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What is the physical meaning of Green-Lagrangian strain and Eulerian-Almansi strain measures?

Sundaraelangovan selvam's picture
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Hello, researchers. I have difficulty in understanding the physical meaning of Green-Lagrangian strain (E) and Eulerian-Almansi strain (A) measures. Mathematically speaking, I can derive the equations of these strains in different ways. But physically speaking, it's a bit harder to understand how these strains (E and A) can be pictured and how to give a proper physical definition for them. In a simple case, considering a uni-axial bar (Please refer the attached file), Engineering strain can be understood easily, but in E and A equations, from where do the squares of the lengths originate? and how does it came into the picture?. or Is E and A are the true strain measures and engineering strain is the linearization of E and A?. How to understand this? I referred several books and online sources, but I failed to understand the clear interpretation of these strain measures (E and A). Would be much appreciated if someone explains me this or give me the useful references to read. Thanks in advance.

Ruobing Bai's picture

Hi Selvam,

I'm assuming you are talking about the case of finite deformation, since in linear elasticity they would be the same. The straightforward answer is: the only real physical meaning they provide is that they all measure the deformation of a material. Of course such deformation can be represented in different mathematical forms, but these various forms do not provide extra physics. For instance, in addition to the three measures you mentioned, you can also measure deformation using the true strain (or logarithmic strain).

I'd like to further make a point that different strain forms surely are related to different stress forms. For example, you can measure or describe the stress state of a material using the nominal stress, or the true stress etc. Again, they all describe the same stress state of the material, but add to no extra physical meanings (you can argue that nominal stress is the force devided by the area in the reference state while true stress is that devided by the area in the deformed state, but such differentiation is just from definition rather than different physics). As long as you keep consistency when applying the constitutive model in your study, you can use any form of the stress/strain.

Back to the three strain forms you mentioned. If you look at them in tensor forms, the engineering strain (corresponding to F - I in tensor) will not exclude rigid body motions, while the other two will. You can even find more strain measures here:

https://en.wikipedia.org/wiki/Finite_strain_theory

Sundaraelangovan selvam's picture

Hi Ruobing,

Thank you so much for sharing your ideas. It is very much useful. I did have an intuition before of whatever you said. But I haven't convinced myself that my understanding is completely true. What you said totally makes sense in many aspects. Thanks again. 

rajan_prithivi's picture

when I thought about this when I took the course - I got this point 

You may think that the strains are mathematically formulated to maintain consistency which is , a rigid body motion or rotaion wont induce any strain in the material. This may have been a reason to start with squares and further go on to get the  strain in form of FF' which chops off the rigid motions.

-Prithivi

Sundaraelangovan selvam's picture

This is the catchy and simple answer that I got till now about this question. It seems totally makes sense. I wish to make some more research into it. Thanks :) 

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