Problem 11 - Biofluid Mechanics: An Introduction to Fluid Mechanics, Macrocirculation, and Microcirculation

Question

Calculate the stress on the cartilage and the change in length of cartilage, assuming that the force on the cartilage is 9875 N and that the diameter of the cartilage is 2 cm (assume that the cartilage has a circular area). The cartilage has a thickness of 1.5 mm and an elastic modulus of 250 MPa.

Step-by-Step Solution

Answer

The stress on the cartilage is approximately 31.44 MPa, and the change in length is approximately 0.188 mm.

Step 1: Calculate the Area of the Cartilage

First, calculate the area of the cartilage. Given that the cartilage is circular, its area can be calculated using the formula for the area of a circle, which is \( A = \pi r^2 \). The diameter of the cartilage is 2 cm, so the radius \( r \) is \( 1 \) cm or \( 0.01 \) m. Thus, the area is:\[A = \pi (0.01)^2 = \pi \times 0.0001 = 0.0001\pi\, \text{m}^2 \]Approximately, \( A \approx 0.000314\, \text{m}^2 \).

Step 2: Calculate the Stress on the Cartilage

Stress is calculated by dividing the force by the area. The given force \( F \) is 9875 N. Using the area \( A \) calculated in Step 1:\[\text{Stress} = \frac{F}{A} = \frac{9875}{0.000314}\, \text{N/m}^2\]Calculating this gives:\[\text{Stress} \approx 31,440,127.4\, \text{Pa} \text{ or } \text{approximately } 31.44\, \text{MPa}\]

Step 3: Calculate the Change in Length of the Cartilage

The change in length \( \Delta L \) of the cartilage is calculated using Hooke's law, represented by \( \Delta L = \frac{F \times L_0}{A \times E} \), where \( L_0 \) is the original thickness of the cartilage, and \( E \) is the elastic modulus. \( L_0 \) is 1.5 mm or 0.0015 m, and \( E \) is 250 MPa or 250,000,000 Pa. Substitute:\[\Delta L = \frac{9875 \times 0.0015}{0.000314 \times 250,000,000}\]Calculating this gives:\[\Delta L \approx 0.000188\, \text{m} \text{ or } 0.188\, \text{mm}\]

Key Concepts

Stress Calculation

When discussing biomechanics, stress calculation is a vital concept that examines how forces are distributed within a material. Stress is defined as the force applied per unit area. In the original exercise, stress is calculated to understand the force's impact on cartilage. We use the formula:

• Stress (\[ \sigma \]) = Force (\[ F \]) / Area (\[ A \])

To compute this, you first calculate the area of the cartilage using the formula for the area of a circle, \( A = \pi r^2 \).
Converting the diameter to meters, the radius is \( 0.01 \) m. The subsequent area calculation results in approximately \( 0.000314 \) m².
Then, by dividing the force (9875 N) by the calculated area, you determine the stress: \[ \sigma = \frac{9875}{0.000314} \approx 31.44 \text{ MPa} \].
This value signifies how much force is exerted per unit area of the cartilage, shedding light on potential structural impacts.

Elastic Modulus

The elastic modulus, often referred to as the modulus of elasticity, is a metric of a material's ability to withstand deformation under stress. It tells us how stiff or flexible a material is. Represented by \( E \), the elastic modulus is the ratio of stress (\( \sigma \)) to strain (\( \varepsilon \)) within the elastic limit:

• Elastic Modulus (\( E \)) = Stress (\( \sigma \)) / Strain (\( \varepsilon \))

In this scenario, we have an elastic modulus value specified as 250 MPa for cartilage.
This means that for every unit of stress applied, cartilage experiences deformation proportional to this modulus.
A higher elastic modulus indicates a stiffer material, requiring more stress to produce the same amount of strain, making it a crucial property in materials science and biomechanics.
Understanding the elastic modulus helps predict how cartilage will behave under mechanical forces, aiding in the design of biomedical devices or in assessing injury risk.

Hooke's Law

Hooke's Law is foundational in understanding how materials behave under mechanical stresses. It states that the deformation of an object, within its elastic limit, is directly proportional to the applied force. The law is formulated as:

• \( \Delta L = \frac{F \times L_0}{A \times E} \)

Here, \( \Delta L \) is the change in length, \( F \) is the applied force, \( L_0 \) is the original length, \( A \) is the cross-sectional area, and \( E \) is the elastic modulus.
In the exercise, Hooke's Law is used to find the change in length of the cartilage when a force of 9875 N is applied.
Given the cartilage thickness (\( L_0 \)) is 1.5 mm, the calculated change in length using Hooke’s Law is approximately 0.188 mm.
This law gives insight into the flexibility and limits of materials, helping to ensure they are used safely in engineering and medical applications.