Development of the Governing Wave Equations for Circumferential Waves

The development of the characteristic equations for a single-layer annulus is a necessary step toward the development of the multilayer cases. Authors such as Viktorov (1967), Liu and Qu (1998a,b), and Zhao and Rose (2004) have made single-layer considerations for circumferential guided waves. Liu and Qu (1998a) and Zhao and Rose (2004) provide detailed derivations of the characteristic equations and wave structures for the single-layer cases of CLT-wave and CSH-wave propagation, respectively. The development presented here is similar to these two cases but is modified in anticipation of developing the multilayer solutions. Specifically, a generalized boundary value problem will be developed such that the phase term is no longer associated with the boundary of the annulus. All dimensionless quantities must also be removed from the formulation.

The solution approach used here utilizes the method of displacement potentials and, as a result, applies only to isotropic materials. Additionally, all generalized plane-strain assumptions prevail. Figure 11.1 shows the theoretical model of a single-layer annulus used for the development of the CSH and CLT characteristic equations.

ê_r ê

r₂ r₁

Figure 11.1. Theoretical model used for the development of the governing equations for CSH-wave and CLT-wave propagation in a single-layer annulus.

The development of the characteristic equations begins with the well-known Navier’s Equation of Motion, Equation (11.1) (Graff 1991):

(λ+2µ)∇ − ∇ × +∆ 2µ ω ρf=ρu , (11.1) where the dilatation of a material, ∆, is given by:

∆ = ∇ ⋅ u, (11.2)

and the rotation vector, ω, is given by:

ω ω= ∇×1

2 u (11.3)

In Equation (11.1), ρ is the material density, λ and µ are Lamé constants, f is the body-force vector, and u is the displacement vector. Note that the form of Navier’s equation shown in Equation (11.1) has the advantage of applicability in any curvilinear coordinate system and the convenience of separated dilatational and rotational components (Graff 1991). From this point forward, it is assumed that no body forces are present.

In the case of circumferential guided wave propagation, the displacement vector is given by Equation (11.4):

u=u rr( , , )θ t er+u rθ( , , )θ t eθ +u rz( , , )θ t ez (11.4) It is seen that, because generalized plane-strain assumptions have been made, the displacement components are functions of r, θ, and t only. This does not imply that displacement cannot occur in the z-direction but, instead, that any displacement in the z-direction must be uniform throughout the entire z-plane. The specific cases of CSH- and CLT-wave propagation are discussed next.

11.2.1 Circumferential Shear Horizontal Waves in a Single-Layer Annulus Shear horizontal (SH-) waves are waves in which the particle motion is in-plane but orthogonal to the direction of propagation. In the case of CSH-waves, particle displacement would be in the z-direction with no displacement in the r- or θ-directions (ur = uθ = 0). Also, from the generalized plane-strain assumption, there must be no variation of any quantity in the z-direction (∂ ∂ =z 0). Under these conditions, Equation (11.1) simplifies to:

µ∇ =ρ∂

∂ ∇ = ∂

2 2 ∂

2

2 2

2 2

u u 1

t or u c

u

z z t

z s

z. (11.5)

Note that in the CSH-wave case, a scalar form of the governing displacement equation of motion has been obtained with no need for the use of Helmholtz decomposition.

Two forms of the wave equation are shown in Equation (11.5), from which it can be determined that bulk shear waves propagate with a velocity given by:

cs = µ

ρ. (11.6)

177 11.2 Development of the Governing Wave Equations for Circumferential Waves

Assuming time-harmonic motion of the form e^{−i tω} and propagation in the θ-direction, one solution of Equation (11.5) is assumed to be:

uz=ψ( )r ei p^{( θ ω−} t⁾, (11.7) where p is the angular wavenumber (Viktorov 1967; Liu and Qu 1998a). It is important to distinguish between the circular wavenumber k and the angular wavenumber p as the terms circular and angular are often used interchangeably. In this work, the angular wavenumber, p, refers specifically to the circular wavenumber, k, multiplied by some radius, R, at which the linear phase velocity is to be determined. It is a dimensionless quantity. Equation (11.8) summarizes this relationship.

p kR= . (11.8)

Equation (11.8) shows that there are two different ways to proceed: either assume a radius and solve for the k-roots of the characteristic equation at the assumed radius or find the p-roots and then calculate the k-roots at any radius. For multilayered annuli, finding the roots of the characteristic equation in the ω-p domain provides the most general solution that can subsequently be used to determine the linear phase velocity, cp, at an arbitrary radius of the multilayered structure by the use of Equations (11.9) and (11.10), first introduced by Viktorov (1967).

α ω

p= . p (11.9)

c R_p( )=α_pR. (11.10)

Equation (11.10) demonstrates that there is a linear increase in phase velocity, cp, from the internal to the external surface of the annulus. This increase in phase velocity with radius is necessary to maintain a constant phase front through the thickness of the annulus. This phenomenon is unique to wave propagation along structures that are curved in the direction of wave propagation. Note that this does not include wave propagation in the axial direction of a hollow cylinder as it is typically assumed that the structure is straight in this direction.

