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 \(9875 \mathrm{~N}\) and that the diameter of the cartilage is \(2 \mathrm{~cm}\) (assume that the cartilage has a circular area). The cartilage has a thickness of \(1.5 \mathrm{~mm}\) and an elastic modulus of \(250 \mathrm{MPa}\).

Step-by-Step Solution

Answer

Stress is 31.43 MPa; change in length is 0.1886 mm.

Step 1: Calculate the Cartilage Area

The cartilage is assumed to have a circular shape. The area \(A\) of the cartilage can be calculated using the formula for the area of a circle, \(A = \pi r^2\), where \(r\) is the radius. The diameter is given as 2 cm, so \(r = \frac{2}{2} = 1 \text{ cm} = 0.01 \text{ m}\). Plug these values into the formula to get \(A = \pi (0.01)^2 = \pi \times 0.0001 \approx 3.14 \times 10^{-4} \text{ m}^2\).

Step 2: Calculate the Stress on the Cartilage

Stress \(\sigma\) is computed using the formula \(\sigma = \frac{F}{A}\), where \(F\) is the force and \(A\) is the area. Substituting the given force \(F = 9875 \text{ N}\) and the calculated area \(A = 3.14 \times 10^{-4} \text{ m}^2\), we find \(\sigma = \frac{9875}{3.14 \times 10^{-4}} \approx 31,434,713.7 \text{ Pa} = 31.43 \text{ MPa}\).

Step 3: Calculate the Strain in the Cartilage

Strain \(\epsilon\) is related to stress and elastic modulus \(E\) by the formula \(\epsilon = \frac{\sigma}{E}\). We know the stress \(\sigma = 31.43 \text{ MPa}\) and the elastic modulus \(E = 250 \text{ MPa}\). Thus, \(\epsilon = \frac{31.43}{250} = 0.1257\).

Step 4: Calculate the Change in Length of the Cartilage

The change in length \(\Delta L\) can be calculated from strain using the formula \(\epsilon = \frac{\Delta L}{L_0}\), where \(L_0\) is the original thickness of the cartilage. The thickness given is \(1.5 \text{ mm} = 0.0015 \text{ m}\). Rearranging the formula gives \(\Delta L = \epsilon \times L_0 = 0.1257 \times 0.0015 \approx 0.0001886 \text{ m} = 0.1886 \text{ mm}\).

Key Concepts

Stress Calculation

In biofluid mechanics, stress calculation is an essential concept, especially when evaluating how much force tissues or materials can withstand. Stress is a measure of force distributed over an area, often calculated using the formula \[ \sigma = \frac{F}{A}\]where \( \sigma \) represents stress, \( F \) is the force applied, and \( A \) is the area over which the force is distributed.
For a tissue with a known cross-sectional area, like cartilage, stress calculation gives insight into the potential for deformation under load. It's measured in Pascals (Pa), a unit comparable to Newton per square meter. Understanding stress is fundamental in designing medical treatments and devices, ensuring they do not exert too much stress on vital structures.
By determining stress, clinicians can predict how cartilage will react to everyday forces, helping to preserve function and structure.

Elastic Modulus

The elastic modulus, also known as Young's modulus, is a key parameter when examining the mechanical properties of materials in biofluid mechanics. It quantifies the ability of a material to resist deformation under stress, acting as a measure of stiffness.
The elastic modulus \( E \) is calculated using the formula:\[ E = \frac{\sigma}{\epsilon}\]Here, \( \sigma \) is the stress applied to the material, while \( \epsilon \) is the strain resulting from the stress. Elastic modulus is measured in Pascals (Pa) or mega Pascals (MPa).

• A high elastic modulus indicates a stiff material that does not deform easily under stress.
• A low elastic modulus suggests a flexible material that will undergo more deformation.

The elastic modulus is vital for selecting materials in biomedical applications, such as implants, where specific mechanical responses are required to mimic or support biological tissues.

Strain in Materials

Strain is a measure of how much a material deforms in response to stress. It is a dimensionless quantity that compares the change in length to the original length of the material.
In biofluid mechanics, strain \( \epsilon \) is described using \[ \epsilon = \frac{\Delta L}{L_0}\]where \( \Delta L \) is the change in length and \( L_0 \) is the original length.

• Positive strain indicates elongation, signifying stretching or tension.
• Negative strain denotes compression, showing the material is being compressed.

The understanding of strain helps in evaluating how biological tissues will behave under physiological loads. Recognizing strain patterns can assist clinicians in assessing injury risk and potential failure of biological or implanted materials.

Circular Area Calculation

Circular area calculation is critical in biofluid mechanics concepts like determining areas of tissue cross-sections to compute stress. For a circle, the area \( A \) can be determined by the equation \[ A = \pi r^2\]where \( r \) represents the radius of the circle.

When working with biological tissues like cartilage, dimensions are crucial, often in centimeters or millimeters, requiring conversion to meters for standard unit consistency.

• For example, if the diameter of cartilage is given as 2 cm, converting to meters gives 0.02 m. The radius, half the diameter, becomes 0.01 m.
• Substituting into the formula yields the area, important for subsequent stress calculations, ensuring precise clinical or mechanical analysis.

Determining the correct circular area ensures accuracy in static and dynamic evaluations of forces acting upon tissues and other circular structures.