A Variational Method for the Homogenous State-vector Equation of Thermoelastic Bodies

LIU Yan-hong,SHEN Guang-shuai,WANG Yan-li

(Aeronautical Engineering College,Civil Aviation University of China,Tianjin 300300,China)

1 Introduction

In recent years,the state space approach of the state-vector equations for the elastic bodies has attracted the attention of many researchers because it is very powerful in analyzing problem related to laminated structures[1-3].Chandrashekara and Santhosh[4]applied the method to obtain natural frequencies of vibration for two-layered cross-ply laminates with varying degrees of orthotropic layers,as well as for thin to very thick plates.Fan and Ye[5]derived the state-vector equations of an orthotropic body by considering three-dimensional elasticity without any initial assumptions.Khdeir et al[6]investigated the static response of cross-ply laminated shallow shells subjected to thermal loadings by the state space approach.Wang et al[7]used state space formulation to study the stress decay in laminates due to edge boundary effects.Lee and Jiang[9]and Chen et al[10]employed the method to analyze the bending of rectangular piezoelectric plates.Ding et al[11],by introducing two displacement functions and two stress functions,established two independent state-vector equations from the three-dimensional piezoelasticity equations for transverse isotropy.Xu et al[12]further presented a general non-axsymmetric exact analysis of the statics of a laminated piezoelectric hollow sphere by introducing three displacement functions and two stress functions.Deü and Ayech[13]proposed a parametric analysis for the free-vibration of simply-supported laminates with embedded transverse shear mode piezoceramic layers by three-dimensional state space solution.Xu et al[14]derived the generalized state-vector equation through governing equations of thermoelectroelastic material and studied the static response and the sensitivity coefficients of multilayered plates.Just as we can use the strategy of generalized state-vector formulations to deal with the problems of the hybrid laminates which include piezoelectric layers,many references,which employed similar method to analyze various problems of the hybrid laminates embodying magnetoelectroelastic layers,can be found in different forms of publication[15-18].

For purpose of dealing with complex plates/shells laminated structures,Sheng and Ye[19]applied a mixed variational principle that includes the variations of both displacements and stresses to establish finite element approximation of state space method for laminated composite plates.Zou[20]and Qing et al[21-25]employed a united modified mixed Hellinger-Reissner variational principle to deduce the state-vector equations of general composite,piezoelectric and magnetoelectroelastic materials and established various 4-node isoparametric element formulations of state-vector equations for laminates problem.

One of advantages using the state space approach to handle laminates is that anisotropic layered materials can be handled,the varying material and geometric properties along the independent spatial variable are allowed[3,26];another outstanding advantage is that,because of the transfer matrix method being employed,the order of the final algebra equation system is independent of the number of layers of a structure and the solution also provides a continuous transverse stresses field across the thickness of multi-layered structures.

Recently,many references[27-30]presented the state space approach to study the steady state temperature problems of hybrid laminated structures by combining the non-homogeneous state-vector equation and second order differential equation derived from the basic control equations of piezothermoelastic material.

In this paper,a variational method is used to deduce the homogeneous state-vector equation of thermoelastic problem at length and the exact solution for the steady state thermal problems of composite laminated shell with simply-supported boundary conditions is investigated.

2 Basic formulations

For the steady state temperature problem of a general thermoelastic body,the basic governing equations can be divided into three groups.

Constitutive relationship is

where σ is the stress column vector(components σ11,σ22,σ33,σ23,σ13and σ12);ε is the strain column vector(components ε11,ε22,ε33,2ε23,2ε13and 2ε12),λ is stress-temperature coefficient column vector(components λii,i=1,2,3);c is the stiffness coefficient matrix(components cij=cji,i,j=1,2,3,4,5,6);T is temperature change.

Gradient relationships

in which,piis heat flux,kiiis thermal conductivity coefficient.

Equilibrium equations

3 Non-homogeneous state-vector equation

In many references[27-30]the similar non-homogeneous equation(4)is presented by the basic governing equations for piezothermoelasticity.If to consider the problem in cylindricalcoordinate system(shown as Fig.1.),the non-homogeneous state-vector equation of thermoelastic materials can be expressed as follows:

Fig.1 An open cylindrical laminated shell

where I is the 3×3 identity matrix,P and Q are symplectic column vectors[31],

Φ11,Φ21and Φ22are the constant coefficients relative to material parameters.The in-plane stresses vector P2=［σxxσθθσxθ］Tcan be evaluated by the following expression:

As the steady state temperature problem of structures is concerned,the solution of Eq.(4)is related to the thermal gradient relationship pi=-kiiT,iand the thermal equilibrium equation pi,i=0.Hence,the following second order differential equation are needed

4 Homogeneous state-vector equation

Assuming the material is isotropic or orthotropic,the constitutive relation of thermoelastic materials can be expressed as follows,which is the matrix form of Eq.(1)

