An approach to multi-body interactions in a continuum-atomistic context: Application to analysis of tension instability in carbon nanotubes

AbstractThe tensile strength of single-walled carbon nanotubes (CNT) is examined using a continuum-atomistic (CA) approach. The strength is identified with the onset of the CNT instability in tension. The focus of this study is on the effects of multi-body atomic interactions. Multiscale simulations of nanostructures usually make use of two- and/or three-body interatomic potentials. The three-body potentials describe the changes of angles between the adjacent bonds – bond bending. We propose an alternative and simple way to approximately account for the multi-body interactions. We preserve the pair structure of the potentials and consider the multi-body interaction by splitting the changing bond length into two terms. The first term corresponds to the self-similar deformation of the lattice, which does not lead to bond bending. The second term corresponds to the distortional deformation of the lattice, which does lead to bond bending. Such a split of the bond length is accomplished by means of the spherical–deviatoric decomposition of the Green strain tensor. After the split, the continuum-atomistic potential can be written as a function of two bond lengths corresponding to the bond stretching and bending independently. We apply an example exponential continuum-atomistic potential with the split bond length to the study of tension instability of the armchair and zigzag CNTs. The results of the study are compared with those obtained by Zhang et al. (2004. J. Mech. Phys. Solids 52, 977–998) who studied tension instability of carbon nanotubes by using the Tersoff–Brenner three-body potential, and with recent experimental results on the tensile failure of single walled carbon nanotubes

Elasticity with energy limiters for modeling dynamic failure propagation

AbstractSeparation of two particles is characterized by a magnitude of the bond energy, which limits the accumulated energy of the particle interaction. In the case of a solid composed of many particles a magnitude of the average bond energy – the failure energy – exists, which limits the energy that can be accumulated in an infinitesimal material volume under strain. The energy limiter controls material softening, which indicates failure. Thus, by limiting the stored energy density it is possible to include a description of material failure in the constitutive model. When the failure energy, i.e. the energy limiter, is introduced in the constitutive model it can be calibrated in macroscopic experiments. Traditional elasticity models do not have energy limiters and they allow for the unlimited energy accumulation under the strain increase, which is physically meaningless because no material can sustain large enough strains without failure. We use elasticity with energy limiters for modeling dynamic failure propagation in brittle solids. Two models of isotropic Hookean solids with energy limiters are introduced and examined in simulations of the penetration of a projectile into a brittle plate in the present work. The first model uses the energy limiter with the overall energy term while the second model has separate energy limiters for the volumetric and deviatoric components. The results of the penetration simulation obtained by using both models are similar qualitatively. It is remarkable that the penetration depth is mesh-independent for fine meshes even without the special regularization procedures. This is the first work where the methods of elasticity with energy limiters are used in dynamic analysis of brittle failure

An Approach to Elastoplasticity at Large Deformations

Finite plasticity theories are still a subject of controversy and lively discussions. Among the approaches to finite elastoplasticity two became especially popular. The first, implemented in the commercial finite element codes, is based on the introduction of a hypoelastic constitutive law and the additive elastic-plastic decomposition of the deformation rate tensor. Unfortunately, the use of hypoelasticity may lead to a nonphysical creation or dissipation of energy in a closed deformation cycle. In order to replace hypoelasticity with hyperelasticity the second popular approach based on the multiplicative elastic-plastic decomposition of the deformation gradient tensor was developed. Unluckily, the latter theory is not perfect as well because it introduces intermediate plastic configurations, which are geometrically incompatible, non-unique, and, consequently, fictitious physically. In the present work, an attempt is made to combine strengths of the described approaches avoiding their drawbacks. Particularly, a tensor of the plastic deformation rate is introduced in the additive elastic-plastic decomposition of the velocity gradient. This tensor is used in the flow rule defined by the generalized isotropic Reiner-Rivlin fluid. The tensor of the plastic deformation rate is also used in an evolution equation that allows calculating an elastic strain tensor which, in its turn, is used in the hyperelastic constitutive law. Thus, the present approach employs hyperelasticity and the additive decomposition of the velocity gradient avoiding nonphysical hypoelasticity and the multiplicative decomposition of the deformation gradient associated with incompatible plastic configurations. The developed finite elastoplasticity framework for isotropic materials is specified to extend the classical -theory of metal plasticity to large deformations and the simple shear deformation is analyzed