Simulation Calculation on Stress of Composite Materials Plate with Elliptical Hole under Compressive Stress

(School of Mechanical Engineering,Zhengzhou University,Zhengzhou 450001,China)

Simulation Calculation on Stress of Composite Materials Plate with Elliptical Hole under Compressive Stress

LI Cheng,TIE Ying,ZHENG Yan-ping

(School of Mechanical Engineering,Zhengzhou University,Zhengzhou 450001,China)

Hole induces the stress concentration on the structure,and decreases the structure’s strength.In this paper,to solve the problem of the composite plate with elliptical hole,the theory of functions of complex variable is used to establish a calculation model.The stress function of complex variable which exactly describes the stress field is obtained by using the method of conformal mapping.The stress analysis on the composite material plate with elliptical hole is carried out,based on the established mathematic model.Aiming at the orthotropic plate with different elliptical hole under compressive stress,the hole-edge stress distribution is discussed.At the same time,the hole-edge stress fields are compared.

stress concentration;orthotropic plate with hole;compressive stress;conformal mapping;analytic calculation;simulation

Biography：LI Cheng(1962-),male,Ph.D.professor of Zhengzhou University.

1 Introduction

Since fibrous composite materials have a lot of merits,including the structure designability,high strength,high rigidity,low denstiy,excellent dynamic performance and fatigue resistance,it has become indispensable to the structure materials in aerospace,aviation,automobile,architecture,chemical industry,naval construction,hydro-structure,biological and Sports&Recreation products,and other fields.At the same time,since this kind of material has significant anisotropic characterics,in its application,a lot of new projects unsolved,such as,the opening problem of structure,are brought forward.Almost any mechanical structures have hole.The shapes of the holes are various,and the functions of the holes are different.However,there is its general property:hole induces break on the continuous structure,it must induce the stress concentration on the structure(Neuber,1985),and lessen the structure’s strength.Therefore,the endurance of structure will be reduced.These holes will become the weak spots on block construction and dangerous point which will induce the structure destoried.And almost all of the accessory are destoried originated from these weak spots and dangerous points.Methods of Series(Lekhnitskii,1968),or approximate solution(Lotfi,et al,2005)were mainly used in former studies to analysize stress distribution problem of orthotropic plate with elliptical or round hole.Other research methods are few.Moreover,there is not enough attention paid to the general effect which different factors have on the hole-edge stress.Generally,this kind of problem is solved with semianalytical approach or finite element method(Yen,Hwu,1991;Ukadgaonker,Rao,2000;Suemasu,et al,2006;Ömer,2005).To these researches,the description of boundary conditions is the main difficulty.The semi-analytical approach is limited in special holeshape.Such as,method of small parameter,which was used to approach the special hole-shape step by step(Engels,et al,2001;Theocaris,1991;Grüber,et al,2007).The limitation of this method is obvious.It can not solve complex boundary condition problem well.Moreover,the calculation results of the finite element method,due to the existence of singular element at hole edge,always have more bigger error.In principle,compared with isotropic material,holeedge stress concentration of anisotropic material has much more effects on material damage.In one side,there are a lot of effect parameters,playing a role.In another side,damage direction and angle must be considered.Different to isotropic material,the maximum stress in anisotropic material is not usually the only one reason which results in the damage.

In order to carry out the accurate and significant analysis of hole-edge stress state of composite materials plate with hole,in this paper,taking the composite materials plate with elliptical hole for example,based on the theory of functions of complex variable,by means of integral equation(Li,2003),the multi-complex variable stress function(Li,2006)is adopted.As the results,the stress analytical solution under accurate boundary condition is obtained.Moreover,the calculation of hole-edge stress state on composite materials plate with elliptical hole under external loading is carried out.Simulation analysis of the effects of external loading change and hole-edge position on hole-edge stress expression equation is also carried out.Through the stress expression equation,the effects of different long axis and short axis of elliptical hole on hole-edge stress concentration factor are analyzed and valuated.

