Simulation of equi-biaxial tension with neo-Hookean material

[zhan.peng]

Dear Prof. R.L. Taylor

    I am trying to simulating equi-biaxial tension with neo-Hookean material with FEAPpv software. The problem is described as follow:

    There is a cuboid with a dimension of 20*1*20 mm. i fixed the 3rd-dire DOF of bottom face, and 1st-dire DOF of left face, and also 2nd-dire DOF of the origin. The cuboid was extended both at right face and top face to 160 mm. The material prop of this cuboid is neo-Hookean hyperelastic, the effective Young's modulus and Poisson's ratio at small strain in the reference configuration was 1.2749575, 0.49995 (to simulate a fully incompressible hyperelastic approximately), respectively. For the solution algorithm, i use Newton iteration method with the command of

loop,iter,100
  tang
  form,converge
  solve,line,0.3
next,iter

    The problem didn't converge at the last iteration. Did i use wrong solution command?

Thanks in advance!
Zhan Peng

[Prof. R.L. Taylor]

If you use a proportional load and increment the displacements with 16 steps it works for one element and using FEAPpv. However, FEAPpv does not have a special element formulation that will allow for solutions with non-constant spatial stresses.  Also if you use many elements it may not converge.

[Prof. R.L. Taylor]

On closer inspection I noted the first time step did not converge.  I did not try many schemes, but setting dt = 0.001 and doing 10 steps, then dt = 0.01 and 9 steps and then dt = 0.1 and 9 steps and finally dt=1 and 9 steps did work with both FEAP and FEAPpv for the displacement element.

[zhan.peng]

Dear Prof. R.L. Taylor

    Thank you very much for your patient reply. i remember for the nearly incompressible hyperelastic material, if the Poisson ratio is greater than 0.495, that is the ratio of initial bulk modulus to initial shear modulus is greater 100, maybe it is better for us to use a mixed formulation (refer to Abaqus documentation). But in the current version of FEAPpv, for the 3D 8-node brick element, only SMALL STRAIN and DISPLACEMENT model is available. But for the 2D FINITE deformation, the MIXED model is available. So in order to verify my thought, i did the 2D equi-axial tension tension test, the input file is given as follows (the unit is mm, mpa, N):

feappv ** neo-Hookean 2D: equibiaxial tension
   4 1 1 2 2 4
   
coord
   1 0 0. 0.
   2 0 20. 0.
   3 0 20. 20.
   4 0 0. 20.
   
mate,1
   solid
      plane stress
      finite
      mixed
         elastic neohookean 1.2749575 0.49995
         
elem
   1 0 1 1 2 3 4
   
bound
   1 0 1 1
   2 0 1 1
   4 0 1 1
   3 0 1 1
   
disp
   2 0 160. 0.
   3 0 160. 160.
   4 0 0 160.
   
end

batch
   loop, iter, 100
      tang,,1
   next, iter
   stress,all
end

inter

     but the result of the stress is 231mpa, which is much bigger than the analytical result of 34.1mpa.

     Thanks in advance.

Zhan Peng

[Prof. R.L. Taylor]

You need to check your analytical solution.  You have a very large volumetric deformation in a nearly incompressible material.  If it was fully incompressible the solution would be infinite.  Remember the strain in the thickness direction is zero, so there is a big thickness stress too.  Look carefully at the stresses from the finite element solution.

