Faculdade de Engenharia da Universidade do Porto
Experimental and Numerical
investigation of the compressive
behaviour of polymer blends at high
strain rates
Pedro José Silva Campos
Thesis submitted under the scope of the Master’s degree in Mechanical Engineering Supervisor: Prof. Dr. Francisco Manuel Andrade Pires co-Supervisor: Eng.º António Luís Galamba de O. F. de Carvalho
Experimental and Numerical investigation of the
compressive behaviour of polymer blends at high
strain rates
Pedro José Silva Campos
Thesis submitted under the scope of the
Master’s degree in Mechanical Engineering
Approved by . . . :
President: Prof. José Dias Rodrigues Referee: Dra. Susana Sousa
Resumo
A presente tese tem como objetivo principal a realização de ensaios mecânicos de materiais poliméricos específicos (neste caso desde Policarbonato a ABS passando por misturas de ambos) a elevadas taxas de deformação numa barra de Split-Hopkinson e comparação
com simulações numéricas em ABAQUSTM de modo a perceber como o modelo material
incorporado no software se compara com o que na realidade acontece.
De modo a alcançar este objetivo, foi feito um estudo aprofundado do comportamento mecânico de Policarbonato, ABS e de misturas de ambos os materiais em diferentes rácios. Estudou-se a variação do comportamento variando a taxa de deformação, variando o rácio dos polímeros primários nas amostras e variando a geometria dos provetes utilizados. A teoria de funcionamento de uma barra de Split-Hopkinson foi também abordada assim como os princípios físicos por trás da obtenção das curvas de deformação/tempo para posterior análise. Foi também realizada uma recolha bibliográfica referente à criação de modelos matemáticos constitutivos de materiais poliméricos a altas taxas de deformação,
uma área ainda em considerável desenvolvimento. A sua integração em ABAQUSTMsob a
forma sub-rotina de utilizador foi estudada, pela sua relevância no mundo de hoje. Estas sub-rotinas aumentam a fiabilidade dos softwares de FEM na análise de cenários em que as taxas de deformação são elevadas, algo que é ainda uma lacuna grande neste tipo softwares. Relativamente aos ensaios experimentais, estes foram realizados no Laboratório de Ótica e Mecânica Experimental dentro do Departamento de Engenharia Mecânica na Fac-uldade de Engenharia da Universidade do Porto. Ao contrário do que seria desejável, apenas se realizou ensaios a uma pequena gama de taxas de deformação devido aos atrasos de todos os trabalhos em curso resultante da pandemia do COVID-19. Os ensaios foram, no entanto úteis para perceber o comportamento real de vários polímeros a altas taxas
de deformação. Foi possível, com ajuda do software SurePulseTM a obtenção das curvas
de tensão/deformação para os materiais analisados a partir dos sinais dos extensómetros aplicados à SHPB e da técnica da Correlação Digital de Imagem. Por conseguinte, estas
curvas foram inseridas no ABAQUSTMpara se ter a melhor representação possível do
ma-terial durante as análises. Estas curvas, quando acopladas à massa volúmica, módulo de Young e coeficiente de Poisson dos materiais, dados estes presentes nas suas fichas técnicas de fabricante, permitem a melhor caracterização material possível dentro do modelo básico
de material no ABAQUSTM sem que seja necessário o uso de modelos constitutivos de
materiais, sejam eles os que já estão incluídos no software ou os criados por sub-rotinas.
As análises em ABAQUSTMdos ensaios reais na SHPB provaram que há de facto
con-siderável semelhança nos resultados obtidos, mas que é possível melhorar o comportamento de materiais poliméricos nele inseridos. São apresentados gráficos de comparação entre o sinal obtido pelos extensómetros nos testes e a sua localização nos elementos no software. Palavras-chave: Policarbonato (PC); Acrilonitrilo Butadieno Estireno (ABS); misturas
de PC-ABS; Barra de Split-Hopkinson (SHPB); Alta taxa de deformação.
Abstract
The main objective of this thesis is to carry out mechanical tests of selected polymeric materials (in this case, from Polycarbonate to ABS through blends of both) at high
strain-rates in a Split-Hopkinson bar and comparison with numerical simulations in ABAQUSTM
in order to understand how the material model incorporated in the software predicts with what actually happens.
In order to achieve this objective, an in-depth study of the mechanical behaviour of Polycarbonate, ABS and blends of both materials in different ratios was carried out. The behaviour variation was studied by varying the strain rate, the ratio of the primary poly-mers in the specimens and the geometry of the specimens themselves. The operating theory of a Split-Hopkinson bar was also addressed as well as the physical principles behind ob-taining the strain/time curves for further analysis. A bibliographic study was also made regarding the creation of mathematical constitutive models of polymeric materials at high
strain rates, an area still under considerable development. Its integration in ABAQUSTM
under the user sub-routine form was studied, due to its relevance in today’s world thus increasing the range of reliability of FEM software in the analysis of scenarios where strain rates are high, something that is still a significant gap in the referred software.
Regarding the experimental tests, these were carried out at the Laboratory of Optics and Experimental Mechanics within the Department of Mechanical Engineering at the Faculty of Engineering of the University of Porto. Contrary to what would be desirable, tests have only been carried out on a small range of strain rates due to delays in all in-progress work resulting from the COVID-19 pandemic. The tests were, however, useful to understand the real behaviour of various polymers at high strain rates. It was possible, with
the help of the SurePulseTM software, to obtain the stress/strain curves for the analysed
materials, based on the strain gauge signals applied to the SHPB and the Digital Image
Correlation technique. Subsequently, these curves were inserted in the ABAQUSTMto have
the best possible representation of the material during the analysis. These curves, when coupled to the density, Young’s modulus and Poisson’s ratio of the materials stated in each of the manufacturer technical sheets, allow for the best possible material characterization
within the basic material model in ABAQUSTM prior to the use of constitutive material
models, whether they are already present in the software or are created and inserted through the previously mentioned user subroutines.
The analysis in ABAQUSTM of the actual tests at SHPB proved that there is
consid-erable similarity in the results obtained but that it is possible to improve the behaviour of polymeric materials inserted in it. Various graphs of comparison are presented between the signal obtained by the strain gauges in the real tests and the signal from the numeric simulations.
Keywords: Polycarbonate (PC), Acrylonitrile Butadiene Styrene (ABS); PC-ABS blends;
Split-Hopkinson Pressure Bar (SHPB); High Strain rate.
Acknowledgments
Firstly, I would like to thank Prof. Dr. Francisco Pires for the opportunity to guide me, not only in this thesis but in the work I conducted for the team in the past four years. It has been very helpful in the quest for jobs and internships and it has contributed a lot to my knowledge.
I would also like to thank Prof. Luís Galamba for his guidance in this thesis and in my scholarship prior and in the learning of CATIA and ABAQUS. Also for the great friendship that came from the work conducted and the after-hours.
