Composite Design Cycle

(1)

Composite Design Cycle

CONSTITUENTS

STRUCTURE UNIT CELL

STRUCTURAL ELEMENT

ELEMENTARY STRUCTURE

Micro Mechanics Experimental Validation

Finite Element Analysis

Laminated Plates

Weaves And Braids

RVE

Local Structural Properties

(2)

A Composite is a Mixture of Two or More Distinct Constituents or Phases

Reinforcements

Polymers BioMaterials

Metals Ceramics

BMCs – Biological Matrix Composites PMCs - Polymer Matrix Composites MMCs - Metal Matrix Composites CMCs - Ceramic Matrix Composites

RBB Composite Materials 3

(3)

Advantage of Fiber Form over Bulk Form

• Plate Glass

 Strength 70 MPa

 Stiffness 70 GPa

• E Glass Fiber

 Strength 3448 MPa

 Stiffness 72 GPa

• If Aspect Ratio is low,

failure will occur due

to slippage between

the fiber and matrix

(4)

Consider Five Bare Fibers Being Loaded

P

F

1f

=0.30 N F

2f

=0 .35 N F

3f

=0 .25 N F

4f

=0 .40 N F

5f

=0 .50 N

P 5

P 5 0.2 P 0.2 P 0.2 P 0.2 P 0.2 P

1.0 N

0.2 N 0.2 N

0.2 N

1.25 N

0.25 N 0.25 N 0.25 N 0.25 N 0.25 N 0.31 N 0.31 N 0.31 N 0.31 N P

4 P 4

P 4

P 4 0.42 N 0.42 N 0.42 N

P 3

BUNDLE FAILS AT 1.25 N

(5)

Consider Five Fibers Surrounded by Matrix Being Loaded

F

1f

=0 .30 N F

2f

=0 .35 N F

3f

=0 .25 N F

4f

=0 .40 N F

5f

=0 .50 N

P 5

P 5 0.2 P 0.2 P 0.2 P 0.2 P 0.2 P

MATRIX

• Good adhesive Shear Strength

• Negligible (zero) Tensile Strength

 

0.25 N 0.25 N 0.25 N 0.25 N 0.25 N

+0.09 N

+0.09 N +0.035 N

+0.035 N

0.285 N 0.34 N 0.34 N 0.285 N

(6)

Classification of Composites

Preferred Orientation Random

Orientation Bidirectional

Reinforcement

(Woven Reinforcements) Unidirectional

Reinforcement

Hybrids Laminates

Discontinuous Fiber Reinforced Composites Continuous Fiber

Reinforced Composites

Multilayered Single-Layer

Preferred Orientation Random Orientation

COMPOSITE MATERIALS

Particle reinforced composites Fiber reinforced

composites

(7)

The Specific Properties of

Composites Are Exceptional

(8)

Stress Tensor

Composite Materials 9

RBB

 

ij zz

zy zx

yz yy

yx

xz xy

xx

ij zz

zy zx

yz yy

yx

xz xy

xx

z zy

zx

yz y

yx

xz xy

x















 







 









 







 









 







 









 

 





 





 





 







xy yz xz z y x





(9)

 _xy

Element with Finite Dimensions

z

y

x Δy

σ _x  _xz σ _z



 _zx

 _zy

 _yx

yz

yz y

y

  ^  



y

y y

y

  ^  



yx

yx y

y

  ^  



xy

xy x

x

  ^  



x

x x

x

  ^  







zy

zy z

z

  ^  



z

z z

z

  ^  



F y

F z

F x

(10)

Equilibrium Equations Result from Equilibrium

RBB

0 0 0



 

 



 





 

 



 





 

 



 



z yz z

xz

y yz

y xy

x xy xz

x

z F y

x

z F y

x

z F y

x

 







 



(11)

Strain Tensor

 

ij yz

yy yx

xz xy

xx

z zx zy

yz y

yx

xy xz x



 



 



 





 







 









 







 









2 2

 

 





 





 





 







 



 







 



 









xy yz xz z y x





 



2

(12)

Strain - Displacements

RBB

w z v y u x

x y x

 



 



 





w y v z

u z w x

u y v x

zy xz xy

 

 

 

 

 

 

 

 

 



(13)

Compatibility

2 2 2

2 2

2 2 2

2 2

2 2 2

2 2

x z

z x

y z

z y

x y

y x

z x

xz

y z yz

x y xy



 



 







 



 







 



 











 



 



 

 







 



 



 



 







 

 

 







 



 



 



