Issue 29

R. Dimitri et alii, Frattura ed Integrità Strutturale, 29 (2014) 266-283; DOI: 10.3221/IGF-ESIS.29.23 280 max max max 1 exp , 1 1 exp N N N N N N T N N N g g g d d g g g                          (34) 2 2 max 1 exp T T T T T g d d g           (35) The damage variables given by Eq. (34,35) increase monotonically from zero, in the undamaged case, up to 1, which corresponds to the fully damaged case. By imposing that the damage variables are non-decreasing, loading and unloading are automatically dealt with monolithically. A further problem is the monolithic handling of decohesion and contact. With the damage approach, an unloading path in the shear or normal direction follows the secant stiffness to the origin of the traction-separation diagram in its associated mode. If the same damage variable was kept under normal compression, the contact penalty stiffness would continuously decrease as the degree of damage under decohesion increases, so that high (and thus unacceptable) interpenetrations would be allowed in the normal direction under contact. Therefore, the Helmoltz energy is decomposed to distinguish between energy contributions from decohesion and closure of crack surfaces, as follows       1 1 1 1 N T N T N N N N T T T d d d d              (36) where 2 1 2 N N N K g    (37)   2 2 1 2 N P N N K g g     (38) and  T always given by Eq. (28b). K N in the last two equations refers to the undamaged stiffness of the spring for g N >0 , while K P is the penalty stiffness for g N <0 . The degradation acts only on the separation part N   such that the resistance to the crack closure is maintained during interface failure. Based on the modified Helmoltz energy, the normal pressure becomes      1 1 N T N N N N N P N N p d d K g K g g      (39) while p T is always defined by Eq. (32b). Numerical implementation and results In the numerical finite element model, the thermodynamically consistent cohesive law illustrated above has been implemented into a generalized contact element based on the node-to-segment strategy illustrated earlier. Two numerical examples are presented hereafter to demonstrate the performance of the improved exponential cohesive algorithm: a patch test, and a matrix-fiber debonding test. - Patch test As first example, we reconsider the patch test of Fig. 15, which is verified in pure modes I, II, and different mode- mixities. A 10x10 mm 2 plate is elongated and compressed at the top and the right sides up to fixed values of horizontal and vertical displacements u and v , respectively, which are then applied in opposite directions during unloading and reversed again during reloading until complete failure. Reference values of 0.05 mm are considered for horizontal and vertical displacements in pure modes I and II, while proportional values between horizontal and vertical displacement components are differently considered for varying mode mixities. An elastic isotropic behaviour is assumed for the plate, with material properties E=200GPa and  =0 . Different cohesive strenghts p Nmax , p Tmax , are assumed in the normal and tangential directions, equal to 0.1N/mm 2 and 0.233N/mm 2 respectively, whereas K N =p Nmax /g Nmax =100N/mm 3 and K T =p Tmax /g Tmax =233N/mm 3 corresponding to maximum separations in the normal and tangential directions g Nmax =g Tmax =10e-3mm . The plate is discretized with 4-node isoparametric plane stress elements, and the cohesive/contact elements are introduced along the diagonal direction. Fig. 16a shows the interfacial normal stresses p N as a function of the