Substituting Equation (11.7) into Equation (11.5) yields Equation (11.11):

r r r

c p

s 2

2

2 0

′′ + ′ + 





 −











 =

ψ ψ ω ψ , (11.11)

which is a second-order ODE, commonly solved using Bessel or Hankel functions (Hayek 2001). The solution of Equation (11.11) in terms of Bessel functions is of the form:

ψ( )r =A J k r1 _p

( )

_s ⁺A Y k r2 _p

( )

_s , (11.12) where:

k_s=ω c_s (11.13)

is the circular wavenumber of a bulk shear wave.

To solve for the unknown coefficients in Equation (11.12), the boundary conditions for the problem of interest must be considered. In the case of a single-layer “free” annulus, traction-free boundary conditions are assumed; that is, stress is required to vanish on the inside and outside surfaces of the annulus. In the case of CSH-waves, shear stress is given by:

σrz µ uz

= ∂r

∂ . (11.14)

Using the recurrence relation for Bessel functions (Abramowitz and Stegun 1964):

2ξν′( )x =ξν−1( )x −ξν+1( ),x where ξ=J or Y, (11.15) and with the use of Equation (11.7) and Equation (11.12), Equation (11.14) can be written as: The e^-iω^t time dependence is inferred but not explicitly written in Equation (11.16) or Equation (11.17), as will be the case for the remainder of this chapter.

In preparation for the case of multiple layers, it is convenient to summarize the relevant stress and displacement eigenfunctions in a single matrix equation, as seen in Equation (11.18): needed, whereas in the multilayer case, both the displacement- and stress-related components will be needed.

In the case of the single-layer annulus, the traction-free boundary conditions require that

σ_{rz r r1 2,} = 0, (11.19)

resulting in the set of linear homogeneous algebraic equations given by Equation (11.20) and Equation (11.21).

D( , )pω A 0= , (11.20)

179 11.2 Development of the Governing Wave Equations for Circumferential Waves

where the determinant of the coefficient matrix, D( , )pω , to zero:

det

(

D p( , )ω

)

^{= 0} . ^(11.22) Equation (11.22) is the characteristic equation for CSH-waves in a single-layer annulus, the eigenvalues of which form the frequency-wavenumber dispersion curves. The angular phase and group velocity dispersion curves can then be possible to solve for the displacement and stress distribution throughout the wall thickness of the annulus for any given mode and frequency combination. The displacement field is given by Equation (11.24):

u_z=

(

J k r_p⁽ _s ⁾−^ΛY k r e_p⁽ _s ⁾

)

^{i p(θ ω− t), (11.24)}

In Equations (11.24), (11.26), and 11.27 the constant A1 has been arbitrarily set to unity, hence the solutions are not unique and multiplication by any constant will yield a valid solution.

This concludes the development of the characteristic equation and the displacement and stress relations for the case of CSH-wave propagation in a single-layer annulus. The next section addresses the development of the Lamb type dispersion equation, displacements, and stresses for the single-layer case.

11.2.2 Circumferential Lamb Type Waves in a Single-Layer Annulus

Lamb waves are waves with displacements in two directions: in-plane and out-of-plane. Unlike SH-waves, the in-plane component in the Lamb wave case is along the line of propagation. Because Lamb waves are technically a type of plate wave, this work refers to circumferential Lamb type, or CLT-, waves when referring to propagation in an annulus. This distinction is important because of the physical differences in the propagation characteristics, although, as the ratio of inner diameter to outer diameter approaches unity, the annulus solution approaches that of a plate and the two cases become equivalent (Liu and Qu 1998a).

Using the theorem introduced by Helmholtz, it is possible to dissect the displacement field, u, into a sum of the gradient of a scalar and the curl of a zero-divergence vector with the use of the scalar and vector potentials, Φ and H (Morse and Feshbach 1953). This is shown in Equation (11.28).

u= ∇ + ∇ ×Φ H,∇⋅ =H 0 , (11.28) where ∇⋅ =H 0 is a necessary condition to determine a unique solution for the three displacement components from the total of four components of Φ and H. Substituting Equation (11.28) into Navier’s equation, Equation (11.1), Equation (11.29) is obtained:

∇

(

^(λ+^2µ)∇^2Φ−^ρΦ

)

+ ∇ × − ∇ × ∇ × −

(

^{µ H ρH}

)

^{=0. (11.29)}

For a nontrivial solution of Equation (11.29), it is required that:

(λ+2µ)∇²Φ=ρΦ , (11.30) and

− ∇ × ∇ × =µ H ρH . (11.31) Recalling Equation (11.6) and also noting that longitudinal waves propagate at a velocity given by

c_l = ( + ) λ µ ,

ρ

2 (11.32)

it is possible to rewrite Equation (11.30) and Equation (11.31) as

∇ = ∂

2 ∂

2 2

2

Φ 1 Φ

c_l t (11.33)

and

∇ = ∂

2 ∂

2 2

2

H 1 H

c_s t , (11.34)

In document Ultrasonic Guided Waves in Solid Media (Page 195-200)