# What is Principal stress: Definition and 19 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:

T

(

n

)

=

n

σ

or

T

j

(
n
)

=

σ

i
j

n

i

.

{\displaystyle \mathbf {T} ^{(\mathbf {n} )}=\mathbf {n} \cdot {\boldsymbol {\sigma }}\quad {\text{or}}\quad T_{j}^{(n)}=\sigma _{ij}n_{i}.}
where,

σ

=

[

σ

11

σ

12

σ

13

σ

21

σ

22

σ

23

σ

31

σ

32

σ

33

]

[

σ

x
x

σ

x
y

σ

x
z

σ

y
x

σ

y
y

σ

y
z

σ

z
x

σ

z
y

σ

z
z

]

[

σ

x

τ

x
y

τ

x
z

τ

y
x

σ

y

τ

y
z

τ

z
x

τ

z
y

σ

z

]

{\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.

View More On Wikipedia.org
1. ### Find the principal stresses in a shaft with torque applied

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3. ### Principal stresses 3D, solving for direction cosines n1,m1,n1

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4. ### Angle of principal stress vs maximum shear stress

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5. ### Angle between principal stress and shear stress

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6. ### How do I interpret multiple principal stresses for 3D loading with shear?

For 3 D loading with shear, if I use the principal stress formula, say for x-y direction, two principal stresses are obtained. If the same is applied to y-z, two more principal are obtained, with one supposed to be common, but not. Thus I obtain six principal values, which cannot be used with...
7. ### (Composite) In plane principal stress or normal stress?

Hey everybody, I went through a discussion with a colleague today about Finite element modeling of composite structures and how to interpret the stress analysis. I understand that for isotropic materials, principal stresses could be used against the allowable stresses to see if failure will...
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9. ### Von Mises stress Vs Principle stress

Hello Folks, May be I am bring up the old topic again., but I've lost enough sleep over this topic. I understand that we use Von-Mises criteria for ductile material failure by comparing to yield limit and max principal stress is used to check failure for brittle materials. My question is bit...
10. ### On the physical meaning of principal stress

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11. ### Principal Stress and Maximum Shear Stress

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12. ### Ansys Workbench- Max Principal Stress Error

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13. ### Principal stress at surface of thin walled pipe

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14. ### What are the shear and principal stresses for given stress components?

id like to think i did this right, but i just want to make sure. i can't do symbols so let d equal sigma. the stress components are d_xx= 200 MPa, d_xy= 50 MPa, d_yy=-100 MPa. I need to find the max shear stress, and the principal stresses, and then a_s and a_p. heres what i got: max...
15. ### What condition defines a principal stress?

What condition defines a principal stress? thx
16. ### Identifying Tensile and Compressive Stress in Principal Stress Extraction

When extracting a Principal stress for the component, how to differentiate/seperate/identify the tensile and compressive stress in it?
17. ### Principal Stress, VonMises Stress , Fatigue

1. I didnt understand why some people extracting Principal stress for aluminum material which is subjected to dynamic loads(acceleration).Why not Von Mises? 2. Which has more advantages in predicting Fatigue Life and how? 3. How will you distinguish between tensile and compressive...
18. ### Principal Stress and Strain on a Fracture

Homework Statement Find the expressions for principal stresses in both plane strain and plane stress conditions sigma(x) = (K/(sqrt(2*pi*r)))*cos(theta/2)*(1-sin(theta/2)*sin(3*theta/2)) sigma(y) = (K/(sqrt(2*pi*r)))*cos(theta/2)*(1+sin(theta/2)*sin(3*theta/2)) T(xy) =...
19. ### Principal Stress, Mohrs Circle

So I understand how to use Mohrs circle and the transformation equations to find principal stresses and stresses for a given plane, but what is the point? Is there a purpose to knowing this other than finding stresses for a given direction, the stress invariants or that shear does not occur on...