$C=F^TF$ is called the "Right" Cauchy-Green tensor, and $b=FF^T$ is called the "Left" Cauchy-Green tensor.

I suppose in $C=F^TF$ the non-transposed $F$ stands on the right, and in $b=FF^T$ it stands on the left, but I guess there's gotta be more to it? Is there any reason for why they are called "right" and "left" specifically?


1 Answer 1


I would guess that it has to do with the polar decomposition of the deformation gradient ($\mathbf{F}$)

$$\mathbf{F} = \mathbf{R} \mathbf{U} = \mathbf{V}\mathbf{R}\, ,$$

with $\mathbf{R}$ an orthogonal tensor, $\mathbf{U} = \mathbf{C}^{1/2}$ the right stretch tensor, and $\mathbf{V} = \mathbf{b}^{1/2}$ the left stretch tensor.

Both, $\mathbf{C}$ and $\mathbf{b}$, should be positive definite tensors for the square root to be well defined.

  • $\begingroup$ Slightly related concept, but definitely more convincing than my version. I'll take it :) $\endgroup$
    – MaxD
    May 27 at 20:19

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