A uniform compressive stress σz is applied to the surface of the metal alloy, and it may be assumed that there is no friction between the alloy and the die. This aluminum alloy has Young’s modulus E=70GPa and a Poisson’s ratio ν=0.33. Determine the value of σz, in MPa, necessary to cause yielding when σyield=400MPa for each of the yield criteria in Rankine, Tresca and Von Mises.

To determine the value of σz necessary to cause yielding for each of the yield criteria (Rankine, Tresca, and Von Mises), we need to consider the given material properties and the equations associated with each yield criteria.

Let's start by understanding the definitions of each yield criterion:

1. Rankine Criterion (also known as Maximum Normal Stress Criterion): According to Rankine's criterion, yielding occurs when the maximum normal stress (σ_max) reaches or exceeds the yield stress (σ_yield).

2. Tresca Criterion (also known as Maximum Shear Stress Criterion): According to Tresca's criterion, yielding occurs when the maximum difference between any two principal stress values (σ1, σ2, σ3) reaches or exceeds the yield stress (σ_yield).

3. Von Mises Criterion (also known as Distortion Energy Criterion): According to Von Mises' criterion, yielding occurs when the equivalent von Mises stress (σ_vonMises) reaches or exceeds the yield stress (σ_yield).

Now, let's calculate the value of σz for each criterion:

1. Rankine Criterion:
In the case of uniaxial stress (σz), the maximum stress is simply σ_max = σz.
For yielding to occur, σ_max = σz ≥ σ_yield.
Substituting the given values, we have:
σz ≥ 400 MPa.

2. Tresca Criterion:
In the case of uniaxial stress (σz), the principal stresses are σ1 = σz, σ2 = 0, and σ3 = 0.
The maximum difference between any two principal stress values is Δσ = |σ1 - σ3|.
For yielding to occur, Δσ = |σ1 - σ3| ≥ σ_yield.
Substituting the given values, we have:
|σz - 0| ≥ 400 MPa.
Therefore, yielding occurs when σz ≥ 400 MPa.

3. Von Mises Criterion:
In the case of uniaxial stress (σz), the equivalent von Mises stress (σ_vonMises) can be calculated using the following equation:
σ_vonMises = √(σz^2 - σz * σy + σy^2/3),
where σy is the yield stress (σ_yield).
Yielding occurs when σ_vonMises ≥ σ_yield.
Substituting the given values, we have:
√(σz^2 - σz * 400 MPa + (400 MPa)^2/3) ≥ 400 MPa.
By solving this equation, we can find the value of σz necessary to cause yielding.

Now that you know the equations and the process to calculate the values of σz for each criterion, you can substitute the given values and calculate the final results.

To determine the value of σz necessary to cause yielding for each of the yield criteria, we can use the equations for each criterion.

1. Rankine Yield Criterion:
The Rankine yield criterion states that yielding occurs when the maximum principal stress reaches the yield strength. For a material with a tensile yield strength of σyield, the yield criterion can be written as:

σ1 = σyield

In this case, since the applied stress is compressive, the maximum principal stress is given by:

σ1 = σz

Setting σ1 equal to the yield strength, we have:

σz = σyield = 400 MPa

So, the value of σz necessary to cause yielding according to the Rankine yield criterion is 400 MPa.

2. Tresca Yield Criterion:
The Tresca yield criterion states that yielding occurs when the maximum difference between any two principal stresses reaches the yield strength. The difference between the maximum and minimum principal stresses is given by:

σ1 - σ3 = 2σyield

In this case, since the applied stress is compressive, the minimum principal stress is given by:

σ3 = -σz

Substituting this into the Tresca yield criterion equation, we have:

σ1 - (-σz) = 2σyield

σ1 + σz = 2σyield

σz = 2σyield - σ1

Substituting σyield = 400 MPa and σ1 = σz, we have:

σz = 2(400 MPa) - σz

2σz = 800 MPa

σz = 400 MPa

So, the value of σz necessary to cause yielding according to the Tresca yield criterion is 400 MPa.

3. Von Mises Yield Criterion:
The Von Mises yield criterion states that yielding occurs when the equivalent stress reaches the yield strength. The equivalent stress is given by:

σeq = √(σ1^2 - σ1σ2 + σ2^2)

In this case, since the applied stress is compressive, the maximum principal stress is given by:

σ1 = σz

The minimum principal stress is related to the maximum principal stress by:

σ3 = -νσ1

Substituting these into the Von Mises yield criterion equation, we have:

σeq = √(σz^2 - σz(-νσz) + (-νσz)^2)

σeq = √(σz^2 + ν^2σz^2 + ν^2σz^2)

σeq = √(σz^2(1 + ν^2 + ν^2))

Now, when yielding occurs, the Von Mises equivalent stress equals the yield strength, so:

σeq = √(σz^2(1 + ν^2 + ν^2)) = σyield

Simplifying the equation:

σz^2(1 + ν^2 + ν^2) = σyield^2

σz^2 = σyield^2 / (1 + ν^2 + ν^2)

Substituting σyield = 400 MPa and ν = 0.33, we have:

σz^2 = (400 MPa)^2 / (1 + 0.33^2 + 0.33^2)

σz^2 ≈ 416.53 MPa^2

Taking the square root, we find:

σz ≈ 20.4 MPa

So, the value of σz necessary to cause yielding according to the Von Mises yield criterion is approximately 20.4 MPa.