In-situ Testing · Microcantilever bending
Microcantilever bending: stiffness and fracture toughness at the microscale
Push down on a tiny FIB-cut beam, watch where it is stressed most, then notch it and watch it crack.
Where is a microcantilever stressed most when you bend it?
An indenter tip pushes near the free end of a silicon beam cut with a focused ion beam (FIB).
Try it: set the tip offset to 1.2 µm: k reads 37% high. Halve h: k drops eight times.
How do you measure fracture toughness with a notched microcantilever?
Notch the beam near the root, push at distance L, and read the load at which it snaps.
Try it: switch Si to WC: the pop load rises from 293 to 2,052 µN.
When is a microcantilever toughness value valid?
K counts only if the plastic zone at the crack tip is small next to the beam.
Try it: press W (micro): the rule needs 58 µm, far more than the 4 µm beam.
What to take away
The root carries the stress
The moment is largest where the beam meets the bulk. Its top surface is in tension.
Toughness from one load
KIQ = (PcL/bh1.5) f(a/h). Measure a and L in the SEM after the test.
Ceramics yes, metals rarely
Brittle ceramics pass the size check at micron sizes. Most metals fail it and need J-integral or CTOD methods.
More detail: the equations, assumptions and limits
Beam theory
A cantilever of length ℓ from the root to the load point, loaded by P, bends to δ = Pℓ³/(3EI), so the stiffness is k = 3EI/ℓ³. I is the second moment of area of the cross-section (how its area is spread away from the neutral axis). For a rectangle I = bh³/12. The moment at the root is M = Pℓ, and the stress at the top surface is σ = Mc/I, where c is the distance from the neutral axis to the top. For a rectangle c = h/2, which gives σ = 6Pℓ/(bh²).
The pentagon here is a rectangle with a triangle below it, with 45° walls, the shape left when the beam is undercut with the ion beam at an angle. The page finds its area, neutral axis and I from the polygon. The neutral axis sits above the middle, so c is smaller than h/2. But I falls even more, so for the same b and h the top stress is higher than in a rectangle, and the stress at the bottom point is higher still.
Support compliance and the tip position
The block the beam is attached to also bends a little under the root moment, so the measured deflection is larger than the beam alone gives, and E from k comes out low. Li et al. (2019), for example, measured about 370 GPa on W-1%Ta beams against a literature value of about 410 GPa. The indenter tip is also placed by eye in the SEM: because k goes as 1/ℓ³, a small error in ℓ gives three times the relative error in k.
Assumptions
Linear elastic, isotropic, small deflection (Euler-Bernoulli beam, no shear). Silicon is anisotropic; E = 169 GPa is the value along <110>, used by Li et al. (2019). For other directions see the elastic anisotropy page.
The fracture toughness equation
For a rectangular beam with a straight notch of depth a from the top, the page uses the widely quoted factor from Di Maio and Roberts (2005): KIQ = (P L / (b h1.5)) f(a/h) with f = 1.46 + 24.36(a/h) − 47.21(a/h)² + 75.18(a/h)³. Brinckmann et al. (2017) use the same form. The 2017 erratum to Di Maio and Roberts corrects their second moment of area for the pentagonal section (their Eq. 3), not this polynomial. As a check, the page compared it with the handbook solution for an edge crack in a bar in pure bending (Tada, Paris and Irwin): 6√(πa/h) F(a/h) with F = 1.122 − 1.40α + 7.33α² − 13.08α³ + 14.0α⁴. The two agree to within 0.7% from a/h = 0.1 to 0.5 and 2.6% at 0.6, so the slider stops at 0.5.
For the pentagon, Matoy et al. (2009) write K = (P L c / I) √(πa) F with a factor fitted to finite element results for that shape. This page uses the same form with the pure-bending factor F above, and I and c of the pentagon. That is an approximation made here, not Matoy's fitted factor; use the published factor for real data.
The subscript Q means "conditional": a FIB notch is not a sharp fatigue crack, and a notch with a round root or a bridge of uncut material reads high (Brinckmann et al. 2017).
The size rule
ASTM E399 accepts a K value as plane-strain KIc only if the crack length a, the thickness b and the ligament h − a are all at least 2.5 (K/σy)². The drawn plastic zone is Irwin's plane-strain estimate, rp = (1/3π)(K/σy)². Wurster, Motz and Pippan (2012) tested micron-sized notched tungsten beams, found this condition not met, and used J-integral and crack tip opening displacement (CTOD) methods instead. The yield strengths for Si, TiN and WC are an assumed 5 GPa, because they break before they yield at room temperature. The verdict depends on that guess: at the default h = 4 µm, Si passes for any value above about 1.2 GPa, but TiN passes only above about 4.2 GPa. The W case uses the yield stress of about 5.2 GPa that Li et al. (2019) measured on micro-beams, and a K of 25 MPa√m, because they found micro-beam values (from elastic-plastic analysis) about five times the bulk value of about 5 MPa√m.
Questions people ask
What is a microcantilever bending test?
A beam a few microns across is cut out of a surface with a focused ion beam, left attached at one end. A nanoindenter or a tip in an SEM pushes down near its free end while load and displacement are recorded. It gives the elastic modulus, strength or, with a notch, fracture toughness of a small volume, such as one grain or one coating.
