Title: Finite element analysis of debonded sandwich beams under compression
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Title: Finite element analysis of debonded sandwich beams under compression
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Language: English
Creator: Narayanan, Manickam, 1973-
Publisher: State University System of Florida
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Publication Date: 1999
Copyright Date: 1999
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Subject: Aerospace Engineering, Mechanics and Engineering Science thesis, M.S   ( lcsh )
Dissertations, Academic -- Aerospace Engineering, Mechanics and Engineering Science -- UF   ( lcsh )
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Summary: ABSTRACT: This thesis highlights research aimed at predicting the compressive failure of sandwich beams with interfacial delaminations. A nonlinear finite element analysis was performed to simulate axial compression of the debonded sandwich beams. The load-deflection diagrams were generated for a variety of specimens used in a previous experimental study. The energy release rate at the crack tip was computed using the J-integral, and plotted as a function of the displacement and load. A detailed stress analysis was performed and the critical stresses in the face sheet and the core were computed. Further, the core was modeled as a elastic, perfectly plastic material and a nonlinear post-buckling analysis was performed. For this model, load-deflection curves, energy release rate plots and von Mises stress plots were generated. By comparing the experimental failure load and the finite element analysis results, the conclusions were reached and the scope of future research in this project was defined.
Summary: KEYWORDS: finite element analysis, sandwich composite, graphite/epoxy, Nomex core, axial compression, post-buckling, delamination, crack propagation, J-Integral, von Mises stress, orthotropic material, elastic perfectly plastic material
Thesis: Thesis (M.S.)--University of Florida, 1999.
Bibliography: Includes bibliographical references (p. 78-79).
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Statement of Responsibility: by Manickam Narayanan.
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General Note: Document formatted into pages; contains v, 80 p.; also contains graphics.
General Note: Vita.
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FINITE ELEMENT ANALYSIS OF DEBONDED SANDWICH BEAMS UNDER
COMPRESSION











By

MANICKAM NARAYANAN


A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF
FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE


UNIVERSITY OF FLORIDA


1999















ACKNOWLEDGMENTS


I would like to acknowledge the support and guidance of Dr. Bhavani Sankar, who has

been my advisor, sponsor and friend during my stay at the University of Florida. He is truly

an intelligent and kind-hearted man from whom many have and will continue to learn.

I would also like to thank Mr. Juan Cruz, from NASA Langley Research Center, who has

supported this grant and has coordinated the cooperation between the University of Florida

and NASA Langley Research Center, Hampton VA.





















TABLE OF CONTENTS


ACKNOW LEDGM ENTS ..............................................


A B ST R A C T . . . . . . . . . . . . . . . . . . . . . . . . . . . .


CHAPTERS


1 INTRODUCTION ...............


Literature Review ..............
M materials Systems ..............
Manufacturing Process ...........
Testing Equipment ..............
In-plane Compression Test Results .
Scope of Present Study ..........


2 FINITE ELEMENT ANALYSIS ....


Geometric Modeling ... . . .....
Finite Element Modeling .........
Loads and Boundary Conditions . .
M material M odeling . . . . . ...
Analytical Sandwich Beam Model .
Finite Element Simulation . . . ..
Riks Algorithm ................
The J-Integral .................


3 RESULTS AND DISCUSSIONS . .


Linear Buckling Analysis . . . ...
Post-buckling Analysis ...........
Energy Release Rate ............
Stress Analysis .................
Effects of Core Plasticity .........


. . . . . . . . . . . . . . . . . . 1 2
. . . . . . . . . . . . . . . . . . 1 4
. . . . . . . . . . . . . . . . . . 1 5
. . . . . . . . . . . . . . . . . . 1 7
. . . . . . . . . . . . . . . . . . 1 9
. . . . . . . . . . . . . . . . . . 2 0
. . . . . . . . . . . . . . . . . . 2 1


. . . . . . . . . . . . . . . . . . 2 3


. . . . . . . . . . . . . . . . . 2 4
. . . . . . . . . . . . . . . . . . 2 8
. . . . . . . . . . . . . . . . . . 3 2
. . . . . . . . . . . . . . . . . . 3 3
. . . . . . . . . . . . . . . . . . 3 7


page


...............................
...............................
...............................
...............................
...............................
...............................











4 CONCLUSIONS AND FUTURE WORK ................................ 49

APPENDICES

A BRIEF OVERVIEW OF ABAQUS . . . ....................... . 55

B RESULTS FROM POST-BUCKLING ANALYSES ....................... 59

LIST OF REFEREN CES .............................................. 78

BIOGRAPHICAL SKETCH ..................................... 80






































iv











Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial
Fulfillment of the Requirements for the Degree of Master of Science

FINITE ELEMENT ANALYSIS OF DEBONDED SANDWICH BEAMS UNDER
COMPRESSION


By

Manickam Narayanan

December 1999



Chairman: Dr. Bhavani Sankar
Major Department: Aerospace Engineering, Mechanics and Engineering Science


This thesis highlights research aimed at predicting the compressive failure of sandwich

beams with interfacial delaminations. A nonlinear finite element analysis was performed to

simulate axial compression of the debonded sandwich beams. The load-deflection diagrams

were generated for a variety of specimens used in a previous experimental study. The energy

release rate at the crack tip was computed using the J-integral, and plotted as a function of

the displacement and load. A detailed stress analysis was performed and the critical stresses

in the face sheet and the core were computed. Further, the core was modeled as a elastic,

perfectly plastic material and a nonlinear post-buckling analysis was performed. For this

model, load-deflection curves, energy release rate plots and von Mises stress plots were

generated. By comparing the experimental failure load and the finite element analysis results,

the conclusions were reached and the scope of future research in this project was defined.















CHAPTER 1
INTRODUCTION


There is a renewed interest in using sandwich construction in aerospace structures mainly

driven by the possibility of reducing weight and cost. Fiber composites such as

graphite/epoxy are favored as the face-sheet material because of their high stiffness and ability

to be co-cured with many core materials. In the field of aerospace structural engineering,

sandwich constructions find application in wing skins and fuselage among other structures.

Debonding of the face-sheet from the core is a serious problem in sandwich constructions.

This may occur during the fabrication process due to inadvertent introduction of foreign

matter at the interface or due to severe transverse loads as in foreign object impact. The

debonded sandwich panels are susceptible to buckling under in-plane compressive loads,

which may lead to the propagation of the delamination, and/or core and face-sheet failure.

Hence there is a need for a systematic study to understand how the core and face-sheet

properties affect the compression behavior of a debonded sandwich composite.

Literature Review

There are many works concerning buckling of delaminated composite beams and plates.

These models were later extended to sandwich beams. Simitses et al. (1985) and Yin et al.

(1986) developed analytical models to study the effects of delamination on the ultimate load

capacity of beam-plates. The latter paper included the post-buckling behavior as well as










2

energy release rate calculations to predict delamination growth. Chen (1993) included the

transverse shear effects on buckling, post-buckling and delamination growth in one-

dimensional plates. A nonlinear solution method was developed by Kassapoglou (1988) for

buckling and post-buckling of elliptical delaminations under compressive loads. This method

employs a series solution approach in conjunction with the perturbation technique to solve

the laminated plate equations for large deflections. Experiments were performed on sandwich

panels containing delaminated face-sheets (note that the delaminations were in between layers

of the face-sheet; the face-sheet/core interface did not contain delaminations). The nonlinear

models were able to predict the onset of delamination and failure loads in the experiments.

Minguet et al.(1987), studied the compressive failure of sandwich panels with a variety

of core materials including honeycomb core. They observed three types of failure modes -

core failure, debond and face-sheet fracture. Based on the test results they developed a

nonlinear model to predict these failures using appropriate failure criteria for each failure

mode. Sleight and Wang (1995), compared various approximate numerical techniques for

predicting the buckling loads of debonded sandwich panels, and compared them with plane

finite element analysis. They concluded that 2-D plane strain FE analysis is necessary in order

to predict the buckling loads accurately. Hwu and Hu (1992), extended the work of Yin et

al. (1986), for the case of debonded sandwich beams. They developed formulas for buckling

loads in terms of sandwich beam properties and debond length. Kim and Dharan (1992), used

a beam on elastic foundation model and computed the energy release rate in debonded

sandwich panels. Based on fracture mechanics they predicted critical debond lengths for crack











3

propagation. They used their model to predict failure in plastic-foam core sandwich panels.

An extensive experimental study was conducted by Kardomateas (1990), to understand the

buckling and post-buckling behavior of delaminated Kevlar/epoxy laminates. The

experimental program documented the load-deflection diagrams, deformation shape in post-

buckling and growth of delamination.

