(a) For a fiber-reinforced composite, the efficiency of reinforcement \(\eta\) depends on fiber length \(l\) according to $$ \eta=\frac{l-2 x}{l} $$ where \(x\) represents the length of the fiber at each end that does not contribute to the load transfer. Make a plot of \(\eta\) versus \(l\) to \(l=50 \mathrm{~mm}\) (2.0 in.), assuming that \(x=1.25 \mathrm{~mm}(0.05 \mathrm{in} .)\). (b) What length is required for a \(0.90\) efficiency of reinforcement?

Understanding the reinforcement efficiency formula is critical when working with fiber-reinforced composites. This formula is fundamental for optimizing materials to ensure they are strong enough for their intended use. The formula is given by:
\[\begin{equation} \[\eta=\frac{l-2 x}{l}\]\end{equation}\]
Here, \(\eta\) represents the efficiency of reinforcement, \(l\) is the length of the fiber, and \(x\) is the length at each end of the fiber that does not contribute to load transfer. This could be due to factors such as inadequate bonding or fiber end conditions.
To put this into practical terms, let's say a certain fiber has a negligible load contribution region of 1.25 mm at each end. If we want to calculate the efficiency of this fiber when it is 50 mm long, we plug the values into the formula to get: \[\begin{equation} \eta = \frac{50 - 2(1.25)}{50}\end{equation}\]By computing this, we find out the efficiency of the reinforcement for a given fiber length.

The relationship between fiber length and reinforcement efficiency can be visualized by plotting \(\eta\) versus \(l\). Plotting this relationship helps us understand how increasing the fiber length affects the efficiency of load transfer in a composite material.
To create such a plot, we evaluate the efficiency formula using a range of fiber lengths. For instance, with an \(x\) value of 1.25 mm, the plot will show the efficiency values from \(l = 0\) mm up to \(l = 50\) mm. This graphic representation allows us to observe the trend that, in general, longer fibers will have higher reinforcement efficiency, up until a certain point where increases in length have diminishing returns due to practical limits such as fiber waviness or other composite structure characteristics.

Visualization and Interpretation

Visualizing the data makes it easier to understand the significance of each fiber length. In the plotted graph, the efficiency curve rises sharply with an increase in fiber length at the beginning and then levels off as the length continues to increase. This leveling off indicates that after a certain fiber length, there is no substantial gain in reinforcement efficiency.

To determine the fiber length required for a specific reinforcement efficiency, we rearrange the reinforcement efficiency formula to solve for \(l\). For instance, if an efficiency of 0.90 is needed, we set up the formula with such efficiency in mind:
\[\begin{equation} 0.90 = \frac{l - 2 x}{l}\end{equation}\]
After inserting the non-contributing length, \(x\), and rearranging for \(l\), we find that a longer fiber length increases the efficiency, up to a certain limit. In the case of our exercise, if \(x = 1.25\) mm, the equation to determine the required fiber length would be, after simplification through algebraic steps:\[\begin{equation} l = \frac{2.5}{0.1} = 25\, mm\end{equation}\]
This calculation is vital for engineers and designers who wish to achieve a certain performance level with their composite materials. Knowing the necessary fiber length for the desired efficiency enables them to design more effectively and predict the behavior of the composite under different load conditions.