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Derivative of Logarithmic Strain

Some of you probably work on problems that involve moderately large strains. An useful strain measure for such problems in the logarithmic or Hencky strain. In particular, if you deal with the numerics of large strain simulations, you will often need to compute the material time derivatives of logarithmic strains.

In the mid 1980s, Anne Hoger , Donald Carlson , Morton Gurtin and their coworkers wrote a series of papers on issues involving such strain measures, how to take their derivatives, and how to solve linear equations containing tensor quantities. Most of this work originated at the Carnegie Mellon University in Pittsburgh, Pennsylvania.

This post is meant as a reminder that the formulae for the material derivatives of the logarithmic strain depend on the number of independent eigenvalues of the stretch tensors. It should also act as a quick reference for looking up some of those formulae.

Recall that the deformation gradient tensor, because it has a positive determinant, can be multiplicatively decomposed into a symmetric part and an orthogonal part, i.e.,
 

$\displaystyle \ensuremath{\boldsymbol{F}}= \ensuremath{\boldsymbol{R}}\cdot\ens...<br />
			...{\boldsymbol{U}}= \ensuremath{\boldsymbol{V}}\cdot\ensuremath{\boldsymbol{R}}~.$

 

In terms of the stretches, the generalized n-th order material and spatial measures of strain are
 

$\displaystyle \ensuremath{\boldsymbol{E}}^{(n)} = \cfrac{1}{n}\left(\ensuremath...<br />
			...\ensuremath{\boldsymbol{\mathit{1}}}- \ensuremath{\boldsymbol{V}}^{-n}\right)~.$

 

The spectral decompositions of the stretches can be written as
 

$\displaystyle \ensuremath{\boldsymbol{U}}= \sum_{i=1}^{3} \lambda_i~\ensuremath...<br />
			...th{\mathbf{n}}_i = \ensuremath{\boldsymbol{R}}\cdot\ensuremath{\mathbf{N}}_i ~.$

 

Therefore we may also write the strain measures as

 

$\displaystyle \ensuremath{\boldsymbol{E}}^{(n)} = \sum_{i=1}^3 \cfrac{1}{n}\lef...<br />
			...math{\ensuremath{\mathbf{n}}_i\boldsymbol{\otimes}\ensuremath{\mathbf{n}}_i} ~.$

 

If you take the limit of these as n goes to zero, you will get the material and spatial logarithmic strains (prove this for fun)
 

$\displaystyle \ensuremath{\boldsymbol{E}}^{(0)} = \sum_{i=1}^3 \ln\lambda_i~\en...<br />
			...oldsymbol{\otimes}\ensuremath{\mathbf{n}}_i} = \ln\ensuremath{\boldsymbol{V}}~.$

 

Note that both the material and spatial strains have the same eigenvalues in this case.

Recall that the Cauchy stress and the rate of deformation tensor (better called the stretching tensor?) are power conjugate. So the question that arises is: what quantity is power conjugate to the material derivative of the Hencky strain, i.e., find the stress measures such that

\ensuremath{\boldsymbol{d}}~.$

 

 

It turns out that when the principal axes of strain remain fixed then
 

$\displaystyle \dot{\overline{\ln\ensuremath{\boldsymbol{V}}}} = \ensuremath{\bo...<br />
			...\widetilde{\ensuremath{\boldsymbol{\tau}}} = \ensuremath{\boldsymbol{\sigma}}~.$

 

However, this is not true in general as the following formulae from [1] show. The detailed expressions are quite complicated and you can find them in Hoger's paper. 

  1. Case 1: $ \lambda_1 = \lambda_2 = \lambda_3$
     

    $\displaystyle \dot{\overline{\ln\ensuremath{\boldsymbol{V}}}} = \ensuremath{\boldsymbol{d}}~.$

     

  2. Case 2: $ \lambda_1 = \lambda_2 \ne \lambda_3$
     

    $\displaystyle \dot{\overline{\ln\ensuremath{\boldsymbol{V}}}} = \theta_1~\ensur...<br />
	...ldsymbol{V}}- \ensuremath{\boldsymbol{V}}\cdot\ensuremath{\boldsymbol{\Omega}})$

     

  3. Case 3: $ \lambda_1 \ne \lambda_2 \ne \lambda_3$
     

    \begin{equation*}\begin{aligned}\dot{\overline{\ln\ensuremath{\boldsymbol{V}}}} ...<br />
				...symbol{V}}\cdot\ensuremath{\boldsymbol{\Omega}}) ~. \end{aligned}\end{equation*}

