What parameter to lower elastic elongation of a rod under tension

Let's assume I have a rod under tension that is only elastically deformed. What parameter can I change to decrease is elongation. $$T_{current} < T_{critical}$$

• decrease Temperature
• increase hardness
• increase Young's modulus
• increase tensile strength
• increase the diameter of the rod
• increase yield strength

I figured Young's modulus because a higher one means more stiffness? Does a higher yield strength change the elongation for a given applied force?

But I am really not sure about the other parameters.

• What formula defines the elongation? How can the variables be changed? Jan 17 '21 at 14:33
• @SolarMike Is it Hooke's law? Then it should be Young's modulus (E) and the cross-sectional area? Jan 17 '21 at 14:42

As insinuated by @Solar Mike's comment, a good first step is to think about the equation which defines elongation, Hooke's Law:

\begin{align} \sigma &= \frac{F}{A} = E\epsilon \\ \therefore \epsilon &= \frac{F}{EA} \end{align}

Now, you don't state this clearly, but I assume you're asking about a case of a rod under a constant force. Therefore, if we want to reduce the elongation, the only variables we have in hand are $$E$$ and $$A$$. If we increase either of these, the elongation decreases.

So, to decrease elongation, choose a material with a higher modulus of elasticity and/or increase its diameter.

• temperature: we usually talk about materials as having a constant modulus of elasticity, but that's actually a simplification. Most (all?) materials have an inverse relationship between temperature and modulus of elasticity (after all, over a certain temperature, the material simply melts, at which point it doesn't resist elongation all that much). So if you lower the temperature, you increase the modulus of elasticity. However, lowering the temperature also causes most materials to shrink, which decreases the cross-sectional area. So the effect of temperature changes will depend on the material (and will usually be minuscule for "normal" temperatures).
• hardness: this is a meaningless term in structural engineering. Do you mean the Mohr hardness scale (which ranks materials by what can scratch what)? This is irrelevant (though maybe correlated with Young's modulus, not sure). A more often used term is "stiffness", which is equal to $$EI$$ for bending and $$EA$$ for axial loads. We've already explored how this influences elongation.
• tensile and yield strength: these are irrelevant. Elongation is a function of the stress applied and the elastic modulus. Tensile and yield strength simply define maximum thresholds for the stress.

Although I agree with wasabi, I'd like to point out that IMHO yield strength can play a role in some cases.

Hooke's law can be resrotten as: $$\Delta L =\frac{F L}{EA}$$

However this only applies to linear / elastic region. When the material is subjected to higher loads then out enters the plastic region.

In steels entering the plastic region is "announced" by reaching the yield strength. In steels reaching the yield strength means that the material gets into a Region where effectively the tangent modulus drops. I.e for the same increase in load there is comparably more deformation.

So for a material with a given young's modulus ( and mainly I am thinking about steel) increasing the yield strength could result into lower deformation. There are some asterisks involved though.

• Yield strength makes no difference in the elastic range. Jan 18 '21 at 22:15
• I agree with you. I'm talking about forces in the plastic range though. However for the stress stain diagram in my post applying a stress of 350MPa will result in a markedly different deformation between the mild steel and the other types of steel. Jan 19 '21 at 5:33

Increase the diameter . You can't change modulus unless you can get something like a tungsten bar. Steel modulus is very close to 30 million psi regardless of all the other factors listed. Correction ; cast iron can have different moduli mostly depending on graphite morphology .

• Steel modulus is only "close to 30" in weird American units (the other 97% of the world population thinks it's about 200 or 210 GPa). Also, there is nothing in the question which says the rod was steel, or even a metal. Jan 17 '21 at 16:00
• What ever units you use, for a uniform metal , the modulus does not change with hardness, yield, or tensile in the elastic range . It changes if you get a different bar. Jan 18 '21 at 0:34
• ASTM made an effort to go metric in the mid 1970's. There were two problems : hard or soft conversion , and which metric system MKS or CGS ? So that gave four options and the program died . Jan 18 '21 at 22:13

Apart from all the other answers, I would add that one other consideration is the anisotropy of the material the rod is made of. Even isotropic materials can exhibit anisotropy according to how they are processed.

More pronounced cases are for orthotropic or anisotropic materials, such as carbon fiber rods. The elastic elongation you get on the rod under tension would depend a lot on the winding direction and type of carbon fiber you use (woven, UD, etc.)

At the micro- and meso- scale, the orientation of the rod's axis relative to the crystalline structure and whether the rod is poly or monocrystalline also makes a difference. For example the paper below shows differences between polycrystalline CVD diamond and elastic theory.

https://cds.cern.ch/record/275239/files/ecp-94-020.pdf