Also to the LOME staff, for the availability for the tests and all subsequent doubts from the data treatment goes a big acknowledgment.
A big acknowledgment goes to Prof. Fenando Macedo who I had previously worked with and, despite not directly supervising this thesis, has always showed himself available when possible and also provided a lot of information and graphs for this thesis.
To all the teachers and professors, that crossed my path as a student and as a person, thank you as you have helped shape what I am today.
Bruno Augusto, whom I have had great fun with over the two years we have worked together, in university projects and in work projects, and in PC building which, as we know, as very funny sides to it.
To Carla Monteiro, the great organizer, a big thank you as well, as without he, the beginning of my work in DesignStudio would not have been as amusing and coffee-less.
To all the DesignStudioFEUP crew, specially Igor, Fernado, Catarina, Nina and Cecília, a thank you for the warm welcome into the team back in 2016, when I was just a sec-ond/third year student and you were full-fledge working teams.
To all my freshman year in civil engineering in FEUP, Rafael, Rodrigo, Miguel, Manuela, Inês, Pedro, Gerardo and so on, a big hug for you made my entry to university amazing.
To all my colleagues in Universidade do Minho, thank you for the support in a year away from home.
Lastly, to my good friends in the mechanical engineering department in FEUP, Bruno, Pedro Pinto, Pedro Severino, Gabriel, Inês, Gonçalo, Matos, Nuno and others, for all the projects together, meals and football games, goes a big thanks for the last years in my degree.
To all the people in my home town Póvoa de Varzim, a special mention as my future was shaped through 18 years there before university. Gonçalo, Guilherme, Mata, Francisco, Alex, Daniela, Débora, Diogo and many others, thank you.
To my family, my parents in particular and brother, Antonieta and Armando and Nuno, my grandparents Fernanda, Bento, Isabel and Armando, and my uncles, a strong appreciation and affection for accompanying me until today and to look after my good and ensure everything that was necessary so that I could reach the end of the course without problem or interruption. To all of you, a big and heartfelt thank you.
vi
To my family from Gaia, also a big hug and thanks for all the availability and help that they have also provided me since 2014.
Finally, for Sofia, my companion in happiness, who I met in Civil Engineering in 2013, who always helped and motivated me even when nothing seemed to be right, for all her strength and courage, and for all the love with which they accompanied me for seven years, a special thanks.
To all that helped me since 1995, a big thank you,
Pedro José Silva Campos
“Extinction is the rule. Survival is the exception”
Carl Sagan
Contents
Resumo i
Abstract iii
Acknowledgements v
Document Notations xvii
1 Introduction 1 1.1 Motivation . . . 1 1.2 Dissertation Goals . . . 2 1.3 Thesis Layout . . . 2 2 Polymers 5 2.1 Concept of polymer . . . 5 2.1.1 Types of Polymer . . . 7 2.2 Polycarbonate - PC . . . 8
2.3 Acrylonitrile Butadiene Styrene - ABS . . . 9
2.4 Blends of PC and ABS . . . 9
2.4.1 Effect of the blend ratio . . . 10
2.5 High speed polymer testing . . . 11
2.5.1 Mechanics of polymer deformation . . . 11
2.6 Material characterization . . . 12
2.6.1 Theory of Viscoplasticity . . . 12
2.6.2 Flow stress models . . . 12
2.6.3 The Johnson-Cook constitutive model . . . 12
2.7 Constitutive modelling of polymers . . . 13
2.7.1 The Ree-Eyring model - Non-Newtonian Flow in solid plastic system 14 2.7.2 The Haward and Thackray model for glassy thermoplastics . . . 15
2.7.3 The Mulliken and Boyce model - elastic-plastic deformation . . . 16
2.7.4 The Safari et al. model . . . 17
3 The Split-Hopkinson Pressure Bar 19 3.1 Brief Historical Reference . . . 19
3.2 SHPB constitutive parts . . . 20
3.3 Workings of an SHPB . . . 21
3.4 FEM modelling of SHPB tests . . . 22
3.5 Specimen geometry effect in testing . . . 23
3.5.1 Impact in the presence of oil film . . . 23
x CONTENTS
3.6 The Pulse Shaping technique . . . 24
3.6.1 Fluid shapers . . . 25
3.6.2 Solid shapers . . . 26
3.6.3 Pulse shaping FEM tests . . . 27
3.7 Data Acquisition . . . 38
3.7.1 Half Wheatstone Bridge - Stain Gauges . . . 38
3.7.2 Digital Image Correlation . . . 41
4 Specimen fabrication and SHPB tests 47 4.1 Design of a SHPB tests set . . . 47
4.1.1 Specimen materials . . . 47
4.1.2 Specimen machining specifications . . . 48
4.1.3 Specimen Numbers . . . 52
4.2 Data Analysis . . . 52
5 Results and Discussions 69 5.1 Stress-Strain Curves . . . 69
5.2 Strain-Time Curves . . . 77
6 Conclusions 83 6.1 Goals accomplishment and conclusions . . . 83
6.2 Future work perspectives . . . 83
7 Glossary 85
List of Figures
1.1 Both major commercial airplane companies that use a large percentage of
advanced composites in their structures (Justin Hale,2006) (Ramadier,2020). 1
2.1 Materials’ density vs. their Young’s modulus. (CES EduPack 2019) . . . 5
2.2 Yield Strength in various materials (logarithmic scale). (CES EduPack 2019) 6 2.3 Relative price of various materials (logarithmic scale). (CES EduPack 2019) 7 2.4 Thermosets (dark blue) and Thermoplastics (lighter blue) as seen in the polymer group. (CES EduPack 2019) . . . 8
2.5 Chemical Structure of Polycarbonate. . . 8
2.6 Chemical Structure of (from left to right) Acrylonitrile, 1.3-Butadiene and Styrene (Smith,1996). . . 9
2.7 Impact of the PC content on an ABS/PC blend (Jwu and Yean,2005). . . . 10
2.8 Haward and Thackray(1968)’s model for large extensions in polymers bellow TG. . . 15
2.9 One-dimensional Mulliken and Boyce(2005) rheological model. . . 16
2.10 Mulliken and Boyce(2005)’s model predictions. . . 17
2.11 One-dimensional rheological interpretation of the Safari (2012) model . . . . 18
2.12 PC storage modulus (solid line) and loss modulus (dashed line) as a function of temperature at 1 Hz (Safari,2012). . . 18
3.1 Architecture of a Split-Hopkinson Pressure Bar (Chen and Song,2011). . . 20
3.2 Actual SHPB used in the tests located in LOME. . . 21
3.3 Wave signal transmission and reflection representation (Chen and Song,2011). 22 3.4 Testing section detail in an SHPB (Chen and Song,2011). . . 22
3.5 Technical drawing of both Types A and B Specimens that will be used. . . . 24
3.6 Simulation and test of impact in the using an oil film (Adrian et al.,2007). 24 3.7 Close up geometry of fluid pulse shaper used by Chen et al.(2014). . . 25
3.8 Contact forces between bars and shaper inChen et al.(2014)’s work. . . 25
3.9 Deformation of the pulse shaper during Chen et al.(2014)’s tests. . . 26
3.10 Strain-time signals of the test (Frew et al.,2005). . . 26
3.11 Forces at the interfaces between bar and specimen (Frew et al.,2005). . . . 27
3.12 General model of the FEM SHPB test. . . 28
3.13 Boundary Conditions and predefined fields applied to the SHPB FEM model. 28 3.14 True stress-strain compression for ABS at 0,1 s−1 (F. P. B. Macedo,2020). . 30
3.16 Comparison between changes in pulse shaper diameter and thickness (Ameri et al.,2019). . . 30
3.15 ABAQUSTM test with an ABS Type A specimen and no pulse shaper. . . . 31
