# Calculating Poisson's Ratio: Step-by-Step Guide

• saravanan_n
In summary, this formula is used in deriving the gauge factor for materials in strain gauges. It is derived from the relationship between resistance and area. Resistance is differentiated and then divided by the area to arrive at the gauge factor.
saravanan_n
please help on this formula how it has been arrived?

dA=A(1-m dL/L)^2-A

where m= Poisson's ratio
A=area of cross section
L=length

In what context have you seen this? I can't recall seeing this form. I'll try to work out a proof. It looks like it's in regards to a tensile test...

gauge factor

While deriving the formulae for Gauge factor of materials(resitors) in strain gauges it comes.In between they have given this formula.

First they told

R=PL/A

R=RESISTANCE
P=SPECIFIC RESISTANCE
L=LENGTH
A= AREA

Then,

dR=a(PdL+LdP) - P L da
-------------------
a^2

Divide L.H.S by R and R.H.S by PL/A we get

dR dL dP dA
-- = --- + ---- - ----- (eqn 1)
R L P A

And suddenly they give this formula

as

dA= A(1-m dL/L)^2 -A

m=poisson's ratio

they substituted this in equation (1)

and finally arrived at gauge factor (dR/R)/(dL/L)

Please I need a answer How this formula is obtained?

da=a(1-m dl/l)^2 - a

I can not find anything in my references that has that particular form. I am trying to derive it but not having much luck right now. I am still trying though. BTW...who is "they"? What book/reference states this?

Book reference

It is from "Transducer Engineering" by S.Renganathan

please provide me the proof for this formulae:

da=a(1-m dl/l)^2-a
i.e., da/a=(1-m dl/l)^2 -1
=1+m^2 (dl/l)^2-2m dl/l -1
=(m dl/l)^2 - 2m dl/l

so da/a= (m dl/l)^2 - 2m dl/l
=m dl/l(m dl/l -2)

so

(da/a)/(dl/l)=m^2 (dl/l)-2m

now atleast you can say me how we arrive at this formula.Because I don't know any of the mechanical concepts and also about Poisson ratio.If you find any material regarding these please send me.My e-mail Id is

saravanan_n@msn.com

Pretty impatient people these days...Maybe "at least" I can tell you this:

Start with the relationship for an electrical conductor:

$$R = \frac{\rho L}{A}$$

Where:
$$R$$ = Resistance
$$\rho$$ = Resistivity
$$A$$ = Cross sectional Area

Now differentiate it:

$$dR = \frac{A(L d\rho + \rho dL) - \rho L dA)}{A^2}$$

Now go back and divide the last equation by the first and have fun with the algebra. Remember the definition of Poisson's Ratio: $$\nu = \frac{dA/A}{dL/L}$$

Sorry for being impatient,Since I have test on this I have to hurry.
Now we get
DR/R=DL/L+DP/P-DA/A

Then we divide whole eqn by dl/l

(dr/r)/(dl/l)=1+(dp/p)/(dl/l)-m

so we can put
Gauge Factor=1+Piezoresistivity-Poisson ratio

Is this equation right?

Equation Not Agrreable

If you say Poisson Ratio =(da/a)/(dl/l)

then do you agree with this equation also

da=a(1-((poisson ratio) dl/l)^2)-a
(in whatever context it comes)

Last edited:
Poisson's ratio is the lateral contraction per unit breadth divided by the longitudinal extension per unit length.

But A is proportional to the square of the length, i.e. a square has area, A = l2, where l is side length, or a circle has area $\pi$r2, where r is radius.

Now looking in three dimensions, if lx and ly contract by $\nu\,(\frac{\Delta{l_z}}{l_z})$ then the new lengths are

lx($1 - \nu (\frac{\Delta{l_z}}{l_z})$) and ly($1 - \nu (\frac{\Delta{l_z}}{l_z})$),

and the Area is then given by the product. If Ao = lx ly, then the new area is

A = Ao * ($1 - \nu\,\frac{\Delta{l_z}}{l_z}$)2

and dA = A - Ao, which defines dA.

Thanks for the reply.I am very much pleased.Thank you!

## What is Poisson's Ratio and why is it important in science?

Poisson's Ratio is a material property that describes the relationship between a material's strain (deformation) in one direction and its strain in a perpendicular direction. It is important in science because it can help predict how a material will behave under stress and can also be used to determine the elastic modulus of a material.

## How do you calculate Poisson's Ratio?

Poisson's Ratio is calculated by dividing the negative lateral strain by the axial strain. The formula is ν = -εl/εa, where ν represents Poisson's Ratio, εl represents the lateral strain, and εa represents the axial strain.

## What are the units of measurement for Poisson's Ratio?

Poisson's Ratio is a dimensionless quantity, meaning it does not have any units of measurement. It is simply a ratio of two strains.

## What is the range of values for Poisson's Ratio?

Poisson's Ratio can range from -1 to 0.5, with most common materials falling between 0.25 and 0.35. A value of -1 indicates that the material has no lateral strain, while a value of 0.5 indicates the material has no axial strain.

## Can Poisson's Ratio be negative?

Yes, Poisson's Ratio can be negative for certain materials. This indicates that the material experiences an increase in lateral strain when compressed, rather than a decrease. Materials with negative Poisson's Ratio are known as auxetic materials and are rare in nature.

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