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Chapter 8: Slip

Prof. Wenjea J. Tseng (曾文甲)Department of Materials Engineering

National Chung Hsing Universitywenjea@dragon.nchu.edu.tw

Reference: W. F. Hosford (Cambridge, 2010)N. E. Dowling (Pearson, 2007)

Mechanical Properties of Materials

OHP 1

Introduction• Plastic deformation of crystalline materials usually

occurs by slip, which is the sliding of planes of atoms over one another by dislocation movements. The planes on which slip occurs are called slip planes and the directions of the shear are the slip directions. The slip planes and directions are characteristic of the crystal structure of materials.

Macroscopic

Microscopic

(see next pages)

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Introduction (microscopic view)

Slip direction

slipped

un-slippedSlip plane

• In microscopic viewpoints, slip is caused by the motion of edge and/or screw dislocations.

Introduction (microscopic view)

Shear/slip steps able to be seen macroscopically when multiple dislocations were moved to the surface ends.

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• Figures illustrate shear/slip bands and shear/slip steps of multiple dislocations upon stress loading.

Shear Bands and Shear Steps

Shear bands

Shear steps

Slip Systems• The slip planes and directions, combined to called the slip

systems, for several common crystals are summarized in Table. The slip directions are the crystallographic directions with theshortest distance between like atoms or ions and the slip planesare usually densely packed planes.

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Schmid’s Law• E. Schmid discovered that if a crystal is stressed, slip

begins when shear stress on a slip system reaches a critical value, c, often called the critical resolved shear stress (CRSS). In uniaxial tension, Schmid’s law is written as

where is the angle between theslip direction and the tensile axis,and is the angle between the tensile axis and the slip-planenormal.

coscosc

coscosm

m : Schmid factor. mmax = 0.5

Schmid’s Law• The Schmid’s law can be shorten to

where m is the Schmid factor, m = cos cos.

A larger m indicates a smaller for slip to occur.

m/

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Schmid’s Law• In a general form, the shear stress induced by a

uniaxial tension along x-direction can be found from the stress transformation, i.e.

where ls are the direction cosines. Therefore, the condition for yielding (i.e., for the slip to occur) under a general stress state is

zxdxnzdznxyzdznydynz

xydynxdxnyzzdznzyydynyxxdxnx

zxdxnzdznxyzdznydynz

xydynxdxnyzzdznzyydynyxxdxnxndc

llllllll

llllllllll

llllllll

llllllllll

)()(

)(

)()(

)(

mnjnimmnn

jnm

imij llll

3

1

3

1

Example: Schmid’s

Law in Multiaxial

Stress State

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Strains Produced by Slip• The incremental strain transformation equations

may be used to find the shape change that results from slip when the strains are small, i.e., when the lattice rotation are negligible. For infinitesimal strains,

With slip on a single slip system in the d direction and on the n plane, the only strain term is dnd, therefore,

In Schmid’s notation, this is

mddd xx coscos

mnjnimij dlld

ndxdxnxx dlld

Strains Produced by Slip• Similarly, the other strain components, referred to

the x, y, and z axes, are

dlllld

dlllld

dlllld

dlld

dlld

ynxdydxnxy

xnzdxdznzx

znydzdynyz

zdznzz

ydynyy

)(

)(

)(

[010]

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Remarks• For a polycrystalline solid to have appreciable

ductility, each of its grains must be able to undergo the same shape change as the entire body. Therefore, each grain in a polycrystalline solid must deform with the same external strains as the whole. For an individual grain, five strain components, 1, 2, 12, 23, 31, exist independently. This indicates that at least 5 independent slip systems are needed for ductility. This means that if a material has less than 5 independent slip systems, a polycrystal solid will have a limited ductility unless another deformation mechanism supplies the number of freedom.

Slip in FCC Crystals• Strain hardening of fcc single crystals

The underlying figure shows a typical stress-strain curve for an fcc single crystal. Slip occurs on a single plane initially and the rate of strain hardening is very low, called easy glide or stage I. Slip is then observed on other slip systems, resulting in dislocation on different slip systems intersect which gives rise to strain hardening in stageII. At stage III, the rate of strain hardening decreases.

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Slip in FCC Crystals• Strain hardening of fcc single crystals (continue)

When slip in fcc single crystals occurs simultaneously on many slip systems (e.g., <111>, <110>). Easy glide region becomes insignificant and the initial strain hardening rate is rapid, even comparable with that in polycrystal.

• Strain hardening of fcc polycrystals

No observable easy-glide region.

The shaded region indicates strain hardening in orientations with only a single slip system.

