Applications of lattice method in the simulation of crack path in heterogeneous materials

The simulation of critical and subcritical crack propagation in heterogeneous materials is not a simple
problem in computational mechanics. These topics can be studied with different theoretical tools. In the crack
propagation problem it is necessary to lead on the interface between the continuum and the discontinuity, and
this region has different characteristics when we change the scale level point of view. In this context, this work
applies a version of the lattice discrete element method (LDEM) in the study of such matters. This approach lets
us to discretize the continuum with a regular tridimensional truss where the elements have an equivalent stiffness
consistent with the material one wishes to model. The masses are lumped in the nodes and an uni-axial bilinear
relation, inspired in the Hilleborg constitutive law, is assumed for the elements. The random characteristics of the
material are introduced in the model considering the material toughness as a random field with defined statistical
properties. It is important to highlight that the energy balance consistence is maintained during all the process.
The spatial discretization lets us arrive to a motion equation that can be solved using an explicit scheme of
integration on time. Two examples are shown in the present paper; one of them illustrates the possibilities of this
method in simulating critical crack propagation in a solid mechanics problem: a simple geometry of grade material.
In the second example, a simulation of subcritical crack growth is presented, when a pre-fissured quasi-brittle
body is submitted to cyclic loading. In this second example, a strategy to measure crack advance in the model is
proposed. Finally, obtained results and the performance of the model are discussed.