8
Section
Composite Laminates
Typically, the basic building block block of a composite material structural component is a
_______________________________________________________________. These individual
_____________________________________________ are stacked and processed together (at
various orientations) to form a _________________________________________ of the desired
properties. The laminate orientation code is used by the composites community to describe the layup patterns of composite laminates.
Laminate Orientation Code1
1. Plies are listed in sequence starting with the
________________________________________, with
__________________________________________ used to indicate the beginning and
end of the code
2. For a symmetric laminate, only the plies on one side of the midplane are shown, and a
subscript ________________________________________ follows the closing bracket.
Subscript T is used to indicate that the ______________________________________ is
shown (although the T is often omitted)
3. Each ply is denoted by a number representing its
__________________________________________________________ as measured
from the geometric x-axis of the laminate to the lamina principal material coordinate direction
(1-axis).
4. When two or more plies of identical properties and orientation are adjacent to each other, a
single number representing _______________________________________________,
with a numerical subscript indicating the number of
_____________________________________________ is used.
5. If the angles of otherwise identical plies are different, or if the angles are the same but the
material properties are different, the plies are separated in the code by a
________________________________.
1
Adopted from Adams, Carlsson, and Pipes, Experimental Characterization of Advanced Composite Materials (2003)
6. When a symmetric laminate contains an odd number of plies of the same material, e.g. -30,
90, 45, 90, and -30◦, the center ply is designated with an __________________________, i.e.,
________________________ .
7. When adjacent plies are at angles of the same magnitude but of opposite sign, the appropriate
use of plus and minus signs (or minus plus signs) is employed, e.g. +20,+20, -30, and +30◦ is
designated _______________________________________________ .
8. Repeating sequences of plies are called sets and are enclosed in
_______________________________________. A set is coded by the same rules that
apply to a single ply. For example, a six-ply 45, 0, 90, 45, 0, and 90◦ laminate would be
designated as [(45/0/90)2]T or [(45/0/90)]2T
9. If a laminate contains plies of more than one type of
_____________________________________________________, a distinguishing
subscript (or superscript) is used with each ply angle to define the characteristics of that ply.
For example, [0g/90k/45c]S for a glass, Kevlar®, and carbon/fiber laminate.
Example
Lay-up the following composite with your partner, [90/ ± (0/45)]3S
68
Lamina Stress Strain Relations
In laminate plate theory, each lamina is assumed to behave as a homogeneous orthotropic material.
The principal material directions (1, 2, & 3) correspond to the ___________________________ (1),
in plane transverse direction (2), and _____________________________ direction (3) as shown
below.
The stress strain relationship is
where γij = 2εij (engineering shear strain is equal to 2 times the tensorial strain). This same relationship
is represented in the ASM handbook as2:
For plane stress (thin samples) we can assume that σ13= σ23= σ33 which is the same as saying τ13=
τ23=σ3 Then the linear equations above simplify to
2
Note, I’ll switch between these two notations (σ1= σ11, σ12=τ12, γ12=2ε12) regularly so don’t get confused.
69
Note that we keep the S66 notation not S33 to match the non-reduced compliance matrix. The inverse
of the compliance matrix (S) above is defined Q, _______________________________________.
⎡σ 1 ⎤ ⎡Q11 Q12
⎢σ ⎥ = ⎢Q
⎢ 2 ⎥ ⎢ 12 Q22
⎢⎣τ 12 ⎥⎦ ⎢⎣ 0
0
0 ⎤ ⎡ ε1 ⎤
0 ⎥⎥ ⎢⎢ ε 2 ⎥⎥
Q66 ⎥⎦ ⎢⎣γ 12 ⎥⎦
Transformation of Stresses and Strains
Given the stress strain relation in the material coordinate system
(1,2,3), what is the stress strain relationship in the global coordinate
system (x, y, z)
70
Note that the engineering strain must be converted to tensorial strain in order to use the
________________________________________ [θ].
Alternately, the strain can be transformed directly by using a different transformation matrix:
It is now possible to determine the stress strain relationship in terms of the x-cordinate:
71
Note that there are additional terms in the Q bar matrix (Q16, and Q26) which relate
______________________________________________________________________ and vice
versa. This effect of a shear strain resulting from an extensional stress is shown below.
72
Example
Calculate the Q bar matrix for each ply in a [0/90]S laminate if each ply in the laminate is a
glass/epoxy lamina with a 70% fiber volume fraction. Assume the modulus of glass and matrix to be
Ef=85 GPa, Em=3.4 GPa, νLTf=0.36, νm=0.30, Gf=35.42 GPa, Gm=1.308 GPa
73
Variation of Ex with angle of rotation and
G12 for typical graphite-epoxy materials
74
where
75
Laminate elastic constants for
high-modulus graphite-epoxy
system, [±θ]s
76
Example
Assuming the composite in the previous example, with each ply thickness equal to 0.3 mm and a
loading of:
⎧500 ⎫
{N } = ⎪⎨700⎪⎬ N/m
⎪300 ⎪
⎩
⎭
⎧2 ⎫
{M } = ⎪⎨3 ⎪⎬ N − m/m
⎪4 ⎪
⎩ ⎭
o Calculate the laminate stiffness [A,B,D] Matrix.
o Find the midplane strains [εx0, εy0, γxy0] and curvatures [κx,
κy, κxy].
o Find the maximum strains and stresses in the material
direction (1, 2, 3) for each ply.
