Mooney–Rivlin solid explained

is a linear combination of two invariants of the left Cauchy–Green deformation tensor

\boldsymbol{B}

. The model was proposed by Melvin Mooney in 1940 and expressed in terms of invariants by Ronald Rivlin in 1948.

The strain energy density function for an incompressible Mooney–Rivlin material is^[3] ^[4]

W=C₁(\bar{I}_1-3)+C₂(\bar{I}_2-3),

where

C₁

and

C₂

are empirically determined material constants, and

\barI₁

and

\barI₂

are the first and the second invariant of

\bar\boldsymbolB=(\det\boldsymbolB)^-1/3\boldsymbolB

(the unimodular component of

\boldsymbolB

^[5]):

\begin{align} \bar{I}₁&=J^-2/3~I_1, I₁=

	2
λ
	1

	2+
λ
	2

	2,
λ
	3

\\ \bar{I}₂&=J^-4/3~I_2, I₂=

	2
λ
	1

	2
λ
	2

	2
λ
	2

	2
λ
	3

	2
λ
	3

	2
λ
	1

\end{align}

where

\boldsymbol{F}

is the deformation gradient and

J=\det(\boldsymbol{F})=λ_1λ_2λ₃

. For an incompressible material,

J=1

Derivation

The Mooney–Rivlin model is a special case of the generalized Rivlin model (also called polynomial hyperelastic model^[6]) which has the form

	N
\sum
	p,q=0

C_pq(\bar{I}₁-

	p~(\bar{I}
3)
	2

-3)^q+

	M
\sum
	m=1

	1
	D_m

~(J-1)^2m

with

C₀₀=0

where

C_pq

are material constants related to the distortional response and

D_m

are material constants related to the volumetric response. For a compressible Mooney–Rivlin material

N=1,C₀₁=C_2,C₁₁=0,C₁₀=C_1,M=1

and we have

W=C₀₁~(\bar{I}₂-3)+C₁₀~(\bar{I}₁-3)+

	1
	D₁

~(J-1)²

C₀₁=0

we obtain a neo-Hookean solid, a special case of a Mooney–Rivlin solid.

For consistency with linear elasticity in the limit of small strains, it is necessary that

\kappa=2/D₁~;~~\mu=2~(C₀₁+C₁₀)

where

\kappa

is the bulk modulus and

\mu

is the shear modulus.

Cauchy stress in terms of strain invariants and deformation tensors

The Cauchy stress in a compressible hyperelastic material with a stress free reference configuration is given by

\boldsymbol{\sigma}=\cfrac{2}{J}\left[\cfrac{1}{J^2/3

}\left(\cfrac + \bar_1~\cfrac\right)\boldsymbol - \cfrac~\cfrac~\boldsymbol \cdot\boldsymbol \right] + \left[\cfrac{\partial{W}}{\partial J} - \cfrac{2}{3J}\left(\bar{I}_1~\cfrac{\partial{W}}{\partial \bar{I}_1} + 2~\bar{I}_2~\cfrac{\partial{W}}{\partial \bar{I}_2}\right)\right]~\boldsymbol For a compressible Mooney–Rivlin material,

\cfrac{\partial{W}}{\partial\bar{I}_1}=C₁~;~~\cfrac{\partial{W}}{\partial\bar{I}_2}=C₂~;~~\cfrac{\partial{W}}{\partialJ}=

	2
	D₁

(J-1)

Therefore, the Cauchy stress in a compressible Mooney–Rivlin material is given by

\boldsymbol{\sigma}=\cfrac{2}{J}\left[\cfrac{1}{J^2/3

}\left(C_1 + \bar_1~C_2\right)\boldsymbol - \cfrac~C_2~\boldsymbol \cdot\boldsymbol \right] + \left[\frac{2}{D_1}(J-1)- \cfrac{2}{3J}\left(C_1\bar{I}_1 + 2C_2\bar{I}_2~\right)\right]\boldsymbol It can be shown, after some algebra, that the pressure is given by

p:=-\tfrac{1}{3}tr(\boldsymbol{\sigma})=-

	\partialW
	\partialJ

	2
	D₁

(J-1).