From this literature survey it is clear that a systematic study of compression behavior of

sandwich panels with debonded face-sheets, especially failure in the post-buckling regime, is

overdue. Any modeling should be preceded by a testing program to understand the effects of

various parameters such as face-sheet stiffness, core stiffness and core thickness, and debond

length on the buckling and post-buckling behavior. In a previous experimental study (Avery,

1998 and Avery and Sankar, 1999) compression tests were performed to understand the

effects of core and face-sheet properties, and delamination length on the compression strength

of debonded sandwich composites. In the present study, an attempt is being made to use finite

element simulation of the compression tests to explain the failures observed in the

experiments. For the purpose of completeness a brief description of the experimental program

is presented in the following sections.

Materials System

The sandwich composites tested during the experimental study have two components: two

sets of face-sheets to carry the majority of the load and the core material that is sandwiched

between them. The face-sheet was made from Fiberite carbon fiber/epoxy plain-woven












prepregs (Product no: HMF 5-322/97714AC). Some manufacturer listed properties are

shown in Table 1.1.



Table 1.1 Manufacturer's data for the graphite/epoxy face-sheet material


Compressive Strength 77 ksi
Tensile Strength 97 ksi
Tensile Modulus 7.7 msi
Tensile strength (max. strain) 12, 658 gi&
Flatwise Tensile Strength 693 psi
Pre-cured Resin Content 41.0 %


w


JL


Figure 1.1 Principal directions of the Nomex honeycomb











5

The core material used was the Nomex7 honeycomb manufactured by Euro-Composites.

This material is made from an aramid fabric bonded together to form small hexagonal cells.

This structure is then coated with a phenolic resin. The honeycomb structure has orthotropic

properties and its principal directions are denoted by L, W, and t. The principal directions are

shown in Fig. 1.1. The W-direction is in the general plane of the material and is perpendicular

to the L-direction. The t-direction is the through-the-thickness direction (perpendicular to the

plane of this paper). The core properties can be found in Avery (1998).

Manufacturing Process

The manufacturing of the sandwich composite for these tests was to simulate the actual

construction of a sandwich composite panel for an aircraft wing. The curing procedure was

designed to allow large wing sections of an aircraft to be manufactured at a reasonable cost.

Since an autoclave this size would substantially increase the cost of production, it was decided

to use only vacuum bag lay-ups. Also, the panels were co-cured to reduce the cost and time

to manufacture the specimens. That is, the face-sheets and core material were bonded while

the skins were being cured. The majority of the panels manufactured for these experiments

use only the excess epoxy from the face-sheets to bond to the core material. Each set of

specimens was manufactured from a flat sandwich plate. Artificial delaminations were

introduced into the specimens by inserting non-porous Teflon' strips between the prepreg and

the honeycomb. After curing, the plate was cut into individual specimens with a water-

injecting, diamond coated masonry saw. Each specimen was then dried, labeled, and

measured to prepare it for testing.











6

Testing Equipment

The compression tests were performed on a 12,000 pound capacity Tinius Olsen testing

machine. The loading was controlled using a personal computer based controller. A Schaevitz

4-inch stroke, Model 2000-HR, Linear Variable Differential Transformer (LVDT), was used

to measure the displacement of the specimen. One of the LVDT brackets was mounted

directly above the load cell, eliminating errors from load cell deflections. The other LVDT

bracket was mounted above the loading fixture. A Sensotec 15,000 pound capacity load cell,

Model RM/7672-03, was used for the load measurements. The load and displacement data

were measured using data acquisition cards installed on the computer. The load, crosshead

deflection, relative time and displacement of the LVDT were recorded by the data acquisition

software.

In-plane Compression Test Results

The compression tests were conducted in a displacement controlled mode. Rectangular

sandwich specimens of length 4 inches and width 2 inches were clamped at the ends and

subjected to axial compression. The tests were stopped after substantial load reduction due

to catastrophic failure of the specimen. A sample compression test is illustrated by actual test

photographs in Fig. 1.2. Sixteen different types of specimens were used to understand the

effects of core thickness, core density, face sheet thickness and delamination length on the

compressive load carrying capacity of the sandwich beam column. Six repeat tests were

conducted for each specimen type. The experimental results are summarized in Table 1.2.











7

In Table 1.2, the face-sheet thickness, core thickness, core density and delamination length

for each set of specimens are provided. The maximum load at which the specimen failed is

listed in the 6th column. The mode shape of each specimen attained during loading is given in

the last column. The face-sheet thickness is expressed in terms of the number of plies in each

face-sheet. The thickness of each ply is 0.0087 in. The failure load is given as load per unit

width of the specimen (lb/in).

Scope of Present Study

In this chapter, the other research that has been done on sandwich composites was

discussed and the previous experimental study was described. In the following chapter, the

various steps involved in the finite element modeling, such as the modeling of the geometry,

defining the material properties, applying the loads and boundary conditions and the finite

element simulation are described. Also, the analytical model of Hwa and Hu (1992), the

nonlinear solution strategy used for the post-buckling FE analysis and the J-integral evaluation

are explained. In Chapter 3, the results from the linear buckling analysis is discussed. These

results are compared to the experimental results and the results from the analytical model

(Hwa and Hu, 1992). These results are found to be inadequate in explaining the failure

behavior observed in the tests. Then, the results of the post-buckling nonlinear analysis are

presented and the load-deflection relationship, the stresses in the face-sheets and the core, the

energy release rate calculations and the effect of the elasto-plastic modeling of the core are

discussed. These results appear to provide a direction for future work in the study of







8
debonded sandwich beams. The conclusions from the results and discussions are summarized
in Chapter 4 and the scope of future work is discussed.


Figure 1.2 Compression test for Specimen 1: local anti-symmetric buckling


I CIM,














Table 1.2. Dimensions and configuration of the sandwich beams and their failure loads in
the compression tests


Specimen Plies per Core Core Delamination Experimental Mode
Number face-sheet Thickness Density Length Failure load Shape
(in.) (lb./ ft3) (in.) (lb/in)
1 1 0.250 1.8 0.5 98 LA
2 1 0.375 3.0 1.0 162 LS
3 1 0.500 3.0 1.5 164 LS
4 1 0.375 6.0 2.0 194 LS
5 3 0.375 3.0 0.5 1210 LS
6 3 0.250 6.0 1.0 497 LS
7 3 0.375 1.8 1.5 361 LS
8 3 0.500 3.0 2.0 439 LS
9 5 0.375 3.0 0.5 2528 GS
10 5 0.500 1.8 1.0 1215 GA
11 5 0.375 6.0 1.5 1385 GA
12 5 0.250 3.0 2.0 893 GS
13 7 0.500 6.0 0.5 4528 FF
14 7 0.375 3.0 1.0 2319 GA
15 7 0.250 3.0 1.5 1688 GS
16 7 0.375 1.8 2.0 1583 GA


LA: Local Anti-symmetric, LS: Local Symmetric, GS: Global
symmetric, FF: Face-sheet material failure


Symmetric, GA: Global Anti-















CHAPTER 2
FINITE ELEMENT ANALYSIS


The finite element analysis was used to (a) estimate the linear buckling loads and mode

shapes of the delaminated sandwich beams and (b) simulate the actual compression tests by

performing a nonlinear post-buckling analysis of the sandwich beam under axial compression.

As will be seen later the results from the linear buckling analysis are needed for the post-

buckling analysis also. The post-buckling analysis simulated the compression tests (Avery,

1998) as closely as possible.

The FE analysis was performed using the FE package ABAQUSTM. First, a geometric

model of the specimen was created using the FE pre-processor MSC/PATRANC. This was

converted to a finite element model by meshing the surfaces of the geometric model to create

nodes and elements. In the next stage, the material properties of the structural components

of the specimen were defined. This was followed by specifying the loads and boundary

constraints that would simulate the conditions of an actual compression test. Finally, the

input file was generated with this information and the analysis was performed using

ABAQUS. A brief introduction to the ABAQUS finite element package is presented in

Appendix A. The various stages of the FE analysis are discussed in detail in this chapter.










11

Geometric Modeling

This stage of the FE analysis involved the creation of a geometric model that would

represent the shape and dimensions of the test specimen (Fig. 2.1). The sandwich composite

specimen was 4 inches long and 2 inches wide. The thickness of the specimen was dependent

on the number of face-sheet plies and the height of the core. Each face-sheet ply was 0.0087

inches thick. The core was of three different thicknesses: 0.25, 0.375 or 5.0 inches. The

delamination was of four different lengths: 0.5, 1.0, 1.5 or 2.0 inches. The complete

dimensions of each design set are listed in Table 1.2 in Chapterl. The geometry and the finite

element model were created using the pre-processing software, MSC/PATRANC. The

sandwich composite specimen was modeled as a 2-D or plane solid.


SCore
Facesheet


Figure 2.1 Sandwich composite specimen approximated as a 2-D or plane solid model



First, points were created using the spatial coordinates that defined the outer boundary

of the 2-D rectangular solid (in Fig. 2.1) and these points were then joined by curves. Next,

surfaces were created using these curves as the surface boundaries. Finally, three 2-D solids

were created using these surfaces two of these solids representing the top and bottom face-

sheets and the other solid forming the central NomexTM core. As mentioned earlier, the











12

honeycomb core was modeled as a continuum with equivalent mechanical properties. The

delamination is created in the FE modeling stage and will be discussed in the following

section.

