mechanical behavior of materials 1. overview of mechanical behavior 2. dislocations

1. 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. fracture mechanics 6. High temperature deformation of crystalline materials 7. high temperature fracture 8. fatigue of engineering materials text book: mechanical behavior of materials, thomas H. courtney reference: the principles of engineering materials

2. 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

3. 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 τ Shear strain 剪應變 γ  =  = 為材料特性，稱為shear modulus. is a measure of resistance to bond distortion.

4. “bonding” of atomic bonds.

5. 1.3 Permanent Deformation ( plastic deformation ) A. Tensile test gauge length V : crosshead speed Lp : gage length cross head speed

6. 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

7. 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 ）

8. necking Work hardening: Plastic flow (flow stress) 隨strain 增加而增高。 Work hardening → dislocation 增殖 (multiplication)

9. 工程應變 ( engineering strain ) 真應變 ( true strain ) l0 = 2 2l0 = 2 4l0 = 22 =2 Total elongation = 200%

11. 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) l0li 定義真應力( true stress ) = 有別於 Engineering stress • = 同樣，真應變true strain ( based on instantaneous sample length ) =……..=  () =

12.  = () Because l0li li l0 = = 1+ = () =(1+ ) = (1+) >  (1+) < (1+) <

13.  < Eu, flow stress at work hardening > work softening due to the decreasing of cross section Work hardening Criterion for necking dF = 0 T = F/ Ai F=TAi dF = 0 = TdAi + AidT dT/T = -dAi/Ai Fraction increase in flow stress (dT/T) = Decrease in load-bearing area (dAi/Ai

14. 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」

15. 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% lfis 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

17. Before necking εEuεT < εE after necking εT> εE But always T > E

18. 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=0.02-0.5 K : strength coefficient (K: true stress ateT=1, )

20. 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.

21. 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

22. 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

23. Rankine yield criterion for the cast iron

24. （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 ,

25. 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 ,

26. C. Yielding Under Multiaxial Loading Condition 多軸負荷之降伏 Tresca yield criterion max - min = y I = y, II = III = 0 y = y/2 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)2 + 1 - 3)2 + 2 - 3)2]1/2 = y y = y/ 15% larger than that of Tresca criterion

27. D. Mohr's Circle 1 = F/A1 Fs (shear force) = Fcos(/2-θ) = Fcosθ Ft (tensile stress) As = A1/cosθ Shear stress   = Fs/As = F/A1 sin θcosθ = ½1 sin 2θ  = F/A1 cos2θ = 1cos2θ = ½ 1(1 + cos2θ)

28. For biaxial tension  = ½(1 - 2)sin2θ = ½1 (1+cos2θ) + ½2 (1-cos2θ) = ½(1 + 2) + ½(1 - 2) cos2θ

31. 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

32. 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.)

33. 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.

42. 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 .

43. For elastic deformation, the shear stress varies linearly with radius.  = twisting moment (force x distance, N-m)

44. 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

45. A. Fracture toughness 大部分材料都含有flaws-cracks, voids F= = F 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)