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 (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