Introduction

Functionally graded materials (FGMs) are advanced classes of nonhomogeneous materials having advantages over conventional metallic materials and laminated composites, especially in high temperature applications. The first FGM concept was proposed in 1984 by Niino et al. at the National Aerospace Laboratory of Japan [1, 2]. Limitations of layered laminated composite shaft, such as thermomechanical property mismatch and inter-laminar stresses could be reduced by the use of FG shafts in design of rotating machinery components. Some works are already available in existing literatures dealing with structures made of FGMs [3,4,5,6,7] along with a few works on FG shafts. Studies on parametric instability of FG beams subjected to a periodic load are also reported along with the effects of system parameters on instability [8, 9]. Gayen and Roy [10] reported the FE-based stability analysis of a rotating FG shaft.

Studying the influence of crack is very important, as the presence of crack may lead to structural failure sometimes associated with loss of economic and human life. Exhaustive literature reviews [11] have been reported which provided valuable information and knowledge about the vibration of cracked structures. A number of works [12, 13] has been reported where local flexibility matrixes are derived for cracked structures. Sekhar and Prabhu [14] have presented the vibration studies of cracked shaft along with identification and detection of crack. The dynamic analysis and influence of crack parameters on homogeneous structural members have been reported by Sinou and Lees [15]. Darpe et al. [16] reported analytical solutions of cracked rotor and obtained various dynamic responses with the surface crack and breathing crack conditions.

Theoretical studies of free vibration and buckling responses for FG cracked beam were presented by Yang and Chen [17]. Ke et al. [18] analytically solved buckling and vibration problem for FG beams. Free vibrations of a cracked FG beam were performed analytically by Ferezqi et al. [19]. Wei et al. [20] analytically solved vibration problem of cracked beams made of FGM with multiple cracks. Aydin [21] studied the effects of crack’s depth and location, material gradient and end conditions on natural frequency of cracked beam made of FGMs.

From the literature review, it could be seen that a good number of published literatures are available in the analysis of structures made of FGM and in addition a few works are also available on FG shafts. However, works on the analysis of cracked FG shafts are scanty even though there are a few works available for such FG beams. Due to its increasing potential demands in various important engineering sections, it is important to analyze transverse vibration of cracked FG shafts. Thus, in this work an attempt is made to develop an FE model capable of analyzing free vibration of cracked FG shaft and to understand the influences of crack’s depth and location, slenderness ratio, material gradient parameter and thermal gradient on free vibration of the shaft.

FE Modeling of an FG Cracked Shaft

Timoshenko beam elements have been used for modeling the non-spinning FG shaft with simply supported end conditions where the properties of material are graded radially. Each element has two nodes and four degrees of freedom at each node. A transverse surface crack is considered and crack is assumed as open. Transverse vibration is studied where damping is not considered. Figure 1a–c shows the FE model of the shaft, a shaft element along with general loads and the cracked cross-section geometry, respectively. Here, the total length is \(L\), \(L_{\text{e}}\) is the element length and \(\alpha\) is the crack depth at distance \(L_{c}\) from the left end support of the shaft. The shaft element is loaded with bending moments P4, P5, P10 and  P11, shear forces P2, P3, P8 and P9. R and b are the shaft radius and half-width of the crack, respectively.

Fig. 1
figure 1

FG cracked shaft: a finite element discretization, b shaft element, c cracked cross-section geometry

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In presence of crack on FG shaft, local flexibility coefficients (LFCs) are derived by applying Paris’ equation [22] in Castigliano’s theorem where stress intensity factors (SIFs) are expressed as [12].

