^{1,a)}, Mark P. Blodgett

^{2}, Eric A. Lindgren

^{2}, Gary J. Steffes

^{2}and Jeremy S. Knopp

^{2}

### Abstract

Prior work has proposed the use of ultrasonic angle-beam shear wave techniques to detect cracks of varying angular location around fastener sites by generating and detecting creeping waves. To better understand the nature of the scattering problem and quantify the role of creeping waves in fastener site inspections, a 3D analytical model was developed for the propagation and scattering of an obliquely incident plane shear wave from a cylindrical cavity with arbitrary shear wavepolarization. The generation and decay of the spiral creeping waves was found to be dependent on both the angle of incidence and polarization of the plane shear wave. A difference between the angle of displacement in 3D and the direction of propagation for the spiral creeping wave was observed and attributed to differences in the curvature of the cavity surface for the tangential and vertical (*z*) directions. Using the model, practical insight was presented on measuring the displacement response in the far-field from the hole. Both analytical and experimental results highlighted the value of the diffracted and leaky spiral creeping wave signals for nondestructive evaluation of a crack located on the cavity. Last, array and signal processing methods are discussed to improve the resolution of the weaker creeping wave signals in the presence of noise.

Support was provided by the Air Force Research Laboratory–NDE Branch and the Air Force Office of Scientific Research. David Judd of Mistras Services (formally integrated Technologies, Inc.) and Bart Drennen of WesDyne International provided support for the acquisition of the experimental data.

I. INTRODUCTION

II. THEORY

III. MODEL RESULTS

A. Magnitude of creeping wave signals

B. Time-of-flight difference and creeping wave speed calculation

C. Displacement profile on the cavity

D. Displacement profile in the far-field

IV. EXPERIMENTAL RESULTS

V. CONCLUSIONS AND RECOMMENDATIONS

### Key Topics

- Transducers
- 33.0
- Polarization
- 17.0
- Elastic waves
- 16.0
- Elasticity
- 14.0
- Ultrasonics
- 13.0

## Figures

(Color online) Schematic diagram of an ultrasonic inspection of a fastener site for corner cracks in a multilayer structure. The mode conversion of an incident shear wave to a spiral creeping wave to detect corner cracks in the shadow region of the hole is indicated.

(Color online) Schematic diagram of an ultrasonic inspection of a fastener site for corner cracks in a multilayer structure. The mode conversion of an incident shear wave to a spiral creeping wave to detect corner cracks in the shadow region of the hole is indicated.

Diagram of an obliquely incident place shear wave at a cylindrical hole with arbitrary polarization.

Diagram of an obliquely incident place shear wave at a cylindrical hole with arbitrary polarization.

Transient total (absolute) displacement response at *r* = *a* and *z* = 0 for 13 select angular location, θ, varying from 0° to 180° for the following incident wave conditions: (a) normal incidence shear wave inspection with φ = 0° and γ = 0°, and (b) an angled-beam shear wave inspection with φ = 45° and γ = 0°.

Transient total (absolute) displacement response at *r* = *a* and *z* = 0 for 13 select angular location, θ, varying from 0° to 180° for the following incident wave conditions: (a) normal incidence shear wave inspection with φ = 0° and γ = 0°, and (b) an angled-beam shear wave inspection with φ = 45° and γ = 0°.

3D image plot of the transient total displacement (magnitude) response (normalized with respect to the incident displacement response) at locations around the cylindrical hole and in the *y* = 0 plane before and after the cylinder (at *z* = 0), for an oblique incident plane shear wave (φ = 45°) with varying polarization: (a) γ = 0°, (b) γ = 90°.

3D image plot of the transient total displacement (magnitude) response (normalized with respect to the incident displacement response) at locations around the cylindrical hole and in the *y* = 0 plane before and after the cylinder (at *z* = 0), for an oblique incident plane shear wave (φ = 45°) with varying polarization: (a) γ = 0°, (b) γ = 90°.

(Color online) (a) Magnitude of the displacement response for the first isolated creeping wave signal far in time at select points around the fastener site and varying angle of polarization: θ = 180° and γ = 90° (solid line with square), θ = 0° and γ = 90° (solid line with diamonds), 180° and γ = 0° (dashed line with circles), θ = 0° and γ = 0° (dashed line with triangles). (b) Displacement ratio representing the change in magnitude of creeping wave signal after propagating π radians in θ for γ = 90° (solid line with square) and γ = 0° (dashed line with circles).