     

Also,

  1. Case 1: $ \lambda_1 = \lambda_2 = \lambda_3$
     

    $\displaystyle \dot{\overline{\ln\ensuremath{\boldsymbol{U}}}} = \ensuremath{\boldsymbol{R}}^T\cdot\ensuremath{\boldsymbol{d}}\cdot\ensuremath{\boldsymbol{R}}~.$

     

  2. Case 2: $ \lambda_1 = \lambda_2 \ne \lambda_3$
     

    $\displaystyle \dot{\overline{\ln\ensuremath{\boldsymbol{U}}}} = \ensuremath{\bo...<br />
				...ol{d}}\cdot\ensuremath{\boldsymbol{V}}\right]\cdot\ensuremath{\boldsymbol{R}}~.$

     

  3. Case 3: $ \lambda_1 \ne \lambda_2 \ne \lambda_3$
     

    \begin{equation*}\begin{aligned}\dot{\overline{\ln\ensuremath{\boldsymbol{U}}}} ...<br />
				...bol{V}}^2)\right]\cdot\ensuremath{\boldsymbol{R}}~. \end{aligned}\end{equation*}

     

Please comment if you know of any further simplifications and developments in this regard in the last 20 years or if you need further clarification on any of the quantities used in these equations.
 


Bibliography
 

1
A. Hoger.

The material time derivative of logarithmic strain.

Int. J. Solids Structures, 22(9):1019-1032, 1986.

Comments

Andrew Norris's picture

Hi Biswajit,

The main development in the past 20 years IMO is the proof of Xiao et al. [1] that the relation

always holds if you allow the time derivative to become corotational and objective.   There are two parts to this that scare most people: corotational and objective.   But corotation is reasonable - it just means that the rate is taken w.r.t. a frame that is rotating relative to where you are sitting.  The objective constraint means that this extra rate of rotation, or spin, should be defined only by the underlying spins in the problem and by V.  This again is reasonable, since what else are you going to use if it is to have general meaning.   By underlying spins I mean things like W, which is the skew part of L, the tensor of which d is the symmetric part, L = \dot{F}F^{-1} = d+W (sorry - no time for latex today). 

Xiao et al. [1] show that there is a unique spin - they call the logarithmic spin - which ensures the equivalence above for every possible V.   In a series of papers they explore this in many directions.  The issue of conjugate stress is best dealt with in [2] (a rather obscure paper, but well worth reading).   They show that the whole idea of conjugate stress and strain, which is normally considered in the Lagrangian sense (see eg. the extensive discussion in Ogden's book - the best primary source for this topic) can be generalized to Eulerian stress and strain.
 

Andy 

refs:

[1] Xiao, H. and Bruhns, O. T. and Meyers, A., Logarithmic strain, logarithmic spin and logarithmic rate. Acta Mechanica 124, 89--105, 1997. doi = 10.1007/BF01213020

[2] Xiao, H. and Bruhns, O. T. and Meyers, A., Objective corotational rates and unified work-conjugacy relation between Eulerian and Lagrangean. Archives of Mechanics,

50, 1015--1045, 1998.

 

Andy,

Thanks for pointing me to the Xiao, Bruhns, Meyers papers.   A nice overview of how their work ties in the finite deformation elastoplasticity can be found in their review article "Elastoplasticity beyond small deformations", Acta Mechanica, 182, 31-111 (2006), DOI 10.1007/s00707-005-0282-7.

One thing that I notice is that these authors start off with an addtive decomposition of the Eulerian stretching tensor (rate of deformation tensor) when they do elastic plastic deformation.  Many other authors say that such an decomposition is unphysical and one should strictly use a multiplicative decomposition of the deformation gradient and then derive an additive decomposition of the stretching tensor accordingly.  Which is the correct approach?

Also, people try to avoid rate form equations of elasticity (particularly Truesdell's hypoelasticity) and favor approaches such as hyperelasticity or equations of state (which can be derived from a potential in most cases).  Do you have any insights into which is better and why?

Biswajit 

Amit Acharya's picture

Biswajit,

 This is a must-read to get the right perspective on these matters - Advances in Applied Mechanics. It does not deal with the eignevalue coalescence issue but there is so much more that is well worth it.

 As for objective rates - what really matters is the transformation between such rates so any can be converted to the other. Ultimately, it is a matter of physics that decides what description is best. For example, you would have noticed that hyperelasticity corresponds to the Truesdell rate, when the stress constitutive equation is posed in terms of rates.

