Steady Load Failure TheoriesSteady Load Failure Theories(Distortion Energy Theory)(Distortion Energy Theory)

Lecture 6Lecture 6

Engineering 473Engineering 473Machine DesignMachine Design

DistortionDistortion--Energy TheoryEnergy Theory

Postulate: Yielding will occur when the distortion-energy per unit volume equals the distortion-energy per unit volume in a uniaxial tension specimen stressed to its yield strength.

Strain EnergyStrain Energy

σ

ε

iσ

iε

U332211 εσ21εσ

21εσ

21U ++=

[ ] [ ] [ ] [ ]32 ininlb

inin

inlbU −=⋅=

UnitsThe strain energy in a tensile test specimen is the area under the stress-strain curve.

The strain energy per unit volume is given by the equation

Strain EnergyStrain Energy

Strain EnergyStrain Energy(Elastic Stress(Elastic Stress--Strain Relationship)Strain Relationship)

( )

( )

( )2133

3122

3211

νσνσσE1ε

νσνσσE1ε

νσνσσE1ε

−−=

−−=

−−=

��

��

�

��

��

�

���

���

�

−−−−−−

=��

��

�

��

��

�

3

2

1

3

2

1

σσσ

1ννν1ννν1

E1

εεε

Algebraic Format Matrix Format

An expression for the strain energy per unit volume in terms of stress only can be obtained by making use of the stress-strain relationship

Strain EnergyStrain Energy(Stress Form of Equation)(Stress Form of Equation)

( )

( )

( )���

��� −−+

���

��� −−+

���

��� −−=

++=

2133

3122

3211

332211

νσνσσE1σ

21

νσνσσE1σ

21

νσνσσE1σ

21

εσ21εσ

21εσ

21U

( )[ ]13322123

22

21 σσσσσσ2νσσσ

2E1U ++−++=

Distortion and Hydrostatic Distortion and Hydrostatic Contributions to Stress StateContributions to Stress State

1σ

2σ

3σ

Principal Stresses Principal Stresses Acting on Principal Acting on Principal

PlanesPlanes

hσ

hσ

hσ

h1 σσ −

h2 σσ −

h3 σσ −

3σσσσ 321

h++=

Hydrostatic StressHydrostatic StressDistortional StressesDistortional Stresses

= +

The distortional stress components are often called the deviatoric stress components.

Physical SignificancePhysical Significance(Hydrostatic Component)(Hydrostatic Component)

hσ

hσ

hσ

3σσσσ 321

h++=

The hydrostatic stress causes a change in the volume.

strain volumetriceModulusBulk K

Keσh

≡≡=

The cube gets bigger in tension,smaller in compression.

Physical SignificancePhysical Significance(Distortional Stresses)(Distortional Stresses)

h1 σσ −

h2 σσ −

h3 σσ −

These unequal stresses act to deform or distort the material element.

There is no change in volume, but there is a change in shape.

These stresses try to elongate or compress the material more in one direction than in another.

Strain Energy Associated with the Strain Energy Associated with the Hydrostatic StressHydrostatic Stress

( )[ ]( )[ ]

[ ]( ) 2

hh

2h

2h

hhhhhh2h

2h

2hh

13322123

22

21

σE2ν-1

23U

σ6νσ32E1

σσσσσσ2νσσσ2E1U

σσσσσσ2νσσσ2E1U

=

⋅−=

++−++=

++−++=

This term is equal to the strain energy per unit volume from the hydrostatic stress components.

Distortional Strain EnergyDistortional Strain Energy

( )[ ]( ) ( )

( )[ ]

( )����

�

�

����

�

�

+++

+++

++−−

++−++=

++−−

++−++=

−=

323123

322122

312121

13322123

22

21

2321

13322123

22

21

hd

σσσσσ

σσσσσ

σσσσσ

3E2ν1

21

σσσσσσ2νσσσ2E1

9σσσ

E2ν1

23

σσσσσσ2νσσσ2E1

UUU

The distortional strain energy is equal to the difference between the total strain energy and the hydrostatic strain energy.