3.17 True Stress-Strain curve for annealed copper (Ramezani et al.,2009). . . 31
xii LIST OF FIGURES 3.18 Comparison between a pulse shaper absence and the set of results for a
cop-per pulse shacop-per with a thickness ePS=0,4 mm and a diameter φPSranging
from 6 mm to 12 mm. . . 32
3.19 Comparison between pulse shaper absence and the set of results for a copper pulse shaper with a thickness ePS=0,4 mm and a diameter φPSranging from 6 mm to 12 mm - focus on the incident pulse. . . 32
3.20 Comparison between pulse shaper absence and the set of results for a copper pulse shaper with a thickness ePS=0,4 mm and a diameter φPSranging from 6 mm to 12 mm - focus on the reflected pulse. . . 33
3.21 Comparison between pulse shaper absence and the set of results for a copper pulse shaper with a thickness ePS=0,8 mm and a diameter φPSranging from 6 mm to 12 mm. . . 34
3.22 Comparison between pulse shaper absence and the set of results for a copper pulse shaper with a thickness ePS=0,8 mm and a diameter φPSranging from 6 mm to 12 mm - focus on the incident pulse. . . 34
3.23 Comparison between pulse shaper absence and the set of results for a copper pulse shaper with a thickness ePS=0,8 mm and a diameter φPSranging from 6 mm to 12 mm - focus on the reflected pulse. . . 35
3.24 Comparison between pulse shaper absence and the set of results for a copper pulse shaper with a thickness ePS=1,5 mm and a diameter φPSranging from 6 mm to 12 mm. . . 35
3.25 Comparison between pulse shaper absence and the set of results for a copper pulse shaper with a thickness ePS=1,5 mm and a diameter φPSranging from 6 mm to 12 mm - focus on the incident pulse. . . 36
3.26 Comparison between pulse shaper absence and the set of results for a copper pulse shaper with a thickness ePS=1,5 mm and a diameter φPSranging from 6mm to 12mm - focus on the reflected pulse. . . 36
3.27 Comparison between pulse shaper absence and the set of results for a copper pulse shaper with a thickness ePS=2,5 mm and a diameter φPSranging from 6 mm to 12 mm. . . 37
3.28 Comparison between pulse shaper absence and the set of results for a copper pulse shaper with a thickness ePS=2,5 mm and a diameter φPSranging from 6 mm to 12 mm - focus on the incident pulse. . . 37
3.29 Comparison between pulse shaper absence and the set of results for a copper pulse shaper with a thickness ePS=2,5 mm and a diameter φPSranging from 6 mm to 12 mm - focus on the reflected pulse. . . 38
3.30 Characterization of a linear conductor in terms of electrical resistance (Gomes and Vaz,2003). . . 39
3.31 Example of an electrical strain gauge (Gomes and Vaz,2003). . . 40
3.32 Schematics of a Wheatstone bridge (Chen and Song,2011). . . 40
3.33 Strain gauges set up in the bars of the LOME SHPB. . . 41
3.34 The three phases that make up the DIC process (LePage,2009). . . 41
3.35 Phases of recognition from reference pattern to points displacement (LePage, 2009). . . 42
3.36 The three phases that make up the Digital Image Correlation process from a software point of view (LePage,2009). . . 43
3.37 Possible patterns divided by quality and dispersion. . . 44
LIST OF FIGURES xiii
3.39 A type B specimen between the bars just before a test takes place. . . 46
3.40 The LOME SHPB that was used in the tests, with data acquisition system, recording camera and adjustable torch. . . 46
4.1 Polymer force-displacement curves obtained at a strain-rate of 0,001 s−1. . . 48
4.2 Lathe in the Mechanical Engineering Department shop used to machine the specimens. . . 49
4.3 Closer look at the injected cylinders placed in the lathe’s spindle. . . 49
4.4 Original Polycarbonate injected cylinder and the resulting specimens. . . 49
4.5 Process of sanding the ends of the specimens before the heat treatment. . . 50
4.6 Before and after stages of specimen sanding. . . 50
4.7 Specimen heat treatment cycle in ceramic oven. . . 51
4.8 Various specimens on their way to the heat treatment. . . 51
4.9 Oven used in the heat treatment and its display. . . 52
4.10 Full pulse signal from both strain gauges in a PC type A specimen. . . 53
4.11 Pulse signal from the area of interest, from both strain gauges in a PC type A specimen. . . 53
4.12 User interface in PFV4 when opening a .*chix file. . . 55
4.13 First part of the Digital Image Correlation process - grid generation. . . 56
4.14 Second part of the Digital Image Correlation process - image processing. . . 56
4.15 DIC points after the correlation process in MatLab on a PC Type A specimen. 57 4.16 3D strain representation (last frame from DIC) of the tracking points applied to the PC Type A specimen. . . 58
4.17 1D DIC results for a PC Type A specimen. . . 59
4.18 Data analysis, truncation and zeroying in SurePulseTM. . . . 60
4.19 Download windows of the SurePulseTM software. . . . 61
4.20 Stress/strain curves of the PC Type A specimen extracted from SurePulseTM via the strain curves obtained experimentally. . . 62
4.21 Results obtained from ABAQUSTMof the strain pulses in the incident and transmission bars without the use of a pulse shaper, using a Type A PC specimen. . . 65
4.22 Four steps of deformation of a the rubber PS in an ABAQUSTM simulation. 66 4.23 Four steps of deformation of a PC type A specimen in an ABAQUSTM simulation. . . 67
4.24 Results obtained from ABAQUSTMof the strain pulses in the incident and transmission bars featuring a rubber pulse shaper, using a Type A PC spec-imen. . . 68
4.25 Comparison of the strain pulses from a simulation without a pulse shaper, a simulation with the rubber pulse shaper (with the BWF25 treatment) and the SHPB test. . . 68
5.1 Stress/strain curves of the PC Type B specimen obtained from SurePulseTM via the strain curves obtained experimentally. . . 70
5.2 Stress/strain curves of the ABS Type A specimen obtained from SurePulseTM via the strain curves obtained experimentally. . . 70
5.3 Stress/strain curves of the ABS Type B specimen obtained from SurePulseTM via the strain curves obtained experimentally. . . 71
5.4 Stress/strain curves of the 80%PC/20%ABS blend Type A specimen ob-tained from SurePulseTM via the strain curves obtained experimentally. . . 71
xiv LIST OF FIGURES 5.5 Stress/strain curves of the 80%PC/20%ABS blend Type B specimen
ob-tained from SurePulseTM via the strain curves obtained experimentally. . . 72
5.6 Stress/strain curves of the 65%PC/35%ABS blend Type A specimen
ob-tained from SurePulseTM via the strain curves obtained experimentally. . . 72
5.7 Stress/strain curves of the 65%PC/35%ABS blend Type B specimen
ob-tained from SurePulseTM via the strain curves obtained experimentally. . . 73
5.8 Stress/strain curves of all the Type A specimen materials studied. . . 74
5.9 Stress/strain curves of all the Type B specimen materials studied. . . 74
5.10 Augmented initial section of figures5.8 and5.9. . . 75
5.11 PC stress/strain curves by strain-rate, all obtained experimentally by
com-pression tests, the three lowest ones kindly provided by F. P. B. Macedo
(2020). . . 75
5.12 ABS stress/strain curves by strain-rate, all obtained experimentally by
com-pression tests, the three lowest ones kindly provided byF. P. B. Macedo(2020). 76
5.13 PC80ABS20 stress/strain curves by strain-rate, all obtained experimentally
by compression tests, the three lowest ones kindly provided by F. P. B.
