# Effect of Variable Thermal Conductivity on the Generalized Thermoelasticity Problems in a Fiber-Reinforced Anisotropic Half-Space

Authors: Chun-Bao Xiong; Li-Na Yu; Yan-Bo Niu
Journal: Advances in Materials Science and Engineering
 (2019)

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## Abstract

Fiber-reinforced materials have widespread applications, which prompt the study of the effect of fiber reinforcement. Research studies have indicated that thermal conductivity cannot be considered as a constant, which is closely related to temperature change. Based on those studies, we investigate the fiber-reinforced generalized thermoelasticity problem under thermal stress, with the consideration of the effect of temperature-dependent variable thermal conductivity. The problem is assessed according to the L-S theory. A fiber-reinforced anisotropic half-space is selected as the research model, and a region of its surface is subjected to a transient thermal shock. The time-domain finite element method is applied to analyze the nonlinear problem and derives the governing equations. The nondimensional displacement, stress, and temperature of the material are obtained and illustrated graphically. The numerical results reveal that the variable conductivity significantly influences the distribution of the field quantities under the fiber-reinforced effect. And also, the boundary point of thermal shock is the most affected. The obtained results in this paper can be applied to design the fiber-reinforced anisotropic composites under thermal load to satisfy some particular engineering requirements.


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## Body

## 1. Introduction

Fiber reinforcement is an inherent property of materials that is considered an effect rather than a form of inclusion in it [1]. In an elastic state, the components of fiber-reinforced composites act as a single anisotropic unit without relative displacement [2–6]. Fiber-reinforced material produces higher specific strength and a larger specific modulus in the direction of fiber reinforcement. The material performance of fiber-reinforced composites is designable, and its corrosion resistance and durability are good. Those outstanding features inherent in fiber-reinforced composites have led to their widespread applications in aerospace, building engineering, automotive industries, and so on [7, 8]. Therefore, the fiber-reinforced effect of materials should be considered when studying mechanical behavior.The assumption of infinite propagation speed of the thermal signal in classical thermoelasticity theory is inconsistent with the real phenomenon. Several generalized thermoelasticity theories have been developed to eliminate this paradox [9–12], such as L-S theory. It firstly used the Maxwell–Cattaneo law of heat conduction instead of the conventional Fourier’s law and presented the generalized thermoelastic theory with one relaxation time, which has been proved to be well investigated and well established. For fiber-reinforced generalized thermoelasticity problems, Othman and Said [4] investigated the thermal shock of 2D fiber-reinforced materials and found that the temperature, displacement, and stress components change drastically at the front of the heat wave. Othman and Lotfy [13] compared coupling theory, G-L theory, and L-S theory to prove that the fiber reinforcement and the magnetic field significantly influenced the physical quantities, such as stress and strain. And also, the results from the three theories are in accordance with each other. Abouelregal and Zenkour [14] analyzed the effects of the fractional parameter, reinforcement, and rotation on the variations of different field quantities inside the elastic medium and found that fiber reinforcement plays an important role in the distributions of the field quantities. Abbas [15] investigated the generalized thermoelastic interaction of an infinite fiber-reinforced anisotropic plate containing a circular hole and found that field quantities are significantly varied in the presence and absence of reinforcement.Thermal conductivity is an important parameter of a material which is typically considered constant. However, several experimental and theoretical studies have indicated that thermal conductivity is closely related to temperature change [16–22]. Xiong and Guo [23] validated the effects of variable temperature-dependent properties on field quantities based on a one-dimensional generalized magneto-thermoelastic problem. Wang et al. [24] studied generalized thermoelasticity with variable thermal material properties and found that variable thermal material properties significantly affect the thermoelastic behaviors, particularly the magnitude of thermoelastic response. Ezzat and El-Bary [25] examined the effects of variable thermal conductivity in a problem of a thermo-viscoelastic infinitely long hollow cylinder and discovered that all functions for the generalized theory with a variable thermal conductivity distinctly differ from those obtained for the generalized theory with a constant thermal conductivity. Abo-Dahab and Abbas [26] evaluated the thermal shock problem of generalized magneto-thermoelasticity and concluded that as the variable thermal conductivity increases, the temperature increases, whereas the radial and hoop stresses decrease. These studies demonstrated that variable thermal conductivity significantly influences the material properties and the distribution of field quantities [27]. Instantaneous changes of temperature can markedly change the thermal conductivity of a material. Therefore, the influence of temperature-dependent variable thermal conductivity must be considered in solving fiber-reinforced generalized thermoelasticity problems suffered from thermal stress.Normal mode analysis is applicable only for solving the steady state problem, whereas integral transformation and time-domain finite element method are suitable for solving dynamic problems. Integral transforms, such as Fourier and Laplace transforms, have been widely used for processing the related generalized thermoelastic problems [28–30]. In considering the effect of temperature-dependent variable conductivity in the fiber-reinforced generalized thermoelasticity problem, the governing equation has a nonlinear form. Given that the governing equations contain higher-order terms and nonlinear coupling terms, integral transformation is difficult to perform, and inverse transformation is needed. This process inevitably produces truncation and discrete errors [31–33]. The governing equations can be directly solved in the time domain by using the finite element method, thereby avoiding the tedious processes in integral transformation. This method may be more efficient and may have higher precision. Furthermore, the time history of variables in constitutive relations can be reflected. Tian et al. [34] solved 2D generalized thermoelasticity problems by using a direct finite element method, which decreases the solving difficulty in the 2D model due to integral transformation. Li et al. [35] analyzed the nonlinear transient response under the generalized thermal diffusion theory based on the time-domain finite element method and obtained a good effect.This paper investigates the transient thermal shock problem for fiber-reinforced materials with a time-dependent variable thermal conductivity according to the L-S theory. Time-domain finite element method is applied to derive the nonlinear governing equations. Numerical examples are presented to clarify the transient thermal shock response on a half-space. Field quantities are obtained for different thermal conductivities and illustrated graphically.


