Failure theories

Failure theoriesLecture 6 – failure theories

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Failure theories

Introduction

• Uniaxial tensile test provides information about the constitutive response of a material under simple stress state.

• Real components are subjected to complex stress state: multiaxial state of stress

• The possibility to experimentally probe the material under multi-axial state of stress is limited.

• We need to follow a different approach: to find, if exists, the way to calculate an “equivalent” uniaxial stress that causes the same effect as for the given multi-axial state of stress

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Failure theories

Introduction

• From the mathematical point of view, this means finding a relationship such as:

• To determine such function, the following approach is followed:

1. Make a hypothesis of equivalence

2. Derive the expression of seq for the generic three dimensional stress state

3. Derive the expression of seq for a biaxial (plane) stress state as a function of in plane components (sx, sy, txy)

4. Calculate the ratio between the limit states sL/tL

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𝜎𝑒𝑞 = 𝑓 𝜎𝑖𝑗

• Limit state indicates the maximum stress state that the material can tolerate before failure

• These values also indicated as “allowables”

• Depends on the material behavior

• For static loads,

• Ductile materials:

• Brittle materials

𝜎𝐿 = 𝜎𝑌

𝜏𝐿 = 𝜏𝑌

𝜎𝐿 = 𝜎𝑅

𝜏𝐿 = 𝜏𝑅

Failure theories

Failure theories

• These relationships or criteria are also known as failure theories since they provide the equivalence relationship between two “critical” stress states: the uniaxial and the multiaxial.

• Different equivalence relationship have been proposed based on material response observed in experiments

• Non of them is better than the other: some fit better than other for specific material classes!

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Failure theories

Failure theories: Maximum normal stress (or Rankine criterion)

1. Assumption: the limit state is predicted to occur at the material point when the maximum principal stress reaches the limit value sL

2. For the generic multiaxial state of stress the critical condition becomes:

Similar condition is obtained for the uniaxial stress state (only one stress component)

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𝜎1 = 𝜎𝐿

𝜎3 = 𝜎𝐿

tensile

compression

𝜎𝑒𝑞 = 𝜎1

𝜎𝑒𝑞 = 𝜎3

tensile

compression

Therefore, the limit state (failure) is predicted to occur when:

3. In the case of plane stress, the equivalent stress as a function of the in plane stress components is obtained from the Mohr circle:

4. For torsion or simple shear:

At failure:

𝜎𝑒𝑞 ≥ 𝜎𝐿

𝜎𝑒𝑞 =𝜎𝑥 + 𝜎𝑦

2±

𝜎𝑥 + 𝜎𝑦

2

2

+ 𝜏𝑥𝑦2

𝜎𝑒𝑞 = 𝜏𝑥𝑦

𝜎𝑒𝑞 = 𝜏𝐿 = 𝜎𝐿 →𝜎𝐿𝜏𝐿

= 1

Failure theories

Failure theories: Maximum normal stress (or Rankine criterion)

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𝜎1

𝜎2

safe

unsafe

𝜎𝐿

𝜎𝐿

Failure theories

Failure theories: Maximum deformation (or Saint-venant criterion)

1. Assumption: the limit state is predicted to occur at the material point when the maximum principal deformation reaches the limit value eL

2. For the generic multiaxial state of stress the critical condition becomes:

For the uniaxial case:

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𝜀1 =1

𝐸𝜎1 − 𝜈 𝜎2 + 𝜎3 = 𝜀𝐿

𝜀3 =1

𝐸𝜎3 − 𝜈 𝜎2 + 𝜎1 = 𝜀𝐿

tensile

compression

𝜀1 = 𝜎𝑒𝑞/𝐸 → 𝜀𝐿 = 𝜎𝐿/𝐸

Therefore, the limit state (failure) is predicted to occur when:

3. In the case of plane stress:

4. For torsion or simple shear:

At failure:

