1 27-301 Microstructure-Properties Fracture Toughness: how to use it, and measure it Profs. A. D. Rollett, M. De Graef Microstructure Properties Processing

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127-301

Microstructure-PropertiesFracture Toughness:

how to use it, and measure itProfs. A. D. Rollett, M. De Graef

Microstructure Properties

ProcessingPerformance

Last modified: 22nd Nov. ‘15


2Objective

• 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.

Examinable


3Key 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.

Examinable


4Questions & Answers

1. Why does fracture toughness matter in design (as opposed to strength)? Fracture toughness matters because brittle failure is often catastrophic and can happen at loads below yielding.

2. Which of the Griffith Eq and the stress concentration equation is more conservative as an estimate of breaking strength for blunt cracks? (slides 7 & 8) The Griffith Eq. is more conservative for blunt cracks because it does not include stress concentration.

3. What difference does the existence of plastic deformation make to the Griffith Eq.? Plastic deformation substantially 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.

4. 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 deformation.

5. 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 Non-Destructive Evaluation (e.g. ultrasonics).

6. 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.

7. 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.

8. 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.

9. 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.


5Toughnesses 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.

Examinable


6Map of

toughness vs. strength

[Ashby]

Glass-likebrittleness

Tough

Design with care below this line!


7Use 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, and a crack length of 2.5 µm.The answer is sbreak = √(2Eg/πc) = √(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:sbreak = √(Egr/4a0c) = √(Eg8a0/4a0c) = √(2Eg/c)

√(2.1011.1/2.5.10-6)= 283 MPa• So, we see that, even for a fairly sharp crack, the Griffith (energy

balance) equation sets the lower limit on fracture strength.


8Which 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.


9Application 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.

• Dieter discusses a method for measuring Gc = gsurface + gplastic in section 11-2.

Examinable


10Toughness

• 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)

Examinable


11Effect 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.

Examinable


12Plastic 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, , to the yield stress, y :

rp /y

[Dowling]

rp

See supplemental slides for an eq. for the theoretical elastic stress

Examinable


13Crack Tip

[McClintock, Argon]

Different length scales at whichto view a crack tip


14Effective 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

Examinable


15Plasticity corrections, contd.

• Re-arrange so that we obtain the following modified form:

s0 syield

• Square both sides and re-arrange:

⇔⇔

s sf sbreak

Examinable


16Effective 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.

Examinable


17

Example problem

Courtney, p. 431Examinable


18Measuring 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).


19Compact 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.

DowlingFatigue crack;grown beforethe fracture expt. in order to obtain a sharp crack


20Charpy Test

Examinable

• 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.


21Charpy - 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).

Examinable

https://www.vpn.cmu.edu/+CSCO+dh756767633A2F2F6E6363662E7666767861626A79727174722E70627A++/full_record.do?product=WOS&search_mode=CitationReport&qid=2&SID=4CDgDmpJdCk7MApnbgG&page=7&doc=66




22Fractography

• 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.

Examinable


23Sample scale

• Example of high strength steel from a compact tension test.

Crack tip

CrackpropagationShearLips

Examinable

Dowling


24Grain scale

• These micrographs contrast the appearance of ductile and brittle fractures at the microstructural scale.

Dowling

Brittle (cleavage)Ductile (tearing)

Examinable


25Ductile fracture

• 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.

Cup and cone fracture - each dimple is a void (which may or may not have a particle in it)

Examinable

Dieter


26Summary (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.


27Case Study:

Failure Analysis of a Rocket Motor CaseA 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:

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, , 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, , 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.

Examinable


28References

• 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, McGraw-

Hill.• 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.


29Current 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 non-hardening 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 A-PHYSICAL METALLURGY AND MATERIALS SCIENCE Volume: 36A Issue: 3A Pages: 725-739 Published: MAR 2005

























30Supplemental Slides


31Theoretical 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.

rp

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1 27-301 Microstructure-Properties Fracture Toughness: how to use it, and measure it Profs. A. D. Rollett, M. De Graef Microstructure Properties Processing