MSE 2203
Mechanical Behavior of Materials
Fracture Mechanics
Jahirul Islam
Lecturer
Dept. of Materials Science and Engineering
Khulna University of Engineering & Technology
Email: jahirul@mse.kuet.ac.bd
Fracture Mechanics
Reference:
Chapter-11: Mechanical Metallurgy By G. E. Dieter
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Introduction
➢ It was shown in Chapter-7, that microcracks can be formed in
metallurgical systems by a variety of mechanisms and that the
critical step usually is the stress (σf) required to propagate the
microcracks to a complete fracture.
➢ The first theoretical approach to find the fracture stress (σf) has
shown by Griffith as:
1/2
2𝐸𝛾𝑠
𝜎=
𝜋𝑐
➢ But the limitation of Griffith theory was to avoid plastic
deformation factor during fracture.
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Introduction
➢ E. Orowan (1952) suggested to allow for the degree of plasticity
in Griffith equation that would be made more compatible with
brittle fracture in metal.
➢ Therefore he suggested to include the term γp expressing the
plastic work required to extend the crack wall. Therefore
2𝐸(𝛾𝑠 + 𝛾𝑝 )
𝜎𝑓 =
𝜋𝑐
1/2
𝐸𝛾𝑝
≈
𝜋𝑐
1/2
(1)
➢ The surface-energy term can be neglected since estimates of the
plastic-work term are about 102 to 103 J/m2 compared with
values of γs of about 1 to 2 J/m2.
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Introduction
➢ In this chapter we denote crack length by the symbol a, as is
customary in the literature of fracture mechanics, rather than
the symbol c, therefore
𝐸𝛾𝑝
𝜋𝑐
1/2
𝐸𝛾𝑝
→ 𝜎𝑓 =
𝜋𝑎
1/2
(1)
➢ Later on the eqn. (11-1) was modified by G.R. Irwin (1958) to
replace the hard to measure γp with a term, ξ that was directly
measurable i.e.
1/2
𝐸𝜉𝑐
𝜎𝑓 =
(2)
𝜋𝑎
➢ Where ξc means the critical value of the crack extension force.
Sometime it is also called fracture toughness of the material.
𝜋𝑎𝜎 2
𝜉=
(3)
𝐸
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Strain-energy Release Rate
❑ We know,
𝐸𝜉𝑐
𝜎𝑓 =
𝜋𝑎
1/2
𝜋𝑎𝜎 2
→𝜉=
𝐸
(3)
➢ For any value of ξ, other than critical
value is also called the strain-energy
release rate, i.e., the rate of transfer of
energy from the elastic stress field of
the cracked structure to the inelastic
process of crack extension. Now we
will learn how ξ can be measured.
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Strain-energy Release Rate
❑ Now we will learn how ξ can be measured.
➢ A single-edge notched specimen (with a sharp notch, usually
done by applying fatigue load) is loaded axially.
➢ A strain gage is placed at the entrance to the notch to measure
the displacement/propagation of the crack.
➢ Now, for different length of
notches, determine the Load
(P) vs displacement (δ) curves.
➢ Here P = Mδ. M is the
stiffness of a specimen with a
crack of length a.
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Strain-energy Release Rate
➢ The elastic strain energy is given by the area under the curve to a
particular value of P and δ.
1
𝑃2
𝑈0 = 𝑃𝛿 =
(4)
2
2𝑀
➢ As the specimen is rigidly gripped so that an increment of crack
growth da results in a drop in load from P1 to P2.
𝑃1
𝑃2
𝛿1 = 𝛿2 =
=
𝑀1 𝑀2
➢ Since P/M constant, therefore after partial different equation, we
obtain
𝜕𝑃 1
𝜕(1/𝑀)
+𝑃
=0
𝜕𝑎 𝑀
𝜕𝑎
𝜕𝑃
𝜕 1/𝑀
= −𝑃𝑀
𝜕𝑎
𝜕𝑎
(5)
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Strain-energy Release Rate
➢ But, the crack extension force is defined as
𝜕𝑈0
𝜉=
𝜕𝑎
𝛿
1 2𝑃 𝜕𝑃
𝜕(1/𝑀)
2
=
+𝑃
2 𝑀 𝜕𝑎
𝜕𝑎
(6)
➢ Combining eqn. (5 ) and (6):
1 2 𝜕 1/𝑀
𝜉=− 𝑃
(7)
2
𝜕𝑎
➢ The fracture toughness, or critical strain-energy release rate, is
determined from the load, Pmax, at which the crack runs unstably
to fracture.
