1
27-301
Microstructure-Properties
Fracture Toughness:
how to use it, and measure it
Profs. A. D. Rollett, M. De Graef
Processing
Performance
Microstructure Properties
Last modified: 22nd Nov. ‘15
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2
Examinable
Objective
• The objective of this lecture is to build upon the basic
concepts of fracture toughness and stress intensity
introduced in part A. Realistic approaches to fracture
toughness are considered with information on how to
measure toughness.
• Modifications to the Griffith Eq. based on a) plasticity and b)
plastic zone size are explained.
• Fractography is introduced, which relates the morphology of
fracture surfaces to the fracture mode.
• Part of the motivation for this lecture is to prepare the class
for a Lab on the sensitivity of mechanical properties to
microstructure.
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Examinable
Key Points, Definitions
•
•
•
•
•
•
•
•
•
The Griffith equation applies to technological materials, albeit with an adjustment to use
toughness instead of surface energy.
Stress intensity (MPa√m), K, measures the local driving force for crack extension and depends
on the crack geometry (mainly its length).
Fracture Toughness (MPa√m), Kc, is the critical value of stress intensity, above which sudden
(brittle) fracture is expected to occur. This quantity is often written as KIc in order to denote
Mode I plane strain conditions (which are the most conservative).
Toughness (Jm-2), G, scales with modulus, as does strength.
Toughness is highly dependent on material type: the most important issue is the presence
(toughness) or absence (brittleness) of plasticity.
Plasticity makes a large contribution to the energy absorbed in crack propagation because
plastic deformation at the crack tip blunts the tip (lower stress concentration) and
substantially increases the amount of work required per unit crack advance, which we
previously identified as the surface energy, g.
Measurement of toughness uses many methods: two basic methods measure critical stress
intensity in plane strain, KIC, and the energy absorbed in impact (Charpy Test).
Fractography, i.e. classification+quantification of the fracture surfaces, is useful as a
microstructural diagnostic for toughness, in addition to the quantitative measures of
mechanical behavior.
A highly counter-intuitive fact is that a material can exhibit substantial ductility in a tensile
test and yet can fail in a brittle manner in, say, a Charpy test. This phenomenon is known is
notch-sensitivity.
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Questions & Answers
1.
2.
3.
4.
5.
Why does fracture toughness matter in design (as opposed to 6.
strength)? Fracture toughness matters because brittle failure is
often catastrophic and can happen at loads below yielding.
Which of the Griffith Eq and the stress concentration equation
is more conservative as an estimate of breaking strength for
7.
blunt cracks? (slides 7 & 8) The Griffith Eq. is more conservative
for blunt cracks because it does not include stress
concentration.
What difference does the existence of plastic deformation
make to the Griffith Eq.? Plastic deformation substantially
8.
increases the energy cost to lengthen a crack. Is it reasonable
to use fracture toughness to measure surface energies?! (slide
9, e.g.) No, because very few solids are ideally brittle.
What is the plastic zone? What happens inside it? (slides 12,
13) The plastic zone is the region around a crack tip within
which the (deviatoric) stress is high enough to cause plastic
9.
deformation.
How do we calculate the trade-off between strength and
toughness for a pressure vessel? (slide 18) The design
philosophy is leak-before-break, so that means that one
designs to ensure that the load/pressure at which the vessel
yields (leak) is lower than the load at which it bursts (break),
based on an assumed flaw size. The assumed flaw size is
generally controlled by what is possible to detect by NonDestructive Evaluation (e.g. ultrasonics).
How do we correct for the finite size of the plastic zone? (slides
15,16) The finite size of the plastic zone extends the effective
size of the crack. Note that this is a small correction compared
to the increase in toughness from plasticity.
Why does absolute size matter in a fracture toughness
specimen? (slide 23) What are shear lips? If the thickness of
the specimen is small then the material near the sides yields
and the crack changes shape. This change in shape leads to a
deviation of the cracks at ~45°, which is called a shear lip.
What are some characteristic differences between ductile and
brittle fracture surfaces? Brittle fracture surfaces are relatively
smooth (macroscopically) and may exhibit cleavage markings.
Ductile fracture surfaces typically exhibit tearing, often in the
form of small cup-and-cone fractures around 2nd phase
particles.
What connection is there between facture toughness and a
Charpy test? (slide 18, onwards) There is no rigorous,
mathematical relationship. Roughly, the higher the fracture
toughness, the higher the Charpy impact energy. However,
keep in mind that the Charpy test is dynamic i.e. the effective
strain rate is high compared to a fracture toughness test, which
means that a material close to its DBTT may appear ductile
according to fracture toughness but brittle according to Charpy.
