is the period- and thickness-averaged flux of energy through a plane normal to the direction of propagation, defined by the wave vector

n

; and,

W

is the period- and thickness-averaged strain energy.³⁴ Now, in view of equations (1.11), (1.12), the

I G V_{I I}

condition splits into three following conditions

\begin{array}{l} I G V_{I I_{A}} : 0 < W < \infty & J \cdot n \to \infty \\ I G V_{I I_{B}} : 0 < J \cdot n < \infty & W \to 0 \\ I G V_{I I_{C}} : J \cdot n = O (W) @ W \to 0 \end{array} .

(1.13)

Condition

I G V_{I I_{A}}

is unlikely, as an infinite flux of energy would require either infinite particle velocity, or infinite strain, but energy

W

is finite by supposition. Condition

I G V_{I I_{B}}

is also unlikely, since

W \to 0

implies vanishing strain

ε \to 0

, which is inconsistent with the assumed finite

J \cdot n

. Finally, condition

I G V_{I I_{C}}

is consistent with vanishing strain

ε \to 0

, since

| J \cdot n | = O (W)

W \to 0

, implies:

I G V_{I I_{C}} : | J \cdot n | = O (W) a t ε \to 0 .

(1.14)

Actually, condition (1.14) means that at the

I G V_{I I_{C}}

, the wave amplitudes become infinite, causing the appearance of a dynamic resonance.

Problem statement

The current research is aimed at finding for IGV at the propagation of Lamb waves in stratified isotropic plates with contrast physical properties of the layers; Figure 3.

It is revealed that the IGV arises in the fundamental symmetric branch $S_{0}$ of Lamb waves propagating in a three-layered plate with symmetric layout (Figure 3) and a soft inner core layer. Such composite plates, being vulnerable to the appearance of IGV, are widely used in aerospace, automotive, marine engineering, etc.^35–39

The ongoing analysis is based on the Cauchy sextic formalism and the exponential fundamental matrix method,⁴⁰ allowing constructing a dispersion equation and, using the geometrical condition (1.10), finding both the finite frequency and phase velocity related to large group velocity values belonging to the fundamental branches.

Dispersion equation

Herein, the dispersion equation is derived by using Cauchy sextic formalism and the exponential fundamental matrix method.⁴⁰

Equations of motion

The hyperbolic equation of motion in a homogeneous layer can be written, as⁴¹

{div}_{x} C \cdot \cdot \nabla_{x} u (x, t) = ρ \partial_{t t}^{2} u (x, t),

(2.1)

where

u (x, t)

is the displacement field;

x

is the spatial coordinate;

t

is the time;

ρ

is the material density; and,

C

is the strongly elliptic fourth-order elasticity tensor, which for an elastic medium takes the form⁴²

C = λ I \otimes I + 2 μ 1 .

(2.2)

In (2.2)

λ

and

μ

are Lame’s constants;

I

is the unit

3 \times 3

-matrix; and,

1

is the unit

6 \times 6

-matrix. It is also assumed that the strain obeys the infinitesimal Cauchy strain-displacement relation⁴¹

ε = sym (\nabla_{x} u) .

(2.3)

Consider a harmonic plane wave, propagating in a layer⁴¹

u (x, t) = m (x) \exp i (r n \cdot x - ω t),

(2.4)

where

m

is the unit vector defining the wave polarization;

x

is the complex transverse coordinate

x = i r ν \cdot x,

(2.5)

and

i = \sqrt{- 1}

. Substituting representation (2.4) into equation of motion (2.1), yields a matrix ordinary differential equation

(A_{1} \partial_{x x}^{2} + A_{2} \partial_{x} + A_{3}) \cdot m (x) = 0,

(2.6)

where

A_{1} = ν \cdot C \cdot ν, A_{2} = ν \cdot C \cdot n + n \cdot C \cdot ν, A_{3} = n \cdot C \cdot n - ρ c^{2} I .

