Failure Criteria for Yielding
Failure Criteria for Yielding
Dr. Andri Andriyana
Centre de Mise en Forme des Matériaux, CEMEF UMR CNRS 7635
École 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 :
σ11 ≥ σy
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
where : τmax =
σmax −σmin
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 :
σmax − σmin = σy
Failure Criteria for Yielding
Tresca Criterion
Special case : Plane stress
Let σ1 , σ2 and σ3 be the principale stresses (σ3 = 0) :
When σ1 and σ2 are of opposite sign : τmax =
The yield condition is given by :
|σ1 − σ2 | = σy
or
|σ1 −σ2 |
2
σ1
σ2
−
= ±1
σy
σy
When σ1 and σ2 carry the same sign :
if
|σ1 | > |σ2 | ,
τmax =
|σ1 − σ3 |
|σ1 |
=
2
2
and
|σ1 | = σy
if
|σ1 | < |σ2 | ,
τmax =
|σ2 − σ3 |
|σ2 |
=
2
2
and
|σ2 | = σy
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 :
Wd,max
=
Wd,y
=
i
1 h
2
2
2
(σxx − σyy )2 + (σyy − σzz )2 + (σzz − σxx )2 + 6 τxy
+ τyz
+ τzx
12G
1 2
σ
6G y
Thus, general form of Von Mises Criterion is :
1/2
1 2
2
2
√ (σxx − σyy )2 + (σyy − σzz )2 + (σzz − σxx )2 + 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 :
i1/2
1 h
√ (σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2
= σy
2
Failure Criteria for Yielding
Von Mises Criterion
Special case : Plane stress
Let σ1 , σ2 and σ3 be the principale stresses (σ3 = 0) :
σvm
i1/2
1 h
2
2
2
√
=
(σ1 − σ2 ) + (σ2 − 0) + (0 − σ1 )
2
q
=
σ12 − σ1 σ2 + σ22
Von Mises yield criterion becomes :
σ12 − σ1 σ2 + σ22 = σy2
In σ1 − σ2 plane, this equation represents an ellipse
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 :
σr = 0
1
σθ =
pr
t
σz =
Tresca criterion
t
σθ − 0 = σy → pmax = σy
r
2
Von Mises criterion
2t
σθ2 − σθ σz + σz2 = σy2 → pmax = √ σy
3r
pr
2t