Mechanical Behavior · Fracture

Fracture toughness: when does a crack run?

Start with 7075 aluminium below and drag the stress up until the crack runs. Then pick glass and see how little stress it takes.

1 K against KIc 2 Griffith energy balance 3 Plastic zone and ASTM E399 4 Largest tolerable crack

What is the stress intensity factor K?

A crack concentrates stress at its tip. One number, the stress intensity factor K = Yσ√(πa), sets how strong that stress field is (σ is the applied stress, a the crack length, Y a shape factor near 1). The crack runs when K reaches the material's fracture toughness KIc.

Left: the plate, crack drawn on a log scale, and K against KIc. Right: opening stress σyy = K/√(2πr)·f(θ) in a window ±a/4 around the tip; the dashed line is where σyy reaches the yield strength.
Material (KIc, yield: Callister Table 8.1)
K = Yσ√(πa)
KIc
Stress that breaks it
Crack that breaks it
Try it: 7075-T651 with a 5 mm edge crack: raise the stress. The crack runs at about 171 MPa, a third of its 495 MPa yield strength. Switch to glass: now 5.3 MPa is enough.

Why do brittle solids break far below their theoretical strength?

Griffith's energy balance. A growing crack frees stored elastic energy (πσ2a2/E per metre of crack front, for a centre crack of length 2a) but costs energy to make new surfaces (2a·Gc). The total has a peak. Past the peak, growing is downhill and the crack runs by itself.

Energy per metre of crack front against crack half-length a. Blue: cost of new surface. Orange: elastic energy released. Dark: the sum. The peak is the critical length a*. The ball rolls downhill.
Gc (energy per m2 of crack)
Critical length a*
Strength with this crack
Ideal strength ≈ E/10
Try it: glass at 50 MPa: a* is about 72 µm. Double the stress to 100 MPa and a* falls to a quarter, about 18 µm. The message below the readouts shows that a crack only about 4 nm long would already bring glass below its ideal strength.

How big is the plastic zone, and when is a KIc test valid?

In a metal the tip stress is capped by yielding, so a small plastic zone forms. Its size is rp = (1/2π)(K/σy)2 in plane stress (thin sheet) and about one third of that in plane strain (thick plate, where the material around the tip stops it from thinning). ASTM E399 accepts a result as the plane-strain KIc only if a, B and W−a are all at least 2.5(K/σy)2.

Left: plastic zone at the tip, to scale (von Mises shapes; ticks mark rp and rp/3). Right top: the crack front across the thickness B, same scale both ways (sketch of the zone depth). Right bottom: the three E399 size checks.
rp plane stress
rp plane strain
Try it: 7075-T651 needs B of at least 5.9 mm; drag B below that and the verdict turns invalid. Set B to 6 mm and switch to Ti-6Al-4V: now it is too thin (it needs 9.1 mm). Lower KQ: both zones shrink with K2.

How big a crack can each material tolerate?

Turn K = Yσ√(πa) around: the largest crack a part can carry at stress σ is ac = (1/π)(KIc/Yσ)2. It goes with the square of the toughness, so a ten times tougher material tolerates a hundred times longer crack.

Largest tolerable crack ac at your stress (log scale). A grey row: the stress is above that material's yield strength, so it yields before a crack of any size matters.
MaterialKIc (MPa√m)ac
Try it: at 100 MPa with an edge crack, glass fails from a 14 µm flaw, 7075-T651 carries 15 mm and 4340 steel (tempered at 260 °C) 63 mm. PMMA turns grey: 100 MPa is above its yield strength.

What to take away

K sets the tip stressOne number, K = Yσ√(πa), describes the stress field at every crack tip. Double the crack and K grows by √2.
KIc is a material propertyThe crack runs when K reaches KIc. Glass is near 0.75 MPa√m, high-strength steels 50 to 90 MPa√m.
Flaws set brittle strengthA crack a few micrometres long drops glass from about 7 GPa ideal strength to under 200 MPa. Griffith's energy balance explains why.
Thickness mattersThin plates show a higher, thickness-dependent toughness. Only a specimen thick enough for plane strain gives a valid KIc.
More detail: where the formulas come from and where they stop working

The crack-tip field. For an opening (mode I) crack in a linear elastic solid, the stresses near the tip are σij = K/√(2πr)·fij(θ). For the opening stress, fyy = cos(θ/2)[1 + sin(θ/2)sin(3θ/2)]. This is the leading term only. It describes the field well for r much smaller than a; far from the tip the stress returns to the applied σ.

Shape factor Y. Y = 1 is exact for a centre crack of length 2a in an infinite plate and Y = 1.12 for a short edge crack in a semi-infinite plate. Finite widths, bending, and surface or corner cracks need their own Y from a handbook (Tada, Paris and Irwin).

Griffith. For a centre crack of length 2a in a plate under plane stress, the released elastic energy per unit thickness is πσ2a2/E. Setting its derivative equal to the surface cost 2Gc gives σf = √(EGc/πa), which is the same as K = KIc with KIc2 = EGc. For an ideally brittle solid Gc = 2γs, twice the surface energy. In metals and polymers Gc is hundreds to thousands of times larger, because plastic work is done at the tip (Orowan). Here Gc is taken from the measured KIc, so it includes that work. In plane strain E becomes E/(1−ν2).

Plastic zone. Setting σyy = σy on the crack line gives r = (1/2π)(K/σy)2 (the first Irwin estimate). Allowing for the load the yielded material can no longer carry roughly doubles it. In plane strain the stress along the crack front raises the stress needed to yield, and the zone is often taken as one third of the plane-stress value. The drawn shapes use the von Mises rule on the elastic field with ν = 0.3; they are first estimates, not a plasticity solution.

