ch3 micromechanics assist.prof.dr. ahmet erklig. objectives o find the nine mechanical: o four...

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