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

4
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Failure Criterion: A criterion used to hypothesize the failure
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Why Need Failure Theories?
(a) To design structural components and calculate margin of safety
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(c) To determine weak and strong directions
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Failure in metallic materials is characterized by Yield Strength
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(b) Maximum principal strain theory
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4
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Macromechanical Failure Theories in Composite Materials
a
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Maximum Strain Theory
c
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Tsai-Wu Theory (Interactive tensor polynomial theory)
4
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Application of Failure Theory
First step is to calculate the stresses/strains in the material principal directions
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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
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3
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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
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005
ε1tu = 0
...
28
ν21 = 0
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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
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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
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3
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Azzi-Tsai extended this equation to anisotropic fiber reinforced composites
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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
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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
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3
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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
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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
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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
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15 even numbers only
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stress

σ1



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