Introduction to composite materials

Source eFunda.com :

http://www.efunda.com/formulae/solid_mechanics/composites/comp_intro.cfm

Composite Materials

A typical composite material is a system of materials composing of

two or more materials (mixed and bonded) on a macroscopic scale.

For example, concrete is made up of cement, sand, stones, and

water. If the composition occurs on a microscopic scale (molecular

level), the new material is then called an alloy for metals or a polymer for plastics.

Generally, a composite material is composed of reinforcement (fibers, particles, flakes, and/or fillers) embedded in a matrix

(polymers, metals, or seramics). The matrix holds the reinforcement to form the desired shape while the reinforcement improves the

overall mechanical properties of the matrix. When designed

properly, the new combined material exhibits better strength than would each individual material.

Top of Page

Common Categories of Composite Materials

Based on the form of reinforcement, common composite materials

can be classified as follows:

1. Fibers as the reinforcement (Fibrous Composites):

a. Random fiber (short fiber) reinforced composites

b. Continuous fiber (long fiber) reinforced composites

2. Particles as the reinforcement (Particulate composites):

3. Flat flakes as the reinforcement (Flake composites):

4. Fillers as the reinforcement (Filler composites):

Top of Page

Benefits of Composites

Different materials are suitable for different applications. When

composites are selected over traditional materials such as metal alloys or woods, it is usually because of one or more of the following

advantages:

Cost: o Prototypes

o Mass production

o Part consolidation o Maintenance

o Long term durability

o Production time o Maturity of technology

Weight:

o Light weight o Weight distribution

Strength and Stiffness:

o High strength-to-weight ratio o Directional strength and/or stiffness

Dimension:

o Large parts o Special geometry

Surface Properties:

o Corrosion resistance o Weather resistance

o Tailored surface finish

Thermal Properties: o Low thermal conductivity

o Low coefficient of thermal expansion

Electric Property: o High dielectric strength

o Non-magnetic

o Radar transparency

Note that there is no one-material-fits-all solution in the engineering

world. Also, the above factors may not always be positive in all applications. An engineer has to weigh all the factors and make the

best decision in selecting the most suitable material(s) for the

project at hand.

Composition of Fiber Reinforced Composites

Common fiber reinforced composites are composed of fibers and a

matrix. Fibers are the reinforcement and the main source of strength

while the matrix 'glues' all the fibers together in shape and transfers stresses between the reinforcing fibers. Sometimes, fillers or

modifiers might be added to smooth manufacturing process, impart

special properties, and/or reduce product cost.

Top of Page

Fibers of Fiber Reinforced Composites

The primary function of the fibers is to carry the loads along their longitudinal directions. Common fiber reinforcing agents include

Aluminum, Aluminum oxide, Aluminum silica

Asbestos

Beryllium, Beryllium carbide, Beryllium oxide

Carbon (Graphite) Glass (E-glass, S-glass, D-glass)

Molybdenum

Polyamide (Aromatic polyamide, Aramid), e.g., Kevlar 29 and Kevlar 49

Polyester

Quartz (Fused silica) Steel

Tantalum

Titanium Tungsten, Tungsten monocarbide

Top of Page

Matrix of Fiber Reinforced Composites

The primary functions of the matrix are to transfer stresses between

the reinforcing fibers (hold fibers together) and protect the fibers

from mechanical and/or environmental damages. A basic requirement for a matrix material is that its strain at break must be

larger than the fibers it is holding.

Most matrices are made of resins for their wide variation in

properties and relatively low cost. Common resin materials include

Resin Matrix

o Epoxy

o Phenolic o Polyester

o Polyurethane

o Vinyl Ester

Among these resin materials, polyesters are the most widely used. Epoxies, which have higher adhesion and less shrinkage than

polyesters, come in second for their higher costs.

Although less common, non-resin matrices (mostly metals) can still

be found in applications requiring higher performance at elevated

temperatures, especially in the defense industry.

Metal Matrix

o Aluminum o Copper

o Lead

o Magnesium o Nickel

o Silver

o Titanium Non-Metal Matrix

o Ceramics

Top of Page

Modifiers of Fiber Reinforced Composites

The primary functions of the additives (modifiers, fillers) are to

reduce cost, improve workability, and/or impart desired properties.

