Isotropic Elastic Material
- Linear elastic materials with Hooke’s law,
- Nonlinear elastic materials with Ogden, Mooney-Rivlin and Arruda-Boyce laws.
Linear Elastic Material (LAW1)
This material law is used to model purely elastic materials, or materials that remain in the elastic range. The Hooke's law requires only two values to be defined; the Young's or elastic modulus EE, and Poisson's ratio, υυ. The law represents a linear relation between stress and strain.
Ogden Materials (LAW42, LAW69 and LAW82)
Where, λ1λ1, ith principal stretch
λi=1+εiλi=1+εi, with εiεi being the ith principal engineering strain
αpαp and μpμp are the material constants.
pp is order of Ogden model and defines the number of coefficients pairs (αp,μp)(αp,μp).
- If pp=1, then one pair (α1,μ1)(α1,μ1) of material constants is needed andin this case if α1=2α1=2 then it becomes a Neo-hookean material model.
- If pp=2 then two pairs (α1,μ1),(α2,μ2)(α1,μ1),(α2,μ2) of material constants are needed and in this case if α1=2α1=2 and α1=-2α1=−2 then it becomes a Mooney-Rivlin material model
With:
ˉW(ˉλ1,ˉλ2,ˉλ3)=∑pμpαp(ˉλ1αp+ˉλ2αp+ˉλ3αp−3)U(J)=K2(J−1)2
Since λ1 ∂J∂λi=J and ∂ˉλj∂λi=23J−13 for i=j and ∂ˉλj∂λi=13J−13λjλi for i≠j
- The rubber is incompressible and isotropic in unstrained state
- The strain energy expression depends on the invariants of Cauchy tensor
- μ2=−2⋅C01
- α1=2
- α2=−2
The model can be generalized for a compressible material.
Viscous Effects in LAW42
τi are relaxation times: τi=ηiGi
Rate effects are modeled through visco-elasticity using convolution integral using Prony series. This corresponds to extension of small deformation theory to finite deformation.
This viscous stress is added to the elastic one.
- M
- Order of the Maxwell model
- F
- Deformation gradient matrix
- ˉF=J−13F
- dev(ˉFˉFT)
- Denotes the deviatoric part of tensor ˉFˉFT
LAW69, Ogden Material Law (Using Test Data as Input)
This law, like /MAT/LAW42 (OGDEN) defines a hyperelastic and incompressible material specified using the Ogden or Mooney-Rivlin material models. Unlike LAW42 where the material parameters are input this law computes the material parameters from an input engineering stress-strain curve from a uniaxial tension and compression tests. This material can be used with shell and solid elements.
- law_ID =1, Ogden law (Same as LAW42):
- W(λ1,λ2,λ3)=5∑p=1μpαp(ˉλ1αp+ˉλ2αp+ˉλ3αp−3)+K2(J−1)2
- law_ID =2, Mooney-Rivlin law
- W=C10(I1−3)+C01(I2−3)
After reading the stress-strain curve (fct_ID1), Radioss calculates the corresponding material parameter pairs using a nonlinear least-square fitting algorithm. For classic Ogden law, (law_ID =1), the calculated material parameter pairs are μp and αp where the value of p is defined via the N_pair input. The maximum value of N_pair = 5 with a default value of 2.
For the Mooney-Rivlin law (law_ID =2), the material parameter C10 and C01 are calculated remembering that μp and αp for the LAW42 Ogden law can be calculated using this conversion.
μ1=2⋅C10, μ2=−2⋅C01, α1=2 and α2=−2.
The minimum test data input should be a uniaxial tension engineering stress strain curve. If uniaxial compression data is available, the engineering strain should increate monotonically from a negative value in compression to a positive value in tension. In compression, the engineering strain should not be less than -1.0 since -100% strain is physically not possible.
