Tensorial shear strain is given by its tensorial definition: e_ij = 1/2(u_i,j+u_j,i) (here, "," means derivative).

Engineering shear strain is two times tensorial shear strain, which represents the total shape change in the x_i-x_j plane.

I'm not sure about the "true shear strain", maybe a correction for engineering shear strain in measurement, something like true elongation strain vs engineering elongation strain?

Dear Xijia

Thanks for contributing to this question, I agree with your answer.  Nevertheless, would you mind if I turn it around and give the answer as:

The Tensorial Shear Strain is 1/2 of the Engineering* and True Shear Strains.  

*Applies for small strains at which Engineering Shear Strain ~ True Shear Strain.  

Do you think the above is a correct definition of Tensorial Shear Strain, at least without needing to go into the detail of what constitutes a valid strain tensor?  

Next we need to define what we mean by Engineering Shear Strain and True Shear Strain.  Here I am not sure if there is a analogy between the axial strain definitions where Engineering Strain is extension/original length and where True Strain and Logarithmic Strain are used interchangeably.  

In my mind there is confusion because I have a metallurgists definition of shear strain rather than a structural mechanics definition:  I think there may to two schools of thought:  

1. Sometimes with axial strain people think of Engineering Strain as an approximation to True Strain which applies when total strains are small.  Is this also the way in which people think of True Shear Strain and Engineering Shear Strain.  If so is:

Shear Strain = Change in angle in radians between two initially perpendicular lines

Engineering Shear Strain = x/L  , which is a approximation that holds for small values

                                             where x is the deflection in x-direction and L is the length in the y-direction (pictures to follow).  

and

True Shear Strain = arctan(x/L) ,   which holds for small and large values.  

In this first school of thought Engineering and True Shear Strains are clearly not the same thing and the term Tensorial Shear Strain could apply to either 1/2(Engineering Shear Strain) or 1/2(True Shear Strain) but not both at the same time.  Somebody with this shcool of thought would be confused by the statement "that Engineering shear strain is two times tensorial shear strain".  Particularly, when what is often presented as gamma_xy=du_x/dy+du_y/dx actually looks to them as a definition of True Shear Strain rather than Engineering Shear Strain!  

In which case I would suggest that for people who's minds are set in School 1 would be much happier with the statement that

"Tensorial Shear Strain is 1/2 of the True Shear Strain"

And actually forget the term Engineering Shear Strain which in this case I think is confusing as metallurgists and general engineers might think of as = x/L   (rather than the more general definition of gamma_xy=du_x/dy+du_y/dx, which for me is better suited to define True Shear Strain).  

Michael

Your view is interesting. It is more vivid in real sense about the change in angle.

Coming back to "tensorial" strain, I would like to point out that it comes from the description of movement of a point in a 3D solid body in the Lagrangian coordinate system, where any displacement can be expressed as the first-order derivatives times the coordinate delta plus the second-order derivatives times the delta square and son on...The first-order derivatives then split into two parts: the first part is rigid body rotation, and the second part is "deformation", the latter has tensorial properties by nature. So, in a sense, tensorial definition only applies to small deformation, or where the displacements are strictly linear functions of the coordinates. Within a small material element, this can be the case, so you have constant strains in a small element (think of the tetrahedral element in FEM).

In engineering, when we measure the strain in a coupon, we have to be sure that the measurement is done in a region that is deformed uniformly. It gives people a real sense of deformation, and it is often used to characterize material property relationships,

In my opinion, the definition of whether "engineering" or "true"  all regards to measurement, for better characterization of material property. For elongation, I see the need of a "true" property since the true stress changes with elongation. I'm not sure in shearing there is a need to. The shear strain definition may be interpreted as  the change in angle, giving a vivid picture; but actually, understanding where it comes from, it is not about the angle per se. So, I think, the arctan definition is questionable.

Dear Zheng-Kun Liu

This has the beginning of a good answer, however, I guess your device did not save all of what you wrote.  Can you please expand your answer.  

Many thanks

Dear Xijia

Thank you for your answer.  I can see all of the points that you made.  I guess I am coming at it from the  point of view of somebody who's experience is in manipulating the results of mechanical tests.  For example, in a torsion plus tension test on a thin cylinder it is possible to measure the axial extension and therefore calculate both engineering and true axial strains.  It is also possible to measure the angular rotation and therefore calculate the shear strain.  In this case also there are options to use engineering (small displacement approximations for) shear strain or true (large displacement formulae for) shear strain, which take the change of axial length into account.  This all looks deceptively straight forward but it is necessary to be careful to avoid errors.  

The problem then gets worse we try to compare the results of such tests with FEA predictions.   The logical thing to do for such multiaxial cases is to compare von Mises Stress and Strain as determined from tests and as predicted by FEA.  

However, it then gets confusing because text book formulae for von Mises strain differ in their form for adding in the shear strain components, either 6(sum squares of shear strains) or 3/2(sum squares of shear strains) depending on whether the formulae have been written down for tensorial shear strain or for engineering/true shear strains.  This is clearly because the tensorial shear strain is 1/2 of the other definitions of shear strain.  However, as it is not always clear which is which there is the potential for errors to be made.  

For example, some FEA codes output tensorial shear strain and others output engineering shear strain.   It is an error trap waiting to catch the unwary.  

Hence, my question I want to clear up the confusion and avoid this error trap.  

Hi Zhengkun

Thanks for the clarification.  What does Konstantin think about this question?  

Michael, thanks for bring the question up. I learned something from it.