Dynamic responses of unsaturated half-space soils to a strip load at different boundary conditions

Abstract

Based on the mixed theory of three-phase porous media, the dynamic response of unsaturated soils under a strip load is addressed under the two different boundary conditions of water/air permeable and water/air impermeable. Using the Fourier transform, the dynamic governing equations of unsaturated soils are established, and the general solution is derived in the frequency domain. The effects of the saturation, frequency, and boundary conditions on the dynamic behavior of the unsaturated soil are analyzed. The results show that for the unsaturated soil these parameters have an important influence on the dynamic characteristic.

Appendix

The elements J_ij and ${\overline{J}}_{ij}$ (i, j = 1, 2, 3, 4) in the matrix [J]and $\left[\overline{J}\right]$ can be expressed as follows:

$$ {J}_{11}=2\mu {\lambda}_0,\kern0.5em {J}_{12}=2\mu {S}_1{\lambda}_1^2+\lambda - a\chi {f}_{w1}-a\left(1-\chi \right){f}_{a1},\kern0.5em {J}_{13}=2\mu {S}_2{\lambda}_2^2+\lambda - a\chi {f}_{w2}-a\left(1-\chi \right){f}_{a2},\kern0.5em {J}_{14}=2\mu {S}_3{\lambda}_3^2+\lambda - a\chi {f}_{w3}-a\left(1-\chi \right){f}_{a3};\kern0.5em {J}_{21}=- i\mu {\lambda}_0^2/\xi - i\mu \xi, \kern0.5em {J}_{22}=-2\mu {S}_1 i\xi {\lambda}_1,\kern0.5em {J}_{23}=-2\mu {S}_2 i\xi {\lambda}_2,\kern0.5em {J}_{24}=-2\mu {S}_3 i\xi {\lambda}_3;\kern0.5em {J}_{31}=0,{J}_{32}={f}_{w1},{J}_{33}={f}_{w2},{J}_{34}={f}_{w3};{J}_{41}=0,{J}_{42}={f}_{a1},{J}_{43}={f}_{a2},{J}_{44}={f}_{a3}. $$

$$ {\overline{J}}_{11}=2\mu {\lambda}_0,\kern0.5em {\overline{J}}_{12}=2\mu {S}_1{\lambda}_1^2+\lambda - a\chi {f}_{w1}-a\left(1-\chi \right){f}_{a1},\kern0.5em {\overline{J}}_{13}=2\mu {S}_2{\lambda}_2^2+\lambda - a\chi {f}_{w2}-a\left(1-\chi \right){f}_{a2},\kern0.5em {\overline{J}}_{14}=2\mu {S}_3{\lambda}_3^2+\lambda - a\chi {f}_{w3}-a\left(1-\chi \right){f}_{a3};\kern0.5em {\overline{J}}_{21}=- i\mu {\lambda}_0^2/\xi - i\mu \xi, \kern0.5em {\overline{J}}_{22}=-2\mu {S}_1 i\xi \lambda, \kern0.5em {\overline{J}}_{23}=-2\mu {S}_2 i\xi {\lambda}_2,\kern0.5em {\overline{J}}_{24}=-2\mu {S}_3 i\xi {\lambda}_3;\kern0.5em {\overline{J}}_{31}=0,\kern0.5em {\overline{J}}_{32}=-{\lambda}_1{f}_{w1},\kern0.5em {\overline{J}}_{33}=-{\lambda}_2{f}_{w2},\kern0.5em {\overline{J}}_{34}=-{\lambda}_3{f}_{w3};\kern0.5em {\overline{J}}_{41}=0,\kern0.5em {\overline{J}}_{42}=-{\lambda}_1{f}_{a1},\kern0.5em {\overline{J}}_{43}=-{\lambda}_2{f}_{a2},\kern0.5em {\overline{J}}_{44}=-{\lambda}_3{f}_{a3}. $$