Micromecanica.xls

5/24/2018 Micromecanica.xls

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1 Fractiune volumetrica si gravimetrica

2

3

4

5

6

7 Rezistenta la intindere pe directie longitudinala (fLt)

8 Rezistenta la compresiune pe directie longitudinala (fLc)

9

10

11

Rezistenta la intindere pe directie transversala (fTt)

Rezistenta la compresiune pe directie transversala (fTc)

Rezistenta la forfecare (fLTs)

MICROMECANICA MATERIALELOR COMPOZITE

Modulul de elasticitate transversal (E2=ET)

Modulul de elasticitate longitudinal (E1=EL)

Modulul de elasticitate la forfecare (G12=GLT)

Coeficientul lui Poisson - major (12=LT)

Coeficientul lui Poisson minor (21=TL)

dilong

)1(L


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directietransversala

irectieitudinala

)3(

Lamela compozita armataunidirectional


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vc=vm+vf mc= mf+ mm mc= mf+ mm

1 = vm/vc+ vf/vc 1 = mm/mc+ mf/mc c*vc=m*vm+f*vf

1 = Vm+ Vf 1 = Mm+ Mf c=m*Vm+f*Vf

mc=c*vc

mf=f*vf

Fraciune volumetric, fraciune gravimetric (de mas)

volum de

compozit,

masa de

compozit, fibre


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regula amestecurilor

mf/mc=f/c*vf/vc


5/24

prima pag

Ef

(N/mm2)

Em

(N/mm2)

Vf Vm EL (N/mm2)

72400.00 2000.00 0.00 1.00 2000.00 213000.00 2000.00

72400.00 2000.00 0.10 0.90 9040.00 213000.00 2000.00

72400.00 2000.00 0.20 0.80 16080.00 213000.00 2000.00

72400.00 2000.00 0.30 0.70 23120.00 213000.00 2000.00

72400.00 2000.00 0.40 0.60 30160.00 213000.00 2000.00

72400.00 2000.00 0.50 0.50 37200.00 213000.00 2000.00

72400.00 2000.00 0.60 0.40 44240.00 213000.00 2000.00

72400.00 2000.00 0.70 0.30 51280.00 213000.00 2000.00

72400.00 2000.00 0.80 0.20 58320.00 213000.00 2000.00

72400.00 2000.00 0.90 0.10 65360.00 213000.00 2000.00

72400.00 2000.00 1.00 0.00 72400.00 213000.00 2000.00

380.00 3.30 0.55 0.45 210.49

Modul de elasticitate longitudinal

lamela compozita armata unidirectional

Regula amestecu ri lor Relaii de calcul


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0.00 1.00 2000.00

0.10 0.90 23100.00

0.20 0.80 44200.00

0.30 0.70 65300.00

0.40 0.60 86400.00

0.50 0.50 107500.00

0.60 0.40 128600.00

0.70 0.30 149700.00

0.80 0.20 170800.00

0.90 0.10 191900.00

1.00 0.00 213000.00

0

50000

100000

150000

200000

250000

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

EL

[MPa]

Vf[%]

ELvs. Vf

CPAFS

CPAFC

Em

Ef


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prima pag

Ecuatia

Ef

(N/mm2)

Em

(N/mm2)

Vf Vm ET (N/mm2) ET/Em

231000.00 2000.00 0.00 1.00 2000.00 1.00

231000.00 2000.00 0.10 0.90 2220.09 1.11

231000.00 2000.00 0.20 0.80 2494.60 1.25

231000.00 2000.00 0.30 0.70 2846.58 1.42

231000.00 2000.00 0.40 0.60 3314.20 1.66

231000.00 2000.00 0.50 0.50 3965.67 1.98

231000.00 2000.00 0.60 0.40 4935.90 2.47

231000.00 2000.00 0.70 0.30 6534.65 3.27

231000.00 2000.00 0.80 0.20 9665.27 4.83231000.00 2000.00 0.90 0.10 18554.22 9.28

231000.00 2000.00 1.00 0.00 231000.00 115.50

380.00 3.30 0.55 0.45 7.26 2.20

Ecuatia

Ef

(N/mm2)

