1 Introduction

As typical structures,plates are widely used in aerospace,civil and construction engineering,and mechanical engineering. It is inevitable to make holes in the plates to meet the engineering designs. The holes will reduce the loading capacity and life time of the structures. Therefore,many experts and scholars studied the problems on the static and dynamic stress concentration around the holes ^{[1, 2, 3, 4, 5]} . Elastic wave methods can be used to describe and simulate the stress-strain states produced by a variety of dynamic loads in solid media or structures ^{[6, 7]}. The elastic wave propagation,scattering and dynamic stress concentration,and localization of vibration in the plates with holes are important frontier problems in the realm of mechanics. The investigations on these problems can promote the innovation and development of classical structural dynamics and their solving methods.

As far back as the 19th century,the classical theory of thin plates has been proposed by Lagrange-German ^[8]. There is a limitation for the classical theory when it is used to analyze the elastodynamics problems under free boundary conditions^[6]. In the middle of the 20th century,Reissner^[9] put forward the equation of thick plates,in which the effects of transverse shear deformations were accounted. Mindlin ^[10] made a systematic investigation on the dynamics of thick plates,in which two rotations were considered as the field variables and the transverse displacement and the rotary inertia and shear effects were included. Theoretical analyses and experimental tests indicate that the applications of the two plate theories give rise to many limitations,especially on dynamic mechanics problems ^{[8, 11]}. It is necessary to search new deduced methods and derive modern refined dynamic theories of plates so as to better describe the dynamic stress strain progress,determine the dynamic response of structures accurately,and realize the light-weight trend of structure designs in the context of meeting the requirement of the structural strength. Hu et al. ^{[12, 13, 14, 15]} presented the refined dynamic equations of plate bending by using the general solution of Boussinesq-Galerkin and the operator theory of partial differential equations. Since the derivation of dynamic equations was conducted without any assumptions,the proposed equation of plate bending was exact,and it could be used to analyze high frequency vibration of plate bending and to evaluate the applicable condition of the engineering plate theory.

In this paper,based on the refined dynamic equation of plate bending^{[9, 10]},using Liu’s complex variable method^[4],the elastic wave scattering and dynamic stress concentrations in the plates with two holes are studied. As examples,the dynamic stress concentration factors in the plates with two holes are computed and analyzed with different parameters. 2 Wave motion equation of bending plate and its solution

The wave motion equation of the bending plate obtained by the refined theory^[12] is

where W,F,and f are the generalized displacement functions of plate bending vibration,ν and ρ are Poisson’s ratio and the density,respectively,h is the thickness of the plate,and

c₁ and c₂ are the compression wave velocity and the shear wave velocity,respectively,which are expressed by

where λ and µ are the Lame constants. C and D are the shear rigidity and the flexural rigidity,respectively,which are defined by

∇² is the Laplace operator,i.e.,

Without loss of generality,the vibration harmonic solution of the problem is studied. Set

where ω is the angular frequency of the plate bending,and i is the imaginary unit.

In the following analysis,the time factor and the symbol “∼” in the generalized displacement functions are omitted. Substituting Eq. (2) into Eq. (1) yields

where α_j (j= 1,2) are wave numbers satisfying

and

Based on the refined theory of plate bending,the expression of the generalized forces in the plates is

where δ_j (j= 1,2) are proportionality coefficients of the displacement potential function

The general solutions of the scattering wave described by the vibration equation (3) can be written as follows:

where H_n⁽¹⁾(·) denotes the Hankel function. K_n (·) denotes the Bessel function of the imaginary argument. A_nm (m = 1,2) and Bn are mode coefficients of the scattered wave,which can be determined by the scatter boundary conditions.

The generalized forces of plate bending vibration can be expressed as follows:

Use complex functions,and introduce the complex variables

Then,Eq.(6) can be converted into where

Employing the conformal mapping method,the exterior region of the noncircular hole boundary in the z-plane can be mapped into a unit circle in the η-plane by the mapping function z = Ω(η). The conformal mapping function can be taken as

where Φ(η) is a holomorphic function.

In the polar coordinates (r,β),Eq. (7) becomes

In the η-plane,Eq.(9) can be written as follows: 3 Excitation of incident waves and total wave fields

When we investigate a plate with two holes,we consider a flexural wave propagating along the positive x-axis. Based on the constructive interference theory,the incident wave can be expressed as

The total wave field of the flexural wave of plates can be expressed as

4 Determination of mode coefficients satisfying boundary conditions

The case with the free boundary condition is investigated. In the η-plane,there are six boundary conditions,i.e.,

Substituting Eqs. (9) and (12) into Eq. (13) yields the infinite linear algebraic aligns as follows:

which can be used to determine the six mode coefficients A_n1^m,A_n2^m,and B_n^m (m = 1,2).

