Or Specially Orthotropic Lamina
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Transcript Or Specially Orthotropic Lamina
Micromechanics
Macromechanics
Fibers
Lamina
Laminate
Structure
Matrix
Macromechanics
Study of stress-strain behavior of composites
using effective properties of an equivalent
homogeneous material. Only the globally
averaged stresses and strains are considered,
not the local fiber and matrix values.
Stress-Strain Relationships for
Anisotropic Materials
First, we discuss the form of the stress-strain
relationships at a point within the material,
then discuss the concept of effective moduli
for heterogeneous materials where properties
may vary from point-to-point.
General Form of Elastic - Relationships
for Constant Environmental Conditions
ij Fij (11 , 12 , 13 ,...),
i, j 1, 2,3... (2.1)
Each component of stress, ij, is related to
each of nine strain components, ij
(Note: These relationships may be
nonlinear)
Expanding Fij in a Taylor’s series and
Retaining only the first order terms,
ij Cijkl kl ,
i, j, k , l 1,2,3
for a linear elastic material
ij 9 com ponents
kl 9 com ponents
Cijkl 81 com ponents
3D state of stress
Generalized Hooke’s Law for
Anisotropic Material
11
22 C1111
33 C 2211
C 3311
23 C 2311
31 = C 3111
C1211
12 ...
C 3211
32 C
13 1311
C 2111
21
C
C
C
C
C
C
1122
2222
3322
2322
3122
1222
...
C
C
C
3222
1322
2122
C
C
C
C
C
C
1133
2233
3333
2333
3133
1233
...
C
C
C
3233
1333
2133
C
C
C
C
C
C
1123
2223
3323
2323
3123
1223
...
C
C
C
3223
1323
2123
C
C
C
C
C
C
1131
2231
3331
2331
3131
1231
...
C
C
C
3231
1331
2131
C
C
C
C
C
C
1112
2212
3312
2312
3112
1212
...
C
C
C
3212
1312
2112
..
C
C
C
C
C
C
1132
2232
3332
2332
3132
1232
...
C
C
C
3232
1332
2132
C
C
C
C
C
C
1113
2213
3313
2313
3113
1213
...
C
C
C
3213
1313
2113
2221
3321
2321
3121
1221
...
C 3221
C1321
C 2121
C
C
C
C
C
C
1121
11
22
33
23
31
12
32
13
21
(2.2)
Symmetry Simplifies the Generalized
Hooke’s Law
1. Symmetry of shear stresses and strains:
21
2
1
O
12
Static Equlibrium
M
0 im plies 12 21
or in general,
0
ij ji or ij ji
Same condition for shear strains, ij
ji
2. Material property symmetry – several types will
be discussed.
Symmetry of shear stresses and shear strains:
ij ji and ij ji
Thus, only 6 components of ij are
independent, and likewise for ij.
This leads to a contracted notation.
Stresses
Tensor Notation
Contracted
Notation
11
1
22
2
33
3
23= 32
4
13= 31
5
12= 21
6
Strains
Tensor Notation
Contracted
Notation
11
1
22
2
33
3
2 23= 2 32= 23= 32
4
2 13= 2 31= 13= 31
5
2 12= 2 21= 12= 21
6
Geometry of Shear Strain
xy
xy
2
xy = Engineering Strain
xy = Tensor Strain
xy
xy
2
Total change in original angle = xy
Amount each edge rotates = xy/2 = xy
Using contracted notation
i Cij j
i, j 1,2,...,6
or in matrix form C
(2.3)
(2.4)
where and are column vectors
and [C] is a 6x6 matrix (the stiffness
matrix)
Alternatively,
i Sij j
i, j 1,2,...,6
or S
(2.5)
(2.6)
where [S] = compliance matrix
and
S C
1
Expanding:
1 S11
S
2 21
3 S31
4 S 41
5 S51
6 S 61
S12
S13
S14
S15
S 22
S 23
S 24
S 25
S32
S 42
S33
S 43
S34
S 44
S35
S 45
S52
S53
S54
S55
S 62
S 63
S 64
S 65
S16 1
S 26 2
S36 3
S 46 4
S56 5
S 66 6
• Up to now, we only considered the stresses
and strains at a point within the material,
and the corresponding elastic constants at a
point.