The stress can then be expressed in the form

\boldsymbol{\sigma}=-p~\boldsymbol{I}+\cfrac{1}{J}\left[\cfrac{2}{J^2/3

}\left(C_1 + \bar_1~C_2\right)\boldsymbol - \cfrac~C_2~\boldsymbol\cdot\boldsymbol -\cfrac\left(C_1\,\bar_1 + 2C_2\,\bar_2\right)\boldsymbol\right] \,.

The above equation is often written using the unimodular tensor

\bar{\boldsymbol{B}}=J^-2/3\boldsymbol{B}

\boldsymbol{\sigma}=-p~\boldsymbol{I}+\cfrac{1}{J}\left[2\left(C₁+\bar{I}_1~C_{2\right)\bar{\boldsymbol{B}}}- 2~C_{2~\bar{\boldsymbol{B}} ⋅ \bar{\boldsymbol{B}}}-\cfrac{2}{3}\left(C_1\bar{I}₁+2C_2\bar{I}_{2\right)\boldsymbol{I}\right]}.

For an incompressible Mooney–Rivlin material with

J=1

there holds

p=0

and

\bar\boldsymbolB=\boldsymbolB

. Thus

\boldsymbol{\sigma}=2\left(C₁+I_1~C_{2\right)\boldsymbol{B}}- 2C_{2~\boldsymbol{B} ⋅ \boldsymbol{B}}-\cfrac{2}{3}\left(C_1I₁+2C_2I_{2\right)\boldsymbol{I}.}

Since

\detJ=1

the Cayley–Hamilton theorem implies

\boldsymbol{B}^-1=\boldsymbol{B} ⋅ \boldsymbol{B}-I_{1~\boldsymbol{B}}+I_{2~\boldsymbol{I}.}

Hence, the Cauchy stress can be expressed as

\boldsymbol{\sigma}=-p^*~\boldsymbol{I}+2C_{1~\boldsymbol{B}}-

	-1
2C
	2~\boldsymbol{B}

where

p^*:=\tfrac{2}{3}(C_1~I₁-C_2~I_2).

Cauchy stress in terms of principal stretches

In terms of the principal stretches, the Cauchy stress differences for an incompressible hyperelastic material are given by

\sigma₁₁-\sigma₃₃=λ_{1~\cfrac{\partial{W}}{\partial}λ_1}-λ_{3~\cfrac{\partial{W}}{\partial}λ_3}~;~~ \sigma₂₂-\sigma₃₃=λ_{2~\cfrac{\partial{W}}{\partial}λ_2}-λ_{3~\cfrac{\partial{W}}{\partial}λ_3}

For an incompressible Mooney-Rivlin material,

W=C_1(λ

	2

	1

	2+
λ
	2

	2
λ
	3

-3)+C_2(λ

	2

	1

	2
λ
	2

	2
λ
	2

	2
λ
	3

	2
λ
	3

	2
λ
	1

-3)~;~~λ_1λ_2λ₃=1

Therefore,

λ_{1\cfrac{\partial{W}}{\partial}λ_1}=2C_1λ

	2

	1

+2C_2λ

	2)

	3

~;~~ λ_{2\cfrac{\partial{W}}{\partial}λ_2}=2C_1λ

	2

	2

+2C_2λ

	2)

	3

~;~~ λ_{3\cfrac{\partial{W}}{\partial}λ_3}=2C_1λ

	2

	3

+2C_2λ

	2)

	2

Since

λ_1λ_2λ₃₌₁

. we can write

\begin{align} λ_{1\cfrac{\partial{W}}{\partial}λ_1}&=2C_1λ

	2

	1

+2C_{2\left(\cfrac{1}{λ}

	2}\right)