Macedo(2020). . . 76
5.14 PC65ABS35 stress/strain curves by strain-rate, all obtained experimentally
by compression tests. . . 77
5.15 Strain/time curves of the PC Type B specimen obtained from ABAQUSTM
simulation and the real SHPB recorded pulses. . . 78
5.16 Strain/time curves of the ABS Type A specimen obtained from ABAQUSTM
simulation and the real SHPB recorded pulses. . . 78
5.17 Strain/time curves of the ABS Type B specimen obtained from ABAQUSTM
simulation and the real SHPB recorded pulses. . . 79
5.18 Strain/time curves of the 80%PC/20%ABS blend Type A specimen obtained
from ABAQUSTM simulation and the real SHPB recorded pulses. . . . 79
5.19 Strain/time curves of the 80%PC/20%ABS blend Type B specimen obtained
from ABAQUSTM simulation and the real SHPB recorded pulses. . . . 80
5.20 Strain/time curves of the 65%PC/35%ABS blend Type A specimen obtained
from ABAQUSTM simulation and the real SHPB recorded pulses. . . . 80
5.21 Strain/time curves of the 65%PC/35%ABS blend Type B specimen obtained
List of Tables
2.1 PC/ABS blend formulations (Jwu and Yean,2005). . . 10
3.1 Steel properties used in the SHPB bars during the PS tests in ABAQUSTM. 29 3.2 ABS properties used in the SHPB bars during the PS tests in ABAQUSTM. 29 4.1 Commercial names of polymers and corresponding PC/ABS ratios. . . 47
4.2 Specifications of the SHPB parts used in the tests. . . 54
4.3 Mechanical properties of Grade 5 Titanium - Ti-6Al-4V. . . 54
4.4 Calibrated properties of the used strain gauges. . . 54
4.5 Various strain rates achieved by the different specimens during the SHPB tests. . . 60
4.6 Specifications of the SHPB parts used in the tests. . . 62
4.7 Polymer properties inserted into ABAQUSTM for the simulations. . . . 63
4.8 Mechanical properties of rubber pulse shaper used in ABAQUSTM. . . . 64
Document Notation
Abbreviations and Acronyms
ABS Acrylonitrile Butadiene Styrene
ASTM American Society for Testing and Materials
BC Boundary Condition
BCF Bridgman correction factor
BWF25 Butterworth Filter with a cut off frequency of 25kHz
C3D8R 8 node brick (finite element in ABAQUSTM) with reduced integration
CM2S Computational Multi-Scale Modelling of Solids and Structures
.*csv Comma-separated values (file format)
DIC Digital Image Correlation
DEM Department of Mechanical Engineering
DMA Dynamic mechanical analyzer
DMTA Dynamic mechanical thermal analysis
DSF Design Studio FEUP
FE Finite Element
FEA Finite Element Analysis
FEM Finite Element Method
FEUP Faculdade de Engenharia da Universidade do Porto - Faculty of
Engi-neering University of Porto
J-C Johnson-Cook
L/D Specimen legth to diameter ratio
LET Laboratório de ensaios tecnológicos - Laboratory of mechanical tests
LOME Laboratório de Ótica e Mecânica e Mecânica Experimental - Optics and
Experimental Mechanics Laboratory
ODB Output Database (from ABAQUS)
PC Polycarbonate
PFV4 Photron Fastcam Viewer 4
PMMA Poly(methyl methacrylate)
SAN Styrene acrylonitrile resin
SCS Shear compression specimen
SS Stress/Strain (curve)
SHPB Split Hopkinson Pressure Bar
SMPT Section of materials and technological processes
TiG5 Grade 5 Titanium
USD U.S. (United States) dollar
xviii Abbreviations, Notations, Letters, Symbols and Operators
Roman letters
A Cross-sectional area
BB isochoric left Cauchy–Green tensor
c elastic wave speed of the striker or bar
CR rubbery modulus
ePS pulse shaper thickness
espe specimen thickness or height
E Young’s modulus or modulus of elasticity
I second order identity tensor
k Boltzmann’s constant
l length or first component of an orthogonal vector in cartesian coordinates
L Langevin function
m second component of an orthogonal vector in cartesian coordinates
n third component of an orthogonal vector in cartesian coordinates
n number of chains per unit volume
ˆ
n unit vector to a plane
Ox,y,z referes to the axis of each direction x, y and z
s cut plane on a random 3D body
t time in seconds [s] unless specified otherwise
T temperature
T striking time during SHPB operation
TG Polymeric glass transition temperature
w width
Greek letters
α primary polymer transition/process
αv volumetric expansion coefficient
β secondary polymer transition/process
γ tertiary polymer transition/process
Γ border of a continuous domain
strain or strain pulse
ζ third natural coordinate (FEM)
η dynamic viscosity (Ree-Eyring model) or second natural coordinate
(FEM)
θ absolute temperature
κ bulk modulus
λ chain stretch on the 8-chain network
µ shear modulus
ν Poisson’s ratio
ρ relative density [kg/m−3] or electrical resistivity [Ω.m]
σ normal stress
τ shear stress
φ diameter
Abbreviations, Notations, Letters, Symbols and Operators xix
Symbols
∂ partial derivate
L fourth order modulus tensor
J fourth order identity tensor
∇ differential operator
Subscripts and Superscripts
0 initial state
b of the SHPB bar
ib of the incident bar
m melting
PS of the pulse shaper
r reflected
R reference
sg of the strain-gauge
spe of the specimen
st of the striker
t transmitted
tb of the transmission bar
Operators
⊗ tensor product
≡ equivalent to
• scalar or dot product
· matrix product
× vector or cross product
∼
= congruent to
Notations unless specified otherwise
A standalone value
A matrix
A function
A’ deviatoric component or second location of a variable
A* refers to natural coordinates
{A} vector e
A average value
[A] matrix
||A|| vector norm − → A vector ˆ A unit vector ˙
A first derivate in time
¨
Chapter 1
Introduction
This chapter presents this thesis’ subject matter. It explains its goals, background and pertinence. A brief preview of its layout and structure is also presented.
