The Use of Peridynamic Virtual Fibres to Simulate Yielding and Brittle Fracture

Abstract

The forces in the ‘arms’ joining the particles in a peridynamic analysis depend upon the state of stress in the equivalent continuum and the orientation, length and density of the arms. Short and long arms carry less force than medium length arms as controlled by the weighting kernel. We introduce an intermediate step of imagining a mat of long fibres in which the fibre forces only depend upon the stress, the fibre orientation and the length of fibres per unit volume without the added complexity of the arm lengths. The effect of the arm lengths can then be considered as a separate exercise, which does not involve the continuum properties. The arm length is proportional to size of the particles and the separation of length from the state of stress allows for modelling of variable particle density in the discretisation of a problem domain, which enables computationally efficient accurate analysis. We then introduce the concept of arm elongation to fracture in order to model surface energy in fracture mechanics. This means that shorter arms have a larger strain to fracture than longer arms. Numerical implementation demonstrates that this produces a fracture stress that is inversely proportional to the square root of the crack length as predicted by the Griffith theory [1, 2].

The numerical simulation of standard concrete tests and steel reinforcement using force flux peridynamics

“…This section will present the most important features of the Force Flux Peridynamcis (FFPD) theory that is used for the simulations presented in section 2 and 4. For the full derivation of the theory, the reader is referred to Reference 20.…”

“…FFPD presents an alternative formulation of the original Peridynamics theory and simplifies the use of variable particle sizes and irregular particle distribution. This is achieved by merging concepts from Smooth Particle Hydrodynamics (SPH) 30 with the Peridynamics theory 7,31 as shown in Reference 20. The bond force in FFPD is reformulated to allow for different size particles while enabling an arbitrary Poisson's ratio and modeling of ideal plastic deformation.…”

“…Therefore, has the dimensions of stress [Pa]. The tension T in a single fiber as shown in Figure 7b, is then calculated as: according to equation (14) in Reference 20 for the 3D case. To prepare for the discrete setup illustrated in Figure 2a, the state of stress needs to be formulated based on a direction vector .…”

“…We also want to separate the deviatoric and volumetric parts of the stress, and , respectively. Following the derivation in Reference 20 (section 3), Equation () takes the form: …”

“…In the study presented here, a numerical meshless method called Force Flux Peridynamics (FFPD) derived in Reference 20 based on SPH and Peridynamics, is extended to 3D simulations and applied to concrete fracture modeling for the first time. The varying particle sizes capabilities with FFPD makes it an attractive choice in the real design scenario, where different parts of a model can be represented with suitable accuracy.…”