PSI - Issue 71

Ayub Khan et al. / Procedia Structural Integrity 71 (2025) 461–468

465

= − ( ) ( ) ( ) − 1 ( ) The convergence of stress is achieved using a Newton-type algorithm as +1 ( ) = ( ) − − 1 [ ]

(14)

(15)

where = ( ) − ( )+∑ = ⊗ + ∑ =1 ∑ =1

=1 ∑

=1 ⊗ ( )

(16) Once the stress is converged, it is evaluated explicitly in the Second Level . Since the Nye's dislocation tensor is a non-local quantity, it is evaluated by a two-step procedure, similar to that used by Roters and Raabe (2006). In the first step, the integration point values of plastic deformation gradient ( ) are extrapolated to the nodes using the shape functions of the isoparametric element. Taking into account the contribution of all the elements attached to a particular node, the extrapolated ( ) values are then averaged at that node. In the second step, the averaged nodal ( ) values are again interpolated to the Gauss integration points using the shape functions by ( ( )) = ∑ =1 ( ( )) (17) The Nye's dislocation tensor is then evaluated as ( ) = ∑ =1 ( )( ( )) (18) Using Eqs. 6, 7, 9, and 18, is evaluated as ( ) = ( )+∑ ℎ | ( ( ), ( ))| + 0 ( ̂ 2 2 ) 2( ( ) − 0 ) ∑ ( ) | ( ( ), ( ))| (19) For the convergence of stress in the First Level , a tolerance check is put on all components of (Eq.16(a)). In the Second Level , a tolerance is defined to check the value of =( ( )− 2 ( ) ) . If it fails the tolerance check, the iteration restarts at First Level with a lower time increment. 3. Results and discussion We have considered 4 different cases to demonstrate the performance of the diffused interface size dependent CPFEM model. In the first case, a bicrystal is considered and simulated with a stepped and diffused interface. The effect of GNDs is not considered. In the second case, a bicrystal with the diffused interface and GNDs is considered to quantify the additional effect on strain hardening. In the third case, the cubic RVE shown in Fig. 2(b) is simulated as a BCC polycrystal, with the `SSD only' model, to calibrate the simulation input parameters by comparing its macroscopic response with that of pure Iron. In the fourth case, the same polycrystal mesh is considered with diffused grain boundaries and GNDs. In all the simulations, roller supports are used on the bottom, left, and back faces, and the top face is pulled upwards with a constant displacement rate. 1.5. Case I: Bicrystal with Stepped and Diffused Interface In this case, hardening is considered due to SSDs only. The RVE is simulated as FCC crystals with properties given in Table 1. A comparison between the results obtained from stepped and diffused interface (Fig. 3(b)) shows that the diffused interface model can reduce the stress concentrations near the GB. Also, the response in the bulk almost remains unchanged when a diffused representation is used. In the absence of GNDs, the macroscopic response obtained from both models is the same (Fig. 3(c)).