	2

~;~~ λ_{2\cfrac{\partial{W}}{\partial}λ_2}=2C_1λ

	2

	2

+2C_{2\left(\cfrac{1}{λ}

	2}\right)

	1

\\ λ_{3\cfrac{\partial{W}}{\partial}λ_3}&=2C_1λ

	2

	3

+2C_{2\left(\cfrac{1}{λ}

	2}\right)

	1

\end{align}

Then the expressions for the Cauchy stress differences become

\sigma₁₁-\sigma₃₃=2C_1(λ

	2)

	3

-2C_{2\left(\cfrac{1}{λ}

	2}\right)~;~~

	3

\sigma₂₂-\sigma₃₃=2C_1(λ

	2)

	3

-2C_{2\left(\cfrac{1}{λ}

	2}\right)

	3

Uniaxial extension

For the case of an incompressible Mooney–Rivlin material under uniaxial elongation,

λ₁=λ

and

λ₂=λ₃=1/\sqrt{λ}

. Then the true stress (Cauchy stress) differences can be calculated as:

\begin{align} \sigma₁₁-\sigma₃₃&=

	2-\cfrac{1}{λ}\right)
2C
	1\left(λ

	2}
-2C
	2\left(\cfrac{1}{λ

-λ\right)\\ \sigma₂₂-\sigma₃₃&=0 \end{align}

Simple tension

In the case of simple tension,

\sigma₂₂=\sigma₃₃=0

. Then we can write

\sigma₁₁=\left(2C₁+\cfrac{2C_2}{λ}\right)\left(λ²-\cfrac{1}{λ}\right)

In alternative notation, where the Cauchy stress is written as

\boldsymbol{T}

and the stretch as

\alpha

, we can write

T₁₁=\left(2C₁+

	2C₂
	\alpha

\right)\left(\alpha²-\alpha^-1\right)

and the engineering stress (force per unit reference area) for an incompressible Mooney–Rivlin material under simple tension can be calculated using

	eng
T
	11

=T₁₁\alpha_2\alpha₃=\cfrac{T₁₁

} . Hence

	eng
T
	11

=\left(2C₁+

	2C₂
	\alpha

\right)\left(\alpha-\alpha^-2\right)

If we define

	*
T
	11

	eng
\cfrac{T
	11

} ~;~~ \beta := \cfrac then

	*
T
	11

=2C₁+2C_2\beta~.

The slope of the

	*
T
	11

versus

\beta

line gives the value of

C₂

while the intercept with the

	*
T
	11

axis gives the value of

C₁

. The Mooney–Rivlin solid model usually fits experimental data better than Neo-Hookean solid does, but requires an additional empirical constant.

Equibiaxial tension

In the case of equibiaxial tension, the principal stretches are

λ₁=λ₂=λ

. If, in addition, the material is incompressible then

λ₃=1/λ²

. The Cauchy stress differences may therefore be expressed as

\sigma₁₁-\sigma₃₃=\sigma₂₂-\sigma₃₃=

	2-\cfrac{1}{λ
2C
	1\left(λ

^4}\right)-

	2}
2C
	2\left(\cfrac{1}{λ

-λ^4\right)

The equations for equibiaxial tension are equivalent to those governing uniaxial compression.

Pure shear

A pure shear deformation can be achieved by applying stretches of the form ^[7]

λ₁=λ~;~~λ₂=\cfrac{1}{λ}~;~~λ₃=1

The Cauchy stress differences for pure shear may therefore be expressed as

\sigma₁₁-\sigma₃₃=

	2-1)
2C
	1(λ

	2}-1\right)
2C
	2\left(\cfrac{1}{λ

~;~~ \sigma₂₂-\sigma₃₃=

	2}
2C
	1\left(\cfrac{1}{λ

-1\right)-

	2
2C
	2(λ

-1)

Therefore

\sigma₁₁-\sigma₂₂=2(C_1+C

	2

	2)\left(λ

-\cfrac{1}{λ^2}\right)

