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Is peridynamics a superset of Continuum Mechanics?

Mario Juha's picture

Recently, I have found a new theory, called Peridynamics, used to solve, mainly, fracture mechanics problems in materials. But, I am confused about the issue if it is a superset of continuum mechanics or is it a totally new theory that reformulate our previous understanding of continuum mechanics? How do you measure material properties with this theory? Do we need to reformulate our theories to deal with fracture mechanics problems? Is it a totally accepted scientific theory?




Mario J. Juha


Erkan Oterkus's picture

 Hello Mario,

Below are some answers to your questions. Please let me know, if you have other questions about peridynamics.

is a superset of continuum mechanics or is it a
totally new theory that reformulate our previous understanding of
continuum mechanics?

- Actually, it is both. Peridynamics is both a superset of continuum mechanics and at the same time it is a new formulation of continuum mechanics. Classical continuum mechanics is a special case of peridynamics. Classical continuum mechanincs has been very useful and been utilized to solve many different solid mechanics problems. However, because of the partial spatial derivatives in its equation of motion, we can not use it if there is some type of discontinuity in the structure. On the other hand, equation of motion of peridynamics does not consist of spatial derivatives, instead it is integral form. So, you can use it whether or not there is a discontinuity in your structure.

How do you measure material properties with this

- Peridynamic theory has its own material constants, but it is possible to relate material properties of peridynamics to the material properties that we are familiar with in classical continuum mechanics.

Do we need to reformulate our theories to deal with fracture
mechanics problems?

- Failure is defined within the constitutive relation of peridynamics. So, you don't need to come up with or use additional failure criteria to figure out the crack direction, crack initiation etc.

Is it a totally accepted scientific theory? 

-It is a new theory and as you may know people are always resistant to new things. I think it will take some time for the wide acceptance of peridynamics within the solid mechanics community. However, there are many papers published in many different journals about the peridynamics.



Mario Juha's picture

Thank you for your answer. But what do you mean by having your own material constants? How do you measure them? What kind of experiement do you use? Do you follow the ASTM code for your experiments?



Erkan Oterkus's picture

Dear Mario,

Because peridynamic theory has a different approach of expressing interactions between material points, it is possible to use material constants different than the ones used in the classical or conventional continuum mechanics. But, as I said earlier, you can relate the material constants of peridynamic theory to the classical ones. So, you don't need to do additional experiments to determine the peridynamic material constants. Please see the following paper for more information about this:

Peridynamic modeling of plain and reinforced concrete
W Gerstle, N Sau, S Silling -
SMiRT18: 18th lnt. Conf. Struct. Mech

In addition to this, you can also incorporate material models of classical continuum mechanics into the state based peridyanmic framework.If you want more information about how to do this, please look at the following papers:

1) A non-ordinary state-based peridynamic method to model solid material
deformation and fracture


Journal of Solids and Structures

Thomas L. Warrena,  , Stewart A. Sillingb, Abe Askaric, Olaf Wecknerc, Michael A. Eptonc and Jifeng Xuc


2) Peridynamic
States and Constitutive Modeling

of Elasticity

S. A. Silling1 ,
M. Epton2, O. Weckner2, J. Xu2 and
E. Askari2


3) Viscoplasticity using peridynamics

Journal for Numerical Methods in Engineering

J. T. Foster 1 *, S. A. Silling 1, W. W. Chen 2"


Best Regards,


HI Erkan

I am now begin to work on the dynamic fragmentation of single crystal silicon, which is orthotropic. I would like to ask that if peridynamics can be conveniently applied in anistropic materials. Are there any mature thoeries existed for anistropic materials? I have read some papers on isotropic materials and composite materials. But they seems to get around the problem. Could you recomend some papers on micromodulus function of anistropic materials ?

Best regards!



HI Bo,

We have extend the peridynamics to model anistropic materials (with crystal plasticity). This work seems perfectely match you need.  It is the first peridynamic simulation that can be compared with the cyrstal plasticity finite element method. If you have interest, you can find the paper at the following link:


This paper presents the first application of peridynamics theory for crystal plas-

ticity simulations. A state-based theory of peridynamics is used (Silling et al.

(2007)) where the forces in the bonds between particles are computed from stress

tensors obtained from crystal plasticity. The stress tensor at a particle, in turn,

is computed from strains calculated by tracking the motion of surrounding par-

ticles. We have developed a quasi{static implementation of the peridynamics

theory. The code employs an implicit iterative solution procedure similar to a

non{linear nite element implementation. Peridynamics results are compared

with crystal plasticity nite element (CPFE) analysis for the problem of plane

strain compression of a planar polycrystal. The stress, strain eld distribution

and the texture formation predicted by CPFE and peridynamics were found to

compare well. One particular feature of peridynamics is its ability to model ne

shear bands that occur naturally in deforming polycrystalline aggregates. Peri-

dynamics simulations are used to study the origin and evolution of these shear

bands as a function of strain and slip geometry.



crystal plasticity; peridynamics; simulation; theory

Mario Juha's picture

Thank you, Erkan. I will be busy reading the papers. If I have more questions then I will tell you.

 Have a happy research,


Erkan Oterkus's picture

Dear Bo,

First of all, welcome to the world of peridynamics. I hope
you will enjoy it.

There are various approaches to represent anisotropic behaviour
in peridynamics. Amongst those I recommend you to look at the following
references due to their simplicity:

Askari, A., Nelson,
K., Weckner, O., Xu, J. and Silling, S., “Hail Impact Characteristics of a
Hybrid Material by Advanced Analysis Techniques and Testing,” Journal of Aerospace Engineering, Vol. 24, 2011, pp. 210-217.

Oterkus, E. and Madenci, E., “Peridynamic Analysis of Fiber Reinforced Composite
Materials,” Journal of Mechanics of
Materials and Structures,
Vol. 7,
No. 1, 2012, pp. 45-84.

These two references use original bond-based peridynamic
formulation which has limitations on the number of independent material
constants that you can specify due to the equal and opposite pairwise
interaction assumption. However, that will be a good start. To overcome this limitation,
you need to use “ordinary state based” peridynamic formulation. This formulation
(for composite materials) is explained in detail in our upcoming book with
Prof. Erdogan Madenci of University of Arizona which we are hoping to be available

Since you
are working on crystal structure, this following publication can be useful as

E. Askari,
F. Bobaru, R. B. Lehoucq, M.L. Parks, S.A. Silling, O. Weckner (2008)
“Peridynamics for multiscale materials modeling”, Scientific Discovery through
Advanced Computing Program (SciDAC 2008), Seattle, 10 pages

We are also
working on a similar problem that you described. If you and your group are interested,
we are open to collaboration.   






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