Investigating the effect of thickness profile of a rotating functionally graded disc on its creep behavior

Abstract

Creep behavior of rotating discs made of functionally graded materials with linearly varying thickness has been investigated. The discs contain silicon carbide particles in a matrix of pure aluminum. The effect of varying disc thickness gradient (TG) has been investigated on the stresses and strain rates in the composite disc. The study shows that with the increase in disc TG, the radial, tangential and effective stresses decrease throughout the disc. The strain rates in the disc also reduce significantly with the increase in TG of the disc.

Introduction

Rotating disc is a widely used structural component in steam and gas turbine rotors, turbo generators, pumps, compressors, flywheels, automotive braking systems, ship propellers and computer disc drives. ^{1 ,2} In most of these applications, it has to operate at high temperature and under high stresses caused by disc rotation at high speed. ³ Due to severe thermomechanical loadings, the disc undergoes significant creep deformations, which may severely affect its performance. ^{1 ,4–6}

Under high temperature, rotating disc made of monolithic material may not perform well. The excellent mechanical properties like high specific strength and stiffness and high temperature stability offered by aluminum (Al) matrix composites reinforced with silicon carbide (SiC) particles, whiskers or fibers make them suitable for rotating disc applications exposed to elevated temperature. ^7–9

The problem of creep in rotating disc made of functionally graded materials (FGMs) subjected to high stresses and high temperature has attracted the interest of many researchers. In FGMs, the content of two or more constituent materials is gradually varied with respect to certain dimension/dimensions. ¹⁰ FGMs have been designed and developed to sustain high-temperature environments. Singh and Ray ¹¹ investigated steady state creep in an isotropic, constant thickness rotating disc made of functionally graded (FG) Al-SiC_p (subscript ‘p’ refers to particle shape of SiC) and subjected to a constant temperature. The content of SiC_p in the disc was assumed to decrease linearly from the inner to the outer radius. The study indicates that the creep response of the FGM disc is significantly superior when compared with a similar disc having uniform distribution of SiC_p. Gupta et al. ¹ further extended the analysis for a rotating FGM disc having uniform thickness but operating under a radial thermal gradient, which results due to braking action of the disc. The study further established the superiority of FGM disc over uniform composite disc having uniform distribution of SiC_p. Bayat et al. ¹² performed thermoelastic analysis of a FG rotating disc of constant thickness with small and large deflections. They observed that for particular values of the grading index (n) of material properties, mechanical responses in the FG disc can be smaller than in a homogenous disc. Bayat et al. ⁶ further extended their work to obtain elastic solutions for rotating FG discs having variable thickness.

Some of the studies ^13–17 indicate that the stresses in rotating disc having variable thickness, with thickness at the center being more than the periphery, are much lower than those observed in a uniform thickness disc operating at the same angular velocity. The use of variable thickness disc increases its plastic limit angular velocity and reduces the magnitude of stresses and deformations in the disc. ¹⁵ The study of Jahed et al. ¹⁶ indicates that the use of variable thickness disc is helpful in minimizing the weight of the disc in aerospace applications. Our earlier work ¹⁸ also indicates that the creep stresses and creep rates in a rotating disc made of uniform composite are significantly reduced, when the disc has linearly varying thickness profile when compared with hyperbolic or constant thickness profile.

The literature consulted so far reveals that the studies pertaining to creep analysis in variable thickness rotating disc made of FG composite are rather scant. Therefore, it is decided to investigate steady state creep in a rotating disc made of FG Al-SiC_p and having linearly varying thickness. In order to bring out the effect of imposing different types of linear thickness gradients (TGs) on the creep response, the results obtained for variable thickness FGM discs are compared with a similar FGM disc but having uniform thickness.

Disc profile

In the present work, the creep response has been estimated for rotating discs having two different linearly varying thickness profiles and for a constant thickness disc, while keeping volume of the discs same. For composite discs having linearly varying thickness, the thickness h(r) at any radius r is given by

$h(r)=h_b+2c(b-r),$

(1)

where the constant ${\displaystyle c=\frac{(h_a-h_b)}{2(b-a)}}$ , h_a and h_b are disc thickness, respectively, at the inner (a) and outer (b) radii (refer Appendix A for notations used).

