PSI - Issue 47

Ilia Nikitin et al. / Procedia Structural Integrity 47 (2023) 617–622 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

620

4

3. Multi-regime two-criterion fatigue failure model and numerical simulation results The multi-regime fatigue failure model was proposed by authors in Nikitin (2021) and developed into two criteria model in Nikitin (2022). The model is based on damage theory and kinking the elastic moduli of the material with the value of damage function. It is assumed that left and right branches of bi-modal fatigue curve could be described by Baquin-like equation in the similar way. Here and below all parameters associated with left branch are marked by index ‘ L ’, and the right - one by index ‘ V ’ The damage accumulation is described by the following kinematic equation for damage function  :

1 ) / (1 )       −   =  − ( , N B

1/



− 

 

/ (1 ) /2 L  −

3 10

/ (     − − B u eq u

)

B B

= =



L

(1)

1/



− 

 

/ (1 ) /2 V  −

8 10

/ (     − −

)

B B

= =



V

eq

u

u

u

B  - ultimate tensile strength, u  - fatigue strength at 10

6 cycles,

u  - VHCF fatigue strength at 10

9 cycles,

Where

 - power of kinematic equation,  - power of fatigue curve. The coefficients of this equation ( ) , B    are sensitive to stress-state via equivalent stress. To link the damage accumulation with normal and shear cracks a corresponding multiaxial criterion could be used for equivalent stress calculation. In the present work the Smith – Watson – Topper (SWT) model was used to simulate normal crack opening and Carpinteri – Spagnoli – Vantadori (CSV) to simulate shear cracks. The kinematic equation (1) can be solved and the following expression for damage function calculation at each spatial node at a given time layer can be obtained: ( ) 1/(1 ) 2 1 1 1 1 ( ) 2(1 ) t t t k k B N      − + −   = − − − −      (2) This solution contains the whole loading history taken from the previous time-layer. Therefore, the value of damage function can be calculated in each node of a fine element model. If we associate the value of damage function with degradation of elastic moduli, we will get the model of material weakening under cyclic loading. In the present work the following relation for elastic moduli degradation is used ( ) ( ) ( ) 1 1 1 0 * 1 0.001 t t t k k k E E H    + + + = − − + (3) critical value of damage function. This equation describes the following material behavior. When the damage function is close to zero, the Heaviside function is equal to 1 and the elastic module is the same to virgin material. With damage accumulation the elastic module will degraded as function of ψ with a wight coefficient. As soon as damage function reach a critical value, the elastic module falls to 0.001 of initial value that imitate the failure. The elements with close to zero elastic module are recognized as ‘quasi - crack’. The calculations of damage function are performed for normal and shear equivalent stresses. Therefore, the fall of the elastic module is associated with a tensile or shear mechanism. It makes possible to indicate the spatial locations of shear and normal crack opening on the CAD model. This model was applied to simulate the VHCF torsion loading of smooth specimens made from titanium alloy VT3-1. The computer aided design (CAD) model of the specimen used in the present study is a classical hourglass where 0 E is initial elastic module, ( ) * H   − – Heaviside function, k – wight coefficient of the model, - a