# How to form the stress tensor component from the equilibrium equation?

• lachgar
In summary, the conversation discusses a suggestion for solving a problem involving a diagonal tensor and plane stress, with equilibre equations and boundary conditions. The boundary conditions are given as T(ex)=-p ex, T(-ex)=p ex, T(ey)=-p ey, and T(-ey)=p ey. The question is posed as finding the form of and , with the suggestion being a linear form of f(y.z)=Ay+Bz, although the linearity is uncertain.
lachgar
Homework Statement
An elastic homogene and isotrope cuboid that is under a constant pressure (-p) in its 4 lateral surfaces. We neglect the weight .
Relevant Equations
equilibre equation, bondary conditions, s
Good evening everybody.
This is my suggestion for answer.
The tensor is diagonal and the compression is a plane stress

equilibre equation div(σ)=0
so:

So, does that means that
= f(y.z) = Ay+Bz and
=f(x.z)= Cx+Dz
A,B,C and D are constants.
Is that what the question meant?

What are the boundary conditions?

As shown in the picture, all lateral surfaces are under a constant pressure.
There is no stress on the superior S(z=h/2) and inferior S(z=-h/2) surfaces. (h is the thickness).
The red arrow represent the stress vector, and the black one represent the unit normal vector of the surface.

Thank you.

So, to repeat my question, what does this imply for the boundary conditions?

The bondary conditions are:

T(ex)=-p ex
T(-ex)=p ex
T(ey) =-p ey
T(-ey) = p ey
T is the stress vector.

So how does this relate to your solution?

The question was posed as:
-Using the equilibre equations, give the form of
and
?

Yes, and you started this thread by writing down differential equations for them. To solve differential equations you need boundary conditions.

it's not demanded to solve it, it's bout giving the general form

So,can we just say that

= f(y.z) = Ay+Bz

How can I be sure that it's a linear form, I mean we only know that it depends on y and z.

Thank you

## 1. What is the stress tensor component?

The stress tensor component is a mathematical representation of the distribution of forces within a material or system. It describes the magnitude and direction of internal forces acting on a specific point within the material or system.

## 2. How is the stress tensor component related to the equilibrium equation?

The stress tensor component is directly related to the equilibrium equation, as it is derived from the equations of motion and satisfies the condition of equilibrium. This means that the sum of all forces and moments acting on a system must be equal to zero in order for the system to be in equilibrium.

## 3. What is the general form of the stress tensor component?

The general form of the stress tensor component is a 3x3 matrix, with each element representing the stress in a particular direction. The diagonal elements represent normal stresses, while the off-diagonal elements represent shear stresses.

## 4. How is the stress tensor component calculated?

The stress tensor component is calculated by taking the derivative of the strain energy with respect to the strain tensor. This involves taking the second derivative of the displacement field and multiplying it by the elastic modulus of the material.

## 5. Why is the stress tensor component important in mechanics and engineering?

The stress tensor component is important in mechanics and engineering because it allows us to analyze the behavior of materials under different loading conditions. It helps us understand how forces are distributed within a material and how it responds to external loads. This information is crucial in designing and predicting the behavior of structures and machines.

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