ZHOU Bo，KANG Zetian，MA Xiao，XUE Shifeng*，YANG Jie

1.College of Pipeline and Civil Engineering，China University of Petroleum（East China），Qingdao 266580，P.R.China；2.School of Engineering，RMIT University，Melbourne 3083，Australia

Abstract: This paper focuses on the thermo-mechanical behaviors of functionally graded（FG）shape memory alloy（SMA）composite beams based on Timoshenko beam theory. The volume fraction of SMA fiber is graded in the thickness of beam according to a power-law function and the equivalent parameters are formulated. The governing differential equations，which can be solved by direct integration，are established by employing the composite laminated plate theory. The influences of FG parameter，ambient temperature and SMA fiber laying angle on the thermo-mechanical behaviors are numerically simulated and discussed under different boundary conditions. Results indicate that the neutral plane does not coincide with the middle plane of the composite beam and the distribution of martensite is asymmetric along the thickness. Both the increments of the functionally graded parameter and ambient temperature make the composite beam become stiffer. However，the influence of the SMA fiber laying angle can be negligent.This work can provide the theoretical basis for the design and application of FG SMA structures.

Key words：shape memory alloy（SMA）；shear deformation；thermal effect；laminated plate theory；functionally graded（FG）beam

0 Introduction

In the last decade，functionally graded（FG）materials have been emerged and widely used to sat?isfy special features in engineering designs. Due to continuously changing material properties，F(xiàn)G ma?terials have striking advantages over traditional ho?mogeneous materials，such as high fracture tough?ness，improved stress distribution，the superior stress relaxation，capabilities of withstanding high temperatures， and large thermal gradients［1-3］.Shape memory alloys（SMAs）are a unique group of materials that have the ability to recover their original shapes from large deformations well beyond their elastic strain limits，owing to its reversible martensitic transformation. The deformation recov?ery may occur spontaneously upon unloading or be delayed until the material is heated to a certain tem?perature. The former behavior is referred to as pseu?do-elasticity and the latter is referred to as the shape memory effect［4-8］.These unique properties have ren?dered the alloys wide use in a range of engineering applications，such as structural design［9］，biomedi?cine［10］and micro electromechanical systems（MEMS）［11］.

To meet the requirement of complex engineer?ing applications，it is desirable to create FG SMAs in the specific direction［12］. FG SMAs would be con?sidered as composites with smooth interfaces［13］that are capable of both suffering from lager recoverable deformation because of the pseudo-elasticity of the SMA phase and exhibiting better controllability of metal phase to guarantee the structural stability.Thanks to the extraordinary characteristics of FG SMAs，many researchers have been motivated to fabricate different kinds of FG SMAs and test their mechanical and transformation behaviors experimen?tally. For instance，F(xiàn)u et al.［14］used the magnetron sputtering system to improve the hardness and tribo?logical properties of NiTi-based SMA film by sur?face treatment. Mahmud et al.［15］created the FG near-equiatomic NiTi SMA by anneal within a tem?perature grade after cold work. Tian et al.［16］fabri?cated the multilayer FG NiTi films by the D. C.sputtering method. Employing the temperature sen?sitivity of phase transition and thermodynamic prop?erties of SMA，several methods for manufacturing FG SMA have been put forward，such as laser irra?diation［17］，temperature gradient-anneal［18］and sur?face laser annealing method［19-20］. Yang et al.［21］ex?perimentally and numerically analyzed the tempera?ture profile of a Ti-45Ni-5Cu（at%）alloy wire gen?erated by the Joule heating treatment. Meng et al.［22-23］created compositionally graded NiTi thin plates by utilizing the surface diffusing method.Shariat et al.［24］presented the fabrication and analyti?cal solutions of FG SMA wires with transformation stress and strain grades. They also made experimen?tal and theoretical investigations on the pseudo-elas?tic response of several geometrically graded NiTi SMA bars/strips at different stages of loading cy?cle［25-26］. Mohri et al.［27-28］created FG NiTi bi-layer thin film structures by RF magnetron sputter deposi?tion incorporated with vacuum arcre-melting and an?nealing technologies，and experimentally investigat?ed their mechanical and phase transformation behav?iors. Liu et al.［29］created the FG SMA cylinder us?ing different heat treatment processes and described its phase transformation behaviors. On the basis of previous work，Shariat et al.［30-31］reported the con?cept，design and experimentation of different types of FG NiTi alloys as well as various fabrication tech?niques. They also experimentally investigated their deformation behaviors under different thermo-me?chanical loads［32］. Khaleghi et al.［33］produced a com?positionally graded high temperature SMA by diffu?sion annealing of palladium into NiTi alloys with dif?ferent chemical compositions and investigated their shape recovery properties on both micro and macro scales experimentally.

