Understand the basic assumptions and procedures used for for the analysis of composite materials

1. MDPN452
Dr. Mohammad Tawfik
Mechanics of Composite
Materials
MDPN452
Dr. Mohammad Tawfik
E-mail: Mohammad.tawfik@gmail.com

2. MDPN452
Dr. Mohammad Tawfik
Micromechanics
• The basic question of micromechanics is:
what is the relationship of the composite
material properties to the properties of the
constituents?

3. MDPN452
Dr. Mohammad Tawfik
The Design Question!
• How can the percentages of the
constituent materials be varied so as to
arrive at the desired composite stiffness
and strength?

4. MDPN452
Dr. Mohammad Tawfik
Definitions
• Macromechanics: The study of composite
material behavior wherein the material is
assumed homogeneous and the effects of
the constituent materials are detected only
as averaged apparent properties of the
composite material.

5. MDPN452
Dr. Mohammad Tawfik
Definitions
• Micromechanics: The study of composite
material behavior wherein the interaction
of the constituent materials is examined in
detail as part of the definition of the
behavior of the heterogeneous composite
material.

6. MDPN452
Dr. Mohammad Tawfik
Approaches to Micromechanics
• Mechanics of Materials
• Elasticity

7. MDPN452
Dr. Mohammad Tawfik
Micromechanics Objective
• To answer the questions set by
micromechanics, the objective becomes:
– Determine the elastic moduli or stiffness or
compliances of a composite material in terms
of the elastic moduli of the constituent
materials
( )mimfifii VPVPPP ,,,=

8. MDPN452
Dr. Mohammad Tawfik
Micromechanics Objective
( )mimfifii VPVPPP ,,,=
MaterialCompositeofVolume
FibresofVolume
Vf =
fm VV −= 1

9. MDPN452
Dr. Mohammad Tawfik
Basic Assumptions
• The lamina:
– Initially stress-free, macroscopically homogeneous,
linearly elastic, macroscopically orthotropic
• The fibers:
– Homogeneous, regularly spaced, linearly elastic,
perfectly aligned, Isotropic, perfectly bonded
• The matrix:
– Homogeneous, Isotropic, linearly elastic, void-free

10. MDPN452
Dr. Mohammad Tawfik
Representative Volume
• The smallest region or piece of material
over which the stresses and strains can be
regarded as macroscopically uniform and
yet the volume still has the correct
proportions of fiber and matrix

11. MDPN452
Dr. Mohammad Tawfik
Mechanics of Materials
Approach

12. MDPN452
Dr. Mohammad Tawfik
Most Prominent Assumptions
• In fibre direction, strain in fibres and matrix
are equal
• Normal to fibre direction, stresses in matric
and fibre are equal

13. MDPN452
Dr. Mohammad Tawfik
Determination of E1
Rule of Mixture

14. MDPN452
Dr. Mohammad Tawfik
Rule of Mixture

15. MDPN452
Dr. Mohammad Tawfik
Experimental Results

16. MDPN452
Dr. Mohammad Tawfik
Determination of E2

17. MDPN452
Dr. Mohammad Tawfik
Determination of E2
• Nondimensionalizing:

18. MDPN452
Dr. Mohammad Tawfik
Experimental Results

19. MDPN452
Dr. Mohammad Tawfik
Determination of Poisson’s
Ratio
• We have two values for the Poisson’s
Ratio ν12 & ν21.

20. MDPN452
Dr. Mohammad Tawfik
Determination of G12

21. MDPN452
Dr. Mohammad Tawfik
Determination of G12

22. MDPN452
Dr. Mohammad Tawfik
Strength

23. MDPN452
Dr. Mohammad Tawfik
Strength
• “A chain is as strong as its weakest link”
• When the composite material gets loaded,
all its components get loaded with different
stresses
• Nevertheless, strength prediction of the
composite materials have not yet reached
the accuracy level of the stiffness
prediction. i.e. Open research area!

24. MDPN452
Dr. Mohammad Tawfik
Longitudinal Tensile Strength
• Main assumptions:
– All fibres have the same strength
– Fibres and matrix behave linearly
– Fibres are brittle when compared to matrix
– Fibres are stiffer than matrix

25. MDPN452
Dr. Mohammad Tawfik
Longitudinal Tensile Strength
• The stress that will cause the fibres to fail:
• Where is the stress in the matrix as
the fibres reach their ultimate stress.
• From that:
mmffut VV *
1 σσσ +=
f
m
fum
E
E
σσ =*








+= m
f
m
ffut V
E
E
Vσσ 1

26. MDPN452
Dr. Mohammad Tawfik
Longitudinal Tensile Strength

27. MDPN452
Dr. Mohammad Tawfik
Longitudinal Tensile Strength

28. MDPN452
Dr. Mohammad Tawfik
Minimum and Critical Vf
• From the graphs, and using the previous
equations, we may calculate:
*
*
mfu
mmu
criticalfV
σσ
σσ
−
−
=−
*
*
max
mmufu
mmu
fV
σσσ
σσ
−+
−
=

29. MDPN452
Dr. Mohammad Tawfik
Longitudinal Compressive
Strength
• In compression, the failure is dominated
by buckling of the fibres!
• Long fibres that are supported by softer
matrix material tend to buckle causing
what is called Microbuckling
• Microbuckling causes great reduction in
the compression stiffness of the composite
material

30. MDPN452
Dr. Mohammad Tawfik

31. MDPN452
Dr. Mohammad Tawfik

32. MDPN452
Dr. Mohammad Tawfik
Compressive Strength
• The compressive strength is greatly
affected by:
– Buckling modes
– Initial misalignment
– Curing procedure

33. MDPN452
Dr. Mohammad Tawfik

34. MDPN452
Dr. Mohammad Tawfik
Homework #3
• Read the papers provided to you
• Read section 4.4 in textbook
• Collect more papers evolving the same topics (at
least 3 per topic)
• Write down a report on each topic
• Due dates:
– Tensile Failure; axial and transverse
– Compressive failure ; axial and transverse
(19/10/2013 23:59)

35. MDPN452
Dr. Mohammad Tawfik
Report should include
1. Introduction to the topic (need, problems, history, etc...)
2. Models available in literature (Including the detailed
derivation of at least one of the models)
3. Results obtained from different models compared to
experimental results (direct from available literature)
4. Conclusions about the models and their accuracy
5. References and bibliography