# How is non-uniaxial strain defined?

1. Nov 5, 2006

### scott_alexsk

Hello,

How is non-uniaxial strain defined? How would Nitinol's 8% maximum strain before failure be defined as in terms of other than stretch? How does one define the 8% in terms of bending the wire? So in other words how much bending of the wire equates to an 8% strain limit?

Thanks,
-scott

2. Nov 5, 2006

I know I'm not sounding very specific, but you may want to look at strain tensors.

3. Nov 5, 2006

### scott_alexsk

Just, looking at that right now on Wikipedia, it does not seem to concern bending as a type of strain, just shear strain and uniaxial strain distortions.

-scott

4. Nov 5, 2006

Well, you'd need some laboratory tests to get the relation between stress and strain in the case of bending.

5. Nov 5, 2006

### Drachir

When a material bends, the outer curved suface is in tension and the inner curved suface is in compression. There is a neutral position between those two surfaces where there is no stress. For most materials the compressive strength is greater than the tensile strength. Therefore most bent materials fail in tension. Bending failure ususally occurs where tensile force is greatest. Stress-raisers, such as surface defects, can cause up to a three fold increase in stress.

Failure in shear is less common, but that is what makes cutting operations possible. Failure in torsion is often a a shear failure.

The measurement of stresses, the design of structures, and the analysis of their failures are the subjects included in courses on the mechanics of materials.

I hope this points you in the right direction.

6. Nov 5, 2006

### Gokul43201

Staff Emeritus
Flexural strain is defined (typically) as the strain in the outer (convex) surface, the surface which is under tension (for the reason stated by drachir, above).

Note: You can estimate the curvature that makes an 8% strain from the cross-sectional geometry and making the neutral axis lie along the cetroidal axis of the wire.

If you tell us the geometry of your wire, someone here can show you how to do the calculation.

Last edited: Nov 5, 2006
7. Nov 6, 2006

### scott_alexsk

The lenght of my wire is 25 cm and its diameter is 0.3175 cm. The 8 percent strain is actually not the point at which the Nitinol wire fails but the point that the wire is plastically deformed. I have also seen the point of plastic deformation written as 70 MPa. Does this basically mean that 70 MPa will create an 8% strain in most cases?

I am going to be measuring the force differently treated samples of nitinol can pull with. The key is that I do not get above 70 MPa's pulling from the spring balance, so that the wire is not plastically deformed. The spring balance is measured in Newtons. How can I convert the 70 MPa's into newtons pulling uniaxially on the sample of Nitinol wire previously described?

Thanks,
-scott

8. Nov 6, 2006

### Gokul43201

Staff Emeritus
This is confusing! Are you looking at two different modes (i) flexure (bending), and (ii) tension (stretching) ?

If you want to use the nitinol to "pull", then you are interested in tensile failure (yield point in tension). To convert from force in N to uniaxial stress in Pa, simply divide the force by the relevant area in sq. meters. A MPa is just 10^6 Pa. For tensile (stretching) loads, the area of interest is the cross-sectional area of the wire.

PS: If 70MPa and 8% are the yield point and the strain at yield for the exact same alloy, then yes, the one produces the other (though only accurately when you have the same conditions as those for which the numbers are specified).

9. Nov 6, 2006

### scott_alexsk

So the relevant area is simply the area is square meters of the single cross section?

Sorry for being confusing, I am testing a lot of different things. Yes I am looking at two different modes. (I had a feeling I should not have posted that second set of questions) For one I am going to see what kind of force a straight annealed wire, strained less than 8%, will pull with.

After that I intend to see what force a wire annealed in a curve will pull with. Using the information I find on calculating strains with bent materials, hopefully I can calculate what force from the spring will provide slightly less than an 8% strain for that sample also.

Finally I hope to test what force another sample, trained with the two way memory effect, can pull with, also being strained less than the critical 8% (although testing this one will be messy.) The information I am asking about strain with curves most particularly applies to this third test, because when training the alloy for the two way memory effect it is critical when training to exceed the 8% strain.

