PSI - Issue 45

Thi D. Le et al. / Procedia Structural Integrity 45 (2023) 109–116 "Thi D. Le" / Structural Integrity Procedia 00 (2022) 000 – 000

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shows the cross section of cylinder with the inner and outer radius (a and b respectively), and the coordinate system. Figure 1 (c) illustrates general state of stresses in the principal directions (under internal pressure) of RTP (excluding the inner HDPE tube), while shear stresses are eliminated. The displacement of field in the polar coordinate system is a function of radial, circumferential, and longitudinal directions for general three-dimensional elasticity issues. But in this circumstance, z dependency vanishes as the cylinder has both ends fixed. Moreover, θ should be considered independent because of symmetric boundary conditions through the body. Therefore, the displacement functions in this case can be reduced to = ( ), = 0 =0 . Since only displacement u r remains, the general displacement u can be replaced with u r ( u = u r ) for convenience. Also, strains can be calculated from the displacement in these conditions based on = , = =0 . As mentioned above, the shear strains are ignored in principal directions , , and are zero. Then, the general relation of strain and stress for orthotropic cylinders is demonstrated in the below compliance matrix (1). [ ] = [ 1 − − − 1 − − − 1 ] [ ] (1) Since it is a symmetric compliance matrix, the equation ( = ) can be written like the equation = which is called Maxwell Relations. In this case, the Maxwell Relations equation can be expressed as: ν ν = = n ν ij = n ν ji = (2) Where R i are the elastic moduli ratios and identify the orthotropy degree, E i or E j is the orthotropic modulus of elasticity in each direction and are the orthotropic Poisson’s ratios in each plane . From the equation (2), some elastic moduli and Poisson’s ratios can be reduced by rearranging the compliance matrix of equation (1). Thus, only three Poisson’s ratios ( , ) and the radial elasticity modulus ( ) are needed to determine. In addition, 3 can also be reduced as the ratio of 2 over 1 after applying algebraic manipulations in equation (2). Then inversing the stress-strain equation, the new stress strain relation is illustrated as the below matrix: [ ] = [ ( 2 2 − 1 ) 1 −( 1 + 2 ) − 2 ( + ) −( 1 + 2 ) 1 ( 2 2 −1) − 2 ( 1 + ) − 2 ( + ) − 2 ( 1 + ) 2 ( 1 2 −1) ] [ ] (3) Where = 1 ( 1 2 −1)+ 1 2 ( +2 )+ 2 2 1 (4) By applying Maxwell relations, the equilibrium equation in Equation (5) needs to be satisfied. After further simplifying, it can be shown that by introducing coefficients C 1 and C 2 , the following expressions for displacement and stresses can be derived: + 1 ( − ) + 2 =0 (5)