 - Amit

Thanks Amit.  Hill's paper does indeed clarify a number of issues and deserves a careful read.  As I understand it, the basic issue is whether to start with a nonlinear elastic model from the get go or to use an extension of the linear Hooke's law.  The inconsistencies in the various objective rates arise because Hooke's law is applied inappropriately.   I would prefer to start with a hyperelastic model and then derive rate equations from that (if necessary).  However, there is no unique way of making such rate equations objective.  Which is why many people avoid rate forms altogether.

I didn't realize that the Truesdell rate leads to hyperelasticity.  Since the Jaumann and Green-Naghdi rates are just particular cases of the Truesdell rate and they don't lead to hyperelasticity, I don't see how the Truesdell rate can.  There is a good discussion (with references) on the matter in Simo and Hughes, Computational Inelasticity, p. 258.  In fact, there is a strong set of Bernstein compatibility conditions that must be satisfied for an objective rate equation to be derivable from a potential.

One the other hand, to quote Lin, Brooks, and Betten, Int. J. Plasticity, 2006, 22, p. 1830, "In the works of Xiao et al. it is demonstrated that only the hypoelastic equations  using the logarithmic corotational rate of the Kirchhoff (or Cauchy) stress is consistent with elasticity."  That's a pretty strong claim and requires further examination.  Note that the physical meaning of the logarithmic corotational rate suffers from the same inqdequacies as all other objective rate equations of elasticity.

Amit Acharya's picture

Biswajit,

 I do not find the time to keep up with imechanica so this late response, as I just saw this note of yours. The Truesdell rate is simply the time derivative of the pull back of the Kirchhoff stress with the deformation gradient, pushed forward by the deformation gradient multiplied by J (or J^-1 I forget which) - in other words, it is basically th Lie derivative of the Kirchhoff stress upto a mutiple of j or J^-1.

 If you take the standard hyperelastic constitutive equation for Cauchy stress, take a material time derivative, and then look for what objective rate you can pose it as most naturally, I believe you will end up with the Truesdell rate.

where is free energy per unit mass

then

therefore  

and if you write out the left hand side in terms of cauchy stress and J you will get the Truesdell rate times J (i believe), so you divide by a J, and you are done. The rhs is the consitutive eqn. for the truesdell rate (again mutliplied of divide dby a suitable factor) as suggested by hyperelasticity

 

 

Amit,

Thanks for the comment.  This is probably going to be drowned in the spate of comments by Henry Tan's students but I'd like to mention that I've converted some of the notes from my finite elements course into wiki form and put in on Wikiversity.  The link is

http://en.wikiversity.org/wiki/Nonlinear_finite_elements/Objective_stress_rates

(n.b.  I converted my LaTeX files using an automatic LaTeX to Wiki convertor that I wrote.  There may still be bugs in the convertor.  If you find anything that looks nonsensical, please correct it.) 

Regarding  corotational Eulerian strain rates, the following paper by Andy Norris is a great resource and puts a number of things in context.

Eulerian conjugate stress and strain , [ps]
arXiv:0708.2736

 

Biswajit 

Amit Acharya's picture

Biswajit,

So do you agree that the natural objective rate suggested by isotropic hyperleasticity is the truesdell rate of Kirchhoff stress?

- Amit

Amit,

As you had mentioned earlier, the Truesdell rate is the Lie derivative of the Kirchhoff stress (with the push-forward/pull-back operations being conducted with the Jacobian of the total motion).  So your question can be restated as: Does that particular Lie derivative lead to a "natural" objective spatial rate?   I'm not sure.

John Dienes has, over many years (e.g. here and here) , advocated a Lie derivative that uses only the rotational part of the motion in the push-forward/pull-back (i.e., a corotational rate which is equivalent to the Green-Naghdi rate).   Many codes used by the national labs in the US use Dienes' approach.  To my naive mind, that approach seems more natural because I have a feeling I can visualize what's happening and it seems intuitive.  However, just using the rotation seems mathematically unnatural in the context of the Lie derivative of the motion.

Regarding isotropic hyperelasticity, I feel that we can choose any rate that we find convenient.  The issue is not the rate per se but how we can find the appropriate material constants for a spatial rate equation.  If we start off with an St. Venant-Kirchhoff material, then the spatial rate equation depends on the  deformation gradient.   Or we can start with a total constitutive equation in the spatial configuration and get a material rate equation that depends on the deformation gradient.

