hamza report

The Institute of Space Technology

An Analytical Modeling and Damage Analysis of Honeycomb Sandwich Structure

under Low Velocity Impact Load

By

Syed Hamza Ali Tirmizi

Muhammad Umer Khurshid

A PROJECT SUBMITTED IN PARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE OF BACHELOR OF SCIENECE

IN AEROSPACE ENGINEERING

Project Supervisor’s Name: Dr. Asif Israr

Project Supervisor’s Signature:

Islamabad, PakistanSeptember, 2013

1Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid

Certificate

This is to certify that the research work described in this thesis is the original work of

author(s) and has been carried out under my direct supervision. I have personally gone

through all the data/results/materials reported in the manuscript and certify their

correctness/authenticity. I further certify that the material included in this thesis is not

plagiarized and has not been used in part or full in a manuscript already submitted or in

the process of submission in partial/complete fulfillment of the award of any other degree

from any institution. I also certify that the thesis has been prepared under my supervision

according to the prescribed format and I endorse its evaluation for the award of Bachelor

of Science in Aerospace Engineering degree through the official procedures of the

Institute.

(Dr. Asif Israr)


Copyright © 2013This document is jointly copyrighted by the author(s) and the Institute of Space Technology (IST). Both IST and author(s) can use, publish or reproduce this document in any form. Under the copyright law no copyright of this document can be reproduced by anyone, except copyright holders, without the permission of author(s).


ABSTRACT

An Analytical Modeling and Damage Analysis of Honeycomb Sandwich Structure under

Low Velocity Impact Load

By

Syed Hamza Ali Tirmizi

Muhammad Umer Khurshid

B.Sc. Aerospace EngineeringInstitute of Space Technology

Sep 2013

Honeycomb sandwich structure is of great importance now-a-days and a lot of research

work has been carried out in the field of aerospace and allied disciplines. Honeycomb

sandwich structure provides high strength and high resistance by being low in weight in

air and space vehicles. Practically, honeycomb structures face different types of forces,

stresses, fatigue and damages due to the collision of any space debris that causes

permanent indentation or failure and reduces the working life and strength of structures.

The most common mode of partial and complete failure is impact on the structure. Up till

now, more practical and less theoretical work has been carried out on the low velocity

impact damages because of the complexity of analytical as well as dynamic modeling of

honeycomb structures under given initial and boundary conditions. The energy-balance

and spring-mass are two different mathematical approaches that are employed in this

study for analyzing the structural response analytically. ANSYS and LSDYNA is the FE

analysis tool in which the impact damages are simulated. Experimentally, three points

bend and drop weight impact test of the structure is also performed on different

standardized specimens. Non-linear high order couple homogeneous ordinary differential

equation is obtained which is then solved by using the approximate method of

perturbation under given boundary and initial conditions. Finally, the results obtained

from analytical, numerical and experimental tests are compared with each other to check

the usefulness of analytical model. The obtained result is in the form of different plots.


Dedication

We pay our humble dedication of this research work to our parents!


ACKNOWLEDGEMENTS

First of all we would like to thank ALLAH Almighty who blessed us the courage to do our

work with sincerity and determination by which we had completed our project. Then, we

would like to thank our parents, who are the most respectful in our eyes and made us able

today so that we are able to make their name high up in the sky. We would like to pay a

special thanks to our supervisor Dr. Asif Israr who helped us a lot in our project and stay

with us through every thick and thin. Then we would like to thank each and every faculty

member especially Dr. Zubair Khan, Mr. Adnan Munir for providing us the necessary

guidelines in our project. We would also like to thank our senior Ms. Wajeeha Siddiqui in

guiding us about the project and the AWC and NESCOM organizations for using their

experimental testing facilities and also to Institute of Space Technology!


Table of Contents

CERTIFICATE 2

ABSTRACT 4

DEDICATION 5

ACKNOWLEDGEMENTS 6

TABLE OF CONTENTS 7

LIST OF FIGURES 10

LIST OF TABLES 13

1. INTRODUCTION 14

1.1. MOTIVATION 141.2. PREVIOUS WORK DONE BY THE EARLIER COLLEAGUES 141.3. INTRODUCTION ABOUT OUR PROJECT 141.4. OBJECTIVES OF THE STUDY 141.5. OUTCOMES OF THE STUDY 151.6. SANDWICH THEORY 15

1.6.1. Honeycomb Sandwich Structure 161.6.1.1. Core 161.6.1.2. Face sheets 171.6.1.3. Adhesive Material 18

1.6.2. Modes of Deformation of Sandwich Structure 191.6.2.1. Strength 20

1.1.1. Stiffness 201.6.2.2. Panel Buckling 201.6.2.3. Shear Crimpling 201.6.2.4. Skin Wrinkling 211.6.2.5. Intra-cell Buckling 211.6.2.6. Local Compression 211.6.2.7. Delamination 21

1.6.3. Applications of Honeycomb Sandwich Structures 221.6.3.1. Aircraft 221.6.3.2. Spacecraft 221.6.3.3. Others 23

1.7. OUTLINE OF MAIN CHAPTERS 231.7.1. Introduction 231.7.2. Literature Review 241.7.3. Analytical Modeling 241.7.4. Impact Analysis using LS-DYNA 251.7.5. Preparation of working samples 261.7.6. Experimentation 261.7.7. Results and Discussions 26

2. LITERATURE REVIEW 27


3. ANALYTICAL MODELING 34

3.1. BASIC PROPERTIES OF HONEYCOMB SANDWICH STRUCTURES 343.1.1. Selection of E1 393.1.2. Selection of E2 403.1.3. Selection of E3 413.1.4. Selection of G12 423.1.5. Selection of G13 433.1.6. Selection of G23 443.1.7. Selection of ν12 453.1.8. Selection of ν13 463.1.9. Selection of ν23 473.1.10. Optimum Model for the selection of Elastic Constants 48

3.2. MODELING USING SPRING-MASS SYSTEM 483.2.1. Linear Modeling 493.2.2. Solution of Non-linear impact using Newton’s Forward Difference Method 51

3.2.2.1. Mathematical Formulation 513.2.2.2. Iteration Procedure 53

3.2.3. Solution of Non-linear Oblique Impact on Honeycomb Sandwich Structures 553.2.4. Solution for Oblique Impact 603.2.5. Free Response of Honeycomb Sandwich Structure 60

3.2.5.1. Qualitative Analysis 603.2.5.2. Appropriate Numerical Technique-Perturbation Method 64

3.2.6. Damage Inclusion in Honeycomb 88

4. IMPACT ANALYSIS USING LS-DYNA 90

4.1. MODELING OF HONEYCOMB SANDWICH STRUCTURE IN ANSYS 904.2. IMPACT MODEL IN LS-DYNA 964.3. FREQUENCY RESPONSE OF STRUCTURE 1054.4. MESH INDEPENDENCE, NUMBER OF EQUATIONS AND TIME 1154.5. FINAL MODEL 116

5. PREPARATION OF WORKING SAMPLES 117

5.1. FABRICATION OF THE SPECIMENS 1175.1.1. Construction of the core 1175.1.2. Cutting of Metal Face sheets 1175.1.3. Choice of Adhesive 1185.1.4. Gluing the face sheets with the core 1185.1.5. Polishing of the Honeycomb Sandwich Structures 118

5.2. ASTM STANDARD USED 1195.2.1. Significance and Use 1195.2.2. Scope 120

6. EXPERIMENTATION 121

6.1. FLEXURAL BENDING TEST 1216.1.1. Placing of sheet on the supports 122


6.1.2. Adjustment of weight 1226.1.3. Removal of errors 1236.1.4. Calculation of Flexural Modulus 123

6.2. THREE POINTS BEND TEST 124

7. RESULTS & DISCUSSION 125

7.1. RESULTS OF OPTIMIZATION OF ELASTIC CONSTANTS 1257.2. RESULTS THROUGH LINEAR MODEL 1257.3. RESULTS THROUGH NON-LINEAR IMPACT MODEL 1267.4. RESULTS OF QUALITATIVE ANALYSIS/UV-PLANE 1277.5. RESULTS OF PERTURBATION THEORY 1277.6. RESULTS OF IMPACT ANALYSIS USING LS-DYNA 1287.7. EXPERIMENTAL RESULTS 129

7.7.1. Testing Parameters 1297.7.2. Three point bend test results 129

7.8. FAILURE ANALYSIS APPROACH 132

8. PROVES FOR DEVELOPED THEORIES 135

8.1. PROVE OF DEVELOPED ITERATION METHOD 1358.1.1. Results 1368.1.2. Conclusion of developed iteration technique 138

8.2. PROVE OF DEVELOPED OBLIQUE MODEL 1388.3. PROVE OF NATURAL FREQUENCY 140

8.3.1. Formulation 1408.3.2. Results 1418.3.3. Conclusion 141

9. CONCLUSION OF IMPACT ANALYSIS 142

10. FUTURE RECOMMENDATIONS 144

REFERENCES 145

APPENDIX A-MATLAB PROGRAM 150

AMPLITUDE VS. FREQUENCY 150FAILURE ANALYSIS (ADJUSTING VALUES OF ELASTIC CONSTANTS) 151PHASE PLANE 155


List of FiguresFIG. 1.2 SANDWICH STRUCTURE 16FIG. 1.3 CORE 17FIG. 1.4 FACE SHEET-CORE COMBINATION 18FIG. 1.5 ADHESIVE MATERIAL LAYER SCHEMATIC DIAGRAM 19FIG. 1.6 EXCESSIVE DEFLECTION 20FIG. 1.7 STRENGTH 20FIG. 1.8 PANEL BUCKLING 20FIG. 1.9 SHEAR CRIMPLING 20FIG. 1.10 LOAD COMPRESSION 21FIG. 1.11 DELAMINATION 22FIG. 1.12 AIRCRAFT 22FIG. 1.13 SPACECRAFT 22FIG. 1.14 2-DOF GENERALIZED SPRING-MASS SYSTEM 25FIG. 3.1 CONTRIBUTION OF DIFFERENT RESEARCHERS 35FIG. 3.2 NUMBER OF OPTIONS AVAILABLE 35FIG. 3.3 TREE DIAGRAM 36FIG. 3.4 MEASUREMENT OF ELASTIC PROPERTY (E1) 39FIG. 3.5 MEASUREMENT OF ELASTIC PROPERTY (E2) 40FIG. 3.6 MEASUREMENT OF ELASTIC PROPERTY (E3) 41FIG. 3.7 MEASUREMENT OF ELASTIC PROPERTY (G12) 42FIG. 3.8 MEASUREMENT OF ELASTIC PROPERTY (G13) 43FIG. 3.9 MEASUREMENT OF ELASTIC PROPERTY (G23) 44FIG. 3.10 MEASUREMENT OF ELASTIC PROPERTY (Ν12) 45FIG. 3.11 MEASUREMENT OF ELASTIC PROPERTY (Ν13) 46FIG. 3.12 MEASUREMENT OF ELASTIC PROPERTY (Ν23) 47FIG. 3.13 OPTIMUM MODEL FOR THE SELECTION OF ELASTIC CONSTANTS 48FIG. 3.14 2-DOF MODEL OF HONEYCOMB SANDWICH STRUCTURE 49FIG. 3.15 FREE BODY DIAGRAM 49FIG. 3.16 OBLIQUE IMPACT DIAGRAM 55FIG. 3.17 4-DOF MODEL FOR OBLIQUE IMPACT ON HONEYCOMB SANDWICH STRUCTURE 55FIG. 3.18 FREE BODY DIAGRAM OF INDIVIDUAL MASSES 56FIG. 3.19 REPRESENTATION OF VERTICAL FORCES 56FIG. 3.20 COMBINATION OF FIG. 3.16 AND FIG. 3.19 56FIG. 3.21 RESOLVING X1 FIG. 3.22 RESOLVING X2 57FIG. 3.22 RESULTANT DIAGRAM FOR FRICTIONAL AND NORMAL FORCES 57FIG. 3.23 FAILURE CRITERIA USED 88FIG. 3.24 STRESS-STRAIN GRAPH FOR FACE SHEET 89FIG. 4.1 ANSYS MECHANICAL APDL PRODUCT LAUNCHER 90FIG. 4.2 ANSYS MAIN MENU 91FIG. 4.3 CREATING COORDINATES IN ONE FACE OF HONEYCOMB 91FIG. 4.4 STEPS TO MAKE STRAIGHT LINES 92FIG. 4.5 CREATING STRAIGHT LINE BETWEEN THE KEYPOINTS 92FIG. 4.6 JOINING THE INNER KEYPOINTS 93FIG. 4.7 JOINING THE OUTER KEYPOINTS 93


FIG. 4.8 STEPS TO CREATE AREA BETWEEN THE LINES 94FIG. 4.9 AREA FORMATION 94FIG. 4.10 CORE FORMATION FIG. 4.11 STEPS TO ADD ALL THE AREAS 95FIG. 4.12 ADD AREAS MENU FIG. 4.13 EXTRUDE AREA ALONG NORMAL 95FIG. 4.14 SELECTION OF MATERIAL MODEL 97FIG. 4.15 MATERIAL PROPERTY CARD 97FIG. 4.16 DEFINING THE SECTIONS-SOLID 98FIG. 4.17 DEFINING THE SECTIONS-SHELLS 98FIG. 4.18 ASSIGNING PROPERTIES TO PARTS 99FIG. 4.19 CONTACT AUTOMATIC GENERAL 99FIG. 4.20 CONTACT AUTOMATIC SURFACE TO SURFACE 100FIG. 4.21CONTACT TIED SURFACE TO SURFACE FAILURE 100FIG. 4.22 CONTACT INTERIOR 101FIG. 4.23 CONTACT FORCE TRANSDUCER PENALTY 101FIG. 4.24 SELECTION OF NODES 102FIG. 4.25 SELECTION OF BOUNDARY AND NODES 102FIG. 4.26 INITIAL VELOCITY GENERATION 103FIG. 4.27 CONTROL CARD_SHELL CARD 103FIG. 4.28 CONTROL CARD_SOLID CARD 104FIG. 4.29 TERMINATION CARD 104FIG. 4.30 ELASTIC MODEL_001 105FIG. 4.31 ORTHOTROPIC ELASTIC_002 106FIG. 4.32 SECTION_SOLID 106FIG. 4.33 SECTION_SHELL 107FIG. 4.34 ASSIGNING PROPERTIES TO THE CREATED PARTS 107FIG. 4.35 CONTACT_GENERAL 108FIG. 4.36 CONTACT_SURFACE TO SURFACE 108FIG. 4.37 CONTACT_TIED_SURFACE TO SURFACE_FAILURE 109FIG. 4.38 CONTACT_INTERIOR 109FIG. 4.39 CONTACT FORCE TRANDUCER PENALTY 110FIG. 4.40 SELECTING THE NODES 110FIG. 4.41 ENTERING OPTIONS AFTER SELECTING NODES IN THE BOUNDARY 111FIG. 4.42IMPLICID DYNAMICS 111FIG. 4.43 IMPLICIT EIGENVALUE 112FIG. 4.44 IMPLICIT GENERAL 112FIG. 4.45 IMPLICIT SOLUTIONS 113FIG. 4.46 IMPLICIT SOLVER 113FIG. 4.47 CONTROL_SHELL 114FIG. 4.48 CONTROL_SOLID 114FIG. 4.49 CONTROL_TERMINATOR 115FIG. 5.1 CUTTING OF FACE SHEETS AND REMOVAL OF DENTS 117FIG. 5.3 LARGE HONEYCOMB STRUCTURE 118FIG. 5.4 POLISHED HONEYCOMB SANDWICHED STRUCTURE 119FIG. 5.5 HONEYCOMB SANDWICH SPECIMEN IN ACCORDANCE WITH ASTM 119FIG. 6.1 PLACING THE ALUMINIUM SHEET ON THE SUPPORTS 122FIG. 6.2 PLACING THE WEIGHT 122


FIG. 6.3 REMOVING ERRORS FROM DIAL 123FIG. 6.4 CALCULATING FLEXURAL MODULUS 123FIG. 7.1 DEFLECTION VS. TIME 125FIG. 7.2 FORCE VS. DEFLECTION 126FIG. 7.3 VELOCITY VS. DEFLECTION 127FIG. 7.4 VARIATION OF AMPLITUDE VS. FREQUENCY 127FIG. 7.5 LOAD VS. POSITION PLOT FOR SPECIMEN (S1) 130FIG. 7.6 LOAD VS. POSITION PLOT FOR SPECIMEN (S2) 131FIG. 7.7 LOAD VS. POSITION PLOT FOR SPECIMEN (S3) 131FIG. 7.8 LOAD VS. POSITION PLOT FOR SPECIMEN (S4) 131FIG. 7.9 LOAD VS. POSITION PLOT FOR SPECIMEN (S5) 132FIG. 7.10 DAMAGED MODEL AT 150 M/S 133FIG. 7.11 PLOT BETWEEN THE DAMAGED, LS-DYNA NATURAL AND PERTURBATION METHOD 134FIG. 8.1 PLOT FOR DISTANCE OF INDENTER VS. TIME 137FIG. 8.2 PLOT FOR DISTANCE VS. TIME FOR SANDWICH PLATE 137FIG. 8.3 PLOT FOR CONTACT FORCE VS. TIME 138FIG. 8.4 PLOT BETWEEN X AND T BY USING PERTURBATION METHOD 141


List of Tables

TABLE 3.1 DIFFERENT INPUT PROPERTIES FOR HONEYCOMB SANDWICH STRUCTURES 36TABLE 3.2 DIFFERENT PROPERTIES ILLUSTRATED BY DIFFERENT RESEARCHERS 37TABLE 4.1MEST STRUCTURE AND ITS RESULTS 115TABLE 4.2 FINAL MODEL 116TABLE 7.1 DIMENSION USED FOR THE SPECIMEN 125TABLE 7.2 ELASTIC MODULUS RESULTED 125TABLE 7.3 REFERENCE VALUES 126TABLE 7.4 RESULTS THROUGH NON-LINEAR IMPACT MODEL 126TABLE 7.5 SECTIONAL VIEW OF THE DAMAGED MODEL 128TABLE 7.6 TESTING PARAMETERS 129TABLE 7.7 TESTING RESULTS 130TABLE 7.8 YOUNG MODULUS CALCULATED FROM FLEXURAL TEST 132TABLE 7.9 FREQUENCIES THROUGH PERTURBATION, DAMAGED AND UNDAMAGED LS-DYNA MODEL 133TABLE 8.1 PROPERTIES AND THEIR VALUES USED IN THE PROCESS 136TABLE 8.2 RESULTS SHOWING THE PROVE OF DEVELOPED OBLIQUE MODEL 139TABLE 8.3 INPUT PROPERTIES 140TABLE 9.1 EQUIPMENT PRICE 143


1. Introduction

1.1. Motivation

Following are the major factors in selecting this project:

i. Interest in the structures and design of the aircrafts

ii. Honeycomb sandwich structures are largely being used in the aerospace and other

allied disciplines

iii. Many of the accidents occur due to the collision of the bird on wings of the

aircrafts, so we want to transform this practical study in the theoretical one

iv. Our sound background in mathematics motivated us to choose this project

1.2. Previous work done by the earlier colleagues

Our colleague of Aero-06 “Tanzeel” did static analysis on honeycomb sandwich structure

by computing the result in terms of stresses and verified the buckling phenomena through

ANSYS. Similarly, our colleague of Aero-07 “Wajeeha” did damage analysis on

honeycomb sandwich structure due to low velocity impact by comparing the results of

already developed impact analysis theory with the results in ABAQUS. Both of these

works are unable to work out the damage occurring in the honeycomb sandwich

structures due to impact. Therefore, a new theory along with its verification is required.