Cost Reduction:

o Low cost to weight ratio, may fill up to 40% (65% in some cases) of the total weight

Workability Improvement:

o Reduce shrinkage o Help air release

o Decrease viscosity

o Control emission o Reduce coefficient of friction on surfaces

o Seal molds and/or guide resin flows

o Initiate and/or speed up or slow down curing process Property Enhancement:

o Improve electric conductivity

o Improve fire resistance o Improve corrosion resistance

o Improve ultraviolet resistance

o Improve surface toughness

o Stabilize heat transfer

o Reduce tendency of static electric charge o Add desired colors

Common materials used as additives include

Filler Materials:

o Feldspar o Glass microspheres

o Glass flakes

o Glass fibers, milled o Mica

o Silica

o Talc o Wollastonite

o Other microsphere products

Modifier Materials: o Organic peroxide, e.g., methylethylketone peroxide

(MEKP)

o Benzoyl peroxide o Tertiary butyl catechol (TBC)

o Dimethylaniline (DMA)

o Zinc stearate, waxes, silicones o Fumed silica, clays

Independent Material Constants

Hooke was probably the first person that suggested a mathematical

expression of the stress-strain relation for a given material.

The most general stress-strain relationship (a.k.a. generalized

Hooke's law) within the theory of linear elasticity is that of the materials without any plane of symmetry, i.e., general anisotropic

materials or triclinic materials. If there is a plane of symmetry, the

material is termed monoclinic. If the number of symmetric planes increases to two, the third orthogonal plane of material symmetry

will automatically yield and form a set of principal axes. In this case,

the material is known as orthotropic. If there exists a plane in which the mechanical properties are equal in all directions, the material is

called transversely isotropic. If there is an infinite number of planes

of material symmetry, i.e., the mechanical properties in all directions are the same at a given point, the material is known as

isotropic.

Please distinguish 'isotropic' from 'homogeneous.' A material is

isotropic when its mechanical properties remain the same in all

directions at a given point while they may change from point to point; a material is homogeneous when its mechanical properties

may be different along different directions at given point, but this

variation is consistant from point to point. For example, consider three common items on a dining table: stainless steel forks, bamboo

chopsticks, and swiss cheese. Stainless steel is isotropic and

homogeneous. Bamboo chopsticks are homogeneous but not isotropic (they are transversely isotropic, strong along the fiber

direction, relatively weak but equal in other directions). Swiss

cheese is isotropic but not homogeneous (The air bubbles formed during production left inhomogeneous spots).

Both stress and strain fields are second order tensors. Each component consists of information in two directions: the normal

direction of the plane in question and the direction of traction or

deformation. There are nine (9) components in each field in a three dimensional space. Since they are symmetric, engineers usually

rewrite them from a 3×3 matrix to a vector with six (6) components

and arrange the stress-strain relations into a 6×6 matrix to form the generalized Hooke's law. For the 36 components in the stiffness or

compliance matrix, not every component is independent to each

other and some of them might be zero. This information is summarized in the following table.

Independent

Constants

Nonzero

On-axis

Nonzero

Off-axis

Nonzero

General

Triclinic (General

Anisotropic)

21 36 36 36

Monoclinic 13 20 36 36

Orthotropic 9 12 20 36

Transversely

Isotropic 5 12 20 36

Isotropic 2 12 12 12

A more detailed discussion of stress, strain, and the stress-strain

relations of materials can be found in the Mechanics of Materials

section.

Macromechanics of Lamina

From control surfaces of modern aircrafts, to hulls and keels of yachts, to racing car bodies, to tennis rackets, fishing rods, golf

shafts and heads, laminated fiber reinforced composite is one of the the most widely used composites in industry.

Unless otherwise noted, the following assumptions are made in our discussion of the macro-mechanics of laminated composites.

1. The matrix is homogeneous, isotropic, and linear elastic. 2. The fiber is homogeneous, isotropic, linear elastic, continuous,

regularly spaced, and perfectly aligned.

3. The lamina (single layer) is macroscopically homogeneous, macroscopically orthotropic, linear elastic, initially stress-free,

void-free, and perfectly bonded.

4. The laminate is composed of two or more perfectly bonded laminae to act as an integrated structural element.

Stress-Strain Relations for Principal Directions

Before discussing the mechanics of laminated composites, we need to understand the mechanical behavior of a single layer -- lamina.