• What do we do in the case of a composite
material, where the properties may vary
from point to point?
• Use the concept of effective moduli of an
equivalent homogeneous material.
Concept of an Effective Modulus of an Equivalent
Homogeneous Material
Heterogeneous composite
under varying stresses and
strains
x3
d
2 L
x3
2
2
Stress, 2
Equivalent homogeneous x
3
material under average
stresses and strains
2
2
Strain, 2
x3
2
2
Stress
Strain
2
Effective moduli, Cij
i Cij j
where,
i average stress
(2.9)
i dv
v
(2.7)
dv
v
i average strain
i dv
v
(2.8)
v
dv
3-D Case
General Anisotropic Material
• [C] and [S] each have 36 coefficients, but
only 21 are independent due to symmetry.
• Symmetry shown by consideration of strain
energy.
• Proof of symmetry:
Define strain energy density
1
W i i i 1,2,...,6
2
1
1
1
W 11 11 ... 6 6
2
2
2
but
i Cij j
1
W Cij i j
2
(2.12)
Now, differentiate:
but
j
j
W 1
1
Cij j Cij i
i 2
2
i
ij Kronecker delta 1 if i=j
i
0 if ij
ij i j
(show)
W
Cij j
i
2W
Cij
i j
(2.11)
(2.13)
But if the order of differentiation is reversed,
W
C ji
j i
2
(2.14)
Since order of differentiation is immaterial,
Cij C ji (Symmetry)
Similarly,
1
W S ij i j
2
and
Sij S ji
Only 21 of 36 coefficients are
independent for anisotropic material.
Stiffness matrix for linear elastic
anisotropic material with no material
property symmetry
(2.15)
3-D Case, Specially Orthotropic
3
2
1
1, 2 , 3 principal
material coordinates
1
12
2
2
1
(a)
12
(b)
(c)
Simple states of stress used to define lamina
engineering constants for specially orthotropic
lamina.
Consider normal stress 1 alone:
3
1
1
1
2
Resulting strains,
1
1
E1
; 2 121 12
1
E1
(2.19)
Typical stress-strain curves from
ASTM D3039 tensile tests
Stress-strain data from longitudinal tensile test of carbon/epoxy composite.
Reprinted from ref. [8] with permission from CRC Press.
Similarly,
3 131 13
1
E1
where E1 = longitudinal modulus
ij = Poisson’s ratio for strain
along j direction due to
loading along i direction
Now consider normal stress 2 alone:
Strains:
2
2
E2
2
3
;
2
1 21 2 21
E2
2
3 23 2 23
E2
Where E2 = transverse modulus
Similar result for 3 alone
1
2
2
(2.20)
• Observation:
All shear strains are zero under pure
normal stress (no shear coupling).
12 13 23 0
For
1, 2 , 3 alone
Now, consider shear stress
12 alone,
3
1
Strain
12
12
2
12 12
G12
Where G12 = Shear modulus in 1-2 plane
1 2 3 13 23 0
(No shear coupling)
(2.21)
Similarly, for 13 alone
13
13
G13
;
1 2 3 12 23 0
and for 23 alone
23
23
G23
;
1 2 3 13 12 0
Now add strains due to all stresses using
superposition
Specially Orthotropic 3D Case
1
E
1
12
1 E1
2 13
3 E1
23 0
31
12
0
0
21
E2
31
E3
0
0
1
E2
32
E3
0
0
23
E2
1
E3
0
0
0
0
1
G23
0
0
0
0
1
G31
0
0
0
0
0
0
1
2
0
3
0 23
31
0 12
1
G12
12 coefficients, but only are 9 independent
(2.22)
Symmetry:
Sij S ji
ij
Ei
ji
Ej
Only 9 independent coefficients.
