### Stress - Dr. Brian Sullivan

```Micromechanics
Macromechanics
Fibers
Lamina
Laminate
Structure
Matrix
Micromechanics
The analysis of relationships between
effective composite properties (i.e., stiffness,
strength) and the material properties, relative
volume contents, and geometric arrangement
of the constituent materials.
Micromechanics - Stiffness
1. Mechanics of materials models –
Simplifying assumptions make it
unnecessary to specify details of stress
and strain distribution – fiber packing
geometry is arbitrary. Use average
stresses and strains.
Micromechanics - Stiffness
2. Theory of elasticity models -
“Actual” stress and strain distributions are
used – fiber packing geometry taken into
account.
a) Closed form solutions
b) Numerical solutions such as finite element
c) Variational methods (bounds)
Volume Fractions
vf 
Where
Vf
 fiber volume fraction
Vc
Vm
vm 
 matrix volume fraction
Vc
Vv
vv   void volume fraction
Vc
v f  vm  vv  1
(3.2)
Vc  V f  Vm  Vv  composite volume
Weight Fractions
wf 
Wf
Wc
 fiber weight fraction
Wm
wm 
 matrix weight fraction
Wc
Where
Wc  Wf  Wm  composite weight
Note: weight of voids neglected
Densities
W
   density
V
Wc  Wf  Wm
 cVc   f V f  mVm
 c   f v f  mvm
“Rule of Mixtures” for density
(3.6)
Alternatively,
c 
1
wf
f

wm
(3.8)
m
Eq. (3.2) can be rearranged as
Wf
vv  1 
f

( Wc  W f )
Wc
c
m
(3.9)
Above formula is useful for void fraction
estimation from measured weights and densities.
Typical void fractions:
Autoclaved cured composite: 0.1% - 1%
Press cured w/o vacuum: 2 - 5%
Measurements typically involve weight fractions,
which are related to volume fractions by
 f Vf  f
wf 


vf
Wc
cVc c
(3.10)
Wm mVm m
wm 


vm
Wc
cVc c
(3.12)
Wf
and
Representative area elements for idealized square
and triangular fiber packing geometries.
Fiber
s
s
d
s
d
s
Square array
Triangular array
Fiber volume fraction – packing
geometry relationships
Square array:
vf 
 d
2
 
4s
When s=d, v f  v f max 
(3.14)

4
 0.785
(3.15)
Fiber volume fraction – packing
geometry relationships
Triangular Array:
vf 
 d
2
 
2 3 s
When s=d, v f  v f max 
(3.16)

2 3
 0.907
(3.17)
Fiber volume fraction – packing
geometry relationships
• Real composites:
Random fiber packing array
Unidirectional: 0.5  v f  0.8
Chopped: 0.05  v f  0.4
Filament wound: close to theoretical
Photomicrograph of carbon/epoxy composite showing
actual fiber packing geometry at 400X magnification
Voronoi cell and its approximation. (From Yang, H. and Colton,
J.S. 1994. Polymer Composites, 51, 34–41. With permission.)
Random nature of fiber packing geometry in real composites can be
quantified by the use of the Voronoi cell. Each point within the space
of a Voronoi cell for a particular fiber is closer to the center of that fiber
than it is to the center of any other fiber
s
Voronoi cells
Equivalent square
cells, with Voronoi
cell size, s
Typical histogram of Voronoi distances and corresponding Wiebull distribution for
a thermoplastic matrix composite. (From Yang, H. and Colton, J.S. 1994. Polymer
Composites, 51, 34–41. With permission.)
Elementary Mechanics of Materials
Models for Effective Moduli
• Fiber packing array not specified – RVE
consists of fiber and matrix blocks.
• Improved mechanics of materials models
and elasticity models do take into account
fiber packing arrays.
Assumptions:
1. Area fractions = volume fractions
2. Perfect bonding at fiber/matrix interface –
no slip
3. Matrix is isotropic, fiber can be
orthotropic
4. Fiber and matrix linear elastic
5. Lamina is macroscopically homogeneous,
linear elastic and orthotropic
Concept of an Effective Modulus of an
Equivalent Homogeneous Material.