In the presence of crack, strain energy \(U_{c}\) and displacement \(u_{i}^{\text{c}}\) are expressed as,

$$\begin{aligned} & u_{i}^{\text{c}} = \frac{{\partial U_{\text{c}} }}{{\partial P_{i} }} ,\quad U_{\text{c}} = \int\limits_{0}^{{\mathop \alpha \nolimits_{x} }} {J(y){\text{ d}}y} \, \\ & {\text{and}}\quad J(y) = \frac{1}{E(y,T)}\left[ {\left( {\sum\limits_{i = 2}^{5} {\mathop K\nolimits_{{{\rm I}i}} } } \right)^{2} + \left( {\sum\limits_{i = 2}^{5} {\mathop K\nolimits_{{{\rm I}{\rm I}i}} } } \right)^{2} + \{ 1 + \nu (y)\} \left( {\sum\limits_{i = 2}^{5} {\mathop K\nolimits_{{{\rm I}{\rm I}{\rm I}i}} } } \right)^{2} } \right], \\ \end{aligned}$$
(1)

where strain energy release rate \(J(y)\) is a function of geometry of crack and loads applied, \(E(y,T)\) is Young’s modulus, Poisson ratio is \(\nu (y)\). KI, KII and KIII are mode I, II and III SIFs, respectively and \(i = 2 , { 3, 4, 5 }\) are load indices. Referring to Fig. 1c, finally the cracked shaft direct and cross-couple LFCs are computed as

$$C_{ij}^{\text{c}} = \frac{{\partial u_{i}^{c} }}{{\mathop {\partial P}\nolimits_{j} }} = \frac{{\mathop \partial \nolimits^{2} }}{{\mathop {\partial P}\nolimits_{i} \mathop { \, \partial P}\nolimits_{j} }}\int\limits_{ - b}^{b} {\int\limits_{0}^{{\mathop \alpha \nolimits_{x} }} {J(y){\text{ d}}y {\text{d}}z} } ,$$
(2)

where \(\alpha_{x} = (\alpha - R) + \sqrt {\mathop R\nolimits^{2} - \mathop z\nolimits^{2} }\) and \(b = \sqrt {\mathop R\nolimits^{2} - \mathop {(R - \alpha )}\nolimits^{2} }\).

While performing the integrations in Eq. (2) for determination of the direct and cross-couple terms of LFCs, \(E(y,T)\) and \(\nu (y)\) are considered temperature dependent as well as function of radial direction.

In the presence of crack, the local flexibility matrix \(\mathop C\nolimits^{\text{c}}\) is represented as

$$\mathop C\nolimits^{\text{c}} = \left[ {\begin{array}{*{20}c} {C_{22}^{\text{c}} } & 0 & 0 & 0 \\ 0 & {C_{33}^{\text{c}} } & 0 & 0 \\ 0 & 0 & {C_{44}^{\text{c}} } & {C_{45}^{\text{c}} } \\ 0 & 0 & {C_{45}^{\text{c}} } & {C_{55}^{\text{c}} } \\ \end{array} } \right].$$
(3)

The elements of \(\mathop C\nolimits^{{\mathbf{c}}}\) are given in detail in Appendix A. Total flexibility matrix is given as

$$C{\mathbf{ = }}\mathop C\nolimits^{uc} + \mathop C\nolimits^{c} ,$$
(4)

where \(\mathop C\nolimits^{{{\text{u}}{\mathbf{c}}}}\) is the un-cracked flexibility matrix obtained using energy method as

$$\mathop C\nolimits^{\text{uc}} = \frac{{\mathop L\nolimits_{\text{e}} }}{6E(y,T)I}\left[ {\begin{array}{*{20}c} {\mathop {2L}\nolimits_{\text{e}}^{2} } & 0 & 0 & {\mathop {3L}\nolimits_{\text{e}}^{{}} } \\ 0 & {\mathop {2L}\nolimits_{\text{e}}^{2} } & {\mathop { - 3L}\nolimits_{\text{e}}^{{}} } & 0 \\ 0 & {\mathop { - 3L}\nolimits_{\text{e}}^{{}} } & {\mathop 6\nolimits_{{}} } & 0 \\ {\mathop {3L}\nolimits_{\text{e}}^{{}} } & 0 & 0 & {\mathop 6\nolimits_{{}} } \\ \end{array} } \right],$$
(5)

where \(I\) denotes area moment of inertia.

