# Peridynamics is a new continuum mechanics formulation. It was originally developed by Dr. Stewart Silling in 2000.

## Peridynamics

In order to determine the deformation response of materials and structures subjected to external loading conditions, classical continuum mechanics (CCM) was introduced by disregarding the atomistic structure. CCM has been successfully applied to numerous challenging problems. However, its governing equation faced a difficulty when there is any discontinuity in the structure such as a crack, since spatial partial derivatives in its governing equation are not defined for such a condition. In order to overcome this problem, a new continuum mechanics approach, Peridynamics (PD), was recently introduced with the intention that its governing equation is always valid whether there is any discontinuity in the structure or not. Moreover, PD can be considered as the continuum version of molecular dynamics. This character of PD makes this new approach a suitable candidate for multi-scale analysis of materials. Furthermore, PD formulation can also be extended to other fields such as thermal, moisture, etc., so that it can be used as a single platform for multiphysics analysis of materials.

Below, we will try to answer some frequently asked questions about peridynamics:

**What is peridynamics? **

- Peridynamics is a new continuum mechanics formulation developed by Dr. Stewart Silling in 2000. The governing equations of peridynamics are integro-differential equations and do not contain spatial derivatives which makes this new theory very attractive for problems including discontinuities such as cracks.

**What is the meaning of "peridynamics"?**

- The word "peridynamics" is originated from combination of two Greek words "peri" and "dynamics". In Greek, "peri" means "horizon" and "dynamics" means "force". "Horizon" is an important parameter in peridynamics.

**What is horizon? How do you choose horizon size?**

- Horizon mainly refers to the domain of influence which defines the range of interactions between material points. It can also be considered as a length-scale parameter which gives peridynamics a "non-local" character. The size of the horizon changes depending on the nature of the problem. If the problem does not show any "non-local" physical behaviour, it should converge to "zero". In this special case, peridynamics and classical continuum mechanics are equivalent to each other. We normally use numerical techniques to solve the governing equations of peridynamics. By using a very small horizon value can make the computational time significantly large. Therefore, in practice, a sufficiently large horizon value can be chosen which doesn't show any significant non-local behaviour. This value can be obtained by performing a simple convergence analysis. On the other hand, some problems may show non-classical and non-local physical behaviour. In these cases, horizon can be used as a length scale parameter and can be adjusted in such a way that the required physical behaviour can be accurately represented.

**What is state-based peridynamics?**

- In the original formulation of Dr. Silling in 2000, the interaction forces between material points are assumed to be equal in magnitude and opposite to each other. Although, this assumption significantly simplifies the calculations, it also introduces several limitations including constraints on material constants, incapable of capturing incompressibility for plastic deformations, etc. In order to overcome these limitations, Dr. Silling and his co-workers developed an advanced version of peridynamics named as "state-based peridynamics". In order to distinguish this new formulation with the original formulation, the original formulation was then named as "bond-based peridynamics". In state-based peridynamics, the interaction forces do not need to be equal in magnitude and opposite to each other.

**Is peridynamics computationally expensive?**

- It is difficult to give a unique answer to this question. In many problems that a regular finite element analysis can provide an accurate outcome, peridynamics can be more computationally expensive. However, it is important to note that peridynamics is especially powerful for particular problems that conventional techniques including finite element analysis has a difficulty in solving or incapable of solving. Moreover, it is very suitable to be implemented in a parallel programming environment which can allow significant reduction in run times depending on the computing resources.

**Is peridynamics a finite element technique?**

- Peridynamics is a non-local type continuum mechanics formulation. In other words, it is a continuum formulation rather than a numerical approach in general. However, it is normally not easy to obtain closed form solutions of peridynamic equations. Therefore, in practice, numerical techniques are used for the solution process. The most common approach for spatial discretisation is "meshless scheme". "Finite element" discretisation is also possible and available in the literature.

**Is peridynamics same as Eringen's non-local elasticity formulation?**

- No. Although both peridynamics and Eringen's formulation belongs to "non-local continuum mechanics" class. However, Eringen is using same set of equations as in classical continuum mechanics when the equations are expressed in terms of stresses. The non-locality in Edingen's formulation comes into the picture when the stress of a point is expressed in terms of strains. The stress in Eringen's formulation depends on not only strain of the material point of interest, but also strain values of other material points. On the other hand, the governing equations of peridynamics are different than peridynamics and do not use concepts like "stress" and "strain" as in the classical continuum mechanics.

**Is peridynamics a numerical technique?**

- The answer is both yes and no. Peridynamics is a non-local continuum mechanics formulation. In that sense, it is not a numerical technique. However, for most problems, we cannot obtain a closed a form solution. Hence, we use numerical techniques for both solution of space and time. Based on this approach, it can also be considered as a numerical technique.

**Can I use implicit time integration in peridynamics?**

- Yes. Although in many applications presented in the literature, explicit time integration is used for numerical time integration, implicit time integration can also be used.

**Can I apply peridynamics for other fields such as heat transfer?**

- Yes. Although peridynamics is originally introduced for structural mechanics problem especially to predict fracture, it is also possible to use peridynamics for other types of problems including heat transfer, porous flow, etc. This also makes peridynamics a suitable platform for multiphysics analysis.

**Can I use a finite element software to perform peridynamic simulation?**

- Yes, although regular finite element analysis is solving different sets of equations than peridynamics, it is possible to implement peridynamic logic in a commercial finite element software by using truss elements, mass points etc. and assigning suitable parameters through a calibration procedure.

**Can I use a molecular dynamics software such as LAMMPS to perform peridynamic simulation?**

- Yes. Peridynamics is a continuum mechanics formulation. In other words, we do not individually model atoms and molecules. If we use meshless scheme for numerical discretisation, the peridynamic equations take a very similar form with those of molecular dynamics equations. Therefore, by following a suitable calibration procedure, a molecular dynamics code can be used for peridynamic simulations. LAMMPS can be used for this purpose and there was already a peridynamic implementation in this platform.