Failure Theory for piping material

7/21/2019 Failure Theory for piping material

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Thermal Strains and Element of the

Theory of Plasticity


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Thermal Strains

Thermal strain is a special class of Elastic strain that

results fromexpansion with increasing temperature, or

contraction with decreasing temperature

Increased temperature causes the atoms to vibrateby

large amount. In isotropic materials, the effect is the

same in all directions.

!er a limited range of temperatures, the thermal

strains at a gi!en temperature T, can be assumed to beproportional to the change, T.

( ) ( )TTT == " #$%&'(


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where T"is the reference temperature #) " at T"(. The

coefficient of thermal expansion, , is seen to be in units

of '*o

+, thus maing strain dimensionless. Since uniform thermal strains occur in all directions in

isotropic material, -ooes law for /&0 can be

generali1ed to include thermal effects.

( )[ ] ( )TE

zyxx ++= '

( )[ ] ( )TE zxyy ++=

'

( )[ ] ( )TE

yxzz ++= '

#$%&2(


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The theory of plasticity is concerned with a number of

different types of problems. It deals with the beha!ior

of metals at strains where -ooes law is no longer!alid.

3rom the !iewpoint of design, plasticity is concerned

with predictingthe safe limitsfor use of a materialunder combined stresses. i.e., the maximum loadwhich

can be applied to a body without causing4

Excessi!e 5ielding

3low 3racture

Plasticity is also concerned with understanding the

mechanism of plastic deformation of metals.


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Plastic deformation is not a re!ersible process, and

depends on the loading path by which the final state is

achie!ed. In plastic deformation, there is no easily measured

constant relating stress to strain as with 5oungs modulus

for elastic deformation.

The phenomena of strain hardening, plastic

anisotropy, elastic hysteresis, and Bauschinger effect

can not be treated easily without introducing

considerable mathematical complexity.


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3igure %&'#a(. Typical true stress&strain cur!es for a ductile metal.

-ooes law is followed up to the yield stress ", and beyond ",

the metal deforms plastically.


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3igure %&'b. Same cur!e as %&'a, except that it shows what happensduring unloading and reloading & -ysteresis. The cur!e will not be

exactly linear and parallel to the elastic portion of the cur!e.


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3igure %&'c. Same cur!e as %&'a, but showing 6auschinger effect.

It is found that the yield stress in tension is greater than the yield

stress in compression.


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3igure %&2. Ideali1ed flow cur!es. #a( 7igid ideal plastic material


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3igure %&2b. Ideal plastic material with elastic region


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3igure %&2c. Piecewise linear # strain&hardening( material.


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$ true stress&strain cur!e is fre8uently called a flow

cur!e, because it gi!es the stress re8uired to cause the

metal to flow plastically to any gi!en strain. The mathematical e8uation used to describe the stress&

strain relationship is a power expression of the form4

where 9 is the stress at ) '." and n, the strain&

hardening coefficient, is the slope of a log&log of

E8. %&'

That is,

n

k = #%&'(

logloglog nK += #%&2(


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3$I:;7E +7ITE7I$4 3:


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In applying a yielding criterion, the resistance of a

material is gi!en by its yield strength.

In applying a fracture criterion, the ultimate tensilestrength is usually used.

3ailure criterion for isotropic materials can be

expressed in the following mathematical form4

where failure #yielding or fracture( is predicted to occur

when a specific mathematical functionf of the principal

normal stresses is e8ual to the failure strength of the

material, c, from a uniaxial tension test.

( ) cf =/2' ,,#%&/(


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The failure strength is either the yield strength o, or the

ultimate strength u, depending on whether yielding or

fracture is of interest.

:et us define an effecti!e stress, , which is a single

numerical !alue that characteri1es the state of applied

stress. If

where cis a nown material property

3ailure is not expected if

The safety factoragainst failure is gi!en as4

That is the applied stress can be increased by a factor of

= before failure occurs.

(#>

occursfailurec =

(#>

failurenoc

c

X =


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Maximum Normal Stress Criterion (Rankine

5ielding #Plastic flow( taes place when the greatest

principal stress in a complex state of stress reachestheflow stress in a uniaxial tension.

