5 Discussion
In this study, theoretical solutions based on the wave function
expansion method and Fourier transform were obtained for an inclusion in
infinite space when subjected to a plane SH-wave. First, the
steady-state response was analyzed using the wave function expansion
method. Then, the Ricker wavelet was introduced as the transient
disturbance. Finally, the Fourier transform was used to determine the
distribution of transient dynamic stress concentration around the
elliptical inclusion. The numerical results indicated that the dynamic
stress concentration generated by the steady-state and transient
incident waves was different. However, the dynamic stress concentration
distribution in the two states was dependent on the elliptical axis
ratio, incident wave number, difference in material properties between
the medium and inclusion.
As reported in literature, the main content is analyzing angular stress
concentration, while the radial stress concentration is often ignored.
In this study, the angular and radial stress expressions under the
steady-state incidence and transient incidence were obtained
theoretically, and two types of distribution of dynamic stress
concentration around the elliptical inclusion was calculated. It is
found that regardless of the steady-state or transient incidence, for
the inclusion stiffer than the medium, significant radial stress
concentration appeared at both ends of the elliptical major axis. At the
elliptical axis ratio of 10, the maximum steady-stateαξDSCF attained 3.081, and the
maximum transient αξDSCFreached 7.869. The angular stress concentration was observed at both
ends of the elliptical minor axis, but the value ofαηDSCF was slightly small. The
maximum value of steady-stateαηDSCF was 0.764, and the
maximum value of transientαηDSCF was 1.969. The closer
the inclusion shape to crack, the larger the value ofαDSCF and the more significant the dynamic stress
concentration. Therefore, the structural failure was likely to occur at
both ends of the major axis and the minor axis of elliptical inclusion,
and the radial stress concentration at both ends of the major axis of
inclusion was more likely to cause structural failure. At this moment,
the closer the inclusion shape to crack, the higher the possibility of
the structural failure. However, for the inclusion softer than the
medium, significant angular stress concentration was observed at both
ends of the elliptical minor axis. The steady-stateαηDSCF achieved the maximum
value of 2.079 at the elliptical axis ratio was 2.16, and the transient
state αηDSCF attained the
maximum value of 4.588 when elliptical axis ratio of 1.1. The radial
stress concentration was noted at both ends of the elliptical major
axis. When the axial ratio was 1.1, the maximum value of steady-stateαξDSCF attained 1.472, and the
maximum value of transient stateαξDSCF reached 2.476. The more
approximate the inclusion shape to circle, the larger the value ofαDSCF and the more significant the dynamic stress
concentration. Therefore, the structural failure of inclusion was likely
to occur at both ends of the major axis and the minor axis of elliptical
inclusion, and the angular stress concentration at both ends of the
minor axis of inclusion was more likely to result in the structural
failure. At this time, the closer the shape of inclusion to circle, the
greater the possibility of structural failure.
Compared with the wave function expansion method based on the
Fourier-Bessel expansion and conformal
transformation32, the Mathieu function was more
convenient to deal with elliptical boundary problems, because it can
avoid complicated boundary mapping, and the mathematical form was more
concise. Moreover, the correctness of the theoretical derivation was
verified by calculating distribution of dynamic stress concentration
with the same material properties parameters between inclusion and
medium. The results indicated that the spatial distribution of angular
stress concentration was only related to the phase difference of the
incident wave, and the maximum value of 1 was obtained in the vertical
direction of the incident angle, which was consistent with the previous
literature.10 The steady-stateαηDSCF gradually decreased with
increasing the wave number for stiffer inclusion than medium. But when
the inclusion was softer than the medium, the steady-stateαηDSCF oscillated with the
change in the wave number, the greater the difference of material
properties between the inclusion and medium, the more intense the
oscillation. This phenomenon indicated thatαηDSCF had a high sensitivity
to wave number, and the greater the difference in material properties
between inclusion and medium, the higher the
sensitivity.26
However, during the process of realizing the algorithm by mathematical
software, the calculation of the Mathieu function was complex in some
aspects. For example, under the limitation of boundary conditions, only
the case of 0° incidence can be calculated. The wave number k of
the incident wave was related to the q variable in the Mathieu
function, the q value was related to the truncation series of the
Mathieu function. In order to ensure accuracy, the truncation series
must be changed when the k value was changed. In addition, the
Mathieu function with no primitive function or the primitive function
was difficult to express. As a result, only numerical integration
methods can be used in Fourier integration, which greatly increased the
amount of calculation. In order to ensure the accuracy of the results,
this study compared the error between the sum of N and N+1 term. The
Mathieu functions were generally truncated to the 8th or 9th to ensure
accuracy. The Mathieu function was determined N=8-16, and the error
between N=12 and N=16 was found to be less than 0.01%. Therefore, N=12
was selected to reduce unnecessary calculations while ensuring accuracy.