Discretized virtual internal bond model for nonlinear elasticity

• A general discrete system is developed to represent the nonlinear elasticity.
• The micro bond parameters in a unit cell are related to macro material constant.
• A parameter calibration method is provided for lattice spring model.
• An efficient approach is developed to simulate fracture.

The virtual internal bond (VIB) is a micro–macro constitutive model. Although this model is based on a postulated discrete microstructure, it ultimately returns to a continuum constitutive relation through a homogenization process. The homogenization process can reduce the internal degrees of freedom, but it omits the effect of the individual micro bond that may play an important role in the fracture process. The present research develops a discrete system to represent the nonlinear elasticity by discretizing the continuous VIB. This discrete system is composed of unit cells, which can adopt any geometry with any number of bonds. The system is characterized by the force–displacement, not the stress–strain constitutive relationship. The nonlinear properties of this discrete system are governed by the micro-bond potential. The micro bond properties are related to Young’s modulus of the material, the volume and the bond number of the unit cell. For a given material, the unit cell has a certain topological structure and configuration. A discussion of two specific cases (the 2D triangular and 3D tetrahedral unit cells) suggests that the discrete system converges with decreasing unit cell size. In the unstructured unit cell scheme, the discrete system can almost precisely represent the initial Young’s modulus and the Poisson ratio of a nonlinear continuum. A mixed fracture example demonstrates that the present method can efficiently simulate the fracture propagation. The present paper provides a theory for developing a lattice-type mechanical model for nonlinear elasticity and provides new method for the fracture simulation of a nonlinear elastic material.