PSI - Issue 5

Anastasiia Kostina et al. / Procedia Structural Integrity 5 (2017) 302–309 Anastasiia Kostina et al. / Structural Integrity Procedia 00 (2017) 000 – 000

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Figure 3 shows simulation results of the defect density when both conditions are satisfied. It can be seen that there is no equilibrium defect concentration and the dissipative structure is localized on the spatial scale equal to the half of the critical distance. To obtain critical distance equal to 0.085 mm, we have used the following values of dimensionless material parameters included in (13): k = 4.7*10 -7 , q =50.9, s =1, β =2, a =10.1. Evolution of the 11 p in the plane perpendicular to the applied loading and located at the center of the stress concentrator is presented in fig.4. It can be seen that uniform distribution of 11 p (fig. 4(a)) becomes localized during the deformation process (fig. 4(b)). According to the proposed model, function 11 max 11 11 /   β σ σ ap qp defines equilibrium state of the system. When 11 σ reaches critical value c σ ( 11 с  p p ) the system is out of equilibrium state which is expressed in a sharp increase in the density of the defects on the spatial scale defined by (14).

Fig. 3. Values of p 11 versus distance from the notch ( 11 σ < c σ and 11 p < c / 2 L ).

(a)

(b)

Fig. 4. Evolution of the spatial distribution of p 11 component in the cross-sectional area perpendicular to the loading direction (a) uniform distribution; (b) localization.

5. Conclusion

This work is devoted to the theoretical justification of the critical distance theory. According to this theory, fracture of the notched structures occurs at some distance from the stress concentrator. This length scale parameter is often introduced empirically which makes difficult to analyze its connection with material structure. Introduction of the structural-sensitive parameter let us to propose constitutive equation which does not require explicit incorporation of the parameter with the dimension of length to the fracture model. It has been shown that localization of the defect ensemble can be observed when two requirements are fulfilled: existence of the area where stresses are higher than ultimate tensile strength and the spatial size of this area is equal to the half of the critical distance. From the physical point of view, critical distance can be considered as a length of dissipative structure growing in a blow-up regime. The application of this model is illustrated by numerical simulation of the quasistatic loading of the notched Grade 2 titanium specimen.