Physics:Yeoh (hyperelastic model)

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300px|right|thumb|Yeoh model prediction versus experimental data for natural rubber. Model parameters and experimental data from PolymerFEM.com The Yeoh hyperelastic material model[1] is a phenomenological model for the deformation of nearly incompressible, nonlinear elastic materials such as rubber. The model is based on Ronald Rivlin's observation that the elastic properties of rubber may be described using a strain energy density function which is a power series in the strain invariants I1,I2,I3 of the Cauchy-Green deformation tensors.[2] The Yeoh model for incompressible rubber is a function only of I1. For compressible rubbers, a dependence on I3 is added on. Since a polynomial form of the strain energy density function is used but all the three invariants of the left Cauchy-Green deformation tensor are not, the Yeoh model is also called the reduced polynomial model.

Yeoh model for incompressible rubbers

Strain energy density function

The original model proposed by Yeoh had a cubic form with only I1 dependence and is applicable to purely incompressible materials. The strain energy density for this model is written as

W=i=13Ci(I13)i

where Ci are material constants. The quantity 2C1 can be interpreted as the initial shear modulus.

Today a slightly more generalized version of the Yeoh model is used.[3] This model includes n terms and is written as

W=i=1nCi(I13)i.

When n=1 the Yeoh model reduces to the neo-Hookean model for incompressible materials.

For consistency with linear elasticity the Yeoh model has to satisfy the condition

2WI1(3)=μ(ij)

where μ is the shear modulus of the material. Now, at I1=3(λi=λj=1),

WI1=C1

Therefore, the consistency condition for the Yeoh model is

2C1=μ

Stress-deformation relations

The Cauchy stress for the incompressible Yeoh model is given by

σ=p1+2WI1B;WI1=i=1niCi(I13)i1.

Uniaxial extension

For uniaxial extension in the 𝐧1-direction, the principal stretches are λ1=λ,λ2=λ3. From incompressibility λ1λ2λ3=1. Hence λ22=λ32=1/λ. Therefore,

I1=λ12+λ22+λ32=λ2+2λ.

The left Cauchy-Green deformation tensor can then be expressed as

B=λ2𝐧1𝐧1+1λ(𝐧2𝐧2+𝐧3𝐧3).

If the directions of the principal stretches are oriented with the coordinate basis vectors, we have

σ11=p+2λ2WI1;σ22=p+2λWI1=σ33.

Since σ22=σ33=0, we have

p=2λWI1.

Therefore,

σ11=2(λ21λ)WI1.

The engineering strain is λ1. The engineering stress is

T11=σ11/λ=2(λ1λ2)WI1.

Equibiaxial extension

For equibiaxial extension in the 𝐧1 and 𝐧2 directions, the principal stretches are λ1=λ2=λ. From incompressibility λ1λ2λ3=1. Hence λ3=1/λ2. Therefore,

I1=λ12+λ22+λ32=2λ2+1λ4.

The left Cauchy-Green deformation tensor can then be expressed as

B=λ2𝐧1𝐧1+λ2𝐧2𝐧2+1λ4𝐧3𝐧3.

If the directions of the principal stretches are oriented with the coordinate basis vectors, we have

σ11=p+2λ2WI1=σ22;σ33=p+2λ4WI1.

Since σ33=0, we have

p=2λ4WI1.

Therefore,

σ11=2(λ21λ4)WI1=σ22.

The engineering strain is λ1. The engineering stress is

T11=σ11λ=2(λ1λ5)WI1=T22.

Planar extension

Planar extension tests are carried out on thin specimens which are constrained from deforming in one direction. For planar extension in the 𝐧1 directions with the 𝐧3 direction constrained, the principal stretches are λ1=λ,λ3=1. From incompressibility λ1λ2λ3=1. Hence λ2=1/λ. Therefore,

I1=λ12+λ22+λ32=λ2+1λ2+1.

The left Cauchy-Green deformation tensor can then be expressed as

B=λ2𝐧1𝐧1+1λ2𝐧2𝐧2+𝐧3𝐧3.

If the directions of the principal stretches are oriented with the coordinate basis vectors, we have

σ11=p+2λ2WI1;σ22=p+2λ2WI1;σ33=p+2WI1.

Since σ22=0, we have

p=2λ2WI1.

Therefore,

σ11=2(λ21λ2)WI1;σ22=0;σ33=2(11λ2)WI1.

The engineering strain is λ1. The engineering stress is

T11=σ11λ=2(λ1λ3)WI1.

Yeoh model for compressible rubbers

A version of the Yeoh model that includes I3=J2 dependence is used for compressible rubbers. The strain energy density function for this model is written as

W=i=1nCi0(I¯13)i+k=1nCk1(J1)2k

where I¯1=J2/3I1, and Ci0,Ck1 are material constants. The quantity C10 is interpreted as half the initial shear modulus, while C11 is interpreted as half the initial bulk modulus.

When n=1 the compressible Yeoh model reduces to the neo-Hookean model for incompressible materials.

References

  1. Yeoh, O. H., 1993, "Some forms of the strain energy function for rubber," Rubber Chemistry and technology, Volume 66, Issue 5, November 1993, Pages 754-771.
  2. Rivlin, R. S., 1948, "Some applications of elasticity theory to rubber engineering", in Collected Papers of R. S. Rivlin vol. 1 and 2, Springer, 1997.
  3. Selvadurai, A. P. S., 2006, "Deflections of a rubber membrane", Journal of the Mechanics and Physics of Solids, vol. 54, no. 6, pp. 1093-1119.

See also