Finite Element Modeling

The development of the FE model involved meshing of the surfaces of the geometric

model and resulted in the creation of approximately 800 isoparametric elements with 2500

nodes. Eight-node, biquadratic, plane strain elements were used to mesh the model. The

*ELEMENT command and the *NODE command in ABAQUS were used to create the

elements and nodes. The detailed set of commands and procedures used in the finite element

modeling are described in Appendix A.

After meshing the surfaces, duplicate nodes were removed using the "equivalence"

command in PATRAN. To create the delamination at the interface between the top face-sheet

and the core, the nodes on the crack were excluded during the equivalence process. This

omission of the nodes along the crack creates a free edge at the interface simulating the

behavior of an interfacial delamination. The number of nodes thus excluded depended on the

length of the delamination in the specimen. The mesh in the vicinity of the crack-tips were

modeled differently from the rest of the geometry to create a mesh fine enough to capture the

crack-tip singularities. Bimaterial fracture mechanics studies (Zak and Williams, 1963, Rice,

1988, Hutchinson, 1990) have indicated that the stresses in the vicinity of an interfacial crack-

tip at the interface between dissimilar materials are singular and vary as


U ~r (2.1)










13

where r is the distance from the crack-tip of points where the stresses are calculated and ais

the bimaterial parameter (Dundurs, 1969) that depends on the elastic constants of the face

sheet and core


Figure 2.2 Finite element mesh in the vicinity of the crack-tip



materials. FE modeling of such bimaterial interfacial crack problems has been studied by

several researchers, e.g., Charalambides,1990. These studies have suggested that a self-

focusing mesh as shown in Fig 2.2 is very efficient in capturing the aforementioned singular

behavior at the crack-tip.



I'
I
N: -

2 __________________
N-
/ A'
',
,~rX ~ >
xx

L










14

Loads and Boundary Conditions

The compression testing of the sandwich composite specimen involved the loading of the

specimen in the axial direction under displacement control (Avery, 1998). The fixture used

in the test is shown in Fig. 2.3. To simulate the compressive loading under displacement

control, a multi-point constraint was created at one end of the sandwich beam. This multi-

point constraint established a linear relationship between 'u' degree-of-freedom of one of the

nodes on the loading end of the beam (Fig. 2.3) and those of the remaining nodes at the same

end. A unit displacement in the axial direction was imposed on this node. This displacement

will be distributed among the remaining nodes appropriately so as to produce a constant

displacement at all the nodes on the loading face. This is accomplished by the multi-point

constraint imposed among the nodes.


Figure 2.3 Modeling of the boundary conditions of the compression test











15

Also, in the experiments the fixture acted like a clamp around the specimen for a length of

0.5" at both the ends. To simulate this in the finite element model, zero-displacement was

imposed in the appropriate directions on the nodes along this clamped length of the beam as

shown in Fig. 2.3.

Material Modeling

The face-sheet was modeled as a homogeneous linear elastic orthotropic material

throughout this study. It should be mentioned that in the tests graphite/epoxy plain-weave

composite laminates were used as face-sheets. This assumption is justified as the face-sheet

did not undergo any delamination or other significant failure. The dominant failure

mechanisms of interest are the core failure and the interfacial fracture. The properties used

for the face-sheet material are given in Table 2.1. The homogeneous properties were derived

from the data provided by the manufacturer (FiberiteTM) of the plain-weave composite. The

1-direction is parallel to the longitudinal beam axis and the 2-direction is the thickness

direction of the sandwich beam. A state of plane strain is considered in the 1-2 plane, and the

beam width in the 3-direction is assumed to be unity.

Although honeycomb core was used in the experimental study, it was decided to model

the core as a homogeneous continuum. This assumption is justifiable only if the characteristic

dimensions of the problem are much larger than the cell size. For example, in the current

problem the crack length is much larger than the cell size. Also, as one of the objectives was

to understand the effect of core properties on the buckling and post-buckling behavior, it was









16

decided that it was not necessary to model the microstructure of the core in detail at this

stage.

There were two core models used in the study. In the first part of the study wherein our

interest was in the energy release rate at the crack-tip, the core was modeled as a linear elastic

orthotropic material. In the second part of the study our interest was in understanding the

effects of the core's elasto-plastic nonlinear behavior on the post-buckling of the sandwich

beam. In that case the core was modeled as an isotropic elastic-perfectly plastic material. The

two sets of properties are listed in Tables 2.1 and 2.2, respectively. The orthotropic material

properties were estimated based on the test results of Avery (1998) and the manufacturer's

(Nomex) material data. The isotropic elasto-plastic material properties were chosen

arbitrarily, but representative of the core properties.



Table 2.1 Properties of face-sheet and core materials used in the FE analysis. The Young's
moduli (E) and shear moduli (G) are in psi. i 's are the Poisson's ratios. The core density fi
is in lb/ft3.



Material Ell E22 E33 i 12 i 23 i31 G12 G23 G31

Face-sheet 7.70E6 1.55E6 7.70E6 0.37 0.37 0.13 6.30E5 6.74E5 6.30E5

fi=1.8 336.00 181.09 15.1E3 0.01 0.01 0.01 3.9E3 3.9E3 3.9E3
C
o fi=3.0 560.00 301.82 27.1E3 0.01 0.01 0.01 7.0E3 7.0E3 7.0E3
r
e
fi=6.0 1121.0 603.64 51.0E3 0.01 0.01 0.01 13.9E3 13.9E3 13.9E3












Table 2.2 Properties of the core material as an isotropic elastic perfectly plastic material,
while the face-sheet properties remain the same. Ec is the Young's modulus, 6y is the
maximum yield stress beyond which the stress-strain curve is parallel to the x-axis, and i is
the Poisson's ratio.


Analytical Sandwich Beam Model

A buckling model for delaminated sandwich composites by Hwu and Hu (1992) was

found to predict satisfactory results for specimens failing in a global buckling mode. This

model uses a combination of laminate theory and shear deformation theory. The face-sheet

is assumed to be thin and carry no out-of-plane shear forces, while the core carries shear. The

core is also assumed to be infinitely stiff in the thickness direction, which is reasonably valid

for the Nomex honeycomb core material. Hwu and Hu (1992) presented the following

equation to predict a critical unit failure load:

[- -

P r (D + D D a + ka + a (2.2)
Lcr + 1tan 1(1- a) 2 tan 2a A3 tan 3a

where,

k 2 1 2 k 2 ( k)P (2.3)
S+ A 3 D (I P S (2.3)
2 3 2 3


Set Ec 6y i
Core EP1 1200 12 0.25
Core EP2 12000 120 0.25










Also,

A B, D,= D B 12,3 (2.4)
S(A), (B All



where A,,, B1, and D,, are the extensional stiffness, coupling stiffness and bending stiffness

of the laminated composite. The symbol a represents half the crack length. Each of the ith

terms represents a section of the composite beam, as shown in the figure below.

h"------ ------

a


3


1 2




Figure 2.5 Delaminated sandwich model in Hwa and Hu (1992)



The equations above were solved iteratively until the critical buckling loads for various

modes were found. A FORTRAN program was written that stepped through loads until the

equations where satisfied. This model was modified from the Hwu and Hu model by adding

an in-plane core stiffness. This additional stiffness slightly increased the critical loads but was

almost negligible. The results from the Hwa and Hu model were compared with the linear









19

buckling results by FEA and the experimental results. These results will be discussed in

Chapter 3.

Finite Element Simulation

As mentioned earlier the FE analysis was performed using the commercial software

ABAQUS. The analyses performed can be broadly classified into two parts: Linear buckling

analysis and nonlinear post-buckling analysis. The main purpose of the linear buckling

analyses was to understand the effects of core thickness, core density, face-sheet thickness

and delamination length on the buckling loads and corresponding mode shapes. Further, as

will be explained later, the linear buckling mode shapes are required in specifying the

imperfections needed to trigger post-buckling during the nonlinear analysis. It should be

mentioned that no gap elements (contact elements) were used in between the nodes on the

delaminated surfaces. Thus interpenetration of the crack surfaces was not prevented in the FE

analysis. However it will be shown that there was no interpenetration in the first buckling

mode shapes, and hence the use of gap elements was not pursued. The results of the linear

buckling analysis are presented in Chapter 3.

The nonlinear post-buckling analysis was performed to simulate the compression tests on

the sandwich specimens. The nonlinear analysis consists of the following steps:

1. An eigen-value buckling analysis was performed on the "perfect" model to obtain the

possible buckling modes.