1.3. Introduction about our project

A new theory for impact analysis is put forth which is easier, more reliable and can be

used widely i.e. it could be applicable to all ranges of velocity. So a theory was

developed using perturbation method and forward difference method which is

independent of velocity division used until now i.e. low velocity and a high velocity

impact.


1.4. Objectives of the study

Following are the main objectives of our study:

i. To perform an impact analysis on honeycomb sandwich structure

ii. To develop a new analytical model and integrating failure criteria in this model

iii. To use the ANSYS/LSDYNA for the dynamic simulation of the impact

iv. To help the researchers in their research in analytical modeling by improving the

numerical integration technique.

v. To develop oblique impact model for a honeycomb sandwich structure.

vi. To perform low velocity impact analysis on honeycomb sandwich structures.

1.5. Outcomes of the study

i. Analytical and numerical techniques are improved, so that if any researcher

wishes to carry previous approach (time domain) then he can do it more

easily.

ii. 4-DoF model is developed which includes the affect of friction and assumes

that no rotation occurs during the collision.

iii. Also the normal 2-DoF impact was extended to the 4-DoF oblique impact

model.

iv. Perturbation technique is applied to convert free response differential

equations to frequency domain since Laplace and Fourier transform were not

applicable and are unable to convert the equations to frequency domain. The

whole study is shifted in time domain.

v. Damages are included through the use of improved numerical technique to

calculate the stiffness of honeycomb sandwich structures.

vi. Finally formula for natural frequency was used to the damage the model.

1.6. Sandwich Theory

Sandwich construction has been well recognized in the industry for well over 40 years.

Engineers take sandwich construction for the reason as they use I-beam or trusses, i.e., to

increase stiffness, strength and low weight.


A typical sandwich structure consists of three layers; two face sheets and a core

sandwiched between them. Sandwich theory is said to describe the behavior of a beam,

plate or shell. The commonly used sandwich theory is linear sandwich theory which is an

extension of first order beam theory. Linear sandwich theory is important in designing

and analysis of sandwich panels. These sandwich panels are used in air vehicles, space

vehicles and building construction. Sandwich structure is shown in Fig. 1.1.

Fig. 1.1 Sandwich Structure

The advantages of sandwich construction are discussed in the proceeding sections:

1.6.1. Honeycomb Sandwich Structure

Sandwich structure is made up of two face sheets; the top and the bottom one and the

core is made up of hexagonal honeycomb cells, sandwiched between them with the layer

of adhesive material to keep core tied in bond with the face sheets. Thickness of the face

sheet is kept very small relative to their span length to maintain weight constraint. The

adhesive or the bonding material (usually an epoxy) between the core and the skin or face

sheet is used to avoid sliding, delamination and the relative motion, thus avoiding local

damages. The core is made up of regular and repeated pattern of the cells and the cells are

generally hexagonal in structure. The core can be available in different types, forms,

shapes, sizes, dimensions, physical and mechanical properties depending upon the

requirements and the usage. The material selection of the core as well as the face sheets is

dependent on the structural requirements of the application. Constituents of the

honeycomb sandwich structures are discussed below:

1.6.1.1. Core

The core is the most significant part of the honeycomb structure on which whole of the

subject matter is based. The honeycomb core in a sandwich laminate plays the same role


as the web does in I-beam by connecting the load bearing skins i.e. the stiffness directly

increases by increasing the height of web and in case of honeycomb, the thickness of

core.

The sandwich construction is often un-debatable; it is due to the fact of the past incidents

occurred by using either inadequate building methods or the wrong core material for the

engineered load.

Fig. 1.2 Core

The fundamental properties are typically:

Density, i.e. the material must be light in weight.

Shear modulus and shear strength for the core to carry the bulk of shear loads

hence high strength and stiffness values are very important to structural

performance.

Compression stiffness and strength; the core materials must be capable to carry

the perpendicular loads on laminate face sheets.

Thermal properties: as often the core has to act as an insulator and heat transfer

must be kept to a minimum.

Although, honeycomb sandwich structure is light in weight, but still, they need local

reinforcement in areas which are load bending and in fixation spots. The open ends of

sandwich panels need reinforcement and protection in order to prevent from impact

damages and moisture access.

1.6.1.2. Face sheets

The sandwich structure consists of thin and stiff face-sheets made from either metal or

composite separated by light weight core. These face sheets are adhesively bonded to the 17

Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid

core (honeycomb) through a secondary bonding process, where the components of the

core do not flow. The metals used for the facesheets are mostly the alloys of aluminum

and the composites usually used are carbon-epoxy and glass-epoxy.

Fig. 1.3 Face Sheet-Core Combination

1.6.1.3. Adhesive Material

In sandwich structure, facesheets and the honeycombs are bonded as two distinct solid-

phases through a secondary bond. Generally, the honeycomb core is bonded to the face-

sheet by either of the two methods explained as under:

An adhesive layer is placed on to the top and bottom of the surfaces of the core,

upon which the prepregs are placed. The adhesive is usually a resin in this case.

The complete assembly is placed in an autoclave to cure the resin. During this

process of curing, the resin from the film plasticizes and thus creates a bond

between the prepregs and the honeycomb wall.

The prepregs that are uncured are then cured by placing them on to the top and

bottom surface of the honeycomb and the resulting assembly is placed in

autoclave for curing. During this process of curing, the resin from the prepregs

gradually flows and a bond is created between the prepregs and the honeycomb

walls.

For the existence of a reliable bond at the interface, the resin should flow from the

adhesive sheet and should create a symmetric fillet on the honeycomb surface

surrounding the interface. In addition to that, the edges of the cell-wall of the honeycomb

should be straight and sharp. The load bearing capacity is massively dependent on the

quality of the adhesive material used between the core and the face sheet. Delamination 18


Face Sheets

Core

of the facesheets of the honeycomb may result in catastrophic failure of the structure.

Thus, it concludes that type, nature and the quality of the bonding between the core and

the face-sheet is important design criterion for sandwich construction.

Fig. 1.4 Adhesive Material Layer Schematic Diagram

Also, due to change in modulus between the core and the face-sheets i.e. honeycomb wall

material, a large amount of shear stress is generated at interface, where the structure is

subjected to bending load. Whenever, a flaw is introduced at the interface, shear stress

may peel off the face-sheets from the honeycomb surface. This flaw is usually initiated

by an impacted force, especially on the exterior structure.

1.6.2. Modes of Deformation of Sandwich Structure

The initiation of the various failure modes depends on the material properties of the

constituents; the facings and core, geometric dimensions and type of loading. The type of

loading condition determines the state of stress throughout the sandwich structure, which

controls the location and modes of failure. For sandwich specimens loaded under bending

moment, shear and axial loading, the failure modes investigated are face sheets

compressive failure, adhesive bond failure, indentation failure, core failure and face

wrinkling. The sandwich beams under quasi static punching loads can fail in several

modes; face yield, face wrinkling, core shear, the bottom face fracture, and interfacial

facture between the core and the faces. Moreover, the parameters like face thickness, cell

size of foam material on the failure and deformation modes can also effect the failure

modes.


Adhesive Material

CoreFace Sheets

During the construction of the sandwich panels, designers must ensure that all the

potential failure modes must be considered in their analysis. Summarized form of the key

failure modes are as follows:

1.6.2.1. Strength

The skin and the core material should be able to withstand the tensile, shear and

compressive stresses induced by the load.

The adhesive material used between the core and the face sheets must be capable of

transferring the compressive stresses between core and the face sheet (Fig. 1.5).

1.1.1.

1.1.1.

Stiffness

The sandwich panel should have sufficient bending and shear stiffness, so that it can

prevent excessive deflection (Fig. 1.6).

1.6.2.2. Panel Buckling

There should be sufficient core thickness and shear modulus to avoid the buckling of the

panel under end compression load (Fig. 1.7).


Fig. 1.6 Strength Fig. 1.5 Excessive Deflection

Fig. 1.7 Panel Buckling Fig. 1.8 Shear Crimpling

1.6.2.3. Shear Crimpling

The core thickness and shear modulus must be adequate to prevent the core from

permanently failing in shear under end compression load (Fig. 1.8).

1.6.2.4. Skin Wrinkling

The compression modulus of the facing skins and the compression strength of the core

must both be high enough to prevent a skin wrinkling failure (Fig. 1.9).

1.6.2.5.

1.6.2.5.Intra-cell Buckling

For a given face sheet material, the core cell size must be small enough to prevent intra-

cell buckling (Fig. 1.10).

1.6.2.6. Local Compression

The core compressive strength must be adequate enough to resist local loads on the panel

surface Fig. 1.11).

Fig. 1.9 Load Compression

1.6.2.7. Delamination

The delamination of the face-sheet as shown in the fig. 1.12 can occur by propagation of

the crack in one or more of the following spaces:


Fig. 1.9 Skin Wrinkling

Fig. 1.10 Intra-cell Buckling

Interface between the resin and fiber in the facesheet i.e. prepregs

Resin layers (between the prepregs and the honeycomb wall) formed during

bonding.

Interface between the resin (from the prepregs/film) and the honeycomb cell-wall

Cell wall (failure in the core)

Fig. 1.10 Delamination

1.6.3. Applications of Honeycomb Sandwich Structures

1.6.3.1. Aircraft

In aircraft, honeycomb structures are used as high static and dynamic load bearing

members. Their major benefit is the light weight and they show good dynamic properties

as well. These are used typically in the areas like vertical and tail, leading and trailing

edges, fuselage sections. In addition to the mechanical properties, these components also

provide good thermal properties, and are used in aircraft as thermal resistant component

as well.


Fig. 1.11 Aircraft

Fig. 1.12 Spacecraft

1.6.3.2. Spacecraft

During space flight, the satellite or the object faces many dynamic, static and thermal

loads of high power. These can damage the structure and its components such as

propulsion, navigation and control units. The solid rocket booster launch vehicle has

several metal based components that require a thermal protective system be applied to the

exterior surface to ensure its structural integrity and to protect the interior from

aerodynamic heating effects. The sandwich structures are being qualified for this

application with a high strength and low thermal conductivity. Major concerns in this

application are:

Dynamic structure for shipping, launch and operation

Detailed component stress and margin calculations

Mechanism analysis for deployment of solar panels and reflector

The spacecraft application is shown in the fig. 1.12.

1.6.3.3. Others

Other applications of honeycomb sandwich structure are listed below:

Automobile structures

Gliders

Rocket Sub-structure

Submarines

Wind tunnels

Trains

Snowboards and furniture

Heating, ventilation, air conditioning equipment and devices

Energy absorption protective structures

Electric shielding enclosures

Acoustic attenuation

Wind turbine blades


1.7. Outline of main chapters

1.7.1. Introduction

A brief introduction about the sandwich theory and different modes of failures is

presented in this chapter. Our motivation in choosing this project is also presented with

the addition of the work done by the past colleagues in this field. The chief objectives of

our study and their possible outcomes have also been outlined in this chapter. Also a brief

introduction of every chapter has also been put forward in this report to make the reader

aware of the study presented in every chapter.

1.7.2. Literature Review

A detailed literature study has been provided in this chapter regarding impact analysis of

honeycomb sandwich structure. A number of researchers used spring-mass and energy-

balance model to get the analytical equations but no one has ever solved those equations

because the resulting equations are non-linear and their analytical solution is impossible.

Thus, many of the researchers either made assumptions to linearize the system of

equations or have moved to the numerical methods to solve the system of equations.

Finite Element Model has been developed by a number of researchers particularly for this

problem using commercially available dynamic software and many of the researchers

have compared the FE results with the experimental ones. Since, less theoretical and

more experimental work has been carried out in this field. So we have tried to describe

the contribution of different researchers in this particular field.

1.7.3. Analytical Modeling

Different analytical approaches are used to model honeycomb sandwich structures

according to the level of complexity and the given conditions. Analytical modeling for

honeycomb core is a complex job as it is done discretely for every case as per the specific

conditions and the end requirements. There is no way to generalize any of the present

models. The degree of complexity, proportional with degree of accuracy required, is also

a highly variable parameter while getting started with the analytical modeling.


Spring-mass systems are simple to work on and provide accurate solutions for the

impacts which are often encountered during the tests on small specimens. The most

complete model consists of one spring representing the linear stiffness of the structure,

another spring for the nonlinear membrane stiffness, a mass representing the effective

mass of the structure, the nonlinear contact stiffness and the mass of the projectile. If the

effect of shear deformation is negligible, the spring constant taken is the combined spring

constant of bending and shear stiffness. It can be one, 2or 3-DoF (degree of freedom)

depending upon the level of accuracy to be achieved and level of complexity of the

problem increases with the accuracy. A simplified 2-DoFs is shown in Fig. 1.15.

Fig. 1.13 2-DOF Generalized spring-Mass System

A damper can also be included in it, if the energy dissipation is to be modeled as well.

1.7.4. Impact Analysis using LS-DYNA

Numerical simulation is usually done to visualize the problem practically with the help of

using Dynamic Finite Element modeling software such as ABACUS and LSDYNA

which are used commercially these days and are user friendly. However, CODAC,

LUSAS, NASTRAN and other software are also in a use for the simulation of the

problem but their uses vary from problem to problem and from the type of analysis.

First of all the problem is visualized how to occur. Then on the basis of that imagination a

model is developed by using different element types and the material properties.

Different element types are used such as shell and solid, and their use is also dependent

on the type of structure. After careful selection of the element type, the material which

can be isotropic, orthotropic or anisotropic is selected according to the type of analysis to


be done. Finite element model is then developed using the inherent properties of the

software.

Thus a dynamic analysis is carried out to analyze the structure for damage analysis

through frequency or stress response. The results obtained from the numerical simulation

are then compared with the analytical or experimental studies to check the validity of the

Finite Element Model.

1.7.5. Preparation of working samples

The sandwich was fabricated according to the ASTM C393 standards. Each aluminium

sheet and the Nomex core were sized according to these standards. The core was first

glued with the face skins by using a suitable epoxy. The specimens were then firmly

pressed so that the face skins make a strong bond with the honeycomb core. The samples

were then polished up to remove the scratches present on the surface of the sheets. The

prepared samples were then used in our subsequent analyses.

1.7.6. Experimentation

Experimental investigation was done in order to compare the theoretical and numerical

results with the experimental ones. And thus the compared results helped us to verify our

analysis. Two different types of experimental tests were conducted such as flexure

bending test and three point bend test. Flexural bend test was conducted on the aluminum

sheet to find out its Young’s modulus so that the measured value of modulus can be used

in our analysis. Compression test was carried out on the honeycomb sandwich structure

to have a plot of force vs. displacement.

1.7.7. Results and Discussions

The results of every method and study done in this project have been compiled in this

chapter resulting into a number of graphs. The perturbation method, forward difference

method, linear and non-linear modeling, LS-DYNA and experimentation results are listed

as well as the comparison of analytical, numerical and experimental study has also been

presented in the form of a graph.


2. Literature Review

Honeycomb sandwich panels and laminated composite structures are being used at a

large scale in aerospace industry. Because of their high stiffness, strength and lightweight

they are of prime importance in aerospace and aircraft industry. Impacts can be occurred

due to a variety of causes. e.g., debris may be propelled at high velocities from runway at

aircrafts landing and takeoffs. Other examples include dropping of the tool on the

structure during maintenance collisions and even by striking of a bird during the flight.

Visually it seems little damage is occurs on the structure but in reality the structure

breaks to a large extent due to the failure of face-sheets and core crushing. This destroys

the structure completely. Reduction of structural strength and stiffness likely to occur and

thus behavior of these structures from strength and failure point of view is of major

importance these days.

Composite structures under impact load are of primary importance these days in aviation

and aerospace industry. A lot of literature review has been done by the researchers in this

field. A thorough literature study has been done by Abrate et. al [1-3]. Honeycomb

sandwich structures are being made of different materials depending upon the

application. Nomex and Aluminum are being used on large scale in aerospace industry.

According to Aberate et. al [3], “a first step towards understanding the effect of impacts

is to develop a model for predicting the contact force history and the overall response of

the structure and thus it involves the modeling of the motion of projectile, the dynamics

of the structure and the local indentation of the structure of the projectile”.

An energy-balance model proposed by Abrate et. al [3] assumes that the quasi-static

behavior of the structure. i.e., when the structure reaches its maximum point of

deflection, the projectile velocity reaches zero and all the initial kinetic energy is used to

deform the structure. A numerical investigation using ABACUS was presented by C. C.

foo et. al [4], in which the impact force and the deflection-time histories were determined

by incorporating an impulse-momentum equation into the energy-balance model and the

results showed that the energy absorbed during impact is independent of the core density.


