Issue 41

V. Rizov, Frattura ed Integrità Strutturale, 41 (2017) 491-503; DOI: 10.3221/IGF-ESIS.41.61 493 2 2 3 3 0 1 2 2 2 4 y z B B B B b h    , (2) where 3 2 2 b b y    , 3 h z h    . (3) In (2), 0 B is the value of B in the centre of the beam cross-section, 1 B and 2 B are material properties which govern the material gradients along axes, 3 y and 3 z , respectively (Fig. 1). Figure 2 : Distribution of the material property, B , in the beam cross-section. It should be mentioned that Eq. (2) is suitable for two-dimensional functionally graded beam configurations which have high mechanical properties at the lower, upper and lateral surfaces of the beam as shown in Fig. 2. The fracture behavior is studied in terms of the strain energy release rate. For this purpose, an elementary increase, da , of the crack length is assumed (the external load is kept constant). The strain energy release rate, G , is expressed via the changes of external work, ext dW , and strain energy, dU , as ext dW dU G dA   (4) where the increase of crack area is dA bda  (5) The change of external work is written as * ext dW dU dU   (6) where * dU is the change of complementary strain energy. By combining of (4), (5) and (6), the strain energy release rate can be expressed as * dU G bda  (7) It should be noted that the present analysis holds for non-linear elastic behavior of material. The analysis is applicable also for elastic-plastic behavior of material if the beam considered undergoes active deformation, i.e. if the external loading increases only [25]. It should also be mention that the present study is based on the small strains assumption.