## 2. Governing Equations

Belfield et al. [5] proposed a constitutive equation for a fiber-reinforced linearly thermoelastic anisotropic medium in studying the deformation of fiber-reinforced composites. Given the reinforced direction a≡a1,a2,a3, a12+a22+a32=1, and with consideration of variable thermal conductivity, the constitutive equations can be expressed as(1)σij=λekkδij+2μTeij+αakamekmδij+aiajekk+2μL−μTaiakekj+ajakeki+βakamekmaiaj−γθδij,(2)ρS=γekk+ρCET0θ,where σij is the stress tensor; δij is the Kronecker delta; eij is the strain tensor; α, β, μL−μT, and γ are reinforcement parameters, with γ=3λ+2μαt, in which λandμ are Lame constants; αt is the coefficient of linear thermal expansion; T0 is the reference temperature; θ=T−T0, in which θ is the temperature difference; S is the entropy density; ρ is the mass density; CE is the specific heat at constant strain; and ekk=u,x+v,y.The equation of motion (in the context of L-S theory) is(3)σij,j=ρu¨i,where ui is the displacement vector.The equation of energy conservation is(4)qi,i=−ρT0S˙,where qi is the heat flux vector.The geometrical equation is(5)eij=12ui,j+uj,i.The fiber-reinforced direction is defined asa≡1,0,0,∂/∂z≡0, and w=0. Thus, equation (1) can be written as(6)σxx=A11u,x+A12v,y−γθ,σyy=A12u,x+A22v,y−γθ,σxy=μLu,y+v,x,(7)σxz=σyz=0,where(8)A11=λ+2α+μT+4μL−μT+β,A12=λ+α,A22=λ+2μT.From equations (5) and (7), equation (3) yields(9)ρ∂2u∂t2=A11∂2u∂x2+C2∂2v∂x∂y+C1∂2u∂y2−γ∂θ∂x,ρ∂2v∂t2=A22∂2v∂y2+C2∂2u∂x∂y+C1∂2v∂x2−γ∂θ∂y,where(10)C1=μL,C2=α+λ+μL.The equation of heat conduction is(11)qi+τ0q˙i=−Kθ,i,where τ0 is the relaxation time.The thermal conductivityK is temperature dependent and assumed to have the following linear form:(12)K=Kθ=K01+K1θ,where K0 is the initial thermal conductivity and K1 is the small quantity for measuring the influence of temperature on thermal conductivity.From equations (2) and (4), equation (11) then yields(13)Kθ,i,i=∂∂t1+τ0∂∂tρCEθ+T0γekk.


