Theories of failure
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This describes the need of various theories of material failure and how they can be represented graphically.
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Theories of failure
- 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 (𝜎1) ≤ Permissible stress (𝜎 𝑝𝑒𝑟, shown below). → 𝜎1 ≤ 𝑆 𝑦𝑡 𝑁 or 𝑆 𝑢𝑡 𝑁 Where: • 𝑆 𝑦𝑡 and 𝑆 𝑢𝑡 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 ( 𝜏 𝑚𝑎𝑥) ≤ Permissible shear stress ( 𝜏 𝑝𝑒𝑟). → 𝜏 𝑚𝑎𝑥 ≤ 𝑆 𝑦𝑡 2𝑁. → For 3D stresses, larger of {|𝜎1 - 𝜎2|, |𝜎2 - 𝜎3|, |𝜎3 - 𝜎1|} ≤ 𝑆 𝑦𝑡 𝑁. → For biaxial state, 𝜎3 = 0, |𝜎1| ≤ 𝑆 𝑦𝑡 𝑁 when 𝜎1, 𝜎2 are like in nature, else, |𝜎1 − 𝜎2| ≤ 𝑆 𝑦𝑡 𝑁.
- 7. Maximum Shear Stress Theory • Safe zone : Area inside the hexagon
- 8. Maximum Principal Strain Theory • Condition for safe design, → Max. Principal Strain (∈1) ≤ Permissible Strain (∈ 𝑌.𝑃 𝑁). Where ∈ 𝑌.𝑃 is strain at yield. → ∈1 ≤ 𝑆 𝑌.𝑇 𝐸𝑁. Where 𝑆 𝑌.𝑇 is yield strength, → 1 𝐸 [𝜎1 - µ(𝜎2 + 𝜎3 )] ≤ 𝑆 𝑌.𝑇 𝐸𝑁 . E is Youngs modulus, → 𝜎1 - µ(𝜎2 + 𝜎3 ) ≤ 𝑆 𝑌.𝑇 𝑁 . N is factor of safety.
- 9. • Safe zone : Area inside the rhombus • For biaxial state of stress , condition for safe design, • 𝜎1 - µ(𝜎2)] ≤ 𝑆 𝑌.𝑇 𝑁 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 2 [𝜎1 ∈1+ 𝜎2 ∈2+ 𝜎3 ∈3] ≤ 1 2 𝜎 𝐸.𝐿 ∈ 𝐸.𝐿. - (1) Where E.L elastic limit, ∈1 = 1 𝐸[𝜎1 - µ(𝜎2 + 𝜎3)] ; similarly ∈2 and ∈3 . - (a), (b), (c) → Using (a), (b), (c) in (1), we get → [LHS] = TSE/vol = 1 2𝐸 [𝜎1 2 + 𝜎2 2 + 𝜎3 2 -2 µ(𝜎1 𝜎2+ 𝜎2 𝜎3 + 𝜎3 𝜎1) ] - (2) → [RHS] = For [TSE/vol] at yield point we can use, 𝜎1 = 𝜎 = 𝑆 𝑌.𝑇 𝑁 , 𝜎2 = 𝜎3 = 0. in (2), to get TSE/vol at yield = 1 2𝐸 ( 𝑆 𝑦𝑡 𝑁)2 • Therefore, condition for safe design : 𝜎1 2 + 𝜎2 2 + 𝜎3 2 -2 µ(𝜎1 𝜎2+ 𝜎2 𝜎3 + 𝜎3 𝜎1) ≤ ( 𝑆 𝑦𝑡 𝑁)2
- 11. • Safe zone : Area inside the ellipse • For biaxial state of stress ( 𝜎3= 0), condition for safe design, • 𝜎1 2 + 𝜎2 2 -2 µ𝜎1 𝜎2 ≤ ( 𝑆 𝑦𝑡 𝑁)2 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) = 1 2 ( 𝜎1+ 𝜎2+ 𝜎3 3 )[(1 −2µ 𝐸 ) (𝜎1 + 𝜎2 + 𝜎3 )] → Substituting the above we get, DE/Vol = 1+ µ 6𝐸 [(𝜎1− 𝜎2)2 + (𝜎2− 𝜎3)2 + (𝜎3− 𝜎1)2 ] - (1) At yield point 𝜎1 = 𝜎 = 𝑆 𝑌.𝑇 𝑁 , 𝜎2 = 𝜎3 = 0 - (2) → Using (2) in (1), DE/Vol @ yield = 1+ µ 3𝐸 ( 𝑆 𝑌.𝑇 𝑁 ) 2 • Therefore, condition for safe design: [(𝜎1− 𝜎2)2 + (𝜎2− 𝜎3)2 + (𝜎3− 𝜎1)2 ] ≤ 2 ( 𝑆 𝑌.𝑇 𝑁 ) 2
- 13. • Safe zone : Area inside the ellipse • For biaxial state of stress ( 𝜎3= 0), condition for safe design, • 𝜎1 2 + 𝜎2 2 - 𝜎1 𝜎2 ≤ ( 𝑆 𝑌.𝑇 𝑁 ) 2 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