PSI - Issue 47

Robert Eriksson et al. / Procedia Structural Integrity 47 (2023) 227–237

231

R. Eriksson, A. Azeez / Structural Integrity Procedia 00 (2023) 000–000

5

σ

σ

Y fl ow σ

ys σ

K EH E + H

Y

K

E

ε

ε

ε pz

ε pz

Fig. 3. The “quasi-ideal plastic” material modeling where the constant σ ys in the ideal-plastic model is taken from the flow stress, σ flow , in a linear isotropic / kinematic hardening model; E and H K are the elastic and plastic moduli.

u y

u x

r= ρ

y

2 π

θ =

Crack

x Plastic zone

Fig. 4. Average strain in the plastic zone is estimated from the displacements on the boundary of the plastic zone, u y , and its size, ρ .

3.3. Estimation of crack tip average strain

In polar coordinates, ( r ,θ ), the displacement in y , u y , around the crack tip becomes

, κ =   3 − ν 1 + ν

K I 2 µ 

2  θ

sin 

θ 2 

2 

plane stress 3 − 4 ν plane strain

r 2 π

E 2(1 + ν )

u y =

κ + 1 − 2cos

(4)

, µ =

where E is the elastic modulus and ν is the Poisson’s ratio. The plastic zone is constrained by the surrounding elastic material. The strain in the plastic zone is thus determined by the displacements at the boundary of the plastic zone. An estimation of the average y -component of the strain in the plastic zone, ϵ pz , can therefore be taken as

u y  r = ρ,θ = ρ

π 2 

K κ 4 µ √ ρπ

(5)

ϵ pz =

=

where ρ is the plastic zone size as illustrated in Fig. 4. This is obviously a rough approximation but it will turn out to be accurate enough for estimating a reasonable flow stress.

3.4. Modeling of the WPS cycle

During the WPS load, P 1 at T 1 , a plastic zone will form. Based on earlier observations that cleavage fracture requires active plastic deformation, fracture is assumed to occur at T 2 when the applied load just causes plastic defor mation outside of the WPS plastic zone; i.e. failure occurs when the plastic zone at T 2 reaches the same size as of that formed during WPS load ( P 1 at T 1 ). For the LCF cycle, this is pretty straight forward: