## You are here

# How to differentiate left and right cauchy deformation tensor in biomechanics modelling?

Hi, dear all,

I am facing problem on differentiating left Cauchy deformation tensor and right Cauchy deformation tensor.

thanks for notes in http://www.engin.umich.edu/class/bme456/ch3strain/bme456straindef.htm, i can derive the them in terms of mathematical equation, but i cannot differentiate them in engineering modelling filed.

From what i read is

right Cauchy deformation tensor is equal to the square of the right stretch tensor?

left Cauchy Deformation tensor is the square of the left stretch tensor ?

What is signifcance between 'left' and 'right' when using it to defining hyperelastic material properties in terms of invariant form?

## Left and right Cauchy-Green tensors

Wei,

The left and right Cauchy-Green tensors are related simply by a rotation. As such, the invariants of the two tensor are the same.

Matt Lewis

Los Alamos, New Mexico

## Hi, Lewis, Thanks

Hi, Lewis,

Thanks for reply and confirmation

## The right Cauchy-Green

The right Cauchy-Green tensor is in the reference configurtion, while left Cauchy-Green tensor is in the current configuration. Cauchy stress (true stress) can only be a function of the left Cauchy-Green tensor.

## tensorial calculus

I have started performing tensorial calculus myself but, admittedly, I am having a hard time doing so. Are there any notes that the experts here think might be not only of theoretical but practical use as well? Also, is there a matlab/mathematica package that allows one to compute derivatives (such as the ones mentioned by the originator of the thread) out?

## To me the best way to do

To me the best way to do tensorial calculus is to take derivative to components, but not a tensor.

## good point

Einstein notion is always garbage, as Biot said

## tensorial calculus

You may take a look this book, if have not read it already:

Mikhail Itskov, 2007, "Tensor Algebra and tensor Analysis for Engineers, with applications to continuum mechanics".

## Left and Right Cauchy-Green Tensors

Wei,

You said you want to take the derivative but you did not say derivative with respect to what, for example time, coordinate variables...

Mohsen

## proof of left cauchy green tensor being in current configuration

Hi

As you said that "The right Cauchy-Green tensor is in the reference configurtion, while left Cauchy-Green tensor is in the current configuration. Cauchy stress (true stress) can only be a function of the left Cauchy-Green tensor"

Please tell me how can I prove that left cauchy green tensor is for the current configuration.

Thank you