## 3. Finite Element Formulations

The finite element method is an approximate method for solving differential equations. The first step in solving the problem is to establish the governing equations, followed by defining the boundary conditions based on the specific problems and then performing the the structural discrete, unit analysis and overall analysis to obtain the numerical solutions. For nonlinear problems, the finite element expression is obtained using the finite element method, which can eliminate the influence of truncation errors and avoid the tediousness of integral transformation. In addition, the time history of the variables in the constitutive relation can be determined to better reflect the wavefront characteristics. FlexPDE is a useful tool for solving partial differential equations, which can form Galerkin finite element integrals, derivatives, and dependencies aiming at the problem description and then build a coupling matrix and solve it. Therefore, FlexPDE is employed to deal with the related partial differential equations generated by the finite element method. For convenience, the constitutive equations of equations (1) and (2) can be expressed in the matrix form as follows:(14)σ=Cee−γθ,ρS=γTe+ρCET0θ.The heat conduction equation of equation (11) can be written as(15)q+τ0q˙=−Kθ′.The basic variables in this study include displacement and temperature. After the elements are divided, the variables are represented by shape functions in each element as follows:(16)u=N1eue,θ=N2eTθe,where ue and θe are the nodal displacement and temperature, respectively. N1e and N2eT are shape functions:(17)N1e=N10N20…Nn00N10N2…0Nn,N2eT=N1N2…Nn,where n denotes the number of nodes in the grid.According to equation (5) and given θ′=θ,i, equation (15) yields(18)e=B1ue,θ′=B2θe,where B1 and B2 are the first-order derivative of the components in N1e and N2e with respect to the material coordinates, respectively.Then, the variational forms of equation (5) are(19)δe=B1δue,δθ′=B2δθe.According to virtual displacement principles, the fiber-reinforced generalized thermoelasticity problem with variable thermal conductivity can be formulated as(20)∫VδeTσ+δθ′Tq+τ0q˙−ρT0δθTS˙+τ0S¨dV=−∫VδuTρu¨dV+∫AσδuTF¯dA+∫AqδθTq¯dA,where F¯ is the traction vector.According to equations (13)–(15), and (19), equation (20) yields(21)∫VδeTσdV=δueTKmmeue−Kmθeθe,∫Vδθ′Tq+τ0q˙dV=−δθeTKθθeθe,−∫VρT0δθTS˙+τ0S¨dV=−δθeTCθmeu˙e+Cθθeθ˙+Mθmeu¨e+Mθθeθ¨,−∫VδuTρu¨dV=−δueTMmmeu¨e,∫AσδuTF¯dA=δueTFme,∫AqδθTθq¯dA=δθeTTθe.These expressions can be summed as the following matrix form:(22)Mmme0MθmeMθθeu¨eθ¨e+00CθmeCθθeu˙eθ˙e+Kmme−Kmθe0Kθθeueθe=Fme−Tθe,where(23)Kmme=∫VB1TCeB1dV,Kmθe=∫VB1TγN2eTdV,Kθθe=∫VB2TKB2dV,Cθme=∫VT0N2eγTB1dV,Cθθe=∫VT0N2eρCET0N2eTdV,Mθme=∫VT0N2eτ0γTB1dV,Mθθe=∫VT0N2eτ0ρCET0N2eTdV,Mmme=∫VN1eTρN1edV,Fme=∫AσN1eTF¯dA,Tθe=∫AqN2eq¯dA.


## 4. Numerical Results and Discussion

### 4.1. Verification

To check the validity of the proposed method, reference [35] is chosen for comparison. Li et al. [35] had investigated the generalized diffusion-thermoelasticity problems with variable thermal conductivity by using the finite element method and had verified its effectiveness. This comparison research is conducted without the consideration of diffusion. In addition to this, the numerical model, initial conditions, and boundary conditions are same with reference [35]. The distribution of the temperature profile at the dimensionless time t = 0.06 is shown graphically in Figure 1, from which a trend consistency can be observed. This guarantees the validity and accuracy of the present method.Figure 1
Distribution of temperature along thex-axis at t = 0.06.