Heterogeneous composite
under varying stresses and
strains
x3
d
2 L
x3
2
2
Stress, 2
Equivalent homogeneous x
3
material under average
stresses and strains
2
2
Strain,  2
x3
2
2
Stress
Strain
2
Representative volume element and simple stress states
used in elementary mechanics of materials models
Representative volume element and simple stress states
used in elementary mechanics of materials models
Longitudinal normal stress
Transverse normal stress
In-plane shear stress
Average stress over RVE:
1
1
   dV   dA
VV
AA
(3.19)
Average strain over RVE:
1
1
   dV   dA
VV
AA
(3.20)
Average displacement over RVE:
1
1
   dV   dA
VV
AA
(3.21)
Longitudinal Modulus
RVE under average stress  c1 governed by
longitudinal modulus E1.
Equilibrium:
 c1 A1   f 1 Af   m1 Am
(3.22)
Note: fibers are often orthotropic.
Rearranging, we get “Rule of Mixtures” for
longitudinal stress
Static
Equilibrium
 c1   f 1v f   m1vm
(3.23)
Hooke’s law for composite, fiber and matrix
Stress – strain
Relations
 c1  E1 c1
 f 1  E f 1 f 1
(3.24)
 m1  Em  m1
So that:
E1 c1  E f 1 f 1v f  Em  m1vm
(3.25)
Geometric
Compatibility
 c1   f 1   m1
(3.26)
Which means that,
E1  E f 1v f  Emvm
(3.27)
“Rule of Mixtures” – generally quite
accurate – useful for design calculations
Variation of composite moduli with fiber
volume fraction
Eq. 3.27
Eq. 3.40
Predicted E1 and E2 from elementary mechanics of materials models
Variation of composite moduli with fiber
volume fraction
Comparison of predicted
and measured E1 for
E-glass/polyester. (From
Engineered Materials
Handbook, Vol. 1,
Composites, 206–217.)
Strain Energy Approach
Uc  U f  Um
(3.28)
Where strain energy in composite, fiber and
matrix are given by,
1
1
2
U c    c1 c1dV  E1 c1 Vc
2 Vc
2
1
1
2
U f    f 1 f 1dV  E f 1 f 1 V f
2 Vf
2
1
1
2
U m    m1 m1dV  Em1 m1 Vm
2 Vm
2
(3.29a)
(3.29b)
(3.29c)
Strain energy due to Poisson strain
mismatch at fiber/matrix interface is
neglected.
Let the stresses in fiber and matrix be
defined in terms of the composite stress as:
 f 1  a1 c1
 m1  b1 c1
(3.30)
Subst. in “Rule of Mixtures” for longitudinal
stress:
 c1   f 1v f   m1vm
(3.23)
 c1  a1v f  b1vm  c1
Or
a1v f  b1vm  1
(3.31)
Combining (3.30), (3.24) & (3.29) in (3.28),
1
2 vf
2 vm
 a1
 b1
E1
Ef1
Em1
(3.32)
Solving (3.31) and (3.32) simultaneously for
E-glass/epoxy with known properties:
 f1
Find a1 and b1, then
 1.00
 m1
Representative volume element and simple stress states
used in elementary mechanics of materials models
Longitudinal normal stress
Transverse normal stress
In-plane shear stress
Transverse Modulus
RVE under average stress  c 2
Response governed by transverse modulus E2
Geometric compatibility:
 c 2   f 2   m2
(3.34)
From definition of normal strain,
 c2   c2 L2
 f 2   f 2Lf
 m2   m2 Lm
(3.35)
Thus, Eq.(3.34) becomes
Or
 c 2 L2   f 2 L f   m 2 Lm
(3.36)
 c 2   f 2v f   m 2vm
(3.37)
Lf
Lm
Where
 vm ,
 vf ,
L2
L2
 c 2  E2  c 2
 f 2  E f 2 f 2
 m2  Em  m2
(3.38)
Where Poisson strains have been neglected.
