EML 4230 Introduction to Composite Materials

Chapter 2 Macromechanical Analysis of a LaminaFailure Theories

Dr. Autar KawDepartment of Mechanical Engineering

University of South Florida, Tampa, FL 33620

Tsai-Wu Failure Theory

Tsai-Wu applied the failure theory to a lamina in plane stress. A lamina is considered to be failed if

Is violated. This failure theory is more general than the Tsai-Hill failure theory because it distinguishes between the compressive and tensile strengths of a lamina.

The components H1 – H66 of the failure theory are found using the five strength parameters of a unidirectional lamina.

H1σ1+H2σ2+H6τ12+H11

+H22

+H66

+2H12σ1σ2< 1

12 211221266

2222

21111262211 HHHHHHH

Components of Tsai-Wu Fail

0,0, 12211 ultT to a unidirectional lamina, the lamina will fail.

Equation(2.152) reduces to

.=σH+σH ultT

ultT 1

211111

a) Apply

0,0, 12211 ultC to a unidirectional lamina, the lamina will fail.

Equation(2.152) reduces to

.=σH+σH ultC

ultC 1

211111

b) Apply

From Equations (2.153) and (2.154),

,σσ

=Hult

Cult

T11

111

,σσ

=Hult

Cult

T11

111

Components of Tsai-Wu Fail

0,,0 12221 ultT to a unidirectional lamina, the lamina will fail.

Equation(2.152) reduces to

.=σH+σH ultT

ultT 1

222222

c) Apply

0,,0 12221 ultC to a unidirectional lamina, the lamina will fail.

Equation(2.152) reduces to

.=σH+σH ultC

ultC 1

222222

d) Apply

From Equations (2.157) and (2.158),

,σσ

=Hult

Cult

T22

211

,σσ

=Hult

Cult

T22

221

Components of Tsai-Wu Fail

ult121221 ,0,0 to a unidirectional lamina, the lamina will fail. Equation

(2.152) reduces to .=H+H ultult 12

1266126

e) Apply

to a unidirectional lamina, the lamina will fail. Equation

(2.152) reduces to

.=H+H ultult 121266126

f) Apply

From Equations (2.157) and (2.158),

,=H 06

,=H

ult

212

66

1

ult121221 ,0,0

Determination of

12 211221266

2222

21111262211 HHHHHHH

Apply equal tensile loads along the two material axes in a unidirectional composite. If

,=τ= σ σ=σ xyyx 0 is the load at which the lamina fails, then

.=σH+H+Hσ+H+H 12 212221121

The solution of the Equation (2.165) gives

.σ)H+H()σH+H-(σ

H 2221121212 1

2

1

Determination of

H1σ1+H2σ2+H6τ12+H11 21 +H22 2

2 +H66 212 +2H12σ1σ2< 1

Take a 450 lamina under uniaxial tensionσ x. The stress σ x at failure is noted.

If this stress is 0σ x then using Equation (2.94), the local stresses at failure are

21

22

212

Substituting the above local stresses in Equation (2.152),

.=H+H+H+Hσ+

σH+H 12

4212662211

2

21

.H+H+Hσ

H + H

σ=H 662211

21

2122

12

Empirical Models of

)σ(H

Tult1212

2

1

)σ()σ(H

ultC

ultT

1112 2

1

)σ ()σ ()σ ()σ( H

ultC

ultT

ultC

ultT

221112

1

2

1

as per Tsai-Hill failure theory8,

as per Hoffman criterion10,

as per Mises-Hencky criterion11.

Example 2.19

Find the maximum value of

0S if a stress

3S- = 2S, = yx and 4S = xy

are applied to a 600 lamina of Graphite/Epoxy. Use Tsai-Wu failure theory. Use the properties of a unidirectional Graphite/Epoxy lamina from Table 2.1.

Example 2.19

Using Equation (2.94), the stresses in the local axes are

S.

.-

.-

.

=

S

S-

S

.-..-

.-..

...

=

τ

σ

σ

1041650

1027140

1017140

4

3

2

500004330043300

866002500075000

866007500025000

1

1

1

12

2

1

Example 2.19

H

12=

,Pa H -1661 0

101500

1

101500

1

,Pa .H -- 18662 100932

10246

1

1040

1

,Pa H -16 0

,Pa.)) ( (

H -- 2166622 1001621

102461040

1

,Pa.))( (

H -- 2196611 1044444

101500101500

1

,Pa.)(

H -- 216

6 266 10162621068

1

.Pa.. ..-H ---- 2181619 2

1

12 1036031001621104444450

Example 2.19

Substituting these values in Equation (2.152), we obtain S.-.+S. - 714210093271410 8

219 7141104444416540 S..+S.-+

216216 1654101626271421001621 S..+S.- .+ -

,S.-S..-+ - 1714271411036032 18

or

MPa.S< 3922

Example 2.19

If one uses the other two empirical criteria for H12 as per Equation (2.171), one obtains

,)σ(

HMPa for.S<ult

T 21

122

14922

.)σ()σ(

HMPa for.S<ult

Cult

T11

121

2

14922

Summarizing the four failure theories for the same stress-state, the value of S obtained is

S = 16.33 (Maximum Stress failure theory), = 16.33 (Maximum Strain failure theory), = 10.94 (Tsai-Hill failure theory), = 16.06 (Modified Tsai-Hill failure theory), = 22.39 (Tsai-Wu failure theory).