Interface Element Method and Its Application Kun SHI Li SONG Junping SHI

School of Mechanical and Precise Instrument Xi’an University of Technology

Xi’an, China shikun@xaut.edu.cn

Abstract—Without any additional element, noncontinuous deformation of body can be simulated efficiently by interface element. The element stiffness matrix and interface stress are derived, which is based on the principle of virtual work. The steps given in the theoretical development were implemented in the computer program. A homogeneous cantilever beam and a composite structure were taken for examples to check the program. The comparisons between the numerical and theoretical results , and between numerical and measured results show that it is accurate and feasible for interface element method to solve the mechanical problem of the continuous and noncontinuous medium.

Keywords- interface element; noncontinuous medium; numerical simulation ; composite structure.

I. INTRODUCTION Interface element method is a new algorithm for solving

static and dynamic problems of the non-uniform, anisotropic, nonlinear discontinuous medium. The method replaces the rigid body spring element, contributed by Kawai [1] for solving homogeneous and linear static problem, with interface element which can reflect the deformation characteristics of components such as elastic, plastic and viscous components. Jiashou Zhuo, Jianhai Zhang et al [2,3] elaborated the basic theory of the two-dimensional and three-dimensional interface element, and derived interface element equation of discrete model. The method is mainly used to solve the interface and fracture problem of geotechnical engineering [4-6], composite materials [7,8] currently.

The interface element method is introduced for analyzing mechanics problems of structural combination in the paper. The corresponding program was written according to the theory of the interface element. A homogeneous cantilever beam and a composite structure were taken as examples to check the program. The comparisons between the numerical and theoretical results, and between numerical and test results show that the method is accurate and feasible.

II. THE THEORY OF INTERFACE ELEMENT METHOD

A. Mechanical Foundation of Interface Element Method The dominant equation of interface element discrete model is

derived by the principle of virtual work. The principle of virtual work, which is equal relationship between the virtual work of internal force Uδ and the virtual work of external force Vδ in the equilibrium state, may be expressed as:

ij ij i i i is

e d f u d p u dsσ

σ δ δ δΩ Ω

Ω = Ω +

or U Vδ δ= 1

In the above equation, ij ijU e dδ σ δ

Ω

= Ω,

i i i is

V f u d p u dsσ

δ δ δΩ

= Ω +.

Because the derivation does not involve the material property, the theorem of virtual work is applied to the elastic and plastic body. The discrete model of interface element is made up of block element and interface element. The deformation of block element accumulates in the interface, so block element itself has no strain and no virtual work of internal force. The surface force of the interface element has done the virtual work in the process of its relative virtual displacement, which is the external virtual work for the block element. Therefore, the virtual work equation of the discrete model may be written as:

0e e e

o

i i i i ie s s

f u d p u ds Ti u dsσ

δ δ δΩ

Ω + + =

or e e eo

i i i i ie es s

Ti u ds f u d p u dsσ

δ δ δΩ

− = Ω + 2

As for the whole discrete model, the negative number of the total virtual work can be regarded as virtual work of the systemic internal force, which is made by the surface force of all interface elements in the process of relative virtual displacement.

B. Dominated Equation of Interface Element Method

In making up the displacement model of block elements, the generalized displacement of its centroid point is regarded as the parameter, and the interfacial stress is linear with the relative movement of the interface. So, the displacement of the block and interface can be defined by the displacement model of block element. Eq.(2) becomes the equation in which the generalized displacement of the centroid point of block elements is regarded

Project supported by the National Program on Key Basic Research Project (973 Program)(No. 2009CB724406), Natural Science Basic Research Plan in Shaanxi Province of China(No. 2010JQ7003)

4150978-1-61284-459-6/11/$26.00 ©2011 IEEE

as the unkown quantity. The equation, similar to the dominated equation of finite elements, may be written as:

K Rδ = 3

In the above equation, δ is the unkown quantity constituted by the generalized displacement of the centroid point of block elements. Kδ is the internal force resisting the deformation. R is the external force acting on the centroid point of block elements. For the elastic problem, K is only related to physical dimension and material properties of the model, and R is only related to external load. They are all known. Therefore, the displacement of any point and the stress of any point in the interface can be respectively obtained from Eq.(3), the displacement model and stress model. The displacement and stress field of the surface structure can be attained finally.