 







 

 

 







 



 



 

 

 







 

z y

x z

y x

z y

x y

x z

z y

x x

z y

zx xy z yx

zx xy yz

y

zx xy x yz

 

 

 





 

 

2 2 2

2

(14)

Transforming the

Stress and Strain Tensor

z

x y

z’

y’

x’

 _x’y

 _x’x

 _x’z

A _x A _z

A _y

RBB

  ^ _x _ _y _ _z _ ^       ^T ^ ^ _xyz ^ ^T ^T

 







 











z z y

z x

z

z y y

y x

y

z x y

x x

x

n n

n

n n

n

n n

n T

, ,

,

, ,

,

, ,

,

  i j j

n i _  cos  _

  ^ _x _ _y _ _z _ ^       ^T ^ ^ _xyz ^ ^T ^T

(15)

Example 1:

Write the transformation Matrix for the following:

- First a positive 45° about z axis

- Second a positive 30° about the new x’ axis

0.7017 0.7017 0 1 0 0

1 0.7017 0.7017 0 2 0 0.866 0.5

0 0 1 0 0.5 0.866

T T

   

   

      

    

   

0.7071 0.7071 0 **2* 1 0.6124 0.6124 0.5**

0.3536 0.3536 0.866 T T

 

 

   

  

 

(16)

Constitutive Equations Based on Fundamental Assumptions

• The stress-strain relations are linear

 the material follows the generalized Hooke’s law

 the coefficients in these linear relations may be either constant (homogeneous body) or variable - functions of position, continuous or

discontinuous (non-homogeneous body)

• Theory is based on classical linear theory of

homogeneous or non-homogeneous elastic bodies

(17)

Relationship Between Stress and Strain

1 1

0 0 0 0 0 0

1 1

0 0 0 0 0 0

1 1

0 0 0 0 0 0

0 0 0 1 0 0 0 0

x x

y y

z z

yz yz

xz xz

E E E E E E

G

   

 

     

 

 

     

 

     

     

     

         

   

        

     

     

2 (1 )

0 0 0

x y

z yz

E

xz



 

 

 

   

   

     

   

     

 

 

 

 

   

(18)

Anisotropic Materials

• ISOTROPIC - material properties are the same in all directions

• ANISOTROPIC - material properties change with direction

• HOMOGENEOUS - material of uniform

composition throughout and whose properties are constant at every point

• HETEROGENEOUS - material uniformity within a body consisting of dissimilar constituents

separately identifiable

(19)

Stress-Strain Relations

Stiffness

Compliance

 





 







 





 













 





 







 





 









 





 







 





 











12 13 23 3 2 1

66 54

64 63

62 61

65 55

54 53

52 51

64 45

44 43

42 41

63 35

34 33

32 31

26 25

24 23

22 21

16 15

14 13

12 11

12 13 23 3 2 1

C C

 







 













 







 









 







 











23 3 2 1

46 45

44 43

42 41

36 35

34 33

32 31

26 25

24 23

22 21

16 15

14 13

12 11

23 3 2 1

S S

S

(20)

Compliance Relations

for an Anisotropic Material

η _ij,k = Mutual Influence Coefficients of the First Kind η _i,jk = Mutual Influence Coefficients of the Second Kind

μ _ij,kl = Chentsov’s Coefficients

23,1 13,1 12,1

23,2 13,2 1

1

2 21 31

1 1 1 1 1

32 12

2 2 2

2

2 2 2

1 1

2 2 13 23

3 4 3 3 3 3

3 3

23 23

13 13 23

12 12

3

1,

,

23

2

23,3 13,3 12,

2,

1

1 1 [ ]

E E E E E E

S

G



  



  

 

   

       

     

     

 



    

     

   

     

   

 



 

  

 



 



 

3 3,23

1,13 2,13 3,13

1,12 2,12 3,12

13,23 12,23

23,13 12,13

23,12 1

1 2 3 23

23 23 23 23 23 13

12

13 13 13 13 13 13

12 12 12 12 1

3

2 12

,12

1

1 G G G G G

G G G G G G

 

 

  

 

      

 

      

  

    

 

 



 

  



 

 

   

 



  



 

(21)

Symmetry of the Stiffness Matrix

• Maxwell-Betti Recipriocal Theorem

• Elastic Potential/Strain Energy Density

 Incremental work per unit volume

 dW= _i d _i

• Using the Stress-Strain Relations

 dW=C _ij  _j d _j

• Work per Unit Volume

 W=1/2 C _ij  _i  _j

• dW/d =C  or dW ² /d d =C thus C = C

(22)