2 Problem of anisotropic plane

In many situations,common engineering composite material structures have some symmetrical characteristic,thus,under these situations,the anisotropic property problem can be transformed into the orthotropic problem to be discussed.To orthotropic plate,generalized Hooke’s law can be simplified as:

To planar gravity problem,stress component is expressed as:

where complex variables z1,z2are introduced:z1=x+s1y and z2=x+s2y(s1=α1+iβ1,s2=α2+iβ2).In formula(2),the partial derivatives of x,y are written as

Thus,stress function also can be written as the form of bi-harmonic function,

Expression of stress function is obtained through integration of the above formula by parts.

Planar stress problem of anisotropic material can be transformed to the problem of searching the stress function which can satisfies the corresponding boundary condition.

Inputting formula(4)to(2),through a series of calculation and simplification,the general expression of hole-edge stress of anisotropic plate can be obtained.

where φ（z1）,（ψ z2）are two analytic functions introduced in order to simplify calculation course.Then,the planar stress problem of anisotropic material can be transformed to the problem of searching these two analytic functions which satisfy the corresponding boundary condition.The following problem is that finding the mapping function which can map the boundary of research hole to an unit circle.Only to search this function can the boundary problem of complex hole be solved.Further,solution of hole-edge stress distribution of anisotropic plate with complex hole can be solved.

According to Riemann theorem,this kind of mapping function exists,however,to each kind of hole shape,the function is unique and the search course is complex.The calculation course is not introduced due to the large calculation content.The results are given directly during the following specific case.

The above analytic course shows that,only finding this specific mapping function may the complex hole-shape boundary problem be solved.Further,the solution of hole-edge stress distribution of anisotropic plate with complex hole can be solved finally.Using the above introduced method,for composite materials plate with different sizes of elliptical hole,under external loading,the analysis of hole-edge stress is carried out.

3 Hole-edge stress distribution calculation of orthotropic plate with elliptical hole under different long axis and short axis

An anisotropic plate with elliptical hole is deformed,under the force distributing at the plate edge and functioning at the middle surface.Supposing the size of hole is much smaller than that of plate,moreover,the hole is not close to plate edge.

Since plate is finite,the effect of external boundary can be neglected so that the problem is simplified.With above introduced method,the function which maps the elliptical hole boundary to an unit circle is found.Through conformal mapping,mapping function of elliptical hole boundary can be obtained.The calculation course is not introduced due to the large calculation content.The results are given directly during the following specific case.

The boundary conditions transformed is

Inputting the obtained mapping function(6)to the boundary condition(7),through a series of calculation and simplification,by using Cauchy integral,the two analytic functions are obtained.So from formula(5),the component of hole-edge stress is obtained.

Through following many specific cases,analysis of orthotropic plate with elliptical hole is carried out.Supposing the given engineering constants are:

With the above introduced method,the accurate analytic solution of hole-edge stress is obtained.All calculations and stress distribution figures are completed in software Mathematica due to its powerful treatment function of the complex function.The expression of stress is not given due to the limitation of the length of this paper.When the ratio of the long and short axis of elliptical hole a/b is respectively 4,3,2,1,0.5,the simulated analytic results of the effects of external loading change and hole-edge positon on hole-edge stress are as following Figs.2-6.

Figs.2-6 describe the situations that the hole-edge stresses change with external loading and hole-edge position,when the geometry sizes of elliptical hole are different.From these figures,it can be seen while external loading increases,the hole-edge stress always decreases.With the adopted point closing to longitudinal axis graduatelly,namely,θ graduatelly is close to π/2,the hole-edge stress also increases,the increasing amplitude of the stress near π/2 is obviously higher than that in other places.It is completely accordant with composite materials characteristic.

When the value of external loading is definite value,to different values of long axis and short axis,according to analytic expression obtained,the corresponding stress field is calculated.In this paper,taking a/b=4,3,2,1,0.5 for example,hole-edge stress distribution respectively is obtained,shown as Figs.7-11.

