Chap 10

• Solid solution strengthening • Precipitation strengthening • Dispersion strengthening

Solid solution

Solid solution hardening The dislocation mobility is restricted by the introduction of solute atoms. Interstitial solute

Substitutional solute

Sources of dislocation-solute interactions • Elastic dislocation-solute interaction due to the size difference between solute and solvent atoms • Interaction due to a difference in modulus between solute and solvent atoms • Electrical interaction • Chemical interaction • Local-order interaction due to the fact that a random atomic arrangement may not be the minimum-energy state in a solid solution

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Substitutional atoms → symmetrical spherical lattice distortion – Do not interact with screw dislocation

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Interstitial atoms → both dilational misfit and tetragonal distortion – Interact with both edge and screw dislocations

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Energy of interaction between an edge dislocation and an oversized (or undersized) solute atoms Volume change induced by solute atom ∆V = ( 4 3)πro3 (1 + ε ) − ( 4 3)πro3 3

∆V ≈ ( 4 3)πro3 3ε = 4πro3ε

The hydrostatic stress associated with an edge dislocation

Increase in stress due to solute atoms ∆τ = F bL = A sin θ r 2bL

σ p = − 1 ( σ 11 + σ 22 + σ 33 ) 3 σp =

1 + υ Gb sin θ 1 − υ 3π r

Interaction energy U misfit = σ p ∆V = A sin θ r

A=

4 1+υ Gbεro3 3 1−υ

Force exerted on dislocation

F = − ∂U misfit ∂r = A sin θ r 2

L: spacing of solute atoms ρ: dislocation density C: concentration of solute atoms

∆τ = A sin θ C r 2bρ If r ~b and sinθ ~1

∆τ = A C b 3 ρ

Mechanical effects associated with solid solutions • Well-defined yield point in the stress-strain curve • Plateau in the stress-strain curve and Lüders band • Strain aging • Serrated stress-strain curve • Snoek effect • Blue brittleness

• Yield point – Dislocations are locked by the interstitial atoms. – A critical stress is needed to unlock the dislocation. – The stress necessary to move the unlocked dislocation is less than the stress necessary to free them.

Cottrell atmosphere

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Lüders band – It corresponds to yield-point elongation. – Deformation is restricted to the moving front (nonhomogeneous deformation). – Propagation of Lüders band: The unlocked dislocations accumulate at grain boundary. The stress concentration at grain boundary can unlock the dislocations in the next grain.

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Ways to avoid Lüders band – remove interstitial atoms – prestraining the sheet to a strain larger than the yield-point elongation

• Strain aging – Pre-strain to a strain above yield point strain →Stop the test and rest for certain time →Reloading, the yield point return

• Applied stress will accelerate the strain-aging process.

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Dynamic strain aging: strain-aging occurs concurrently with plastic deformation. Dynamic-strain-aging results in an enhancement of the workhardening rate, leading to an increased ultimate tensile strength. – This is caused by solute atoms that have a mobility higher than the dislocation and that, therefore, can continue to “drag” them, leading to increased work-hardening.

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Serrated stress-strain curve can be caused by – Solute-dislocation interaction (Portevin-Le Chatelier effect, P-L effect) – Mechanical twining – Stress-assisted martensitic transformations

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The solute-dislocation interaction occurs within a specific range of temperatures and strain rates. – – – –

The solute atoms is able to diffuse fast enough to chase the dislocation. The speed of dislocation is determined by the applied strain rate. Inverse strain-rate sensitivity The P-L effect is dependent on density of dislocations, concentration and mobility of dislocations, strain rate, etc.

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Snoek effect

– Interstitial solute atoms such as C and N can migrate in α-Fe (BCC lattice) under an applied stress. Such short-range migrations of C or N can result in an anelastic or internal friction effect, called Snoek effect. – The movement of interstitials can cause strain to lag behind stress (internal friction). – This phenomenon is related to the tetragonal distortion associated with interstial atoms in BCC lattice

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Snoek effect (internal friction) can be used to measure the concentration of C or N in the lattice of high purity Fe (BCC α-Fe).

Precipitation- and dispersion- hardening

Al-Cu

Al-Li

γ’ in superalloy

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Precipitation treatment – Solubilization (solution treatment) – Quenching – Aging

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General aging sequences: – Supersaturated solid solution → transition phase → aged phase

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Crystallographic relationships between matrix and precipitate – Coherent – Semicoherent – incoherent

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Dislocation-precipitate interaction (1) Dislocations move around the particles in the slip plane (impenetrable particles; Orowan model) x: average spacing between two particles τ: Orowan bowing stress

τ ≈ Gb 2 R = Gb x τm is the critical shear stress for matrix yielding in the absence of particles.

τ y = τ m + Gb x

(2) Dislocations cut through the particles in the slip plane (particles are penetrable.)

τbx( 2ro ) = (πro2 )γ

τ = πroγ 2bx

γ: surface energy f: volume fraction of particles 4  f =  πro3  x 3 3  ro x = [ ( 3 4π ) f ]

τ ∝ f 1/ 3

1/ 3

Comparison between particle shear and dislocation bowing mechanisms

Orowan bowing stress decreases with increasing particle spacing, while particle spacing increases with increasing time or r. The shear strength τ depends on the particle radius r and the particle volume fraction f according to

τ ∝ rf