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# Modulus of [Elasticity|Rigidity] and Shear Modulus

09/04/2017 11:16 AM

There's this material: JIS S45C Steel, Tempered over on matweb that I am trying to make sure that the terms are properly aligned with what I am studying.

Mechanical Properties

Hardness, Brinell201 - 269201 - 269
Hardness, Rockwell C13.8 - 27.613.8 - 27.6
Tensile Strength, Ultimate686 MPa99600 psi
Tensile Strength, Yield490 MPa71100 psi
Elongation at Break17 %17 %
Reduction of Area45 %45 %
Modulus of Elasticity205 GPa29700 ksiTypical steel
Poissons Ratio0.290.29Typical steel
Machinability55 %55 %Based on AISI 1212 steel as 100% machinability
Shear Modulus80.0 GPa11600 ksiTypical steel
Impact8.08.0kg(f)/cm²

Sometimes you'll see a capital G as the modulus of rigidity; is that the same thing as modulus of elasticity? This wikipedia page seems to say so but sometimes they are used differently;

Also shear stress, sometimes written as Fs; is that the same thing as shear modulus? This wikipedia page, seems to refer to shear stress as relating to modulus of rigidity or G from the previous example

Calculating the diameter of a solid shaft based on strength criteria the formula is:

T/J = Fs/R

or based on rigidity criteria the formula is

T/J = G*theta/L

I am uncertain based on the material properties up above if "Fs" == shear modulus == 80GPa and if G == modulus of elasticity == 205GPa.

Is modulus of elasticity the same as modulus of rigidity?

Is shear modulus the same thing as shear stress?

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#1

### Re: Modulus of [Elasticity|Rigidty] and Shear Modulus

09/04/2017 12:00 PM

Is modulus of elasticity the same as modulus of rigidity?

Whenever a force is applied to an elastic object, the object gives a little. The force can be compression, tension, or shear. The applied force is called stress and the amount the object "gives" is called strain. Under compression it gets shorter, under tension it gets longer, and under shear it deforms "sideways". The ratio of stress to strain is called the modulus of elasticity, and this can apply to any kind of stress and strain.

https://en.wikipedia.org/wiki/Elastic_modulus

Modulus of rigidity is the same as shear modulus or (shear stress) / (shear stain)

"In materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is defined as the ratio of shear stress to the shear strain: where = shear stress is the force which acts is the area on which the force acts = shear strain."

https://en.wikipedia.org/wiki/Elastic_modulus

Is shear modulus the same thing as shear stress?

No, shear stress is the amount of sideways force whereas shear modulus is a ratio of shear stress to shear strain. They have the same units because strain is unitless.

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#4

### Re: Modulus of [Elasticity|Rigidty] and Shear Modulus

09/06/2017 8:19 PM

If you apply 1.5x10^3Nm to some material; that force is called the stress and how the material deforms is called the strain.

If you are applying some force or stress to a material; is there standard deformation calculations to see just how much they'll deform?

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#2

### Re: Modulus of [Elasticity|Rigidty] and Shear Modulus

09/04/2017 1:20 PM

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#3

### Re: Modulus of [Elasticity|Rigidty] and Shear Modulus

09/05/2017 5:34 AM

Shear stress and shear modulus are not the same. They're related (for torsion in a shaft etc) by the equation you gave, T/J = Fs/R, where J = modulus of rigidity, R = radius. You know how to calculate J from previous thread!

That equation and T/J = G*theta/L are not either/or, they both apply. theta is the angle of twist over length L.

G and E are not the same, though they have same units, GPa. They are related by the formula G = E/2/(1 + ν) where ν (nu) is Poisson's ratio, usually taken as 0.26 in my recollection, though between that and 0.29 doesn't make a deal of difference.

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#5

### Re: Modulus of [Elasticity|Rigidty] and Shear Modulus

09/09/2017 10:15 PM

thank you; this is what I was trying to refer to in my question #4.

Now I should be able to calculate the needed values and then use those in my formulas to try and sort this issue out.

p.s.

I apologize for the delay, real life calls sometimes!

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#6

### Re: Modulus of [Elasticity|Rigidty] and Shear Modulus

09/10/2017 6:50 AM

Is the shear modulus is actually the modulus of rigidity as denoted by G?

Is the shear stress is denoted by Fs?

That material sheet that I posted in the original post doesn't use modulus of rigidity but uses modulus of elasticity. Looking at this website it seems that modulus of elasticity is another name for Young's modulus or denoted by E.

Although the way modulus of elasticity is described in the link, why would the same behavior be described in so many different ways with different names and symbols?

If that's the case I can get the modulus of rigidity by solving the problem G = E/2(1+v) in this case v [Poisson's ratio] is provided by the material sheet above as 0.29

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#7

### Re: Modulus of [Elasticity|Rigidty] and Shear Modulus

09/10/2017 4:58 PM

Is the shear modulus is actually the modulus of rigidity as denoted by G? Yes

Is the shear stress is denoted by Fs? Yes

That material sheet that I posted in the original post doesn't use modulus of rigidity but uses modulus of elasticity. As far as I can see it doesn't use either of them, but it defines them both, and also bulk modulus. I don't see your problem,

Looking at this website it seems that modulus of elasticity is another name for Young's modulus or denoted by E. That's correct.

Although the way modulus of elasticity is described in the link, why would the same behavior be described in so many different ways with different names and symbols? It's not the same behaviour, the various moduli relate to material being stressed in different ways.

If that's the case I can get the modulus of rigidity by solving the problem G = E/2(1+v) in this case v [Poisson's ratio] is provided by the material sheet above as 0.29. That's right. The table didn't need to list all 3 of E, G and ν as they're related by the formula, but it did and you can check the figures agree near enough (I haven't checked, maybe exact in ksi, a bit out in GPa but near enough in practice).

The formula you gave for angle of twist theta doesn't immediately drop out of the relation between shear stress and shear strain, given in various posts and links. To derive it would need a calculation involving integration. But I'm sure it's correct as it agrees with one in my (old) copy of Roark, and other sources.

Also note that J in the formulas for Fs and theta is only the same for both in the case of a cylindrical shaft. For other shapes it's different and if you get involved in that you need to study the literature.

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#8

### Re: Modulus of [Elasticity|Rigidty] and Shear Modulus

09/12/2017 8:13 PM

thank you, this was very helpful clarifying those questions.

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#9

### Re: Modulus of [Elasticity|Rigidty] and Shear Modulus

09/13/2017 4:24 AM

No problem!

Looking at it again, the formula for angular twist does come out readily from the definitions, I was over-thinking it in my comment.

A year or 2 back I wanted to calculate figures for a helical spring with rectangular "wire". I made the mistake of assuming the torsional J-value could be got from the perpendicular axes theorem (adding the 2 J-values at right angles to each other and to the plane of twisting). This only works for circular sections. Other forum members put me right, we can all help each other!

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