PSI - Issue 14

C. Praveen et al. / Procedia Structural Integrity 14 (2019) 798–805 C. Praveen et al. / Structural Integrity Procedia 00 (2018) 000–000 C. Praveen et al. / Structural Integrity Procedia 00 (2018) 000–000 C. Praveen et al. / Structural Integrity Procedia 00 (2018) 000–000 C. Praveen t al. / Structural Integrity Procedia 00 (20 8) 0 0–000

800

    Ev   Ev   Ev Ev

λ = λ = λ = λ =

where where where where

   

   

1 + v 1 - 2v 1 + v 1 - 2v 1 + v 1 - 2v 1 + v 1 - 2v

For the small increment in elastic strain, the Eq. (2) can be written as   G Tr I   2  e e Δσ Δε Δε For the small increment in elastic strain, the Eq. (2) can be written as   G Tr I   2  e e Δσ Δε Δε For the small increment in elastic strain, the Eq. (2) can be written as   G Tr I   2  e e Δσ Δε Δε For the small increment in elastic strain, the Eq. (2) can be written as   G Tr I   2  e e Δσ Δε Δε

(3) (3) (3) (3)

For the given total strain increment, the elastic strain in Eq. (2) can be calculated by summing the accumulated elastic strain in previous step ( e t ε ) and incremental elastic strain ( e Δε ) and it is given as For the given total strain increment, the elastic strain in Eq. (2) can be calculated by summing the accumulated elastic strain in previous step ( e t ε ) and incremental elastic strain ( e Δε ) and it is given as For the given total strain increment, the elastic strain in Eq. (2) can be calculated by summing the accumulated elastic strain in previous step ( e t ε ) and incremental elastic strain ( e Δε ) and it is given as For the given total strain increment, the elastic strain in Eq. (2) can be calculated by summing the accumulated elastic strain in previous step ( e t ε ) and incremental elastic strain ( e Δε ) and it is given as

= t p ε ε + Δε ε + Δε - Δε e e e e t p ε ε + Δε ε + Δε - Δε e e e e t p ε ε + Δε ε + Δε - Δε e e e e t p ε ε + Δε ε + Δε - Δε = = = = e e e e t t t t = = =

(4) (4) (4) (4)

Using the above Eq. (4), Eq. (2) can be modified as     G Tr + I G    2 2  e e p t t σ ε + Δε ε Δε Δε Using the above Eq. (4), Eq. (2) can be modified as     G Tr + I G    2 2  e e p t t σ ε + Δε ε Δε Δε Using the above Eq. (4), Eq. (2) can be modified as     G Tr + I G    2 2  e e p t t σ ε + Δε ε Δε Δε Using the above Eq. (4), Eq. (2) can be modified as     G Tr + I G    2 2  e e p t t σ ε + Δε ε Δε Δε

(5) (5) (5) (5)

   

   

   

   

is defined as elastic predictor part or trial stress ( is defined as elastic predictor part or trial stress ( is defined as elastic predictor part or trial stress ( is defined as elastic predictor part or trial stress ( p Δε G 2 denotes the plastic corrector part. Equation (5) can also be p Δε G 2 denotes the plastic corrector part. Equation (5) can also be p Δε G 2 denotes the plastic corrector part. Equation (5) can also be p Δε G 2 denotes the plastic corrector part. Equation (5) can also be tr σ ). The last tr σ ). The last tr σ ). The last tr σ ). The last

e e e e ε + Δε ε + Δε ε + Δε ε + Δε

e

In Eq. (5), In Eq. (5), In Eq. (5), In Eq. (5),

Tr + I   ε Δε Tr + I   e ε Δε Tr + I   e ε Δε Tr + I   e ε Δε

G G G G

2 2 2 2

t t t t

t t t t

term in right-hand side of Eq. (5) i.e. term in right-hand side of Eq. (5) i.e. term in right-hand side of Eq. (5) i.e. t rm in right-hand side of Eq. (5) i.e. represented in terms of represented in terms of represented in terms of represented in terms of tr σ as tr σ as tr σ as tr σ as G   2 tr p σ σ Δε G   2 tr p σ σ Δε G   2 tr p σ σ Δε G   2 tr p σ σ Δε

(6) (6) (6) (6)

