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Scattering of flexural wave in a thin plate with multiple circular inclusions by using the

multipole Trefftz method

Wei-Ming Lee1, Jeng-Tzong Chen2, Hung-Ho Hsu1

1 Department of Mechanical Engineering, China University of Science and Technology, Taipei, Taiwan

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

2009年 11月 14日國立聯合大學

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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.

In 1932, Trefftz presented the Trefftz method categorized as the boundary-type solution such as the BEM or BIEM which can reduce the dimension of the original problem by one.

The Trefftz formulation is regular and free of calculating improper boundary integrals.

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Objective

The concept of multipole method to solve multiply scattering problems was firstly devised by Zavika in 1913.

Our objective is to develop an analytical approach to solve the scattering problem of flexural waves in an infinite thin plate with multiple circular inclusions by combining the Trefftz method and the multipole method .

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Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Introduction

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For time-harmonic motion, the governing equation of motion for the plate

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

4 is the biharmonic operator

eΩ is the unbounded exterior plate

w(x)

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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The slope, moment and effective shear operators

slope

Normal moment

effective shear

Tangential moment

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Analytical derivations for flexural wave scattered by multiple circular inclusions

The scattered field of plate can be expressed as an infinite sum of multipole

The displacement field of the plate is

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The displacement field of the pth inclusion is expressed as

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The continuity conditions

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For the pth circular displacement continuity

(1) The higher order derivatives (2) Several different variables

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,k k kx

p

p

pΟ

,p p p x

,p p pk k kr r

x

p pk k rx x

( )( )e ( )e ( )p pk kim i m n inm p m n pk n k

n

J J r J e

( )( )e ( )e ( )p pk kim i m n inm p m n pk n k

n

I I r I e

( )

( )

( )e ( ) ,

( )e

( )e ( ) ,

pk k

p

pk k

i m n inm n pk n k k pk

im nm p

i m n inm n pk n k k pk

n

Y r J e r

Y

J r Y e r

( )

( )

( 1) ( )e ( ) ,

( )e

( 1) ( )e ( ) ,

pk k

p

pk k

i m n innm n pk n k k pk

im nm p

i m n inm nm n pk n k k pk

n

K r I e r

K

I r K e r

k

k

kO

Addition Theorem

kpkpr

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The interface displacement continuity

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The interface slope continuity

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The interface bending moment continuity

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The interface shear force continuity

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The analytical model for the flexural scattering of an infinite plate containing multiple circular inclusions

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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:R1=1mThicknessPlate:0.002m

Distribution of DMCF on the circular boundary (ka=0.005, h/h0=0.0005)

1.8514

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Distribution of DMCF on the circular boundary

ka=0.5 ka=1.0 ka=3.0

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Far-field backscattering amplitude

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Case 2: An infinite plate with two inclusions

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Distribution of DMCF on the circular boundary (L/a=2.1)

ka=0.5 ka=1.0 ka=3.0

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Far-field backscattering amplitude (L/a=2.1)

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Distribution of DMCF on the circular boundary (L/a=4.0)

ka=0.5 ka=1.0 ka=3.0

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Far-field backscattering amplitude (L/a=4.0)

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Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Introduction

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Concluding remarks

An analytical approach to solve the scattering problem of flexural waves

and to determine the DMCF and the far field scattering amplitude in an infinite thin plate with multiple circular inclusions was proposed By using the addition theorem, the Trefftz method can be extended to

deal with multiply scattering problems.

The magnitude of DMCF of two inclusions is larger than that of one when the space of inclusions is small .

The proposed algorithm is general and easily applicable to problems with multiple inclusions which are not easily solved by using the traditional analytical method.

1.

2.

3.

4.

5. The effect of the space between inclusions on the near-field DMCF is

different from that on the far-field scattering amplitude.

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Thanks for your kind attention

The End