Since the volume of disc with linearly varying thickness is equal to the volume of uniform thickness disc, therefore,

$\underset{a}{\overset{b}{\int }}2\mathit{\pi }rh(r)\mathrm{d}r=\mathit{\pi }(b^2-a^2)t,$

(2)

where t is the thickness of constant thickness disc.

Substituting h(r) from Eq. (1) into Eq. (2) and simplifying, we get

$h_a=\frac{3(b+a)t-h_b(2b+a)}{(b+2a)}.$

(3)

Therefore, for given values of $h_b$ , a, b and t, the disc thickness at the inner radius ( $h_a$ ) may be estimated from the above equation.

Distribution of reinforcement

The distribution of SiC_p in the FG disc is assumed to decrease linearly from the inner to the outer radius. The content (vol%) of SiC_p, V(r), at any radius r, is given by

$V(r)=V_{max}-\frac{(r-a)}{(b-a)}(V_{max}-V_{min})={\mathit{\delta }}_1-{\mathit{\delta }}_{2}r,$

(4)

where ${\mathit{\delta }}_1=(V_{max}+a{\mathit{\delta }}_2)$ , ${\mathit{\delta }}_2=\frac{(V_{max}-V_{min})}{(b-a)}$ and $V_{max}$ and $V_{min}$ are the maximum and minimum SiC_p contents at the inner and outer radii of the disc, respectively.

Following the rule of mixture, the density $\mathit{\rho }(r)$ of the FG disc is

$\mathit{\rho }(r)={\mathit{\rho }}_m+\frac{({\mathit{\rho }}_d-{\mathit{\rho }}_m)V(r)}{100},$

(5)

where ${\mathit{\rho }}_m$ (=2698.9 kg/m³) and ${\mathit{\rho }}_d$ (=3210 kg/m³) are, respectively, the densities of pure Al matrix and SiC. ^{19 ,20}

Substituting $V(r)$ from Eq. (4) into Eq. (5),

$\mathit{\rho }(r)=A_{\mathit{\rho }}-B_{\mathit{\rho }}\cdot r,$

(6)

where

$A_{\mathit{\rho }}={\mathit{\rho }}_m+({\mathit{\rho }}_d-{\mathit{\rho }}_m)\frac{{\mathit{\delta }}_1}{100}$

and

$B_{\mathit{\rho }}=\frac{{\mathit{\delta }}_2({\mathit{\rho }}_d-{\mathit{\rho }}_m)}{100}.$

For comparison, it is assumed that the total amount of SiC_p in the FGM disc having variable thickness, h(r), disc is equal to that in a similar disc but having uniform distribution of SiC_p (say V_av ). Therefore,

$V_{av}\underset{a}{\overset{b}{\int }}2\mathit{\pi }rh(r)\mathrm{d}r=\underset{a}{\overset{b}{\int }}2\mathit{\pi }rh(r)V(r)\mathrm{d}r.$

Using Eq. (2) in the above equation, we get

$V_{av}=\frac{\underset{a}{\overset{b}{\int }}2\mathit{\pi }rh(r)V(r)\mathrm{d}r}{\mathit{\pi }(b^2-a^2)t}.$

(7)

Substituting h(r) and $V(r)$ , respectively, from Eqs. (1) and (4) into Eq. (7), we obtain the minimum SiC_p content in the disc as

$V_{\mathrm{m}\mathrm{i}\mathrm{n}}=\frac{V_{max}(h_b{b}^2-2h_b{a}^2+b^{3}c-5a^{2}bc+a{b}^{2}c+ab{h}_b+3a^{3}c)-3V_{av}t(b^2-a^2)}{(ab{h}_b+a{b}^{2}c+a^2{h}_b+a^{2}bc-a^{3}c-2b^2{h}_b-b^{3}c)}.$

(8)

Creep law and creep parameters

Similar to our previous work, ¹⁸ the steady state creep behavior of the composite is described by following threshold stress ( ${\mathit{\sigma }}_0$ )-based law,

$\dot{\overline{\epsilon }}=\left[M(r)(\overline{\sigma }-{\sigma }_0)\right]{}^5,$