Since each lamina is a thin layer, one can treat a lamina as a plane

stress problem. This simplification immediately reduces the 6×6 stiffness matrix to a 3×3 one.

Since each lamina is constructed by unidirectional fibers bonded by a metal or polymer matrix, it can be considered as an orthotropic

material. Thus, the stress-strain relations on the principal axes can

be expressed by the compliance matrix [S] such that

[ ] = [S][ ]

or by the stiffness matrix [C] such that

[ ] = [C][ ]

Please note that the engineering shear strain is used in the stress-strain relations, and, the notation S for the compliance matrix and C

for the stiffness matrix are not misprints. Please consult this page

for more information.

For both stiffness and compliant matrices are symmetric, i.e.,

only four of , , , , and are independent material properties. Again, the shear modulus G12 corresponds to the

engineering shear strain which is twice the tensor shear strain .

Please note that there can be many fibers across the thickness of a lamina and these fibers may not be arranged uniformly in most

industrial practice. However, the combination of the matrix and the

fibers forms an orthotropic and homogeneous material from a marcomechanics standpoint. Some literature therefore schematically

illustrates a lamina with only one layer of uniformly distributed fibers as shown below.

Mechanical Behaviors of a Lamina

A continuous, unidirectional fiber reinforced composite lamina is an

orthotropic material. As discussed in Stress-Strain Relations of Materials, there are 9 independent material constants for an

orthotropic material. For a thin plate such as a lamina, the plane stress assumption holds and the number of independent constants

can be further reduced from 9 to 4 (see this section for details).

The stress-strain relations can be written as

and since

only four of , , , , and are independent material

constants.

Due to the large number of possible fiber-matrix combinations and

their volume fraction ratios, these constants are usually not

available without conducting a series of experiments. Nontheless, estimated values can be obtained, assuming that the properties of

both the matrix and the fibers are known.

Top of Page

Determination of E1

Suppose the bonding between the fibers and the matrix is perfect, the strain of the fibers and the strain of the matrix have to be the

same in the fiber direction (i.e., ) when the lamina is subjected to a uniaxial force along the fiber direction.

The total force applied on the lamina is

where Af and Am are the the cross section areas of the fibers and the matrix, respectively. The Young modulus E1 can then be written

where V is the volume fraction and L is the length of the lamina.

Notice that based on the no-void assumption.

One can visualize the fibers and the matrix as two springs in parallel

as illustrated below.

Top of Page

Determination of E2

Again, assuming perfect fiber-matrix bonding, the stress of the fiber

and the stress of the matrix are the same in the transverse direction

of the fiber ( ) when the lamina is subjected to a uniaxial force:

The transverse strain is the sum of the contributions from the fibers and the matrix which are in proportion to their respective volume

fractions:

The Young's modulus E2 can be calculated using the serial-spring model:

In this case, the fibers and the matrix act like two springs in series:

Top of Page

Determination of 12

The major Poisson's ratio 12 is defined as

As shown in the Determination of E1 section, we have

and,

The major Poisson's ratio can then be written as

Top of Page

Determination of G12

Based on the same argument used in the Determination of E2

section, we assume that the shear stress of the fibers and that of

the matrix are the same, that is, .

The shear strain is the sum of the contributions from the fibers and

the matrix, which are proportional to their respective volume fractions:

The shear modulus G12 can therefore be calculated using the serial-spring model:

Material constants calculated from the above formulae are merely estimates and should not be trusted without further verification. The

true material properties can only be obtained through experiments.

Coordinate Transformation is Necessary

The generalized Hooke's law of a fiber-reinforced lamina for the

principal directions is not always the most convenient form for all

applications. Usually, the coordinate system used to analyze a structure is based on the shape of the structure rather than the

direction of the fibers of a particular lamina.

For example, to analyze a bar or a shaft, we almost always align one

axis of the coordinate system with the bar's longitudinal direction.

However, the directions of the primary stresses may not line up with the chosen coordinate system. For instance, the failure plane of a

brittle shaft under torsion is often at a 45° angle with the shaft. To

fight this failure mode, layers with fibers running at ± 45° are

usually added, resulting in a structure formed by laminae with

different fiber directions. In order to "bring each layer to the same

table," stress and strain transformation formulae are required.

