Hyperelastic model para

QinHe_321

https://help.altair.com/hwsolvers/os/topics/solvers/os/mathe_bulk_r.htm

How's I and J calculated?

Thanks

Qin

Roger DeAngelis_21630

Based on the generalized polynomial form (Mooney model) shown in your image, here is how I and J are calculated or defined:

Strain Invariants (I₁, I₂)

The terms Iˉ1Iˉ1​ and Iˉ2Iˉ2​ in the equation refer to the deviatoric (distortional) strain invariants.

In continuum mechanics, the strain invariants are derived from the right Cauchy-Green deformation tensor C.

1. Standard Invariants:
   • I1=tr(C)=λ12+λ22+λ32I1​=tr(C)=λ12​+λ22​+λ32​
   • I2=12[(trC)2−tr(C2)]=λ12λ22+λ22λ32+λ32λ12I2​=21​[(trC)2−tr(C2)]=λ12​λ22​+λ22​λ32​+λ32​λ12​
   • I3=det⁡(C)=λ12λ22λ32=J2I3​=det(C)=λ12​λ22​λ32​=J2
2. Deviatoric Invariants (Iˉ1,Iˉ2Iˉ1​,Iˉ2​):
   Since the model separates distortional (shape change) and volumetric (volume change) behavior, it uses the "deviatoric" parts of the invariants. These are obtained by multiplying the principal stretches (λiλi​) by J−1/3J−1/3 to remove the volume change.
   • λˉi=J−1/3λiλˉi​=J−1/3λi​
   • Consequently:
     • Iˉ1=J−2/3I1Iˉ1​=J−2/3I1​
     • Iˉ2=J−4/3I2Iˉ2​=J−4/3I2​

Jacobian (J)

The term JJ (referred to as JdasJdas​ in your text, likely a typo or specific variable name for "Jacobian Determinant" or volume ratio) represents the volume ratio.

• Definition: J=det⁡(F)J=det(F), where F is the deformation gradient tensor.
• Physical Meaning: It is the ratio of the deformed volume over the reference (initial) volume.
  • J=VV0J=V0​V​
• In terms of stretches: J=λ1λ2λ3J=λ1​λ2​λ3​
• In the equation: It is used in the volumetric strain energy term (Jdas−1)2p(Jdas​−1)2p. The (J−1)(J−1) term represents the volumetric strain (change in volume).

Internal Calculation (OptiStruct)

As noted in your image description:

• J11,J22J11​,J22​ (likely a mislabeling of I1,I2I1​,I2​ or specific solver notation) are calculated internally by OptiStruct based on the nodal displacements and element formulation.
• The solver takes the displacements, computes the deformation gradient F at integration points, derives the principal stretches, and then computes I1,I2,I1​,I2​, and JJ.