1.1
Motivation
Nowadays, polymers and polymeric materials are being used for pretty much everything. Some applications that were classically taken up by metals and ceramics are now occupied by polymers. There are great examples of that, such as planes. Both AirBus and Boeing showcase in their current lineup planes that are more than 50 % composed of polymeric materials, though they are mainly composites with a thermoset matrix, as can be seen
from Figure 1.1. As such, their study and understanding is vital for the modern society.
Finding ways of accurately describing materials in various states of stress at various rates of application is very important and is in constant development.
(a) Boeing 787 (b) Airbus A350
Figure 1.1: Both major commercial airplane companies that use a large percentage of
advanced composites in their structures (Justin Hale,2006) (Ramadier,2020).
The Split-Hopkinson Pressure Bar (SHPB) is an apparatus that helps to find the be-haviour of materials, particularly at high strain rates which is the area where the polymeric
2 Introduction materials are still quite unknown, thus rendering this thesis pertinent in its theme. ABS and PC are very commonly used polymers and knowledge of the way they behave at certain strain rates is very relevant to the project and development of mechanical parts and parts in general. The desire to find out what is still little known in the industry is fascinating.
1.2
Dissertation Goals
The main goal of this dissertation is to study the behaviour of specific polymers (namely Polycarbonate (PC), Acrylonitrile Butadiene Styrene (ABS) and blends of both) by testing actual specimens in an SHPB and then simulating that exact test in a Finite Element Analysis (FEA) software. By doing so, it is possible to analyse and compare how different polymers react to different strain rates in two types of geometry.
Also, by studying this, the work team in our office will now have at their disposal stress/strain curves of PC, ABS and blends of both to input into simulation software in a more reliable way, something that was previously not possible. This improves the simulations’ results and brings them ever so slightly closer to what actually happens in real life events.
1.3
Thesis Layout
A brief chapter by chapter layout will now be presented as to complement the contents list by explaining in summary what is approached in each chapter. This way, it is easier for the reader to get oriented in the document.
• Chapter 1 - This chapter marks the beginning of the thesis. It shows the reason and pertinence for this work, showcases the goals to be achieved and summarizes what will constitute this work by dividing it into 8 chapters from the preliminary search work to the results and discussion of what was carried out during the semester. • Chapter 2 - In chapter 4, a polymer review is conducted. The concept of a polymer
and, more specifically, PC and ABS are reviewed. Some constitutive polymer models are described because of their importance in the world of FEM simulations. Polymeric behaviour under high strain rates is also referred due to its pertinence to this work. • Chapter 3 - The Split-Hopkinson Pressure Bar is the theme of chapter 5. It is the apparatus used in the specimen tests conducted during this work and is a piece of technology that is very relevant to the characterization of polymers at high strain-rates as well as many other materials.
• Chapter 4 - In this chapter, the machining of the specimens and preparations for the actual tests are described. The actual tests are also conducted during this chapter. The example for PC is shown in this chapter as well as an indication and example
1.3 Thesis Layout 3
for the graph showcasing and discussions to occur in chapter 7.
• Chapter 5 - In this chapter, all the results are showcased and discussed taking into account what was to be expected and what was read in the literature. Considerations are made about the ratio of PC and ABS in the blends.
• Chapter 6 - All the conclusions are drawn in this chapter as well as all the prospects of future work to be conducted on this thesis’ subject.
Chapter 2
Polymers
This chapter presents the polymer material group. It briefly explores specificities of dif-ferent polymeric materials at the macro and micro scales. Additional focus is placed on ABS and PC polymers, as they are the focus of this work. A brief reference to high speed polymer testing is also carried out.
2.1
Concept of polymer
From an etymological perspective, polymer means "many parts". It means that, chemically,
a polymer is a group of connected units (monomers) (Smith,1996). Polymers are a broad
class of materials that include traditional engineering materials, thermoplastics, and most types of adhesives. The polymer group is said by some authors to contain the elastomer group. In this case, they will be treated separately.
Figure 2.1: Materials’ density vs. their Young’s modulus. (CES EduPack 2019)
6 Polymers Starting from the nanoscale, a polymer is a group macromolecules of monomers ar-ranged in a chain-like structure. These monomers are constituted by atoms held as a group by covalent bonds. These bonds share electrons between them resulting in very little electron mobility. This explains polymers’ general weak conductivity of heat and electricity. The polymer is not only the result of these strong bonds. Weak bonds also exist between molecules and the latter are the ones who give polymers their typical ductile properties and low stiffness.
Polymers are in a so-called "sweet spot" in the material world, as Figure2.1suggests.
They have a relatively high Young’s modulus at a relatively low density. Although not as high as metals and ceramics, polymers’ Young’s modulus is above all the foams, elas-tomers and most relevant natural materials while having (in a general concept) a lower density. These characteristics provide the polymer group with a good ability to fulfil a lot of applications, which were once occupied by metals, as an example.
Figure 2.2: Yield Strength in various materials (logarithmic scale). (CES EduPack 2019) For the purposes of this illustration, Gold and Silver were subtracted from the graph
of Figure 2.3. Taking into account that the axis is in a logarithmic scale, it is possible to
see that both ABS and PC are in the region of 2 USD/kg putting them in the cheaper end of the scale, alongside most polymers. Adding this to the relatively high yield strength
shown in Figure 2.2makes the polymer group a very valuable and useful material group.
Industrial substitution of metals by polymers in high-performance applications is rel-atively recent, but growing. Nevertheless, it has brought about many new useful features like elimination of finishing stages, reduction in weight, reduction in noise and vibration and, in some cases, no need for lubrication. In some electric applications, polymers excel due to their inherent low conductivity.