Top of Page

Coordinate Transformation of Stress-Strain Relations

for Lamina

If we define the coordinate transformation matrix as

and

The coordinate transform of plane stress can be written in the following matrix form:

Similarly, the strain transform becomes

Please notice that the tensor shear strain is used in the above

formula. Suppose we define the engineering-tensor interchange matrix [R]

then

The stress-strain relations for a lamina of an arbitry orientation can

therefore be derived as detailed below.

where the stiffness matrix is defined as

The complicance matrix is therefore

The individual components of the stiffness and compliance matrices can be found here.

Strength Needed in More Than One Direction

Considering its light weight, a lamina (ply) of fiber reinforced

composite is remarkably strong along the fiber direction. However, the

same lamina is considerably weaker in all off-fiber directions. To

address this issue and withstand loadings from multiple angles, one would use a lamination constructed by a number of laminae oriented at

different directions.

Basic Assumptions of Classical Lamination Theory

Similar to the Euler-Bernoulli beam theory and the plate theory, the classical lamination theory is only valid for thin laminates (span a and

b > 10×thinckness t) with small displacement w in the transverse

direction (w << t). It shares the same classical plate theory assumptions:

Kirchhoff Hypothesis

1. Normals remain straight (they do not bend)

2. Normals remain unstretched (they keep the same

length)

3. Normals remain normal (they always make a right

angle to the neutral plane)

In addition, perfect bonding between layers is assumed.

Perfect Bonding

1. The bonding itself is infinitesimally small (there is no

flaw or gap between layers).

2. The bonding is non-shear-deformable (no lamina can

slip relative to another).

3. The strength of bonding is as strong as it needs to be

(the laminate acts as a single lamina with special

integrated properties).

Classical Lamination Theory From Classical Plate

Theory

The classical lamination theory is almost identical to the classical plate theory, the only difference is in the material properties (stress-strain

relations). The classical plate theory usually assumes that the material

is isotropic, while a fiber reinforced composite laminate with multiple layers (plies) may have more complicated stress-strain relations.

The four cornerstones of the lamination theory are the kinematic,

constitutive, force resultant, and equilibrium equations. The outcome

of each of these segments is summarized as follows:

Kinematics:

where u0, v0, and w0 are the displacements of the

middle plane in the x, y, and z directions, respectively. Please note that some literature may

define kxy as the total skew curvature which

eliminates the factor of 2. Also note that Kirchhoff's assumptions are introducted to simplify the

displacement fields.

Constitutive:

alternatively,

where the subscript k indicates the kth layer

counting from the top of the laminate.

Resultants:

Again, the subscript k indicates the kth layer from the top of the laminate and N is the total number of

layers. Note that perfect bonding is assumed so we

can move the integration inside the summation.

Equilibrium:

Top of Page

Forming Stiffness Matrices: A, B, and D

The plate is assumed to be constructed by a homogeneous but not necessarily isotropic material and subjected to both transverse and

in-plan loadings. Also, the Cartesian coordinate system is used. The

goal is to develop the relations between the external loadings and the displacements. However, the relations between the resultants

(forces N and moments M) and the strains (strains and curvatures

k) are of most interest in practice.

Replace the stresses in the force and moment resultants with strains

via the constitutive equations, we have

By applying the summation and integration operations to their

respective components, the force and moment resultants can be further simplified to

Combine the above equations we can write:

where A is called the extensional stiffness, B is called the coupling stiffness, and D is called the bending stiffness of the laminate. The

components of these three stiffness matrices are defined as follows:

where tk is the thickness of the kth layer and is the distance from

the mid-plan to the centroid of the kth layer. Forming these three stiffness matrices A, B, and D, is probably the most crucial step in

the analysis of composite laminates.

In some situations, strains expressed in terms of resultants are

more handy. The strain-resultant relations can be derived with

appropriate matrix operations:

where

Note that A, B, D and A*, B*, D* are all symmetric matrices. Among

them, A, B, and D are considered universal notations in the field of

composites, i.e., the same notations appear in almost all literature

of composite materials. A*, B*, and D*, on the other hand, are not.

Calculators Excel workbook:

See http://www.efunda.com/formulae/solid_mechanics/composites/calc_ufrp_abd_layout.cfm

Material properties See http://www.efunda.com/materials/materials_home/materials.cfm

Or http://www.matweb.com/search/MaterialGroupSearch.aspx