Abstract
We present the macroscale three-dimensional numerical solution of anisotropic Biot's poroelasticity, with coefficients derived from a micromechanical analysis as prescribed by the asymptotic homogenisation technique. The system of partial differential equations (PDEs) is discretised by finite elements, exploiting a formal analogy with the fully coupled thermal displacement systems of PDEs implemented in the commercial software Abaqus. The robustness of our computational framework is confirmed by comparison with the well-known analytical solution of the one-dimensional Therzaghi's consolidation problem. We then perform three-dimensional numerical simulations of the model in a sphere (representing a biological tissue) by applying a given constant pressure in the cavity. We investigate how the macroscale radial displacements (as well as pressures) profiles are affected by the microscale solid matrix compressibility (MSMC). Our results suggest that the role of the MSMC on the macroscale displacements becomes more and more prominent by increasing the length of the time interval during which the constant pressure is applied. As such, we suggest that parameter estimation based on techniques such as poroelastography (which are commonly used in the context of biological tissues, such as the brain, as well as solid tumours) should allow for a sufficiently long time in order to give a more accurate estimation of the mechanical properties of tissues.
Highlights
Biot’s theory of poroelasticity deals with the mechanical and hy draulic behaviour of a medium in which the solid skeleton interplays with fluid percolating its pores (Detournay and Cheng (1993); Biot (1955, 1956a, 1956b, 1962))
In order to verify the reliability of our computational framework, we solve the problem for a saturated column of sand under an applied load pp 1⁄4 1000 Paat height z 1⁄4 0, and compare the results with the analyt ical solution given in Ferronato et al (2010), which reads pð0Þðz; tÞ 1⁄4 4 p0 X ∞ 1 π n1⁄40 2 n þ 1
We have exploited the findings reported in Dehghani et al (2018) to compute the corresponding macroscale solution, clearly highlighting the link between relevant poroelastic maps, such as displacements and pres sures, and the pore-scale properties, in particular focusing on the microscale solid matrix compressibility
Summary
Biot’s theory of poroelasticity deals with the mechanical and hy draulic behaviour of a medium in which the solid skeleton interplays with fluid percolating its pores (Detournay and Cheng (1993); Biot (1955, 1956a, 1956b, 1962)). European Journal of Mechanics / A Solids 83 (2020) 103996 computational analysis via computing the relevant macroscale poroe lastic coefficients, i.e. the stiffness tensor, the hydraulic conductivity, Biot’s modulus, and Biot’s coefficients, as prescribed via the asymptotic homogenisation technique via solving elastic-type and Stokes’-type pe riodic cell problems on a connected porous structure Their analysis, framed in the context of tumour modelling, serves as a basis to quantify the response of poroelastic materials in terms of the microstructure, and primarily focuses on the role of the matrix’s porosity and compressibility.
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