The strength characteristics of aluminum honeycomb sandwich panels were studied by

Jeom Kee Paika et. al [5] both theoretically and experimentally. From three-point

bending test it was observed that starting point of the plasticity can be delayed by using

thick honeycomb core cell, results in increase of ultimate strength and also the instability

effects in the structure can be reduced. Also delaminating could occur when the height of

the core become large but core height is not an influential factor on the crushing behavior

of honeycomb core. A low-velocity impact response was investigated by Md. Akil

Hazizan et. al [6] by drop-weight impact tests using an instrumented falling-weight

impactor on two different glass fiber/epoxy aluminum honeycomb structures. It was

found that indentation characteristics can be analyzed using Meyer’s indentation law. The

energy-balance model was used which accounts for energy absorption in bending, shear

and contact effects and was found to have fair agreement between the analytical

experimental study particularly at low energies. Energy breakdown was also identified

during impact by using energy-balance model and it was shown that incident energy

partition strongly depends on the geometry of the impacting projectile. Inés Ivañez et. al

[7] developed a 3-D FE model and Hou failure criteria were used to predict the failure of

facesheets for the low velocity impact. The comparison between the numerical and

experimental results was satisfactory and thus damage evolution study revealed that the

collapse of the foam core under the impact region favored the failure of the upper face

sheet because of its high deflection and the resulting great curvature. A similar study was

done by T. Besant et. al [8], finite element procedure of sandwich panels for predicting

the behavior under low velocity impact consisting of brittle composite skins supported by

a ductile core. In elastic-plastic portion of analysis of metal cores, a non-standard

approach was used and the strategy for modeling was presented.

Analysis revealed that honeycomb is a good absorber of energy and absorbs energy by

the combination of local crush under the impactor and through thickness shear yielding.

For the prediction of the crushing behavior of the honeycomb sandwich structures, a

finite element methodology was developed by Chawla et. al [9]. The crushing strength of

the honeycomb depends on the geometry and the material property. The FE mesh in the

simulations also affects the results substantially so an optimum mesh should be


established until the convergence is obtained. The adhesive bonding between cell faces

were simulated by two different approaches, using glued nodes and using merged nodes.

Dynamic analysis was carried out using PAM-CRASH, an explicit FE code and the

results were verified against the data obtained from the experiments. The methodology

presented here can also be used to study the effect of other parameters like overall size,

cell size, foil thickness, thickness of honeycomb, number of cells and the material

properties. Meo et. al [10] carried out the study and discussed the results from the

experimental and numerical simulations of low-velocity impact and penetration damage

on an aircraft sandwich panel by a solid, round-shaped impactor. The work was focused

on the recent progress on the material modeling and numerical simulation of low-velocity

impact response onto a composite aircraft sandwich panel. FE analysis was carried out

using LS-DYNA3D finite element code. Numerical and experimental results were in

good agreement with each other; particularly numerical simulation was able to predict the

damage and impact energy absorbed by the structure.

The analysis of static and low-velocity impact response of two topologies of aluminum

honeycomb sandwich structures with different cell sizes has been studied by V. Crupi et.

al [11]. Various collapse modes produced by the static bending tests were performed

using servo-hydraulic load machine for panels with the same nominal size, depending on

the support span distance and on the honeycomb cell size. Low-velocity impact tests were

also carried out by means of drop test machine on the structures and energy-balance

model was applied in order to investigate their impact behavior. He assumed a perfect

bond between the faces and the core and eliminated the possibility of delamination for the

development of a theoretical model.

Effect of the amount of adhesive on the bending fatigue strength of adhesively bonded

aluminum honeycomb sandwich beams was analyzed by Yi-Ming Jen et. al [12]. He

experimentally proved that fatigue strength increases as the amount of adhesive increases.

It was also observed from fatigue test that de-bonding at the interface between the

honeycomb core and the face sheet is the main cause of fatigue failure. The predicted

failure locations using the three interfacial parameters were also examined by comparing

the results from fatigue tests. Among the three, the combined interfacial peeling and


shear stress parameter was recommended in fatigue design as it provides good fatigue life

correlations and predicts the correct locations of failure initiation simultaneously.

A low-velocity impact response was predicted using a displacement-based, plate bending,

FE algorithm [13]. They predicted out that fifth-order Hermatian interpolation allows 3-D

equilibrium integration for measuring the transverse stress to be carried out symbolically

through interpolating functions and by using elasto-plastic foundation. Nomex

honeycomb core was modeled and contact load was simulated using Hertzian pressure

distribution. Damage prediction by failure criteria and damage progression via stiffness

reduction were also analyzed. It was demonstrated by [14] the possibility of representing

the Nomex honeycomb core by a grid of nonlinear springs and pointed out both the

structural behavior of the honeycomb and the influence of core-skin boundary conditions.

This discrete approach accurately predicted the static indentation on honeycomb core

alone and the indentation on sandwich structure with metal skins supported on rigid flat

support. They also pointed out that this approach is not valid for sharp projectiles on thin

skins.

Nettles et. al [15] analyzed that a static test method for modeling low-velocity foreign

object impact events to composites would prove to be very beneficial to researchers since

much more data can be obtained from a static test than from an impact test. A series of

static indentation and low-velocity impact tests were carried out and compared to check

whether static tests are better or not than impact tests. Results indicate that static

indentation can be used to represent a low-velocity impact event.

The quasi-static indentation and impact response of very thick sandwich panels is

described in [27] and studied drop weight impact, quasi-static indentation, and quasi-

static core-crushing characteristics of the different core panel configurations. Damages

were evaluated using conventional methods including cross-sectioning and visual

inspection, and dent depth was measured by means of an automated high-precision depth

gage. The indentation profile was monitored by means of the digital image correlation

technique during the quasi-static tests to gain insight in the response of these complex

structures. Paolo Feraboli et. al [28] studied the structural properties for impact energy


values below and above the damage threshold. A modified approach to the classic spring-

mass model, which employs the notions of damaged stiffness and dissipated energy, leads

to the derivation of approximate formulas that describe the peak force-energy curve. A

novel method to assess the residual performance of the damaged plate was also

developed, using nondestructive impact testing, the results from which bear a striking

resemblance with the curves obtained by compression after impact.

J. Gustin et. al [29] investigated impact and compression after impact properties of plain

weave carbon fibers and sandwich composites. Impact tests were conducted on different

sample to obtain information about absorbed energy and maximum impact force. The

impact and compression after impact data provided valuable information to allow for

comparisons between different samples. In conclusion, a two-degrees-of-freedom spring-

mass model was compared with the experimental results.

Finite element modeling is one of the most popular and cost-effective approach involved

in the study of sandwich structures. To attain efficiency in numerical analysis, the core in

sandwich structures, which has a large number of cells, is usually replaced with an

equivalent continuum model. The sandwich panels are analyzed in terms of their

effective properties and stiffness’s. Numerous numbers of experimental and analytical

techniques have been proposed to predict the effective continuum properties of the core

and the face sheets in terms of its geometric and material characteristics. However,

theoretical formulation of the effective elastic constants for the core could be tedious or

almost impossible if the sandwich construction is too complicated. Even if it is possible,

the mathematical derivations for one type of sandwich core might not be applicable to

other types. An equivalent continuum model may seem a convenient way to represent the

real core geometrically, but errors have also been attributed to the continuum model when

it is used to model damage in impact problems. One possible reason is that it may be very

difficult to simulate exact damage or crack propagation since the honeycomb core is

made up of discrete cells. The onset of damage progression and failure in honeycomb

core may be sensitive to detailed local damage distribution along the cells. This limitation

can be overcome by adopting discrete element modeling approaches, so that more

realistic distributions of stresses and strains can be obtained in the detailed core structure.


A great deal of research work from the past many years has been carried out in

experimental and numerical studies, but relatively fewer analytical solutions have been

proposed for sandwich structures because of the complex interaction between the

composite face sheet and core during deformation and failure whenever an impact occurs

on the honeycomb sandwich panel. Such solutions highlight important impact

parameters, and provide benchmark solutions for more refined finite element analysis.

The spring-mass and energy-balance models are two popular mathematical models used

to study the impact dynamics of foreign objects on composite structures.

In the spring-mass model, there is a combination of bending, shear, membrane and

contact springs to represent the transverse load-deformation behavior. The complete force

history is then predicted by solving the equations of motion. Most analytical models

assume elastic behavior and they are unable to model damage growth. In addition, they

neglect core crushing and large face sheet deflection. Some researchers also used

homogeneous and isotropic properties to make their work easy. In addition many

researchers have neglect the membrane and core stiffness’s by simply assuming the mass

of honeycomb to be small as compared to the impactor or indenter mass. Using such

assumptions make the analysis very easy because the resulting non-linear equations are

transformed to the simple linear equations.

Thus, for analytical models to be successful in predicting the impact response of

sandwich panels, large face sheet deflections and core crushing have to be considered as

well, researchers have derived approximate solutions for the dashpot and spring

resistances from the static load-indentation response using the principle of minimum

potential energy, and then adjusted these properties with the dynamic material properties

of the facesheet and core. They incorporated a constant force dashpot in the spring-mass

model to represent the dynamic crushing resistance of the core. A single DOF model

including a damper can also be used as a mechanism to account for material damage on

sandwich structures subjected to low-velocity impact. Also many researchers have used

the quasi-static analysis for the study of low velocity impact on aluminum or Nomex

honeycomb sandwich structure to simplify their problem however impact is a dynamic

analysis.


The nonlinear approaches have also been studied in [15-25] in detail by the researchers.

The two approaches described above are being used today and in the past for the impact

or damage analysis of aluminum or Nomex honeycomb sandwich structure. Another

technique may be used to carry out the research may be as follows. It starts with the

derivation of equations of motions of SDOF system which will form a characteristic

equation in the form of natural frequencies. The frequencies can be calculated of an

undisturbed system and then the natural frequencies of the system can be calculated after

the impact. The comparison can be made between tge initial and final frequencies of the

system to predict the damages caused by the impact.


3. Analytical Modeling

Majority of researchers have used different models for the analytical modeling of

damages caused due to low/high velocity impact on honeycomb sandwich structure. The

two different types of models are energy-balance model and the spring-mass model.

Energy-balance model is used to find the equations in terms of energy by using the

principle of conservation of energy or the principle of conservation of momentum on the

collision of two different objects. The spring-mass apparently being simple but the

resultant equations are tough from the computation point of view. Also two types of

boundary conditions exist for honeycomb sandwiched structures one is fully backed

system and other is simply supported system. General model is developed where both of

these boundary conditions could be used.

The analytical modeling of impact problem is carried iteratively. But before starting the

analytical modeling, some basic properties of honeycomb structure need to be explored.

3.1. Basic Properties of Honeycomb Sandwich structures

Following are the properties of honeycomb sandwich structures:

Young modulus in x, y and z direction.

Shear modulus in x, y and z direction.

Poisson ratio in x, y and z direction.

Figure 3.1 shows the contribution of different researchers. Lorna J Gibson and Michael F.

Ashby worked on all of the 9 important constants. Total 33 formulas are available; since

E3 is same for some authors so total 29 formulas are analyzed in this study.


Fig. 3.14 Contribution of different researchers

Now to select a particular model, the number of options available is:

Fig. 3.15 Number of options available

The more suitable results are selected among above using following approach. Select one

block for each one of the elastic constant. Inside each block different numbers referring

the available options and finally possible combinations are selected. This method is same

as making tree diagram:


Abd al Sayed

E1

E2

E3

ν12

Master and Evans

E1

E2

E3

G12

ν12

E nast

E1

E2

E3

G12

G13

G23

ν12

ν23

ν13

Shi

G13

G23

Qunli

E3

G13

G23

Gibson and Ashby

E1

E2

E3

G12

G13

G23

ν12

ν23

ν13

Ashby

E3

select E1 from four available formulas

select E2 from four available formulas

select E3 from two available formulas

Select G12 from three available formulas

Select G13 from four available formulas

Select G23 from four available formulas

Select ν12 from four available formulas

Select ν13 from two available formulas

Select ν23 from two available formulas

Fig. 3.16 Tree diagram

Thus multiplying all these values we have 24576 possible combinations. Therefore, study

was performed for the selection of best possible elastic constants. Here cell size, height,

thickness of core wall is varied and the results are compared with the experimental results

taken from excel data sheet and other researcher’s experiments. Seven different

compatibility tests were performed, each one with different input properties. The input

properties table is shown below:

Table 3.1 Different input properties for honeycomb sandwich structures

thickness/mm 0.064 0.036 0.102 0.038 0.025 0.064 0.3cell size/mm 19 6.3 13 4.8 3.2 6.3 13l1/mm 10.97 3.6373 7.5056 2.7713 1.8475 3.6373 7.50555l2/mm 11.226 3.7813 7.9136 2.9233 1.9475 3.8933 8.70555Es / (Nm-2) 7E+10 7E+10 7E+10 7E+10 7E+10 7E+10 9E+08Gs/ ( Nm-2) 3E+10 3E+10 3E+10 3E+10 3E+10 3E+10 3.2E+08vs 0.3 0.3 0.3 0.3 0.3 0.3 0.4

Input properties for Honeycomb

These values are substituted into the formulas, we get


4 for E1

4 for E2

2 for E3

3 for G12

4 for G13

4 for G23

4 for ν12

2 for ν13

2 for ν23

Table 3.2 Different Properties illustrated by different Researchers

M and e e.nast abd qunli ashby shi gibson and ashby ExperimentalE1 31643.79 46243.48 0 0 0 0 29409.10198E2 31645.32 76519.11 0 0 0 0 29409.10198E3 6.2E+08 6.2E+08 6.2E+08 6.2E+08 6.2E+08 619789474.1 454241425.5G12 7911.193 8195.226 0 0 0 0 7288.342666 70000000G23 0 1.8E+08 0 90947368 0 90947368.46 88819404.3 120000000G13 0 2.43E+08 0 1.36E+08 0 136421052.7 88819404.3 120000000v12 1 0.751808 1 0 0 0 1v23 0 1.89E-05 0 0 0 0 2.13653E-05v13 0 1.71E-05 0 0 0 0 2.13653E-05

M and e e.nast abd qunli ashby shi gibson and ashby ExperimentalE1 154385.9 225765.3 0 0 0 0 136948.9535E2 154495.6 373574 0 0 0 0 136948.9535E3 1.05E+09 1.05E+09 1.05E+09 1.05E+09 1.05E+09 1051428572 758540993.5G12 38622.01 40009.92 0 0 0 0 33939.52325 170000000G23 0 3.05E+08 0 1.54E+08 0 154285714.3 148320156.3 270000000G13 0 4.11E+08 0 2.31E+08 0 231428571.5 148320156.3 270000000v12 1 0.751808 1 0 0 0 1v23 0 5.44E-05 0 0 0 0 5.95791E-05v13 0 4.92E-05 0 0 0 0 5.95791E-05

M and e e.nast abd qunli ashby shi gibson and ashby ExperimentalE1 399589.1 584442.6 0 0 0 0 339830.8595E2 399945.4 967077.5 0 0 0 0 339830.8595E3 1.44E+09 1.44E+09 1.44E+09 1.44E+09 1.44E+09 1443692308 1026944771G12 99977.07 103574.4 0 0 0 0 84218.95213 200000000G23 0 4.18E+08 0 2.12E+08 0 211846153.8 200802079.7 310000000G13 0 5.65E+08 0 3.18E+08 0 317769230.6 200802079.7 310000000v12 1 0.751808 1 0 0 0 1v23 0.000103 0 0 0 0 0.000109202v13 9.28E-05 0 0 0 0 0.000109202


Test 1

Test 2

Test 3

M and e e.nast abd qunli ashby shi gibson and ashby ExperimentalE1 409808.8 600341.7 0 0 0 0 348590.7789E2 410825.5 993385.7 0 0 0 0 348590.7789E3 1.46E+09 1.46E+09 1.46E+09 1.46E+09 1.46E+09 1456666667 1035693972G12 102696.7 106392 0 0 0 0 86389.88867 225000000G23 0 4.22E+08 0 2.14E+08 0 213750000.2 202512841.2 360000000G13 0 5.7E+08 0 3.21E+08 0 320625000.3 202512841.2 360000000v12 1 0.751808 1 0 0 0 1v23 0 0.000104 0 0 0 0 0.00011107v13 0 9.44E-05 0 0 0 0 0.00011107

M and e e.nast abd qunli ashby shi gibson and ashby ExperimentalE1 393414.2 576954.4 0 0 0 0 335699.4254E2 394821.1 954686.8 0 0 0 0 335699.4254E3 1.44E+09 1.44E+09 1.44E+09 1.44E+09 1.44E+09 1437500001 1022766158G12 98696.24 102247.3 0 0 0 0 83195.07498 230000000G23 0 4.17E+08 0 2.11E+08 0 210937500 199985020.8 370000000G13 0 5.63E+08 0 3.16E+08 0 316406250.1 199985020.8 370000000v12 1 0.751808 1 0 0 0 1v23 0 0.000102 0 0 0 0 0.000108315v13 0 9.2E-05 0 0 0 0 0.000108315

M and e e.nast abd qunli ashby shi gibson and ashby ExperimentalE1 864604.6 1268498 0 0 0 0 704954.3562E2 868057.5 2098984 0 0 0 0 704954.3562E3 1.87E+09 1.87E+09 1.87E+09 1.87E+09 1.44E+09 1437500001 1309724092G12 216980.7 224801.9 0 0 0 0 174706.0796 270000000G23 0 5.42E+08 0 2.74E+08 0 213750000.2 256094902.8 430000000G13 0 7.31E+08 0 4.11E+08 0 320625000.3 256094902.8 430000000v12 1 0.751808 1 0 0 0 1v23 0 0.000172 0 0 0 0 0.000177621v13 0 0.000156 0 0 0 0 0.000177621

M and e e.nast abd qunli ashby shi gibson and ashby ExperimentalE1 129781.8 205752.8 0 0 0 0 132186.1442 448000E2 132725.4 340459.2 0 0 0 0 132186.1442 443000E3 55384615 55384615 55384615 55384615 55384615 55384615.4 41538461.55 120000000G12 33154.93 36463.31 0 0 0 0 32759.17488G23 0 14586895 0 7384615 0 7384615.386 7380135.256G13 0 19692308 0 11076923 0 11076923.08 7380135.256v12 1 0.751808 1 0 0 0 1v23 0 0.001141 0 0 0 0 0.001272904v13 0 0.001032 0 0 0 0 0.001272904


Test 4

Test 5

Test 6

Test 7

This is very crude form. To analyze data Microsoft Excel is used. 56 graphs are plotted.

