© 2003, P. Joyce...1111 11 1122 12 1133 13 1123 14 1131 15 1112 16 23 4 31 5 12 6 11 1 22 2 33 1 23...

MacromechanicalMacromechanical Analysis of a Analysis of a LaminaLamina

© 2003, P. Joyce

Generalized Generalized Hooke’s Hooke’s LawLaw

ijijklij C εσ =

Cijkl is a 9 x 9 matrix!

© 2003, P. Joyce

Hooke’sHooke’s LawLaw

Assume linear elastic behaviorsmall deformations

εσ E=

Uniaxial loading

© 2003, P. Joyce

TriaxialTriaxial Stress StateStress State

Shears are omittedConsider shear separatelySuperimpose three uniaxialstresses

x

z

y

x

z

y Uniaxial: σxUniaxial: σx

σx

x

z

y

σx

xxδ

dxE

dx xxxx

σ=ε=δ

Uniaxial: σx

σx

x

z

y

σx

yxδdy

Edy x

xyxσ

υ−=υε−=δ

dxE

dx xxxx

σ=ε=δ

Uniaxial: σx

σx

x

z

y

σx

zxδ

dxE

dx xxxx

σ=ε=δ

dyE

dy xxyx

συ−=υε−=δ

dzE

dz xxzx

συ−=υε−=δ

x

z

y Uniaxial: σyUniaxial: σy

σy

σx

x

z

y σy

dyE

dy yyyy

σ=ε=δ

yyδ

Uniaxial: σy

σy

σx

x

z

y σy

xyδ

dyE

dy yyyy

σ=ε=δ

dxE

dx yyxy

συ−=υε−=δ

Uniaxial: σy

σy

σx

x

z

y σy

zyδ dyE

dy yyyy

σ=ε=δ

dyE

dy yyxy

συ−=υε−=δ

dzE

dz yyzy

συ−=υε−=δ

x

z

y Uniaxial: σzUniaxial: σz

dzE

dz zzzz

σ=ε=δ

σz

x

z

y

zzδσz

Uniaxial: σz

dxE

dx zzxz

συ−=υε−=δ

dzE

dz zzzz

σ=ε=δ

σz

x

z

y

xzδσz

Uniaxial: σz

dxE

dx zzxz

συ−=υε−=δ

dzE

dz zzzz

σ=ε=δ

σz

x

z

y

yzδ

σz

dyE

dy zzyz

συ−=υε−=δ

xzxyxxxx dx δ+δ+δ=ε=δ xzxyxxxx dx δ+δ+δ=ε=δ

dxE

dxE

dxE

dx zyxxx

συ−

συ−

σ=ε=δ

xzxyxxxx dx δ+δ+δ=ε=δ

EEEzyx

xσ

υ−σ

υ−σ

=ε

dxE

dxE

dxE

dx zyxxx

συ−

συ−

σ=ε=δ

xzxyxxxx dx δ+δ+δ=ε=δ

EEEzyx

xσ

υ−σ

υ−σ

=ε

dxE

dxE

dxE

dx zyxxx

συ−

συ−

σ=ε=δ

( )[ ]zyxx E1 σ+συ−σ=ε

Similarly for the deformations in the y- and z-directions

( )[ ]yxzz E1 σ+συ−σ=ε

( )[ ]zxyy E1 σ+συ−σ=ε

© 2003, P. Joyce

Triaxial Triaxial Stress StateStress State

( )[ ]zyxx E1 σ+συ−σ=ε

( )[ ]yxzz E1 σ+συ−σ=ε

( )[ ]zxyy E1 σ+συ−σ=ε

Strain-Stress Relations

© 2003, P. Joyce

Triaxial Triaxial Stress StateStress State

( )( )( )[ ]zyxx )1(

211E ε+ευ+ευ−

υ−υ+=σ

( )( )( )[ ]zxyy )1(


υ−υ+=σ

( )( )( )[ ]yxzz )1(


υ−υ+=σ

Stress-Strain Relations

© 2003, P. Joyce

Similarly for shear stressesSimilarly for shear stresses

© 2003, P. Joyce

Shear Stress Strain RelationshipsShear Stress Strain Relationships

Shear stresses are independent of each other and all axial stressesShear strains are independent of each other and all axial strains

