Transformation of stress components and of elastic constants Principal material coordinates are ‘natural’ coordinates: Sign convention: It is often necessary to know the stress-strain relationships in non- principal coordinates (‘off-axis’) such as x and y. Therefore: How do we transform stress and strain? How do we transform the elastic constants? Transformation of stress components between coordinate axes This is obtained by writing a force balance equation in a given direction. For example, in the x direction: Fx x dA 1 dA cos 2 2 dA sin 2 2 12 dA sin cos 0 x 1 cos 2 2 sin 2 2 12 sin cos By repeating this, the complete set of stress transformations in xy coordinates can be obtained: x c 2 s 2 2cs 1 2 (*) y s c2 2cs 2 cs cs c 2 s 2 where c = cos and s = sin xy 12 1 1 T 2 12 And in the 12 system 1 x c2 s2 2cs we have: 2 T y with T s 2 c 2 2cs cs cs c2 s2 12 xy Similarly, we have: 1 x (**) 2 T y 12 xy 2 2 Now, remember that for a 2-dimensional lamina in its principal coordinates we showed that: 1 Q11 Q12 0 1 1 Q 1 2 12 Q22 0 2 2 Q 2 (***) 12 0 0 2Q66 12 12 2 12 2 (remember: only 4 independent constants). Then, substituting (**) into (***), and then into (*), we find x x y T 1 Q T y xy xy Q11 Q12 Q16 x Q12 Q22 Q26 y Q16 Q26 Q66 xy x Q y xy and the components of the ‘transformed reduced stiffness matrix’ are as follows: (How many independent constants among these transformed components?) In terms of engineering constants, we have: EXAMPLES: Nylon-reinforced elastomer composite Zinc (hexagonal Carbon AS4/epoxy symmetry) composite Anisotropy of bone (J. Biomechanics 1992) Therefore, from the relationships it is clear that we must know (and find ways to measure and/or predict) the principal elastic constants E 1, E2, G12, n12 ! This is the topic of micromechanics models Micromechanics models for elastic constants 1. Microstructural aspects The elastic constants can be measured through careful experiments. However, it is desirable to be able to reliably predict lamina properties as a function of constituent properties and geometric characteristics (such as fiber volume fraction and geometric packing characteristics). Thus we will be looking for a functional relationship of the following type: C ij f E f , E m ,n f ,n m , V f , Vm , Vvoids , S , A) Typical transverse cross-sections of unidirectional composites: Composites with low volume fractions tend to have a random fiber distribution, whereas with high volume fractions the fibers tend to pack hexagonally. Silicon carbide/glass ceramic, fiber diameter ~ 15 mm, Vf ~ 0.40 Carbon/epoxy, fiber diameter ~ 8 mm, Vf ~ 0.70 Expected volume fractions in composites using representative area elements NOTE - It is assumed that the fiber spacing (s) and the fiber diameter (d) do not change along the fiber length, so that area fractions = volume fractions d 2 vf 2 3 s Max(Vf) @ s=d: Vf = 0.907 d 2 vf Max(Vf) @ s=d: 4s Vf = 0.785 In practice, max(Vf) = 0.5 – 0.8 Fiber content, matrix content, void content For n constituent materials n we must have for volume v i 1 fractions vi = Vi/Vtot i 1 In many cases this simply reduces to v f v m v voids 1 For weight fractions wi = W i/W tot, we have: n w i 1 w f wm 1 i 1 Since weight = (density)*(volume), we obtain immediately n c i vi i 1 thus, a Rule of Mixtures, which for 2-phase composites becomes: c f v f m vm And in terms of weight fractions instead of volume fractions: 1 c w f f w m m The void fraction can W Wf be calculated from W f composite m f measured weights vvoids 1 (not fractions) and W composite composite densities: Other structural aspects to be quantified: The fiber length distribution The fiber orientation and the fiber orientation distribution b sin 1 a The same formula may be used to calculate the real depth of a hole from a (SEM) picture at an angle 2. Elementary mechanics models Longitudinal Young’s modulus - Assume an isostrain situation (Voigt 1910) [implicitely: perfect interfacial adhesion]: f m composite Assuming linear elasticity for the fiber and matrix: f E f composite m E m composite Assume Ac, Af, Am are the cross-sectional areas of the composite, the fiber and the matrix. From balancing the forces in the fiber direction, we have: Pcomposite Pf Pm composite Acomposite f A f m Am Therefore: composite Acomposite ( E f A f E m Am ) composite composite Af Am and: Ecomposite (E f Em ) composite Acomposite Acomposite Thus: E composite E f v f E m v m E11 (Voigt) Also: composite f v f m v m Transverse Young’s modulus - Assume an isostress situation (Reuss 1929): f m composite In this case it is easily shown that: 1 vf vm E22 (Reuss) Ecomposite E f Em composite f v f m v m Voigt bound Reuss bound A few remarks • The longitudinal modulus is always greater than the transverse modulus • The ratio E11/E22 may be considered as a measure of the degree of anisotropy (or orthotropy) of the lamina • Poisson ratio n12 is also predicted by a rule of mixtures • The shear modulus G12 is predicted by an inverse rule of mixtures • Improved (tighter) bounds were developed by various authors (Hashin & Rosen) 3. Semiempirical models • The most successful semiempirical model is that of Halpin & Tsai (1967): p 1 v f pm 1 v f with p pm 1 f p f pm where p represents composite moduli, for example E11, E22, G12 or G23; pf and pm are the corresponding fiber and matrix moduli. is the ‘measure of the degree of reinforcement of the matrix by the fibers’, which depends on fiber geometry and distribution, loading conditions, etc. It is essentially a fitting factor. These equations are quite accurate at low volume fractions, a bit less so at higher volume fractions. They have been modified to include the maximum packing fraction. Specific values include the following: Modulus E11 2(l/d) E22 0.5 G12 1.0 G21 0.5 K 0 Note finally that when 0, the ‘inverse’ rule of mixtures is obtained, and when , the direct rule of mixtures is obtained.
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