Modeling Fracture: Intuition, Formula for Elastic Potential & Nodal Forces

[alienspy77]

Hi, I've been looking into ways to model fracture in real-time and i ran into a few papers on the net, one was by a few people from MIT compsci lab that seemed to fit what i was trying to do very well. However some of the math in it is a bit involved. The link to the paper is, for instance, graphics.cs.yale.edu/julie/pubs/Fracture.pdf .

So the formulation of the model goes approximately like this:

u [u1 u2 u3] - spatial coordinates of the undeformed object point in world coords

deformation is a function p( u ) = [p1, p2, p3] - maps undeformed world to deformed world coords

Green's tensor is Ju(p)*JuTransposed(p) - I.

Ju( p ) is the jacobian of the vector function p with respect to u
hooke's law relates stress sigma to strain e. for isotropic materials it can be expressed as
sigma = 2 mu e + lambda trace(e)* I
sigma is the stress tensor, e is strain tensor

eta = 0.5 trace( sigma * e ) - elastic potential energy

total elastic potential is integral of eta over the body volume

energy conservation states that internal work (elastic potential) has to be equal to the external work done by the external forces

we want to solve for deformation function p given the external force field.

our FEM formulation approximates function p as piecewise linear within tetrahedral elements
linear deformations yield constant strain and stress tensors within each element therefore tensors can be moved outside of any integration over an element's volume
we assume constant strain tetrahedra and use barycentric coords for interpolating within them
m = [m1, m2, m3, m4] - coords of the 4 nodes of a tetrahedron in undeformed material coordinates
x1, x2, x3, x4 - deformed coords for the same points
b = [b1, b2, b3, b4] - barycentric coords in terms of element's nodal positions in the undeformed frame
thus [u,1] = [m,1] * b - mapping from barycentric to undeformed world coordinates
we use the same barycentric coordinates b to identify u with p( u ), given that we know x1...x4
[p,1] = [x,1] * b

we can combine the relations above to form a direct mapping p( u )

p(u) = [x,1]*beta*[u,1], where beta = inv( [m,1] )

this defines the deformation function p, allowing the computation of the strain tensor e, stress tensor sigma and the potential density eta. these turn out to be constant within each element

the elastic force on the ith node fi is defined as the partial derivative with respect to xi of the elastic potential density eta, integrated over an element's volume

this amounts to

fi = nu/2 * beta * G * sigma * Gtranspose * beta_transpose * xi

where nu is the element's volume in the undeformed coord frame and

G = [ [ 1 0 0 ] [ 0 1 0 ] [ 0 0 1 ] [ 0 0 0 ] ] (3x4 matrix)


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while i have this written out there is a number of things i don't quite understand, if someone could help me figure this out i'd appreciate that greatly.

First, what's the intuition behind the elastic potential density? Where does that formula (eta = 1/2 Trace( sigma * e ) come from?

Second, how do they arrive to the formula for the nodal forces? They say they integrate the elastic potential's derivative over the element's volume but how would i go about actually writing that out? It seems the expression for eta ends up to be a fourth degree polynomial of the components of x. Are they integrating eta over the tetrahedron first and then differentiating the result or are they integrating the derivative? Whats the physical meaning of that operation? How do i go about writing that integral out?

I know this might be a lot of questions to ask but i really want to figure this one out :)

 

[alienspy77]

PerennialII

By the way does anyone happen to know how i'd be able to get in touch with PerennialII ?? he seems to be the expert in this field.

 

[Astronuc]

I'll contact him.

Interesting paper and I'm sure PerennialII would be interested.

 

[alienspy77]

I'll contact him.

Interesting paper and I'm sure PerennialII would be interested.

Thank you.

