Isotropic Elastic Material

Two kinds of isotropic elastic materials are considered:
  • 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)

Ogden's law is applied to slightly compressible materials as rubber or elastomer foams undergoing large deformation with an elastic behavior. The detailed theory for Odgen material models can be found in 1. The strain energy WW is expressed in a general form as a function of W(λ1,λ2,λ3)W(λ1,λ2,λ3):(1) W(λ1,λ2,λ3)=5p=1μpαp(ˉλ1αp+ˉλ2αp+ˉλ3αp3)+K2(J1)2W(λ1,λ2,λ3)=5p=1μpαp(¯λ1αp+¯λ2αp+¯λ3αp3)+K2(J1)2

Where, λ1λ1, ith principal stretch

λi=1+εiλi=1+εi, with εiεi being the ith principal engineering strain

JJ is relative volume with:(2) J=λ1λ2λ3=ρ0ρJ=λ1λ2λ3=ρ0ρ
ˉλi¯λi is the deviatoric stretch(3) ˉλi=J13λi¯λi=J13λi

αpαp and μpμp are the material constants.

pp is order of Ogden model and defines the number of coefficients pairs (αp,μp)(αp,μp).

This law is very general due to the choice of coefficient pair (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
For uniform dilitation:(4) λ1=λ2=λ3=λλ1=λ2=λ3=λ
The strain energy function can be decomposed into deviatoric part ˉW(ˉλ1,ˉλ2,ˉλ3)¯¯¯¯W(¯λ1,¯λ2,¯λ3) and spherical part U(J)U(J):(5) W=ˉW(ˉλ1,ˉλ2,ˉλ3)+U(J)W=¯¯¯¯W(¯λ1,¯λ2,¯λ3)+U(J)

With:

ˉW(ˉλ1,ˉλ2,ˉλ3)=pμpαp(ˉλ1αp+ˉλ2αp+ˉλ3αp3)U(J)=K2(J1)2

The stress σi corresponding to this strain energy is given by:(6) σi=λiJWλi
which can be written as:(7) σi=λiJWλi=λiJ(3j=1ˉWˉλjˉλjλi+UJJλi)

Since λ1Jλi=J and ˉλjλi=23J13 for i=j and ˉλjλi=13J13λjλi for i≠j

Equation 7 is simplified to:(8) σi=1J(ˉλiˉWˉλi(133j=1ˉλjˉWˉλjJUJ))
For which the deviator of the Cauchy stress tensor si, and the pressure P would be:(9) si=1J(ˉλiˉWˉλi133j=1ˉλjˉWˉλj) (10) p=133j=1σj=UJ
Only the deviatoric stress above is retained, and the pressure is computed independently:(11) P=KFscaleblkfblk(J)(J1)
Where, fblk(J) a user-defined function related to the bulk modulus K in LAW42 and LAW69:(12) K=μ2(1+ν)3(12ν)
For an imcompressible material (ν0.5), J=1 and no pressure in material.(13) μ=pμpαp2
With μ being the initial shear modulus, and υ the Poisson's ratio.
Note: For an incompressible material you have υ0.5. However, υ0.495 is a good compromise to avoid too small time steps in explicit codes.
A particular case of the Ogden material model is the Mooney-Rivlin material law which has two basic assumptions:
  • The rubber is incompressible and isotropic in unstrained state
  • The strain energy expression depends on the invariants of Cauchy tensor
The three invariants of the Cauchy-Green tensor are:(14) I1=λ12+λ22+λ32 (15) I2=λ12λ22+λ22λ32+λ32λ12
For incompressible material:(16) I3=λ12λ22λ32=1
The Mooney-Rivlin law gives the closed expression of strain energy as:(17) W=C10(I13)+C01(I23)
with:(18) μ1=2C10
μ2=2C01
α1=2
α2=2

The model can be generalized for a compressible material.

Viscous Effects in LAW42

Viscous effects are modeled through the Maxwell model:


Figure 1. Maxwell Model
Where, the shear modulus of the hyper-elastic law μ is exactly the long-term shear modulus G.(19) μ=pμpαp2=G

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.

The visco-Kirchoff stress is given by:(20) τv=Mi=1Git0etsτidds[dev(ˉFˉFT)]ds
Where,
M
Order of the Maxwell model
F
Deformation gradient matrix
ˉF=J13F
dev(ˉFˉFT)
Denotes the deviatoric part of tensor ˉFˉFT
The viscous-Cauchy stress is written as:(21) σv(t)=1Jτv(t)

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.

