MICROMECHANICS

1
MICROMECHANICS

• Dr.Ahmet Erklig

2
Definition

• Micromechanics is the study of composite material
  behaviour wherein the ineraction of constituents
  (matrix and fibres) is examined on a microscopic
  scale.

3
We will discuss the stiffness and strength
analyses of a fibrous composite ply on a
macroscopic scale. The macroscopic stiffness ad
strength analyses depend on the pre-determined
(experimentally or otherwise) composite
engineering elastic values E1, E2, n12, G12 and
composite basic strength F1t, F1c, F2t, F2c, F6.
Thus, the analyses of fibrous composite ply can
be carried out these elastic and strength values
are known.
4

• The study of micromechanics involves two
  approaches
• Strength of materials approach
• Relatively simple model mathematically and
  provides the basic understanding of composite
  material behaviour.
• Theory of elasticty approach
• Involves rigorous mathematical solutions.

5
Assumptions

• The fibers are
• Homogeneous
• Isotropic
• Linearly elastic
• Regularly spaced
• Perfctly aligned

• 2. The matrix is
• Homogeneous
• Isotropic
• Linearly elastic

6
Assumptions

• 3. The composite ply is
• Macroscopically homogeneous
• Macroscopically orthotropic
• Linearly elastic
• Initially stress free
• 4. There is a perfect bonding between the fibers
  and matrix, and no voids in present.

7
Notations

• Subscript f, m, c refer to fiber, matrix,
  composite ply, respectively
• v volume
• V volume fraction
• w weight
• W weigth fractions
• ? density

8
Volume Weight Fractions

• The relative proportions of the fibers and matrix
  is one major factor the ply properties, and
  proportions are usually expressed as volume or
  weight fractions.
• Weight fractions are generally easier to
  determine during the composite material
  fabrication or by test methods after fabrication.
  However it is the volume fractions which are used
  for the micromechanical analysis. Hence, we need
  to know the relationship between the volume
  fractions and the weight fractions of a composite
  material. This relationship is achieved thorugh
  the composite density.

9
Composite Density

• Total composite weigth
• wc wf wm
• Substituting for weights in terms of volumes and
  densities
• ?c.vc ?f.vf ?m.vm
• Dividing through by vc gives,
• ?c ?f.vf ?m.vm
• vc vc
• ?c ?f.Vf ?m.Vm

10
Composite Density

• Total volume (if there is no voids)
• vc vf vm
• Substituting for volumes interms of weigths,
• Dividing through by wc gives

11
Composite Density

• Volume and wiegth fractions relationship

12
Representative Volume Element (RVE)

• This is the smallest ply region over which the
  stresses and strains behave in a macroscopically
  homogeneous behavior. Microscopically, RVE is of
  a heteregeneous behaviour. Generally, single
  force is considered in the RVE.

13
Representative Volume Element (RVE)
RVE
matrix
fibre
14
Coordinate frame

• To begin lets define a coordinate frame

15
Material Properties

• The material properties we will be concerned with
  for composite materials (due to homogeneous and
  anisotropic material) are
• E1 Modulus of elasticty in fiber direction
• E2 Modulus of elasticty in transverse direction
• n12 inplane Poissons ratio
• G12 inplane shear modulus
• G23 out-of-plane shear modulus

16
Longitudinal modulus
What material properties can we calculate here?
E1s1/e1 or E11s1/e1
n12-e2/e1
17
Transverse modulus

• What material properties can we calculate here?

E2s2/e2 or E22s2/e2
n21-e1/e2
18
Reciprocity relationship

• n21n12E22/E11
• Is it better to calculate n21 using direct
  experimental measurement or using reciprocity
  relationship?

19
In-plane Shear Modulus G12
G12t12/g12
pure shear
Iosipescu shear test
20
Examples of laminated composites

• Human intervertebral disc annulus fibrosus

21
Longitudinal Modulus, E1

• It is assumed that there is a perfect bond btw
  the fibers and matrix,i.e. No slippage occurs at
  the interface. This implies that when the RVE is
  subjected to a stress in direction 1, the strains
  in the fiber and matrix are the same in direction
  1
• ef em e1

22
Longitudinal Modulus, E1
Stress in the fibre
Stress in the matrix
23
Longitudinal Modulus, E1
Substitute force in the expression Pc PfPm
If the crossections are constant Af/Ac Vf and
Am/AcVm Therefore,
OR
24
Transverse Modulus, E2

• It is assumed that the same stress s2 acts on the
  matrix as well as on the fibre, and neglecting
  any Poisson effect in direction 1.

25
Transverse Modulus, E2
Transverse strain in the fibre direction
Transverse strain in the matrix direction
Transverse displacements
26
Transverse Modulus, E2
Total transverse displacements in the composite
is dc df dm
Substituting the dc, df and dm terms, gives
Rearranging