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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?

Matt Lewis's picture

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 for reply and confirmation

 

Lihua Jin's picture

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. 

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?

Lihua Jin's picture

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

Einstein notion is always garbage, as Biot said

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". 

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 

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 

 

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