Mooney-Rivlin Model - LS-DYNA MAT_027

In this rubber model, the strain energy density is defined as the function of right Cauchy-Green tensor C=FTF, where F is the deformation gradient.

W=A(I13)+B(I23)+C(I321)+D(I31)2

where A, B are two material constants, and

C=0.5A+BD=A(5ν2)+B(11ν5)2(12ν)ν=Poisson's ratio

I1, I2, and I3 are invariants of C.

I1=λ12+λ22+λ32I2=λ12λ22+λ22λ32+λ32λ12I3=detC=λ12λ22λ32=J2

λ1,λ2,λ3 are the stretch ratio in principal directions.


Parameter Identification

There are two constants in this model. So we can identify them if knowing two test points in uniaxial tension. To get a useful equation, the principal stresses can be derived from strain energy function by the partial derivative as

σ1=λ1Wλ1=λ1(WI1I1λ1+WI2I2λ1+WI3I3λ1)=2λ12A+2λ12(λ22+λ32)B4I32C+4I3(I31)D

σ3=λ3Wλ3=λ3(WI1I1λ3+WI2I2λ3+WI3I3λ3)=2λ32A+2λ32(λ12+λ22)B4I32C+4I3(I31)D

To remove the Jacobian terms, σ1 minus σ3 is :

σ1σ3=2(λ12λ32)A+2(λ12λ22+λ22λ32)B

Since σ3=0,σEng=σ1λ1, λ1=λ, and λ2=λ3=1λ in uniaxial tension by assuming incompressible, this equation is reduced :

σEng=2A(λ1λ2)+2B(11λ3)

Now using this equation, if we know the two engineering stresses at the each stretch ratio, the two constants, A and B can be determined by performing least square fit to the two points.

For example, a material description is obtained like the following table, then you know the engineering stresses, σEng at 100% and 380%, break elongation respectively.

Tensile stress at 100% 2.06 MPa
Tensile stress at break 4.10 MPa
Elongation at break 380%

Using stretch ratio definition, λ=Lo+ΔLLo=1+ϵEng. Therefore the only unknowns are A and B with two equations. A and B can be determined for the curve to fit the two points as :

A=0.318,B=0.540

Engineering stress-strain curve can be predicted from above parameters in uniaxial tension as below:



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