Stiffness and Compliance down from 36 to 21 Constants

23

Stiffness

Compliance

RBB Composite Materials

11 12 13 14 15 16

1 1

22 23 24 25 26

2 2

33 34 35 63

3 3

44 45 64

23 23

55 6

1

5 13 13

6 12

13 23

4 4

6 12 12

24 3

15 25 35 45

16 26 36 46 56

C C C C

C C

C C C

C C C C

C C C C C

C C

C C C C C

C C C C

C C C

C C

 

 

 

   

 

   

 

   

 

   

      

     

    

    



 

 

11 12 13 14 15 16

1 1

22 23 24 25 26

2 2

33 34 35 36

3 3

44 45 46

23 23

55 5

1

6 13 13

6 12

13 23

4 4

6 12 12

24 3

15 25 35 45

16 26 36 46 56

S S S S

S S

S S S

S S S S

S S S S S

S S

S S S S S

S S S S

S S S

S S

 

 

 

   

 

   

 

   

 

   

      

     

    

    



 

 

(23)

One Plane of Elastic Symmetry

Monoclinic

13 Independent Constants

11 12 13 16

1 1

22

4 23 26

2 2

33 36

3 3

44 4

12 13 23

5

5 23 23

55 13 13

0 0

0 0 0 0

C C C C

C C C

C C

C

 

 

 

   

 

   

 

   

 

   

       

     

     

(24)

Three Planes of Elastic Symmetry Orthotropic Body

25

9 Independent Constants

11 12 13

1 1

22 23

2 2

33 3 3

44 23 23

55 13 13

66 1

12 13 23

12 2

0 0 0

0 0 0 0 0

C C C C C C

C

C C

 

 

 

   

 

   

 

   

 

   

       

     

     

     

(25)

C In Terms of Engineering Constants

 

   

 

   

 

23 32 1

11

21 31 23 1 12 32 13 2

12

31 21 32 1 13 12 23 3

13

13 31 2

22

32 12 31 2 23 21 13 3

23

12 21 3

33

1 1

1 1 C E

E E

C

E E

C

C E

E E

C

C E

 



     

 

     

 



     

 



  

 

     

 

 

     

 

 

  

 

     

 

 

  

 

(26)

Three Planes of Elastic Symmetry Orthotropic Body

27

9 Independent Constants

11 12 13

1 1

21 22 23

2 2

31 32 33

3 3

44 23 23

55 13 13

12 66 12

0 0 0 0 0 0 0 0 0

0 0 0 0 0

S S S

S

 

 

 

   

 

   

 

   

 

   

       

     

     

     

(27)

S In Terms of Engineering Constants

11 12 13

1 2 3

21

1 22 23

1 2 3

31 32 33

1 2 3

44 55 66

23 1 2

31

2 3 1

21 3 1

3 23

2

1

1 1 1 1

S S S

E E E

S S S

E E E

S S S

E E E

S S S

G G G



 

    

    

    

  

(28)

Relationship Between S and C

2 22 33 23 13 23 12 33

11 44 12

44 2

33 11 13 12 23 13 22

22 55 13

55 2

12 13 23 11

11 22 12

33 66 23

66 ; 1 ;

; 1 ;

S S S S S S S

C C C

S S S

S S S S S S S

C C C

S S S

S S S S S S S

C C C

S S S

  

  

  

  



 

  

2 2 2

11 22 33 11 23 22 13 33 12 2 12 23 13

S  S S S  S S  S S  S S  S S S

(29)

Restrictions on Engineering Constants - Orthotropic

• S ₁₁ , S ₂₂ , S ₃₃ , S ₄₄ , S ₅₅ , S ₆₆ >0

• C ₁₁ , C ₂₂ , C ₃₃ , C ₄₄ , C ₅₅ , C ₆₆ >0

• (1- ₂₃  ₃₂ )>0, (1- ₁₃  ₃₁ )>0, (1- ₁₂  ₂₁ )>0

• =1-  ₁₂  ₂₁ -  ₂₃  ₃₂ -  ₁₃  ₃₁ - 2 ₂₁  ₃₂  ₁₃ >0

• See Jones pp 68-69

(30)

One Plane in which the Mechanical Properties are Equal

31

Transversely Isotropic 6 Independent Constants

 

11 12 13

1 1

12 11 13

2 2

13 13 33

3 3

23 44 23

13 44 13

11 12

12 12

0 0 0

0 0 0 0 0

0 0 0 0 0 2

C C C C C C C C C

C

C C

 

