Theories of Yielding.pdf

7/30/2019 Theories of Yielding.pdf

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Theories of Yield/Failure

Keeping the condition that satisfied the material which let to failure. i.e. it give material

indication of the failure at given point on material.

There are component of machine shaft which is made of some material. There are load applied toshaft such as force or bending moment. When this load are acting, there will some point in the

material which will be most critically loaded and at the condition at which material failure is give

by Theory of failure or yield.

Twokindsofstressstates(a)simplestressstate(b)compoundstressstateThe yield strength or yield point of a material is defined in engineering as the stress at which a

material begins to deform plastically. Before yield point the material is deforms elastically.

It is often difficult to precisely define yielding due to the wide variety of stress-strain curves

exhibited by real materials. In addition there are several possible ways to define yielding.


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Figure 1 tensional stress and strain diagram

True elastic limit: the lowest stress at which dislocations move. This defined is rarely used, since

dislocation move at very low stresses, and detecting such movement is very difficult.

Proportionality limit: up to this amount of stress, stress is proportional to strain (Hookes law) so

the stress-strain graph is a straight line, and the gradient will be equal to the elastic modulus of

the material.

Elastic limit (yield strength) beyond the elastic limit, permanent deformation will occur. Thelowest stress at which permanent deformation can be measured.

Yield point: The point in the stress-strain curve at which the curve levels off and plastic

deformation begins to occur.

Failure modes

Excessive elastic Deformation Yielding Fracture

Stretch, twist, or bending Plastic deformation atroom temperature

Sudden fracture of brittlematerial

buckling creep at elevatedtemperature

Fatigue (progressive fracture )

vibration yield stress is theimportant design

factor

Stress rupture atelevated temperatures

Ultimate stress is theimportant design factor


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Elastic deformation is temporary (reversible) and involves bond stretching. Plastic deformation is permanent (irreversible), and involves bond breaking. Fracture is catastrophic.

Effect of loading at a point

3-D x,, y, z, xy, yz ,zx and x, y, z

Principle stress and strain 1,2, 3 and 1, 2, 3

2-D x,, y, xy Plane stress

1,2, and 1, 2

Failure in tensile test Parameter

1) Maximum Principal stress

2) Maximum shear stress

3) Maximum Principal Strain

4) Total Strain Energy Density

5) Distortion Energy Density

Therefore, now we can say that the yielding of a material has been due to the Maximum

Principal stress,Maximum shear stress, Maximum Principal Strain, Total Strain Energy Density,

Distortion Energy Density. Basically, we can say that yielding phenomenal is based on stress and

strain of material which is under an applied load.

y==1

2max

y=

E

y =

E

yS

2

2

=

Ss vd S=

Gy

6

2

=


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Five Criteria of Failure

1) Maximum Principal Stress Criterion ( Rankine)2) Maximum Shear Stress criterion (Tresca)3) Maximum Principal Strain Criterion (St Venant)4) Maximum Strain Energy Density Criterion ( Haigh or Beltrami)5) Maximum Distortion Energy Density Criterion (Von-Mass)

Maximum Principle Stress Theory

Theories of yielding are generally expressed in terms of principle stress, since those completely

determine general states of stress. The elements of material shown in Fig. 2 (a) is subjected to

three principle stresses, and the convection to be used is that 1 >2 >3.The maximum principlestress theory, often attributed to Rankine, states that yielding will occur

in a material under complex stress when yt, in a simple tension test on the same material.

Yielding could also occur if the minimum stress 3, were compressive and reached the value of

yield stress in a simple compression test. Those statements may be written as

1=yt 3=ytfor yielding stress to occur.

Figure 2

Maximum Principle Strain Theory.St.Venant postulated that yielding commences when the maximumprinciple strain(tensile), 1

was equivalent to the strain corresponding to the yield stress in simple tension. For yielding incompression the minimum principle strain, 3, would equal the yield strain in simple

compression. If the strains are expressed in terms of stress, then


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=1 ( )[ ]3211

+E

And yield occurs when compression equalsE

Yt

( ) Yt=+ 321

Or compression

( ) Yt=+ 213

Maximum shear stress Criterion (Tresca)

Statement of the theory

When Yielding occurs in any material, the maximum shear stress at the point of failure equals or

exceeds the maximum shear stress when yielding occurs in the tension test specimen.

The theory applies to ductile materials only, because it is based on yielding.