2
𝑃𝑚𝑎𝑥
𝜕 1/𝑀
𝜉𝑐 =
(8)
2
𝜕𝑎
➢ This is the criterion for brittle fracture in the presence of a cracklike defect was that unstable rapid failure would occur when the
stresses at the crack tip exceeded a critical value (Pmax).
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Stress Intensity Factor
➢ The stress distribution/component of an applied nominal stress, σ
at the crack tip in a thin plate (i.e. t, very small) are:
𝑎
𝜎𝑥 = 𝜎
2𝑟
1/2
𝜃
𝜃
3𝜃
𝑐𝑜𝑠
1 − 𝑠𝑖𝑛 𝑠𝑖𝑛
2
2
2
𝑎 1/2
𝜃
𝜃
3𝜃
𝜎𝑦 = 𝜎
𝑐𝑜𝑠
1 + 𝑠𝑖𝑛 𝑠𝑖𝑛
2𝑟
2
2
2
𝑎 1/2
𝜃
𝜃
3𝜃
𝜏𝑥𝑦 = 𝜎
𝑠𝑖𝑛 𝑐𝑜𝑠 𝑐𝑜𝑠
2𝑟
2
2
2
(9)
➢ From the above eqns. we can see that if the crack is oriented
aligned to the loading direction (i.e. θ=0)
𝑎 1/2
𝜎𝑥 = 𝜎𝑦 = 𝜎
𝑎𝑛𝑑
𝜏𝑥𝑦 = 0
2𝑟
and if r=0; i.e. very sharp crack; σx and ay will be infinity.
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Stress Intensity Factor
➢ Also we can see that, the local stresses (i.e. stress component)
near a crack depend on the product of the nominal stress σ and the
square root of the half-flaw length (a/2).
➢ G. R. Irwin (1954) stated this, as stress intensity factor, K which
indicate the localized stress intensity and he stated that for a sharp
elastic crack in an infinitely wide plate, K is defined as
𝐾 = 𝜎 𝜋𝑎
(10)
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Stress Intensity Factor
❑ Features of Stress Intensity Factor (K)/ Fracture Toughness (Kc):
➢ The stress intensity factor K is a convenient way of describing
the stress distribution around a flaw.
➢ It is the inherent resistance of the material to failure in the
presence of a crack-like defect.
➢ Relationship of ξc and K is as follows: K2 = ξcΕ
➢ That’s why, ξc also called fracture toughness as said earlier and
the K determination process is more likely similar to ξc.
➢ Note the K has the unusual dimensions of MNm-3/2 or MPam1/2
➢ As the crack-tip stresses condition can be described by the
stress intensity factor K, a critical value of K (i.e.. Kc) can be
used to define the conditions for brittle failure.
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Stress Intensity Factor
❑ Features of Stress Intensity Factor (K)/ Fracture Toughness (Kc):
➢ If Kc is known, then it is possible to compute the maximum
allowable stress for a given flaw size.
➢ While Kc is a basic material property, in the same sense as
yield strength, it changes with important variables such as
temperature and strain rate.
➢ Kc usually decreases with decreased temperature and increased
strain rate.
➢ If two flaws of different geometry have the same value of Kc,
then the stress fields around each of the flaws are identical.
➢ For a given alloy, Kc is strongly dependent on such
metallurgical variables as heat treatment, microstructure,
melting practice, impurities, inclusions, etc.
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Stress Intensity Factor
❑ Features of Stress Intensity Factor (K)/ Fracture Toughness (Kc):
➢ The magnitude of Kc, depends on the geometry of the solid
containing the crack, the size and location of the crack, and the
magnitude and distribution of the loads imposed on the solid.
➢ Therefore a general eqn. for Kc is:
𝐾 = 𝛼𝜎 𝜋𝑎
(12)
where α is a geometrical factor of the specimen
➢ Due to different loading condition (i.e. σx/ σy/ τxy), different
mode of deformation of crack can be occurred at crack tip.
➢ Therefore, the Kc will also change along with different loading
condition at crack tip ( usually 3 modes deformation occurs).
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Stress Intensity Factor
❑ Modes of Crack Deformation (3modes of loading condition @ crack):
➢ Mode I: the crack-opening
mode, refers to a tensile stress
applied in the y-direction normal
to the faces of the crack.
➢ This is the usual mode for
fracture toughness tests and a
critical value of stress intensity
determined for this mode would
be designated, KIc.
