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].
Highlights
2 Introduction to the Fibre Model. In meshless methods such as aforementioned smoothed particle hydrodynamics (SPH) [9, 13, 14] and peridynamics [3], a fluid or solid continuum is approximated by a system of particles in which each particle is joined by arms to its near neighbours
In order to validate the fracture model the relationship between stress and surface energy was examined numerically and compared with with Griffith’s prediction, Eq (103). It does not matter whether we have an edge crack or an internal crack, as we have studied, since we are primarily concerned with the fact that we expect the failure stress to be proportional to the square root of the surface energy divided by the crack length, and are less concerned with the constant of proportionality
It enables us to simulate a homogeneous material with varying particle sizes which can be useful in the study of stress concentrations and fracture propagation
Summary
We give a brief historical overview of the development of peridynamics, as far as possible using the notation of previous authors. Additions have been made to enable variable material point distribution for homogeneous materials in [4, 5] and [6] with the motivation to reduce computational cost (see Fig. 1) These additions complicates the theory and an argument is presented here for a potentially simpler approach where the bond force is based on the theory of Smooth Particle Hydrodynamics (SPH). Where is the material density, vi is the ith component of the velocity, ij is the stress tensor, xj is the jth cartesian component of the position vector and gi denotes the ith component of the body force per unite mass This expression is developed in [12] Eq(3.1) into, dvia dt. The kernel function Wab − , h which in [13] is defined as a function of , and h is the same for both particles and need to be separated into Wab and Wba , allowing for different values of the particle size parameter h
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