PSI - Issue 5

Terekhina Alena et al. / Procedia Structural Integrity 5 (2017) 569–576 Terekhina Alena et al. / Structural Integrity Procedia 00 (2017) 000 – 000

572 4

  f Z f 

  f

Z 



(3)

  Id K K Z f Z    Id

(4)

where   Id K f Z are functions which can be either estimated experimentally or derived theoretically, Z is loading rate/ strain rate/displacement rate . The length scale parameter also has to change with Z and definition for the dynamic case can be written as (5). To design notched materials against dynamic loading effective stress σ eff can now be rearranged as (6).   f f Z  and

    1 Z      Id K Z 0

2

   

 

 

L Z

L f Z





(5)







  Z

 

  0 Z

r L Z 

(6)

/ 2,

0









y 





eff

2.3. Theory of Critical Distances based on elasto-plastic analysis

It was shown that linear-elastic analysis can be used to assess the quasistatic and dynamic strength of specimens with the stress concentrators, supplementing it the hypothesis that the critical distance depends on the strain rate (Yin et al. (2014, 2015)). In this paper, a simplified Johnson-Cook law in a form of (7) is used to model of the material response, taking into account changes in the response of the material with a change in the strain rate. For determine the value of the critical distance elasto-plastic stress fields will be used. The adoption of these measures is conditioned by the fact that the material behavior being, by nature, highly non-linear and cannot be described in the framework of the linear theory of elasticity.

 

  







n A B 

(7)

1 ln C



 

 

0 

where, A, B, n, C and m are constants, 0  is the reference strain rate. In the simplified Johnson-Cook constitutive model are combined two key material responses are strain hardening and strain-rate effects. The adiabatic heating effect is considered negligible for the tension tests as the material necks down at relatively low strains before any significant adiabatic heating. All the materials constants could be acquired through fitting above equations (7) with all the experimental data and curves obtained under different strain rates.

3. Results

3.1. Materials and experimental conditions

Thirty-eight cylindrical samples of titanium alloy Grade2 were tested under tensile loading with several orders of strain rate. Tensile tests were carried out using a 300 kN electromechanical testing machine Shimadzu AG-X Plus and Gopkinson- Kolskiy’s split bar . For this study three types of cylindrical specimens were used with different stress concentrators such as semi-circular edge notches with radius 1 mm and 2 mm, V-shaped notches and un-notched (plain) specimens. For all specimens, the notches depth was kept constant. The geometry of specimens is shown in Figure 3. Loading tensile specimens occurred with range of strain rate 10 -4 – 10 4 s -1 . During each test the failure force and time of failure were determined. The failure force was taken equal to the maximum force recorded during each test. In order to determine the required local stress/strain states elasto-plastic finite element models were solved using commercial software Abaqus SE.