Issue 47

V. Rizov, Frattura ed Integrità Strutturale, (2047) 468-481; DOI: 10.3221/IGF-ESIS.47.37 472 material properties which describe the hardening behavior of the material (actually, the second term of the right-hand side of (6) models the material non-linearity). Each layer exhibits smooth material inhomogeneity along the width and length of the layer. Thus, it is assumed that the modulus of elasticity in the i -th layer varies continuously along the width according to following cosine law: 1 1 1 1 1 cos 2 i i i i d f i i y y E E E y y             (7) where 1 1 1 1 i i y y y    . (8) In (7), i d E is the value of the modulus of elasticity at the right-hand lateral surface of the layer, i f E is a material property which governs the material gradient along the width. Apparently, the value of the modulus of elasticity at the left-hand lateral surface of the layer is i i d f E E  . The continuous variation of i d E in the length direction of the i -th layer is written as 1 2 3 1 2 cos 2 i i i d g r l l x E E E l l             (9) where   3 1 2 0 2 x l l    (10) The 3 x -axis is shown in Fig. 1. In (9), i g E is the value of i d E at the two end sections, 3 0 x  and   3 1 2 2 x l l   , of the beam, i r E is a material property which governs the material gradient in the length direction. It is obvious that the value of i d E in the mid-span is i i g r E E  . For the Ramberg-Osgood stress-strain relation, * 0 i L u which is needed in order to calculate * L U by (5) can be written as [17, 18]   1 2 * 0 1 2 1 i i i i m m i i i L i m i i m u E m H       (11) By substituting of (7) in (11), one arrives at   1 2 * 0 1 1 1 1 1 1 2 cos 1 2 i i i i i i m m i i i L i m d f i i i i m u y y E E m H y y                        (12) In order to perform the integration in (5), i d E in (12) should be expressed as a function of 1 x . For this purpose, (9) is re- written as 1 1 2 ( ) cos 2 i i i d g r x E E E l l            (13)