Issue 50

K. Meftah et alii, Frattura ed Integrità Strutturale, 50 (2019) 276-285; DOI: 10.3221/IGF-ESIS.50.23 281 In this study the solution is obtained resorting to the modified Newton-Raphson method [1]. In this algorithm the modification consists of computing the tangent stiffness matrix only once in the beginning of each load increment than in each iteration. Q4  finite element The Q4  element shown in Fig. 2 is adopted in the present study. This element contains four nodes at the corners and the associated classical interpolation functions given by [13]:    1 1 1 4 i i i N      ; i = 1, 2, 3 and 4 (27) with     1 2 3 4 , , , 1,1,1, 1        and     1 2 3 4 , , , 1, 1,1,1        . A1 A2 B1  B2 1  2 3 4  1 A   2 A   1 B   2 B   Figure 2 : Q4  quadrilateral isoparametric element [13]. For the Q4  element the transverse field of distortion  is linearly discretizes in the element of reference by side so that: 1 2 1 2 1 1 2 2 1 1 2 2 A A B B                                              (28) By means of then the relations:     1 , 1 0 w d             ;     1 , 1 0 w d             ; for 1    and 1    (29) One establishes that:   1 2 1 1 2 1 2 A w w           ;   2 4 3 3 4 1 2 A w w           (30.a)   1 4 1 1 4 1 2 B w w           ;   2 3 2 2 3 1 2 B w w           (30.b) By deferring the two results above in the statement of  , one from of deduced that: