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eISSN: 2641-9335

Mathematical and Theoretical Physics

Forum Article Volume 1 Issue 2

The effect of temperature-dependent physical properties and fractional thermoelasticity on nonlocal nanobeams

Ahmed E Abouelregal

Department of Mathematics, Mansoura University, Egypt

Correspondence: Ahmed E Abouelregal, Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516 Egypt

Received: November 10, 2017 | Published: April 4, 2018

Citation: Abouelregal AE. The effect of temperature – dependent physical properties and fractional thermoelasticity on nonlocal nanobeams. Open Acc J Math Theor Phy. 2018;1(2):46-55 DOI: 10.15406/oajmtp.2018.01.00009

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Abstract

There are many papers are presented in the field of nanoengineering and nanotechnology. In this work, a new model of heat equation with fractional derivatives and relaxation time is introduced to investigate the thermo elastic vibrations of nonlocal nanobeams. In addition, the thermal conductivity and the modulus of elasticity are taken as a linear function of environmental temperature and the body is due to a ramp−type heating. The basic equations of the considered problem are formulated and solved using Laplace transform along with its numerical inversion. The distributions of the temperature, deflection as well as other variable fields are numerically obtained and illustrated graphically. The effects of the nanoscale, fractional order derivative, ramping-time and relaxation time on the considered variables are concerned and discussed in detail. The introduced model can be reduced to the corresponding classical coupled and Lord and Shulman theories for different values of phase lags and fractional order parameters.

Keywords: fractional heat equation, nanobeams, temperature−dependent properties, ramp−type heating, fractional order.

Introduction

In recent years, many interesting models have been established by using fractional calculus to study the physical manners, particularly in the area of diffusion, viscoelasticity, heat conduction, mechanics of solids and biological systems. The use of fractional system derivatives and integrals leads to the formulation of some physical problems that are more economical and useful than the classical approach.

In the second half of the nineteenth century, Caputo,1 Caputo and Mainardi2 found an agreement between experimental and theoretical results when using fractional derivatives to describe viscoelastic materials. Povstenko3 proposed a fractional time derivative heat conduction equation for a quasi−static uncoupled theory of thermo elasticity.  He also investigated in4 a generalizations of the heat conduction and Fourier law by using space and time fractional derivatives. Jiang & Xu5 obtained a fractional heat equation with a time fractional derivative in the general orthogonal curvilinear and other coordinate system. Samko et al. in their book6 provided an excellent historical review of this area under discussion. Formulae establishing relations between the two types of fractional derivatives of Riemann−Leuville and Caputo are discussed in detail in.7,8 It should be confirmed that if attention is given, the results obtained using the Caputo construction can be modified to the Riemann−Liouville version.

The theory of couple thermo elasticity is extended by Lord & Shulman,9 Green and Lindsay10 by including the thermal relaxation time in constitutive relations. Green & Naghdi11 proposed a new generalized thermo elasticity theory by including the thermal−displacement gradient among the independent constitutive variables. An important feature of this theory, which is not present in other thermo elasticity theories, is that it does not accommodate dissipation of thermal energy.

Recently, a completely new fractional order thermo elasticity model involves one thermal relaxation parameter was introduced by Sherief et al.12 Also, Ezzat13 established a model of fractional heat conduction equation by using the new Taylor series expansion of time−fractional order developed by Jumarie.14 In addition, authors in15 presented a two generalized models of heat conduction with fractional order for an isotropic nonhomogeneous thermo elastic solid.

Modern structural elements are often exposed to temperature changes of such magnitude that their physical properties are not fixed even in an approximate sense. The thermal and mechanical properties of materials vary with temperature, so that the temperature dependence of material properties must be taken into consideration in the thermal stress analysis of these elements.16,17 At high temperatures, material properties such as the thermal conductivity, Poisson’s ratio, the modulus of elasticity and the thermal expansion coefficients are no longer keep constant.18 In numerous of papers,19–24 fractional calculus is often applied in vibration studies of rods, beams, plates as well as in other fields of mechanics.

Micro−scale mechanical resonators have high sensitivity as well as fast response and are widely used as sensors and modulators. Micro and nano−mechanical resonators have attracted considerable attention recently due to their many important technological applications. Accurate analysis of various effects on the characteristics of resonators, such as resonant frequencies and quality factors, is crucial for designing high−performance components. Many authors have studied the vibration and heat transfer process of beams. Some of them are found in the references.25−27

The nonlocal theory of elasticity was used to study applications in nano–mechanics including lattice dispersion of elastic waves, wave propagation in composites, dislocation mechanics, fracture mechanics, surface tension fluids, etc. Of all the nanostructures, the mechanical behaviors of nanotubes and nanobeams have been most widely investigated. The models of the nonlocal beams expected increasing attention in the earlier few years. In 1972, Eringen introduced the theory of nonlocal continuum mechanics,28 in an effort to deal with the small–scale structure problems. The theories of nonlocal continuum consider the state of stress at a point as a function of the states of strain of all points in the body while the classical continuum mechanics assumes the state of stress at a certain point uniquely depends on the state of strain at that same point. Solutions from various problems support this theory.28−35

The current manuscript is an effort to study a thermo elastic problem of a nanobeam loaded thermally by ramp–type heating. Also, a new fractional heat conduction equation model is introduced and the modulus of elasticity is considered to be a linear function of reference temperature. Laplace transform analytical technique is used to calculate the vibration of displacements and temperature. The variations along the axial direction and the through–the–thickness distributions of all fields are investigated. Some comparisons have been also shown graphically to estimate the effects of the small–scale, fractional order and ramping time parameters on all the studied fields.

Mathematical model for nonlocal thermo elasticity with fractional heat conduction

According to the nonlocal elasticity theory of Eringen, the nonlocal differential constitutive equations for a homogenous thermoelastic materials is28−30

( 1ξ 2 ) σ ij = τ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaWdaeaapeGaaGymaiabgkHiTiabe67a4jabgEGir=aa daahaaqabKqbGeaapeGaaGOmaaaaaKqbakaawIcacaGLPaaacqaHdp WCpaWaaSbaaKqbGeaapeGaamyAaiaadQgaaKqba+aabeaapeGaeyyp a0JaeqiXdq3damaaBaaajuaibaWdbiaadMgacaWGQbaajuaGpaqaba aaaa@4965@   (1)

             where σ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHdpWCpaWaaSbaaKqbGeaapeGaamyAaiaadQgaaKqba+aa beaaaaa@3B4F@ and τ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHepaDpaWaaSbaaKqbGeaapeGaamyAaiaadQgaaKqba+aa beaaaaa@3B51@ are the nonlocal and local stress tensors, respectively, 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGHhis0paWaaWbaaKqbGeqabaWdbiaaikdaaaaaaa@3956@ is Laplacian operator and ξ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH+oaEaaa@3868@ is the nonlocal parameter. One may see that when the internal characteristic length is neglected, i.e., the particles of a medium are considered to be continuously distributed, ξ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH+oaEaaa@3868@ is zero, and Eq. (1) reduces to the constitutive equation of classical local thermo elasticity. The classical thermo elasticity is based on the principles of the classical theory of heat conductivity, specifically on the classical Fourier law, which relates the heat flux vector

q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGXbaaaa@379B@ to the temperature gradient as follows: q=Kθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcfaieaa aaaaaaa8qacaWFXbGaeyypa0JaeyOeI0Iaam4saiabgEGirlabeI7a Xbaa@3DA2@ (2)

                  

where K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcfaieaa aaaaaaa8qacaWFlbaaaa@377D@ is the thermal conductivity of a solid, θ=T T 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH4oqCcqGH9aqpcaWGubGaeyOeI0Iaamiva8aadaWgaaqc fasaa8qacaaIWaaajuaGpaqabaaaaa@3DC5@ is the excess temperature distribution, in which T 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGubWdamaaBaaajuaibaWdbiaaicdaaKqba+aabeaaaaa@3943@ is the environmental temperature. Equation (4) together with the energy equation yields the heat conduction equation or the parabolic heat conduction equation and is diffusive with the notion of infinite speed of propagation of thermal disturbances.

ρ C E θ t +γ T 0 t ( div( u ) )=div( q )+Q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHbpGCcaWGdbWdamaaBaaajuaibaWdbiaadweaaKqba+aa beaapeWaaSaaa8aabaWdbiabgkGi2kabeI7aXbWdaeaapeGaeyOaIy RaamiDaaaacqGHRaWkcqaHZoWzcaWGubWdamaaBaaajuaibaWdbiaa icdaa8aabeaajuaGpeWaaSaaa8aabaWdbiabgkGi2cWdaeaapeGaey OaIyRaamiDaaaadaqadaWdaeaapeGaaeizaiaabMgacaqG2bWaaeWa a8aabaacbmWdbiaa=vhaaiaawIcacaGLPaaaaiaawIcacaGLPaaacq GH9aqpcqGHsislcaqGKbGaaeyAaiaabAhadaqadaWdaeaapeGaa8xC aaGaayjkaiaawMcaaiabgUcaRiaadgfaaaa@5A54@ (3)

      

where ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccmqcfaieaa aaaaaaa8qacqWFbpGCaaa@386D@ is the density, C E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcfaieaa aaaaaaa8qacaWFdbWdamaaBaaajuaibaWdbiaa=veaaKqba+aabeaa aaa@3946@ is the specific heat, u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcfaieaa aaaaaaa8qacaWF1baaaa@37A7@ is the displacement vector, γ= Ε α τ 1-2ν = α τ ( 3λ+2μ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcfaieaa aaaaaaa8qacaWFZoGaa8xpamaalaaapaqaa8qacaWFvoGaa8xSd8aa daWgaaqcfasaa8qacaWFepaapaqabaaajuaGbaWdbiaa=fdacaWFTa Gaa8Nmaiaa=17aaaGaa8xpaiaa=f7adaWgaaqcfasaaiaa=r8aaKqb agqaamaabmaapaqaa8qacaWFZaGaa83Udiaa=TcacaWFYaGaa8hVda GaayjkaiaawMcaaaaa@4B2B@ , being Lame's constants, α τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcfaOaa8 xSdmaaBaaajuaibaGaa8hXdaqcfayabaaaaa@39EA@ being the coefficient of linear thermal expansion, E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcfaieaa aaaaaaa8qacaWFfbaaaa@3776@ is Young modulus, ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcfaieaa aaaaaaa8qacaWF9oaaaa@37F1@ is Poisson's ratio, and Q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcfaieaa aaaaaaa8qacaWFrbaaaa@3782@ is the intensity of heat source. The Riemann‒Liouville fractional integral is introduced as a natural generalization of the convolution type integral:36,37

I α f( t )= 0 t ( tτ ) α1 Γ( α ) f( τ )dτ,   ( 0<α<1 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGjbWdamaaCaaajuaibeqaa8qacqaHXoqyaaqcfaOaamOz amaabmaapaqaa8qacaWG0baacaGLOaGaayzkaaGaeyypa0Jaey4kIi =aa0baaKqbGeaacaaIWaaabaGaamiDaaaajuaGdaWcaaWdaeaapeWa aeWaa8aabaWdbiaadshacqGHsisliiaacqWFepaDaiaawIcacaGLPa aapaWaaWbaaKqbGeqabaWdbiabeg7aHjabgkHiTiaaigdaaaaajuaG paqaa8qacqWFtoWrdaqadaWdaeaapeGaeqySdegacaGLOaGaayzkaa aaaiaadAgadaqadaWdaeaapeGae8hXdqhacaGLOaGaayzkaaGaamiz aiab=r8a0jaacYcacaqGGcGaaeiOaiaabckadaqadaWdaeaapeGaaG imaiabgYda8iabeg7aHjabgYda8iaaigdaaiaawIcacaGLPaaacaGG Uaaaaa@641A@ (4)

       

Where Γ( α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbeqcfaieaa aaaaaaa8qacaWFtoWaaeWaa8aabaWdbiaa=f7aaiaawIcacaGLPaaa aaa@3AA2@ is the Gamma function and f( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcfaieaa aaaaaaa8qacaWFMbWaaeWaa8aabaWdbiaa=rhaaiaawIcacaGLPaaa aaa@3A34@ is a Lebesgue integrable continuous function satisfies

lim α1 d α d t α f( t )=f'( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaCbeae aaqaaaaaaaaaWdbiGacYgacaGGPbGaaiyBaaWdaeaapeGaeqySdeMa eyOKH4QaaGymaaWdaeqaa8qadaWcaaWdaeaapeGaamiza8aadaahaa qabKqbGeaapeGaeqySdegaaaqcfa4daeaapeGaamizaiaadshapaWa aWbaaeqajuaibaWdbiabeg7aHbaaaaqcfaOaamOzamaabmaapaqaa8 qacaWG0baacaGLOaGaayzkaaGaeyypa0JaamOzaiaabEcadaqadaWd aeaapeGaamiDaaGaayjkaiaawMcaaaaa@4F40@ (5)

                    

Using the Taylor series expansion of time fractional order as in,13 a new non–Fourier model was constructed by14 in the form