2.1 Concept of polymer 7
Figure 2.3: Relative price of various materials (logarithmic scale). (CES EduPack 2019)
2.1.1 Types of Polymer
The broadest way to divide polymers is to divide them in natural or man-made (synthetic) polymers. Natural polymers can be protein, polysaccharides or even latex, the latter being fitted into the elastomer category as does natural rubber.
Another way to distinguish polymers is by establishing if they hold their final shape when set or can be reheated and reshaped. These are called thermosets and thermoplastics, respectively. This is usually the most common denomination of polymers.
Starting from a microscopic point of view, both thermoplastics and thermosets contain very long principal chains of covalently connected carbon atoms, the only difference being
that the chains in thermoplastics are usually longer (Smith, 1996). Sometimes, oxygen,
nitrogen and sulphur atoms or atom groups are also connected to the chain via covalent connections. The main chains of thermoplastics are connected with secondary connections. Thermoplastic pellets become more fluid when heat is applied and can take a new shape. This process can take place several times without considerable loss or change of its prop-erties. In the case of the thermosets, the polymers cross-link together during the curing process in order to form an irreversible chemical bond. If reheated, they lose properties and degrade, rendering them useless. It is usually not possible to recycle them either. It is
possible to see from Figure2.4 the darker blue representing the thermoset group and the
lighter blue, the thermoplastics, all inside the polymer group.
Finally, they can be differentiated via their crystalline constitution. They can either be amorphous, if their molecules are arranged internally in a random way, or crystalline, where the internals consist of lamelar crystal. A mixture of both phases also exists and is denominated semi crystalline.
8 Polymers
Figure 2.4: Thermosets (dark blue) and Thermoplastics (lighter blue) as seen in the poly-mer group. (CES EduPack 2019)
The polymers that are used in this work are synthetic amorphous thermoplastics. They are PC (Polycarbonate), ABS (Acrylonitrile Butadiene Styrene) and blends of both.
2.2
Polycarbonate - PC
Polycarbonate is, as referred in Section 2.1.1, an amorphous thermoplastic. PC is usually
inserted in the category of structural thermoplastics as it is generally considered a strong polymer and, in some cases, it can be optically transparent. It offers great impact strength and has a relatively high Young’s modulus. PC has high tensile, shear, and flexural strength as well as low deformation under load. Like most polymers, PC also shows a good resistance to thermal and electrical conductivity. It is capable of self-extinguishing which is very desirable in many applications such as helmet visors in motor racing.
PC is obtained by reacting bisphenol A (BPA) and phosgene COCl2. The sodium salt
of bisphenol A is then reacted with phosgene to produce the final polycarbonate. PC is usually difficult to process during manufacturing due to its high melt viscosity.
It’s chemical formula is [OC(OC6H4)2CMe2]n.
2.3 Acrylonitrile Butadiene Styrene - ABS 9
2.3
Acrylonitrile Butadiene Styrene - ABS
Whereas PC is a standalone polymer, Acrylonitrile Butadiene Styrene is composed of three species of polymers. Firstly, the acrylonitrile is copolymerized with styrene to give SAN. This is then extruded together with polybutadiene forming ABS. Like PC, ABS is an amorphous thermoplastic. Unlike PC, ABS is usually put in the category of thermoplastics of general use.
ABS combines interesting properties from three different polymers. Acrylonitrile gives its contribution as it has very good chemical and heat resistance; Butadiene has good impact resistance and holds its properties at lower temperatures; and Styrene introduces rigidity, easy processing and superficial shine. It is possible to alter the ratios of components in the mixture in order to reach higher levels of impact resistance or elongation before break. It is well known for its notch insensitivity and relatively low cost. On the other hand, it
has low thermal stability. According toKrache and Debah(2011) this mechanical property
can be modified by altering the percentage of polybutadiene in its composition. Its general chemical formula is (C8H8.C4H6.C3H3N)n.
Figure 2.6: Chemical Structure of (from left to right) Acrylonitrile, 1.3-Butadiene and Styrene (Smith,1996).
ABS is easy to produce and machine into different parts and it is widely used in tubing (and tubing accessories), car parts, shielding in fridges, boxes and many more.
2.4
Blends of PC and ABS
Creating blends of conventional polymers is a good way of developing or enhancing a set of desired properties from the intervening parts. Blends of PC and ABS are usually used in order to enhance the behaviour of both polymers, or to be more precise, the four basic polymers. The resulting blend has better mechanical properties than PC or ABS alone. The blend has better low temperature toughness than PC and better room temperature toughness than ABS. High resistance to impact at low temperatures is usually the main "selling point" of PC/ABS blends. Glossiness is a characteristic that comes with the blend. It is not useful for this work but it might be for some products manufactured from these types of blend. The blend is also easier to process than pure PC.
10 Polymers True behaviour of these blends is still not fully understood and described due to the complexity of the system.
2.4.1 Effect of the blend ratio
Blends of PC and ABS consist, as previously referred, of four polymers in three different
phases (Krache and Debah, 2011). The composition of the phases and which polymer
assumes each phase depends on the percentage of each one in the blend. The molecular composition of ABS itself is a factor on the blend, as well as rheological properties, thermal treatments and processing conditions.
Adding PC to ABS improves the tensile properties of the blend. Increasing its percent-age in the blend increases tensile strength and the Young’s modulus. However, according to
Jwu and Yean(2005), at 20wt% PC, a minimum in the impact strength occurred. Impacts were tested on an Izod Impact Machine with the specimen clamped vertical as a cantilever. Dimensions of the specimen were done according to the ASTM D256 standard and 5 of
each composition (referred in Table 2.1) were tested and the average values noted.
Table 2.1: PC/ABS blend formulations (Jwu and Yean,2005).
Blend B1 B2 B3 B4 B5 B6
PC (wt %) 0 20 40 60 80 100
ABS (wt %) 100 80 60 40 20 0
(a) Young’s modulus and tensile strength (b) Impact Strength
Figure 2.7: Impact of the PC content on an ABS/PC blend (Jwu and Yean,2005).
When a small amount of PC is added to a blend with ABS, a triple phase blend is formed with SAN being the matrix phase. Once the percentage of PC is above 40wt %, the PC becomes the continuous part and the ABS is dispersed within it and the impact resistance increases but the blend is also increasingly effective on initiating yielding.
2.5 High speed polymer testing 11
An interesting taking is that blends with a percentage of PC lower than 40 have worse behaviour on impact than pure ABS. However, beyond 40wt %, impact strength of the blend increases proportionally with PC content. Pure PC gives the best impact strength,
thus its use in helmet visors as a bid to stop intrusion by projectiles. Jwu and Yean(2005)
conclude this but states that an optimum blend proportion is close 60wt % of PC.