Each obeys a simple linear elastic model

G – Shear Modulus

γ=τ GG – Shear Modulus

γ=τ G)1(2

EGυ+

=

© 2003, P. Joyce

Shear StressShear Stress--Strain RelationshipsStrain Relationships

xyxyxy )1(2EG γ

υ+=γ=τ

xzxzxz )1(2EG γ

υ+=γ=τ

yzyzyz )1(2EG γ

υ+=γ=τ

© 2003, P. Joyce


Strain-stress relations for an isotropic material/matrix form

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

−−

−−

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

xy

zx

yz

z

y

x

xy

zx

yz

z

y

x

G

G

G

EEE

EEE

EEE

τττσσσ

υυ

υυ

υυ

γγγεεε

100000

010000

001000

0001

0001

0001

Compliance matrix

© 2003, P. Joyce


Stress-strain relations for an isotropic material /matrix form

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−−

+−+−

+−+−−

+−

+−+−+−−

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

xy

zx

yz

z

y

x

xy

zx

yz

z

y

x

GG

Gv

Ev

Ev

Ev

Ev

Ev

Ev

Ev

Ev

E

γγγεεε

νυ

νυ

νυ

νυ

νυ

νυ

νυ

νυ

νυ

τττσσσ

000000000000000

000)1)(21(

)1()1)(21()1)(21(

000)1)(21()1)(21(

)1()1)(21(

000)1)(21()1)(21()1)(21(

)1(

Stiffness matrix

© 2003, P. Joyce

StressStress--Strain Relations in Strain Relations in Composite MaterialsComposite Materials

Material No. of independent elastic constants

General anisotropic material 81

Anisotropic material (considering symmetry of stress and strain tensors)

36

Anisotropic material with energy considerations

21

General orthotropic material 9

Transversely orthotropic material 5

Isotropic material 2

© 2003, P. Joyce

StressStress--Strain Relations in Strain Relations in Composite MaterialsComposite Materials

Orthotropic material ~ has three mutually perpendicular plans of material symmetrySpecially orthotropic material ~ when the reference system of coordinates is selected along principal planes of material symmetryTransversely isotropic material ~ one of its principal planes is a plane of isotropy (properties are the same in all directions.)

© 2003, P. Joyce

Contracted NotationContracted NotationThanks to symmetry of the stress and strain tensors, the compliance matrix reduces to a6 x 6 matrix, introduce a contracted notation.

262212252231242223232233222222212211

161112151131141123131133121122111111

612531423

133222111

612531423

333222111

2,2,2,,,2,2,2,,,

2,2,2,,,

,,,,,

CCCCCCCCCCCCCCCCCCCCCCCC

============

======

======

εεεεεεεεεεεε

σσσσσσσσσσσσ

© 2003, P. Joyce

StressStress--Strain RelationsStrain Relationsfor Thin UD Laminafor Thin UD Lamina

Assumed to be under a state of plane stress

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

6

2

1

66

2212

1211

6

2

1

0000

γεε

τσσ

QQQQQ

Fully characterized by 4 independent constants,

Qij ~ reduced stiffnesses

© 2003, P. Joyce

Reduced Stiffness Matrix?Reduced Stiffness Matrix?

If the stiffness matrix is the inverse of the compliance matrix, what is the reduced stiffness matrix?

)6 ,2 ,1,( 33

33 =−= jiC

CCCQ ji

ijij

© 2003, P. Joyce

Relations Between Mathematical Relations Between Mathematical and Engineering Constantsand Engineering Constants

1266

2112

212

2112

12112

2112

222

2112

111

11

1

1

GQ

EEQ

EQ

EQ

=−

=−

=

−=

−=

υυυ

υυυ

υυ

υυ

© 2003, P. Joyce

You Said Four Independent You Said Four Independent Constants?Constants?

From symmetry of the compliance matrix

j

ji

i

ij

EEυυ

=

The above can also be deduced from Betti’s reciprocal law according to which transverse deformation due to a stress applied in the longitudinal direction is equal to the longitudinal deformation due to an equal stress applied in the transverse direction.

© 2003, P. Joyce

StressStress--Strain RelationsStrain Relations

Also expressed in terms of compliances

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

6

2

1

66

2212

1211

6

2

1

0000

τσσ

γεε

SSSSS

© 2003, P. Joyce

Relations between Mathematical Relations between Mathematical and Engineering Constantsand Engineering Constants

1266

2

21

1

1212

222

111

1

1

1

GS

EES

ES

ES

=

−=−=

=

=

υυ

That’s Better!

© 2003, P. Joyce

Sample CalculationSample CalculationFor a graphite/epoxy UD laminate, find the following:

1.) Compliance matrix2.) Minor Poisson’s ratio3.) Reduced Stiffness Matrix4.) Strains in the 1-2 coordinate system if the applied stresses are

σ1 = 2 MPa, σ2 = -3MPa, τ12 = 4 MPa

σ1

σ2

τ12

© 2003, P. Joyce

Sample DataSample DataProperty Symbol Units Glass/

epoxyBoron/epoxy

Graphite/epoxy

Fiber volume fraction

Vf - 0.45 0.50 0.70

Long. elastic modulus

E1 GPa 38.6 204 181

Trans. elastic modulus

E2 GPa 8.27 18.50 10.30

Major Poisson’s ratio

ν12 - 0.26 0.23 0.28

Shear Modulus G12 GPa 4.14 5.59 7.17

© 2003, P. Joyce

Sample CalculationSample Calculation

)10(1395.0)10(17.7

11

)10(9709.0)10(3.10

11

)10(1547.0)10(181

28.0

)10(5525.0)10(181

11

99

1266

109

122

119

1

1212

119

111

−

−

−

−

===

===

−===

===

GS

ES

ES

ES

ν

The compliance matrix elements arecalculated as follows:

(all terms have units of Pa-1)

© 2003, P. Joyce


01593.0)10)(3.10()10(181

)28.0( 9921

1

12

2

21

==

=

ν

ννEE

From Betti’s reciprocal law:

© 2003, P. Joyce


)10(17.7

)10(35.10)01593.0)(28.0(1

)10(3.101

)10(897.2)01593.0)(28.0(1

)10)(3.10)(28.0(11

)10(8.181)01593.0)(28.0(1

)10(1811

91266

99

2112

222

99

2112

121

2112

21212

99

2112

111

==

=−

=−

=

=−

=−

=−

=

=−

=−

=

GQ

EQ

EEQ

EQ

νν

ννν

ννν

νν

The stiffness matrix elements are calculatedas follows:

(all terms have units of Pa)

© 2003, P. Joyce


[ ]

[ ] Pa )10(17.700

0)10(35.10)10(897.20)10(897.2)10(8.181

)10(1395.0000)10(9709.0)10(1547.00)10(1547.0)10(5525.0

9

99

99

1

9

1011

1111

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

=

−

−

−−

−−

Q

Q

The stiffness matrix can also be calculatedby inverting the compliance matrix of Part 1:

(all terms have units of Pa)

© 2003, P. Joyce


)(10 9.5574.294

69.15

)10(4)10(3

)10(2

)10(1395.0000)10(9709.0)10(1547.00)10(1547.0)10(5525.0

0000)

6-

12

2

1

6

6

6

9

1011

1111

12

2

1

12

2

1

66

2212

1211

12

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−−

−−

γεε

γεε

τσσ

γεε

SSSSS

The strains in the 1-2 coordinate system arecalculated as follows:

(10-6 is microstrain)

© 2003, P. Joyce

StressStress--Strain RelationsStrain Relationsfor Thin Angle Laminafor Thin Angle Lamina

Generally, a laminate does not consist only of UD laminae becauseof their stiffness and strength properties in the transverse direction.

Hence, in most laminates, some laminae are placed at an angle.

x

y 12

Global and material axes of an angle lamina.

© 2003, P. Joyce


The axes in the x-y coordinate system are called the global axes of the off-axes.The axes in the 1-2 coordinate system are called the material axes or the local axes, where direction 1 is parallel to the fibers (also called the longitudinal direction) and direction 2 is is perpendicular to the fibers (also called the transverse direction.)The angle between the two axes is denoted by the angle θ.

© 2003, P. Joyce


The stress-strain relationship in the 1-2 coordinate system has already been established.From Mechanics of Materials, the stresses in the global and material axes are related to each other through the angle of the lamina, θ.

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

12

2

11)(

τσσ

θτσσ

T

xy

y

x

Where [T(θ)] is called the transformation matrix and is defined as

[ ] [ ] [ ])T(- 22

)( thus22

)(22

22

22

1

22

22

22

θθθ =⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−= −

scscscsccsscsc

Tscscsc

sccsscsc

T

© 2003, P. Joyce


Using the stress-strain equation in the material axes together with the transformation equation we obtain:

[ ] [ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

12

2

11

γεε

τσσ

QT

xy

y

x

© 2003, P. Joyce


Similarly, the strains in the global and material coordinate axes are related through the transformation matrix

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

xy

y

x

Tγεε

γεε

21

1221

2

1

which can be rewritten as

[ ][ ][ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

xy

y

x

RTRγεε

γεε

21

1

1221

2

1

where [R] is the Reuter matrix and is defined as

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

200010001

R

© 2003, P. Joyce


Multiplying out the first five matrices on the RHS of the previous equation we obtain the transformed reduced stiffness matrix, Thus, Summarizing,

66222

1222

2222

1122

6633

1233

223

113

6633

1233

223

113

6622

1244

2222

1122

6622

1222

224

114

6622

1222

224

114

)(2

)(2)(

)(2)(

4)(

42

42

QscQscQscQscQ

QcsscQcsscsQcQcsQ

QsccsQsccsQcssQcQ

QscQscQscQscQ

QscQscQcQsQ

QscQscQsQcQ

ss

ys

xs

xy

yy

xx

−+−+=

−+−+−=

−+−+−=

−+++=

+++=

+++=

[ ]xyQ[ ] [ ] [ ] yxyxyx Q ,,, εσ =

The subscript s corresponds to shear stress or strain components in the x-y system, i.e., xyxys γγττ == s and

© 2003, P. Joyce

StrainStrain--Stress Relations for Thin Stress Relations for Thin Angle LaminaAngle Lamina

Similarly, the transformed compliance can be obtained:

66222

1222

2222

1122

6633

1233

223

113

6633

1233

223

113

6622

1244

2222

1122

6622

1222

224

114

6622

1222

224

114

)(844

)()(222

)()(222

4)(

42

42

SscSscSscSscS

ScsscScsscsScScsS

SsccsSsccsScssScS

SscSscSscSscS

SscSscScSsS

SscSscSsScS

ss

ys

xs

xy

yy

xx

−+−+=

−+−+−=

−+−+−=

−+++=

+++=

+++=

Thus, [ ] [ ] [ ] yxyxyx S ,,, σε =

© 2003, P. Joyce


How about in terms of Engineering Constants?If we imagine a series of simple experiments on an element with sides parallel to the x- and y-axes, we obtain:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

s

y

x

xyy

ys

x

xs

xy

sy

yx

xy

xy

sx

y

yx

x

s

y

x

GEE

GEE

GEE

τσσ

ηη

ην

ην

γεε

1

1

1

© 2003, P. Joyce


What is η ? shear coupling coefficientηxs , the first subscript denotes normal loading in the x-direction; the second subscript denotes shear strain.

sysy

sxsx

ysys

xsxs

γεηγεη

εγηεγη

==

==

© 2003, P. Joyce


Comparison of equivalent strain-stress relations yields the following relationships:

xy

sy

y

yssyys

xy

sx

x

xssxxs

y

yx

x

xyyxxy

xyss

yyy

xxx

GESS

GESS

EESS

GS

ES

ES

ηη

ηη

νν

===

===

−=−==

=

=

=

1

1

1

ss

ysyx

yy

syys

ss

xssx

xx

sxxs

yy

xyyx

xx

yxxy

ssxy

yyy

xxx

SS

SS

SS

SS

SS

SS

SG

SE

SE

==

==

−=−=

=

=

=

ηη

ηη

νν

;

;

;

1

1

1

© 2003, P. Joyce

Sample CalculationSample CalculationFind the following for a 60° angle lamina of graphite/epoxy.

