Density Of Material Given Circumferential Stress In Solid Disc Calculator

Formula Used:

\[ \rho = \frac{((\frac{C1}{2}) - \sigma_c) \times 8}{(\omega^2) \times (r_{disc}^2) \times ((3 \times \nu) + 1)} \]

Constant at boundary condition (C1):

N/m²

Circumferential Stress (σc):

Pascal

Angular Velocity (ω):

rad/s

Disc Radius (r_disc):

meter

Poisson's Ratio (ν):

(unitless)

Density Of Disc (ρ):

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1. What is the Density Calculation Formula?

The formula calculates the density of a solid disc material based on circumferential stress, boundary conditions, angular velocity, disc radius, and Poisson's ratio. It provides a method to determine material density from mechanical properties and stress distribution.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \rho = \frac{((\frac{C1}{2}) - \sigma_c) \times 8}{(\omega^2) \times (r_{disc}^2) \times ((3 \times \nu) + 1)} \]

Where:

\( \rho \) — Density of disc material (kg/m³)
\( C1 \) — Constant at boundary condition (N/m²)
\( \sigma_c \) — Circumferential stress (Pascal)
\( \omega \) — Angular velocity (rad/s)
\( r_{disc} \) — Disc radius (meter)
\( \nu \) — Poisson's ratio (unitless)

Explanation: The formula relates material density to stress distribution and rotational dynamics in a solid disc.

3. Importance of Density Calculation

Details: Accurate density calculation is crucial for material selection, structural analysis, and predicting the behavior of rotating discs under stress conditions.

4. Using the Calculator

Tips: Enter all values in appropriate units. Ensure angular velocity and disc radius are positive values. Poisson's ratio should be between 0 and 0.5 for most materials.

5. Frequently Asked Questions (FAQ)

Q1: What is the physical significance of the constant C1?
A: C1 represents the integration constant obtained from boundary conditions when solving the stress equilibrium equations for a rotating disc.

Q2: Why is Poisson's ratio important in this calculation?
A: Poisson's ratio accounts for the material's tendency to deform in directions perpendicular to the applied stress, affecting stress distribution.

Q3: What are typical density values for common materials?
A: Steel: ~7850 kg/m³, Aluminum: ~2700 kg/m³, Titanium: ~4500 kg/m³, but values vary with alloy composition.

Q4: When is this formula most accurate?
A: The formula is most accurate for homogeneous, isotropic materials undergoing elastic deformation in rotating disc applications.

Q5: Can this formula be used for non-circular discs?
A: No, this formula is specifically derived for solid circular discs with axisymmetric stress distribution.