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:
{\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.
Homework Statement
Homework Equations
M = Fd
σ/y = M/I
T/J = τ/r
σalternating (σa) = σmax - σmin / 2
σmean (σm) = σmax + σmin / 2
soderberg: σa/σ'e + σm/σy = 1/FoS
The Attempt at a Solution
I am still unsure whether my progress so far is correct but..
I have calculated J to be 38.35e-9 m4...
Homework Statement
The problem asks to find out whether the rod yields at points in section A according to the tresca and von mises criteria
P is 120N
Homework Equations
Shear stress= VQ/It
Stress= Mx/I
shear stress= Tp/J
The Attempt at a Solution
I picked 2 points at section A.
The...
Homework Statement
-4.882L1+M1+2N1=0
L1-2.882M1=0
2L1-0.882N1=0
L1^2+M1^2+N1^2=1
How to solve for L1 ,M1 and N1 ?
0<L1,M1,N1<1Homework EquationsThe Attempt at a Solution
Homework Statement
I'm having problem with the angle of $$theta_s1$$ . My ans is +35 (as in my working) , but the ans provided is $$theta_s1$$ = -55[/B]
Homework EquationsThe Attempt at a Solution
As in the picture posted
Homework Statement
I know that an element that is subjected to max shear stress is 45 degree away from position of an element that is subjected to principal stress.I couldn't understand why the θ s= θ p -45 degree?
Cant it be θ s= θ p +45 degree ?
Homework EquationsThe Attempt at a Solution...
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...
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...
HI everybody
I am in dire need of help. I have started a mechanical systems module as part of a distance learning foundation degree. The notes seem quite sparse to me and I can't seem to get to grips with it from the start. I have attached a photo of the tutorial questions but the are no...
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...
Hi everyone, I was looking up Mohr's circle on planar stress and stress/strain relation and all that
and I read a context on principal stress. It said that if there is both normal (tensile) stress and shear stress in a given area, then the principal stress at that location is greater than...
Homework Statement
Hi all,
For my CW I have a question on a simple beam, ABCD and its cross-section. Please see attachment for figures
The material of the beam is steel, where modulus of elasticity, E = 210 GN/m^2
I have been asked to calculate the principle stresses and the maximum shear...
Hi all,
I did simple problem, in which assembly is subjected to single load & it is fixed at other end. I used Higher Order Tet element (& tried with Hexdominent Method also) for the casting body.
The vonmises stress shows the true results but when i was looking for the Max Principal...
Homework Statement
See attached jpg for problem statement and diagram.
I know we didn't discuss this type of problem in class. The rest of this homework set has been solving stress transformations with Mohr's circles for a given state of stress. I know how to find τxy (that was supposed to...
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...
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...
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) =...
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...