mechanical behavior of materials

1. overview of mechanical behavior2. dislocations3. plastic deformation in single and polycrystalline materials4. strengthening of crystalline materials5. high temperature deformation of crystalline materials6. fracture mechanics7. high temperature fracture8. fatigue of engineering materials

text book: mechanical behavior of materials, thomas H. courtneyreference: the principles of engineering materials

chapter 1. overview of mechanical behavior

彈性變形Elastic deformation

Hooke's Law

The extension depends on sample dimensions

normalized force F/A 定義為 stress

normalized extension(dimensionless) 定義為 strain

l - l0 = l F

l - l0 = l F

ll 0

ε =

E 為材料特性，稱為 Young’s Modulus ，為 stiffness 的指標。E is a measure of a material’s bond strength resistant to tensile deformation.

Shear stress 剪應力 τShear strain 剪應變 γ

為材料特性，稱為 shear modulus. is a measure of resistance to bond distortion.

=

=

“bonding” of atomic bonds.

1.3 Permanent Deformation ( plastic deformation )

A. Tensile test

cross head speed

gauge length

V : crosshead speedLp : gage length

理想化之応力応変曲線　 (a) 完全弾性体、 (b) 完全剛塑性体、 (c) 直線硬化剛塑性体、 (d) 完全弾塑性体、 (e) 直線硬化弾塑性体

Ｐ： 比例限Ｅ： 弾性限σyu ： 上降伏応力（ upper yield stress ）σyl ： 下降伏応力（ lower yield stress ）σB ： 引張強度（ tensile strength ）ｅ u ： 均一伸長（ uniform elongation ）ｅ l ： 局部伸長（ local elongation ）ｅ f ： 全伸長（破断伸長； total elongation ）

Beginning of necking

Work hardening: Plastic flow (flow stress) 隨strain 增加而增高。

Work hardening → dislocation 增殖 (multiplication)

necking

工程應變 ( engineering strain )真應變 ( true strain )

l0

2l0

4l0

2 l 0− l 0l 0

=𝟏𝟎𝟎%

4l 0−2 l 02 l 0

=𝟏𝟎𝟎%

4 l 0− l 0l 0

=𝟑𝟎𝟎%

Total elongation = 200%

𝑻=𝒍𝒏l il0

= 2

= 2

𝟐 𝒍𝒏𝟐

= 22 =2

Const-Volume Condition 拉伸過程中，截面積 ( cross section ) 減少，( 體積不變條件下，長度增加，截面積減 )

定義 真應力 ( true stress )

有別於Engineering stress

同樣，真應變 true strain ( based on instantaneous sample length )

=

=

= ……..= ()

= ∫𝑙 0

𝑙𝑑𝑙𝑙=𝑙𝑛 𝑙

𝑙𝑜

l0 li

= () Because l0 li

li l0

= = 1+

= () =(1+ ) = (1+)

>

(1+) <

(1+) <

Work hardening

< Eu, flow stress at work hardening > work softening due to the decreasing of cross section

Criterion for necking dF = 0T = F/ Ai F=TAi

dF = 0 = T dAi + AidT

dT/T = -dAi/Ai

Fraction increase in flow stress (dT/T) =Decrease in load-bearing area (dAi/Ai

塑性不安定 (plastic instability)

局所変形産生、只有長度Ｌ的区域有 L + dL （ dL>0 ）的変形、断面積也減少 A + dA （ dA<0 ）。同時、加工硬化 ( strain hardening, work-hardening) 使此区域再産生塑性変形之応力将増加 σ + dσ （ dσ>0 ）、 Necking 開始条件、

体積一定条件（ AL = 一定 ）及由　　　定義、

塑性不安定条件

左辺為加工硬化率（ strain hardening rate ）

「加工硬化越大、 Necking越不容易発生、材料的塑性変形安定進行」

dT/T = -dAi/Ai plastic instability

Engineering - curve e > Eu : of little fundamental value fracture strain (f) : percentage of elongation is used. f = [(lf-l0 )/ l0 ]x100% lf is sample length at fracture

Material ductility

Eu is more of an inherent material property than f

resistance to neck development. Work hardening capability.

Reduction in area (R. A.) %RA = [(Ao-Af)/Ao]x100%

RA is no relative to the sample gage length.

index of material ductility

Before necking εEu εT < εE

after necking εT > εE

But always T > E

ε = /E 為 elastic tensile loading - 相當於 chemical bond strength.

