Schmid Factor Simulator — Slip System Activation in Single Crystals
Evaluate the Schmid factor m = cos(lambda) cos(phi) in real time to predict slip activation. From sigma and tau_crss the tool reports tau_r, yield stress sigma_y = tau_crss/m and the m(lambda) curve.
Parameters
Tensile-axis to slip-direction angle lambda
deg
Tensile-axis to slip-plane-normal angle phi
deg
Applied stress sigma
MPa
Critical resolved shear stress tau_crss
MPa
With the defaults (lambda = 45 deg, phi = 45 deg, sigma = 200 MPa, tau_crss = 50 MPa) the tool reports m = 0.500, resolved shear stress tau_r = 100 MPa, single-crystal yield stress sigma_y = 100 MPa and safety factor SF = sigma_y/sigma = 0.50. SF < 1 means slip is already occurring. As lambda or phi approach 0 or 90 deg, m collapses and sigma_y rises sharply.
Results
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Schmid factor m
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Resolved shear tau_r
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Yield stress sigma_y
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Safety factor SF
Single-crystal sketch
Blue arrow: tensile axis sigma (vertical). Translucent blue plane: slip plane inside the crystal. Green arrow: slip direction. Red arrow: slip-plane normal. Yellow arc: lambda (angle between axis and slip direction). Purple arc: phi (angle between axis and normal). With the slip plane tilted at 45 deg both lambda and phi reach 45 deg and the Schmid factor reaches its maximum 0.5. When the plane is normal or parallel to the axis, either the slip direction or the normal is perpendicular to the axis and m = 0.
Schmid factor m(lambda) curve
Horizontal axis: lambda (deg, 0 to 90). Vertical axis: Schmid factor m. Blue solid line: m(lambda) = cos(lambda) times cos(45 deg) cosine curve at fixed phi = 45 deg. Yellow marker: current operating point. Green dashed line: theoretical maximum m_max = 0.5. At phi = 45 deg the curve gives m = 0.7071 cos(lambda), so m = 0.5 at lambda = 45 deg and m = 0 at lambda = 90 deg. The absolute maximum 0.5 of the surface m = cos(lambda) cos(phi) is reached only at lambda = phi = 45 deg.
Theory & Key Formulas
Schmid factor:
$$m = \cos\lambda \, \cos\phi$$
Resolved shear stress and Schmid's law for slip activation:
$\lambda$ is the angle between the tensile axis and the slip direction, $\phi$ is the angle between the tensile axis and the slip-plane normal, $\sigma$ is the applied stress (MPa) and $\tau_{\mathrm{crss}}$ is the critical resolved shear stress (MPa). The physical bound is $0 \leq m \leq 0.5$, with the maximum $m = 0.5$ reached at $\lambda = \phi = 45^{\circ}$.
What is the Schmid Factor Simulator?
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I have never heard of the Schmid factor before. What is it for?
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It is the geometric factor that tells you how easily a single-crystal metal will yield when pulled along a chosen direction. The German physicist Erich Schmid introduced it in 1924. The formula is m = cos(lambda) cos(phi), where lambda is the angle between the tensile axis and the slip direction, and phi is the angle between the tensile axis and the slip-plane normal. With the tool defaults (lambda = 45 deg, phi = 45 deg, sigma = 200 MPa, tau_crss = 50 MPa) you should see m = 0.500, resolved shear stress tau_r = 100 MPa, single-crystal yield stress sigma_y = 100 MPa and safety factor SF = 0.50. SF = 0.5 means the applied stress is twice the yield, in other words slip is already happening.
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Why is the maximum value 0.5 and not 1.0?
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Good question. From Mohr's circle you may remember that the maximum shear stress on a plane in a uniaxial test is sigma over 2 and occurs on planes inclined at 45 deg to the loading axis. The Schmid factor is exactly the same idea: when both the slip plane and the slip direction are tilted at 45 deg, cos(45 deg) cos(45 deg) = 0.7071 times 0.7071 = 0.500. If the plane is normal or parallel to the load axis, either the slip direction or the normal becomes perpendicular to the axis, the cosine vanishes and m = 0, so no slip is resolved. On the m(lambda) chart on the right, sweep lambda from 0 to 90 deg and watch the symmetric bell-shaped cosine curve.