Then for optimum results theoretical formula closest to experimental formula is selected:

3.1.1. Selection of E1

M and e e.nast abd qunli ashby shi gibson and ashby

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

Series1


0

50000

100000

150000

200000

250000

Series1


0

100000

200000

300000

400000

500000

600000

700000

Series1


0

100000

200000

300000

400000

500000

600000

700000

Series1

M an

d ee.n

ast

abd

qunli

ashby sh

i

gibso

n and as

hby

Exper

imen

tal0

50000

100000

150000

200000

250000

300000

350000

400000

450000

500000

Series1


0

100000

200000

300000

400000

500000

600000

700000

Series1


0

200000

400000

600000

800000

1000000

1200000

1400000

Series1

Fig. 3.17 Measurement of Elastic Property (E1)


Test 1 Test 2

Test 3Test 4

Test 5 Test 6

Test 7

From these graphs E.nast formula for E1 is closest to experimental.



0

10000

20000

30000

40000

50000

60000

70000

80000

90000

Series1

M and e e.nast abd qunli ashby shi gibson and

ashby

0

50000

100000

150000

200000

250000

300000

350000

400000

Series1


0

200000

400000

600000

800000

1000000

1200000

Series1


0

200000

400000

600000

800000

1000000

1200000

Series1

M and e

e.nast abd qunli ashby shi gibson and

ashby

0

200000

400000

600000

800000

1000000

1200000

Series1

M and e


ashby

0

500000

1000000

1500000

2000000

2500000

Series1

M an

d e

e.nas

tab

dqunli

ashby

shi

gibso

n and as

hby

Exper

imen

tal0

50000

100000

150000

200000

250000

300000

350000

400000

450000

500000

Series1

Fig. 3.18 Measurement of elastic Property (E2)

E. Nast formula for E2 is more accurate than the rest.


Test 1 Test 2

Test 3 Test 4

Test 5Test 6

Test 7



0

100000000

200000000

300000000

400000000

500000000

600000000

700000000

Series1


0

200000000

400000000

600000000

800000000

1000000000

1200000000

Series1


0

200000000

400000000

600000000

800000000

1000000000

1200000000

1400000000

1600000000

Series1


0

200000000

400000000

600000000

800000000

1000000000

1200000000

1400000000

1600000000

Series1


0

200000000

400000000

600000000

800000000

1000000000

1200000000

1400000000

1600000000

Series1


0

200000000

400000000

600000000

800000000

1000000000

1200000000

1400000000

1600000000

1800000000

2000000000

Series1

M and e

e.nast ab

dqunli

ashby sh

i

gibso

n and as

hby

Experi

mental

0

20000000

40000000

60000000

80000000

100000000

120000000

140000000

Series1

Fig. 3.19 Measurement of Elastic Property (E3)

Gibson and Ashby give same result. Let’s select E. Nast formula for E3.


Test 1 Test 2

Test 3 Test 4

Test 5 Test 6

Test 7

3.1.4. Selection of G12

M an

d e

e.nas

tab

dqunli

ashby

shi

gibso

n and as

hby

Exper

imen

tal0

10000000

20000000

30000000

40000000

50000000

60000000

70000000

80000000

Series1

M an

d e

e.nas

tab

dqunli

ashby

shi

gibso

n and as

hby

Exper

imen

tal0

20000000400000006000000080000000

100000000120000000140000000160000000180000000

Series1

M a

nd e

e.nas

tab

dqunli

ashby

shi

gibso

n and a

shby

Exper

imen

tal0

50000000

100000000

150000000

200000000

250000000

Series1

M an

d ee.n

ast

abd

qunli

ashby sh

i

gibso

n and as

hby

Exper

imen

tal0

50000000100000000150000000200000000250000000300000000

Series1

M an

d e

e.nas

tab

dqunli

ashby

shi

gibso

n and as

hby

Exper

imen

tal0

50000000

100000000

150000000

200000000

250000000

Series1

M an

d e

e.nas

tab

dqunli

ashby

shi

gibso

n and as

hby

Exper

imen

tal0

50000000

100000000

150000000

200000000

250000000

Series1


0

5000

10000

15000

20000

25000

30000

35000

40000

Series1

Fig. 3.20 Measurement of Elastic Property (G12)

All formulas give extremely poor results so experimental value is preferred.


Test 1 Test 2

Test 3 Test 4

Test 5Test 6

Test 7


M an

d ee.n

ast

abd

qunli

ashby sh

i

gibso

n and as

hby

Exper

imen

tal0

50000000

100000000

150000000

200000000

250000000

300000000

Series1

M an

d e

e.nas

tab

dqunli

ashby

shi

gibso

n and as

hby

Exper

imen

tal0

50000000

100000000

150000000

200000000

250000000

300000000

350000000

400000000

450000000

Series1

M a

nd e

e.nas

tab

dqunli

ashby

shi

gibso

n and a

shby

Exper

imen

tal0

100000000

200000000

300000000

400000000

500000000

600000000

Series1

M an

d e

e.nas

tab

dqunli

ashby

shi

gibso

n and as

hby

Exper

imen

tal0

100000000

200000000

300000000

400000000

500000000

600000000

Series1

M an

d e

e.nas

tab

dqunli

ashby

shi

gibso

n and as

hby

Exper

imen

tal0

100000000

200000000

300000000

400000000

500000000

600000000

700000000

800000000

Series1

M an

d e

e.nas

tab

dqunli

ashby

shi

gibso

n and as

hby

Exper

imen

tal0

100000000

200000000

300000000

400000000

500000000

600000000

Series1


0

5000000

10000000

15000000

20000000

25000000

Series1


Shi formula for G13 gives more précised value as compare to experimental result.


Test 1Test 2

Test 3 Test 4

Test 5 Test 6

Test 7


M an

d e

e.nas

tab

dqunli

ashby

shi

gibso

n and as

hby

Exper

imen

tal0

20000000

40000000

60000000

80000000

100000000

120000000

140000000

160000000

180000000

200000000

Series1

M an

d e

e.nas

tab

dqunli

ashby

shi

gibso

n and as

hby

Exper

imen

tal0

50000000

100000000

150000000

200000000

250000000

300000000

350000000

Series1

M an

d e

e.nas

tab

dqunli

ashby

shi

gibso

n and as

hby

Exper

imen

tal0

50000000

100000000

150000000

200000000

250000000

300000000

350000000

400000000

450000000

Series1

M an

d e

e.nas

tab

dqunli

ashby

shi

gibso

n and as

hby

Exper

imen

tal0

50000000

100000000

150000000

200000000

250000000

300000000

350000000

400000000

450000000

Series1

M an

d e

e.nas

tab

dqunli

ashby

shi

gibso

n and as

hby

Exper

imen

tal0

50000000

100000000

150000000

200000000

250000000

300000000

350000000

400000000

450000000

Series1

M an

d e

e.nas

tab

dqunli

ashby

shi

gibso

n and as

hby

Exper

imen

tal0

100000000

200000000

300000000

400000000

500000000

600000000

Series1


0

2000000

4000000

6000000

8000000

10000000

12000000

14000000

16000000

Series1


Shi formula for G23 gives more precise value as compared to experimental result.


Test 5

Test 2

Test 3 Test 4

Test 1

Test 6

Test 7

3.1.7. Selection of ν12


0

0.2

0.4

0.6

0.8

1

1.2

Series1


0

0.2

0.4

0.6

0.8

1

1.2

Series1


0

0.2

0.4

0.6

0.8

1

1.2

Series1


0

0.2

0.4

0.6

0.8

1

1.2

Series1


0

0.2

0.4

0.6

0.8

1

1.2

Series1


0

0.2

0.4

0.6

0.8

1

1.2

Series1


0

0.2

0.4

0.6

0.8

1

1.2

Series1

Fig. 3.23 Measurement of Elastic Property (ν12)

Here Gibson and Ashby, Abd Al Sayed and Masters and Evans give the same results.

While only E. Nast differs. E. Nast is preferred since it gives the value less than 1.


Test 1 Test 2

Test 3Test 4

Test 5Test 6

Test 7



0

0.00001

0.00002

0.00003

0.00004

0.00005

0.00006

0.00007

Series1


0

0.00001

0.00002

0.00003

0.00004

0.00005

0.00006

0.00007

Series1

e.nast abd qunli ashby shi gibson and ashby

0

0.00002

0.00004

0.00006

0.00008

0.0001

0.00012

Series1


0

0.00002

0.00004

0.00006

0.00008

0.0001

0.00012

Series1


0

0.00002

0.00004

0.00006

0.00008

0.0001

0.00012

Series1


0

0.00002

0.00004

0.00006

0.00008

0.0001

0.00012

0.00014

0.00016

0.00018

0.0002

Series1


0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

Series1


Gibson, Ashby and E. Nast are the possible solution in this case. However E. Nast is

preferred since it gives the value less than 1.


Test 1 Test 2

Test 3 Test 4

Test 5 Test 6

Test 7



0

0.00001

0.00002

0.00003

0.00004

0.00005

0.00006

0.00007

Series1


0

0.00001

0.00002

0.00003

0.00004

0.00005

0.00006

0.00007

Series1


ashby

0

0.00002

0.00004

0.00006

0.00008

0.0001

0.00012

Series1


0

0.00002

0.00004

0.00006

0.00008

0.0001

0.00012

Series1


0

0.00002

0.00004

0.00006

0.00008

0.0001

0.00012

Series1


0

0.00002

0.00004

0.00006

0.00008

0.0001

0.00012

0.00014

0.00016

0.00018

0.0002

Series1


0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

Series1


Gibson and Ashby and E. Nast are the available models. However, in this case E. Nast is

preferred since it gives the value less than 1.


Test 1 Test 2

Test 3 Test 4

Test 5Test 6

Test 7

3.1.10. Optimum Model for the selection of Elastic Constants

Fig. 3.26 Optimum Model for the selection of Elastic constant

An optimization has been done and also an optimized model has been developed on the

basis of the study done by different researchers to select the 9 different elastic constants,

which are necessary to be calculated and plugged in the simulation and analytical

modeling.

3.2. Modeling using Spring-Mass system

Many researchers use spring-mass system for analyzing the honeycomb sandwich

structures. An effort has been made in the study to analyze the impact behavior of

honeycomb sandwich structures by the use of same approach. Now consider a 2-Dof


E1 by E. Nast

E2 by E.nast

E3 by E.nast

G12 experimental

G13 by shi

G23 by shi

v12 by E. Nast

v13 by E. Nast

v23 by E. Nast

system consists of impact mass and contact stiffness and mass of the sandwich structure

along with their stiff nesses which is shown as follows:

Fig. 3.27 2-DOF Model of Honeycomb Sandwich Structure

The free body diagram of the system is shown below:

a. Impactor b. Honeycomb structurec.

Fig. 3.28 Free Body diagram

m1 x1+K c ( x1−x2 )32=0 Equation (3.2.1.1)

m2 x2+k bs x2+K m x23−K c (x1−x2 )

32=0

Putting value of k bs=Kbs+K core in above equation

m2 x2+(Kbs+K core) x2+K m x23−K c (x1−x2 )

32=0 Equation (3.2.1.2)

3.2.1. Linear Modeling

As a first approximation linearize the above system of equation reduces to:


m1 x1+K c ( x1−x2 )=0 Equation (3.2.1.3)

m2 x2++(Kbs+K core) x2+Km x2−K c (x1−x2 )=0 Equation (3.2.1.4)

The above system of equations is coupled and exact solution can only be obtained if this

system could be decoupled.

Equation (3.2.1.5)

Now to decouple such a system either K c=0 or m2 x2 ≈ 0 or m1 x1 ≈ 0. According to

literature, if m1≥ 3 m2, thenm2 x2 ≈ 0. So now substitute this condition in equation (3.2.1.4)

and obtain value of x2

In terms ofx1,

Equation (3.2.1.6)

Now substitute equation (3.2.1.6) into equation (3.2.1.3)

Equation (3.2.1.7)

The solution for this equation is

Equation (3.2.1.8)

Where

Equation (3.2.1.9)

To calculate the values of constants ( A∧B), the initial conditions are:

x1 (0 )=0 , x1 (0 )=V 0

Using x1 (0 )=0 in (3.2.1.8) we getB=0


Using x1 (0 )=V 0in derivative of (3.2.1.8) we get A=V 0

ω

Substitute these initial conditions in Equation (3.2.1.8):

x1(t)=(V 0

ω )sin (ωt) Equation (3.2.1.10)

Substitute Equation (3.2.1.10) in Equation (3.2.1.6)

Equation (3.2.1.11)

The contact force could be calculated as shown below:

Equation (3.2.1.12)

Substitute Equation (3.2.1.10) & Equation (3.2.1.11) in Equation (3.2.1.12), we get the

final form of contact form as:

Equation (3.2.1.13)

The contact duration can be calculated as:

T=( πω ) Equation (3.2.1.14)

3.2.2. Solution of Non-linear impact using Newton’s Forward Difference Method

The analytical solution of differential Equations (3.2.1.1) and (3.2.1.2) do not exist.

Therefore, numerical integration technique is used to solve for the two displacements.

Newton forward difference formula and hence rectangular integration method is used

here. For time being let’s call x1= y1 &x2= y2.

3.2.2.1. Mathematical Formulation

y1p=

y1p+1− y1

p

∆ t

Equation (3.2.3.a.1)

Now this equation can be simplified to usable form:


y1p ∆ t+ y1

p= y1p+1 Equation (3.2.2.a.1.a)

Forward difference formula for second derivative would be:

y1p=

y1p+2−2 y1

p+1+ y1p

(∆ t )2

Equation (3.2.2.a.1.b)

Substituting, Eqn. (3.2.2.a.1.b) in Eqn. (3.2.1.1) and simplifying:

m1 y1+K c ( y1− y2 )32=0

m1( y1p+2−2 y1

p+1+ y1p

(∆ t )2 )+K c ( y1− y2 )32=0

y1p+2−2 y1

p+1+ y1p

(∆ t )2=−( K c

m1) ( y1

p− y2p)3 /2

y1p+2−2 y1

p+1=( ∆ t )2(−K c

m1) ( y1

p− y2p )3 /2− y1

p

y1p+2=2 y1

p+1+(∆ t )2(−K c

m1)( y1

p− y2p )3/2− y1

p

Now substitute (3.2.2.a.1.a)

y1p+2=2 { y1

p ∆ t+ y1p }+(∆ t )2(−K c

m1)( y1

p− y2p )3/2− y1

p


Let’s do the same procedure for y2.

y2p=

y2p+1− y2

p

∆ t


y2p ∆ t+ y2

p= y2p+1 Equation (3.2.2.a.3.a)

y2p=

y2p+2−2 y2

p+1+ y2p

(∆ t )2

Equation (3.2.2.a.3.b)

Simplify Eqn. (3.2.1.2)

m2 y2+K bs y2+Km y23−K c ( y1− y2 )

32=0

m2 y2=−Kbs y2−Km y23+K c ( y1− y2 )

32=0


y2p=

K c

m2( y1

p− y2p )3 /2−Kb

m2

y2p−

Km

m2

y2p3

Now substitute Eqn. (3.2.2.a.3.b) in above Equations, we get

y2p+2−2 y2

p+1+ y2p

(∆ t )2=( K c

m1)( y1

p− y2p )3/2− Kb

m2

y2p−

Km

m2

y2p3

y2p+2−2 y2

p+1=[( K c

m1)( y1

p− y2p )

32−

Kb

m2

y2p−

Km

m2

y2p3] ( ∆ t )2+ y2

p

Substituting Eqn. (3.2.2.a.3.a) in above equation leads to the following form of equations:

y2p+2−2( y2

p ∆ t+ y2p)=[( K c

m1) ( y1

p− y2p )

32−

Kb

m2

y2p−

K m

m2

y2p3] (∆ t )2+ y2

p

y2p+2=[(K c

m1) ( y1

p− y2p )

32−

Kb

m2

y2p−

Km

m2

y2p3] (∆ t )2+ y2

p+2( y2p ∆ t+ y2

p) Equation (3.2.3.a.4)

3.2.2.2. Iteration Procedure

The following procedure is used:

1) Initial conditions before impact at p=1 are y11(0)=V 0, y2

1(0)=0,y11(0)=0,y2

1(0)=0

2) Decide ∆ t as per computation power

3) Substituting initial conditions into Eqn. (3.2.2.a.1.a), we get

y1p+1= y1

p ∆ t+ y1p

y12= y1

1 ∆ t+ y11

y12=V 0∆ t Result (3.2.2.b.i)

4) Substituting initial conditions into Equation (3.2.2.a.2):

y1p+2=2 { y1

p ∆ t+ y1p }+(∆ t )2(−K c

m1)( y1

p− y2p )3/2− y1

p

y13=2 { y1

1 ∆ t+ y11 }+( ∆t )2(−K c

m1) ( y1

1− y21 )3/2− y1

0

y13=2 {V 0 ∆ t } Result (3.2.2.b.ii)


5) Similarly for Equation (3.2.2.a.3.a):

y2p+1= y2

p ∆ t+ y2p

y22= y2

1 ∆ t+ y21

y22=(0)∆ t+(0)

y22=0 Result (3.2.2.b.iii)

6) And for Equation (3.2.2.a.4):

y2p+2=[(K c

m1) ( y1

p− y2p )

32−

Kb

m2

y2p−

Km

m2

y2p3] (∆ t )2+ y2

p+2( y2p ∆ t+ y2

p)

y23=[( K c

m1) ( y1

1− y21 )

32−

K b

m2

y21−

Km

m2

y213] (∆ t )2+ y2

1+2( y21 ∆t+ y2

1)

y23=[( K c

m1) (0−0 )

32−

K b

m2

(0)−K m

m2

(0)3] (∆ t )2+0+2 {(0)∆ t+0 }

y23=0 Result (3.2.2.b.iv)

7) Substitute these new values obtained in step 3 for p=2, we get

y1p=

y1p+1− y1

p

∆ t

y12=

y13− y1

2

∆ t

y12=

2 {V 0 ∆ t }−V 0 ∆ t

∆ t

y12=V 0 Result (3.2.2.b.v)

8) Substitute initial conditions & result (3.2.2.b.v) in (3.2.2.a.2)

y1p+2=2 { y1

p ∆ t+ y1p }+(∆ t )2(−K c

m1)( y1

p− y2p )3/2− y1

p


y14=2 { y1

2 ∆ t+ y12 }+ (∆ t )2(−K c

m1)( y1

2− y22 )3 /2− y1

2

y14=2 {(V 0)∆ t+V 0 ∆ t }+ (∆ t )2(−K c

m1) (V 0 ∆ t−0 )3/2−V 0 ∆ t

y14=3 V 0 ∆ t+ (∆ t )2(−K c

m1) (V 0 ∆ t )3 /2

Result (3.2.2.b.vi)

9) Substitute initial conditions in (3.2.2.a.3)

y2p=

y2p+1− y2

p

∆ t

y22=

y23− y2

2

∆ t

y22=0−0

∆ t

y22=0 Result (3.2.2.b.vii)

10) Substitute initial conditions & result (3.2.2.b.vii) in equation (3.2.2.a.4)

y2p+2=[(K c

m1) ( y1

p− y2p )

32−

Kb

m2

y2p−

Km

m2

y2p3] (∆ t )2+ y2

p+2( y2p ∆ t+ y2

p)

y24=[(K c

m1) ( y1

2− y22 )

32−

Kb

m2

y22−

Km

m2

y223] ( ∆ t )2+ y2

2+2( y22 ∆ t+ y2

2)

y24=[(K c

m1) ( y1

2−0)32−

Kb

m2

(0)−Km

m2

(0)3] (∆ t )2+0+2 {(0)∆t+0 }

y24=[(K c

m1) ( y1

2)32 ] (∆ t )2

y24=[(K c

m1) (V 0 ∆ t )

32 ] (∆ t )2

Result (3.2.2.b.viii)

11) Now repeat the steps from 6 to 10 by changing the value of p.