(2.7)

Note that in view of the assumed strong ellipticity of the elasticity tensor, matrix

A_{1}

is not singular, and matrix

A_{3}

is not singular, if

ρ c^{2} \notin Sp (n \cdot C \cdot n)

Cauchy sextic formalism

The general solution of the matrix ordinary differential equation (2.6) can be constructed by introducing a new 3-vector

v (x) = \partial_{x} m (x),

(2.8)

and a new 6-vector

Y (x) = (\begin{array}{l} m (x) \\ v (x) \end{array}) .

(2.9)

Now, by rearranging the terms in equation (2.6), it can be rewritten, as a first order differential equation

\partial_{x} Y (x) = G \cdot Y (x),

(2.10)

where

G = (\begin{array}{l} 0 & I \\ - A_{1}^{- 1} \cdot A_{3} & - A_{1}^{- 1} \cdot A_{2} \end{array}) .

(2.11)

Note that in view of the supposition

ρ c^{2} \notin Sp (n \cdot C \cdot n)

\det G \neq 0

; thus, equation (2.10) is not singular. The matrix

G

is essential to construct the dispersion equation, following⁴³ it is called the fundamental matrix.

Exponential fundamental solution

The general solution of equation (2.10) can be written in the form⁴⁴

Y (x) = \exp (G x) \cdot Y_{0},

(2.12)

where

\exp (G x)

is the exponential fundamental matrix; and

Y_{0}

is an arbitrary 6-vector, specified by boundary conditions. To construct matrix

\exp (G x)

, the Jordan normal form decomposition of the fundamental matrix

G

is needed⁴⁵

G = W \cdot J \cdot W^{- 1},

(2.13)

where

W

is a non-singular

6 \times 6

-matrix containing either proper (right) eigenvectors of matrix

G

, if it is semisimple, or

W

contains both proper and generalized eigenvectors, if

G

is non-semisimple; and,

J

is either diagonal matrix, containing eigenvalues of matrix

G

, if it is semisimple, or

J

contains both diagonal part and Jordan blocks, if

G

is non-semisimple. The case of a non-semisimple

G

was studies in,⁴⁶ and herein will not be considered.

With decomposition (2.13) and that $G$ is semisimple, the exponential fundamental matrix $\exp (G x)$ becomes

\exp (G x) = W \cdot diag (\exp (λ_{1} x), . . ., \exp (λ_{6} x)) \cdot W^{- 1},

(2.14)

where

λ_{1}, . . ., λ_{6}

are eigenvalues of matrix

G

. Note also, that neither eigenvalues, nor eigenvectors stored columnwise in

W

need to be real.^46–48

Boundary and interfacial conditions

The surface traction field acting on a plane $x = c o n s t$ , can be written (up to a multiplier $\exp (i ω t)$ ) in terms of the 6-vector (2.9)

t_{ν} (x) \equiv ν \cdot C \cdot \cdot \nabla_{x} u = (A_{4}; A_{1}) \cdot Y (x),

(2.15)

where

A_{4} = ν \cdot C \cdot n .

(2.16)

With equation (2.15), the traction-free boundary conditions at the plate outer surfaces

x = \pm i r H

, where

2 H

is the overall plate thickness, become

{(A_{4}; A_{1}) \cdot Y (x) |}_{x = \pm i r H} = 0 .

(2.17)

Consider now an interface with perfect mechanical contact, it can be written in terms of vectors

Y_{k} (x)

and impedance

6 \times 6

-matrices

Z_{k}

{Z_{k} \cdot Y_{k} (x) |}_{x = - i r h_{k}} = {Z_{k + 1} \cdot Y_{k + 1} (x) |}_{x = + i r h_{k + 1}}

(2.18)

where lower indices are referred to the adjacent layers, and

Z = (\begin{array}{l} I & 0 \\ A_{4} & A_{1} \end{array}) .

