m m s s v v 0 scattering of flexural wave in a thin plate with multiple inclusions by using the...

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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

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Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Introduction

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Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Introduction

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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.

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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.

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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.

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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.

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Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Introduction

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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)

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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

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Problem Statement

Problem statement for an infinite plate containing multiple circular

inclusions subject to an incident flexural wave

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The integral representation for the plate problem

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Kernel function

The kernel function is the fundamental solution which satisfies

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The slope, moment and effective shear operators

slope

moment

effective shear

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Kernel functions

In the polar coordinate of

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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

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Oxr

f

Degenerate kernel (separate form)

iUeU

f

r

x

q

s

R

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Fourier series expansions of boundary data

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Boundary contour integration in the adaptive observer system

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Adaptive observer system

Source pointSource point

Collocation pointCollocation point

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Vector decomposition

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Transformation of tensor components

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Linear system

where H denotes the number of circular boundaries

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Techniques for solving scattering problems

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The systems for surrounding plate and each inclusion

The continuity conditions

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Dynamic moment concentration factor

Scattered far field amplitude

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Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Introduction

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Case 1: An infinite plate with one inclusion

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

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

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

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Far field backscattering amplitude versus the dimensionless wave number

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Case 2: An infinite plate with two inclusions

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

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

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

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Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Introduction

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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.

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Thanks for your kind attention

The End