( 1+ τ 0 α α! α t α )q=Kθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaWdaeaapeGaaGymaiabgUcaRmaalaaapaqaaGGaa8qa cqWFepaDpaWaa0baaKqbGeaapeGaaGimaaWdaeaapeGaeqySdegaaa qcfa4daeaapeGaeqySdeMaaiyiaaaadaWcaaWdaeaapeGaeyOaIy7d amaaCaaabeqcfasaa8qacqaHXoqyaaaajuaGpaqaa8qacqGHciITca WG0bWdamaaCaaabeqcfasaa8qacqaHXoqyaaaaaaqcfaOaayjkaiaa wMcaaGqadiaa+fhacqGH9aqpcqGHsislcaWGlbGaey4bIeTaaeiUda aa@5189@   (6)

                     Taking divergence of both sides of Eq. (6), we get

( 1+ τ 0 α α! α t α )( div( q ) )=K 2 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaWdaeaapeGaaGymaiabgUcaRmaalaaapaqaaGGaa8qa cqWFepaDpaWaa0baaKqbGeaapeGaaGimaaWdaeaapeGaeqySdegaaa qcfa4daeaapeGaeqySdeMaaiyiaaaadaWcaaWdaeaapeGaeyOaIy7d amaaCaaajuaibeqaa8qacqaHXoqyaaaajuaGpaqaa8qacqGHciITca WG0bWdamaaCaaajuaibeqaa8qacqaHXoqyaaaaaaqcfaOaayjkaiaa wMcaamaabmaapaqaa8qacaqGKbGaaeyAaiaabAhadaqadaWdaeaaie WapeGaa4xCaaGaayjkaiaawMcaaaGaayjkaiaawMcaaiabg2da9iab gkHiTiaadUeacqGHhis0paWaaWbaaeqajuaibaWdbiaaikdaaaqcfa OaaeiUdaaa@595E@ (7)

           From Eqs. (3) and (7), we can get the fractional ordered generalized heat conduction equation as

K 2 θ=( 1+ τ 0 α α! α t α )[ ρ C E θ t +γ T 0 t ( div( u ) )Q ]. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGlbGaey4bIe9damaaCaaabeqcfasaa8qacaaIYaaaaKqb akaabI7acqGH9aqpdaqadaWdaeaapeGaaGymaiabgUcaRmaalaaapa qaaGGaa8qacqWFepaDpaWaa0baaKqbGeaapeGaaGimaaWdaeaapeGa eqySdegaaaqcfa4daeaapeGaeqySdeMaaiyiaaaadaWcaaWdaeaape GaeyOaIy7damaaCaaabeqcfasaa8qacqaHXoqyaaaajuaGpaqaa8qa cqGHciITcaWG0bWdamaaCaaabeqcfasaa8qacqaHXoqyaaaaaaqcfa OaayjkaiaawMcaamaadmaapaqaa8qacqaHbpGCcaWGdbWdamaaBaaa juaibaWdbiaadweaaKqba+aabeaapeWaaSaaa8aabaWdbiabgkGi2k abeI7aXbWdaeaapeGaeyOaIyRaamiDaaaacqGHRaWkcqaHZoWzcaWG ubWdamaaBaaajuaibaWdbiaaicdaa8aabeaajuaGpeWaaSaaa8aaba WdbiabgkGi2cWdaeaapeGaeyOaIyRaamiDaaaadaqadaWdaeaapeGa aeizaiaabMgacaqG2bWaaeWaa8aabaacbmWdbiaa+vhaaiaawIcaca GLPaaaaiaawIcacaGLPaaacqGHsislcaWGrbaacaGLBbGaayzxaaGa aiOlaaaa@707B@ (8)

        The introduced new model of thermo elasticity using the methodology of fractional calculus with wide range 0<α1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIWaGaeyipaWJaeqySdeMaeyizImQaaGymaaaa@3C71@ covering two cases of conductivity, 0<α<1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIWaGaeyipaWJaeqySdeMaeyipaWJaaGymaaaa@3BC0@ corresponds to weak conductivity, α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHXoqycqGH9aqpcaaIXaaaaa@3A04@ for normal conductivity. For the Lord and Shulman (LS) theory α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHXoqycqGH9aqpcaaIXaaaaa@3A04@ and for the calssical coupled theory of thermoelasticity (CTE) α=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHXoqycqGH9aqpcaaIWaaaaa@3A03@ and τ 0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcfaieaa aaaaaaa8qacqWFepaDpaWaaSbaaKqbGeaapeGaaGimaaqcfa4daeqa a8qacqGH9aqpcaaIWaaaaa@3C03@ . Constitutive equations:

τ ij =2μ e ij +λ e ij γθ δ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHepaDpaWaaSbaaeaajuaipeGaamyAaiaadQgaaKqba+aa beaapeGaeyypa0JaaGOmaiabeY7aTjaadwgapaWaaSbaaKqbGeaape GaamyAaiaadQgaa8aabeaajuaGpeGaey4kaSIaeq4UdWMaamyza8aa daWgaaqcfasaa8qacaWGPbGaamOAaaWdaeqaaKqba+qacqGHsislcq aHZoWzcqaH4oqCcqaH0oazpaWaaSbaaKqbGeaapeGaamyAaiaadQga a8aabeaaaaa@517C@ (9)

                    Equation of motion:

σ ji,j + F i =ρ u ¨ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHdpWCpaWaaSbaaKqbGeaapeGaamOAaiaadMgacaGGSaGa amOAaaWdaeqaaKqba+qacqGHRaWkcaWGgbWdamaaBaaajuaibaWdbi aadMgaa8aabeaajuaGpeGaeyypa0JaeqyWdiNabmyDa8aagaWaamaa BaaajuaibaWdbiaadMgaaKqba+aabeaaaaa@4677@ (10)

                   Equations (1) and (8) describe the nonlocal thermo elasticity theory with fractional order derivative. It can be seen that the corresponding local thermoelasticity with fractional order is recovered by putting ξ=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH+oaEcqGH9aqpcaaIWaaaaa@3A27@ in equation (1). It is known that the values of physical properties, such as the moduli of elasticity and thermal expansion coefficients, cannot be assumed constant over a wide range of high–temperature applications. Consider a thermoelastic body of material having temperature–dependent properties on the form16,38,39

( E,γ,K )=( E 0 , γ 0 , K 0 )F( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaWdaeaapeGaamyraiaacYcacqaHZoWzcaGGSaGaam4s aaGaayjkaiaawMcaaiabg2da9maabmaapaqaa8qacaWGfbWdamaaBa aajuaibaWdbiaaicdaa8aabeaajuaGpeGaaiilaiabeo7aN9aadaWg aaqcfasaa8qacaaIWaaapaqabaqcfa4dbiaacYcacaWGlbWdamaaBa aajuaibaWdbiaaicdaa8aabeaaaKqba+qacaGLOaGaayzkaaGaamOr amaabmaapaqaa8qacqaH4oqCaiaawIcacaGLPaaaaaa@4DE4@   (11)

        where E 0 , γ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGfbWdamaaBaaajuaibaWdbiaaicdaaKqba+aabeaapeGa aiilaiabeo7aN9aadaWgaaqcfasaa8qacaaIWaaajuaGpaqabaaaaa@3D5F@ and K 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGlbWdamaaBaaajuaibaWdbiaaicdaaKqba+aabeaaaaa@3939@ are considered to be constants; F( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGgbWaaeWaa8aabaWdbiabeI7aXbGaayjkaiaawMcaaaaa @3ACD@ is given in a nondimensional function of temperature. In the case of temperature-independent modulus of elasticity, F( θ )=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaIaaGPaVN qbacbaaaaaaaaapeGaamOramaabmaapaqaa8qacqaH4oqCaiaawIca caGLPaaacqGH9aqpcaaIXaaaaa@3E47@ and ( E,γ,K )=( E 0 , γ 0 , K 0 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaWdaeaapeGaamyraiaacYcacqaHZoWzcaGGSaGaam4s aaGaayjkaiaawMcaaiabg2da9maabmaapaqaa8qacaWGfbWdamaaBa aajuaibaWdbiaaicdaaKqba+aabeaapeGaaiilaiabeo7aN9aadaWg aaqcfasaa8qacaaIWaaajuaGpaqabaWdbiaacYcacaWGlbWdamaaBa aajuaibaWdbiaaicdaaKqba+aabeaaa8qacaGLOaGaayzkaaGaaiOl aaaa@4A6D@

The variation of some or all of the these mechanical and thermal properties with temperature can be approximated by linear, exponential, quadratic laws or any other law appropriately approximating the experimental data. We will consider that16,38,39

F( θ )= e βθ 1βθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGgbWaaeWaa8aabaWdbiabeI7aXbGaayjkaiaawMcaaiab g2da9iaadwgapaWaaWbaaKqbGeqabaWdbiabgkHiTiabek7aIjabeI 7aXbaajuaGcqGHfjcqcaaIXaGaeyOeI0IaeqOSdiMaeqiUdehaaa@4830@ (12)

               where β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHYoGyaaa@3845@ is called the empirical material constant measured in 1/K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIXaGaai4laiaabUeaaaa@38E0@ . Equation (12) introduces an empirical model that usually works for a limited range of temperatures.

Since only the infinitesimal temperature deviations from reference temperature are considered and for linearity of the governing partial differential equations of the problem, we have to take into account the condition | T T 0 |/ T 0 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaabdaWdaeaaieWapeGaa8hvaiabgkHiTiaa=rfapaWaaSba aKqbGeaapeGaaGimaaqcfa4daeqaaaWdbiaawEa7caGLiWoacaGGVa Gaa8hva8aadaWgaaqcfasaa8qacaaIWaaapaqabaqcfa4dbiablQMi 9iaaigdaaaa@43CF@ ,which give us the approximating function of F( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGgbWaaeWaa8aabaWdbiabeI7aXbGaayjkaiaawMcaaaaa @3ACD@ to be in the form.40,41

F( θ )F( T 0 )=1β T 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGgbWaaeWaa8aabaWdbiabeI7aXbGaayjkaiaawMcaaiab gwKiajaadAeadaqadaWdaeaapeGaamiva8aadaWgaaqcfasaa8qaca aIWaaajuaGpaqabaaapeGaayjkaiaawMcaaiabg2da9iaaigdacqGH sislcqaHYoGycaWGubWdamaaBaaajuaibaWdbiaaicdaaKqba+aabe aaaaa@480E@ (13)

                 

Formulation of the problem and modeling of beam structures

We consider a thermo elastic thin nanobeam initially at temperature T 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGubWdamaaBaaajuaibaWdbiaaicdaa8aabeaaaaa@38B4@ such that axis is drawn along the axial direction of the beam and y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG5baaaa@37A2@ , z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG6baaaa@37A3@ axes correspond to the width and thickness, respectively (Figure 1). The small flexural deflections of the nanobeam with dimensions of length L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGmbaaaa@3775@ ,width b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGIbaaaa@378B@ and thickness h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGObaaaa@3791@ are considered.

The displacement components are given by

u=z w x , v=0,  w( x,y,z,t )=w( x,t ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG1bGaeyypa0JaeyOeI0IaamOEamaalaaapaqaa8qacqGH ciITcaWG3baapaqaa8qacqGHciITcaWG4baaaiaacYcacaqGGcGaam ODaiabg2da9iaaicdacaGGSaGaaeiOaiaabckacaWG3bWaaeWaa8aa baWdbiaadIhacaGGSaGaamyEaiaacYcacaWG6bGaaiilaiaadshaai aawIcacaGLPaaacqGH9aqpcaWG3bWaaeWaa8aabaWdbiaadIhacaGG SaGaamiDaaGaayjkaiaawMcaaiaacYcaaaa@56CE@ (14)

                         where w is the lateral deflection.

Figure 1 Schematic diagram for the nanobeam.