Em

(N/mm2)

Vf Vm x h1 ET (N/mm2)231000.00 2000.00 0.00 1.00 2.00 0.97 2000.00

231000.00 2000.00 0.10 0.90 2.00 0.97 2647.81231000.00 2000.00 0.20 0.80 2.00 0.97 3452.43

231000.00 2000.00 0.30 0.70 2.00 0.97 4478.65

231000.00 2000.00 0.40 0.60 2.00 0.97 5832.64

231000.00 2000.00 0.50 0.50 2.00 0.97 7701.24

231000.00 2000.00 0.60 0.40 2.00 0.97 10446.72

231000.00 2000.00 0.70 0.30 2.00 0.97 14875.50

231000.00 2000.00 0.80 0.20 2.00 0.97 23220.08

231000.00 2000.00 0.90 0.10 2.00 0.97 44788.93

231000.00 2000.00 1.00 0.00 2.00 0.97 231000.00

380.00 3.30 0.55 0.45 1.00 0.98 11.06

MODULUL DE ELASTICITATE TRANSVERSAL PENTRU MATERIALUL

COMPOZIT ARMAT CU FIBRE UNIDIRECTIONAL

REGULA INVERSA A AMESTECURILOR

relatia Halpin & Tsai


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0

20

40

60

80

100

120

0.

ET/Em

0.00 0.20 0.40 0.60 0.80 1.00

ET

Vf

ETvs. Vf (Halpin & Tsai)

0.00 0.20 0.40 0.60 0.80 1.00

ET

Vf

ETvs. Vf (Inverse rule of mixtures)

Em

Ef


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0 0.20 0.40 0.60 0.80 1.00

Vf

ET/Emvs. Vf


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prima pag

Ecuatii

Gf (N/mm2)

Gm

(N/mm2)

Vf VmGLT

(N/mm2)

15700.00 1700.00 0.00 1.00 1700.00

15700.00 1700.00 0.10 0.90 1866.43

15700.00 1700.00 0.20 0.80 2068.99

15700.00 1700.00 0.30 0.70 2320.87

15700.00 1700.00 0.40 0.60 2642.57

15700.00 1700.00 0.50 0.50 3067.82

15700.00 1700.00 0.60 0.40 3656.16

15700.00 1700.00 0.70 0.30 4523.73

15700.00 1700.00 0.80 0.20 5931.11

15700.00 1700.00 0.90 0.10 8609.68

15700.00 1700.00 1.00 0.00 15700.00

158.33 1.20 0.55 0.45 2.64

Ecuatii

Gf

(N/mm2)

Gm

(N/mm2)

Vf Vm x h2 GLT(N/mm

2)

231000.00 2000.00 0.00 1.00 2.00 0.97 2000.00

231000.00 2000.00 0.10 0.90 2.00 0.97 2647.81

231000.00 2000.00 0.20 0.80 2.00 0.97 3452.43

231000.00 2000.00 0.30 0.70 2.00 0.97 4478.65231000.00 2000.00 0.40 0.60 2.00 0.97 5832.64

231000.00 2000.00 0.50 0.50 2.00 0.97 7701.24

231000.00 2000.00 0.60 0.40 2.00 0.97 10446.72

231000.00 2000.00 0.70 0.30 2.00 0.97 14875.50

231000.00 2000.00 0.80 0.20 2.00 0.97 23220.08

231000.00 2000.00 0.90 0.10 2.00 0.97 44788.93

231000.00 2000.00 1.00 0.00 2.00 0.97 231000.00

158.33 1.20 0.55 0.45 1.00 0.98 4.04

Modulul de elasticitate la forfecare


relatia Halpin & Tsai

Regula inversa a amestecu ri lor

GLT

G


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EM EF M F GM GF

3.30 380.00 0.38 0.20 1.20 158.33

0.00 0.20 0.40 0.60 0.80 1.00

Vf

GLTvs. Vf

mGf


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First page

Equation

f m Vf Vm LT

0.20 0.30 0.00 1.00 0.30

0.20 0.30 0.10 0.90 0.29

0.20 0.30 0.20 0.80 0.28

0.20 0.30 0.30 0.70 0.27

0.20 0.30 0.40 0.60 0.26

0.20 0.30 0.50 0.50 0.25

0.20 0.30 0.60 0.40 0.24

0.20 0.30 0.70 0.30 0.23

0.20 0.30 0.80 0.20 0.22

0.20 0.30 0.90 0.10 0.21

0.20 0.30 1.00 0.00 0.20

0.20 0.38 0.55 0.45 0.28

Coeficientul lui Poisson - major


Rule of mixtures


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0.00 0.20 0.40 0.60 0.80 1.00

