SAFE Formulation for Cylindrical Structures

directions, whereas in isotropic plates, Lamb waves only have displacements in the x and z directions while SH waves have y direction displacements only. Nevertheless, analogies between the guided wave modes in composite plates and the wave modes in isotropic plates can still be observed. For instance, the wave structure of mode 1 in the composite plate has a dominant out-of-plane displacement (uz) that makes mode 1 similar to the Lamb wave mode A0 in isotropic plates. Similarly, mode 2 and mode 3 have analogies with the SH0 mode and the S0 mode, respectively. Because of the analogy based on wave structures, the fundamental wave modes in the composite plate possess somewhat similar characteristics as the A0, S0, and SH0 modes in isotropic plates when considering wave excitabilities, which are discussed elsewhere in the text.

The stress distributions across the plate thickness for the three fundamental modes of the composite plate at 500 kHz are shown in Figure 9.6.

The stress distributions are calculated from the wave structures using Equation (9.7) and Equation (9.9). As has been shown, the calculated stress components σxz , σyz, and σzz vanish at the plate surfaces, which satisfies the traction-free boundary conditions. They also satisfy the continuity interface conditions between different layers of the composite plate. The correctness of the SAFE calculations is thus verified.

The skew angle dispersion curves calculated based on Equation (9.36) are shown in Figure 9.7. The integrations in Equation (9.36) are evaluated numerically.

The wave vector direction is zero degrees.

Although the composite plate is of a quasi-isotropic stacking sequence, the skew angles introduced by the material anisotropy actually vary in a large range roughly from –45 degrees to 38 degrees. For guided wave modes with nonzero skew angles, the wave energies propagate in the directions away from the wave vector direction.

Hence, a careful investigation of wave skew effects is necessary in guided wave applications when studying anisotropic waveguides.

9.8 SAFE Formulation for Cylindrical Structures

The SAFE formulation for cylindrical structures also starts with the governing equation provided by the virtual work principle. Equation (9.37) holds for a stress-free hollow cylinder. Linear elastic and viscoelastic material behavior is considered here.

δu^T ρu δε σ

V

T V

dV dV

⋅ + ⋅ =

∫

^

∫

^{0, (9.37)}

where T represents matrix transpose, ρ is density, and u is the second derivative of displacement u with respect to time t. dΓ

∫

_Γ ^and

∫

V^dV are the surface and volume Table 9.1. Material properties of Cytec CYCOM 977–3

carbon epoxy prepreg

E1 172 GPa

E₂ = E3 9.8 GPa

G23 3.2 GPa

G₁₂ = G13 6.1 GPa

v23 0.55

v₁₂ = v13 0.37

0 0.5 1 1.5 2 0

5 10 15 20

Frequency (MHz)

Phase Velocity (km/sec.)

0 0.5 1 1.5 2

0 5 10 15 20

Frequency (MHz)

Phase Velocity (km/sec.)

0 0.5 1 1.5 2

0 5 10 15 20

Frequency (MHz)

Phase Velocity (km/sec.)

1 2 3

4

5

6

(a)

(b)

(c)

Figure 9.3. Phase velocity dispersion curves of the eight-layer quasi-isotropic plate for the wave vector directions of (a) 0°, (b) 30°, and (c) -30°.

147 9.8 SAFE Formulation for Cylindrical Structures

0 0.5 1 1.5 2

0 1 2 3 4 5 6 7

Frequency (MHz)

Group Velocity (km/sec.)

0 0.5 1 1.5 2

0 1 2 3 4 5 6 7

Frequency (MHz)

Group Velocity (km/sec.)

0 0.5 1 1.5 2

0 1 2 3 4 5 6 7

Frequency (MHz)

Group Velocity (km/sec.)

(a)

(b)

(c)

Figure 9.4. Group velocity dispersion curves of the eight-layer quasi-isotropic plate for the wave vector directions of (a) 0°, (b) 30°, and (c) -30°.

0.8

–1 –0.5 0 0.5 1

Normalized Particle Displacment

Plate Thickness (mm)

(a) –0.8

–0.6 –0.4 –0.2 0.2 0.4

u_x u_y u_z 0.6

0

0.8

–1 –0.5 0 0.5 1

Normalized Particle Displacment

Plate Thickness (mm)

(b) –0.8

–0.6 –0.4 –0.2 0.2 0.4

ux u_y u_z 0.6

0

0.8

–1 –0.5 0 0.5 1

Normalized Particle Displacment

Plate Thickness (mm)

(c) –0.8

–0.6 –0.4 –0.2 0.2 0.4

u_x u_y u_z 0.6

0

Figure 9.5. Example wave structures of the three fundamental modes at 500 kHz. (a) mode 1, (b) mode 2, (c) mode 3.