This material law is stable when (with p=1,…5) is satisfied for parameter pairs for all loading conditions. By default, Radioss tries to fit the curve by accounting for these conditions (Icheck=2). If a proper fit cannot be found, then Radioss uses a weaker condition (Icheck=1: ) which ensures that the initial shear hyperelastic modulus (μ) is positive.
Once the material parameters are calculated by the Radioss Starter in LAW69, all the calculations done by LAW69 in the simulation are the same as LAW42.
LAW82
- If ν=0, D1 should be entered.
- If ν≠0, D1 input is ignored and will be recalculated
and output in the Starter output using:
(23) D1=3(1−2v)μ(1+v) - If ν=0 and D1=0, a default value of ν=0.475 is used and D1 is calculated using Equation 23
Unloading can be represented using an unloading function, FscaleunL, or by providing hysteresis, Hys and shape factor, Shape, inputs to a damage model based on energy.
If the unloading function, FscaleunL, is entered, unloading is defined based on the unloading flag, Tension and the damage model is not used.
Arruda-Boyce Material (LAW92)
LAW92 describes the Arruda-Boyce material model, which can be used to model hyperelastic behavior. The Arruda-Boyce model is based on the statistical mechanics of a material with a cubic representative volume element containing eight chains along the diagonal directions. It assumes that the chain molecules are located on the average along the diagonals of the cubic in principal stretch space.
Where, Material constant ci are:
- μ
- Shear modulus
- μ0
- Initial shear modulus
λm is the limit of stretch which describes the beginning of hardening phase in tension (locking strain in tension) and so it is also called the locking stretch.
Arruda-Boyce is always stable if positive values of the shear modulus, μ, and the locking stretch, λm are used.
with ˉλi=J−13λi
- Uniaxial test
(32) σ=λ∂W∂λ=2μ(λ2−1λ)5∑i=1i⋅ciλm2i−2(ˉλ12+ˉλ22+ˉλ32)i−1 with λ1=λ and λ2=λ3=λ−12, then ˉI1=λ2+2λ
and nominal stress is:(33) Nth=∂W∂λ=2μ(λ−λ−2)5∑i=1i⋅ci(λm)2i−2(λ2+2λ)i−1 - Equibiaxial test
(34) σ=λ∂W∂λ=2μ(λ2−1λ4)5∑i=1i⋅ciλm2i−2(ˉλ12+ˉλ22+ˉλ32)i−1 with λ1=λ2=λ and λ3=λ−2, then ˉI1=2λ2+1λ4
and the nominal stress is:(35) Nth=∂W∂λ=2μ(λ−λ−5)5∑i=1i⋅ci(λm)2i−2(2λ2+1λ4)i−1 - Planar test
(36) σ=λ∂W∂λ=2μ(λ2−1λ2)5∑i=1i⋅ciλm2i−2(ˉλ12+ˉλ22+ˉλ32)i−1 with λ1=λ , λ3=1 and λ2=λ−1, then ˉI1=λ2+1+λ−2
and nominal stress is:(37) Nth=∂W∂λ=2μ(λ−λ−3)5∑i=1ici(λm)2i−2(λ2+1+λ−2)i−1
Yeoh Material (LAW94)
- ˉI1=ˉλ21+ˉλ22+ˉλ23
- First strain invariant
- ˉλi=J−13λi
- Deviatoric stretch
For incompressible materials with i=1 only and D1 are input and the Yeoh model is reduced to a Neo-Hookean model.
The material constant specify the deviatoric part (shape change) of the material and parameters D1, D2, D3 specify the volumetric change of the material. These six material constants need to be calculated by curve fitting material test data. RD-E: 5600 Hyperelastic Material with Curve Input includes a Yeoh fitting Compose script for uniaxial test data. The Yeoh material model has been shown to model all deformation models even if the curve fit was obtained using only uniaxial test data.
The initial shear modulus and the bulk modulus are computed as:
μ=2⋅C10 and K=2D1
LAW94 is available only as an incompressible material model.