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)