PSI - Issue 42

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

427

3

The crack length was measured by electrical potential drop method. Heat dissipation rate at the crack tip was recorded by original contact heat flux sensor based on Seebeck effect which was developed in Institute of Continuous Media Mechanics (Vshivkov et al. (2016)). Based on these values the dependence of energy dissipation per cycle on crack length was obtained and used for the model verification.

Table 1. Chemical composition of Ti-5Al-2V in wt %. Titanium alloy Ti Al Fe V

C

Si

Ti-5Al-2V

base

4.6

0.11

1.77

0.012

0.014

Table 2. Chemical composition of Grade 2 in wt %. Titanium alloy Ti Fe C O 2

N 2

Si

Grade 2

base

0.25

0.07

0.2

0.04

0.1

3. Mathematical model To derive mathematical model of energy dissipation during fatigue crack propagation we will apply thermodynamic constitutive theory based on multiple dissipation potentials (Murakami (2012)) and introduce several internal variables: structural parameter responsible for the additional strain induced by initiation and coalescence of the defects p (Naimark (2003)), isotropic r and kinematic j α hardening parameters. Constitutive relations for these parameters and main equations of the model are given below. We assume that during cyclic loading additional strain induced by structural changes is small and can be neglected. In these circumstances p is an internal variable responsible for the energy accumulation. This hypothesis is based on the presumption that most part of the energy dissipates. Therefore, total strain rate ε  can be represented as the sum of elastic e ε  and plastic strain rates p ε  : e p   ε ε ε    . (1) In isothermal case free energy F is a function of elastic strain, structural parameter, isotropic and kinematic hardening parameters (Naimark (2003)), (Murakami (2012)):

e

e e

( ) F r F F     p ( ) r p

α ,

( , , , ) ε p α

( ) ε

( )

F F r 

F



j

j

( ) e e F ε is the elastic free energy,

( ) r F r is the free energy due to isotropic plastic deformation,

( ) p F p is the

where

( ) j F  α is the free energy responsible for the kinematic hardening

part of the energy which is stored in the material,

in plastic deformation. We assume that considered material exhibits isotropic linear elastic behaviour. Thus, the Hook’s law in the following form can be used:

0 0 e K      ,

(2)

2 e d G  S ε   ,

(3)

where 0  , S signify the hydrostatic(spherical) and the deviatoric stress tensor, 0 e  , e d ε denote hydrostatic(spherical) and the deviatoric elastic strain tensor, K is the bulk modulus, G is the shear modulus. Using quasi-standard thermodynamic approach (Murakami (2012)), we will postulate the existence of the convex function  in the form: