Tresca criterion in terms of invariants

[sandon]

Hi,

The Tresca Critrion is given in the form of non continuous equations:

Max(½|σ₁-σ₂|,½|σ₂-σ₃|,½|σ₃-σ₁|) = k

How did they come up with the invarient equation

f(J₂,θ) = 2√J₂ * sin(θ+⅓π)-2k, θ from (0 to 60)

 

[Illmatic1]

The invarient equation f(J2,θ) is derived from the Tresca Criterion by expressing it in terms of J2 (the second invariant of the deviatoric stress) and θ (the angle between the principal stresses). The expression for f(J2,θ) can be derived as follows:σ1, σ2, and σ3 are the principal stresses. Using the formula for the second invariant of the deviatoric stress, we can write J2 asJ2 = ½(σ1-σ2)² + ½(σ2-σ3)² + ½(σ3-σ1)²Substituting this expression for J2 into the Tresca Criterion, we getMax(½|σ1-σ2|,½|σ2-σ3|,½|σ3-σ1|) = kWe can re-write this expression by substituting in the following trigonometric identities:|σ1-σ2| = 2√J2*sinθ|σ2-σ3| = 2√J2*sin(θ+120°)|σ3-σ1| = 2√J2*sin(θ+240°)Using these identities, we can rewrite the Tresca Criterion asMax(2√J2*sinθ,2√J2*sin(θ+120°),2√J2*sin(θ+240°)) = kSimplifying this expression yields2√J2 * sin(θ+⅓π) - 2k, θ from (0 to 60)which is the form of the invariant equation f(J2,θ).