Thermo-elastic Stresses in a Functional Graded Material under Thermal Loading,Pure Bending and Thermo-mechanical Coup ling

Pengcheng Ni

1　Introduction

Functionally graded materials(FGMs),as one kind of the functionally materials,are formed of two or more constituent phases with a continuously variable composition.FGMs shows many excellent merits in engineering applications such as the reduction of the in-plane stresses and stress in tensity factors of a composite,and improved thermal properties,residual stress and the fracture toughness[Aboudi,Pindera,and Arnold(1994);Chakraborty,Gopalakrishnan and Reddy(2003);Sarkar,Datta and Nicholson(1997);Zimmerman and Lutz(1999)].Due to this interest,a number of researches on both theoretical and experimental work dealing with various aspects of FGM have been published in recent years including mechanics,manufacturing,applications and the thermal properties[Bishay and Atluri(2013);Birman and Byrd(2007);Dong,El-Gizawy,Juhany and Atluri(2014);Dong,El-Gizawy,Juhany and Atluri(2014);El-Hadek and Tippur(2003);Huang,Yao and Wang(2011);Liu,Dui and Yang(2013);Sun,Hong and Yuan(2014);Wu,He and Li(2002);Yang,Gao and Chen(2010)].

The designation and fabrication of the FGMs to achieve unique microstructures have already been launched[Miyamoto,Kaysser,Rabin,Kawasaki and Ford(1999)],and many analytical investigations have been conducted on the behaviors of FGM materials under thermal or mechanical loadings.As for the research on FGM beams,Librescu et al.[Librescu,Oh and Song(2005)]studied the behavior of thin walled beams made of FGM operating at high temperatures,in which the vibration and instability analysis along with the effects of volume fraction and temperature gradients were considered.The averaging technique of composite was utilized in[Liu,Dui and Yang(2013)],the effective stresses of each phase in a FGM beam under thermal loadings were calculated to judge the plasticity of the whole system,and the stress distributions of the whole material were obtained.[Yang,Chen,Xiang and Jia(2008)]studied both the free and forced vibrations of an FGM beam with variable thickness under thermally induced initial stresses based on the Timoshenko beam theory.Based on the nonlinear first-order shear deformation beam theory and the physical neutral surface concept,both the static and dynamic behaviors of FGM beams subjected to uniform in-plane thermal loading were derived by[Ma and Lee(2011)].The thermo-elastic stresses in a three layered composite beam system with FGM layer were researched by[Nirmala,Upadhyay,Prucz and Lyons(2006)].As for the research on the FMG plates,[Zhang and Zhou(2008)]presented a theoretical analysis to the FGM thin plates based on the physical neutral surface,in which the classical nonlinear lam inated plate theory and the concept of physical neutral surface were employed to formulate the basic equations of the FGM thin plate.The elastic analysis for a thick cylinder made of FGMs was carried out by[Chen and Lin(2008)],and stress distributions along the radial direction were studied.The analytical solutions for the rotation problem of an inhomogeneous hollow cylinder with variable thickness under plane strain assumption was developed by[Zenkour(2010)],and for different types of the hollow cylinders,the analytical solutions of the elastic properties are given.Numerical investigations have also been carried out to study the thermo-elastoplastic behaviors of metal-ceramic FGMs by utilizing the method of finite element(FE)techniques[Giannakopoulos,Suresh,Finot and Olsson(1995);Finot and Suresh(1996);Weissenbek,Pettermann and Suresh(1997)].

It is worth mention that there is seldom work focus on the analytical solutions of the FGM plates subjected to the thermo-mechanical coupling.It has been reported by[Ekhlakov,Khay,Zhang,Sladek,Sladek and Gao(2012)]that the research on thermo-mechanical behaviors of FGMs subjected to thermo-mechanical coupling is important.Therefore,there is a strong need for an accurate analytical formulation to predict the thermo-mechanical behaviors of FGMs under different loadings such as thermal loading,pure bending and thermo-mechanical coupling.In order to get the theoretical solution,the averaging technique of composites are used to demonstrate the thermo-elastic behaviors of a three-layered FGM system under thermal loading,pure bending and thermo-mechanical coupling,respectively.When the temperature gradient and the bending moment vanish,the results can be degenerated to Liu’s work[Liu,Dui and Yang(2013)].Besides,this is one of the two parts work,this paper is focus on the thermo-elastic behaviors of the FGM system under thermal mechanical coupling,and the plastic work will be show in the next job.