77
Special Laminates:
Recall that the strains and curvatures are related to applied loads through the ABD matrix:
⎧ N x ⎫ ⎡ A11
⎪N ⎪ ⎢
⎪ y ⎪ ⎢ A12
⎪⎪ N xy ⎪⎪ ⎢ A16
⎬=⎢
⎨
⎪ M x ⎪ ⎢ B11
⎪ M y ⎪ ⎢ B12
⎪ ⎢
⎪
⎪⎩M xy ⎪⎭ ⎢⎣ B16
A12
A16
B11
B12
A22
A26
B12
B22
B26
A26
A66
B16
B26
B66
B12
B16
D11
D12
D16
B22
B26
D12
D22
D26
B16 ⎤ ⎧ ε 0 x ⎫
⎪
⎪
B26 ⎥⎥ ⎪ ε 0 y ⎪
B66 ⎥ ⎪⎪γ 0 xy ⎪⎪
⎥⎨
⎬
D16 ⎥ ⎪ κ x ⎪
D26 ⎥ ⎪ κ y ⎪
⎥⎪
⎪
D66 ⎥⎦ ⎪⎩κ xy ⎪⎭
In the general case this can result in __________________________________________ between
bending, extension, torsion, and shear as illustrated below:
(from Barbero Intro to Composite
Design)
In the general case, application of a load in one direction can make all 6 strains and curvatures terms
non-zero. Several special laminates can be designed based on ____________________________ to
eliminate some of these couplings by forcing the respective ABD matrix elements to be zero.
78
_________________________________________ Laminates
•
Description: For every layer at +θ, there is another at -θ, and for each 0° layer there is a 90°
layer
•
Example: __________________________________________
•
Characteristics:
– Qbar16(θ )= -Qbar16(-θ)
–
•
Important characteristics:
–
•
Qbar26(θ )= -Qbar26(-θ)
A16=A26=0
Mechanical characteristic:
–
no ____________________________________ coupling.
Symmetric Laminates
•
Layers of the same material, thickness, and orientation are symmetrical w.r.t. to
_____________________________________.
•
Example: [+θ/- θ /- θ /+ θ]
–
•
Important characteristics:
–
•
both balanced and symmetric
Bij=0
Mechanical characteristics
–
no ___________________________________________ coupling
Antisymmetric Laminates
•
Pairs of layers of opposite orientation but the same material and thickness symmetrical w.r.t
.middle surface of laminate.
79
•
Example: [+θ /- θ /+ θ /- θ]
•
Important characteristics:
•
–
______________________
–
______________________
Not easy to analyze since B16 and B26 coupling terms are non-zero
Quasi-isotropic Laminates
•
A laminate that behaves like an isotropic plate
•
in-plane is similar to isotropic plates
•
bending stiffness is different from isotropic plates
•
Given by
θk =
–
•
kπ
+ θ0
N
where k is the layer number; N=total layers (>=3), q0 is arbitrary initial angle
Equal number of plies at
–
0, 45, -45, 90 or
–
0, 60, -60
•
A matrix _________________________________
•
B & D matrices are not
•
Mechanical uses…..
80
Cross-Ply and Angle-Ply Laminates
•
•
cross-ply: 0° and 90° layers only
–
16 and 26 entries for A and D =0
–
Easy for analysis if symmetric also, i.e., B=0
angle-ply: ± θ ° layers only
–
symmetric or antisymmetric balanced laminate
–
symmetric: __________________________
–
antisymmetric: _________________________
81
Lamina Failure Criterion:
Many different failure theories have been suggested based on measured strengths in the __________
___________________________________________. Three of the most popular strength criteria
are:________________________ ________________________ ______________________ .
These failure theories recognize that most composite materials have different strengths in
_________________ and __________________.
82
Lamina failure theories can be classified as:
(1) ______________________________________________________
•
No interaction among different stress components on failure is considered
•
Examples: Max stress and Max strain
(2) ______________________________________________________
•
All stress components are included in one expression (failure criterion)
•
Examples: Tsai-Hill and Tsai-Wu theories
Maximum Stress Failure Criterion
The max stress criterion assumes that failure occurs when any one of the in-plane stresses attains its
limiting value independent of the other components of stress.
83
Maximum Strain Failure Criterion
The max strain criterion assumes that failure occurs when any one of the in-plane reaches its ultimate
value in uniaxial tension, compression, or pure shear.
Tsai-Wu Failure Criterion
Analogous to von-Mises stress criterion for isotropic materials, Tsai and Wu proposed a theory by
assuming the existence of a ___________________________ in the stress space. The criterion takes
the following form for plane stress:
84
Failure under combined stress is assumed to occur when _________________________ of the
above equation is equal to or greater than one. All of the parameters (except F12) can be expressed in
terms of basic strengths.
F12 can be estimated from the following relationship.
The Tsai-Wu criterion has been found to
be quite accurate in predicting strength
and is widely used.
85