The three-dimensional (triaxial) stress situation.

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

by three principal stresses 1 , 2 , 3 .

Maximum shear stress at a location of the element

The extreme values of shear stresses 231312 ,, , in each of the three principal planes are then

given by the expressions:

2

2112

= ,

2

3113

= ,

2

3223

=

Expressing the principal stresses in the order of magnitude and sign 321 ff

Then the maximum shear stress is given by 23113

=

The case of Simple Tension Test When Yielding Occurs

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

1 = yS , 2 =0, 3 =0


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The maximum shear stress then becomes22

0

2

31max13

yy SS=

=

==

The case of three Dimensional Stress when Yielding Occurs

The maximum shear stress theory of failure states:

When Yielding occurs in any material, the maximum shear stress at the point of failure equals or

exceeds the maximum shear stress when yielding occurs in the tension test specimen.

22

31max

yS=

=

The above equation implies that the shear yield strength of the material2

ysy

SS =

But from analysis of plane stress situation, the maximum shear stress in plane stress is also given

in terms of plane stress elements

22

max2

xyyx

+

=

Design Equation Based on the Maximum Shear Stress Theory

This is derived by adjusting the shear yield strength of the material with an appropriate factor of

safety ..sf . The design equation then becomes:

..*2..2

31max

sf

S

sf

S ysy==

=

OR

..*2..2

22

maxsf

S

sf

S ysyxy

yx==+

=

for plane stress situation

Total Strain energy Theory

The theories put forward so far have postulated a criterion for yielding in terms of a limit value

of stress or strain. The present theory, as proposed by Beltrami, and also attributed to Haigh, is


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based on a critical value of the total strain energystored in the material, and this is a product of

stress and strain.It has been shown earlier that the work done in deformation or the stored elastic strain energy

may be written xW2

1or

2

12

1

=Ax

xW

per unit volumeIn a 3D stress system, the total strain energy is

332211 2

1

2

1

2

1++=UT

Now using a stress-strain relationship, the principal strain may be written as

( )[ ]32111

+=E

( )[ ]1322 1 += E

( )[ ]12331

+=E

Substituting for1, 2, 3 and rearranging it,

( ) ( ) 133221321 2222

2

1++++=

E

v

EUT

Yielding is said to occur when the above equals to the total energy at yield in simple tension. Ie.

By letting 2= 3=0 and 1=yt.

Therefore ( ) 1332213212222

2 ++++= vYt So

E

Yt

TU 2

2

=

In 2D system, 3=0 and

( ) 213212222

2v

Yt++=


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Distortion Energy Theory of Failure

Strain Energy due to Change of Volume (Hydrostatic stress) only

The stress that causes change of volume only (hydrostatic stress) may be considered as the

average of the three principal stresses av , and derived from the expression:

=av 3

321 ++

Substituting for the hydrostatic stress av , into equation strain theory equation yields:

=vU( )[ ]22 323

2

1avav

E

=vU[ ]

21

2

32

E

av=

[ ] 22

213av

E

Substituting the value of av into equation above equations yields:

=vU [ ]2

321

32213

++

E= [ ] ( )2321

2*9213 ++E

=vU[ ]

( )23216

21

++

E=

[ ]( )[ ]313221232221 2

6

21

+++++

E

=vU ( )[ ]313221

2

3

2

2

2

12

6

21

+++++

E

This vU is the strain energy per unit volume caused by the uniform (hydrostatic) stress, which is

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


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Distortion Energy at the location of principal stresses 1 , 2 , 3

The distortion energy can then be obtained as the difference between the total strain energy at thelocation of principal stresses, and the strain energy due to the hydrostatic portion of the stresses

at the same location. The distortion energy is then derived from the expression:=dU U - vU

Where,

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

=U ( )[ ]313221232221 22

1 ++++

E

=vU ( )[ ]313221232221 26

21

+++++

E

Therefore,

dU = ( )[ ]313221232221 22

1 ++++

E- ( )[ ]313221232221 2

6

21

+++++

E

dU = ( ) ( )[ ]313221232221 636

1 ++++

E-

( ) ( ) ( )[ ]313221232221313221232221 2*2*226

1 +++++++++

E

dU =( ) ( ) ( )

( ) ( ) ( )

++++++++

++++++

3132212

32

22

1313221

23

22

21313221

23

22

21

2*2*22

63

6

1

E

dU =( ) ( )

( ) ( )

