Difference between revisions of "Contrib:KeesWouters/solids/mooney-rivlin"

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The bulk modulus K is defined in terms of Youngs modulus en poisson ratio: K = E/(3*(1-2*nu)). For nu approaches 0.5, K approaches to infinity, ie incompressible material.
 
The bulk modulus K is defined in terms of Youngs modulus en poisson ratio: K = E/(3*(1-2*nu)). For nu approaches 0.5, K approaches to infinity, ie incompressible material.
  
==='''Results'''===
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==''Results''==
  
 
We applying a pressure load of 6 [Pa] on the top area in 20 steps.<br/>
 
We applying a pressure load of 6 [Pa] on the top area in 20 steps.<br/>

Revision as of 22:25, 30 September 2011

Material behaviour Mooney-Rivlin

Simple block of Mooney-Rivlin material

september 2011 - Salome 6.3 - Code Aster VERSION DE DEVELOPPEMENT 11.00.10 - Ubuntu 11.04

This is a simple example of a block of Mooney-Rivlin material. As usual, the input and partly the output files can be found at the end of this contribution.
Errors are all mine.
Creative criticism welcome (of course I rather have positive than negative remarks).
Have fun.

Geometry

This is really annoying. The block has dimensions 2Xmax*2Ymax*Lz = 5x12x43 [mm3] (Xmax = 2.5, Ymax = 6, Lz = 43 [mm]). My only excuse is that it does show the behaviour of the material.
Four groups are defined on the block: two areas, top and bottom areas, and two node groups: centre node of the bottom plane and two nodes on the extreme positon of the y-axis. The are used to define boundary conditions and pressure (load).

Kw mooneyrivlin block.png : Kw mooneyrivlin groups.png

Loads and boundary conditions
A pressure of 6 [MPa] is applied to the top area Atop.
The boundary conditions are applied to the bottom area: all z displacements of the bottom area are restricted.
On node Nfixx the displacement in y direction is restricted.
On nodes Nfixy (two nodes on the extremes of the x axis) the displacement in x direction is restricted.

Material properties

The material properties are defined by Mooney-Rivlin.
This material behaviour is defined by a number of parameters:

  • C01 = 2.3456;
  • C10 = 0.709;
  • C20 = 0.0;
  • NU = 0.499
  • K = (6*(C10+C01))/(3*(1-2*NU))

and in Code Aster called by the hyper elastic material module:

RUBBER1=DEFI_MATERIAU(ELAS_HYPER=_F(C10=C10,
                                  C01=C01,
                                  C20=C20,
                                  K=K,
                                  RHO=1000.0),);

The parameters Cxy are coefficients of the two invariants of the Strain energy function, see eg [wiki].
For small strains the shear modulus G can be expressed as twice the sum of C01 and C10: G = 2(C01 + C10). And Youngs modulus E is equal to E = 2G(1+nu). So we have E = 4(C01+C10)(1+nu). For incompressible material the possion ratio nu --> 0.5. Hence we use nearly incompressible material here using nu = 0.499. Note that the numerical value of the Youngs modulus is E = 18.3 [MPa] for the values above. Later on we will use this to verify the results.
The bulk modulus K is defined in terms of Youngs modulus en poisson ratio: K = E/(3*(1-2*nu)). For nu approaches 0.5, K approaches to infinity, ie incompressible material.

Results

We applying a pressure load of 6 [Pa] on the top area in 20 steps.
The picture below shows the displacements in x, y and z direction. The displacements are scaled to 1.

Kw mooneyrivlin displacement.png

In this figure the displacement in each direction are given.The maximum values are:

  • dux = 0.63600 (both directions)
  • duy = 1.52641 (both directions)
  • duz = 43.441 (only positive direction)
Kw mooneyrivlin dxdydz.png

Volume change

Since we put the poisson ratio nearly equal to 1/2 (nu ~ 0.5) we expect that the volume does not change. The undeformed state has a volume Vo = 2Xmax * 2Ymax * Lz = 2580 [mm3]. The deformed state has a volume of Vdef = 2(Xmax-dux)*2(Ymax-duy)*(Lz+duz) = 2583.1 [mm3], a change of 3.1 [mm3]. Taking into account the effect of the poisson ratio (nu=0.499) the volume change may be dV = Vo*eps*(1-2NU). Depending on the definition of eps this change of volume is 4.1 [mm3] (Note that maybe we have to take more digits of the displacements into account ...).

Initial stiffness

The small strain axial stiffness of the block is duz/Fz = duz/PA = Lz/AE. This yields for small strains duz = P*Lz/E. The pressure P is 6 [Pa], the length of the block in z direction is 43 [mm] and the Youngs modulus E = 6(C01+C10)*(1+nu) = 14.09 [mm]. The bottom figure shows that at 0.6 [MPa] (also calculated using 20 steps) the displacement is 1.496 [mm], slightly more than the predicted 14.09/10 ~ 1.41 [mm]

Kw mooneyrivlin topplane.png
Kw mooneyrivlin displacement p6pa.png