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Engineering strain vs logarithmic strain

Engineering Strain vs Logarithmic strain I would like to have your (experts) interpretation on Engineering strain and Logarithmic strain.  Based on what I’ve studied, I would request your comments on what I state below in regard to Engineering strain and logarithmic strain: 

  1. Engineering strain: We define “Engineering strain” as the change  in length by original length- assuming a bar of initial length 100 cms which stretches by 1 cm and becomes 101 cms.Then,

 Engineering strain = (101-100)/ 101 = 0.01 

  1. Whereas, we define the logarithmic strain as:

 Logarithmic strain = ln (L/Lo) = ln (101/100) = 0.00995 IS my interpretation correct if I say: Logaritmic strain is more realistic than engineering strain, because here we take the strain as the summation of numerous small differential segments and thus express the final strain whereas in Engineering strain it is just the strain over the whole length (just 1 segment) compared to numerous segments in Logarithmic strain. Please advise.

Zhigang Suo's picture

I discussed various definitions of stress and strain in my notes on finite deformation.  In particular, I wrote:  

When we call e the engineering strain, we do not mean that e is quick-and-dirty, or unscientific, or unnatural. We just wish to name the quantity (l-L)/L. Later on, we will describe motivations for some of the definitions, but these motivations are just elaborate ways to express preferences of individual people. The motivations, however elaborate, should not obscure a simple fact: you can use any one-to-one function of l/L to define strain.

There are several earlier, related, discussions, for example,

Eng. strain considers only the final state compared to the
initial state although the log. strain takes into account the continuous
variation of length. 

In fact, the first one can be seen as a simplification of the second one (even if eng strain cannot be always used):

If you define the strain as the relative variation of length: dEps = dl / l, and you can integrate it from l0 to l0 + Δl. You get the logarithmic strain Eps = ln( (l0 + Δl)/l0 ) = ln(1 + Δl/l0).

Then, if you consider small deformations, this expression can be reduced to EpsEng =  Δl/l0 because the limit of ln (1+X) is X when X -> 0.

So eng. strain and log. strain give the same value for small deformations but they differ a lot if you consider large deformation.

When large deformation are encountered, the initial length can be no more taken as reference to compute every strain increments (like in eng. strain), one have to take in account the value of length just before each strain increment (that is the idea of the log. strain).


1) So, can we say, like this:

in case of true strain, the sequential strains can be added, i.e.

ln(L1/Lo) + ln(L2/L1) = ln(L2/L0)

wherein, the same is not the casae in case with engineering strain,

i.e. L1-L0/L0 + L2-L1/L1 is not equal to L2-L0/ L0

2)Further, can anyone please help us here:

arash_yavari's picture

I agree with Zhigang that calling (l-L)/L "engineering" strain is a bit misleading.

I think the confusion arises in the 1D deformation of a rod when one defines strain for the final deformed configuration versus looking at intermediate states. First of all,  in the example of a rod in tension (or compression for small enough loads) there is a uniform state of strain and (l-L)/L is a valid measure of strain even in the nonlinear regime. Now having a constitutive equation in terms of (l-L)/L it can be converted to one in terms of ln(l/L) and vice versa as "ln" is a one-to-one function (in agreement with Zhigang's statement). So, either one can be used.

In the 3D case, one starts with a local description of motion in terms of deformation gradient (or other measures of strain that are related to deformation gradient). One can define a logarithmic strain but I don't think it's necessary. Perhaps one may simplify certain things using a logarithmic strain but the point is that there is nothing fundamental about such a choice. I would like to know other opinions in this regard. One can define functions of matrices (or operators) for some classes of functions (let's say smooth) and matrices (let's say diogonalizable). Such definitions can be useful, e.g. exponential of a matrix, in solution of systems of ODEs.


I recently had to rethink the issue of appropriate strain measures while trying to develop a constiutive model.  My conclusion is that we should try to develop these modesl using true stress and "engineering" strain (not small strain) and then do appropriate conversions during application to retain energy conjugacy.

Some of my observations and questions regarding log strains are:

1) One of the problems I have run into involves dealing with pure shear experiments. Shear strains (change in angle) observed in these experiments don't seem to have a straightforward logaritmic version that can be translated into Hencky strains for 3D models.  Does anyone know of a map between angle changes and "logarithmic" shear strains?

2) One of the conveniences of logarithmic strain measures is the possibility of additive separation of volumetric and isochoric deformations.  The Hencky strain measure is also energy conjugate to the Kirchhoff stress and therefore can be convenient in working out dissipation inequalities.  But is the extra effort worth it?

Please comment.

-- Biswajit

Sir, can anyone of you explain the physical significance of Green's strain?

Agreed, logarithmic strain is the true strain and has a mathematical foundation i.e.integral [dL / L] limits- Lo - L gives logarithmic strain.Similarly, what is the mathematical background of Green's strain?

Green-Lagrange strain, as a function of new length and old length, closely approximates logarithmic strain for sufficiently small strains.  However, its main advantage arises in the corresponding strain-displacement relations.

The definitions of axial and shear strain, as presented in an engineering mechanics course such as "Strength of Materials", are a linearized form of Green-Lagrange strain, only the linearization is with respect to changes in displacement*.  As a consequence, pure rotations in the linear case may predict artificial strains, and through Hooke's Law, stresses.  Green-Lagrange strain accounts for the rotation and returns zero strains for rotation without deformation.


*as opposed to the conception of engineering strain as being linear with respect to the deformed length.  The two are the same for an axially loaded 1D continuum (e.g. bar, rod, truss member, beam, shaft, etc.) but not necessarily for 2D & 3D continua.


 For the analysis of finite deformation problems, we often use Green-Lagrange strain as a strain meausre. One of the advantages of using Green strain is that we can also model rigid body motions as well. However, the question is how much it is suitable to use Green strain for large deformation problems, where we know that it shows flexible respense as compare to other strain measures. The same unsuiatability can be seen, when we see stiff response on mesh refinement.

 Any Comments on the subject will be welcomed 


Hello, everyone, I am writing my Bachelor thesis. It's about the Simulation using ANSYS Workbench to calculate the Strain of Model.

My question is: I can use ANSYS Workbench to calculate the Equivalent Strain von Mises, but the Thesis requires the "Logarithmic Strain".

So I wonder, is there a connction between these two Strains? Or is thers any convertion bwtween these two Strains.


Thank your for your Help.

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