For a pure shear deformation

I₁=

	2
λ
	1

	2
λ
	2

	2
λ
	3

=λ²+\cfrac{1}{λ^2}+1~;~~ I₂=

	2}
\cfrac{1}{λ
	1

	2}
\cfrac{1}{λ
	2

	2}
\cfrac{1}{λ
	3

=\cfrac{1}{λ^2}+λ²+1

Therefore

I₁=I₂

Simple shear

The deformation gradient for a simple shear deformation has the form^[7]

\boldsymbol{F}=\boldsymbol{1}+\gamma~e_{1 ⊗ e}₂

where

e_1,e₂

are reference orthonormal basis vectors in the plane of deformation and the shear deformation is given by

\gamma=λ-\cfrac{1}{λ}~;~~λ₁=λ~;~~λ₂=\cfrac{1}{λ}~;~~λ₃=1

In matrix form, the deformation gradient and the left Cauchy-Green deformation tensor may then be expressed as

\boldsymbol{F}=\begin{bmatrix}1&\gamma&0\ 0&1&0\ 0&0&1\end{bmatrix}~;~~ \boldsymbol{B}=\boldsymbol{F} ⋅ \boldsymbol{F}^T=\begin{bmatrix}1+\gamma²&\gamma&0\ \gamma&1&0\ 0&0&1\end{bmatrix}

Therefore,

\boldsymbol{B}^-1=\begin{bmatrix}1&-\gamma&0\ -\gamma&1+\gamma²&0\ 0&0&1\end{bmatrix}

The Cauchy stress is given by

\boldsymbol{\sigma}=\begin{bmatrix}-p^*+2(C_1-C_2)+2C

	2

	1\gamma

&2(C_1+C_2)\gamma&0\ 2(C_1+C_2)\gamma&-p^*+2(C₁-C₂₎-

	2
2C
	2\gamma

&0\ 0&0&-p^*+2(C₁-C₂₎\end{bmatrix}

For consistency with linear elasticity, clearly

\mu=2(C_1+C₂₎

where

\mu

is the shear modulus.

Rubber

Elastic response of rubber-like materials are often modeled based on the Mooney–Rivlin model. The constants

C_1,C₂

are determined by fitting the predicted stress from the above equations to the experimental data. The recommended tests are uniaxial tension, equibiaxial compression, equibiaxial tension, uniaxial compression, and for shear, planar tension and planar compression. The two parameter Mooney–Rivlin model is usually valid for strains less than 100%.^[8]

Notes and references

Mooney, M., 1940, A theory of large elastic deformation, Journal of Applied Physics, 11(9), pp. 582–592.
Rivlin, R. S., 1948, Large elastic deformations of isotropic materials. IV. Further developments of the general theory, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 241(835), pp. 379–397.
Boulanger, P. and Hayes, M. A., 2001, "Finite amplitude waves in Mooney–Rivlin and Hadamard materials", in Topics in Finite Elasticity, ed. M. A Hayes and G. Soccomandi, International Center for Mechanical Sciences.
C. W. Macosko, 1994, Rheology: principles, measurement and applications, VCH Publishers, .
Unimodularity in this context means
\det\bar\boldsymbolB=1

.
Book: Bower, Allan . Applied Mechanics of Solids . 2009 . CRC Press . 978-1-4398-0247-2 . 2018-04-19.
Ogden, R. W., 1984, Nonlinear elastic deformations, Dover
Hamza. Muhsin. Alwan. Hassan. Hyperelastic Constitutive Modeling of Rubber and Rubber-Like Materials under Finite Strain. Engineering and Technology Journal. 2010. 28. 13. 2560–2575. 10.30684/etj.28.13.5 . free.

Mooney–Rivlin solid explained

Derivation

Cauchy stress in terms of strain invariants and deformation tensors

Cauchy stress in terms of principal stretches

Uniaxial extension

Simple tension

Equibiaxial tension

Pure shear

Simple shear

Rubber

Notes and references

See also