# Strength of materials - Chapter 5: Failure theories - Nguyễn Sỹ Lâm

CHAPTER 5: FAILURE THEORIES
A ductile material deforms significantly before fracturing. Ductility is measured by %
elongation at the fracture point. Materials with 5% or more elongation are considered
ductile.

FAILURE THEORIES – DUCTILE MATERIALS
Yield strength of a material is used to design components made of
ductile material 16 trang thamphan 26/12/2022 1160
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1. CHAPTER 5: FAILURE THEORIES Failure – A part is permanently distorted (bị bóp méo) and will not function properly. A part has been separated into two or more pieces. Material Strength Sy = Yield strength in tension, Syt = Syc Sys = Yield strength in shear Su = Ultimate strength in tension, Sut Suc = Ultimate strength in compression Sus = Ultimate strength in shear = 0.67 Su
2. FAILURE THEORIES – DUCTILE MATERIALS Yield strength of a material is used to design components made of ductile material • Maximum shear stress theory (Tresca 1886) (Thuyết bền: ứng suất tiếp lớn nhất – TB3) (max )component > ( )obtained from a tension test at the yield point Failure  Sy  = Sy  = 2 To avoid failure S ( ) < y max component 2  = Sy Sy  = n = Safety factor max 2 n Design equation  =Sy
3. FAILURE THEORIES – DUCTILE MATERIALS • Distortion energy theory (von Mises-Hencky) (Thuyết bền: Thế năng biến đổi hình dáng lớn nhất – TB4) Simple tension test → (S ) y t Hydrostatic state of stress → (Sy)h t (Sy)h >> (Sy)t h Distortion contributes to failure much more than h change in volume. h t (total strain energy) – (strain energy due to hydrostatic stress) = strain energy due to angular distortion > strain energy obtained from a tension test at the yield point → failure
4. FAILURE THEORIES – DUCTILE MATERIALS Distortion strain energy = total strain energy – hydrostatic strain energy Ud = UT – Uh 1 2 2 2 UT = (1 + 2 + 3 ) - 2v (12 + 13 + 23) (1) 2E Substitute 1 = 2 = 3 = h 1 U = ( 2 +  2 +  2) - 2v (  +   +   ) h 2E h h h h h h h h h Simplify and substitute 1 + 2 + 3 = 3h into the above equation 2 2 3h (1 + 2 + 3) (1 – 2v) Uh = (1 – 2v) = 2E 6E Subtract the hydrostatic strain energy from the total energy to obtain the distortion energy 1 + v 2 2 2 (2) U = U – U = (1 – 2) + (1 – 3) + (2 – 3) d T h 6E
5. FAILURE THEORIES – DUCTILE MATERIALS 2 2 2 (1 – 2) + (1 – 3) + (2 – 3) ½ < Sy 2 2D case, 3 = 0 2 2 ½ (1 – 12 + 2 ) < Sy =  Where  is von Mises stress Sy ′ = Design equation n
6. FAILURE THEORIES – DUCTILE MATERIALS Distortion energy theory Maximum shear stress theory Sy Sy ′ =  = n max 2n • Select material: consider environment, density, availability → Sy , Su • Choose a safety factor n Size Weight Cost The selection of an appropriate safety factor should be based on the following:  Degree of uncertainty about loading (type, magnitude and direction)  Degree of uncertainty about material strength  Uncertainties related to stress analysis  Consequence of failure; human safety and economics  Type of manufacturing process  Codes and standards
7. FAILURE THEORIES – DUCTILE MATERIALS Design Process • Select material, consider environment, density, availability → Sy , Su • Choose a safety factor • Formulate the von Mises or maximum shear stress in terms of size. • Use appropriate failure theory to calculate the size. Sy Sy ′ =  = n max 2n • Optimize for weight, size, or cost.
8. FAILURE THEORIES – BRITTLE MATERIALS Modified Coulomb-Mohr theory (Thuyết Bền Morh – TB5) 3 or 2 3 or 2 S ut Sut Safe Safe S I ut Sut 1 S uc Safe II -Sut Safe III S uc Suc Cast iron data Three design zones