12) Stop the process when total time becomes equal to contact time, given in Equation

(3.2.1.14)


3.2.3. Solution of Non-linear Oblique Impact on Honeycomb Sandwich Structures

Now include the effect of friction and impact is at an angle. The 2-DOF system of

equations described earlier is converted to 4-DOF system and ignoring the rotational

effects. It is assumed that neither the ball rotates after impact nor the plate rotates after

impact. It is concluded from figure 3.17 Kbs=K bs+K core. In calculations Left hand side

i.e. Kbs∧K core is used.

Fig. 3.29 Oblique Impact Diagram

The spring-mass model redefined for this case is:

Fig. 3.30 4-DoF Model for Oblique Impact on Honeycomb Sandwich Structure

The friction effect is introduced by the use of damper. The FBD for vertical forces as

defined in Fig. 3.17.


a. Impactor including friction b. Honeycomb structure including frictionFig. 3.31 free body diagram of individual masses

Now let’s draw another useful figure 3.19, Combining figure 3.16 and figure 3.19, we get

Fig. 3.32 Representation of Vertical forces

Also Hertz contact law is modified by considering that A1 and A2 are distances along

reaction direction, therefore, Fc is in reaction direction.

So Fc = reaction

F c=K c ( A1−A2 )3 /2


Now resolve X1,


Fig. 3.33 Combination of Fig. 3.16 and Fig. 3.19

Fig. 3.34 Resolving X1 Fig. 3.22 Resolving X2

A1=X1cos (θ1+β−900 ) Equation (3.2.3.a.2)

Now resolve X2, we get (figure 3.22)

A2=X2 cos (θ2+β−900 ) Equation (3.2.3.a.3)

Substituting equation (3.2.3.a.2) and (3.2.3.a.3) in (3.2.3.a.1)

F c=K c¿¿ Equation (3.2.3.a.4)

Since cos90 = 0 and sin 90 = 1

F c=K c¿¿ Equation (3.2.3.a.5)

Consider Figure 3.23

Fig. 3.35 Resultant Diagram for Frictional and Normal Forces

Consider Fig. 3.23 and applying Pythagoras theorem:

(Fc )2=( μN )2+( N )2 Equation (3.2.3.a.6)


Since N is the normal force determined from Hertz contact law with displacement y1 and

y2 as shown in fig. 3.18.

N=K c ( y1− y2)3/2


Substitute equation (3.2.3.a.7) into equation (3.2.3.a.6), we get

(Fc )2=[K c ( y1− y2 )

32 ]

2

(1+μ2) c Equation (3.2.3.a.8)

From fig. 3.21

N=Fc sin β Equation (3.2.3.a.9)

F r=μN=Fc cos β Equation (3.2.3.a.10)

Dividing Eqns. (3.2.3.a.9) and (3.2.3.a.10), following form of

equation is obtained:

NμN

=Fc sin β

Fc cos β

tan β=1μ


Let’s solve vertical spring mass system s from fig. 3.17:

m1 y1=−Fc sin β Equation (3.2.3.a.12)

m2 y2+K m y23+Kbs y2+K core y2−Fc sin β=0 Equation (3.2.3.a.13)

Substitute equation (3.2.3.a.8) in (3.2.3.a.12) and (3.2.3.a.12), we get

m1 y1=−K c¿¿ Equation (3.2.3.a.14)

m2 y2+K m y23+Kbs y2+K core y2−K c¿¿ Equation (3.2.3.a.15)

For horizontal component of displacement, we get

m1 x1=−Fc cos β Equation (3.2.3.a.16)

m2 x2+K plate x2+Kbending x2−Fc cos β=0 Equation (3.2.3.a.17)

Substitute equation (3.2.3.a.8) in (3.2.3.a.16) and (3.2.3.a.17), we get


m1 x1=−K c¿¿ Equation (3.2.3.a.18)

m2 x2+K plate x2+Kbending x2−K c¿¿ Equation (3.2.3.a.19)

Now there are 5 variables, x1, x2, y1, y2 andβ. Equation (3.2.3.a.11), (3.2.3.a.14),

(3.2.3.a.15), (3.2.3.a.18) and (3.2.3.a.19) could be solved to get the desired values.

These equations are very complex and processor would take much time and memory to

solve. In practice if these equations are not used for the simulation purposes, simplified

normal impact without angle is preferred by researchers.

Now to avoid solving Eqn. (3.2.3.a.18) and Eqn. (3.2.3.a.19) for getting the values of y1,

y2, we developed a simple methodology. Let’s substitute Equations (3.2.3.a.8) in

(3.2.3.a.12) and (3.2.3.a.13).

m1 y1=−K c ( y1− y2 )32 (sinβ√1+μ2 ) Equation (3.2.3.b.1)

m2 y2+K m y23+Kbs y2+K core y2−K c ( y1− y2 )

32 (sinβ√1+μ2 )=0

Equation (3.2.3.b.2)

Now from Equation (3.2.3.a.11), we obtain

tan β=1μ

β=tan−1 1μ


Insert Equation (3.2.3.b.3) into (3.2.3.b.1) and (3.2.3.b.2),

m1 y1=−K c ( y1− y2 )32 (sin( tan−1( 1

μ ))√1+μ2) Equation (3.2.3.b.4)


32 (sin( tan−1( 1

μ ))√1+μ2)=0 Equation (3.2.3.b.5)

Summary of oblique Impact Model

Variables Equations to solve for variables


x1, x2,β β=tan−1 1μ


m1 x1=−K c¿¿ Equation (3.2.3.a.18)

m2 x2+K plate x2+Kbending x2−K c¿¿ Equation (3.2.3.a.19)

y1, y2

m1 y1=−K c ( y1− y2 )32 (sin( tan−1( 1

μ ))√1+μ2) Equation (3.2.3.b.4)


32 (sin( tan−1( 1

μ ))√1+μ2)=0 Equation (3.2.3.b.5)

F c F c=K c ( y1− y2 )32√1+μ2


The first three equations could be used to get values of x1, x2 andβ. The next two

equations give values of y1 and y2.The last equation gives contact force Fc.

3.2.4. Solution for Oblique Impact

To obtain solution for y1, y2 repeat the same procedure as done in iteration method for

normal impact. But in this case there is an additional step called zero step, where β value

is found using Equation (3.2.3.a.11). To obtain the solution of x1, x2, simple method is

used. The matrix used for this purpose is:

[m1 00 m2] [ x 1

x 2]+[0 00 kplate+kbending] [x 1

x 2]=[−Fc cosβFc cosβ ]

3.2.5. Free Response of Honeycomb Sandwich Structure

Till date most of the researchers have done experimental analysis for getting the natural

frequency of honeycomb structures and used the data obtained from experimental

analysis for their analysis. Therefore, an appropriate analytical solution is proposed in

this study for finding the natural frequency of honeycomb sandwich structures before and

after the impact for comparison.

3.2.5.1. Qualitative Analysis

The general equation for the free response of the honeycomb sandwich panel is:



Integrate the equation w.r.t time.

Since x2(t) is not known and above integral could not be solved. To solve use the

procedure described as follows:

Multiply Equation (3.2.4.a.1) by x2 on both sides.

Integrate the above equation

Now cancel dt term in second integral in Equation, we get

Now the first integral term is reduced as:


Cancel dt term in first integral in Equation, we get

Put x2=v2 in above equation

Integrating equation, we obtained


Further integration is not possible using the same technique. Let’s analyze what we have

found:

1) Since this is a mechanical system, the first term is essentially the kinetic energy

2) The second and third term is potential energy

3) h represents energy level

So Equation (3.2.4.a.2) represents the conservation of energy. For a given value of h, the

Equation (3.2.4.a.2) in uv-plane (phase plane) is called level curve, or a curve of constant

energy, or integral curve; the branches of these level curves are called trajectories. Here u

represents x2.The Equation (3.2.4.a.2) can be expressed as:

For real solution of this system to exist, following condition must be satisfied



Equation (3.2.4.a.1) can be expressed as:

It shows that singular and equilibrium points exist at point where left hand side of the

above equation is zero or infinity i.e. gradient of level curve defined by Equation

(3.2.4.a.2) is zero or infinity.

A specific potential energy vs. displacement curve is also essential. It is useful to know

that whether equilibrium points obtained from level curves are stable or not.

Let’s derive a formula for time period.




the upper and lower limits are found out using level curves in phase plane. These values

correspond to x-intercepts in phase plane.Here

3.2.5.2. Appropriate Numerical Technique-Perturbation Method

Numerical methods were used by many investigators to solve for these types of non-

linear equations. Einaudi (1975) used an iterative method: Argyris, Dunne, and

Angelopoulos (1973) used a finite element technique; and Susemihl and Laura (1975)

used a collocation technique. Here using a new technique, perturbation method is used

for analyzing the system response.

Initially, differential equation is expressed as follows:


The free response equation for honeycomb structure is written below. The subscripts are

not used here; also x is vertical displacement.

m x+(kbs+kcore) x+Km x3=0

x+(k bs+kcore)

mx+

Km

mx3=0


Equation (3.2.4.b.1) can be further simplified as


Compare Equation (3.2.4.b.2) and Equation (3.2.4.b.3), we get 65



Now introducing a new variable,



Taking derivative on both sides



Substitute equation (3.2.4.b.8) into Equation (3.2.4.b.3)


Now we can express ω∧xas,



Substitute Equation (3.2.4.b.10) & (3.2.4.b.11) in Equation (3.2.4.b.9)



Expand the terms and taking the co-efficient ofϵ , ϵ 2 , ϵ 3 , ϵ 4 , ϵ 5. The table below gives the

values of co-efficient:

Term Co-efficient

ϵ

ϵ 2

ϵ 3

ϵ 4


ϵ 5

ϵ 6

In this case we only have α 3 terms as per in Equation (3.2.4.b.4)


Now from the above values the coefficients are equated to zero, Equation (3.2.4.b.13) is

used and the differential equations formed are shown below:


Terms Differential Equations

ϵ Equation (3.2.4.b.14)

ϵ 2 Equation (3.2.4.b.15)






Before proceeding further, useful cosine identities are listed below:






3.2.5.2.i Solution for x1

Assume Equation (3.2.4.b.22)


Differentiating Equation (3.2.4.b.25)




The general solution of Equation (3.2.4.b.14) becomes,


Substitute Equation (3.2.4.b.25) in Equation (3.2.4.b.28), we get


3.2.5.2.ii Solution for x2 and w1

Substituting Equation (3.2.4.b.28) in (3.2.4.b.15)


Substitute Equations (3.2.4.b.25), (3.2.4.b.26), and (3.2.4.b.27) in Eqn. (3.2.4.b.30)


Removing secular term in Equation (3.2.4.b.31) and equating coefficient of cosine term

against zero


Since neither aor {ω} rsub {0} are non-zero, so then solving Equation (3.2.4.b.32) for ω1

ω1=0 Equation (3.2.4.b.33)

Substitute Equation (2.2.4.b.33) in Equation (2.2.4.b.31)71



Now use equation (3.2.4.b.24) in (3.2.4.b.34)


Solve the differential equation; we are only interested in particular solution


3.2.5.2.iii Solution for x3 and w2

Substitute equation (3.2.4.b.29), (3.2.4.b.36) in Equation (3.2.4.b.16), we get



Substitute Equation (3.2.4.b.35) in Eqn. (3.2.4.b.37)


Substitute equation (3.2.4.b.23) in Equation (3.2.4.b.38), we get



Substitute Equation (3.2.4.b.24) in Equation (3.2.4.b.39), we get


Removing secular term and equating co-efficient of cos (ϕ) against zero i.e.


Substitute Equation (3.2.4.b.33) in (3.2.4.b.41)


Substitute Equation (3.2.4.b.33), Equation (3.2.4.b.42) in Eqn. (3.2.4.b.40)



Solve the above differential equation for a particular solution only.


Equation (2.2.4.b.44*)

3.2.5.2.iv Solution for x4 and w3

x4 and w3 are found using the same procedure as described above. Equation (3.2.4.b.17) is

written below for further simplification:

Substitute Equations (3.2.4.b.29), (3.2.4.b.36) and (3.2.4.b.44) in Equation (3.2.4.b.17),

we get



Substitute Equation (3.2.4.b.25) in (3.2.4.b.45), we get



Substituting Equation (3.2.4.b.22) in Equation (3.2.4.b.46) and then substituting Eqn.

(3.2.4.b.23) in the result of first substitution and finally substituting Equation (3.2.4.b.24)

into the result of second substitution. Forms the following Equation:



Simplifying the above Equation (3.2.4.b.47)


Now equate the coefficient of cos (ϕ) from Equation (3.2.4.b.48) against zero


Substitute Equation (3.2.4.b.33) in Equation (3.2.4.b.49)



Now solve the differential Equation (3.2.4.b.48), the particular solution is


3.2.5.2.v Solution for x5 and w4

The same procedure is carried out. Note that by increasing the terms the calculations

become more complex. The Equation (3.2.4.b.18) is repeated below for further

simplification:

Substituting Equations (3.2.4.b.29), (3.2.4.b.36), (3.2.4.b.44) and (3.2.4.b.51) in Equation

(3.2.4.b.18) also replace ϕ using Equation (3.2.4.b.25), we get


Equation (3.2.4.b.25) into above equation to convert the terms of ϕ


Substituting Equation (3.2.4.b.21) in Equation (3.2.4.b.52), then substitute Equation

(3.2.4.b.22) in result and after that substitute Equation (3.2.4.b.23) in the new result

obtained. Finally substitute equation (3.2.4.b.24) in the result from previous substitutions,

we get


Equate the co-efficient of secular terms against zero, the Equations become


Substitute Equations (3.2.4.b.33), (3.2.4.b.42) and (3.2.4.b.50) in Equation (3.2.4.b.54) 82



Now solve the differential Equation (3.2.4.b.53) taking only the particular solution of the

equation:



Substitute Equations (3.2.4.b.33), (3.2.4.b.42), (3.2.4.b.50) and (3.2.4.b.55) into Equation

(3.2.4.b.56), we get84



3.2.5.2.vi Solution of x6 and w5

The same procedure is repeated for finding the solution of x6 and w5.The differential

equation is solved after substituting values of x1, x2, x3, x4, and x5 in equation (3.2.4.b.19).

The secular term is equated against zero:


Substitute Equations (3.2.4.b.33), (3.2.4.b.42), (3.2.4.b.50) and (3.2.4.b.55) into Equation

(3.2.4.b.58)


After substituting the Equations (3.2.4.b.33), (3.2.4.b.42), (3.2.4.b.50), (3.2.4.b.55) and

Equation (3.2.4.b.58) particular solution of Equation (3.2.4.b.19) obtained is:



3.2.5.2.vii Using initial conditions

The appropriate solution is of the form:


Substitute values found in Equations (3.2.4.b.29), (3.2.4.b.36) and (3.2.4.b.44) in

Equation (3.2.4.b.61). The result is:


This expression can be shown in terms of time using equation (3.2.4.b.5) as


Substitute t = 0 here




Differentiate Equation (3.2.4.b.63), substitute t = 0 and initial velocity v (0):


There are two options either substitute first Equation (3.2.4.b.4) in Equations (3.2.4.b.66)

and (3.2.4.b.64) or vice versa. To solve for the solution it would require expanding “a”

and “β” in terms of “ε” and equating like powers of ε.

To solve this first approach is used.




3.2.5.2.viii Application on Honeycomb Structures

Substituting Equation (3.2.4.b.4) in Eqns. (3.2.4.67) & (3.2.4.68), we get



Substitute Equation (3.2.4.b.4) in Equation (3.2.4.b.63):


Substitute Equations (3.2.4.b.33), (3.2.4.b.42), and (3.2.4.b.50) in Equation (3.2.4.b.10)88



Substitute equation (3.2.4.b.13) & (3.2.4.b.4) into equation (3.2.4.b.72)


Equation (3.2.4.b.73) is the required form which shows that frequency is a function of

amplitude.

3.2.6. Damage Inclusion in Honeycomb

The damage was included by modifying the constants. Experimental results of stress

strain graph of face sheet and whole Sample is used for finding new parameters. Failure

criteria given throughout literature are used.

