macromechanics ver. 2 1

Macromechanics -1

Generalized Hooke’s law for anisotropic lamina

15/11/2006 Properties of laminated structures 2

Three steps in composites design

micromechanics

macromechanics

macromechanics

Macromechanics

1. Generalized Hooke’s law for anisotropic lamina

2. Classical lamination theory (CLT)3. Hygrothermal stresses in laminates4. Prediction of failure: failure criteria5. Strength of laminates

Properties of laminated structures 3

Properties of laminated structures 4

Properties of laminated structures

• Properties of single ply– Generalised Hooke’s law for anisotropic media – Stress-strain relationship in plane of orthotropy– Stress-strain relationship in arbitrary coordinate system

• Properties of a laminate

Properties of laminated structures 5

Hooke’s law of linear anisotropic elasticity

• The constitutive equation of a linear anisotropic solid is given by

ij = components of the stress tensorkl = components of the strain tensorCijkl = components of the elastic property tensori,j,k,l = 1,2,3

• It can be shown that

• This means that a general anistropic solid has 21 independent elastic constants Cijkl

klijklij C

klijijlkjiklijkl CCCC

12

31

23

33

22

11

121231122312331222121112

123131312331333122311131

122331232323333222231123

123331332333333322331133

122231222322332222221122

111211311123113311221111

12

31

23

33

22

11

222

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

Properties of laminated structures 6

Hooke’s law of linear anisotropic elasticity

• This equation is written in the coordinate system xi relative to base vectors ei

• In a different coordinate system x’i relative to base vectors e’i:

where

• Changing from tensor to compact notation:

klijklij ''C'

jiij

mnpqlqkpjnimijkl

pqlqkpkl

mnjnimij

e'ea

Caaaa'C

aa'

aa'

klijklij C

126315234

333222111

126315234

333222111

222

;;;;;;;;

Properties of laminated structures 7

Hooke’s law of linear elastic anisotropy

• Generalised Hooke’s law reads

Where the stiffness coefficients Cij are given by identification, e.g.

• The stiffness matrix [Cij] is symmetric. It does not transform like a tensor

• The compliance matrix [Sij] is the inverse of the stiffness matrix:

• A General anisotropic material has 21 independent elastic constants that describe the stress-strain behaviour in the linear elastic regime– 21 stiffness coefficients– 21 compliance coefficients

1 CSS jiji

jiji C

Properties of laminated structures 8

Material symmetries reduce the number of independent elastic

coefficients• Materials are classified according to symmetries:

– Triclinic: no symmetries• 21 independant elastic constants

– Monoclinic: one plane of symmetry– Orthotropic: three orthogonal planes of symmetry– Transversely isotropic: one plane of isotropy– Isotropic

jiji

ij

'C'

a

x'x;x'x;x'x

100010001

332211

One plane of symmetry (x1-x2): monoclinic material

• Elastic coefficients are invariant to the following transformation

Properties of laminated structures 9

Monoclinic material• Using transformation law for tensor components, we find

• We thus have

• We now have 13 independent elastic constants

• In another coordinate system, the stiffness matrix is in general fully populated, but only 13 coefficients are independent

5,4ifor0CCCC6,3,2,1ifor0CC

6j3j2j1j

5i4i

6

5

4

3

2

1

66362616

5545

4544

36332313

26232212

16131211

6

5

4

3

2

1

C00CCC0CC0000CC000C00CCCC00CCCC00CCC

Properties of laminated structures 10

Orthotropic material• Three orthogonal planes of symmetry• Define the coordinate axes xi by the symmetry

planes• In these axes of orthotropy, Hooke’s law

reduces to

• Only 9 independent elastic constants for orthotropic materials• In the orthotropy axes:

– No coupling between normal stresses 1, 2, 3 and shear strains 4, 5, 6– No coupling between shear stresses 4, 5, 6 and normal strains 1, 2, 3

• Coupling will occur in any coordinate system other than orthotropy axes!

