§ 7 Dislocation & strengthening Mechanism in Metals → vacancy diffusion (substutional) Dislocation & plastic deformation

§ 7.2 Basic concept

Plastic deformation → motion of larger number of dislocation Motion of edge dislocation Fig 7.1 (The process by which plastic deformation is produced by dislocation motion is tensile slip) It form an edge dislocation finally → called edge dislocation Slip plane : The crystallographic plane along which the dislocation line traverses is the slip plane Dislocation density (number of dislocation)

Total dislocation length per unit volume. Number of dislocation that intersect a unit area of a random section

Unit mm/mm3 ,or /mm2 For carefully solidified metals 103mm-2 For heavily deformed metals 109 ~ 1010 mm-2

Often heat treatment 105 ~ 106 mm-2 Estimate dislocation density : 109 mm/mm3

§ 7.3 Characteristics of Dislocations Extra half plane : lattice strains 有 compression, tensile & shear → affect the mobility of dislocation & ability to multiply Dislocation interaction：

Two edge dislocation of the same sign → Repulsive

compression

tension

§ 7.4 slip system Dislocation 在不同平面不同方向的滑移能力不同 有 preferred place : slip plane, slip direction Slip system 包含 slip plane & slip direction Slip system depends on crystal structure (fcc : 111) → atomic distortion 因 dislocation motion 最小 For FCC {1 1 1} family, i.e. Slip direction <1 1 0> with {1 1 1} plane Slip system {1 1 1} <1 1 0> 一個 plane 有可能有很多的 slip direction

Dislocation in FCC & BCC slip easier than in HCP ∴ FCC & BCC metals 較 soft. HCP → brittle Microstructure ←→ properties

[0 1 -1]

(1 0 -1)

(-1 1 0)

§ 7.5 Slip in single crystal →Question : slip system 最簡單的 case : single crystal Although an applied stress may be more tensile or compressive, shear component exists.

Force θcos' FF = , shear force , area

θθστ cossin'=

θθσθθθθτ cossincossin

cossin

000

====AF

AF

AF

pure 拉伸應力有可能在電子平面產生 shear ☆ Resolved shear stress 分解剪應力

ψ：angle between the normal to the slip plane and the applied stress direction. λφστ coscos=R ......two ways to drive this equation.

有一個最大分的 slip system max(max) )cos(cos λφστ =R

當 Rτ 大到使 slip 開始的 stress 於 Critical resolved shear stress λ：the angle between the slip & stress direction

°≠+ 90λφ 當 single crystal 開始 deform plastically, stress=σy , yield stress

crssR ττ =(max)

( )maxcoscos λφτ

σ crssy =

當 °== 45λφ 時， crssy τσ 2=

1. use φφστ cossin'=

投影到 σ 上 → φ

φφσsin

1cossin

→ 再投影到 slip direction, λφσλφ

φφσ cossincossin

1cossin =

2. use force. F 在 slip system 分力 F”

λcos" FF = , 再除以 area：

φ

λ

φ cos

cos

cos

""00 A

FA

FAF

==

7.13 This problem asks that we compute the critical resolved shear stress for silver. In order to

do this, we must employ Equation(7.3), but first it is necessary to solve for the angles λ

and φ from the sketch below.

If the unit cell edge length is a, then

°=

= − 45tan 1

aaλ

For the angle ψ, we must examine the triangle OAB. The length of line OA is just a, whereas, the length of AB is aa . Thus,

°=

= − 452tan 1

aaλ

And, finally

( )λφστ coscosycrss =

( ) ( ) ( )[ ] (65.4psi) 45.045cos7.54cos1.1 MPaMPa =°°= Example for slip Ex. 7.13 single crystal

Slip occurs on a (1 1 1) plane i=and in a [-1 0 1] direction and it initiated at an applied tensile stress of 1.1 MPa (160psi). Compute the critical resolved shear stress. <Solution> The angle θ between the plane (h1 k1 l1) & (h2 k2 l2)

