PSI - Issue 42

A. Kostina et al. / Procedia Structural Integrity 42 (2022) 425–432 A. Kostina / Structural Integrity Procedia 00 (2019) 000–000

429

5

To derive constitutive equation for structural parameter we apply quasi-standard approach for rate-dependent material (Murakami (2012)). More detailed information can be found in (Kostina and Plekhov (2018)):

v

p

p

 , p     P

p 

,





k

P P  

where

denotes Macauley

bracket.

Under

an

assumption

that

1 v  ,

,





 1



  

  

  

,

H p





,

2 1 p G Exp       

k p

    we can obtain constitutive equation in the following form:

a

 

1



F      p

p 

  S



,

(11)



 1



    

   

,

H p

2 1 p G Exp

   

  

  

a

 

  1 1 n f p m p  ;



   1 f p     ;

: p  p p ;

p  is the characteristic relaxation time;

where

, H p

3 : 2

  S S ; 1  , a , 1 m , n are the material constants. To describe the energy balance in metals the first law of thermodynamics was applied and the following relation was obtained:   : : p p p s F Q W E        σ ε p p p       , (12) p W  is the plastic work rate; s E  is the stored energy rate. It should be noted that in case of the small structural strain compared to the plastic strain the plastic work rate depends only on plastic strain rate tensor. 4. Numerical algorithm Numerical simulation was carried out by finite-element method in Comsol Multiphysics software using plane stress formulation. Geometry of the specimen is shown in Fig. 1. Cyclic load with the maximum force and stress ratio corresponded to the experimental values was applied at the upper boundary of the specimen. The lower boundary of the specimen was fixed. Stationary crack approach was used to calculate energy dissipation per loading cycle during fatigue crack propagation. According to this method, several discrete crack lengths were chosen. Then, the stress-strain state for each crack length was calculated using equations (1)-(10) of the proposed model and equilibrium equation. It should be noted that this step was performed using stationary solver. For each crack length several loading cycles were considered and integral value of the plastic work for the last (stabilized) cycle was obtained. To evaluate stored energy per cycle for each crack length time-dependent solver was applied. For this purpose, interpolation of all stress tensor components obtained on the previous step within the considered time interval was carried out and p-tensor components was calculated by ODE interface of Comsol Multiphysics software using equation (11). Next, the integral value of stored energy per loading cycle was obtained using the second term in the right-hand side of (12). The dissipated energy per loading cycle was calculated as the difference between plastic work and stored energy. The dependence of energy dissipation per cycle on crack length was got by interpolation of the results between the obtained points for each of the considered crack lengths. Material parameters for the considered titanium alloys are given in Tables 3-4. where p Q  is the energy dissipation rate,