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Examinable
Toughnesses in Materials
• Before looking at the influence of microstructure on fracture
toughness, it is useful to review the range of toughnesses
observed in real materials.
• We find that to a first (crude!) approximation, toughness
scales with strength.
• An immediate refinement is to consider the bonding type in
the various classes of materials: metals tend to have simpler
structures with easier dislocation motion, i.e. more energy
absorbed in crack propagation. Ceramics have covalent or
ionic bonding with much higher resistance to dislocation
motion, especially at ambient conditions.
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Map of
toughness vs.
strength
Tough
Glass-like
brittleness
Design with
care below this
line!
[Ashby]
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Use of the Griffith equation
• The Griffith equation can be applied immediately to practical
problems.
• Problem: estimate the fracture strength of a brittle material
(meaning that we can ignore plastic yield) with properties,
E = 100 GPa, g = 1 J.m-2,
2gE
and a crack length of 2.5 µm.
s break =
The answer is sbreak = (2Eg/πc) =
pc
(2.1011.1/π/2.5.10-6)= 160 MPa
• Now it is instructive to compare this result with that from the stress
concentration equation, with the crack tip radius set equal to, say,
8a0:
gE r
sbreak = (Egr/4a0c) = (Eg8a0/4a0c) = (2Eg/c)
s break =
11
-6
(2.10 .1/2.5.10 )= 283 MPa
4a0 c
• So, we see that, even for a fairly sharp crack, the Griffith (energy
balance) equation sets the lower limit on fracture strength.
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Which equation controls?
The paradox: although the Griffith equation (black line) appears to be a necessary but not
sufficient condition for fracture because one also needs for the stress at the crack tip to exceed
the breaking stress (the red line), as a matter of practical experience, it does successfully
predict when fracture will actually occur.
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Examinable
Application to structural materials
• Notwithstanding the previous slides on energy balance (Griffith)
versus stress concentration, the experimental fact is that the Griffith
equation works well for many different materials.
• It works well, not in its literal form with the surface energy
determining the energy consumed, but with an additional energy
term that accounts for the effect of plasticity (crack bridging, phase
transformation….). This was one of Orowan’s (many) contributions
to the field.
s break
2(g surface + g plastic)E
=
pc
• Dieter discusses a method for measuring Gc = gsurface + gplastic in section 11-2.
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Examinable
Toughness
• Recall that we define a stress intensity as K=s c.
• Cracking is defined by K > Kc, where Kc is a critical stress intensity or
fracture toughness, and is a material property.
sbreak = Kc/ (πc)
• We can also define a toughness, Gc, which is given by
sbreak = (EGc/πc)
and allows us to modify (increase) the apparent surface energy to
account for plastic work at the crack tip.
• The toughness can be thought of as the combination of surface
energy and plastic work done at the crack tip noted on the previous
slide. By definition: Gc = 2(gsurface + gplastic)
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11
Examinable
Effect of plasticity, plastic zone
• How important is the additional term?
• In metals, very important: compared to typical surface
energies between 0.5 and 2 J.m-2, the plastic work term
ranges up to 103 J.m-2. Therefore the surface energy term
can be neglected in most metal alloys.
• Again, we cannot use the Griffith equation in its basic
form, even with the addition of the plastic work, however.
• The plasticity results in a plastic zone immediately in front
of the crack tip. This is the zone within which significant
yielding has occurred. Remember that the stress
concentration leads to locally higher stresses and so, only
in the vicinity of the crack will the yield stress be
exceeded.
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Examinable
Plastic Zone
•
The plastic zone is a simple concept to visualize. Within a certain radius of the
crack tip, rp, the yield stress is exceeded and the material
has deformed (consuming energy
thereby and contributing to
toughness). Clearly the lower
the yield strength, the larger
the plastic zone, rp. Actually the
size depends on the ratio of the applied
stress, s, to the yield stress, sy :
rp  s/sy
rp
[Dowling]
See supplemental slides for an eq. for the theoretical elastic stress
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Crack Tip
Different length
scales at which
to view a crack tip
[McClintock, Argon]
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14
Examinable
Effective crack length: plasticity corrections
• An important but slightly counter-intuitive idea is that the effective
crack length is longer than the actual value as a result of the plastic
zone, i.e. ceffective = cactual + rp.
• Size, rp, of the plastic zone?
• Substituting this relationship into the standard Griffith equation,
we obtain:
as
sf  sbreak
s0  syield
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15
Examinable
Plasticity corrections, contd.
