Elastic wave scattering by a pair of parallel semi-infinite cracks in mechanical metamaterials with multi resonators

Abstract

As emerging artificial structures, elastic wave metamaterials can show some unique properties in special frequency regions. In this study, the elastic wave scattering by a pair of parallel semi-infinite cracks in mechanical metamaterials with local resonators is studied. According to the discrete Wiener–Hopf method, the far field displacement solution is obtained. At the same time, the dynamic negative effective mass and band gaps can be observed with local resonators. Our attention is focused on how the internal microstructure affects the crack faces. Numerical results show that because of the stop band, the crack can be prevented from being disturbed by incident waves in specific frequency regions. As a result, the possibility of crack initiation and instability propagation is reduced considerably. This present work is expected to provide a way to improve arrest performance of elastic waves metamaterials.

Appendices

Appendix A

Substitution of Eq. (4) into Eqs. (1–3) yields the homogeneous equation for amplitudes as

$$ {\mathbf{DA}}\,{ = }\,{\mathbf{0,}} $$

(A1)

where ${\text{A = }}\{ A_{1} {, }A_{2} {, }A_{3} \}^{{\text{T}}}$ is the amplitude vector and

$$ {\mathbf{D}}\,{ = }\,\left[ {\begin{array}{*{20}c} {2k_{1} \left[ {2\, - \,\cos \left( {k_{x} } \right)\, - \,\cos \left( {k_{y} } \right)} \right]\, - \,m_{1} \omega^{2} \, + \,k_{2} } & { - \,k_{2} } & 0 \\ { - k_{2} } & {k_{2} \, + \,k_{3} \, - \,m_{2} \omega^{2} } & { - \,k_{3} } \\ 0 & { - \,k_{3} } & {k_{3} \, - \,m_{3} \omega^{2} } \\ \end{array} } \right]. $$

(A2)

The linear algebraic equations with respect to A_j (j = 1–3) have nontrivial solutions when det[D] = 0. Then, the determinant can result in the dispersion relation as Eqs. (5 and 6). And the terms can be expressed as

$$ a\,{ = }\, - \,m_{1} m_{2} m_{3} , $$

(A3)

$$ b\,{ = }\,{4}k_{{1}} m_{{2}} m_{{3}} \,{ + }\,k_{{2}} m_{{1}} m_{{3}} \,{ + }\,k_{{3}} m_{{1}} m_{{2}} \,{ + }\,k_{{2}} m_{{2}} m_{{3}} \,{ + }\,k_{{3}} m_{{1}} m_{{3}} \, - \,{2}k_{{1}} m_{{2}} m_{{3}} {\text{cos}}\left( {k_{x} } \right)\, - \,{2}k_{{1}} m_{{2}} m_{{3}} {\text{cos}}\left( {k_{y} } \right){,} $$

(A4)

$$ c\,{ = }\,{2}k_{{1}} k_{{2}} m_{{3}} {\text{cos}}\left( {k_{x} } \right)\,{ + }\,{2}k_{{1}} k_{{3}} m_{{2}} {\text{cos}}\left( {k_{x} } \right)\,{ + }\,{2}k_{1} k_{3} m_{{3}} {\text{cos}}\left( {k_{x} } \right)\,{ + }\,{2}k_{1} k_{2} m_{3} {\text{cos}}\left( {k_{y} } \right)\,{ + }\,{2}k_{{1}} k_{{3}} m_{{3}} {\text{cos}}\left( {k_{y} } \right) $$

$$ { + }\,{2}k_{{1}} k_{{3}} m_{{2}} {\text{cos}}\left( {k_{y} } \right)\, - \,{4}k_{{1}} k_{2} m_{3} \, - \,{4}k_{1} k_{3} m_{2} \, - \,k_{2} k_{3} m_{1} \, - \,4k_{1} k_{3} m_{3} \, - \,k_{2} k_{3} m_{2} \, - \,k_{2} k_{3} m_{3} , $$

(A5)

$$ d\,{ = }\,{4}k_{1} k_{2} k_{3} \, - \,{2}k_{{1}} k_{2} k_{3} {\text{cos(}}k_{x} {)}\, - \,{2}k_{1} k_{2} k_{3} {\text{cos}}\left( {k_{y} } \right){,} $$

(A6)

$$ A\,{ = }\,b^{{2}} \, - \,{3}ac, $$

(A7)

$$ B\,{ = }\,bc\, - \,9ad, $$

(A8)

$$ C\,{ = }\,c^{{2}} \, - \,{3}bd, $$

(A9)

$$ T\,{ = }\,\frac{{{2}Ab\, - \,{3}aB}}{{2\sqrt {A^{3} } }}, $$

(A10)

$$ S\,{ = }\,\arccos (T). $$

(A11)

Appendix B

The discrete Fourier transform of function f(z) in the complex plane can be shown as

$$ f^{{\text{F}}} (z)\, = \,f_{ + }^{{\text{F}}} (z)\, + \,f_{ - }^{{\text{F}}} (z), $$

(B1)

$$ f_{ + }^{{\text{F}}} (z)\, = \,\sum\limits_{x = 0}^{x = + \infty } {f(x)z^{ - x} } , $$

(B2)

$$ f_{ - }^{{\text{F}}} (z)\, = \,\sum\limits_{x = - \infty }^{x = - 1} {f(x)z^{ - x} } , $$

(B3)

where $f_{ + }^{{\text{F}}} (z)$ and $f_{ - }^{{\text{F}}} (z)$ are analytic at |z|> R₊ and |z|< R_–, respectively. As a result, $f^{{\text{F}}} (z)$ is analytic at R₊ <|z|< R_–.

On the other hand, f(x) can be obtained by the following inverse Fourier transform:

$$ f(x)\, = \,\frac{1}{{2\pi {\text{i}}}}\oint\limits_{{C_{a} }} {f(z)z^{x - 1} } {\text{d}}z, $$

(B4)

where C_a is a closed curve with the counter clockwise at R₊ <|z|< R_–.

Appendix C

The wave motion equations of the lower crack face are shown as

$$ - \,m_{1} \omega^{2} w_{x,N - 1}^{(1)} \, + \,k_{1} \left( {3w_{x,N - 1}^{(1)} \, - \,w_{x - 1,N - 1}^{(1)} \, - \,w_{x + 1,N - 1}^{(1)} \, - \,w_{x,N - 2}^{(1)} } \right)\, + \,k_{2} \left( {w_{x,N - 1}^{(1)} \, - \,w_{x,N - 1}^{(2)} } \right) $$

$$ - \,k_{1} v_{x,N}^{{}} H(x)\, = \, - \,k_{1} v_{x,N}^{{{\text{inc}}}} H( - \,x\, - \,1), $$

(C1)

$$ - \,m_{2} \omega^{2} w_{x,N - 1}^{(2)} \, + \,k_{2} \left( {w_{x,N - 1}^{(2)} \, - \,w_{x,N - 1}^{(1)} } \right)\, + \,k_{3} \left( {w_{x,N - 1}^{(2)} \, - \,w_{x,N - 1}^{(3)} } \right)\, = \,0, $$

(C2)

$$ - \,m_{3} \omega^{2} w_{x,N - 1}^{(3)} \, + \,k_{3} \left( {w_{x,N - 1}^{(3)} \, - \,w_{x,N - 1}^{(2)} } \right)\, = \,0. $$

(C3)