Mechanical Behavior · Schmid factor and slip systems

Schmid factor and slip systems: which slip system goes first

Start with the crystal below: drag φ and watch the shear stress on the slip plane rise to its highest value at 45° and fall again. Then pick any tensile axis in widget 2 and see all 12 slip systems ranked.

What is the Schmid factor?

A crystal slips when the shear stress on one slip plane, along one slip direction, reaches a fixed value, the critical resolved shear stress (CRSS, τc). Pulling with stress σ along the axis gives that plane a shear stress τ = σ cos φ cos λ. The product m = cos φ cos λ is the Schmid factor.

The crystal. Red: the slip plane. Blue n: its normal, at φ from the axis. Red arrow: the slip direction, at λ from the axis. Drag the picture to turn it.
Yield stress σy = τc/m against φ. Curve: λ = 90° − φ, the lowest possible. Above it: other λ. Red band: angles that slip at your σ.

Crystal and load

Try it: keep λ at its lowest value (90° − φ; the slider stops there) and drag φ: the yield stress is lowest, 2 × CRSS, at φ = 45°. Then press Ramp stress and watch the top half slide along the slip direction.

Which slip system activates first in FCC, BCC and HCP?

A real crystal has many slip systems (a slip plane plus a slip direction in it). Each has its own φ and λ to the same tensile axis, so each has its own Schmid factor. The one that reaches its CRSS first slips first. Click in the triangle to choose the tensile axis in crystal coordinates.

Tensile axis in the crystal (stereographic triangle). Colour: Schmid factor m of the system that goes first, contours every 0.05. Click or drag to move the axis.
Slip systems ranked by the stress that starts slip, σ = τc/m. Bar length: m. Top row goes first.

Crystal

Structure
Load
Colour the triangle by

Tensile axis

Jump to
Try it: FCC: click the [213] chip: one system has m = 0.467 and the next only 0.350 (single slip). Click [419] and press Stretch: the axis turns toward the slip direction [011] until it meets the [001]-[111] edge near [113], where a second (conjugate) system ties; the axis then runs along the edge toward [112], the end point of tension in FCC. Then pick HCP and click near [0001]: basal slip has m near 0, so the crystal twins in tension, or needs ⟨c+a⟩ slip in compression.

Which grains yield first in a polycrystal?

In a polycrystal each grain has its own orientation, so each has its own highest Schmid factor. Under one uniaxial stress the grains with m near 0.5 reach their CRSS first. The map is like an EBSD map of randomly oriented grains.

Grain map (random orientations). Full colour: grain has yielded at the present σ. Faded: still elastic. Arrow badge: load direction. Hover or tap a grain to read its numbers.
Share of grains yielded against σ. Lines: first possible grain (m = 0.5), Sachs average of 1/m, and Taylor's M = 3.06 for FCC.

Polycrystal

Structure
Colour grains by
Hover or tap a grain.
Try it: FCC: slide σ up slowly from 1.30 MPa. Nothing yields below 2 × CRSS (1.30 MPa); the bright yellow grains (m near 0.5) go first; by the Taylor line (1.99 MPa) nearly every grain would be slipping if it were free. Switch to HCP Mg: no grain yields below 1.0 MPa (2 × the basal CRSS) and about half have yielded by 1.5 MPa, but grains with their c-axis near the load need 10 MPa or more, because their basal Schmid factor is small.

What to remember

τ = m σ, m ≤ 0.5The shear stress on a slip system is the Schmid factor times the applied stress. m is largest, 0.5, when the plane normal and slip direction both sit at 45° to the load.
Lowest yield stress winsWith many systems, the one with the lowest τc/m slips first. In FCC and BCC all systems of a family share one CRSS, so the highest m wins.
HCP depends on orientationIn magnesium at room temperature basal slip is about 80 times easier than the other slip systems. A crystal loaded near its c-axis cannot use basal slip, so it twins or needs very high stress.
Polycrystals need more stressRandom grains have an average 1/m of about 2.23 (computed here from 20 000 grains). Grains must also fit together, so Taylor's factor 3.06 (FCC) relates polycrystal yield to the CRSS.
More detail: the derivation and the limits of the model

Where cos φ cos λ comes from. A force F along the axis of a rod with cross-section A0 gives σ = F/A0. The slip plane is tilted, so its area is A0/cos φ. The part of F along the slip direction is F cos λ. So τ = F cos λ / (A0/cos φ) = σ cos φ cos λ. In vector form, with unit plane normal n, unit slip direction d and unit load axis l, m = (n·l)(d·l).

