# Elastic and Plastic deformation gradients in AB Viscoplastic model

Dr. Jorgen,

I am trying to write a VUMAT to implement the Arruda-Boyce viscoplastic model. After assuming linear elasticity at the beginning of the analysis, the total deformation gradient (**F**) can be easily obtained from defgradNew.

I am a little bit confused about how to decompose **F **into its elastic and plastic components. Should I obtain the plastic components from the state variables and then, of course, find the elastic part via the product of **F** and the inverse of the plastic part?

How about the plastic flow rate, the rate of the plastic deformation, time-derivative of the deformation gradient, and the strain softening. How to incorporate these to capture the material response?

My reference is Chapter 8 of your book (Mechanics of Solid Polymers).

Thank you very much,

Warm regards,

Melly

Good question. You need to save either the elastic or the plastic components of **F** as state variables. I typically save **Fp**. Then you need to write an equation for the time-derivative of **Fp**, and calculate **Fp** at time t+dt. Once you have **Fp** at t+dt, you can calculate **Fe** at t+dt, which you can use to calculate the new **stress** at t+dt.

Best of luck,

Jorgen

I appreciate. Let me try and figure it out.

The time derivative of **Fp **is given in the book **(Equation 8.38). **I believe the deviatoric part of stress driving the plastic flow required to get the rate of plastic deformation is calculated from the **Fe **at time **t**. I also believe that I don't have to explicitly state the time increment **dt **as it is taken care of by Abaqus.

Suppose the **Fp ** (t+dt) = **Fp**(t) + dt**Fp**(time-derivative), How about the total **F**(t+dt) so that I can get **Fe**(t+dt).

Thank you very much.

Warm regards,

Melly.

Dr. Jorgen,

Many many thanks. I realized that the total **F** at the beginning and at the end of the time step is given by defgradOld and defgradNew, respectively.

Regards,

Melly.

Dr. Jorgen,

Once again, I appreciate your guidance. It means everything to my research.

I have written the code for AB viscoplastic model. I must have missed a few important details as there is an excessive distortion of elements.

The only part of the code that I borrowed is the one that utilizes Jacobi method for eigen values and vectors (In order to calculate the Hencky strain).

If you have a minute, kindly have a look at my code (attached) and (if possible) give me some hints on where I could be wrong.

Warm regards,

Melly

At first sight, your code looks reasonable. Unfortunately it is very time consuming to carefully review code, and my schedule is too busy to spend hours reviewing the code. I'm happy to answer any questions, however 😀

-Jorgen

I understand. Thank you.

I have narrowed down the problem. Both the **F** and **Fp **turn out to be identity matrices leading to zero Hencky strains and zero Cauchy stresses. Eventually, the plastic deformation rate **Dp** and the time rate change of **Fp(dot)** including the plastic deformation gradient at **t+dt** turns out to be **NaN**.

I am probably missing out on how to properly define the total deformation gradient at the beginning and end of the increment. Also on the plastic deformation gradient at the beginning of the increment.

Importantly, how to define the beginning of plastic deformation? using the **totalTime** or **stepTime**.

Many thanks,

Melly

I'm not quite sure I understand. The plastic deformation at the start of an increment needs to be obtained from the state variables. Similarly, you need to save the new plastic deformation at the end of an increment as the state variables.

-Jorgen

Many thanks.

My code is now working perfectly. I made numerous mistakes in the previous code that I posted to you. For instance, I was saving the plastic flow stress instead of the Total(Cauchy) stress as the new stress at the end of the increment.

Now, I will be comparing my simulation results with some published experimental work to check the accuracy of the code.

Soon, I will be inquiring about the MCalibration software.

I appreciate your guidance.

Warm regards,

Melly

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