149 9.8 SAFE Formulation for Cylindrical Structures

0.8

Figure 9.6. Stress distributions of the three fundamental modes at 500 kHz. (a) mode 1, (b) mode 2, (c) mode 3.

integrals of the element, respectively. For wave propagation in cylindrical structures, the wave equation needs to be solved in cylindrical coordinates, dV rdrd dz= θ . The first and second terms on the left-hand side are the corresponding increments of kinetic energy and potential energy.

We adopted an exact analytical solution e^inθ in the circumferential direction. Exact analytical harmonic solutions are therefore used in both the θ and z directions. The finite element approximation reduces to only one dimension, r. This 1-D SAFE formulation not only improves the accuracy in the calculation of flexural modes with higher circumferential orders, but also greatly reduces the computational cost compared to the 2-D SAFE formulation for cylindrical structures. For a harmonic wave propagating in the z direction, the displacement at any point u( , , , )rθz t can be represented by function in the thickness direction r. For a two-node element, U^j is a six-element vector and N( )r is a 3 × 6 matrix. The shape function matrix is chosen as follows,

N=

using linear shape functions

N₁ 1 N₂

2 1 1

2 1

=

(

-^ξ

)

^, =

(

+^ξ

)

^{, (9.40)}

where - ≤ ≤1 ξ 1 is the natural coordinate in the r direction. The strain-displacement relations in cylindrical coordinates are:

0 0.5 1 1.5 2

Figure 9.7. Skew angle dispersion curves of the composite plate for the 0° wave vector direction. Modes 1–6 are labeled.

151 9.8 SAFE Formulation for Cylindrical Structures

εrr ur

Substituting Equation (9.38) and Equation (9.40) into the strain-displacement relationships Equation (9.41), the six strain components can be expressed as:

ε

B₁

Substituting the strain components obtained in Equation (9.42) into the constitutive relation for strain and stress yields the expression for stress components:

σ =^Cε=^{C B}

(

1+ik^{B U}2

)

^{j i kz n}e^{( + -θ ωt)}, (9.47) where C is the stiffness matrix. The values in the matrix C are real for elastic materials and complex for viscoelastic materials according to the correspondence principle (Christensen 1982).

Substituting the displacements Equation (9.38), strains Equation (9.42), and stresses Equation (9.47) into the governing Equation (9.37), one obtains Equation (9.48) for element j after simplification.

K₁^j+ikK₂^j+k²K₃^j U^{j 2}M U^{j j} 0

153 9.10 Exercises

Similarly, Equation (9.49) can be assembled into the system of equations given in Equation (9.21) with vector Q replaced by U. It can also be further reduced to the first-order eigensystem as given in Equation (9.23) and solved using a standard eigenvalue solution routine.

9.9 Summary

This chapter discusses the guided wave solutions to multilayered plate and cylindrical structures using the SAFE method. Both isotropic and multilayer anisotropic composite plate and pipe structures can be modeled by the SAFE technique. Mode sorting algorithms are implemented into the dispersion curve calculations based on the wave mode orthogonality that is derived in the SAFE formulation for both plate and pipe structures. Important characteristics of free guided waves including group velocity, energy velocity, wave structure, stress distribution, Poynting vector, and skew angle are discussed. The wave characteristics form a theoretical foundation for guided wave applications. Example calculations on the guided wave characteristics are performed for an eight-layer, quasi-isotropic, fiber-reinforced composite plate. It is demonstrated that the influence of material anisotropy on guided wave propagation remains strong, even when a quasi-isotropic stacking sequence is used. Complete dispersion curves for a viscoelastic coated pipe are calculated. The solution convergence of the SAFE technique is also discussed. A physically based explanation for the differences in the solution convergence of different wave modes and frequencies is given based on a wave structure analysis.

9.10 Exercises

1. What are the differences between the SAFE method and the conventional finite element method?

2. What are the major benefits of using the SAFE method for ultrasonic guided wave applications?

3. What are the physical meanings of the real and imaginary parts of a wavenumber calculated in a SAFE calculation? How to determine the wave vector directions from the wavenumbers?

4. What are guided wave skew effects? Are there skew effects in composite plates with quasi-isotropic layups?

5. Explain how to use a 1-D SAFE to calculate guided wave dispersion curves for flexural modes in a hollow cylinder.

6. For a calculation with a particular mesh size, why does the SAFEM calculation match an analytical solution at relatively low frequencies, but fail to give an accurate answer at relatively high frequencies?

7. Make a list of several potential waveguide problems with different geometrical cross sections. Indicate whether each problem may be solved using a 1-D or 2-D SAFEM approach.

8. Assume you have a computer code that calculates modes in a plate using the 1-D SAFEM approach. Describe the changes necessary to develop this code to solve 2-D problems with the SAFEM approach.

9. For the eight-layer quasi-isotropic plate mentioned in Section 9.7, why would you expect to have some mode-frequency combinations with a higher skew angle than others?

10. Why does orthogonality-based mode sorting work?

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