Chapter 4: Failure Theories 4.0 Introduction 4.1 Macromechanical Failure Theories 4.1.1 Maximum Stress Theory 4.1.2 Maximum Strain Theory 4.1.3 Tsai-Hill Theory Theory (Deviatoric energy theory) 4.1.4 Tsai-Wu Theory Theory (Interactive tensor theory) 4.2 How to Apply a Failure Theory 4.3 Description of Failure Failure Theories 4.3.1 Maximum Stress Theory 4.3.2 Maximum Strain Theory 4.3.3 Tsai-Hill Theory Theory (Deviatoric energy theory) 4.3.4 Tsai-Wu Theory Theory (Interactive tensor theory) 4.4 Comparison of Failure Failure Theories 4.5 Application Structural Analysis
4.0 Introduction Failure: Every material has certain strength, expressed in terms of stress or strain, beyond which it fractures or fails to carry the load. Failure Criterion: A criterion used to hypothesize the failure.
Failure Theory: A Theory behind a failure criterion.
Why Need Failure Theories? (a) To design structural components and calculate margin of safety. (b) To guide in materials development. (c) To determine weak and strong directions.
Failure Theories for Isotropic Materials: Strength and stiffness are independent of the direction. Failure in metallic materials is characterized by Yield Strength.
σ ult Stress
σ ys
ε ys
ε ult
Strain
Theories: (a) Maximum principal stress theory. (b) Maximum principal strain theory. (c) Quadratic or Distortional Energy Theory.
4.1. Macromechanical Failure Theories in Composite Materials a. Maximum Stress Theory b. Maximum Strain Theory c. Tsai-Hill Theory (Deviatoric strain energy theory) d. Tsai-Wu Theory (Interactive tensor polynomial theory) 4.2. Application of Failure Theory First step is to calculate the stresses/strains in the material principal directions. This can be done by transformation of stresses from the global coordinates to local material coordinates of the ply.
Ply Stresses:
{σ } x − y
=
[Tσ ]{σ }1
−2
or
{σ }1− 2
=
[T σ ]
−1
{σ } x − y
Ply strains:
{ε }1− 2
= [Q]1− 2 {σ }1− 2
Now apply the failure criteria in the material coordinate system.
4.3.1 Maximum Stress Criterion Failure occurs when at least one stress component along the principal material axes exceeds the corresponding strength in that direction. σ2
Tensile stresses:
≥ F 1t σ 2 ≥ F 2t σ 1
σ2
≤
σ 2
≤
σ1
Fiber break Matrix crack
σ2
F 2t
Compressive stresses:
σ 1
σ1
F 1c
Fiber crushing
F 2c
Matrix yielding
F 1c
σ1
No failure
F 2c
F 1t
Shear stresses:
σ 12
≥ F6
or σ 6
≥ F 6
Shear crack
Note there is no interaction between the stress components.
Failure of an Angle Ply Laminate Material: E-Glass/Epoxy F1t = 1,080 MPa F1c = 620 MPa F2t = 39 MPa F2c = 128 MPa F6 = 89 MPa u u ε1t = 0.028 ε2t = 0.005 ν12 = 0.28 ν21 = 0.06
y
x1
σx
σx
x2 x
1. Maximum Stress Theory 2
F 1t 2 Cos θ F or σ x = 22t Sin θ
σ1
= σ x Cos
θ
@ failure σ 1
=
F1t or σ x
σ2
= σ x Sin
θ
@ failure σ 2
=
F2t
σ1
= σ x Cos 2θ
@ failure σ 1
= F1c
or σ x
= −
σ2
= σ x Sin 2θ
@ failure σ 2
= F2c
or σ x
= −
τ6
= −σ x CosθSinθ
2
@ failure τ 6
= F6
=
F 1c Cos 2θ F 2c Sin 2θ
⇒
⇒
Longitudinal Tension
Transverse Tension
⇒ Longitudinal Compression ⇒ Transverse Compression
or σ x
= ±
F 6 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
4.3.2 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. Tensile strain:
u
ε1
≥ ε 1t
ε2
≥ ε 2t
Compressive strain:
ε1
u
ε2
u
ε 2t u
≤ ε 1c
ε 2 ≤ ε 2uc
ε 1c u
No failure
ε 2c u
Shear strain:
γ 12
≥ γ 6u
or γ 6
≥ γ 6u
ε 1t u ε1
Maximum Strain Theory Expressed in Stresses
= (σ 1 − ν 12σ 2 ) / E 1 ε 2 = (σ 2 − ν 21σ 1 ) / E 2 γ 6 = τ 6 / G12 ε1
Maximum strains:
= ε 1ut or ε 2 = ε 2ut or u γ 6 = γ 6
@ Failure
σ2
u
ε1
- ε 1c
σ2
- ε 2uc
σ 2 σ 1 − ν 12 σ 2
ε 1ut =
F 1t
ε 2ut =
F 2t
γ 6u
=
E 1 E 2
u
and ε 1c
=
u
=
and ε 2c
F6 / G12
F 1c E 1 F 2c E 2
σ2
σ 1 − ν 12σ 2
= − F1c
σ 2
σ1
− ν 21σ 1 = F2 t
No failure