The mechanical behaviors of SMA composite structures subjected to complex loads have been de?scribed and predicted through theoretical analysis.For instance，Shariyat et al.［34］studied the eccentric impact behavior of SMA wire reinforced composite plates using composite laminate plate theory. Sama?dpour et al.［35］investigated the nonlinear free vibra?tion behavior of composite plates embedded with SMA wire by the extended polycrystalline microme?chanics method and composite laminate plate theo?ry. Kamarian et al.［36］analyzed the thermal buckling behavior of SMA wire reinforced composite plates and presented the optimal design method. Soltanieh et al.［37］studied the impact behavior of SMA fiber reinforced composite laminates and developed the corresponding numerical calculation program.

The abovementioned studies focus on the SMA composite structures without considering the graded distribution of material parameters，material component and geometrical dimension. With the rapid increase of requirement for FG SMAs in the high and new technology field，it is urgent to com?prehensivly understand the mechanical and transfor?mation properties of FG SMA structures，and to de?scribe their constitutive relations correctly. In recent years，the analytical solution for FG SMA struc?tures has been taken up by many researchers. For examples，Shariat et al.［38］analytically described the deformation behavior of property graded NiTi plates subjected to the thermal-mechanical loads. Liu et al.［39］analytically described the thermal-mechanical behavior of the FG SMA composite subjected to thermal loading. Liu et al.［40］analytically solved the mechanical and phase transformation problems of FG SMA composites with Young’s modulus and the thermal expansion coefficient varying along the thickness direction subjected to the thermo-mechani?cal coupling. Liu et al.［29］created a FG SMA cylin?der and derived the analytical solution to predict its mechanical behaviors. Xue et al.［41］proposed a con?stitutive model to describe the mechanical behaviors of the FG porous SMA and applied the constructed model to the finite element calculation of the FG po?rous SMA cylinder. Asadi et al.［42］investigated the nonlinear thermal instability of FG SMA sandwich plates subjected to a moving speed based on the geo?metrically nonlinear third-order deformation theory.Liu et al.［43］investigated the stress-induced phase transformation in geometrically graded SMA layers.

To the authors’knowledge，no previous study on the bending behaviors of transverse FG SMA composite beams considering the shear deformation and thermal effect has been reported in the open lit?erature，although it is an important issue. There?fore，the main goal of this paper is to investigate the thermo-mechanical behaviors of FG SMA compos?ite Timoshenko beams with volume fraction of SMA fiber varying along the thickness. To accom?plish this，the nonlinear transformation evolution law of SMA is simplified to be linear，and the equiv?alent parameters of the FG SMA composite beam is obtained by the extended multi-cell micromechanics approach. The governing differential equations are established by the composite laminate plate theory.For further investigation， FG SMA composite beams with simply supported and clamped-free boundary conditions are numerically simulated. The influence of FG parameter，ambient temperature and SMA fiber laying angle on the thermo-mechani?cal behaviors of the FG SMA composite beams are discussed in detail. This work can provide a theoreti?cal guidance and basis for the design and application of smart beam structures in the relevant field.