Why m ≤ 0.5. The slip direction lies in the slip plane, so φ + λ ≥ 90°. For φ + λ = 90°, m = cos φ sin φ = ½ sin 2φ, which is at most 0.5, at φ = 45°. That is why widget 1 will not let λ go below 90° − φ.

Lattice rotation. Slip on one system changes a line along the tensile axis from L to L + γ(n·L)d, where γ is the shear. In tension the grips keep the axis fixed in space, so the crystal turns: the axis moves toward the slip direction. In compression the platens keep a plane fixed, and its normal moves as N − γ(d·N)n, toward the slip-plane normal. Widget 2 uses these two rules. When a second system reaches the same τ/τc, both slip equally.

Limits. Schmid's law holds well for FCC metals and for basal slip in HCP metals at room temperature. In BCC metals the CRSS depends on the sense of shear on {112} planes (twinning and antitwinning) and on stresses other than the resolved shear stress, so the equal-CRSS BCC ranking here is a first guess. Real FCC crystals such as α-brass keep slipping on the primary system for a while past the [001]-[111] edge (overshoot), because the conjugate system has been hardened by the primary slip; the page assumes no hardening and no overshoot.

Magnesium numbers. Basal 0.5 MPa and about 40 MPa for prismatic and pyramidal ⟨c+a⟩ slip are the room-temperature single-crystal values summarised by Wang et al. (2019). The pyramidal ⟨a⟩ value is set equal to prismatic here as an assumption. The {1012} tension twin value, about 2.5 MPa, is the estimate of Fukuda et al. (2017). Twinning is polar: it acts only when the shear has the right sense, which is why the ranking changes between tension and compression.

Polycrystal. Widget 3 treats each grain as if it were free (the Sachs picture): a grain "yields" when its own τ/τc reaches 1. Real grains are held by their neighbours, which is why yield in a polycrystal is closer to Taylor's Mτc (M = 3.06 for random FCC, Taylor 1938, Bishop and Hill 1951). Measured polycrystal yield stresses are higher again because of grain boundaries and work hardening; see strengthening mechanisms.

On this site: EBSD 1: from Kikuchi pattern to orientation · EBSD 2: reading IPF maps · EBSD 3: misorientation and boundaries · EBSD 4: texture, pole figures and ODFs · EBSD 8: KAM and GNDs · Stereographic projections lab · Indentation 3: beyond hardness · Indentation 5: when Oliver-Pharr fails
In this chapter: Dislocations and Burgers vectors · Strengthening mechanisms · Reading a tensile curve · Elastic anisotropy · Fracture toughness · Fatigue and S-N curves · Creep and deformation maps

Questions people ask

What is the Schmid factor?

It is cos φ cos λ, where φ is the angle between the load axis and the slip-plane normal and λ is the angle between the load axis and the slip direction. Multiply it by the applied stress to get the shear stress on that slip system. It runs from 0 to 0.5.

Why is the maximum Schmid factor 0.5?

The slip direction lies in the slip plane, so φ and λ cannot both be small. The best case is φ = λ = 45°, which gives cos 45° × cos 45° = 0.5.

What is the critical resolved shear stress (CRSS)?

The shear stress on a slip system at which dislocations start to move on it. It is a property of the material and the slip system, not of the orientation. For pure copper single crystals at room temperature it is below 1 MPa; for iron it is tens of MPa.

How many slip systems do FCC, BCC and HCP metals have?

FCC: 12, four {111} planes with three ⟨110⟩ directions each. BCC: 12 on {110}, plus 12 on {112} and 24 on {123}, all along ⟨111⟩. HCP: 3 basal, 3 prismatic and 6 pyramidal ⟨a⟩ systems, which give only 4 independent ways to change shape, plus 6 pyramidal ⟨c+a⟩ systems that can stretch the c-axis.

Why is magnesium hard to deform at room temperature?

Basal slip is easy (CRSS about 0.5 MPa) but cannot change the length along the c-axis. The other slip systems need about 40 MPa. Grains loaded along c must twin or use ⟨c+a⟩ slip, so textured Mg sheet behaves very differently in different directions and has limited ductility at room temperature.

