failure theory for piping material
TRANSCRIPT
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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