Hi everybody! can anyone help me with this problem?
HW9_2: BLOCK BETWEEN RIGID WALLS
A cube L0 * L0 * L0 of isotropic linear elastic material (properties
E and ν, with ν>0) is placed between two rigid frictionless walls at fixed distance
L0 and compressed by a lateral pressure p along the x-axis, as indicated in the figure. No loads or constraints are acting along the z-axis. Under the effect of the applied lateral pressure, the cube elongates in the direction normal to the plane of the figure (z-direction) and the strain in this direction is measured to be ϵzz=ϵ0.
HW9_2_1
Obtain symbolic expressions for the applied pressure p and for the constraining normal stress along the y
direction σyy exerted by the walls on the block. Write your expressions in terms of the measured strain along the
z-axis, ϵ0 (enter as epsilon_0) and of the elastic properties E, ν (enter ν as nu).
p = ?
σyy = ?
To solve this problem, we need to use the theory of linear elasticity and apply the principles of stress and strain. Let's break down the problem step by step:
1. Define the given information:
- Cube dimensions: L0 * L0 * L0
- Material properties: E (elastic modulus) and ν (Poisson's ratio)
- Lateral pressure: p
- Measured strain along the z-axis: ϵzz = ϵ0
2. Determine the relationship between stress (σ) and strain (ϵ):
In linear elasticity, stress and strain are related through Hooke's law:
σ = E * ϵ
3. Calculate the strain in the y-direction (ϵyy) using Poisson's ratio:
Poisson's ratio (ν) relates the lateral strain to the longitudinal strain:
ν = -ϵyy / ϵzz
Rearranging the equation gives:
ϵyy = -ν * ϵzz
4. Substitute the given strain (ϵzz = ϵ0) and Poisson's ratio (ν) into the equation:
ϵyy = -ν * ϵ0
5. Determine the normal stress along the y-direction (σyy):
Stress (σ) is force per unit area. Since we have a cube with fixed dimensions, the force acting on the cube due to the pressure p is given by:
Force = Area * Pressure
The area of the y-z plane is L0 * L0, and the force is equal to the stress (σyy) times the area.
Therefore, σyy * (L0 * L0) = p * (L0 * L0 * L0)
Rearranging the equation gives:
σyy = p * L0
6. Substitute the given cube length (L0) into the equation to get the final expression for σyy:
σyy = p * L0
Now, let's summarize the obtained symbolic expressions:
p = ?
σyy = ?
Using the equations derived, we can write the final expressions for p and σyy:
p = σyy / L0
Substitute the expression for σyy: p = (p * L0) / L0
σyy = p * L0
Now, you can use these equations to calculate the values of p and σyy by substituting the appropriate values for ϵ0, E, ν, and L0.