# FE method, how ? (software Comsol)

1. Jun 23, 2014

### 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 :

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

Last edited: Jun 23, 2014
2. Jun 23, 2014

### 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

Last edited: Jun 23, 2014
3. Jun 23, 2014

### 6Stang7

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 here.