PSI - Issue 2_A

Sami Holopainen et al. / Procedia Structural Integrity 2 (2016) 2718–2725 S. Holopainen et al. / Structural Integrity Procedia 00 (2016) 000–000

2720

3

σ 1

σ 1

d s

β > 0 ˙ β ≥ 0 ˙ α̸ = 0 ˙ D ≥ 0

β > 0 ˙ β < 0 ˙ α = 0 ˙ D = 0

d s

s

s

d α

α

α

σ 2

σ 3

σ 2

σ 3

(a)

(b)

Figure 6.9: Ottosen’s HCF model. (a) Movement of the endurance surface and damage growth when the stress is outside the endurance surface and moving away from it. (b) When the stress is outside the endurance surface, damage and back stress does not evolve. Fig. 1. (a) Movement of the endurance surface and damage growth when the stress is outside the endurance surface and moving away from it. (b) When the stress state is outside the endurance surface but moving towards it, damage and back stress do not evolve. where σ 0L and σ 0T are the fatigue stress amplitudes under pulsating loading condition in the longitudinal and trans verse directions, respectively. The scalar variable ζ reflects the average loading direction and is defined as ζ = σ L : σ L σ : σ n = 2 I 5 − I 2 4 2 I 2 . (7) The endurance surface, β = 0, moves in the stress space driven by the back stress which memorizes the load history. Contrarily to plasticity theory, the stress states out of the endurance surface, β > 0, are allowed. A linear evolution equation ˙ α = C ( s − α ) ˙ β, (8) is chosen for the the back stress type tensor α , where C is a non-dimensional material parameter and the superim posed dot denotes time rate. Equation (8) in similar to the Ziegler’s kinematic hardening rule in plasticity theory. The evolution equation for the damage D is

K (1 − D ) k

Version February 26, 2016 exp( L β ) .

(9)

g ( β, D ) =

In the original formulation by Ottosen et al. (2008) the value k = 0 has been used, which results in a linear damage growth in a cyclic, constant amplitude loading after the initial phase. Parameters K , L , k can be estimated from the S-N curve of the material. From the evolution equation (9) it can be concluded that damage and backstress only develope when the stress state is moving away from the endurance surface, that is β ≥ 0 and ˙ β > 0.

4. Example

The model described above has been implemented in the finite element program Abaqus using the user material subroutine. As an example, a plate in plane strain conditions with a circular inclusion under uniaxial cyclic loading is considered. Diameter of the inclusion is 40 % of the side length of the domain. Due to symmetry only one quarter is discretized. Analytical solution for an infinite plate with a circular inclusion is given by Sezewa and Nishimura (1931). The surprising result of their solution is that irrespectively of the sti ff ness di ff erence bewteen the base material and the inclusion, all the stressess in the inclusion are constant along a radius but not along an azimuthal direction. Elastic properties for the base material are E = 210 GPa and ν = 0 . 3 , and E = 375 GPa, ν = 0 . 22 for the aluminium oxide inclusion, respectively. Only the base material is accounted for in the fatigue analysis. Typical Al 2 O 3 inclusion size for the AISI-SAE 4340 steel is 5.5 micrometers. Uniform normal stress along the upper boundary is prescribed