For a one–dimensional problem, the differential form of the constitutive Eq. (9) after using Eqs. (1) and (14) can be expressed as:32,33

σ x ξ 2 σ x x 2 = E 0 F( T 0 )[ 2 w x 2 + α T θ ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHdpWCpaWaaSbaaKqbGeaapeGaamiEaaqcfa4daeqaa8qa cqGHsislcqaH+oaEdaWcaaWdaeaapeGaeyOaIy7damaaCaaajuaibe qaa8qacaaIYaaaaKqbakabeo8aZ9aadaWgaaqcfasaa8qacaWG4baa juaGpaqabaaabaWdbiabgkGi2kaadIhapaWaaWbaaeqajuaibaWdbi aaikdaaaaaaKqbakabg2da9iabgkHiTiaadweapaWaaSbaaKqbGeaa peGaaGimaaWdaeqaaKqba+qacaWGgbWaaeWaa8aabaWdbiaadsfapa WaaSbaaKqbGeaapeGaaGimaaqcfa4daeqaaaWdbiaawIcacaGLPaaa daWadaWdaeaapeWaaSaaa8aabaWdbiabgkGi2+aadaahaaqabKqbGe aapeGaaGOmaaaajuaGcaWG3baapaqaa8qacqGHciITcaWG4bWdamaa Caaabeqcfasaa8qacaaIYaaaaaaajuaGcqGHRaWkcqaHXoqypaWaaS baaKqbGeaapeGaamivaaWdaeqaaKqba+qacqaH4oqCaiaawUfacaGL Dbaaaaa@62E8@ (15)

                        where σ x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHdpWCpaWaaSbaaKqbGeaapeGaamiEaaqcfa4daeqaaaaa @3A6F@ is the nonlocal axial stress, and α T = α t /( 12ν ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHXoqypaWaaSbaaKqbGeaapeGaamivaaqcfa4daeqaa8qa cqGH9aqpcqaHXoqydaWgaaqcfasaaiaadshaaeqaaKqbakaac+cada qadaWdaeaapeGaaGymaiabgkHiTiaaikdacqaH9oGBaiaawIcacaGL Paaaaaa@4529@ .For transversely vibration of nanobeams, the equilibrium conditions of Euler–Bernoulli theory can be written as

2 M x 2 =ρA 2 w t 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaWdaeaapeGaeyOaIy7damaaCaaabeqcfasaa8qacaaI YaaaaKqbakaad2eaa8aabaWdbiabgkGi2kaadIhapaWaaWbaaeqaju aibaWdbiaaikdaaaaaaKqbakabg2da9iabeg8aYjaadgeadaWcaaWd aeaapeGaeyOaIy7damaaCaaabeqcfasaa8qacaaIYaaaaKqbakaadE haa8aabaWdbiabgkGi2kaadshapaWaaWbaaKqbGeqabaWdbiaaikda aaaaaaaa@4A7E@ (16)

                           where A=bh MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGbbGaeyypa0JaamOyaiaadIgaaaa@3A44@ is the cross–section area.

With aid of Eq. (15), the flexure moment is given by

M( x,t )ξ 2 M x 2 =I E 0 F( T 0 )[ 2 w x 2 + α T M T ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWaaeWaa8aabaWdbiaadIhacaGGSaGaamiDaaGaayjk aiaawMcaaiabgkHiTiabe67a4naalaaapaqaa8qacqGHciITpaWaaW baaeqajuaibaWdbiaaikdaaaqcfaOaamytaaWdaeaapeGaeyOaIyRa amiEa8aadaahaaqabKqbGeaapeGaaGOmaaaaaaqcfaOaeyypa0Jaey OeI0IaamysaiaadweapaWaaSbaaKqbGeaapeGaaGimaaWdaeqaaKqb a+qacaWGgbWaaeWaa8aabaWdbiaadsfapaWaaSbaaKqbGeaapeGaaG imaaWdaeqaaaqcfa4dbiaawIcacaGLPaaadaWadaWdaeaapeWaaSaa a8aabaWdbiabgkGi2+aadaahaaqcfasabeaapeGaaGOmaaaajuaGca WG3baapaqaa8qacqGHciITcaWG4bWdamaaCaaabeqcfasaa8qacaaI YaaaaaaajuaGcqGHRaWkcqaHXoqypaWaaSbaaKqbGeaapeGaamivaa WdaeqaaKqba+qacaWGnbWdamaaBaaajuaibaWdbiaadsfaaKqba+aa beaaa8qacaGLBbGaayzxaaaaaa@6321@ (17)

                           Where I=b h 3 /12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGjbGaeyypa0JaamOyaiaadIgapaWaaWbaaeqajuaibaWd biaaiodaaaqcfaOaai4laiaaigdacaaIYaaaaa@3E30@ is the inertia moment of the cross–section, I E 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGjbGaamyra8aadaWgaaqcfasaa8qacaaIWaaapaqabaaa aa@3973@ is the flexural rigidity of the beam and M T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWdamaaBaaajuaibaWdbiaadsfaaKqba+aabeaaaaa@395A@ is the thermal moment,

M T = 12 h 3 h/2 h/2 θ( x,z,t )zdz MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWdamaaBaaajuaibaWdbiaadsfaa8aabeaajuaGpeGa eyypa0ZaaSaaa8aabaWdbiaaigdacaaIYaaapaqaa8qacaWGObWdam aaCaaabeqcfasaa8qacaaIZaaaaaaajuaGcqGHRiI8daqhaaqcfasa aiabgkHiTiaadIgacaGGVaGaaGOmaaqaaiaadIgacaGGVaGaaGOmaa aajuaGcqaH4oqCdaqadaWdaeaapeGaamiEaiaacYcacaWG6bGaaiil aiaadshaaiaawIcacaGLPaaacaWG6bGaamizaiaadQhaaaa@51C4@ (18)

                    Substituting Eq. (17) into Eq. (16), one can get the motion equation of the nanobeam as

4 w x 4 + ρ I E 0 F( T 0 ) ( 2 w t 2 ξ 4 w t 2 x 2 )+ α t 2 M T x 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaWdaeaapeGaeyOaIy7damaaCaaabeqcfasaa8qacaaI 0aaaaKqbakaadEhaa8aabaWdbiabgkGi2kaadIhapaWaaWbaaKqbGe qabaWdbiaaisdaaaaaaKqbakabgUcaRmaalaaapaqaa8qacqaHbpGC a8aabaWdbiaadMeacaWGfbWdamaaBaaajuaibaWdbiaaicdaaKqba+ aabeaapeGaamOramaabmaapaqaa8qacaWGubWdamaaBaaajuaibaWd biaaicdaaKqba+aabeaaa8qacaGLOaGaayzkaaaaamaabmaapaqaa8 qadaWcaaWdaeaapeGaeyOaIy7damaaCaaabeqcfasaa8qacaaIYaaa aKqbakaadEhaa8aabaWdbiabgkGi2kaadshapaWaaWbaaeqajuaiba WdbiaaikdaaaaaaKqbakabgkHiTiabe67a4naalaaapaqaa8qacqGH ciITpaWaaWbaaeqajuaibaWdbiaaisdaaaqcfaOaam4DaaWdaeaape GaeyOaIyRaamiDa8aadaahaaqabKqbGeaapeGaaGOmaaaajuaGcqGH ciITcaWG4bWdamaaCaaajuaibeqaa8qacaaIYaaaaaaaaKqbakaawI cacaGLPaaacqGHRaWkcqaHXoqypaWaaSbaaeaapeGaamiDaaWdaeqa a8qadaWcaaWdaeaapeGaeyOaIy7damaaCaaajuaibeqaa8qacaaIYa aaaKqbakaad2eapaWaaSbaaKqbGeaapeGaamivaaqcfa4daeqaaaqa a8qacqGHciITcaWG4bWdamaaCaaabeqcfasaa8qacaaIYaaaaaaaju aGcqGH9aqpcaaIWaaaaa@73EC@ (19)

                     Also, it is exactly seen that the flexure moment of the nonlocal nanobeams is given by

M( x,t )=ξAρ 2 w t 2 I E 0 F( T 0 )[ 2 w x 2 + α t M T ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWaaeWaa8aabaWdbiaadIhacaGGSaGaamiDaaGaayjk aiaawMcaaiabg2da9iabe67a4jaadgeacqaHbpGCdaWcaaWdaeaape GaeyOaIy7damaaCaaajuaibeqaa8qacaaIYaaaaKqbakaadEhaa8aa baWdbiabgkGi2kaadshapaWaaWbaaeqajuaibaWdbiaaikdaaaaaaK qbakabgkHiTiaadMeacaWGfbWdamaaBaaajuaibaWdbiaaicdaa8aa beaajuaGpeGaamOramaabmaapaqaa8qacaWGubWdamaaBaaajuaiba WdbiaaicdaaKqba+aabeaaa8qacaGLOaGaayzkaaWaamWaa8aabaWd bmaalaaapaqaa8qacqGHciITpaWaaWbaaeqajuaibaWdbiaaikdaaa qcfaOaam4DaaWdaeaapeGaeyOaIyRaamiEa8aadaahaaqabKqbGeaa peGaaGOmaaaaaaqcfaOaey4kaSIaeqySde2damaaBaaajuaibaWdbi aadshaa8aabeaajuaGpeGaamyta8aadaWgaaqcfasaa8qacaWGubaa juaGpaqabaaapeGaay5waiaaw2faaaaa@6500@  (20)

                        Substituting the Euler–Bernoulli assumption and Eq. (14) into Eq. (8), gives the geralized heat conduction equation with fractional order derivative without the heat source ( Q=0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaWdaeaapeGaamyuaiabg2da9iaaicdaaiaawIcacaGL Paaaaaa@3AE2@ , as 2 θ x 2 + 2 θ z 2 =( 1+ τ 0 α α! α t α )[ 1 k θ t γ 0 T 0 K 0 z t ( 2 w x 2 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaWdaeaapeGaeyOaIy7damaaCaaabeqcfasaa8qacaaI YaaaaKqbakaabI7aa8aabaWdbiabgkGi2kaadIhapaWaaWbaaKqbGe qabaWdbiaaikdaaaaaaKqbakabgUcaRmaalaaapaqaa8qacqGHciIT paWaaWbaaeqajuaibaWdbiaaikdaaaqcfaOaaeiUdaWdaeaapeGaey OaIyRaamOEa8aadaahaaqabKqbGeaapeGaaGOmaaaaaaqcfaOaeyyp a0ZaaeWaa8aabaWdbiaaigdacqGHRaWkdaWcaaWdaeaaiiaapeGae8 hXdq3damaaDaaajuaibaWdbiaaicdaa8aabaWdbiabeg7aHbaaaKqb a+aabaWdbiabeg7aHjaacgcaaaWaaSaaa8aabaWdbiabgkGi2+aada ahaaqcfasabeaapeGaeqySdegaaaqcfa4daeaapeGaeyOaIyRaamiD a8aadaahaaqabKqbGeaapeGaeqySdegaaaaaaKqbakaawIcacaGLPa aadaWadaWdaeaapeWaaSaaa8aabaWdbiaaigdaa8aabaWdbiaadUga aaWaaSaaa8aabaWdbiabgkGi2kaabI7aa8aabaWdbiabgkGi2kaads haaaGaeyOeI0YaaSaaa8aabaWdbiabeo7aN9aadaWgaaqcfasaa8qa caaIWaaajuaGpaqabaWdbiaadsfapaWaaSbaaKqbGeaapeGaaGimaa qcfa4daeqaaaqaa8qacaWGlbWdamaaBaaajuaibaWdbiaaicdaa8aa beaaaaqcfa4dbiaadQhadaWcaaWdaeaapeGaeyOaIylapaqaa8qacq GHciITcaWG0baaamaabmaapaqaa8qadaWcaaWdaeaapeGaeyOaIy7d amaaCaaabeqcfasaa8qacaaIYaaaaKqbakaadEhaa8aabaWdbiabgk Gi2kaadIhapaWaaWbaaeqajuaibaWdbiaaikdaaaaaaaqcfaOaayjk aiaawMcaaaGaay5waiaaw2faaaaa@8102@  (21)

                  

Solution of the problem

There is no heat flow across the upper and lower surfaces of the nanobeam (thermally insulated), so that θ z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaWdaeaapeGaeyOaIylcbmGaa8hUdaWdaeaapeGaeyOa IyRaa8NEaaaaaaa@3C02@  should be vanish at the upper and lower surfaces of the nanobeam z=±h/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcfaieaa aaaaaaa8qacaWF6bGaeyypa0JaeyySaeRaa8hAaiaac+cacaaIYaaa aa@3CF8@ . For a very nanobeam, assuming that the increment temperature varies in a sinusoidal form along the thickness direction. That is

θ( x,z,t )=Θ( x,t )sin( πz h ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH4oqCdaqadaWdaeaapeGaamiEaiaacYcacaWG6bGaaiil aiaadshaaiaawIcacaGLPaaacqGH9aqpiiaacqWFyoqudaqadaWdae aapeGaamiEaiaacYcacaWG0baacaGLOaGaayzkaaGaci4CaiaacMga caGGUbWaaeWaa8aabaWdbmaalaaapaqaa8qacqaHapaCcaWG6baapa qaa8qacaWGObaaaaGaayjkaiaawMcaaaaa@4D9E@ (22)

                            

Substituting Eq. (22) into Eq. (19), one can get the motion equation of the nanobeams as

