Deformation gradient f(3,3) vs Coordinates

[AccessTUD]

Dear,

I have a task to model the behaviour of certain interphase material. Let's say that functions which describe the change of material parameters are known.
i.g. change of the Young's modulus as function of distance from neighbouring material (or (0,0) origin) - PAR=PAR(x)

Furthermore, based on geometry (nodal coordinates) it is easy to estimate a deformation gradient f(3,3) at any point.
I am trying to obtain the stresses by integrating at Guass points which coordinates are also known.

Taking into consideration that Sigma=Youngsmodulus * Epsilon,
standard and time-wasting procedure would be something like: - find coordinates of Gauss points - read E value at that specific point - multiply by deformation gradient

Can I skip this just by using common sense approach? If F-deformation gradient is function of Gauss point coordinate, does that mean that I can find the function which describes F-E relation instead of already known x-E relation!?

*(In this way, I can directly get the value of E just by knowing F at that point)

Thank you

 

[AccessTUD]

Somewhat like this..

 

[muzialis]

AccessTUD,

I am unsure I understand the problem.
So you know already the deformation gradient (as a tensor, not only the F(3,3) component I assume) on your domain: your elastic problem is solved already! You wonder then about how to compute the stress tensor: I do not understand why would you bother mproving the "standard practice" which admittedly takes some time, but certainly much, much less than the time needed to solve your problem, i.e. find F.
Anyhow, knowing F(x) and E(x) (F a second-order tensor, E a fourth-order tensor in general, x a 3-d vector) it seems unplausible you will be able to build a function E(F) as it seems to be it would be multi-valued (in general you might have an infinity of points with different E but same F).
I apologise if I missed the point.

 

[AccessTUD]

Dear muzials,

first of all I have to correct myself and say that by F(3,3) I meant F with dimension 3x3 with all its elements already known.

I was going the opposite way of course. The purpose of my study is estimating the values/functions of parameters. But let us take it like it's not a problem to obtain F-deformation gradient. (p.s. I have no clue how it's obtained. I guess from geometry.. If possible, maybe you can give me a hint or two!?)

Finding a way to obtain E(F) would be a soultion of my problem. I can see you got my point.. I just wanted to see if it's possible at all?

I was thinking of it in very simple way of simple beam under tension. If we fix one end and have deformation from x0 length to x1.
Epsilon= delthax/x0 delthax=x1-x0
-> x0=x1/(epsilon+1)

and then get this E(x0) as E(x1/epsilon+1) *((--->E(eps)))

Is it possible to something like this on 3D level with respecting all continuum mechanics principles?

 

[muzialis]

AccessTUD,

finding F is the problem of elasticity itself! Analytically, you have to solve the Navier or Cauchy PDE, not always easy. Numerically, it is the deliverable of a FE computation, for example.
Now back to your point: your example confuses me. In the beam under tension, the strain is constant: in this case you only have one component of F, and you see easily that it equals (strain + 1). If I understood you E, Young's modulus, is assigned as a value of the distance from x0, for example. So your E(F) "function" in this case would look like a bizarre object assigning a range of values to a constant! And more bizarelly, the same range of values is assigned to a different constant (if you pull more). Maybe if you describe your problem more concretely I would foloow better...

 

[AccessTUD]

I hope I managed to show how I came up to this silly idea.
But even there I have the problems with this x0,x1,deltx .. Can deformation gradient provide us info about new position of the point? If I remember correct from the Continuum Mechanics classes, the answer is NO!? (F gives the ratio of ''coordinate differences'' just like epsilon) ..But knowing Epsilon in the given example can provide us the new coordinates, I mean x1. ??
Is it wrong to think like this?

Young modulus should be a function of initial configuration .. in this case x0

 

[AccessTUD]

I forgot the pic... Sorry

 

[muzialis]

Both F and strain tensor are a function only of gradients of displacement, as is clear from their definition.
Physically this is an essential requirement, the principle of objectivity: moving a body by a rigid motion can not alter its energetic content (which is supposed to be a function of its strain state). Seems obvious, but is not, it is still a debated topic.
So, the position of a material point can not be inferred from F or any strain tensor unless the boundary conditions are used: in your case the procedure uses the fact that x0 is fixed.
Anyhow, in your example the Young Modulus is constant, is the value at x0. I thought you were dealing with examples where E varies accoding to the position (for a example, a beam that is progressively stiffer). This is the case where all breaks down, let alone the full 3D case.

 

[AccessTUD]

I really appreciate your prompt responds. Thank you for that.

Before closing the topic I have to make everything clear, if possible.
I have a 'interphase material' which becomes progressively less stiffer. So starting from material 1. with really high stiffness (i.g. E=1000000) , via interphase (E=?) and coming to material 2. with low stifness (i.g. E=1000).
What happens in that 'interphase' is up to me .. I mean to find a function which describes the behaviour... (I can easily get it from experimental results) .. I am doing all this with FEAP (or Finite Elemebt Analysis Program) which is already really advanced, so I can get all this outputs like F tensor and so on. I just wanted to use F to read the value of my Elasticity modulus in interphase. Any other clue how to do it?

Thank you once again!

 

[muzialis]

No worries for the quick replies, boring day in the office...
Ok maybe I get it know, i was confused by the fact you claimed in the first post wanted to read the value of the Elasticity Modulus in the interphase.
I interpreted this as, I do not know such value in advance.
Well then, how can you then use FEAP and get an elastic solution and F? You must be inputting some material properties for the interphase!
Unless...maybe your interphase material starts from a given modulus, and then gets damaged according to the deformation (is that the case? How is then damage driven?)
If that is the case, you are sorted: plot the axial strain as a function of X, the stress is constant because of equilibrium (this is a general requirement) so the modulus as a function of X is simply sigma / epsilon (x), that is your function (in the 1D case you shown, where all the problem depends mainly on the length coordinate). In 3 D the same can be achieved with vectorial formulas, slighlty more complex, but in 1D, is this easy (this is what confused me, as you mentioned computing the stress in the first post, while the stress is constant, unless you are using non-linear elasticity, is that the case?.

 

[AccessTUD]

OK. Time for one more Paint-piece of art of mine.
The function which describes E will be predetermined .. From experimental data. I still have no clue which mathematical level of approximation I'll use.. (Linear, 2nd order polynomials, some step funciton...whatever)
In the example that I am currently working with, I just assigned a random values. Just to make it run.. Like divide the length of interphase by ten and assigned values (like linear... 1000, 900, 800, 700 ..). Just to have something to do the programming with. I will know the values from the beginning.

''In 3 D the same can be achieved with vectorial formulas, slighlty more complex, but in 1D'' This will be 'interesting' -.- and this is the reason all this conversation was started.

And yes non-linear elasticity is used. Neo-Hookean material model.. I guess it's hyperelastic type of nonlinearty!

 

[muzialis]

I am more and more confused...
So you know Young's modulus! (actually you know the constant of the Neo-Hookean model, is not a Young's modulus strictly speaking, also the formula you reported, Sigma=Youngsmodulus * Epsilon, is not valid at all), you know the stress (you are applying it and it has to be continuous, in your example), you know everything there is to know!
In the most general 3D case (even your model is 3D, but is simpler in some sense because all unkwons are the same on a transversal section, hence it is equivalent to a 1D problem) you will run into massive problems (my opinion) because you could have the same F for points with different constant E, you will not be able to build your function. But look at the bright side, computing stresses knowing already F and material propertis is easy, I would understandf your worries if you hd to do it manually but we have computers for that...