FE method, how ? (software Comsol)

[trabo]

Hi all,

The plastic flow rule for large strains in a continuum medium can be written as :

$-\dfrac{1}{2} \Big ( \dfrac{d}{dt}(F_p^{-1} ). ^t F_p^{-1} + F_p^{-1} . ^t \dfrac{d}{dt} ( F_p^{-1} ) \Big)= \lambda F^{-1} \dfrac{\partial f}{\partial \tau} F F_p^{-1} . ^t F_p^{-1}$
​

where $F_p$ is the plastic part of the total deformation gradient $F$, $f$ the yield surface and $\tau$ the Kirchhoff stress tensor.
According to Comsol notes, we denote $M=F_p^{-1}$ and discretize the law by using 'variations with $\Delta$' :

$-\dfrac{1}{2} \Big [ 2M. ^t M -M_{old} . ^t M - M . ^t M_{old} \Big ]= \lambda \Delta t F^{-1} \dfrac{\partial f}{\partial \tau} FM. ^t M$​

but I don't understand what is stated after. I quote :

For each Gauss point, the plastic state variables (ε_p and M, respectively) and the plastic multiplier,$\Lambda =\lambda \Delta t$, are computed by solving the above time-discretized flow rule together with the complementarity conditions

$\Lambda \ge 0 \quad , \quad f \le 0 \quad , \quad \Lambda f = 0$​

This is done as follows (Ref. 4):
1- Elastic-predictor: Try the elastic solution $M= M_{old}$ and $\Lambda=0$. If this satisfies $f \le 0$ it is done.
2- Plastic-corrector: If the elastic solution does not work (this is $f > 0$), solve the nonlinear system consisting of the flow rule and the equation $f=0$ using a damped Newton method.

Few things I don't understand or I'm not sure of :

1/ Gauss points stand for the mesh nodes, right ?
2/ What does "For each Gauss point the plastic strain variables are computed" mean ? We have matrices that describe the body state at all his points, that is, there is not a matrix for each point of the body, so I don't understand the beginning of their sentence.
3/ The total deformation gradient $F$ is also a variable, we don't know its value, so how the computing is done for $M$

Regards

 

[trabo]

For the third point, does F depend on the boundary conditions ? For instance, if we consider a round tensile specimen with prescribed displacement at the top and below surface and with free stress on the lateral surface, can we infer F ?
Sure, we know its impqct on points at the top and below surface, but can we infer all its components ? If the total displacement vector is known at the two outer surfaces, then yes we can know all components of F, but if only the radial displacement is prescribed, some components can not be determined

 

[6Stang7]

Few things I don't understand or I'm not sure of :

1/ Gauss points stand for the mesh nodes, right ?
2/ What does "For each Gauss point the plastic strain variables are computed" mean ? We have matrices that describe the body state at all his points, that is, there is not a matrix for each point of the body, so I don't understand the beginning of their sentence.
3/ The total deformation gradient $F$ is also a variable, we don't know its value, so how the computing is done for $M$

Regards

No, the Gauss points are no the mesh nodes. The Gauss Points, or Gaussian Integration Points, are internal to the elements. More can be found here (last paragraph of the page marked 11) and http://www.colorado.edu/engineering/cas/courses.d/IFEM.d/IFEM.Ch28.d/IFEM.Ch28.pdf.