### 4.2. Results and Discussion

Consider the problem of a fiber-reinforced anisotropic elastic half-space (x≥0) with variable thermal conductivity. As shown in Figure 2(a), the boundary surface is assumed to be without traction, and the banded area on x=0 is subjected to a time-dependent transient thermal shock.Initial conditions:(24)ux,y,0=υx,y,0=θx,y,0=0,∂ux,y,0∂t=∂υx,y,0∂t=∂θx,y,0∂t=0.Boundary conditions:(25)σxx0,y,t=σxy0,y,t=0,(26)θ0,y,t=HtHL−y,where H⋅ is the Heaviside unit step function.Figure 2
(a) Diagram of a half-space under thermal shock. (b) Simplifiedxy plane diagram of the half-space.
(a)
(b)Under the given conditions, the half-space model can be simplified as axy plane model, as shown in Figure 2(b). The components of displacement and temperature can be simplified as follows:(27)ux=ux,y,t,uy=υx,y,t,uz=0,θ=θx,y,t.Copper material is selected for numerical evaluation, and the parameters are presented in Table1 [36].Table 1
Material parameters of the copper material.
λ
=
7.76
×
10
10
N
/
m
2
α
t
=
1.78
×
10
−
5
⋅
K
−
1
η
=
8886.73
s
/
m
2
C
E
=
383.1
J
/
kg K
ρ
=
8954
kg
/
m
3
K
0
=
386
W
/
m K
T
0
=
293
K
α
=
−
1.28
×
10
10
N
/
m
2
β
=
0.32
×
10
10
N
/
m
2
μ
L
=
2.45
×
10
10
N
/
m
2
μ
T
=
1.89
×
10
10
N
/
m
2For convenience, the following nondimensional quantities are introduced:(28)x∗,y∗,u∗,v∗=c1ηx,y,u,v,t∗=c12ηt,τ0∗=c12ητ0,θ∗=γλ+2μTθ,K1∗=λ+2μTγK1,i,j=1,2,σij∗=σijλ+2μT,η=ρCEK0,c12=λ+2μTρ.According to equation (28), equations (7), (9), and (12) can be written as follows (the asterisk is removed for brevity):(29)σxx=h11∂u∂x+h12∂v∂y−θ,σyy=h22∂v∂y+h12∂u∂x−θ,σxy=μLλ+2μT∂u∂y+∂v∂x,(30)∂2u∂t2=h11∂2u∂x2+h2∂2v∂x∂y+h1∂2u∂y2−∂θ∂x,∂2v∂t2=h22∂2v∂y2+h2∂2u∂x∂y+h1∂2v∂x2−∂θ∂y,(31)∂∂t1+τ0∂∂tε∂u∂x+∂υ∂y+θ=1+K1θ∂2θ∂x2+∂2θ∂y2+K1∂θ∂x2+∂θ∂y2,where(32)h11,h22,h12,h1,h2=A11,A22,A12,C1,C2λ+2μT,ε=γ2T0ρCEλ+2μT,Under the given conditions of equations (24)–(27), the nonlinear governing equations (29)–(31) can be solved directly in the time domain. Under the given conditions, the research model is symmetrical about the x-axis. Thus, the rectangular area OABD is used for the subsequent analysis for simplification.The dimensionless distributions of temperature, displacement, and stress are illustrated graphically in Figures3–8. The K1 value is expressed in four cases to discuss the effects of variable thermal conductivity as follows:(1)
Case 1:K1 = −0.6(2)
Case 2:K1 = −0.3(3)
Case 3:K1 = 0.Figure 3
Distribution of nondimensional temperature alongOA.Figure 4
Distribution of nondimensional temperature alongOD.Figure 5
Distribution of nondimensional horizontal displacement alongOA.Figure 6
Distribution of the nondimensional horizontal displacement alongOD.Figure 7
Distribution of the nondimensional vertical displacement alongOD.Figure 8
Distribution of the nondimensional stressσyy along OD.Figures3 and 4 show the distributions of nondimensional temperature along OA and OD. The dimensionless temperature is equal to 1 at x=0 and 0≤y≤0.2 region, which agrees with the boundary conditions that were previously assigned. As the thermal conductivity increases, the temperatures along OA and OD increase. This result indicates that the coefficient of thermal conductivity is positively correlated with the temperature change, and that variable thermal conductivity significantly affects the temperature distribution. Moreover, the wavefront effect is more pronounced as the thermal conductivity increases. There is a diverse trend at x=0.5 in Figure 3, which is mainly caused by the effect of wavefront and the calculation errors.Given that the research model is symmetrical about thex-axis, the vertical displacement along uy−OA is zero and need not be considered. Figure 5 depicts the distribution of the nondimensional horizontal displacement ux−OA, which shows that the closest section to the origin undergoes expansion, the next undergoes compression, and the rest away from the origin is undisturbed. Figure 5 demonstrates that variable thermal conductivity is positively correlated with the distribution of the horizontal displacement along OA.Figures6 and 7 show the distribution of the nondimensional horizontal displacement ux−OD and the vertical displacement uy−OD, respectively. As shown, the variable thermal conductivity obviously affected the distribution of displacement. In addition, y=0.2 is the most affected thermal shock boundary point. The negative value in Figure 6 indicates that the particles tend to move toward the unconstrained direction.Figure8 shows the distribution of the nondimensional stress σyy−OD. The stress values show a violent oscillation in the 0<y<0.2 zone, and variable thermal conductivity obviously affects the distribution of stress along OD, particularly in the 0.2<y<0.8 zone.