(9)

where $M(r)=\frac{1}{E}{\left(A^{\prime }exp\frac{-Q}{RT}\right)}^{1/5}$ is a creep parameter, in which the symbols A′, Q, E, R and T, respectively, denote the structure-dependent parameter, true activation energy, temperature-dependent Young’s modulus, gas constant and operating temperature. The creep parameters M and ${\mathit{\sigma }}_0$ depend on the dispersoid-size (P) and content (V) and the operating temperature (T). The values of parameters M and ${\mathit{\sigma }}_0$ at any radius (r) of the FGM disc, in terms of P, V(r) and T are estimated from the following regression equations developed in our earlier work ²¹

$M(r)=0.0288-\frac{0.0088}{P}-\frac{14.0267}{T}+\frac{0.0322}{V(r)}$

(10)

${\mathit{\sigma }}_0(r)=-0.084P-0.023T+1.185V(r)+22.207$

(11)

In this study, the constant values of P (=1.7 μm) and T (=623 K) are used. Therefore, the creep parameters will depend only on the content of SiC_p in the FG disc, which is a function of radius alone. The values of M(r) and ${\mathit{\sigma }}_0$ (r) at any radius (r) are estimated by substituting the value of V(r), as estimated from Eq. (4), at the corresponding locations into regression Eqs. (10) and (11), respectively.

Mathematical formulation

This section develops a mathematical model to describe the steady creep behavior of the FG disc having linearly varying thickness. The analysis carried out is based on the following assumptions.

• Material of the disc is incompressible and locally isotropic.

• Stresses at any point in the disc remain constant with time.

• Elastic deformations in the disc, being small, are neglected when compared with creep deformations.

• The disc is under plane stress condition, that is, axial stress ( ${\mathit{\sigma }}_z$ ) is zero.

The generalized constitutive equations for creep in an isotropic composite disc under plane stress condition, when the reference frame is along the principal directions r, θ and z, are ¹

$\begin{array}{l}{\dot{\epsilon }}_r=\frac{\dot{\overline{\epsilon }}}{2\overline{\sigma }}\left[2{\sigma }_r-{\sigma }_{\theta }\right],\\ {\dot{\epsilon }}_{\theta }=\frac{\dot{\overline{\epsilon }}}{2\overline{\sigma }}\left[2{\sigma }_{\theta }-{\sigma }_r\right]\text{and}\\ {\dot{\epsilon }}_z=\frac{\dot{\overline{\epsilon }}}{2\overline{\sigma }}\left[-{\sigma }_r-{\sigma }_{\theta }\right]\end{array}$

(12)

where ${\dot{\mathit{\epsilon }}}_r,{\dot{\mathit{\epsilon }}}_{\mathit{\theta }}$ and ${\dot{\mathit{\epsilon }}}_z$ are the strain rates in r, $\mathit{\theta }$ and z directions, respectively, and ${\mathit{\sigma }}_r$ and ${\mathit{\sigma }}_{\mathit{\theta }}$ are, respectively, the stresses in r and $\mathit{\theta }$ directions.

The effective stress ( $\bar{\mathit{\sigma }}$ ) in an isotropic rotating disc under plane stress condition may be expressed according to von Mises yield criterion as ²²

$\bar{\mathit{\sigma }}=\frac{1}{\sqrt{2}}{\left[{\mathit{\sigma }}_{\mathit{\theta }}^2+{\mathit{\sigma }}_r^2+({\mathit{\sigma }}_r-{\mathit{\sigma }}_{\mathit{\theta }}{)}^2\right]}^{1/\phantom{12}2}.$

(13)

Substituting $\dot{\overline{\epsilon }}$ and $\bar{\mathit{\sigma }}$ , respectively, from Eqs. (9) and (13) into the first equation among the set of equations in Eq. (12), one gets

${\dot{\mathit{\epsilon }}}_r=\frac{\mathrm{d}{\dot{u}}_r}{\mathrm{d}r}=\frac{[2x(r)-1]}{2[\{x(r){\}}^2-x(r)+1){]}^{1/2}}{\left[M(r)\{\bar{\mathit{\sigma }}-{\mathit{\sigma }}_0(r)\}\right]}^5.$

(14)

where $x={\mathit{\sigma }}_r/\phantom{{\mathit{\sigma }}_r{\mathit{\sigma }}_{\mathit{\theta }}}{\mathit{\sigma }}_{\mathit{\theta }}$ and ${\dot{u}}_r=\mathrm{d}u/\mathrm{d}t$ are the radial deformation rate and u is the radial deformation of the disc.