2. In the second step of the analysis, an imperfection in the geometry was introduced by

adding a fraction of deflections from the eigen modes (buckling mode shapes) to the

"perfect" geometry to create a perturbed mesh. The choice of the scale factors of the









20

various modes was dependent on the face-sheet thickness. Usually, 10% of the face-sheet

thickness was assumed to be the scale factor for the major buckling mode. In the present

study only the first mode shape was included in the imperfection.

3. Finally, a geometrically nonlinear load-displacement analysis of the structure was

performed using the Riks method (Riks, 1979, Crisfield, 1981). A brief explanation of the

Riks method is presented in the next section.

During the post-buckling analyses the following quantities were computed at each load step:

(a) total load and displacement (end shortening), (b) stresses 6,x, 6y and ky in the face-sheets,

(c) stresses 6,x, 6y and ky in the core and (d) J-integral around one of the crack tips. These

results are discussed in Chapter 3. The J-integral is a widely accepted quasi-static fracture

mechanics parameter used to study the onset of cracking for linear material response and,

with limitations, for nonlinear material response. A brief description of the J-integral and its

relevance to the present study is presented in Section 2.8.

Riks Algorithm

The Riks algorithm is an incremental iterative solution strategy to solve geometric

nonlinear problems of buckling and post-buckling analysis. An incremental iterative approach

is based on the Newton-Raphson method of iterations. The modified Newton-Raphson

method is more accurate and minimizes computation time and hence it is preferred in

nonlinear finite element analysis. The equilibrium equations are obtained by applying the

minimum potential energy principles at each load increment. But these equilibrium equations

at the target point are algebraically equivalent to those equations that are obtained by applying

the second variation to the total potential at the known initial point of the load increment. The









21

resulting algebraic equations are solved by the Gauss elimination method to get a converged

solution for each load increment. The Newton method is good in its convergence, if the initial

solutions used for iteration are close to the solution paths. In order to obtain the approximate

initial solution, a known adjacent point is usually required. Thus the approach to solve the

problem is a process to follow the equilibrium paths step by step, starting from a known point

which is usually the unloaded state of the structure.

In general, an ideal nonlinear solution technique has to trace the entire pre- and post-

critical static load path of the structure under loading including the instability points.

Conventional algorithms are carried out at a constant load level, and hence at the limit points

(instability points), when the tangent stiffness matrix becomes singular, these iterations cannot

traverse them. To overcome this, Riks algorithm uses the load magnitude as an additional

unknown; it solves simultaneously for loads and displacements. Therefore another quantity

must be used to measure the progress of the solution. This quantity is the "arc length" 1, along

the static equilibrium path in load-displacement space. This approach provides solutions

regardless of whether the response is stable or unstable.

The J-Integral

The J-integral is usually used in rate-independent quasi-static fracture analysis to

characterize the energy release associated with crack growth. It can be related to the stress

intensity factor if the material response is linear. The J-integral is defined in terms of the

energy release rate associated with crack advance. In fact, it can be shown that the J-integral

is equal to the energy release rate G in elastic materials and also in inelastic materials as long

as there is no unloading. The J-integral is computes as











J = Uonx nu )ds i= 1,2 j= 1,2 (2.5)
F



where F is the path around which the integral is evaluated.

The J-integral evaluations are possible at each location along the crack front. In a finite

element model each evaluation can be though of as the virtual motion of a block of material

surrounding the crack tip. Each such block is defined by contours; each contour is a ring of

elements completely surrounding the crack tip. ABAQUS automatically finds the elements

that form each ring from the node sets given as the crack tip. Each contour provides an

evaluation of the J-integral. The command used for a J-integral evaluation is *CONTOUR

INTEGRAL. This command and the associated procedure is discussed in Appendix A.

The J-integral should be independent of the domain used, but J-integral estimates from

different rings may vary because of the approximate nature of the finite element solution.

Therefore, there is a need for mesh refinement near the crack tip. Numerical tests suggest that

the estimate from the first ring of elements abutting the crack tip does not provide a high

accuracy result. Also, due to the face-sheets being very thin in the specimens, later J-integral

paths would also give erroneous results as they might traverse out of the geometry of the

specimen. In the present study, the J-integral was computed along 20 contours around the

crack tip. The value of the energy release rate from the 7th, 8th or the 9th contour was used

in the analysis and discussions (Chapter 3) due to the above reasons.















CHAPTER 3
RESULTS AND DISCUSSION


The FE simulation of a compression test was performed on 16 models, by varying the

following parameters: face-sheet thickness, core thickness, core density and delamination

length. Also, this set of 16 FE runs was repeated for three different material models of the

core (a) an orthotropic linear elastic model with properties as listed in Table 2.1 and (b) two

elastic perfectly plastic models with different values of elastic moduli and yield stress with

properties as listed in Table 2.2.

The results from these finite element analyses are discussed in the following sections. The

linear buckling analyses are presented first and compared with the experimental results and

the results from the analytical model of Hwa and Hu (1992). Next, the post-buckling

nonlinear analyses results are presented and the critical loads are compared with the

experimental failure loads. The stresses obtained from the post-buckling analyses are

tabulated and their relevance to the failure phenomena is discussed. This is followed by an

explanation of the results from the energy release rate calculations. The plastic material model

results are presented next and the effects of nonlinear elastic constitutive properties of the

core on the load carrying capacity of the sandwich beams is discussed.










24

Linear Buckling Analysis

The results of the linear buckling analysis are presented in Table 3.1. The specimen

numbers correspond to those given in Table 1.2 in Chapter 1. The properties of the face sheet

and the core, and the specimen dimensions are also given in Table 1.2. The buckling loads

presented in Table 3.1 are loads per unit width (1 inch) of the beam. The results include

buckling loads and mode shapes for the first three buckling modes. A sample mode shape for

Specimen 8 is shown in Fig.3.1. In some of the specimens penetration between the core and

the delaminated face sheet occurred, especially in second and third modes, and hence they do

not represent realistic buckling loads. This penetration behavior in Specimen 1 is shown in

Figs. 3.2 and 3.3.

The experimental failure loads are given in the last column of Table 3.1. The purpose of

the linear buckling analysis was to see if the specimens failed near the lowest buckling load.

Further, the FE results for buckling loads can be compared to those obtained from sandwich

beam models (Hwa and Hu, 1992) to see if the specimens can be modeled as sandwich beams.

A comparison of FE results for the lowest buckling loads and the corresponding

experimental failure loads in Table 3.1 lead to the following conclusions. There seems to be

no correlation between the linear buckling loads and the ultimate compressive strength as

measured by the tests. When the buckling loads are smaller, as is the case with longer

delaminations, the experimental failure load is higher than the buckling load indicating that

the beam goes under post-buckling. When the buckling loads are higher, as in beams with

shorter delmainations and/or thicker face sheets, the experimental failure load is closer to the










25

buckling load. The cases wherein the experimental failure load is lower than the buckling load,

indicate that other factors such as core failure, core instability, delamination propagation

would have played a role in the failure, and lowered the failure load below the buckling load.

In one of the beams (Specimen 13) with thick face sheets and high density core, the load

carrying capacity was limited by the face-sheet failure which cannot be predicted by a linear

buckling analysis. It should be reminded that the properties of the core used are the best

estimates of the actual core properties, and further the cellular core is assumed to be a linear

elastic continuum in both the analytical and FE models.

Another interesting observation from the results in Table 3.1 is the comparison of the FE

and analytical results (Hwa and Hu, 1992) for the buckling loads. Since both models use the

same material properties, the differences can only be attributed to the applicability of

sandwich beam models to the present problem. The FE model, which uses plane solid

elements, is applicable irrespective of the core thickness or delamination length. Further the

boundary conditions can be modeled exactly in the FE analysis. Thus the results can be

thought of as an evaluation of the beam model in the present context.

The analytical and FE buckling loads have reasonable agreement when the face sheets are

thin (Specimens 1-4) or when the beam undergoes global buckling (short delamination,

thicker face-sheets as in specimens 9 and 13). In most other cases the analytical buckling load

was greater than the FE results indicating the beam model is much stiffer, and a plane model

is required for accurate prediction of buckling loads. No comparison was made for the second

and third buckling loads as the analytical model considered only symmetric modes whereas













the FE analysis considered the full beam. Further, we did not use gap elements between the

delaminated surfaces, and interpenetration of the delaminated surfaces occurred in these

modes (see Figs. 3.2 and 3.3) invalidating the results.


