Formal HW 4

Sample Solution

I'd be happy to help you with your civil engineering exercise, but I cannot complete it entirely as it requires data and specific software for plotting and calculations. However, I can guide you through the process and provide sample calculations using hypothetical data:

Data Requirements:

• Stress (MPa): Please provide the stress values for each data point in both tests.
• Strain (mm/mm): Please provide the strain values for each data point in both tests.

Plotting the Stress-Strain Diagram:

Full Answer Section

Plotting the Stress-Strain Diagram:

• Use a plotting software like Excel or MATLAB to create a graph with stress on the y-axis and strain on the x-axis.
• Plot both datasets on the same graph for comparison.
• Clearly label the axes with units (MPa and mm/mm).

Material Properties:

1. Modulus of Elasticity (E):

   • The slope of the linear portion of the stress-strain curve represents the modulus of elasticity.
   • Select two points within the linear portion and calculate the slope:

     E = (σ2 - σ1) / (ε2 - ε1)
     

     where σ and ε are stress and strain values, respectively, and 1 and 2 represent the chosen points.
   • Repeat for both datasets and compare the results.

2. Proportional Limit:

   • The proportional limit is the highest stress at which the stress-strain relationship remains linear.
   • Identify the point where the curve starts deviating from linearity.
   • Read the stress value at that point, representing the proportional limit for each dataset.

3. Yield Stress:

   • The yield stress is the stress at which the material exhibits a significant permanent deformation (yield point).
   • For some materials, this is a clear point on the curve (offset yield). For others, it's defined by a specific strain offset (e.g., 0.2% offset).
   • Refer to the relevant standards or material specifications for the appropriate definition of yield stress.
   • Determine the yield stress using the specified definition for each dataset.

4. Ultimate Stress:

   • The ultimate stress is the maximum stress sustained by the material before failure.
   • Identify the highest point on the stress-strain curve for each dataset.
   • Read the stress value at that point, representing the ultimate stress.

5. Modulus of Resilience:

   • The modulus of resilience represents the elastic energy stored per unit volume under the proportional limit.
   • Calculate the area under the stress-strain curve up to the proportional limit for each dataset.
   • Divide the area by the gauge volume of the test specimen:

     Modulus of Resilience = Area / Volume
     

6. Modulus of Toughness:

   • The modulus of toughness represents the total energy absorbed per unit volume up to the point of failure.
   • Calculate the area under the entire stress-strain curve for each dataset:

     Modulus of Toughness = Total Area / Volume
     

Sample Calculations (using hypothetical data):

Data:

Calculations:

• Using points (100 MPa, 0.004 mm/mm) and (200 MPa, 0.008 mm/mm):

  E_hypothetical = (200 MPa - 100 MPa) / (0.008 mm/mm - 0.004 mm/mm) = 25,000 MPa
  

• Repeat similar calculations for other properties using the hypothetical data and appropriate definitions/formulas.

Remember to replace the hypothetical data with your actual data points and conduct the necessary calculations based on the specific standards or material specifications applicable to your exercise.