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In the crystal sketch I see blue, green and red arrows and a translucent blue plane. What does each one mean?
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The vertical blue arrow is the tensile axis sigma, the translucent blue plane is the slip plane inside the crystal, the green arrow lying on the plane is the slip direction and the red arrow sticking out of the plane is the slip-plane normal n. lambda is the angle between the blue and green arrows, phi is the angle between the blue and red arrows. As you slide phi towards 0 the plane lies flat with its normal parallel to the axis, and as you slide phi towards 90 deg the plane becomes vertical and the normal becomes horizontal. In a real FCC crystal such as Cu or Al the (111) family of slip planes and the [110] slip directions form 12 slip systems, each with its own m, and Schmid's law says slip starts on whichever system has the largest m at a given orientation.
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What about tau_crss? Show me how the slider changes sigma_y.
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tau_crss is the critical resolved shear stress, the minimum shear stress a slip system needs before it can move. It is a material property: about 0.5 MPa for pure annealed copper, up to 50 MPa after work hardening, 30 to 100 MPa for BCC iron and only 0.5 to 2 MPa for the basal slip of HCP magnesium. Schmid's law sigma m = tau_crss inverts to sigma_y = tau_crss / m. With the defaults tau_crss = 50 MPa and m = 0.5, you get sigma_y = 100 MPa. Double tau_crss to 100 MPa and sigma_y doubles to 200 MPa; halve it to 25 MPa and sigma_y halves to 50 MPa. In practice, work hardening, precipitation strengthening and grain refinement all act by raising tau_crss.
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The right chart marks lambda = 45 deg with m = 0.5. How should I read the safety factor SF = 0.50?
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SF = sigma_y / sigma is the ratio of the yield stress to the applied stress. SF greater than 1 means you are still elastic, SF = 1 means you are exactly at yield, and SF less than 1 means you are already plastic. With the defaults sigma_y = 100 MPa and sigma = 200 MPa give SF = 0.50, i.e. the applied load is twice the yield. Drop sigma to 100 MPa and SF rises to 1.00; drop it to 50 MPa and SF becomes 2.00, a comfortable two-times margin. Or rotate to lambda = 60 deg with phi = 45 deg, which gives m = 0.354 and sigma_y = 141 MPa with SF = 0.71, illustrating how strength changes purely from orientation, the essence of crystal plasticity.
Frequently Asked Questions
The Schmid factor m is a geometric factor that gives the fraction of axial stress sigma resolved as shear stress on a slip plane in a single-crystal tensile test, m = cos(lambda) cos(phi). lambda is the angle between the tensile axis and the slip direction; phi is the angle between the tensile axis and the slip-plane normal. Physically m lies between 0 and 0.5, the maximum 0.5 occurring at lambda = phi = 45 deg. With the tool defaults (lambda = 45 deg, phi = 45 deg, sigma = 200 MPa, tau_crss = 50 MPa) you get m = 0.500, tau_r = 100 MPa, sigma_y = 100 MPa and a safety factor of 0.50, the threshold at which slip just begins.
The product cos(lambda) cos(phi) is largest when both lambda and phi equal 45 deg, giving m_max = 0.7071 times 0.7071 = 0.500. This is the same classical result as Mohr's circle: the maximum shear stress in a uniaxial test occurs on planes inclined at 45 deg to the loading axis. In real polycrystals plastic deformation typically begins in grains whose most active slip system is oriented near this 0.5 value, the soft orientation.
The critical resolved shear stress (CRSS) is the minimum shear stress that must be reached on a slip system before slip can occur. It depends on material, temperature and dislocation density. Typical values are 0.5 MPa for FCC pure copper, up to 50 MPa after work hardening, 30 to 100 MPa for BCC iron and only 0.5 to 2 MPa for the basal slip of HCP magnesium. Schmid's law states that slip begins on a system as soon as the resolved shear stress tau_r = sigma m reaches tau_crss. The tool lets you sweep tau_crss from 5 to 300 MPa to see the effect on yield stress.
