PSI - Issue 14

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

tr tr tr tr tr tr tr tr

e G p M Gb      0 3     G p M Gb      0 3     G p M Gb      0 3     G p M Gb      0 3     G p M Gb      0 3     G p M Gb      0 3     G p M Gb      0 3     G p M Gb      0 3     Applying consistency condition Applying consistency condition Applying consistency condition Applying con istency condition Applyi g con istency c ndition e Applyi g consistency c ndition e Applying consistency condition e Applyi g consistency c ndition Newton-Raphson method.   0 ,    f f p  Newton-Raphson method.   0 ,    f f p  Newton-Raphson method.   0 ,    f f p  Newton-Raphson method.   0 ,    f f p  Newton-Raphson method.   0 ,    f f p  Newton-Raphson method.   0 ,    f f p  Newton-Raphson ethod.   0 ,    f f p  Newton-Raphson method.   0 ,    f f p  f f f f e e e e f f f f f f f f

(16) (16) (16) (16) (16) (16) (16) (16)

802

0   f , Eq. (16) becomes a non-linear equation in Δp and it can be solved using 0   f , Eq. (16) becomes a non-linear equation in Δp and it can be solved using 0   f , Eq. (16) becomes a non-linear equation in Δp and it can be solved using 0   f , Eq. (16) becomes a non-linear equation in Δp and it can be solved using 0   f , Eq. (16) becomes a non-linear equation in Δp and it can be solved using f 0   f , Eq. (16) becomes a non-linear equation in Δp and it can be solved using f 0   f , Eq. (16) becomes a non-linear equation in Δp and it can be solved using f 0   f , Eq. (16) beco es a non-linear equation in Δp and it can be solved using

(17) (17) (17) (17) (17) (17) (17) (17) (18) (18) (18) (18) (18) (18) (18) (18)

Neglecting higher order terms, Eq. (17) can be defined as Neglecting higher order terms, Eq. (17) can be defined as Neglecting higher order terms, Eq. (17) can be defined as Neglecting higher order terms, Eq. (17) can be defined as Neglecting higher order terms, Eq. (17) can be defined as Neglecting higher order terms, Eq. (17) can be defined as Neglecting higher order terms, Eq. (17) can be defined as Neglecting higher order ter s, Eq. (17) can be defined as f f

            f  f  f  f  f f f f

       

0 0 0 0 0 0 0 0

f f f f f f f f

d p d p d p d p d p d p d p d p

d d d d d d d d

       

               

f         f f f f f f

       

f   f  f  f  f  f  f  f f f f

p p p p p p p p

       

The complete expression of Eq. (18) can be represented as The complete expression of Eq. (18) can be represented as The complete expression of Eq. (18) can be represented as The complete expression of Eq. (18) can be represented as The complete expression of Eq. (18) can be represented as f The complete expression of Eq. (18) can be represented as f The co plete expression of Eq. (18) can be represented as f f The complete expression of Eq. (18) can be represented as

                       

                       

M Gb M  M Gb M  M Gb M  M Gb M  M Gb M  M Gb M  M Gb M  b 

tr tr tr tr tr tr tr tr

0 0 0 0 0 0 0 0

    3 G p     3 G p     3 G p     3 G p     3 G p     3 G p     3 G p     3 G p 0 0 0 0 0 0 0 0

M Gb    M Gb    M Gb    M Gb    M Gb    M Gb       M Gb    b

   3 Gd p    3 Gd p    3 Gd p    3 Gd p    3 Gd p    3 Gd p    3 d p    3 Gd p

2 2 2 2 2 2 2 2 K d p  K d p  K d p  K d p  K d p  K d p  K d p  K d p  f f f f f f f f

       

       

               

(19) (19) (19) (19) (19) (19) (19) (19)

e e e e e e e e

f f f f f f f f

bL bL bL bL bL bL bL bL

f f         f f f

2 2 2 2 2 2 2 2

From Eq. (19), the small incremental plastic strain can be obtained and it is given as From Eq. (19), the small incremental plastic strain can be obtained and it is given as From Eq. (19), the small incremental plastic strain can be obtained and it is given as From Eq. (19), the small incremental plastic strain can be obtained and it is given as From Eq. (19), the small incremental plastic strain can be obtained and it is given as f From Eq. (19), the small incremental plastic strain can be obtained and it is given as f From Eq. (19), the small incremental plastic strain can be obtained and it is given as f Fro Eq. (19), the s all incre ental plastic strain can be obtained and it is given as

tr tr tr tr tr tr tr tr

    3 G p     3 G p     3 G p     3 G p     3 G p     3 G p     3 G p     3 p 0 0 0 0 0 0 0 0 M Gb M  M Gb M  M Gb M  M Gb M  M Gb M  M Gb M  M b M  M Gb  f f

M Gb    M Gb    M Gb    M Gb    M Gb    M Gb    M Gb    M b   

       

e e e

f f f f f f f f

(20) (20) (20) (20) (20) (20) (20) (20)

d p d p d p d p d p d p d p d p

               