Fig. 3.36 Failure criteria used

The graph shown in Fig. 3.24 represents stress-strain graph for face sheet used in the

honeycomb sandwich structure and is plotted using MATLAB. While stress-strain graph

for the whole honeycomb sandwich structure including the face sheets and the

honeycomb core can be obtained from section 3.5.


Fig. 3.37 Stress-strain graph for face sheet


4. Impact Analysis using LS-DYNA

Finite element analysis is carried in ANSYS/LS-DYNA ANSYS APDL is used for

modeling of the honeycomb sandwich structure while impact analysis is performed in

LSDYNA.

4.1. Modeling OF Honeycomb sandwich Structure in ANSYS

A core and face sheets are modeled through a step by step procedure defined below:

Step 1: First open Mechanical APDL launcher and click to High performance

computation setup:

Fig. 4.38 ANSYS Mechanical APDL Product Launcher


Step 2: Select Shared Memory Parallel (SMP) and enter Number of processors. In our

case, there are four processors. Click on Run.

Step 3: Now to create key points, go to pre-processor Modeling Create

Keypoints In active CS

Fig. 4.39 ANSYS Main Menu

Step 4: Now enter 12 coordinates for one face of honeycomb and click ok.

Fig. 4.40 Creating Coordinates in one face of Honeycomb

Step 5: Now join the inner keypoints with line. Go to pre-processor Modeling

Create lines lines Straight lines92


Fig. 4.41 Steps to make Straight Lines

Step 6: Now select any two keypoints, ANSYS will create a line between these

keypoints.

Fig. 4.42 Creating Straight Line Between the Keypoints

Step 7: Now join other inner keypoints, we get the following shape as shown in Fig. 4.6


Fig. 4.43 Joining the inner Keypoints

Step 8: Now join outer keypoints to get the complete layout of honeycomb core.

Fig. 4.44 Joining the Outer Keypoints

Step 9: Now make area between these lines. Go to pre-processor Modeling Create

Area Arbitrary Through KPs


Fig. 4.45 Steps to create Area between the Lines

The area formed in step 9 is shown in Fig. 4.9

Fig. 4.46 Area Formation

Step 10: Now make rest of the core by following the same steps


Fig. 4.47 Core Formation Fig. 4.48 Steps to add all the Areas

Step 11: Add all areas. Go to pre-processor Modeling Operate Booleans Add

Areas (Fig. 4.11).

After following step 11, additional window appears. Click Pick all Ok

Fig. 4.49 Add Areas Menu Fig. 4.50 Extrude Area along Normal


Step 12: Go to pre-processor Modeling Extrude Areas Along Normal

Areas.

Step 13: After following step 12, additional window appears. Click Pick all Ok. If Pick

all option does not appears, then select the Area and click ok.

Step 14: Following window appears as shown in above Fig. 4.13.

In Length of extrusion, enter the depth thickness of core.

Step 15: After doing all these steps described above the honeycomb core is formed. In the

same way two plates at faces and spherical can be formed.

4.2. Impact Model in LS-DYNA

Import the model into LS-DYNA as “iges” format. LS- DYNA performs dynamic impact

analysis of the model created in the previous section. To define the problem in LS-

DYNA, it uses cards. The cards used in the process are:

Material Cards, Section Card, Part Card, Contact card, Initial Card, Control card, SetD,

Boundary, Dbase Card.

Following steps are followed to define the problem in LS-DYNA:

Step 1: Go to File Import iges. Here import iges file made in ANSYS. Any other

modeling software like solid works, Pro Engineering could be used.

Step 2: Go to page 3, Select Material card and material model. Mat 001-Elastic

and Mat 002-orthotroppic elastic is used. The following two figures (4.14 and

4.15) shows the properties used:


Fig. 4.51 Selection of Material Model

Fig. 4.52 Material Property Card

Step 3: Two different types of sections are defined: Solid and Shell


Fig. 4.53 Defining the sections-Solid

Fig. 4.54 Defining the sections-shells

Step 4: Now assign these properties to part, as an example:


Fig. 4.55 Assigning properties to parts

Step 5: Now go to page 5, select SetD, select create, select set part then select

core. Click Apply and then done.

Step 6: Automatic General, Automatic Surface to Surface, Tied Surface to

Surface, Interior, and force transducer penalty.

Fig. 4.56 Contact Automatic General


Fig. 4.57 Contact Automatic Surface to Surface

Fig. 4.58Contact Tied Surface to Surface Failure


Fig. 4.59 Contact Interior

Fig. 4.60 Contact Force Transducer Penalty

Step 7: Again go to Page 5, select SetD, and then select set node. Now select

those boundaries where simply supported condition is required. In our case the


outer boundary of the structure is selected. The Fig. 4.24 depicts the nodes

selected:

Fig. 4.61 Selection of nodes

Step 8: Go to page 3, select Boundary. In Boundary select nodes and enter the

following options:

Fig. 4.62 Selection of Boundary and Nodes


Step 9: On page 3, in initial card select velocity generation and enter following

options in the card:

Fig. 4.63 Initial Velocity Generation

Now enter the value of Vz velocity.

Step 10: In control card, select three different cards; Shell card, solid card and

termination card. The following figure appears for entering the required values:

Fig. 4.64 Control card_shell card


Fig. 4.65 Control card_solid card

Fig. 4.66 Termination card

Step 11: Now in Dbase card select Binary D3plot and ASCII option.

Step 12: Save .k file and run the input file from Mechanical APDL launcher.

Step 13: After normal termination, open Binary D3plot from LS prepost.


4.3. Frequency Response of Structure

Step by step procedure is defined as follows:

Step 1: Go to File Import iges and import the required iges file. Any other

modeling software like solid works, Pro Engineering could be used.

Step 2: Go to page 3, Select Material card and the material model. Mat 001-

Elastic and Mat 002-orthotroppic elastic is used. The following two figures (4.30

and 4.31) shows the properties used:

Fig. 4.67 Elastic Model_001


Fig. 4.68 Orthotropic Elastic_002

Step 3: Two different types of sections are defined: Solid and Shell

Fig. 4.69 Section_Solid


Fig. 4.70 Section_shell

Step 4: Now assign these properties to part, as an example:

Fig. 4.71 Assigning properties to the created parts

Step 5: Now go to page 5, select SetD, select create, select set part then select

core. Click Apply and then done.


Step 6: Automatic General, Automatic Surface to Surface, Tied Surface to

Surface, Interior, and force transducer penalty.

Fig. 4.72 Contact_general

Fig. 4.73 Contact_surface to surface


Fig. 4.74 Contact_tied_surface to surface_failure

Fig. 4.75 Contact_Interior


Fig. 4.76 Contact force tranducer penalty

Step 7: Again go to Page 5, select SetD, and then select set node. Now select

those boundaries where simply supported condition is required. In my case the

outer boundary of the structure is selected. The figure below shows the nodes

selected:

Fig. 4.77 Selecting the nodes


Step 8: Now go to page 3, select Boundary. In Boundary select nodes and enter

the following options:

Fig. 4.78 Entering options after selecting nodes in the boundary

Step 9: Now in control card, select three different cards; Shell card, solid card,

implicit Dynamics, Implicit Eigen Value, Implicit General, Implicit Solution,

Implicit Solver and termination card. The following figure shows the values

entered:


Fig. 4.79Implicid Dynamics

Fig. 4.80 Implicit Eigen-value


Fig. 4.81 Implicit General

Fig. 4.82 Implicit Solutions


Fig. 4.83 Implicit Solver

Fig. 4.84 Control_Shell


Fig. 4.85 Control_Solid

Fig. 4.86 Control_Terminator

Step 10: Now in Dbase card select Binary D3plot and ASCII option.

Step 11: Save .k file and run the input file from Mechanical APDL launcher.

Step 12: After normal termination, open Binary D3 plot from LS-prepost.

Step 13: Open Eigen plot, and view the Eigen values.


4.4. Mesh Independence, Number of Equations and Time

The mesh of the structure was refined and the results are shown below:

Table 4.3Mest Structure and its results

Element ¿¿ f Number of Equations

Time Required for w

Time Required for Impact

2.8 859.9 58665 3 minutes 1 hour 14 minutes

1.4 827.5 306276 5 minutes 3 hours 56 minutes

0.7 826.2 1512204 13 minutes 7 hours 03 minutes

4.5. Final Model

The table below shows the corresponding experimental model and software model.

Table 4.4 Final Model

Top

Left


Front

5. Preparation of Working Samples

5.1. Fabrication of the Specimens

Fabrication of the honeycomb sandwich panel is not an easy task that requires a lot of

accuracy and perfection. The construction of the honeycomb sandwich panel includes the

following main steps:

i. Construction of the core

ii. Cutting of the metal face sheets

iii. Choice of the adhesive

iv. Gluing the face sheets with the core

v. Polishing of the panel


5.1.1. Construction of the core

Since the fabrication of the Nomex core is itself a very had and tedious work and was not

the requirement of our project. In this study, the first step is omitted by utilizing the

Nomex core got from the open market.

5.1.2. Cutting of Metal Face sheets

We choose the Aluminium 5052 grade for the face sheets. The Aluminium face sheets

were then cut according with the ASTM standards from the large sheet with the suitable

equipment available in IST workshop.

Fig. 5.87 Cutting of face sheets and removal of dents

The sheets were then hammered by a wooden hammer to remove any dents present on it.


5.1.3. Choice of Adhesive

A suitable adhesive was chosen which should be compatible with the aluminium and the

Nomex core, because Nomex is and Aramid fibrous material whereas aluminium is a

metal.

5.1.4. Gluing the face sheets with the core

Since it is hard for both of these materials to be in contact with one another, so, in that

case, an epoxy was used as a gluing material between the aluminium sheet and the

Nomex core. After applying adhesive on one of the sides of the face sheets, the side on

which epoxy was layered was then glued with the core and was set to be wetted for a

suitable amount of time so that both the face sheets and the core should make a strong

bond with one another. The large honeycomb prepared until this step is shown in the Fig.

5.3

Fig. 5.88 Large Honeycomb Structure

5.1.5. Polishing of the Honeycomb Sandwich Structures

Since, the sheets were bought from the metal shop; they were not in the suitable form to

be used for the testing, because impact analysis is sensitive to even a small scratch on the

sheet as well. Thus just to nullify that, the prepared honeycomb sandwich panels were

polished to remove the scratches from the surface of the aluminium face sheets. This was

an important step to be done and was performed in a firmly manner. The polished

honeycomb sandwich structure is shown in the Fig. 5.4.


Fig. 5.89 Polished Honeycomb Sandwiched Structure

Fig. 4.4.

The final product is shown in the Fig. 5.5.

Fig. 5.90 Honeycomb Sandwich Specimen in accordance with ASTM

5.2. ASTM standard used

From the fabrication and experimentation point of view of the honeycomb sandwich

structures, the ASTM C393 standard was used.

5.2.1. Significance and Use

Flexure tests on flat sandwich construction may be conducted to determine the sandwich

flexural stiffness, the core shear strength and shear modulus, or the facings compressive

and tensile strengths. Tests to evaluate core shear strength may also be used to evaluate

core-to-facing bonds.

This test method is limited to obtaining the core shear strength or core-to-facing shear

strength. This test method can be used to produce core shear strength and core-to-facing

shear strength data for structural design allowable, material specifications, and research

and development applications; it may also be used as a quality control test for bonded

sandwich panels.


Factors that influence the shear strength and shall therefore be reported include the

following: facing material, core material, adhesive material, methods of material

fabrication, core geometry (cell size), core density, adhesive thickness, specimen

geometry, specimen preparation, specimen conditioning, environment of testing,

specimen alignment, loading procedure, speed of testing, and adhesive void content.

Further, core-to-facing strength may be different between precured/bonded and co-cured

facings in sandwich panels with the same core and facing material.

5.2.2. Scope

This test method covers determination of the core shear properties of flat sandwich

constructions subjected to flexure in such a manner that the applied moments produce

curvature of the sandwich facing planes. Permissible core material forms include those

with continuous bonding surfaces (such as balsa wood and foams) as well as those with

discontinuous bonding surfaces (such as honeycomb).


6. Experimentation

An experimental investigation on honeycomb sandwich plates subjected to quasi-static

indentation or compression test is presented in this section. The test specimens consist of

Nomex paper (Aramid fiber) for the honeycomb core with aluminium alloy 5052 for the

face sheets. Each of the specimens is measured to be 14 cm x 5 cm with a core thickness

of 5 mm and a thickness of 1.4 mm for both the top and bottom face sheets. All tests are

conducted at a temperature of 25.5 0C and humidity of 47%. The static indentation test

results are shown in the Figs (7.5-7.9), respectively.

The static indentation was performed using the Load frame system, shown in the Fig.

operating under displacement control at a constant cross-head speed of 1mm/min. The

specimen was positioned between the top and bottom clamp plates, with the mid-point of

the plate directly located underneath the indenter. The two clamped plates were then

bolted in place manually.

The analytical solution presented in the previous chapters is validated with the results of

indentation test performed on 0041L-HC-FLEX samples. A total of five structural

configurations (summarized in the Table 7.7) were tested by varying the laminate

thickness. The reference configuration S1 is a honeycomb sandwich panel with a nominal

thickness of 7.8 mm, supported over a flat support.

The instrumented machine consists of a Load Frame model 8502, and the software used

for data recording/analysis is the X version. To accomplish this task, two different types

of test are performed. The tests are:

1. Flexural bending test

2. Three point bend test

6.1. Flexural Bending Test

The flexural bending test was performed on the aluminium sheet to measure the Young’s

Modulus. The measured value was then used for further investigation of other properties.


This test is performed by using different weights and as per ASTM standards under

normal environmental conditions and are shown in the Figs. (6.1-6.4)

6.1.1. Placing of sheet on the supports

The first step in flexural bed test is to place the aluminium sheet properly on the two

sports of the apparatus. Only properly adjusted and justified placement of the testing

material gives the correct value of the young modulus.

Fig. 6.91 Placing the Aluminium sheet on the supports

6.1.2. Adjustment of weight

The second step in bend test is to adjust the weight at the centre of aluminium sheet is of

prior importance. Properly adjusted weight will give us the proper value of the Young’s

Modulus.

Fig. 6.92 Placing the weight


6.1.3. Removal of errors

The third step is to have a slight contact of the reading dial with the sheet to be test and is

placed at the center. The reading shown on the dial presently is set to zero just to remove

any error. This removal of error is necessary and will then not affect our subsequent

calculations.

Fig. 6.93 Removing errors from dial

6.1.4. Calculation of Flexural Modulus

The weight on the sheet produced bending in it. As the bending begins, the dial begins to

move with it. The dial stops at the maximum point of bending showing the amount of

bend produced in the sheet in SI units.

Fig. 6.94 Calculating Flexural Modulus


Same procedure is repeated for a series of tests at different weights. And at each value of

weight, corresponding value of young’s modulus is generated. The average value gives us

the required young’s modulus. Te results are shown in the table 6.9 of chapter 6.

6.2. Three Points Bend Test

Here a compression (three point bend test) test is performed using a load Frame (8502)

under room temperature and as per ASTM standards. In three points bend test the

specimen is place horizontally on the supports. The supports are at an equal distance from

the center of the specimen. The indenter applies indentation or compression on the

specimen at an equal strain rate. The purpose of the testing is to find out the Young’s

Modulus of the overall sample. The results are shown in chapter 6.


7. Results & Discussion

The dimension of the specimen used is as under:

Table 7.5 Dimension used for the Specimen

Length of specimen/mm

Width of specimen/mm

Thickness of each face sheet/mm

Thickness of core/mm

Thickness of cell wall/mm

140 50 1.4 5.0 0.08

7.1. Results of optimization of Elastic constants

Following are the result of elastic modulus that is found using the study presented in

section 3.1.

Table 7.6 Elastic Modulus resulted

Ex Ey Ez G12 G13 G23 v21 v13 v23

0.024585 0.024585 4.2791 0.006093 0.80571 0.80571 0.33 0.001896 0.001896

7.2. Results through linear model

For a velocity of 30m/s, x1, x2 shown in Fig 7.1 are the displacements and Fc shown in

Fig. 7.2 shows the contact force.

Fig. 7.95 Deflection vs. Time


Fig. 7.96 Force vs. Deflection

The table 7.3 shows reference values:

Table 7.7 Reference values

V0 X1 (max) X2 (max) Fc (max) ω Tc

5.0 1.1891e-005 3.0686e-006 6.7024e+004 4.2050e+005 7.4711e-006

10.0 2.3781e-005 6.1371e-006 1.3405e+005 4.2050e+005 7.4711e-006

15.0 3.5672e-005 9.2057e-006 2.0107e+005 4.2050e+005 7.4711e-006

20.0 4.7562e-005 1.2274e-005 2.6809e+005 4.2050e+005 7.4711e-006

25.0 5.9453e-005 1.5343e-005 3.3512e+005 4.2050e+005 7.4711e-006

30.0 7.1343e-005 1.8411e-005 4.0214e+005 4.2050e+005 7.4711e-006

35.0 8.3234e-005 2.1480e-005 4.6917e+005 4.2050e+005 7.4711e-006

40.0 9.5124e-005 2.4549e-005 5.3619e+005 4.2050e+005 7.4711e-006

50.0 1.0701e-004 2.7617e-005 6.0321e+005 4.2050e+005 7.4711e-006

7.3. Results through non-linear impact model

Table 7.8 Results through non-linear impact model

V0 X1 (max) X2 (max) Fc (max) ω Tc

5.0 1.1203e-004 4.3672e-007 8.9564e+003 9.5634e+004 3.2850e-005

10.0 1.9527e-004 9.7344e-007 2.0586e+004 1.0965e+005 2.8650e-005

15.0 2.7029e-004 1.5898e-006 3.3483e+004 1.1900e+005 2.6400e-005

20.0 3.4044e-004 2.3101e-006 4.7261e+004 1.2592e+005 2.4950e-005

25.0 4.0718e-004 3.0742e-006 6.1734e+004 1.3172e+005 2.3850e-005

30.0 4.7131e-004 3.8712e-006 7.6790e+004 1.3659e+005 2.3000e-005

35.0 5.3338e-004 4.6935e-006 9.2356e+004 1.4088e+005 2.2300e-005

40.0 5.9371e-004 5.5361e-006 1.0837e+005 1.4444e+005 2.1750e-005

50.0 7.1018e-004 7.2654e-006 1.4161e+005 1.5104e+005 2.0800e-005


7.4. Results of Qualitative Analysis/uv-plane

Fig. 7.97 Velocity vs. Deflection

7.5. Results of Perturbation Theory

Equation (3.2.4.b.73) and properties given in Table 7.3 and 7.4 are used to calculate

natural frequency of honeycomb sandwich structure. Frequency of honeycomb sandwich

structure changes with amplitude. But it changes at very large amplitude. This shift is

significant where amplitude of vibration is very large e.g. in BOEING 737.