(Color online) (a) Magnitude of the displacement response for the first isolated creeping wave signal far in time at select points around the fastener site and varying angle of polarization: θ = 180° and γ = 90° (solid line with square), θ = 0° and γ = 90° (solid line with diamonds), 180° and γ = 0° (dashed line with circles), θ = 0° and γ = 0° (dashed line with triangles). (b) Displacement ratio representing the change in magnitude of creeping wave signal after propagating π radians in θ for γ = 90° (solid line with square) and γ = 0° (dashed line with circles).

(Color online) Displacement ratio representing the change in magnitude for the first isolated creeping wave signal far in time after propagating π radians in θ: *f* = 2.5 MHz (solid line with squares), *f* = 5.0 MHz (dashed line with circles), and *f* = 7.5 MHz (dash-dotted line with x).

(Color online) Displacement ratio representing the change in magnitude for the first isolated creeping wave signal far in time after propagating π radians in θ: *f* = 2.5 MHz (solid line with squares), *f* = 5.0 MHz (dashed line with circles), and *f* = 7.5 MHz (dash-dotted line with x).

(Color online) Diagram of an obliquely incident shear wave at a cylindrical hole showing pertinent paths and angles for the creeping wave time-of-flight difference calculation.

(Color online) Diagram of an obliquely incident shear wave at a cylindrical hole showing pertinent paths and angles for the creeping wave time-of-flight difference calculation.

Ratio of creeping wave group velocity (*c* _{ R′ }) with respect to the shear wave speed (*c* _{ T }) for varying plane wave incident angle at 5.0 MHz (squares).

Ratio of creeping wave group velocity (*c* _{ R′ }) with respect to the shear wave speed (*c* _{ T }) for varying plane wave incident angle at 5.0 MHz (squares).

Path of displacement response for the first isolated creeping wave signal from an angled-beam shear plane wave (with φ = 45° and γ = 0°) at *r* = *a*, θ = 270° and *z* = 0 (a) in 3D, and (b) in tangential (*t*) and *z*-directions plane.

Path of displacement response for the first isolated creeping wave signal from an angled-beam shear plane wave (with φ = 45° and γ = 0°) at *r* = *a*, θ = 270° and *z* = 0 (a) in 3D, and (b) in tangential (*t*) and *z*-directions plane.

(Color online) (a) Displacement response at *r* = *a*, θ = 75° and *z* = 0 representing the incident signals at virtual crack locations of θ_{ c } = 15° and θ_{ c } = 165°. Three displacement components are shown: the tangential component (solid line), the radial component (dashed line), and the z-component (dash-dotted line). (b) Normalized peak to peak displacement response on the cavity surface for varying angular locations of a virtual crack, θ_{ c }. The displacement results are presented for the tangential (squares), radial (circles), and *z*-components (×).

(Color online) (a) Displacement response at *r* = *a*, θ = 75° and *z* = 0 representing the incident signals at virtual crack locations of θ_{ c } = 15° and θ_{ c } = 165°. Three displacement components are shown: the tangential component (solid line), the radial component (dashed line), and the z-component (dash-dotted line). (b) Normalized peak to peak displacement response on the cavity surface for varying angular locations of a virtual crack, θ_{ c }. The displacement results are presented for the tangential (squares), radial (circles), and *z*-components (×).

(Color online) Displacement response in the far field at three locations: (a) *x* = 3*a*, *y* = *a*, *z* = 0; (b) at *x* = 3*a*, *y* = 0, *z* = 0; and (c) *x* = −3*a*, *y* = *a*, *z* = 0. Three displacement components are shown: the *x*-component (solid line), the *y*-component (dashed line), and the *z*-component (dash-dotted line).

(Color online) Displacement response in the far field at three locations: (a) *x* = 3*a*, *y* = *a*, *z* = 0; (b) at *x* = 3*a*, *y* = 0, *z* = 0; and (c) *x* = −3*a*, *y* = *a*, *z* = 0. Three displacement components are shown: the *x*-component (solid line), the *y*-component (dashed line), and the *z*-component (dash-dotted line).

(Color online) (a) Maximum peak to peak displacement response and (b) traducer position (*x*-direction: circles, and *y*-direction: squares) of the maximum peak to peak reflected signal as a function of EDM notch angular location, θ_{ c }, around the hole. The transducer positions are relative to a “reference” provided by the maximum peak to peak response from the fastener hole.

(Color online) (a) Maximum peak to peak displacement response and (b) traducer position (*x*-direction: circles, and *y*-direction: squares) of the maximum peak to peak reflected signal as a function of EDM notch angular location, θ_{ c }, around the hole. The transducer positions are relative to a “reference” provided by the maximum peak to peak response from the fastener hole.

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