Generally orthotropic 3-D case –
similar to anisotropic with 36 nonzero
coefficients, but 9 are independent as with
specially orthotropic case
Specially Orthotropic – Transversely Isotropic
3
2
1
Fibers randomly packed in 2-3
plane, so properties are invariant to
rotation about 1-axis (2 same as 3)
Specially orthotropic, transversely isotropic
(2 and 3 interchangeable)
G13 G12 , E2 E3 , 21 31
E2
G23
2(1 32 )
(2.23)
Now, only 5 coefficients are independent.
Isotropic
G13 G23 G12 G
E1 E2 E3 E
12 23 13
E
G
2(1 )
2 independent coefficients
Usually measure E, υ – calculate G
Isotropic – 3D case
1
E
1 E
2
3 E
4 0
5
6 0
0
E
1
E
E
E
E
1
E
0
0
0
0
0
0
0
0
1
G
0
0
0
0
1
G
0
0
0
0
0
0
1
2
0
3
0 4
5
0 6
1
G
Same form for any set of coordinate axes
3-D Isotropic – stresses in terms of strains
E
(1 ) x ( y z )
x
(1 )(1 2)
E
(1 ) y ( x z )
y
(1 )(1 2)
E
(1 ) z ( x y )
z
(1 )(1 2)
xy G xy
E
xy
2(1 )
xy G xy
E
xy
2(1 )
xy G xy
E
xy
2(1 )
3-D Case, Generally Orthotropic
z
y
x
Material is still orthotropic,
but stress-strain relations
are expressed in terms of
non-principal xyz axes
Generally Orthotropic
x S 11
y S 21
z S 31
yz S 41
zx S 51
xy S 61
S 12
S 22
S 32
S 42
S 52
S 62
S 13
S 23
S 33
S 43
S 53
S 63
S 14
S 24
S 34
S 44
S 54
S 64
S 15
S 25
S 35
S 45
S 55
S 65
S 16 x
S 26 y
S 36 z
S 46 yz
S 56 zx
S 66 xy
Same form as anisotropic, with 36 coefficients, but 9
are independent as with specially orthotropic case
Elastic coefficients in the stress-strain relationship for different
materials and coordinate systems
Material and coordinate system
Number of nonzero
coefficients
Number of
independent coefficients
Anisotropic
36
21
Generally Orthotropic
(nonprincipal coordinates)
36
9
Specially Orthotropic (Principal coordinates)
12
9
Specially Orthotropic, transversely isotropic
12
5
Isotropic
12
2
Anisotropic
9
6
Generally Orthotropic
(nonprincipal coordinates)
9
4
Specially Orthotropic (Principal coordinates)
5
4
Balanced orthotropic, or square symmetric
(principal coordinates)
5
3
Isotropic
5
2
Three – dimensional case
Two – dimensional case (lamina)
2-D Cases
Use 3-D equations with,
3 13 23 0
Plane stress,
1 , 2 , 12 , 0
Or
x , y , xy , 0
Specially
Orthotropic
Lamina
1 S11
2 S 21
0
12
S12
S 22
0
0 1
0 2 (2.24)
S 66 12
Or
2
1
1 Q11 Q12
2 Q21 Q22
0
0
12
0 1
(2.26)
0 2
Q66 12
5 Coefficients - 4 independent
Specially Orthotropic Lamina in Plane Stress
1 S11
2 S 21
0
12
S12
S 22
0
0 1
0 2
S 66 12
5 nonzero coefficients
4 independent coefficients
(2.24)
Or in terms of ‘engineering constants’
1
S11
E1
1
S 22
E2
21
12
S12 S 21
E2
E1
1
S66
G12
(2.25)
Experimental Characterization of
Orthotropic Lamina
• Need to measure 4 independent elastic
constants
• Usually measure E1, E2, υ12, G12
(see ASTM test standards later in Chap. 10)
Stresses in terms of tensor strains,
1 Q11 Q12
2 Q21 Q22
0
0
12
0 1
0 2 (2.26)
2Q66 12 / 2
where Q S
1
Inverting [S]:
S 22
E1
Q11
2
1 12 21
S11 S 22 S12
S12
12 E2
Q12
2
1 12 21
S11 S 22 S12
S11
E2
Q22
2
1 12 21
S11 S 22 S12
1
Q66
G12
S66
Off – Axis Compliances:
S ij f ij all Sij and angle
Off – Axis Stiffnesses:
Q ij f ij ' all Qij and angle
Where fij and fij’ are found from transformations
of stress and strain components from 1,2 axes to
x, y axes
Sign convention for lamina orientation
y
y
1
2
2
x
x
Positive
Negative
1
Stress Transformation:
2 dAsin
Y
2
1
12 dAsin
X
12dAcos
1dAcos
F
x
xydA
0 and Fy 0
x dA
dA
F
x
x dA 1dAcos 2 dAsin
2
2
2 12 dAsin cos 0
x 1 cos 2 sin 212 sin cos
2
F
y
2
0
xy 1 cos sin 2 cos sin
12 (cos sin )
2
2
Equations used to generate Mohr’s circle.