1 Simplified Constitutive Law of SMA

The constitutive law of the SMA fiber and the SMA-matrix mixture must be derived before deploy?ing the governing equations of the FG SMA com?posite beam.

In order to replace experimental results for SMA behavior，certain constitutive models were de?veloped.［44-45］As the phase variance in SMA are re?lated to the axis of loading，SMAs are utilized pre?dominantly in one-dimensional form. As shown in Fig.1，based on the thermo-mechanical model de?posed by Zhou et al.［45］for SMAs，the correspond?ing pseudo-elastic stress-strain relation with the ther?mal strain can be given as

Fig.1 Stress-strain curves of SMA during the loading pro?cess

where σ is the uniaxial stress；ε the uniaxial strain；η the loading direction coefficient with η = 1 for ten?sile state while η = -1 for compressive state；ξ the martensitic volume fraction；εtrthe maximum trans?formation strain；and ΔT the temperature incre?ment. Based on the rule of mixtures，the Young’s modulus EFand thermal expansion coefficient αFof SMA fiber can be expressed as

where the subscripts“ma”and“au”stand for mar?tensite and austenite phases，respectively.

For the sake of simplicity，the nonlinear phase transformation evolution law of SMA during the loading process is simplified to be linear，and the martensitic volume fraction ξ during the direct trans?formation process can be described as follows

where εmsand σmfare the martensite starting and the finishing strains，respectively. According to Brin?son［44］，the martensite starting stress σmsand the fin?ishing stress σmfduring the direct transformation pro?cess can be expressed as

where σscr，σfcr，T，Msand Cmaare the initial values of the martensite starting and the finishing stresses，the temperature，the martensite starting tempera?ture and the stress influence coefficient of martensite phase，respectively. The equivalent strain can be given as［40］

where εijdenotes the strain components and δijthe Kronecker delta function.

It is assumed that martensite and austenite have the same Poisson’s ratio，namely μma= μau=μF［40］，then the shear modulus of the SMA fiber can be obtained as

where μFdenotes the Poisson’s ratio of the SMA fi?ber.

2 Establishment of Governing Dif?ferential Equations

2.1 Equivalent parameter formulations

Consider a SMA fiber reinforced composite beam in the Cartesian coordinate system as shown in Fig.2（a）. The length，the width and the thick?ness are presented as L，b and h，respectively. The SMA fibers are defined in the material coordinate system 1-2-3 as schematically shown in Fig.2（b）.Furthermore，SMA fibers are located parallel to the material coordinate 1-axis. As shown in Fig.2（b），β is the 1-axis orientation counterclockwise from the x axis，which is named as SMA fiber laying angle in this paper. Assume that the elastic matrix material and SMA fiber are isotropic，then the non-homoge?neous material properties of the FG SMA compos?ite monolayer are obtained by the extended multicell micromechanics approach［42］as follows

Fig.2 Schematic diagram of FG SMA composite beams and SMA fiber reinforced elastic matrix monolayer

where EM，GM，μMand αMstand for the Young’s modulus，the shear modulus，the Poisson’s ratio and the thermal expansion coefficient of the elastic matrix，respectively；fFdenotes the volume fraction of the SMA fiber. The subscript“1”“2”and“3”de?note the fiber direction，the in-plane transverse di?rection of the fiber and the out-of-plane transverse direction of the fiber，respectively［34］.

2.2 Governing differential equations

According to the Timoshenko beam theory，the displacement components can be written as

where u0and w0are the axial and the transverse dis?placements of a point on the middle plane of the beam，respectively；θ is the rotation angle；and z the distance of a point on the transverse section from the middle plane of the beam. Hence，the non-zero strains for the beam can be given by

where the subscript“，x”denotes the differentiation with respect to x.

