A piece of material
subjected to three mutually
perpendicular stresses of 70,
56 and 84 MPa. If E= 200
Gpa, Poissions ratio= 0.28
Determine 1) Principal strains
2) Shear modulus 3) Bulk
modulus.
To determine the principal strains, shear modulus, and bulk modulus, we need to use the given information and formulas related to stress, strain, and material properties.
1) Principal strains:
The principal strains are the strains that occur along the principal axes (directions) of the stress state. The principal strains, ε1, ε2, and ε3, can be calculated using the formula:
ε1 = (σ1 / E) - ν (σ2 / E) - ν (σ3 / E)
ε2 = -ν (σ1 / E) + (σ2 / E) - ν (σ3 / E)
ε3 = -ν (σ1 / E) - ν (σ2 / E) + (σ3 / E)
where σ1, σ2, and σ3 are the three mutually perpendicular stresses (70 MPa, 56 MPa, and 84 MPa), E is the Young's modulus (200 GPa = 200,000 MPa), and ν is the Poisson's ratio (0.28).
Substituting the given values into the formula, we can calculate the principal strains.
ε1 = (70 / 200,000) - 0.28 * (56 / 200,000) - 0.28 * (84 / 200,000)
ε2 = -0.28 * (70 / 200,000) + (56 / 200,000) - 0.28 * (84 / 200,000)
ε3 = -0.28 * (70 / 200,000) - 0.28 * (56 / 200,000) + (84 / 200,000)
Calculating the values will give you the principal strains.
2) Shear modulus:
The shear modulus (G) is a measure of a material's resistance to shear deformation. It can be calculated using the formula:
G = E / (2 * (1 + ν))
Substituting the given values of E (200 GPa) and ν (0.28) into the formula will give you the shear modulus.
3) Bulk modulus:
The bulk modulus (K) measures a material's resistance to uniform compression. It can be calculated using the formula:
K = E / (3 * (1 - 2 * ν))
Substituting the given values of E (200 GPa) and ν (0.28) into the formula will give you the bulk modulus.
By following these steps, you will be able to determine the principal strains, shear modulus, and bulk modulus.