2.5
High speed polymer testing
Polymer testing by traction and compression is a very common procedure that is used in order to find the mechanical properties of the materials being tested. However, these tests are conducted in a very low strain rate. The rate-dependence of the elastic, plastic and failure behaviour of polymers is relatively well-known. Above the value of strain-rate of
100 s-1 the behaviour of polymers is still not yet well understood.
Due to improved crash-worthiness of polymeric materials, its uses now imply rapid loading and unloading. Whereas previous understanding of low-speed material mechanics sufficed, it is now important to properly describe and predict how these materials behave at high speeds, filling in the knowledge gaps.
One way to test materials at high rates of strain is through a Split Hopkinson Pressure Bar, or, as usually referred to, a Kolsky bar. This thesis’ work puts much focus on this
tool. In Chapter3it will be reviewed with detail.
2.5.1 Mechanics of polymer deformation
The way a polymeric material reacts under loads is dependent on its composition and microstructure. External factors also play a big roll on the matter. Pressure, temperature
and strain-rate are the most relevant ones (Siviour and Jordan,2016).
Works on the response of materials to dynamic loading began as early as 1949 with
Kolsky’s work (Chen and Song,2011). Over the next years, work has been put into studying
a number of polymers on a larger strain rate interval (10-4s-1 up to 105s-1) proving that
polymers exhibit a time dependent mechanical behaviour evidenced by the rate dependent elastic moduli, yield strength and post-yield behaviour. As said before, temperature and pressure are also bi factors since a polymer can go from a rubbery behaviour to plastic ductile and all the way up to brittle.
A common feature of polymers is that the strain-rate sensitivity is larger at higher
rates (Siviour, 2017). The cause of this phenomenon is the effect of underlying polymer
transitions which would normally be observed below room temperature (typically the β transition) but switch to room temperature as the strain rate increases.
Specimen heating is a big difference between static loading and dynamic loading. A portion of the plastic work happening on the specimen is converted to heat. At low strain rates this heat has time to conduct out of the specimen and the experiment is isothermal. At sufficiently high strain rates, the heat remains in the specimen and the experiment is
12 Polymers adiabatic. The transition between both depends on the thermal diffusivity of the material and the specimen geometry, and it is usually quite subtle.
2.6
Material characterization
2.6.1 Theory of ViscoplasticityResearch on plasticity began as early as 1864 by Henri Tresca and Saint Venant in 1870. Von Mises developed an improved model on plasticity in 1913 which is now known as the Von Mises yield criterion. This model features inherent strain-rate independence.
In the studies presented in this work, strain-rate dependency is key due to the high strain rates to which the specimens are subjected. Viscoplasticity is a theory in continuum
mechanics that describes the rate-dependent inelastic behaviour of solids (Perzyna,1966).
Viscoplastic material models differ from their rate-independent counterparts in the sense that they not only showcase permanent deformation after load application but continue to undergo creep flow as a function of time under the applied loads.
Rate-dependence means that the deformation of the material depends on the rate at which the loads are applied. Thus, it is important to use a proper model to describe the material behaviour under high strain rates.
2.6.2 Flow stress models
When plasticity occurs in a material, the instantaneous level of stress necessary to keep deforming the specimen plastically is called flow stress. Flow stress models are strain-rate dependent plasticity models that follow, preferably, high strain-strain-rates. Usually, these models take into account the temperature effect as well. There are five well know flow stress models but the most widely used is known as the Johnson-Cook model. This model
is referred to in Section 2.6.3.
2.6.3 The Johnson-Cook constitutive model
The Johnson-Cook model is one of the most widely used of the flow stress constitutive models. It exhibits an unrealistic small strain-rate dependence at high temperatures but this is a neglectable problem since this work relates only to high strain-rates at room temperature. It is also originally thought to be used in metals but can be used in polymeric materials.
The J-C model is purely empirical and relies on a set of parameters called material
constants. Originally proposed by Johnson and Cook(1983), the model for the von Mises
flow stress is expressed by the following equation:
2.7 Constitutive modelling of polymers 13
The parameter represents the equivalent plastic strain, ˙∗ = ˙
˙0 is the dimensionless
plastic strain rate for ˙0 = 1, 0 s−1 and T∗ is the homologous temperature. A, B, C, n and
m are the previously referred material constants.
A is the value of the yield stress and B and n represent the effects of strain hardening. These three constants can be derived from a true stress/true strain curve acquired by a tension test when it is converted to the equivalent tensile flow stress. The equivalent flow stress curve is obtained by using the Bridgman correction factor and is then adjusted using
FE simulations of the tension test. According toJohnson and Cook(1983), the BCF gives
acceptable results.
Temperature T∗ is the dimensionless temperature parameter that brings the
tempera-ture effect to the equation. Its input can be one of the following:
T∗ = 0 for T < TR T−TR Tm−TR for TR ≤T ≤ Tm 1 for T > Tm (2.2)
where T is the current temperature, Tm is the melting temperature of the material
and TR is the reference temperature. In this context, the reference temperature TR is
interpreted as the stress free temperature. Room temperature is usually elected for this
role. This leads to the use of the value 0 for the parameter T∗ and thus neglecting the
effect of the temperature in the tests as the last group between parenthesis is equal to 1, something that was always intended. The J-C model now features the form:
σ = [A + B.n][1 +C.ln˙∗
]. (2.3)
The J-C model is more suitable for metals and therefore, there are ways of describing a polymer in a more accurate way. Such ways will be dug into in the following chapter.
2.7
Constitutive modelling of polymers
Constitutive models for polymers depend on the type of polymer, whether it is amorphous or semi-crystalline. It is also necessary to know if the polymer is glassy over the temperature range of interest. These models require an extensive set of parameters which are usually found experimentally in order to accurately complete the model. They require a great deal of work to parameterize but once completed, they can be implemented relatively easily
14 Polymers
2.7.1 The Ree-Eyring model - Non-Newtonian Flow in solid plastic sys-tem
In 1955, in the beginning of material study at more elevated strain rates, Ree and Eyring
(1955) proposed a model in which they stated that the flow rate of a system is a function of
the relaxation times of the flow units, the distribution of the referred relaxation times and the deformation of the system with stress. By introducing these factors, a general equation for flow is obtained. This model is an evolution of the original Eyring model which was criticized by the simplicity of its relaxation theory for viscous flows.
The model considers the existence of n groups of flow units which differ in relaxation times and geometrical dimensions. The assumptions are the fractional area on a shear
surface of the various units to be x1, x2,..., xn and the shear stresses per unit area acting
on the areas to be f1, f2,..., fn, and that all the units on the same shear plane are obliged
to move with the same shear rate ˙s. This shear rate is represented in Eyring’s equation:
˙s = λ
λ1
n
2k0 sinh(αn)fn, (2.4)
where k0 is the rate constant for the flow process of a unit which belongs to the nth
group of units, αn= (λλ2λ3)n/(2kT )with λ being the molecular parameters in this theory
of viscosity and n is the group number to which the parameters belong.