2　Theoretical solutions for the FGMs under thermal loading

The coordinate axes and dimensions of the three-layered plate system can be illustrated in Fig.1.The upper layer is the metallic phase,the lower layer is the ceramic layer and the middle layer of the system is the FGMs.Here we assumed that the interfaces of the FG-layers atz=-aandz=+aare continuous and be perfectly bonded at all times.

Figure 1:(a)Three-layered structural model and(b)coordinate axes and dimensions of the three-layered plate system.

In order to get the theoretical solution,a power functionV(z)which represents the volume fraction of the metallic phase is assumed to be the compositional gradation function of the FGM layer with a parameter‘m’.Hence,for a simple example,the following function ofV(z)can be considered as

And for different values of the parameter‘m’,different graded distributions can be obtained.According to(1),for the boundary condition at the layers’interfaces,exist that,

According to the small strain kinematics,the total strain of the thermo-elastic phase of the system can be composed of the elastic strain,,and the thermal component,,as

For the three layers system,under equal biaxial stress condition,exist that

Then the stress tensor and the strain tensor can be described as

When the total strain of the system can be considered as a function of the out-of plane coordinate,z,here we can define thatε0tis the strain at the m id-plane of the FGM layer atz=0,and it can be shown that the small strain compatibility equations lead to a linear relation between the total strain and curvature(K0t),and the subscript‘0t’represent the result of the system under thermal loading.

Under plane stress conditions,the only non-zero stress componentσ(z)of the system can be given by

whereE(z),a(z)andυrepresent the different Young’s modulus,the coefficient of thermal expansion and the Poisson’s ratio through the thickness of the threelayered system,respectively.Assumed that the number 1 represents the ceramic phase and the number 2 represents the metallic phase.Then for the ceramic phase,exist thatE(z)=E1,a(z)=a1,υ=υ1,and for the metallic phase,exist thatE(z)=E2,a(z)=a2,υ=υ2.In order to get the analytical solutions,here we assumed thatυ1=υ2.Assume that?T(z)=T0-T(z)represents temperature distributions through the thickness of the system,and if we assume a steady state distribution of temperature,T(z)satis fies the difference principle,and assumed that the different temperatures in the multilayer system as

whereThe expressions forε0tandK0tcan be derived by the boundary condition

Leads

Substitute function(9)into(11),the follow ing expressions can be obtained by

Results to

where

where

And substituted(15-17)into(14),results can be got as

For any application of the FGM system,the stress distribution of this model with different gradation parameter“m”can be obtained by using the general expressions of(9),(13)and(18)with the given material parameters.When the temperature distributions through the thickness of the system are assumed to be homogeneous,the work can be degenerate to Liu’s work(2013).

3　Theoretical solutions for the FGMs under pure bending

This part is focus on the analytical solutions for the three layered system under pure bending.Assumed that the system is under isothermal environment with no thermal loading(?T(z)=0).According to(6),the stress componentσ(z)can be expressed by

whereε0b,K0bhave the same physical significance as(6),while the subscript‘0b’represent the corresponding values of the system under pure bending.Then in the similar way,the stress distribution of the system under pure bending can be expressed by

Due to the pure bending,the externally applied force is zero while the bending moment is not,then the expressions forε0bandK0bcan be derived by

which lead to

Substituted Eq.(20)into Eq.(22),

where

and whereH(z)can be expressed as

And substituted(25)into(24),the solutions can be got as

As a simple example for cases of different variations ofV(z)with different‘m’,results of stress distributions of the system under pure bending can be obtained by(20,23,26)with the given material parameters.

4　Theoretical solutions for the FGMs under thermo-mechanical coup ling

For the case of the system under thermo-mechanical coupling,with the different min-plane strainε0cand the laminate curvatureK0c,the similar non-zero stress componentσ(z)can be given by

The subscript‘0c’represents the corresponding values of the system under pure bending.The homogeneous temperature distribution in the multilayer system is the same as Eq.(8).Then the stress distribution of the system under varying temperature and the bending loads can be expressed as

Because the bending load is not zero,so the expressions forε0candK0ccan be derived by

which lead to

Substitute Eq.(28)into Eq.(30),these can be integrated to produce

where

and

And substituted(33-35)into(26),the solutions can be got as

Then for cases of different variations ofV(z)with different‘m’,results of stress distributions for the system under thermo-mechanical coupling can be obtained by(28,31,36)with the given material parameters.

