PSI - Issue 13

M. Hrstka et al. / Procedia Structural Integrity 13 (2018) 1123–1128

1125

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

3

coupled DOF, force load redistributed

l

30

clamped

ØD

Ød

2

F

pure mode II (A) pure mode III (B)

30°

a

6.5 p

symmetry boundary condition

R0.15

submodel

full model

Fig. 2: Geometry and boundary conditions of the finite element model in ANSYS software and specimen dimensions. There is only a half of the submodel depicted.

This paper is focused on detailed analysis of the sizes and shapes of plastic zones in dependence on loading and it contributes to the explanation of the paradox of large plastic zones at threshold loading. Results from both analytical and numerical methods are compared.

2. Material properties

Linear-elastic properties of ARMCO iron are characterized by Young modulus E = 210 GPa and Poisson’s ratio ν = 0 . 3. Plastic material behaviour is adopted from Horn´ıkova´ et al. (2014) by stress-plastic strain multi-linear definition. Due to the smooth stress-strain representation which admits only an approximative yield condition, the yield strength was defined in the o ff set yield point at which 0 . 2% of plastic deformation occurs, concretely σ 0 . 2 y = 189 MPa. Inputs for stress-strain material law are Ramberg-Osgood parameters α and n . These values for a considered mate rial are extracted form elastic-plastic data by employing procedure based on linear regression in U.S. Department of Defense (1998). Assuming the yield stress defined by yield o ff set point σ y 0 . 2 , Ramberg-Osgood hardening parameter n for ARMCO iron n = 11 . 42 and α = 2 . 224. The specimen geometry is depicted in Fig. 2. It is a notched cylindrical bar with an outer diameter D = 25 mm, an inner diameter d = 12 mm and a length of l = 100 mm. At the notch root a precrack of the length of p = 0 . 5 mm in the transversal plane is situated. The bar is loaded so that pure shear crack modes can be achieved. Ends of the bar of the length of e = 30 mm are vertically shifted while the bending moment is zero at the central point of the bar.

3. Methods

3.1. Irwin’s solution Each loading mode of a crack produces the 1 / √ r singularity at the crack tip. In the following text, only pure loading modes are considered while mode I is not induced. The asymptotic stress field in the vicinity of a crack tip can be written in a compact form as

K II √ 2 π r

K III √ 2 π r

f II

f III

i j ( θ ) +

i j ( θ )

(3)

σ i j =

where indices i , j represent the global Cartesian coordinates. This equation includes both in-plane (mode II) and anti plane fields (mode III). If plane state is considered and anti-plane components depends only on x , y , each mode can be treated separately. Assuming shape functions f II i j ( θ ) , f III i j ( θ ) as functions on polar coordinate θ only and von Mises