[zhan.peng]

Dear Prof. R.L. Taylor

   Thank you very much for your kindly reply. I have read your reply, but i have some questions:

• If the material is fully incompressible, then volume stays unchanged, since for the incompressible constraint dV/dV0=J=lambda1*lambda2*lambda3=1, Does this mean the volumetric deformation is small?
• Since this is a plane stress condition, so the stress in the thickness direction is zero, am I right? (but the stress for the 33-direction of the FEAPpv output is not zero, here is the output result of PEAPpv)
  Elmt  Matl  11-Stress  22-Stress  33-Stress  12-Stress  23-Stress  13-Stress
                 1-Stress   2-Stress      Angle  log(lam1)  log(lam2)  log(lam3)
                  1-Coord    2-Coord    3-Coord    log-J     eff-ep
       1     1  2.310E+02  2.310E+02  2.305E+02  0.000E+00  0.000E+00  0.000E+00
                2.310E+02  2.310E+02  4.500E+01  2.197E+00  2.197E+00  0.000E+00
                 90.00000   90.00000    0.00000  4.394E+00  0.000E+00
• Here is the analytical calculation i did:
  Since the displacement load in the 1- and 2- direction is same, then lambda=lambda1=lambda2 and sigma=sigma1=sigma2, where lambda1 and lambda2 are the stretch ratio in the 1- and 2- direction, and sigma1 and sigma2 are the stresses in the 1- and 2- direction.
  For the plane stress condition, sigma3=0. For the fully incompressible constraint, J=lambda1*lambda2*lambda3=1, then lambda3=1/(lambda^2).
  As for the neo-Hookean hyperelastic material, the three principal Cauchy stresses can be calculated as:
  sigma_i = -p+mu*lambda_i^2, i=1,2,3.
  where mu is the initial shear modulus, and p is the hydrostatic pressure which can be determined from sigma3=0.
  Finally the stresses in the 1- and 2- direction can be calculated as
  sigma=sigma1=sigma2=mu*(lambda^2-lambda^(-4)).
  When the stretch ratio is lambda=lambda1=lambda2=9, then the stress is sigma=sigma1=sigma2=34.42mpa.
  

     Hope to get your reply soon.

     Thanks in advance

Zhan Peng

[Prof. R.L. Taylor]

Feap does not do plane stress in finite deformation, only feap.  So you get plane strain solution

[FEAP_Admin]

I think you meant to say FEAPpv does not do finite deformation plane stress.  (Probably we should print an warning, if someone tried to do this).

[Prof. R.L. Taylor]

Yes, FEAPpv does not do plane stress in finite deformation.  At present no warning is printed.

[zhan.peng]

Dear Prof. R.L. Taylor and FEAP_Admin,

    Thank for very much for your patience on my questions! In FEAPpv,since for the problem with large displacement, hyperelastic material model, and mixed method, 2D plane strain element is the only available choice, then i tried 2D uniaxial tension with plane strain, neo-Hookean hyperelastic material model. Here is the input file:

feappv ** neo-Hookean 2D: uniaxial tension
   4 1 1 2 2 4
   
coord
   1 0 0. 0.
   2 0 0.01 0.
   3 0 0.01 0.01
   4 0 0. 0.01
   
mate,1
   solid
      plane strain
      finite
      mixed
         elastic neohookean 1.2749575 0.49995
         
elem
   1 0 1 1 2 3 4
   
bound
   1 0 1 1
   2 0 1 0
   4 0 1 0
   3 0 1 0
   
disp
   2 0 0.08 0.
   3 0 0.08 0.
   
end

batch
   loop, iter, 50
      tang,,1
   next, iter
   stress,all
end

inter

    But the solution didnot converge. *WARNING* Check for NO CONVERGENCE at time =    0.00000E+00

    Hope to receive your reply soon. Thanks in advance.

Zhan Peng

[FEAP_Admin]

800% deformation in one load step can be a bit challenging.
Try prop,,1 with default values, dt,,0.1, then
loop,,10
  time
  loop,,30
    tang,,1
  next
next

[zhan.peng]

Dear Prof. FEAP_Admin,

   Thank you very much for your detailed and precise suggestion for the solution command. With the help of your solution command, i finally obtained the correct results of the problem of 2D plane strain, uniaxial tension, large deformation, nearly incompressible hyperelastic material properties by using 2D solid element and mixed formulation method.

    Thank you very much again!

Zhan Peng