5　Numerical results

In order to get the thermo-mechanical behaviors of the system,as a simple example,the system can be treated as a FGM thin plate made up of Ni-FGM-AL2O3layers,the upper layer is the isotropic elastic Ni,the lower layer is the isotropic AL2O3.In order to get the theoretical solutions,the FGM layer is assumed to be in the FGM layer with linear and quadratic and square variation ofV(z)by substitutingm=1,2 and 0.5 into(18,26,36),respectively.For different cases of the system under thermal loading,pure bending and thermo-mechanical coupling,numerical results can be obtained by(9),(13),(18),(20),(23),(28),and(28),(31),(36).Here we assumed that all layers are isotropic elastic material,free of damage and having the temperature independent properties in Table 1[Giannakopoulos,Suresh,Finot and Olsson(1995);Weissenbek,Pettermann and Suresh(1997)].

Table 1:Properties for the metallic(Ni)and ceramic(AL2O3)phases[Giannakopoulos,Suresh,Finot and Olsson(1995);Weissenbek,Pettermann and Suresh(1997)].

Figure 2:Analytical thermo-elastic stress distributions of the system with m=0.5,1,2 at

Fig.2 shows the stress distributions through the thickness of the three-layered system under the pure thermal loading.The initial temperature is isotherm during the whole system with T0=T1=200°C,and the finial state is T1=60°Cand T0=200°C.The temperature gradient between the upper side and the lower side of the system can be determined by the different composition pro files withm=1,2 and 0.5,respectively.As seen in Fig.2,the case withm=2 shows a higher value of the stress in the Ni and FGM layer and a lower value of the stress in the AL2O3layer than the other cases.So one can choose a smaller value of‘m’to lower the thermo-elastic stress of the FGM and metal layers,and choose a higher value of‘m’to lower the thermo-elastic stress of the AL2O3layer.The results withm=1 from T1=180°Cto T1=60°Ccan be shown in Fig.3.As seen in Fig.3,with the increasing thermal loadings,the stress distribution of the system shows a higher value.

Figure 3:Analytical thermo-elastic stress distributions of the system with m=1 under different T1.

Fig.4 shows the stress distributions through the thickness of the three-layered system under different bending moment at uniform temperature circumstance withm=1.As seen in Fig.4,the stress shows a higher value with the increasing bending moment.The stress distributions through the thickness of the three-layered system under pure bending withM=300 for the different composition pro files atm=1,2 and 0.5 can be shown in Fig.5.It shows that there is little effect by the gradient function on the stress distribution of the FGM system under pure bending.

Figure 4:Analytical thermo-elastic stress distributions of the system with m=1 under different M.

Figure 5:Analytical thermo-elastic stress distributions of the system with m=0.5,1,2 under M=300N·M.

Figure6:Analytical thermo-elastic stress distributions of the system under different bending moment with m=1 at T0=200°C,T1=60°C.

Figure 7:Analytical thermo-elastic stress distributions of the system with m=0.5,1,2 under M=1000N·M at T0=200°C,T1=60°C.

Fig.6 shows the stress distributions through the thickness of the system under different bending moments at T0=200°Cand T1=60°C.As seen in Fig.6,there is a higher stress with the increasing bending moment.The stress distributions through the thickness of the three-layered system with constant bending momentM=1000 and at T0=200°Cand T1=60°Cfor the different composition pro files withm=1,2 and 0.5 can be shown in Fig.7.Results of the system under the same bending momentM=1000 withm=1,T0=200°Cand different T1can be shown in Fig.8.When the temperature gradient and the bending moment vanish,this model can be degenerated to Liu’s model,and the stress distribution for the system withm=1 can be obtained in Fig.9.The dot curve is Liu’s work and the solid one is the present result.As shown in Fig.9,they agree very well with each other.

Figure 8:Analytical thermo-elastic stress distributions of the system with m=1,M=1000 N·Munder different temperature distribution T1.

6　Conclusions

The analytical solutions on the thermo-elastic stress solutions for FGMs under thermal loading,pure bending and thermo-mechanical coupling are studied in this work,respectively.The proposed relations for the stress distributions within a generic metal-FGM-ceramic system can predict accurately complex stress distributions induced by thermal loading,pure bending and thermo-mechanical coupling,respectively.By choosing the different appropriate FGM compositional gradation with linear,quadratic and square variations,the stress distribution within the sys-tem can be controlled so that undesirable stresses at critical locations are minimized or avoided.The analytical thermo-elastic solutions presented here may be accounted for in many potential FGM composites design and can provide a simple,yet accurate tool for the prediction of thermally induced stresses in an FGM layer sandwiched between two homogeneous materials.

Figure 9:Analytical thermo-elastic stress distributions of the system with m=0.5,1,2 at?T=200°C.

Acknowledgement:This work is supported by National Natural Science Foundation of China and Civil Aviation Administration of China jointly funded project(Grant#U1233106),the Fundamental Research Funds for the Central Universities funded project of Civil Aviation University of China(Grant#3122014C015).

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