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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)
2Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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).
3Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
4Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Dedication
We pay our humble dedication of this research work to our parents!
5Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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!
6Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
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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
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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
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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
10Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
11Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
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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
13Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
14Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
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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
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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
Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
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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).
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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:
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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.
22Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
23Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
24Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
25Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
26Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
27Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
28Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
29Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
30Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
31Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
32Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
33Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
34Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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:
35Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
36Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
37Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
38Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
M and e e.nast abd qunli ashby shi gibson and ashby
0
50000
100000
150000
200000
250000
Series1
M and e e.nast abd qunli ashby shi gibson and ashby
0
100000
200000
300000
400000
500000
600000
700000
Series1
M and e e.nast abd qunli ashby shi gibson and ashby
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
M and e e.nast abd qunli ashby shi gibson and ashby
0
100000
200000
300000
400000
500000
600000
700000
Series1
M and e e.nast abd qunli ashby shi gibson and ashby
0
200000
400000
600000
800000
1000000
1200000
1400000
Series1
Fig. 3.17 Measurement of Elastic Property (E1)
39Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
3.1.2. Selection of E2
M and e e.nast abd qunli ashby shi gibson and ashby
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
M and e e.nast abd qunli ashby shi gibson and ashby
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
e.nast abd qunli ashby shi gibson and
ashby
0
200000
400000
600000
800000
1000000
1200000
Series1
M and e
e.nast abd qunli ashby shi gibson and
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.
40Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Test 1 Test 2
Test 3 Test 4
Test 5Test 6
Test 7
3.1.3. Selection of E3
M and e e.nast abd qunli ashby shi gibson and ashby
0
100000000
200000000
300000000
400000000
500000000
600000000
700000000
Series1
M and e e.nast abd qunli ashby shi gibson and ashby
0
200000000
400000000
600000000
800000000
1000000000
1200000000
Series1
M and e e.nast abd qunli ashby shi gibson and ashby
0
200000000
400000000
600000000
800000000
1000000000
1200000000
1400000000
1600000000
Series1
M and e e.nast abd qunli ashby shi gibson and ashby
0
200000000
400000000
600000000
800000000
1000000000
1200000000
1400000000
1600000000
Series1
M and e e.nast abd qunli ashby shi gibson and ashby
0
200000000
400000000
600000000
800000000
1000000000
1200000000
1400000000
1600000000
Series1
M and e e.nast abd qunli ashby shi gibson and ashby
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.
41Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
M and e e.nast abd qunli ashby shi gibson and ashby
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.
42Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Test 1 Test 2
Test 3 Test 4
Test 5Test 6
Test 7
3.1.5. Selection of G13
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
M and e e.nast abd qunli ashby shi gibson and ashby
0
5000000
10000000
15000000
20000000
25000000
Series1
Fig. 3.21 Measurement of Elastic Property (G13)
Shi formula for G13 gives more précised value as compare to experimental result.
43Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Test 1Test 2
Test 3 Test 4
Test 5 Test 6
Test 7
3.1.6. Selection of G23
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
M and e e.nast abd qunli ashby shi gibson and ashby
0
2000000
4000000
6000000
8000000
10000000
12000000
14000000
16000000
Series1
Fig. 3.22 Measurement of Elastic Property (G23)
Shi formula for G23 gives more precise value as compared to experimental result.
44Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Test 5
Test 2
Test 3 Test 4
Test 1
Test 6
Test 7
3.1.7. Selection of ν12
M and e e.nast abd qunli ashby shi gibson and ashby
0
0.2
0.4
0.6
0.8
1
1.2
Series1
M and e e.nast abd qunli ashby shi gibson and ashby
0
0.2
0.4
0.6
0.8
1
1.2
Series1
M and e e.nast abd qunli ashby shi gibson and ashby
0
0.2
0.4
0.6
0.8
1
1.2
Series1
M and e e.nast abd qunli ashby shi gibson and ashby
0
0.2
0.4
0.6
0.8
1
1.2
Series1
M and e e.nast abd qunli ashby shi gibson and ashby
0
0.2
0.4
0.6
0.8
1
1.2
Series1
M and e e.nast abd qunli ashby shi gibson and ashby
0
0.2
0.4
0.6
0.8
1
1.2
Series1
M and e e.nast abd qunli ashby shi gibson and ashby
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.
45Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Test 1 Test 2
Test 3Test 4
Test 5Test 6
Test 7
3.1.8. Selection of ν13
M and e e.nast abd qunli ashby shi gibson and ashby
0
0.00001
0.00002
0.00003
0.00004
0.00005
0.00006
0.00007
Series1
M and e e.nast abd qunli ashby shi gibson and ashby
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
M and e e.nast abd qunli ashby shi gibson and ashby
0
0.00002
0.00004
0.00006
0.00008
0.0001
0.00012
Series1
M and e e.nast abd qunli ashby shi gibson and ashby
0
0.00002
0.00004
0.00006
0.00008
0.0001
0.00012
Series1
M and e e.nast abd qunli ashby shi gibson and ashby
0
0.00002
0.00004
0.00006
0.00008
0.0001
0.00012
0.00014
0.00016
0.00018
0.0002
Series1
M and e e.nast abd qunli ashby shi gibson and ashby
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
Series1
Fig. 3.24 Measurement of Elastic Property (ν13)
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.
46Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Test 1 Test 2
Test 3 Test 4
Test 5 Test 6
Test 7
3.1.9. Selection of ν23
M and e e.nast abd qunli ashby shi gibson and ashby
0
0.00001
0.00002
0.00003
0.00004
0.00005
0.00006
0.00007
Series1
M and e e.nast abd qunli ashby shi gibson and ashby
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
M and e e.nast abd qunli ashby shi gibson and ashby
0
0.00002
0.00004
0.00006
0.00008
0.0001
0.00012
Series1
M and e e.nast abd qunli ashby shi gibson and ashby
0
0.00002
0.00004
0.00006
0.00008
0.0001
0.00012
Series1
M and e e.nast abd qunli ashby shi gibson and ashby
0
0.00002
0.00004
0.00006
0.00008
0.0001
0.00012
0.00014
0.00016
0.00018
0.0002
Series1
M and e e.nast abd qunli ashby shi gibson and ashby
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
Series1
Fig. 3.25 Measurement of Elastic Property (ν23)
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.
47Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
48Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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:
49Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
50Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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:
51Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
Equation (3.2.3.a.2)
Let’s do the same procedure for y2.
y2p=
y2p+1− y2
p
∆ t
Equation (3.2.3.a.3)
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
52Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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)
53Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
54Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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)
55Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
56Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
Equation (3.2.3.a.1)
Now resolve X1,
57Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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)
58Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
Equation (3.2.3.a.7)
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
Equation (3.2.3.a.11)
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
59Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
Equation (3.2.3.b.3)
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)
m2 y2+K m y23+Kbs y2+K core y2−K c ( y1− y2 )
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
60Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
x1, x2,β β=tan−1 1μ
Equation (3.2.3.a.11)
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)
m2 y2+K m y23+Kbs y2+K core y2−K c ( y1− y2 )
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
Equation (3.2.3.a.8)
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:
61Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Equation (3.2.4.a.1)
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:
62Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Cancel dt term in first integral in Equation, we get
Put x2=v2 in above equation
Integrating equation, we obtained
Equation (3.2.4.a.2)
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
63Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Equation (3.2.4.a.3)
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.
