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1Interlaminar Stresses
In classical lamination theory, each ply is assumed to be in plane stress (x, y , xy), and interlaminar stresses (z, xz, yz) are neglected.
Even under in plane loading however, interlaminar stresses may exist near free edges.
Interlaminar stresses may lead to delamination. This is particularly true for tensile stresses z and shear stresses xz and yz.
Pipes and Pagano model for analysis of interlaminar stresses in a laminate under uniaxial extension.
From Pipes and Pagano, 1970.
2Equilibrium Considerations in Free edge Effect
Interior element of lamina
Element near free edge
at free edge
in equilibrium = 0zM
xxy
xxy
0zM0=xy
must have shear stress xz on top and bottom surfaces of element to have = 0zM
xN
x z
y
3 D Stress Equilibrium Equations for a Differential Element
,0 =xF or 0=
++
zyxxzxyx (7.100)
,0 =yF or 0=
++
zyxyzyyx (7.101)
,0 =zF or 0=
++
zyx
zzyzx (7.102)
3From Equation (7.100), if we assume that stresses do not vary along the loading direction (x axis),
we have and the interlaminar stress,0=
xx
( )
=z
t
xyxz dzy
z2
(7.103)Free edge:
as ,by 0xyand
yxy
increases.
Therefore, xz increases.(See Figure)
Schematic representation of in plane shear stress and interlaminar shear stress distributions at ply interface
Rule of Thumb: Boundary layer thickness roughly equal to laminate thickness
4Similarly, the other interlaminar stresses can be found from
( )
=z
t
yyz dzy
z2
(7.104)
( )
=z
t
yzz dzy
z2
(7.105)and
Pipes and Pagano Solution:
3 D stress equilibrium equations:
0=+
+
zyxxzxyx
0=+
+
zyxyzyyx
0=+
+
zyxzzyzx
5 Strain displacement relations:
;xu
x = ;
yv
y =
zw
z =
;yw
zv
yz +
= ;zu
xw
zx +
=xv
yu
xy +
= 3-D stress strain relations for kth lamina
{ } [ ] { }kkk C =where is the full 3-D stiffness matrix for the kth lamina
kC
Combining those equations, get a set of coupled, second order partial differential equations in the displacements u, v, w.
Solved these equations subject to boundary conditions for a 4 layer [45/-45]s laminate of graphite/epoxy.
xNxN
Used finite difference solution (See Figure)
6Distributions of All Stresses From Pipes and Pagano Analysis. From Pipes and Pagano, 1970.
Effect of stacking sequence on interlaminar shear stress. From Pipes and Pagano, 1974. Reprinted by permission of
the American Society of Mechanical Engineering.
Conclusion: higher when layers of same orientation stacked togetherxy
7Effect of ply orientation on interlaminar shear stress in angle ply laminates (After Pipes and Pagano)
Distribution of Interlaminar normal stress in boundary layer region vs. z. (After Pagano and Pipes)
Conclusion: stacking sequence has significant effect on interlaminar stresses
83 D finite Element Modelquarter domain finite element model of laminate used by Hwang and Gibson to analyze Pipes Pagano problem.
From Hwang and Gibson, 1992.
Comparison of stress distributions near the free edge. From Hwang and Gibson, 1992.
9Laminate Strength Analysis
Failure due to in plane stresses First ply failure predicted based on CLT and
multiaxial strength criteria for laminae Subsequent ply failure and final failure predicted
in sequential process involving degradation of ply properties after each ply failure
Failure due to interlaminar stresses Delamination initiation predicted by mechanics of
materials models Delamination growth and failure predicted by
fracture mechanics analysis (ME 7720)
Failure due to in plane stresses
Example: Symmetric cross ply laminate like [0/90/90/0]
xNxN
10
xN
x)(+
Te)(+
Le
Ultimate laminate failure (0o plies fail)
First ply failure (90o plies fail)
Load strain curve for uniaxially loaded laminate showing multiple ply failures leading up to ultimate
laminate failure.
11
After first ply failure and subsequent ply failures, the stiffnesses are degraded, and the force deformation equations are given as
=
)(
)(
)()(
)()(
)(
)(
n
n
nn
nn
n
n
DBBA
MN
(7.108)
Where [A(n)], [B(n)], [D(n)] are modified stiffness matrices and the total forces and moments are
=
=
k
nn
n
total MN
MN
1)(
)(
(7.106)
and the corresponding strains and curvatures are
=
=
k
nn
n
total1
)(
)(
(7.107)
First ply failure analysis:
=
)1(
)1(
)1()1(
)1()1(
)1(
)1(
DBBA
MN
Where superscript (1) refers to the first section of the stress strain curve
12
Aij(1), Bij(1), Dij(1) are laminate stiffnesses before first ply failure
Stresses in laminae:
{ } [ ] { } { }( ) zQ kk +== lamina stiffnesses before
first ply failure[ ])1(Q
Procedures for Modifying, or Degrading Stiffness Matrices
a) Set all ply stiffnesses equal to zero for the failed plies, then recalculate laminate stiffness matrix.
b) Base the ply degradation on the failure mode. For example, longitudinal shear failure.