 

 

   

 

   

 

   

 

   

       

     

     

      

     

(31)

Material Properties Equal in all Directions

Isotropic

2 Independent Constants

 

11 12 12

1 1

12 11 12

2 2

12 12 11

3 3

11 12

23 23

11 12

13 13

11 12

12 12

0 0 0

0 0 0 2 0 0

0 0 0 0 2 0

0 0 0 0 0 2

C C C

C C

 

 

 

   

 

   

 

   

 

   

       

      

     

      

     

      

     

(32)

Matrix Form of Stress-Strain Relations

RBB

1 1

0 0 0 0 0 0

1 1

0 0 0 0 0 0

1 1

0 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0

0 0 0 0 0 1

x x

y y

z z

yz yz

xz xz

xy xy

E E E E E E

G G

G

   

 

     

 

 

     

 

     

     

     

         

   

        

     

     

   

 

 

2 (1 )

0 0 0

2 (1 )

0 0 0 0 0

2 (1 )

0 0 0 0 0

x y

z yz

xz xy

E



 

  



 

 

   

   

     

   

     

 

 

 

 

     

   

 

 

 

 

(33)

Restrictions on Engineering Constants - Isotropic

• Shear Modulus

 G=E/2(1+)

  > -1

• Hydrostatic Pressure  _x =  _y =  _z =-p

 The sum of the normal or extensional strains

  =  _x +  _y +  _z = p/ (E/ 3(1-2n)) = p/K

 K  Bulk Modulus = E / 3(1-2)

  < 1/2

(34)

35

Typical Composite Properties

Graphite-

Polymer Glass-Polymer Aluminum

E

1

(Gpa) 155.0 50.0 72.4

E

2

(Gpa) 12.10 15.20 72.4

E

3

(Gpa) 12.10 15.20 72.4



23

0.458 0.428 0.3



13

0.248 0.254 0.3



12

0.248 0.254 0.3

G

23

(Gpa) 3.20 3.28 27.8

G

13

(Gpa) 4.40 4.70 27.8

G

12

(Gpa) 4.40 4.70 27.8



1

(10

^-6

/

^o

C) -0.018 6.34 22.5



2

(10

^-6

/

^o

C) 24.3 23.3 22.5



3

(10

^-6

/

^o

C) 24.3 23.3 22.5



1

(10

^-6

/%M) 146.0 434. 0



2

(10

^-6

/%M) 4770 6320 0



3

(10

^-6

/%M) 4770 6320 0

(35)

Free Thermal Strains

• Material Expands and Contracts when Heated and Cooled

• Rate of Expansion Different in the 3 Principal Material Directions

 Graphite fibers contract along their length when heated

 Polymers, aluminum, boron, ceramics and most

other matrix materials expand when heated

(36)

37

Free Thermal Strain for Composites

• Directly Related to Change in

Length

1 2 3

1 3

1 2

2 3

 ₁

 ₃

 ₂



₁

+ 

₁



₁

+ 

₁



₂

+ 

₂



2

+  

2



3

+  

3



3

+  

3

 

1 1 1

1 2

2 2

2 3

3 3

3 , , ,

T

ref

T

ref

T

ref

T T T

  

    



    



    



  

(37)

Example

1 2 3

For the unconstrained state, determine the final dimensions of a 50mm x 50mm x 50mm cube of material made out of a Graphite-Reinforced-Plastic subject to a 50C temperature change.

 ₁

 ₃

 ₂

Graphite-Polymer

₁(10^-6/^oC) -0.018

₂(10^-6/^oC) 24.3

₃(10^-6/^oC) 24.3

 

6 9

1

9 1

6 3

2 3

3

2 3

0.018 10 1 50 900 10

50 900 10 50 49.99996

24.3 10 1 50 1.215 10 50 1.215 10 50 50.061

T

T T

C C

l mm mm mm

C C

l l mm mm mm



 

 



 



        

     

       

     

(38)

39

Notes on Linear Thermal Expansion Coefficients

•  ₁ ,  ₂ , and  ₃ , are referred to as the LINEAR coefficients of thermal expansion

• If expansion effects are not linearly proportional to temperature, ’s have no meaning

• Free thermal strains do not involve any shearing deformations

• For an element of material isolated in space, the thermal strains do not result in stresses

(39)

Concept of Mechanical Strain

 

