Abstract
Wave numbers about the three types of waves in saturated soils are firstly given in this paper. The lengths of the pipe piles are much larger than their diameters, so the isolation problem about SV waves by discontinuous barriers composed of a row of pipe piles can be simplified as a two-dimensional scattering problem. The expansion method of wave functions is adopted, the stresses and displacements at the boundaries between the pipe piles and adjacent soils are considered as continuous and the inner sides of the pipe piles are free, and then the theoretical solutions are obtained about this two-dimensional scattering problem. Normalized displacements are introduced, which are the displacements behind the barriers caused by both the incident and scattered waves to those only by the incident SV waves, contours and curves of the normalized displacements are drawn, and the influences of wall thickness of pipe piles, modulus ratio of pipe piles to soils, spacing distance between the pipe piles and pipe pile numbers on the isolation effectiveness are analyzed.
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant No. 51278467), China Postdoctoral Science Foundation (Grant Nos. 2015M582204 and 2016T90681), Program for Science and Technology Innovation Talents in Universities of Henan Province (Grant No. 14HASTIT050) and Outstanding Young Talent Research Fund of Zhengzhou University (Grant No. 1421323078).
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Appendices
Appendix A: Total Potential Functions of P\(_{1}\) Waves, P \(_{2}\) Waves and SV Waves in the Saturated Soils
where \(T_{nm}^+ \left( \cdot \right) \) and \(T_{nm}^- \left( \cdot \right) \) are
Appendix B: Components of Stress and Displacements in Saturated Soils
The components of stresses, displacements and water pressures in the saturated soils, soil skeletons and water are listed in radial coordinate system as follows:
-
1.
Normal stress \(\sigma _r \)
where \(\Gamma _{11}^{\left( l \right) } =\left[ {n^{2}+n-\left( {1+{\lambda _c }/{2\mu }+\gamma _1 \alpha M/{2\mu }} \right) k_1^2 r^{2}} \right] Z_n^{\left( l \right) } \left( {k_1 r} \right) -k_1 rZ_{n-1}^{\left( l \right) } \left( {k_1 r} \right) \).
where \(\Gamma _{12}^{\left( l \right) } =\left[ {n^{2}+n-\left( {1+{\lambda _c }/{2\mu }+\gamma _2 \alpha M/{2\mu }} \right) k_2^2 r^{2}} \right] Z_n^{\left( l \right) } \left( {k_2 r} \right) -k_2 rZ_{n-1}^{\left( l \right) } \left( {k_2 r} \right) \).
where \(\Gamma _{13}^{\left( l \right) } =n\left[ {-\left( {1+n} \right) Z_n^{\left( l \right) } \left( {k_s r} \right) +k_s rZ_{n-1}^{\left( l \right) } \left( {k_s r} \right) } \right] \).
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2.
Shear stress \(\tau _{r\theta } \)
where \(\Gamma _{21}^{\left( l \right) } =n\left[ {-\left( {1+n} \right) Z_n^{\left( l \right) } \left( {k_1 r} \right) +k_1 rZ_{n-1}^{\left( l \right) } \left( {k_1 r} \right) } \right] \).
where \(\Gamma _{22}^{\left( l \right) } =n\left[ {-\left( {1+n} \right) Z_n^{\left( l \right) } \left( {k_2 r} \right) +k_2 rZ_{n-1}^{\left( l \right) } \left( {k_2 r} \right) } \right] \).
where \(\Gamma _{23}^{\left( l \right) } =-\left( {n^{2}+n-{k_s^2 r^{2}}/2} \right) Z_n^{\left( l \right) } \left( {k_s r} \right) +k_s rZ_{n-1}^{\left( l \right) } \left( {k_s r} \right) \).
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3.
Radial displacement \(u_r \) in the soil skeletons
where \(\Gamma _{31}^{\left( l \right) } =k_1 rZ_{n-1}^{\left( l \right) } \left( {k_1 r} \right) -Z_n^{\left( l \right) } \left( {k_1 r} \right) \).
where \(\Gamma _{32}^{\left( l \right) } =k_2 rZ_{n-1}^{\left( l \right) } \left( {k_2 r} \right) -Z_n^{\left( l \right) } \left( {k_2 r} \right) \).
where \(\Gamma _{33}^{\left( l \right) } =nZ_n^{\left( l \right) } \left( {k_s r} \right) \).
