On Nonlocal Lagrangian-based Models and Applications to Material Failure

Abstract

Through this study, we present a discrete nonlocal Lagrangian approach called ”tridynamics” which is designed at a length scale of interest to characterize the response of the body. As a basic unit to describe the interaction, instead of two particles required to define a bond in conventional discrete frameworks, we introduce three particles at the vertices of a triangular surface. The main idea is to understand the dynamics of a deformable body via a macro potential corresponding to a coupled interaction of rigid particles in the reduced dimension. Because the continuum limit is not taken, the framework automatically relaxes the requirement of differentiability of field variables. The discrete Lagrangian based approach is illustrated to derive equivalent Euler–Bernoulli beam model based upon the corresponding potential function. We also present a set of physical quantities that explain the deformation of Timoshenko beam and Mindlin plate, which help to derive the potential energy.

Although the construction of potential functions for basic elements such as beam and plate might be possible, it is challenging to create it in the generic case. For example, the behavior of carbon nanotube or a graphene sheet is very dependent upon molecular structure. Therefore a derivative-free balance law pertaining to a higher scale of interest has been developed based on the molecular level information, which might be useful in a continuum or discrete setting. Derived using a probabilistic projection technique, the law exploits certain microstructural information in a weakly unique manner. The projection generalizes the notion of directional derivative and, depending on the application, may be interpreted as a discrete Cauchy–Born map with the structure of the classical deformation gradient emerging in the infinitesimal limit. As an illustration, we use the Tersoff–Brenner potential and obtain a discrete macroscopic model for studying the deformation of a singlewalled carbon nanotube (SWCNT). The macroscopic (or continuum) model shows the effect of chirality – a molecular phenomenon – in its deformation profile. We also demonstrate the deformation of a fractured SWCNT, which is a first-of-its-kind simulation, and predict crack branching phenomena in agreement with molecular dynamics simulations. As another example, we have included simulation results for fractured SWCNT bundle with a view to establishing our claim regarding the efficacy of the proposed method.

The discrete Cauchy-Born rule with the principle of virtual work done are employed to formulate a generalized model with the hope to unify local and nonlocal continuum frameworks. We also found a compact mapping matrix which converts surface-based forces (stresses) to the nonlocal body-based forces. The transformation matrix allows reconstructing continuum models at a lower length scale in a discrete setting. Despite the conventional mapping of the microscopic bond from the undeformed configuration, the consistent derivation requires a transformation on the Average Deviation of Lattice (ADL) vector in the region of influence. The new conversion proffers flexibility to the framework for the analysis of nonuniform distribution of particles in the field. To see the credibility of the model, fracture evolution in SCB specimen made of Polymethyl methacrylate (PMMA) is simulated, and the results are compared with experiments.