Schmid's law sigma m = tau_crss directly inverts to sigma_y = tau_crss / m. So a larger Schmid factor (orientation closer to 45 deg) means the single crystal yields at a lower applied stress. With the tool defaults (lambda = 45 deg, phi = 45 deg, tau_crss = 50 MPa) sigma_y = 50 / 0.5 = 100 MPa. As lambda or phi approach 0 or 90 deg, m goes to zero and sigma_y diverges to infinity, the so-called hard orientation that explains why the ease of plastic deformation in a grain depends so strongly on its crystallographic orientation.
Real-world Applications
Nickel-base single-crystal turbine blades: the high-pressure turbine blades of jet engines made of CMSX-4 or Rene N5 are grown as single crystals with a controlled [001] orientation aligned with the principal stress in the airfoil. The maximum Schmid factor of the FCC {111}<110> slip systems for [001] tension is only 0.408, giving roughly 20 percent higher yield stress than the polycrystal-equivalent orientation (m about 0.5). The tool lets you sweep lambda and phi to feel this anisotropy and to understand why crystal-orientation control matters in real engines.
Dislocation control in silicon single-crystal wafers: Czochralski (CZ) and float-zone (FZ) silicon wafers used in semiconductor manufacturing are prone to {111}<110> slip introduced by thermal stresses during pulling, which degrades device characteristics. Engineers compute the Schmid factor distribution for the principal stress axis [100] to identify the most active slip systems and optimise pull parameters such as temperature gradient and pull speed. This tool is a good entry point for that analysis, mapping orientation to m.
Anisotropy of magnesium-alloy sheet: the HCP structure of Mg-Al-Zn alloys (AZ31, AZ91), of interest for lightweighting, has a basal-slip CRSS of only 0.5 to 2 MPa whereas non-basal pyramidal and prismatic slip systems sit at 30 to 80 MPa, an enormous anisotropy. Combining the rolling-texture orientation distribution with the angle to the tensile axis, engineers compute the Schmid factor for each slip system to predict yield-point elongation and the r-value. The sliders show how a small orientation change can swing sigma_y by orders of magnitude in HCP metals.
Teaching aid for Crystal Plasticity Finite Element (CPFEM): modern CPFEM codes such as the ABAQUS UMAT or DAMASK assign many slip systems to every integration point and explicitly solve Schmid factors, resolved shear stresses and dislocation density evolution. This tool visualises the most fundamental ingredient of CPFEM, the Schmid factor and resolved shear stress on a single slip system, in real time, helping students and lecturers grasp the physical foundation of CPFEM before opening a full simulation.
Common Misunderstandings and Cautions
The most common misconception is that Schmid's law applies directly to polycrystalline materials. It is a single-crystal, single-slip-system law. In a polycrystal each grain is differently oriented and must satisfy compatibility with its neighbours, so the Taylor polycrystal model with M about 3.06 or the von Mises five-slip-system criterion is needed. Only after multiplying the single-crystal estimate by M does the tool become predictive for a polycrystal: sigma_polycrystal about M times tau_crss. For full polycrystal analysis use a dedicated crystal-plasticity FEM tool.
The second misconception is that lambda + phi always equals 90 deg. lambda is the angle between the slip direction and the axis, phi is the angle between the slip-plane normal and the axis. The slip direction lies in the plane and the normal is out of the plane, so they are mutually perpendicular, but cos(lambda) and cos(phi) are independent quantities described by spherical trigonometry; in general lambda and phi are independent, subject only to lambda + phi greater than or equal to 90 deg (Schwarz inequality). The tool does not enforce that constraint visually, so unphysical combinations like lambda = 10 deg with phi = 10 deg are accepted; treat such inputs as purely sensitivity-analysis exercises.
The last misconception is that a large Schmid factor means a weak material. The correct reading is that an orientation with a large m reaches tau_crss at a smaller axial stress, so the single-crystal sigma_y is lower for that orientation. That is not weakness of the material itself, only easy slip in a soft orientation. In practice work hardening, precipitation strengthening and grain refinement (Hall-Petch) all raise tau_crss itself, on top of the geometric Schmid effect. The tool keeps tau_crss fixed and shows only the orientation effect; remember that real strengthening combines both.