𝜎𝑒𝑞 ≥ 𝜎𝐿

𝜎𝑒𝑞 = 1 − 𝜈𝜎𝑥 + 𝜎𝑦

2+ 1 + 𝜈

𝜎𝑥 + 𝜎𝑦

2

2

+ 𝜏𝑥𝑦2

𝜎𝑒𝑞 = 1 + 𝜈 𝜏𝑥𝑦

𝜎𝑒𝑞 = 1 + 𝜈 𝜏𝐿 = 𝜎𝐿 →𝜎𝐿𝜏𝐿

= 1 + 𝜈

Failure theories

Failure theories: Maximum deformation (or Saint-venant criterion)

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𝜎1

𝜎2

safe

unsafe

𝜎𝐿

𝜎𝐿

Failure theories

Failure theories: Maximum shear (or Tresca criterion)

1. Assumption: the limit state is predicted to occur at the material point when the maximum shear stress reaches the limit value tL

2. For the generic multiaxial state of stress the critical condition becomes:

For the uniaxial case:

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𝜏𝑚𝑎𝑥 =1

2𝜎1 − 𝜎3 = 𝜏𝐿

𝜏𝑚𝑎𝑥 = 𝜎𝑒𝑞/2 → 𝜏𝐿 = 𝜎𝐿/2

Therefore, the limit state (failure) is predicted to occur when:

3. In the case of plane stress:

4. For torsion or simple shear:

At failure:

𝜎𝑒𝑞 ≥ 𝜎𝐿

𝜎𝑒𝑞 = 𝜎𝑥 − 𝜎𝑦2+ 4𝜏𝑥𝑦

2

𝜎𝑒𝑞 = 2𝜏𝑥𝑦

𝜎𝑒𝑞 = 2𝜏𝐿 = 𝜎𝐿 →𝜎𝐿𝜏𝐿

= 2

Failure theories

Failure theories: Maximum shear stress (or Tresca criterion)

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𝜎1

𝜎2

safe

unsafe

𝜎𝐿

𝜎𝐿

Failure theories

Failure theories: Maximum distortion energy (or Von Mises criterion)

1. Assumption: the limit state is predicted to occur at the material point when the distortion energy reaches the limit value EL

2. For the generic multiaxial state of stress the critical condition becomes:

For the uniaxial case:

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𝐸𝑒𝑞 =1

12𝐺𝜎𝑒𝑞

2→ 𝐸𝐿 =

1

12𝐺𝜎𝐿

2

Therefore, the limit state (failure) is predicted to occur when:

3. In the case of plane stress:

4. For torsion or simple shear:

At failure:

𝜎𝑒𝑞 ≥ 𝜎𝐿

𝜎𝑒𝑞 = 𝜎𝑥2 + 𝜎𝑦

2 − 𝜎𝑥𝜎𝑦 + 3𝜏𝑥𝑦2

𝜎𝑒𝑞 = 3𝜏𝑥𝑦

𝜎𝑒𝑞 = 3𝜏𝐿 = 𝜎𝐿 →𝜎𝐿𝜏𝐿

= 3

𝐸 =1

12𝐺𝜎1 − 𝜎2

2 + 𝜎2 − 𝜎32 + 𝜎3 − 𝜎1

2 = 𝐸𝐿

Failure theories

Failure theories: Maximum shear stress (or Tresca criterion)

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𝜎1

𝜎2

safe

unsafe

𝜎𝐿

𝜎𝐿

Failure theories

Failure theories: Westergaard rapresentation

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𝜎1

𝜎2

safe

unsafe

𝜎𝐿

𝜎𝐿Max strain

Max stress

Von Mises

Tresca

Failure theories

Failure theories application to materials

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DUCTILE

BRITTLE

TRESCAVON MISES

Max stressMohr

Failure theories

Failure theories application to materials

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• Failure theories do not address any specific mechanism of failure

• They are so called “abrupt criteria”

• Do not take into account of the progressive deterioration of the material (damage)

• Are simple but uncoupled with the dissipative processed (i.e. plastic deformation)

• Good for simple, conservative design

Failure theories

Suggested reading

• Brnic, Josip. Analysis of Engineering Structures and Material Behavior. John Wiley & Sons, 2018.

Mechanical Engineering Design - N.Bonora 2018