多孔質弾性体と流体の連成解析 (非線形現象の数理解析と実験解析)

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1 (Hiroshi Iwasaki) Faculty of Mathematics and Physics Kanazawa University quasi-static Biot 1 : ( ) (coup iniury) (contrecoup injury)

2 49 [9] Navier-Stokes $\rho(\frac{\partial v}{\partial t}+v\cdot\nabla v)=-\nabla p+\eta\nabla^{2}v+\eta \nabla(\nabla\cdot v)_{:}$ $( \frac{\partial\rho}{\partial t}+v\cdot\nabla\rho)=-\rho\nabla\cdot v$ $( or\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho v)=0)$ $\rho=\rho(p)$. $\rho$ $vp$ $\eta $ $\eta$ $\rho_{0}\frac{\partial v}{\partial t}=-\nabla p+\eta\nabla^{2}v+\eta \nabla(\nabla\cdot v)$ (2.1) $\frac{1}{k_{f}}\frac{\partial p}{\partial t}=-\nabla\cdot v$ (2.2) $\rho_{0}=\rho(p_{0})$ $\frac{1}{k_{f}}=\frac{1}{\rho}\frac{\partial\rho}{\partial p} _{p_{o}}$ (2.3) Stokes 2 $K_{f}$

3 $\lambda$ 50 $u$ $\mu$ $\hat{\epsilon}$ $\hat{\sigma}$ Lam\ e $\rho\frac{\partial^{2}u}{\partial t^{2}}-\nabla\cdot\hat{\sigma}=0$ (2.4) $\hat{\sigma}=2\mu\hat{\epsilon}+\lambda(i;\hat{\epsilon})i$ (2.5) $\hat{\epsilon}=\{(\nabla u)+(\nabla u)^{t}\}/2$ (2.6) 1 Navier 2 Hooke Navier 2.2 Darcy $q$ Darcy $q+ \frac{k}{\eta}\nabla p=0$ (2.7) $k$ $\eta$ (porocity) $\phi$ $v$ $q$ porocity $q=\phi v$ (2.8) $\frac{\partial(\phi\rho_{f})}{\partial t}+\nabla\cdot(\phi\rho_{f}v)=0$ $q$ $\frac{1}{\rho_{f}}\frac{\partial(\phi\rho_{f})}{\partial t}+\nabla\cdot q=0$ (2.9) $\frac{\phi}{k_{f}}\frac{\partial p}{\theta t}+\nabla\cdot q=0$

4 51 $q+ \frac{k}{\eta}\nabla p=0$ (210) $\frac{\phi}{k_{f}}\frac{\partial p}{\partial t}+\nabla\cdot q=0$ (211) Darcy 2.3 Biot Darcy Navier Biot [2]. Biot [3][8][5]. $(\hat{\sigma}\hat{\epsilon})$ $(p \phi)$ $\Psi_{s}=\Psi_{s}(\hat{\epsilon} \phi)$ (212) $d\psi_{s}=\hat{\sigma}$ : $d\hat{\epsilon}+pd\phi$ (213) $G_{s}=\Psi_{s}-p\phi$ (214) $G_{s}=G_{s}(_{\vee} \hat{\epsilon}p);\hat{\sigma}=\frac{\partial G_{s}}{\partial\hat{\epsilon}}(\hat{\epsilon}p);\phi=-\frac{\partial G_{s}}{\partial p}(\hat{\epsilon}p)$ (215) $d\sigma_{i_{\dot{j}}}=\frac{\partial\sigma_{ij}}{\partial\epsilon_{kl}}d\epsilon_{kl}+\frac{\partial\sigma_{ij}}{\partial p}dp$ $= \frac{\partial^{2}g_{s}}{\partial\epsilon_{kl}\partial\epsilon_{ij}}d\epsilon_{kl}+\frac{\partial^{2}g_{s}}{\partial p\partial\epsilon_{ij}}dp$ $d\phi=\frac{\partial\phi}{\partial\epsilon_{ij}}d$ $+ \frac{\partial\phi}{\partial p}dp$ $=- \frac{\partial^{2}g_{s}}{\partial\epsilon_{ij}\partial p}d\epsilon_{ij}-\frac{\partial^{2}g_{s}}{\partial p\partial p}dp$

