PSI - Issue 13

M. Hrstka et al. / Procedia Structural Integrity 13 (2018) 1123–1128 Hrstka, M., Zˇ a´k, S., Vojtek, T. / Structural Integrity Procedia 00 (2018) 000–000

1126

4

yield condition, the plastic zone shape for each mode:

K II σ 0 . 2 y

2

1 2 π

4 1 − ν + ν 2 + 3 cos θ 2

2

3 1 − 2 sin

2

θ 2

3 θ

3 θ

θ 2 +

θ 2

θ 2 +

r II

sin 2

cos 2

cos

2 cos

sin

(4a)

p ( θ ) =

,

K III σ 0 . 2 y

1 2 π

2

r III

p ( θ ) =

(4b)

.

Stress intensity factors K II , K III determining stress amplitude are computed by using empiric relations from Vojtek et al. (2015) for the considered specimen.

3.2. HRR solution - In-plane and anti-plane field

By considering a nonlinear elastic-plastic material behaviour, a formalism based on Ramberg-Osgood stress-strain approximation was derived in Hutchinson (1968a) and Rice and Rosengren (1968). Hardening of such material is characterised by two parameters α and n . In those papers, stress and strain field for pure mode I have been derived. Further studies were devoted to the elastic-plastic analysis of the crack in pure mode II (Hutchinson (1968b), Shih (1974), Varolomeyev and Klepaczko (1992)), and pure mode III (Rice (1968), Pan and Shih (1990), Pan and Shih (1992), Zhang and Gross (1993)). Shape of a plastic zone can be expressed by

JE

JG

r II

˜ σ n + 1 e

III p , HRR =

˜ τ n + 1

r

(5)

e ,

p , HRR =

ασ 2

ατ 2

0 . 2 y I n

0 . 2 y I n

where for stress potentials ˜ σ i j ( θ ), ˜ τ rz ( θ ), ˜ τ θ z ( θ ) and ˜ τ e ( θ ) it follows

˜ σ θθ = s ( s − 1) φ, ˜ σ r θ = ˜ τ r θ = (1 − s ) φ , ˜ σ e = 3 4

2

1 2

(2 s − s 2 ) φ + φ 2

+ 3 (1 − s ) φ

˜ σ rr = s φ + φ ,

(6)

.

Unknown eigenfunctions φ , ψ are solutions of fourth order di ff erential equations for in-plane or anti-plane problem with the eigenvalues s = (2 n + 1) / ( n + 1) and t = n / ( n + 1) and following boundary conditions: traction-free crack surfaces and symmetry conditions at θ = 0. There is a relation between yield stress in tension and shear defined by τ 0 . 2 y = σ 0 . 2 y / √ 3.

3.3. FEM solution

The finite element model was constrained and loaded according to installation in the gripping device in Vojtek et al. (2015). The tensile force F should induce mode II in the cut A and mode III in the cut B (see Fig. 2) without super position with mode I. In the longitudinal cuts a general plane state occurs. However, considered analytical approaches can cover only plane strain or plane stress conditions. Since plane stress is defined by σ z = 0 which correspond to a free surface, for following comparative analyses plane strain condition ( ε z = 0) will be considered. Numerical simulation was performed by using submodelling technique in order to achieve su ffi ciently fine mesh in the vicinity of a crack. Due to the character of loading, the model was considered as symmetric with respect to a plane including the symmetry axis and the vector of the applied force. Loading was ramped from zero to 12000 N. Linear elastic and plastic material properties are defined in the paragraph 2 by multi-linear stress-plastic strain data.

4. Results

Values I n for mode II were I n = 0 . 724 which does not correspond with values in Hutchinson (1968b), while for mode III the value is I n = 1 . 709 which can be verified in Zhang and Gross (1993). Fig. 3 and Fig. 4 summarize plastic zones for mode II and mode III determined by Irwin’s solutions, HRR theory described in sections 3.1 and 3.2 and reference FEM solution for the cylindrical specimen. Plastic zone is represented by a contour of von Mises stress for yield stress σ 0 . 2 y . Iwrin’s and HRR solutions are under a conditions of plane