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Stress Intensity Factor
❑ Modes of Crack Deformation (3modes of loading condition @ crack):
➢ Mode II: the forward shear
mode or sliding mode, refers to a
shear stress applied normal to
the leading edge of the crack but
in the plane of the crack.
➢ Mode III: the parallel shear
mode or tearing mode, is for
shearing stresses applied parallel
to the leading edge of the crack.
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Fracture Toughness (KIc) and Design
➢ A properly determined fracture
toughness, KIc independent of
crack length, geometry, or loading
system (i.e. shape factor α is
negligible).
➢ It is a material property in the
same sense that yield strength is a
material property.
➢ If the material is selected, KIc is
fixed.
➢ For the presence of a relatively
large stable crack, then the design
stress is fixed and must be less
than KIc.
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Fracture Toughness (KIc) and Design
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Fracture Toughness (KIc) and Design
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Effects of Specimen Thickness on Stress and
Mode of Fracture
➢ Depends on material thickness, crack deformation Mode-I can be
two type.
➢ With thin plate-type specimens the stress state is plane stress (i.e.
stress is uniform) while with thick specimens there is a planestrain (strain is uniform) condition.
Plane stress:
𝐾 2 = 𝜉𝐸
(14)
Plane-strain:
𝐾 2 = 𝜉𝐸/(1 − 𝜐 2 )
(15)
➢ A notch in a thick plate is far more damaging than in a thin plate
because it leads to a plane-strain state of stress with a high
degree of triaxiality and the values of KIc are lower than for
plane-stress specimens.
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Effects of Specimen Thickness on Stress and
Mode of Fracture
➢ From Fig. a mixed-mode, ductile brittle fracture with 45° shear
lips is obtained for thin specimens.
➢ Once the specimen has the critical thickness, the fracture surface
is flat and the fracture stress is constant with increasing specimen
thickness.
➢ The minimum thickness to
achieve plane-strain conditions
and valid KIc measurements is
𝐾𝐼𝑐
𝐵 = 2.5
𝜎0
2
(18)
where σ0 is the 0.2 percent offset
yield strength
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Measurement of KIc Plane-strain Condition
➢ The KIc measurement process is not a straight forward single test
process through which KIc can be measured directly.
➢ It is more likely, assume a starting value and run the test process,
then check.
➢ This trial and error process is running until certain criteria meet at
the end of the process.
➢ Assume an expected KIc value and determine the specimen
thickness using the eqn.
𝐾𝐼𝑐
𝐵 = 2.5
𝜎0
2
(18)
where σ0 is the 0.2 percent offset
yield strength
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Measurement of KIc Plane-strain Condition
➢ A notch is machined in the specimen. The sharpest possible crack
is produced at the notch root by fatiguing the specimen typically
1,000 cycles with a strain of 0.03.
➢ The test is carried out and a continuous autographic record of
load P and relative crack displacement value are obtained from
machine.
➢ Usually three types of load crackdisplacement curves are obtained
for different materials.
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Measurement of KIc Plane-strain Condition
➢ Type-I load-displacement curve represents the behavior for a
wide variety of ductile metals in which the crack propagates by a
tearing mode with increasing load.
➢ This curve contains no characteristic features to indicate the load
corresponding to the onset of unstable fracture.
➢ The ASTM procedure is to
first draw the secant line
OPs from the origin with a
slope that is 5 percent less
than the tangent OA. This
determines Ps.
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Measurement of KIc Plane-strain Condition
➢ Next draw a horizontal line at a load equal to 80 percent of Ps and
measure the distance x1 along this line from the tangent OA to the
actual curve.
➢ If x1 exceeds one-fourth of the corresponding distance xs at Ps, the
material is too ductile to obtain a valid KIc value.
➢ If the material is not too
ductile, then the load Ps
is designated PQ and used
to calculate a conditional
value
of
fracture
toughness denoted KQ,
using
the
equation
described latter.
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Measurement of KIc Plane-strain Condition
➢ The type II load-displacement curve has a point where there is a
sharp drop in load followed by a recovery of load.
➢ The load drop represents a "pop in" which arises from sudden
unstable, rapid crack propagation before the crack slows down to a
tearing mode of propagation.
➢ The same criteria for
excessive ductility is
applied to type II curves,
but in this case PQ is the
maximum recorded load.
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Measurement of KIc Plane-strain Condition
➢ The type III curve shows complete "pop in" instability where the
initial crack movement propagates rapidly to complete failure.
➢ This type of curve is characteristic of a very brittle “elastic
materials”.
