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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
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and the notch
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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).
Different approximations can be used for the description of the first
antisymmetric mode A0, a flexural wave. Simplified equations can either be
developed in terms of the physical parameters taken into account or the full
three-dimensional equations can be developed in terms of a dimensionless
parameter e,
describing the relation of wavelength to plate thickness. This asymptotic
approach was further investigated and applied to the scattering at a circular
cavity. However, the simplest approach to describe flexural waves in plates is
using classical plate theory (CPT), taking only inertia and bending stiffness
into account according to
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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
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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).
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
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The scattered wave is formulated in terms of three potentials and consists of the flexural wave
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and a shear boundary layer with the two components
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with
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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]. The
theory 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.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
Yx,
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.
To calculate the scattering at a hole with a defect, the stress-free boundary
conditions are implemented. The hole is approximated with a right-angled
contour. The defect is modeled as two stress-free parallel lines and one point
at the tip of the notch. This simulates a notch with a width of
Dx and
a blunt tip. The sharp edge of a crack and effects like crack closure are not
considered. No numerical problems or instabilities were found, adhering to the
usual stability criteria.
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
calculation.
The main influence on the change in the scattered field due to the notch was
found to be the ratio between notch length and the wavelength of the incident
wave. In the figure above the maximum change of the complex magnitude (including
phase information) due to the notch, normalized with the amplitude of the
incident wave, is shown for two different geometric relations. It can be seen
that the curves show a similar influence and that no further significant
increase in amplitude, and thus a better detection of the defect, occurs for a
notch larger than about one fifth of the wavelength. With the current
experimental setup higher frequencies are more difficult to handle, especially
concerning the generation of a sufficiently high excitation amplitude and the
relative positioning of excitation transducer and measurement spot. Due to these
experimental difficulties the use of frequencies that are higher than necessary
should be avoided.
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