PSI - Issue 39

D. Fedotova et al. / Procedia Structural Integrity 39 (2022) 419–431 Author name / Structural Integrity Procedia 00 (2019) 000–000

422

4

established in the work by Fleck and Hutchinson (1997). It should be noted that the flow stresses  flow and the material property yield strength at uniaxial tension  y have different definitions and do not coincide with each other. In order to simplify the defining relations and exclude high-order terms from consideration, Huang et al. (2004) proposed a viscoelastic analog of the formulation of the CMSG gradient plasticity theory in the form of the following constitutional equations of material behavior:

m

   

 

m

  

   

e 

e 

p      







(6)

  2 p



p  

f

l  



flow



ref

   

   

m

  

  

3



2            kk ij ij K ij

ij  

 

e

  

(7)

2

 

e

flow

where  e is the equivalent stress, i j    is the deviatoric tensor of strain rate, and m is the rate-sensitivity exponent. The CMSG theory, as in the case of other continuum strain gradient plasticity models, is designed to model the collective behavior of linear lattice defects and is therefore inapplicable at scales smaller than the distance between dislocations, which is about 30 nm, so that the flow stress in (4) is maintained at a scale above 100 nm. Huang et al. (2004) compared the simplified version of CMSG with the theory of higher-order plasticity of the deformation gradient based on the mechanism, created on the basis of the same Taylor dislocation model. As a result, it was found that the stress distributions predicted by both theories differ only in a very thin layer with a thickness of about 10 nm. 3. Experimental crack paths and finite element method In the present study, series of experiments and corresponding numerical calculations were performed over a range of materials by employing the CTS specimen, Fig. 1a. CTS specimens were made of steels 34X and P2M, Ti-6Al-4V and aluminum 7050 alloys. The main mechanical properties for all tested materials are listed in Table 1. In this table, E , σ y , n , and N denote the Young’s modulus, monotonic tensile yield stress and static strain hardening exponent.

Table 1. Main mechanical properties for tested materials. Material E (GPa) σ y (MPa) n

N

σ y /E

Steel P2M Steel 34X

226.90 216.21

362.4 714.4 471.6 885.5

4.13 7.89

0.242 0.127 0.092 0.079

0.001597 0.003304 0.006683 0.007504

Al-alloy 7050

70.57 118.0

10.85 12.59

Ti-6Al-4V

Tests for CTS specimens under pure mode I, mode II and subsequent mixed mode was carried out. Pure mode I and initial pure mode II loading conditions are obtained by the applied force F direction α (Fig. 1b) listed in Table 2. The tests were performed on a servo-hydraulic test system with a maximum capacity of 100 kN at a frequency of 10 Hz and a stress ratio R = 0.1. An optical microscope and the drop potential method were used to determine the actual position of the crack tip on the curvilinear crack path monitoring and continuous measurements of the crack size were made along the path in the CTS specimen. As a result of the experiments, curvilinear crack paths based on the periodically measured increments of crack length for all tested materials in the CTS samples were obtained, Fig. 1c. In accordance with these experimental observations, the main feature of the crack growth under pure mode II loading conditions is the direction of crack propagation almost never coincides with the plane of its initial orientation. As seen in Fig. 1c, curvilinear crack path in specimen made of aluminum 7050 alloy differs significantly from the experimental crack paths of other materials under consideration under the same loading conditions.