• Square both sides and re-arrange:
⇔
⇔
• Re-arrange so that we obtain the following modified
form:
s f pc
K effective =
s  sf  sbreak
2
s0  syield
1æ s ö
1- çç
÷÷
2 è s yield ø
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16
Examinable
Effective crack length, contd.
•
•
This second version is an empirical generalization of the first one: sf is the
fracture strength, s is the operating stress in the material, and syield is the yield
stress of the material. KIc is the plane strain fracture toughness (i.e. the critical
stress intensity). A, B and  are dimensionless constants that depend on crack
geometry (of order unity). In the next slides, B is written as a function of c/a,
the ratio of the (elliptical) crack
(semi-)length, a, to its depth, c.
One can either calculate a fracture strength for a given set of parameters,
calculate a maximum operating stress similarly, or, determine whether the
fracture toughness dictated by the quantities on the RHS is higher than the
actual fracture toughness of the material.
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Example
problem
Courtney, p. 431
Examinable
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Measuring Fracture Toughness
• How do we measure fracture toughness?
• Two examples:
A - measure the critical stress intensity (KIC) in plane
strain by measuring the stress required to propagate a
sharp crack.
B - measure the energy absorbed in a rapid fracture of a
bar - the Charpy test.
• The first method measures a quantity corresponding to
the values in the equations discussed (but a pre-existing
crack is used).
• The second test is a more macroscopic test but it includes
the effect of crack nucleation (which may be difficult
enough to raise the effective toughness).
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Compact Tension test
•
The load is increased until crack propagation starts: for a large enough
specimen, the stress intensity at this point is the critical stress intensity, KIC.
P is the load, t is the specimen thickness, b is the distance from the loading
point to the right-hand face, and Fp is a function of the crack geometry.
Fatigue crack;
grown before
Dowling
the fracture
expt. in order
to obtain a
sharp crack
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Examinable
Charpy Test
• The Charpy test uses a square bar
with a small notch in it.
• The further the pendulum swings
after breaking the specimen, the
less energy was absorbed in the
impact, and vice versa.
• Higher toughness results in higher
energy absorbed.
• The test is effectively a dynamic
test because the strain rates are
much higher than in a fracture
toughness test.
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21
Examinable
Charpy - fracture toughness correlation
• Here is an example of a correlation between Charpy impact
energy and critical stress intensity in PMMA-based bone
cements, from Lewis and Mladsi, Biomaterials 2000.
see also: Determination of the
fracture toughness of a low alloy
steel by the instrumented Charpy
impact test
Author(s): Rossoll A, Berdin C,
Prioul C, International Journal Of
Fracture 115, 205-226 (2002).
See also reports by Rolfe &
Barsom, ASTM STP 466 (1970) and
by Rolfe & Novak, ASTM STP 464
(1970).
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22
Examinable
Fractography
• Fractography is the practice of characterizing fracture
surfaces.
• Surface preparation is not needed - one needs to examine
the surfaces as fractured, which means that it should be
done promptly so as to avoid changes from oxidation,
corrosion etc.
• The rough, irregular nature of fracture surfaces means that
optical microscopy is of little use.
• Scanning electron microscopy is most useful in fractography.
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Examinable
Sample scale
• Example of high strength steel from a compact tension
test.
Dowling
Crack
propagation
Shear
Lips
Crack tip
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Examinable
Grain scale
• These micrographs contrast the appearance of ductile and
brittle fractures at the microstructural scale.
Dowling
Ductile (tearing)
Brittle (cleavage)
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Examinable
Ductile fracture
Cup and cone fracture - each
dimple is a void (which may or
may not have a particle in it)
• In contrast to brittle
fracture, which is a
cleavage process
(and, in crystalline
materials typically
follows low index
planes), ductile
fracture only occurs
after much plastic
deformation.
Dieter
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Summary (part B)
• The Griffith equation has been extended to technological
materials.
• Fracture Toughness scales with modulus, as does strength.
• Fracture Toughness is highly dependent on material type: the
most important issue is the presence (toughness) or absence
(brittleness) of plasticity.
• Plasticity makes a large contribution to the energy absorbed
in crack propagation.
• Measurement methods contrasted between KIC and impact
testing (Charpy).
• Fractography introduced as a diagnostic for toughness, in
addition to the quantitative measures.
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27
Examinable
Case Study:
Failure Analysis of a Rocket Motor Case
A rocket motor case was made of a material that had a yield strength of 215 ksi (= 1485 MPa) and
a KIC of 53 ksi(in)1/2 (= 58 MPa.m3/2) and it failed at a stress of 150 ksi. Examination of the
failed component showed that there was an elliptical surface crack with a depth of 0.039
inches (= 0.99 mm) and a length of 1.72 in (= 43.7 mm). Could this flaw have been
responsible for the failure?
