Mech Comp

Micromechanical Behavior of a Lamina

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

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 = 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


Solution:E1 = Ef1Vf1 + Ef2Vf2 + EmVm


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

Macromechanical Behavior of a Lamina

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)

Derivation of Compliance and Stiffness Symmetry

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

13 independent constants

Two Orthogonal Planes of SymmetryOrthotropic Material

9 independent constants

Stress-Strain Relations for Plane Stress in anOrthotropic Lamina Material

(7 independent constants)

(4 independent constants)

The Q Matrix

Engineering Constants for OrthotropicMaterials

Macromechanics of a Lamina

Lamina Coordinate System

Stress-Strain Relations for a Lamina of Arbitrary Orientation

Expression for the General Case Becomes

Invariant Properties of An OrthotropicLamina

Invariants

Macromechanical Behavior of a Laminate

Laminate Mechanical Behavior Derived From Lamina Building Blocks

The “Building Block”

Classical Lamination Theory

Displacements

(small angle)

(plane sections remain plane --?is slope of mid-surface)

Strains -- Linear Elasticity

(small strains)

Stresses

?1?2?3?4

Strains 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



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

Consequences of Stacking SequenceThe 16 and 26 Terms

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)

Classical Orthotropic Laminates

Classical Anisotropic Laminates

Pseudo Orthotropic Laminates

Ratios of Bending Coefficients

Unsymmetric Cross-Ply Laminates

Unsymmetric Angle Ply Laminates

General Laminates

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

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: [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

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 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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