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.

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To determine the principal strains, shear modulus, and bulk modulus, we need to use the given stresses and the material properties.

(a) Principal Strains:
Principal strains are the strains that occur in the principal directions of stress. To find the principal strains, we can use Hooke's Law:

ε = σ/E

where ε is the strain, σ is the stress, and E is the Young's modulus.

Given the stresses σ₁ = 70 MPa, σ₂ = 56 MPa, and σ₃ = 84 MPa, we can calculate the principal strains ε₁, ε₂, and ε₃.

ε₁ = σ₁/E = 70 MPa / (200 GPa) = 0.35 x 10⁻³
ε₂ = σ₂/E = 56 MPa / (200 GPa) = 0.28 x 10⁻³
ε₃ = σ₃/E = 84 MPa / (200 GPa) = 0.42 x 10⁻³

So, the principal strains are ε₁ = 0.35 x 10⁻³, ε₂ = 0.28 x 10⁻³, and ε₃ = 0.42 x 10⁻³.

(b) Shear Modulus:
The shear modulus (G) relates to the shear stress and shear strain. It can be calculated using the following formula:

G = E / (2(1 + ν))

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

Given that E = 200 GPa and ν = 0.28, we can calculate the shear modulus.

G = (200 GPa) / (2(1 + 0.28)) = 71.43 GPa

Therefore, the shear modulus is 71.43 GPa.

(c) Bulk Modulus:
The bulk modulus (K) relates the volumetric stress and volumetric strain. It can be calculated using the following formula:

K = E / (3(1 - 2ν))

Given that E = 200 GPa and ν = 0.28, we can calculate the bulk modulus.

K = (200 GPa) / [3(1 - 2(0.28))] = 115.38 GPa

Therefore, the bulk modulus is 115.38 GPa.

In summary, the answers are:
(a) Principal strains: ε₁ = 0.35 x 10⁻³, ε₂ = 0.28 x 10⁻³, ε₃ = 0.42 x 10⁻³.
(b) Shear modulus: 71.43 GPa.
(c) Bulk modulus: 115.38 GPa.