Mech Comp

 

Micromechanic Microm echanical al Behavior Behavior of a Lamin Laminaa

 

Defini Def initio tion n of Micro Micromech mechanic anicss

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 tthe Approaches he study of Micromecha Micromechanics: nics: • Mech Mechanic anicss of Mate Material rialss

• El Elas asti tici city ty - Bou Boundi nding ng P Princ rinciple ipless - Ex Exac actt Sol Soluti ution on - Appr Approxim oximate ate Solu Solutions tions

 

Mechanics of Materials Approach to Stiffness Determination of E1

 

Mechanics of Materials Approach to Stiffness Determination of E2

 

Determination of E2

 

Mechanics of Materials Approach to Stiffness Determination of ? 12

 

Mechanics of Materials Approach to Stiffness Determination of G12

 

Equations to Approximate Lamina Properties from Constituents

E1 = Ef Vf  + EmVm ? 12

= ? f Vf  + ? mVm

E2 = Ef Em/(Ef V m + EmVf ) G12 = Gf Gm/(Gf V m + GmVf )

 

Micromechani Microm echanics cs of Lamina Lamina Exam Examples ples

Solution: E1 = 40M(0.4 40M(0.4)) + 0.5 0.5M(0. M(0.6) 6) = 16 16.3M .3M psi E2 = 40M(0.5M)/[40M(0.6) 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) = 0.27 G12 = 14M(0.2M)/[14M(0.6) 14M(0.2M)/[14M(0.6) + 0.2M(0.4)] = 2.8/8.48 = 0.33M psi

 

Micromechani Microm echanics cs of Lamina Lamina Exam Examples ples

Solution:

E1 = Ef1Vf1 + Ef2Vf2 + EmV m

 

Micromechani Microm echanics cs of Lamina Lamina Exam Examples ples

Solution: E2 must be stiffer than the matrix modulus Em. The matrix modulus modu lus is the same in an any y direc direction tion,, and Ef  serves to increase E2 according to the equation:

or  If Ef  > Em, and kn knowi owing ng tthat hat Vf  + Vm = 1, then E2 > Em

 

Macromechanic Macrom echanical al Behavior Behavior of a Lamin Laminaa

 

Defini Def initio tion n of Macrom Macromech echani anics cs

The study of composite material behavior where the material is  presumed homogeneous and the effects of constituent c onstituent materials material s are detected only as averaged “apparent” properties of the composite material

 

Gene Ge nera rali lize zed d Hook Hooke’ e’ss La Law w - Anis Anisot otro ropi picc Mate Materi rial al

36 constants (6x6 matrix) 21 independent constants (symmetry)

 

Derivation of Compliance and Stiffness Symmetry

 

One Plane of Material Symmetry (z = 0) Monoclinic Material

13 independent constants

 

Two Orthogonal Planes of Symmetry Ortho Ort hotro tropic pic Mat Materi erial al

9 independent constants

 

Stress-Strain Relations for Plane Stress in an Orthot Ort hotrop ropic ic Lam Lamina ina Mat Materi erial al

(7 independent constants)

(4 independent constants)

 

The Q Matrix

 

Engineerin Engi neering g Con Constants stants for Orthotro Orthotropic pic Materials

 

Macrome Macr omecha chanic nicss of a L Lami amina na

 

Lamina Coordinate System

 

Stress-Strain Relations for a Lamina of Arbitrary Orientation

 

Expression for the General Case Becomes

 

Invaria Inv ariant nt Pro Proper perties ties of of An Orth Orthotro otropic pic Lamina

 

Invariants

 

Macromechani Macrom echanical cal Beha Behavior vior of a Lamin Laminate ate

 

Laminate Mechanical Behavior Derived From Lamina Building Blocks

 

The “Building Block”

 

Classical Lamination Theory

 

Displacements

(small angle) (plane sections remain plane --? is slope of midsurface)

 

Strain Str ainss -- Lin Linear ear Elasti Elasticit city y (small strains)

 

Stresses

Strains ?1 ?2 ?3 ?4

Stresses

 

Force and Moment Resultants

Running Loads (unit width)

Running Moments (unit width)

 

Force and Moment Resultants

 

Equation Manipulation

 

The A,B, D Matrices

 

Determination of Laminate Ex, Ey, Gxy, ? xy

 

Determination of Laminate Ex, Ey, Gxy, ? xy

 

Determination of Laminate Ex, Ey, Gxy, ? xy

 

Laminate Lam inate Terminology Refresher

Symmetric Laminate: Laminate composed of plies such that  both geometric geometric and material pr properties operties are symm symmetric etric 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

 

Consequences of Stacking Sequence The 16 and 26 Terms

 

Conseq Con sequen uences ces of of Sta Stacki cking ng S Sequ equenc encee --- Bendin Bending g

[A]:

? ?Zk  -

[B]:

? ?Zk 2 2

0

45

??

?

??

?

?

??