Considering Eqs.(7),(2b)and(3b),one can establish a new constitutive relationships of thermoelastic materials

Exchange the rows and columns of Eq.(8),we have

hence,

Express the generalized strain-displacement relation as vector forms

Hence,based on the method of references[20-26],a new variational principle for the thermoelastic bodies can be expressed as follows:

Performing the operation of variation to Eq.(12),we can obtain a homogenous state-vector equation

Compare Eqs.(13)and(4),the order of Eq.(13)is higher than that of Eq.(4).But appar-ently the solution program of the homogeneous equation(13)is much simpler than that of Eq.(4)since Eq.(13)needs to consider neither convolution operation nor Eqs.(14)and(15).Hence,Eq.(13)can be employed to analyze the steady state temperature problem of closed shells and open shells of thermoelastic material independently.

5 Exact solution of the simply supported laminated shells

The boundary conditions of a simply supported laminated cylindrical shell(Fig.1)are

The state variables,which exactly satisfy Eq.(23),can be expressed as follows:

where η=mπ/a,ζ=nπ/b.

Substituting Eq.(15)into Eqs.(13)and(4),two systems of ordinary differential equations of a layer can be written as the following matrix form

The general solution to Eq.(16a)for a layer of a n-layered shell is

where riand roare the inside and outside radiuses of a layer,respectively;d=ro-ri,R（ri）=is the initial value.

It must be mentioned that the value of the variable T in Eq.(16a)will be obtained by solving Eq.(6).

The general solution to Eq.(16b)for a layer is

In this paper,the precise integration method[31]is used to evaluateIn K of a layer,r=ri+d/2.

At the interface between layers,the compatibility conditions can be defined as follows:

Therefore,one can obtain the following recurrence formulations:

Example 1

Material properties:Stiffness coefficientsGPa, GPa;Thermal conductivity coefficientsStress-temperature coefficientsGeometry parameters(the denotation is shown in Fig.1)d=0.005

Load case:Outer Surface of Shelltemperature change

Solutions to non-homogeneous and homogenous equations are listed in Tab.1 and Tab.2.The error formulation isRn and Rh denote result of the non-homogenous and homogenous equations,respectively.In tables 1 and 2,the dimensionless quantities are defined as:

Tab.1 Comparison of displacements and heat flux pr

Tab.2 Comparison of generalized displacements and out-plane stresses of neutral plane

Example 2

Consider a three-layered laminated shell[0/90/0](Fig.1),the material of each layer is the same as Example 1.Geometry parameters thickness of three layer H=0.01 m,top layer ht=0.3H,middle layer hm=0.4H,bottom layer hb=0.3H,a=100×H,r=1 m,b=π/4.Load case:Outer surface of shell σrr=100sin（ξx）sin（ηy）,temperature change T=1/1 000sin（ξx）sin（ηy）,m=n=1.

It is obvious that if we use homogenous equation approach to analyze this example,the problems of variables along thickness direction become very easy.

The solutions are shown in Figs.2-13.

Some of phenomena can be found in Figs.2－13,for instance,out-plane stresses σxr,σθr,σrrand heat flux prare continuous and nonlinear along the thickness of shell when the shell subjects to combined loading.However in-plane stresses σxx,σθθ,and heat flux pxare discontinuous between layers.

Fig.2 （0,b,r）through thickness of shell

Fig.4 （a/2,b,r）through thickness of shell

Fig.5 （a/2,b,r）through thickness of shell

Fig.6 （0,b,r）through thickness of shell

Fig.7 （a/2,0,r）through thickness of shell

Fig.8 （a/2,b,r）through thickness of shell

Fig.9 （a/2,b,r）through thickness of shell

Fig.10 （a/2,b,r）through thickness of shell

Fig.11 （a/2,b,r）through thickness of shell

Fig.12 （a/2,b,r）through thickness of shell

Fig.13 （a/2,b,r）through thickness of shell

6 Conclusions

In this paper,one variational method has been used to derive the homogenous state-vector equation of the thermoelastic bodies in the cylindrical coordinate system.The homogeneous state-vector equation method can greatly simplify the solution programs,especially for the static analysis of the thermoelastic structures.The homogenous state-vector equation is a straightforward extension of the state-vector equation.Hence,the present homogenous state-vector equation continues to have all merits using the strategy of state-vector equation to investigate other problems of laminated structures.

On the other hand,the variational method in this paper is propitious to establish a semianalytical solution[20-25]for the steady state temperature problem of singular elastic and smart hybrid lamina with complex boundary conditions and complex geometric shapes.

Acknowledgment

This work was supported by the National Natural Science Foundation of China(Grant No.60979001)and the Major Project of Civil Aviation University of China(Grant No.2011kyE07).

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