3 Distortion Energy Theory Derivation

8/12/2019 3 Distortion Energy Theory Derivation

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DISTORTION ENERGY THEORY OF FAILURE

A THEORY OF FAILURE APPLICABLE TO DUCTILE MATERIALS

Statement of the theory

When Yie!in"occurs in any material, the !i#tortion #train ener"y $er %nit &o%meat the

point of failure equals or exceeds the !i#tortion #train ener"y $er %nit &o%mewhen

yie!in"occurs in theten#ion te#t #$e'imen.

The theory applies to ductile materials only, because it is based on yie!in".

The three !imen#iona (tria)ia* #tre## #it%ation+

In the three dimensional stress situation, the state of stress at a particular location is fully

defined by three principal stress 1 , 2 , 3 .

Strain ener"y at a o'ation of the eement

The strain energy at a particular location of the element can be segregated into threecategories, namely

!a" Total strain energy per unit #olume of the stressed element, arising from the principal

stresses 1 , 2 , 3 .

!b" $train energy per unit #olume arising from the change of #olume caused by ahydrostatic stress, which is uniform in all three directions

!c" $train energy per unit #olume arising from distortion of the element, and which can be

considered as being the difference between !a" and !b".

ELASTIC STRESS,STRAIN RELATIONS

Uni,A)ia #tre##

This is the case of a single principal stress 1 .

%rincipal strains are then gi#en by the expressions

EEE

13

12

11 ,,

===

Where,=1 %rincipal strain in the direction of the principal stress= %oisson&s ratio for the material=E 'odulus of elasticity for the material


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University of Nairobi Engineering Design II

Bi,A)ia #tre## #it%ation

In this case the stress situation consists of two principal stresses 1 , 2 , and the strains1are

gi#en by

=1 ( )211

E

, =2 ( )121

E

, and =3 ( )211

+E

Tri,A)ia #tre## #it%ation

This is the case of three principal stresses 1 , 2 , 3 , and the strains in the directions of the

principal stresses are then gi#en by

=1 ( )[ ]3211

+E

!1"

=2 ( )[ ]3121

+E

!2"

=

3 ( )[ ]213

1 +

E

!3"

ENERGY PER UNIT -OLUME AT STRESS LOCATION

Tota #train ener"y U

The total strain energy is the strain energy caused by the three principal stresses 1 , 2 , 3 .

It is gi#en by

=U 112

1 ( 22

2

1 ( 33

2

1 !)"

$ubstituting the three strains 321 ,, and in equations !1",!2" and !3" into equation !)"

yields

=U ( )[ ]313221232221 22

1 ++++

E!*"

Strain Ener"y !%e to Chan"e of -o%me (Hy!ro#tati' #tre##* ony

The stress that causes change of #olume only !hydrostatic stress" may be considered as the

a#erage of the three principal stresses av , and deri#ed from the expression

=av3

321 ++

!+"

$ubstituting for the hydrostatic stress av , into equation !*" yields

=vU ( )[ ]22 3232

1avav

E !"

1'echanical -ngineering esign/ $higley, 0oseph, pg 12), 'craw ill, $e#enth -dition, 2)

Nyangasi %age 2 of + ./ 0an%ary 1//2


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=vU [ ]

212

3 2

E

av 4 [ ] 22

213av

E

!5"

$ubstituting the #alue of av from equation !+" into equation !5" yields

=vU [ ]

2

321

32

213

++

E4 [ ] ( )

2

321267

213

++

E

=vU [ ]

( ) 2321+

21

++

E4

[ ]( )[ ]313221232221 2

+

21

+++++

E

=vU ( )[ ]313221232221 2+

21

+++++

E!7"

This vU is the strain energy per unit #olume caused by the uniform !hydrostatic" stress,

which is part of the three principal stresses 1 , 2 , 3 .

