演講者：蕭錫源

A new BEM formulation for transient axisymmetric

poroelasticity via particular integrals

K.H. Park a, P.K. Banerjee b,*

Abstract

A simple particular integral formulation is presA simple particular integral formulation is presented for the first time in a purely axisymmetriented for the first time in a purely axisymmetric poroelastic analysis. c poroelastic analysis.

The axisymmetric elastostatic and steady-statThe axisymmetric elastostatic and steady-state potential flow equations are used as the come potential flow equations are used as the complementary solution.plementary solution.

The particular integrals for displacement, tractThe particular integrals for displacement, traction, pore pressure and flux are derived by inteion, pore pressure and flux are derived by integrating three-dimensional formulation alongtgrating three-dimensional formulation alongthe circumferential direction leading to elliptic he circumferential direction leading to elliptic integrals.integrals.

1. Introduction

The general theory of poroelasticity is gThe general theory of poroelasticity is governed by two coupled differential equoverned by two coupled differential equations: the Navier equation with pore prations: the Navier equation with pore pressure body force and the pore fluid floessure body force and the pore fluid flow equation as (Banerjee, 1994)w equation as (Banerjee, 1994)

是位移 是有效滲透率 是位移 是有效滲透率

是孔隙壓力是孔隙壓力

和 和 Lame’sLame’s 的常數 的常數 the undrainedthe undrained 不透水的不透水的

該排水體積彈性模量 該排水體積彈性模量

是在對堅實組成部分的是在對堅實組成部分的 BULKBULK 模數某些情況經驗得知的常數模數某些情況經驗得知的常數 是 是 body forcebody force 和和 sourcesource （如果存在的話）（如果存在的話）

for 2D (3D)for 2D (3D)

iup

2

u

u

1s

K

K

2

3K

sK

if 1,2(3)i

常數　和　也可以表示在不排水體積的常數　和　也可以表示在不排水體積的彈性模數 　彈性模數 　 (Rice and Cleary, 1976)(Rice and Cleary, 1976)

　　是著名的　　是著名的 skemptonskempton 係數的孔隙壓力。係數的孔隙壓力。

uK

Park and Banerjee (2002a) first proPark and Banerjee (2002a) first proposed the particular integral formulposed the particular integral formul

ation ation

左為２００６左為２００６下為２００２下為２００２

2.Three-dimensional particular inte2.Three-dimensional particular integral formulationgral formulation

位移 位移 曳引力曳引力孔隙壓力孔隙壓力流量 流量

iu

it

qp

將右式將右式global shape fuglobal shape fu

nctionnction代入代入

將上三頁的式子代入２００６的將上三頁的式子代入２００６的非均值式子中得下列係數非均值式子中得下列係數

3. Axisymmetric particular integral f3. Axisymmetric particular integral formulationormulation

For axisymmetric problems, use of such For axisymmetric problems, use of such polynomial functions as functions of r apolynomial functions as functions of r and z coordinates have been discussed in nd z coordinates have been discussed in Henry et al. (1987). Henry et al. (1987).

It is of considerable interest to note that It is of considerable interest to note that Wang and Banerjee (1988, 1990) in their Wang and Banerjee (1988, 1990) in their developments of particular integrals in fdevelopments of particular integrals in free-vibration analysis of axisymmetric sree-vibration analysis of axisymmetric solids also observed the same to be true.olids also observed the same to be true.

為方便起見，所界定的軸對稱為方便起見，所界定的軸對稱 casecase

考慮到在圓柱坐標系的純粹的軸對稱體考慮到在圓柱坐標系的純粹的軸對稱體

thenthen

分別為在分別為在 XX 點在點在 RR 和和 Z -Z - 方向方向 的的 normal vectornormal vector 。。

4. Numerical implementation

軸對稱彈性力學和穩定狀態勢流方程的根本解

分別代表 jump terms resulting

所產生的奇異性質的 和

和

離散↓離散↓

↓ ↓

因考慮而加入因考慮而加入

有限的數量、時間、位移、牽引、有限的數量、時間、位移、牽引、

孔隙壓力和流量孔隙壓力和流量

5. Numerical examples5. Numerical examplesExample 1Example 1

Example 2 Example 2

其中　　是　　　　　　　　　　　　的根其中　　是　　　　　　　　　　　　的根

Example 3 Example 3

6. Conclusions

The simple particular integral formulation has The simple particular integral formulation has been developed for axisymmetric coupled porbeen developed for axisymmetric coupled poroelastic analysis.oelastic analysis.

The equations of axisymmetric elastostatic anThe equations of axisymmetric elastostatic and steady-state potential flow have been used ad steady-state potential flow have been used as the complementary functions.s the complementary functions.

The particular integrals of displacement, tractiThe particular integrals of displacement, traction, pore pressure and flux are obtained by inteon, pore pressure and flux are obtained by integrating three-dimensional BEM formulation algrating three-dimensional BEM formulation along the circumferential direction and convertiong the circumferential direction and converting them into elliptic integrals.ng them into elliptic integrals.