# Dependence of the stress vector on surface orientation

AndersF
Homework Statement:
According to Cauchy's stress theorem, the stress vector at some point in a continuous medium depends on the direction of the surface to which it is referred. Therefore, at the same point, there are infinitely many stress vectors, each of which corresponds to a different surface. How is it possible that the stress vector at the same point is not unique?
Relevant Equations:
##\mathbf{T}^{(\mathbf{n})}=\mathbf{n} \cdot \boldsymbol{\sigma}##
According to Cauchy's stress theorem, the stress vector ##\mathbf{T}^{(\mathbf{n})}## at any point P in a continuum medium associated with a plane with normal unit vector n can be expressed as a function of the stress vectors on the planes perpendicular to the coordinate axes, i.e., in terms of the components ##\sigma_{ij}## of the stress tensor σ.

##\mathbf{T}^{(\mathbf{n})}=\mathbf{n} \cdot \boldsymbol{\sigma}##

This means that, at the same point, infinite stress vectors are defined, each corresponding to a different orientation of the chosen surface. That is something that confuses me, because in the force fields I knew so far (gravitational, electromagnetic...) the force associated with each point was unique.

What is the reason for this behaviour in the case of fluids and continuous mediums?

Homework Helper
2022 Award
This is fundamentally because the stress tensor is a tensor. When you choose a different direction ##\hat n## you get a different answer. I'm not really sure I even understand your expectation of uniqueness...the result is a vector

Homework Helper
2022 Award
Homework Statement:: According to Cauchy's stress theorem, the stress vector at some point in a continuous medium depends on the direction of the surface to which it is referred. Therefore, at the same point, there are infinitely many stress vectors, each of which corresponds to a different surface. How is it possible that the stress vector at the same point is not unique?
Relevant Equations:: ##\mathbf{T}^{(\mathbf{n})}=\mathbf{n} \cdot \boldsymbol{\sigma}##

According to Cauchy's stress theorem, the stress vector ##\mathbf{T}^{(\mathbf{n})}## at any point P in a continuum medium associated with a plane with normal unit vector n can be expressed as a function of the stress vectors on the planes perpendicular to the coordinate axes, i.e., in terms of the components ##\sigma_{ij}## of the stress tensor σ.

##\mathbf{T}^{(\mathbf{n})}=\mathbf{n} \cdot \boldsymbol{\sigma}##

This means that, at the same point, infinite stress vectors are defined, each corresponding to a different orientation of the chosen surface. That is something that confuses me, because in the force fields I knew so far (gravitational, electromagnetic...) the force associated with each point was unique.

What is the reason for this behaviour in the case of fluids and continuous mediums?

The stress vector is a force per unit area, not a force. The resulting force depends not just on the size of the area, but also its normal.

The force due to stress experienced by a finite volume $V$ of material with boundary $\partial V$ is $\int_{\partial V} T_{ij} n_j\,dS.$ This is unique once you define the volume. But why should we choose one volume rather than another?

The equivalent of Newton's Seond Law for the medium is essentially "density times acceleration equals force per unit volume". By the divergence theorem we have $$\int_{\partial V} T_{ij} n_j\,dS = \int_V \frac{\partial T_{ij}}{\partial x_j}\,dV$$ for an arbitrary volume $V$, so at each point in the medium the force per unit volume due to stress is the divergence of the stress tensor, $\frac{\partial T_{ij}}{\partial x_j}$. This is uniquely defined at each point.

AndersF
Homework Helper
Gold Member
2022 Award
This means that, at the same point, infinite stress vectors are defined, each corresponding to a different orientation of the chosen surface. That is something that confuses me, because in the force fields I knew so far (gravitational, electromagnetic...) the force associated with each point was unique.
Maybe this will help…

First think about a simpler situation: some physical quantity is represented by a vector (a 1st order tensor) ##\vec V##.

We can find the projection of ##\vec V## onto some other vector ##\vec W##. The projection is a effectively a scalar (we say it is the component of ##\vec V## in the direction of ##\vec W##). Note that there are infinitely many possible projections because there are infinitely many choices for ##\vec W##. I’ll assume you are OK with that.

In general, the stress at a point can't be represented by a vector. Don't compare it to a force (gravitational, electromagnetic or other). At a point, the physical stress is represented by a 2nd order tensor, commonly expressed as a 3x3 matrix.

We can project this tensor onto ##\vec W## - and this produces a vector rather than a scaler. There are infinitely many possible projections because there are infinitely many choices for ##\vec W##.

AndersF
Mentor
The state of stress in a deformed material is not typically isotropic, unlike to force fields that you referred to. As a result of this anisotropy, the stress vector ay different surface orientations is different.