Transformed compliance matrixTransformed reduced stiffness matrixGlobal strainsLocal strainsIf the applied stresses are σx = 2 MPa, σy = -3MPa, τxy = 4 MPa

© 2003, P. Joyce


From the previous example:

)10(1395.0

)10(9709.0

)10(1547.0

)10(5525.0

966

1022

1112

1111

−

−

−

−

=

=

−=

=

S

S

S

S

(all terms have units of Pa-1)

© 2003, P. Joyce

Sample CalculationSample CalculationThe transformed compliance matrix elements are calculated as follows:

)10(1141.0)(844

)10(4696.0)()(222

)10(3475.0)()(222

)10(3234.04)(

)10(7878.042

)10(8053.042

966

22212

2222

2211

22

1066

3312

3322

311

3

1066

3312

3322

311

3

1066

2212

4422

2211

22

1166

2212

2222

411

4

1066

2212

2222

411

4

−

−

−

−

−

−

=−+−+=

−=−+−+−=

=−+−+−=

−=−+++=

−=+++=

=+++=

SscSscSscSscS

ScsscScsscsScScsS

SsccsSsccsScssScS

SscSscSscSscS

SscSscScSsS

SscSscSsScS

ss

ys

xs

xy

yy

xx

(All terms are in Pa-1)

© 2003, P. Joyce


Next, invert the transformed compliance matrix [S] to obtain the transformed reduced stiffness matrix [Q].

[ ] [ ]

[ ] )10(3674.05419.02005.05419.0094.13246.02005.03246.02365.0

)10(1141.0)10(4696.0)10(3234.0)10(4696.0)10(3475.0)10(7878.0)10(3234.0)10(7878.0)10(8053.0

11

1

91010

101010

101010

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−−−

==

−

−−−

−−−

−−−

−

Q

SQ

(all terms in Pa)

© 2003, P. Joyce


The global strains in the x-y plane are given by [ ] [ ] [ ] yxyxyx S ,,, σε =

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−−−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−

−

−−−

−−−

−−−

)10(5328.0)10(3078.0

)10(5534.0

)10(43

2

)10(1141.0)10(4696.0)10(3234.0)10(4696.0)10(3475.0)10(7878.0)10(3234.0)10(7878.0)10(8053.0

3

3

4

6

91010

101010

101010

xy

y

x

xy

y

x

γεε

γεε

© 2003, P. Joyce

Sample CalculationSample CalculationThe local strains in the lamina can be calculated using the Transformation equation.

[ ]

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−

−

−

−

−

)10(5809.0)10(2662.0

)10(1367.0

2/)10(5328.0)10(3078.0

)10(5534.0

60sin60cos60sin60cos60sin60cos60sin60cos260cos60sin

60sin60cos260sin60cos

3

3

4

12

2

1

3

3

4

22

22

22

1221

2

1

21

1221

2

1

γεε

γεε

γεε

γεε

xy

y

x

T

© 2003, P. Joyce

Transformation ofTransformation ofEngineering ConstantsEngineering Constants

Flow chart for determination of transformed elastic constants of UD lamina.

[S]1,2

[E]1,2

[S]x,y

[Q]1,2[Q]x,y

θ [E]x,y

© 2003, P. Joyce

MacromechanicalMacromechanical Strength Strength ParametersParameters

From a macromechanical POV, the strength of a lamina is an anisotropic property.It is desirable, for example, to correlate the strength along an arbitrary direction to some basic strength parameters (analogous to micromechanic definitions before.)

© 2003, P. Joyce

Strength Failure Theories of an Strength Failure Theories of an Angle LaminaAngle Lamina

Various theories have been developed for studying the failure of an angle lamina.Generally based on the normal and shear strengths of a UD lamina.Need to consider tension and compressionUD lamina has 2 material axes, 1-direction parallel to the fibers and 2-direction which is perpendicular to the fibers.Hence there are 4 normal strength parameters for UD lamina.

Tensile strength in fiber directionTransverse tensile strengthCompressive strength in fiber directionTransverse compressive strength

The fifth strength parameter is the shear strength

© 2003, P. Joyce

Strength Failure Theories of an Strength Failure Theories of an Angle LaminaAngle Lamina

Unlike the stiffness parameters, these strength parameters cannot be transformed directly for an angle lamina.Hence, the failure theories are based on first finding the stresses in the material axes and then using these five strength parameters of a UD lamina to find whether the lamina has failed.

© 2003, P. Joyce


Also predict transverse compressive strength and in-plane shear strength using micromechanics. . .Failure mechanisms vary greatly with material properties and type of loading.Even when predictions are accurate with regard to failure initiation at critical points, they are only approximate as far as global failure of the lamina is concerned.Furthermore, the possible interaction of failure mechanisms makes it difficult to obtain reliable strength predictions under a general type of loading.A macromechanical or phenomological approach to failure analysis may be preferable.

