Calculation of shear modulus

[lavecchiasignora]

Hi,

How do I calculate the shear modulus G12, G13 and G23 of a lamina/ply if I know the :

Thickness
Resin Wt
Fiber WT
Laminate Wt
Density
Vf
Ex, 1-direction
Ey, 2-direction

 

[RPstress]

Strictly, you cannot. The nature of orthotropic materials means that, unlike the G=E/(2(1+nu)) for more or less isotropic material like steel or aluminium, there is no predictable relationship between Ex, Ey and Gxy.

For conventional material combinations, such as woven glass or carbon fibres and a polymer matrix, the in-plane direct moduli (the E's) are largely a function of fibre longitudinal stiffness (not true of E2 of unidirectional material), and the in-plane shear is largely a combination of matrix shear (also matrix extension for a roughly isotropic matrix such as most polymers) and fibre shear (for glass fibre, which is roughly isotropic, this is also more or less a function of longitudinal stiffness; not true for carbon fibres, which are themselves orthotropic).

So, if you know the sort of fibres and matrix present it is possible to make a reasonable guess for the Gxy, or just use Gxy for a similar material.

Finding the through-thickness shear moduli is similar. Even if you have the through-thickness E values there is no simple relationship with the G's. However, once you know the in-plane Gxy (or G12—strictly, x, y and xy apply to laminates and 1, 2 and 12 apply to laminae) you can make a reasonable estimate of the out-of-plane Gs.

 

[SWComposites]

Agree with RP; you do not have enough data to determine G12. Buy a copy of Mil-Handbook-17 volume 2 and find a similar material and use that G12 value.

What is your specific material?

 

[lavecchiasignora]

Thank you for the answers.

My material is GFRP, i.e glassfiber reinforced polymer.

I just have a material datasheet with information on

Laminate - Vacuum infused
Thickness 1.45 mm
Resin Wt 0.83 kg/sq.m
Fiber Wt 1.90 kg/sq.m
Density 1.90 g/cc
Vf 51.95 by Vol
0 degree Modulus, Ex
90 degree modulus, Ey
And the ultimate stresses in 0 and 90 degree

I need to now shear modulus G12, G13, G23 and poissons ratio aswell.

Any ideas or do I need more Information about the material to obtain these?

 

[SWComposites]

polymer?? what specific one? epoxy? urethane? what?
what cure temp?

as we said above, you need more info.

or, just use these approximate values
G12 = G12 = 0.6 msi
G13 = 0.4 msi
v12 = 0.3 if it is a uni tape material, v12 = 0.06 if it is a fabric

SW

 

[RPstress]

Info has quite few implications; however, it does depend a bit on the polymer. And also as SW says, MIL-HDBK-17 vol. 2 would help if it's a thermoset polymer, especially if it's epoxy. However, polyester or vinyl ester will be similar.

It's likely to be continuous fibre with those weights in kgsm and that Vf of 52%, and quite possibly woven. It's also likely to be E-glass (not S- or R-glass or something fancy). Verification of this sort of thing would be sensible.

http://www.cmh17.org/

($$$)

http://assist.daps.dla.mil/quicksearch/

(needs some fiddling about to find it: Enter MIL-HDBK-17 in the 'Document ID' box for a starter)

http://snap.lbl.gov/pub/bscw.cgi/98090

http://www.lib.ucdavis.edu/dept/pse/resources/fulltext/HDBK17-3F.pdf

MIL-HDBK-17 only has shear modulus for one E-glass material that I can spot; SW's numbers are of course fine. There is one wet knockdown if that's of interest (comes to 0.65 times at RT).

 

[lavecchiasignora]

Thank you all again. Really appreciate it.

Do you have any idea why I cant use the Huber formula for orthotropic lamina in a single plane :

G12= sqrt(E1*E2)/2(1+sqrt(ny12*ny21))

 

[lavecchiasignora]

The polymer is Polysterene by the way

 

[lavecchiasignora]

Sorry I mean polyester

 

[RPstress]

Polyester is a fairly classic thermoset matrix and should behave well in terms of ply stiffness behaviour.

I'm not too sure about the 'Huber formula.' I last saw something similar used for bodged equations for skin wrinkling of carbon on honeycomb (on the discontinued Ariane V H10 interstage). The skin there was sort of not-quite quasi-isotropic.