1
2
13
G12 and E2 would be affected more by failure than E1. Thus, we could set G12 = E2 = 0, but leave E1 unchanged.
Experimental data usually does not show as sharp a knee as predicted curves because actual failure occurs over a finite strain range, not instantaneous ply failure at a certain strain.
Also, different types of behavior would be predicted after ply failure, depending on what is controlled during test.
Strain
Stress
Gradual failure
Load control
Displacement control
14
Comparison of predicted and measured stress
strain response of [0/ 45/90]s glass/epoxy laminate. From Halpin, 1984.
Note: knee in curve for 45o ply failure more distinct than knee for 90o ply failure because of greater no. of 45o plies
15
7.31
16
7.31
7.31
17
18
19
7.327.31
Angle ply laminates [ ] No knee in curve all plies fail simultaneously if tensile and compressive strengths are the same
Failure ofplies
Stress
Strain
20
Failure of Angle Ply Laminates
From Jones, Mechanics of Composites Materials
Comparison of predicted and measured uniaxial strength and stiffness of glass/epoxy angle ply laminates. From Tsai, 1965.
21
22
23
Prediction of Delamination Initiation or Onset
1. Mechanics of Materials approach (ME5720)2. Fracture Mechanics approach (ME7720)
Fracture mechanics approach (ME7720)
Prediction of Delamination Growth and Failure
Graphical interpretation of average interlaminar normal stress near free edge
according to Kim Soni Criterion
24
Kim Soni CriterionDelamination begins once( )()( ++ == TZz SS for transversely
isotropic material )(7.109)
where
( ) ==
b
bbz
oz
o
dyyb
0,1 average stress near free edge
(See Figure)and )(+
ZS = interlaminar tensile strengthob = averaging dimension
OK when is dominant stress, not in general case with and
zxz yz
Quadratic Delamination Criterion (Brewer and Lagace)
12
)(
2
)(
22
=
+
+
+
+Z
cz
Z
tz
YZ
yz
XZ
xz
SSSS (7.110)
= average interlaminar shear stresses = average interlaminar tensile and
compressive normal stresses = interlaminar shear strengths = interlaminar tensile and compressive
strengths
whereyzxz ,cz
tz ,
YZXZ SS ,)()( , + ZZ SS
25
Average Stresses for Quadratic Delamination Criterion (QDC)
davg
ijavg
ij =0
1 (7.111)
= averaging dimensionSimplified QDC
12
)(
2
=
+
+Z
tz
XZ
xz
SS
(7.112)
avg and SXZ used as curve fitting parameters sz(+)assumed to be = ST(+) (Transversely isotropic)
Tensile test Coupon Configuration
26
Predicted and measured delamination initiation
stresses for [ 15n]s laminates. From Brewer and Lagace, 1988.
OBrien analysis of stiffness reduction due to delamination in symmetric laminates
Youngs Modulus of symmetric laminate: (Fig. 7.36 (a))
11'1
tAEx = (7.113)
For totally delaminated laminate (Fig. 7.36(b))
t
tEE
m
iixi
td
== 1 (7.114)
For partially delaminated laminate (Fig. 7.36(b))
( ) xxtd EbaEEE += (7.115)
27
Rule of Mixtures Analysis of Stiffness LossEx
(Eq. 7.113)Etd
(Eq. 7.114) E
(Eq. 7.115)
Stiffness as a function of delamination size.From OBrien, 1982
28
Interlaminar stresses occur at a variety of discontinuities in composite structures. From Newaz, 1991.
Reduction of in plane compressive strength of laminate after transverse impact
29
Compression after impact (CAI) fixture. (From Nettles and Hodge, 1991. Reprinted by permission of the Society for the
Advancement of Material and Process Engineering.)
30
31
Methods for Improving Delamination Resistance
Toughened matrix materials Laminate design
Stacking sequence Ply thickness
Stitching through the thickness
3 D braiding (no distinct plies)
stitches
Methods for Improving Delamination Resistance
Z pinning
Edge cap reinforcement
Tough adhesive interleaf
Pins
Interleaf adhesive layer
cap
32
Use of composites in Boeing 777 airliner(Courtesy of Boeing Company)