4 w x 4 + ρA E 0 F( T 0 ) ( 2 w t 2 ξ 4 w t 2 x 2 )+ 24 α t h π 2 2 Θ x 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaWdaeaapeGaeyOaIy7damaaCaaabeqcfasaa8qacaaI 0aaaaKqbakaadEhaa8aabaWdbiabgkGi2kaadIhapaWaaWbaaeqaju aibaWdbiaaisdaaaaaaKqbakabgUcaRmaalaaapaqaa8qacqaHbpGC caWGbbaapaqaa8qacaWGfbWdamaaBaaajuaibaWdbiaaicdaa8aabe aajuaGpeGaamOramaabmaapaqaa8qacaWGubWdamaaBaaajuaibaWd biaaicdaaKqba+aabeaaa8qacaGLOaGaayzkaaaaamaabmaapaqaa8 qadaWcaaWdaeaapeGaeyOaIy7damaaCaaajuaibeqaa8qacaaIYaaa aKqbakaadEhaa8aabaWdbiabgkGi2kaadshapaWaaWbaaeqajuaiba WdbiaaikdaaaaaaKqbakabgkHiTiabe67a4naalaaapaqaa8qacqGH ciITpaWaaWbaaeqajuaibaWdbiaaisdaaaqcfaOaam4DaaWdaeaape GaeyOaIyRaamiDa8aadaahaaqabKqbGeaapeGaaGOmaaaajuaGcqGH ciITcaWG4bWdamaaCaaabeqcfasaa8qacaaIYaaaaaaaaKqbakaawI cacaGLPaaacqGHRaWkdaWcaaWdaeaapeGaaGOmaiaaisdacqaHXoqy daWgaaqcfasaaiaadshaaKqbagqaaaWdaeaapeGaamiAaiabec8aW9 aadaahaaqabKqbGeaapeGaaGOmaaaaaaqcfa4aaSaaa8aabaWdbiab gkGi2+aadaahaaqabKqbGeaapeGaaGOmaaaaiiaajuaGcqWFyoqua8 aabaWdbiabgkGi2kaadIhapaWaaWbaaKqbGeqabaWdbiaaikdaaaaa aKqbakabg2da9iaaicdaaaa@7962@ (23)

              

Also, the flexure moment can be determined from Eqs. (20) and (22) as M( x,t )=ξAρ 2 w t 2 I E 0 F( T 0 )[ 2 w x 2 + 24 α t h π 2 Θ ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWaaeWaa8aabaWdbiaadIhacaGGSaGaamiDaaGaayjk aiaawMcaaiabg2da9iabe67a4jaadgeacqaHbpGCdaWcaaWdaeaape GaeyOaIy7damaaCaaabeqcfasaa8qacaaIYaaaaKqbakaadEhaa8aa baWdbiabgkGi2kaadshapaWaaWbaaeqajuaibaWdbiaaikdaaaaaaK qbakabgkHiTiaadMeacaWGfbWdamaaBaaajuaibaWdbiaaicdaa8aa beaajuaGpeGaamOramaabmaapaqaa8qacaWGubWdamaaBaaajuaiba Wdbiaaicdaa8aabeaaaKqba+qacaGLOaGaayzkaaWaamWaa8aabaWd bmaalaaapaqaa8qacqGHciITpaWaaWbaaeqajuaibaWdbiaaikdaaa qcfaOaam4DaaWdaeaapeGaeyOaIyRaamiEa8aadaahaaqabKqbGeaa peGaaGOmaaaaaaqcfaOaey4kaSYaaSaaa8aabaWdbiaaikdacaaI0a GaeqySde2damaaBaaajuaibaWdbiaadshaa8aabeaaaKqbagaapeGa amiAaiabec8aW9aadaahaaqabKqbGeaapeGaaGOmaaaaaaaccaqcfa Oae8hMdefacaGLBbGaayzxaaaaaa@69C2@ (24)

                Now, multiplying Eq. (21) by means of 12z/ h 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIXaGaaGOmaiaadQhacaGGVaGaamiAa8aadaahaaqabKqb GeaapeGaaG4maaaaaaa@3BE7@ and integrating it with respect to z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG6baaaa@37A4@ through the beam thickness from h/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGHsislcaWGObGaai4laiaaikdaaaa@39EE@ to h/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGObGaai4laiaaikdaaaa@3901@ , yields

2 Θ x 2 π 2 h 2 2 Θ z 2 =( δ+ τ 0 α α! α t α )[ 1 k Θ t γ 0 T 0 π 2 h 24 K 0 t ( 2 w x 2 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaWdaeaapeGaeyOaIy7damaaCaaabeqcfasaa8qacaaI YaaaaGGaaKqbakab=H5arbWdaeaapeGaeyOaIyRaamiEa8aadaahaa qabKqbGeaapeGaaGOmaaaaaaqcfaOaeyOeI0YaaSaaa8aabaWdbiab ec8aW9aadaahaaqabKqbGeaapeGaaGOmaaaaaKqba+aabaWdbiaadI gapaWaaWbaaeqajuaibaWdbiaaikdaaaaaaKqbaoaalaaapaqaa8qa cqGHciITpaWaaWbaaeqajuaibaWdbiaaikdaaaqcfaOaeuiMdefapa qaa8qacqGHciITcaWG6bWdamaaCaaabeqcfasaa8qacaaIYaaaaaaa juaGcqGH9aqpdaqadaWdaeaapeGaaeiTdiabgUcaRmaalaaapaqaa8 qacqaHepaDpaWaa0baaKqbGeaapeGaaGimaaWdaeaapeGaeqySdega aaqcfa4daeaapeGaeqySdeMaaiyiaaaadaWcaaWdaeaapeGaeyOaIy 7damaaCaaabeqcfasaa8qacqaHXoqyaaaajuaGpaqaa8qacqGHciIT caWG0bWdamaaCaaabeqcfasaa8qacqaHXoqyaaaaaaqcfaOaayjkai aawMcaamaadmaapaqaa8qadaWcaaWdaeaapeGaaGymaaWdaeaapeGa am4AaaaadaWcaaWdaeaapeGaeyOaIyRaeuiMdefapaqaa8qacqGHci ITcaWG0baaaiabgkHiTmaalaaapaqaa8qacqaHZoWzpaWaaSbaaKqb GeaapeGaaGimaaqcfa4daeqaa8qacaWGubWdamaaBaaajuaibaWdbi aaicdaaKqba+aabeaapeGaeqiWda3damaaCaaabeqcfasaa8qacaaI YaaaaKqbakaadIgaa8aabaWdbiaaikdacaaI0aGaam4sa8aadaWgaa qcfasaa8qacaaIWaaapaqabaaaaKqba+qadaWcaaWdaeaapeGaeyOa Iylapaqaa8qacqGHciITcaWG0baaamaabmaapaqaa8qadaWcaaWdae aapeGaeyOaIy7damaaCaaabeqcfasaa8qacaaIYaaaaKqbakaadEha a8aabaWdbiabgkGi2kaadIhapaWaaWbaaeqajuaibaWdbiaaikdaaa aaaaqcfaOaayjkaiaawMcaaaGaay5waiaaw2faaaaa@8D9E@  (25)

                   Now, for simplicity we will use the following non-dimensional variables:

{ x , z , u , w , L , h }= c 0 η 0 { x,z,u,w,L,h },   { t ,τ ' 0 , ξ }= c 0 2 η 0 { t, τ 0 , η 0 ξ }, Θ = Θ T 0 ,    σ x = σ x E 0 ,     M = M c 0 η 0 I E 0 F( T 0 ) ,   c 0 = E 0 ρ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGqa aaaaaaaaWdbmaacmaapaqaa8qaceWG4bWdayaafaWdbiaacYcaceWG 6bWdayaafaWdbiaacYcaceWG1bWdayaafaWdbiaacYcaceWG3bWday aafaWdbiaacYcaceWGmbWdayaafaWdbiaacYcaceWGObWdayaafaaa peGaay5Eaiaaw2haaiabg2da9iaadogapaWaaSbaaKqbGeaapeGaaG imaaWdaeqaaKqba+qacqaH3oaApaWaaSbaaKqbGeaapeGaaGimaaqc fa4daeqaa8qadaGadaWdaeaapeGaamiEaiaacYcacaWG6bGaaiilai aadwhacaGGSaGaam4DaiaacYcacaWGmbGaaiilaiaadIgaaiaawUha caGL9baacaGGSaGaaiiOaiaacckacaGGGcWaaiWaa8aabaWdbiqads hapaGbauaapeGaaiilaGGaaiab=r8a0jaacEcapaWaaSbaaKqbGeaa peGaaGimaaWdaeqaaKqba+qacaGGSaGafqOVdG3dayaafaaapeGaay 5Eaiaaw2haaiabg2da9iaadogapaWaa0baaKqbGeaapeGaaGimaaWd aeaapeGaaGOmaaaajuaGcqaH3oaApaWaaSbaaKqbGeaapeGaaGimaa qcfa4daeqaa8qadaGadaWdaeaapeGaamiDaiaacYcacqWFepaDpaWa aSbaaKqbGeaapeGaaGimaaqcfa4daeqaa8qacaGGSaGaeq4TdG2dam aaBaaajuaibaWdbiaaicdaaKqba+aabeaapeGaeqOVdGhacaGL7bGa ayzFaaGaaiilaaqaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UafuiMdeLbau aacqGH9aqpdaWcaaWdaeaapeGaeuiMdefapaqaa8qacaWGubWdamaa BaaajuaibaWdbiaaicdaaKqba+aabeaaaaWdbiaacYcacaGGGcGaai iOaiaacckacuaHdpWCpaGbauaadaWgaaqcfasaa8qacaWG4baajuaG paqabaWdbiabg2da9maalaaapaqaa8qacqaHdpWCpaWaaSbaaKqbGe aapeGaamiEaaqcfa4daeqaaaqaa8qacaWGfbWdamaaBaaajuaibaWd biaaicdaaKqba+aabeaaaaWdbiaacYcacaGGGcGaaiiOaiaacckaca GGGcGabmyta8aagaqba8qacqGH9aqpdaWcaaWdaeaapeGaamytaaWd aeaapeGaam4ya8aadaWgaaqcfasaa8qacaaIWaaapaqabaqcfa4dbi abeE7aO9aadaWgaaqcfasaa8qacaaIWaaajuaGpaqabaWdbiaadMea caWGfbWdamaaBaaajuaibaWdbiaaicdaaKqba+aabeaapeGaamOram aabmaapaqaa8qacaWGubWdamaaBaaajuaibaWdbiaaicdaa8aabeaa aKqba+qacaGLOaGaayzkaaaaaiaacYcacaGGGcGaaiiOaiaadogapa WaaSbaaKqbGeaapeGaaGimaaqcfa4daeqaa8qacqGH9aqpdaGcaaWd aeaapeWaaSaaa8aabaWdbiaadweapaWaaSbaaKqbGeaapeGaaGimaa qcfa4daeqaaaqaa8qacqaHbpGCaaaabeaacaGGUaaaaaa@D410@ (26)

                    

So, the basic equations in nondimensional forms are simplified as (dropping the primes for convenience)

4 w x 4 + A 1 ( 2 w t 2 ξ 4 w t 2 x 2 )= A 2 2 Θ x 2 ( 2 x 2 A 4 )Θ=( 1+ τ 0 α α! α t α )[ Θ t A 5 t ( 2 w x 2 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaqaaaaa aaaaWdbiaaykW7juaGcaaMc8+aaSaaa8aabaWdbiabgkGi2+aadaah aaqabKqbGeaapeGaaGinaaaajuaGcaWG3baapaqaa8qacqGHciITca WG4bWdamaaCaaajuaibeqaa8qacaaI0aaaaaaajuaGcqGHRaWkcaWG bbWdamaaBaaajuaibaWdbiaaigdaaKqba+aabeaapeWaaeWaa8aaba Wdbmaalaaapaqaa8qacqGHciITpaWaaWbaaeqajuaibaWdbiaaikda aaqcfaOaam4DaaWdaeaapeGaeyOaIyRaamiDa8aadaahaaqabKqbGe aapeGaaGOmaaaaaaqcfaOaeyOeI0IaeqOVdG3aaSaaa8aabaWdbiab gkGi2+aadaahaaqabKqbGeaapeGaaGinaaaajuaGcaWG3baapaqaa8 qacqGHciITcaWG0bWdamaaCaaabeqcfasaa8qacaaIYaaaaKqbakab gkGi2kaadIhapaWaaWbaaeqajuaibaWdbiaaikdaaaaaaaqcfaOaay jkaiaawMcaaiabg2da9iabgkHiTiaadgeapaWaaSbaaKqbGeaapeGa aGOmaaqcfa4daeqaa8qadaWcaaWdaeaapeGaeyOaIy7damaaCaaabe qcfasaa8qacaaIYaaaaKqbakabfI5arbWdaeaapeGaeyOaIyRaamiE a8aadaahaaqabKqbGeaapeGaaGOmaaaaaaaajuaGbaWaaeWaa8aaba Wdbmaalaaapaqaa8qacqGHciITpaWaaWbaaeqajuaibaWdbiaaikda aaaajuaGpaqaa8qacqGHciITcaWG4bWdamaaCaaabeqcfasaa8qaca aIYaaaaaaajuaGcqGHsislcaWGbbWdamaaBaaajuaibaWdbiaaisda aKqba+aabeaaa8qacaGLOaGaayzkaaGaeuiMdeLaeyypa0ZaaeWaa8 aabaWdbiaaigdacqGHRaWkdaWcaaWdaeaapeGaeqiXdq3damaaDaaa juaibaWdbiaaicdaa8aabaWdbiabeg7aHbaaaKqba+aabaWdbiabeg 7aHjaacgcaaaWaaSaaa8aabaWdbiabgkGi2+aadaahaaqabKqbGeaa peGaeqySdegaaaqcfa4daeaapeGaeyOaIyRaamiDa8aadaahaaqabK qbGeaapeGaeqySdegaaaaaaKqbakaawIcacaGLPaaadaWadaWdaeaa peWaaSaaa8aabaWdbiabgkGi2kabfI5arbWdaeaapeGaeyOaIyRaam iDaaaacqGHsislcaWGbbWdamaaBaaajuaibaWdbiaaiwdaaKqba+aa beaapeWaaSaaa8aabaWdbiabgkGi2cWdaeaapeGaeyOaIyRaamiDaa aadaqadaWdaeaapeWaaSaaa8aabaWdbiabgkGi2+aadaahaaqcfasa beaapeGaaGOmaaaajuaGcaWG3baapaqaa8qacqGHciITcaWG4bWdam aaCaaabeqcfasaa8qacaaIYaaaaaaaaKqbakaawIcacaGLPaaaaiaa wUfacaGLDbaaaaaa@A9CF@ (27)