(2.29)
Resulting stress Transformation:
2
2
c
s
2cs 1
X
1
2
1
2
c
2cs 2 T 2
Y s
cs cs c 2 s 2
XY
12
12
Where c cos , s sin
or
1
X
2 T Y
12
XY
(2.31)
(2.30)
Where
c2 s2
2cs
2
2
[T ] s
c
2cs
cs cs c 2 s 2
(2.32)
Strain Transformation:
x
1
2 T y
/ 2
/ 2
12
xy
(2.33)
Recall: Tensor shear strain
xy
1
xy
2
Where xy = engineering shear strain
or
x
1
1
y T 2
/ 2
/ 2
12
xy
Substituting (2.33) into (2.26), then substituting
the resulting equations into (2.30)
x
x
(2.34)
1
y [T ] [Q][T ] y
/ 2
xy
xy
Carrying out matrix multiplications and
converting back to engineering strains,
x Q11 Q12 Q16 x
y Q12 Q 22 Q 26 y
Q
Q
Q
26
66 xy
xy 16
(2.35)
Where
Q11 Q11c Q22 s 2(Q12 2Q66 )s c
4
4
2 2
Q11
(2.36)
Q 66
Alternatively
x S 11
y S 12
S 16
xy
Where
S
1
S 12
S 22
S 26
[Q]
S 16 x
S 26 y
S 66 xy
(2.37)
Generally
Orthotropic
Lamina (Off
Axis)
x S 11
y S 12
S 16
xy
S 12
S 22
S 26
S 16 x
S 26 y
S 66 xy
(2.37)
Or
x
2
1
y
x Q11 Q12
y Q12 Q 22
Q
xy 16 Q 26
Q16 x
Q 26 y
Q 66 xy
9 Coefficients - 6 independent
In expanded form:
S11 S11c 4 2S12 S66 s 2c 2 S 22 s 4
S12 S12 s 4 c 4 S11 S 22 S66 s 2c 2
S22 S11s 4 2S12 S66 s 2c 2 S 22c 4
(2.38)
S16 2S11 2S12 S66 sc3 2S 22 2S12 S66 s 3c
S26 2S11 2S12 S66 s 3c 2S 22 2S12 S66 sc 3
S66 2 2S11 2S22 4S12 S66 s 2c 2 S66 s 4 c 4
Off-axis lamina engineering constants
Young’s modulus, Ex
2
y
x
x
1
or
Ex
x
Ex
x
When x 0, y xy 0
x
1
Ex
(2.39)
S 11 x S 11
1
1 4 212
1 2 2 1 4
c
s
c s
E1
G12
E2
E1
(2.40)
Complete set of transformation equations for lamina
engineering constants
1
1
2
1 4
Ex c 4
12 s 2 c 2
s
E1
E2
G12
E1
1
1 4 1
212 2 2 1 4
Ey s
c
s c
E
G
E
E
1
1
2
12
1
1
1
1 212
1
Gxy
s4 c4 4
E1
2G12
G12
E1 E2
12 4
1
1
1 2 2
4
xy Ex
s c
s c
E
E
E
G
1
2
12
1
(2.40)
2 2
s c
1
Variations of off-axis engineering constants with lamina orientation for
unidirectional carbon/epoxy, boron/aluminum and glass/epoxy composites.
