A Bond-Based Peridynamic Modeling With Arbitrary Poisson’s Ratio

The peridynamic (PD) equation of motion is an integro-differential equation, and the integrand is free of spatial derivatives of the displacement field.  It explicitly considers the nonlocal interactions between the material points through a set of bonds in an interaction domain (horizon).  When a particular failure criterion is satisfied, the bonds can be removed to describe the initiation and arbitrary damage paths at multiple sites.  It is capable of simulating the expected failure modes under general loading conditions.  These novel features distinguish PD from other mesh-free particle methods. 

The internal bond force can be expressed through a micro-potential between the material points.  The micro-potential is defined in terms of a single material parameter referred to as “bond constant or micro-modulus” and bond stretch.  Therefore, the bond force develops due to only extensional deformation.  The micro-modulus is determined through calibration against the classical SED, and it presents only one independent elastic constant with a fixed value of Poisson’s ratio.  Therefore, it cannot include the effect of arbitrary Poisson’s ratio; thus, it presents a major limitation. 

Therefore, there exist many modifications to the original bond-based (BB) PD in order to remove the restriction on the value of Poisson’s ratio.  Primarily, these modifications introduce additional modes of deformation.  The micro-polar PD models permit the bonds to experience stretch and bending deformations through pairwise PD forces and moments.  The conjugated BB PD models account for bond stretch and a series of relative rotation of a pair of conjugated bonds. The BB PD models with axial and transverse pairwise forces account for axial and transverse displacements as well as particle rotation.  The BB PD models with the rotation include the effect of rigid body rotation and shear deformation.  However, they fail to provide accurate results in the presence of nonuniform deformations.  This discrepancy can be resolved by measuring the bond rotation based on “local strain” state. 

This study presents a new bond-based peridynamic approach for modeling the elastic deformation of isotropic materials with bond stretch and rotation; thus removing the constraint on the Poisson’s ratio.  The resulting PD equilibrium equation is derived under the assumption of small deformation and can be solved by employing implicit techniques.  The bond constants associated with stretch and shear deformation are directly related to the constitutive relations of stress and strain components in continuum mechanics.  Also, the expressions for the critical stretch and critical angle are derived in terms of the critical energy release rate.  Lastly, it does not require a surface correction procedure and the displacement and traction type boundary conditions are directly imposed without introducing fictitious regions in the domain.  The capability of this approach is first demonstrated by capturing the correct deformation under general loading conditions.  Subsequently, its capability for failure prediction will be established by simulating the response of a Double Cantilever Beam (DCB) under Mode I type loading.