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Definition of Micromechanics
The study of composite material behavior where the interaction of constituent material is examined in detail and used to predict and define the behavior of the heterogeneous composite material
Approaches to the study of Micromechanics:• Mechanics of Materials• Elasticity
- Bounding Principles- Exact Solution- Approximate Solutions
Equations to Approximate Lamina Properties from Constituents
E1 = EfVf + EmVm
?12 = ? fVf + ? mVm
E2 = EfEm/(EfVm + EmVf)
G12 = GfGm/(GfVm + GmVf)
Micromechanics of Lamina Examples
Solution:E1 = 40M(0.4) + 0.5M(0.6) = 16.3M psiE2 = 40M(0.5M)/[40M(0.6) + 0.5M(0.4)] = 20/24.2 = 0.83M psi?12 = 0.45(0.4) + 0.3(0.6) = 0.27G12 = 14M(0.2M)/[14M(0.6) + 0.2M(0.4)] = 2.8/8.48 = 0.33M psi
Micromechanics of Lamina Examples
Solution:E2 must be stiffer than the matrix modulus Em. The matrix modulus is the same in any direction, and Ef serves to increase E2according to the equation:
or
If Ef > Em, and knowing that Vf + Vm = 1, then E2 > Em
Definition of Macromechanics
The study of composite material behavior where the material is presumed homogeneous and the effects of constituent materials are detected only as averaged “apparent” properties of the compositematerial
Generalized Hooke’s Law - Anisotropic Material
36 constants (6x6 matrix)21 independent constants (symmetry)
Stress-Strain Relations for Plane Stress in anOrthotropic Lamina Material
(7 independent constants)
(4 independent constants)
Laminate Terminology Refresher
Symmetric Laminate: Laminate composed of plies such that both geometric and material properties are symmetric about the middle surface (mid-plane)
Balanced Laminate: For every +? ply there exists a -? ply of the same thickness and material property
Cross-ply Laminate: Laminate composed of 0° and 90° plies
Angle-ply Laminate: Laminate composed of +? and -? plies
Consequences of Stacking Sequence -- Bending
0?????
45????
[A]: ? ?Zk - Zk-1) = ? tk = 4 ? tk = 4 (equal)
[B]: ? ?Zk2 - Zk-1
2) = ? ?Zk2 - Zk-1
2) = 0 (symmetric)(-12 - (-2)2) + (02 - (-1)2)
+ (12 - 02) + (22 - 12) = 0
[D]: 0.333? Qij?Zk3 - Zk-1
3) = 0.667[7(Qij)45 + (Qij)0]0.667[7(Qij)0 + (Qij)45]
Consequences of Stacking Sequence -- Bending
0?????
45????
[D] = 87 5 3 [D] = 40 20 195 12 3 20 29 193 3 7 19 19 22
2X better in on-axis 2X better in off-axisbending (D11) bending (D22)
3X better in torsion (D66)
Conclusions
• Stacking sequence does not affect the [A] matrix
• [B] = 0 as long as symmetry is preserved
• [D] matrix most affected by stacking sequence
• For balanced laminates A16 = A26 = 0
• Generally, D16 and D26 are insignificant with
respect to D11 for > 16 plies
Which [ABD] Terms Are Zero For a [0,45,-45,90]s Laminate?
Assume all identical tape plies of same thickness
Solution:Symmetric laminate: [B] = 0Balanced laminate: A16 = A26 = 0
Determine if the Following Statements are True or False
Adding plies to a laminate will always increase the axial stiffness, E, in either the X or Y direction
For mechanical loading, the A matrix is independent of stacking sequence
For a balanced laminate, the D16 and D26 terms are always zero
The axial stiffness Ex of a 9010 laminate is greater than the axial stiffness Ex of a 904 laminate
A symmetric laminate will always have the same value for D11 and D22
Solution:False
True
False
False
False
How Would You Change the Stacking Sequence For the Laminate Shown to Get the Maximum D66?
Solution: The 45° plies have the highest Q66, then the 22.5° plies, then the 0° and the 90°, thus to maximize D66 one should use [-45,45,-22.5,22.5,0,90]s
What Plies Would You Add to the Following Laminate to Eliminate Shear Deformation Resulting
From Extensional Loading?
Solution: Add 22.5°, 45° and -30° plies to balance the laminate, so that A16 = A26 = 0
“Real World” Analyses• Many analyses governed by failure other than ply by ply• Effective properties determined for range of families
Aluminum 60/30/10 45/45/10 25/60/15Density 0.101 0.056 0.056 0.056Ftu 74.0 69.8 58.4 43.6Fcu 65.0 37.9 35.3 29.6Fsu 45.0 11.7 16.9 22.0E 10.3 13.9 11.2 7.7G 3.9 2.0 2.7 3.4
Ftu/? 733 1246 1043 779Fcu/? 644 677 630 529Fsu/? 446 209 302 393E/? 102 248 200 138G/? 39 36 48 61
Family Properties Are Only Valid For Specific ThicknessesThickness Potential Stiffness (msi) Poisson’s(# plies) Families Axial Transverse Shear Ratio17 -- 13.23 5.30 2.79 0.4218 -- 13.23 5.30 2.79 0.42 Current methods19 -- 13.23 5.30 2.79 0.42 optimize thickness20 50.0/40.0/10.0 13.23 5.30 2.79 0.42 but use constant21 -- 13.23 5.30 2.79 0.42 material properties22 -- 13.23 5.30 2.79 0.4223 -- 13.23 5.30 2.79 0.42
17 41.2/47.1/11.8 11.59 5.30 2.79 0.4217 47.1/47.1/5.9 12.56 5.30 2.79 0.4218 44.4/44.4/11.1 12.19 5.30 2.79 0.4219 42.1/42.1/15.8 11.78 5.30 2.79 0.4219 47.4/42.1/10.5 12.74 5.30 2.79 0.42 Revised methods use19 52.6/42.1/5.3 13.61 5.30 2.79 0.42 material properties20 50.0/40.0/10.0 13.23 5.30 2.79 0.42 appropriate for the21 42.9/38.1/19.0 11.90 5.30 2.79 0.42 specific thickness21 47.6/38.1/14.3 12.80 5.30 2.79 0.4222 45.5/36.4/18.2 12.38 5.30 2.79 0.4222 54.5/36.4/9.1 14.07 5.30 2.79 0.4223 47.8/34.8/17.4 12.82 5.30 2.79 0.4223 52.2/34.8/13.0 13.65 5.30 2.79 0.4223 56.5/34.8/8.7 14.44 5.30 2.79 0.42
+12% +29% +12% +29%
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