A Computational Model for Transverse Thermal Displacements in Symmetric Sandwich Beam by Using Higher Order Shear Deformation Theory

In the mathematical framework, the concept of virtual work principle is used to derive the equations of motion and boundary conditions. The integration by parts is used for further solution. The simply supported symmetric sandwich beam is considered for thermal analysis. The effect of thermal load is included in strain. The Hook’s law and corresponding constitutive relationship is taken in to consideration during analysis. The thermal load or temperature variation is assumed to be linear across the thickness of the beam. The z axis is along the thickness of sandwich beam and assumed as positive in downward direction. Further, close-form Navier’s technique is adopted to develop analytical solution. A computer program in FORTRAN-90 has been developed to obtain thermal central displacements in sandwich beam when aspect ratio, modular ratio and coefficient of thermal expansion ratio changes.

The displacement field or kinematic of the PSDT is based on the following assumption:

1. The components of displacement along x axis and z axis are represented by u and w respectively.
2. Since the sandwich beam is a dimensional problem the displacement along y is considered as zero.
3. The axial displacement (u) along x axis consists of stretching $\left(u_0\right)$, bending $\left(-Z \frac{\partial w(x)}{\partial x}\right)$, and shear component $z\left(1-\frac{4 z^2}{3 h^2}\right)$.
4. The shear component in the TSDT is $\frac{h}{\pi} \sin \left(\frac{\pi z}{h}\right)$.
5. The shear component in the CBT is zero.
6. The transverse thermal displacement (w) is considered as a function of x only.
7. The beam is acted upon by thermal load only.
8. No body forces are considered in the thermal analysis.
9. The perfect bond is assumed between layers of beam.

Based on the above assumptions the kinematic or displacement field of the PSDT can be written as below

$u(x, z)=u_0(x)-z \frac{\partial w(x)}{\partial x}+z\left(1-\frac{4 z^2}{3 h^2}\right) \varphi_x(x)$                 (2)

$w(x, z)=w(x)$                  (3)

where, $u(x, z)$ represents the displacement along x axis and $w(x, z)$ represents transverse displacement along z direction. The axial displacement along x is the function x and z, whereas the transverse displacement along z is the function of x. The mid-plane displacement $u_0(x)$ and shear slope $\varphi_x$ are the functions of x only. The displacement field has cubic term representing the effect of shear deformation. The present theory (PSDT) excludes the transverse normal effect. The transverse displacement along z direction is the function of x only. This is the limitation of the present theory.

4.1 Strain displacement relation

The strains (normal and shear) are evaluated from fundamentals of theory of elasticity. These strains are shown by following equations

$\varepsilon_x=\frac{\partial u}{\partial x}=\frac{\partial u_0}{\partial x}-z \frac{\partial^2 w}{\partial x^2}+z\left(1-\frac{4 z^2}{3 h^2}\right) \frac{\partial \varphi_x}{\partial x}$                    (4)

$\gamma_{z x}=\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}=\left(1-\frac{4 z^2}{h^2}\right) \varphi_x$                   (5)

4.2 Constitutive relations (CR)

The constitutive relation for sandwich beam is expressed by the following equation

$\left\{\begin{array}{c}\sigma_x \\ \tau_{z x}\end{array}\right\}_k=\left[\begin{array}{cc}\bar{Q}_{11} & 0 \\ 0 & \bar{Q}_{55}\end{array}\right]_k\left\{\begin{array}{c}\varepsilon_x-\alpha_x T \\ \gamma_{z x}\end{array}\right\}_k$                      (6)

The reduced stiffness coefficients in the above equation are represented by $\bar{Q}_{i j}^{(k)}$. These coefficients are computed by using the following equation

$\bar{Q}_{11}^{(k)}=\frac{E_1^{(k)}}{\left(1-\mu_{12}^{(k)} \mu_{21}^{(k)}\right)}, \bar{Q}_{55}^{(k)}=\bar{G}_{13}^{(k)}$                 (7)

The Young’s modulus, shear modulus and Poisson’s ratio are represented by E, G and $\mu_{i j}$, respectively.

4.3 Thermal variation (Through thickness)

The temperature distribution across the thickness (h) of the soft core sandwich beam is considered as given below

$T(x, z)=\frac{2 z}{h} T_1(x)$               (8)

The change in temperature is represented by T in the above Eq. (8). The temperature change is considered as a function of x and z. The thermal load T₁ is linearly varying through the depth of sandwich beam. The thermal load T₁ is considered as a function of x. The sinusoidal distribution of temperature change is as given below

$T(x, z)=\left(\frac{2 z}{h} T_1\right) \sin \left(\frac{m \pi x}{a}\right)$                 (9)

The positive integer of sine series is represented by m in the above Eq. (9). This is further used in Navier’s solution of simply supported sandwich beam.

