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# Terminology for discrete approaches to modelling fracture

It appears to me that there is more and more research done on discrete approaches to modelling fracture.

Especially for brittle or quasi-brittle materials these methods are undergoing a revival.

I am not sure why this is the case. I am also not in the position to judge if this revival is useful.

I just had recently a look at some of these models and I found these discrete models to be suitable

for description of fracture in heterogeneous materials, where cracks appear at many different positions.

However, I am confused by literature in the field, even if I just focus on fracture.

There are models called lattice, particle, disctinct element, discrete element etc.,

which belong all to the group of discrete approaches.

These names seem to be used in different ways, which makes it difficult to communicate developments in this field.

In a recent paper we have tried to distinguish between lattice and particle models for fracture in the following way:

... discrete approaches are divided in the group of

particle and lattice models.

In particle models, the arrangement of particles can evolve, so that

neighbours of particles might change during analysis.

Therefore, particle models are suitable to describe processes involving

large displacements.

On the other hand, in lattice models the connectivity between nodes is

not changed during the analysis, so that contact determination is not

required.

Consequently, lattice models are mainly suitable for analysis involving

small strains.

I was wondering if anyone would agree with the above classification.

I think one should come up with something more refined and complete.

Any suggestions?

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## Comments

## re: discrete models

Dear Peter:

I have looked at lattice models (for other purposes), and especially the interesting work of Prof. Sukumar. I can't comment on these methods (perhaps Suku can tell us why lattice models are useful) but just wanted to make the following comment. Other that the numerical methods you mentioned, there are fracture models that assume a minimum crack propagation length (to avoid the nonphysical conclusion of Grifith's theory that predicts infinite strength for a vanishing crack size). In other words, these models assume that crack propagation has a discrete nature. The idea started by the work of Novozhilov (1969) and has recently been investigated by several groups. In the literature these models are refereed to as "finite fracture", "quantized fracture", "discrete fracture", etc.

Regards,

Arash

## Re.: Parenthetically concerning the unphysicality...

Dear Mr. Arash Yavari,

Perhaps you attribute to Griffith's theory what is due to Inglis' theory...

After all, isn't it right in the classical analytical solid mechanics itself that a crack is rather unphysically defined as a hole that has zero as the radius of curvature?... Inglis simply followed the classical paradigm of analytical solution, defining the hole as above... Years later, Griffith simply took Inglis's solution as a given... The conclusion you mention, then, was a direct result---out of Inglis' i.e. analytical/mathematical theories...

So, isn't the awkward abstraction used in the mathematical analysis really to be blamed?

As an aside, I think in classical mathematics there is no solution to this issue, and there cannot be---you simply cannot model a situation like "one thing becomes two things" or "two infinitesimally close points become separated by a finite distance" within any continuum theory at all...If so, why attribute this shortcoming to Griffith?

(We could take this discussion to another thread if necessary... Thanks in advance, everybody!)

## Re: More on discrete models

Dear Arash

thank you for your comment. I studied some of Bolander's and Sukumar's papers and tried to follow their approaches in some of my work as well. Their work is indeed very interesting.

I will have a look at these finite fracture approaches.

Maybe one can categorise numerical approaches for fracture modelling (at a large scale) in

The finite fracture approaches would then belong to group 2, but might be very similar to certain lattice approaches in group 3? (I will need to have a close look at some of the papers to understand this finite fracture approach.) One could then try to subdivide each group further (lattice versus particles).

Peter

## Re: More on discrete lattices

Dear Peter,

if my understanding is correct, the finite/quantized/discrete

fracture models Arash is referring to are based on purely continuum

description, but the minimum crack propagation length is set to a

given length (perhaps related to lenghtscale parameters of non-local

approaches). Therefore, they should belong to the first group.

Nevertheless, I fail to see other connection to particle/lattice

models.

Jan

## Discrete models

Dear Jan

thank you for pointing this out.

Back to the original comment regarding the difference between particle and lattice models. I had written that

A

rrangement of particles can evolve, so thatneighbours of particles might change during analysis.

This is not the case for lattice models. However, I guess that there are other methods, in which the connectivity changes and which are not considered to be particle models.

1) Maybe particle models should be limited to the group of models in which there are particles of finite size for which the contact arrangement is determined (and might change). Contact detection would then be characteristic for these models.

2) Lattice models are those for which the connectivity remains the same.

3) For other "discrete" models, the connectivity might change but contact detection between "particles" is not required.

Peter

## Distinction between discrete and meshless models

Just to hammer this nail once more, the main difference between these methods can be distilled to continuum versus discontinuum perspectives.

In continuum mechanics, particle methods commonly refer to meshless methods. Smoothed Particle Hydrodynamics (SPH) and Element-Free Galerkin (EFG) are two popular incarnations of these. These methods allow arbitrarily large strains (through dynamic "remeshing" via neighbor-sorting and compact support potentials) at the expense of fidelity. Lattice methods (to which Lagrangian finite element methods would arguably belong) are typically higher order methods but are often restricted to relatively small strains and often (though not always) do not admit changes in nodal connectivity (i.e., potential fracture surfaces must be predefined). Continuum methods, of course, require constitutive equations based on combinations of empirical and mechanistic models to capture the physical behavior.

In discontinuum mechanics, the emergent behavior (which would be the behavior characterized by constitutive equations in continuum mechanics) is captured by the interactions of the constituent solid materials, where the contacts between bodies are characterized by a contact law. This is often necessary for highly heterogeneous materials (e.g., sand under dynamic loading). The popular methods for discontinuum analysis are the Discrete Element Method (synonymous with the Distinct Element Method ... these are often referred to simply as DEM) and Discontinuous Deformation Analysis (DDA). The main attraction of these methods is the natural treatment of multi-body contact and comminution.

There are also methods that span both physical levels, including combined finite element - discrete element methods (FEM-DEM) and quasi-continuum methods, which resolve the area at the crack tip using a molecular dynamics (MD) representation coupled with a finite element (FE) representation away from the crack tip. Both of these methods provide the efficiency of FE in the continuous material while allowing for arbitrary discontinuities to appear and evolve.