Fig. 7.98 Variation of Amplitude vs. Frequency


It can be observed that at very large amplitude, frequency up to 2 decimal places remains the same.

7.6. Results of impact analysis using LS-DYNA

After the impact is simulated and damages are observed as shown in Table 7.5.

Table 7.9 Sectional view of the damaged model

Velocity (m/s)

Failure type Section view

10

Skin Core Delamination

2030405060

70Core Crush+ Core

Buckling

80 Top face sheet yielding

90Core crush+plastic

buckling100


110 Bottom face yield

120 Core crush

130Core Crush+ plates peel

140

150

7.7. Experimental Results

Young’s modulus of the aluminium face sheet is obtained by flexural bending of the

specimen, E facesheet=64 GPa. The results of 3 point bend test are given below:

7.7.1. Testing Parameters

Following are the main parameters needed to perform the testing of the specimen.

Table 7.10 Testing Parameters

Testing Machine Rate of Loading Test Type Temperature 0C Humidity %

Load Frame

(8502)

1mm/min Compression (3

Point Bend Test)

25.5 47

7.7.2. Three point bend test results

The results obtained from the three point bend test performed on the given plate of

honeycomb sandwich structure are summarized in Table 7.7:


Table 7.11 Testing Results

Sample Sandwich

Average

Thicknes

s d (mm)

Core

Average

Thickness

c (mm)

Sandwich

Average

Width b

(mm)

Facing

Thickness

t (mm)

Span

Length

L (mm)

Break

Load P

(N)

Core

Shear

Stress

(MPa)

P/(d+c)

b

Facing

Bending

Stress

(MPa)

PL/2t(d+

c)b

S1 7.80 5.00 50.37 2.8 94 319.73 0.50 8.32

S2 7.82 5.00 50.12 2.8 94 952.87 1.448 24.89

S3 7.80 5.00 50.98 2.8 94 407.02 0.62 10.47

S4 7.83 5.00 50.43 2.8 94 1066.21 1.65 27.66

S5 7.80 5.00 51.12 2.8 94 366.63 0.56 9.41

It is observed from the test results that the samples S1, S3 and S5 having same thickness

(d) and have very close values of the core shear stress and facing bending stress.

However, the results of the samples S2 and S4 deviate largely from the other specimens.

Following are the plots obtained from the three point bend test performed on the

Universal Test Machine.

Fig. 7.99 Load vs. Position plot for Specimen (S1)



Table 7.12 Young Modulus calculated from Flexural Test

S1 20.92 GPa

S3 17.82 GPa

S5 19.37 GPa

Average

19.37 GPa

Table 7.8 shows the Young modulus of honeycomb sandwich structure obtained from the

tests obtained above. Young modulus of face sheet is measured from flexural testing.

While the core elastic modulus is extracted using Table 7.7.

7.8. Failure Analysis Approach

Different impacts are made on the specimen and the results are saved as .k file. Then

frequency analysis on the damaged model is performed to get the natural frequency.

These are compared with the results obtained previously.

Damaged model at 150 m/s is shown below:


Fig. 7.104 Damaged Model at 150 m/s

Table 7.13 Frequencies through Perturbation, damaged and undamaged LS-DYNA model

Velocity

(m/s)

Frequency through

Perturbation

Frequency of undamaged through LS-

DYNA Model

Frequency of damaged through

perturbation

Frequency of damaged through LS-DYNA Model

10 839.2 826.2 839.2 828.7

20 839.2 826.2 839.2 830.3

30 839.2 826.2 839.2 832.2

40 839.2 826.2 839.2 833.9

50 839.2 826.2 839.2 835.1

60 839.2 826.2 839.2 836.3

70 839.2 826.2 845.2 847.3

80 839.2 826.2 845.2 849.7

100 839.2 826.2 845.2 865.2

110 839.2 826.2 872.2 875.5

120 839.2 826.2 872.2 881.5


130 839.2 826.2 892.1 903.5

150 839.2 826.2 892.1 904.1

The graph of the result is plotted below:

20 40 60 80100

120140

780

800

820

840

860

880

900

920

DamagedLs Dyna NaturalPerturbation NaturalDamged Perturbation

Fig. 7.105 Plot between the Damaged, Ls-DYNA natural and Perturbation Method

In Fig. 7.11 a comparison is shown between the frequencies obtained from the damaged,

LS-DYNA natural and Perturbation natural. This graph shows that the results of the

natural frequencies obtained from the LS-DYNA and Perturbation method are parallel to

each other. However, the frequency of the damaged model varies with the increase of

velocity.


f (Hz)

Velocity (m/s)

8. Proves for Developed Theories

8.1. Prove of developed iteration method

A detailed literature is available on the experimental investigation of the honeycomb

sandwich structures in comparison with analytical techniques. Finite Element (FE)

analyses of the honeycomb structures using different commercial software have also been

performed by many researchers to predict the usefulness of the structures for future

applications. Analytical modeling of honeycomb sandwich structure is still needed to be

explored more for better understanding of these structures. Exact solutions for impact on

honeycomb sandwich structure are usually obtained using Jacobbian elliptical integral.

Since, for less computation and complexity, different numerical techniques have been

employed instead of Jacobbian elliptical integral. Keeping in view the present work

proposes a simple numerical technique which could be employed to solve the impact

phenomenon on honeycomb sandwich structure both efficiently and accurately in terms

of computation and time respectively. A number of researchers have used spring-mass

model on a large scale to study the failure, dynamic and impact response of honeycomb

sandwich structure. Different numerical studies were performed by Serge Abrate et.al

[29-33] on spring-mass model. For quick accurate results Yang Mijia et. al [34] used the

linearization technique. Akil Hazizan and Cantwell et. al [35] uses spread sheet to solve

for force and displacement time response of sandwich structure. Although it is a simple

technique but to obtain one value, iterations were carried until both left and right hand

side of energy-balance equation becomes equal. G.B. Chai et. al [36] first solved energy-

balance equations for load and velocity histories, and then they integrated the velocity-

time response to calculate deflection. This was indirect technique and requires heavy

computation power. Daiva Zeleniakiene did experimental investigation and the results

were obtained using equations of laminate theory and the modified beam theory

equations were compared with experimental ones, this methodology was used for the

strength analysis of the investigated structure.

Unlike other numerical methods, here using an entirely different methodology is

introduced Newton Forward Difference technique containing velocity term. Classical


techniques are also applied to convert the obtained equations into equations that are

independent of differentiation. By introducing the velocity term, the method gains an

advantage that only four initial conditions and no assumption of higher terms are

required.

After employing initial conditions into the Runge-Kutta-Fulberg 4-5 th order , the results

were obtained in the form of graphs of distance of indenter vs. Time, distance vs. Time

for sandwich plate and contact force vs. Time. The proposed technique is then compared

with more complex numerical techniques and the results are verified using past literature

and shows that the proposed technique produces accurate results. This technique requires

less computation using simple code and could be used as an alternative of finite element

method. A MATLAB and MAPLE code is presented at appendix to give the results.

8.1.1. Results

Table 8.1 shows the values used in the developed model:

Table 8.14 Properties and their values used in the process

Properties Values

Km 1000

K c 1.54 × 1010

K core 1.0 ×1008

Kbs 3.84 × 1006

k bs(¿K bs+K core) 1.0384 × 1008

V 0 30

m1 0.5

m2 0.1

Below are the graphs ofx1 , x2 , Fc vs . time, the values are compared with two numerical

methods:138


Fig. 8.106 Plot for Distance of indenter vs. time

Figure 8.1 shows a plot against displacement of indenter vs. time.

Fig. 8.107 Plot for Distance vs. time for Sandwich plate

The difference between x1 and x2 gives the indentation produced during the impact.

Figure 8.2 shows a plot against distance vs. time for honeycomb sandwich plate. Thus it

explains the motion of the honeycomb sandwich plate during impact.


Fig. 8.108 Plot for Contact Force vs. Time

Plot of contact force vs. time is shown in the Figure 8.3. The area under the graph

represents the total change in momentum and can also be used to calculate the impact

energy absorbed by the sandwich structure.

8.1.2. Conclusion of developed iteration technique

The precision of other methods is almost the same as that of the Newton forward

difference method developed in this study. So this method can be used instead of other

complex methods. It is computationally less tedious. Once implemented, it does not

requires emergency exits, like other numerical techniques; since it is defined even at

those points where Runge-Kutta 4-5th order, Cash-Karp 4-5th order; Dverk 7-8th order are

undefined.

8.2. Prove of Developed Oblique Model

Oblique impact model is not developed till date therefore; it is very difficult to verify the

model with the previous studies (which is not available). To validate whether the theory

developed is correct or not, just put friction (coefficient of friction) as zero. Check

whether it reduces to 2-DOF spring-mass system already presented in this thesis.


Without friction oblique impact would be same as normal impact, therefore, transfer of

force could not be carried out. The results are shown in tabular form below:

Table 8.15 Results showing the prove of Developed Oblique Model

Original Equations At Zero Friction

β=tan−1 1μ

β=n π2

m1 x1=−K c¿¿ m1 x1=0

m2 x2+K plate x2+Kbending x2−K c¿¿ m2 x2=0

m1 y1=−K c ( y1− y2 )32 (sin( tan−1( 1

μ ))√1+μ2) m1 y1=−K c ( y1− y2 )32

m2 y2+K m y23+Kbs y2+K coℜ y2−K c ( y1− y2 )

32 (sin( tan−1( 1

μ ))√1+μ2)=0 m2 y2+K m y23+Kbs y2+K core y2−K c ( y1− y2 )

32=0

F c=K c ( y1− y2 )32√1+μ2 F c=K c ( y1− y2 )

32

It can be seen that the last three equations of the second column reduces to normal

impact equations. Thus it is proved that the oblique impact model is correct.

8.3. Prove of natural frequency

No such formulation is available for finding the natural frequency of the system

presented earlier. So indirect method of verification is made, here instead of natural

frequency, displacement vs. time is compared. If the wavelengths are same, then

frequencies are same. So for that matter, Jacobbian elliptical integral is used.


8.3.1. Formulation

The input properties used tabulate is Table 8.3:

Table 8.16 Input Properties

Properties Values

Km 1000

K c 1.54 × 1010

K core 1.0 ×1008

Kbs 3.84 × 1006

k bs(¿K bs+K core) 1.0384 × 1008

V 0 30

m2 1

The free response equation becomes:

The Jacobbian of the equation is:

Substituting same initial conditions as used in iteration procedure to get values of

constants, these are: ,

The Jacobbian integral of the free response of equation is:

x (t )=0.05000000003 Jacobbian(9.999975983√1038405 t+1.570761681 , 2.40754426810−7 I √207681ian constants , theyfor finding the natural frequency of the system presented earlier .)


8.3.2. Results

Equation (3.2.4.b.71) and (7.3.a.) are plotted on the same graph, as shown here is:

Fig. 8.109 Plot between x and t by using Perturbation Method

The wavelength of our free response equation and that of Jacobbian elliptical integral

exactly overlaps. This shows that our perturbation study is correct.

9. Conclusion of Impact Analysis

The ω analysis, first provide platform where the analysis of damage become easier.

Traditionally following approach is used to complete the project:


The procedure we adopted in this project is:

Through this procedure, there are numerous advantages. First and most important, the

experimental cost has been reduced. Table below shows the equipment price:

Table 9.17 Equipment Price

Equipment Velocity range (m/s) Price (GBP)


Instron Dynatup 0−8 120,000

Integerated Frequency

measuring equipments

0−∞ < 500

Secondly, frequency analysis is less complex. It requires few computations. Also the

number of equations, time and properties required are less in frequency analysis, as

shown in section 4.4. Damage in model can be predicted through frequency analysis.

From section 7.8 it can be seen that as damage is introduced in the structure, there is shift

in natural frequency of the sample. By using 3 point bend test and flexural tests, even the

effect of adhesives were considered in the project.

Presently, only three failure criteria are available: First core is crushed, then upper face

sheet is broken and finally the bottom face sheet is broken. These failures are shown as

three times rising of frequency. From during impact analysis, to pre and post impact

analysis is performed.

10. Future Recommendations


The present theories only shows three complete failure modes, core crushing

followed by upper plate breaking and lastly bottom plate breaking. But actually,

in these modes there is shifting of failure criteria i.e. the above mentioned modes

occur in 7 stages instead of three as shown by LS-DYNA results in section 6.8.

Thus theories regarding improved model should be present

Then the study of natural frequency should be extended to cambered and tapered

honeycombs. These honeycombs are used in the wing of airplane and thus

calculating there natural frequencies has many advantages like avoiding

resonance

Optimizing the design of wing could be another study which could be performed

REFERENCES

[1] Abrate S. Impact of composite laminates. Appl Mech Rev1991; 44(4):155±90


[2] Abrate S. Impact of laminated composites: recent advances. Appl Mech Rev

1994;47(11):517±44

[3] Abrate S. Impact on composite structures. Cambridge: Cambridge University Press;

1998

[4] C.C. Foo *, L.K. Seah, G.B. Chai. Low-velocity impact failure of aluminium

honeycomb sandwich panels, School of Mechanical and Aerospace Engineering,

Nanyang Technological University, Singapore Available online 17 October 2007,

Composite Structures 85 (2008) 20–28

[5] Jeom Kee Paika,*, Anil K. Thayamballib, Gyu Sung Kima, The strength

characteristics of aluminum honeycomb sandwich panels, Department of Naval

Architecture and Ocean Engineering, Pusan National University, Pusan 609-735, South

Koreab Chevron Shipping Co., San Francisco, CA 94105, USA, Thin-Walled Structures

35 (1999) 205–23

[6] Md. Akil Hazizan, W.J. Cantwell*, the low velocity impact response of an aluminium

honeycombSandwich structure, Materials Science and Engineering, University of

Liverpool, Liverpool L69 3GH, UK, Composites: Part B 34 (2003) 679–687

[7] Inés Ivañez, Carlos Santiuste, Sonia Sanchez-Saez *, FEM analysis of dynamic

flexural behaviour of composite sandwich beams with foam core, Department of

Continuum Mechanics and Structural Analysis, Carlos III University of Madrid, Avda.

Universidad, 30, 28911 Leganes, Spain

[8] T. Besant*, G.A.O. Davies, D. Hitchings, finite element modelling of low velocity

impact of composite sandwich panels, department of aeronautics, Imperial college of


science, technology and medicine, prince consort road, London SW7 2BY, UK,

Composites: Part A32 (2001) 1189-1196

[9] A Chawla, S Mukherjee, Dileep Kumar, T. Nakatani and M. Ueno, PREDICTION OF

CRUSHING BEHAVIOUR OF HONEYCOMB STRUCTURES

[10] M. Meo a,*, A.J. Morris a, R. Vignjevic a, G. Marengo b, Numerical simulations of

low-velocity impact on an aircraft sandwich panel, Composite Structures 62 (2003) 353–

360

[11] V. Crupi*, G. Epasto, E. Guglielmino, COLLAPSE MODES IN ALUMINIUM

HONEYCOMB SANDWICH PANELS UNDER BENDING AND IMPACT

LOADING, DCIIM, University of Messina, Contrada Di Dio, Sant’Agata (ME), 98166

Messina, Italy

[12] Yi-Ming Jen *, Chih-Wei Ko, Hong-Bin Lin, Effect of the amount of adhesive on

the bending fatigue strength of adhesively bonded aluminum honeycomb sandwich

beams, International Journal of Fatigue 31 (2009) 455–462

[13] A.N. Palazotto *, E.J. Herup, L.N.B. Gummadi, Finite element analysis of low-

velocity impact on composite sandwich Plates, Composite Structures 49 (2000) 209-227

[14] B. Castanie´ a,_, C. Bouveta, Y. Aminandab, J.-J. Barrauc, P. Thevenet, Modelling

of low-energy/low-velocity impact on Nomex honeycomb sandwich structures with

metallic skins, International Journal of Impact Engineering 35 (2008) 620–634.

[15] Nettles, A.T. and Douglas, M.J. (2000) A Comparison of Quasi-static Indentation to

Low-velocity Impact, NASA TP 210481, August 2000

[17] Feraboli P., Kedward K.T., “Enhanced evaluation of the low velocity impact

response of composite plates” – AIAA Journal – 42/10, 2004, pp. 2143-2152


[18] Feraboli P., Kedward K.T., “A new Composite Structures Impact Performance

Assessment Program” – Composites Science and Technology – 66/10, 2006, pp. 1336-

1347

[19] Feraboli, P., “Some Recommendations for the characterization of the impact

performance of composite panels by means of drop tower impact testing” – Journal of

Aircraft, in press Winter 2006

[20] Feraboli, P., “Modified SDOF models for improved representation of the impact

response of composite plates” – Journal of Composite Materials, in press Fall 2006

[21] Feraboli P.J., Ireland, D.R., Kedward, K.T., “On the role of Force, Energy and

Stiffness in Low Velocity Impact events” – 18th ASC Technical Conference, Gainesville,

FL – 2003

[22] Feraboli P., Kedward K., “A multi-parameter approach to impact performance

characterization” – 19th ASC/ ASTM D30 Joint Technical Conference, Atlanta, GA –

2004

[23] Feraboli P.J., Ireland, D.R., Kedward, K.T., “The role of Peak Force and Impact

Energy in Low Velocity Impact events” – 45thAIAA/ASME/ASCE/AHS/ASC

Structures, Dynamics and Materials Conference, No. 2004-1841, Palm Springs, CA –

2004

[24] Ambur, D.R., Prasad, C.B., Rose, C.A., Feraboli, P., Jackson, W.C., “Scaling the

nonlinear impact response of flat and curved anisotropic composite plates” – 46th

AIAA/ASME/ASCE/AHS/ASC Structures, Dynamics and Materials Conference, No.