( )( )22

22

22

21

21

21

212121coslkhlkh

llkkhh

++++

++=θ

Direction normal to (111) plane

(111) plane / slip plane

y

x [-101] Direction / slip direction

[001]Direction

ψ

λ

O

A B

z

§ 7.6 Plastic deformation of polycrystalline materials The direction of slip varies from one grain to another . Fig 7.10 On polished → deformed Slip lines are visible Two slip systems operated. 拉伸時會產生大規模的 plastic deformation → 鄰近的 grain 一起變形 → grain boundary : barer for the motion of dislocation

(100)

(111) (110)

§ 7.7 Deformation by twinning

Mechanisms of strengthening in metals ☆ 材料工程師常常會被要求設計改良材料強度。

High strength → low ductility The ability of a metal to plastically deform depends on the ability of dislocations to move. Strengthening mechanism → understand the relationship between dislocation motion & mechanical behavior of metals. Strengthening techniques → Restricting or hindering dislocation motion renders a material harder and stronger

Strengthening mechanism for single-phase metals.

(a) Grain size reduction (b) Solid-solution alloying (c) Strain hardening

§ 7.8 Strengthening by grain size reduction Grain boundary : barrier to dislocation motion

For two reasons：

(1) A dislocation asses into grain B has to change its direction of motion Misorientation ↑, difficult to move

(2) The atomic disorder in G.B results in a discontinity of slip plane from one grain into another.

For high angle G.B.s,

Dislocation 穿過 G.B.會在 G.B.處產生應力集中。 → active new dislocation in an adjacent grain

Fine-grained material → harder & stronger ∵ has greater total G.B area to impede dislocation. ☆ Hall-Petch equation (要背!)

σy = σ0 + kyd-1/2 σ0、κy : constant for a particular material

☆ How to get smaller grain size (1) Higher rate of solidification (2) Plastic deformation, followed by heat treatment

§ 7.9 Solid- solution strengthening Add impurity atoms that go into lighter substutional or interstitial → yield strength & tensile strength ↑

Mechanism :

Impurity impose lattice strain on the surrounding host atoms

吸引小的 solute atoms 到 dislocation 上方

→ reduce overall strain energy Stable 後，要再移動 dislocation 必須有較高的能量

→ hard to move dislocation after the formation of solid-solution

Tensile stress imposed on hold atoms

compressive

§ 7.10 Strain hardening Strain hardening : A ductile metal becomes harder and stronger when it is plastically deformed. Also called work hardening 冷加工 ∵在室溫(低溫)下加工。 如果拉一條金屬 → 會越來越硬或越軟？

Cold work, %cw %1000

0 ×−

=A

AA d

Cold working → yield strength ↑ 付出的代價 ductility ↓ Strengthen mechanism

dislocation → dislocation strain field interaction cold working → dislocation density ↑,→ distance between dislocation ↓ ∵ dislocation-dislocation strain interactions are repulsive

→ motion of dislocation becomes harder

Cross section after deformation Original cross section

§ 7.11 Recovery 回覆

Plastic deformation 造成 (1) change in grain shape (2) strain hardening (3) an increase in dislo density

→ The properties & structure may change back to the precold-worked states by appropriate heat treatment / annealing . 發生 : Recovery, recrystallization, grain growth Recovery : (1) some of internal strain energy is relieved. (2) dislocation number ↓ (3) physical properties, like electrical & thermal conductivities are recovered to their

precold-worked states. - but grain is still in highs strain energy state.

§ 7.12 Recrystallization (再結晶)

- formation of a new set of strain-free & equiaxed grains - low dislocation density - mechanical properties recovered (become softer, ductile) tailor mechanical properties

Driving force ? Strain materials (high internal energy) → unstrained material (low internal energy)

Recrystallization : function (temp, time) Recrystallization temperature : the temp at which recrystallization just reaches completion in 1

hr. ～ 1/3 ~ 1/2 of absolute melting temperature Ex. Recrystalline melting temp Al 80 ℃ (353 K) 660 ℃ (933 K)

§ 7.13 Grain Growth Keep at elevated temp → Grain Growth 不需經過 recovery & recrystallization 一般的 polycrystals 在室溫中也會 grow Driving force for grain growth : lower G.B

Grain boundary (G.B) : high energy Grain size ↑, total G.B ↓, → total energy ↓

ktdd nn =− 0

D0 : grain diameter at t = 0 K, n : constant , n ≧ 2