6

5

4

3

2

1

66

55

44

332313

232212

131211

6

5

4

3

2

1

C000000C000000C000000CCC000CCC000CCC

Properties of laminated structures 11

Transversely isotropic material

• One plane of isotropy• Every plane containing x1 axis is plane of symmetry• Plane (x2-x3) is the isotropy plane• Hooke’s law reduces to

In all systems of coordinate such that (x2-x3) is the isotropy plane

• Only 5 independent elastic constants for transversely isotropic solids

6

5

4

3

2

1

66

66

2322

222312

232212

121211

6

5

4

3

2

1

C000000C0000

00CC21000

000CCC000CCC000CCC

Properties of laminated structures 12

Isotropic material

• Any plane is a plane of symmetry• Stiffness matrix [Cij] is independent of coordinate system• Hooke’s law reads in any coordinate system :

• Only 2 independent elastic coefficients for isotropic materials• Other notation: classical Lamé coefficients

6

5

4

3

2

1

1211

1211

1211

111212

121112

121211

6

5

4

3

2

1

CC2100000

0CC210000

00CC21000

000CCC000CCC000CCC

ijmmijij

121112

2

CC21C

Properties of laminated structures 13

Engineering constants for orthotropic materials

Tension Young’s moduli and Poisson’s ratios

• In the orthotropy axes, Hooke’s law reads:

• Simple tension in direction 1: 1 = constant and i = 0 for i = 2, 3, …, 6

• Hooke’s law gives

• Only normal strains are induced by tension in an orthotropy direction

6

5

4

3

2

1

66

55

44

332313

232212

131211

6

5

4

3

2

1

S000000S000000S000000SSS000SSS000SSS

0654

SSS 113311221111

Properties of laminated structures 14

Engineering constants for orthotropic materials-2

• Young’s modulus in orthotropy direction 1:

• Poisson’s ratios :

• Simple tension in orthotropy directions 2 and 3 yields :

111

11

1S

E

1311

313

1211

212

SE

SE

323232212121313131

2333213331

232231222133

322

211

EEEEEESESESESE

SE

SE

Properties of laminated structures 15

Engineering constants for orthotropic materials-3

Shear Shear moduli

• Uniform shear s6 applied to (x1-x2) coordinate plane :

• Only shear deformation is induced in the orthotropy axes

• Associated shear modulus :

• Similarly, for shear tests applied to (x2-x3) and (x1-x3) planes :

52106666 ,...,,iS i

666

612

1S

G

555

513

444

423

11S

GS

G

• Graphic representation of the engineering constants

Properties of laminated structures 16

Engineering constants for orthotropic materials-4

Properties of laminated structures 17

Stress-strain relations for orthotropic materials in terms of engineering

constants• In the axes of orthotropy, Hooke’s law reads

6

5

4

3

2

1

12

13

23

32

23

1

132

23

21

121

13

1

12

1

6

5

4

3

2

1

100000

010000

001000

0001

0001

0001

G

G

G

EEE

EEE

EEE

Properties of laminated structures 18

Stress-strain relations for orthotropic materials in terms of engineering

constants-2• By inversion, we get the stiffness coefficients Cij in terms of

engineering constants :

126613552344

1

2212

2

3223

1

3213231312

1

31

212

1

2333

131222311

33223

213

1

3222

13231233113

231331222112

223

2

3111

21

1

1

1

GCGCGCEE

EE

EE

EEDwhere

DEEEC

DEEEECC

DEEEC

DECCDEECC

DEEEC

Properties of laminated structures 19

Stress-strain relations for (transversely) isotropic materials in terms of engineering

constants

• Transversely isotropic with (x2-x3) as isotropy plane :

And thus

• For isotropic solids :

232244

665513123322

2 SSSSSSSSS

23

2231312

131232

12

EGGG

EE

12231312

231312

321

EGGGG

EEEE

Properties of laminated structures 20

Hooke’s law for orthotropic materials under state of plane stress

• Applies to thin orthotropic plies or laminae• If (1-2) is orthotropy plane, state of plane stress means

• Stress-strain relations reduce to and

where

0543 22

231

1

133

EE

6

2

1

66

2212

1211

6

2

1

6

2

1

1

1

1

6

2

1

0000

00

0

0

12

2112

112

1

QQQQQ

G

EE

EE

1266

122

12

21212

122

12

222

122

12

111

GQ

EE1

EQ

EE1

EQ

EE1

EQ

Properties of laminated structures 21

Stress-strain relations for orthotropic ply of arbitrary orientation

• Goal: write stress-strain relation in coordinate system (x-y) other than orthotropy axis (1-2)

• Angle between x and 1 is

• Tensor transformation laws can be derived– for stresses: equilibrium of forces on unit plane– for strains: projection of displacement vectors

sinncosm

nmmnmnmnmnmnnm

Tnmmnmnmnmnmnnm

T

TT

xy

y

x

xy

y

x

22

22

22

22

22

22

12

2

1

12

2

1

2222

22

15/11/2006 Properties of laminated structures 22

Compliance tensor for orthotropic ply of arbitrary orientation-2

• In the (x-y) system, Hooke’s law reads

– Where

Stress in (x,y) -> Stress in 1,2) -> strain in (1;2) -> strain in (x,y)