Ultimate strains are calculated from Uniaxial & Shear tests:
σ1
− ν 21σ 1 = − F2 c
σ1
= F1 t
Application of Maximum Strain Theory to Angle-ply Laminate
ε1
Strains
ε2 Tension Loaded: σ x
=
σ x
=
=
(σ 1 − ν 12σ 2 ) / E 1 (
)
= −ν 21σ 1 + σ 2 / E 2
x1
F 1t Cos 2θ − ν 12 Sin 2θ
σx
F 2t
σ x
=−
2
σ x
=−
F 1c Cos 2θ − ν 12 Sin 2θ F 2c Sin θ − ν 21Cos θ 2
2
x
⇒ Longitudinal ⇒ Transverse
Shear Loaded: =±
F 6 CosθSinθ
⇒
σx
x2
Sin θ − ν 21Cos θ 2
Compression Loaded:
σ x
y
Shear
Uniaxial Strength of an Off-Axis Lamina Maximum Strain Theory y
L-Tension
1200 1000
σx
800
MPa
σx
x2
Shear
600 σx
x1
x
400 200
T-tension
0 -200
T-Compression
-400
Shear
-600
L-Compression
-800 0
10
20
30
40
50 θ , deg
60
70
80
90
4.3.3 Tsai-Hill Theory Hill extended the von Mises criterion for ductile anisotropic material. Azzi-Tsai extended this equation to anisotropic fiber reinforced composites. Failure occurs when the LHS of the following equation is equal to or greater than one. 2
Aσ 1
+
2
Bσ 2
+ Cσ 1σ 2 +
2
Dτ 6 = 1
From longitudinal, transverse, and shear tests on a uniaxial laminate, A, B, and D are determined. A=
1
B=
, 2 F 1
1
, and 2 F 2
D=
1 2
F 6
From Equal Biaxial test: Failure occurs when the transverse stress (σ2) reaches F2. C1=-1/F12
Tsai-Hill failure criterion: σ 12 + 2 F1
σ 22 − 2 F2
2 σ 1σ 2 τ 6 + 2 = 2 F1 F6
σ 12 1
2 +
F1
σ 22
σ 1σ 2
F2
2 F 1
2 −
=
1 − κ
2
Note: No distinction is made between tensile & compression strengths.
κ =
τ 6 F 6
Application of Tsai-Hill Failure Criterion to Angle-Ply Laminate Substitute for σ1, σ2, and τ6 in
y
x1
terms of σx in:
σ 12 2
+
F1
σ 22 2
F2
−
σ 1σ 2 2
F1
+
τ 6 2 2
=
F6
1
σx
σx
x2 x
We get the failure stress:
1
σ x2 1
σ x2
= =
Cos 4θ 2
F 1t
Cos 4θ 2
F 1c
+ +
Sin 4θ 2
F2t
Sin 4θ 2
F2c
1 1 + 2 − 2 Cos 2θSin 2θ F6 F 1t
For Tensile Stresses
1 1 + 2 − 2 Cos 2θSin 2θ F6 F 1c
For Compressive Stresses
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
4.3.4 Tsai-Wu Theory Tsai-Wu theory is a simplification of GolÕdenblat and KapnovÕs generalized failure theory for anisotropic materials. It is stated as
fiσ i
+ f ij σ iσ j = 1
I,j=1,2,3,4,5,6
For plane-stress condition: 2
f1σ 1 + f 2σ 2 + f6 τ 6 + f11σ 1
2
2
+ f22σ 2 + f66 τ 6 +
+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. Therefore we get f1σ 1 + f 2σ 2 + f11σ 12 + f 22σ 22 + f66 τ 6 2 + 2 f12σ 1σ 2 = 1
Now we will evaluate all six constants for tests:
(a) Longitudinal tension & compression tests:
f 1
=
1 F1t
−
1 F 1c
and f 11
=
1 F1t F 1c
(b) Transverse tension & compression tests: 1 1 1 f 2 = and f 22 = − F2t F 2c F2t F 2c (c) Shear tests:
f 66 =
1 2
F 6
(d) Interaction coefficient f12 is assumed as
f12
≅ − 12
f11 f 22
or
f 12
= − 21
1 F1t F1c F2t F2 c
Application of Tsai-Wu Failure Criterion to Angle-Ply Laminate
f1σ 1 + f 2σ 2
+ f11σ 12 + f 22σ 22 +
f66 τ 6 2 + 2 f12σ 1σ 2 = 1
y
x1
Substituting for σ1, σ2, and τ6 in σx in the above eqn. We get
aσ x2
+ bσ x − 1 = 0
σx
σx
x2
Where
x
4 4 2 2 2 2 a = f11Cos θ + f22 Sin θ + 2 f12 Cos θSin θ + f66 Cos θ Sin θ 2
2
b = f1Cos θ + f2 Sin θ Solution is:
σ x
=
−b ±
b
2
2a
+
4a
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
3.4 Comparison of Failure Theories Theory
Physical basis
Operational convenience
Required operational convenience
Maximum stress
Tensile behaviour of brittle material
Inconvenient
Few parameters by simple testing
Conservative Design σ2
Tensile behaviour of brittle material Maximum strain Some stress interaction
Inconvenient
Few parameters by simple testing
Max. strain F2t -F1c F1t
Ductile behavior of Can be programmed anisotropic Deviatoric Different functions materials strain energy required for tensile "Curve fitting" for (Tsai-Hill) and compressive 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 4.5 to 4.15 even numbers only.
-F2c Interactive theory
Max. stress
σ1
