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 of a material subjected to three mutually perpendicular stresses, we can use the following formulas:
1) Principal strains:
The formula for principal strains is given as:
ε1 = σ1 / E
ε2 = σ2 / E
ε3 = σ3 / E
where ε1, ε2, and ε3 are the principal strains, σ1, σ2, and σ3 are the three mutually perpendicular stresses, and E is the Young's modulus.
2) Shear modulus:
The formula for shear modulus is given as:
G = (σ1 - σ2) / (2 * γ)
where G is the shear modulus, σ1 and σ2 are the two principal stresses, and γ is the shear strain.
3) Bulk modulus:
The formula for bulk modulus is given as:
K = (σ1 + σ2 + σ3) / (3 * εv)
where K is the bulk modulus, σ1, σ2, and σ3 are the three principal stresses, and εv is the volumetric strain.
Now, let's calculate the values:
Given:
σ1 = 70 MPa
σ2 = 56 MPa
σ3 = 84 MPa
E = 200 GPa (convert to MPa by dividing by 1000)
Poisson's ratio (ν) = 0.28
1) Principal strains:
Substituting the values in the formula:
ε1 = 70 / (200 * 1000)
ε2 = 56 / (200 * 1000)
ε3 = 84 / (200 * 1000)
Solving the above calculations will give you the values of ε1, ε2, and ε3.
2) Shear modulus:
Substituting the values in the formula:
G = (70 - 56) / (2 * γ)
Since the value of γ is not given in the given information, we are unable to calculate the shear modulus.
3) Bulk modulus:
Substituting the values in the formula:
K = (70 + 56 + 84) / (3 * εv)
Since the value of εv is not given in the given information, we are unable to calculate the bulk modulus.
In summary, we can determine the principal strains using the given stresses and Young's modulus. However, without information about the shear strain or volumetric strain, we cannot calculate the shear modulus or bulk modulus.