© 2003, P. Joyce


This characterization recognizes the fact that most composite materials have different strengths in tension and compression.By convention the sign of the shear stress is immaterial, as long as the shear strength is referred to the principal material directions.Exception, refers to the case when the shear stress is applied at an angle wrt the principal material directions.Since most composites have different tensile and compressive strengths and they are weakest in transverse tension, it follows that in this case the lamina would be stronger under positive shear.

© 2003, P. Joyce

MacromechanicalMacromechanical Strength Strength ParametersParametersτ6

σy = τ6

σx = -τ6Positive shear stress

Shear stress acting along principal material axes.

© 2003, P. Joyce

MacromechanicalMacromechanical Strength Strength ParametersParametersτ6

σy = τ6

σx = -τ6Negative shear stress

Shear stress acting along principal material axes.

© 2003, P. Joyce

MacromechanicalMacromechanical Strength Strength ParametersParametersτs

σ1 = τ6

σ2 = -τsPositive shear stress

Shear stress acting at 45° wrt principal material axes.

© 2003, P. Joyce

MacromechanicalMacromechanical Failure Failure TheoriesTheories

Given a state of stress, the principal stresses and their directions are obtained by stress transformation (independent of material properties.)The principal strains and their directions are obtained by using the appropriate anisotropic stress-strain relations and strain transformation.In general, the principal stress, principal strain, and material symmetry directions do not coincide.Since strength varies with orientation, maximum stress alone is not the critical factor in failure.

© 2003, P. Joyce

Macromechanical Macromechanical Failure Failure TheoriesTheories

An anisotropic failure theory is needed.Failure criteria for homogeneous isotropic materials, such as

Maximum normal stress (Rankine), Maximum shear stress (Tresca), Maximum distortional energy (von Mises), and so forth are well established.

More than 40 anisotropic theories have been proposed – look at the four most widely used.

© 2003, P. Joyce

Maximum Stress Failure TheoryMaximum Stress Failure TheoryRelated to the Maximum Normal stress theory by Rankine and the Maximum Shear stress theory by Tresca.The stresses acting on a lamina are resolved into the normal and shear stresses in the material axes.Failure is predicted in a lamina, if any of the normal or shear stresses in the material axes are equal to or greater than the corresponding ultimate strengths of a UD lamina.

Each component of stress is compared with the corresponding strength and hence does not have an interaction with the others.

( ) ( ) ( ) ( ) ( ) ( )ultultultT

ultC

ultT

ultC

121212222111 , , τττσσσσσσ <<−<<−<<−

© 2003, P. Joyce

Sample CalculationSample CalculationFind the off-axis shear strength of a 60° graphite/epoxy lamina using the Maximum Stress failure criteria.Assume the following stress state

,,0 ,0 ττσσ === xyyx

Find the stresses along the principal material axes, using the Transformation Equation.

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ττ

τ

τσσ

ττσσ

500.0866.0

866.0

00

5000.04330.04330.08660.02500.07500.0

8660.07500.02500.0

12

2

1

12

2

1

© 2003, P. Joyce

Sample CalculationSample CalculationApplying the Maximum Stress Failure Criteria together with strength data for graphite/epoxy composites from the Data Sheet,we have

68500.06840866.0246

1500866.01500

<<−<−<−<<−

τττ

or

0.1360.1361.28419.46

17321732

<<−<<−

<<−

ττ

τ

© 2003, P. Joyce

Sample CalculationSample CalculationThe off-axis shear strength of a lamina is defined as the minimum of the positive and negative shear stress which can be applied to an angle lamina before failure.Calculations show that τxy = 46.19 MPa is the largest magnitude of shear stress one can apply to the 60° graphite/epoxy composite.However, the largest positive shear stress one could apply is 136.0 MPa, and the largest negative shear stress one could apply is –46.19 MPa.This shows that the maximum magnitude of allowable shear stress in other than the material axes direction depends on the sign of the shear stress.This is because the tensile strength perpendicular to the fiber direction is much lower than the compressive strength perpendicular to the fiber direction.