The strain energy density formulation used depends on the law_ID.
law_ID =1, Ogden law (Same as LAW42):
W(λ1,λ2,λ3)=5p=1μpαp(ˉλ1αp+ˉλ2αp+ˉλ3αp3)+K2(J1)2
law_ID =2, Mooney-Rivlin law
W=C10(I13)+C01(I23)
Curve Fitting

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=2C10, μ2=2C01, 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

The Ogden model used in LAW82 is:(22) W=Ni=12μiαi2(ˉλ1αi+ˉλ2αi+ˉλ3αi3)+Ni=11Di(J1)2i
The Bulk Modulus is calculated as K=2D1 based on these rules:
  • If ν=0, D1 should be entered.
  • If ν0, D1 input is ignored and will be recalculated and output in the Starter output using:(23) D1=3(12v)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
LAW88, A simplified hyperelastic material with strain rate effects
This law utilizes tabulated uniaxial tension and compression engineering stress and strain test data at different strain rates to model incompressible materials. It is only compatible with solid elements. The material is based on Ogden’s strain energy density function but does not require curve fitting to extract material constants like most other hyperelastic material models. Strain rate effects can be modeled by including engineering stress strain test data at different strain rates. This can be easier than calculating viscous parameters for traditional hyperelastic material models. The following Ogden strain energy density function is used but instead of extracting material constants via curve fitting this law determines the Ogden function directly from the uniaxial engineering stress strain curve tabulated data. 5(24) W=3i=1mj=1μjαj(ˉλiαj1)deviatoricpart+K(J1lnJ)sphericalpart

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.

When using the damage model, the loading curves are used for both loading and unloading and the unloading stress tensor is reduced by:(25) σ=(1D)σ (26) D=(1Hys)(1(WcurWmax)Shape)

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.

The strain energy density function is:(27) W=μ5i=1ci(λm)2i2(ˉIi13i)W(ˉI1)+1D(J212+ln(J))U(J)

Where, Material constant ci are:

c1=12,c2=120,c3=111050,c4=197000,c5=519673750
μ
Shear modulus
μ0
Initial shear modulus
(28) μ0=μ(1+35λm2+99175λm4+513875λm6+4203967375λm8)

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.

ˉI1 is deviatoric part of first strain invarient I1(29) ˉI1=ˉλ12+ˉλ22+ˉλ32=J2/3I1

with ˉλi=J13λi

D is a material parameter for the bulk modulus computation given as:(30) D=2K
The Cauchy stress corresponding to above strain energy is:(31) σi=λiJWλi
For incompressible materials, the Cauchy stress is then given by:
  • Uniaxial test(32) σ=λWλ=2μ(λ21λ)5i=1iciλm2i2(ˉλ12+ˉλ22+ˉλ32)i1

    with λ1=λ and λ2=λ3=λ12, then ˉI1=λ2+2λ

    and nominal stress is:(33) Nth=Wλ=2μ(λλ2)5i=1ici(λm)2i2(λ2+2λ)i1
  • Equibiaxial test(34) σ=λWλ=2μ(λ21λ4)5i=1iciλm2i2(ˉλ12+ˉλ22+ˉλ32)i1

    with λ1=λ2=λ and λ3=λ2, then ˉI1=2λ2+1λ4

    and the nominal stress is:(35) Nth=Wλ=2μ(λλ5)5i=1ici(λm)2i2(2λ2+1λ4)i1
  • Planar test(36) σ=λWλ=2μ(λ21λ2)5i=1iciλm2i2(ˉλ12+ˉλ22+ˉλ32)i1

    with λ1=λ,λ3=1 and λ2=λ1, then ˉI1=λ2+1+λ2

    and nominal stress is:(37) Nth=Wλ=2μ(λλ3)5i=1ici(λm)2i2(λ2+1+λ2)i1

Additional information about Arruda-Boyce model. 2 3

Yeoh Material (LAW94)

The Yeoh model (LAW94) 4is a hyperelastic material model that can be used to describe incompressible materials. The strain energy density function of LAW94 only depends on the first strain invariant and is computed as:(38) W=3i=1[Ci0(ˉI13)iW(ˉI1)+1Di(J1)2iU(J)]
Where,
ˉI1=ˉλ21+ˉλ22+ˉλ23
First strain invariant
ˉλi=J13λi
Deviatoric stretch
The Cauchy stress is computed as:(39) σi=λiJWλi

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=2C10 and K=2D1

LAW94 is available only as an incompressible material model.

If D1=0, an incompressible material is considered, where ν=0.495 and D1 is calculated as:(40) D1=3(12v)μ(1+v)
1
Ogden R.W., “Nonlinear Elastic Deformations”, Ellis Horwood, 1984.
2
Arruda, E.M. and Boyce, M.C., “A three-dimensional model for the large stretch behavior of rubber elastic materials”, J. Mech. Phys. Solids, 41(2), pp. 389–412, 1993.
3
Jörgen Bergström, “Mechanics of solid polymers: theory and computational modeling”, pp. 250-254, 2015.
4
Yeoh, O. H., “Some forms of the strain energy function for rubber”, Rubber Chemistry and Technology, Vol. 66, Issue 5, pp. 754-771, November 1993.
5
Kolling S., Du Bois P.A., Benson D.J., and Feng W.W., "A tabulated formulation of hyperelasticity with rate effects and damage." Computational Mechanics 40, no. 5 (2007).