ANSYS problem

							                                  ANSYS problem
Situation:

I have a grid of little pines that can only move vertically. Those pines are controlled to
create a desired shape or can be moved by user’s hand to display a desired shape. In
order to improve the appearance of the surface, I put a hyper elastic covering membrane.

My purpose is to determine the influence of the membrane over the underneath pines.
The idea is to create an analytical model considering that pines are linked by non linear
springs. Thus, if we can determine the non linear spring characteristic curve or a
polynomial fitting curve, and knowing the elongations of the non linear springs between
each pines thanks to their positions, we could get the force applied on each pine due to
our “analytical membrane”.

To determine the non linear spring characteristic curve, I use a finite element code,
ANSYS, to run non linear analysis and get the reaction forces to a certain load. With
ANSYS results, I can compute the forces in each non linear spring that links 2
neighboring pines.

My problem with ANSYS is that, with a hyper elastic material, the membrane should be
able to be largely stretched. And with my set, I can only stretched it out of its initial flat
plane by 30%.

ANSYS settings:

I am currently using ANSYS 6.0 on a Windows 2000 platform.

The piece of membrane that I consider for the moment is a hexagon with edges of 3 mm.
There is a pine at each submit of the hexagon and an additional one in the center of the
hexagon. Membrane is supposed to be attached to the pines. Thus I just create a model
of the membrane and defined zones where the pines are supposed to be attached. For the
central pine, its zone of attachment is a disc and for the other, it is the piece of disc that it
included in the hexagon, the discs are centered at each submit. Discs have a diameter of
1.5 mm.

As my membrane is hyper elastic, I use the non linear mechanics of material model:
Mooney-Rivlin. As I could not find some known values for the constants of that model, I
just used the fact that they are related to the Young Modulus E as follows:

6*(C1+C2) = E

and I fix the quotient C1/C2.
Material parameters:

Young Modulus: E = 107 Mpa
Poisson’s ratio: ν = 0.49
Mooney-Rivlin constants quotient: C2/C1 = varying

Meshing:

Element: Shell181 with a thickness of 1 mm. This shell element allows large strain. I
have not yet used a 3-D hyper elastic element as Hyper58 or Hyper86.
I use triangular element because the circular zones of attachment are better meshed like
this. Otherwise, quadratic elements are too stretched at some places, even in the initial
configuration, and I have an error message.
I tried different meshing sizes: 1/5, 1/10, 1/20 of the reference length 3 mm.

Solving settings:

Due to the non linearity of the analysis, I define the following parameters.
Number of substeps: 40, max = 100, min = 26                    (nsubst,40,100,25)
Number of iterations at each substep: 20                       (nquit,20)
Large deformation allowed                                      (nlgeom,on)
Optimized nonlinear solution                                   (!solcontrol,on)
Convergence criterion                                          (cnvtol,f,,,,0.001)

Load configuration:

For the moment, I fix all the attachment areas around the hexagon to zero, UX, UY and
UZ are fixed to zero. And I rise the central attachment area. I tried different values, and
the maximum value I can reach is 1 mm before divergence of the non linear analysis.

Analysis:

I ran several analysis to find the influence of my different parameters.

For the Mooney-Rivlin constants quotient, I tried some values from 0 to 1.
With a meshing of 1/10, the solving settings in brackets and a vertical displacement of 1
mm for the central attachment area, it converges for a quotient less or equal to 1/3.

The changes in meshing size are not very significant for the same settings. I get almost
the same results.
When I increase the meshing to 1/20, I can reach a vertical displacement of 1.1mm.

I cannot reach a higher vertical displacement when I increase the number of substeps.
I can get almost the same results when I use a different convergence criterion:
displacement instead of force. If I want to use moment criterion, I have to increase the
minimum reference number from 0.001 to 0.005.

If I change the optimized nonlinear solution (solcontrol), it still converges and I get
almost the same results.

I have to use large deformation with the Mooney-Rivlin model.

Questions:

First, I do not know how to find the values of the Mooney-Rivlin constants for hyper
elastic materials as natural rubber. I tried to search in several material references but they
never give values and always refer to experimental tests that are the only way to get
them.
Would you know where I could find those Mooney-Rivlin constants values? Or if it not
directly, is there a book that gathers experimental results of different mechanical tests run
on elastomer material as natural rubber?

Second and most important, why I cannot reach higher vertical displacement of the
central attachment area? Why it diverges above 1 mm? Can I get higher displacement?

Appendix: my text program

/filname,test,1

/prep7
/title,hyper elastic membrane

S=0.003                                ! hexagonal edge length
ES1=S/10                               ! meshing size for the attachment areas
ES2=S/10                               ! meshing size for the rest of the model
E=1E7                                  ! Young Modulus
RAB=1/3                                ! Mooney-Rivlin constants quotient
A=E/(6*(1+RAB))                        ! first M-R constant
B=RAB*A                                ! second M-R constant
cS=(sqrt(3)/2)*S
sS=0.5*S
M=(sqrt(3)/2)*S*0.5
N=(sqrt(3)/2)*S*(sqrt(3)/2)


et,1,shell181                          ! element
r,1,0.001                              ! element thickness

k,1,,-S
k,2,cS,-sS
k,3,cS,sS
k,4,,S
k,5,-cS,sS
k,6,-cS,-sS
k,7,,


l,1,2
l,2,3
l,3,4
l,4,5
l,5,6
l,6,1

a,1,2,3,4,5,6

R=S/4

circle,1,R,,2,120
circle,2,R,,3,120
circle,3,R,,4,120
circle,4,R,,5,120
circle,5,R,,6,120
circle,6,R,,1,120
circle,7,R

l,1,8
l,13,2
l,2,11
l,16,3
l,3,14
l,19,4
l,4,17
l,22,5
l,5,20
l,25,6
l,6,23
l,10,1

al,23,7,8,34
al,24,25,9,10
al,26,27,11,12
al,28,29,13,14
al,30,31,15,16
al,32,33,17,18
al,19,20,21,22

asba,1,2,,keep,keep
asba,9,3,,delete,keep
asba,10,4,,delete,keep
asba,9,5,,delete,keep
asba,10,6,,delete,keep
asba,9,7,,delete,keep
asba,10,8,,delete,keep

mshape,1                 ! triangular meshing
esize,ES1                ! meshing of the attachment areas
amesh,2,8

esize,ES2                ! meshing of the rest of the model
amesh,9

mp,nuxy,1,0.49           ! Poisson’s ratio
tb,mooney,1
tbdata,1,A,B

nsubst,40,100,25
nlgeom,on

da,2,uz,0                ! attachment areas fixed to zero
da,3,uz,0
da,4,uz,0
da,5,uz,0
da,6,uz,0
da,7,uz,0

da,2,ux,0
da,3,ux,0
da,4,ux,0
da,5,ux,0
da,6,ux,0
da,7,ux,0
da,8,ux,0

da,2,uy,0
da,3,uy,0
da,4,uy,0
da,5,uy,0
da,6,uy,0
da,7,uy,0
da,8,uy,0
da,8,uz,0.001          ! vertical displacement of the central attachment area

neqit,20

nsel,all
!nsel,s,node,,91,117

finish

/solution
nsel,all
outres,all,all
!solcontrol,on
cnvtol,f,,,,0.001             ! convergence criterion
solve
/out
finish