The purpose of this ongoing blog series is to make the Finite Element Analysis software within SolidWorks, more accessible to a wider audience of not-necessarily-engineering users.
In the SolidWorks Simulation dialog for Material Properties, the fields colored RED are the ones that are absolute must-have to perform the analysis. Last month, I blathered at length about the Young’s Modulus. This month we will explore the next-most-important material property for doing a linear, static Stress and Strain analysis.
Must-Have: Poisson’s Ratio, u
If you pull very hard on an elastic band, you’ll notice that in addition to getting longer, the band gets noticeably thinner. This is a behavior that is common to most engineering materials, (though seldom to the same degree that you see in rubber). The Simulation software MUST know this property for each material in your study, in order to figure out how the forces acting on the outside of your solid bodies, turns into stress distributed within the interior.
Poisson’s ratio is a ratio of two strains. Since measurement of Strain has no units, the Poisson’s ratio is unitless, as well. In the case of a test specimen subjected to a tug in a single direction, the Poisson’s ratio is the proportion between; How much the specimen shrinks down, transverse to the pull direction, divided by how much it grows, in the pull direction.
Getting Test Data
In the discussion of Young’s Modulus last month, we used this same picture at right of a dog-bone test specimen being pulled in the jaws of a testing machine. We were mostly focused upon the change in length between some marks on the specimen, (the circles at right).
In order to measure a property that is about the material only, independent of the size of our test piece, we convert the resulting stretch values, in inches or millimeters, into Strain, by dividing the total change in length, by the original length:
Engineering Strain = Stretch,(the change in the length) / Original Length
eL = DL / L (strain in the length direction)
We use the same test rig and the same specimen shape to measure the Poisson’s ratio. We measure the original thickness of the specimen before the load is applied. Then, as the load is gradually ramped-up, we record the Thickness of the test-section, at each point where we recorded the Length. Again, we want to record these number as a unitless strain, so that the resulting property is independent of the size of our particular test-specimen;
eT = DT / T (strain in the thickness direction)
Poisson’s ratio is the ratio of the strain in the transverse direction, divided by the strain in the primary direction:
Poisson’s Ratio = u = - eT / eL
You’ll notice I snuck in a minus sign in the definition above – the convention when measuring strains is to report Stretch as a positive number, and shrinkage as a negative number. So you’d think that Poisson’s Ratio would have to usually be a negative number. The vast majority of materials are going to shrink down transverse to the load. But everyone knows that, and we’re all too lazy to carry the minus sign around in every report and table.. So we just add the minus sign as part of the definition, so that we usually get a positive number.
For most materials, we find that the ratio of these two strains, length-strain and thickness-strain, are wonderfully consistent – especially for lower values of strain. So for the useful working range of a material, the Poisson’s ratio is a single fixed number, (ranging usually between 0 and 0.5) .
Range of Values for Poisson’s Ratio
The more brittle a material is, generally the lower the Poisson’ s Ratio. For very stretchy materials like rubber, the Poisson’s Ratio is very high, approaching 0.5. Imagine you had a balloon, filled with water. Anywhere you squeeze on the balloon, it is going to bulge out somewhere else. The water is pretty near incompressible, and because it has no solid structure, it can flow anywhere, take any shape, without building up any strain. That’s what a Poisson’s Ratio of 0.5 means – the material is incompressible.
In fact, our FEA analysis cannot handle a material that flows like water; it can only understand solid materials that obey Hooke’s Law, where stress is proportional to strain. So we can’t actually ever enter a Poisson’s ratio of 0.5 – it would crash your study. But you can get close. In a linear, static study with rubbery materials, I can usually get away with a value of 0.485, but pushing higher will tease the stability monster. If you need to analyze a true rubber, at very high stretch ratios, you have better use the Non-Linear solver, which has a specialized material type called the Mooney-Rivelin model, which can tip-toe much close to the brink of 0.5 than the linear, static solvers can.
Some cork materials have been found to have a Poisson’s Ratio at or near zero. That means that as you stretch it in one direction, there is practically no change in size the other two directions.
Some typical values of Poisson’s Ratio
Nylon 6 .35 (unfilled)
Poisson’s Ratio values can usually be obtained from internet sources for metals and most engineering materials. But it can be difficult to obtain for specific grades of plastics. Your best bet is to look for a supplier for a particular grade of plastic, and then call their support line – most plastics houses have an engineering-sales support function whose job is to make it easier to spec their grade of plastic, than a competitor’s… So you should be able to get good advice straight from the horse’s mouth.
If you are shopping around for different classes of plastic, not yet ready to investigate a particular grade or supplier yet, then the job gets tougher. Personally, I pay $35 a year to belong to MATWEB.COM, and mostly just for looking up plastics. Now, if you tunnel down the search path here to a very specific formulation of plastic, you will frequently find there is no Poisson’s Ratio listed on the data sheets. But, if you then back out from the search to the generic listing for that family of plastic, you usually DO find a Poisson’s value listed, and you can assume that the value holds true within grade formulations of that species. What you cannot assume is that a glass-filled grade will behave the same as the unfilled plastic – filler content sometimes changes the Poisson Ratio.
There is a class of material called “auxetics” – these are materials that have a negative Poisson’s Ratio. That is, you stretch them in one direction, and they thicken in the transverse direction(s). Fortunately for you, this class of behavior applies to practically nothing. The only auxetic materials in our common experience is: paper; Gore-Tex; some materials engineered for use in dissipative body armor; and a few naturally occurring minerals. If you’re simulating something that uses an auxetic material, call in help from an expert.
The bulk of the discussion above applies to doing a Linear, Static analysis of a material that is isotropic – that the material has the same strength properties no matter which way you cut it. But if you are analyzing wood, (which has a grain direction), or FR-4 circuit-board material, (which is laminated), then the material could have THREE Poisson’s Ratios, one in each direction. Such materials are called Orthotropic.
What I said earlier, that the Poisson’s Ration should never exceed (or really, even be equal to) 0.5, is no longer true for orthotropics. If the material has a different amount of ‘give’ in the other two directions – the compatibility-of-strains requirement can be satisfied even with Poisson’s Ratio values up to 1 and even beyond. And a very high Poisson’s Ratio in one direction can sometimes be offset by a negative Poisson’s ratio in the other two directions, without the material being considered auxetic.
Some strength data for loblolly pine:
Parallel to the Grain: E = 13,500 GPa u = 0.382
Parallel to the growth rings: E = 1,500 GPa u = 0.292
Paralle to the Grain: E = 1,050 GPa u = 0.328
Who is this guy, Von Mises, and why is his stress better than anybody else’s?