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Differences of various strain measures
Tue, 2009-07-07 18:03 - Upasara Wulung
For large deformation, we have various deformation measures such as Green-Lagrange tensor, Almansi strain tensor, left Cauchy-Green deformation tensor, and right Cauchy-Green deformation tensor. Appreciate very much any explanation in respect to physical differences of those measures, why they are necessary, conditions where one is more accurate than another! Thank you.
Upa-
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Re: Differences of various strain measures
In the beginning pages of my notes on finite deformation, I try to explain to myself some of the basic ideas in a one-dimensional setting. So far as I can tell, there is no real merit to introduce so many measures of strain. A student has to learn several measures simply because they are all in use. The situation is similar to that one needs to know the word "bathroom" in several languages if one travles around in the world a lot.
I'd like to know how other people think about the question raised here.
Re: Re: Differences of various strain measures
In his 1878 paper (pp.184-218), Gibbs characterized the state of strain by using the deformation gradient. The deformation gradient has nine components. He then noted (p. 204) that "the state of strain of any element considered withouit reference to directions in space is capable of only six independent variations." He went on to introduce six quantities to characterize the state of strain.
We now say that a material model should be invariant to rigid-body rotation. From this consieration, the deformation gradient is not a suitable measure of strain. Instead, we should use the Green deformation tensor, or something equivalent. This development is described in many modern textbooks, and is also described in my notes on finite deformation: general theory.
measures of strain
Dear Upasara:
This is a good question and I agree with Zhigang's opinion.
In continuum mechanics, there are two configurations that are physically important and different: 1) material configuration (manifold), which is where the body is stress free, and 2) current configuration (ambient space manifold) where the body lives in. Continuum quantities, and in particular, strain measures are defined with respect to these two configurations. It is possible to define a measure of strain with respect to either or both and this is why you would see several measures of strain. There are also conventions like subtracting an "identity matrix", etc from a given strain that make the choices even more (one should be careful with these, in general, when either of the two manifolds is non-Euclidean).
In short, all these measures of strains are intended to locally describe a deformation. In this sense, there is no preferred strain measure as long as things are defined consistently. When there are different and valid choices for local description of deformation, it is hard to say which one is preferred. I think all one needs is to know the exact connection between these equivalent measures of strain.
Hope this helps.
Regards,
Arash
How to get material properties for different stress and strain?
Hi all,
I have a related question. How can we get/modify the constitutive models (numerical values for the parameters) for use with various stress and strain measures? Generally, the experimental data (say Young's Modulus) relates infinitesimal strain to Cauchy stress. I feel it is not correct to use these values directly for other stress and strain measures... Thanks in advance for any suggestions/comments.
Jayadeep
Other stress and strain measures from experimental data
Hi,
I do not have any expertise in Non-linear Mechanics, but, I have started studying (learning) recently.
Refering to Jaydeep's question, experimentally (using UTM) one can estimate large strains (for example Green-Lagrange = (final length^2-initial lentgh^2)/(2*initial lenth^2) or Almansi strain) and Cauchy stress (Jaydeep's point). From these, one can compute the other stress and strain measures.
If I am wrong, please correct.
Thanks and regards,
- Ramadas