Continuous Yielding of a Material Under Constant Stress
We have already seen that in a simple aligned fibre composite, loaded parallel to the fibres that both the matrix and the fibre experience the same strain (amount of stretch). It would be logical therefore to expect the composite to break at the lower of the matrix fracture strain or the fibre fracture strain. There are two cases to consider, firstly, where the matrix fails first and secondly, where the fibre fails first. The former situation is common in polymer matrix composites with low strength brittle matrices such as polyesters, epoxies or bismelamides, the latter case is observed in metal matrix composites or thermoplastic polymer composites where, because of plastic deformation in the matrix, the failure strain of the fibre is the smaller value.
Elastic-Brittle, Matrix Fails First
The first and simplest case that we shall consider is that where the both the fibre and the matrix are linear elastic to failure but the matrix fails at a lower strain that the fibre. Strain is the important factor in determining the failure strength of the composite when testing parallel to the fibres because both the fibre and the matrix experience the same strain.
When the strain in the composite reaches the fracture strain of the matrix, the matrix will fail. Immediately before the matrix breaks, the stress being carried by the composite is
What actually happens to the composite as the matrix breaks depends on how the composite is being loaded and how much fibre is present in the composite. There are two different, but distinct ways that load is imposed on a material, constant deflection and constant load.
Loading under "constant deflection"
In a conventional tensile test the material being tested is actually stretched, i.e. a slowly increasing displacement is applied to one end of the material, the other end remains fixed in place. What is measured is the resistance that the material is imposing against being stretched. If part of the material breaks, like the matrix in the composite is just about to do, the deflection at that instant does not change. The fibres are still stretched by the same strain so the stress in the fibres remains as it was just prior to the matrix failing. Hence, the fibres will not break. However, the load will fall as will the stress on the composite as a whole. The stress in the fibres when the matrix breaks is 
, the force required to sustain that stress is 
, where f is the area [=volume] fraction of fibres and A the cross sectional area of the composite. The actual stress on the composite is thus
Since the fibres run the whole length of the composite, the fibres continue to resist the imposed deflection and we can continue to stretch the fibres to their failure strain at which point the stress being carried by the composite is 
If 
then clearly the ultimate tensile strength of the composite is 
otherwise it's 
.
Loading under "constant load"
In the real world, and cases where materials are tested by imposing slowly increasing loads, what happens when the matrix breaks differs from the case of constant deflection loading. The fibres that remain when the matrix cracks now instantantly have to support the imposed load. The stress in the fibre now jumps to [] because the fibres only occupy a fraction of the original cross-sectional area. Because the stress in the fibres increases, the strain in the fibres also increases and the material will exhibit an increase in deflection (strain) with no additional increase in load. If the increased stress is higher than the failure strength of the fibres then the fibres will break, if not we can continue to increase the load until the fibres break. The stress on the composite at this point is still , as before!
Thus we can envision two strengths- a "working strength" equal to the stress applied to the composite at the strain at which the matrix fails and an "ultimate tensile strength" equal to the larger of the working strength,, or . In the graph below we plot both equations together with the larger of the two as a function of the volume fraction of fibres.
We can see that when the volume fraction of fibres is low that the composite breaks when the matrix fails but when the fibre fraction is above a critical value the composite can remain intact to higher stresses than those required to break the matrix. The concept of "working strength" is important in composites because if the material is unloaded after the matrix has failed, but the fibres remain intact, it can be reloaded. However, the elastic modulus on retesting will not be the same because the matrix takes no part in resisting the applied loads so the modulus just scales as f.Ef which is less than before. The composite could be described as damaged - but it is damage tolerant in that even when the matrix has failed it can continue to support an applied load. In real composites the fibres reinforcing the composite are far from perfect and have a range of strengths due to surface imperfections or geometrical (thickness/diameter) variations. Thus the fibres fail at a range of strengths (and strains) and we often observe a progressive onset of failure.
Elastic - Brittle, Fibres Fail First
When a composite in which the elastic strain to failure of the fibres is less than the strain to failure of the matrix is tested, the fibres willbe the first compoent of the composite to break. The "working strength" of the composite is simply the stiffness of the composite multiplied by the strain to failure of the fibres.
Again, once the fibres have broken what happens depends on whether the compsoite is being loaded under conditions of constant deflection or constant load.
Loading under "constant deflection"
Once the fibres break, the strain remains the same and since the cross-sectional area of the sample has effectively been reduced by that of the fibres, the load required to maintain that deflection/strain is less than before and the load/stress drops. When the deflection is increased, only the matrix resists and can ultimately be stretched to its failure strain. The stress on the composite is then
The ultimate tensile strength of the composite is then the larger of the two stresses.
Loading under "constant load"
When the fibres break, the load is immediately transferred to the matrix, and since the cross sectional area of the matrix is less than that of the composite, the stress in the matrix instantly increases, as does the strain. The increased stress is given by
which, if greater than , will result in fracture of the matrix (and the composite). If the increase in stress is small, such as would be eppected at low volume fractions of fibre then the matrix can continue to be loaded until it fails.
Again the final stress being carried by the composite is
and like before, the ultimate tensile strength of the composite is the larger of this and the working strength.
Again we can examine the concept of a working strength and an ultimate tensile strength coupled with damage tolerance. When the volume fraction of fibres is low, failure of the fibres does not result in failure of the composite as the matrix alone is capable of supporting the imposed stress. However, just like the case where the matrix failed first, unloading and reloading the composite will result in failure at the reduced matrix strength and the stiffness of the composite will be much reduced. We should also note that when the fibres fail at a lower tensile strain than the matrix, that adding small amounts of fibre produces a composite material with a strength less than that of the matrix alone.
Let's see how we can implement a generic design tool to determine the strength of aligned fibre composites using Mathcad - In the exercise we will learn a few more useful techniques
- Using the max() and min() functions
- Using a logical "IF" statement to automatically choose between two conditions
We'll also demostrate how to use the tool to plot out the tensile load - deflection curves for a composite loaded under either an increasing deflection or an increasing load. Jump to the Mathcad exercise.
Source: https://sites.google.com/site/compositematerialsdesign/home/strength-parallel-to-aligned-continuous-fibres
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