• Algebra yields :

xy

y

x

ssysxs

ysyyxy

xsxyxx

xy

y

x

SSSSSSSSS

2

TSTS 1

sinn,cosmwithSnmSSnmSSnmS

SnmmnSSnmSSmnS

SnmmnSSmnSSnmS

SmSSnmSnS

nmSSSSnmS

SnSSnmSmS

ss

ys

xs

yy

xy

xx

66222

221222

121122

6622

22123

12113

6622

22123

12113

224

661222

114

4412662211

2222

46612

2211

4

44

22

22

2

2

Properties of laminated structures 23

Stiffness tensor for orthotropic ply of arbitrary orientation

• In the (x-y) system, Hooke’s law reads– Where

– Strain in (x,y) -> strain in (1,2) -> stress in (1,2) -> stress in (x,y)

• Algebra yields :

• Shear-extension coupling occurs if (x-y) is different from (1-2)

xy

y

x

ssysxs

ysyyxy

xsxyxx

xy

y

x

QQQQQQQQQ

2

TQTQ 1

sinn,cosmwithQnmQQQnmQ

QnmmnQQnmQQmnQ

QnmmnQQmnQQnmQ

QmQQnmQnQ

nmQQQQnmQ

QnQQnmQmQ

ss

ys

xs

yy

xy

xx

66222

12221122

6622

22123

12113

6622

22123

12113

224

661222

114

4412662211

2222

46612

2211

4

2

2

2

22

4

22

Properties of laminated structures 24

Stress-strain relations for orthotropic ply of arbitrary orientation

isotropic orthotropic general orthotropic

loaded or anisotropic// orthotropy axis

Properties of laminated structures 25

Engineering constants for orthotropic ply of arbitrary orientation

• Pure tension x yields

• This defines the apparent engineering constants :

• Similarly, pure tension along the direction of the y-axis yields

xxsxyxxyyxxxx SSS 2

xx

xs

x

xyxxy

xx

xy

x

yxy

xxx

xx

SS

SS

SE

2

1

, Due to shear-extension coupling

yy

ysyxy

yy

xyyx

yyy S

SSS

SE ,

1

Properties of laminated structures 26

Engineering constants for orthotropic ply of arbitrary orientation-2

• Simple shear xy yields

• This defines the different apparent engineering constants :

• The coupling coefficients satisfy the following relations :

xyssxyxyysyxyxsx SSS 2

ss

ys

xy

yxyy

ss

xs

xy

xxyx

ssxy

xyxy

SS

SS

SG

22

12

,,

xy

xy,y

y

y,xy

xy

xy,x

x

x,xyGEGE

Properties of laminated structures 27

Engineering constants for orthotropic ply of arbitrary orientation-3

• In terms of apparent engineering constants, Hooke’s law reads

xy

y

x

xyy

y,xy

x

x,xy

y

y,xy

yx

xyx

x,xy

x

xy

x

xy

y

x

GEE

EEE

EEE

1

1

1

2

Properties of laminated structures 28

Engineering constants for orthotropic ply of arbitrary orientation-4

• Directional dependence of apparent engineering constants :

12

22

121

12

21

22

121

12

2

3

121

12

1

3,

121

12

2

3

121

12

1

3,

2

4

1

12

12

22

1

4

1221

2244

1

12

2

4

1

12

12

22

1

4

1142221

122122

122122

12111

111

12111

Gnm

GEEEnm

G

GEEnm

GEEmnE

GEEmn

GEEnmE

Em

EGnm

En

E

GEEnmnm

EE

En

EGnm

Em

E

xy

yyxy

xxxy

y

xxy

x

• Illustration : Variation of engineering constants as a function of the loading angle

– For glass-epoxy (Vf= 15%)

Properties of laminated structures 29

Engineering constants for orthotropic ply of arbitrary orientation

• Illustration : Variation of engineering constants as a function of the loading angle

– For carbon-epoxy (Vf= 15%)

Properties of laminated structures 30

Engineering constants for orthotropic ply of arbitrary orientation

Properties of laminated structures 31

Polar plot

Glass-epoxy carbon-epoxy

15/11/2006 Properties of laminated structures 32

Engineering constants for orthotropic ply of arbitrary orientation

• Illustration : Variation of the tensile modulus as a function of for carbon-epoxy (T8OO, Vf=80%), absolute values