PSI - Issue 14

Ritam Chatterjee et al. / Procedia Structural Integrity 14 (2019) 251–258 Author name / Structural Integrity Procedia 00 (2018) 000 – 000

254

4

where A is a constant, n is a factor multiplied with the ratio of Burgers vector and average spacing between debris. The dislocation forest contribution to slip resistance is given by: = √ (7) where χ is dislocation interaction coefficient between 0.1 to 1 and μ is the shear modulus. The initial slip resistance is evaluated separately for each slip mode and is of the form: = ( − ) (8) where A is a constant and B is a temperature. These have been calculated via curve fitting data obtained from literature. The dislocation density evolution for each slip mode is updated based on the expression provided by Essmann and Mughrabi (1979) that has been further developed by Mecking and Kocks (1981, 2003): = 1 √ − 2 ( ̇, ) (9) Here, k 1 is a material constant and k 2 is a function of temperature and strain which describes dynamic recovery via thermally activated mechanisms. k 2 is evaluated as: 2 ( ̇, ) 1 = (1 − 3 ̇ ̇ ) (10) where, g m is the normalized activation energy.

2.3. Simulation Algorithm

A few critical material properties and model parameters for Ti are shown in Table 1 (Tromans (2011)).

Table 1: A few critical VPSC input parameters C 11 (GPa) C 33 (GPa) C 44 (GPa)

C 12 (GPa)

C 13 (GPa)

μ (GPa)

χ

160

181

46.5

90

66

38.5

0.7

b (nm)

ρ in (m

k

k 1 (basal) (m -1 )

k 1 (pyramidal) (m -1 )

n

ρ deb (m

-2 )

-2 )

1 (prismatic) (m -1 )

0.295 12 Here, C i j are the elements of the elastic stiffness matrix, is the shear modulus, χ is an interaction coefficient that strongly affects initial work hardening, b is the Burgers vector, ρ in is the initial dislocation density for both old and new grains, ρ deb is the initial debris density, k 1 is a parameter which defines the rate of dislocation storage due to strain hardening and n is the strain hardening exponent that is set to a large value so that slip only occurs when the resolved shear stress is close to critical value. The complete modelling process is summarized in a flow chart in Fig.1. The complete process shown in Fig.1 is within one time step of numerical integration. Within one time step, nucleation occurs several times (for various grains) based on the probability criteria. The threshold value for nucleation to occur viz. P random has been assumed to be 0.5. The probability for each grain is calculated as: 10 12 1*10 6 1*10 6 6*10 7 20