Equation (3.2.4.a.4)
64Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Equation (3.2.4.a.5)
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:
Equation (3.2.4.b.1)
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.2)
Equation (3.2.4.b.1) can be further simplified as
Equation (3.2.4.b.3)
Compare Equation (3.2.4.b.2) and Equation (3.2.4.b.3), we get 65
Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Equation (3.2.4.b.4)
Now introducing a new variable,
Equation (3.2.4.b.5)
Equation (3.2.4.b.6)
Taking derivative on both sides
Equation (3.2.4.b.7)
Equation (3.2.4.b.8)
Substitute equation (3.2.4.b.8) into Equation (3.2.4.b.3)
Equation (3.2.4.b.9)
Now we can express ω∧xas,
Equation (3.2.4.b.10)
Equation (3.2.4.b.11)
Substitute Equation (3.2.4.b.10) & (3.2.4.b.11) in Equation (3.2.4.b.9)
Equation (3.2.4.b.12)
66Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
67Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
ϵ 5
ϵ 6
In this case we only have α 3 terms as per in Equation (3.2.4.b.4)
Equation (3.2.4.b.13)
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:
68Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Terms Differential Equations
ϵ Equation (3.2.4.b.14)
ϵ 2 Equation (3.2.4.b.15)
ϵ 3 Equation (3.2.4.b.16)
ϵ 4 Equation (3.2.4.b.17)
ϵ 5 Equation (3.2.4.b.18)
69Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
ϵ 6 Equation (3.2.4.b.19)
Before proceeding further, useful cosine identities are listed below:
Equation (3.2.4.b.20)
Equation (3.2.4.b.21)
Equation (3.2.4.b.22)
Equation (3.2.4.b.23)
Equation (3.2.4.b.24)
3.2.5.2.i Solution for x1
Assume Equation (3.2.4.b.22)
Equation (3.2.4.b.25)
Differentiating Equation (3.2.4.b.25)
Equation (3.2.4.b.26)
70Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Equation (3.2.4.b.27)
The general solution of Equation (3.2.4.b.14) becomes,
Equation (3.2.4.b.28)
Substitute Equation (3.2.4.b.25) in Equation (3.2.4.b.28), we get
Equation (3.2.4.b.29)
3.2.5.2.ii Solution for x2 and w1
Substituting Equation (3.2.4.b.28) in (3.2.4.b.15)
Equation (3.2.4.b.30)
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)
Equation (3.2.4.b.31)
Removing secular term in Equation (3.2.4.b.31) and equating coefficient of cosine term
against zero
Equation (3.2.4.b.32)
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
Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Equation (3.2.4.b.34)
Now use equation (3.2.4.b.24) in (3.2.4.b.34)
Equation (3.2.4.b.35)
Solve the differential equation; we are only interested in particular solution
Equation (3.2.4.b.36)
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
72Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Equation (3.2.4.b.37)
Substitute Equation (3.2.4.b.35) in Eqn. (3.2.4.b.37)
Equation (3.2.4.b.38)
Substitute equation (3.2.4.b.23) in Equation (3.2.4.b.38), we get
73Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Equation (3.2.4.b.39)
Substitute Equation (3.2.4.b.24) in Equation (3.2.4.b.39), we get
Equation (3.2.4.b.40)
Removing secular term and equating co-efficient of cos (ϕ) against zero i.e.
Equation (3.2.4.b.41)
Substitute Equation (3.2.4.b.33) in (3.2.4.b.41)
Equation (3.2.4.b.42)
Substitute Equation (3.2.4.b.33), Equation (3.2.4.b.42) in Eqn. (3.2.4.b.40)
74Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Equation (3.2.4.b.43)
Solve the above differential equation for a particular solution only.
Equation (3.2.4.b.44)
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
75Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Equation (3.2.4.b.45)
Substitute Equation (3.2.4.b.25) in (3.2.4.b.45), we get
76Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Equation (3.2.4.b.46)
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:
77Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Equation (3.2.4.b.47)
Simplifying the above Equation (3.2.4.b.47)
Equation (3.2.4.b.48)
Now equate the coefficient of cos (ϕ) from Equation (3.2.4.b.48) against zero
Equation (3.2.4.b.49)
Substitute Equation (3.2.4.b.33) in Equation (3.2.4.b.49)
78Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Equation (3.2.4.b.50)
Now solve the differential Equation (3.2.4.b.48), the particular solution is
Equation (3.2.4.b.51)
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
79Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Equation (3.2.4.b.25) into above equation to convert the terms of ϕ
80Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Equation (3.2.4.b.52)
81Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
Equation (3.2.4.b.53)
Equate the co-efficient of secular terms against zero, the Equations become
Equation (3.2.4.b.54)
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
Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Equation (3.2.4.b.55)
Now solve the differential Equation (3.2.4.b.53) taking only the particular solution of the
equation:
83Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Equation (3.2.4.b.56)
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
Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Equation (3.2.4.b.57)
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:
Equation (3.2.4.b.58)
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)
Equation (3.2.4.b.59)
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:
85Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Equation (3.2.4.b.60)
3.2.5.2.vii Using initial conditions
The appropriate solution is of the form:
Equation (3.2.4.b.61)
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:
Equation (3.2.4.b.62)
This expression can be shown in terms of time using equation (3.2.4.b.5) as
Equation (3.2.4.b.63)
Substitute t = 0 here
86Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Equation (3.2.4.b.64)
Equation (3.2.4.b.65)
Differentiate Equation (3.2.4.b.63), substitute t = 0 and initial velocity v (0):
Equation (3.2.4.b.66)
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.
