Download - Macromechanics Ver. 2 1
![Page 1: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/1.jpg)
Macromechanics -1
Generalized Hooke’s law for anisotropic lamina
![Page 2: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/2.jpg)
15/11/2006 Properties of laminated structures 2
Three steps in composites design
micromechanics
macromechanics
macromechanics
![Page 3: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/3.jpg)
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
![Page 4: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/4.jpg)
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
![Page 5: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/5.jpg)
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
![Page 6: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/6.jpg)
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
;;;;;;;;
![Page 7: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/7.jpg)
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
![Page 8: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/8.jpg)
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
![Page 9: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/9.jpg)
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
![Page 10: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/10.jpg)
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
![Page 11: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/11.jpg)
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
![Page 12: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/12.jpg)
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
![Page 13: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/13.jpg)
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
![Page 14: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/14.jpg)
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
![Page 15: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/15.jpg)
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
![Page 16: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/16.jpg)
• Graphic representation of the engineering constants
Properties of laminated structures 16
Engineering constants for orthotropic materials-4
![Page 17: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/17.jpg)
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
![Page 18: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/18.jpg)
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
![Page 19: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/19.jpg)
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
![Page 20: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/20.jpg)
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
![Page 21: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/21.jpg)
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
![Page 22: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/22.jpg)
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
![Page 23: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/23.jpg)
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
![Page 24: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/24.jpg)
Properties of laminated structures 24
Stress-strain relations for orthotropic ply of arbitrary orientation
isotropic orthotropic general orthotropic
loaded or anisotropic// orthotropy axis
![Page 25: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/25.jpg)
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
![Page 26: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/26.jpg)
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
![Page 27: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/27.jpg)
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
![Page 28: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/28.jpg)
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
![Page 29: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/29.jpg)
• 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
![Page 30: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/30.jpg)
• 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
![Page 31: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/31.jpg)
Properties of laminated structures 31
Polar plot
Glass-epoxy carbon-epoxy
![Page 32: Macromechanics Ver. 2 1](https://reader035.vdocuments.us/reader035/viewer/2022062316/577c83761a28abe054b50c9e/html5/thumbnails/32.jpg)
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