671017.pdf

1c

2c

c

element)

Ve(thin-layer

d

Figure 3: Thin-layer element method.

[퐷]푒푝is defined using the following constitutive law of an
interface [ 6 ]:

{

Δ휎푛

Δ휏

}=[퐷]푒푝{

[Δ푢푛]

[Δ푢푠]

},

[퐷]푒푝=[

푘푛 푘푛푠

푘푠푛 푘푠

],

(5)

where휎푛,휏are the normal and shear stresses,푢푛,푢푠are
the normal and shear displacements,푘푛,푘푠are the normal
and shear stiffness, and푘푠푛,푘푛푠are the coupling stiffnesses
between normal and shear deformations.
Goodman et al. [ 2 ] did not consider the coupling effect
between normal and shear deformations. They took the
material matrix as

[퐷]푒푝=[

푘푛 0

0푘푠

] (6)

and the shear stiffness푘푠as

푘푠=푘 1 훾푤(

휎푛

푃푎

)

푛 1

(1 −

푅푓1휏

휎푛푡푔휙

)

2

, (7)

where푘 1 and푛 1 are two parameters,휎푛is the normal stress
on the interface,휏is the shear stress along the interface,푃푎is
the atmospheric pressure,훾푤is the unit weight of water,푅푓1
is the failure ratio, and휙is the angle of internal friction.푘 1 ,
푛 1 ,푅푓1,and휙are the four parameters to be determined from
direct shear tests. The normal stiffness푘푛is usually given a
large number when the interface element is in compression
and a small number when in tension.

2.3. Thin-Layer Element Method.In this method, an interface
is treated as a thin-layer solid element (Figure 3 ). This thin
layer is given a relatively low modulus and can experience
large deformation [ 8 – 11 ]. The problem shown in Figure 1 with
athinlayerhasthefollowingweakform:

휋 1 +휋 2 +휋thin=0, (8)

wherethetermonthethinlayer,휋thin,isgivenby

휋thin=∫

푉퐿

{훿휀}푇{휎}d푉, (9)

1c

2c

c

P P

A

B

Figure 4: Contact analysis method.

with푉퐿denoting the domain of interfaceΓ푐.If푉퐿has a finite

thickness of푑, the element stiffness of thin layer element푉푒

is

퐾th푒=∫

푉푒

퐵푇퐷퐵d푉≅

푑≪푆푒

푑∫

푆푒

퐵푇퐷퐵d푆, (10)

where퐵isthestrainmatrix,퐷is the material matrix, and

푆푒is the element length. Previous studies revealed that the

accuracy of element stiffness is sensitive to the aspect ratio

푑/푆푒. When the aspect ratio varies in the range of 0.01–0.1,

slippage is modeled quite accurately [ 8 – 11 ].

2.4. Contact Analysis Method

2.4.1. Contact of Two Deformable Bodies.As shown in

Figure 4 ,thepotentialcontactboundariesareΓ1푐inΓ 1 andΓ2푐

inΓ 2 , while the exact contact boundary is denoted as interface

Γ푐, which is usually unknown beforehand. The weak form of

each deformable body is expressed individually as follows.

For deformable bodyΩ 1

{∫

Ω 1

{훿휀}푇{휎}dΩ−∫

Ω 1

{훿푢}푇{푏}dΩ−∫

Γ1푡

{훿푢}푇{푡}dΓ}

−∫

Γ푐

{훿푢}푇{푃}dΓ=0.

(11)

For deformable bodyΩ 2

{∫

Ω 2

{훿휀}푇{휎}dΩ−∫

Ω 2

{훿푢}푇{푏}dΩ−∫

Γ2푡

{훿푢}푇{푡}dΓ}

−∫

Γ푐

{훿푢}푇{푃}dΓ=0,

(12)

where{푃}is the interaction force. Upon discretizing the weak

forms in ( 11 )and( 12 ), the following discrete system equation

is obtained for each deformable body:

퐾 11 푢 1 +퐾 12 푢 12 +퐿 1 푃=푓 1 forΩ 1 , (13)

퐾 22 푢 2 +퐾 21 푢 21 +퐿 2 푃=푓 2 forΩ 2 , (14)