5 $\lambda$ 52 $C_{-kl} \equiv\frac{\partial^{2}g_{8}}{\partial\epsilon_{kl}\partial\epsilon_{ij}}$ $b_{ij} \equiv-\frac{\partial^{2}g_{8}}{\partial\epsilon_{ij}\partial p}=-\frac{\partial^{2}g_{s}}{\partial p\partial\epsilon_{ij}}$ $\frac{1}{k_{\phi}}\equiv-\frac{\partial^{2}g_{s}}{\partial p\partial p}$ Cijkl $=\lambda\delta_{ij}\delta_{kt}+\mu(\delta_{ik}\delta_{jl}+\delta_{it}\delta_{jk})$ $b_{ij}=b_{w}\delta_{ij}$ $d\hat{\sigma}=d\hat{\sigma}_{s}-b_{w}$dpi (2.16) $\hat{\sigma}_{s}\equiv\mu\{(\nabla u)+(\nabla u)^{t}\}+\lambda(\nabla\cdot u)i$ $\mu$ $d\phi=b_{w}\nabla\cdot du+\frac{1}{k_{\phi}}dp$ (2.17) Lam\ e $B_{w}$ $K_{\phi}$ Biot-Wilhs [3] Biot-Willis (217) $\phi$ (2.9) $\frac{1}{\rho_{f}}\frac{\partial(\phi\rho_{f})}{\partial t}=\frac{\partial\phi}{\partial t}+\frac{\phi}{\rho_{f}}\frac{\partial\rho_{f}}{\partial t}$ $=(B_{w} \nabla\cdot\frac{\partial u}{\partial t}+\frac{1}{k_{\phi}}\frac{\partial p}{\partial t})+\frac{\phi}{\rho_{f}}\frac{\partial\rho_{f}}{\partial t}$ $\approx B_{w}\nabla\cdot\frac{\partial u}{\partial t}+(\frac{1}{k_{\phi}}+\frac{\phi_{0}}{k_{f}})\frac{\partial p}{\partial t }$ $\frac{1}{k}\frac{\partial p}{\partial t}+\nabla\cdot q+b_{w}\nabla\cdot\frac{\partial u}{\partial t}=0$ (2.18) $\frac{1}{k}\equiv\frac{1}{k_{\phi}}+\frac{\phi_{0}}{k_{f}}$ (2.19) $\phi_{0}$ $K $ Navier Hooke (2.5) Hooke (216) $\rho\frac{\partial^{2}u}{\partial t^{2}}-\nabla\cdot\hat{\sigma}_{s}+b_{w}\nabla p=0$ (2.20) $\hat{\sigma}_{s}\equiv\mu\{(\nabla u)+(\nabla u)^{t}\}+\lambda(\nabla\cdot u)i$ (2.21)

6 53 Biot $q+ \frac{k}{\eta}\nabla p=0$ (2.22) $\frac{1}{k}\frac{\partial p}{\partial t}+\nabla\cdot q+b_{u\prime}\nabla\cdot\frac{\partial u}{\partial t}=0$ (2.23) $\rho\frac{\partial^{2}u}{\partial t^{2}}-\nabla\cdot\hat{\sigma}+\underline{b_{w}\nabla p}=0$ (2.24) Biot $\hat{\sigma}=\mu\{(\nabla u)+(\nabla u)^{t}\}+\lambda(\nabla\cdot u)i$ (2.25) Darcy Navier $B_{w}\nabla\cdot u$ $B_{w}\nabla p$ $B_{v)}$ quasi-static Biot $\rho$ $1/K $ quasi-static Biot $q+ \frac{k}{\eta}\nabla p=0$ (2.26) $\nabla\cdot q+b_{w}\nabla\cdot\frac{\partial u}{\partial t}=0$ (2.27) $-\nabla\cdot\hat{\sigma}+b_{w}\nabla p=0$ (2.28) $\hat{\sigma}=\mu\{(\nabla u)+(\nabla u)^{t}\}+\lambda(\nabla\cdot u)i$. (2.29) 3 quasi-static Biot quasi-static Biot $\Omega$ 1