2.1.2 Yield criteria The yield surface f, defines a surface for a given stress state that separates the elastic zone f < 0 and plastic zone f > 0. Von-Mises 'J2 plasticity' yield criterion has been employed in the present formulation and it is defined as 2.1.2 Yield criteria The yield surface f, defines a surface for a given stress state that separates the elastic zone f < 0 and plastic zone f > 0. Von-Mises 'J2 plasticity' yield criterion has been employed in the present formulation and it is defined as 2.1.2 Yield criteria The yield surface f, defines a surface for a given stress state that separates the elastic zone f < 0 and plastic zone f > 0. Von-Mises 'J2 plasticity' yield criterion has been employed in the present formulation and it is defined as 2.1.2 Yield criteria The yield surface f, defines a surface for a given stress state that separates the elastic zone f < 0 and plastic zone f > 0. Von-Mises 'J2 plasticity' yield criterion has been employed in the present formulation and it is defined as

3 2 3 2 3 2 3 2

           

           

 tr tr σ :σ  tr tr σ :σ  tr tr σ :σ  tr tr σ :σ

   

(7) (7) (7) (7)

f f f f

                   

   

   

e e e e

f f f f

f f f f

2.1.3 Flow rule The incremental plastic strain is defined by the flow rule as 2.1.3 Flow rule The incremental plastic strain is defined by the flow rule as 2.1.3 Flow rule The incremental plastic strain is defined by the flow rule as 2.1.3 Flow rule The incremental plastic strain is defined by the flow rule as

f f f f σ σ σ σ

      

= Δp = Δp = Δp = Δp

Δε p Δε p Δε p Δε p

(8) (8) (8) (8)

In Eq. (8), Δp is the plastic multiplier which is equivalent plastic strain. The direction of flow is the same as the direction of the outward normal to the yield surface ( f) i.e. ( f    ). Differentiating f with respect to σ from Eq. (7) and substituting in Eq. (8), incremental plastic strain is represented in terms of deviatoric stresses as In Eq. (8), Δp is the plastic multiplier which is equivalent plastic strain. The direction of flow is the same as the direction of the outward normal to the yield surface ( f) i.e. ( f    ). Differentiating f with respect to σ from Eq. (7) and substituting in Eq. (8), incremental plastic strain is represented in terms of deviatoric stresses as In Eq. (8), Δp is the plastic multiplier which is equivalent plastic strain. The direction of flow is the same as the direction of the outward normal to the yield surface ( f) i.e. ( f    ). Differentiating f with respect to σ from Eq. (7) and substituting in Eq. (8), incremental plastic strain is represented in terms of deviatoric stresses as In Eq. (8), Δp is the plastic multiplier which is equivalent plastic strain. The direction of flow is the same as the direction of the outward normal to the yield surface ( f) i.e. ( f   ). Differentiating f with respect to σ from Eq. (7) and substituting in Eq. (8), incremental plastic strain is represented in terms of deviatoric stresses as

   

tr σ tr σ tr σ

3 2  3 2  3 2  3 2  t σ

p p p p

(9) (9) (9) (9)

Δε Δε Δε Δε

p p p p

       

tr e tr e tr e tr e

2.1.4 Hardening rule 2.1.4 Hardening rule 2.1.4 Hardening rule 2.1.4 Hardening rule

The f  in Eq. (7) is defined as the flow stress and the value of flow stress depends on the internal-state-variable i.e. forest dislocation density. According to the internal-state-variable approach, the interrelationship between forest dislocation density and flow stress is given as The f  in Eq. (7) is defined as the flow stress and the value of flow stress depends on the internal-state-variable i.e. forest dislocation density. According to the internal-state-variable approach, the interrelationship between forest dislocation density and flow stress is given as The f  in Eq. (7) is defined as the flow stress and the value of flow stress depends on the internal-state-variable i.e. fore t disl cation density. According to the internal-state-variable approach, the interrelationship between forest dislocation density and flow stress is given as The f  in Eq. (7) is defined as the flow stress and he value of flow stress depends on the internal- tate-variable i.e. forest dislocation density. According to the internal-state-variable approach, the interrelationship between forest dislocation density and flow stress is given as

(10) (10) (10) (10)

M Gb M Gb M Gb M Gb

  0      0      0      0   

f     f f f

f f f f