© 2003, P. Joyce

Failure EnvelopesFailure Envelopes

A failure envelope is a 3D plot of the combinations of normal and shear stresses which can be applied to an angle lamina before failure.Drawing 3D graphs is time consuming. . .One may develop failure envelopes for constant shear stress, τxy, and then use the 2 normal stresses σx and σy as the 2 axes.If the applied stress is within the failure envelope, the lamina is safe; otherwise it has failed.

© 2003, P. Joyce

Failure EnvelopesFailure EnvelopesFor a UD lamina at a given shear stress loading, the failure envelope takes the form of a rectangle as shown.

For a 60° lamina at a given shear stress loading, the failure envelope takes the form of a rectangle as shown.

σ2τxy

σ2σ2 σ1

( )ultC1σ− ( )ult

T1σ−

( )ultT2σ−

( )ultC2σ−


T1σ−

( )ultT2σ−

( )ultC2σ−

σ1σ1

© 2003, P. Joyce

Maximum Strain Failure TheoryMaximum Strain Failure TheoryBased on the Maximum Normal Strain Theory by St. Venant and the Maximum Shear Stress Theory by Tresca.The strains applied to a lamina are resolved into the normal and shear stresses in the material axes.Failure is predicted in a lamina, if any of the normal or shear strains in the material axes are equal to or greater than the corresponding ultimate strains of a UD lamina.

The ultimate strains can be found directly from the ultimate strength parameters and the elastic moduli, assuming the stress-strain response is linear until failure.Each component of strain is compared with the corresponding ultimate strain and hence does not have an interaction with the others.Yields different results from Maximum Stress Failure Theory, because the local strains in a lamina include the Poisson’s ratio effect (allows some interaction of stress components.)

( ) ( ) ( ) ( ) ( ) ( )ultultultT

ultC

ultT

ultC

121212222111 , , γγγεεεεεε <<−<<−<<−

© 2003, P. Joyce

Maximum Strain Failure TheoryMaximum Strain Failure TheoryAssume a general biaxial state of stresson an angle lamina.

Obtain the stress components along the principal material axes by stress transformation.

σ1

σ2

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

12

2

11)(

τσσ

θτσσ

T

xy

y

x

© 2003, P. Joyce

Maximum Strain Failure TheoryMaximum Strain Failure TheoryThen the corresponding strain components can be calculated by means of the lamina stress-strain relations:

Next calculate the ultimate strains as follows12

66

1

112

2

22

2

221

1

11

G

EE

EE

τγ

σνσε

σνσε

=

−=

−=

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )12

1212

2

22

2

22

1

11

1

11 , , , ,

GEEEEult

ultult

C

ultCult

T

ultTult

C

ultCult

T

ultT τ

γσ

εσ

εσ

εσ

ε =====

© 2003, P. Joyce

Maximum Strain Failure TheoryMaximum Strain Failure Theory

Failure subcriteria restated in terms of the stresses:

( )( )( )( )

( ) ( )ultult

ultC

ultT

ultC

ultT

121212

22

221212

11

112121

0 when 0 when

0 when 0 when

τττ

εσεσ

σνσ

εσεσ

σνσ

<<−⎩⎨⎧

<−>

=−

⎩⎨⎧

<−>

=−

© 2003, P. Joyce

Maximum Strain Failure TheoryMaximum Strain Failure Theory

For a 2D state of stress with τ6 = 0, the failure envelope takes the form of a parallelogram with its center off the origin.

σx

σ1

σ2

σy


T1σ−

( )ultT2σ−

( )ultC2σ−

( )C12121 σσνσ −=−

( )T12121 σσνσ =−

( )C11212 σσνσ −=−

( )T11212 σσνσ =−

© 2003, P. Joyce

TsaiTsai--Hill Failure TheoryHill Failure TheoryBased on the deviatoric or distortional energy failure theory of von Mises.Adapted to anisotropic materials by Hill.Then adapted to a UD lamina by Tsai.

Given the global stresses in a lamina, one can find the local stresses in a lamina and apply the above failure theory to determine whether or not the lamina has failed.

( ) ( ) ( ) ( ) 12

12

12

2

2

22

1

21

2

1

1 <⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡

ultultT

ultT

ultT τ

τσσ

σσσ

σσ

© 2003, P. Joyce

TsaiTsai--Hill Failure TheoryHill Failure TheoryThe failure envelope described by the Tsai-Hill criterion is a closed surface in the σ1, σ2, τ12 space.