                       

                       

3 3 3 3 e 3 e 3 e 3 e 3 e

G G G G G G G

MK MK MK MK MK MK MK MK

       

       

f         f f f f f

2 2 2 2 2 2 2 2

bL bL bL bL bL bL bL bL

f        

2 2 2 2 2 2 2 2

k p  ), the current incremental plastic strain can k p  ), the current incremental plastic strain can k p  ), the current incremental plastic strain can k p  ), the current incremental plastic strain can k p  ), the current incremental plastic strain can k p  ), the current incremental plastic strain can k p  ), the current incremental plastic strain can k p  ), the current incre ental plastic strain can

By using d p  and incremental plastic strain in previous step ( By using d p  and incremental plastic strain in previous step ( By using d p  and incremental plastic strain in previous step ( f By using d p  and incr mental plastic strain in previous step ( f By using d p  and incremental plastic strain in previous step ( f By using d p  and incremental plastic strain in previous step ( f f By using d p  and incremental plastic strain in previous step ( f f By using d p  and incre ental plastic strain in previous step ( be updated and it is written as be updated and it is written as be updated and it is written as be updated and it is written as be updated and it is written as be updated and it is written as be updated and it is written as be updated and it is written as k

k p d p       1 k k p d p       1 k k p d p       1 k k p d p       1 k k p d p       1 k k p d p       1 k k p d p       1 k k p d p       1 p p p p p p p p

(21) (21) (21) (21) (21) (21) (21) (21)

Equation (20) is iterated continuously with the updated state variables such as f  , L ,  p . Once the required tolerance is reached (i.e.numerator in Eq. (21)  1  10  6 ) the iteration would be stopped. The obtained p  values at the last increment is substituted in Eq. (9) to calculate p Δε . 2.3 Stress - strain update algorithm The implementation of elasto-plastic material model into finite element scheme is performed with user material subroutine UMAT. The implicit Euler backward algorithm with radial-return scheme has been chosen for integration in order to update stress and internal-variables. Since this method is unconditionally stable and enables large increments, convergence towards solution can be achieved rapidly. For the implicit procedure, it becomes necessary to provide the material tangent stiffness matrix (material Jacobian) in addition to the integration of the plasticity constitutive equations. The material Jacobian is derived as (Dunne and Petrinic, 2005) Equation (20) is iterated continuously with the updated state variables such as f  , L ,  p . Once the required tolerance is reached (i.e.numerator in Eq. (21)  1  10  6 ) the iteration would be stopped. The obtained p  values at the last increment is substituted in Eq. (9) to calculate p Δε . 2.3 Stress - strain update algorithm The implementation of elasto-plastic material model into finite element scheme is performed with user material subroutine UMAT. The implicit Euler backward algorithm with radial-return scheme has been chosen for integration in order to update stress and internal-variables. Since this method is unconditionally stable and enables large increments, convergence towards solution can be achieved rapidly. For the implicit procedure, it becomes necessary to provide the material tangent stiffness matrix (material Jacobian) in addition to the integration of the plasticity constitutive equations. The material Jacobian is derived as (Dunne and Petrinic, 2005) Equation (20) is iterated continuously with the updated state variables such as f  , L ,  p . Once the required tolerance is reached (i.e.numerator in Eq. (21)  1  10  6 ) the iteration would be stopped. The obtained p  values at the last increment is substituted in Eq. (9) to calculate p Δε . 2.3 Stress - strain update algorithm The implementation of elasto-plastic material model into finite element scheme is performed with user material subroutine UMAT. The implicit Euler backward a gorithm with radial-return s heme has been chosen for integration in order to update stress and internal-variables. Since this method is unconditionally stable and enables large increments, converg nce tow rds solution can be achieved rapidly. For the implicit procedure, it becomes ecessary to provide the material tangent stiffness matrix (material Jacobian) in addition to the integration of the plasticity constitutive equations. The material Jacobian is derived as (Dunne and Petrinic, 2005) Equation (20) is iterated continuously with the updated state variables such as f  , L ,  p . Once the required tolerance is reached (i.e.numerator in Eq. (21)  1  10  6 ) the iteration would be stopped. The obtained p  values at the last increment is substituted in Eq. (9) to calculate p Δε . 2.3 Stress - strain update algorithm The implementation of elasto-plastic material model into finite element scheme is performed with user material subroutine UMAT. The implicit Euler backward algorithm with radial-return scheme has been chosen for integration in order to update stress and internal-variables. Since this method is unconditionally stable and enables large increments, convergence tow rds solution can be achieved rapidly. For the implicit procedure, it becomes ecessary to provide the material tangent stiffness matrix (material Jacobian) in addition to the integration of the plasticity constitutive equations. The material Jacobian is derived as (Dunne and Petrinic, 2005) Equation (20) is iterated continuously with the updated state var ables such as f  , L ,  p . Once the required tolerance is reached (i.e.numerator in Eq. (21)  1  10  6 ) the iteration would be stopped. The obtained p  values at the last increment is substituted in Eq. (9) to calculate p Δε . 2.3 Stress - strain update algorithm The implementation of elasto-plastic material model into finite element scheme is performed with user material subroutine UMAT. The implicit Euler backward algorithm with radial-return scheme has been chosen for integration in order to update stress and internal-variables. Since this method is unconditionally stable and enables large increments, convergence towards solution can be achieved rapidly. For the implicit procedure, it becomes necessary to provide the material tangent stiffness matrix (material Jacobian) in addition to the integration of the plasticity constitutive equations. The material Jacobian is derived as (Dunne and Petrinic, 2005) Equation (20) is iterated continuously with the updated state variables such as f  , L ,  p . Once the required tolerance is reached (i.e.numerator in Eq. (21)  1  10  6 ) the iteration would be stopped. The obtained p  values at the last increment is substituted in Eq. (9) to calculate p Δε . 2.3 Stress - strain update algorithm The implementation of elasto-plastic material model into finite element scheme is performed with user material subroutine UMAT. The implicit Euler backward algorithm with radial-return sche e has been chosen for integration in order to update stress and internal-variables. Since this method is unconditionally stable and enables large increments, convergence towards solution ca be achieved rapidly. For the implicit procedure, it becomes necessary to provide the material tangent stiffness matrix (material Jacobian) in addition to the integration of the plasticity constitutive equations. The material Jacobian is derived as (Dunne and Petrinic, 2005) Equation (20) is iterated continu usly with the updated state variables such as f  , L ,  p . Once the required tolerance is reached (i.e.nu erator in Eq. (21)  1  10  6 ) the iteration would be stopped. The obtained p  values at the last incre nt is substituted in Eq. (9) to calculate p Δε . 2.3 Stress - strain upda e alg rithm The imple ntation of elasto-plastic aterial odel into fini e element scheme is performed wit user m terial subroutine U AT. The i plicit Euler backward algorith with radial-return sche e has been chosen for int gration in order to update stress and internal-variables. Since this ethod is unconditionally stable and enables large increments, convergence towards solution ca be achieved rapidly. For the implicit procedure, it becomes necessary to provide the material ta gent stiffness matrix (material Jacobian) in addition to the integration of the plasticity constitutive equations. The aterial Jacobian is derived as (Dunne and Petrinic, 2005) Equation (20) is iterated continu usly with the updated state variables such as f  , L ,  p . Once the required tolerance is reached (i.e.numerator in Eq. (21)  1  10  6 ) the iteration would be stopped. The obtained p  values at the last increment is substituted in Eq. (9) to calculate p Δε . 2.3 Stress - strain update algorithm The implementation of elasto-plastic material model into finite element scheme is performed wit us r material subroutine UMAT. The implicit Euler backward algorithm with radial-return scheme has been chosen for integrati n in order to update stress and internal-variables. Since this method is unconditionally stable and enables large incre ents, convergence tow rds solution can be achieved rapidly. For the i plicit procedure, it beco es necessary to provide the material ta gent stiffness matrix (material Jacobian) in addition to the integration of he plasticity constitutive equations. The material Jacobian is derived as (Dunne and Petrinic, 2005)  tr tr σ σ 