Zk-1) = ? tk  = 4

- Zk-12) = 2

 

? ?Zk 2

  2

? tk  =

4

(equal)

- Zk-12) = 0

(symmetric)

2

-2()-1 2 -1 +(-1 (12--(0-22)) +) + (2(0 =)0) [D]: 0.333? Q ij?Zk 3 - Zk-13) = 0.667[7(Qij)0 + (Qij)45]

0.667[7(Qij)45 + (Qij)0]

 

Conseq Con sequen uences ces of of Sta Stacki cking ng S Sequ equenc encee --- Bendin Bending g 0

45

??

?

??

?

?

??

[D] = 87 5 3

5 12 3

3 3 7

2X better in on-axi on-axiss  bending  bendin g (D11)

[D] = 40 20 19

20 29 19

19 19 22

2X better in off-axis bending (D22) 3X better in torsion (D66)

 

Clas Cl assi sica call Or Orth thot otro ropi picc La Lami mina nate tess

 

Clas Cl assi sica call An Anis isot otro ropi picc La Lami mina nate tess

 

Pseu Ps eudo do Orth Orthot otro ropi picc La Lami mina nate tess

 

Ratios of Bending Coefficients

 

Unsymmetr Unsy mmetric ic Cross Cross-Pl -Ply y Laminat Laminates es

 

Unsymmet Unsy mmetric ric Angle Angle Ply Lami Laminates nates

 

General Laminates

 

Conclusions

• Stacking sequence does not affect th thee [A] matrix • [B] = 0 as long as symmetr symmetry y is prese preserved rved • [D] ma matrix trix most affected by stack stacking ing sequ sequence ence

• For ba balanc lanced ed lami laminate natess A16 = A2 A26 6=0 • Generally, D16 and D26 are insignificant with respect to D11 for > 16 plies

 

Laminate Example Problems

 

Which [ABD] Terms Are Zero For a [0,45,-45,90]s Laminate? Assume all identical tape plies of same thickness

Solution: Symmetric laminate: laminate: [B] = 0 Balanced 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

Solution: False

For mechanical loading, the A matrix is independent of stacking sequence

True

For a balanced laminate, the D16 and D26 terms are always zero

False

The axial stiffness Ex of a 9010 laminate is greater than the axial stiffness Ex of a 904 laminate

False

A symmetric laminate will always have the same value for D 11 and D22

False

 

For the Laminate Shown, Circle the Correct Answer 

 

For the Laminate Shown, Circle the Correct Answer 

 

How Would You Change the Stacking Sequence For the Laminate Shown to Get the Maximum D ? 66

Solution: The 45° plies have the highest Q 66, then the 22.5° plies,

then the 0° and the 90°, thus to maximize D 66 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 analy analyses ses govern governed ed by fai failure lure other than ply by ply • Effecti Effective ve prope properties rties deter determined mined fo forr range of fa families milies Aluminum 0.101

60/30/10 0.056

45/45/10 0.056

25/60/15 0.056

Ftu Fcu Fsu E G

74.0 65.0 45.0 10.3 3.9

69.8 37.9 11.7 13.9 2.0

58.4 35.3 16.9 11.2 2.7

43.6 29.6 22.0 7.7 3.4

Ftu/? Fcu/? Fsu/? E/?

733 644 446 102

1246 677 209 248

1043 630 302 200

779 529 393 138

G/?

39

36

48

61

Density

 

Family Properties Are O Only nly Valid Valid For For Specific Specific Thicknesses Thickness

Potential

(# plies) 17 18 19 20 21 22

Families ---50.0/40.0/10.0 ---

Axial 13.23 13.23 13.23 13.23 13.23 13.23

Transverse 5.30 5.30 5.30 5.30 5.30 5.30

Shear 2. 79 2. 79 2. 79 2. 79 2. 79 2. 79

Ratio 0.42 0.42 0.42 0.42 0.42 0.42

-41.2/47.1/11.8 47.1/47.1/5.9 44.4/44.4/11.1 42.1/42.1/15.8 47.4/42.1/10.5 52.6/42.1/5.3 50.0/40.0/10.0 42.9/38.1/19.0 47.6/38.1/14.3 45.5/36.4/18.2 54.5/36.4/9.1 47.8/34.8/17.4 52.2/34.8/13.0 56.5/34.8/8.7

13.23 11.59 12.56 12.19 11.78 12.74 13.61 13.23 11.90 12.80 12.38 14.07 12.82 13.65 14.44

5.30 5.30 5.30 5.30 5.30 5.30 5.30 5.30 5.30 5.30 5.30 5.30 5.30 5.30 5.30

2. 79 2. 79 2. 79 2. 79 2. 79 2. 79 2. 79 2. 79 2. 79 2. 79 2. 79 2. 79 2. 79 2. 79 2. 79

0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42

+12%

+29%

+12%

+29%

23 17 17 18 19 19 19 20 21 21 22 22 23 23 23

Stiffness (msi)

Poisson’s

Current methods op optimize thickness but use constant material properties

Revised methods use material properties appropriate for the specific thickness