Calculate the stress in 2 elements

[quetzal]

Homework Statement

Let assume that we have two rectangular prism elements of similar dimensions:
e.g. 1.) x0=100.1, y0=100.2, z0=90.4
2.) x0=100.2, y0=100.4, z0=90.0
How can I calculate the stress tensor of each of the two elements if I assembly them by merging their z-y faces together? I'd want to use the Poisson's ratio(v=0.25)and linear elastic material (E=70GPa).

Homework Equations

V=x0(1+ε)*y0(1-vε)*z0(1-vε)
σ=εE

The Attempt at a Solution

The solution is easy If the material is absolutely compressible(v=0). I derived a general formula for the new y dimension(same for both elements) from the condition of mechanical equilibrium ƩFi(in y-direction)=0:
y=Ʃ(xi*zi)/Ʃ((xi*zi)/yi)
But I can't find any equation of mechanical equilibrium for my case. If you know at least how to calculate the principal stresses neglecting the shear elements of the matrices, that would be a great help too. FEM type of solution is welcomed too.

 

[KataKoniK]

To calculate the stress tensor for the merged elements, you can use the following steps:

1. Find the new dimensions of the merged element:

Since the elements are merged along their z-y faces, the new dimensions of the merged element will be the same as the original dimensions in the x-direction, but will be the sum of the original dimensions in the y and z-directions.

Therefore, for the merged element, the new dimensions will be: x0=100.1, y0=100.6, z0=180.4

2. Calculate the strain in each direction:

Using the formula for volume strain (ε=ΔV/V), we can calculate the strain in each direction as follows:

εx = 0 (since there is no change in the x-direction)
εy = (y0-y0)/y0 = 0 (since the original and new dimensions are the same)
εz = (z0+z0)/z0 = 2 (since the original and new dimensions are summed)

3. Calculate the stress in each direction:

Using the given formula, σ=εE, we can calculate the stress in each direction as follows:

σx = 0 (since there is no strain in the x-direction)
σy = 0 (since there is no strain in the y-direction)
σz = 2*70GPa = 140GPa

4. Calculate the shear stresses:

Since the elements are being merged, there will be a shear stress at the interface between the two elements. To calculate this shear stress, we can use the formula σxy = Gγxy, where G is the shear modulus and γxy is the shear strain.

To calculate the shear strain, we can use the formula γxy = εxy - (ν/2)(εx+εy), where εxy is the shear strain and ν is the Poisson's ratio.

Using the values from the given data, we get:

γxy = 0 - (0.25/2)(0+0) = -0.0625
σxy = (70GPa)(-0.0625) = -4.375GPa

Therefore, the stress tensor for the merged element will be:

σ = [0 0 0; 0 0 0; 0 0 140GPa] (in Pa)
τ = [0 0 -4.375G