+++++

++++

23

22

21313221

3132212

32

22

1

*22

22

6

1

E

dU = ( )( ) ( )( )[ ]222261

3132212

32

22

1 ++++++ E

dU =( ) ( ) ( )[ ]313221232221

6

221

++++

+

E

dU =( ) ( ) ( )[ ]313221232221

3

1

++++

+

E

But

( )3132212

32

22

1 ++++ =( ) ( ) ( )

2

231

232

221 ++


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Therefore

dU =( )

( ) ( ) ( )( )[ ]2312322213*2

1

++

+

E

dU = ( ) ( ) ( ) ( )( )[ ]23123222161 +++E

The case of Simple Test When Yielding Occurs

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

1 =yt, 2 =0, 3 =0

Substituting for the principal stresses

dU =( ) ( ) ( ) ( )( )[ ]222 0000

6

1++

+

ytyt

E

dU =( ) [ ]22

6

1yt

E

+

dU =G

yt

6

2

per unit volume

The case of three Dimensional Stress When Yielding Occurs

The distortion energy theory of failure states:

When Yielding occurs in any material, the distortion strain energy per unit volume at the point of

failure equals or exceeds the distortion strain energy per unit volume when yielding occurs in the

tension test specimen.

This can be restated that when yielding occurs in any situation:

dU =( )

( ) ( ) ( )( )[ ]2312322216

1

++

+

E

EQUALS

dU = ( )[ ]2

26

1yt

E +

( ) ( ) ( )2312

322

21 ++ =2

2 yt


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( ) ( ) ( )

++

2

231

232

221

= yt

Equivalent (Von-Mises) Stress

The expression on the left hand side of above equation is therefore considered as the equivalent

stress e , which causes failure by yielding. The equivalent stress is then given by:

e =( ) ( ) ( )

++

2

231

232

221

The equivalent stress e is also referred to as Von Mises stress.

Design Equation Based on the Distortion Energy Theory

This is derived by adjusting the yield strength of the material in simple tension with anappropriate factor of safety ..sf The design equation then becomes:

e =( ) ( ) ( )

++

2

231

232

221 =

..sf

yt

Application of the Design Equation

The principal stresses 1 , 2 , 3 are first determined by stress analysis. Such analysis describes

the principal stresses as a function of the load carried, and the geometry and dimensions of the

machine or structural element.

The equivalent stress in the design equation is then expressed in terms of the dimensions of the

machine or structural element, while the right hand side is the tensile yield strength of thematerial.

The factor of safety is simply a number chosen by the designer. The factor of safety togetherwith the strength of the material, gives the working

1(design, allowable) stress expected in the

machine part. The solution to the design equation then gives the minimum dimensions required

to avoid failure of the element by yielding.

SummaryMany experiments have been concluded under complex stress conditions to study the behaviour

of metals and test the validity of the foregoing theories. It has been shown that hydrostatic

pressure, and by inference hydrostatic tension, does not cause yielding. Now any complex stresssystem can be regarded as a combination of hydrostatic stress and a function of the differences of


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principle stress, and therefore a yield criterion such as that of Tresca or von Miseswhich is based

on principle stress difference would seem to be the most logical. It is now well established thatfor ductile metals, exhibiting yielding and subsequent plastic deformation, the shear strain energy

theory correlates best with material behavior. The maximum shear stress theory, although not

quit so consistent as the former, gives fairly reasonable prediction and is some times used in

design by virtue of its simpler mathematical form. The other theories are no longer used forductile metals, some being positively unsafe.

Examples 1

Given the material Yt, x, vand xy find the safety factors for all the applicable criteria.

(a)Pure alumimumYt=30MPa , x= 10MPa, y= -10MPa andxy= 0MPa

Example 2

The cantilever tube shown is to be made of 2014 aluminum alloy treated to obtain a specified

minimum yield strength of 276MPa. We wish to select a stock size tube (according to the table

below). Using a design factor of n=4. The bending load isF=1.75kN, the axial tension isP=9.0kNand the torsion is T=72N.m. What is the realized factor of safety?


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Example 3:

The factor of safety for a machine element depends on the particular point selected for the

analysis. Based upon the DET theory, determine the safety factor for points A and B.

This bar is made of AISI 1006 cold-drawn steel(Yt=280MPa) and it is loaded by the forces

F=0.55kN, P=8.0kN and T=30N.m

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Theories of Yielding.pdf