Deriving displacement tensor from Hencky (true) strain tensor

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Hencky strain, also known as true or logarithmic strain, is related to the displacement tensor through the equation E = ln(U). It is primarily applied to principal strains, but calculating the full displacement tensor using this definition can yield inaccurate results, particularly for shear terms. When E is less than 1, especially in strain rate scenarios, the displacement tensor components tend to approximate 1, which is not reflective of the actual principal components. There is a noted lack of literature addressing the necessary corrections for shear terms in the context of Hencky strain. The discussion highlights the need for further exploration into the relationship between displacement and strain tensors.
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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?
 
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What about expanding exp(U) in a Taylor series using the Caley-Hamilton theorem?
 
FQVBSina_Jesse said:
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.
 
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