Since '? 2? /, 3low occurs when

"#tension( ) '

+ompressi!e strength is usually greater than tensile

strength.

3low stress in uniaxial tension

@aximum normal stress in a

complex stress state.

#%&A(

( ) ( )ncompressiotension "'"


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Experiment to determine the yield stress of the shrimp

#defined as the stress at which the amplitude of the tail

wiggling would ha!e becomes less than a critical !alue(

when crushed between two fingers showed that it occurredat a stress of about '"&CD*m&2#'A.C psi(.

-ence,

7anines criterion predicts that shrimp failure would occur at

This corresponds to a depth of only '"m. 3ortunately for all

lo!ers of crustaceans, this is not the case, and hydrostatic

stresses do not contribute to plastic flow.

2C

" *'" mN=

2C" *'" mNp =


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Maximum!Shear!Stress or Tresca Criterion

This yield criterion assumes that yielding occurswhen

the maximum shear stress in a complex state of stresse8uals the maximum shear stress at the onset of flow in

uniaxial&tension.

3rom E8,#2.2'(, the maximum shear stress is gi!en by4


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3or uniaxial tension, , and the maximum

shearing yield stress is gi!en by4

Substituting in E8. #%./(, we ha!e

Therefore, the maximum&shear&stress criterion is gi!en by4

"/2,"' === "

2"

"

=

22"

"/'

max

==

=

"/' = #%.G(


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This criterion corresponds to taing the differences

between '

and /

and maing it e8ual to the flow stressin uniaxial tension.

This criterion does not predict failure under hydrostatic

stress, because we would ha!e ') /) p and no

resulting shear stress.


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von Mises" or #istortion!$nergy Criterion

This criterion is usually applied to ductile material !on @ises proposed that yielding would occur when the

second in!ariant of the stress de!iator H2exceeds some

critical !alue.

where

for yielding in uniaxial tension

22 kJ =( ) ( ) ( )[ ]2'/

2

2'

2

/22G

' ++=J

"B /2"' ===

22"

2" Gk=+

#%.F(


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Substituting E8. %&% into %&F, we obtain the usual form of

!on @ises yield criterion.


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%dditional &ailure Criteria

ctahedral Shear Stress 5ield +riteria4 This is another

yield criteria often used for ductile metals. It states thatyielding occurs when the shear stress on the octahedral

planes reaches a critical !alue.

@ohr&+oulomb 3ailure +riterion4 This is used for

brittle metals, and is a modified Tresca criterion.

Jriffith 3ailure +riterion4 $nother criterion used forbrittle metals. It simply states that failure will occur

when the tensile stress tangential to an ellipsoidal ca!ity

and at the ca!ity surface reaches a critical le!el ".


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@c+lintoc&


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$xample

$ region on the surface of a G"G'&TA aluminum alloy

component has strain gage attached, which indicate thefollowing stresses4

'' ) F" @Pa

22 ) '2" @Pa

'2 ) G" @Pa

0etermine the yielding for both Trescas and !on @ises

criteria, gi!en that " ) 'C" @Pa #the yield stress(.


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Solution

Since we were gi!en the !alue of '2, we must therefore

first establish the principal stresses. In!oe E8. A&/F.

-ence,

' ) 'G" @PaB 2 ) /" @PaB ' ) "

$ccording to Tresca, max ) #'G" & "(*2 ) %" @Pa

3or yielding in uniaxial tension4

"*2 ) FC @Pa

Since the %" @Pa ? FC @Pa, Tresca criterion would be

unsafe.

+

+= 2'2

222''22''

2'22

,


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The !on @ises criterion can be in!oed from E8. %&.

The :.-.S. of the abo!e E8. gi!es a !alue of 'FC @Pa.

This criterion predicts that the material will not fail #flow(,unlie the Tresca criterion, which predicts that the material

will flow.

Therefore, the Tresca criterion is more conser!ati!e than

the !on @ises criterion in predicting failure.

[ ] 2*'2'/

2

/2

2

2' (#(#(#

2

'

++=o

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Failure Theory for piping material