Stress Principles Flashcards
Cauchy’s Postulate
t^(n) = sigma*n
What is the traction vector?
This is the stress vector that appears over a surface that has been cut.
What does Cauchy’s postulate do?
It performs a linear mapping between the traction vector and the normal vector to a surface.
Equation for sigma’
s’ = R s R^T
Where R is the rotation matrix with unit vectors of the “primed coordinate system”
How to define sigma_nn and sigma_nr
Use cauchy’s postulate:
s_nn = t^(n).n = (s.n).n
s_nr = t^(n).r = (s.n).r = (s.r).n
Does the sum of the stresses add to zero?
Nope. Only sum of the forces adds to zero, not stresses
Index notation for Sigma_nn
s_nn =s_ij ni nj
Optimization equation for Sigma
Sn=(lambda)n
Characteristic equation to find principle stresses
L^3-I1L^2+I2L-I3=0
I1, I2, and I3 are invariants and represent the three principle stresses
Equations for the three invariants
I1 = trace (sigma) = sigma_ii
I2=Sum of the principle minors = 0.5*[(tr(sigma))^2 - tr(sigma^2)]
I3=Det(sigma)
Equation for shear stress in terms of normal stress
Tau^2 = t^(n).t^(n) - (sigma_nn)^2
Normal stress in octahedral coordinates
sigma_nn = trace(sigma)/3
This is the mean or hydrostatic component of the stress
Shear stress in octahedral coordinates
(Tau^2) = t^(oct).t^(oct) - (sigma_nn)^2
Von mises stress
Sqrt[ (s1-s2)^2 + (s1-s3)^2 + (s2-s3)^2) / 2 ]
Strong form of differential equation of equilibrium
sigma_ij,j + bi = 0