From the conditions of equilibrium of shaft element,

$$\left\{ {P_{2} , \, P_{8} , \, P_{3} , \, P_{9} , \, P_{4} , \, P_{10} , \, P_{5} , \, P_{11} } \right\}^{\text{T}} = \mathop {\varvec{\Pi}}\nolimits^{{\mathbf{T}}} \left\{ {P_{8} , \, P_{9} , \, P_{10} , \, P_{11} } \right\}^{\text{T}} ,$$
(6)

where the transformation matrix is

$$\varPi = \left[ {\begin{array}{*{20}c} { - 1} & 1 & 0 & 0 & 0 & 0 & {\mathop { - L}\nolimits_{\text{e}} } & 0 \\ 0 & 0 & { - 1} & 1 & {\mathop L\nolimits_{\text{e}} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & { - 1} & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & { - 1} & 1 \\ \end{array} } \right].$$

For the cracked element, the stiffness matrix is derived applying the virtual work principle as

$$K_{\text{c}} = \mathop \varPi \nolimits^{\text{T}} \mathop C\nolimits^{ - 1} \varPi .$$
(7)

System Equation of Motion

Having a crack on the FG shaft in fixed coordinate, the system equations of motion are

$${\mathbf{{\rm M}\ddot{p}}}(t){\mathbf{ + Kp}}(t){\mathbf{ = f}}(t),$$
(8)

where M represents the rotary and translational mass matrix, K represents the structural stiffness matrix and the vectors of nodal displacement and external forces are presented by p(t) and f(t), respectively.

Material Property Variation of FGMs

In thermal environment, material properties \(C\) [23] are considered as

$$C = C_{0} (C_{ - 1} T^{ - 1} + 1 + C_{1} T + C_{2} T^{2} + C_{3} T^{3} ),$$
(9)

where temperature (T in Kelvin) coefficients for each constituent material is presented by \(C_{0} , \, C_{ - 1} , \, C_{1} , \,\)\(C_{2} {\text{ and }}C_{3}\).

According to the power law gradation, material properties \(C(y,T)\) vary with radial coordinate and temperature [4] as

$$C(y,T) = C_{\text{m}} (T) + \{ C_{\text{c}} (T) - C_{\text{m}} (T)\} \left( {\frac{{y - R_{\text{m}} }}{{R_{\text{c}} - R_{\text{m}} }}} \right)^{k} ,$$
(10)

where \(C\) represents the properties [\(E, \, \nu\) \(K\) (thermal conductivity) and \(\rho\) (density)]. m and c represent subscripts for metal and ceramic, respectively. For fully ceramic material gradient index \(k \to 0\) and for fully metal \(k \to \infty\).

Temperature distribution obtained in the radially graded FG shaft using Fourier equation of heat conduction (steady-state one-dimensional) without heat generation is,

$$\frac{\text{d}}{{{\text{d}}y}}\left[ {K(y)\frac{{{\text{d}}T}}{{{\text{d}}y}}} \right] = 0,$$
(11)

where at \(y = R_{\text{m}}\), \(T = T_{\text{m}}\) and at \(y = R_{\text{c}}\), \(T = T_{\text{c}}\). The solution of Eq. (11) was obtained following [5] as,

$$T(y) = T_{\text{m}} + (T_{\text{c}} - T_{\text{m}} )\zeta (y).$$
(12)

Solution Procedure

For cracked elements, the element stiffness matrix is replaced by \(K_{\text{c}}\) in the global stiffness matrix \(K\). Thus, eigen frequencies are computed by solving for eigenvalue of \(K - \omega_{i}^{{{\text{uc}}2}} \, M \, = \, 0\) and \(K_{\text{c}} - \omega_{i}^{{{\text{c}}2}} \, M \, = \, 0\), for the un-cracked and cracked shaft system, respectively. \(\omega_{i}^{\text{c}}\) and \(\omega_{i}^{\text{uc}}\) are natural frequencies for FG cracked and un-cracked shaft, respectively, and the mode number indicates by \(i\).

Results and Discussions

Using the formulation discussed in “FE Modeling of an FG Cracked Shaft”, a finite element code is developed in MATLAB environment. Aluminum oxide (Al2O3) and steel are considered to form the FGM. Material properties for steel, stainless steel (SS) and aluminum oxide are listed in Tables 1 and 2. A shaft (both the homogeneous as well as the FG shaft) with simply supported boundary conditions is modeled using 19 elements as shown in Fig. 1a. In this study, shaft diameter, \(D = 0.08{\text{ m}}\) and slenderness ratio \(L/D = 12.5,{ 11} . 5 ,\) \(1 0. 5\) are considered.

