PSI - Issue 61

Sandipan Baruah et al. / Procedia Structural Integrity 61 (2024) 180–187 Baruah and Singh/ Structural Integrity Procedia 00 (2024) 000 – 000

183

4

4 D T      = − − +   (5) The linearization of the primary variable and state variables is performed through one step of Taylor’s series as per Eqs. (6)-(9), in order to use an incremental and iterative solution procedure. ( ) ( 1) δ i i t t − = + u u u (6) ( ) ( 1) pl pl pl ( ) ( ) δ( ) i i t t − = + ε ε ε (7) ( ) ( 1) δ i i t t    − = + (8) ( ) ( 1) δ i i t t D D D − = + (9) The finite element equations of gradient damage for elastoplasticity are derived by weakening the strong forms of the governing equations and then by performing the linearization process (Engelen et al., 2003). They are given in a compact form, as shown in Eq. (10) below. The stiffness matrices and load vectors of Eq. (10) are given below from Eq. (11)-(16). The elastoplastic modulus matrix Z can be found in Engelen et al. (2003). d d T T T uu P   = −    B ZSS ZB K B ZB (11) max pl max pl pl d T y u D P         =−     K B ZSN (12) d T T u P   =−   S Z K N B (13) max pl 2 max pl pl 1 d y T T D l P           =− − +           K N N B B (14) pl 1 exp ( ) int int pl       ext u F F u uu K K K K u u      −                =   F ε (10)

int  F B σ T u =

d



(15)



   

max pl       max D   

y  P

 

 

int F N  =

N

B B

T

2 T

1

d

d

l

(16)

−

 +



 

pl

pl

In the stiffness matrices, the expression of P is given as, max pl  

pl (17) The solution is found by iteratively solving Eq. (10) until convergence is obtained and updating the primary variables. The strains, stresses, and damage are obtained as state variables. Through a large number of incremental load-steps, the stress-strain plot for the full simulation at all Gauss points in the domain can be obtained. 3. Example Problems and Results The temperature-dependent damage law combined with the gradient damage method for elastoplasticity is applied to two different materials, as discussed below. (1 ) P D H = − 3 G  + 