What is the Taylor factor?

The ratio of the polycrystal flow stress to the CRSS, M = σ/τc, when every grain must follow the overall shape change. For randomly oriented FCC (and BCC {110}⟨111⟩) metals M = 3.06. It is larger than the average 1/m (about 2.23) because grains need several slip systems at once to fit their neighbours.

Why does the tensile axis rotate during single-crystal tension?

Slip shears the crystal, but the grips keep the load line straight, so the crystal lattice turns. The tensile axis moves toward the slip direction. In FCC it reaches the [001]-[111] edge of the standard triangle, where a second system starts.

How do I get Schmid factor maps from EBSD data?

Each EBSD point gives the crystal orientation. Rotate the sample load direction into crystal coordinates, compute m = (n·l)(d·l) for every slip system, and keep the largest (or the lowest τc/m). MTEX and the vendor software do this; widget 3 does the same for its grains.

References

Show the 12 references
  1. E. Schmid and W. Boas, Kristallplastizität, Springer, Berlin (1935); English translation Plasticity of Crystals, F. A. Hughes, London (1950). The resolved shear stress law and the orientation dependence of yield in Zn and Cd crystals.
  2. G. E. Dieter, Mechanical Metallurgy, SI metric edition, McGraw-Hill (1988), chapter 4 (plastic deformation of single crystals; table of critical resolved shear stresses for metal single crystals). Source of the widget 1 and cubic CRSS values.
  3. D. Hull and D. J. Bacon, Introduction to Dislocations, 5th ed., Butterworth-Heinemann (2011), chapter 3 (slip, Schmid law) and chapter 6 (slip in HCP and BCC metals).
  4. W. F. Hosford, Mechanical Behavior of Materials, 2nd ed., Cambridge University Press (2010), chapters on slip and on lattice rotation in single crystals.
  5. G. I. Taylor, Plastic strain in metals, Journal of the Institute of Metals 62, 307 to 324 (1938).
  6. J. F. W. Bishop and R. Hill, A theoretical derivation of the plastic properties of a polycrystalline face-centred metal, Philosophical Magazine 42, 1298 to 1307 (1951).
  7. G. Sachs, Zur Ableitung einer Fließbedingung, Zeitschrift des Vereines Deutscher Ingenieure 72, 734 (1928).
  8. R. von Mises, Mechanik der plastischen Formänderung von Kristallen, Zeitschrift für Angewandte Mathematik und Mechanik 8, 161 to 185 (1928).
  9. J.-Y. Wang, N. Li, R. Alizadeh, M. A. Monclús, Y. W. Cui, J. M. Molina-Aldareguía and J. LLorca, Effect of solute content and temperature on the deformation mechanisms and critical resolved shear stress in Mg-Al and Mg-Zn alloys, Acta Materialia 170, 155 to 165 (2019), doi:10.1016/j.actamat.2019.03.027. Summarises pure Mg at room temperature: basal about 0.5 MPa, prismatic and pyramidal about 40 MPa.
  10. K. Fukuda, Y. Koyanagi, M. Tsushida, H. Kitahara, T. Mayama and S. Ando, Activation stress for slip systems of pure magnesium single crystals in pure shear test, Materials Transactions 58, 587 to 591 (2017), doi:10.2320/matertrans.M2016402. Basal 0.7 MPa measured; {1012} twinning about 2.5 MPa.
  11. T. Obara, H. Yoshinaga and S. Morozumi, {1122}⟨1123⟩ slip system in magnesium, Acta Metallurgica 21, 845 to 853 (1973).
  12. U. F. Kocks, C. N. Tomé and H.-R. Wenk, Texture and Anisotropy, Cambridge University Press (1998), chapters on Taylor and Sachs models.
Cite this page: Tripathy, Manisha. “Schmid Factor and Slip Systems.” untethered atom, 2026, https://untetheredatom.com/mechanical-behavior/schmid-factor-and-slip-systems.
BibTeX 
@misc{tripathy2026schmid,
  author = {Tripathy, Manisha},
  title  = {Schmid Factor and Slip Systems},
  year   = {2026},
  howpublished = {\url{https://untetheredatom.com/mechanical-behavior/schmid-factor-and-slip-systems}},
  note   = {Interactive web tool}
}
Last updated 24 September 2026.