(From Sun, C.T. 1998. Mechanics of Aircraft Structures. John Wiley & Sons, New York.
With permission.)
Shear Coupling Ratios, or Mutual
Influence Coefficients
• Quantitative measures of interaction
between normal and shear response.
• Example: when x 0,
y xy 0,
Shear Coupling Ratio
xy S 16 x S 16
x, xy
x S 11 x S 11
Analogous to Poisson’s Ratio
(2.41)
Example of off-axis strain in terms of
off-axis engineering constants
yx
xy , x
1
x
x
y
xy
Ex
Ey
Gxy
(2.43)
Compliance matrix is still symmetric for
off-axis case, so that, for example
S12 S21
and
yx
Ey
xy
Ex
Balanced Orthotropic Lamina
(Ex: Woven cloth, cross-ply)
E1 E2
2
Q11 Q22
S11 S22
1
Only 3 independent
coefficients
Lamina Stiffness Transformations
Q11 c 4
4
Q 22 s
Q c 2 s 2
12
2 2
Q 66 c s
Q c 3 s
16 3
Q 26 cs
s4
2c 2 s 2
4c 2 s 2
4
2 2
2 2
Q
c
2c s
4c s
11
Q
2 2
4
4
2 2
c s
c s
4c s
22
2 2
2 2
2
2 2
c s
2c s
(c s ) Q12
3
3
3
3
3
cs cs c s 2(cs c s ) Q66
3
3
3
3
3
c s c s cs 2(c s cs )
Use of Invariants
The lamina stiffness transformations can be
written as:
Q11 U1 U 2 cos 2 U 3 cos 4
Q12 U 4 U 3 cos 4
Q 22 U1 U 2 cos 2 U 3 cos 4
U2
sin 2 U 3 sin 4
Q16
2
U2
sin 2 U 3 sin 4
Q 26
2
(2.44)
Where the invariants are
1
U1 (3Q11 3Q22 2Q12 4Q66 )
8
1
U 2 (Q11 Q22 )
8
1
U 3 (Q11 Q22 2Q12 4Q66 )
8
1
U 4 (Q11 Q22 6Q12 4Q66 )
8
1
U 5 (Q11 Q22 4Q66 2Q12 )
8
(2.45)
Alternatively, the off-axis compliances can be
expressed as
S11 V1 V 2 cos 2 V 3 cos 4
S12 V 4 V 3 cos 4
S 22 V1 V 2 cos 2 V 3 cos 4
S16 V 2 sin 2 2V 3 sin 4
S 26 V 2 sin 2 2V 3 sin 4
S 66 2(V1 V 4 ) 4V 3 cos 4
(2.46)
where the invariants are
1
V1 (3S11 3S 22 2S12 S 66 )
8
1
V 2 (S11 S 22 )
2
1
V 3 (S11 S 22 2S12 S 66 )
8
1
V 4 (S11 S 22 6S12 S 66 )
8
(2.47)
Example: decomposition of Q11 using invariants
Q11
U
Q11
0
2
1
0
U3 cos4
U 2 cos2
U1
U2
0
2
U3
0
2
2
References: Mechanics of Composite Materials, Jones
Introduction to Composite Materials, Tsai
& Hahn
Invariants in transformation of stresses:
Mohr’s circle
Ex :
x
I R cos2
R
x
2 p
Where
I
I
x y
2
p
(2.48)
Invariant
x y
2
xy Invariant
R
2
2
Similar graphical interpretation of stiffness
transformations
Ex: Q11 U1 U2 cos2 U3 cos4
Isotropic
Part
Orthotropic Part
(2.49)
U1 and U4: First
order invariants
U2 and U3:
Second order
invariants
Radii of circles indicates degree of orthotropy.
(i.e., if U2=U3=0, we have isotropic material)