M( x,t )= A 1 ( ξ 2 w t 2 2 w x 2 ) A 2 Θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWaaeWaa8aabaWdbiaadIhacaGGSaGaamiDaaGaayjk aiaawMcaaiabg2da9iaadgeapaWaaSbaaKqbGeaapeGaaGymaaWdae qaaKqba+qadaqadaWdaeaapeGaeqOVdG3aaSaaa8aabaWdbiabgkGi 2+aadaahaaqabKqbGeaapeGaaGOmaaaajuaGcaWG3baapaqaa8qacq GHciITcaWG0bWdamaaCaaabeqcfasaa8qacaaIYaaaaaaajuaGcqGH sisldaWcaaWdaeaapeGaeyOaIy7damaaCaaabeqcfasaa8qacaaIYa aaaKqbakaadEhaa8aabaWdbiabgkGi2kaadIhapaWaaWbaaeqajuai baWdbiaaikdaaaaaaaqcfaOaayjkaiaawMcaaiabgkHiTiaadgeapa WaaSbaaKqbGeaapeGaaGOmaaWdaeqaaKqba+qacqqHyoquaaa@59C6@ (28)

                               where

A 1 = 12 F( T 0 ) h 2 , A 2 = 24 T 0 α t π 2 h ,  A 3 = ξ A 1 , A 4 = π 2 h 2 , A 5 = γ 0 k π 2 h 24 K 0    MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGbbWdamaaBaaajuaibaWdbiaaigdaa8aabeaajuaGpeGa eyypa0ZaaSaaa8aabaWdbiaaigdacaaIYaaapaqaa8qacaWGgbWaae Waa8aabaWdbiaadsfapaWaaSbaaKqbGeaapeGaaGimaaqcfa4daeqa aaWdbiaawIcacaGLPaaacaWGObWdamaaCaaabeqcfasaa8qacaaIYa aaaaaajuaGcaGGSaGaamyqa8aadaWgaaqcfasaa8qacaaIYaaajuaG paqabaWdbiabg2da9maalaaapaqaa8qacaaIYaGaaGinaiaadsfapa WaaSbaaKqbGeaapeGaaGimaaqcfa4daeqaa8qacqaHXoqypaWaaSba aKqbGeaapeGaamiDaaqcfa4daeqaaaqaa8qacqaHapaCpaWaaWbaaK qbGeqabaWdbiaaikdaaaqcfaOaamiAaaaacaGGSaGaaiiOaiaadgea paWaaSbaaKqbGeaapeGaaG4maaqcfa4daeqaa8qacqGH9aqpcaGGGc GaeqOVdGNaamyqa8aadaWgaaqcfasaa8qacaaIXaaapaqabaqcfa4d biaacYcacaWGbbWdamaaBaaajuaibaWdbiaaisdaaKqba+aabeaape Gaeyypa0ZaaSaaa8aabaWdbiabec8aW9aadaahaaqcfasabeaapeGa aGOmaaaaaKqba+aabaWdbiaadIgapaWaaWbaaKqbGeqabaWdbiaaik daaaaaaKqbakaacYcacaWGbbWdamaaBaaajuaibaWdbiaaiwdaaKqb a+aabeaapeGaeyypa0ZaaSaaa8aabaWdbiabeo7aN9aadaWgaaqcfa saa8qacaaIWaaajuaGpaqabaWdbiaabUgacqaHapaCpaWaaWbaaeqa juaibaWdbiaaikdaaaqcfaOaamiAaaWdaeaapeGaaGOmaiaaisdaca WGlbWdamaaBaaajuaibaWdbiaaicdaaKqba+aabeaaaaWdbiaabcka caqGGcaaaa@7E23@

Initial and boundary conditions

To solve the problem, the initial and boundary conditions must be taken into consideration. The homogeneous initial conditions are taken as

Θ( x,0 )= Θ( x,0 ) t =0=w( x,0 )= w( x,0 ) t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHyoqudaqadaWdaeaapeGaamiEaiaacYcacaaIWaaacaGL OaGaayzkaaGaeyypa0ZaaSaaa8aabaWdbiabgkGi2IGaaiab=H5arn aabmaapaqaa8qacaWG4bGaaiilaiaaicdaaiaawIcacaGLPaaaa8aa baWdbiabgkGi2kaadshaaaGaeyypa0JaaGimaiabg2da9iaabEhada qadaWdaeaapeGaamiEaiaacYcacaaIWaaacaGLOaGaayzkaaGaeyyp a0ZaaSaaa8aabaWdbiabgkGi2kaabEhadaqadaWdaeaapeGaamiEai aacYcacaaIWaaacaGLOaGaayzkaaaapaqaa8qacqGHciITcaWG0baa aaaa@58BF@ (29)

                 We will assume that the two ends of the nanobeam are clamped i.e.

w( 0,t )=w( L,t )=0= 2 w( 0,t ) x 2 = 2 w( L,t ) x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqG3bWaaeWaa8aabaWdbiaaicdacaGGSaGaaeiDaaGaayjk aiaawMcaaiabg2da9iaabEhadaqadaWdaeaapeGaamitaiaacYcaca qG0baacaGLOaGaayzkaaGaeyypa0JaaGimaiabg2da9maalaaapaqa a8qacqGHciITpaWaaWbaaeqajuaibaWdbiaaikdaaaqcfaOaae4Dam aabmaapaqaa8qacaaIWaGaaiilaiaabshaaiaawIcacaGLPaaaa8aa baWdbiabgkGi2kaadIhapaWaaWbaaeqajuaibaWdbiaaikdaaaaaaK qbakabg2da9maalaaapaqaa8qacqGHciITpaWaaWbaaeqajuaibaWd biaaikdaaaqcfaOaae4Damaabmaapaqaa8qacaWGmbGaaiilaiaabs haaiaawIcacaGLPaaaa8aabaWdbiabgkGi2kaadIhapaWaaWbaaeqa juaibaWdbiaaikdaaaaaaaaa@5E35@  (30)

                    Also, we consider the nanobeam is loaded thermally by ramp-type heating, which give

Θ( x,t )= Θ 0 { 0,                t0, t t 0 ,     0t t 0 1,                 t>0 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcfaieaa aaaaaaa8qacqWFyoqudaqadaWdaeaapeGaamiEaiaacYcacaWG0baa caGLOaGaayzkaaGaeyypa0Jae8hMde1damaaBaaajuaibaWdbiaaic daaKqba+aabeaapeWaaiqaa8aabaqbaeqabmqaaaqaa8qacaaIWaGa aiilaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckaca GGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaa cckacaWG0bGaeyizImQaaGimaiaacYcaa8aabaWdbmaalaaapaqaa8 qacaWG0baapaqaa8qacaWG0bWdamaaBaaajuaibaWdbiaaicdaa8aa beaaaaqcfa4dbiaacYcacaGGGcGaaiiOaiaacckacaGGGcGaaiiOai aaicdacqGHKjYOcaWG0bGaeyizImQaamiDa8aadaWgaaqcfasaa8qa caaIWaaapaqabaaajuaGbaWdbiaaigdacaGGSaGaaiiOaiaacckaca GGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaa cckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaWG0bGaey Opa4JaaGimaaaaaiaawUhaaiaacYcaaaa@8481@ (31)

                     Where t0 is a non–negative constant called ramp–type parameter and Θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcfaieaa aaaaaaa8qacqWFyoqupaWaaSbaaKqbGeaapeGaaGimaaqcfa4daeqa aaaa@39E5@ is a constant. In addition, the temperature at the end boundary should satisfy the following relation

Θ x =0       on      x=L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaWdaeaapeGaeyOaIylccaGae8hMdefapaqaa8qacqGH ciITcaWG4baaaiabg2da9iaaicdacaGGGcGaaiiOaiaacckacaGGGc GaaiiOaiaacckacaGGGcGaae4Baiaab6gacaGGGcGaaiiOaiaaccka caGGGcGaaiiOaiaacckacaWG4bGaeyypa0Jaamitaaaa@5182@ (32)

         

Solution of the problem in the Laplace transform domain

The closed form solution of the governing and constitutive equations can be possible by adapting the Laplace transformation method. Taking the Laplace transform defined by the relation

f ¯ ( x,t )= 0 f( x,t ) e st dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWGMbGbaebadaqadaWdaeaapeGaamiEaiaacYcacaWG0baa caGLOaGaayzkaaGaeyypa0Jaey4kIi=aa0baaKqbGeaacaaIWaaaba GaeyOhIukaaKqbakaadAgadaqadaWdaeaapeGaamiEaiaacYcacaWG 0baacaGLOaGaayzkaaGaamyza8aadaahaaqcfasabeaapeGaeyOeI0 Iaam4CaiaadshaaaqcfaOaamizaiaadshaaaa@4DAF@ (33)

          to both sides of Eqs. (27) and (28) and using the homogeneous initial conditions (29), one gets the field equations in the Laplace transform space as

( d 4 d x 4 A 3 s 2 d 2 d x 2 + A 1 s 2 ) w ¯ = A 2 d 2 Θ ¯ d x 2 ,  ( 2 x 2 B 1 ) Θ ¯ = B 2 2 w ¯ x 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGqa aaaaaaaaWdbmaabmaapaqaa8qadaWcaaWdaeaapeGaamiza8aadaah aaqabKqbGeaapeGaaGinaaaaaKqba+aabaWdbiaadsgacaWG4bWdam aaCaaabeqcfasaa8qacaaI0aaaaaaajuaGcqGHsislcaWGbbWdamaa BaaajuaibaWdbiaaiodaaKqba+aabeaapeGaam4Ca8aadaahaaqabK qbGeaapeGaaGOmaaaajuaGdaWcaaWdaeaapeGaamiza8aadaahaaqa bKqbGeaapeGaaGOmaaaaaKqba+aabaWdbiaadsgacaWG4bWdamaaCa aabeqcfasaa8qacaaIYaaaaaaajuaGcqGHRaWkcaWGbbWdamaaBaaa juaibaWdbiaaigdaa8aabeaajuaGpeGaam4Ca8aadaahaaqabKqbGe aapeGaaGOmaaaaaKqbakaawIcacaGLPaaaceWG3bGbaebacqGH9aqp cqGHsislcaWGbbWdamaaBaaajuaibaWdbiaaikdaaKqba+aabeaape WaaSaaa8aabaWdbiaadsgapaWaaWbaaeqajuaibaWdbiaaikdaaaqc fa4daiqbfI5arzaaraaabaWdbiaadsgacaWG4bWdamaaCaaabeqcfa saa8qacaaIYaaaaaaajuaGcaGGSaGaaiiOaaqaaiaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8+aaeWaa8aa baWdbmaalaaapaqaa8qacqGHciITpaWaaWbaaeqajuaibaWdbiaaik daaaaajuaGpaqaa8qacqGHciITcaWG4bWdamaaCaaabeqcfasaa8qa caaIYaaaaaaajuaGcqGHsislcaWGcbWdamaaBaaajuaibaWdbiaaig daaKqba+aabeaaa8qacaGLOaGaayzkaaWdaiqbfI5arzaaraWdbiab g2da9iabgkHiTiaadkeapaWaaSbaaKqbGeaapeGaaGOmaaqcfa4dae qaa8qadaWcaaWdaeaapeGaeyOaIy7damaaCaaabeqcfasaa8qacaaI YaaaaKqbakqadEhagaqeaaWdaeaapeGaeyOaIyRaamiEa8aadaahaa qabKqbGeaapeGaaGOmaaaaaaqcfaOaaiilaaaaaa@97F8@ (34)            