I think the easiest way to proceed is to do experiments to find a total constitutive equation in the material  or spatial configuration and then use whatever rates and measures are needed for work conjugacy in the spatial/material configuration.    

Biswajit 

Amit Acharya's picture

Biswajit,

I agree with most what you say in terms of content except the rephrasing of my question: NO, I was not addressing any question of a 'natural' objective rate. There is none as the transformations between the rates is what matters (as I had said in an earlier post on imechanica) and the relationships between them are linear in the velocity gradient etc., etc.

So I am absolutely not interested in the question of a natural  objective rate - I think it is a non-question. But I do think that isotropic hyperleasticity leads most naturally to the Truesdell rate in the first instance (and probably the reason why Truesdell suggested it).

On the matter of total constitutive equations, I differ with you. It may absolutely not be possible to determine a total consitutive equation for an initially stressed body, the origin of whose initial stress is unknown. Hypoleasticity for such a body can still be measured - and this was one of the main reasons behind Biot's fundamental work on incremental deformations from arbitrarily stressed states.

- Amit

Amit,

I was confused by the term "natural" that you had used.  I went back and re-read one of your earlier comments where you showed that the Truesdell rate can be derived consistently from a hyperelastic constitutive relation.    I agree that the Truesdell rate appears to be natural.  

I was not aware of the reasons behind Biot's work on incremental defomations.  In fact, I have not read any papers by Biot other that those on poroelasticity (though I know that Biot is cited quite a bit in the literature on material stability).  Can you point me to some of the relevant papers by Biot - or are you talking about the 1965 book called Mechanics of incremental deformations?

 

I am more concerned about isotropic materials with two elastic parameters which are independent of  the deformation gradient (or the Cauchy stress - for hypoelastic materials).    If you look at the literature, most rate equations for elasticity use a hypoelastic model of grade 0 - which is clearly not elastic, i.e., the model cannot be derived from a stored energy potential.

Clearly, if we need a rate equation to describe our material and also expect our material to be elastic, an easy way out is to use a hypoelastic model of grade 1 (see, for example, eq. 99.19, p. 404 in Truesdell and Noll, Nonlinear field theories of mechanics).  I just cannot figure out how to design an experiment to determine the five dimensionless material constants needed by that model using known experimental stress-strain data.

Biswajit 

I've read your comments with interest, and being fairly new to problems of large elastic strain would welcome some clarification on which is the appropriate conjugate stress for logarithmic strain in the context of large (i.e. stretch ratios up 1.4) rate-independent elastic strain.

hy

i am trying to get strain of my model and compare it to the experimental result obtained by strain gauge.

so,i dont know get which kind of strain  from output history.

thanks.

 

Andrew Norris's picture

Biswajit,

 I am really not qualified to give definite answers as I have not thought or done much on either topic. My general feeling though is that if there are proponents of both multiplicative and additive decomposition, then there must be some advantage in each.  Every different decomposition is just another kinematic description, and should be welcomed!   That reminds me of a very nice paper from last year: 

He, Q. C. and Zheng, Q. S., Decomposition of Large Incompressible Deformations, J. Elasticity, 2006. DOI 10.1007/s10659-006-9080-2

The authors show that every incompressible deformation can be decomposed into three perpendicular simple shears preceded or followed by a rotation.  Thats an example of a decomposition that provides new insight into the kinematics of deformation.  Another nice instance of that is the beautiful result of Boulanger and hayes that extends the whole notion of polar decomposition: 

Boulanger, Ph and Hayes, M., Unsheared triads and extended polar decompositions of the deformation gradient, International Journal of Non-Linear Mechanics, 36, 399--420, 2001.

 They show that polar decomposition is simply a specical case of a much more general type of decomposisiotn based on triads of material directions: unsheared triads. 

 As for hypoelasticity, in my limited experience it can arise from microstructural models, e.g. granular assemblages (soil models), so it seems natural to use it in those cases.  Its obviously not appropriate for most situations. 

 Maybe someone who has first hand experience in these matters can comment. 

 Andy 

Andy,

Thanks for the references.  I've posted a blog entry on hypoelasticity and its problems at http://imechanica.org/node/1652 .   It's not comprehensive but should give our readers an idea about the issues at stake.

I love to do those confusing problems and it is amazing to know that there is a method

to compute the level of complexity and confusion developed. I am going to try this

method to compute the compute the complexity of my favorite problems.

 

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