2005-2224, Austin, TX – 2005


[25] Feraboli, P., “Damage resistance characteristics of thick-core honeycomb composite

panels” – 47th AIAA/ASME/ASCE/AHS/ASC Structures, Dynamics and Materials

Conference, No. 2006-2169, Newport, RI – 2006

[26] Damage Resistance Characteristics of Thick-Core Honeycomb Composite Panels

Paolo Feraboli1, 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics,

and Materials Conference 1 - 4 May 2006, Newport, Rhode Island

[27] Enhanced Evaluation of the Low-Velocity Impact Response of Composite Plates,

Paolo Feraboli ∗ and Keith T. Kedward† University of California, Santa Barbara, Santa

Barbara, California 93106, AIAA JOURNAL Vol. 42, No. 10, October 2004

[28] Low-velocity Impact of Sandwich Composite Plates, by J. Gustin, M. Mahinfalah,

G. Nakhaie Jazar and M.R. Aagaah

[29] Abrate S. Impact of composite laminates. Appl Mech Rev 1991;44:155-90

[30] Abrate S. Impact of laminated composite: recent advances. Appl Mech Rev

1994;47(11):517-44

[31] Abrate S. Localized Impact on sandwich structures with laminated facings. Appl

Mech Rev 50(2):69-82

[32] Abrate S. Impact on composite structure. Cambridge University Press; 1998.

[33] Abrate S. Modelling of impacts on composite structures

[34] Mijia Yang, Impact mechanics of elastic and elastic plastic sandwich structures, May

2006

[35] A. Hazizan, J. Cantwell, The low velocity response of an Aluminium honeycomb

sandwich structure


[36] C.C. Foo, L.K. Seah, G.B. Chai, A modified energy-balance model to predict low

velocity impact response for sandwich composites


Appendix A-MATLAB Program

Amplitude vs. frequencyclose all;clear all;clc;%------ calculation of stifness parameters----%Ra= 10*10^(-3);%input (' what is the value of radius of indentor = '); Ezz_plate= 69*10^9;%input (' what is the young modulus of face plate contacting the impactor in thickness direction = '); Ez_steel=200*10^9; val=0.33; vst=0.27; E_star=((1-val^2)/(Ezz_plate)+(1-vst^2)/(Ez_steel))^(-1); Ef=0.024585*10^9; kc = 4/3 * E_star * (Ra)^(1/2); b=50*10^(-3);%input(' what is width of sandwich beam = '); L=140*10^(-3);%input( ' what is length of the sandwich beam = '); t_top_plate=1.4E-3; %input (' what is the thickness of the top plate = '); t_bottom_plate=1.4E-3;%input(' what is the thickness of the bottom plate = '); t_core=5E-3;%input(' what is the thickness of the core = '); Ic=1/12*(b)*(t_core)^3; Iplate=1/12*(b)*(t_top_plate)^3+(b*(t_top_plate)*((t_core/2)+(t_top_plate/2))^2); I=2*Iplate*Ic; Ecore_in_plane=69*10^9; Ezz_core=69*10^9;%input(' enter the young modulus of core in impact direction = '); D=0*Ezz_plate*(2*Iplate)+Ecore_in_plane*(Ic); kbs=48*D/(L)^3; Volume_of_core=7913.55*10^(-9); Area=Volume_of_core/t_core; kcore = Ezz_core*Area/t_core;% % calculate km % km=010;%supposed%%------Function to plot amplitude vs frequency-------------%%kbs=3.84E6;%(1/kb+1/ks)^(-1);%kcore=1E8;%input(' Enter the value of core stiffness = ');%km =10^3;%input ('enter the value of membrane stiffness = ');


m2=0.0759846*10336; %input('mass of plate');w0=((kbs+kcore)/m2)^(1/2);x2_at_min_p.e=0;%kc=1.54*10^10;m1=0.031878;for i=1:1:2000s0(i)=0.005+i*0.01;v0(i)=0;a0(i)=((s0(i))^2+(v0(i)/w0)^2)^(1/2);beta0(i)=acos(s0(i)/a0(i));R(i)=1/2*w0*m2*(a0(i))^2*(1/(kbs+kcore))*((-48*km*(sin(beta0(i)))^2*(19)^(1/2))/m2+(136*km*m2*(sin(beta0(i)))^4)/m2+(32*km*(sin(beta0(i)))^4*(19)^(1/2))/m2-(204*km*(sin(beta0(i)))^2)/m2+51*km/m2+(12*km*(19)^(1/2))/m2)^(1/2);R1(i)=1/4*(km)*((64*(cos(beta0(i)))^4-8-(19)^(1/2)-32*(cos(beta0(i)))^(2)-4*(19)^(1/2)*((cos(beta0(i)))^(2))+8*(19)^(1/2)*(cos(beta0(i)))^(4))*(a0(i))^3)/(kbs+kcore);A(i)=a0(i)-(R(i))-R1(i);W(i)=(1/8)*((8*kbs)+(8*kcore)+3*km*(A(i))^2)/((w0)*(m2));f(i)=W(i)/(2*pi);endfigure(1);plot(f,a0,'marker','+','markersize',2)hold on;grid on;xlabel(' frequency/ Hz ');ylabel(' amplitude of vibration ')

Failure Analysis (adjusting values of elastic constants)clear all;close all;clc;Ra= 10*10^(-3);%input (' what is the value of radius of indentor = '); Ezz_plate= 69*10^9;%input (' what is the young modulus of face plate contacting the impactor in thickness direction = '); Ez_steel=200*10^9; val=0.33; vst=0.27; E_star=((1-val^2)/(Ezz_plate)+(1-vst^2)/(Ez_steel))^(-1); Ef=0.024585*10^9; kc = 4/3 * E_star * (Ra)^(1/2); b=50*10^(-3);%input(' what is width of sandwich beam = ');


L=140*10^(-3);%input( ' what is length of the sandwich beam = '); t_top_plate=1.4E-3; %input (' what is the thickness of the top plate = '); t_bottom_plate=1.4E-3;%input(' what is the thickness of the bottom plate = '); t_core=5E-3;%input(' what is the thickness of the core = '); Ic=1/12*(b)*(t_core)^3; Iplate=1/12*(b)*(t_top_plate)^3+(b*(t_top_plate)*((t_core/2)+(t_top_plate/2))^2); I=2*Iplate*Ic; Efx= Ezz_plate; Ecore_in_plane=69*10^9; Ezz_core=69*10^9;%input(' enter the young modulus of core in impact direction = '); D=0*Ezz_plate*(2*Iplate)+Ecore_in_plane*(Ic); kbs=48*D/(L)^3; Volume_of_core=7913.55*10^(-9); Area=Volume_of_core/t_core; kcore = Ezz_core*Area/t_core;% % calculate km % km=0;%supposed% m1=0.031878;m2=0.0759846;tf=t_top_plate;thickness_of_core=t_core;Gcxz=6.0930*10^6;d=t_core+t_top_plate;h=t_core+t_top_plate*2;% % kc=9200000000.00000;%1.54*10^10;%input(' Enter the value of contact stiffness = ');% kb=input( ' Enter the value of bending stiffness = ');% ks=input (' Enter the value of shear stiffness = ');% kbs=7360274.27200000;%3840000.00;%(1/kb+1/ks)^(-1);% kcore=0;%1*10^8;%input(' Enter the value of core stiffness = ');% km =0;%320837415000.000;%1000;%input ('enter the value of membrane stiffness = ');y1(1)=0;%input (' enter the initial value of indentor = ');y1_dot(1)=50;% input (' enter the value of indentor velocity at contact = ');y2(1)=0;%input(' enter the initial value of plate distance= ');y2_dot(1)= 0;%input(' enter the initial velocity of honeycomb = ');


delta_t= 1E-7 ;%input(' enter the value of time-step = ');p=1; % initialize value of p %t(1)=0;%input(' enter the value of initial time = ');%--------Equations---------%Fc(p)= kc*(y1(p)-y2(p))^(3/2);y1(p+2) = -(Fc(p)/m1)*(delta_t)^2-y1(p)+2*(y1_dot(p)*(delta_t)+y1(p));y1(p+1)=y1_dot(p)*(delta_t)+y1(p);%-------Equations for y2 -------%A(p)=kbs*y2(p)+kcore*y2(p)+km*(y2(p))^3-Fc(p);y2(p+2)= -A(p)/m2*(delta_t)^2-y2(p)+2*(y2_dot(p)*(delta_t)+y2(p));y2(p+1)=y2_dot(p)*(delta_t)+y2(p);%-------------------------------------%%--------------loop--------------------------%%while (y1(p) >= 0)&&(y2(p)>0) for p=2:4000 t(p)=delta_t*(p-1)+t(1); y1_dot(p)=(y1(p+1)-y1(p))/(delta_t); y2_dot(p)=(y2(p+1)-y2(p))/(delta_t); Fc(p)= kc*(y1(p)-y2(p))^(3/2); A(p)=kbs*y2(p)+kcore*y2(p)+km*(y2(p))^3-Fc(p); y1(p+2) = -Fc(p)/m1*(delta_t)^2-y1(p)+2*(y1_dot(p)*(delta_t)+y1(p)); y2(p+2)= -A(p)/m2*(delta_t)^2-y2(p)+2*(y2_dot(p)*(delta_t)+y2(p)); %p=p+1; k(p)=y1(p)-y2(p); if k(p)<0 l(p)=p; end end%endt(p)=delta_t*(p-1)+t(1);t(p+1)=delta_t*(p)+t(1);t(p+2)=delta_t*(p+1)+t(1);Fc(p)=kc*(y1(p)-y2(p))^(3/2);Fc(p+1)=kc*(y1(p+1)-y2(p+1))^(3/2);Fc(p+2)=kc*(y1(p+2)-y2(p+2))^(3/2);idx = find(l~=0, 1, 'first');tm=t(idx);x1_max=max(y1(1:idx));x2_max=max(y2(1:idx));Fc_max=max(Fc(1:idx));


thetaar=L/(thickness_of_core)*(Gcxz/(2*Efx)*(thickness_of_core/tf)*(1+3*d^2/tf^2))^(1/2);del_max=x2_max;zeta=thetaar*(tf^5/9+(tf^3*d^2)/3)/(h*tf^3*(thetaar-1)/3+tf^4/3+tf^2*d^2);d=(thickness_of_core)+(t_top_plate);del_distance=del_max-del_linear;compressive_strength_of_core=276*10^6;sigmasc=compressive_strength_of_core;

%------------------- Experimental Analysis -----------------------------%strain=[0 0.008 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06];str=[0 480 545 580 630 670 710 730 755 770 780 790 792];stress=10^6.*str;strain1=6*x2_max*(thickness_of_core)/(L^2);stress1=3*Fmax*L/(2*b*d^2);strain_2=interp1(stress,strain,stress1);stress2 =stress1+10*10^6;stress3 =stress1-10*10^6;strain_3=interp1(stress,strain,stress2);strain_4=interp1(stress,strain,stress3);del_strain =abs(strain_4-strain_3);E_damged_plate = (stress2-stress3)/(del_strain);plot(strain,stress);grid on;xlabel('Strain');ylabel('Stress Nm-2');title(' stress strain graph for Aluminium Facesheet ');%------------------- 7 failure Criterias --------------------------------%% (a). Top skin Yield %Ffy= 4*sigmafy*(tf/L)*zeta*width_of_simply_supported_beam;W1=Ffy;if (W1<Fmax) k1=((m*v^2-del_linear*W1^2)/(del_distance)-W1)/(del_max);end% (b). Intra cell buckling% Fib= 8/(1-vf^2)*(tf/alpha)^2*(Ef)*(tf/L)*zeta*width_of_simply_supported_beam;% (c). Face Wrinkling %


Ffw= 4*B1*(Ef)^(1/3)*(Es)^(2/3)*(tf/L)*(rhoc/rhos)^(2/3)*zeta*width_of_simply_supported_beam;% (d). Core Shear %Fcs= 2*(A)*Es*d*(rhoc/rhos)^3*width_of_simply_supported_beam;% (e). Indentation %Fi= 3.25*(sigmasc)*(rhoc/rhos)^(5/3)*(indentation)*width_of_simply_supported_beam;W2=Fi;if (W2<Fmax) k2=((m*v^2-del_linear*W2^2)/(del_distance)-W2)/(del_max);end%------------------------------------------------------------------------%% if failure has occured, get kc, find Ez.... Before this check which% criteria occurs... use gradient between Fc and indentation graph to% calculate equivalent young modulus.... Convert system to 1 dof....then% use D=Ezz_plate*(2*Iplate)+Ecore_in_plane*(Ic) to calculate for Ezz_plate% and Ecore_in_plane, check... that whether you need to modify Ezz_plate or% Ecore_in_plane or both. Calculate kcore using kcore =% Ezz_core*Area/t_core

Phase plane%-----Phase plane------% clear all;close all;clc;Ra= 10*10^(-3);%input (' what is the value of radius of indentor = '); Ezz_plate= 69*10^9;%input (' what is the young modulus of face plate contacting the impactor in thickness direction = '); Ez_steel=200*10^9; val=0.33; vst=0.27; E_star=((1-val^2)/(Ezz_plate)+(1-vst^2)/(Ez_steel))^(-1); Ef=0.024585*10^9; kc = 4/3 * E_star * (Ra)^(1/2);


b=50*10^(-3);%input(' what is width of sandwich beam = '); L=140*10^(-3);%input( ' what is length of the sandwich beam = '); t_top_plate=1.4E-3; %input (' what is the thickness of the top plate = '); t_bottom_plate=1.4E-3;%input(' what is the thickness of the bottom plate = '); t_core=5E-3;%input(' what is the thickness of the core = '); Ic=1/12*(b)*(t_core)^3; Iplate=1/12*(b)*(t_top_plate)^3+(b*(t_top_plate)*((t_core/2)+(t_top_plate/2))^2); I=2*Iplate*Ic; Ecore_in_plane=69*10^9; Ezz_core=69*10^9;%input(' enter the young modulus of core in impact direction = '); D=0*Ezz_plate*(2*Iplate)+Ecore_in_plane*(Ic); kbs=48*D/(L)^3; Volume_of_core=7913.55*10^(-9); Area=Volume_of_core/t_core; kcore = Ezz_core*Area/t_core;% % calculate km % km=0;%supposed% m1=0.031878;m2=0.0759846;linestyles = cellstr(char('-',':','-.','--','-',':','-.','--','-',':','-',':',...'-.','--','-',':','-.','--','-',':','-.'));MarkerEdgeColors=jet(9); % n is the number of different items you haveMarkers=['o','x','+','*','s','d','v','^','<','>','p','h','.']; % % kc=9200000000.00000;%1.54*10^10;%input(' Enter the value of contact stiffness = ');% kb=input( ' Enter the value of bending stiffness = ');% ks=input (' Enter the value of shear stiffness = ');% kbs=7360274.27200000;%3840000.00;%(1/kb+1/ks)^(-1);% kcore=0;%1*10^8;%input(' Enter the value of core stiffness = ');% % km =0;%320837415000.000;%1000;%input ('enter the value of membrane stiffness = '); for nm=1:9


y1(1)=0;%input (' enter the initial value of indentor = ');y1_dot(1)=input (' enter the value of indentor velocity at contact = ');y2(1)=0;%input(' enter the initial value of plate distance= ');y2_dot(1)= 0;%input(' enter the initial velocity of honeycomb = ');delta_t= 1E-7 ;%input(' enter the value of time-step = ');p=1; % initialize value of p %t(1)=0;%input(' enter the value of initial time = ');%--------Equations---------% Fc(p)= kc*(y1(p)-y2(p))^(3/2);y1(p+2) = -(Fc(p)/m1)*(delta_t)^2-y1(p)+2*(y1_dot(p)*(delta_t)+y1(p));y1(p+1)=y1_dot(p)*(delta_t)+y1(p); %-------Equations for y2 -------%A(p)=kbs*y2(p)+kcore*y2(p)+km*(y2(p))^3-Fc(p);y2(p+2)= -A(p)/m2*(delta_t)^2-y2(p)+2*(y2_dot(p)*(delta_t)+y2(p));y2(p+1)=y2_dot(p)*(delta_t)+y2(p);%-------------------------------------% %--------------loop--------------------------% %while (y1(p) >= 0)&&(y2(p)>0) for p=2:4000 t(p)=delta_t*(p-1)+t(1); y1_dot(p)=(y1(p+1)-y1(p))/(delta_t); y2_dot(p)=(y2(p+1)-y2(p))/(delta_t); Fc(p)= kc*(y1(p)-y2(p))^(3/2); A(p)=kbs*y2(p)+kcore*y2(p)+km*(y2(p))^3-Fc(p); y1(p+2) = -Fc(p)/m1*(delta_t)^2-y1(p)+2*(y1_dot(p)*(delta_t)+y1(p)); y2(p+2)= -A(p)/m2*(delta_t)^2-y2(p)+2*(y2_dot(p)*(delta_t)+y2(p)); %p=p+1; k(p)=y1(p)-y2(p); if k(p)<0 l(p)=p; end end%endt(p)=delta_t*(p-1)+t(1);t(p+1)=delta_t*(p)+t(1);


t(p+2)=delta_t*(p+1)+t(1);Fc(p)=kc*(y1(p)-y2(p))^(3/2);Fc(p+1)=kc*(y1(p+1)-y2(p+1))^(3/2);Fc(p+2)=kc*(y1(p+2)-y2(p+2))^(3/2);idx = find(l~=0, 1, 'first');plot(y1_dot(1:idx),y1(1:idx),[linestyles{nm} Markers(nm)],'Color',MarkerEdgeColors(nm,:));hold on;H(nm)=1/2*m1*(y1_dot(1))^2; end grid on; xlabel('displacement'); ylabel('velocity');legend('v=05m/s h=0.3985 J','v=10m/s h=1.5939 J','v=15m/s h=3.5863 J','v=20m/s h=6.3756 J','v=25m/s h=9.9619 J','v=30m/s h=14.3451 J','v=35m/s h=19.5253 J','v=40m/s h=25.5024 J','v=50m/s h=39.8475 J')


hamza report

Documents