1 1

2 2

3 3

23 13 12

, 1

, 2

, 3

23 13 12

T

ref T

ref

T T Mechanical Strain Tota

T T Mechanical Strain Mechanical Strain T T

Mechanical Strain Mechanical Strain Mechanical Strain

 



    

 

    

 

      

   

   

 

   

 

1 1

2 2

3 3

23 13 12

l Strain Thermal Strain Total Strain Thermal Strain Total Strain Thermal Strain

Total Strain Total Strain Total Strain

  

  

 

  

 

 

 

 

T

T T

 

  

 



    

(40)

41

- Relations Including LINEAR Thermal Strains (Compliance)

11 12 13

1 1 1

21 22 23

2 2 2

31 32 33

3 3 3

44 23 23

55 13 13

12 66 12

0 0 0

0 0 0 0 0

S S S

T

S S S

T

S S S

T

S

  

 

   

   

 

       

   

 

     

       

   

 

   

 

   

 

   

 

   

     

(41)

- Relations Including Linear Thermal Strains (Stiffness)

11 12 13

1 1 1

21 22 23

2 2 2

31 32 33

3 3 3

23 44 23

55 13 13

66 12 12

0 0 0

0 0 0 0 0

C C C T

C

  

 

   

   

 

       

   

 

     

       

   

 

   

 

   

 

   

 

   

     

(42)

Example

For a completely constrained state, determine the final state of stress in the 50mm x 50mm x 50mm cube of material made out of a Graphite-

Reinforced-Plastic. Subject to a 50C temperature change

1 2 3

 ₁

 ₃

 ₂

Graphite-Polymer

₁(10^-6/^oC) -0.018

₂(10^-6/^oC) 24.3

₃(10^-6/^oC) 24.3

11 12 13

1 1

21 22 23

2 2

31 32 33

3 3

44 23

55 13

66 12

0 0 0

0 0

3 1

0 0 0

6

0

0 0

3

0 0

1.57796 1 563 90738 5636390738 0 0 0

0 0

0 0 0 0 0 0

56

C C C T

C C

E

C

 



   

   

 

      

   

 

    

   

   

 

   

 

   

 

   

   

   

  







0.0000009

6390738 15513210895 7214171115 0 0 0 0.001215

5636390738 7214171115 15513210895 0 0 0 0.001215

0 0 0 3.2 09 0 0 0

0 0 0 0 4400000000 0 0

0 0 0 0 0 4400000000 0

13.55 27.61 27.61 0

E

   

   

   

   

   

 





 







 





 

0 0

MPa

 

 

 

 

 

(43)

Hygroscopic Strains

• Polymers exposed to liquids will

 absorb moisture

 expand, typically linearly with %Moisture

• Free Moisture Strains

 

1 1

2 2

3 3

, , ,

M

ref M

ref

M M M

 

  

(44)

45

Thermal/Mechanical Properties- Compliance

11 12 13

1 1 1 1

21 22 23

2 2 2 2

31 32 33

3 3 3 3

44 23 23

55 13 13

66 12 12

0 0 0

0 0 0 0 0

S S S

T M

S S S

T M

S S S

T M

S

   

 

     

   

 

         

   

 

       

       

   

 

   

 

   

 

   

 

   

     

(45)

Thermal/Mechanical Properties- Stiffness

11 12 13

1 1 1 1

21 22 23

2 2 2 2

31 32 33

3 3 3 3

44 23 23

55 13 13

66 12 12

0 0 0

0 0 0 0 0

C C C T M

C

   

 

   

 

   

 

         

   

 

       

       

   

 

   

 

   

 

   

 

   

     

(46)

Composite Design Cycle

CONSTITUENTS

STRUCTURE UNIT CELL

STRUCTURAL ELEMENT

ELEMENTARY STRUCTURE

Micro Mechanics

E x

E y

 G

Experimental Validation

Finite Element Analysis

Laminated Plates

Weaves And Braids

RVE

Local Structural Properties

(47)

Stresses and Deformations

(48)

Plane-Stress Assumption

• Fiber-reinforced materials are utilized in beams, plates, cylinders and other structures

• Typically one characteristic geometric dimension is an order of magnitude less than the other two

• Three of the six components of stress are generally much smaller than the other three

49

(49)

Plane Stress Inaccuracies

• Errors in analysis near edges

 The stresses  ₃ ,  ₂₃ ,  ₁₃ lead to delamination

 Bonded joints can not be modeled

 Adhesive interface can not be evaluated

• These stress components equated to zero are forgotten and no attempt is made to estimate their magnitude

• erroneously assumed that 