Answer:
The value of f(c/a) (=B) for this flaw is 1.38. Rearranging the equation that relates fracture
toughness to yield strength and operating stress, we obtain:
f (c a ) - 0.212(s s y )
2
s fracture =
1.38 - 0.212(s s y )
2
K IC =
K IC
1.20pc
1.20pc
Now we estimate the fracture stress iteratively by substituting values of KIC and the crack depth,
c, (not the half-length!) and assume the operating stress value, s, of 150 ksi, in order to
estimate the RHS; then we compare the value on the RHS with the known fracture stress on
the LHS. The answer turns out to be 156 ksi, which is not far off the actual fracture stress of
150 ksi. Substituting 156 ksi as the operating stress value, s, into the RHS produces 156 ksi
as the computed fracture stress. At this point the iteration has converged well enough for
our purposes. The close agreement between the actual and the computed fracture stresses
suggests that the flaw was very likely to have been the cause of the failure.
Source: Courtney: Mechanical Behavior of Materials, Ch. 9.
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References
• Materials Principles & Practice (1991), Butterworth Heinemann, Edited by
C. Newey & G. Weaver.
• G.E. Dieter (1989), Mechanical Metallurgy, McGraw-Hill, 3rd Ed.
• T.H. Courtney (2000). Mechanical Behavior of Materials. Boston, McGrawHill.
• R.W. Hertzberg (1976), Deformation and Fracture Mechanics of Engineering
Materials, Wiley.
• J.F. Knott (1973), Fundamentals of Fracture Mechanics, Wiley.
• N.E. Dowling (1998), Mechanical Behavior of Materials, Prentice Hall.
• M.F. Ashby and H.J. Frost, Deformation-Mechanism Maps: The Plasticity
and Creep of Metals and Ceramics, Pergamon, ISBN 0080293379.
• D.J. Green (1998). An Introduction to the Mechanical Properties of
Ceramics, Cambridge Univ. Press, NY.
• A.H. Cottrell (1964), The Mechanical Properties of Matter, Wiley, NY.
• J.A. Collins (1981), Failure of Materials in Mechanical Design, Wiley, NY.
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Current Articles
•
•
•
•
•
•
Nanostructured diamond-TiC composites with high fracture toughness, Wang, Haikuo; He,
Duanwei; Xu, Chao; Tang, Mingjun; Li, Yu; Dong, Haini; Meng, Chuanmin; Wang, Zhigang; Zhu,
Wenjun, Journal of Applied Physics, 113, pp. 043505-043505-4 (2013).
Fracture toughness of alpha- and beta-phase polypropylene homopolymers and random- and
block-copolymers Author(s): Chen HB, Karger-Kocsis J, Wu JS, et al., Source: POLYMER Volume:
43 Issue: 24 Pages: 6505-6514 Published: NOV 2002
EVALUATION OF DYNAMIC FRACTURE-TOUGHNESS PARAMETERS BY INSTRUMENTED CHARPY
IMPACT TEST,
Author(s): KOBAYASHI T, YAMAMOTO I, NIINOMI M, Source: ENGINEERING FRACTURE
MECHANICS Volume: 24 Issue: 5 Pages: 773-782 Published: 1986
On the effects of irradiation and helium on the yield stress changes and hardening and nonhardening embrittlement of similar to 8Cr tempered martensitic steels: Compilation and analysis
of existing data , Author(s): Yamamoto T, Odette GR, Kishimoto H, et al.
Source: JOURNAL OF NUCLEAR MATERIALS Volume: 356 Issue: 1-3 Pages: 27-49 Published:
SEP 15 2006
The influence of ductile tearing on fracture energy in the ductile-to-brittle transition
temperature range , Author(s): Hausild P, Nedbal I, Berdin C, et al. Source: MATERIALS SCIENCE
AND ENGINEERING A-STRUCTURAL MATERIALS PROPERTIES MICROSTRUCTURE AND
PROCESSING Volume: 335 Issue: 1-2 Pages: 164-174 Published: SEP 25 2002
Correlation of microstructure and fracture properties of API X70 pipeline steels, Author(s):
Hwang B, Kim YM, Lee S, et al., Source: METALLURGICAL AND MATERIALS TRANSACTIONS APHYSICAL METALLURGY AND MATERIALS SCIENCE Volume: 36A Issue: 3A Pages: 725739 Published: MAR 2005
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Supplemental Slides
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Theoretical Elastic Stress
• The stress distribution near a sharp crack tip
in a plate in a linear elastic solid is given by
the following equations, reproduced from
Dieter 11-3.
r
p
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