What is Cauchy stress: Definition and 11 Discussions
In continuum mechanics, the Cauchy stress tensor
σ
{\displaystyle {\boldsymbol {\sigma }}}
, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components
σ
i
j
{\displaystyle \sigma _{ij}}
that completely define the state of stress at a point inside a material in the deformed state, placement, or configuration. The tensor relates a unit-length direction vector n to the traction vector T(n) across an imaginary surface perpendicular to n:
{\displaystyle {\boldsymbol {\sigma }}=\left[{\begin{matrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\\\end{matrix}}\right]\equiv \left[{\begin{matrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{yx}&\sigma _{yy}&\sigma _{yz}\\\sigma _{zx}&\sigma _{zy}&\sigma _{zz}\\\end{matrix}}\right]\equiv \left[{\begin{matrix}\sigma _{x}&\tau _{xy}&\tau _{xz}\\\tau _{yx}&\sigma _{y}&\tau _{yz}\\\tau _{zx}&\tau _{zy}&\sigma _{z}\\\end{matrix}}\right]}
The SI units of both stress tensor and stress vector are N/m2, corresponding to the stress scalar. The unit vector is dimensionless.
The Cauchy stress tensor obeys the tensor transformation law under a change in the system of coordinates. A graphical representation of this transformation law is the Mohr's circle for stress.
The Cauchy stress tensor is used for stress analysis of material bodies experiencing small deformations: It is a central concept in the linear theory of elasticity. For large deformations, also called finite deformations, other measures of stress are required, such as the Piola–Kirchhoff stress tensor, the Biot stress tensor, and the Kirchhoff stress tensor.
According to the principle of conservation of linear momentum, if the continuum body is in static equilibrium it can be demonstrated that the components of the Cauchy stress tensor in every material point in the body satisfy the equilibrium equations (Cauchy's equations of motion for zero acceleration). At the same time, according to the principle of conservation of angular momentum, equilibrium requires that the summation of moments with respect to an arbitrary point is zero, which leads to the conclusion that the stress tensor is symmetric, thus having only six independent stress components, instead of the original nine. However, in the presence of couple-stresses, i.e. moments per unit volume, the stress tensor is non-symmetric. This also is the case when the Knudsen number is close to one,
K
n
→
1
{\displaystyle K_{n}\rightarrow 1}
, or the continuum is a non-Newtonian fluid, which can lead to rotationally non-invariant fluids, such as polymers.
There are certain invariants associated with the stress tensor, whose values do not depend upon the coordinate system chosen, or the area element upon which the stress tensor operates. These are the three eigenvalues of the stress tensor, which are called the principal stresses.
Why do the Cauchy Stress Tensor & the Energy Momentum Tensor have the same SI units? Shouldn't adding time as a dimension changes the Energy Momentum Tensor's units?
Did Einstein start with the Cauchy Tensor when he started working on the right hand side of the field equations of GR?
If so, What...
I am in a course in applied strength of materials and we often use the 3D stress tensor for stress analysis of materials i.e. Mohr's circles, bending, torsion, etc. Is the stress-energy tensor in relativity basically a 4-d extension to the Cauchy stress tensor commonly used in mechanical...
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...
During lecture today, we were given the constitutive equation for the Newtonian fluids, i.e. ##T= - \pi I + 2 \mu D## where ##D=\frac{L + L^T}{2}## is the symmetric part of the velocity gradient ##L##. Dimensionally speaking, this makes sense to me: indeed the units are the one of a pressure...
Hi everyone,
I'm trying to understand the rationale behind the boundary condition for the problem "Finite bending of an incompressible elastic block". (See here from page 180).Here we have as Cauchy Stress tensor (see eq. (5.82)):
##T = - \pi I + \mu (\frac{l_0^2}{4 \bar{\theta}^2 r^2} e_r...
I am studying the finite bending of a rubber-like block, assuming Neo-Hookean response. In the following, ##l_0##,##h##, ##\bar{\theta}## are parameters, while the variables are ##r## and ##\theta##.
The Cauchy stress tensor is
##T= - \pi I + \mu(\frac{l_0^2}{4 \bar{\theta}^2 r^2} e_r \otimes...
I have been trying to fully grasp the concept of the Cauchy stress tensor and so I thought I'd make a post where I clear up my confusion. There may be subsequent replies as I pose more questions.
I am specifically confused at how the stress tensor relates to the control volume in the image...
Hi, I've been trying to figure out how to get Cauchy Stress Tensor components (~9) from a surface force for a while now. My background in this subject is not too deep, but I'm trying to build a renderer simulation in my free time.
I can get surface traction from a Stress Tensor:
t =...
Homework Statement
https://en.wikipedia.org/wiki/Cauchy_stress_tensor[/B]
I don't understand the difference between τxy . τyx , τxz , τzx , τyz , τzy ..What did they mean ?
Homework EquationsThe Attempt at a Solution
taking τxy and τyx as example , what are the difference between them ? They...
Hello,
I am not sure what the first indice in the cauchy stress tensor indicates
For example,
σ_xy means that the stress in the y direction, but does x mean the cross sectional area is normal to the x direction?
First of all, thanks for all the helpful comments to my previous posts.
I'm trying to get a grasp of stress tensors and have been doing some studying.
In the literature I've been looking at, it says something about the eigenvalues of
stress tensors and the principle stresses. This is...