MMSS VV

1

Scattering of flexural wave in a thin plate with multiple inclusions by using the null-field integral

equation approach

Wei-Ming Lee1, Jeng-Tzong Chen2, Rui-En Jiang1

1 Department of Mechanical Engineering, China Institute of Technology, Taipei, Taiwan

2 Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan

2009年 06月 06日北台科技大學

National Taiwan Ocean UniversityMSVLAB ( 海大河工系 )

Department of Harbor and River Engineering

MMSS VV

2

Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Introduction

MMSS VV

3

Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Introduction

MMSS VV

4

Introduction

Inclusions, or inhomogeneous materials, usually take place in shapes of discontinuity such as thickness reduction or strength degradation .

The deformation and corresponding stresses caused by the dynamic force are induced throughout the structure by means of wave propagation.

MMSS VV

5

Scattering

At the irregular interface of different media, stress wave reflects in all directions scattering

The near field scattering flexural wave results in the dynamic stress concentration which will reduce loading capacity and induce fatigue failure.

Certain applications of the far field scattering flexural response can be obtained in vibration analysis or structural health-monitoring system.

MMSS VV

6

Literature review

From literature reviews, few papers have been published to date reporting the scattering of flexural wave in plate with more than one inclusion.

Kobayashi and Nishimura pointed out that the integral equation method (BIEM) seems to be most effective for two-dimensional steady-state flexural wave.

Improper integrals on the boundary should be handled particularly when the BEM or BIEM is used.

MMSS VV

7

Objective

For the plate problem, it is more difficult to calculate the principal values to treat the improper integral.

Our objective is to develop a semi-analytical approach to solve the scattering problem of flexural waves in an infinite thin plate with multiple circular inclusions by using the null-field integral formulation, degenerate kernels and Fourier series.

MMSS VV

8

Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Introduction

MMSS VV

9

Flexural wave of plate

4 4( ) ( ),u x k u x xÑ = Î WGoverning Equation:

u is the out-of-plane displacement k is the wave number

4 is the biharmonic operator

is the domain of the thin plates

u(x)

24

3

12(1 )

hk

D

E hD

wr

m

=

=-

ω is the angular frequency

D is the flexural rigidityh is the plates thickness

E is the Young’s modulus

μ is the Poisson’s ratio

ρ is the surface density

MMSS VV

10

Problem Statement

Problem statement for an infinite plate containing multiple circular

inclusions subject to an incident flexural wave

MMSS VV

11

The integral representation for the plate problem

MMSS VV

12

Kernel function

The kernel function is the fundamental solution which satisfies

MMSS VV

13

The slope, moment and effective shear operators

slope

moment

effective shear

MMSS VV

14

Kernel functions

In the polar coordinate of

MMSS VV

15

Direct boundary integral equations

Among four equations, any two equations can be adopted to solve the problem.

displacement

slope

with respect to the field point x

with respect to the field point x

with respect to the field point x

normal moment

effective shear force

MMSS VV

16

Oxr

f

Degenerate kernel (separate form)

iUeU

f

r

x

q

s

R

MMSS VV

17

Fourier series expansions of boundary data

MMSS VV

18

Boundary contour integration in the adaptive observer system

MMSS VV

19

Adaptive observer system

Source pointSource point

Collocation pointCollocation point

MMSS VV

20

Vector decomposition

MMSS VV

21

Transformation of tensor components

MMSS VV

22

Linear system

where H denotes the number of circular boundaries

MMSS VV

23

MMSS VV

24

Techniques for solving scattering problems

MMSS VV

25

The systems for surrounding plate and each inclusion

The continuity conditions

MMSS VV

26

MMSS VV

27

Dynamic moment concentration factor

Scattered far field amplitude

MMSS VV

28

Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Introduction

MMSS VV

29

Case 1: An infinite plate with one inclusion

Geometric data:a =1mThicknessPlate:0.002mInclusion: 0.001m

MMSS VV

30

MMSS VV

31

Distribution of dynamic moment concentration factors by using the present method and FEM

MMSS VV

32

The far field scattering pattern for a flexible inclusion with h1=h/2

MMSS VV

33

Far field backscattering amplitude versus the dimensionless wave number

MMSS VV

34

Case 2: An infinite plate with two inclusions

MMSS VV

35

MMSS VV

36

Distribution of dynamic moment concentration factors by using the present method and FEM( L/a = 2.1)

MMSS VV

37

Far field scattering pattern for two flexible inclusions with h1=h/2 and L/a=2.1

MMSS VV

38

Far field scattering pattern for two flexible inclusions with h1=h/2 and L/a=10.0

MMSS VV

39

Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Introduction

MMSS VV

40

Concluding remarks

A semi-analytical approach to solve the scattering problem of flexural

waves and to determine the DMCF and the far field scattering pattern in an infinite thin plate with multiple circular inclusions was proposed The present method used the null BIEs in conjugation with the degenerat

e kernels, and the Fourier series in the adaptive observer system.

DMCF of two inclusions is apparently larger than that of one when two

inclusions are close to each other.

Fictitious frequency of external problem can be suppressed by using the more number of Fourier series terms.

1.

2.

3.

4.

5.

The space between two inclusions has different effects on the near field

DMCF and the far field scattering pattern.

MMSS VV

41

Thanks for your kind attention

The End