Homogenization of very rough three-dimensional interfaces for the poroelasticity theory with Biot's model
Keywords:homogenization, homogenized equations, very rough interfaces, fluid-saturated porous media
In this paper, we carry out the homogenization of a very rough three-dimensional interface separating two dissimilar generally anisotropic poroelastic solids modeled by the Biot theory. The very rough interface is assumed to be a cylindrical surface that rapidly oscillates between two parallel planes, and the motion is time-harmonic. Using the homogenization method with the matrix formulation of the poroelasicity theory, the explicit homogenized equations have been derived. Since the obtained homogenized equations are totally explicit, they are very convenient for solving various practical problems. As an example proving this, the reflection and transmission of SH waves at a very rough interface of tooth-comb type is considered. The closed-form analytical expressions of the reflection and transmission coefficients have been derived. Based on them, the effect of the incident angle and some material parameters on the reflection and transmission coefficients are examined numerically.
W. Kohler, G. Papanicolaou, and S. Varadhan. Boundary and interface problems in regions with very rough boundaries. In Multiple Scattering and Waves in Random Media, (1981), pp. 165–197.
J. Nevard and J. B. Keller. Homogenization of rough boundaries and interfaces. SIAM Journal on Applied Mathematics, 57, (6), (1997), pp. 1660–1686. https://doi.org/10.1137/s0036139995291088.
R. P. Gilbert and M.-J. Ou. Acoustic wave propagation in a composite of two different poroelastic materials with a very rough periodic interface: a homogenization approach. International Journal for Multiscale Computational Engineering, 1, (4), (2003), pp. 431–440. https://doi.org/10.1142/9789812704405 0024.
P. C. Vinh and D. X. Tung. Homogenized equations of the linear elasticity in two-dimensional domains with very rough interfaces. Mechanics Research Communications, 37, (3), (2010), pp. 285–288. https://doi.org/10.1016/j.mechrescom.2010.02.006.
P. C. Vinh and D. X. Tung. Homogenization of rough two-dimensional interfaces separating two anisotropic solids. Journal of Applied Mechanics, 78, (4), (2011), p. 041014. https://doi.org/10.1115/1.4003722.
P. C. Vinh and D. X. Tung. Homogenized equations of the linear elasticity theory in two-dimensional domains with interfaces highly oscillating between two circles. Acta Mechanica, 218, (3-4), (2011), pp. 333–348. https://doi.org/10.1007/s00707-010-0426-2.
D. X. Tung, P. C. Vinh, and N. K. Tung. Homogenization of an interface highly oscillating between two concentric ellipses. Vietnam Journal of Mechanics, 34, (2), (2012), pp. 113–121. https://doi.org/10.15625/0866-7136/34/2/926.
P. C. Vinh and D. X. Tung. Homogenization of very rough interfaces separating two piezoelectric solids. Acta Mechanica, 224, (5), (2013), pp. 1077–1088. https://doi.org/10.1007/s00707-012-0804-z.
P. C. Vinh, V. T. N. Anh, D. X. Tung, and N. T. Kieu. Homogenization of very rough interfaces for the micropolar elasticity theory. Applied Mathematical Modelling, 54, (2018), pp. 467–482. https://doi.org/10.1016/j.apm.2017.09.039.
P. C. Vinh, D. X. Tung, and N. T. Kieu. Homogenization of very rough two-dimensional interfaces separating two dissimilar poroelastic solids with time-harmonic motions. Mathematics and Mechanics of Solids, 24, (5), (2019), pp. 1349–1367.
M. A. Biot. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range. The Journal of the Acoustical Society of America, 28, (2), (1956), pp. 168–178. https://doi.org/10.1121/1.1908239.
M. A. Biot. Mechanics of deformation and acoustic propagation in porous media. Journal of Applied Physics, 33, (4), (1962), pp. 1482–1498. https://doi.org/10.1063/1.1728759.
J. L. Auriault. Dynamic behaviour of a porous medium saturated by a Newtonian fluid. International Journal of Engineering Science, 18, (6), (1980), pp. 775–785. https://doi.org/10.1016/0020-7225(80)90025-7.
J. L. Auriault, L. Borne, and R. Chambon. Dynamics of porous saturated media, checking of the generalized law of Darcy. The Journal of the Acoustical Society of America, 77, (5), (1985), pp. 1641–1650. https://doi.org/10.1121/1.391962.
A. Bensoussan and J. L. Lions. Asymptotic analysis for periodic structures, Vol. 5. North-Holland, Amsterdam, (1978). https://doi.org/10.1090/chel/374.
T. C. T. Ting. Anisotropic elasticity: Theory and applications. Oxford University Press, New York, (1996).
R. D. Borcherdt. Viscoelastic waves in layered media. Cambridge University Press, (2009).