Failure envelopes for constant values of

have the form

Where:

Graphically represents 4 different elliptical arcs joined at the σ1, σ2 axes.

( )ultk 1212 ττ=2

21

212

2

22

21

21 1 k

FFF−=−+

σσσσ

( )( )( )( )⎩

⎨⎧

<>

=

⎩⎨⎧

<>

=

0 when 0 when

0 when 0 when

22

222

11

111

σσσσ

σσσσ

ultC

ultT

ultC

ultT

F

F(Modified Tsai-Hill Criterion)

© 2003, P. Joyce

TsaiTsai--Hill Failure TheoryHill Failure Theory

Considers the interaction between the 3 UD lamina strength parameters, unlike the Maximum Stress and Maximum Strain Theories.Tsai-Hill Failure Theory is a Unified Theory and hence does not give the mode of failure like the Maximum Stress and Maximum Strain Theories.

© 2003, P. Joyce

TsaiTsai--Wu Failure TheoryWu Failure Theory

Based on a general failure theory for anisotropic materials first proposed by Gol’denblat and Kopnov (1965).Capable of predicting strength under general states of stress for which no experimental data are available.Uses the concept of strength tensors.Has the form of an invariant formed from stress and strain tensor componentsHas the capability to account for the difference between tensile and compressive strengths

© 2003, P. Joyce

TsaiTsai--Wu Failure TheoryWu Failure TheoryTsai and Wu (1971) proposed a modified tensor polynomial theory by assuming the existence of a failure surface in the stress space of the form —

12 21122

12662

2222

1111262211 <++++++ σστσστσσ HHHHHHH

The coefficients are obtained by applying elementary loading conditions to the lamina. Thus —

( ) ( )

( ) ( )ultC

ultT

ultC

ultT

H

H

1111

111

1

11

σσ

σσ

=

−= ( ) ( )

( ) ( )ultC

ultT

ultC

ultT

H

H

2222

222

1

11

σσ

σσ

=

−=

( ) 212

66

6

10

ult

H

H

τ=

=

© 2003, P. Joyce

TsaiTsai--Wu Failure TheoryWu Failure TheoryThe remaining coefficient H12 must be obtained by some type of biaxial testing.Direct biaxial testing is not easy or practical to perform.An easier test producing a biaxial state of stress is the off-axis tensile test.For θ = 45° we can measure the off-axis tensile strength, σ.Then,Again produces an elliptical failure envelope.

( ) ( )66221121

12 212 HHHHHH ++−

+−=

σσ

© 2003, P. Joyce

Comparison with Experimental ResultsComparison with Experimental Results

Observations —The difference between the Maximum Stress and Maximum Strain Failure Theories and the experimental results is quite pronounced.Tsai-Hill and Tsai-Wu Failure Theories are in good agreement with experimentally obtained results.The cusps observed in the Maximum Stress and Maximum Strain Failure Theories correspond to the change in failure mode.The variation of the strength as a function of angle is smooth in the Tsai-Hill and Tsai-Wu Failure Theories.

© 2003, P. Joyce

Macromechanical Macromechanical Failure TheoriesFailure TheoriesTheory Physical Basis Operational

ConvenienceReq’d experimental characterization

Maximum Stress Tensile behavior of brittle material

Inconvenient Few parametersBy simple testing

No stress interaction

Maximum Strain Tensile behavior of brittle material

Inconvenient Few parameters by simple testing

Some stress interaction

Deviatoric strain energy

Ductile behavior of anisotropic materials

Can be programmed Biaxial testing is needed in addition to uniaxial testing

(Tsai-Hill) “Curve fitting”for heterogeneous brittle composites

Different functions required for tensile and compressive strengths

Interactive tensorpolynomial

Mathematically consistent

General and comprehensive; operationally simple

Numerous parameters

Tsai-Wu Reliable “curve fitting” Comprehensive experimental program needed.

© 2003, P. Joyce

ReferencesReferencesEngineering Mechanics of Composite Materials, Daniel, I.M. and Ishai, O., 1994.Mechanics of Composite Materials, Kaw, A.K., 1997.Introduction to Composite Materials, Tsai, S. W. and Hahn, H. T., 1980.“Application of Advanced Composites in Mechanical Engineering Designs,” Zweben, C., Proceedings of the 31st International SAMPE Technical Conference, 1999.

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