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Measurement of KIc Plane-strain Condition
➢ If the material is not too ductile, then the load Ps is designated PQ
and used to calculate a conditional value of fracture toughness
denoted KQ, using the equation described below:
𝑃𝑄
𝐾𝑄 =
𝐵𝑊 1/2
𝑎
29.6
𝑊
1/2
3
2
𝑎
𝑎
− 185.5
+ 655.7
𝑊
𝑊
7
9
𝑎 2
𝑎 2
−1017.0
+ 638.9
𝑊
𝑊
5
2
(19)
➢ The crack length, a used in the equations is measured after
fracture.
➢ Next, use the value of KQ to calculate the material thickness, B
using the eqn.
2
𝐾𝑄
𝐵 = 2.5
𝜎0
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Measurement of KIc Plane-strain Condition
➢ The value of PQ determined from the load-displacement curve is
used to calculate a conditional value of fracture toughness denoted
KQ, using the equation described below:
𝑃𝑄
𝐾𝑄 =
𝐵𝑊 1/2
𝑎
29.6
𝑊
1/2
3
2
𝑎
𝑎
− 185.5
+ 655.7
𝑊
𝑊
7
9
𝑎 2
𝑎 2
−1017.0
+ 638.9
𝑊
𝑊
5
2
(19)
➢ The crack length, a used in the equations is measured after
fracture.
➢ Next, use the value of KQ to calculate the material thickness, B
using the eqn.
2
𝐾𝑄
𝐵 = 2.5
𝜎0
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Measurement of KIc Plane-strain Condition
➢ If this quantity, B is less than both the thickness and crack
length of the specimen, then KQ is equal to KIc and the test is
valid.
➢ Otherwise it is necessary to us a thicker specimen to determine
KIc.
➢ The measured value of KQ can be used to estimate the new
specimen thickness through eqn. (11-18).
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Ductile to Brittle Transition Temperature
(DBTT) Phenomenon
Liberty ship broken in shipyard
Broken in harbour
Reference:
Chapter-14: Mechanical Metallurgy By G. E. Dieter
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Introduction
➢ During WWII, the brittle fracture of mild steel (ductile material)
ship draw a great deal of attention of researchers because some of
these ships broke completely into two pieces.
➢ Most of the failure occurred during the winter months. Failures
occurred both when the ships were in heavy seas and when they
were anchored at dock.
➢ These calamities focused attention on the fact that normally ductile
mild steel can become brittle under certain conditions. These are:
➢ a triaxial state of stress (such as exists at a notch),
➢ a low temperature, and
➢ a high strain rate or rapid rate of loading.
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Introduction
➢ All three of these factors do not have to be present at the same
time to produce brittle fracture.
➢ Steels which have identical properties when tested in tension or
torsion at slow strain rates can show pronounced differences in
their tendency for brittle fracture when tested in a notched-impact
test.
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Notched-bar Impact Tests
➢ Two classes of specimens have
been standardized for notchedimpact testing.
➢ Charpy bar specimens are used
most commonly in the United
States, while the Izod specimen
is favored in Great Britain.
➢ The Charpy specimen has a
square cross section (10 x 10
mm) and contains a 45° V notch,
2 mm deep with a 0.25-mm root
radius.
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Notched-bar Impact Tests
➢ The specimen is supported as a
beam in a horizontal position
and loaded behind the notch by
the impact of a heavy swinging
pendulum (the impact velocity
is approximately 5 m/s).
➢ The specimen is forced to bend
and fracture at a high strain
rate.
➢ The principal measurement
from the impact test is the
energy absorbed, Cv (in J) in
fracturing the specimen.
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Important Findings from Impact Test
➢ Energy measured by the Charpy test is only a relative energy
absorbed due to fracture therefore this types of test result cannot
be used directly to the design requirement.
➢ However, a common measurement obtained from the Charpy test
whether the fracture is fibrous (ductile fracture), granular
(cleavage fracture), or a mixture of both.
➢ The flat facets of cleavage fracture provide a high reflectivity and
bright appearance, while the fibrous surface provides a lightabsorptive surface and dull appearance.
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Important Findings from Impact Test
➢ Usually an estimate is made of the percentage of the fracture
surface that is cleavage or fibrous fracture.
➢ The test is used for comparing the influence of alloy studies and
heat treatment on notch toughness.
Figure: Photograph of fracture surfaces of A36 steel Charpy Vnotch specimens tested at indicated temperatures (in 0C).
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Important Findings from Impact Test
➢ The notched-bar impact test is most meaningful when conducted
over a range of temperature so that the temperature at which the
ductile-to-brittle transition takes place can be determined.
