Composite Materials

Composite materials are mixtures of two or more fundamental materials. Usually one forms a continuous material, known as a matrix, within which a material, usually microscopic, is embedded.

The term composite materials are used to describe situations where two constituent materials have been mixed together physically, rather than different components appearing naturally as different phases in an alloy, such as spontaneous mixture of ferrite and cementite that appears on cooling of steel. Composite materials can be based on matrices of metals, ceramics and polymers, but by far the most common composites employed, and the most relevant to motorsport are those based on polymer matrices.

Usually it is the filler particles that have the most attractive mechanical properties and the function of the matrix is to act as a glue to hold the particles together.

Looking at the lump of composite material, in Figure 5.6, of total mass m = mm+ mf and total volume v = vm + vf, where the subscripts m and f referring to the matrix and the fibres respectively.

Figure 5.6 Lump of composite material ^[4]

The Rule of Mixtures (RoM):







_f



_f

(1 

_f

) 

_m(5.1) where



f v_f

v and

(1

f

)

m v_m v

This basically uses the volume fraction of fibres in the material to superpose the densities of both the matrix and the fibres in order to get the overall density of the combination of the two. The Rule of Mixtures can be used for more properties of the composite material, such as the stress, strain,

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Poisson’s ratio and the Young’s Modulus. These can vary depending on the orientation or the axis, which the material is being measure, owing to the anisotropic behaviour of composite materials.

Due to the composite material being utilised in motorsport, the filler used will take the form of continuous cylindrical fibres which are embedded in a matrix which needs to be capable of bonding to the fibres in either a mechanical or a chemical manner. In unidirectional (UD) continuous fibre composites, all of the fibres are oriented in the same, single direction. For now, analysis of a UD lamina will be considered, building up to the analysis of a laminate, which consists of multiple UD laminae oriented in different directions to try and create more isotropic behaviour.

Three orthogonal reference axes are used to define the directions: with the 1-direction being parallel to both the fibres and the lamina, the 2-direction being perpendicular to the fibres and parallel to the lamina and the 3-direction being perpendicular to both the fibres and the lamina.

Using the theory outlined in An Introduction to Composite Materials ^[5], the composite material properties can be calculated. These properties, detailed below, are to be calculated and used in the analysis of the chassis, but first the equations need to be formed for the varying properties of interest.

Axial Stiffness

As the composite materials are bonded together, a no sliding condition can be applied so that they have the same lengths parallel to the bonded surface, resulting in both constituents exhibiting the same strain.





1

 

f1

 

_{f 1}

E

_f

 

m1

 

_m1

E

_m

For the composite material of interest, the reinforcing fibres are much stiffer and put under much higher stresses than the matrix, so there is a redistribution of load. Therefore the overall stress can be expressed, in terms of the contributions from both the fibres and matrix:

₁_f_{f 1}(1_f)_m1

Combining these two equations and after some rearrangement, the Young’s Modulus is found:

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

E₁ 

 ^_fE_f (1_f)E_m (5.2)

This is a proof of the Rule of Mixtures being utilised to find another property of the composite material as a whole in similar fashion.

Transverse Stiffness

Calculating the transverse stiffness is more difficult due to the higher concentration of stresses in the matrix between the fibres, as the stresses aren’t redistributed as well as in the axial state. The limiting case will be considered of equal stress, so when a stress is applied in the 2-direction



₂ _{f 2} _{f 2}E_f _{m 2} _{m 2}E_m

the overall strain is the combination of the total strains of the matrix and fibres in the 2-direction:



The Shear Moduli are estimated in a similar fashion to the axial and transverse stiffness. The shear modulus is the ratio of the shear stress to the shear strain. Considering 2 directions:



G₁₃fG_f (1f)G_m

(5.4)

Poisson’s Ratios

The Poisson’s ratio ij is the ratio the negative strain (contraction) in the j-direction when a stress is

applied in the i-direction, shown by



ij  



j



i. This leads on to two more useful relationships:

 summing them, the Poisson’s ratios for the 2 directions parallel to the lamina can be produced:

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The relationship between ij and ij can be expressed using Einstein Notation: ij=Cijklkl where Cijkl

is a stiffness tensor. However in practice, it can be more useful to express observed strains in terms of applied stress: ij=Sijklkl where Sijkl is the compliance tensor. If plane stress conditions are

applied, When the lamina is loaded in a direction oriented at



 to the 1-direction, the compliance tensor needs to be both pre and post-multiplied by appropriate rotation tensors to give one transformed compliance tensor laminate is stacked with adjacent plies at 45˚ orientation to each other. So this has been chosen as the orientation for the stacking of the laminae in this design to try and create as isotropic behaviour as possible, as a racing car is exerted to varying orientations of force whilst in performance – bias isn’t needed in just a single direction.

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Figure 5.7 Variation of loading angle



˚ of Young’s Modulus E and Poisson’s ratio 12 for a single UD lamina, a cross ply laminate [0˚/90˚] and a [0˚/45˚/90˚/135˚] laminate ^[7]

Composite Material Strengths

Using the theory outlined in the B8 Composites Notes⁸, the tensile and compressive strengths of composite materials can be estimated. The strength depends on the failure strains of both the matrix and the fibres. Usually, in the case of reinforced fibres in a polymer matrix, the fibres are more brittle than the matrix as this is the ‘stiffener’ in the composite material; so therefore have a

smaller failure strain compared to the matrix. If the volume fraction of fibres is small (



_f 

1

_{), the} effect of the fibres actually weakens the matrix, so there is a critical fibre fraction below which the matrix is stronger on its own, and above which the fibres start to have a beneficial effect.

Figure 5.8 Graph showing how the strength of composite materials varies with fibre fraction ^[9]

The right hand side of the graph is shaded in as this is the area of



_f 

.7

, which is geometrically impossible to pack fibres in a matrix of this fraction region. It is only of interest to look into high, but

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realistic, volume fractions of

.3





_f 

.7

, and in this case the maximum stress of the composite occurs at the maximum stress of the fibres, i.e. when the fibres break. (*) denotes point of failure.

Figure 5.9 Graph showing tensile strength of composite material with high volume fraction ^[8]

When the fibres fail, the total stress is carried by the composite, which can be calculated by equating forces in the 1-direction. So the tensile strength through superposition of stresses at f* is



1

*

 

_f



^*_f

 (1 

_f

) 

_m^'

(5.10) This is the equation for the majority of the bold line in Figure 5.8.

The compressive strength of composite materials is usually lower than the tensile strength due to the matrix normally having a fairly low shear modulus so “kink bands” ^[9] can form in the matrix between the fibres. These kink bands impose an axial shear on the fibres, which are notoriously susceptible to shear failure due to their very high aspect ratio. The transverse compressive and tensile strengths are much less than the axial ones, due to the presence of fibres with different elastic properties resulting in local stresses greatly exceeding the average stress. The fibre-matrix interface is weaker than the matrix as there is a great discontinuity of properties here. In this case the compressive strength is slightly greater than the close to almost irrelevant tensile strength.

When stresses are applied along non-principal axes, a maximum stress criterion needs to be applied to all varieties of failure modes that are assumed to be independent of one another. Failure will occur at the lowest failure stress.

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In document Oxford Octane Formula Student Report (Page 76-82)