1. Ken Youssefi Mechanical Engineering Dept., SJSU 1 Failure Theories – Static Loads Static load – a stationary load that is gradually applied having an unchanging magnitude and direction Failure – A part is permanently distorted and will not function properly. A part has been separated into two or more pieces. Material Strength S y = Yield strength in tension, S yt = S yc S ys = Yield strength in shear S u = Ultimate strength in tension, S ut S uc = Ultimate strength in compression S us = Ultimate strength in shear =.67 S u

2. Ken Youssefi Mechanical Engineering Dept., SJSU 2 Ductile and Brittle Materials 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. Brittle material yields very little before fracturing, the yield strength is approximately the same as the ultimate strength in tension. The ultimate strength in compression is much larger than the ultimate strength in tension.

3. Ken Youssefi Mechanical Engineering Dept., SJSU 3 Failure Theories – Ductile Materials Maximum shear stress theory (Tresca 1886) Yield strength of a material is used to design components made of ductile material  = S y  SySy 2 = (  max ) component > (  ) obtained from a tension test at the yield point Failure (  max ) component < SySy 2 To avoid failure  max = SySy 2 n n = Safety factor Design equation

4. Ken Youssefi Mechanical Engineering Dept., SJSU 4 Failure Theories – Ductile Materials Distortion energy theory (von Mises-Hencky) Hydrostatic state of stress → ( S y ) h hh hh hh tt tt Simple tension test → ( S y ) t (Sy)t(Sy)t (Sy)h(Sy)h >> Distortion contributes to failure much more than change in volume. (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

5. Ken Youssefi Mechanical Engineering Dept., SJSU 5 Failure Theories – Ductile Materials The area under the curve in the elastic region is called the Elastic Strain Energy. Strain energy U = ½  ε 3D case U T = ½  1 ε 1 + ½  2 ε 2 + ½  3 ε 3 ε1ε1 = 11 E 22 E 33 E v v ε2ε2 = 22 E 11 E 33 E v v ε3ε3 = 33 11 E 22 E v v Stress-strain relationship E U T = (  1 2 +  2 2 +  3 2 ) - 2 v (  1  2 +  1  3 +  2  3 ) 2E2E 1

6. Ken Youssefi Mechanical Engineering Dept., SJSU 6 Failure Theories – Ductile Materials U d = U T – U h Distortion strain energy = total strain energy – hydrostatic strain energy Substitute  1 =  2 =  3 =  h U h = (  h 2 +  h 2 +  h 2 ) - 2 v (  h  h +  h  h +  h  h ) 2E2E 1 Simplify and substitute  1 +  2 +  3 = 3  h into the above equation U h = ( 1 – 2 v ) = 2E2E 3h23h2 6E6E (1 – 2v)(1 – 2v) (1 + 2 + 3)2(1 + 2 + 3)2 U d = U T – U h = 6E6E 1 + v (  1 –  2 ) 2 + (  1 –  3 ) 2 + (  2 –  3 ) 2 Subtract the hydrostatic strain energy from the total energy to obtain the distortion energy U T = (  1 2 +  2 2 +  3 2 ) - 2 v (  1  2 +  1  3 +  2  3 ) 2E2E 1 (1) (2)

7. Ken Youssefi Mechanical Engineering Dept., SJSU 7 Failure Theories – Ductile Materials Strain energy from a tension test at the yield point  1 = S y and  2 =  3 = 0 Substitute in equation (2) 3E3E 1 + v (S y ) 2 U test = To avoid failure, U d < U test (  1 –  2 ) 2 + (  1 –  3 ) 2 + (  2 –  3 ) 2 2 ½ < S y U d = U T – U h = 6E6E 1 + v (  1 –  2 ) 2 + (  1 –  3 ) 2 + (  2 –  3 ) 2 (2)

8. Ken Youssefi Mechanical Engineering Dept., SJSU 8 Failure Theories – Ductile Materials ½ 2D case,  3 = 0 (  1 2 –  1  2 +  2 2 ) < S y =  Where  is von Mises stress  ′ = SySy n Design equation (  1 –  2 ) 2 + (  1 –  3 ) 2 + (  2 –  3 ) 2 2 ½ < S y

9. Ken Youssefi Mechanical Engineering Dept., SJSU 9 Pure torsion,  =  1 = –  2 (  1 2 –  2  1 +  2 2 ) = S y 2 Failure Theories – Ductile Materials 32 = Sy232 = Sy2 S ys = S y / √ 3 → S ys =.577 S y Relationship between yield strength in tension and shear (  x ) 2 + 3(  xy ) 2 = SySy n 1/2 If  y = 0, then  1,  2 =  x / 2 ± [(  x )/ 2 ] 2 + (  xy ) 2 the design equation can be written in terms of the dominant component stresses (due to bending and torsion)

10. Ken Youssefi Mechanical Engineering Dept., SJSU 10 Design Process  ′ = SySy n  max = SySy 2n2n Maximum shear stress theoryDistortion energy theory Select material: consider environment, density, availability → S y, SuSu Choose a safety factor 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  Type of manufacturing process  Uncertainties related to stress analysis  Consequence of failure; human safety and economics  Codes and standards n CostWeightSize

11. Ken Youssefi Mechanical Engineering Dept., SJSU 11 Design Process  Use n = 1.2 to 1.5 for reliable materials subjected to loads that can be determined with certainty.  Use n = 1.5 to 2.5 for average materials subjected to loads that can be determined. Also, human safety and economics are not an issue.  Use n = 3.0 to 4.0 for well known materials subjected to uncertain loads.

12. Ken Youssefi Mechanical Engineering Dept., SJSU 12 Design Process Formulate the von Mises or maximum shear stress in terms of size. Optimize for weight, size, or cost. Select material, consider environment, density, availability → S y, SuSu Choose a safety factor Use appropriate failure theory to calculate the size. ′ =′ = SySy n  max = SySy 2n2n

13. Ken Youssefi Mechanical Engineering Dept., SJSU 13 Failure Theories – Brittle Materials One of the characteristics of a brittle material is that the ultimate strength in compression is much larger than ultimate strength in tension. S uc >> S ut Mohr’s circles for compression and tension tests. Compression test S uc Failure envelope The component is safe if the state of stress falls inside the failure envelope.  1 >  3 and  2 = 0 Tension test   S ut 33 11 Stress state

14. Ken Youssefi Mechanical Engineering Dept., SJSU 14 Failure Theories – Brittle Materials 11 S ut S uc S ut S uc Safe -S ut Cast iron data Modified Coulomb-Mohr theory 11  3 or  2 S ut S uc -S ut I II III Three design zones  3 or  2

15. Ken Youssefi Mechanical Engineering Dept., SJSU 15 Failure Theories – Brittle Materials 11 33 S ut S uc -S ut I II III Zone I  1 > 0, 2 > 02 > 0 and  1 >  2 Zone II  1 > 0, 2 < 02 < 0 and  2 < S ut Zone III  1 > 0, 2 < 02 < 0 and  2 > S ut 1 (1 ( 1 S ut 1 S uc – ) – 22 = 1 n Design equation 1 =1 = S ut n Design equation 1 =1 = S ut n Design equation