Stereography (review)• The stereographic projection is used to find angular

relations between directions and planes in a crystal.

For Cubic structure

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Slip in FCC Crystals• Tensile deformation of fcc crystals

For all orientations of fcc crystals within the basic stereographic triangle with [100], [110], and [111] corners, the Schmid factor for slip in the [101] direction on the (111) plane is higher than that for any other slip system.

coscosc

The tensile axis lies on the great circle b/t slip direction and the silp-plane normal with and both equal to 45o.

Slip in BCC Crystals• The slip direction in bcc metals is always the direction

of close packing, <111>. Slip has been reported on various planes, {110}, {123}, and {112}. All of these planes contain at least one <111> direction. The underlying figure shows the orientation dependence of the Schmid factors for uniaxial tension with <111>-pencil glide.

The basic orientation triangle is divided into two regions, with a different <111> slip direction in each.

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Slip in HCP Crystals• The most common slip direction of hcp metals is

<1120>, which is the direction of close contact between atoms in the basal plane. The underlying figure shows shear stresses required for several slip systems in Be metals.

The stresses required

for pyramid slip are

much higher than the

stresses to cause basal

and prism slip and are

often high enough to

cause fracture.

Lattice Rotation in Tension• Slip normally causes a gradual lattice rotation or orientation

change. The figure shows the elongation of a long single crystal by slip on a single slip system. The slip plane and slip direction are represented as being fixed in space with the tensile axis rotating relative to these. Note that in real tension test, the tensile axis remains vertical and the crystal elements rotate instead.

(a) w/o constraint

(b) w/ constraint

Slip plane

Slip direction

Tensile axis

Dropping normalsto slip direction through O

Dropping normals to slip plane through O

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Lattice Rotation in Tension• In previous figures, slip causes translation of point P

parallel to the slip direction to a new position P’. Points C and C’ are constructed by extending the slip direction through point O and dropping normals from P and P’. Points B and B’ are constructed by dropping normals from P and P’ to the slip plane through O. The lattice rotation follows a geometrical relation:

= cos / cos - coso / coso

where is the shear strain. In the figure, increases and decreases during tensile extension. The orientation change in tension is hence a simple rotation of the slip direction toward the tensile axis.

decreases

Lattice Rotation in Tension for FCC• As shown in the figure below, primary slip in the [110]

direction causes the tensile axis to rotate toward the [101]. Once the tensile axis reaches the [100]-[111] symmetry line; at this point, slip starts on [110](111) which is in the conjugate triangle, a simultaneous slip in the [101] and [110] directions causes a rotation toward [211].

For fcc single crystals

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Lattice Rotation in Compression for FCC• For the compression of thin flat single crystal, the

compression axis is rotates toward the slip-plane normal. As shown in the figure, compression causes a rotation of the slip-plane normal toward the compression axis, which is equivalent to a rotation of the compression axis toward the slip-plane normal.

• For an fcc crystal, the compression axis rotates toward [111] until it reaches the [100]-[110] boundary, where duplex slip will cause a net rotation toward [110].

For fcc single crystals

Lattice Rotation in Tension for BCC• The tensile axis of a bcc crystal deforming by gliding

rotates toward the active <111> slip direction. For orientations near [110] and [111] (region A in figure) the tensile axis will rotate toward [111]. For crystal orientations in the basic triangle

near [100], the rotation will be

toward [111]. Once the tensile

axis enters region A, the rotation

will be toward [111]. When the

[100]-[110] boundary is reached,

combined slip in the [111] and

[111] directions will cause

rotation toward [110].For bcc single crystal

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Lattice Rotation in Compression for BCC• For gliding mechanism,

rotation toward the compression axis is equivalent to rotation away from the active slip direction. The figure shows that for bcc single crystals, orientations initially in region A will end up rotating to [111], whereas those initially in region B will rotate toward [100].

Texture Formation in Polycrystals• In polycrystalline metals, the grains undergo similar

rotation and these lead to crystallographic textures or preferred orientations. The table shows the experimentally observed textures developed in tensile extension and compression of polycrystals .

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Summary• Slip occurs when the shear stress arising from the

external loading exceeds a critical resolved shear stress. At the time, dislocations often move along a particular slip system (called the easy glide region). Strain hardening follows when interaction between multiple dislocations begins to prevail. When slip system is unavailable for the dislocations to move, strain hardening may occur without going through the easy glide region.

• Slip can be illustrated by the stereographic projection, so as the lattice distortion.

• Multiple microscopic slips eventually become macroscopically visible, i.e., the shear/slip bands, and the shear/slip steps.