Search for notes by fellow students, in your own course and all over the country.

Browse our notes for titles which look like what you need, you can preview any of the notes via a sample of the contents. After you're happy these are the notes you're after simply pop them into your shopping cart.

My Basket

You have nothing in your shopping cart yet.

Title: Stress and Strain Failure Theories
Description: Stress and strain failure theories provide critical insights into how materials behave under various loads and conditions. These theories, such as Maximum Stress Theory, Maximum Strain Theory, Von Mises Stress, and Tresca Criterion, help engineers predict when a material will fail due to excessive deformation or fracture. By analyzing stress and strain relationships, these theories enable the safe design of structures and components, ensuring they can withstand applied forces without failure. Essential in fields like mechanical engineering and materials science, these theories aid in optimizing material performance and structural integrity

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above


Chapter 4: Failure Theories
4
...
1 Macromechanical Failure Theories
4
...
1 Maximum Stress Theory
4
...
2 Maximum Strain Theory
4
...
3 Tsai-Hill Theory (Deviatoric energy theory)
4
...
4 Tsai-Wu Theory (Interactive tensor theory)
4
...
3 Description of Failure Theories
4
...
1 Maximum Stress Theory
4
...
2 Maximum Strain Theory
4
...
3 Tsai-Hill Theory (Deviatoric energy theory)
4
...
4 Tsai-Wu Theory (Interactive tensor theory)
4
...
5 Application Structural Analysis

4
...

Failure Criterion: A criterion used to hypothesize the failure
...


Why Need Failure Theories?
(a) To design structural components and calculate margin of safety
...

(c) To determine weak and strong directions
...

Failure in metallic materials is characterized by Yield Strength
...

(b) Maximum principal strain theory
...


4
...
Macromechanical Failure Theories in Composite Materials
a
...
Maximum Strain Theory
c
...
Tsai-Wu Theory (Interactive tensor polynomial theory)
4
...
Application of Failure Theory
First step is to calculate the stresses/strains in the material principal directions
...


Ply Stresses:

{σ } x − y = [Tσ ]{σ }1− 2

or

{σ }1− 2 = [Tσ ] {σ } x − y
−1

Ply strains:

{ε }1− 2 = [Q]1− 2 {σ }1− 2
Now apply the failure criteria in the material coordinate system
...
3
...

σ2

σ2

Tensile stresses:

σ 1 ≥ F1t

Fiber break

σ 2 ≥ F2t

Matrix crack

Compressive stresses:

σ 1 ≤ F1c

Fiber crushing

σ 2 ≤ F2c

Matrix yielding

σ1

σ1
σ2

F 2t
F 1c

σ1

No failure

F 1t
F 2c

Shear stresses:

σ 12 ≥ F6

or

σ 6 ≥ F6

Shear crack

Note there is no interaction between the stress components
...
005
ε1tu = 0
...
28
ν21 = 0
...
Maximum Stress Theory
F1t
⇒ Longitudinal Tension
Cos 2θ
F
or σ x = 22t
⇒ Transverse Tension
Sin θ

σ 1 = σ x Cos 2θ @ failure σ 1 = F1t or σ x =
σ 2 = σ x Sin 2θ @ failure σ 2 = F2t

F1c
⇒ Longitudinal Compression
Cos 2θ
F
or σ x = − 2c2 ⇒ Transverse Compression
Sin θ

σ 1 = σ x Cos 2θ @ failure σ 1 = F1c or σ x = −
σ 2 = σ x Sin 2θ @ failure σ 2 = F2c

τ 6 = −σ x CosθSinθ @ failure τ 6 = F6 or σ x = ±

F6
CosθSinθ

⇒ Shear

Uniaxial Strength of an Off-Axis Lamina
Maximum Stress Theory
y

L-Tension

1200

σx

1000
800

MPa

σx

x2

Shear

600

σx

x1

x

400
200

T-tension

0
Shear

-200

T-Compression

-400
L-Compression

-600
-800
0

10

20

30

40

50

θ , deg

60

70

80

90

4
...
2 Maximum Strain Theory:
Failure occurs when at least one of the strain components along the
principal material axis exceeds that of the ultimate strain in that direction
...
3
...

Azzi-Tsai extended this equation to anisotropic fiber reinforced composites
...

Aσ 12 + Bσ 22 + Cσ 1σ 2 + Dτ 62 = 1
From longitudinal, transverse, and shear tests on a uniaxial laminate,
A, B, and D are determined
...

C1=-1/F12

Tsai-Hill failure criterion:
σ 12 σ 22 σ 1σ 2 τ 62
+ 2 − 2 + 2 =1
2
F1 F2
F1
F6

σ 12 σ 22 σ 1σ 2
2
2 + 2 −
2 = 1−κ
F1 F2
F1

Note: No distinction is made between tensile & compression strengths
...
3
...
It is stated as

fiσ i + fij σ iσ j = 1

I,j=1,2,3,4,5,6

For plane-stress condition:

f1σ 1 + f 2σ 2 + f6 τ 6 + f11σ 12 + f 22σ 22 + f66 τ 62 + +2 f12σ 1σ 2 + 2 f16 σ 1τ 6 + 2 f 26 σ 2τ 6 = 1

Shear strength is independent of sign of the shear stress, therefore all
liner shear stress terms must vanish
...
We get

aσ x2

+ bσ x − 1 = 0

y

σx

x1

σx

x2

Where

x

a = f11Cos 4θ + f 22 Sin 4θ + 2 f12 Cos 2θSin 2θ + f66 Cos 2θSin 2θ
b = f1Cos 2θ + f 2 Sin 2θ
Solution is:

− b ± b 2 + 4a
σx =
2a

Uniaxial Strength of an Off-AxiLamina
Tsai-Hill & Tsai-Wu Theories
y

1200

x1

1000
800

σx

600

σx
MPa

Tsai-Hill

400

σx

x2

Tsai-Wu

x

200
0
-200
-400

Tsai-Hill
Tsai-Wu

-600
-800
0

10

20

30

40

50

θ, deg

60

70

80

90

3
...
strain
F2t
-F1c
F1t

Ductile behavior of
Can be programmed
anisotropic
Different functions
Deviatoric
materials
required for tensile
strain energy
"Curve fitting" for
and compressive
(Tsai-Hill)
heterogeneous
strenghts
brittle composites

Interactive
tensor
polynomial

Mathematically
consistent
Reliable "curve
fitting"

Biaxial testing is
needed in addition to
uniaxial testing

Numerous parameters
General and
Comprehensive
comprehensive;
experimental program
operationally simple
needed

Home work:Problems 4
...
15 even numbers only
...
stress

σ1


Title: Stress and Strain Failure Theories
Description: Stress and strain failure theories provide critical insights into how materials behave under various loads and conditions. These theories, such as Maximum Stress Theory, Maximum Strain Theory, Von Mises Stress, and Tresca Criterion, help engineers predict when a material will fail due to excessive deformation or fracture. By analyzing stress and strain relationships, these theories enable the safe design of structures and components, ensuring they can withstand applied forces without failure. Essential in fields like mechanical engineering and materials science, these theories aid in optimizing material performance and structural integrity