Strain energy, which represents the energy stored within a deformable body due to its deformation, can be measured using different approaches, each tailored to specific scenarios. The measurement depends primarily on whether the stress is proportional to strain (elastic behavior). Below are ways to measure strain energy.
Measurement Methods
1. When Stress is Proportional to Strain (Elastic Behavior)
When the material behaves elastically, meaning stress (σ) is directly proportional to strain (ϵ), we can use the following formula to measure strain energy (U):
- Formula: U = σ2 / 2E × V
- Where:
- U = Strain Energy
- σ = Stress (defined in references as 'stress')
- E = Modulus of Elasticity (Young's Modulus)
- V = Volume of the material
- Where:
This formula indicates that the strain energy is directly proportional to the square of the stress and the volume of the material, and inversely proportional to the material's stiffness (Young's Modulus).
2. Using Force and Deformation
Another way to measure strain energy is using the applied force and the deformation of the object, especially in a simple loading case:
- Formula: U = Fδ / 2
- Where:
- U = Strain Energy
- F = Force applied
- δ = Compression (or deformation) (defined in the references as 'compression')
- Where:
This method essentially calculates the work done by the force during deformation, which is equal to the strain energy stored.
Example Calculation:
Let’s use the example of the rod from the reference: "A rod of area 90 mm2 has a length of 3 m. Determine the strain energy if the stress of 300 MPa is applied when stretched." We can use the first formula:
- Given Values:
- σ (Stress) = 300 MPa = 300 x 106 N/m2
- Area (A) = 90 mm2 = 90 x 10-6 m2
- Length (L) = 3m
- We need the material's Young's Modulus (E) , Assuming it is a steel with E = 200 GPa = 200 x 109 N/m2
- Volume (V) = Area x Length = 90 x 10-6 m2 * 3 m = 270 x 10-6 m3
- Calculation:
- U = (300 x 106 N/m2)2 / (2 200 x 109 N/m2) (270 x 10-6 m3)
- U = 9 x 1016 / 4 x 1011 * 270 x 10-6
- U ≈ 60.75 Joules
Practical Insights
- Strain energy is vital in structural design and analysis. It helps engineers understand how much energy a structure can absorb before it fails.
- Materials with high strain energy capacity are used in applications where energy absorption is critical (e.g., springs, shock absorbers).
- The methods mentioned are for linear elastic conditions. In cases where materials deform plastically, more complex methods are needed.
In conclusion, you can measure strain energy by calculating the work done by an external force to deform an object or by using the material's stress, elastic modulus and volume, and each method provides valuable insights into a material’s capacity to store energy under stress.
