Issue 48

T. Profant et alii, Frattura ed Integrità Strutturale, 48 (2019) 503-512; DOI: 10.3221/IGF-ESIS.48.48 511 influenced by the sheer stresses and mode II loading. It is worth mentioning that the maximum value of the angle 50 q   is due to the value of the stress singularity exponent l , which becomes complex for the 50 q >  . The complex value of l requires different approach of the stress field description. The graph also shows that for some combination of the external stresses 1 T getting some specific value from the range ( ) 1 80, 60 T Î - - MPa it is difficult to assess the crack path direction, because the dependence of the energy release rate G e is rather flat without the clear local extremal value. The Fig. 7 shows the position of the crack initiation at the interface between the interfacial zone and matrix represented by the angle a and the crack inclination q . The elastic moduli of the used materials ( ) 1 1 , E n , ( ) 2 2 , E n , ( ) 3 3 , E n , the finite crack size e , geometry 1 R , 2 R and degree of Laurent series are again given in the graph. The external load 1 T is zero in this case, while 2 100MPa T  . The local maximum value of G e is for 40 a   and the crack path direction 0 q is near the zero value, i.e. the crack growth under the mode I loading. Similarly to the previous example, the maximum values of G e are limited due to the real values of the stress singularity exponent l . The influence of the interfacial zone on the crack initiation position and potentially crack growing direction shows Fig. 8. The width of the interfacial zone 2 W plays an unimportant role in the crack behavior and it is not illustrated there. On the other hand, the material properties of 2 W influence the point of the crack initiation as well as the crack path direction. The grey curves represent the material of the interfacial zone with elastic moduli similar to those of the stiffer particle 3 W , the black ones similar to the material softer then the matrix 1 W . On can see that the position of the point of the crack initiation represented by the angle a tends to the lower values for the stiffer material of 2 W and the crack path direction 0 q moves to the positive values. C ONCLUSIONS he energy release rate associated with the arbitrary oriented finite small crack initiating at the interface between the interfacial zone of the circular inclusion and the matrix was evaluated. The scheme of the energy release rate evaluation was adopted from [9] but modified to the presented problem. The proposed procedure combines traditional as well as modern analytical approaches with a numerical one. The most important part of the analysis is the establishing of the fundamental solution given by the interaction between the edge dislocation or point force and circular inclusion with an interfacial zone. A numerical example of the convergence of the approximating series of the fundamental solution was shown. The asymptotic analysis, where the crack perturbs the matrix only and the crack has to have a finite small length, allows one to apply the fundamental solution to model as the cracked infinity as the uncrack finite loaded matrix containing the inclusion. The numerical results showed the influence of the external loads on the crack path direction but also the position of its point of initiation along the interface between the interfacial zone and matrix. The influence of the material properties of the interfacial zone on the cracking of the matrix was also shown. Even that the character of the presented results is theoretical only and their validation is desirable, the applied procedure offers an interesting tools to solve the fracture problems of the composites with sound mathematical background. A CKNOWLEDGMENTS his research has been financially supported by the Czech Science Foundation through the Grant 16-18702S and by the Ministry of Education, Youth and Sports of the Czech Republic under the project CEITEC 2020 [LQ1601]. R EFERENCES [1] Krepl, O., Klusák, J. (2017). The influence of non-singular terms on the precision of stress description near a sharp material inclusion tip, Theor. Appl. Fract. Mech., 90, pp. 85–99, DOI: 10.1016/J.TAFMEC.2017.03.007. [2] Krepl, O., Klusák, J. (2016). Reconstruction of a 2D stress field around the tip of a sharp material inclusion, Procedia Struct. Integr., 2, pp. 1920–7, DOI: 10.1016/J.PROSTR.2016.06.241. T T