Similarly, the second equation among the set of equations in Eq. (12) becomes

${\dot{\mathit{\epsilon }}}_{\mathit{\theta }}=\frac{{\dot{u}}_r}{r}=\frac{[2-x(r)]}{2[\{x(r){\}}^2-x(r)+1){]}^{1/2}}{\left[M(r)\{\bar{\mathit{\sigma }}-{\mathit{\sigma }}_0(r)\}\right]}^5.$

(15)

Dividing Eq. (14) by Eq. (15) and integrating the resulting equation between limits a to r, we get

${\dot{u}}_r={\dot{u}}_{a}exp\left[\underset{a}{\overset{r}{\int }}\frac{\mathit{\varphi }(r)}{r}dr\right],$

(16)

Where ${\dot{u}}_a$ is the radial deformation rate at the inner radius and ${\displaystyle \mathit{\varphi }(r)=\frac{[2x(r)-1]}{[2-x(r)]}.}$

Substituting ${\dot{u}}_r$ from Eq. (16) into Eq. (15) and simplifying, one gets the tangential stress ( ${\mathit{\sigma }}_{\mathit{\theta }}$ ) as

${\mathit{\sigma }}_{\mathit{\theta }}=\frac{{\dot{u}}_a^{1/5}{\mathit{\psi }}_1(r{)}^{}}{M(r)}+{\mathit{\psi }}_2(r),$

(17)

where

${\mathit{\psi }}_1(r)=\frac{\mathit{\psi }(r{)}^{1/5}}{[\{x(r){\}}^2-x(r)+1){]}^{1/2}},{\mathit{\psi }}_2(r)=\frac{{\mathit{\sigma }}_0(r)}{[\{x(r){\}}^2-x(r)+1){]}^{1/2}}$

(18)

and

$\mathit{\psi }(r)=\frac{2[\{x(r){\}}^2-x(r)+1){]}^{1/2}}{r[2-x(r)]}exp\left(\underset{a}{\overset{r}{\int }}\frac{\mathit{\varphi }(r)}{r}\mathrm{d}r\right).$

(19)

The equilibrium equation for a rotating disc having variable thickness is given by ²²

$\frac{\mathrm{d}}{\mathrm{d}r}[h(r)r{\mathit{\sigma }}_r]-h(r){\mathit{\sigma }}_{\mathit{\theta }}+\mathit{\rho }(r){\mathit{\omega }}^2{r}^{2}h(r)=0.$

(20)

Assuming that the disc is connected to the shaft by means of splines, where small axial movement is permitted, the boundary conditions ¹² are

${\mathit{\sigma }}_r=0\mathrm{a}\mathrm{t}\mathrm{(}\mathrm{i}\mathrm{)}r=a\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{(}\mathrm{i}\mathrm{i}\mathrm{)}r=b.$

(21)

Integrating the equilibrium Eq. (20) between limits a to b under the boundary conditions given in Eq. (21), we obtain

${\mathit{\sigma }}_{\mathit{\theta }\mathrm{a}\mathrm{v}}=\frac{{\mathit{\omega }}^2}{A_0}\left[A_{\mathit{\rho }}{I}_0-B_{\mathit{\rho }}\left(\frac{(h_b+2cb)(b^4-a^4)}{4}-\frac{2c(b^5-a^5)}{5}\right)\right],$

(22)

where ${\mathit{\sigma }}_{\mathit{\theta }\mathrm{a}\mathrm{v}}\left(=\frac{1}{A_0}\underset{a}{\overset{b}{\int }}h(r){\mathit{\sigma }}_{\mathit{\theta }}\mathrm{d}r\right)$ is the average tangential stress in the disc area, A ₀ is the area and J ₀ is the polar moment of area of the disc, as defined in Appendix B.

