M M S S V V 0 Scattering of flexural wave in thin plate with multiple holes by using the null-field integral equation method Wei-Ming Lee 1, Jeng-Tzong Chen 2 Ching-Lun Chien 1, Yung-Cheng Wang 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 時 間 : 2008 年 05 月 24 日 地 點 : 台北科技大學

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 Intrduction Circular holes can reduce the weight of the whole structure or to increase the range of inspection. Geometric discontinuities result in the stress concentration, which reduce the load carrying capacity. The deformation and corresponding stresses produced by the dynamic force are propagated through the structure in the form of waves.

M M S S V V 4 Scattering At the irregular interface of different media, stress wave reflects in all directions scattering The scattering of the stress wave results in the dynamic stress concentration

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 hole. 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 Our objective is to develop a semi-analytical approach to solve the scattering problem of flexural waves and dynamic moment concentration factors in an infinite thin plate with multiple circular holes by using the null-field integral formulation in conjunction with 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 ω is the angle frequency ρ is the surface density D is the flexural rigidity h is the plates thickness E is the Young’s modulus ν is the Poisson ratio u(x)u(x)

M M S S V V 9 Problem Statement Problem statement for an infinite plate with multiple circular holes 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 x s O x Expansion Degenerate kernel (separate form) Fourier series expansions of boundary data

M M S S V V 16 Boundary contour integration in the adaptive observer system

M M S S V V 17 Vector decomposition

M M S S V V 18 Transformation of tensor components

M M S S V V 19 Linear system where H denotes the number of circular boundaries

M M S S V V 20

M M S S V V 21 Techniques for solving scattering problems

M M S S V V 22 Outlines 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Introduction

M M S S V V 23 Case 1: An infinite plate with one hole Geometric data: a =1m thickness=0.002m Boundary condition: Inner edge : free

M M S S V V 24

M M S S V V 25 Distribution of DMCF on the circular boundary by using different methods, the present method, analytical solution and FEM

M M S S V V 26

M M S S V V 27

M M S S V V 28 Case 2: An infinite plate with two holes

M M S S V V 29

M M S S V V 30 Distribution of DMCF on the circular boundary by using different methods, the present method and FEM

M M S S V V 31

M M S S V V 32

M M S S V V 33 Outlines 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Introduction

M M S S V V 34 Concluding remarks A semi-analytical approach to solve the scattering problem of flexural waves and to determine DMCF in an infinite thin plate with multiple circular holes was proposed The present method used the null BIEs in conjugation with the degenerate kernels, and the Fourier series in the adaptive observer system. The improper integrals in the direct BIEs were avoided by employing the degenerate kernels and were easily calculated through the series sum. The DMCFs have been solved by using the present method in comparison with the available exact solutions and FEM results using ABAQUS Numerical results show that the closer the central distance is, the larger the DMCF is.

M M S S V V 35 Thanks for your kind attention The End