PSI - Issue 61

Enes Günay et al. / Procedia Structural Integrity 61 (2024) 34–41

36

E. Gu¨nay et al. / Structural Integrity Procedia 00 (2024) 000–000

3

2. Numerical Analysis

2.1. Crystal Plasticity Framework

A rate-dependent lower order strain gradient crystal plasticity model is employed in the finite element simula tions. The strain gradient framework allows the inclusion of intrinsic size e ff ects through geometrically necessary dislocations (GNDs). The 12 slip systems of a face-centered cubic material are present and active in the framework. The slip rate is expressed in power law form as,

T n

0 τ α g α

(1)

˙ γ α = ˙ γ

sign ( τ α )

where ˙ γ 0 is the reference slip rate, τ α is the total slip resistance, n is the rate sensitivity exponent, and sign ( x ) is the sign function. The total slip resistance is a function of slip resistance resulting from statistically stored dislocations (SSDs) and geometrically necessary dislocations (GNDs). α is the resolved shear stress, g T

T = g

2

2

(2)

g α

+ g α

α ssd

gnd

The SSD slip resistance is described by the following relations,

˙ g α ssd = β h αβ ˙ γ

β

(3)

where h αβ is the hardening modulus between slip systems α and β . The self and latent hardening law is expressed as,

2 h 0 γ

g s − g 0

(4)

h αα = h

0 sech

h αβ = q αβ h αα

(5)

Here, h 0 is the initial hardening modulus, g s is the saturation slip resistance, g 0 is the initial slip resistance and q αβ is the latent hardening coe ffi cient. The strain gradient framework assumes that, initially, the density of GNDs in the model is zero. The GND slip resistance is a function of initial slip resistance g 0 , internal length scale parameter l , and the GND density η gnd .

gnd = g 0 l η α

g α

(6)

gnd

The intrinsic length scale parameter l in the strain gradient formulation is a function of α T , the Taylor coe ffi cient related to the dislocation mechanisms of the crystal (see e.g., Mughrabi (2016)), µ s , the shear modulus of the material, and b , the Burger’s vector length.

α 2

2 s b

T µ g 2 0

l =

(7)

The GND density is calculated as,

gnd = n

α · m β ∇ γ β × n β

α ×

β m

η α

(8)

2.2. Finite Element Method Model

A single crystal, and three polycrystal geometries with average grain diameters g d of 5 µ m , 15 µ m , and 50 µ m are created. All three polycrystals have 16 grains and the same grain morphologies, as shown in Fig. 1a. Hence, the only di ff erence between the samples is the geometrical dimensions. The finite element mesh consists of 90,000 eight-node hexahedral elements with a finer mesh at the scratching region compared to the rest of the sample, as seen in Fig. 1b. The specimen undergoes deformation applied by a rigid Berkovich indenter. The deformation process is carried out in two steps: First, an indentation step occurs until the prescribed scratching depth is reached. Afterwards, the Berkovich