Issue 41

V. Rizov, Frattura ed Integrità Strutturale, 41 (2017) 491-503; DOI: 10.3221/IGF-ESIS.41.61 501 The crack location along the height of the beam cross-section is characterized by 1 / 2 h h ratio. In the calculations, 0 B is kept constant. Therefore, 1 B is varied in order to generate various 1 0 / B B ratios. The strain energy release rate in non- dimensional form is plotted against 1 0 / B B ratio at three 1 / 2 h h ratios and 2 0 / 2.0 B B  in Fig. 6. The curves in Fig. 6 indicate that the strain energy release rate decreases with increasing of 1 / 2 h h ratio. This finding is attributed to the increase of the lower crack arm stiffness. It can be observed also in Fig. 6 that the strain energy release rate decreases with increasing of 1 0 / B B ratio. The influence of non-linear behavior of the material on the fracture is also analyzed. For this purpose, the strain energy release rate in non-dimensional form is presented as a function of 2 0 / B B ratio at 1 / 2 0.5 h h  and 1 0 / 0.5 B B  in Fig. 7. It can be observed that the strain energy release rate decreases with increasing of 2 0 / B B ratio (Fig. 7). The strain energy release rate obtained assuming linear-elastic behavior of the two-dimensional functionally graded material is also presented in Fig. 7 for comparison with the non-linear solution (the linear-elastic solution is derived by substituting of m =1 in formula (22)). The curves in Fig. 7 indicate that material non-linearity leads to increase of the strain energy release rate. Figure 7 : The strain energy release rate in non-dimensional form presented as a function of 2 0 / B B ratio (curve 1 – at non-linear elastic behavior of material, curve 2 – at linear-elastic behavior of material). C ONCLUSIONS he longitudinal fracture behavior of two-dimensional functionally graded cantilever beam that exhibits material non-linearity is investigated analytically. The beam is loaded by a bending moment applied at the free end of the lower crack arm. The fracture is studied in terms of the strain energy release rate. The beam mechanical behavior is described by using a power-law stress-strain relation. The material property, B , varies continuously in the beam cross- section according to a quadratic law. The solution derived is applicable for longitudinal crack located arbitrary along the height of the beam cross-section. In order to verify the solution, the fracture is analyzed also by applying the J -integral approach. The distribution of the J - integral value along the crack front is investigated. The physical meanings of the dependence of the J -integral value on material gradient are discussed. The effects of material gradients and crack location along the beam height on the fracture behavior are also analyzed. The analysis revealed that the strain energy release rate decreases with increasing of the lower crack arm thickness. It is found also that the strain energy release rate decreases with increasing of 1 0 / B B and 2 0 / B B ratios. This finding is attributed to the increase of the beam stiffness. The influence of material non-linearity on the fracture is studied too. It is found that the material non-linearity leads to increase of the strain energy release rate. Therefore, non-linear behavior of the material has to be taken into account in fracture mechanics based safety design of structural members and components made of two-dimensional functionally graded materials. The results obtained in the T