Equation (3.2.4.b.67)
87Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Equation (3.2.4.b.68)
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
Equation (3.2.4.b.69)
Equation (3.2.4.b.70)
Substitute Equation (3.2.4.b.4) in Equation (3.2.4.b.63):
Equation (3.2.4.b.71)
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
Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Equation (3.2.4.b.72)
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)
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.
89Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Fig. 3.37 Stress-strain graph for face sheet
90Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
91Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
93Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
94Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
95Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
96Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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:
97Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
98Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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:
99Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
100Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Fig. 4.57 Contact Automatic Surface to Surface
Fig. 4.58Contact Tied Surface to Surface Failure
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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
102Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
103Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
104Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
105Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
106Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Fig. 4.68 Orthotropic Elastic_002
Step 3: Two different types of sections are defined: Solid and Shell
Fig. 4.69 Section_Solid
107Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
108Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
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Fig. 4.74 Contact_tied_surface to surface_failure
Fig. 4.75 Contact_Interior
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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
111Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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:
112Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Fig. 4.79Implicid Dynamics
Fig. 4.80 Implicit Eigen-value
113Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Fig. 4.81 Implicit General
Fig. 4.82 Implicit Solutions
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Fig. 4.83 Implicit Solver
Fig. 4.84 Control_Shell
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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.
116Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
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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
118Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
119Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
120Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
121Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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).
122Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
123Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
124Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
125Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
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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
127Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
128Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
129Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
130Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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:
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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)
132Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Fig. 7.100 Load vs. Position plot for Specimen (S2)
Fig. 7.101 Load vs. Position plot for Specimen (S3)
Fig. 7.102 Load vs. Position plot for Specimen (S4)
133Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
Fig. 7.103 Load vs. Position plot for Specimen (S5)
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:
134Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
135Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
136Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
137Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
139Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
140Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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.
141Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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 .)
142Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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:
143Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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)
144Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
145Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
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[1] Abrate S. Impact of composite laminates. Appl Mech Rev1991; 44(4):155±90
146Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
[2] Abrate S. Impact of laminated composites: recent advances. Appl Mech Rev
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[3] Abrate S. Impact on composite structures. Cambridge: Cambridge University Press;
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[8] T. Besant*, G.A.O. Davies, D. Hitchings, finite element modelling of low velocity
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147Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
science, technology and medicine, prince consort road, London SW7 2BY, UK,
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[9] A Chawla, S Mukherjee, Dileep Kumar, T. Nakatani and M. Ueno, PREDICTION OF
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[11] V. Crupi*, G. Epasto, E. Guglielmino, COLLAPSE MODES IN ALUMINIUM
HONEYCOMB SANDWICH PANELS UNDER BENDING AND IMPACT
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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
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[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
148Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
[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
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[20] Feraboli, P., “Modified SDOF models for improved representation of the impact
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[21] Feraboli P.J., Ireland, D.R., Kedward, K.T., “On the role of Force, Energy and
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[22] Feraboli P., Kedward K., “A multi-parameter approach to impact performance
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[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 –
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[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
149Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
[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
150Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
[36] C.C. Foo, L.K. Seah, G.B. Chai, A modified energy-balance model to predict low
velocity impact response for sandwich composites
151Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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 = ');
152Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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 = ');
153Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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 = ');
154Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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));
155Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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 %
156Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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);
157Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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
158Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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);
159Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid
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')
160Senior Design Project September, 2013Syed Hamza Ali Tirmizi Muhammad Umer Khurshid