PSI - Issue 14

Ritu. J. Singh et al. / Procedia Structural Integrity 14 (2019) 549–555 Ritu .J.Singh / Structural Integrity Procedia 00 (2018) 000–000

551

3

1100

1000

900

800

700 True Stress (MPa)

Axial Direction

600

0.0

0.1

0.2

0.3

0.4

Plastic Strain

b

a

Fig. 1. (a) A plane stress FE model of cylinder with boundary condition (b) True stress – plastic strain plot in axial direction for Zr 2.5% Nb

elasticity stiffness matrix can be defined using only two constants viz. young’s modulus and poisons ratio, however for an anisotropic material, there are 21 constants which are required to be defined. The existence of symmetries in the internal structure of the material simplifies the structure of the stiffness tensor. The anisotropy of Zr 2.5%Nb is considered as orthotropic symmetry consisting of three principal coordinates such i.e. radial, transverse and longitudinal direction. There are 9 independent components in elastic tensor for orthotropic material viz young’s modulus, poisson’s ratio and shear modulus in the three directions. For the isotropic material model, the elastic behavior is approximated by considering elastic properties of Zr 2.5% Nb in axial direction i.e. axial direction young’s modulus and poison ratio is used to describe the elastic behavior and true stress –plastic strain plots in axial direction given by Christodoulou et al. (2000) as shown in Fig. 1b. is used to describe the plastic behavior. For orthotropic material model, the elastic behavior is described by 9 elasticity constants as given in table 1 evaluated by Cheong, Kim, and Kim (2002). The yield criteria proposed by R. Hill (1948) is considered in the analysis. This is one of the most widely used criteria of yielding of anisotropic metals better known as Hill’s anisotropic yield criteria given as equation below ܨ ሺ ߪ ஺ െ ߪ ் ሻ ଶ ൅ ܩ ሺ ߪ ் െ ߪ ோ ሻ ଶ ൅ ܪ ሺ ߪ ோ െ ߪ ஺ ሻ ଶ ൅ ʹ ߪܮ ஺் ଶ ൅ ʹ ߪܯ ்ோ ଶ ൅ ʹܰ ߪ ோ஺ ଶ ൌ ߪ ௔ ௒ ଶ hills parameters F,G,H, L, M and N as determined by Christodoulou et al. (2000) for Zr 2.5%Nb are used and are reproduced in table 1. A reference curve based on true stress- strain in axial direction, is used. The stress ratio in each direction is evaluated based on hills anisotropy constants and used in the analysis.

Table 1.Anisotropy related elastic and plastic constants Elasticity Constants of Zr 2.5%Nb ( Cheong, Kim, and Kim 2002) Young’s Modulus, GPa

Shear Modulus, GPa

Poisons ratio

Transverse

104.275

36.95 33.80 34.24

0.3440 0.3158 0.3615

Radial Axial

96.15 96.80

Hills anisotropy coefficient (Christodoulou et al. 2000) F G H L

M

N

0.623

0.1

0.378

1.479

1.065

2.558

2.2. Benchmarking Methodology of analysis to obtain residual stress in autofrettage problem is benchmarked using work done by Hu and Puttagunta (2012). Stress distribution for different autofrettage pressure and residual stress developed was validated using results obtained by Hu and Puttagunta (2012) and analytical solution by Gao (1992) . A thick walled cylinder with inner radius ‘a’ and outer radius ‘b’ subjected to internal pressure ‘p w ’ is considered. Assuming von mises yield criteria, the maximum internal pressure at which yielding begins at inner surface is obtained for isotropic material model by using following equation. It can be verified from the FE results that yielding starts at 194 MPa.