mechanical behavior of materials
1. overview of mechanical behavior
2. dislocations
3. plastic deformation in single and polycrystalline materials
4. strengthening of crystalline materials
5. high temperature deformation of crystalline materials
6. fracture mechanics
7. high temperature fracture
8. fatigue of engineering materials
text book: mechanical behavior of materials, thomas H. courtney
reference: the principles of engineering materials
chapter 1. overview of mechanical behavior
彈性變形
Elastic deformation
Hooke's Law
l - l0 = l  F
The extension depends on
sample dimensions
l
l - l0 = l  F( 0)
𝑨
normalized force F/A
定義為 stress 
normalized
extension(dimensionless)
定義為 strain 
l
l0
ε=

𝑬
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.
 =
l
l0
為材料特性，稱為shear modulus.
 is a measure of resistance to bond distortion.
Shear stress 剪應力 τ
Shear strain 剪應變 γ
=


“bonding” of atomic bonds.
1.3 Permanent Deformation ( plastic deformation )
A. Tensile test
gauge
length
V : crosshead speed
Lp : gage length
cross head
speed
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
Beginning of necking
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 ）
ｅu： 均一伸長（ uniform elongation ）
ｅl： 局部伸長（ local elongation ）
ｅf： 全伸長（破断伸長； total elongation ）
necking
Work hardening:
Plastic flow (flow stress)
隨strain 增加而增高。
Work hardening →
dislocation 增殖
(multiplication)
工程應變 ( engineering strain )
真應變 ( true strain )
li
𝑻 = 𝒍𝒏
l0
l0
2l0
𝒍𝒏
𝟐l 0
= 𝒍𝒏2
l0
𝟒l
𝒍𝒏 𝟐l0 = 𝒍𝒏2
0
4l0
𝟐𝒍𝒏𝟐
2l0 − l0
= 𝟏𝟎𝟎%
l0
4l0 − 2l0
= 𝟏𝟎𝟎%
2l0
Total elongation = 200%
4l0 − l0
= 𝟑𝟎𝟎%
l0
4l
𝒍𝒏 l 0 = 𝒍𝒏22 =𝟐𝒍𝒏2
0
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 )
Const-Volume Condition
Aol0 = Aili
定義 真應力
( true stress )
T = 𝐅/Ai
有別於
Engineering stress
E = 𝐅/Ao
同樣，真應變true strain ( based on instantaneous sample length )
𝒍 𝒍 𝒍 𝒍 𝒍 𝒍
𝒍
εT = l + l + l + l + l + l ……..=  ( l )
0
1
𝟐
3
4
5
i
𝒅𝒍
𝒅εT = l
𝑙
𝑙0
𝑑𝑙
𝑙
= 𝑙𝑛
𝑙
𝑙𝑜
T = 𝑙𝑛
T =
𝐹
𝐴𝑖
𝐹 𝐴𝑜
𝐴
𝑜 𝑖
=𝐴
𝑙
𝑙𝑜
𝐴
= 𝐸 ( 𝐴𝑜)
Because 𝐨𝐟 Aol0 = Aili
li = l0 + 𝒍
𝑖
𝐴𝑜
𝐴𝑖
=
l0+l
l
= 1+
l0
l0
𝐴
l
T = 𝐸 ( 𝐴𝑜) =𝐸 (1+ l ) = 𝐸 (1+E)
𝑖
0
T > 𝐸
𝑓𝑜𝑟 𝑎 𝑡𝑒𝑛𝑠𝑠𝑖𝑙 𝑡𝑒𝑠𝑡
𝑻 = 𝒍𝒏(1+𝑬 ) < 𝑬 𝒊𝒏 𝒂 𝒕𝒆𝒏𝒔𝒊𝒍𝒆 𝒕𝒆𝒔𝒕
𝒍𝒏(1+𝐱 ) < 𝐱
Work
hardening
 < Eu, flow stress at work
hardening > work softening
due to the decreasing of
cross section
Criterion for necking
dF = 0
T = F/ Ai
F=TAi
dF = 0 = T dAi + AidT
dT/T = -dAi/Ai
Fraction increase in
flow stress (dT/T)
=
Decrease in loadbearing 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」
dT/T = -dAi/Ai
plastic instability
Engineering - curve
 > 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
after necking
But always
εT < εE
εT > εE
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(T)n
n : strain-hardening coefficient
is a measure of the material’s
work hardening behavior，
應變硬化指數 n=0.02-0.5
K : strength coefficient
(K: true stress at T =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 constant
0 < 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.
T = K' (
)m
而 T = K(T)n
描述strain-rate hardening in absence of strain-hardening.
描述 strain hardening without strain-rate hardening.
因此當both effect are important ，可用下式表示
T = K” (T)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 stress
Compressive 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．
Sear strain energy density at position (x0, y0, z0)
Aluminum ,
C. Yielding Under Multiaxial Loading Condition
多軸負荷之降伏
Tresca yield criterion
max - min = y
The algebraic difference between the
maximum and minimum normal stresses
is equal to the material’s tensile yield
strength , y
Von Mises yield criterion
𝟏
√𝟐
[(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
= ½1 sin 2θ
 = F /A1 cos2 θ = 1 cos2 θ
= ½ 1(1 + cos 2θ)
For biaxial tension
 = ½(1 - 2)sin2θ
 = ½1 (1+cos2θ) + ½2 (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 =
𝟐𝑭
𝝅𝑫[𝑫− 𝑫𝟐−𝒅𝟐
/
𝟏 𝟐
]
=
𝑭
𝝅𝑫𝒅
(N/m𝟐)
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 length
θ’: θ/L , angle of twist per unit length .
For elastic deformation, the shear stress varies linearly with radius.
=
𝟑𝟐𝑴𝒕 𝒓
𝝅𝑫𝟒
𝑴𝒕: twisting moment (force x distance, N-m)
1.4 Fracture
Failure
permanent deformation
fracture
E → deflection
→ plastic deformation
Kc→ fracture
A. Fracture toughness
B. Tensile fracture
C. Creep fracture
D. Fatigue fracture
E. Embrittlement
A. Fracture toughness
大部分材料都含有flaws-cracks, voids
F =
Kc /
𝒄 𝟏 𝟐
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)
Kc = F 𝒄