A piece of material subjected to three mutually perpendicular stresses of 70,

56 and 84 Mpa. If E = 200GPa, Poisson's ratio = 0.28, determine (10)
(a) Principal strains (b) Shear modules (c) Bulk modules.

answer

A piece of material subjected to three mutually perpendicular stresses of 70,

56 and 84 Mpa. If E = 200GPa, Poisson's ratio = 0.28, determine (10)
(a) Principal strains (b) Shear modules (c) Bulk modules.

To determine the principal strains, shear modulus, and bulk modulus, we first need to understand the equations and formulas involved.

(a) Principal Strains:
Principal strains are the strains that occur in directions aligned with the principal stresses. They can be calculated using the following equations:

ε1 = (σ1/E) - ν(σ2/E) - ν(σ3/E)
ε2 = (σ2/E) - ν(σ1/E) - ν(σ3/E)
ε3 = (σ3/E) - ν(σ1/E) - ν(σ2/E)

where ε1, ε2, and ε3 are the principal strains, σ1, σ2, and σ3 are the principal stresses, E is the Young's modulus, and ν is the Poisson's ratio.

(b) Shear Modulus:
The shear modulus, also known as the modulus of rigidity, is a measure of a material's resistance to shear deformation. It is calculated using the following equation:

G = E / (2 * (1 + ν))

where G is the shear modulus, E is the Young's modulus, and ν is the Poisson's ratio.

(c) Bulk Modulus:
The bulk modulus is a measure of a material's resistance to uniform compression. It is calculated using the following equation:

K = E / (3 * (1 - 2 * ν))

where K is the bulk modulus, E is the Young's modulus, and ν is the Poisson's ratio.

Now, let's plug in the given values to calculate the required quantities:

Given:
σ1 = 70 MPa
σ2 = 56 MPa
σ3 = 84 MPa
E = 200 GPa (convert to MPa by dividing by 1000)
ν = 0.28

(a) Principal strains:
Substituting the values in the equations for ε1, ε2, and ε3:

ε1 = (70/200) - (0.28 * 56/200) - (0.28 * 84/200)
ε2 = (56/200) - (0.28 * 70/200) - (0.28 * 84/200)
ε3 = (84/200) - (0.28 * 70/200) - (0.28 * 56/200)

Simplifying the above equations will give you the values of ε1, ε2, and ε3.

(b) Shear modulus:
Substituting the values in the equation for G:

G = (200 / (2 * (1 + 0.28)))

Simplifying this equation will give you the value of G.

(c) Bulk modulus:
Substituting the values in the equation for K:

K = (200 / (3 * (1 - 2 * 0.28)))

Simplifying this equation will give you the value of K.

By following these steps and performing the necessary calculations, you can determine the principal strains, shear modulus, and bulk modulus for the given material.