Issue 49

M. Tashkinov et alii, Frattura ed Integrità Strutturale, 49 (2019) 396-411; DOI: 10.3221/IGF-ESIS.49.39 399 Progressive failure model The progressive failure model allows to evaluate the performance of the strength criteria in every ply of composite material at each step of deformation, which makes it possible to change the properties of finite elements during the analysis, depending on whether the criterion is fulfilled for them or not [22–31]. Considering the material with initially linear elastic behavior, the ratio of stresses of the initial and damaged material is expressed, respectively, as follows:   ˆ : : ˆ C C D       (6) where ˆ C denotes an effective (undamaged) stiffness tensor,   C D is the stiffness tensor depending on the damage tensor D. The relationship between the effective stress ˆ  and conditional stress  is written using the damage tensor D:     1 1 : : ˆ ˆ ˆ D S S          (7) Using the principle of compatibility (equivalence) of strains, the tensor   S D is expressed as follows:   1111 1122 1133 11 2222 2233 22 3333 33 1212 12 2323 23 1313 13 0 0 0 1 0 0 0 1 0 0 0 1 . 0 0 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 0 1 1 S S S D S S D S D S D S sym D S D S D                                               (8) To calculate the damage variables ij D , it is necessary to specify the criterion of failure and the damage evolution law. In this work, the damage accumulation in plies is modeled using the multicomponent fracture criterion and the Hashin criterion. The multicomponent fracture criterion includes five fracture indicators corresponding to mechanical response of the composite ply when the specimen is loaded in different directions: 11 11 , 0; A t f X     11 11 , 0; B c f X      22 С 22 ,  0; t f Y     22 22 ,  0; D c f Y      12 E f S   , (9) where t X is tensile strength in direction 1; c X is compressive strength in direction 1; t Y is tensile strength in direction 2; c Y is compressive strength in direction 2; S is shear strength in the plane (1,2). Hereinafter, it is assumed that the plane 1- 2 coincides with the surface of the woven ply.