The performance or efficiency of parabolic (PSDT) shear deformation beam theory is examined by applying it to sandwich beam subjected temperature field. A three-layer soft core sandwich beam (simply supported) is considered for thermal analysis. The central transverse displacements are evaluated for different aspect ratios and modular ratios. The effect of change in coefficient of thermal expansion ratio on transverse displacement is also examined for different aspect ratios. The material properties are as shown below:

Material properties for orthotropic layer (0⁰)

The Graphite-Epoxy material is considered for orthotropic layer [19].

$\frac{E_L}{E_T}=25, \frac{G_{L T}}{E_T}=0.5, \frac{G_{T T}}{E_T}=0.2, \mu_{L T}=\mu_{T T}=0.25, \frac{\alpha_T}{\alpha_L}=1125$

The direction parallel to the fibre is represented by L (Longitudinal) and the direction perpendicular to the fibre is represented by T (Transverse). The thermal expansion coefficient in the fibre direction is shown by $\alpha_L$. The thermal expansion coefficient in the transverse direction is represented by $\alpha_T$.

The properties of material are as given below [13]:

For $0^0$ layers: $\left(Q_{11}=25\right),\left(Q_{55}=0.5\right)$

For soft core: $\left(Q_{11}=4\right),\left(Q_{55}=0.06\right)$

The thermal expansion coefficients:

$\left(\frac{\alpha_L}{\alpha_0}=0.333\right),\left(\frac{\alpha_T}{\alpha_0}=1\right)$

For soft core: $\alpha^{\text {core }}=\alpha_L=\alpha_T=1.36 \alpha_0$.

The normalization factor is $\alpha_0$ for the thermal expansion coefficients.

6.1 Closed-form Navier technique

Below mentioned are the edge conditions used for sandwich beam (0/core/0) along the edges $x=0$ and $x=a$.

$w=0, M_x=0, N_x=0, M_x^s=0$         (21)

Navier’s technique is used to get displacement variables. The solution form for $\left(u_0\right),(w)$ and $\left(\varphi_x\right)$ that satisfies the boundary conditions exactly as given below

$u_0(x)=\sum_{m=1,3,5}^{\infty} u_{0 m} cos \left(\frac{m \pi x}{a}\right)$                (22)

$w(x)=\sum_{m=1,3,5}^{\infty} w_m sin \left(\frac{m \pi x}{a}\right)$                     (23)

$\varphi_x(x)=\sum_{m=1,3,5}^{\infty} \varphi_{x m} cos \left(\frac{m \pi x}{a}\right)$          (24)

where, $u_{0 m}, w_m$ and $\varphi_{x m}$ are to be determined and known as series coefficients. The thermal load is expanded given below.

$T_1(x)=\sum_{m=1}^{\infty} T_{1 m} sin \left(\frac{m \pi x}{a}\right)$          (25)

where, m is positive integer and $T_{1 m}$ is the coefficient of single Fourier sine series expansion for thermal load as follows

$\left(T B_{11}, T D_{11}, T F_{11}\right)=\sum_{k=1}^N\left(\alpha_x\right) \int_{z_k}^{z_{k+1}} \bar{Q}_{11}^{(k)}\left(z, z^2, z^4\right) d z$          (26)

The intensity of thermal load is denoted by $T_1$ in the above Eq. (25). Substitution of Eqs. (22) through (25) into equations of motion (11), (12) and (13) gives simultaneous equations as given below

$\left(T B_{11}, T D_{11}, T F_{11}\right)=\sum_{k=1}^N\left(\alpha_x\right) \int_{z_k}^{z_{k+1}} \bar{Q}_{11}^{(k)}\left(z, z^2, z^4\right) d z$            (27)

The stiffness coefficients $\left(k_{11}, k_{12}, \ldots.\right)$ of the stiffness matrix $[k]$ in Eq. (27) are defined as follows

$\begin{gathered}k_{11}=A_{11} \frac{m^2 \pi^2}{a^2}, k_{12}=k_{21}=-B_{11} \frac{m^3 \pi^3}{a^3}, \\ k_{13}=k_{31}=\left(B_{11}-\frac{4}{3 h^2} E_{11}\right) \frac{m^2 \pi^2}{a^2} \\ k_{22}=D_{11} \frac{m^4 \pi^4}{a^4}, k_{23}=k_{32}=-\left(D_{11}-\frac{4}{3 h^2} F_{11}\right) \frac{m^3 \pi^3}{a^3} \\ k_{33}=\left(D_{11}+\frac{16}{9 h^4} H_{11}-\frac{8}{3 h^2} F_{11}\right) \frac{m^2 \pi^2}{a^2}+\left(A_{55}-\frac{4}{h^2} D_{55}\right)-\left(D_{55}-\frac{4}{h^2} F_{55}\right) \frac{4}{h^2}\end{gathered}$

The force vector elements $\left\{f_1, f_2, f_3\right\}$ in Eq. (27) are as follows

$\begin{gathered}f_1=-T B_{11} \frac{2}{h} \frac{m \pi}{a} T_{1 m}, f_2=\frac{2}{h} T D_{11} \frac{m^2 \pi^2}{a^2} T_{1 m} \\ f_3=-\left(T D_{11}-\frac{4}{3 h^2} T F_{11}\right) \frac{m \pi}{a}\left(\frac{2}{h}\right) T_{1 m}\end{gathered}$

After solving the above set of algebraic equations, the values of $u_{0 m}, w_m$ and $\phi_{x m}$ can be determined. Once obtained the values of $u_{0 m}, w_m$ and $\varphi_{x m}$ one can then calculate all central thermal displacements within the beam by using Eqs. (5)-(9) and (11).

6.2 Numerical results and discussion

Thermal central displacements are computed in the soft core sandwich beam. The boundary condition is simply supported and the sandwich beam is subjected to pure thermal load varying linearly through the thickness of sandwich beam.

The central thermal transverse displacements are obtained for different aspect ratios, i.e., length to thickness ratios $\left(S=\frac{a}{h}\right)$, various modular ratios $\left(\frac{E_1}{E_2}\right)$ and various coefficient of thermal expansion ratios $\left(\frac{\alpha_2}{\alpha_1}\right)$. These central thermal displacements are given in following dimensionless form for the discussion purpose

$\bar{w}\left(\frac{a}{2}, 0\right)=\frac{h w}{\alpha_L T_1 a^2}$

The effect of change of aspect ratios on transverse displacement is shown in the Table 1. The results of transverse thermal displacements computed by PSDT and TSDT shows a significant difference within aspect ratios 2 to 20. This is due to shape function of parabolic and TSDT. The results obtained by TSDT are higher as compared to results obtained by PSDT for thick sandwich beam. These displacements are more or less similar after aspect ratio 20. The results computed by CBT are irrespective of aspect ratio. The results obtained by these three theories are same for aspect ratio 100. This indicates that higher order theories (PSDT and TSDT) are applicable to thin

and moderately thick beams, whereas CBT is applicable to thin beam. The assumptions of CBT ignore the shear deformation effect, whereas actually the shear deformation effect is very prominent in laminated sandwich beams. The higher order theories (PSDT and TSDT) include the shear deformation effect in the theory; therefore, these theories are applicable to thick and thin plate. The use of higher order theory is practical take away and becomes necessary for the analysis of sandwich beams. Figure 2 shows the effect of the change of aspect ratio on transverse thermal displacement under sinusoidal thermal load. The transverse thermal displacements obtained by PSDT, TSDT, and CBT are compared with the results available in the literature [7] wherever possible, as shown in Table 1.

Table 1. Normalized transverse central thermal displacements for different aspect ratios under sinusoidal and uniform thermal load in symmetric sandwich beam

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Figure 2. Effect of change of aspect ratio on central transverse thermal displacement

Table 2. Normalized central transverse displacement under sinusoidal thermal load for various modular ratios for aspect ratio 4

Table 3. Normalized central transverse displacement under sinusoidal thermal load for various modular ratios for aspect ratio 10

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Figure 3. The effect of change of modular ratio on transverse displacement beam subjected to single sine thermal load for aspect ratio 4

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Figure 4. The effect of change of modular ratio on transverse displacement of beam subjected to single sine thermal load for aspect ratio 10

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Figure 5. The effect of change of coefficient of thermal expansion ratio on transverse displacement of beam subjected to single sine thermal load for aspect ratio 4

Table 4. Dimensionless transverse thermal displacements for various ratios of coefficients of thermal expansions under sinusoidal thermal load for aspect ratio (S) 4

Note: Modular ratio is considered as 25

Table 5. Dimensionless transverse thermal displacements in beam subjected to single sine thermal load for various coefficients of thermal expansions ratios for aspect ratio (S) 10

Note: Modular ratio is considered as 25

The change in modular ratio, i.e., material orthotropy affects the transverse displacement as shown in Tables 2 and 3. The results of transverse displacements obtained by TSDT for aspect ratio 4 shows a considerable variation as compared to PSDT results when modular ratio changes from 5 to 20. However, these results are more or less similar for aspect ratio 10. This is clearly noted from Figures 3 and 4. It is noted that for low modular ratio the transverse displacement is high and for high modular ratio the transverse displacement is low. Thus, these results are useful in selection of material for sandwich beam. The sandwich beams used in automotive engineering need low thermal transverse displacement and high stiffness. The modular ratio between 25 to 40 shows lower thermal deformations. This high modulus is preferable in automotive engineering for better thermal performance.

The effect of change of thermal expansion coefficient ratio on transverse displacement is shown in Tables 4 and 5. The ratio of coefficient of thermal expansion is considered from 2 to 20. It is noted that the transverse thermal displacements are directly proportional to coefficient of thermal expansion ratios. This linear variation is shown in Figure 5. The transverse displacement is low for low ratio of coefficient of thermal expansion. The sandwich beams used in satellite structures, automotive engineering, suspension components in vehicle need low or minimum thermal expansion and high stiffness. This practical take away is useful for future study.

The author acknowledges the continuous support and encouragement from the Symbiosis Institute of Technology, Pune.