Some question about framework of J2 model in feappv

[wjs]

Dear Prof. Taylor/ FEAP admin/users of feappv:
    I recently read the book you recommended,maybe it was a long time ago. And I also re-read the relevant code in feappv and the updated content in FEAP in its official website .There are some questions Iwant to ask for help:
    1. When we use FEM to discrete the space of Elastoplastic IBVP,then,it comes to time discretization and we use a incremental precedure.
        So the numerical method of this incremental process can be Eular method(forward) and N-R method and our book use N-R method.
         Is this my idea right?
    2. In N-R method, it seems do not accumulate displacement values of each iteration in one time step in the book ,which gives :d(k)(n+1)=d(n)+delta_d(k)(n+1),
        but should delta_d(n+1)=sum(delta_d(1)(n+1), delta_d(2)(n+1), delta_d(3)(n+1),...delta_d(k)(n+1)) be right？
        That really confuse me.
    3.  Assume that the numerical method of this incremental process above can be Eular method(forward),  so , in my idea within the scope of my knowledge, the specific process is as follows：
         use the “continuum” elastoplastic tangent Cep(sigma(n)) in each time step,and solute the equation of Cep(sigma(n))*delta_u=delta_F(load).
         use the “continuum” elastoplastic tangent Cep(sigma(n)) to calculate the sigma（n+1) = Cep(sigma(n)):delta_strain       ,      or use reture-mapping
         no iteration and go next time step
         Is something wrong above?How to evaluate this method?
    4. If material parameters vary with temperature like intial yield stress, E and so on, without thermal coupled,will the framework of solution of IBVP be changed?
        Like time discretization' method , the integration of the consitutive equation with temperature and so on. Like the yield stress will decrease when temperature increases and thus N-R method will not be suitable?
        And will the return-mapping method be changed by introduces a partial derivative of temperature or is there some other method?
        Sincerely hope for your help.
         


       

[Prof. R.L. Taylor]

1. Typically we do not use the Euler forward (explicit) method as it is conditionally stable and you need to continually assess the stability limitation -- which for many problems is very small compared to the backward (implicit) Euler method. 

2. Depends on how you do the integration.  In feap/feappv we store the value at t_n (last converged step) and also at t_n+1 (current solution step).  If you always start with the t_n step then you are doing the current solution from the last converged point and compute the full increment.  If you start from t_n+1 data you are only doing the solution from where the last iteration was.  In return map you always are solving the equation from t_n to t_n+1 though.

3. If you were to use the Euler forward, then if the iteration for the current solution is the one from the last step, then the continuum and return map consistent tangent should be the same.  That is what consistency requires.

4. For coupled algorithms the constitution will depend on how you set up the problem -- you need to study different papers to see which form you like.  I am not familiar with any except the one by Simo and Armero which was a split algorithm.

[wjs]

Dear Prof. Taylor：
    Thank you very much for your reply. I am sorry that my expression of the question may not be clear enough.
    For point 1 and 3, the numerical method is not for the local integration of the consitutive equation,but for the time discretization(Solving Nonlinear Equations). What I mean is as follows:
    When we do the  time discretization of the IBVP, a method of iteration is given in the book and the code. One of the most important is how to determine the delta_u(incremental deformation). And it is clear that the it is a nonlinear equation for the displacement increment.
    As I know we can use the two methods to solve Nonlinear Equations, one is N-R method in the book,the other one is Eular Forward method. What confuse me is that:
    I. Using Eular forward method to solve Nonlinear Equations has two main process ,one is solving the Cep(sigma(n))*delta_u=delta_F(load) to determine delta_u and no iteration in each tiome step. The other is update the stress.So how to determine the Cep in each time step? I think it is the  “continuum” elastoplastic tangent Cep(sigma(n)),right?
    II. How to update the stress in Eular forward method' second process. I think we can use sigma（n+1) = Cep(sigma(n)):delta_strain + sigma(n),where the Cep is the  “continuum” elastoplastic tangent at sigma(n). We can also use the return-mapping,but just once in each time step, no iteration, sounds like a hybridization method.  Is this my idea right?
    Follow my understanding，the solution of Elastoplastic IBVP can be divide in two process, one is the solution of  Nonlinear Equations, the other is the local integration of the consitutive equation, and they have very close relationship.
    I am just a tiro and my understanding may be wrong. So I hope you can correct anything wrong above.
    Thank you very much again.

[Prof. R.L. Taylor]

When you use an explicit method the increment to the displacements for the step is known.  The explicit method solves for the acceleration using the momentum equation.

Thus, in feap it is necessary for plasticity to set the material parameter to start the solution using an inelastic step -- by default our plasticit starts with an elastic predictor -- such that the return map gives the correct stress increment for the known incremental strain.

We never use the continuum tangent.

End of discussion

[wjs]

    Thank you very much!