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.

To determine the value of σz, we need to consider the yield criteria for the given aluminum alloy material. Two commonly used yield criteria are the maximum normal stress criterion and the maximum shear stress criterion. We will calculate both values and compare them to find the minimum stress necessary for yielding.

1. Maximum Normal Stress Criterion:
According to this criterion, yielding occurs when the maximum normal stress at any point in the material reaches the yield strength (σyield = 400 MPa).

First, let's consider a stress element in the material. We assume that the stress distribution is uniform in this case, so all the stresses acting on the element are normal stresses, σx, σy, and σz.

The equation for the maximum normal stress is given by:
σmax = (σx + σy)/2 + ((σx - σy)/2)^2 + σz^2)^0.5

We know that the normal stresses (σx and σy) and the uniform compressive stress applied (σz) are related in terms of Poisson's ratio (ν). The relationship is given by:
(σx - σy)/2 = ν * σz

Now, substituting this expression for (σx - σy)/2 in the equation for σmax, we get:
σmax = (σz + ν * σz)/2 + ((σz - ν * σz)/2)^2 + σz^2)^0.5

Since we have the value of ν (0.33) and σyield (400 MPa), we can solve this equation to find the value of σz that causes yielding.

2. Maximum Shear Stress Criterion:
According to this criterion, yielding occurs when the maximum shear stress at any point in the material reaches the yield strength (σyield = 400 MPa).

The equation for the maximum shear stress is given by:
τmax = σmax - σmin = ((σx - σy)^2 + (σy - σz)^2 + (σz - σx)^2)^0.5 / 2

Using the relationship between normal stresses and ν, we can express σx, σy, and σz in terms of σz as:
σx = σz + 2 * ν * σz = (2 + 2ν) * σz
σy = σz - ν * σz = (1 - ν) * σz

Now substituting these values in the equation for τmax, we get:
τmax = [((2 + 2ν) * σz - (1 - ν) * σz)^2 + ((1 - ν) * σz - σz)^2 + (σz - (2 + 2ν) * σz)^2]^0.5 / 2

Again, we can solve this equation using the given values of ν (0.33) and σyield (400 MPa) to find the value of σz that causes yielding.

By comparing the minimum values of σz obtained from both criteria, we can determine the necessary compressive stress to cause yielding.

Note: The equations provided are simplified versions of the actual equations used in material mechanics, considering only uniaxial loading conditions and neglecting other factors. The actual behavior may be more complex in real-world applications.