PSI - Issue 60

B.P. Kashyap et al. / Procedia Structural Integrity 60 (2024) 494–509 B.P. Kashyap et al. / Structural Integrity Procedia 00 (2023) 000 – 000 3 (4) Here, σ(ε) is the flow stress, and σ 0 (ε) and k(ε) are the constantsdepending on the strain and temperature. 2.1. Constitutive relationship for high temperature deformation At high temperatures, on the other hand, stress is presumably taken to be independent of strain when representing the steady state behavior between stress and strain rate, viz. = ̇ (5) Here K H is the proportionality constant between stress and strain rate depending on test temperature T and grain size d ;and the above relationship (Eq. 5) in its full form by making use of normalization of the parameters so as to get the proportionality constant in dimensionless form is written as (Courtney, 2000) ̇ = 0 ( ) ( ) = 1 (− ) (6) Here, A is material constant, b is the Burgers vector, G is the shear modulus, σ is the flow stress, k B is the Boltzmann constant, p is the inverse grain size exponent, n H is the stress exponent, d is the grain size, = 0 − / is diffusion coefficient with D 0 being the frequency term, Q is the activation energy for deformation, R the universal gas constant and T the test temperature. The flow stress σ , in Eq. (6), is the applied stress for deformation, but when considering certain materials containing particles or other obstacles to deformation or the obstacles that might develop concurrently with deformation, σ is replaced by effective stress, σ e . Thus, instead of σ , σ e = ( σ - σ TH ) or σ e = ( σ - σ i ) or σ e = ( σ - σ TH - σ i ) is used to make Eq. (6) for high temperature deformation more realistic for such materials. σ TH is the threshold stress or the flow stress taken at zero strain rate. The values of m = 1/ n H , Q , and p in Eq. (6) are calculated from the stress-strain rate plots at different temperatures and for different grain sizes, as applicable. The equations used for m = 1/ n H , Q , and p are listed below. = [ ̇ ] , = [ ̇ ] , (7) = [ ( 1 ) ] ̇, = [ ̇ ( 1 ) ] , (8) =[ ] , ̇ = [ ̇ ] , (9) The values of m = 1/ n H , Q and p are used in Eq. (6) to determine the material and mechanism based constant A , all of which are then compared with the corresponding parameters given by different models to evaluate the mechanisms operating for high temperature deformation. Presented in Table 1 are the values of these parameters, as proposed by different models for high temperature deformation.It may however be noted that it is possible that several mechanisms for deformation may take place in parallel or in sequence in materials varying with their locations within the test sample (Gifkins, 1976). ( ) = 0 ( ) + ( ) −0.5

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Table 1. Magnitude of parameters of constitutive relationships according to different models for high temperature deformation.

Mechanisms for high temperature deformation

Parameters for constitutive relationships

Reference

n H

p

Q

Nabarro-Herring creep

1

2

(Nabarro, 1948; Herring, 1950; Courtney 2000; Cao et al., 2013) (Courtney 2000; Cao et al., 2013) Courtney 2000; Cao et al., 2013)

Q L

Coble creep

1

3

Q gb

Grain boundary sliding by lattice diffusion controlled

2

2

Q L