## 4.1. Verification

To check the validity of the proposed method, reference [35] is chosen for comparison. Li et al. [35] had investigated the generalized diffusion-thermoelasticity problems with variable thermal conductivity by using the finite element method and had verified its effectiveness. This comparison research is conducted without the consideration of diffusion. In addition to this, the numerical model, initial conditions, and boundary conditions are same with reference [35]. The distribution of the temperature profile at the dimensionless time t = 0.06 is shown graphically in Figure 1, from which a trend consistency can be observed. This guarantees the validity and accuracy of the present method.Figure 1
Distribution of temperature along thex-axis at t = 0.06.


## 4.2. Results and Discussion

Consider the problem of a fiber-reinforced anisotropic elastic half-space (x≥0) with variable thermal conductivity. As shown in Figure 2(a), the boundary surface is assumed to be without traction, and the banded area on x=0 is subjected to a time-dependent transient thermal shock.Initial conditions:(24)ux,y,0=υx,y,0=θx,y,0=0,∂ux,y,0∂t=∂υx,y,0∂t=∂θx,y,0∂t=0.Boundary conditions:(25)σxx0,y,t=σxy0,y,t=0,(26)θ0,y,t=HtHL−y,where H⋅ is the Heaviside unit step function.Figure 2
(a) Diagram of a half-space under thermal shock. (b) Simplifiedxy plane diagram of the half-space.
(a)
(b)Under the given conditions, the half-space model can be simplified as axy plane model, as shown in Figure 2(b). The components of displacement and temperature can be simplified as follows:(27)ux=ux,y,t,uy=υx,y,t,uz=0,θ=θx,y,t.Copper material is selected for numerical evaluation, and the parameters are presented in Table1 [36].Table 1
Material parameters of the copper material.
λ
=
7.76
×
10
10
N
/
m
2
α
t
=
1.78
×
10
−
5
⋅
K
−
1
η
=
8886.73
s
/
m
2
C
E
=
383.1
J
/
kg K
ρ
=
8954
kg
/
m
3
K
0
=
386
W
/
m K
T
0
=
293
K
α
=
−
1.28
×
10
10
N
/
m
2
β
=
0.32
×
10
10
N
/
m
2
μ
L
=
2.45
×
10
10
N
/
m
2
μ
T
=
1.89
×
10
10
N
/
m
2For convenience, the following nondimensional quantities are introduced:(28)x∗,y∗,u∗,v∗=c1ηx,y,u,v,t∗=c12ηt,τ0∗=c12ητ0,θ∗=γλ+2μTθ,K1∗=λ+2μTγK1,i,j=1,2,σij∗=σijλ+2μT,η=ρCEK0,c12=λ+2μTρ.According to equation (28), equations (7), (9), and (12) can be written as follows (the asterisk is removed for brevity):(29)σxx=h11∂u∂x+h12∂v∂y−θ,σyy=h22∂v∂y+h12∂u∂x−θ,σxy=μLλ+2μT∂u∂y+∂v∂x,(30)∂2u∂t2=h11∂2u∂x2+h2∂2v∂x∂y+h1∂2u∂y2−∂θ∂x,∂2v∂t2=h22∂2v∂y2+h2∂2u∂x∂y+h1∂2v∂x2−∂θ∂y,(31)∂∂t1+τ0∂∂tε∂u∂x+∂υ∂y+θ=1+K1θ∂2θ∂x2+∂2θ∂y2+K1∂θ∂x2+∂θ∂y2,where(32)h11,h22,h12,h1,h2=A11,A22,A12,C1,C2λ+2μT,ε=γ2T0ρCEλ+2μT,Under the given conditions of equations (24)–(27), the nonlinear governing equations (29)–(31) can be solved directly in the time domain. Under the given conditions, the research model is symmetrical about the x-axis. Thus, the rectangular area OABD is used for the subsequent analysis for simplification.The dimensionless distributions of temperature, displacement, and stress are illustrated graphically in Figures3–8. The K1 value is expressed in four cases to discuss the effects of variable thermal conductivity as follows:(1)
Case 1:K1 = −0.6(2)
Case 2:K1 = −0.3(3)
Case 3:K1 = 0.Figure 3
Distribution of nondimensional temperature alongOA.Figure 4
Distribution of nondimensional temperature alongOD.Figure 5