The formula is demonstrably wrong for the limiting case of woven lamina material: E1 = E2 and nu12 = nu21, so you'd get E/(2(1+nu)). For E1 = E2 = 25 GPa and nu = 0.06, G12predicted = 25/(2*1.06) ~= 12 GPa, fc. ~4 GPa. As the laminate becomes more isotropic it will behave more in accordance with E/(2(1+nu)), but the basic orthotropic material will never do so.

For UD glass E1 ~= 45 GPa, E2 ~= 10 GPa, nu12 ~= 0.3, nu21 ~= 0.02 and G12predicted would be ~10 GPa; a fair bit out from the actual 4 GPa. (G12 is roughly the same for woven and UD).

[G12predicted = sqrt(45*10)/2/(1+sqrt(0.3*0.02))]

So no, this won't work. It might be acceptable for a small difference from isotropic, such as a rolled metallic plate or even the formula for skin wrinkling that I used for an almost QI laminate. I confess I've not heard of Huber other than just looking him up via Google.

If you require some sort of reason for a number rather than quoting the (free!) MIL-HDBK-17 data, you could try a micro-mechanics approach. The glass fibre G is about 72000/(2(1+0.3)) = 28000 MPa (like aluminium) and the polyester E will be about 3000 MPa, giving a G of maybe 3000/(2*(1+0.35)) = 1100 MPa; I think fibers in series with a matrix for G12glasspolyester would then be, er, 1100*[(28000*(1+0.52) + 1100*0.48) / (28000*0.48 + 1100*(1+0.52))] = 3200 MPa??? (460 ksi.)

(See

http://www.hearne.co.uk/attachments/MicromechanicsFormulations.pdf

.)

Well, maybe. You're better off with MIL-HDBK-17.

 

[lavecchiasignora]

Thank you RPstress. You are helping me alot.

I just have one question. When I'm using the equation for G12 on the last link u posted, it says G12,f which means shear modulus in the 12 direction of the fiber? But the glass-fiber is isotropic I suppose? Why is there an index of the fiber shear modulus when its isotropic?

Best regards

 

[lavecchiasignora]

Or why cant I just use that a lamina is transversily isotropic E22=E33, nu12=nu13, G12=G13 and use the equation

G23=E22/2(1+nu23)

By the way, Why does e-glass have a minor poissons ratio and a minor poissons ratio when it is isotropic?

 

[RPstress]

Gf12 is for any fibre, perhaps orthotropic carbon, as well as the usually-assumed-to-be-isotropic glass. I don't know much about Hearne, but they've probably made their equations as widely applicable as possible. They just popped up in Google when I looked for a micromechanics formula that looked sort of reasonable, which makes their provision of free basic equations a good marketing ploy (assuming the equation is right).

Until today I hadn't thought about G23 = G33/(2(1+nu23)). You're deforming fibres and matrix in series so it should work: G33 ~= G22unidirectional ~= 10e3 MPa for glass, and nu23 ~= nu12unidirectional = 0.3, so G23 ~= 10e3/2.6 = 3850 MPa, not a mile out. Thanks. Note: the out-of-plane G13 should be a bit less than the in-plane G12 as through the thickness the shear stress drops to zero at the free surface and varies through the thickness in the classic way assumed for a rectangular section, so G13 ~= 5/6 * G12 for metallics (SW's value of 2/3 is quite typical for composite laminate, where the plate is not uniform through its thickness).

(You wanted G23 = E22/(2(1+nu23)), which assumes E33 = E22; pretty true for unidirectional material, not anywhere near true for woven.)

It may get complicated, as G23 is only approximately = G13 (shouldn't be too bad for glass) and G13 is quite different from G31 (think G12 vs G21), whereas G32 should be similar to G23. (All for UD; it's different for woven.)

I think that this addresses your last question "...why does e-glass have a minor poissons ratio and a minor poissons ratio when it is isotropic?".

I think that usually for the glass itself it's usual to assume isotropy. The differences come for the fibre-matrix composite, where stretching UD composite along the fibres will reduce the thickness with a nu of about 0.3, as will stretching the material across the fibres; however, stretching the material through its thickness will not reduce the in-plane fibre-direction length very much (stiff fibres), whereas the across-fibres in-plane dimension will reduce by about 0.3 again. I don't really understand your question, so this may not be much help.