Calculate the Schmid factor from phi and lambda angles or solve for either angle, and add an applied stress to get resolved shear stress (Schmid’s law).
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Schmid Factor Formula
The Schmid factor, often written as SF or m, is the orientation factor that converts an applied normal stress into the shear stress acting on a crystal slip system. It is a core concept in materials science because slip begins on the slip system that experiences the highest resolved shear stress.
Under uniaxial loading, the resolved shear stress on the slip system is:
Slip is expected when the resolved shear stress reaches the critical resolved shear stress for the material:
What the Angles Mean
| Quantity | Meaning | Typical Units |
|---|---|---|
| Schmid Factor (SF) | Dimensionless geometric multiplier that describes how favorably the slip system is oriented relative to the applied load | None |
| Phi angle (ฯ) | Angle between the applied stress direction and the slip-plane normal | Degrees |
| Lambda angle (ฮป) | Angle between the applied stress direction and the slip direction | Degrees |
| Applied stress (ฯ) | External normal stress applied to the crystal | Pa, MPa, psi, etc. |
| Resolved shear stress (ฯRSS) | Shear component of the applied stress acting along the slip system | Same as ฯ |
How to Use the Calculator
- Select Schmid factor, Phi angle, or Lambda angle as the quantity to solve.
- Enter the two known values. Angles use degrees from 0 to 180; select Calculate.
- The calculator evaluates the cosine of each angle and multiplies them together.
- The result is the Schmid factor for that loading orientation.
To solve an angle, the calculator uses the principal inverse cosine from 0 to 180 degrees. A known angle of 90 degrees with zero factor does not determine a unique missing angle. Physically compatible angles must satisfy cosยฒ(ฯ) + cosยฒ(ฮป) โค 1; incompatible pairs are reported as arithmetic results with a warning.
How to Interpret the Result
| Schmid Factor Range | Interpretation |
|---|---|
| 0 | No resolved shear stress acts on that slip system for the chosen loading direction. |
| Between 0 and 0.5 | The slip system is partially favorably oriented. Larger values mean greater tendency for slip under the same applied stress. |
| 0.5 | Maximum physically compatible value for a slip system under uniaxial loading; this occurs when both angles are 45°. |
| Greater than 0.5 | The arithmetic may be correct for the entered angles, but the angle pair does not represent a physically compatible slip-plane/slip-direction combination. |
| Negative | The shear acts in the opposite sense on the selected system. If you only care about favorability, compare magnitudes. |
Example
For a crystal loaded so that ฯ = 45° and ฮป = 45°, the Schmid factor is:
If the applied stress is 80 MPa, then the resolved shear stress on that slip system is:
This means half of the applied normal stress is being resolved as shear on the selected slip system, which is the most favorable orientation under simple uniaxial loading.
Why the Schmid Factor Matters
- Predicts slip activation: The slip system with the highest resolved shear stress is typically the first to deform plastically.
- Connects geometry to yielding: Two crystals made of the same material can yield at different applied stresses if they are oriented differently.
- Useful in crystal plasticity: It is widely used when analyzing single crystals, textured polycrystals, and anisotropic deformation behavior.
- Helps compare orientations: A higher Schmid factor means the same external load produces more shear on that slip system.
Important Notes
- Enter angles in degrees unless your calculator explicitly states otherwise.
- For most textbook slip-system problems, angles are taken between 0° and 90°.
- The Schmid factor is dimensionless; it has no units.
- The resolved shear stress has the same units as the applied stress.
- A high Schmid factor does not guarantee yielding by itself; the material must still reach its critical resolved shear stress.
Common Questions
- Why is 0.5 treated as the maximum?
- For a physically valid slip system under uniaxial loading, the slip direction lies in the slip plane, which constrains the geometry. The largest possible orientation factor occurs at 45° and 45°.
- Can two different slip systems have different Schmid factors in the same crystal?
- Yes. Each slip system has its own plane normal and slip direction, so each one can have a different orientation relative to the load.
- What does a Schmid factor of zero mean?
- It means the applied load produces no shear on that particular slip system, so that system is not favorably oriented for slip under the chosen loading direction.
