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

• 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


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


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

;;;;;;;;


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


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


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


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


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


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


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


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



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




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


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


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


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


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


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


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


Stress-strain relations for orthotropic ply of arbitrary orientation

isotropic orthotropic general orthotropic

loaded or anisotropic// orthotropy axis


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


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



• 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



• 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%)



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

– For carbon-epoxy (Vf= 15%)




Polar plot

Glass-epoxy carbon-epoxy



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

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