       

                       

                       

                 

                       

2 3 3 3 3 3 3 3 3

e e e  e e  e e  e  

    

tr tr e e tr tr e e tr tr σ σ tr tr e e    tr tr σ σ tr tr e e    tr tr σ σ tr tr e e    tr tr σ σ tr tr e e    tr tr σ σ tr tr e e    tr tr σ σ tr tr e e    t t σ σ    

e tr e tr e tr e tr tr e tr e tr e tr

e tr e e tr e e tr e e tr e e tr e tr e e tr e e tr e

K G K G 2 K G 2 K G 2 K G 2 K G 2 K G 2 K G 2

: II δε II δε II δε II δε II δε II δε II δε II δε : : : : : : :

δσ δσ δσ δσ δσ δσ δσ δσ

GA GA GA GA GA GA GA GA

G G G G G G G G

       

       

                     

2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2

(22) (22) (22) (22) (22) (22) (22) (22)

       

       

       

                                                                 

                                                      

3 3 3 3 3 3 3 3 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1

e e e e  e  e e  e  

1 1

e tr e tr e tr e tr e tr e tr e tr e tr

          

where A = where A = where A = where A = where A = where A = where A = where A =

(23) (23) (23) (23) (23) (23) (23) (23)

f  G M Gb M Mk bL     3 2   G M Gb M Mk bL     3 2   G M Gb M Mk bL     1 3 2   G M Gb M Mk bL     1 3 2   G M Gb M Mk bL     1 3 2   G M Gb M Mk bL     1 3 2   G M Gb M Mk bL     1 3 2   G M Gb M Mk bL     3 2  1

       

       

   

                       

f         f f f f f f f

2 2 2 2 2 2 2 2

f f f f f f f