ASTM E399. A test gives a candidate value KQ. It is reported as KIc only if, among other checks, a, B and W−a are at least 2.5(KQ/σys)2 and the maximum load is not more than 1.10 times the load used to find KQ. That keeps the plastic zone about 50 times smaller than the specimen. Recent editions of E399 have revised some of these checks, so read the edition you test to.

Limits. Linear elastic fracture mechanics fails when the plastic zone is not small compared with the crack and the part (then J-integral or CTOD methods apply). KIc values here are room-temperature handbook values; real values depend on heat treatment, orientation, temperature and loading rate. Ceramics and glass also crack slowly in moist air below KIc.

Questions people ask

What is fracture toughness?

It is the value of the stress intensity factor K at which a crack starts to grow fast. Its unit is MPa√m. A high value means a part can carry a longer crack, or a higher stress, before it breaks.

What is the difference between K and KIc?

K is the load on a crack: it depends on the stress, the crack length and the shape. KIc is the material's limit, measured in plane strain in mode I (opening). The crack runs when K reaches KIc.

Why is glass so much weaker than its theoretical strength?

Its surface carries tiny cracks, a few micrometres long, from handling and chemical attack. With KIc near 0.75 MPa√m, a 5 µm crack breaks it at about 170 MPa, far below the ideal strength of about E/10 ≈ 7 GPa. Fresh, flawless glass fibres come much closer to the ideal value.

Why are stronger steels often less tough?

Raising the yield strength shrinks the plastic zone at the crack tip, so less energy is used up in plastic work. In Callister's data, 4340 steel tempered at 260 °C has a yield strength of 1640 MPa and KIc of 50 MPa√m; tempered at 425 °C it has 1420 MPa and 87.4 MPa√m.

What does plane strain fracture toughness mean?

In a thick specimen, the material around the tip stops it from thinning, so the tip is in plane strain. The measured toughness then stops changing with thickness. That lowest value is KIc. Thinner sheets give a higher value that depends on thickness.

How thick does a KIc specimen need to be?

At least 2.5(KIc/σys)2 under ASTM E399. For 7075-T651 (24 MPa√m, 495 MPa) that is 5.9 mm; for Ti-6Al-4V (55 MPa√m, 910 MPa) it is 9.1 mm.

How do you calculate the critical crack length?

ac = (1/π)(KIc/Yσ)2. For 7075-T651 at 100 MPa with an edge crack (Y = 1.12): (1/π)(24/112)2 m ≈ 15 mm.

Is fracture toughness the same as toughness from a tensile test?

No. The area under a tensile curve is energy per volume for an uncracked bar. Fracture toughness is resistance to a crack that is already there. A material can be strong and ductile in a tensile test and still fail from a crack at low stress.

Related: Reading a tensile curve (yield strength and ductility) · Fatigue and S-N curves (how cracks grow to ac) · Strengthening mechanisms · Dislocations and Burgers vectors · Scratch test critical loads (coatings cracking and flaking) · Scratch vs indentation hardness · Indentation beyond hardness (toughness from indent cracks) · FIB specimen prep artifacts (milling notches and micro-cantilevers)

References

Show the 8 references
  1. W. D. Callister and D. G. Rethwisch, Materials Science and Engineering: An Introduction, Wiley: Table 8.1 (room-temperature yield strength and plane strain fracture toughness, from ASM Advanced Materials and Processes, 1990), Appendix B, Table B.2 (elastic modulus: 7075 aluminium 71, 4340 steel 207, Ti-6Al-4V 114, soda-lime glass 69 and PMMA 2.24 to 3.24 GPa) and Table 12.5 (alumina, 393 GPa). Source of every KIc, yield strength and E on this page. Ranges (glass 0.7 to 0.8, alumina 2.7 to 5.0, PMMA 0.7 to 1.6 MPa√m, PMMA yield 53.8 to 73.1 MPa) are used at their middle.
  2. A. A. Griffith, The phenomena of rupture and flow in solids, Philosophical Transactions of the Royal Society A 221, 163 to 198 (1921). doi:10.1098/rsta.1921.0006
  3. G. R. Irwin, Analysis of stresses and strains near the end of a crack traversing a plate, Journal of Applied Mechanics 24, 361 to 364 (1957).
  4. T. L. Anderson, Fracture Mechanics: Fundamentals and Applications, 4th ed., CRC Press (2017): chapter 2 (crack-tip fields, Griffith, plastic zone) and chapter 7 (KIc testing).
  5. ASTM E399, Standard Test Method for Linear-Elastic Plane-Strain Fracture Toughness of Metallic Materials, ASTM International (size requirement 2.5(KQ/σys)2).
  6. H. Tada, P. C. Paris and G. R. Irwin, The Stress Analysis of Cracks Handbook, 3rd ed., ASME Press (2000). Shape factors Y.
  7. M. F. Ashby and D. R. H. Jones, Engineering Materials 1, 4th ed., Butterworth-Heinemann (2012): chapters 13 to 15 (fast fracture, toughness, case studies).
  8. R. W. Hertzberg, R. P. Vinci and J. L. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 5th ed., Wiley (2012): chapter 8 (plastic zone, thickness effect on toughness).
Cite this page: Tripathy, Manisha. “Fracture Toughness Lab.” untethered atom, 2026, https://untetheredatom.com/mechanical-behavior/fracture-toughness.
BibTeX 
@misc{tripathy2026fracturetoughness,
  author = {Tripathy, Manisha},
  title  = {Fracture Toughness Lab},
  year   = {2026},
  howpublished = {\url{https://untetheredatom.com/mechanical-behavior/fracture-toughness}},
  note   = {Interactive web tool}
}
Last updated 24 September 2026.