Based on the composite laminate plate theory，the constitutive relation for the thin layer of the FG SMA composite beam with distance z from the mid?dle plane is considered as［35-36］

where

The stiffness matrix［37］can be expressed as

where

where Q denotes the reduction stiffness matrix，and

where Tzdenotes the coordinate transformation ma?trix.

In this paper，the sum algorithm of the overall stiffness matrix in the traditional laminate plate theo?ry is improved into an integral form. Hence，the stiffness matrix of the FG SMA composite beam is given as

The higher-order term in Eq.（25）is omitted，and it is rewritten as

In view of Eq.（26），substituting Eq.（8）into Eq.（19）and integrating Eq.（19）along the height of the beam cross section，it yields

In view of Eq.（26），substituting Eq.（8）into Eq.（19a）and taking the moment about the neutral plane，it yields

where Nx，Mx，Q stand for the axial resultant force，the resultant moment and the resultant shear force in the FG SMA composite beam section due to the ap?plied force，respectively. The differential Eq.（27）de?scribes the bending deformation of FG SMA Timosh?enko beams.

It should be noted that the governing differen?tial Eq.（27）can be solved by the direct integration method. Plugging Eq.（27a）into Eq.（27c）and inte?grating them with respect to x yields

where c1and c2are unknown constant coefficients，which can be determined in view of the boundary conditions. Substituting Eq.（28）into Eq.（27b）and integrating them with respect to x yields

where c3is an unknown constant coefficient，which can also be determined in view of the boundary con?ditions.

For validating the mentioned method in this work，some numerical simulations on the mechani?cal and transformation behaviors of FG SMA com?posite beams subjected to a concentrated load with simply supported and clamped-free boundary condi?tions are transmitted and delivered in the succeeding section.

3 Numerical Simulations and Anal?ysis

As shown in Fig.3，the FG SMA composite Timoshenko beam with two different boundary con?ditions，such as clamped-free and simply supported boundary conditions，are assumed. The mathemati?cal expression of these types of boundary conditions may be written as follows.

Simply supported boundary condition

and

at the loading position.

Clamped-free boundary condition

Fig.3 Schematic diagram of FG SMA composite subjected to a concentrated load

For the clamped-free boundary condition，as shown in Fig.3（a），the left end of the composite beam is fixed and the right end is subjected to a con?centrated force P. The axial resultant force，the re?sultant moment and the resultant shear force in the FG SMA composite beam section due to the ap?plied force P can be written as

The rotation angle and transverse deflection of the FG SMA composite cantilever beam can be ob?tained by plugging Eqs.（32，33）into Eqs.（28，30）.

For the simply supported boundary condition，a FG SMA composite simply supported beam is un?der a concentrated load as shown in Fig.3（b）. The distance between the loading point and the left end of the beam is a. The axial resultant force，the resul?tant moment and the resultant shear force in the FG SMA composite beam section due to the applied force P can be written as

The rotation angle and the transverse deflec?tion of the FG SMA composite simply supported beam can be obtained by plugging Eqs.（31，34）into Eqs.（28，30）.

For both boundary conditions，the volume frac?tion of SMA fiber varies continuously through the thickness of the beam and obey a power-law type of graded function as given by

where the non-negative parameter k is the powerlaw index that dictates the material component varia?tion profile through the thickness of the FG SMA composite beam，which is defined as the FG param?eter in this paper.

The geometric size of the FG SMA composite beam is L=500 mm，h=100 mm and b=50 mm.Besides，the material properties of matrix material and SMA are provided in Table 1 and Table 2，re?spectively.

Table 1 Material parameters of matrix[40]

Table 2 Material parameters of SMA[40]

Next，the martensitic transformation and bend?ing behavior of FG SMA composite beams with two different boundary conditions are simulated and analyzed. The concentrated load is set as -20 kN，the FG parameter k as 1，the ambient temperature T as 50 ?C and the fiber laying angle β as 0°.