Table 1 Mechanical properties of composition of FGM
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Table 2 Temperature coefficients for mechanical properties [3]
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Validation of the FE Code for Homogeneous Material

For validation, a shaft with \(L = 1{\text{ m, }}D = 0.08{\text{ m}}\) and made of steel is considered. The compliance \(\overline{C}_{55}\) in dimensionless form determined with crack size \(\alpha /R\) and Table 3 shows an excellent agreement when compared with published results [13]. The code has also been validated by showing excellent agreements while comparing the natural frequencies obtained with those obtained from closed form solutions for the metal and ceramic shafts, as shown in Table 4.

Table 3 Dimensionless compliance \(\overline{C}_{55}\) with \(\alpha /R\)
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Table 4 Natural frequencies (in Hz) of simply supported, homogenous, un-cracked shaft
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The first four dimensionless natural frequencies \((\omega_{i}^{\text{c}} /\omega_{i}^{\text{uc}} )\) with \(\alpha /R\) and \(L/D,\) are computed for centrally located crack, and compared with published work by Sekhar and Prabhu [14] for cracked metal shaft as shown in Fig. 2a–c and an excellent agreement could be observed.

Fig. 2
figure 2

Variation of \(\omega_{i}^{\text{c}} /\omega_{i}^{\text{uc}}\) with \(\alpha /R\) and \(L/D\): a first, b second, c third

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Validation of the FE Code for Cracked FG Beam

After validating the FE formulation for homogenous cracked shaft, the code has also been validated for a cracked FG component. In the absence of numerical results for a cracked FG shaft, validation has been done by comparing the results obtained from the present code for a cracked FG beam with the already published results [17, 20].

For this, a cracked FG beam with different end conditions (cantilever, simply supported, fixed–fixed), and with dimensions \((L/D = 10.0, \alpha /R = 0.4, \, k = 2.0)\) and material properties \((E_{1} = 70{\text{ GPa, }}\rho_{1} = 2780{\text{ kg}}/{\text{m}}^{3} , { }\nu = 0.33)\) similar to those used by Yang and Chen [17] and Wei et al. [20] are used. Computed dimensionless natural frequencies are compared with published analytical solutions [17, 20] and are tabulated in Table 5. It shows an excellent agreement with those of published results thus validating the developed code.

Table 5 Variation in dimensionless frequencies for a FG cracked beam with end conditions
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Variation of LFCs with \(\alpha /R\) and \(k\)

In the presence of crack on FG shaft, dimensionless direct and cross-couple LFCs are determined as \(\overline{C}_{22} = C_{22} /\pi E_{\text{m}} R,\) \(\, \overline{C}_{33} = C_{33} /\pi E_{\text{m}} R,\) \(\overline{C}_{44} = C_{44} /\pi E_{\text{m}} \mathop R\nolimits^{3} ,\) \(\overline{C}_{45} = C_{45} /\pi E_{\text{m}} \mathop R\nolimits^{3} ,\) \(\bar{C}_{55} = C_{55} /\pi E_{\text{m}} \mathop R\nolimits^{3}\). Variations of dimensionless LFCs with \(\alpha /R\) and different material gradient indices \(k\) have been plotted in Fig. 3a–e. It could be observed from Fig. 3a–e that as \(\alpha /R\) increases dimensionless LFCs increase and for a given \(\alpha /R\), as \(k\) increases the LFCs increase. This is because the metal content increases as \(k\) increase.

