# Elasticity: Determine zero normal stress plane

• Zipi Damn
In summary, to determine the zero normal stress plane for a given stress tensor at a point, one can use the eigenvalues of the tensor to calculate the normal stress and shear stress. If the normal stress is zero, the maximum shear stress can be calculated using the equation (σ1-σ3)/2. This shear stress value can be found on Mohr's circle, but it is not the maximum value. To find the maximum shear stress, one must use the equation of the major circumference of center (0,2) and radius R=3 from Mohr's circle, with the normal stress set to zero. This will give the value of the maximum shear stress, which can then be used to determine the normal direction of the
Zipi Damn
Given the stress tensor in a point, determine the zero normal stress plane.
...2 3 0
T= 3 2 0
...0 0 5
----------------------
Eigenvalues: σ1=σ2=5, σ3=-1
It must be simple, but I don't know how to determine the normal vector of that plane analytically.

I know σ=0.
t=Tn=σ+τ=τ
If normal stress σ equals zero, ¿the shear stress will be the maximum value of τ ((σ1-σ3)/2)?

Taking a look to Mohr's circle I think τ in that plane must be the intersection between the circunference and the τ axis, but that's not τmax.

I'm confused.

Okay, I think I got it.

Using the equation of the mayor circunference of center (0,2) and radius R=3 from Mohr's circle, we make σ=0. Then we isolate the shear stress.

(σ-2)2+(τ-0)2=32

τ=√5

Now that I have the modulus, I want to know the normal direction of the plane in which shear stress equals √5.

## 1. What is elasticity?

Elasticity is the property of a material to return to its original shape and size after being deformed by an external force.

## 2. How is elasticity measured?

Elasticity is typically measured using the Young's modulus, which is the ratio of stress to strain in a material. It is expressed in units of pressure, such as pascals (Pa).

## 3. What is the zero normal stress plane?

The zero normal stress plane is a plane within a material where there is no normal stress acting on the surface. This means that the material does not experience any change in length or shape in that particular plane.

## 4. How is the zero normal stress plane determined?

The zero normal stress plane can be determined by finding the point in a material where the normal stress is equal to zero, using the equation σz = 0. This can also be visualized by plotting the normal stress on a stress-strain curve and finding where it intersects with the x-axis.

## 5. Why is the zero normal stress plane important?

The zero normal stress plane is important because it represents a direction within a material where there is no change in length or shape. This information is useful for engineers and scientists in designing and analyzing structures and materials under different loads and forces.

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