In view of boundary condition Eq.（32）and the martensitic volume fraction Eq.（4），the critical heights for the transformation starting and finishing of the FG SMA composite cantilever beam can be obtained by substituting Eqs.（28—30，33）into Eqs.（18，7）. The height of neutral plane can be deter?mined with the equivalent strain Eq.（7）being zero.Fig.4（a）shows the distribution patterns of martens?itic volume fraction in the beam section and Fig.4（b）shows the height distribution curves of the neu?tral plane and martensitic transformation critical lay?ers along the axial direction of the FG SMA com?posite cantilever beam（shown in Fig.3（a））.

Fig.4 Martensitic and neutral plane of the FG SMA com?posite cantilever beam with P = -20 kN

As can be seen from Fig.4（a），when the con?centrated end load P is -20 kN，the part of the SMA fiber in the cantilever beam has undergone an incompletely martensitic transformation. The mar?tensitic volume fraction distribution in the cantilever beam section is asymmetrically along the z axis，and it is larger in the SMA fiber on the tension side than that on the compressive side of the cantilever beam. The position of the neutral plane does not co?incide with that of the middle plane，which is the re?sult of the graded distribution of SMA fiber along the beam thickness.

It can be seen from Fig.4（b）that the FG SMA composite cantilever beam is divided into three sec?tions along the axial direction when the concentrated end load P is -20 kN. At the free end of the cantile?ver beam，no martensitic transformation occurs in the SMA fiber，and the height of the neutral plane znpremains to be -11.37 mm. The critical position for the martensitic transformation starting in the SMA fiber on the upper surface is x = 466 mm，while the critical position for the martensitic transfor?mation starting in the SMA fiber on the bottom sur?face is x = 447 mm. From the critical position where the martensitic transformation occurs to the fixed end of the cantilever beam，the height of the martensitic transformation starting critical layer un?der the tensile condition zstdecreases，while the height of the martensitic transformation starting criti?cal layer under the compressive condition zscincreas?es along the axial direction. Due to the incompletely martensitic transformation in the SMA fiber，the height of the neutral plane znpgradually decreases and then tends to be stabilized from the critical posi?tion where the martensitic transformation occurring to the fixed end of the cantilever beam.

In view of boundary condition Eq.（31） and martensitic volume fraction Eq.（4），substituting Eqs.（28—30，34）into Eqs.（18，7），the critical heights for the transformation starting and finishing of the FG SMA composite simply supported beam can be obtained. Fig.5 shows the distribution pat?terns of martensitic volume fraction in the section of the FG SMA composite simply supported beam（shown in Fig.3（b））with the distance between the loading point and the left end of the beam a = L/4 and a = L/2，respectively.

It can be found from Fig.5（a）that the martens?itic volume fraction distribution in the simply sup?ported beam section is asymmetrical about the load?ing position with a = L/4，and it is larger in the right side than that in the left side of the loading po?sition. As can be seen from Fig.5（b），the martensit?ic volume fraction distribution in the simply support?ed beam section is symmetrical about the loading po?sition with a = L/2. The martensitic volume frac?tion distribution in the simply supported beam sec?tion is asymmetrical along the z axis，and it is larger in the SMA fiber on the compressive side than that on the tensile side of the simply supported beam.The position of the neutral plane does not coincide with that of the middle plane，which is the result of the graded distribution of SMA fiber along the beam thickness.

Fig.5 Distribution of martensitic volume fraction in FG SMA composite simply supported beam with a = L/4 and a = L/2

Fig.6 show the height distribution curves of the neutral plane and martensitic transformation critical layers along the axial direction of the FG SMA com?posite simply supported beam when a is L/4 and L/2，respectively.