 

[alienspy77]

By the way it's possible that my company (video game industry) could offer a contract or a full-time position to someone capable of fully understanding and implementing a system similar to that described in the paper

 

[Vanechka]

Hi alienspy77,

was away from PF a while and been trying to get my account active and working again (as you can see working on a different username as of now). Astronuc let me know there'd be something of interest here and so it seems -- I'll take a look at it sometime today and get back to you,

-p2-

 

[Vanechka]

alienspy77,

I'm late but the topic is so interesting it merits a reply even if you don't need it anymore . I've seen some real time simulations of "normal" materials ("stiff" by the terminology used by Muller et al, materials where the constitutive relationship is complex enough to typically prevent a real-time solution without overly simplifying things like working on a limited 2D discretization of a linear problem) but most haven't certainly included fracture or contact within the simulation. And those are the features would think are some of the greatest benefits of a real-time analysis (I myself work on non-real-time fracture analysis so this is kind of "cool" stuff). And at least yields the best animations . Good read and a nice paper.

But to your questions ...

1) the elastic strain energy density, or elastic potential density depending on what term you wish to use: This can be answered in a somewhat lengthy manner, so please ask for clarification to the direction you're interested in since I'm trying to keep this simple if possible. First, the whole formulation of continuum mechanics is based on balance and equilibrium equations (or conservation equations might be the best term -- different people, different books etc. use to some extent different terminology). These are derived from thermodynamics (thermomechanics to be a tad more precise). The underlying concept is that the system we're interested in has to follow the first fundamental theorem of thermodynamics, which in particular states that there must be a balance of energy. In that paper they talk about internal work = external work, which is the very idea here (conservation of energy). In general, if we'd take a couple of steps back from where the paper starts, we'd write something like: internal energy + kinetic energy = external work (this is the "what comes in must stay there or come out" - sort of principle for energy). This is the balance equation used to formulate the equations for the finite-element solution of the partial differential equations. Now, naturally, the different terms, especially the one for internal energy are dependent on the material we're studying. Particularly the material model (constitutive relationship) we're using. And for a linear-elastic material the internal energy density is the elastic potential density using the terminology of Muller et al (i.e. it's the energy done for the deformation of a material point when we generate stresses, and once we integrate it over the volume we're interested in we've got the internal energy associated with the process as a whole). The clue why the density has the form that is has is that it needs to complete the energy balance in order to be considered "valid" (this can be worked on in a far more lengthy manner with lots of stuff about the fundamental building blocks of continuum- & thermomechanics). Stress and strain measures which fulfill this criterion are often referred to as conjugate measures, and hence make some "thermodynamical sense" as building blocks of theories.

2) About the expression for the nodal forces... the treatment of nodal forces is complicated in this paper a bit by the fact that they need to do the treatment incorporating finite deformations (they use Green's strain tensor), which makes the expressions themselves more lengthy, but in general the principles remain the same as in the stuff one comes across more often (can naturally go deeper here if you wish). They seem to use (at the paper right above Eq. (9)) a fairly common form for attaining the internal nodal forces. What makes this a bit difficult is that the relations arise from the weak form of the FE balance equations (if you're deriving FE using for example Galerkin's method, or principle of virtual work)(this involves derivation of the weak form typically from Cauchy's equation of motion (or from conservation of momentum)). The internal work arising from the balance equations can be written as
$$\delta W_{internal} = \int{\delta u_{,x} \sigma }dV,$$
so the internal work arising from stresses and strains is written as a function of displacements and the (true) stress tensor. For a linear-elastic material, it should turn out that the integrand is actually (for this element type) the partial derivative of the elastic potential with respect to $x_{i}$ (hope it is, I didn't check the expressions in the paper ), which then needs to equal to the internal work done by nodal forces and displacements, $f_{i} u_{i}$ over the element nodes, and from this equality you can solve the nodal forces as they've done (alternatively you could just compute the energy associated with the stress and strain state of an element, and set this equal to the work done by internal nodal forces and displacements and solve the nodal forces from there).

I'm not quite sure how well all this hit the target but that's why iteration has been invented .