Di#tortion Ener"y at the o'ation of $rin'i$a #tre##e# 1 3 2 3 3

The distortion energy can then be obtained as the difference between the total strain energy at

the location of principal stresses, and the strain energy due to the hydrostatic portion of the

stresses at the same location. The distortion energy is then deri#ed from the expression=dU U 8 vU

Where,=dU istortion energy in the element at the location of principal stresses 1 3 2 3 3

=U ( )[ ]313221232221 22

1 ++++

E

!*"

=vU ( )[ ]313221232221 2+

21

+++++

E!7"

Therefore,

dU 4 ( )[ ]313221232221 22

1 ++++

E8

( )[ ]313221232221 2+

21

+++++

E

dU 4 ( ) ( )[ ]313221232221 +3+

1 ++++

E8

( ) ( ) ( )[ ]313221232221313221232221 262622+

1 +++++++++

E

Nyangasi %age 3 of + ./ 0an%ary 1//2


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dU

4

( ) ( )( )( ) ( ) ( )

++++++++

++++++

313221

2

3

2

2

2

1313221

2

3

2

2

2

1313221

2

3

2

2

2

1

262622

+3

+

1

E

dU 4

( ) ( )( ) ( )

+++++

++++

2

3

2

2

2

1313221

313221

2

3

2

2

2

1

622

22

+

1

E

dU 4 ( )( ) ( )( )[ ]2222+

1313221

2

3

2

2

2

1 ++++++ E

dU 4

( )

( ) ( )[ ]313221

2

3

2

2

2

1+

221

++++

+

E

dU 4( ) ( ) ( )[ ]3132212322213

1

++++

+

E!1"

9ut

( )3132212

32

22

1 ++++ 4( ) ( ) ( )

2

231

232

221 ++

Therefore

dU 4( ) ( ) ( ) ( )( )[ ]231232221362

1

++

+

E!11"

dU 4( )

( ) ( ) ( )( )[ ]231232221+

1

++

+

E!12"

THE CASE OF SIMPLE TENSION TEST 4HEN YIELDING OCCURS

Nyangasi %age ) of + ./ 0an%ary 1//2


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:or the simple tension test specimen, the three principal stresses when yielding occurs are

1 4 yS , 2 4, 3 4

$ubstituting for the principal stresses in equation !12" yields

dU 4( ) ( ) ( ) ( )( )[ ]222 +

1++

+yy SS

E

dU 4( ) [ ]22+

1

ySE

+!13"

THE CASE OF THREE DIMENSIONAL STRESS 4HEN YIELDING OCCURS

The distortion energy theory of failure states

When Yie!in"occurs in any material, the !i#tortion #train ener"y $er %nit &o%meat the

point of failure equals or exceeds the !i#tortion #train ener"y $er %nit &o%mewhen

yie!in"occurs in theten#ion te#t #$e'imen.

This can be restated that when yielding occurs in any situation

dU 4( )

( ) ( ) ( )( )[ ]231232221+

1

++

+

E!12"

E5UALS

dU 4( ) [ ]22+

1

ySE

+!13"

( ) ( ) ( ) 2312

32

2

21 ++ 4 2

2 yS

( ) ( ) ( )

++

2

2

31

2

32

2

21

4 yS !1)"

E5UI-ALENT (-on,Mi#e#* STRESS

The expression on the left hand side of equation !1)" is therefore considered as the

e6%i&aent #tre## e , which causes failure by yielding. The equi#alent stress is then gi#en

by

e

4

( ) ( ) ( )

++

2

2

31

2

32

2

21

!1*"

The equi#alent stress e is also referred to as -on Mi#e#stress.

DESIGN E5UATION BASED ON THE DISTORTION ENERGY THEORY

This is deri#ed by ad;usting the yield strength of the material in simple tension with an

appropriate factor of safety ..sf The design equation then becomes

Nyangasi %age * of + ./ 0an%ary 1//2


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e 4 ( ) ( ) ( )

++

2

2

31

2

32

2

21

4..sf

Sy!1+"

APPLICATION OF THE DESIGN E5UATION

The principal stresses 1 3 2 3 3 are first determined by stress analysis. $uch analysis

describes the principal stresses as a function of the oa!carried, and the "eometryand

!imen#ion#of the machine or structural element.

The equi#alent stress in the design equation is then expressed in terms of the !imen#ion#ofthe machine or structural element, while the right hand side is the ten#ie yie! #tren"thof

the material.

The fa'tor of #afetyis simply a number chosen by the designer. The factor of safety togetherwith the strength of the material, gi#es the wor

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3 Distortion Energy Theory Derivation