As can be seen from Fig.6，no martensitic transformation occurs in the SMA fiber at the sup?ported ends of the beam，and the height of the neu?tral plane znpremains -11.37 mm. The critical posi?tions for the martensitic transformation starting in the SMA fiber on the upper surface are x = 78 mm and x = 266 mm，while the critical positions for the martensitic transformation starting in the SMA fiber on the bottom surface are x = 120 mm and x =141 mm with a = L/4. The critical positions for the martensitic transformation starting in the SMA fiber on the upper surface are x = 135 mm and x =365 mm，while the critical positions for the martens?itic transformation starting in the SMA fiber on the bottom surface are x = 219 mm and x = 281 mm with a = L/2.

Fig.6 Curves of heights of the neutral plane and martensitic transformation critical layers along the axial direction of the FG SMA composite simply supported beam with a = L/4 and a = L/2

It is obvious that from the critical position where the martensitic transformation occurs to the loading position，the height of the martensitic trans?formation starting critical layer in tensile condition zstincreases while the height of the martensitic trans?formation starting critical layer in compressive condi?tion zscdecreases along the axial direction. Due to the incompletely martensitic transformation in the SMA fiber，the height of the neutral plane znpgradu?ally decreases from the critical position where the martensitic transformation occurs to the lading posi?tion of the simply supported beam before the com?pletely martensitic transformation occurs in the beam.

In view of boundary condition Eqs.（31，32），the deflection and rotation angle of the FG SMA composite simply supported beam can be obtained by plugging Eq.（34）into Eqs.（28，30），while the deflection and rotation angle of the FG SMA com?posite cantilever beam can be obtained by plugging Eq.（33）into Eqs.（28，30），respectively.

Fig.7 shows the deflection curves of the FG SMA composite beams with two different boundary conditions. Among the curves in Fig.7，it is obvious that the deflection curve of the FG SMA composite cantilever beam forms a semi-arch shape whose peak occurs in the free end. The deflection curve of the FG SMA composite simply supported beam forms a sinusoidal hump whose peak occurs at the loading position.

Fig.7 Deflection curves of FG SMA composite beams with two different boundary conditions

Fig.8 shows the rotation angle curves along the axial direction of the FG SMA composite beams with two different boundary conditions. As can be seen in Fig.8，the rotation angle curve of the FG SMA composite cantilever beam forms a semi-arch shape whose peak occurs in the free end. The rota?tion angle curve of the FG SMA composite simply supported beam is presented with a wave-shape form，in which the critical position for the rotation angle being zero coincides with the loading position.

Fig.8 Rotation angle curves along the axial direction of FG SMA composite beams with two different boundary conditions

To verify the influence of the FG parameter，ambient temperature and SMA fiber laying angle，a FG SMA composite cantilever beam subjected to a concentrated end load is then analyzed.

3.1 Influence of FG parameter

The FG parameter is an important factor for the design and optimization of FG SMA composite beams. And the influence of FG parameter on the thermo-mechanical behaviors of the FG SMA com?posite cantilever beam is numerically investigated in this section. The concentrated end load P is -20 kN，the ambient temperature is 50 ?C and the SMA fiber laying angle β is 0°.

Fig.9 shows the volume fraction distribution of SMA fiber along the thickness of the FG SMA com?posite cantilever beam with respect to different val?ues of FG parameter k. The height curves of neutral plane and martensitic transformation critical layers along the axial direction，deflection and rotation an?gle curves of the FG SMA composite cantilever beam with respect to different values of FG parame?ter k are presented in Figs.10—13.

Fig.9 Volume fraction distribution of SMA fiber along the thickness of the FG SMA composite cantilever beam with respect to different FG parameters

Fig.10 shows the height curves of the neutral plane znpalong the axial direction of the FG SMA composite cantilever beam with respect to different values of FG parameter k. It is found that no mar?tensitic transformation occurs in the SMA fiber at the free end of the beam in all cases. The heights of the neutral plane znpremain -11.37 mm，-9.07 mm and -7.39 mm at the free end of the cantilever beam while the FG parameter k is set to be 1，2 and 3，respectively. And the height of the neutral plane of the beam increases with the increase of FG pa?rameter k. In all curves，the height of the neutral plane znpfirst remains constant，then decreases，and finally tends to be stabilized from the free end to the fixed end of the cantilever beam.

