fracture analysis

1. Fracture and Failure Analysis
MME 4228

2. Basics of vector algebra
Fracture and Failure Analysis (MM4228), S. Guha, 2024 2
e2
e1
e3
T
T
1
T
2
T
3
ei are the unit base vectors
Properties of the unit base vectors
O
A

3. Traction vectors
Fracture and Failure Analysis (MM4228), S. Guha, 2024 3
Traction vector (stress vector) definition
Cauchy’s Lemma
1
2
3
}

4. Cauchy Stress (in 2D)
Fracture and Failure Analysis (MM4228), S. Guha, 2024 4
1
2
Traction
Vector on a
plane with
unit normal
‘n’ Cauchy
Stress
Tensor
Unit normal
to the plane
considered
Plane Stress Condition

5. Equilibrium Equations in 2D in absence of body forces
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1
2
Force Balance in respective directions
Some useful identity
Along with appropriate boundary conditions solutions for simple problems could be obtained using
analytical methods

6. Equilibrium Equations in 2D
Fracture and Failure Analysis (MM4228), S. Guha, 2024 6
1
2
Moment Balance about ‘C’
B
A
C
D

7. Solution for simple 2D problems
Fracture and Failure Analysis (MM4228), S. Guha, 2024 7
Steps
● Use Cauchy Stress Principle and traction boundary conditions
● Solve equilibrium equations to obtain the stress fields
Traction boundary condition on CD
1
2
Uniaxial loading of a
rectangular block
B
A
C
D
Cauchy Stress Principle
Note that on the boundary CD, n1 = 0 and n2 = 1, therefore
on boundary CD
Constant traction σ applied on the boundaries
AB and CD

8. Methodology for simple 2D problems
Fracture and Failure Analysis (MM4228), S. Guha, 2024 8
Equilibrium Equation on boundary CD
Along the boundary CD, σ12 is zero, therefore,
M
1
2
B
A
C
D
M′
δ
x2
C
B
δ
M′
● Same conditions are valid at an arbitrary section MM′ located at a distance of δ
● For any value of δ the above conditions should satisfy which leads to
along x1 for all x2
} This condition
holds everywhere
in the domain

9. Methodology for simple 2D problems
Fracture and Failure Analysis (MM4228), S. Guha, 2024 9
Traction boundary condition on BC and AD
1
2
Uniaxial loading
B
A
C
D
Cauchy Stress Principle
Note that on the boundary BC, n1 = 1 and n2 = 0, therefore
on boundary BC
, everywhere in the domain
State of stress under uniaxial tension of an uniform section

10. Solution for simple 2D problems
Fracture and Failure Analysis (MM4228), S. Guha, 2024 10
1
2
pure shear of a
rectangular block
B
A
C
D
Constant traction て applied on the
boundaries AB and CD

11. Constitutive Relations and Strain-Displacement relations
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Plane Stress Condition
Generalized Hooke’s Law for isotropic linear elasticity
Loading only in x-y plane (in-
plane loading)

12. Constitutive Relations and Strain-Displacement relations
Fracture and Failure Analysis (MM4228), S. Guha, 2024 13
Plane Strain Condition

13. Solution approach for plane elasticity problems
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● Three independent components
● Two equilibrium equations
Need one more equation - Compatibility
Condition
Stress Approach
Displacement Approach
Replace constitutive law in equilibrium
equations
Replace strain-displacement relation
Differential equation in terms of
displacement component u1 and u2

14. Compatibility Equations in plane elasticity
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Strain Compatibility
Stress Compatibility

15. Airy’s Stress Function
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