M ¯ ( x,t )= A 1 ( ξ s 2 w ¯ d 2 w ¯ d x 2 ) A 2 Θ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWGnbGbaebadaqadaWdaeaapeGaamiEaiaacYcacaWG0baa caGLOaGaayzkaaGaeyypa0Jaamyqa8aadaWgaaqcfasaa8qacaaIXa aapaqabaqcfa4dbmaabmaapaqaa8qacqaH+oaEcaWGZbWdamaaCaaa beqcfasaa8qacaaIYaaaaKqbakqadEhagaqeaiabgkHiTmaalaaapa qaa8qacaWGKbWdamaaCaaajuaibeqaa8qacaaIYaaaaKqbakqadEha gaqeaaWdaeaapeGaamizaiaadIhapaWaaWbaaeqajuaibaWdbiaaik daaaaaaaqcfaOaayjkaiaawMcaaiabgkHiTiaadgeapaWaaSbaaKqb GeaapeGaaGOmaaqcfa4daeqaaiqbfI5arzaaraaaaa@5447@ (35)

                      where                 B 1 =s( δ+ s α τ 0 α α! )+ A 4 ,     B 2 =s( δ+ s α τ 0 α α! ) A 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGcbWdamaaBaaajuaibaWdbiaaigdaaKqba+aabeaapeGa eyypa0Jaam4Camaabmaapaqaa8qacaqG0oGaey4kaSYaaSaaa8aaba WdbiaadohapaWaaWbaaeqajuaibaWdbiabeg7aHbaajuaGcqaHepaD paWaa0baaKqbGeaapeGaaGimaaWdaeaapeGaeqySdegaaaqcfa4dae aapeGaeqySdeMaaiyiaaaaaiaawIcacaGLPaaacqGHRaWkcaWGbbWd amaaBaaajuaibaWdbiaaisdaa8aabeaajuaGpeGaaiilaiaacckaca GGGcGaaiiOaiaacckacaWGcbWdamaaBaaajuaibaWdbiaaikdaa8aa beaajuaGpeGaeyypa0Jaam4Camaabmaapaqaa8qacaqG0oGaey4kaS YaaSaaa8aabaWdbiaadohapaWaaWbaaeqajuaibaWdbiabeg7aHbaa juaGcqaHepaDpaWaa0baaKqbGeaapeGaaGimaaWdaeaapeGaeqySde gaaaqcfa4daeaapeGaeqySdeMaaiyiaaaaaiaawIcacaGLPaaacaWG bbWdamaaBaaajuaibaWdbiaaiwdaaKqba+aabeaaaaa@6974@ Elimination Θ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafuiMde Lbaebaaaa@3813@ or< w ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWG3bGbaebaaaa@37B8@ from Eqs. (34), one obtains:

( D 6 A D 4 +B D 2 C ){ Θ ¯ , w ¯ }( x )=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaWdaeaapeGaamira8aadaahaaqcfasabeaapeGaaGOn aaaajuaGcqGHsislcaWGbbGaamira8aadaahaaqabKqbGeaapeGaaG inaaaajuaGcqGHRaWkcaWGcbGaamira8aadaahaaqabKqbGeaapeGa aGOmaaaajuaGcqGHsislcaWGdbaacaGLOaGaayzkaaWaaiWaa8aaba accaGaf8hMdeLbaebapeGaaiilaiqadEhagaqeaaGaay5Eaiaaw2ha amaabmaapaqaa8qacaWG4baacaGLOaGaayzkaaGaeyypa0JaaGimai aacYcaaaa@4FA5@ (36)

                     where the coefficients A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGbbaaaa@376A@ , B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGcbaaaa@376B@ and C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGdbaaaa@376C@ are given by

A= A 3 s 2 + A 2 B 2 + B 1 ,    B= A 1 s 2 + B 1 A 3 s 2 , C= A 1 B 1 s 2 ,D= d dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGbbGaeyypa0Jaamyqa8aadaWgaaqcfasaa8qacaaIZaaa juaGpaqabaWdbiaadohapaWaaWbaaKqbGeqabaWdbiaaikdaaaqcfa Oaey4kaSIaamyqa8aadaWgaaqcfasaa8qacaaIYaaajuaGpaqabaWd biaadkeapaWaaSbaaKqbGeaapeGaaGOmaaqcfa4daeqaa8qacqGHRa WkcaWGcbWdamaaBaaajuaibaWdbiaaigdaa8aabeaajuaGpeGaaiil aiaacckacaGGGcGaaiiOaiaacckacaWGcbGaeyypa0Jaamyqa8aada Wgaaqcfasaa8qacaaIXaaapaqabaqcfa4dbiaadohapaWaaWbaaeqa juaibaWdbiaaikdaaaqcfaOaey4kaSIaamOqa8aadaWgaaqcfasaa8 qacaaIXaaapaqabaqcfa4dbiaadgeapaWaaSbaaKqbGeaapeGaaG4m aaqcfa4daeqaa8qacaWGZbWdamaaCaaabeqcfasaa8qacaaIYaaaaK qbakaacYcacaGGGcGaam4qaiabg2da9iaadgeapaWaaSbaaKqbGeaa peGaaGymaaqcfa4daeqaa8qacaWGcbWdamaaBaaajuaibaWdbiaaig daaKqba+aabeaapeGaam4Ca8aadaahaaqabKqbGeaapeGaaGOmaaaa juaGcaGGSaGaamiraiabg2da9maalaaapaqaa8qacaWGKbaapaqaa8 qacaWGKbGaamiEaaaaaaa@6DAB@ (37)

                     Equation (36) can be moderated to ( D 2 m 1 2 )( D 2 m 2 2 )( D 2 m 3 2 ){ Θ ¯ , w ¯ }( x )=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaWdaeaapeGaamira8aadaahaaqabKqbGeaapeGaaGOm aaaajuaGcqGHsislcaWGTbWdamaaDaaajuaibaWdbiaaigdaa8aaba WdbiaaikdaaaaajuaGcaGLOaGaayzkaaWaaeWaa8aabaWdbiaadsea paWaaWbaaeqajuaibaWdbiaaikdaaaqcfaOaeyOeI0IaamyBa8aada qhaaqcfasaa8qacaaIYaaapaqaa8qacaaIYaaaaaqcfaOaayjkaiaa wMcaamaabmaapaqaa8qacaWGebWdamaaCaaabeqcfasaa8qacaaIYa aaaKqbakabgkHiTiaad2gapaWaa0baaKqbGeaapeGaaG4maaWdaeaa peGaaGOmaaaaaKqbakaawIcacaGLPaaadaGadaWdaeaaiiaacuWFyo qugaqea8qacaGGSaGabm4DayaaraaacaGL7bGaayzFaaWaaeWaa8aa baWdbiaadIhaaiaawIcacaGLPaaacqGH9aqpcaaIWaGaaiilaaaa@5B37@ (38)

              where m n 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGTbWdamaaDaaajuaibaWdbiaad6gaa8aabaWdbiaaikda aaqcfaOaaiilaaaa@3B11@ n=1,2,3,4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGUbGaeyypa0JaaGymaiaacYcacaaIYaGaaiilaiaaioda caGGSaGaaGinaaaa@3D9F@ are roots of

m 6 A m 4 +B m 2 C=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGTbWdamaaCaaabeqcfasaa8qacaaI2aaaaKqbakabgkHi TiaadgeacaWGTbWdamaaCaaajuaibeqaa8qacaaI0aaaaKqbakabgU caRiaadkeacaWGTbWdamaaCaaabeqcfasaa8qacaaIYaaaaKqbakab gkHiTiaadoeacqGH9aqpcaaIWaGaaiilaaaa@462C@ (39)

              The solution of the governing equations (39) in the Laplace transformation domain can be represented as { w ¯ , Θ ¯ }( x )= n=1 3 ( { 1, β n } C n e m n x +{ 1, β n+3 } C n+3 e m n x ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaGadaWdaeaapeGabm4DayaaraGaaiilaGGaa8aacuWFyoqu gaqeaaWdbiaawUhacaGL9baadaqadaWdaeaapeGaamiEaaGaayjkai aawMcaaiabg2da9maawahabeqcfaYdaeaapeGaamOBaiabg2da9iaa igdaa8aabaWdbiaaiodaaKqba+aabaWdbiabggHiLdaadaqadaWdae aapeWaaiWaa8aabaWdbiaaigdacaGGSaGaeqOSdi2damaaBaaajuai baWdbiaad6gaaKqba+aabeaaa8qacaGL7bGaayzFaaGaam4qa8aada Wgaaqcfasaa8qacaWGUbaapaqabaqcfa4dbiaabwgapaWaaWbaaeqa juaibaWdbiabgkHiTiaad2gajuaGpaWaaSbaaKqbGeaapeGaamOBaa Wdaeqaa8qacaWG4baaaKqbakabgUcaRmaacmaapaqaa8qacaaIXaGa aiilaiabek7aI9aadaWgaaqaaKqbG8qacaWGUbGaey4kaSIaaG4maa qcfa4daeqaaaWdbiaawUhacaGL9baacaWGdbWdamaaBaaabaqcfaYd biaad6gacqGHRaWkcaaIZaaajuaGpaqabaWdbiaabwgapaWaaWbaae qajuaibaWdbiaad2gajuaGpaWaaSbaaKqbGeaapeGaamOBaaWdaeqa a8qacaWG4baaaaqcfaOaayjkaiaawMcaaiaac6caaaa@6E7C@       (40) Where the compatibility between these two equations and Eq. (34), gives

β n = m n 2 B 2 m n 2 B 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHYoGypaWaaSbaaKqbGeaapeGaamOBaaqcfa4daeqaa8qa cqGH9aqpcqGHsisldaWcaaWdaeaapeGaamyBa8aadaqhaaqcfasaa8 qacaWGUbaapaqaa8qacaaIYaaaaKqbakaadkeapaWaaSbaaKqbGeaa peGaaGOmaaqcfa4daeqaaaqaa8qacaWGTbWdamaaDaaajuaibaWdbi aad6gaa8aabaWdbiaaikdaaaqcfaOaeyOeI0IaamOqa8aadaWgaaqc fasaa8qacaaIXaaajuaGpaqabaaaaaaa@4A07@ (41)

       where C n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGdbWdamaaBaaajuaibaWdbiaad6gaaKqba+aabeaaaaa@396A@ and β n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHYoGypaWaaSbaaKqbGeaapeGaamOBaaqcfa4daeqaaaaa @3A43@ are parameters depending on s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Caa aa@377C@ .

The axial displacement after using Eq. (40) takes the form u ¯ ( x )=z d w ¯ dx =z n=1 3 m n ( C n e m n x C n+3 e m n x ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWG1bGbaebadaqadaWdaeaapeGaamiEaaGaayjkaiaawMca aiabg2da9iabgkHiTiaadQhadaWcaaWdaeaapeGaamizaiqadEhaga qeaaWdaeaapeGaamizaiaadIhaaaGaeyypa0JaamOEamaawahabeqc faYdaeaapeGaamOBaiabg2da9iaaigdaa8aabaWdbiaaiodaaKqba+ aabaWdbiabggHiLdaacaWGTbWdamaaBaaajuaibaWdbiaad6gaa8aa beaajuaGpeWaaeWaa8aabaWdbiaadoeapaWaaSbaaKqbGeaapeGaam OBaaqcfa4daeqaa8qacaqGLbWdamaaCaaabeqcfasaa8qacqGHsisl caWGTbqcfa4damaaBaaajuaibaWdbiaad6gaa8aabeaapeGaamiEaa aajuaGcqGHsislcaWGdbWdamaaBaaajuaibaWdbiaad6gacqGHRaWk caaIZaaajuaGpaqabaWdbiaabwgapaWaaWbaaeqajuaibaWdbiaad2 gajuaGpaWaaSbaaKqbGeaapeGaamOBaaWdaeqaa8qacaWG4baaaaqc faOaayjkaiaawMcaaiaac6caaaa@6482@ (42)

             

Substituting the expressions of w ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWG3bGbaebaaaa@37B8@ and Θ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafuiMde Lbaebaaaa@3813@ from (40) into (36), we get at the solution for the bending moment M ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcfaOab8 xtayaaraaaaa@3776@ as follows:

M ¯ ( x )= n=1 3 ( A 3 s 2 m n 2 A 2 β n )( C n e m n x C n+3 e m n x ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWGnbGbaebadaqadaWdaeaapeGaamiEaaGaayjkaiaawMca aiabg2da9maawahabeqcfaYdaeaapeGaamOBaiabg2da9iaaigdaa8 aabaWdbiaaiodaaKqba+aabaWdbiabggHiLdaadaqadaWdaeaapeGa amyqa8aadaWgaaqcfasaa8qacaaIZaaapaqabaqcfa4dbiaadohapa WaaWbaaeqajuaibaWdbiaaikdaaaqcfaOaeyOeI0IaamyBa8aadaqh aaqcfasaa8qacaWGUbaapaqaa8qacaaIYaaaaKqbakabgkHiTiaadg eapaWaaSbaaKqbGeaapeGaaGOmaaqcfa4daeqaa8qacqaHYoGypaWa aSbaaKqbGeaapeGaamOBaaWdaeqaaaqcfa4dbiaawIcacaGLPaaada qadaWdaeaapeGaam4qa8aadaWgaaqcfasaa8qacaWGUbaapaqabaqc fa4dbiaabwgapaWaaWbaaeqajuaibaWdbiabgkHiTiaad2gajuaGpa WaaSbaaKqbGeaapeGaamOBaaWdaeqaa8qacaWG4baaaKqbakaadoea paWaaSbaaKqbGeaapeGaamOBaiabgUcaRiaaiodaa8aabeaajuaGpe Gaaeyza8aadaahaaqabKqbGeaapeGaamyBaKqba+aadaWgaaqcfasa a8qacaWGUbaapaqabaWdbiaadIhaaaaajuaGcaGLOaGaayzkaaGaai Olaaaa@6B25@ (43)

