Theories of failure

This describes the need of various theories of material failure and how they can be represented graphically.

1. Theories of Material Failure Akshay Mistri Wayne State University

2. What are theories of failure? Material strengths are determined from uni-axial tension tests. Thus, the strengths obtained from those tension tests cannot be directly used for component design since, in actual scenarios components undergo multi-axial stress conditions. Hence, to use the strengths determined from tension tests to design mechanical components under any condition of static loading, theories of failure are used.

3. How many theories are there? Maximum Principal Stress Theory (Rankine’s Theory). Maximum Shear Stress Theory (Tresca-Guest Theory). Maximum Principal Strain Theory (Venant’s Theory). Total Strain Energy Theory (Haigh’s Theory). Maximum Distortion Energy Theory (Von-Mises and Hencky’s Theory).

4. Maximum Principal Stress Theory Designed for brittle materials. Can be used for ductile materials in special occasions: Uni-axial or bi-axial loading (if principal stresses are of similar nature). Hydrostatic stress condition (no shear stresses). Condition for safe design: → Maximum principal stress ( ) ≤ Permissible stress ( , shown below). → ≤ or Where: and are yield strength and ultimate strength, respectively. N is the factor of safety.  

5. Maximum Principal Stress Theory Safe zone : Area inside the square

6. Maximum Shear Stress Theory Suitable for ductile materials as these are weak in shear. Gives over safe design which could be uneconomic sometimes. Condition for safe design: → Max. shear stress ( ) ≤ Permissible shear stress ( ). → ≤ . → For 3D stresses, larger of {| - |, | - |, | - |} ≤ . → For biaxial state, = 0, ≤ when , are like in nature, else, ≤ .  

7. Maximum Shear Stress Theory Safe zone : Area inside the hexagon

8. Maximum Principal Strain Theory Condition for safe design, → Max. Principal Strain ( ) ≤ Permissible Strain ( ). Where is strain at yield. → ≤ . Where is yield strength, → [ - µ( + )] ≤ . E is Youngs modulus, → - µ( + ) ≤ . N is factor of safety.  

9. Safe zone : Area inside the rhombus For biaxial state of stress , condition for safe design, - µ( )] ≤   Maximum Principal Strain Theory

10. Total Strain Energy Theory Condition for safe design , → Total Strain Energy per unit volume (TSE/vol) ≤ Strain energy per unit volume at yield point . → [ + + ] ≤ . - (1) Where E.L elastic limit , = [ - µ( + )] ; similarly and . - (a), (b), (c) → Using (a), (b), (c) in (1), we get → [LHS] = TSE/vol = [ + + -2 µ( + + ) ] - (2) → [RHS] = For [TSE/vol] at yield point we can use , = = , = = 0 . in (2), to get TSE/vol at yield = Therefore, condition for safe design : + + -2 µ( + + ) ≤  

11. Safe zone : Area inside the ellipse For biaxial state of stress ( = 0 ), condition for safe design, + -2 µ ≤   Total Strain Energy Theory

12. Total Distortion Energy Theory Condition for safe design, → Max Distortion Energy per unit volume (DE/vol) ≤ Distortion energy per unit volume at yield point (DE/Vol @ yield). → Now, TSE/vol = Volumetric SE/Vol + DE/Vol → DE/Vol = TSE/vol - Volumetric SE/Vol ; Here we already know TSE/Vol from equation 2 in slide 7. → Volumetric SE/Vol = ½ (Avg. Stress)(Volumetric Strain) = )[( ) ( )] → Substituting the above we get, DE/Vol = [ + + ] - (1) At yield point = = , = = 0 - (2) → Using (2) in (1), DE/Vol @ yield = Therefore, c ondition for safe design: [ + + ] ≤ 2  

13. Safe zone : Area inside the ellipse For biaxial state of stress ( = 0 ) , condition for safe design, + - ≤   Total Distortion Energy Theory

14. References The Gate Academy: http://thegateacademy.com/files/wppdf/Theories-of-failure.pdf Book: Introduction to Machine Design by VB Bhandari NPTEL: https://nptel.ac.in/course.html