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4.
Circumferential displacement \(u_\theta \) in the soil skeletons
where \(\Gamma _{41}^{\left( l \right) } =nZ_n^{\left( l \right) } \left( {k_1 r} \right) \).
where \(\Gamma _{42}^{\left( l \right) } =nZ_n^{\left( l \right) } \left( {k_2 r} \right) \).
where \(\Gamma _{43}^{\left( l \right) } =-\left[ {k_s rZ_{n-1}^{\left( l \right) } \left( {k_s r} \right) -nZ_n^{\left( l \right) } \left( {k_s r} \right) } \right] \).
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5.
Radial displacement \(w_r \) in the water
where \(\Gamma _{51}^{\left( l \right) } =\gamma _1 \Gamma _{31}^{\left( l \right) } =\gamma _1 \left[ {k_1 rZ_{n-1}^{\left( l \right) } \left( {k_1 r} \right) -Z_n^{\left( l \right) } \left( {k_1 r} \right) } \right] \).
where \(\Gamma _{52}^{\left( l \right) } =\gamma _2 \Gamma _{32}^{\left( l \right) } =\gamma _2 \left[ {k_2 rZ_{n-1}^{\left( l \right) } \left( {k_2 r} \right) -Z_n^{\left( l \right) } \left( {k_2 r} \right) } \right] \).
where \(\Gamma _{53}^{\left( l \right) } =\gamma _s \Gamma _{33}^{\left( l \right) } =\gamma _s nZ_n^{\left( l \right) } \left( {k_s r} \right) \).
The parameter l in \(Z_n^{(l)} (\cdot )\) of Eqs. (B.1)–(B.15) is valued 1–3, and \(Z_n^{(l)} (\cdot )\) denotes the different special functions: \(Z_n^{\left( 1 \right) } (\cdot )=J_n (\cdot )\), \(Z_n^{\left( 2 \right) } (\cdot )=N_n (\cdot )\) and \(Z_n^{\left( 3 \right) } (\cdot )=H_n^{\left( 1 \right) } (\cdot )\).
Appendix C: Theoretical Solutions About the Undefined Complex Coefficients
The undefined complex coefficients \(A_{1n}^l \sim Q_n^l \) (\(1\le l\le N)\) can be finally determined from the linear equation group [Eqs. (C.1)–(C.6)] as follows:
where
where \(x_1^l =k_1 b_l \), \(x_2^l =k_2 b_l \), \(x_s^l =k_s b_l \), \(\tilde{x}_p^l =\tilde{k}_p b_l \), \(\tilde{x}_s^l =\tilde{k}_s^l b_l \), \(\tilde{y}_p^l =\tilde{k}_p a_l \) and \(\tilde{y}_s^l =\tilde{k}_s a_l \); \(\hat{{\delta }}_{ij} \) is opposite to \(\delta _{ij} \): \(\hat{{\delta }}_{ij} =1(i\ne j)\) and \(\hat{{\delta }}_{ij} =0(i=j)\); \(\mu _l^*={\tilde{\mu }_l }/\mu \), which is the modulus ratio of the \(l\hbox {th}\) pipe pile to the solid skeleton of saturated soils, and \(\mu _l^*=1\) when \(\kappa =3,4\); the parameter of \(\kappa \) in valued 1–5 in \(\Gamma _{\kappa 1}^{\left( l \right) } \sim \Gamma _{\kappa 3}^{\left( l \right) } \) and 1–4 in \(\Lambda _{\kappa 1}^{\left( l \right) } \sim \Lambda _{\kappa 2}^{\left( l \right) } \), the expressions of \(\Gamma _{\kappa 1}^{\left( l \right) } \) are given in Eq. (B.1)–(B.15), and the expressions of \(\Lambda _{11}^{\left( l \right) } \sim \Lambda _{42}^{\left( l \right) } \) are given as follows:
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Xu, P. Analysis of Isolation Effectiveness of Shear Waves by a Row of Hollow Pipe Piles in Saturated Soils. Transp Porous Med 120, 415–432 (2017). https://doi.org/10.1007/s11242-017-0931-z
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DOI: https://doi.org/10.1007/s11242-017-0931-z