Deriving displacement tensor from Hencky (true) strain tensor

[FQVBSina_Jesse]

TL;DR Summary

Calculating full displacement tensor from strain tensor using Hencky (true) strain definition.

The Hencky strain, AKA true strain, logarithmic strain, can be related to displacement tensor as follows:

$$
E = ln(U)
$$

However, Hencky strain is typically done only for principal strains. This can be easily shown by actually trying to calculate the full displacement tensor using the above definition:

$$
U = e^E
$$

Typically we have E<1, and if I am looking at a strain rate, then it is E<<1, the above equation will return a displacement tensor with a value around 1 in all components when only the principal components should be close to 1. I haven't found any literature discussing the proper correction to calculate the shear terms, specifically the Hencky definition of strain and displacement. Does anyone know more about this topic?

 

[Chestermiller]

What about expanding exp(U) in a Taylor series using the Caley-Hamilton theorem?

 

[Chestermiller]

TL;DR Summary: Calculating full displacement tensor from strain tensor using Hencky (true) strain definition.

The Hencky strain, AKA true strain, logarithmic strain, can be related to displacement tensor as follows:

$$
E = ln(U)
$$

However, Hencky strain is typically done only for principal strains. This can be easily shown by actually trying to calculate the full displacement tensor using the above definition:

$$
U = e^E
$$

Typically we have E<1, and if I am looking at a strain rate, then it is E<<1, the above equation will return a displacement tensor with a value around 1 in all components when only the principal components should be close to 1. I haven't found any literature discussing the proper correction to calculate the shear terms, specifically the Hencky definition of strain and displacement. Does anyone know more about this topic?

The displacements determine the components of the strain tensor, not the other way around.