Table 3.1 Buckling loads per unit width for first three modes. Analytical results are from
Hwu and Hu (1992)


Mode 1 Mode 2 Mode 3 Expt. Error % Error % of
Set Failure ofFEA Analytical
Method Method Method Method
FEA of Hwa FEA of Hwa FEA of Hwa Load compared compared to
and Hu and Hu and Hu (lb/in.) to test FEA

1 111 142 177 567 278 700 98 13 28

2 32 36 67 143 138 320 162 -80 11

3 15 16 31 63 64 142 164 -91 6

4 8 9 17 36 34 80 194 -96 6

5 1280 2465 1631 2600 2926 2615 1210 6 93

6 712 956 1101 1480 1259 3133 497 43 34

7 304 426 540 654 582 1441 361 -16 40

8 193 240 392 736 640 961 439 -56 25

9 2545 2551 3422 2614 4531 2621 2528 1 0

10 1699 1928 1762 1947 2738 1948 1215 40 13

11 1389 1969 2308 2773 2551 5098 1385 0 42

12 789 1103 972 1176 1487 1743 893 -12 40

13 6678 6752 8724 6919 10897 6937 4528 47 1

14 3756 2591 4012 2620 6225 2622 2319 62 -31

15 2489 1728 3661 1747 3825 1748 1688 47 -31

16 1647 1456 2141 1462 3133 1462 1583 4 -12





















Figure 3.1. First buckling mode for Specimen 8


Figure 3.2. Second buckling mode for Specimen 1


Figure 3.3. Third buckling mode for Specimen 1











28

Post-Buckling Analysis

As described in Chapter 2, a geometrically nonlinear analysis of the sandwich beam was

performed using the Riks algorithm. The purpose of the analysis was to see if the

experimental failure loads correspond to the maximum loads attained in the post-buckling

analysis. Sample load-deflection curves for Specimens 4, 5 and 9 are shown in Figs. 3.4

through 3.6. From Fig. 3.4 it can be seen that the load reaches a plateau for Specimen 4, and

this maximum load seems to correspond to the failure load. However this was not the case

for all specimens. For example in Specimen 9 (Fig. 3.5) the plateau occurs at a load of about

3,200 lb/in, but the specimen failed at 2,528 lb/in.




250


200


150

o 100


50

0 ------------------------------
0.00 0.01 0.02 0.03 0.04 0.05
Displacement (in.)

Figure 3.4. Post-buckling load-displacement curve for Specimen 4














4000
3500
3000


2500
2000
1500
1000
500
0"
0.00


0.01 0.02 0.03


0.04


Displacement (in.)


Figure 3.5. Post-buckling load-displacement curve for Specimen 9





2500 -

2000 -..-.,v

1500 -

1000 -


500

0
0.00 0.01 0.01


0.02 0.02 0.03 0.03


Displacement (in.)


Figure 3.6. Post-buckling load-displacement curve for Specimen 5










30

Thus we find that the maximum post-buckling load cannot be considered as the ultimate

load in compression. Further as seen in Fig. 3.6 some specimens do not exhibit a clear plateau

at least in the range of load steps used in the analysis indicating the maximum post-buckling

load could be much higher than the experimental failure load. Thus, there are three types of

load-deflection behavior associated with the post-buckling of the delaminated sandwich

specimens: Type 1 when the load reaches a plateau and corresponds to the experimental

failure load, Type 2, when the load reaches a plateau but does not correspond to the

experimental failure load, and Type 3 when there is no clear plateau and the post-buckling

load might be higher than the experimental failure load. The post-buckling load-deflection

plots for all the 16 specimens used in the experimental study are presented in Appendix B.

The summary of maximum loads attained in the FE analysis are presented in Table 3.2. The

FE post-buckling loads are compared to the experimental failure load by computing the

percentage difference and it is presented in the last column of Table 3.2.

From Table 3.2 the following observation can be made. The maximum load predicted by

the FE post-buckling analysis is approximately equal to or higher than the experimental failure

load. The values are closer in specimens 4, 8, 11, 12 and 16. In these specimens the

delamination length was either 1.5 inches or 2.0 inches. That is, when the delamination length

is longer post- buckling will occur relatively sooner, and the post-buckling analysis is able to

predict the load carrying capacity with reasonable accuracy. However in other specimens the

actual failure occurred earlier than the post-buckling instability indicating that some other

failure mechanisms triggered the collapse of the specimens. It should be noted that in the case











31

Table 3.2 Results of post-buckling analysis

Set Delamination Pcr by FEM Pcr by experiment Error
(in.) (lb.) (lb.) (%)

1 0.5 91.0 98.5 -8

2 1 270.0 161.7 67

3 1.5 330.0 163.8 101

4 2 191.0 193.6 -1

5 0.5 1182.0 1210.0 -2

6 1 985.0 496.6 98

7 1.5 430.0 361.0 19

8 2 404.0 439.1 -8

9 0.5 3200.0 2528.0 27

10 1 1860.0 1215.0 53

11 1.5 1406.0 1385.0 2

12 2 815.0 892.6 -9

13 0.5 8100.0 4528.0 79

14 1 3637.0 2319.0 57

15 1.5 1744.0 1688.0 3

16 2 1643.0 1583.0 4


of Specimen 13, which has thickest face-sheets, thickest high-density core and short

delamination, the post-buckling load is the highest (8,100 lb/in.). However its face sheets

failed at a much lower load (4,528 lb/in.). This is because the face-sheets failed in

compression even before the specimen went into the post-buckling regime.











32

In summary, the post-buckling instability alone cannot be used to predict the compression

failure of the debonded sandwich beam specimens. Before the beam goes into the post-

buckling regime other failure mechanisms can initiate failure leading to catastrophic failure

of the specimen. These failure mechanisms, for example, include delamination propagation,

core failure (especially near the crack-tip) and face-sheet failure. These mechanisms will be

investigated in the following sections.

Energy Release Rate

From early on it was suspected that the compressive failure in a debonded sandwich beam

will occur due to delamination buckling followed by catastrophic failure due to unstable

delamination propagation. However a postmortem analysis of failed specimens indicated that

there was no or little crack propagation in most of the failed specimens. In order to check

this, the energy release rate at the crack tip was computed using the J-integral at each load

step of the post-buckling analysis. A typical graph showing the variation of energy release rate

with the load is presented in Fig. 3.7. The energy release rate at the experimental failure load

for each specimen is given in Table 3.3.

In the same table the interfacial fracture toughness for the corresponding specimen is also

given. This fracture toughness was measured using DCB specimens in the experimental study

(Avery, 1998). The amounts of crack extension (increase in the length of delamination) are

also given in the same table. From the results it is clear that the G was considerably lower

than G, in most specimens and delamination propagation could not have been the trigger

mechanism that caused the failure.



















2.0


1.5


1.0


S0.5

SAn n


0 100 200 300 400 500
Load (lb.)

Figure 3.7. Energy release rate as a function of load for Specimen 8




Stress Analysis

The stresses in the face sheet and the core were computed at each load step of the

nonlinear analysis. These stresses were compared with corresponding strength values to check

if they could have initiated the failure. Sample plots of stress distribution through the

thickness of the core are presented in Figs. 3.8 and 3.9.

The stresses in Fig. 3.8 are in the vicinity of the crack-tip, whereas Fig. 3.9 shows stresses

at the center of the specimen. The maximum compressive stresses in the face sheet and the

core corresponding to the experimental failure load are presented in Table 3.3. It must be

noted that the core stresses presented in the table are values either at the mid-span of the













Table 3.3. Comparison of interfacial fracture toughness G, and maximum energy release
rate Gmax, and maximum core and face-sheet stresses with corresponding strength values at
experimental failure load


Interfacial fracture Core Face-sheet

Delam. Crack Core Max. core Max. Face-
b/.) (b/.) length extension Strength stress (psi) sheet stress
(in.) (in.) (psi) & location* (ksi)

1 1.71 0.07 0.5 0.125 2.75 2.30 (CT) 19.4

2 1.43 0.21 1.0 0.000 6.10 8.77 (CC) 17.6

3 1.22 0.08 1.5 0.000 6.10 6.50 (CC) 18.9

4 1.05 Gmax>Glc 2.0 0.125 22.9 65 (CC) 21.1

5 4.99 1.65 0.5 1.000 6.10 10.2 (CT) 63.4

6 3.31 0.05 1.0 1.000 22.9 3.9 (CT) 13.9

7 6.59 0.14 1.5 0.625 2.75 3.5 (CT) 20.0

8 4.26 1.95 2.0 0.000 6.10 32 CC) 24.6

9 7.17 0.29 0.5 0.250 6.10 4.3 (CT) 32.9

10 8.05 0.11 1.0 0.125 2.75 2.9 (CT) 55.3

11 5.25 2.12 1.5 0.250 22.9 22.5 (CC) 65.2

12 6.47 Gmax>Glc 2.0 0.000 6.10 11 (CT) 23.4

13 4.96 0.12 0.5 0.000 22.9 15 (CT) 54.1

14 7.94 0.13 1.0 0.875 6.10 6.5 (CT) 25.9

15 7.16 4.70 1.5 0.125 6.10 25 (CT) 53.3

16 10.5 1.70 2.0 0.750 2.75 7.5 (CT) 28.4


The strain energy release rate values increase rapidly when the load reaches the plateau.
* The core stresses were monitored at two locations, CT: below the crack-tip; CC: center of
the beam.
Compressive strength of graphite/epoxy face sheet material is 77 ksi.











