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 can use the equations related to stress and strain in a material. Here's how you can calculate each of these values:
1) Principal Strains:
The principal strains (ε1, ε2, ε3) are the strains corresponding to the principal stresses (σ1, σ2, σ3). The equations to calculate the principal strains are:
ε1 = (σ1/E) - (v/E) * (σ2 + σ3)
ε2 = (σ2/E) - (v/E) * (σ1 + σ3)
ε3 = (σ3/E) - (v/E) * (σ1 + σ2)
where E is the Young's modulus (200 GPa) and v is the Poisson's ratio (0.28). Substituting the given values, we get:
ε1 = (70/200) - (0.28/200) * (56 + 84)
ε2 = (56/200) - (0.28/200) * (70 + 84)
ε3 = (84/200) - (0.28/200) * (70 + 56)
Calculating these expressions will give you the principal strains.
2) Shear Modulus:
The shear modulus (G) is a measure of a material's resistance to shearing deformation. It can be calculated using the formula:
G = (E / (2 * (1 + v)))
where E is the Young's modulus and v is the Poisson's ratio. By substituting the given values, we get:
G = (200 * 10^9) / (2 * (1 + 0.28))
Evaluating this expression will give you the shear modulus.
3) Bulk Modulus:
The bulk modulus (K) represents a material's resistance to uniform compression. It can be determined using the following equation:
K = (E / (3 * (1 - 2v)))
By substituting the given values, we get:
K = (200 * 10^9) / (3 * (1 - 2 * 0.28))
Calculating this expression will give you the bulk modulus.
Using these formulas and values, you can determine the principal strains, shear modulus, and bulk modulus of the given material.