➢ The energy absorbed decreases with decreasing temperature but
that for most cases the decrease does not occur sharply at a
certain temperature.
➢ The material with the lowest
transition temperature is to be
preferred.
➢ E.g. Steel-A shows higher notch
toughness at room temperature;
yet its transition temperature is
higher than that of steel-B.
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Important Findings from Impact Test
❑ Limitations
➢ The results obtained from notched-bar tests are not readily
expressed in terms of design requirements, since it is not
possible to measure the components of the triaxial stress (σx,
σy, σz) condition at the notch and difficult to correlate Cv data
with service performance.
➢ Moreover, there is no correlation of Charpy data with flaw
size.
➢ In addition, the large scatter inherent in the test may make it
difficult to determine well-defined transition-temperature
curves.
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Significance of DBTT Curve
➢ The transition-temperature behavior of a wide spectrum of
materials falls into the three categories:
➢ Medium and low-strength fcc metals and most hcp metals have
such high notch toughness that brittle fracture is not a problem.
➢ High-strength materials (σ0 >
E/150) have such low notch
toughness that brittle fracture
can occur at nominal stresses in
the elastic range at all
temperatures e.g. High strength
steel, Ti and Al- alloy.
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Significance of DBTT Curve
➢ The notch toughness of low- and medium-strength bcc metals, as
well as Be, Zn, and ceramic materials is strongly dependent on
temperature.
➢ At low temperature the fracture occurs by cleavage while at high
temperature the fracture occurs by ductile rupture.
➢ Thus, there is a transition from notch brittle to notch tough
behavior with increasing temperature.
➢ In metals this transition occurs at 0.1 to 0.2 Tm (Tm = melting
temperature), while in ceramics the transition occurs at about 0.5
to 0.7 Tm.
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Design Philosophy Using Transitiontemperature Curves
➢ Using a DBTT sensitive material, the design strategy is to select a
temperature above which brittle fracture will not occur.
➢ It is preferable to have low DBTT than the operating temperature
with high fracture toughness.
➢ But the realistic problem is
that there is no single criterion
that can define the transition
temperature.
➢ The various definitions of
transition temperature obtained
from an energy vs temperature
curve or a fracture appearance
vs. temperature curve are
illustrated in Fig.
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Design Philosophy Using Transitiontemperature Curves
➢ Criterion-I: A more strict criteria for transition temperature is
fracture transition plastic (FTP ) temperature where above the
temperature the fracture is 100% fibrous (i.e. 0% Brittle fracture).
➢ The probability of brittle fracture is negligible above the FTP.
➢ Criterion-II: @50% Cleavage
➢ Criterion-Ill: Average of upper
and lower shelf temperature.
➢ Criterion-IV: @ Cv = 20 J,
ductility transition temperature;
(after a huge investigation during
WWII, it was established that
brittle fracture would not initiate
if Cv = 20 J).
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Design Philosophy Using Transitiontemperature Curves
➢ Criterion-V: @ 100% Cleavage (NDT= Nil Ductility
Temperature).
➢ Below the NDT the probability of ductile fracture is negligible.
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Metallurgical Factors Affecting Transition
Temperature
❑ Compositional Effects on DBTT
➢ Carbon, (C)↑, DBTT ↑; DBTT ↑ @ 14 °C / 0.1%C
➢ Manganese (Mn)↑, DBTT ↓; DBTT ↓ @ 5 °C / 0.1%Mn
➢ Phosphorus, (P)↑, DBTT ↑: DBTT ↑ @ 7 °C / 0.01%P
➢ Ni ↑ , DBTT ↓; Ni >2% effective
➢ Silicon, (Si)↑, DBTT↑; Si>0.25%,
DBTT ↑
➢ Oxygen, (O)↑, DBTT ↑; -15 °C @
0.001% O2 & 340 °C @ 0.0057%
O2
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Metallurgical Factors Affecting Transition
Temperature
➢ Rimmed steel, with its high iron oxide content, generally shows a
transition temperature above room temperature.
➢ Semi-killed steels, which are deoxidized with silicon, have a
lower transition temperature, while for steels which are fully
killed with silicon plus aluminum the 20 J transition temperature
will be around - 60°C.
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Effect of Grain Size DBTT
➢ Grain size ↓(ASTM grain size number ↑),
DBTT↓
➢ ASTM Grain number @ 5 DBTT = 20 oC
➢ ASTM Grain number @ 20 DBTT = -50 oC
➢ Cooling rate ↑, Grain size ↓, DBTT ↓
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Thank You