PhD thesis: Defect detection in plates using guided waves
The geometry of the scattering problem studied can be seen above. The incident flexural wave
with a straight wave front is scattered by a through hole in a plate. The defect is represented by a notch of length a at angle j0 relative to the propagation direction of the incident wave. The origin of the coordinate system is chosen at the center of the hole with radius r0. Cylindrical coordinates (r,j) are introduced. To satisfy the stress-free boundary conditions at the hole
and the notch
a scattered wave must occur. The introduced scattered waves propagate radially outwards from the hole and have an angular dependence. The boundary conditions can be fulfilled with different degrees of accuracy, depending on the selected theory to describe the wave propagation.
The propagation of guided
waves in isotropic, homogeneous plates can be described by the theory of Lamb.
There are two types of possible wave modes: symmetric and antisymmetric. Their
dispersion relation can be deduced either from considering multiple
reflections through the thickness of the plate or by the formulation of a
standing wave mode. These modes are in general dispersive, i.e., the
propagation velocity is dependent on the frequency (Lamb
wave dispersion diagram).
with out-of-plane displacement w of the plate. This approach is valid only at low frequencies when the wavelength is large compared to the plate thickness. For higher frequencies, corresponding to shorter wavelengths, the effects of shear and rotatory inertia have to be considered. Therefore, the theory of Mindlin [J. Appl. Mech. 18, 31-38 (1951)] is used
Three types of waves can exist in the plate, a propagating flexural wave (real wave number k1), a flexural boundary layer (imaginary wave number k2) and a shear wave (wave number k3).
Scattering at a circular hole
Using Mindlin theory and following the work of Pao and Chao [AIAA J. 2(11), 2004-2010 (1964)], the boundary conditions at a circular hole without defect can be fulfilled as an average over the plate thickness with
The scattered wave is formulated in terms of three potentials and consists of the flexural wave
and a shear boundary layer with the two components
Evaluating the boundary conditions in polar coordinates, the coefficients of the scattered waves are calculated. Good agreement between the different approximate theories and experiments is found for the range of validity of the different theories [JASA 2002]. Thetheory of Mindlin has a wide range of validity and can be easily implemented numerically. Therefore it was widely used in this work and implemented numerically to describe the scattering at a hole with a defect.
Scattering at a hole with a defect
Finite difference methods (FDM)
are used to calculate the propagation and scattering characteristics of a
flexural wave in a plate with a defect, employing explicit time marching.
Mindlin’s equation of motions are discretized on a Cartesian, staggered grid.
The bending and twisting moments Mx, My, Mxy
per unit of length, the transverse shear forces Qx, Qy per
unit of length, the displacement w, and the rotation angles
Yy are calculated
at different grid points, half a grid step (Dx/2,
Dy/2) apart. They are
arranged such that the respective centered first derivatives in x and y can be
calculated from neighboring points (half grid step). Calculating moments and
forces besides the displacement uses more memory space, but allows an easy
implementation of stress-free boundary conditions.
The modeling of the influence of a notch on the scattered field is compared to experimental results in the figure above. After a first measurement for an undamaged hole, a notch of 2 mm length was cut at an angle of 90° on the hole boundary using a fine saw blade. The scattered field was measured again (a) and the difference in amplitude due to the notch is shown in (b). At the free surface of the notch an additional scattered wave is generated, which changes the scattered field significantly up to about 30% in amplitude and allows the detection of such a notch from an amplitude measurement. The calculation of the amplitude (c) and the change in amplitude (d) show good agreement with the measurements.
When inspecting large structures, the crack orientation is not necessarily known a priori. Therefore it is important to quantify the measurement method for defects at arbitrary angles to the propagation direction of the incident wave. The difference in amplitude for a defect at the side of the hole is significantly larger than for a defect oriented in the propagation direction of the incident wave, as a larger aspect of the notch is ‘visible’ to the incident wave. Therefore in practical applications for NDT purposes, one should aim at positioning the excitation transducer such that the expected crack position is more or less vertical to the propagation direction. The change in amplitude is then significantly larger and allows a detection of smaller defects. The main focus of the further parameter and detectability study is on a defect at 90°, as this is the angle at which all fatigue cracks in the tensile specimens [NDT Application] were located.
A variety of parameters define
the geometry of the hole, the position and size of the defect and the incident
wave and can have an influence on the minimum detectable crack length. Namely
the plate thickness, hole radius, notch length, and the wavelength (dependent on
the excitation frequency) can vary relative to each other. One of the four
length parameters can be eliminated, as it can be easily shown that only the
relation between the sizes and not the absolute size matter for the theoretical