# Engineering of a FGM Interlayer to Reduce the Thermal Stresses Inside the PFCs
## 2 Analytical Methods for the FGM Engineering
### Hollow Cylinder with Thin Thickness and a Homogeneous Thermal Load
The thickness of the interlayer is therefore a key engineering parameter to couple the total strain of the PFM to the one of the heat sink. The last step for the design of the interlayer is to determine the FGM concentration. The FGM must itself satisfy the condition on its mean thermal strain to be equal to the one of the other sub-components, otherwise the effort of eliminating the stress due to the CTE mismatch would be in vain:  
\[\frac{1}{\delta_{3}}\int_{\delta_{2}}^{\delta_{2}+\delta_{3}}\alpha(T,C)(T-T_ {0})dy=\varepsilon_{T,m3}=\varepsilon_{0} \tag{9}\]  
where \(\varepsilon_{0}\) is the target value of the total strain, defined by the heat sink expansion (\(\varepsilon_{0}=\varepsilon_{T,m1}=\varepsilon_{T,m2}=\varepsilon_{T,m3}\)). From the infinite set of \(\alpha\), solutions of Equation (9), one chooses the element for which the thermal strain of the FGM is equal to \(\varepsilon_{0}\) everywhere in the interlayer thickness:  
\[\alpha(T,C)(T-T_{0})=\varepsilon_{0} \tag{10}\]  
In this way, one has tailored the thermal expansion in the FGM so that no thermal stresses arise in the material. In fact, as the thermal strain field is a constant, it automatically satisfies the congruence equations in the whole solid domain. Therefore, even the contribution due to the thermal gradient vanishes, resulting in a stress-free FGM. Combining Equations (1) and (10), the FGM concentration is derived as:  
\[C=\frac{1}{\alpha_{1}(T)-\alpha_{2}(T)}\bigg{[}\frac{\varepsilon_{0}}{T-T_{0} }-\alpha_{2}(T)\bigg{]} \tag{11}\]  
Finally, one obtains the ideal FGM concentration, resulting in the minimization of the thermal stresses in the PFC. The armor and heat sink will be subjected only to the loads due to the thermal gradient, while the interlayer will be stress free. One notices, that \(C\) is not defined for \(T=T_{0}\). Therefore, one imposes that inside the FGM the temperature field must have values either strictly greater or strictly lower than the stress-free temperature. The results of a computation using the methodology we described are reported in Figure 2, that shows the trend over the cylinder thickness of the main parameters. The study has been carried out having a 1-mm-thick AISI316L cylindrical heat sink and 2 mm of W armor. The FGM is developed as a mixture of these two materials. To carry out the computation, the temperature function \(T(y)\) must be given. In our case we use:  
\[T(y)=\frac{b+\sqrt{b^{2}-2a\Big{[}c+qR\ln\Bigl{(}1+\frac{y}{R}\Bigr{)}\Big{]}}}{a} \ ;\ c=-\frac{a}{2}\Bigl{(}\frac{q}{h}+T_{water}\Bigr{)}^{2}-b\Bigl{(}\frac{q}{h}+T _{water}\Bigr{)} \tag{12}\]  
which is the analytical solution of the steady state heat equation for a homogeneously loaded hollow cylinder with inner radius \(R\) and having a thermal conductivity \(\lambda(T)=aT+b\). The coefficients \(a\) and \(b\) are chosen to be the one of AISI316L [6]. \(q\) is the heat flux absorbed at the wall interface with water, \(T_{water}\) is the water bulk temperature and \(h\) is the heat transfer coefficient due to convection. Figure 2 shows that the design rationale of having a constant total strain over the cylinder thickness is satisfied. The ideal \(W\) concentration is computed in such a way that the FGM is stress-free, since it is derived by keeping \(\varepsilon_{T}=\varepsilon_{0}\) in the whole interlayer. One observes that \(C(y)\), computed using Equation (11), does not have as boundary conditions \(C=0\) at the interface with the heat sink or \(C=1\) next to the armor. In previous FGM design studies [10; 20] these conditions were applied to the concentration function, but we find this to be deleterious for the thermal stresses. In fact, to match the total strain we showed that the mean thermal strain of the bodies must be matched, and no need for the continuity of the thermal strain field is required even in an ideal case. The material distribution was also found discontinuous in [11], and here a thermomechanical justification of this effect is found.