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Submitted by Biswajit Banerjee on Fri, 2007-09-14 01:10.

Following Andy's recommendation I have been reading Ellis Dill's Continuum Mechanics[1]. In page 75 of the book, we find the well known result that the constitutive equation for an isotropic hypoelastic material can be derived from a stored energy function only if

$\displaystyle \lambda + \mu = 0<br /> $

 

where $ \lambda$ and $ \mu$ are the Lame constants.

The derivation of this result can be traced to "a straightforward calculation" mentioned in an important 1984 paper by Simo and Pister [2]. Unfortunately, the calculation is not turning out to be particularly straightforward for me. Any help in deriving this result will be appreciated.

Let me state the problem.

Typically, for isotropic hypoelastic materials, we assume that the spatial elasticity tensor has the form

$\displaystyle \mathsf{c}_{ijkl} = \lambda \delta_{ij} \delta_{kl} +<br /> \mu (\delta_{ik} \delta_{jl} + \delta_{jk} \delta_{il})<br /> $

 

If $ \boldsymbol{\mathsf{C}}$ is the material elasticity tensor, then

$\displaystyle \mathsf{c}_{ijkl} = F_{iI} F_{jJ} F_{kK} F_{lL} \mathsf{C}_{IJKL}<br /> $

 

where $ \ensuremath{\boldsymbol{F}}$ is the deformation gradient.

We can invert this relationship to get

$\displaystyle \mathsf{C}_{IJKL} = F^{-1}_{Ii} F^{-1}_{Jj} F^{-1}_{Kk} F^{-1}_{Ll} <br /> \mathsf{c}_{ijkl}<br /> $

 

Plugging in the expression for $ \mathsf{c}_{ijkl}$ we get

$\displaystyle \begin{aligned}<br /> \mathsf{C}_{IJKL} = &<br /> \lambda (\ensuremath{\bolds...<br /> ...{\boldsymbol{F}}^{-1}\cdot\ensuremath{\boldsymbol{F}}^{-T})_{JK}]<br /> \end{aligned}$

 

If we define

$\displaystyle \ensuremath{\boldsymbol{C}}= \ensuremath{\boldsymbol{F}}^T\cdot\ensuremath{\boldsymbol{F}}<br /> $

 

we get

$\displaystyle \mathsf{C}_{IJKL} =<br /> \lambda C^{-1}_{IJ} C^{-1}_{KL} +<br /> \mu[C^{-1}_{IK} C^{-1}_{JL} + C^{-1}_{IL} C^{-1}_{JK}]<br /> $

 

Now, if the material law can be derived from a stored energy function, we must have

$\displaystyle \ensuremath{\frac{\partial \mathsf{C}_{IJKL}}{\partial C_{MN}}} = \ensuremath{\frac{\partial \mathsf{C}_{IJMN}}{\partial C_{KL}}}<br /> $

 

Simo writes that if we plug in the expression for $ \mathsf{C}_{IJKL}$and use the above condition, a straightforward calculation leads to

 

$\displaystyle (\lambda+\mu) C^{-1}_{KL}(C_{IM}^{-1} C_{JN}^{-1} + C_{IN}^{-1} C...<br /> ...<br /> (\lambda+\mu) C^{-1}_{MN}(C_{IK}^{-1} C_{JL}^{-1} + C_{IL}^{-1} C_{JK}^{-1})<br /> $

(1)

 

This then gives us the required condition

$\displaystyle \lambda + \mu = 0<br /> $

 

My question is about deriving equation (1).

I proceeded by recalling that

$\displaystyle \ensuremath{\frac{\partial \ensuremath{\boldsymbol{C}}^{-1}}{\par...<br /> ...{C}}^{-1}\cdot\ensuremath{\boldsymbol{T}}\cdot\ensuremath{\boldsymbol{C}}^{-1}<br /> $

 

In index notation

$\displaystyle \ensuremath{\frac{\partial C^{-1}_{IJ}}{\partial C_{KL}}} T_{KL} = -C^{-1}_{IK} T_{KL} C^{-1}_{LJ}<br /> $

 

which gives us the formula (using the symmetry of $ \ensuremath{\boldsymbol{C}}$)

$\displaystyle \ensuremath{\frac{\partial C^{-1}_{IJ}}{\partial C_{KL}}} = -C^{-1}_{IK} C^{-1}_{JL}<br /> $

 

After taking the derivatives I get

$\displaystyle \begin{aligned}<br /> \ensuremath{\frac{\partial \mathsf{C}_{IJKL}}{\pa...<br /> ...} C^{-1}_{LN} C^{-1}_{JK} +<br /> C^{-1}_{IL} C^{-1}_{JM} C^{-1}_{KN}]<br /> \end{aligned}$

 

and

$\displaystyle \begin{aligned}<br /> \ensuremath{\frac{\partial \mathsf{C}_{IJMN}}{\pa...<br /> ...} C^{-1}_{LN} C^{-1}_{JM} +<br /> C^{-1}_{IN} C^{-1}_{JM} C^{-1}_{KL}]<br /> \end{aligned}$

 

Equating the two, we see that the terms that cancel out are

$\displaystyle \lambda C^{-1}_{IJ} C^{-1}_{KM} C^{-1}_{LN} ,  <br /> \mu C^{-1}_{IM} C^{-1}_{JK} C^{-1}_{LN} ;  <br /> \mu C^{-1}_{IK} C^{-1}_{JM} C^{-1}_{LN}<br /> $

 

That leaves us with

$\displaystyle \begin{aligned}<br /> \lambda C^{-1}_{IM} C^{-1}_{JN} C^{-1}_{KL}<br /> + \m...<br /> ...} C^{-1}_{LM} C^{-1}_{JN} +<br /> C^{-1}_{IN} C^{-1}_{JM} C^{-1}_{KL}]<br /> \end{aligned}$

 

I have not been able to get from this point to the required relation

$\displaystyle (\lambda+\mu) C^{-1}_{KL}(C_{IM}^{-1} C_{JN}^{-1} + C_{IN}^{-1} C...<br /> ...<br /> (\lambda+\mu) C^{-1}_{MN}(C_{IK}^{-1} C_{JL}^{-1} + C_{IL}^{-1} C_{JK}^{-1})<br /> $

 

What am I missing?

 


Bibliography

 

1
E. H. Dill.
Continuum Mechanics: Elasticity, Plasticity, Viscoelasticity.
CRC Press, Boca Raton, 2007.

 

2
J. C. Simo and K. S. Pister.
Remarks on rate constitutive equations for finite deformation problems: computational implications.
Comp. Meth. Appl. Mech. Eng., 46:201-215, 1984.

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