35

specimen or in the vicinity of the crack tip. The compressive strength of the graphite/epoxy

face sheet material is given as 7.7x 106 psi by the manufacturer. The core compressive strength

varied from specimen to specimen depending on the core density.

In Table 3.3, the maximum stresses developed in the face sheet are presented in the

last column. Comparing with the face sheet strength provided by the manufacturer, it is clear

that the face sheet stresses were much lower than the corresponding strength values. But the

compressive stresses in the core (Column 7) exceeded the compressive strength of the core

(Column 6) indicating that core failure could have triggered the specimen failure. To

understand the relation between core instability and the failure of the sandwich beam

specimen, it was decided that a thorough investigation of the core was necessary. In a

subsequent experimental study of the core material, it was found that the core exhibited

properties similar to an elastic-perfectly plastic material as shown in the stress-strain plot in


0)
.2

El
LU


14
12
10
" 8

4--

2
8


-15 -10 -5 0
Stress (psi)


Figure 3.8. Through-the-thickness 6x stress distribution in the core in the
vicinity of the crack-tip (Specimen 5) for a compressive load of 1078 lb/in











36

Fig. 3.10. In the next section, the results of the inelastic constitutive modeling of the core are

presented and the effects of such a model on the load carrying capacity are discussed.


0)

.2
'a-

0"-


-40 -30 -20 -10
Stress (psi)

Figure 3.9. Through-the-thickness 6, stress distribution in the core at the
center of the beam (Specimen 8) for a compressive load of 391 lb/in




45



30






n


0.008


0.016


0.024


Strain

Figure 3.10 Stress-strain curve for a core specimen with f = 6.0 pcf
and height = 0.25 in.


A

1 2L

8

4-

E) 4












37

Effects of Core Plasticity

As discussed in the two previous sections neither the energy release rate nor the stress

field in the beam could explain the instability of the sandwich beam and provide an estimate

of the maximum load the beam can carry. Further, inspection of the failed specimens and

evaluation of G corresponding to the failure load indicated that delamination initiation was

not the trigger mechanism for failure. Hence the effects of plasticity of the core on the

instability of the beam was investigated.

In this part of the study, the core was modeled as an isotropic elastic-perfectly material

using two different set of core properties, Core EP1 and Core EP2 (see Table 2.2 in Chapter

2). Sample load-end shortening diagrams for both sets of core are presented in Figs.3.11

through 3.14.



250


200


S150


100 _ -


50
5 0 _ _ _ _ __------------------------------


0
0.00E+00 2.00E-02 4.00E-02 6.00E-02 8.00E-02 1.00E-01
Displacement (in.)

Figure 3.11. Post-buckling load-displacement curve for Specimen 5 with
Core EP1

















1200 -


1000 -


800 -_--- -


-o 600


400


200 -


0
0 -_-----------------------------
0.OOE+00 5.00E-03 1.00E-02 1.50E-02
Displacement (in.)


Figure 3.12. Post-buckling load-displacement curve for Specimen 15 with
Core EP


1000


800


600


400


0 -1 - -
0.OOE+00 5.00E-02 1.00E-01 1.50E-01 2.00E-01 2.50E-01
Displacement (in.)


Figure 3.13. Post-buckling load-displacement curve for Specimen 5 with
Core EP2
















1400 -

1200 -

1000

800 -

600 -

400 -

200 -

0
0.OOE+00 2.00E-03 4.00E-03 6.00E-03
Displacement (in.)


8.00E-03 1.00E-02


Figure 3.14. Post-buckling load-displacement curve for Specimen 15 with
Core EP2



The maximum loads before failure for different specimens are presented in Table 3.4 for

Core EP1 and in Table 3.5 for core EP2. The contour plot of von Mises stresses in the core

are presented in Figs. 3.15 and 3.16.

From the von Mises stress plots and the load-deflection diagrams the following

conclusions can be reached. As the beam is loaded, the deboned face sheet buckles and

creates a stress concentration at the crack tip. The core material yields at these high stresses

forming a small plastic zone or yield zone around the crack tip. As the load increases the

plastic zone size also increases. When the plastic zone size reaches a critical value, the

yielding spreads thorough the entire width of the core making the specimen unstable and

causing a large drop in the load. At this stage the load carrying capacity of the beam is











40

primarily determined by the face-sheet stiffness as the core becomes ineffective in carrying the

load.

From figures 3.11 through 3.14 it may be noted that there are two distinct types of failure

are possible. In the first type the load drops suddenly before the beam goes into the post-

buckling regime (Figs.3.11, 3.13). In the second type there is no distinct load drop but the

beam goes into post-buckling as shown in Figs. 3.12 and 3.14. The amount of load drop is

computed as follows. The deflection corresponding to the maximum load, Fmax, is noted in

each case and this deflection is designated as af (failure deflection). Then the load

corresponding to 1.1 af is determined from the load-deflection diagrams and this is designated

as Fmn. The factor 1.1 was arbitrarily chosen in order to maintain consistency among the load-

deflection diagrams of various specimens. The load drop AF is defined as the difference

between Fmax and Fmjn. Values of Fmax, Fmn and AF for each specimen are given in Tables 3.4

and 3.5. The value of energy release rate corresponding to Fmax are also provided.

From Tables 3.4 and 3.5, it is clear that both Fmax and Fmn. are a strong function of the face

sheet thickness and the core yield strength. Further, the values of Fmn. have a stronger

correlation to the face sheet thickness, indicating that after the core yields the load carrying

capacity depends mainly on the face sheet stiffness. An inspection of energy release rate G at

Fmax show no correlation between G and the failure load. This indicates that small scale

yielding near the crack tip could not explain the instability of the specimens. Hence future

studies should focus on the size of the plastic at the point of failure and its relation to

delamination length and core thickness.












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Figure 3.16 (a) von Mises stresses in the vicinity of the left crack tip for Specimen 10 with Core EP1 at 480 lb. The
maximum load was at 529 lb.


6j606


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5 B05

5360

4.9 15


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Sx i : .. j.r.a 1j.r 1
rIii,, i .7.rjJ .-:h :

A'iufl 7..I.rk3 I, n


Figure 3.16 (c) von Mises stresses in the vicinity of the left crack tip for Specimen 10 with Core EP1 at 490 lb.


, 1


















IS1FCPATIRAN Version 8.5 11-Sep-99 1504 .00
- rings, Dciul1, SBop1,Toi lTir -0.00709g7: S9rB=, CompFon rrrle(NOtJ. .-AVEF D) (VOIIM)
Delorrn: Delul, Siep 1,Toa nlTime=0.0070937: Delormalio,r DEplbacmenis


3


I 2.722
I


+


1 949

SI f


IeI3ufFi \.e
ra 1:5 ..jrjd1 I.X;'
re.,i. -,cI :;.-idJ 17:0
,el3u _eI .I.cT,",i :
=,,,, 0CI;? ,T'I" 1."v,5


Figure 3.16 (d) von Mises stresses in the vicinity of the left crack tip for Specimen 10 with Core EP1 at 470 lb.


12.0M

1 1.


7.361


6 88

15K











47

Table 3.4 Maximum load and drop in load after maximum load for various specimens with
different core thickness, delamination length, and face sheet thickness. The core properties
are those of Core EP1 as listed in Table 2.2



Maximum Load at Drop in
Set hf(in.) (in.) (in.) (lb./in.) Load 1.1 af Load
(in.) (in.) (lb./in.) ( ( (
(lb.) (lb.) (lb.)

1 0.0087 0.250 0.50 .07370 73 49 24

2 0.0087 0.375 1.00 .06827 93 47 46

3 0.0087 0.500 1.50 .04640 126 45 81

4 0.0087 0.375 2.00 .01667 80 39 41

5 0.0261 0.375 0.50 .00749 216 177 39

6 0.0261 0.250 1.00 .01039 198 160 38

7 0.0261 0.375 1.50 .04860 194 190 4

8 0.0261 0.500 2.00 .05640 207 179 28

9 0.0435 0.375 0.50 .00573 584 509 75

10 0.0435 0.500 1.00 .02590 529 526 3

11 0.0435 0.375 1.50 .06460 445 442 3

12 0.0435 0.250 2.00 .06710 482 478 4

13 0.0609 0.500 0.50 .07013 1067 1062 5

14 0.0609 0.375 1.00 .03387 1165 1165 0

15 0.0609 0.250 1.50 .06900 956 956 0

16 0.0609 0.375 2.00 .04195 1175 1175 0


The load at 1.1 af, where af is the displacement at the maximum load











48

Table 3.5 Maximum load and drop in load after maximum load for various specimens with
different core thickness, delamination length, and face sheet thickness. The core properties
are those of Core EP1 as listed in Table 2.2



hf hc a Maximum Load at Drop in
Set (in.) (in.) (in.) G (lb./in.) Load (lb.) 1.1af Load (lb.)