              

In addition, the strain will be

e ¯ ( x )= d u ¯ dx =z n=1 3 m n 2 ( C n e m n x + C n+1 e m n x ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWGLbGbaebadaqadaWdaeaapeGaamiEaaGaayjkaiaawMca aiabg2da9maalaaapaqaa8qacaWGKbGabmyDayaaraaapaqaa8qaca WGKbGaamiEaaaacqGH9aqpcqGHsislcaWG6bWaaybCaeqajuaipaqa a8qacaWGUbGaeyypa0JaaGymaaWdaeaapeGaaG4maaqcfa4daeaape GaeyyeIuoaaiaad2gapaWaa0baaKqbGeaapeGaamOBaaWdaeaapeGa aGOmaaaajuaGdaqadaWdaeaapeGaam4qa8aadaWgaaqcfasaa8qaca WGUbaajuaGpaqabaWdbiaabwgapaWaaWbaaeqajuaibaWdbiabgkHi Tiaad2gajuaGpaWaaSbaaKqbGeaapeGaamOBaaWdaeqaa8qacaWG4b aaaKqbakabgUcaRiaadoeapaWaaSbaaKqbGeaapeGaamOBaiabgUca RiaaigdaaKqba+aabeaapeGaaeyza8aadaahaaqabKqbGeaapeGaam yBaKqba+aadaWgaaqcfasaa8qacaWGUbaapaqabaWdbiaadIhaaaaa juaGcaGLOaGaayzkaaGaaiOlaaaa@6421@ (44)

             After using Laplace transform, the boundary conditions (30)‒(32) take the forms

w ¯ ( 0,s )= w ¯ ( L,s )=0,     2 w ¯ ( 0,s ) x 2 = 2 w ¯ ( L,s ) x 2 =0, Θ ¯ ( 0,s ) x = Θ 0 ( 1 e s t 0 s 2 t 0 )= G ¯ ( s ),    Θ ¯ ( L,s ) x =0   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGqa aaaaaaaaWdbiqadEhagaqeamaabmaapaqaa8qacaaIWaGaaiilaiaa dohaaiaawIcacaGLPaaacqGH9aqpceWG3bGbaebadaqadaWdaeaape GaamitaiaacYcacaWGZbaacaGLOaGaayzkaaGaeyypa0JaaGimaiaa cYcacaGGGcGaaiiOaiaacckacaGGGcaabaWaaSaaa8aabaWdbiabgk Gi2+aadaahaaqabKqbGeaapeGaaGOmaaaajuaGceWG3bGbaebadaqa daWdaeaapeGaaGimaiaacYcacaWGZbaacaGLOaGaayzkaaaapaqaa8 qacqGHciITcaWG4bWdamaaCaaajuaibeqaa8qacaaIYaaaaaaajuaG cqGH9aqpdaWcaaWdaeaapeGaeyOaIy7damaaCaaabeqcfasaa8qaca aIYaaaaKqbakqadEhagaqeamaabmaapaqaa8qacaWGmbGaaiilaiaa dohaaiaawIcacaGLPaaaa8aabaWdbiabgkGi2kaadIhapaWaaWbaae qajuaibaWdbiaaikdaaaaaaKqbakabg2da9iaaicdacaGGSaaabaWa aSaaa8aabaWdbiabgkGi2IGaa8aacuWFyoqugaqea8qadaqadaWdae aapeGaaGimaiaacYcacaWGZbaacaGLOaGaayzkaaaapaqaa8qacqGH ciITcaWG4baaaiabg2da9iab=H5ar9aadaWgaaqcfasaa8qacaaIWa aajuaGpaqabaWdbmaabmaapaqaa8qadaWcaaWdaeaapeGaaGymaiab gkHiTiaadwgapaWaaWbaaeqabaWdbiabgkHiTiaadohacaWG0bWdam aaBaaajuaibaWdbiaaicdaaKqba+aabeaaaaaabaWdbiaadohapaWa aWbaaeqabaWdbiaaikdaaaGaamiDa8aadaWgaaqcfasaa8qacaaIWa aapaqabaaaaaqcfa4dbiaawIcacaGLPaaacqGH9aqppaGabm4rayaa raWdbmaabmaapaqaa8qacaWGZbaacaGLOaGaayzkaaGaaiilaiaacc kacaGGGcGaaiiOaaqaamaalaaapaqaa8qacqGHciITpaGaf8hMdeLb aebapeWaaeWaa8aabaWdbiaadYeacaGGSaGaam4CaaGaayjkaiaawM caaaWdaeaapeGaeyOaIyRaamiEaaaacqGH9aqpcaaIWaGaaiiOaiaa cckaaaaa@98CD@ (45)

                 Substituting Eq. (40) into the above boundary conditions, one obtains six linear equations;

n=1 3 ( C n + C n+1 )=0, n=1 3 ( C n e m n L + C n+1 e m n L )=0,   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGqa aaaaaaaaWdbmaawahabeqcfaYdaeaapeGaamOBaiabg2da9iaaigda a8aabaWdbiaaiodaaKqba+aabaWdbiabggHiLdaadaqadaWdaeaape Gaam4qa8aadaWgaaqcfasaa8qacaWGUbaapaqabaqcfa4dbiabgUca RiaadoeapaWaaSbaaKqbGeaapeGaamOBaiabgUcaRiaaigdaa8aabe aaaKqba+qacaGLOaGaayzkaaGaeyypa0JaaGimaiaacYcaaeaadaGf WbqabKqbG8aabaWdbiaad6gacqGH9aqpcaaIXaaapaqaa8qacaaIZa aajuaGpaqaa8qacqGHris5aaWaaeWaa8aabaWdbiaadoeapaWaaSba aKqbGeaapeGaamOBaaWdaeqaaKqba+qacaqGLbWdamaaCaaabeqcfa saa8qacqGHsislcaWGTbqcfa4damaaBaaajuaibaWdbiaad6gaa8aa beaapeGaamitaaaajuaGcqGHRaWkcaWGdbWdamaaBaaajuaibaWdbi aad6gacqGHRaWkcaaIXaaajuaGpaqabaWdbiaabwgapaWaaWbaaeqa juaibaWdbiaad2gajuaGpaWaaSbaaKqbGeaapeGaamOBaaWdaeqaa8 qacaWGmbaaaaqcfaOaayjkaiaawMcaaiabg2da9iaaicdacaGGSaGa aiiOaiaacckaaaaa@6B4A@  (46)

      

n=1 3 m n 2 ( C n + C n+1 )=0, n=1 3 m n 2 ( C n e m n L + C n+1 e m n L )=0,     MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGqa aaaaaaaaWdbmaawahabeqcfaYdaeaapeGaamOBaiabg2da9iaaigda a8aabaWdbiaaiodaaKqba+aabaWdbiabggHiLdaacaWGTbWdamaaDa aajuaibaWdbiaad6gaa8aabaWdbiaaikdaaaqcfa4aaeWaa8aabaWd biaadoeapaWaaSbaaKqbGeaapeGaamOBaaqcfa4daeqaa8qacqGHRa WkcaWGdbWdamaaBaaajuaibaWdbiaad6gacqGHRaWkcaaIXaaajuaG paqabaaapeGaayjkaiaawMcaaiabg2da9iaaicdacaGGSaaabaWaay bCaeqajuaipaqaa8qacaWGUbGaeyypa0JaaGymaaWdaeaapeGaaG4m aaqcfa4daeaapeGaeyyeIuoaaiaad2gapaWaa0baaKqbGeaapeGaam OBaaWdaeaapeGaaGOmaaaajuaGdaqadaWdaeaapeGaam4qa8aadaWg aaqcfasaa8qacaWGUbaajuaGpaqabaWdbiaabwgapaWaaWbaaeqaju aibaWdbiabgkHiTiaad2gajuaGpaWaaSbaaKqbGeaapeGaamOBaaWd aeqaa8qacaWGmbaaaKqbakabgUcaRiaadoeapaWaaSbaaKqbGeaape GaamOBaiabgUcaRiaaigdaa8aabeaajuaGpeGaaeyza8aadaahaaqa bKqbGeaapeGaamyBaKqba+aadaWgaaqcfasaa8qacaWGUbaapaqaba WdbiaadYeaaaaajuaGcaGLOaGaayzkaaGaeyypa0JaaGimaiaacYca caGGGcGaaiiOaiaacckacaGGGcaaaaa@750C@  (47)

          

n=1 3 m n ( β n C n β n+1 C n+1 )=G( s ), n=1 3 m n ( β n C n e m n L β n+1 C n+1 e m n L )=0,  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGqa aaaaaaaaWdbmaawahabeqcfaYdaeaapeGaamOBaiabg2da9iaaigda a8aabaWdbiaaiodaaKqba+aabaWdbiabggHiLdaacaWGTbWdamaaBa aajuaibaWdbiaad6gaaKqba+aabeaapeWaaeWaa8aabaWdbiabek7a I9aadaWgaaqcfasaa8qacaWGUbaajuaGpaqabaWdbiaadoeapaWaaS baaKqbGeaapeGaamOBaaqcfa4daeqaa8qacqGHsislcqaHYoGypaWa aSbaaKqbGeaapeGaamOBaiabgUcaRiaaigdaa8aabeaajuaGpeGaam 4qa8aadaWgaaqcfasaa8qacaWGUbGaey4kaSIaaGymaaWdaeqaaaqc fa4dbiaawIcacaGLPaaacqGH9aqpcqGHsislcaqGhbWaaeWaa8aaba WdbiaabohaaiaawIcacaGLPaaacaGGSaaabaWaaybCaeqajuaipaqa a8qacaWGUbGaeyypa0JaaGymaaWdaeaapeGaaG4maaqcfa4daeaape GaeyyeIuoaaiaad2gapaWaaSbaaKqbGeaapeGaamOBaaqcfa4daeqa a8qadaqadaWdaeaapeGaeqOSdi2damaaBaaajuaibaWdbiaad6gaaK qba+aabeaapeGaam4qa8aadaWgaaqcfasaa8qacaWGUbaajuaGpaqa baWdbiaabwgapaWaaWbaaeqajuaibaWdbiabgkHiTiaad2gajuaGpa WaaSbaaKqbGeaapeGaamOBaaWdaeqaa8qacaWGmbaaaKqbakabgkHi Tiabek7aI9aadaWgaaqcfasaa8qacaWGUbGaey4kaSIaaGymaaqcfa 4daeqaa8qacaWGdbWdamaaBaaajuaibaWdbiaad6gacqGHRaWkcaaI Xaaapaqabaqcfa4dbiaabwgapaWaaWbaaKqbGeqabaWdbiaad2gaju aGpaWaaSbaaKqbGeaapeGaamOBaaWdaeqaa8qacaWGmbaaaaqcfaOa ayjkaiaawMcaaiabg2da9iaaicdacaGGSaGaaiiOaaaaaa@85CD@  (48)

           

The solution of the above system of linear equations gives the unknown parameters,. In order to determine the studied fields in the physiacl domain, the Riemann-sum approximation method is used to obtain the numerical results. The details of these methods can be found in Honig & Hirdes.33

Special case

The following special cases can be obtained from the system of Eqs. (1), (8) and (9):

  1. The equations of a coupled theory of nonlocal thermoelasticity with temperature-dependent mechanical properties result from (1), (8), in the limiting caseand (11) by lettingand putting.
  2. The equations of a coupled theory of local thermoelasticity with temperature-dependent mechanical properties result from (1), (8), in the limiting caseand (11) by lettingand putting.
  3. The equations of a coupled theory of nonlocal thermoelasticity with temperature-independent mechanical properties result from (1), (8) and (9), in the limiting caseand (11) by letting,and putting.
  4. The equations of a coupled theory of local thermoelasticity with temperature-independent mechanical properties result from (1), (8) and (9), in the limiting caseby lettingand putting.
  5. The equations of a generalized theory of nonlocal thermoelasticity with temperature-dependent mechanical properties without fractional derivatives are obtained from Eqs. (1), (8) and (9) by letting the fractional parameterand putting.
  6. The equations of a generalized theory of nonlocal thermoelasticity with temperature-independent mechanical properties without fractional derivatives are obtained from Eqs. (1), (8) and (9) by letting the fractional parameterand puttingand.
  7. The equations of the generalized nonlocal thermoelasticity without energy dissipation with temperature-dependent mechanical properties without fractional derivatives result from Eqs. (1), (8) and (9) by letting,and.