Fig. 3
figure 3

Variation of LFCs. a \(\overline{C}_{22}\), b \(\overline{C}_{33}\), c \(\overline{C}_{44}\), d \(\overline{C}_{45}\) and e \(\overline{C}_{55}\) with \(\alpha /R\) for different \(k\)

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Variation in \(\omega_{i}^{\text{c}}\) with \(\alpha /R\)

First four natural frequencies are determined for cracked FG (steel/Al2O3) and metal (steel) shafts with \(k = 0.5\) and \(L/D = 12.5\), and for different \(\alpha /R\) and are listed in Table 6. Results in Table 6 show that for each mode, the natural frequency of shaft made of FGM is more compared to that of the metallic shaft and the frequencies are higher for the un-cracked shaft than those for the cracked shaft. Results also show that for a specific \(L/D\), crack location \(L_{\text{c}} /L\) and \(k\) as \(\alpha /R\) increases, natural frequencies decrease for all the modes. However, for the first and third mode, the percentage reduction in frequency is observed to be higher than those for the second and fourth modes. This is due to the fact that mode shapes for first and third modes are anti-nodes at the crack location. It is also noticed that the extent of percentage reduction in frequency is higher in the FG shaft than those of the homogeneous (metal) shaft.

Table 6 Variation in natural frequencies (in Hz) with \(\alpha /R\) for homogeneous (steel) and FG (steel/Al2O3) shaft
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Effect of \(\alpha /R\) and \(k\) on \(\omega_{i}^{\text{c}}\)

For the FG shaft system having \(L/D = 12.5{\text{ and}}\) \(L_{\text{c}} /L = 0.5\), Fig. 4a–d shows the variations of first four frequencies with \(k\) and \(\alpha /R\). From Fig. 4a–d, it is clearly noticed that with increase in \(\alpha /R\), natural frequencies decreases and at particular values of \(\alpha /R\), natural frequencies increase with decrease in \(k\). It could be also seen that at higher values of \(\alpha /R\) reduction in natural frequency with \(\alpha /R\) is more compared to that for lower \(\alpha /R\). It is observed that for all the four modes, variations in natural frequencies are more for \(k < 1\), which represents increase in ceramic content.

Fig. 4
figure 4

Variation of \(\omega_{i}^{\text{c}}\) with \(\alpha /R\) for different \(k\): a first, b second, c third and d fourth

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Effect of \(\alpha /R\) and \(L_{\text{c}} /L\) on Natural Frequency

With a crack on an FG shaft with \(k = 3.0\) and \(L/D = 12.5\), Fig. 5a–d shows the first four dimensionless natural frequencies corresponding to different values of \(\alpha /R\) and \(L_{\text{c}} /L\). Results show that for a given \(L/D\), \(\alpha /R\) and \(k\), the first four natural frequencies (dimensionless) monotonically decrease as \(\alpha /R\) increase and the extent of reduction in natural frequency is maximum for the first mode and is minimum for the third mode. However, for an offset crack located at \(L_{\text{c}} /L = 0.1316\), reduction in the natural frequency is maximum for third mode and is minimum for first mode.

Fig. 5
figure 5

Variation of \(\omega_{i}^{\text{c}}\) with \(L_{\text{c}} /L\) and \(\alpha /R\): a first, b second, c third and d fourth

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Figure 6a–d shows the variation in first four natural frequencies (dimensionless) with crack locations and \(k\), for \(L/D = 12.5\) and \(\alpha /R = 0.6\), it is seen that increase in \(k\) leads to more reduction in all the modes. However, the amount of reduction is dependent on the crack location.

Fig. 6
figure 6

Variation of \(\omega_{i}^{\text{c}}\) with \(L_{\text{c}} /L\) and \(k\): a first, b second, c third and d fourth

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Effect of \(\alpha /R\) and \(L/D\) on Natural Frequency

Figure 7a–d shows the first four natural frequencies \(\omega_{i}^{\text{c}}\) for a cracked FG shaft with \(k = 3.0\) and \(L_{\text{c}} /L = 0.5\) for different \(L/D\) and \(\alpha /R\). These results show that for a given \(L/D,L_{\text{c}} /L\) and \(k\), \(\omega_{i}^{\text{c}}\) decrease with the increase in \(\alpha /R\). However, for first and third modes, the reductions are observed to be more than those for the second and fourth modes. It also shows for the cracked shaft that \(\omega_{i}^{\text{c}}\) increases as \(L/D\) decreases. Results in Table 7 show that with decrease in \(L/D\), natural frequencies increase and in the presence of crack, percentage increase in natural frequencies with decreasing \(L/D\) is more as compared to that of un-cracked FG shaft.