1 0.0087 0.250 0.50 1.8460 360 154 206

2 0.0087 0.375 1.00 0.3977 419 144 275

3 0.0087 0.500 1.50 0.2491 517 190 327

4 0.0087 0.375 2.00 0.0696 413 151 262

5 0.0261 0.375 0.50 1.3850 966 485 481

6 0.0261 0.250 1.00 1.0180 628 417 211

7 0.0261 0.375 1.50 1.3780 862 352 510

8 0.0261 0.500 2.00 1.2110 1162 1146 16

9 0.0435 0.375 0.50 0.0343 2216 743 1473

10 0.0435 0.500 1.00 1.3060 1526 1089 437

11 0.0435 0.375 1.50 1.1690 1063 1014 49

12 0.0435 0.250 2.00 0.4365 794 780 14

13 0.0609 0.500 0.50 0.1854 2163 1632 531

14 0.0609 0.375 1.00 0.3191 2182 1813 369

15 0.0609 0.250 1.50 0.9628 1186 1186 0

16 0.0609 0.375 2.00 1.8280 1766 1662 104


The load at 1.1 af, where af is the displacement at the maximum load















CHAPTER 4
CONCLUSIONS AND FUTURE WORK


A finite element analysis was performed to simulate axial compression of debonded

sandwich beams. A linear buckling analysis was performed to determine the buckling loads

and corresponding mode shapes. The nonlinear analysis modeled the post-buckling behavior

of the sandwich beams. The load-deflection diagrams were generated for a variety of

specimens used in a previous experimental study. The energy release rate at the crack tip was

computed using the J-integral, and plotted as a function of end-shortening and the load. A

detailed stress analysis was performed and the critical stresses in the face sheet and the core

were computed. Further, the core was modeled as a elastic-perfectly plastic material and a

nonlinear post-buckling analysis was performed. For this model, load-deflection curves,

energy release rate plots and von Mises stress plots were generated.

By comparing the experimental failure load and the FEA results the following conclusions

can be reached. The linear buckling analysis is inadequate in predicting the load carrying

capacity of debonded sandwich beams. The specimens go well into post-buckling regime

before failure. From the nonlinear analysis it was found that failure occurs before the

maximum load is attained in post-buckling. Thus the failure has to be initiated by the debond

propagation or stresses in the core and/or face-sheets. However the energy release rate was

considerably lower than the interfacial fracture toughness thus eliminating interface failure as











50

a mechanism for the specimen failure. The stress analysis results show that the face sheet

stresses were much lower than the corresponding strength values. But the compressive

stresses in the core exceeded the compressive strength indicating that core failure could have

triggered the specimen failure.

From the results of the FE analysis of the specimens with an elastic-perfectly plastic core,

it is clear that both Fmax (the maximum load) and Fmn. (the load at 1.1 af, where af is the

displacement at Fmax) are strongly dependent on the face sheet thickness and the core yield

strength. Further, the values of Fmn. have a stronger correlation to the face sheet thickness,

indicating that after the core yields the load carrying capacity depends mainly on the face

sheet stiffness. An inspection of the energy release rate G at Fmax show no correlation between

G and the failure load. This indicates that small scale yielding near the crack tip could not

explain the instability of the specimens. The von Mises stresses in the core before and after

Fmax indicate that the plastic zone size around the crack tip might have an effect on the failure

of the core. The plastic zone size needs to be quantified for the various specimens and a

relation between the plastic zone size and the failure load needs to be established. Currently

the core is being modeled as a 3-D structure with each honeycomb cell being modeled as a

hexagonal solid with 6 faces (Figs. 4.1- 4.3). This model will be used to understand the

elasto-plastic material behavior of the core material to establish this relation between the core

instability and the failure of the sandwich specimen.




































Figure 4.1 (a) Top view of the core in 2-D








Figure 4.1 (a) Top view of the core in 2-D











































Figure 4.1 (b) Hexagonal cellular structure of the core in 2-D
































z
Y X


Figure 4.2: (a) Hexagonal solid structure of the honeycomb cells in the 3-D FE model of core material











































Figure 4.2 (b) Each cell is modeled using 6 plates and each of these plates has 4 plate elements















APPENDIX A
BRIEF OVERVIEW OF ABAQUS


A. 1 Introduction

ABAQUS is one of the leading commercial FE packages available for industrial and

academic use. The ABAQUS finite element system used in this study includes

ABAQUS/Standard, a general purpose finite element program and ABAQUS/Post, an

interactive post-processor that provides X-Y plots, animations, contour plots, and tabular

output of results from ABAQUS/Standard. In order to run a finite element analysis, an input

file is required. An ABAQUS input file is an ASCII data file that can be created by using a

text editor like vi or pico. The input file consists of a series of lines containing ABAQUS

options and data. Most input files have the same basic structure. There are two maj or sections

in an ABAQUS input file the model data section and the history data section.

A.2 Model Data

After the heading, the input file usually contains a model data section to define the nodes,

elements, materials, initial conditions, etc. The following model data must be included in an

input file to define the finite element model:

Geometry: The geometry of a model is described by elements and their nodes. The

rules and methods for defining the nodes and elements are described in Section 2.3.

Cross-sections for structural elements (such as beams) must be defined.











56

Material definitions: A material type must be associated with most portions of the

geometry. The materials used are described in Section 2.5.

Boundary conditions: Zero-valued boundary conditions (including symmetry

conditions) can be imposed on individual solution variables such as displacements or

rotations. The definition of boundary conditions is discussed in Section 2.4.

A.3 History Data

After the model data section, the input file contains history data to define the analysis

type, loading, output requests, etc. The STEP option divides the model data from the history

data in the input file. The following history data must be included in an input file to define an

analysis procedure:

Response type: An option to define the analysis procedure type must appear

immediately after the *STEP option. ABAQUS can perform many types of analyses -

linear or nonlinear, static or dynamic, etc. The type of analysis used in this study is

discussed in Section 2.6.

Loading: Usually some form of external loading is defined. For example, concentrated

or distributed loading can be applied, temperature changes leading to thermal

expansion can be prescribed or contact conditions can be used to apply loads. The

loading is discussed in Section 2.4.

The input file containing the model and the history data sections is then processed by the

"solver input file processor" prior to executing the appropriate solver (in this study,

ABAQUS/Standard). The functions of the solver input file processor are to interpret the











57

ABAQUS options, perform the necessary consistency checking, and prepare the data for the

solver. After the analysis is completed by ABAQUS/Standard, the results are post-processed

using ABAQUS/Post.

A.4 Element and Node definition

The syntax for the commands to define elements and nodes are shown below:

To create an element, the following command is used:

*ELEMENT, TYPE, ELSET

The first line after the above command contains the following information:

1. Element number

2. First node number forming the element

3. Second node number forming the element

4. Etc., up to 15 node numbers

TYPE This parameter is set to equal the element type. The element type used was CPE8

which corresponds to continuum plain strain element with 8 nodes

ELSET This parameter is set to equal the element set to which these elements will be

assigned. For e.g., if the element is to be part of the core, then ELSET = CORE

To create a node, the following command was used:

*NODE, NSET

The first line after the command contains the following information:

1. Node number

2. First spatial coordinate of the node













3. Second spatial coordinate of the node

4. Third spatial coordinate of the node

NSET This parameter is set to equal the name of the node set to which these nodes will be

assigned

A5. Contour Integral

The *CONTOUR INTEGRAL output option is used to compute contour integral

estimates in fracture mechanics studies. ABAQUS automatically finds the elements that form

each ring from the node sets given as the crack tip.. Each contour provides an estimate of the

J-integral. The number of evaluations possible is the number of such rings of elements. The

user must specify the number of contours to be used in calculating the J-integrals by using the

CONTOURS parameter on the *CONTOUR INTEGRAL option. The TYPE parameter i

sused to select the type of contour integral to be calculated. The default is to calculate the J-

integral. It should be noted that the contour integral evaluations can be performed only in two

or three dimensions and can be used only with quadrilateral elements or brick elements.















APPENDIX B
RESULTS FROM POST-BUCKLING ANALYSES


The load deflection plots from the post-buckling analysis of the debonded sandwich beam

specimens are presented in this section. First, the plots of the analyses with the core as an

elastic orthotropic material are presented. This is followed by the plots from the analyses with

the core as an elastic-perfectly plastic material.











60

















Figure B-1. Load-displacement plots with the core as an elastic orthotropic material












Set 1


200
150
100
50


0 0.002 0.004 0.006 0.008 0.01
End shortening (in.)





Set 2


400
300
-* 200
100
0


0 0.005 0.01


0.015 0.02


End shortening (in.)


0.025













Set 3


0 0.005 0.01 0.015 0.02 0.025 0.03
End shortening (in.)






Set 4


0.01 0.02 0.03 0.04


0.05


End shortening (in.)