(2.19)

It is assumed that each layer in equation (2.18) has thickness

2 h_{k}

. Note that

\det Z = A_{1} > 0,

(2.20)

which ensures the non-degeneracy of matrix

Z

Regarding the boundary conditions, or better, the radiation condition at $n \cdot x \to \pm \infty$ , it should be noted that the assumption on harmonicity of the displacement field in the $n \cdot x$ -direction, eliminates the need to impose any conditions in this direction; see.^40,48

Dispersion equation and infinite group velocity condition

The combination of boundary and interfacial conditions (2.17), (2.18) along with the exponential fundamental solution (2.12), yield the desired dispersion equation⁴⁶

F (c, ω) \equiv \det [(0; I) \cdot (\prod_{k = 1}^{3} Z_{k} \cdot \exp (- 2 i r h_{k}) \cdot Z_{k}^{- 1}) \cdot (\begin{array}{l} I \\ 0 \end{array})] = 0,

(2.21)

herein, as before, lower index

k

is referred to the corresponding layer. Equation (2.21) can be considered as an implicit equation for finding dispersion relations between phase velocity and circular frequency. The implicit function theorem allows us to write

\frac{\partial F}{\partial ω} \frac{\partial ω}{\partial c} + \frac{\partial F}{\partial c} = 0 .

(2.22)

Now, combining equation (1.10) with equation (2.22), yields the following equation for the IGV:

\frac{\partial F}{\partial c} {(\frac{\partial F}{\partial ω})}^{- 1} = - \frac{ω}{c} .

(2.23)

Actually, dispersion equation (2.21) means that the mapping from a non-trivial surface displacement field and vanishing surface tractions acting on the upper surface of the plate to a surface traction field acting on the bottom surface is degenerate. The latter implies the existence of a non-trivial displacement field satisfying boundary conditions (2.15); see.⁴⁶

Dispersion analysis

Physical properties

Consider a three-layered plate with symmetric layout (Figure 3) and isotropic layers satisfying Wiechert conditions

\frac{λ_{inner}}{λ_{outer}} = \frac{μ_{inner}}{μ_{outer}} = \frac{ρ_{inner}}{ρ_{outer}} = q; \frac{h_{inner}}{h_{outer}} = g

(3.1)

where lower indices are referred to the corresponding layers;

λ

and

μ

are Lame’s constants;

ρ

is material density;

2 h

is the thickness of a layer. The dimensionless parameter

q

which defines the relative physical properties of the inner core layer varies in a range

q \in [10^{- 3}; 10^{- 1}]

; and, the dimensionless geometrical parameter

g

varies in a range

g \in [10^{- 2}; 10^{+ 2}]

, from a very thin inner core

g = 10^{- 2}

, simulating a crack-like defect at small

q

; to a very thick inner layer

g = 10^{+ 2}

simulating various composite plates that are widely used in aerospace, marine, and automotive engineering.^35,37,38,49 In the current analysis, the Poisson ratio of both inner and outer layers is taken

ν = 0.25

. And a necessary remark should be given for the Wiechert condition (3.1), which may seem too restricted for most composite plates. However, it is known that a fairly large number of aerogels with different porosities have physical properties obeying the Wiechert condition.⁵⁰

Dispersion portraits

An overview

The typical dispersion portrait for a three-layered plate with $q = 0.01$ and $g = 0.1$ is shown in Figure 4, where the relative phase velocity $c *$ and the relative frequency $ω *$ are defined by the following expressions

c * = \frac{c}{c_{S}}; ω * = \frac{ω h_{outer}}{c_{S}} .

(3.2)

Herein,

c_{S}

is the shear bulk wave velocity, which is the same for the inner and outer layers, in view of equation (3.1).

The plots in Figure 4(b) reveal that above the bend zone of the $A_{0}$ -branch, the angles $ψ$ between tangents at points $(c; ω (c))$ and the corresponding straight lines passing through points $(0; 0)$ and $(c; ω (c))$ are relatively small. Note that these angles are invariant under the (same) linear transformation of the axes.⁵¹

Instead of finding angles $ψ$ , more consistent results can be achieved at computing the scalar product of the following unit vectors

w = α \cdot β,

(3.3)

where

α

and

β

are defined as

α = \frac{(1; \frac{\partial ω (c)}{\partial c})}{\sqrt{1 + {(\frac{\partial ω (c)}{\partial c})}^{2}}}; β = \frac{(1; \frac{ω (c)}{c})}{\sqrt{1 + \frac{ω^{2} (c)}{c^{2}}}},

(3.4)

where the symbol “;” is used to separate components of the corresponding vectors. Taking into expressions (3.4), the scalar product in (3.3) becomes

w = \frac{(1 + \frac{ω (c)}{c} \frac{\partial ω (c)}{\partial c})}{\sqrt{(1 + {(\frac{\partial ω (c)}{\partial c})}^{2}) (1 + \frac{ω^{2} (c)}{c^{2}})}} .