Multiplying Eq. (17) by h(r)dr and integrating the resulting equation between limits a to b, one gets,

${\dot{u}}_a^{1/n}=\frac{A_0{\mathit{\sigma }}_{\mathit{\theta }av}-\underset{a}{\overset{b}{\int }}h(r){\mathit{\psi }}_2(r)\mathrm{d}r}{\underset{a}{\overset{b}{\int }}\frac{h(r){\mathit{\psi }}_1(r)}{M(r)}\mathrm{d}r}.$

(23)

Knowing ${\mathit{\sigma }}_{\mathit{\theta }\mathrm{a}\mathrm{v}}$ from Eq. (22) and ${\dot{u}}_a^{1/5}$ from Eq. (23), the tangential stress in the disc may be estimated from Eq. (17).

Integrating equilibrium Eq. (20) again between limits a to r, the radial stress ( ${\mathit{\sigma }}_r$ ) in the disc is obtained as

${\mathit{\sigma }}_r=\frac{1}{h(r)r}\left[\underset{a}{\overset{r}{\int }}h(r){\mathit{\sigma }}_{\mathit{\theta }}\mathrm{d}r-{\mathit{\omega }}^2\left\{A_{\mathit{\rho }}{I}_{}-B_{\mathit{\rho }}\left(\frac{(h_b+2cb)(r^4-a^4)}{4}-\frac{2c(r^5-a^5)}{5}\right)\right\}\right].$

(24)

where J is the polar moment of area of the disc element with the inner and outer radii as a and r, respectively (Appendix B).

The distribution of stresses, ${\mathit{\sigma }}_{\mathit{\theta }}$ and ${\mathit{\sigma }}_r$ , in the disc are obtained by an iterative numerical scheme. Following which, the strain rates ${\dot{\mathit{\epsilon }}}_r$ and ${\dot{\mathit{\epsilon }}}_{\mathit{\theta }}$ in the disc are estimated from Eqs. (14) and (15), respectively.

Results and discussions

A computer code, based on the mathematical analysis presented in previous section, has been developed to obtain the distribution of stresses and steady state creep rates in the FGM discs having two different linear TG as described in Table 1. The values of a, b and t are taken, respectively, as 31.75, 152.4 and 25.4 mm. These dimensions are similar to those used in earlier study by Bhatnagar et al. ¹³ The results obtained for FGM discs D2 and D3 are compared with a similar FGM disc but having uniform thickness, while keeping volume of all the discs equal.

Table 1.

Description of rotating discs.^a

Disc (notation)	Disc thickness (mm)		Thickness gradient, (TG) = (ha  − hb ) (mm)	V min (vol%)
	ha	hb
Uniform thickness FG disc (D1)	25.40	25.40	0	10.37
Linearly varying thickness FG disc (D2)	35.41	19.62	14.72	8.59
Linearly varying thickness FG disc (D3)	43.22	13.97	29.25	6.53

FG: functionally graded.

^a For all the discs V _max = 35% and V _av = 20%.

Variation of creep parameters

Figure 1a and b show the variation of creep parameters M and ${\mathit{\sigma }}_0$ , respectively, with radial distance in the FG discs. The parameter M observed in all the FG discs (D1–D3) increases with increasing radial distance. The increase observed in M is attributed to decrease in particle content V(r) in the FG discs on moving from the inner to the outer radius, as evident from Figure 2. With the increase in TG, defined as the difference between maximum and minimum thickness of the disc, the content of reinforcement decreases toward the outer radius (Figure 2), therefore increasing the value of M. The distribution of M becomes steeper with the increase in TG. On the other hand, the threshold stress ( ${\mathit{\sigma }}_0$ ) in the FG discs (D1–D3) decreases with increasing TG except at the inner radius, where all the discs have equal amount of reinforcement (V _min = 35%; Figure 1b). The threshold stress is higher in disc having relatively higher amount of SiC_p compared with the disc containing relatively lesser amount of SiC_p. Similar to the creep parameter M, the variation of ${\mathit{\sigma }}_0$ also becomes steeper with the increase in TG.