Fig.10 Curves of height of the neutral plane of the FG SMA composite cantilever beam with respect to dif?ferent FG parameters

Figs.11 shows the height distribution curves of martensitic transformation starting critical layers along the axial direction of the FG SMA composite cantilever beam with respect to different values of FG parameter k.

Fig.11 Curves of heights of martensitic transformation starting critical layers of the FG SMA composite cantilever beam in tensile and compressive sides with respect to different FG parameters

As can be seen from Fig.11（a），the upper sur?face of the FG SMA composite cantilever beam，which is in tensile state，is divided into two regions along the axial direction，including the non-trans?formed region and the partly transformed region.The critical coordinates for martensitic transforma?tion starting in the SMA fiber on the upper surface are 466 mm，439 mm and 418 mm and the heights of the martensitic transformation starting critical lay?er in tensile state zstare -9.97 mm，-3.07 mm and 0.66 mm while the FG parameter k is set to be 1，2 and 3，respectively.

As can be seen from Fig.11（b），the bottom surface of the FG SMA composite cantilever beam，which is in compressive state，is also divided into two regions along the axial direction. The critical co?ordinates for martensitic transformation starting in the SMA fiber on the bottom surface are 447 mm，408 mm and 387 mm and the heights of the martens?itic transformation starting critical layer in tensile state zstare -17.15 mm， -18.09 mm and-17.86 mm while the FG parameter k is set to be 1，2 and 3，respectively.

It can be concluded that the FG effects of mar?tensitic transformation behavior in SMA fibers are obvious. The area of the martensitic transformed re?gion and the martensitic volume fraction in SMA fi?bers all decrease with the increase of FG parame?ter k.

Fig.12 indicates the deflection curves of FG SMA composite cantilever beam with respect to dif?ferent values of FG parameter k. Furthermore，rota?tion angle along the axial direction of the FG SMA composite cantilever beam with respect to different values of FG parameter k is examined through Fig.13.

Fig.12 Deflection curves of the FG SMA composite canti?lever beam with respect to different FG parameters

Fig.13 Rotation angle curves along the axial direction of the FG SMA composite cantilever beam with re?spect to different FG parameters

It can be obviously seen from the curves in Figs.12，13 that each curve of the deflection and ro?tation angle of the FG SMA composite cantilever beam forms a different semi-arch shape whose peak occurs in the free end of the cantilever beam. The absolute values of the maximum deflection of the cantilever beam are 54.80 mm，41.27 mm and 36.68 mm，and the corresponding absolute values of maximum rotation angle are 9.10°，7.22° and 6.42°while the FG parameter k is 1，2 and 3，respective?ly. It is concluded that the FG effect of the bending behavior of the FG SMA composite cantilever beam is obvious. The absolute value of the deflection and rotation angle of the FG SMA composite cantilever beam all decrease with the increase of FG parameter k. This illustrates that the larger the value of FG pa?rameter is，the stiffer the FG SMA composite canti?lever beam is.

It can be noticed from above mentioned simula?tion and discussion that the FG parameter k has a significant effect on the martensitic transformation and bending deformation of the FG SMA composite beam and therefore the static bending deformation may be controlled by choosing an appropriate value of k. For the sake of simplicity，the FG parameter k is set to be 1 in the following numerical examples.

3.2 Influence of ambient temperature

In order to study the influence of ambient tem?perature on the bending behavior of the FG SMA composite cantilever beam，the deflection and rota?tion angle curves of the cantilever beam with differ?ent ambient temperatures are presented in Figs.14，15，respectively. The concentrated end load P is-20 kN，and the SMA fiber laying angle β is 0°.