LT

Vf

ELvs. Vf

m

f


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prima pagina

Ef

(N/mm2)

Em

(N/mm2)

Vf Vm EL (N/mm2) f m LT

231000.00 2000.00 0.00 1.00 2000.00 0.20 0.30 0.300

231000.00 2000.00 0.10 0.90 24900.00 0.20 0.30 0.290

231000.00 2000.00 0.20 0.80 47800.00 0.20 0.30 0.280

231000.00 2000.00 0.30 0.70 70700.00 0.20 0.30 0.270

231000.00 2000.00 0.40 0.60 93600.00 0.20 0.30 0.260

231000.00 2000.00 0.50 0.50 116500.00 0.20 0.30 0.250

231000.00 2000.00 0.60 0.40 139400.00 0.20 0.30 0.240

231000.00 2000.00 0.70 0.30 162300.00 0.20 0.30 0.230

231000.00 2000.00 0.80 0.20 185200.00 0.20 0.30 0.220

231000.00 2000.00 0.90 0.10 208100.00 0.20 0.30 0.210

231000.00 2000.00 1.00 0.00 231000.00 0.20 0.30 0.200

231000.00 2000.00 1.00 0.00 231000.00 0.20 0.30 0.200

Coeficientul lui Poisson - minor


Ecuatii


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x h1 ET (N/mm2) TL2.00 0.97 2000.00 0.300

2.00 0.97 2647.81 0.031

2.00 0.97 3452.43 0.020

2.00 0.97 4478.65 0.017

2.00 0.97 5832.64 0.016

2.00 0.97 7701.24 0.017

2.00 0.97 10446.72 0.018

2.00 0.97 14875.50 0.021

2.00 0.97 23220.08 0.028

2.00 0.97 44788.93 0.045

2.00 0.97 231000.00 0.200

2.00 0.97 231000.00 0.200


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First page

Ef

(N/mm2)

Em

(N/mm2)

Vf Vm EL (N/mm2)

fft

(N/mm2)

fLt

(N/mm2)

72400.00 2000.00 0.00 1.00 2000.00 2250.00 62.15

72400.00 2000.00 0.10 0.90 9040.00 2250.00 280.94

72400.00 2000.00 0.20 0.80 16080.00 2250.00 499.72

72400.00 2000.00 0.30 0.70 23120.00 2250.00 718.51

72400.00 2000.00 0.40 0.60 30160.00 2250.00 937.29

72400.00 2000.00 0.50 0.50 37200.00 2250.00 1156.08

72400.00 2000.00 0.60 0.40 44240.00 2250.00 1374.86

72400.00 2000.00 0.70 0.30 51280.00 2250.00 1593.65

72400.00 2000.00 0.80 0.20 58320.00 2250.00 1812.43

72400.00 2000.00 0.90 0.10 65360.00 2250.00 2031.22

72400.00 2000.00 1.00 0.00 72400.00 2250.00 2250.00

REZISTENTA LA INTINDERE PE DIRECTIE LONGITUDINALA

Ec

0

1000

2000

3000

4000

5000

0.00 0.10 0.20 0.30 0.40

fLt

[MPa]

)1(

)1)](1([ 3/1

fmff

muffmff

LcVV

VVEVEf

)]1([ f

fL

mfftLt V

E

EVff


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Ef

(N/mm2)

Em

(N/mm2)

Vf VmEL

(N/mm2)

fft

(N/mm2)

fLt

(N/mm2)

