Dependence of the stress vector on surface orientation

In summary, the stress vector at a point in a continuous medium is not unique because it depends on the orientation of the surface to which it is referred, due to the anisotropy of the material's stress state.
  • #1
AndersF
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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?
 
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  • #2
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
 
  • #3
AndersF said:
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 [itex]V[/itex] of material with boundary [itex]\partial V[/itex] is [itex]\int_{\partial V} T_{ij} n_j\,dS.[/itex] 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 [tex]\int_{\partial V} T_{ij} n_j\,dS = \int_V \frac{\partial T_{ij}}{\partial x_j}\,dV[/tex] for an arbitrary volume [itex]V[/itex], so at each point in the medium the force per unit volume due to stress is the divergence of the stress tensor, [itex]\frac{\partial T_{ij}}{\partial x_j}[/itex]. This is uniquely defined at each point.
 
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  • #4
AndersF said:
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##.
 
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  • #5
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.
 

1. What is the dependence of the stress vector on surface orientation?

The dependence of the stress vector on surface orientation refers to how the direction and magnitude of the stress acting on a surface is influenced by the orientation of that surface. This is an important concept in materials science and engineering, as it can affect the mechanical properties and behavior of materials.

2. How does surface orientation affect stress distribution?

The orientation of a surface can affect the distribution of stress by changing the direction and magnitude of the stress acting on that surface. For example, a surface oriented perpendicular to a tensile stress will experience a higher stress magnitude compared to a surface oriented parallel to the same stress.

3. What factors influence the dependence of the stress vector on surface orientation?

There are several factors that can influence the dependence of the stress vector on surface orientation, including the type of loading (e.g. tensile, compressive, shear), the material properties, and the geometry of the object. Additionally, the presence of any surface defects or imperfections can also affect the stress distribution.

4. How is the dependence of the stress vector on surface orientation measured?

The dependence of the stress vector on surface orientation can be measured through various experimental techniques, such as strain gauges, optical methods, and X-ray diffraction. These techniques allow for the visualization and quantification of stress distribution on a surface at different orientations.

5. What are the practical applications of understanding the dependence of the stress vector on surface orientation?

Understanding the dependence of the stress vector on surface orientation is crucial in designing and engineering materials for specific applications. It can help predict the mechanical behavior and failure mechanisms of materials under different loading conditions, and aid in optimizing material properties for desired performance. This knowledge is also important in industries such as aerospace, automotive, and construction, where materials are subjected to various stresses and orientations.

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