The Young's modulus for steel is 2x1011 and its rigidity modulus is 8x1010 Nm-2. Find the Poisson's ratio and its bulk modulus.
For homogeneous isotropic materials, simple relations exist between elastic constants (Young's modulus E, shear modulus G, bulk modulus K, and Poisson's ratio í) that allow calculating them all as long as two of them are known:
See http://en.wikipedia.org/wiki/Young's_modulus for the formulas you need
The Young's modulus for a material is 11.0 x 1010 N/m2. The material is stretched to a strain of 3.0 x 10-3. How much elastic energy will be expended?
answer is 495000.0
To find the Poisson's ratio and bulk modulus, we need to use the formulas derived from the definitions and relationships between Young's modulus, rigidity modulus, Poisson's ratio, and bulk modulus.
1. Poisson's ratio (ν):
Poisson's ratio represents the ratio of lateral strain to the longitudinal strain in a material. It is defined as the negative ratio of lateral strain (ε_y) to axial (longitudinal) strain (ε_x):
ν = -ε_y / ε_x
To find Poisson's ratio, we need to determine the longitudinal strain and lateral strain separately.
For a material under tensile stress in the x-direction (longitudinal direction):
ε_x = stress_x / Young's modulus = σ_x / E
For a material under shear stress in the yz-plane (lateral direction):
ε_y = stress_y / rigidity modulus = σ_y / G
Given:
Young's modulus (E) = 2x10^11 Nm^(-2)
Rigidity modulus (G) = 8x10^10 Nm^(-2)
Using the given values, we can now calculate the Poisson's ratio:
ε_x = σ_x / E = σ_x / (2x10^11)
ε_y = σ_y / G = σ_y / (8x10^10)
Since we don't have the values of the applied stresses (σ_x and σ_y), we cannot directly calculate the Poisson's ratio. Additional information is required to find the Poisson's ratio.
2. Bulk modulus (K):
Bulk modulus represents the ratio of the change in volume to the applied pressure on a material. It is defined as:
K = -V ΔP / ΔV
Where:
K is the bulk modulus
ΔP is the change in pressure
ΔV is the change in volume
V is the initial volume
Unfortunately, we don't have sufficient information to directly calculate the bulk modulus using the given values.