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

614

3

where    is shear rate and α P is the Schmid tensor for each slip system. In the framework of rate dependent plasticity, a commonly used description of shear rate on each α slip system is given by,

m

        g  







sgn( )  

(3)





o

The resolved shear stress (   ) is assumed to obey Schmid’s law on each slip system. If σ is the Cauchy stress applied, the resolved shear stress on each slip system with plane normal  n and slip direction  m is given as: (4) As mentioned in the previous section, the slip resistance (  g ) is modeled using Taylor’s dislocation density model is given as: . .    m n : α σ σ P   

(5)

      s g b

g

The statistically stored dislocations evolve based on the KM equation as

  

  

k

1

     



    b





(6)

s

a k b

b

Eq. 6 comprises of two terms on RHS. The first term indicates the dislocation generation and the second term with negative sign indicates the dislocation annihilation. The values of various parameters used in the model are shown in Table 1. In modeling framework, the SSDs evolve along with GNDs. However, in experiments it is not possible to distinguish between SSDs and GNDs. The introduction of GNDs also leads to the size effect that is observed in the experiments.

Table 1. List of parameters. Parameter

Value

10 20

6 mm/mm 3

ρ o M

0.258*10 -6 mm

b

100

k a k b

10

48000 MPa

µ

2. Evolution of GNDs The formation of GNDs comes from the notion of preserving the lattice continuity during inhomogeneous plastic deformation. These dislocations drive gradient of slip due to the incompatibility of deformation field in the intermediate configuration (Nye (1953)). The formulations based on strain gradient plasticity are broadly classified into higher order and lower order. In the higher order formulation by Aifantis (1984), Fleck and Hutchinson (2001), Gurtin and Anand (2004) to name a few, higher order stress conjugate is considered with higher order strains. Thus, the conventional plasticity boundary conditions do not suffice to solve the equations. These models have proved to be very successful in