Fig. 7
figure 7

Variation of \(\omega_{i}^{\text{c}}\) with \(\alpha /R\) and \(L/D\): a first, b second, c third and d fourth

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Table 7 Variation in \(\omega_{i}\) (in Hz) with \(L/D\)
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Effect of Temperature Gradient on Natural Frequency

Table 8 shows the first four natural frequencies for cracked FG (SS/Al2O3) shaft system considering temperature-dependent material properties (Table 2) for different \(\alpha /R\) and temperature gradient \(\Delta T\), with \(k = 0.5\), \(L/D = 12.5\) and \(L_{\text{c}} /L = 0.5\). Further, with the variation of \(L_{\text{c}} /L\) and temperature gradient, for \(\alpha /R\), \(k = 0.5\) and \(L/D = 12.5\), results are tabulated in Table 9. Results from both Tables 8 and 9 show that temperature has little effect on natural frequencies for lower \(\alpha /R\) and \(L_{\text{c}} /L\) but for higher values of \(\alpha /R\) and \(L_{\text{c}} /L\), natural frequencies are observed to decrease, while \(\Delta T\) increases. This is because the material becomes softer at higher values of \(\Delta T\).

Table 8 Variation in \(\omega_{i}^{\text{c}}\) (in Hz) with \(\alpha /R\) and \(\Delta T\)
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Table 9 Variation in \(\omega_{i}^{\text{c}}\) (in Hz) with \(L_{\text{c}} /L\) and \(\Delta T\)
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Influence of End Conditions on \(\omega_{1}^{\text{c}}\)

To understand the importance of end conditions on \(\omega_{1}^{\text{c}}\) of the shaft made of FGM, cantilever, simply supported and fixed–fixed are considered for the a non-spinning shaft made of FGM and frequency with \(k , { }L_{\text{c}} /L\), for \(\alpha /R = 0.6\) and \(L/D = 12.5\) are computed using the present code and listed in Table 10. As expected for simply supported and fixed–fixed end conditions, fundamental frequency is same for symmetrically located cracks with respect to the mid span of the shaft. However, for cantilever, fundamental frequency values decrease as crack location is moved from fixed end towards the free end. In addition, for the end conditions, \(\omega_{1}^{\text{c}}\) decrease with the increase in material gradient index.

Table 10 Variation in fundamental frequency (in Hz) with different end conditions and surface crack at different \(k{\text{ and }}L_{\text{c}} /L\,\,{\text{for}}\,\alpha /R = 0.6\)
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Conclusions

In this work, a FE-based formulation is put forward to study the free vibration characteristics for a simply supported shaft considering transverse crack on FG shaft. Stiffness of the cracked FG shaft is modified by calculating the direct and cross-couple terms of LFCs. Using the FE code developed, the effects of crack depths, locations, slenderness ratios, temperature gradients and material gradient indices on the free vibration for the shaft made of FGM are determined. The outcomes of the present work are stated as

  • In an FG shaft, where a crack is developed, LFCs are significantly influenced by the material gradient parameter for a given crack depth. Thus, by deciding a suitable material gradient index, the increase in flexibility due to the presence of a crack could be minimized.

  • For a given \(L/D\) and location of crack, as crack size increase, the natural frequency decreases. This reduction in natural frequencies is dependent on material gradient parameter \(k\). Thus, choosing a proper material gradient parameter may limit the extent of reduction in \(\omega\) for a surface crack to appear on the FG shaft.

  • For all the four modes, the studied maximum changes in natural frequency is observed corresponding to material gradient index less than 1 (one).

  • Even though the natural frequency decrease with the increase in slenderness ratio, it is possible to allow higher slenderness ratio for designing an FG shaft by choosing a proper material gradient parameter.

  • Temperature gradient has little effect on the variation in \(\omega\), however, with increase in crack size the reduction increases.

  • Decrease in fundamental frequency with the increase in material gradient is more pronounced in case of fixed–fixed end condition compared to those in the case of simply supported and cantilever cases.