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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