(3.5)

Thus defined parameter

w

varies in a range

w \in [- 1; 1]

, with its extreme value

w = + 1

yielding the IGV according to the geometrical condition given by equation (1.10). At fixed

q

and

g

, the introduced scalar product

w

can be considered as a function of the phase velocity

c

, which allows us to pose the following optimization problem

W \equiv \max_{c} w (c) .

(3.6)

Another remark concerns the geometrical representation of

W

, which according to equations (3.3)–(3.6), yields the desired minimal angle:

Ψ = \arccos (W) .

(3.7)

Equation (3.7) is used in the next subsection for analyzing the behavior of

Ψ

at varying

q

and

g

parameters.

Variation of $W$ and $Ψ$ at varying parameters $q$ and $g$

The plots in Figure 5 show the variation of $W$ and $Ψ$ at the varying parameters $q \in [10^{- 3}; 10^{- 1}]$ and $g \in [10^{- 2}; 10^{+ 2}]$ , revealing the gradual increase of $W$ and the corresponding decrease of $Ψ$ as $q$ decreases, which take place at almost any $g$ .

Moreover, as these plots indicate, at small $q = 10^{- 2}$ the difference in orientation of the unite vectors $α$ and $β$ becomes small, not exceeding $0.99$ in terms of $W$ , meaning that the corresponding group velocity at $q \to 0$ approaches IGV. The dependencies of $W$ and the corresponding (acute) $ψ$ -angle, which are shown in Figure 5, follow from the dispersion portraits, exhibiting bends in the fundamental $S_{0}$ branch at any $q < 1$ and virtually any $g$ ; see Figure 4. Meanwhile, at $q > 1$ no bend in the $S_{0}$ branch takes place (Figure 2) and hence, $ψ$ -angle becomes obtuse.

Concluding remarks

The performed analysis on the IGV in three-layered plates with soft inner core layer reveals that at the decreasing parameter $q$ , which is responsible for a relative Young’s modulus and density of the inner core layer, the difference in orientation of unit vectors $α$ and $β$ (Figure 4(b)) also gradually decreases (Figure 5(a)), meaning that at small $q \to 0$ , the corresponding group velocity, which satisfies condition (3.6) tends to the IGV. This result is virtually independent of the inner layer thickness, as the plots in Figure 5 demonstrate.

The IGV, defined by equations (1.9), (1.10), corresponds to the vanishing strain energy, according to equations (1.12)–(1.14) and, hence, to the appearance of a new type of resonance occurring at a finite frequency and finite phase velocity, where the symmetric fundamental branch undergoes a bend (Figure 4(b)). It should also be noted that as the performed analysis shows, the discussed resonance phenomenon is absent in the three-layered plates with a symmetric layout, but having a more rigid inner core layer compared to the outer layers. That is because of the absence of an acute bend in the fundamental dispersion curve $S_{0}$ ; see Figure 6.

Another remark concerns the relation between the discussed resonance phenomenon associated with the IGV occurring at a finite phase velocity and the proper resonance known also as the “long-wave high-frequency resonance,” which appears in both homogeneous and layered plates at the infinite phase velocity values; see.^52–56

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Sergey V Kuznetsov

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On infinite group velocity in composite plates

Abstract

Keywords

Introduction

An overview

Group velocity

Group slowness, group velocity, and zero group velocity

Infinite group velocity

Group velocity and flux of energy

Problem statement

Dispersion equation

Equations of motion

Cauchy sextic formalism

Exponential fundamental solution

Boundary and interfacial conditions

Dispersion equation and infinite group velocity condition

Dispersion analysis

Physical properties

Dispersion portraits

An overview

Variation of W and Ψ at varying parameters q and g

Concluding remarks

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

Variation of $W$ and $Ψ$ at varying parameters $q$ and $g$