但當達 plastic deformation 時，σ=Eε 已不成立。

Empirical equations

T = K(eT)n

n : strain-hardening coefficient is a measure of the material’s work hardening behavior ，

應變硬化指數 n=0.02-0.5K : strength coefficient (K: true stress at eT =1, )

B. Strain-Rate Sensitivity ( 應變速率敏感指數 ) m

Strain rate 增加， flow stress of material 增高， is a strong function of the temperature and is a specific to the materials.

經驗方程式 T = K' ( )m

: true strain rate ， m : strain rate sensitivity ， K' : a constant0 < m < 1 ( m = 0 ， not strain rate sensitivity ； m = 1 ， a viscous solid ， stress increase linearly with )

Metals and alloys 在一般溫度下， strain-rate sensitivity 不明顯，例如 : 大部分材料 m=0.00-0.1 at room temperature ，但 strain-rate sensitivity 隨溫度增高而變大，某些金屬可高達 0.8

” Superplasticity ” at these strain-temperature combination.

描述 strain-rate hardening in absence of strain-hardening.

而 T = K(eT)n 描述 strain hardening without strain-rate hardening.

因此當 both effect are important ，可用下式表示

T = K' ( )m

T = K” (eT)n( )m

最大主応力説

材料受任意作用力，材料某一位置 (x0; y0; z0) 之最大主応力超過其材料固有値以上時，或最小主応力低於材料固有値以下時，此位置 (x0; y0; z0) 発生降伏．

Tensile yield stressCompressive stress

Brittle materials

Rankine yield criterion for the cast iron

（２）最大剪応力説 (Tresca 説 )

材料受任意作用力，材料某一位置 (x0; y0; z0) 之最大剪応力超過其材料固有値以上時，此位置 (x0; y0; z0) 発生降伏．

The maximum shear stress at position (x0, y0, z0)

Aluminum ,

剪応変能説 (von Mises 説 )

材料受任意作用力，材料某一位置 (x0; y0; z0) 之最大剪応変能密度超過其材料固有値以上時，此位置 (x0; y0; z0) 発生降伏．

Aluminum ,

Sear strain energy density at position (x0, y0, z0)

C. Yielding Under Multiaxial Loading Condition 多軸負荷之降伏

Tresca yield criterion

The algebraic difference between the maximum and minimum normal stresses is equal to the material’s tensile yield strength , y

Von Mises yield criterion

max - min = y

1 - 2)2 + 1 - 3)2 + 2 - 3)2] 1/2

= y

D. Mohr's Circle

1 = F/A1

Fs (shear force) = Fcos(/2-θ) = Fcos θ

Ft (tensile stress)

As = A1/cos θ

Shear stress

= Fs /As = F/A1 sin θ cos θ = ½1 sin 2θ

= F /A1 cos2 θ = 1 cos2 θ = ½ 1(1 + cos 2θ)

For biaxial tension = ½(1 - 2)sin2θ = ½1 (1+cos2θ) + ½2 (1-cos2θ) = ½(1 + 2) + ½(1 - 2) cos2θ

E. The Hardness Test

a measure of a material's resistance to surface penetration by an indenter having a force applied to it 。因為產生塑變，因此硬度可視為 materials' plastic flow resistance 。Brinell hardness test 最普遍 , 10mm steel ball, applied mass 3000kg,BHN is defined as the load, F, divided by the surface area of indentation.

BHN = (N/m) Stress unit

Hardness = (2.5~3.0) y

考慮 work hardening 時，一般取 3 ，Hardness is a measure of a material's plastic flow resistance.Vickers hardness test: square-based diamond pyramid indenter, applied loads 1~120 kg.

DPH(diamond pyramid hardness) 或 VHN(Vickers hard. No.)

Micro Vickers hardness test,Rockwell hardness test

…R A, RB, RC, ..…….

Rockwell hardness number has no units, and does not relate unambiguously to material yield strength.

F. The Torsion Test

useful for studying material flow at large plastic strains.The shear strain

γ = tanΦ = rθ/L = rθ’

r: radial position , θ: displacement angle , L: axial lengthθ’: θ/L , angle of twist per unit length .

For elastic deformation, the shear stress varies linearly with radius.

= twisting moment (force x distance, N-m)

1.4 Fracture

Failure permanent deformation fractureE → deflection → plastic deformationKc→ fracture

A. Fracture toughnessB. Tensile fractureC. Creep fractureD. Fatigue fractureE. Embrittlement

A. Fracture toughness 大部分材料都含有 flaws-cracks, voids

F : stress required to propagate a surface crack of length c

α: a constant on the order of unity and dependent on the precise crack shape.Kc: 材料特性 fracture toughness,(N/m3/2)

F = = F