400
350
300
250
200
150
100
50
0


250
200
150
100
50
0













Set 5

2500
2000
. 1500 --- ----

o 1000 -
500 -
5 0 0 _ __--------------------
0 __---__-__---
0 0.005 0.01 0.015 0.02 0.025 0.03
End shortening (in.)






Set 6

600
500 ----
400 --
-o 300 ----
6 200 -
2100
-j 100 ________-------
0
0 0.002 0.004 0.006 0.008


End shortening (in.)














Set 7

1200
1000 -
-. 800 -----,--
600 ----_- -
o 400 ____
S 200 ----
0 _ _
0 0.005 0.01 0.015 0.02
End shortening (in.)




Set 8

500
400 ...........-
, 300
" 200
o 100
0
0 0.005 0.01 0.015


End shortening (in.)













Set 9


4000
3000
-o 2000
o
1000
0


0.01


0.02


End shortening (in.)





Set 10


0.005


0.01


End shortening (in.)


0.03


0.04


2500
2000
1500
1000
500
0


0.015


0.02















Set 11


0.005


0.01


End shortening (in.)





Set 12


0.005


0.01


0.015


End shortening (in.)


2000

1500

1000

500


0.015


800
700
600
500
400
300
200
100
0


0.02













Set 13


10000
8000
6000
4000
2000
0


0
0


0.02 0.04
End shortening (in.)


Set 14


4000
3000
-o 2000
o
1000
0


0 0.005 0.01 0.015
End shortening (in.)


0.02 0.025


i_- 1-e< *


'K


0.06













Set 15


0.005


0.01


End shortening (in.)





Set 16


0 0.01 0.02


End shortening (in.)


2000
1500
1000
500


0.015


2000
1500
1000
500


0.03































Figure B-2. Load-displacement plots with the core as an elastic-perfectly plastic material
(Core EP 1)










70



Set 1


0 0.005


0.01


0.015


0.02


End shortening (in.)






Set 2


End shortening (in.)


0.025


0.03


100


0 0.005 0.01 0.015 0.02 0.025 0.03













Set 3

150

- 100

o 50

0
0 0.02 0.04 0.06
End shortening (in.)




Set 4

100
S80
60 _
- 40 _
20
20 _--- ---_-- ---_-___-----
0 0
0 0.02 0.04 0.06 0.08


End shortening (in.)














Set 5

250 -
200
8 150 -

o 100 _
-J
50 -
0
0 0.05 0.1 0.15 0.2
End shortening (in.)






Set 6

250
200 -
8 150 ______+ +

o 100 -
50
0 /
0 0.002 0.004 0.006 0.008
0 0.002 0.004 0.006 0.008


End shortening (in.)













Set 7

250
200
S150 _-
m 100
50
0
-' 50 __-----
0 _--------------
0 0.005 0.01 0.015 0.02 0.025
End shortening (in.)





Set 8

250
200
. 150 _
o 100 _'
-J t
50 _
0
0 0.01 0.02 0.03 0.04 0.05 0.06


End shortening (in.)














Set 9












0 0.005 0.01 0.015 0.02
End shortening (in.)








Set 10

600
500 ------------------

.6 400 -
"0 300 -
o 200
-j
100 -
0 A,
0 0.002 0.004 0.006 0.008
End shortening (in.)
















Set 11


500
400
300

200
100
0


0.000


0.005


0.010


0.015


0.020


End shortening (in.)





Set 12


600
500 -- - ,----- -----
400 -------
300
200
100 ,
0
0.000 0.010 0.020 0.030 0.040 0.050 0.060


End shortening (in.)















Set 13


0.000 0.050 0.100 0.150 0.200 0.250 0.300
End shortening (in.)





Set 14


End shortening (in.)


1200
1000
800
600
400
200
0


1400
1200
1000
800
600
400
200
0


0 0.002 0.004 0.006 0.008 0.01 0.012
















Set 15


1200
1000
800
600
400
200
0


0 0.005 0.01 0.015


End shortening (in.)





Set 16


1400
1200
1000
800
600
400
200
0


0 0.05


0.1 0.15 0.2 0.25


End shortening (in.)


0.02


0.3















LIST OF REFERENCES


ABAQUS/Standard User's Manual, 1998. Hibbitt, Karlsson & Sorensen, Inc. Pawtucket, RI.

Avery, J.L., and B.V. Sankar, 1999. "An Experimental Study of Post-buckling Behavior of
Debonded Sandwich Composites," J. Composite Materials (under review).

Avery, J.L., 1998. "Compressive Failure of Delaminated Sandwich Composites," Master of
Science thesis, Department of Aerospace Engineering, Mechanics & Engineering Science,
University of Florida, Gainesville, Florida.

Charalambides, P.G., H.C.Cao, J.Lund, and A.G.Evans, 1990. "Development of Test Method
for Measuring the Mixed Mode Fracture Resistance of Bimaterial Interfaces," Mechanics of
Materials, 8: 268-283.

Chen, H.P., 1993. "Transverse Shear Effects on Buckling and Post-buckling of Laminated
and Delaminated Plates," AIAA Journal, 31(1):163-169

Crisfield, M.A., 1981. "A Fast Incremental/Iterative Solution Procedure That Handles Snap-
Through", Comput. Struct., 13: 55-62.

Dundurs, J., 1969. "Edge-Bonded Dissimilar Orthogonal Elastic Wedges," J. Appl. Mech. 36:
650-652.

Hutchinson, J.W., 1990. "Mixed Mode Fracture Mechanics of Interfaces," Metal-Ceramic
Interfaces (M.Ruhle, A.G. Evans, M.F. Ashby, and J.P. Hirth, eds.), Pergamon Press, New
York, pp. 295-306.

Hwu, C., and J.S. Hu., 1992. "Buckling and Post-buckling of Delaminated Composite
Sandwich Beams," AIAA Journal, 30(7): 1901-1909.

Kardomateas, G.A., 1990. "Post-buckling Characteristics in Delaminated Kevlar/Epoxy
Laminates: An Experimental Study." J. Composites Technology & Research, 12(2): 85-90.

Kassapoglou, C., 1988. "Buckling, Post-buckling and Failure of Elliptical Delaminations in
Laminates under Compression," Composite Structures, 1 9:139-159













Kim, W.C., and C.K.H. Dharan, 1992. "Face-Sheet Debonding Criteria for Composite
Sandwich Panels Under In-Plane Compression," Engineering Fracture Mechanics,
42(4):642-652.

Minguet, P., J. Dugundji and P.A. Lagace, 1987. "Buckling and Failure of Sandwich Plates
with Graphite-Epoxy Faces and Various Cores," J. Aircraft, 25(4):372-379.

Rice, J.R., 1998. "Elastic Fracture Concepts for Interfacial Cracks," J. Appl. Mech., 55: 98-
103.

Riks, E., 1979. "An Incremental Approach to the Solution of Snapping and Buckling
Problems," Int. J Solids. Struct., 15: 529-551.

Sankar, B.V., M.Narayanan, and J.L.Avery, 1999. "Post-buckling Behavior of Debonded
Sandwich Composite Beams Under Compression," Paper No. AIAA-99-1295, AIAA SDM
Conference, St Louis, MO, April.

Simitses, G.J., S. Sallam and W.L. Yin, 1985. "Effect of Delamination of Axially Loaded
Homogeneous Laminated Plates," AIAA Journal, 23(9): 1437-1444.

Sleight, D.W., and J.T. Wang, 1995. "Buckling Analysis of Debonded Sandwich Panel Under
Compression," NASA Tech Memorandum 4701.

Yin, W.-L., S.N. Sallam and G.J. Simitses, 1986. "Ultimate Axial Load Capacity of a
Delaminated Beam-Plate," AIAA Journal, 24(1):123-128.

Zak, A.R., and Williams, M.L., 1963. "Crack Point Singularities at a Bimaterial Interface,"
J. Appl. Mech. 30: 142-143.















BIOGRAPHICAL SKETCH


I was born in the city of Madras (now known as Chennai), India, on November 2, 1973.

I was the first of two sons born to my parents who are both professors at the University of

Madras. My younger brother named Elaya Manickam has just completed his Master of

Science degree in computer applications from Anna University, Madras, and is now working

as a software developer in Madras. I have been married to Aarthi Rao since January 1999.

She is working as a software developer in Cincinnati, Ohio.

I attended schools in Madras throughout my pre-college life and then I attended the Birla

Institute of Technology and Science at Pilani, India. After earning a bachelor's degree in civil

engineering and a minor in physics, I joined the National Aerospace Laboratory, Bangalore,

India, as an engineer-trainee in the Structures Division. I was part of a team that was

developing an in-house finite element analysis package. After a year at Bangalore, Ij oined the

Aerospace Engineering Department at the University of Florida for my graduate studies. After

two years of research on delaminated sandwich composites, I defended my master's thesis in

September 1999.

I am looking forward to a new phase in my life. I hope to join the engineering industry

and be successful in all my endeavors.




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