12.25. Compute Schmid Factors

Group (Subgroup)

Statistics (Crystallography)

Description

This Filter calculates the Schmid factor of each Feature given its average orientation and a user-defined loading axis. The Schmid factor combines the component of the axial force F that lies parallel to the slip direction with the component perpendicular to the slip plane. The equation is:

Schmid Factor = (cos φ cos λ)

The angle φ is the angle between the tensile axis and the slip plane normal, and λ is the angle between the tensile axis and the slip direction in the slip plane.

The Filter evaluates this equation for every enumerated slip system in each Feature’s crystal structure. It stores the largest Schmid factor and the corresponding slip-system index. Critical resolved shear stress is not considered.

The user-supplied Loading Direction is normalized before use, so only its direction matters — [1, 2, 3] and [3, 6, 9] give identical results. It is a sample-frame direction; the Filter rotates it into each Feature’s crystal frame using that Feature’s average orientation before evaluating the slip systems. The loading direction must be non-zero. When Override Default Slip System is enabled, the supplied slip-plane normal and slip direction must also be non-zero and perpendicular.

Feature 0 is the conventional “unassigned” Feature and is never computed; all of its output values are zero. A Feature whose crystal-structure value is outside EbsdLib’s supported Laue-group range is skipped, and all of its output values are also zero. For a valid Laue class that has no enumerated slip systems, Schmids, Slip Systems, Phis, and Lambdas are zero, while Poles still records the transformed loading direction.

Ties Between Slip Systems

Candidate slip systems are compared with a strict greater-than, in the fixed order in which they are enumerated for the crystal structure. When two or more slip systems share the maximum Schmid factor — which happens for high-symmetry loading directions such as [0, 0, 1] or [1, 1, 1] in a cubic crystal — the lowest-numbered of the tied systems is reported. This is a property of the enumeration order, not a physical preference: the tied systems are equally favoured.

Slip Systems Numbering Depends on Override Default Slip System

The meaning of the Slip Systems output changes with the Override Default Slip System toggle:

Override Default Slip System

Slip Systems value

Off (default)

Index into the crystal structure’s built-in slip system list — numbered from 0 (0–11) for Cubic-High and Cubic-Low, and from 1 (1–6) for Hexagonal-High and Hexagonal-Low

On

Index of the symmetry operator (0–23 for Cubic-High) that maps the user-supplied slip plane and slip direction onto the winning variant

The two numbering schemes are not comparable, and neither is the base of the off numbering comparable between the cubic and hexagonal classes.

A Slip Systems value of 0 can mean “no slip system found” in two situations:

  • With Override Default Slip System on and no symmetry-operator variant producing a non-zero Schmid factor — for example a loading direction parallel to the slip plane normal, which puts the loading axis perpendicular to every slip direction.

  • With the toggle off on a Hexagonal-High or Hexagonal-Low phase, where the built-in systems are numbered 1–6 and no candidate exceeds the initial Schmid factor of 0. Because that numbering starts at 1, 0 falls outside the valid range and is unambiguous. (For the cubic classes, whose numbering starts at 0, the reported 0 in this situation is indistinguishable from a genuine win on system 0; check the reported Schmid factor, which is 0 in the degenerate case.)

In every one of these cases the reported Schmid factor is 0.

With Override Default Slip System on, the reported index is relative to the symmetry-operator table of the orientation library, and several operators typically tie at the maximum Schmid factor — for a (001)[100] system under [1,2,3] loading, six of the twenty-four do. The reported index is simply the first of those in table order. It identifies a maximizing variant, not a uniquely determined one, and the same input run through DREAM3D 6.5.x reports a different index for the same physical answer because that version’s table is ordered differently. Treat this value as a handle into the current library’s table, not as a portable label.

Phis and Lambdas Units Depend on Override Default Slip System

The Phis and Lambdas arrays change units with the same toggle, which is easy to miss because the array names do not change:

Override Default Slip System

Phis / Lambdas contents

Off (default)

The direction cosines cos φ and cos λ (dimensionless, in [0, 1])

On

The angles φ and λ themselves, in radians

With the toggle off, the reported Schmid factor is the product of the two stored values. With the toggle on, it is the product of their cosines. If you need consistent units across both modes, convert explicitly rather than assuming.

Poles Is Not a Miller Index

Despite its name, the Poles array does not contain a crystallographic index. It is the unit loading direction expressed in the Feature’s crystal frame, multiplied by 100 and truncated toward zero to an integer — a compact fixed-point encoding of that unit vector, retaining two decimal places. Components can be negative, and the sum of squares is approximately 10 000 rather than 1. Because the conversion truncates rather than rounds, a component whose scaled value falls near an integer can differ by one from the value you would get by rounding. Reducing the triplet to a Miller index requires dividing by the greatest common divisor yourself, and the truncation means the result is only approximate.

Required Input Sources

Input Parameter(s)

Parameter Name

Parameter Type

Parameter Notes

Description

Loading Direction

Vector of Float32 Values

Order=X,Y,Z

A non-zero loading direction in the sample reference frame

Store Angle Components of Schmid Factor

Bool

Whether to store the angle components for each Feature

Override Default Slip System

Bool

Allows the user to manually input the slip plane and slip direction

Slip Plane

Vector of Float32 Values

Order=X,Y,Z

A non-zero vector defining the slip plane normal

Slip Direction

Vector of Float32 Values

Order=X,Y,Z

A non-zero vector defining the slip direction

Input Feature Data

Parameter Name

Parameter Type

Parameter Notes

Description

Phases

Array Selection

Allowed Types: int32 Comp. Shape: 1

Specifies the Ensemble to which each Feature belongs

Average Quaternions

Array Selection

Allowed Types: float32 Comp. Shape: 4

Specifies the average orientation of each Feature in quaternion representation

Input Ensemble Data

Parameter Name

Parameter Type

Parameter Notes

Description

Crystal Structures

Array Selection

Allowed Types: uint32 Comp. Shape: 1

Enumeration representing the crystal structure for each Ensemble

Output Feature Data

Parameter Name

Parameter Type

Parameter Notes

Description

Schmids

DataObjectName

The name of the array containing the value of the Schmid factor for the most favorably oriented slip system (i.e., the one with the highest Schmid factor)

Slip Systems

DataObjectName

The name of the array containing the enumeration of the slip system that has the highest Schmid factor

Poles

DataObjectName

The name of the array containing the normalized loading direction in the crystal frame, multiplied by 100 and truncated toward zero

Phis

DataObjectName

The name of the array containing cos(phi) when the default slip systems are used or phi in radians when they are overridden

Lambdas

DataObjectName

The name of the array containing cos(lambda) when the default slip systems are used or lambda in radians when they are overridden

Example Pipelines

  • (04) Small IN100 Crystallographic Statistics

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