PSI - Issue 14

B V S S Bharadwaja et al. / Procedia Structural Integrity 14 (2019) 612–618 B V S S Bharadwaja, A.Alankar/ Structural Integrity Procedia 00 (2018) 000–000

615

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predicting the size effect on micro beams, torsion of thin wires, strain localization near voids etc. where conventional plasticity theories fail. Due to the complexity of higher order models and their numerical implementation, lower order models were more popular earlier. Several strain-gradient plasticity models have been proposed by Acharya and Bassani (2000), Gao et al. (1999), Huang et al. (2000) without invoking the higher order stresses. As per Huang et al. (2004) these models are different in the sense that GNDs enter directly into the strengthening equations by preserving the same conventional crystal plasticity modeling structure. In the present case we adopted the model developed by Huang et al. (2004) which is considered as lower order and the strain gradient effects are included directly in the Taylor based equation for dislocation hardening. The expressions used to establish the evolution of GNDs are as following. Edge GNDs can be obtained by projecting the gradient of shear strain along slip direction.

1 .     s  





(7)

, GND edge

b

and screw GNDs can be obtained by projecting the gradient of shear along line direction.

1 .    l  





(8)

, GND screw

b

where  s and  l denote the slip direction and line direction respectively. The symbol  is the gradient operator. The net GNDs are given as the resultant of edge and screw GNDs.

2

2







(9)











, GND edge

, GND screw

GND

3. Simulations In this section we describe the salient features of the simulation results. A planar double slip model (Fig. 1) derived by Asaro (1979) from a 3D configuration of single crystal is adopted here for the study of plane strain pure bending. According to Asaro (1979), two slip systems of type (1 1 1) [-1 -1 2] and (-1 -1 1) [1 1 2] will be equally active if the crystal is loaded along [-110] direction. Thus, following Asaro (1979) and Dai (1997) a finite element planar double slip model is setup with zero strain direction as [-110]. Finally the crystal [110] direction is aligned with global [100], crystal [001] direction is aligned with global [010] and crystal plane strain direction [-110] is aligned with global [001]. A 2D beam with 1000 elements of CPE8R elements is modeled for the study. Here, our primary objective is to capture the gradient of plastic slip for evolution of GNDs. As per Mitsutoshi (2011), quadratic elements are more suitable than linear elements due to calculations involving gradients. The integration points in CPE8 and CPE8R are 8 and 4 in number respectively. In order to reduce the computation time, we chose CPE8R over CPE8 in the present model.

Fig. 1. Planar double slip model (Asaro (1979)).