CH3MİCROMECHANİCS
Assist.Prof.Dr. Ahmet Erklig
Objectives
o Find the nine mechanical: o four elastic moduli, o five strength parameters
o Four hygrothermal constants: o two coefficients of thermal expansion, and o two coefficients of moisture expansion
of a unidirectional lamina
Micromechanics
Determining unknown properties of the composite based on known properties of the fiber and matrix
Micromechanics
Uses of Micromechanics
Predict composite properties from fiber and matrix data
Extrapolate existing composite property data to different
fiber volume fraction or void content
Check experimental data for errors
Determine required fiber and matrix properties to produce
a desired composite material .
Limitations of Micromechanics
Predicted composite properties are only as good as fiber and matrix properties used
Simple theories assume isotropic fibers many fiber reinforcements are orthotropic
Some properties are not predicted well by simple theories more accurate analyses are time consuming and expensive
Predicted strengths are upper bounds
Notations
Subscript f, m, c refer to fiber, matrix, composite ply, respectively
v volume
V volume fraction
w weight
W weigth fractions
ρ density
Terminology Used in Micromechanics
• Ef, Em – Young’s modulus of fiber and matrix
• Gf, Gm – Shear modulus of fiber and matrix
• υf, υm – Poisson’s ratio of fiber and matrix
• Vf, Vm – Volume fraction of fiber and matrix
Micromechanics and Assumptions
Approaches: Mechanics of materials approach, Semi-empirical approach; Involves rigorous mathematical
solutions.
Assumption: the lamina is looked at as a material whose properties are different in various directions, but not different from one location to another.
Volume Fractions
Fiber Volume Fraction
Matrix Volume Fraction
Mass Fractions
Fiber Mass Fraction
Matrix Mass Fraction
Density
Total composite weigth:
wc = wf + wm
Substituting for weights in terms of volumes and densities
Dividing through by vc gives,
Density
When more than two constituents enter in the composition of the composite material
where n is the number of constituent.
Void Content
Effects of Voids on Mechanical Properties
Lower stiffness and strength� Lower compressive strengths� Lower transverse tensile strengths� Lower fatigue resistance� Lower moisture resistance� A decrease of 2-10% in the preceding matrix-�
dominated properties generally takes place with every 1% increase in void content .
Void Content
Evaluation of Four Elastic Moduli
There are four elastic moduli of a unidirectional lamina:
Longitudinal Young’s modulus, E1
Transverse Young’s modulus, E2
Major Poisson’s ratio, υ12
In-plane shear modulus, G12
Strength of Materials Approach
Assumptions are made in the strength of materials approach
The bond between fibers and matrix is perfect. The elastic moduli, diameters, and space between fibers are
uniform. The fibers are continuous and parallel. The fiber and matrix follow Hooke’s law (linearly elastic). The fibers possess uniform strength. The composites is free of voids.
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 heterogeneous behavior. Generally, single force is considered in the RVE.
RVE
RVE
fibre
matrix
Longitudinal Modulus, E1
Total force is shared by fiber and matrix
Assuming that the fibers, matrix, and composite follow Hooke’s law and that the fibers and the matrix are isotropic, the stress–strain relationship for each component and the composite is
The strains in the composite, fiber, and matrix are equal (εc = εf = εm);
Longitudinal Modulus, E1
Longitudinal Modulus, E1
The ratio of the load taken by the fibers to the load taken by the composite is a measure of the load shared by the fibers.
Predictions agree well with experimental data
Longitudinal Modulus, E1
Transverse Young’s Modulus, E2
The fiber, the matrix, and composite stresses are equal.
σc = σf = σm
the transverse extension in the composite Δc is the sum of the transverse extension in the fiber Δf , and that is the matrix, Δm.
Δc = Δf + Δm
Δc = tc εc
Δf = tf εf
Δm = tm εm
tc,f,m = thickness of the composite, fiber and matrix, respectively
εc,f,m = normal transverse strain in the composite, fiber, and matrix, respectively
Transverse Young’s Modulus, E2
By using Hooke’s law for the fiber, matrix, and composite, the normal strains in the composite, fiber, and matrix are
Transverse Young’s Modulus, E2
Transverse Young’s Modulus, E2
Transverse Young’s Modulus, E2
Major Poisson’s Ratio, ν12
Major Poisson’s Ratio, ν12
Major Poisson’s Ratio, ν12
Major Poisson’s Ratio, ν12
In-Plane Shear Modulus, G12
Apply a pure shear stress τc to a lamina
In-Plane Shear Modulus, G12
In-Plane Shear Modulus, G12
FIGURE 3.13Theoretical values of in-plane shear modulus as a function of fiber volume fraction and com-parison with experimental values for a unidirectional glass/epoxy lamina
In-Plane Shear Modulus, G12
Halphin-Tsai Equation
Longitudinal Young’s Modulus
Major Poisson’s Ratio
Transverse Young’s Modulus, E2
For a fiber geometry of circular fibers in a packing geometry of a square array, ξ = 2. For a rectangular fiber cross-section of length a and width b in a hexagonal array, ξ = 2(a/b), where b is in the direction of loading.
Transverse Young’s Modulus, E2
In-Plane Shear Modulus, G12
For circular fibers in a square array, ξ = 1. For a rectangular fiber cross-sectional area of length a and width b in a hexagonal array, ξ = , where a is the direction of loading. Hewitt and Malherbe suggested choosing a function
In-Plane Shear Modulus, G12
Elasticity Approach
Elasticity accounts for equilibrium of forces, compatibility, and Hooke’s law relationships in three dimensions.
The elasticity models described here are called composite cylinder assemblage (CCA) models. In a CCA model, one assumes the fibers are circular in cross-section, spread in a periodic arrangement, and continuous.
Composite Cylinder Assemblage (CCA) Model
CCA Model
Longitudinal Young’s Modulus, E1
Major Poisson’s Ratio
Transverse Young’s Modulus, E2
The CCA model only gives lower and upper bounds of the transverse Young’s modulus of the composite.
Transverse Young’s Modulus, E2
Transverse Young’s Modulus, E2
Transverse Young’s Modulus, E2
FIGURE 3.21Theoretical values of transverse Young’s modulus as a function of fiber volume fraction and comparison with experimental values for boron/epoxy unidirectional lamina
Transverse Young’s Modulus, E2
In-Plane Shear Modulus, G12
In-Plane Shear Modulus, G12