Fig.14 Deflection curves of FG SMA composite cantilever beam with different ambient temperatures

Fig.15 Rotation angle curves along the axial direction of the FG SMA composite cantilever beam with differ?ent ambient temperatures

It can be obviously seen from the curves in Figs.14，15 that the absolute values of the maxi?mum deflection of the cantilever beam are 54.80 mm，53.72 mm and 52.77 mm，and the corre?sponding absolute values of maximum rotation angle are 9.10°，8.89° and 8.72°，when the ambient tem?peratures are 50 ?C，60 ?C and 70 ?C，respectively.This illustrates that with the increase of ambient temperature，the absolute value of the deflection and rotation angle of the FG SMA composite canti?lever beam decreases and stiffness of the cantilever beam increases. This is because that the critical mar?tensitic transformation stresses increase with the in?crease of ambient temperature，which leads to a smaller martensitic transformation strain in the SMA fiber at the same stress level.

3.3 Influence of fiber laying angle

In order to study the influence of SMA fiber laying angle on the bending behavior of the FG SMA composite cantilever beam，the deflection and rotation angle curves of the cantilever beam with dif?ferent SMA fiber laying angles are presented in Figs.16， 17. The concentrated end load P is-20 kN，and the ambient temperature is 50 ?C.

As can be seen from the curves in Figs.16，17，the absolute values of the maximum deflection of the cantilever beam are 54.80 mm，55.10 mm，55.70 mm and 56.08 mm，and the corresponding ab?solute values of maximum rotation angle are 9.10°，9.14°，9.25° and 9.31°，while the SMA fiber laying angles are 0°，15°，30° and 45°，respectively. It is concluded that the influence of fiber laying angle on the deflection and rotation angle of the FG SMA composite cantilever beam is not obvious，and the influence of the SMA fiber laying angle can be ne?glected in designing the smart beam structures.

Fig.16 Deflection curves of FG SMA composite cantilever beam with different fiber laying angles

Fig.17 Rotation angle curves along the axial direction of the FG SMA composite cantilever beam with differ?ent fiber laying angles

4 Conclusions

A thermo-mechanical investigation on the func?tion graded shape memory alloy（FG SMA）Ti?moshenko composite beams is presented by consid?ering the shear deformation and thermal effect. The volume fraction of the SMA fiber is assumed to vary along the thickness according to a power-law type function，and the equivalent parameters of the FG SMA composite beam are obtained by the extended multi-cell micromechanics approach. The governing differential equations are established by the compos?ite laminated plate theory，and the martensitic trans?formation and bending deformation of FG SMA composite beams subjected to a concentrated load with two different boundary conditions are numeri?cally simulated. Furthermore，the influences of FG parameter，ambient temperature and SMA fiber lay?ing angle on the martensitic transformation and bending deformation of FG SMA composite beams are numerically analyzed. Some important conclu?sions are summarized as follows：

（1）The graded distribution of SMA fiber vol?ume fraction along the beam thickness results in that the position of the neutral plane does not coincide with that of the middle plane. The height of the neu?tral plane remains constant in the non-transformed region and decreases to be stable in the partly trans?formed region，but increases in the completely trans?formed region of SMA fiber.

（2）The FG parameter has a significant effect on the martensitic transformation and bending defor?mation of the FG SMA composite beam. The area of the martensitic transformed region and the mar?tensitic volume fraction in SMA fiber all decrease with the increased value of FG parameter. The larg?er the value of FG parameter is，the stiffer the FG SMA composite beam is.

（3）The ambient temperature is an important factors related to the martensitic transformation of SMA fiber and bending deformation of the FG SMA composite beam. With the ambient tempera?ture increases，the absolute value of deflection and the rotation angle of the FG SMA composite beam decrease and the stiffness of the FG SMA compos?ite beam increases.

（4）The influence of SMA fiber laying angle on the martensitic transformation of SMA fiber and bending deformation of the FG SMA composite beam is not obvious and can be negleted.