Why are some microcantilevers pentagonal?
To free a beam from the bulk you have to cut under it. With the ion beam at an angle, the undercut leaves two sloped walls, so the section becomes a rectangle on top of a triangle. It is quicker to make than a fully rectangular beam, but the neutral axis moves up and it needs its own geometry factor.
What does KIQ mean?
It is a conditional mode I fracture toughness. The Q says that not every condition of the standard (a sharp fatigue crack, enough size) was met. It becomes KIc only if those checks pass.
Can you measure fracture toughness of metals with microcantilevers?
Usually not with the simple K equation. The plastic zone in a tough metal is often larger than the whole beam, so the beam bends and the crack blunts instead of popping. Groups use the J-integral, often from the unloading stiffness during the test, or the crack tip opening displacement instead.
How sharp does the FIB notch need to be?
As sharp as the ion beam allows, usually a notch root of tens of nanometres. A blunt notch needs a higher load to start the crack, so KIQ comes out too high. Gallium implanted near the notch can also change the material there.
Why not just measure toughness from indentation cracks?
Indentation crack length methods rely on fitted constants and a complex residual stress field, and they scatter a lot. A notched beam has a known geometry and a known K equation. It takes longer to make but is closer to a standard test.
On this site: Fracture toughness · Micropillar compression · FIB specimen prep artifacts · Indentation beyond hardness · Elastic anisotropy
References
Show the 11 references
- D. Di Maio and S. G. Roberts, Measuring fracture toughness of coatings using focused-ion-beam-machined microbeams, Journal of Materials Research 20, 299 to 302 (2005). doi:10.1557/JMR.2005.0048
- K. Matoy, H. Schönherr, T. Detzel, T. Schöberl, R. Pippan, C. Motz and G. Dehm, A comparative micro-cantilever study of the mechanical behavior of silicon based passivation films, Thin Solid Films 518, 247 to 256 (2009).
- S. Wurster, C. Motz and R. Pippan, Characterization of the fracture toughness of micro-sized tungsten single crystal notched specimens, Philosophical Magazine 92, 1803 to 1825 (2012). doi:10.1080/14786435.2012.658449
- F. Iqbal, J. Ast, M. Göken and K. Durst, In situ micro-cantilever tests to study fracture properties of NiAl single crystals, Acta Materialia 60, 1193 to 1200 (2012). doi:10.1016/j.actamat.2011.10.060
- S. Brinckmann, K. Matoy, C. Kirchlechner and G. Dehm, On the influence of microcantilever pre-crack geometries on the apparent fracture toughness of brittle materials, Acta Materialia 136, 281 to 287 (2017). doi:10.1016/j.actamat.2017.07.014
- B. N. Jaya, C. Kirchlechner and G. Dehm, Can microscale fracture tests provide reliable fracture toughness values? A case study in silicon, Journal of Materials Research 30, 686 to 698 (2015): KIc about 0.8 MPa√m.
- B.-S. Li, T. J. Marrow, S. G. Roberts and D. E. J. Armstrong, Evaluation of fracture toughness measurements using chevron-notched silicon and tungsten microcantilevers, JOM 71, 3378 to 3389 (2019). doi:10.1007/s11837-019-03696-1: Si E = 169 GPa; W E = 410 GPa; bulk W KIc about 5 MPa√m, micro-beam KQ about five times higher, yield about 5.2 GPa, measured modulus about 370 GPa.
- J. Buchinger, L. Löfler, J. Ast, A. Wagner, Z. Chen, J. Michler, Z. L. Zhang, P. H. Mayrhofer, D. Holec and M. Bartosik, Fracture properties of thin film TiN at elevated temperatures, Materials & Design 194, 108885 (2020). The page uses a typical room-temperature value for TiN films, 2.9 MPa√m.
- L. Ortiz-Membrado, N. Cuadrado, D. Casellas, J. J. Roa, L. Llanes and E. Jiménez-Piqué, Measuring the fracture toughness of single WC grains of cemented carbides by means of microcantilever bending and micropillar splitting, International Journal of Refractory Metals and Hard Materials 98, 105529 (2021): 5.6 MPa√m.
- Elastic moduli: TiN 450 to 590 GPa (G. Abadias, Surface and Coatings Technology 202, 2223 to 2235 (2008)), 450 GPa used; WC 530 to 700 GPa (A. S. Kurlov and A. I. Gusev, Tungsten Carbides: Structure, Properties and Application in Hardmetals, Springer (2013)), 700 GPa used.
- H. Tada, P. C. Paris and G. R. Irwin, The Stress Analysis of Cracks Handbook, 3rd ed., ASME Press (2000): edge crack in pure bending; ASTM E399, Standard Test Method for Linear-Elastic Plane-Strain Fracture Toughness of Metallic Materials, ASTM International: the 2.5(K/σy)² size rule.
BibTeX
@misc{tripathy2026microcantilever,
author = {Tripathy, Manisha},
title = {Microcantilever Bending Lab},
year = {2026},
howpublished = {\url{https://untetheredatom.com/insitu/microcantilever-bending}},
note = {Interactive web tool}
}