7 $r.\backslash _{:}:i$ $!^{:}$ $ $ $\Omega=\{(xy) x <\frac{l_{x}}{2}0<y<l_{y}\}$ $ $ $\Gamma_{2}=\{(xy)\in\partial\Omega x >\frac{l_{x}}{5}y=l_{y}\}$ 54 $y\cdot\frac{111fll1^{\triangleleft 0}\ulcorner}{}$ :: :: :.:!1 ::.!: $\Gamma_{3}=\partial\Omega\backslash :: :: : $\Gamma_{1}=\{(xy)\in\partial\Omega x \leq\frac{l_{x}}{5}y=l_{y}\}$ : (\Gamma_{1}\cup\Gamma_{2})$. $r\cdot oi.:::x\overline{--\omega J2nx\cdot*t}x\prime 2$ 1 $x\in\partial\omega$ $p(xt)=0$ $\sigma_{yy}(xt)=-\sigma_{0}$ $x\in\gamma_{1}$ $\sigma_{yy}(xt)=0$ $x\in.\gamma_{2}$ $\sigma_{xy}(x t)=0$ $x\in\gamma_{1}\cup\gamma_{2}$ $u(xt)=0$ $x\in\gamma_{3}$ $x\in\omega_{:}$ $p(xt=0)=0$ $x\in\omega$ $u(xt=0)=0$ $L_{x}=L_{y}=1$ $\mu=\underline{e}$ $2(1+\nu)$ $k=\eta=b_{w}=1$ $\sigma_{0}=1$ $\lambda=\frac{\nu E}{(1+\nu)(1-2\nu)}$ $E=3$ $\nu=0.2$ $100\cross 100$ Euler

55 1 $p:t=$ 9e-05 $p:t=$ 0.0009 1 0.8 os o.s 0.6 0.7 0.6 0.6 0.4 0.4 0.5 0.4 0.3 0.2 0.2 0.2 0.1 $0$ $0$ $0$ $-0.1$ $0$ 0.2 0.4 0.6 0.8 1 $0$ 0.2 0.4 0.6 0.8 1 2 3 $p:t=$ $0.

8 55 1 $p:t=$ 9e-05 $p:t=$ os o.s $0$ $0$ $0$ $-0.1$ $0$ $0$ $p:t=$ $ $ $p:t=$ $-$ o.s $0.s$ $0$ $0$ $0$ $0.s$ 1 $0$ $0.s$

$56 4 [6]. LB (Ladyzhenskaya $B$ $-B$ abuska-brezzi) $su\triangleright\inf$ 1 [4][7]. 2 LBB [1] [1] S. Badia A. Quaini and A.$

9 56 4 [6]. LB (Ladyzhenskaya $B$ $-B$ abuska-brezzi) $su\triangleright\inf$ 1 [4][7]. 2 LBB [1] [1] S. Badia A. Quaini and A. Quarteroni Coupling Biot and Navier-Stokes equations for modelling fluid-poroelastic media interaction J. Comput. Phys. 228 (2009) [2] M. A. Biot General Theory of three-dimensional consolidation J. Appl. Phys.. 12 (1941) [3] M. A. Biot and D. G. Willis The elastic coefficients of the theory of the consolidation J. Appl. Mech. 24 (1957) [4] F. Brezzi and J. Douglas Jr. Stabilized mixed methods for the Stokes problem : Numer. Math. 53 (1988) [5] O. Coussy Poromechanics John Wiley & Sons 2004.

10 57 [6] M. A. Murad and A. F. D. Loula On stability and convergence offinite element approximation of Biot s consolidation problem Internat. J. Numer. Methods Engrg. 37 (1994) [7] [8] J. R. Rice and M. P. Cleary Some basic stress-diffusion solutions for fluid saturated elastic porous media with compressible constituents Rev. Geophys. Space Phys. 14 (1966) [9] R. E. Showalter. Poroelastic filtmtion coupled to Stokes flow : Published in Control Theory of Partial Differential Equation Lecture Notes in Pure and Applied Mathematics 242 (2005)

MD $\text{ }$ (Satoshi Yukawa)* (Nobuyasu Ito) Department of Applied Physics, School of Engineering, The University of Tokyo Lennar

$MD $\text{ }$ (Satoshi Yukawa)* (Nobuyasu Ito) Department of Applied Physics, School of Engineering, The University of Tokyo Lennar$ 1413 2005 36-44 36 MD $\text{ }$ (Satoshi Yukawa)* (Nobuyasu Ito) Department of Applied Physics, School of Engineering, The University of Tokyo Lennard-Jones [2] % 1 ( ) *yukawa@ap.t.u-tokyo.ac.jp ( )