213000.00 2000.00 0.00 1.00 2000.00 3700.00 34.74

213000.00 2000.00 0.10 0.90 23100.00 3700.00 401.27

213000.00 2000.00 0.20 0.80 44200.00 3700.00 767.79

213000.00 2000.00 0.30 0.70 65300.00 3700.00 1134.32

213000.00 2000.00 0.40 0.60 86400.00 3700.00 1500.85

213000.00 2000.00 0.50 0.50 107500.00 3700.00 1867.37

213000.00 2000.00 0.60 0.40 128600.00 3700.00 2233.90

213000.00 2000.00 0.70 0.30 149700.00 3700.00 2600.42

213000.00 2000.00 0.80 0.20 170800.00 3700.00 2966.95

213000.00 2000.00 0.90 0.10 191900.00 3700.00 3333.47

213000.00 2000.00 1.00 0.00 213000.00 3700.00 3700.00

atii

0.50 0.60 0.70 0.80 0.90 1.00

Vf[%]

fLtvs. Vf

CPAFS

CPAFC


19/24

Sa se determine:

1 fractiunile volumetrice de fib

2 densitatea compozitului

3 caracteristicile elastice

Mf=0,75-0,03n 4rezistenta la intindere a com

f = 2500 kg/m3

m= 1250 kg/m3

Ef=72 Gpa

Em=4 Gpa

Rtf=3500 Mpa

f=0,22

m=0,38

Se considera bara realizata din materiale

compozite, utilizata la armarea elementelor

structurale din beton armat


20/24

ra si matrice

pozitului in directie longitudinala


21/24

1

3

5

7

8

9

10

11

12

13

14

15

16

17

18

19

20.1

20.2

*cedare transversale din intindere transverse

fracture due to Poisson strain

Ecuaii

* regula inversa a amestecurilor

* regula amestecurilor

* flambajul fibrelor prin extensie

*Modulul de elasticitate la

forfecare (G12=GLT)

* regula amestecurilor

*relatia Halpin&Tsai

*Halpin&Tsai correction

2

4

6

mffm

mf

TVEVE

EEE

c f m

v v v

c

f

fv

vV

c

m

mv

vV

mfc www

c

f

fw

wW

c

m

mw

wW

gvgvgv mmffcc

n

i

ii

c

W1

1

i

c

i

i VW

ii

c

i WV

n

i

iV1

1

mmffcL VEVEEE 11

f

f

mTV

VEE

1

11

1

1

h

hx

11

1

x

h

mf

mf

EE

EE

mffm

mf

LTVGVG

GGG

2

2

1

xh

mf

mf

GG

GG

f

f

mLTV

VGG

2

22

1

1

h

hx

mmffLT VvVvv

L

TLTTLE

E

)]1([ ffL

m

fftLt VEEVff

)1(

)1)](1([3/1

fmff

muffmff

LcVV

VVEVEf

f

fmf

fLcV

EEVVf

132


22/24

20.3 * flambajul fibrelor prin forfecare

])1([2f

mfffsLc

E

EVVff


23/24

tensile

*microbuckling of fibres in either extensional or shear mode

*shear failure of fibres without buckling

*transverse tensile fracture due to Poisson strain

)(2 mLTsmfLTsfLc VfVff


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vf

vm

vcwf

wm

wc

V

W

E1=EL

E2=ET

G12=GLT

12=LT21=TL

fLt

fLc

fTt

fTc

fLTs

fu

mu

fLTsf

fLTsm rezistenat la forfecare a matricei

rezistenta la forfecare

rezistenta la forfecare a fibrelor

alungirea la rupere a fibrei, deformatia specifica ultima a fibrei

coeficientul lui Poisson major

alungirea la rupere a matricei, deformatia specifica ultima a matricei

rezistenta la intindere longitudinala

rezistenta la compresiune longitudinala

rezistenta la intindere transversala

fractiunea de masa, gravimetrica

rezistenta la compresiune transversala

coeficientul lui Poisson minor

volumul de material compozitmasa de fibre

modulul de elasticitate longitudinal

modulul de elsticitate la forfecare

modulul de elsticitate transversal

Symbols

volumul de fibre

volumul de matrice

masa de matrice

masa de compozit

fractiunea volumetrica

Documents

Micromecanica.xls