Fig.1 The model of interface element

For the given figure, Fig.1 shows the interface element model (

1G and 2G is the centre of rigid element

1e and 2e

respectively; 1h and

2h is the distance of between the interface and

1G and 2G respectively). The relevant parameters of Eq.(3) are

shown as follows:

(1) (1) (1) *

* * (1) (1) *

(loop sum of the unit

(loop sum of the interface

eo

j

T T Te eS

e

T T Te e jS

j

K C N L DL N dSC e

C N L DL N dSC S

∗

∗

=

=

4

In the above formula, ( ) ( )

( ) ( ) ( )

( ) ( )

1 0 0 00 1 0 00 0 1 0

k kg g

k k kg g

k kg g

z z y y

N z z x xy y x x

− −= − −

− −

;

* (1) (2)N N N= − ; (x y, z) is the coordinate of any point in the

interface jS ; ( ( )k

gx , ( )kgy , ( )k

gz )is the coordinate of the centroid point

of rigid unit 1(k=1) and 2(k=2) respectively; TeC is the selection

matrix which establishs the connection between the element displacement matrix and overall displacement matrix; L is the direction cosine matrix of the local coordinate of the interface; D is the constitutive matrix which reflects the characteristic of the different interface element.

e

T T T Te ee S

e e

R C N fd C N pdSσΩ

= Ω + 5

{ }1 1 1 1 1 1, ,T

g g g x y z gn gn gn xn yn znu v w u v wδ θ θ θ θ θ θ= 6

III. CALCULATION PROCEDURE OF INTERFACE ELEMENT METHOD

The calculation procedure of interface element method is as follows:

1) Build the discrete model and generate the element mesh. The block element, node, interface element and material category are numbered. The arrays of information about the element, coordinate, material, constraint and load condition are set up.

2) Establish the shape function matrix N and the elasticity matrix D respectively according to the block displacement mode and material constitutive.

3) Establish the load column matrix R. The element equivalent load matrixs are established in accordance with block elements one by one, then these matrixs are combined into a system equivalent load matrix R, as follows:

eRe e

T T

S

N fd N pdSσΩ

= Ω + 7

eRT

ee

R C= 8

4) Establish the stiffness matrix K. The individual interface element stiffness matrixs are established in accordance with interface elements one by one, then these matrixs are combined into a system stiffness matrix K, as follows:

(1) (1)

j

T Tj

s

k N L DL N dS∗ ∗= 9

* *Te j e

j

K C k C= 10

5) Slove the generalized displacement matrixs of the individual centroid point of block elements

gU . The system displacement matrixs U can be sloved according to the dominant equation KU = R. Then, the generalized displacement matrixs of the individual centroid point of block elements

gU may be obtained in accordance with block elements one by one, and the relative displacement of the individual interface (1) (2)

l lδ δ+ can be gotten in accordance with the interfaces.

6) Slove the interfacial stress of the individual interface elements

lT . The interfacial stress of the individual interface

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elements lT can be sloved by the interface stress formula in

accordance with interface elements one by one, as follows:

(1) (2)( )l l lT D δ δ= + 11

7) Describe the structural displacement and the stress field with corresponding graphs.

IV. EXAMPLE

A. Solve the problem of continuum mechanics A homogeneous cantilever beam[2] is shown in Fig.2. The length is 12m, the width is 2m, the elastic modulus E is 1.0× 510 Mpa, and the poisson ratio μ is 0.33. The load exerted on the cantilever beam P is 60KN. The mesh of the cantilever beam is shown in Fig.3, which is made up of 16 block elements and 24 interface elements. The analytical and theoretical solution of the cantilever beam's displacements is shown in Fig.4. From the figure, it is seen that the analytical solution and theoretical solution is very close. The precision of solving the problem of the continuous medium with the interface element is high.

Fig.2 The model of homogeneous cantilever beam

Fig. 3 Mesh of homogeneous cantilever beam

Fig.4 Displacements of center line with analytical and theoretical solution

B. Solve the mechanical problem of discontinuous medium A composite structure [9-13] is shown in Fig.5. The model is composed by two parts of Steel 45. The elastic modulus E is

2.06× 510 Mpa,and the poisson ratio μ is 0.26. Contact surfaces between Part and Part is polished, so that the surface roughness is less than 0.8 m. There is not oil on the contact surfaces. The lower surface of Part is fixed. The uniformly distributed ring load q, exerted on the supine surface of Part , is 74.66 10 Pa× . Fig.6 shows six test points of a,b,c,d,e,f, located on the supine surface of Part .

Fig.5 The model of structural assembly

Fig.6 Test point on the surface A of structural assembly

The mesh of the model was made up of 456 block elements and 1008 interface elements. The analytical and theoretical solution of the model displacements is shown in Fig.7. From the figure, it is seen that the analytical solution and measured value is subequal. The method of solving the mechanical problem of the discontinuous medium with the interface element is feasible .

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Fig.7 Displacements of test point on the surface A with

analytical and measured solution

V. CONCLUSION From the example of sloving the problem of the

homogeneous cantilever beam and the composite structure, it is seen that the precision of interface element method is high. The analytical solution is subequal with theoretical solution and measured value. When analyzing the mechanical problem of continuous or discontinuous medium, the deformation of body can be simulated efficiently without any additional element to reslove the deformation of adjacent blocks. So the method can easily simulate discontinuous deformation between multiple blocks, and have a wide range of application prospect.

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