Understanding Axial Strain Calculations

[Regards]

Hi all,

I have a problem with respect to axial strain.As far as I know the axial strain is calculated by using the formula given below.

Strain=(L-Lo)/Lo

where Lo : Orginal length and L : current length.

However, another formula is written in one of the program manuals I have been using, which is given below.

Strain= (L-Lo) / [(0.5)*(Lo+L)]

where Lo : Orginal length and L : current length.


My mind has a little bit complicated now. Has anybody ever seen the second formula ? And can anybady explain what the second formula is used for ?

I appreciate if someone could explain this post.

Have a nice day...

 

[AlephZero]

There are two standard measures for strain.

Linear (engineering) strain is (L-Lo)/Lo
Logarithmic strain (sometimes used for large strains) is ln(1 + L/Lo)

Your second formula is a pretty good approximation to log strain - see picture.

Why anybody would want to use the approximation, I don't know - unless it's meant for people who are not too happy about maths, and/or might get confused between common logs (to base 10) and natural logs.

 

[Regards]

Thanks for your response sir. Do you have the source for the picture you have attached. As I said, it is mentioned in a program manual which is Particular Flow Code in 3 Dimension(PFC3D). That is why it is used in logaritmic scale in my opinion.

Is there anybody else who knows the explanation of the second formula ?

Regards,

 

[AlephZero]

I plotted the graphs in a spreadsheet, using the formulas given in the thread.

The idea of logarithmic strain is based on adding up strain increments based on the current length, not the original length. E.g. imagine a 1in long bar of flexible material that is stretched to 2in long. Logarithmic strain says that an additional stretch from 2.00 to 2.02 is "the same increment of strain" as the initial stretch from 1.00 to 1.01. Engineering strain says a stretch of 2.00 to 2.02 is "twice as much strain " as a stretch from 1.00 to 1.01. If you take this to the limit and add up very small increments, you get the ln(1 + l/l0) strain formula. (See any textbook on strain measures for large strains for a reference)

Measuring the strain based on the average length not the original length, makes sense as a physical approximation to log strain. It's similar to evaluation an integral numerically, using just one increment of the trapezium rule. (You could derive it mathematically using a standard curve-fitting technique like Pade approximations, but that doesn't explain why it makes sense).

I don't know anything about the PFC3D code, so I've no idea why the code uses the approximation.

 

[Regards]

You are right sir. When I read the manual of the program again, I realized that it is used for the large starins indeed. I hadn't realized that before.Just because I don't know what is the logaritmic strain.

Let me explain the situation now. According to the model of the PFC code in the manual, one core sample which has 2 meter in diameter and 4 m in height was created.It is just to carry out the triaxial compressive strength test. And it needs to measure large strains for these dimensions. That is why the code has used logaritmic starin. However, according to my model, the dimensions of the core sample will be 13,5mm in diameter and 27 mm in height.So I will use the linear strain formula.

Thanks for your attention sir. Have a nice day...

Regards,

 

[FredGarvin]

One other thing you may want to check on other than the large deformation assumption is the rate at which the strain is applied. The log version may also help if there is a large strain rate.