2.1 Spectral Analysis
Fig. 1 illustrates the spectral analysis of wave scattering at an edge crack induced by
incident waves. When incident waves $\left(A_{I}^{+}\right)$, whether bending $\left(A_{p}^{+}\right)$
or axial $\left(A_{a}^{+}\right)$, encounter a crack, they are reflected and transmitted
as three types of wave motions: propagating, mode-converted, and evanescent (Yuan et al., 2008). Among the wave motions, there are two types of wave motions that result from the
mode conversion, distinguishing them from the original incident waves. For example,
if the incident waves is axial, the wave motions resulting from the mode conversion
are mode-converted bending waves and evanescent bending waves.
Fig. 1. Spectral Representation of Wave Scattering at the Edge Crack Induced by Incident
Waves in an Infinite Beam
The spectral solutions for the axial displacement at the neutral axis of the beam,
calculated on both sides of the crack, follow the principles of elementary rod theory
(Doyle, 1989). These spectral solutions are influenced by the type of incident waves. The spectral
solutions for the axial displacement at the left and right sides of the crack due
to the bending incident waves are written as follows:
where $A_{a}^{-}$ and $B_{a}^{+}$ denote the reflected and transmitted axial waves
during the mode conversion from bending to axial waves, respectively. Meanwhile, the
spectral solutions for axial displacement at the left and right sides of the crack
induced by axial incident waves are expressed as follows:
where $A_{a}^{-}$ and $B_{a}^{+}$ represent two axial waves which are directly reflected
and transmitted without mode conversion (i.e., axial to axial), respectively. The
parameter $k_{a}$ in both spectral solutions above denotes the axial wave number,
which is determined by the dispersion equation of elementary rod theory, as follows:
where $\rho$, $E$, and $\omega$ are the beam density, elastic modulus, and angular
frequency, respectively.
The spectral solutions for the rotational angle and transverse displacement are based
on Timoshenko beam theory (Mei and Mace, 2005). These spectral solutions, calculated on both sides of the crack, are also affected
by the type of incident waves. The spectral solutions for rotational angle and transverse
displacement generated by bending incident waves at the left and right sides of the
crack are expressed as follows:
where $P$, $N$, $k_{p}$, and $k_{e}$ denote the rotational angle coefficient, transverse
displacement coefficient, propagating wave number, and evanescent wave number, respectively.
It should also be noted that $A_{p}^{-}$ and $B_{p}^{+}$ in the equation above represent
the reflected and transmitted propagating bending waves, respectively, while $A_{e}^{-}$
and $B_{e}^{+}$ correspond to the reflected and transmitted evanescent bending waves,
respectively.
The spectral solutions for rotational angle and transverse displacement generated
by axial incident waves at the left and right sides of the crack are written as follows:
where $A_{p}^{-}$ and $B_{p}^{+}$ are the reflected and transmitted bending waves
during the mode conversion from axial to bending waves, respectively, while $A_{e}^{-}$
and $B_{e}^{+}$ denote the reflected and transmitted evanescent bending waves, respectively.
The parameters $P$, $N$, $k_{p}$, and $k_{e}$ in both spectral solutions above can
be determined as follows:
where $\kappa$, $G$, $I$, and $A$ represent the shear coefficient, shear modulus,
second moment of inertia, and beam cross-sectional area, respectively. It should be
noted that Eq. (6) is valid when the corresponding angular frequency is below the cut-off frequency
$\left(\dfrac{c_{q}}{r}\right)$.
As explained above, each incident wave generates six unknowns, represented as $A_{a}^{-}$,
$B_{a}^{+}$, $A_{p}^{-}$, $B_{p}^{+}$, $A_{e}^{-}$, and $B_{e}^{+}$, as outlined in
Eqs. (1) and (4) for bending incident waves, and Eqs. (2) and (5) for axial incident waves. The proposed closed-form solutions for wave scattering
coefficients, labeled as $R_{a}$, $T_{a}$, $R_{p}$, $T_{p}$, $R_{e}$, and $T_{e}$
in Fig. 1, are obtained by dividing these unknowns by the respective incident bending or axial
wave.
2.2 Compatibility and Equilibrium Conditions at the Edge Crack
Six conditions at the edge crack were considered to derive the wave scattering coefficients:
three compatibility conditions and three equilibrium equations. Fig. 2 demonstrates the three compatibility conditions for axial displacement, rotational
angle, and transverse displacement. These compatibility conditions assume that the
local behavior of the beam at the edge crack follows linear fracture mechanics (Park and Lim, 2020), as follows:
The three equilibrium equations correspond to the equilibrium of axial force, moment,
and shear force as follows:
It should be noted that $'$ represent the spatial differential operator. The local
flexibility coefficients $F_{aa}$, $F_{ab}$, $F_{ba}$, and $F_{bb}$ are mode I crack
(opening crack mode) (Park and Lim, 2020) and are defined in terms of the configuration correction factors in linear fracture
mechanics (Tada et al., 1973).
Fig. 2. Three Compatibility Conditions at the Edge Crack. (a) Axial Displacement,
(b) Rotational Angle, (c) Transverse Displacement