M M S S V V 0 Scattering of flexural wave in a thin plate with multiple inclusions by using the null-field integral equation approach Wei-Ming Lee 1, Jeng-Tzong Chen 2, Rui-En Jiang 1 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 University MSVLAB ( 海大河工系 ) Department of Harbor and River Engineering

M M S S V V 1 Outlines 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Introduction

M M S S V V 2 Outlines 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Introduction

M M S S V V 3 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.

M M S S V V 4 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.

M M S S V V 5 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.

M M S S V V 6 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.

M M S S V V 7 Outlines 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Introduction

M M S S V V 8 Flexural wave of plate Governing Equation: is the out-of-plane displacement is the wave number is the biharmonic operator is the domain of the thin plates u(x)u(x) ω is the angular frequency D is the flexural rigidity h is the plates thickness E is the Young’s modulus μ is the Poisson’s ratio ρ is the surface density

M M S S V V 9 Problem Statement Problem statement for an infinite plate containing multiple circular inclusions subject to an incident flexural wave

M M S S V V 10 The integral representation for the plate problem

M M S S V V 11 Kernel function The kernel function is the fundamental solution which satisfies

M M S S V V 12 The slope, moment and effective shear operators slope moment effective shear

M M S S V V 13 Kernel functions In the polar coordinate of

M M S S V V 14 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 normal moment effective shear force

M M S S V V 15 O x Degenerate kernel (separate form) x s

M M S S V V 16 Fourier series expansions of boundary data

M M S S V V 17 Boundary contour integration in the adaptive observer system

M M S S V V 18 Adaptive observer system Source point Collocation point

M M S S V V 19 Vector decomposition

M M S S V V 20 Transformation of tensor components

M M S S V V 21 Linear system where H denotes the number of circular boundaries

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M M S S V V 23 Techniques for solving scattering problems

M M S S V V 24 The systems for surrounding plate and each inclusion The continuity conditions

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M M S S V V 26 Dynamic moment concentration factor Scattered far field amplitude

M M S S V V 27 Outlines 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Introduction

M M S S V V 28 Case 1: An infinite plate with one inclusion Geometric data: a =1m Thickness Plate:0.002m Inclusion: 0.001m

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M M S S V V 30 Distribution of dynamic moment concentration factors by using the present method and FEM

M M S S V V 31 The far field scattering pattern for a flexible inclusion with h 1 =h/2

M M S S V V 32 Far field backscattering amplitude versus the dimensionless wave number

M M S S V V 33 Case 2: An infinite plate with two inclusions

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M M S S V V 35 Distribution of dynamic moment concentration factors by using the present method and FEM( L/a = 2.1)

M M S S V V 36 Far field scattering pattern for two flexible inclusions with h 1 =h/2 and L/a=2.1

M M S S V V 37 Far field scattering pattern for two flexible inclusions with h 1 =h/2 and L/a=10.0

M M S S V V 38 Outlines 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Introduction

M M S S V V 39 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 degenerate 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 The space between two inclusions has different effects on the near field DMCF and the far field scattering pattern.

M M S S V V 40 Thanks for your kind attention The End