I have seen the two phrases of 8% strain and 70 MPa used interchangably to represent the point of plastic deformation and the point at which the memory properties are degraded. However looking on the fact sheet, 70 MPa is defined as the Yield Strength.

Is it only when the wire is being uniaxially strained do the 8% and the 70 MPa equate? Otherwise when dealing with strain with curved surfaces do I simply go with the 8% strain for determining the point of plastic deformation?

Thanks,
-scott

10. Nov 9, 2006

### scott_alexsk

So does anyone know how to find the strain of a curve or know where it might be discussed?

Thanks,
-scott

11. Nov 9, 2006

### Gokul43201

Staff Emeritus
There's no such thing as "strain of a curve", so it's not clear what you're asking. Do you mean strain in flexure (bending)? Or are you simply talking about tensile strain in a curved (and constrained) wire?

Is the applied force trying to bend/unbend the wire or is it trying to stretch a wire bent around something?

http://www.diracdelta.co.uk/science/source/f/l/flexure mode/image001.jpg <figure on left is flexure>

http://kosmoi.com/Science/Physics/Machines/pulley.gif [Broken] <rope bent around pulley is in tension>

Last edited by a moderator: May 2, 2017
12. Nov 9, 2006

### scott_alexsk

I am refering to flexural strain.

-scott

13. Nov 9, 2006

### Gokul43201

Staff Emeritus
Let the angle (in radians) subtended at the center be $\theta$, when the strain is $\epsilon$. The length of the wire is L and its radius is r. Then, $\theta = \epsilon L /r$.

14. Nov 11, 2006

### scott_alexsk

Thanks Gokul, but what unit is strain in? Is it percent lenght? Would 8% strain be 0.08, 1.08, 108, 8? I assume it is 0.08 but I am not sure.

-scott

15. Nov 11, 2006

### Gokul43201

Staff Emeritus
Yup, that's the right one. It's a unitless ratio.

16. Nov 11, 2006

### scott_alexsk

Thanks again. I get a pretty big number from that but from what I have seen, it seems true.

-scott

17. Jan 3, 2007

### Chris Hillman

Sources for theory of elasticity?

Sigh... I complained long ago in WP talk pages that the articles on elasticity tended to be unreadable, disorganized, featured messy and uneven coverage of possible topics, and so on. The principle author took this well (the subject wasn't his thing, he was just trying to fill a gap in WP as best he could) and invited me to contribute improvements, but I never had time and eventually stopped editing the WP entirely (long story). I haven't had the heart to take a look to see if there has been any improvement.

Anyway, I'd advise that in general, WP should never be your only source for anything important, that you bear in mind that anyone can edit the WP (with all the consquences that implies for the instability and questionable reliability of this "encyclopedia"), and that you move on to real textbooks published by reputable publishers as quickly as possible.

Alas, in this particular area I don't know of any books which do a really good job of explaining the fundamental ideas of the theory of elasticity in a reasonably brief space in a way suitable for contemporary students. Landau and Lifschitz, Theory of Elasticity, Vol. 7 in their Course of Theoretical Physics, Pergamon, 1986, might be your best choice, if you can find it. If your interest comes from physics rather than engineering, which seems not be the case here... If you read between the lines and can handle a somewhat idiosyncratic notation, Jeffreys and Jeffreys, Methods of Mathematical Physics, 3rd Ed., Cambridge University Press, 1972, offer a very brief overview. I'd be happy to hear recommendations from others! Gokul?

Last edited: Jan 3, 2007
18. Jan 3, 2007

### Gokul43201

Staff Emeritus
Chris,

L&L v7 is, as you've pointed out, aimed at the physicist, rather than the engineer. And even by the standards of L&L, I believe this one sets the record for brevity, or if you wish, succinctness. It took me many times as long to digest one page from that text, as I'd spend on a page from most other texts. But all this is based on no more, sadly, than the most fleeting relationship with this book.

I was raised on Timoshenko (which is aimed at engineers - and will likely serve as a good reference to the OP) and would recommend most any of his books on the mechanics of solids.

https://www.amazon.com/s/103-396551...rank&field-author-exact=Timoshenko, Stephen P.