Contrib:KeesWouters/solids/mooney-rivlin
Contents
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. A block of 5x12x43 [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).
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 [MPa] on the top area in 20 steps.