Numerical results

In the present work, the thermoelastic coupling effect is analyzed by considering a beam made of silicon (Table 1).

Material properties/ Material

Metal (Aluminum)

Thermal conductivity K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGlbaaaa@3774@ (W m (1) K (1) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaGGOaGaam4vaiaad2gapaWaaWbaaeqajuaibaWdbiaacIca cqGHsislcaaIXaGaaiykaaaajuaGcaWGlbWdamaaCaaajuaibeqaa8 qacaGGOaGaeyOeI0IaaGymaiaacMcaaaqcfaOaaiykaaaa@4297@

          156

Young' modulus E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGfbaaaa@376E@ (GPa) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiikai aadEeacaWGqbGaamyyaiaacMcaaaa@3A64@

          169

Density ρ( Kgm 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHbpGCdaqadaWdaeaapeGaae4saiaabEgacaqGTbWdamaa Caaabeqcfasaa8qacqGHsislcaaIZaaaaaqcfaOaayjkaiaawMcaaa aa@3F5B@

          2330

Thermal expansion α t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHXoqypaWaaSbaaKqbGeaapeGaaeiDaaWdaeqaaaaa@39B7@ ( K 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaGGOaGaae4sa8aadaahaaqabKqbGeaapeGaeyOeI0IaaGym aaaajuaGcaGGPaaaaa@3B70@

2.59× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIYaGaaiOlaiaaiwdacaaI5aGaey41aqRaaGymaiaaicda paWaaWbaaeqajuaibaWdbiabgkHiTiaaiAdaaaaaaa@3F3C@

Thermal diffusivity k ( m 2 s 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGRbGaaiiOamaabmaapaqaa8qacaqGTbWdamaaCaaabeqc fasaa8qacaaIYaaaaKqbakaabohapaWaaWbaaeqajuaqbaWdbiabgk HiTiaaigdaaaaajuaGcaGLOaGaayzkaaaaaa@40C4@

84.18× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaI4aGaaGinaiaac6cacaaIXaGaaGioaiabgEna0kaaigda caaIWaWdamaaCaaajuaibeqaa8qacqGHsislcaaI2aaaaaaa@3FFB@

Poisson's ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH9oGBaaa@385C@

           0.22

Specific heat C E  ( J/kgK  ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGdbWdamaaBaaajuaibaWdbiaadweaaKqba+aabeaapeGa aiiOamaabmaapaqaa8qacaqGkbGaai4laiaabUgacaqGNbGaae4sai aabckaaiaawIcacaGLPaaaaaa@4166@

            713

Table 1 Mechanical and thermoelastic properties parameter of the graded nanobeam

Also, the length−to−thickness ratio of the nanobeam is fixed asand the dimensional parameters carried out in the numerical simulations areand. For all numerical calculations Mathematica programming Language has been used. The numerical calculations of the flexure momentthermodynamical temperature, displacement, lateral vibrationand strainhave been considered for various values of the nonlocal parameterfractional orderthickness of nanobeam and periodic frequency. The results are investigated graphically in Figures 2−17 in the wide range ofat different positions of. Some plots consider the present quantitis throug the length of the beam and others take into account both the length and thickness directions.

Figure 2 The transverse deflection with different fractional order parameter .

Figure 3 The temperature with different fractional order parameter .

Figure 4 The displacement with different fractional order parameter .

Figure 5 The flexure moment with different fractional order parameter .

Figure 6 The transverse deflection with different nonlocal parameter .

!--Figure start here-->

Figure 7 The temperature with different nonlocal parameter .

Figure 8 The displacement with different nonlocal parameter .

Figure 9 The flexure moment with different nonlocal parameter .

Figure 10 The transverse deflection with different ramping time .

Figure 11 The temperature with different ramping time .

Figure 12 The displacement with different ramping time .

Figure 13 The flexure moment with different ramping time .

Figure 14 The transverse deflection in the case of temperature−independent properties.

Figure 15 The temperature in the case of temperature−independent properties.

Figure 16 The displacement in the case of temperature−independent properties.

Figure 17 The flexure moment in the case of temperature−independent properties.

From the results we obtained it is clear that, the field quantities such as the deflection, temperature, displacement, and bending moment distributions depend not only on the timeand space coordinate, but also depend on the fractional parameter, nonlocal parameter ramping time parameterand effect of reference temperature. Numerical calculations and graphs have been divided into four cases. In this work, the obtained results have been compared with the results obtained previously in.22,23,35,4

The effect of the fractional order parameter

In the first case, we represent the dimensionless lateral vibration, thermodynamic temperature, displacement, and bending moment at different fractional order parameter  to stand on the effect of this parameter on all the studied fields (Figures 2−5). The computations were carried out for wide range of , for different values of the parameter  with wide rangewhen,, and.The different values of the parameter,describes two types of conductivity (weak conductivity,,normal conductivity,), respectively. From the different figures it is observed that the nature of variations of all the field variables for fractional order parameter is significantly different. The difference is more prominent for higher values of fractional derivative. This observation is consistent with previous results in.22We also observes the following important facts from Figures 3−6.

  1. The lateral vibrationdistribution has large change against to fractional order parameter. Also, the lateral vibration increases with the increase of the parameter.
  2. From figure 1, lateral vibration atis zero as shown, which agrees with the boundary condition prescribed. This coincided with the mechanical boundary condition that the two ends of the nanobeam are clamped.
  3. We observed that at the lateral vibrationreaches a maximum value at some distance from the surface of the nanobeam and then it decreases with the increase of distance.
  4. The temperature distribution have small differences for different values of.
  5. Figure 3 indicate that thetemperature has a maximum value at boundaryand decreases smoothly and finally tends to zero after some distance (which satisfy the boundary conditions).
  6. Figure 3 shows that the displacementincreases as It can be observed that the fractional order parameterhas a great effects on the displacement distribution.
  7. From Figure 4 it is observed that the bending momentincreases as the distanceincreases and then it decreases with the increase of distance. Figure 5 shows that the bending moment larg dependent on the variation of the fractional order parameter.
  8. According to these results, we have to construct a new classification to all the materials according to its fractional parameter, where this parameter becomes new indicator of its ability to conduct the thermal energy; confirms the results obtained in.22

The effect of the nonlocal parameter

Figures 6−9 describe the temperature, displacement, lateral vibration and bending moment field distributions for different values of nonlocal parameter. In this case, we notice that when the nonlocal parametervanishingindicates the old situation (local model of elasticity) while other values indicate the nonlocal theories of elasticity and thermo elasticity. The figures 6−9 show that this parameter has significant effect on all the fields. The waves reach the steady state depending on the value of the nonlocal parameter. The conclusion remarks from the Figures can be shortened as follows:

  1. The lateral vibrationdecrease when the value of increases while the values of the bending momentincreases when the value of
  2. It can be seen from Figure 7 that the temperaturesmall dependent on the variation of nonlocal parameter .
  3. From Figure 5, the values of the displacementstart increasing with the nonlocal parameter in the range,thereafter increasing to maximum amplitudes in the range. This shows the difference between the local generalized thermoelasticity and the nonlocal generalized thermoelasticity models.
  4. It can be seen from Figure 9 that the bending moment larg dependent on the variation of nonlocal parameter.
  5. As in,23,35,40 the field variables have pronounced sensitive to the variation of the nonlocal parameter.

The effect of the ramping time parameter

This case illustrating how the field quantities vary with the different values of the ramping time parameterwith constant,, and. The numerical results are obtained and presented graphically in Figures 10−13. We can see the significant effect of the ramping time parameteron all the studied fields. Also, we can conclude that:

  1. The increasing in the value of the ramping time parameter causes decreasing in the values of thelateral vibration, temperatureand bending momentwhich is very obvious in the peek points of the curves.
  2. Also, the values of the displacementstart decreasing with the ramping time parameter in the range, thereafter increasing to maximum amplitudes in the range.
  3. It can be observed that the ramping time parameterhas great effects on the distribution of field quantities. This is also consistent with what has been obtained in.40

The effect of the temperature−independent modulus

This case presents the effect of temperaturedependent properties (reference temperature) on the field variables for fixed values of the ramping time parameterand the nonlocal parameteraccording to the equation. For a comparison of the results, the lateral vibration, the temperature, the displacement, and the bending moment of the nano−beam are shown in Figures 14−17. We found that, the increasing in the value of causes increasing in the values of the lateral vibration, temperature, displacement fields which is very obvious in the peek points of the curves. When the modulus of elasticity is a linear function of reference temperatureand in the case of a temperature−independent modulu. On the other hand, high sensitivity is observed for the effect of the reference temperature on all of the studied variables. This also agrees with what Zenkour has reached in.40

Conclusion

The thermal and mechanical properties of materials vary with temperature, so that the temperature dependent on material properties must be taken into consideration in the thermal stress analysis of these elements. In this work, a new model of nonlocal generalized thermoelasticity based upon Euler–Bernoulli theory for nanobeams is constructed. The vibration characteristics of the deflection, temperature, displacement and bending moment of nanobeam subjected to ramp–type heating are investigated. The effects of the nonlocal parameter, fractional order parameterand the ramping time parameteron the field variables are investigated. Numerical technique based on the Laplace transformation has been used. The effects of the nonlocal parameter, reference temperature and the ramping time parameter on all the studied field quantities have been shown and presented graphically.

According to the results shown in all figures, it is find that:

  1. The nonlocal parameterhas significant effects on all the studied fields.
  2. The effect of the fractional parameter on all the studied fields is very significant.
  3. The thermo elastic stresses, displacement and temperature have a strong dependency on the ramping time parameter.
  4. The parameterand the dimension cause significant changes on all the studied fields and the phenomenon of finite speeds of propagation are manifested in all figures.
  5. New classification of the materials must be constructed according to the fractional parameter which describes the ability of the material to conduct the heat.
  6. This study is very important for micro scale problems because in these cases the material parameters are temperature dependent.
λ,μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH7oaBcaGGSaGaeqiVd0gaaa@3ABF@

Lam´e’s constants

K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4saa aa@3754@

thermal conductivity

α t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde 2aaSbaaKqbGeaacaWG0baabeaaaaa@396B@

thermal expansion coefficient

α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@

Fractional order parameter

γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC gaaa@382B@

coupling parameter

q i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGXbWdamaaBaaajuaibaWdbiaadMgaa8aabeaaaaa@3905@

components of the heat flows vector

T 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaBaaajuaibaGaaGimaaqcfayabaaaaa@38F4@

environmental temperature

δ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH0oazpaWaaSbaaKqbGeaapeGaamyAaiaadQgaaKqba+aa beaaaaa@3B31@

Kronecker's delta function

θ=T T 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaamivaiabgkHiTiaadsfadaWgaaqcfasaaiaaicdaaeqa aaaa@3CE8@

temperature increment

u i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG1bWdamaaBaaajuaibaWdbiaadMgaa8aabeaaaaa@3909@

displacement components

T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivaa aa@375D@

absolute temperature

F i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGgbWdamaaBaaajuaibaWdbiaadMgaa8aabeaacaaMc8oa aa@3A65@

body force components

C E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4qam aaBaaajuaibaGaamyraaqcfayabaaaaa@38F3@

specific heat

Q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGrbGaaGPaVdaa@3905@

heat source

e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyzaa aa@376E@

cubical dilatation

τ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHepaDpaWaaSbaaKqbGeaapeGaaGimaaqcfa4daeqaaaaa @3A2E@

relaxation time

σ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHdpWCpaWaaSbaaKqbGeaapeGaamyAaiaadQgaaKqba+aa beaaaaa@3B4F@

nonlocal stress tensor

h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGObaaaa@378F@

nanobeam thickness

e ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGLbWdamaaBaaajuaibaWdbiaadMgacaWGQbaapaqabaaa aa@39E8@

strain tensor

ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHbpGCaaa@3864@

material density

L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitaa aa@3755@

nanobeam length

b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGIbaaaa@378B@

nanobeam width

A=bh MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyqai abg2da9iaadkgacaWGObaaaa@3A24@

cross-section area

oxyz MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGVbGaamiEaiaadMhacaWG6baaaa@3A92@

Cartesian coordinate

τ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHepaDpaWaaSbaaKqbGeaapeGaamyAaiaadQgaa8aabeaa caaMc8oaaa@3C4E@

local stress tensor

2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGHhis0paWaaWbaaKqbGeqabaWdbiaaikdaaaaaaa@3955@

Laplacian operator

ξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOVdG haaa@3847@

Nonlocal parameter

E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyraa aa@374E@

Young’s modulus

Γ( α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGtoWaaeWaa8aabaWdbiabeg7aHbGaayjkaiaawMcaaiaa ykW7aaa@3C8F@

Gamma function

t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiDaa aa@377D@

the time

Γ( α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGtoWaaeWaa8aabaWdbiabeg7aHbGaayjkaiaawMcaaiaa ykW7aaa@3C8F@

Gamma function

 

 

ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH9oGBaaa@385C@

Poisson's ratio

 

 

Acknowledgements

None

Conflict of interest

The author declares that there is no conflict of interest.

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