In Figs.7-11,it can be seen that to composite materials plate with elliptical hole under compressive stress,with the change of geometrical size of elliptical hole,there are significant differences in the hole-edge stress distribution.With the ratio of long axis and short axis of elliptical hole a/b decreasing,the maximum stress increases gradually.Moreover,the region area of compressive stress gradually increases,while the region area of tensile stress gradually decreases.When a/b is 4,the maximum stress is-1.356p.When a/b is 0.5,the maximum stress is-10.679p,there is an obvious increase.The analyses show that the effects of different geometrical size of elliptical hole on stress distribution can not be neglected.So,engineer personnel should give enough attention to this problem in practice.

4 Hole-edge stress field comparison of orthotropic plate with elliptical hole under different ratio of long axis to short axis

In order to directly show above contention,hole-edge stress distributions in hole edge 0→π/2 region,under the above five kinds of geometrical sizes of elliptical hole,are given.

In Fig.12,hole-edge stress distributions of the orthotropic plate with elliptical hole under different elliptical geometrical sizes are compared.Stress distributions that each curve repre-sents are marked in the figure.It can be seen that when the ratio of long axis and short axis of elliptical hole a/b is 0.5,the stress peak value is obviously higher than other two situations(seen Fig.11).It shows that under allowing condition,the reasonable design of geometrical sizes of elliptical hole can reduce stress concentration,lessen destruction of the structure and component due to high local stress.

5 Comparison of isotropic and orthotropic plate with elliptical hole

With stress component expression(5),making E11equal to E22,the stress distribution around hole in orthotropic plate is obtained.The derivation process is the same as above section and omitted.The stress distributions around hole in isotropic and orthotropic plate under the same ratio of long axis to short axis are given in the same figure as follows.

From Fig.13,it can be seen that for two kinds of material plates,near the long axis of ellipse,the stresses are all tensile,the stresses in orthotropic plate are bigger than the isotropic one.When away from the long axis,firstly,the decreasing speed of orthotropic plate is faster than that of isotropic one.Then,the isotropic plate is faster.So near the short axis of ellipse,the stresses are all compressive,the stress value in isotropic plate is bigger than the orthotropic one.

6 Conclusions

In this paper,the boundary condition problem of complex hole is solved by using the method of conformal mapping.The analytic solution of hole-edge stress is put forward based on integral equation.The calculations of composite materials plate with elliptical hole,under different hole sizes conditions,are carried out.The results can be seen in the Figs.1-13.For different ratio of long axis and short axis of elliptical hole a/b,the effects of the hole-edge stress concentration coefficient are compared.The change situations of hole-edge stress peak value with a/b are given quantitatively.

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含椭圆孔的复合材料板在压应力作用下的应力仿真计算

李 成，铁 瑛，郑艳萍
（郑州大学机械工程学院，郑州450001）

孔使原本连续的结构产生突变而不连续，必然会引起结构上的应力集中，降低其强度，孔的影响对于复合材料要比均质材料大得多。文中对于含椭圆孔的复合材料板，运用复变函数理论建立计算模型。采用保角映射的方法得到准确描述应力场的复变应力函数。按照所建立的数学模型对含有椭圆孔的复合材料板进行应力分析。针对在压应力作用下的含不同几个尺寸椭圆孔的正交各向异性板，讨论了其相应的孔边应力分布，并对所研究的各种情况下孔边应力场进行比较。

应力集中；含孔正交各向异性板；压应力；保角映射；解析计算；仿真

TB33 O346

A

李 成（1962-），男，博士，郑州大学机械工程学院教授；

郑艳萍（1975-），女，硕士，郑州大学机械工程学院讲师。

TB33 O346

A

1007-7294（2010）03-0272-08

date：2008-09-08 Revised date:2009-10-21

Supported by the National Natural Science Foundation of China(No.50975261)

铁 瑛（1978-），女，博士，郑州大学机械工程学院讲师；