A Rigid Body-Spring Model with FE-Based Kinematic Assumptions for the Elastic Analysis of Continua

Rigid Body-Spring Models (RBSM) are a kind of discrete models which are developed mainly for the simulation of quasi-brittle materials ranging from ceramic, concrete, and masonry, to rock and soil. In this approach, material domain is discretized to a set of rigid cells interconnected through a set of translational and rotational springs located at cell interfaces. These cells are constructed over a set of points (seeds) distributed regularly or randomly over the domain. When it comes to heterogeneous materials, the seeds may be located in accord to the geometry and distribution of inclusions. For two-dimensional problems, each rigid cell has normally two translational and one rotational degrees of freedom (DOFs). The springs may be distributed along the interface or lumped at a point called contact/computational point (CP) and activated by the relative movement of connecting cells. As a fundamental issues, before being applicable for the simulation of inelastic behavior of materials, the kinematics of an RBSM and also the force-displacement relations of its springs should be defined in such a way that the model can adequately predict the elastic behavior of continuum at both macro and micro scales. Our review of the literature shows that except one of the RBSMs, used in the current paper for comparison, others suffer from some shortcomings which result in their inaccurate elastic predictions. In the aforementioned model, cells are convex polygons generated by the Voronoi diagram of seeds (cell nucleus) and the spring set of an interface is comprised of two translational (normal and tangential) and one rotational springs located at the midpoint of the interface. Our study shows that, although this RBSM presents generally a reliable predictions, however, there exists some kind of scattering in the predicted micro strain and stress distributions. Accordingly, with the aim of eliminating the observed scatters, this paper borrows the interpolation functions of the conventional finite element method and presents a new kinematic formulation for the RBSM. In the new model, called FE-RBSM, a Delaunay tessellation is constructed over cell nuclei. This results in a network of triangular elements which can be considered as 3-node constant strain triangular finite elements. Two translational DOFs at each nucleus and two CPs per interface with normal and tangential springs are assumed. Next the triangles including the CPs are determined. Finally, the normal, tangential, and lateral strains of each CP are calculated by projecting the constant strain tensor of the associated triangle on the corresponding interface. In order to examine the efficiency and accuracy of the proposed FE-RBSM formulation, two kinds of numerical analyses including constant and variable stress fields are employed. For the case of constant stress field, a 100mm square sample is analyzed in uniaxial tension and pure shear. Besides, for the case of variable stress field, a 300mm square sample including a 10mm diameter hole at its centroid is analyzed in uniaxial and biaxial tension. Also, a 300mm diameter circle sample is analyzed under splitting compression. The results are compared with those of the selected RBSM and also the analytical solutions. They show that, compared to the RBSM, the FE-RBSM can better predict the macro elastic properties and also gives scatter-free microstress fields.