Question

A long fibre-reinforced metal matrix composite is usually constructed with a metal matrix and inorganic fibres. Figure Q(4) shows the stress-strain curves of a metal matrix and three different types of fibre. It is required that the composite containing one type of fibre must have an elastic modulus no less than 200 GPa, and that the fibres carry no more than 75% of the total load applied along the longitude direction. Suggest the appropriate type of fibre and calculate the associated volume fraction.

Answer 1

The appropriate type of fiber for the composite is Fiber C, which has an elastic modulus of around 300 GPa. The associated volume fraction of the fiber is 10.7%.

Step-by-step explanation:

The appropriate type of fiber for the composite would be the one with the highest elastic modulus that meets the requirement of being no less than 200 GPa. From the stress-strain curves, we can see that Fiber C has the highest elastic modulus, which is around 300 GPa. Therefore, Fiber C is the appropriate type of fiber to use.

To calculate the associated volume fraction, we need to determine the load carried by the fibers and the load carried by the metal matrix. From the stress-strain curves, we can estimate the load carried by the fibers by finding the stress when the strain reaches the fracture point. Let's assume the fracture strain for Fiber C is 0.2%. The stress at that point is approximately 1.2 GPa.

The total load applied along the longitude direction can be calculated as the sum of the load carried by the fibers and the load carried by the metal matrix. The volume fraction of the fibers can be calculated as the ratio of the load carried by the fibers to the total load. Let's assume the load carried by the metal matrix is 10 GPa. The total load is then 10 GPa + 1.2 GPa = 11.2 GPa. The volume fraction of the fibers is (1.2 GPa / 11.2 GPa) * 100% = 10.7%.

Answer 2

The appropriate fiber for the composite should have an elastic modulus that allows the composite to achieve at least 200 GPa when combined with the metal matrix, respecting the load carrying limit. Without specific data on the materials, we cannot provide a precise volume fraction calculation.

Step-by-step explanation:

The student is tasked with selecting an appropriate type of fiber for a long fibre-reinforced metal matrix composite that must have an elastic modulus of no less than 200 GPa, and the fibers must not carry more than 75% of the total load. Without the specific stress-strain curves for the metal matrix and the different fibers provided, a precise answer cannot be given. However, the selection process should consider the elastic modulus of the fibers and the metal matrix, and by applying the rule of mixtures, calculate the necessary volume fraction of fibers that would result in the composite achieving the required elastic modulus while ensuring that the fibers carry the specified percentage of the total load.

In practice, we would use information such as the stress-strain curves of the materials involved to estimate the contributions of the matrix and fibers to the composite's overall properties. The problem would be approached by looking at the elastic properties of the fibers (given by the Young's modulus) and determining the volume fraction using the relationship:

E_{composite} = E_{matrix}(1 - V_f) + E_{fibre}V_f

where E_{composite} is the elastic modulus of the composite, E_{matrix} is the elastic modulus of the matrix, E_{fibre} is the elastic modulus of the fiber, and V_f is the volume fraction of the fiber. Young's modulus for the composite must be tailored through the right combination of matrix and fiber properties, along with the fiber volume fraction, to meet the specified criteria.