A three dimension lattice-spring model with rotational degree of freedom and its application in fracture simulation of elastic brittle materials

Abstract

Applying parameter mapping theory, this paper establishes a new three dimension lattice-spring model with rotational degree of freedom (3D-LSMR). This 3D-LSMR contains nearest neighbor, next nearest neighbor and third nearest neighbor which highly increases the accuracy of system. Compared with the general lattice spring model, 3D-LSMR adds the rotational and tangential degrees of freedom, whose spring parameters are obtained by finite element parameter mapping. The traditional lattice spring model only considers the normal interaction between two points (the Poisson’s ratio is fixed), which limits the application of the lattice spring model in a wider range of materials. Some scholars proposed to add shear spring into the traditional model to solve the problem of fixed Poisson’s ratio, but the rotation invariance could not be maintained. The main reason is that the shear spring can’t distinguish the difference of tangential velocity (or displacement) between the two particles due to the common rotation or shear. Therefore, the rotation of the whole rigid body may cause incorrect generation of additional strain energy inside the shear spring. 3D-LSMR model is able to maintain rotation invariance and reproduces different Poisson’s ratios because of the spring parameters with rotational degrees of freedom and the rotation effect of lattice points. In this paper, the finite element stiffness matrix with rotational freedom is derived by using 3D-LSMR as the basic mechanical model, and the parameter mapping theory of spring stiffness coefficient is introduced. By means of numerical simulation, the large deflection of slender cantilever beam, elastic symmetrical collision and the simulation of dynamic fracture for concrete L-specimen are checked and calculated respectively, and then the correctness of the model is verified.