Failure Criteria for Yielding

Failure Criteria for Yielding

Dr. Andri Andriyana

Centre de Mise en Forme des Materiaux, CEMEF UMR CNRS 7635Ecole 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 longerproportional to strain

Plastic (irreversible) deformation (permanent molecularrearrangement) 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 irreversiblestrain, i.e. material must remain elastic

Failure Criteria for Yielding

Introduction

Fracture vs yield

Fracture

Driven by normal stresses, acting to separate one atomic planefrom another

Broken atomic bonds are not allowed to reform in newpositions

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, yieldoccurs 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 thematerial τmax equals the maximum shear stress atyielding in a simple tension test τy

τmax = τy

where : τmax = σmax−σmin2

σmax and σmin are the maximum and minimum principal stressesrespectively

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 = |σ1−σ2|2

The yield condition is given by :

|σ1 − σ2| = σy orσ1

σy− σ2

σy= ±1

When σ1 and σ2 carry the same sign :

if |σ1| > |σ2| , τmax =|σ1 − σ3|

2=

|σ1|2

and |σ1| = σy

if |σ1| < |σ2| , τmax =|σ2 − σ3|

2=

|σ2|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/shearenergy in the material Wd,max equals the maximumdistortion/shear energy at yielding in a simple tensiontest Wd,y

Wd,max = Wd,y

Distortion/shear energy :Part of the strain energy corresponds to volume-preserved shapechange

Failure Criteria for Yielding

Von Mises Criterion

General multiaxial stress states

In terms of the stress components :

Wd,max =1

12G

[

(σxx − σyy)2 + (σyy − σzz)2 + (σzz − σxx)2 + 6(

τ2

xy + τ2

yz + τ2

zx

)

]

Wd,y =1

6Gσ2

y

Thus, general form of Von Mises Criterion is :

1√

2

[

(σxx − σyy)2 + (σyy − σzz)2 + (σzz − σxx)2 + 6

(

τ2

xy + τ2

yz + τ2

zx

)]1/2

= σy

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

[

(σ1 − σ2)2 + (σ2 − σ3)

2 + (σ3 − σ1)2]1/2

= σy

Failure Criteria for Yielding

Von Mises Criterion

Special case : Plane stress

Let σ1, σ2 and σ3 be the principale stresses (σ3 = 0) :

σvm =1√2

[

(σ1 − σ2)2 + (σ2 − 0)2 + (0 − σ1)

2]1/2

=√

σ21 − σ1σ2 + σ2

2

Von Mises yield criterion becomes :

σ21 − σ1σ2 + σ2

2 = σ2y

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 maximumallowable 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) andlongitudinal (σz) stresses are :

σr = 0 σθ =pr

tσz =

pr

2t

1 Tresca criterion

σθ − 0 = σy → pmax =t

rσy

2 Von Mises criterion

σ2θ − σθσz + σ2

z = σ2y → pmax =

2t√3 r

σy