Multiplying Eq. (14) by e ^-isθ and integrating between -π and -π,we can get the expression

where

The elements of E_n and E_s are shown in Appendix A.

The dynamic stress concentration is defined as the ratio of the stress due to the total wave at a point to the stress due to the incident wave (without scatter) at the same point ^[3]. In terms of the definition,the dynamic stress concentration factor at the edges of the holes can be described as

where

Therefore,for the plate with two holes,the dynamic moment concentration around the mth hole can be expressed by

5 Numerical examples

The aforementioned analysis can be used to compute the dynamic stress concentrations in the plates with two holes. We put forward the program to calculate the dynamic stress concentration factor in the plate with two holes. For the numerical calculation,set ν= 0.3,ka ∈ [0.1,5.0],and n = 15.

The numerical results of the dynamic moment concentration factors for different plates with two holes obtained by the Mindlin theory and the refined dynamic equation are depicted in Figs. 1-8,which illustrate the angular distributions of the dynamic moment around the holes with different wave numbers,hole-spacings,and plate thicknesses when = 0. In each figure,the upper half illustrates the dynamic stress distributions around the holes obtained by the Mindlin theory,while the lower half illustrates the dynamic stress distributions obtained by the refined dynamic equation. Figures 9-11 show the dynamic moment concentration factors versus the dimensionless hole-spacing L/a.

6 Discussion and conclusions

From Figs. 1-8,we can see that when the wave number is small and the plate is thin,the numerical results obtained by the refined theory are close to those obtained by the Mindlin theory; when the wave number is large and the plate is thick,the numerical results obtained by the refined theory seam quite different from those obtained by the Mindlin theory. Especially,when the ratio of radius to thickness a/h is 0.1,the dynamic moment factor obtained by the refined equation may approach the maximum value,which is over 10% compared with that obtained by the Mindlin theory. It shows good agreement with the conclusion verified by many dynamicists that the Mindlin plate theory is only useful for medium-thick plate and lower frequency incident wave.

If the ratio of the dynamic stress concentration factor for two holes to that for single hole ^{[14, 15]} is denoted by A,when |A−1|≤2%,the influence between holes can be ignored. From the analysis of the computed results,we can obtain that: when the incidence wave number is small,the minimum spacing to ensure no influence between holes is small; when the incidence wave number is big,the mutual influence between holes is intensified,and the minimum spacing to ensure no influence between holes is large. When ka = 0.1 and a/h = 0.1,L/a = 4.5. When ka = 0.1 and a/h = 1.0,L/a = 5.0. When ka = 0.1 and a/h = 5.0,L/a = 6.0. When ka = 5.0 and a/h = 0.1,L/a = 36. When ka = 0.1 and a/h = 1.0,L/a = 43. When ka = 0.1 and a/h = 5.0,L/a = 52. In other words,the radius-to-thickness ratio a/h has some effects on the minimum spacing to ensure no influence between holes. When the radius-to-thickness is larger,the minimum spacing is a little smaller.

In this paper,based on the refined dynamic equation of plate bending ^[9] ,the elastic wave scattering and dynamic stress concentration in the plates with two holes are investigated with Liu’s complex variable method and the conformal mapping method. According to the analysis of the above numerical results,we can conclude that the parameters such as the incident wave number,the thickness of plates,and the hole-spacing have great effects on the dynamic stress distributions. The detail effects are as follows:

(i) When the plate has a small thickness and under a lower frequency,the numerical results obtained by the refined theory approach those obtained by the Mindlin theory. When the plate has a great thickness and under a higher frequency,the numerical results obtained by the Mindlin theory and the refined theory are different.

(ii) Compared with the situation of single hole,the variations of the dynamic stress concentration factors for two holes are complex due to the mutual influence between the two holes. In most cases,the dynamic stress concentrations are intensified,while sometimes are relieved.

(iii) When the incident waves are at a low frequency,the minimum hole-spacing to ensure no mutual influence between holes is small. When the incident waves are at a high frequency,the minimum hole-spacing to ensure no mutual effect between holes is big.

According to the study in Ref. ^[12],the refined equation can be derived without any engineering hypotheses. Therefore,the obtained results in this paper are more accurate,which can be used in the situation of thick-walled structures and high-frequency vibration. Furthermore,the numerical simulation on two circular holes is a typical example. If a proper mapping function,i.e.,Ω(ζ ),is given,the application range of this method to solve the problem on the scattering of elastic waves can be extended to arbitrarily shaped holes,which provides a unified and standardized method for analyzing the dynamic stress concentrations around the holes. The theory and numerical results in this paper can be used for the dynamic analysis and strength design of engineering thick-walled structures,the precision design of structures,and the light weight design of structures. Appendix A

The elements of vectors E_n and E_s are as follows: