Failure Criteria for Yielding
Failure Criteria for Yielding
Dr. Andri Andriyana
Centre de Mise en Forme des Mat´eriaux, CEMEF UMR CNRS 7635 Ecole´ des Mines de Paris, 06904 Sophia Antipolis, France
Spring, 2008 Failure Criteria for Yielding Outline
Outline
1 Introduction
2 Tresca Criterion
3 Von Mises Criterion
4 Comparison and Example Failure Criteria for Yielding Introduction
Introduction Failure Criteria for Yielding Introduction
Background and definitions
Yielding For ductile material under simple tension, stress no longer proportional to strain Plastic (irreversible) deformation (permanent molecular rearrangement) once a certain level of stress is reached Highly material dependent
Understanding yielding is important for designing a pressure vessel, rotating disc, crank shaft, ... that does not allow any irreversible strain, i.e. material must remain elastic Failure Criteria for Yielding Introduction
Fracture vs yield
Fracture Driven by normal stresses, acting to separate one atomic plane from another Broken atomic bonds are not allowed to reform in new positions
Yield Driven by shear stresses, sliding one plane along another Broken atomic bonds are allowed to reform in new positions Failure Criteria for Yielding Introduction
Stress-strain curve of ductile materials Failure Criteria for Yielding Introduction
Yield criteria
For material stretched uniaxially along e1 direction, yield occurs when : σ σ 11 ≥ y with σy is the yield stress
When does yield occurs in multiaxial stress states...?? Failure Criteria for Yielding Tresca Criterion
Tresca Criterion Failure Criteria for Yielding Tresca Criterion
General multiaxial stress states
Maximum shear stress Yielding starts when the maximum shear stress in the material τmax equals the maximum shear stress at yielding in a simple tension test τy
τmax = τy
σmax−σmin where : τmax = 2
σmax and σmin are the maximum and minimum principal stresses respectively Failure Criteria for Yielding Tresca Criterion
General multiaxial stress states Mohr’s circle for simple tension test :
Thus, general form of Tresca Criterion is :
σ σ = σ max − min y Failure Criteria for Yielding Tresca Criterion
Special case : Plane stress
Let σ1, σ2 and σ3 be the principale stresses (σ3 = 0) :
|σ1−σ2| When σ1 and σ2 are of opposite sign : τmax = 2 The yield condition is given by :
σ1 σ2 σ1 σ2 = σy or = 1 | − | σy − σy ±
When σ1 and σ2 carry the same sign : σ σ σ if σ > σ , τ = | 1 − 3| = | 1| and σ = σ | 1| | 2| max 2 2 | 1| y σ σ σ if σ < σ , τ = | 2 − 3| = | 2| and σ = σ | 1| | 2| max 2 2 | 2| y Failure Criteria for Yielding Tresca Criterion
Tresca yield surface for plane stress problems Failure Criteria for Yielding Von Mises Criterion
Von Mises Criterion Failure Criteria for Yielding Von Mises Criterion
General multiaxial stress states
Maximum distortion/shear energy
Yielding starts when the maximum distortion/shear energy in the material Wd,max equals the maximum distortion/shear energy at yielding in a simple tension test Wd,y
Wd,max = Wd,y
Distortion/shear energy : Part of the strain energy corresponds to volume-preserved shape change Failure Criteria for Yielding Von Mises Criterion
General multiaxial stress states
In terms of the stress components :
1 2 2 2 2 2 2 Wd,max = (σxx − σyy) + (σyy − σzz) + (σzz − σxx) + 6 τ + τ + τ 12G h xy yz zxi 1 2 Wd,y = σ 6G y
Thus, general form of Von Mises Criterion is :
1 2 2 2 2 2 2 1/2 (σxx σyy) +(σyy σzz) +(σzz σxx) + 6 τxy + τyz + τzx = σy √2 − − −
Left hand side : the Von Mises stress σvm Failure Criteria for Yielding Von Mises Criterion
General multiaxial stress states
In terms of the principal stresses σ1, σ2, σ3 :
1/2 1 2 2 2 (σ1 σ2) + (σ2 σ3) + (σ3 σ1) = σy √2 h − − − i Failure Criteria for Yielding Von Mises Criterion
Special case : Plane stress
Let σ1, σ2 and σ3 be the principale stresses (σ3 = 0) :
1/2 1 2 2 2 σvm = (σ1 σ2) + (σ2 0) + (0 σ1) √2 h − − − i = σ2 σ σ + σ2 q 1 − 1 2 2
Von Mises yield criterion becomes :
σ2 σ σ + σ2 = σ2 1 − 1 2 2 y In σ σ plane, this equation represents an ellipse 1 − 2 Failure Criteria for Yielding Von Mises Criterion
Von Misses yield surface for plane stress problems Failure Criteria for Yielding Comparison and Example
Comparison and Example Failure Criteria for Yielding Comparison and Example
Tresca and Von Misses yield surfaces : 2D space Failure Criteria for Yielding Comparison and Example
Tresca and Von Misses yield surfaces : 3D space
[Source : Wikipedia] Failure Criteria for Yielding Comparison and Example
Example : Thin pressurized tube with end caps
Given a thin walled tube (radius r, thickness t) containing gas. Using Tresca and Von Mises yield criteria, determine the maximum allowable gas pressure pmax so that no yielding occurs. Failure Criteria for Yielding Comparison and Example
Example : Thin pressurized tube with end caps
From Strength of Material course, the radial (σr), hoop (σθ) and longitudinal (σz) stresses are : pr pr σ = 0 σ = σ = r θ t z 2t
1 Tresca criterion t σ 0= σ p = σ θ − y → max r y
2 Von Mises criterion 2 2 2 2t σ σθσz + σ = σ pmax = σy θ − z y → √3 r