# Tresca criterion in terms of invariants

• sandon
This equation represents the critical condition for failure under the Tresca Criterion, where k is a constant value.
sandon
Hi,

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

Max(½|σ12|,½|σ23|,½|σ31|) = k

How did they come up with the invarient equation

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

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).

## 1. What is the Tresca criterion in terms of invariants?

The Tresca criterion is a failure criterion used in material science to predict the failure of ductile materials under multiaxial loading. It is defined in terms of the first and second invariants of the stress tensor.

## 2. How is the Tresca criterion different from other failure criteria?

The Tresca criterion differs from other failure criteria such as the von Mises criterion in that it considers the maximum shear stress as the indicator of failure rather than the maximum distortion energy.

## 3. What are the assumptions made in the Tresca criterion?

The Tresca criterion assumes that the material behaves in a perfectly plastic manner and that there is no work hardening.

## 4. How is the Tresca criterion used in practical applications?

The Tresca criterion is commonly used in engineering and design to determine the maximum allowable stress a material can withstand before failure under multiaxial loading. It is also used in finite element analysis to predict the failure of structures and components.

## 5. What are the limitations of the Tresca criterion?

The Tresca criterion has limitations in predicting the failure of materials that exhibit strain-hardening behavior, as it does not consider the effects of work hardening. It also does not take into account the stress state beyond yielding, making it less accurate for materials that undergo significant plastic deformation before failure.

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