Combining (3.37) and (3.38),
 c2
E2

Assuming that
We get
 f2
Ef 2
vf 
 m2
Em
vm
(3.39)
 c 2   f 2   m2
vf
vm
1


E2 E f 2 Em
(3.40)
- “Inverse Rule of Mixtures” – Not very accurate
Assume,
 f 2  a2  c 2
 m2  b2  c2
(3.41)
Substituting in the compatibility equation (Rule
of mixture for transverse strain), we get
a2v f  b2vm  1
(3.42)
Then substituting these expressions for 
 m 2 in
Uc  U f  Um
f2
and
(3.28)
We get
E2  a2 E f 2v f  b2 Emvm
2
2
(3.43)
Solving (3.42) and (3.43) simultaneously for a2
and b2, we get for E-glass/epoxy,
 f2
 5.63
 m2
Representative volume element and simple stress states
used in elementary mechanics of materials models
Longitudinal normal stress
Transverse normal stress
In-plane shear stress
In-Plane Shear Modulus, G12
• Using compatibility of shear displacement
and assuming equal stresses in fiber and
matrix:
vf
vm
1


G12 G f 12 Gm
(3.47)
(Not very accurate)
Major Poisson’s Ratio, υ12
• Using compatibility in 1 and 2 directions:
12   f 12v f mvm
(Good enough for design use)
(3.45)
Design Equations
E1  E f 1v f  Emvm
12   f 12v f mvm
• Elementary mechanics of materials
Equations derived for G12 and E2 are not
very useful – need to develop improved
models for G12 and E2.
Improved Mechanics of Materials
Models for E2 and G12
Mechanics of materials models refined by
assuming a specific fiber packing array.
Example: Hopkins – Chamis method of
sub-regions
RVE
Convert RVE with circular fiber to equivalent
RVE having square fiber whose area is the same
as the circular fiber.
RVE
A
d
s
B
A
sf
sf
A
Sub
Region A
B
Sub
Region B
A
Sub
Region A
Division of representative volume element into
sub regions based on square fiber having
equivalent fiber volume fraction.
Equivalent Square Fiber:
sf 

4
d
(from
sf 
2

4
2
d )
(3.48)
Size of RVE:
s

4v f
d
For Sub Region B:
(3.49)
s
sf
sf
Following the procedure for the elementary
mechanics of materials analysis of transverse
modulus:
but
1
1 sf
1 sm


EB 2 E f 2 s Em s
(3.50)
sm
 1 v f ;
 vf ;
s
s
(3.51)
sf
So that
EB 2
Em

1  v f 1  Em E f 2 
(3.52)
For sub regions A and B in parallel,
E2  E B 2
Or finally
sf
sm
 Em
s
s
(3.53)


vf

E2  Em  1  v f 
1  v f 1  Em E f 2 



(3.54)
Similarly,


vf

G12  Gm  1  v f 
1  v f 1  Gm G f 2 



Simplified Micromechanics Equations (Chamis)
Only used part of the analysis for sub region B
in Eq. (3.52):
 EB 2
Em

1  v f 1  Em E f 2 
(3.52)
Gm
G12 
1  v f 1  Gm G f 12 
Fiber properties Ef2 and Gf12 in tables inferred
from these equations.
Semi empirical Models
Use empirical equations which have a
theoretical basis in mechanics
Halpin-Tsai Equations
Where
E2 1  v f

Em 1  v f

E

E
f
f
Em   1
Em   
(3.63)
(3.64)
And
  curve-fitting parameter
  2 for E2 of square array of
circular fibers
  1 for G12
As
As
    Rule of Mixtures
  0  Inverse Rule of
Mixtures
Tsai-Hahn Stress Partitioning Parameters
let
Get
 m2  2 f 2
(3.65)
 v f  2 vm 
1
1




E2 v f   2 vm  E f
Em 
Where 2
(when
(3.66)
 stress partitioning parameter
2  1.0,
get inverse Rule of Mixtures)
Transverse modulus for glass/epoxy according to Tsai-Hahn equation (Eq.
3.66). (From Tsai, S.W. and Hahn, H.T. 1980. Introduction to Composite
Materials. Technomic Publishing Co., Lancaster, PA. With permission from
Technomic Publishing Co.)
Eq. 3.66
Micromechanical Analysis of Composite
Materials Using Elasticity Theory
Micromechanical analysis of composite materials
involve the development of analytical models for
predicting macroscopic composite properties in terms
of constituent material properties and information on
selection of a representative volume element, or RVE,
which depends on the assumed fiber packing array in
the composite.
Example: Square packing array
RVE
Matrix
Fiber
Due to double symmetry, we only need to
Fiber
Matrix
• The RVE is then subjected to uniform stress
or displacement along the boundary. The
resulting boundary value problem is solved
by either stress functions, finite differences
or finite elements.
• Later in this course we will discuss specific
examples of finite difference solutions and
finite element solutions for micromechanics
problems.
```