Figure 1.

Variation of creep parameters in composite discs.

Figure 2.

Variation of reinforcement content in composite discs.

Distribution of stresses and strain rates

The effect of imposing TG on creep behavior of the FG discs is shown in Figures 3–7. The radial stress in all the discs (Figure 3) increases from zero at the inner radius, reaches maximum before decreasing to zero again at the outer radius, under the imposed boundary conditions given in Eq. (21). The radial stress in FG disc having linearly varying thickness (i.e., D2 and D3) is lower than the uniform thickness disc D1. By increasing TG of the disc, the radial stress is observed to decrease over the entire radius. The FG disc D3 having maximum TG exhibits lower radial stress over the entire radius compared with any other FG disc. The maximum value of radial stress in the FG discs D2 and D3 is lower, respectively, by about 8% and 19%, when compared with FG disc D1.

Figure 3.

Effect of thickness gradient on radial stress.

Figure 4 shows the variation in tangential stress of different FG discs. The tangential stress increases a little on moving from the inner to outer radius of the disc, becomes maximum before decreasing again toward the outer radius. The effect of imposing TG on the tangential stress in FG disc is similar to that observed for radial stress in Figure 3. The tangential stress in FG disc D3 having maximum TG is lower than the FG discs D1 and D2. When compared with FG disc D1, the maximum values of tangential stress in FG discs D2 and D3 are lower, respectively, by about 13% and 24%. The effect of imposing TG on effective stress $(\bar{\mathit{\sigma }})$ , as shown in Figure 5, is similar to those observed for radial (Figure 3) and tangential (Figure 4) stresses. By employing higher TG, the effective stress in the FG disc decreases throughout, the decrease being relatively higher near the inner radius than that observed toward the outer radius.

Figure 4.

Effect of thickness gradient on tangential stress.

Figure 5.

Effect of thickness gradient on effective stress.

The tangential strain rate in the FG disc remains tensile throughout and decreases on moving from the inner to the outer radius (Figure 6). By increasing TG in the FG disc, there occurs a significant reduction in the tangential strain rate. The reduction observed near the inner radius is much higher than that observed near the outer radius. Furthermore, by increasing TG in the FG disc (discs D2 and D3), the distribution of tangential strain rate becomes relatively more uniform, thereby decreasing the chances of distortion in the FG discs. Unlike tangential strain rate, the radial strain rate in the FG disc remains compressive throughout, except for some tensile value in the middle of FG disc D1 having uniform thickness (Figure 7). Furthermore, the magnitude of radial strain rate in the FG disc is relatively lower than the magnitude of tangential strain rate at the corresponding radius, as evident from the comparison of Figures 6 and 7. The effect of imposing TG on the radial strain rate (Figure 7) is similar to those observed for tangential strain rate in Figure 6. The radial strain rate in constant thickness FG disc D1 remains compressive, except for some tensile value in the middle of the disc. But on imposing TG, the radial strain rate in the disc becomes compressive over the entire radius. It is evident from the above discussion that when compared with constant thickness FG disc, the magnitude of stress and strain rates in a similar FG disc but having linearly varying thickness is much lower. The creep performance of the FG disc is further improved with the increase in TG of the FG disc.

Figure 6.

Effect of thickness gradient on tangential strain rate.

Figure 7.

Effect of thickness gradient on radial strain rate.

Conclusions

The study carried out has led to the following conclusions.

1. The magnitude of radial, tangential and effective stresses in the FGM disc is significantly reduced when the disc profile is changed from uniform thickness to linearly varying thickness. The reduction observed in stresses further decreases with the increase in TG of the FGM disc.

2. The steady state creep rates in the FG disc are also reduced significantly when its profile changes from constant thickness to linearly varying thickness. The magnitude of strain rates in the FG disc reduces further with the increase in TG of the disc.

3. By employing FG disc with linearly varying thickness, the distribution of strain rates becomes relatively uniform when compared with a similar FG disc of constant thickness. Therefore, the chances of distortion in the FGM disc are reduced when a uniform thickness FG disc is replaced by a similar FG disc but having linearly varying thickness.