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

[itex]-\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}[/itex]where [itex]F_p[/itex] is the plastic part of the total deformation gradient [itex]F[/itex], [itex]f[/itex] the yield surface and [itex]\tau[/itex] the Kirchhoff stress tensor.

According to Comsol notes, we denote [itex]M=F_p^{-1}[/itex] and discretize the law by using 'variations with [itex]\Delta[/itex]' :[itex]-\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 [/itex]but I don't understand what is stated after. I quote :

Few things I don't understand or I'm not sure of : For each Gauss point, the plastic state variables (ε_p and M, respectively) and the plastic multiplier,[itex]\Lambda =\lambda \Delta t[/itex], are computed by solving the above time-discretized flow rule together with the complementarity conditions[itex]\Lambda \ge 0 \quad , \quad f \le 0 \quad , \quad \Lambda f = 0[/itex] This is done as follows (Ref. 4):

1- Elastic-predictor: Try the elastic solution [itex]M= M_{old}[/itex] and [itex]\Lambda=0[/itex]. If this satisfies [itex] f \le 0 [/itex] it is done.

2- Plastic-corrector: If the elastic solution does not work (this is [itex] f > 0 [/itex]), solve the nonlinear system consisting of the flow rule and the equation [itex] f=0 [/itex] using a damped Newton method.

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 [itex]F[/itex] is also a variable, we don't know its value, so how the computing is done for [itex]M[/itex]

Regards

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# FE method, how ? (software Comsol)

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