Issue 41

V. Rizov, Frattura ed Integrità Strutturale, 41 (2017) 491-503; DOI: 10.3221/IGF-ESIS.41.61 494 Figure 3 : The free end of the lower crack arm (. 1 1 n n  . is the neutral axis). The beam complementary strain energy, * U , is written as   2 2 2 2 * 0 1 2 2 2 2 1 * 0 * 1 1 dz dy u a l dz dy u a U h h b b U h h b b L                               (8) where * 0 L u and * 0 U u are the complementary strain energy densities in the lower crack arm and the un-cracked beam portion   l x a   3 , respectively. Axes, 1 y and 1 z , are shown in Fig. 3, 2 y and 2 z are the centroidal axes of the cross- section of the un-cracked beam portion. In principle, the complementary strain energy density is equal to the area OQR that supplements the area OPQ , enclosed by the stress-strain curve, to a rectangle (Fig. 4). Therefore, the complementary strain energy density in the lower crack arm is written as L L u u 0 * 0    (9) where the strain energy density, L u 0 , is equal to the area OPQ       0 0 )( d u L (10) From (1), (9) and (10), the complementary strain energy density can be obtained as 1 1 * 0    m mB u m L  (11) The strain distribution is analyzed by applying the Bernoulli’s hypothesis for plane sections since the span to height ratio of the beam under consideration is large. Concerning the application of the Bernoulli’s hypothesis in the present study, it should also be noted that since the beam is loaded in pure bending (Fig. 1) the only non-zero strain is the longitudinal strain, ε . Therefore, according to the small strain compatibility equations, ε is distributed linearly along the height of the beam cross-section. Thus, ε in the lower crack arm is written as