Issue34

L. E. Kosteski et alii, Frattura ed Integrità Strutturale, 34 (2015) 226-236; DOI: 10.3221/IGF-ESIS.34.24 233 S ECOND APPLICATION : THE SUBCRITICAL CRACK GROWTH USING LDEM LDEM Model ith the goal of studying the subcritical crack propagation in a rectangular plate built with quasi-brittle material, a model was developed using the Lattice Discrete Element Method (LDEM) presented before. The rectangular plate model has a pre-crack that emanates from a lateral border. The loads and boundary conditions used are illustrated in Fig 6(left). In the present case, plane strain condition was adopted; this condition is implemented in LDEM using only one cubic module of thickness and restraining of displacements in out of plane direction (dir. y). The plate was also fixed in the right boundary as indicated in Fig. 6 (left). A pre-crack was introduced in the model with the goal of favoring subcritical crack growth through the middle of the plate. It is important to highlight that the goal here is not to characterize a specific material, but to explore the capacity of LDEM in simulating the subcritical propagation of a preexistent fissure. With the considered parameters, the ratio between rupture and critical strain will be  r /  p = 60. For this analysis, material toughness G f was considered as a random field characterized by means of a variation coefficient. It will also be considered that the correlation length of the random field is of the same order of the discretization level. Implementations regarding the unlinking of discretization level and correlation length of random field are presented in Puglia et al [13]. Figure 6 : (Left) LDEM model used with details of how the loads and boundary conditions are applied, where b=0.75m (100 modules) and h=0.3075m (41 modules); (Right) Force transmitted to a normal bar element close to pre-crack tip versus percentage of simulation time. The load was applied in the central nodes of the cubic cells that constitute the top and bottom surfaces of the plate, as indicated in Fig. 6 (left). In Fig. 6 (right) the graphic shows the tensile force transmitted to a normal bar element placed in the middle of the plate’s high, h, and close to the pre-crack tip versus percentage of simulation time. In this model, a cubic cell length of L = 0.0075 m was adopted, the model has 100 modules in the x direction, 1 module in y direction and 41 modules in z direction. The material properties are characterized for Poisson coefficient of  = 0.25, the Young Modulus is E = 35 GPa , the density value is  = 2400 kg/m 3 and the critical strain  p = 2.18 10 -4 . The toughness random field is characterized by a mean value of  ( G f ) = 1155 N/m and a variation coefficient of CV( G f )=5%. In Fig. 6 (right), it can be verified that, in a first moment, load rises according to an exponential function, until a stable magnitude is achieved, so that inertial effects are none. After that, oscillation begins and amplitude grows, also according to an exponential function, until stable loading cycle amplitude is reached. This oscillation does not induce considerable dynamic effects in the model, and after 880 cycles loading amplitude is assumed constant. The applied loads produce remote stress acting in the plate of 2.93MPa before oscillation starts, ranging later between a top value of 3.34MPa and lower value of 2.52MPa. Obtained Results In Fig. 7 (left), six crack configurations obtained during propagation process are ilustrated. The elements in darker shade of grey, siding the main crack, are parilly damaged. Broken elementes constitute main crack and are not shown. The applied version of LDEM constitutive law does not take residual strain into account. Constitutive laws that account for such effects are presented in Kosteski [10]. The cluster of microfissures that precedes the main fissure in fatigue crack growth (characteristic in quasi-brittle materials) can be clearly seen in the partial configurations illustrated in Fig. 7 (left). It is also important to observe that crack propagation does not influence in a significant way the discrete element mesh. W