Novel material tailoring method for internally pressurized FG spherical and cylindrical vessels

Abstract

A new material tailoring method for spherical and cylindrical vessels made of functionally graded materials (FGMs) is presented. It is assumed that the FG material is composed of an Al–SiC metallic-matrix composite. A uniform ratio of in-plane shear stress to yield strength [$\varphi \left( r \right)$] is used as the design criterion to utilize the maximum capacity of the vessel. The aim is to find a distribution of SiC particles in the radial direction, i.e., $f\left( r \right)$, that achieves a uniform index $\varphi \left( r \right) =\hbox {const}.$ through the wall thickness of the internally pressurized spherical or cylindrical vessel. Both the Mori–Tanaka and rule-of-mixtures homogenization schemes are used to express the effective elastic module and Poisson’s ratio. Moreover, the strength of the composite is expressed based on the rule of mixtures. Besides, finite element simulation is carried out to verify the accuracy of the analytical solution. The effects of input parameters such as the internal pressure, strength of the SiC particles, ratio of in-plane shear stress to effective yield strength, and choice of homogenization scheme on the tailored distribution of the SiC volume fraction in the radial direction are also investigated.

Appendix A

The constants $Z_1 -Z_9$ in Eqs. (14) and (20) are given as follows

$$\begin{aligned} Z_1= & {} 9C_1 N_1^2,\nonumber \\ Z_2= & {} 9\left( {C_2 N_1 -2C_1 N_1 N_2 } \right) ,\nonumber \\ Z_3= & {} 9\left( {C_1 N_2^2 +C_3 -C_2 N_2 } \right) ,\nonumber \\ Z_4= & {} C_4 N_1^2,\nonumber \\ Z_5= & {} \left( {C_5 N_1 -2C_4 N_1 N_2 } \right) ,\nonumber \\ Z_6= & {} {C_4 N_2^2 +C_6 -C_5 N_2 } ,\nonumber \\ Z_7= & {} C_7 N_1^2 /2,\nonumber \\ Z_8= & {} \left( {C_8 N_1 -2C_7 N_1 N_2 } \right) /2,\nonumber \\ Z_9= & {} \left( C_7 N_2^2 +C_9 -C_8 N_2 \right) /2, \end{aligned}$$

(A1)

where $N_1 -N_2$ and $C_1 -C_9$ are $N_1 =\frac{1}{c\left( {i+1} \right) (Y_\mathrm{p} -Y_\mathrm{m} )}$ ($i=0$ for cylinder and $i=1$ for sphere) and

$$\begin{aligned}&N_2 =\frac{Y_\mathrm{m} }{Y_\mathrm{p} -Y_\mathrm{m} }, \end{aligned}$$

(A2)

$$\begin{aligned}&C_1 =(a_5 a_8 +a_6 )(a_1 a_4 +a_2 ),\nonumber \\&C_2 =(a_1 a_4 +a_2 )a_5 a_7 +(a_5 a_8 +a_6 )a_1 a_3,\nonumber \\&C_3 =a_5 a_1 a_7 a_3,\nonumber \\&C_4 =a_4 a_6 +a_5 a_4 a_8 +3a_1 a_4 a_8 +3a_2 a_8,\nonumber \\&C_5 =a_5 a_7 a_4 +3a_1 a_7 a_4 +a_5 a_8 a_3 +3a_1 a_8 a_3 \nonumber \\ {}&\quad +\,a_6 a_3 +3a_2 a_7,\nonumber \\&C_6 =a_5 a_7 a_3 +3a_1 a_7 a_3,\nonumber \\&C_7 =-\,2a_4 a_6 -2a_5 a_4 a_8 +3a_1 a_4 a_8 +3a_2 a_8,\nonumber \\&C_8=-\,2a_5 a_7 a_4 +3a_1 a_7 a_4 -2a_5 a_8 a_3 +3a_1 a_8 a_3\nonumber \\&\quad -\,2a_6 a_3 +3a_2 a_7,\nonumber \\&C_9 =3a_1 a_7 a_3 -2a_5 a_7 a_3, \end{aligned}$$

(A3)

and $a_1-a_9$ are

$$\begin{aligned} a_1= & {} K_\mathrm{m},\nonumber \\ a_2= & {} (K_\mathrm{p} -K_\mathrm{m} )(3K_\mathrm{m} +4\mu _\mathrm{m} ),\nonumber \\ a_3= & {} 3(K_\mathrm{p} -K_\mathrm{m} )+(3K_\mathrm{m} +4\mu _\mathrm{m} ),\nonumber \\ a_4= & {} -\,3(K_\mathrm{p} -K_\mathrm{m} ),\nonumber \\ a_5= & {} \mu _\mathrm{m},\nonumber \\ a_6= & {} 5G_\mathrm{m} (\mu _\mathrm{p} -\mu _\mathrm{m} )(3K_\mathrm{m} +4\mu _\mathrm{m} ),\nonumber \\ a_7= & {} 5\mu _\mathrm{m} (3K_\mathrm{m} +4\mu _\mathrm{m} )+6(\mu _\mathrm{p} -\mu _\mathrm{m} )(2\mu _\mathrm{m} +K_\mathrm{m} ),\nonumber \\ a_8= & {} -\,6(\mu _\mathrm{p} -\mu _\mathrm{m} )(2\mu _\mathrm{m} +K_\mathrm{m} ). \end{aligned}$$

(A4)