Distortional Strain EnergyDistortional Strain Energy(Continued)(Continued)

[ ]13322123

22

21d σσσσσσσσσ

3Eν1U −−−+++=

( )[ ]( ) ( )( )133221

23

22

21

13322123

22

21

hd

σσσσσσ2σσσ3E

2ν121

σσσσσσ2νσσσ2E1

UUU

+++++−−

++−++=

−=

Distortional Strain Energy in Tension Distortional Strain Energy in Tension Test SpecimenTest Specimen

[ ]2yd

13322123

22

21d

S3Eν1U

σσσσσσσσσ3Eν1U

+=

−−−+++=

Hamrock, Fig. 3.1

Postulate: Yielding will occur when the distortion-energy per unit volume equals the distortion-energy per unit volume in a uniaxial tension specimen stressed to its yield strength.

Distortion Energy Failure TheoryDistortion Energy Failure Theory

[ ]

13322123

22

21eff

yeff

13322123

22

21

2y

2y

13322123

22

21d

σσσσσσσσσσ

Sσ

σσσσσσσσσS

S3Eν1

σσσσσσσσσ3Eν1U

−−−++=

=

−−−++=

+=

−−−+++=

Equating the distortional strain energy at the point under consideration to the distortional strain energy in the tensile test specimen at the yield point yields

Alternate Forms of Effective StressAlternate Forms of Effective Stress

( ) ( ) ( )2

σσσσσσσ

σσσσσσσσσσ

213

232

221

eff

13322123

22

21eff

−+−+−=

−−−++=

theory. the todcontribute whoMises von R. Dr.after stress, Mises von theas to

referredcommonly is stress effective The

Form 1

Form 2

Plane Stress ConditionPlane Stress Condition

( )2

σσσσσ

σσσσσ

21

22

221

eff

2122

21eff

++−=

−+=

0σ3 =

yS

yS

yS-

yS-

1σ

2σ

� As long as the stress state falls within the shaded area, the material will not yield.

� The surface, blue line, at which the material just begins to yield is called the yield surface.

Pure Shear ConditionPure Shear Condition

Mohr�s Circle for Pure Shear

1σ2σ

3σ

1,3τ

13 σσ −=

yS

yS

yS-

yS-

1σ

3σ

ysymax

y2max

21

3123

21eff

SS0.577τ

S3τσ3

σσσσσ

=⋅=

===

−+=

°45

This is an important result.This is an important result.

Yield Surface in 3Yield Surface in 3--D Stress StateD Stress State

Hamrock, Fig. 6.9

Other Names for Distortion Other Names for Distortion Energy TheoryEnergy Theory

( ) ( ) ( )2

σσσσσσσ2

132

322

21eff

−+−+−=

�Shear Energy Theory�Von Mises-Hencky Theory�Octahedral-Shear-Stress Theory

People came up with the same equation using different starting

points.1σ2σ3σ σ

τ1/3τ

1/2τ2/3τ

321 σσσ >>

AssignmentAssignment� Show that the two forms of the equation for the effective stress

are equal.� Show that the effective stress for a hydrostatic stress state is

zero.� Compute the effective stress at the critical location in the

stepped shaft loaded in tension (previous assignment). The yield strength of the material is 30 ksi. Will the material yield at the critical location?

( ) ( ) ( )2

σσσσσσσ

σσσσσσσσσσ

213

232

221

eff

13322123

22

21eff

−+−+−=

−−−++=

AssignmentAssignment(Continued)(Continued)

In the rear wheel suspension of the Volkswagen �Beetle� the spring motion was provided by a torsion bar fastened to an arm on which the wheel was mounted. See the figure for more details. The torque in the torsion bar was created by a 2500-N force acting on the wheel from the ground through a 300-mm lever arm. Because of space limitations, the bearing holding the torsion bar was situated 100-mm from the wheel shaft. The diameter of the torsion bar was 28-mm. Find the von Mises stress in the torsion bar at the bearing.

Hamrock, Fig. 6.12