Distribution of nondimensional horizontal displacement alongOA.Figure 6
Distribution of the nondimensional horizontal displacement alongOD.Figure 7
Distribution of the nondimensional vertical displacement alongOD.Figure 8
Distribution of the nondimensional stressσyy along OD.Figures3 and 4 show the distributions of nondimensional temperature along OA and OD. The dimensionless temperature is equal to 1 at x=0 and 0≤y≤0.2 region, which agrees with the boundary conditions that were previously assigned. As the thermal conductivity increases, the temperatures along OA and OD increase. This result indicates that the coefficient of thermal conductivity is positively correlated with the temperature change, and that variable thermal conductivity significantly affects the temperature distribution. Moreover, the wavefront effect is more pronounced as the thermal conductivity increases. There is a diverse trend at x=0.5 in Figure 3, which is mainly caused by the effect of wavefront and the calculation errors.Given that the research model is symmetrical about thex-axis, the vertical displacement along uy−OA is zero and need not be considered. Figure 5 depicts the distribution of the nondimensional horizontal displacement ux−OA, which shows that the closest section to the origin undergoes expansion, the next undergoes compression, and the rest away from the origin is undisturbed. Figure 5 demonstrates that variable thermal conductivity is positively correlated with the distribution of the horizontal displacement along OA.Figures6 and 7 show the distribution of the nondimensional horizontal displacement ux−OD and the vertical displacement uy−OD, respectively. As shown, the variable thermal conductivity obviously affected the distribution of displacement. In addition, y=0.2 is the most affected thermal shock boundary point. The negative value in Figure 6 indicates that the particles tend to move toward the unconstrained direction.Figure8 shows the distribution of the nondimensional stress σyy−OD. The stress values show a violent oscillation in the 0<y<0.2 zone, and variable thermal conductivity obviously affects the distribution of stress along OD, particularly in the 0.2<y<0.8 zone.


## 5. Concluding Remarks

It is well known that instantaneous changes of temperature can markedly change the thermal conductivity of a material. Therefore, this article investigated the effect of temperature-dependent variable thermal conductivity on a fiber-reinforced generalized thermoelastic half-space. Given the reinforced directiona=1,0,0, a region of its surface is subjected to a transient thermal shock. The problem is studied in the context of L-S theory. The time-domain finite element method is proposed to analyze the nonlinear response.Based on the simulation, we can draw that the time-finite element method is very effective for analyzing nonlinear problems with given initial and boundary conditions, and by which we can capture a pronounced wavefront effect. In consideration of the fiber-reinforced effect, variable thermal conductivity positively affects the distributions of temperature, displacement, and stress. In addition, the boundary point of thermal shock is affected the most.



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*Source: 8625371-2019-09-03.xml*