# A Theoretical Model with the Effect of Cracks in the Local Spalling of Full Ceramic Ball Bearings
## 3 Time-Varying Stiffness of Outer Ring Crack Location  
Assuming that the outer ring of full ceramic bearings contains cracks, as shown in Figure 4, the main causes of cracks are bending load and tensile load. Considering the effect of bearing seat on bearing, ignoring the effect of tensile stress on outer ring, the paper assumes that the main cause of cracks is bending load; then, the applied bending moment will not only promote crack propagation, but also increase the strain energy of structure. The final strain energy generated on the outer ring can be decomposed into the sum of the strain energy of the outer ring and the strain energy generated by the crack propagation in the crack-free state because the external action moment caused by the crack propagation does not change [33]:  
\[W=EN_{C}+\Delta U=2\Delta U \tag{16}\]  
\[EN_{C}=\Delta U \tag{17}\]  
Figure 4: A schematic cross-sectional view of a crack at a spalling fault location in the ceramic bearing outer ring. where \(EN_{\rm C}\) is the energy for crack growth and \(\Delta U\) is the increase of elastic strain energy in a cracked outer ring. The final strain energy of the cracked outer ring is  
\[U_{\rm C}=U+\Delta U=U+EN_{\rm C} \tag{18}\]  
where \(U\) is the strain energy of bearing outer ring in the absence of cracks. For the uncracked outer ring with bending stiffness \(EI\), subjected to a pure bending moment \(M\), the strain energy in the outer ring is given by  
\[U=\frac{1}{2}\int\frac{M^{2}}{EI}dx. \tag{19}\]  
According to the load distribution, the bending moment of the crack at the spalling position can be calculated as follows:  
\[M=\int Q_{\varphi}\cdot r\cos\varphi d\varphi. \tag{20}\]  
The energy consumed for crack growth, based on fracture mechanics considerations, is given by  
\[EN_{\rm C}=\int(SERR)dA \tag{21}\]  
where \(SERR\) is the strain energy release rate. Under the bending moment, the strain energy release rate \(SERR\) could be expressed as  
\[SERR=\frac{{K_{\rm I}}^{2}}{E} \tag{22}\]  
where \(E\) is the modulus of elasticity of the ceramic material; \(K_{\rm I}\) is that gravitational intensity factor of the crack calculated by the crack opening displacement method, the crack developing from the spalling is a mode I open crack, the cross-sectional view of the crack is shown in Figure 5 and according to the fracture mechanics of the ceramic material, the calculation of expression is as follows [30]:  
\[K_{\rm I}=M_{\rm e}S_{b}\sqrt{\frac{\pi a}{Q}} \tag{23}\]  
where \(M_{\rm e}\) is the surface correction factor, which is difficult to calculate: \(S_{\rm b}\)=3\(M\)/\(bt^{2}\). In general, the surface correction factor \(M_{\rm e}\) is related to the ellipticity \(a\)/\(c\) of the crack, the crack-outer ring thickness ratio \(a\)/\(t\), the crack length-outer ring width ratio \(c\)/\(b\), and the crack front position \(\theta\). [PERSON] and [PERSON] give the expression of the surface correction factor [30]:  
\[M_{\rm e}= \begin{array}{cc}H\Big{(}\frac{a}{c},\frac{a}{t},\frac{a}{c},\frac {c}{b},\phi\Big{)}&F\end{array} \Big{(}\frac{a}{c},\frac{a}{t},\frac{a}{c}, \frac{c}{b},\phi\Big{)}. \tag{24}\]  
Figure 5: A schematic local cross-sectional view of the location of cracks in the outer ring. The function \(H(\,)\) and \(F(\,)\) in the formula can be calculated by the reference. In addition, it should be noted that _a/t_, _a/c_, _and c/b_ have corresponding value ranges. \(Q\) is the crack shape factor. According to the literature, we can get:  
\[Q=1+1.464\left(\frac{a}{c}\right)^{1.65}. \tag{25}\]  
Simultaneous Equations (16)-(25) available:  
\[\frac{1}{2}\int\frac{M^{2}}{EI_{C}}dx=\frac{1}{2}\int\frac{M^{2}}{EI}dx+\int( SERR)dA \tag{26}\]  
where \(EI_{C}\) is the flexural stiffness of full ceramic bearing outer rings in the event of cracks. From the above equation, the modified bending stiffness of the cracked outer ring is obtained as  
\[EI_{C}=\mu EI, \tag{27}\]  
where \(\mu\) is the crack damage coefficient.