Mechanisms of deformations

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Mechanisms of deformations

Lecture 3 – defects and dislocations

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following type of deformations:

• Elastic deformations. Fully recoverable, they appears immediately with the application of stress

• Inelastic deformations. Not (all) recoverable, irreversible. They do not disappear when the stress is removed.

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INITIAL LOAD UNLOAD

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Deformation

• The ability to deform depends on material structure

• Deformation can occur along specific crystal planes only

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a crystal plane over another can be estimated from the bond strength:

• No real materials exhibit such strength!

𝜏 = 𝐺

2𝜋

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Deformation and strain

• In 1934 Orowan, Polanyi e Taylor, almost at the same time, understood that the ability of a material to plastically deform was due to the presence of defects in the lattice

• Line defects called: dislocations

• Theory of dislocations was firstly proposed by Vito Volterra in 1907, although the term

dislocation was used by Taylor in 1934.

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• Dislocation climb

• Twinning

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Dislocations can move

• Dislocation glide

• Twinning

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• Dislocation climb

• Twinning

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Dislocations can move

• Dislocation glide

• Twinning

Twinning results when a portion of a crystal takes up an orientation that is related to the orientation of the

untwinned lattice in a definite, symmetrical way.

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• Edge dislocation forms «surface steps»

• Uniaxial deformation by twinning

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Stress state induced by dislocations

• The region deformed by a dislocation affect the ability of dislocation to move and to multiply

• Most of the deformation internal energy is due to dislocations

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• For same sign dislocations, laying on the same plane, the action of the

deformation field is repulsive

• The action on opposite sign dislocations is attractive. When in contact the

restoration of the crystal plane takes place (annihilation)

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Multiplication of dislocations

• The «Frank-Read source» is the mechanism that explain the generation of multiple

dislocation on slip planes when deformation occurs

• Consider the straight dislocation pinned in A and B. Under shear, the dislocation bends.

When

𝜏 = 𝐺

2𝜋

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• The «Frank-Read source» is the mechanism that explain the generation of multiple

dislocation on slip planes when deformation occurs

• Dislocations have to develop to produce a slip in a deformed crystal. This implies that during deformation, dislocations are formed mainly along that sliding plane.

• Hardening increases the number of dislocations according to Frank-Read mechanism

• High dislocation density increases the yield stress and causes material hardening

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Le dislocazioni nella realtà

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• From theory and experimental evidences we know that dislocation density is

function of stress and plastic deformation:

𝜀 = 𝜌𝑏𝑣

𝜌 = 𝛼 𝜎 𝑏

2

𝜌 = 𝜌

₀

+ 𝐶𝜀

_𝑝^𝑛

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Orowan equation

• During hardening dislocations continue to move. This cause a «back stress» che that reduces the effective stress. Assuming

linear hardening we can write

𝜎 = 𝜎

_𝑎𝑝𝑝

− 𝜃𝜀 𝑣 = 𝐴 𝜎

𝜎

₀

𝑚

𝑣 = 𝐴 𝜎

_𝑎𝑝𝑝

− 𝜃𝜀 𝜎

₀

𝑚

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𝜀 = 𝜀

_𝑒𝑙

+ 𝜀

_𝑝

= 𝜎

𝐸 + 𝐴′ 𝜌

₀

+ 𝐶𝜀

_𝑝^𝑛

𝜎

_𝑎𝑝𝑝

− 𝜃𝜀

𝜎

₀

𝑏

𝜎

_𝑎𝑝𝑝

= 𝜃𝜀 + 𝜎

₀

𝜀

𝐴′ 𝜌

₀

+ 𝐶𝜀

_𝑝^𝑛

𝑏

1/𝑚

For hardening only:

At yield:

This explains the increase of the yield stress with strain rate

𝜎

_𝑈𝑃𝑆

= 𝜎

₀

𝜀

𝐴′ 𝜌

₀

𝑏

1/𝑚

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L’equazione di Orowan

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• Plastic deformation occurs on preferred crystal planes

• The number of planes depends on the crystal structure:

• FCC 12 independent slip planes

• BCC 5 independent slip planes

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Temperature effect

• High temperature promotes climbing

• Low temperature limits the capability to slip particularly in BCC

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• Deformation in metals and alloys is related to defects (dislocazions)

• Dilocations motion is the basic

mechanisms for plastic deformation to

• Dilocations can move also at very low stress (elastic at macroscopic scale) – Peierls stress

• Orowan law allow to predict plastic flow and strain rate effect on material yield stress

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Mechanisms of deformations