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TECHNICAL BRIEFS

# On the Modified Virtual Internal Bond Method

[+] Author and Article Information
K. Y. Volokh

Faculty of Civil and Environmental Engineering, Technion, Haifa 32000, Israelcvolokh@tx.technion.ac.il

H. Gao

Max Planck Institute for Metals Research, Heisenbergstrasse 3, Stuttgart 70569, Germanyhjgao@mf.mpg.de

Klein and Gao (2) extended the VIB method to directionally bonded lattices such as Si by introducing an internal degree of freedom. This approach is not applicable to the isotropic case.

Here we mean a simultaneous interaction of many particles, which cannot be described by a sum of the pair potentials of every two particles involved in the interaction as it is often done in the literature.

J. Appl. Mech 72(6), 969-971 (Apr 05, 2005) (3 pages) doi:10.1115/1.2047628 History: Received November 02, 2004; Revised April 05, 2005

## Abstract

The virtual internal bond (VIB) method was developed for the numerical simulation of fracture processes. In contrast to the traditional approach of fracture mechanics where stress analysis is separated from a description of the actual process of material failure, the VIB method naturally allows for crack nucleation, branching, kinking, and arrest. The idea of the method is to use atomic-like bond potentials in combination with the Cauchy-Born rule for establishing continuum constitutive equations which allow for the material separation–strain localization. While the conventional VIB formulation stimulated successful computational studies with applications to structural and biological materials, it suffers from the following theoretical inconsistency. When the constitutive relations of the VIB model are linearized for an isotropic homogeneous material, the Poisson ratio is found equal to $1∕4$ so that there is only one independent elastic constant—Young’s modulus. Such restriction is not suitable for many materials. In this paper, we propose a modified VIB (MVIB) formulation, which allows for two independent linear elastic constants. It is also argued that the discrepancy of the conventional formulation is a result of using only two-body interaction potentials in the microstructural setting of the VIB method. When many-body interactions in “bond bending” are accounted for, as in the MVIB approach, the resulting formulation becomes consistent with the classical theory of isotropic linear elasticity.

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## Figures

Figure 1

“Stretching” (left) and “shearing” (right) forces in the modified VIB formulation

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