If the force acting on the acting on the units of the nth group is x
nfn the stress f comes
as: f = n X n=1 xnfn. (2.5)
By introducing the fn’s from Equation (2.4) into Equation (2.5) the following results:
f = n X n=1 xn αnsinh −1 (βn˙s) (2.6)
where βn = 1/{(λ/λ1)n2kn0)} which is proportional to the relaxation time of the nth
kind. Therefore, viscosity η is given as: η = n X n=1 xnβn αn sinh−1 (βn˙s) βn˙s . (2.7)
This theory allows for multiple rate activated processes that are closely related to specific degrees of freedom in the polymer chains, thus controlling yield. This renders the polymer chains restricted to particular temperatures/strain rates and the effects are seen in the increase of yield strength.
2.7 Constitutive modelling of polymers 15
2.7.2 The Haward and Thackray model for glassy thermoplastics
In 1985, Haward and Thackray developed a new model to describe large recoverable ex-tensions observable in polymers below glass transition. This is the first one-dimensional model to describe this polymer behaviour while being based on the Ree-Eyring model (see
Section2.7.1). The model pioneers a way of looking into polymer deformation, dividing it
into two parts. These parts are a Hookean spring in series with an Eyring dashpot and a
Langevin spring (Haward and Thackray,1968).
Figure 2.8: Haward and Thackray (1968)’s model for large extensions in polymers bellow
TG.
When a typical stress-strain is performed and a constant rate of extension is applied to the specimen, the deformation is initially Hookean. This carries on until the stress level is high enough as to produce plastic deformation at the imposed rate. As the viscous resistance is non-Newtonian, the yield stress is not proportional to the rate of strain. It is important to always employ the use of true strain and true stress. This is due to the fact that as the elongation proceeds, there is a cross-sectional area reduction. In this case the stress registered by the machine is reduced and necking starts to occur. If fracture does not take place, the machine stress increases again and so, it is concluded that these apparently permanent deformations are actually largely reversible on warming the test piece. Therefore, a rubber elasticity element must be included in the model.
In this model, the Hookean spring and Eyring dashpot are able to capture intramolecu-lar resistance to chain segment rotation. The Langevin spring represents the entropic
resis-tance to chain alignment (Siviour and Jordan,2016) and accounts for the strain hardening
post-yield generated by the alignment of the macromolecular network built of entangled polymer molecules. Non linear dashpots are responsible for the rate dependency of yield
16 Polymers in the material.
Variations of this model use multiple dashpots in order to more accurately model molecular processes that affect yield, later called α- and β-transitions.
There was then a need to generalize the model into 3D. Various models came forward after this one and shall be showcased in the following sections.
2.7.3 The Mulliken and Boyce model - elastic-plastic deformation
Between this model and the Haward and Thackray model, a few more appeared where the
3D interpretation models based on the Ree-Eyring viscosity theory seen on Section 2.7.1.
Mary C. Boyce is a renowned author in this area and she participates in all the three major works after the Haward and Thackray model. These works involved the "Evolution of plastic anisotropy in amorphous polymers during finite straining" developed with E. M. Arruda in 1993, her work with renowned author M. Bergstr¨om "Constitutive modelling of the large strain time-dependent behaviour of elastomers" and "Large inelastic deformation of glassy polymers. Part I: rate dependent constitutive model" as co-author with D. M. Parks.
TheMulliken and Boyce(2005) model offers a temperature, pressure and rate-dependent
3D model. Like most other practical applications of the Ree and Eyring (1955) theory of
yield, it is capable of capturing both the primary α processes and the most significant
sec-ondary process (β). The visualization of such component division is shown in Figure 2.9
by splitting one of the branches of the model proposed by Haward and Thackray(1968).
Figure 2.9: One-dimensional Mulliken and Boyce (2005) rheological model.
These study focuses on the examples of polycarbonate (PC) and poly(methyl
methacry-late) (PMMA)at strain rates from 10-4 s-1 up to 104 s-1 using a DMA, a servo-hydraulic
testing machine and an aluminium Split-Hopkinson pressure bar. The model accounts for different molecular motions which become operational at certain at different regimes.
2.7 Constitutive modelling of polymers 17
(a) PC elastic modulus curve at 3,2*105s-1
de-composed into α and β components
(b) Model prediction of the PC elastic modulus curve at five strain-rates: 10-4s-1 up to 104s-1
Figure 2.10: Mulliken and Boyce (2005)’s model predictions.
The possibility to capture the transition into yield behaviour and the prediction of post-yield large strain behaviour at a wide range of temperatures and strain rates is also enabled. This model is empirical and therefore needs extensive testing in order to acquire material constants for the model to work properly.
The kinematic framework of this model follows that of Bergstr¨om and Boyce in 1998, Boyce et al. in 2000 and 2001 with the relevant difference that every intramolecular resistance is in duplicate (the other models used two ramifications only).
Part A of of Figure 2.9 represents the intermolecular contribution to the stress state
of the material and it is related to the deformation by the constitutive laws for the linear elastic springs.
The Arruda and Boyce (1993) 8-chain model is used as interpretation of molecular alignment. The stress is the non-linear hardening component due to entropic resistance to molecular alignment, and it is defined as
In the words ofMulliken and Boyce(2005):" In the most general form of this
constitu-tive model, the strain softening phenomenon may be considered as the sum of the softening in the α and β components.
2.7.4 The Safari et al. model
Keivan H.Safari(2012) et al. developed theMulliken and Boyce(2005) model even further
by proving that the secondary rate-activated process γ important at sufficiently high strain
rates in PC (up to 104 s−1) with good results accomplished by applying a user material
subroutine called VUMAT into ABAQUSTM. The SHPB used in their study confirmed
the results obtained in the FEM tests in ABAQUSTM.
The material parameters were obtained by DMTA and DSR, enabling the authors to have a better understanding of the post-yield behaviour of PC and the material transitions
18 Polymers as well as the linkages between material viscoelastic, yield and stress-strain curve of the material.
Figure 2.11: One-dimensional rheological interpretation of the Safari(2012) model .
(a) Young’s modulus and tensile strength
(b) Impact Strength
Figure 2.12: PC storage modulus (solid line) and loss modulus (dashed line) as a function
Chapter 3
The Split-Hopkinson Pressure Bar
In this chapter, a brief historical reference is carried out. The components of a SHPB are listed and explained. The way a SHPB works and the underlying theory are also explained. A few tests regarding the choice of pulse shaper are also conducted.
3.1
Brief Historical Reference
Originally developed by Herbert Kolsky in 1949 and gaining the same name, it is also called a split-Hopkinson bar in honour of John Hopkinson and his son who developed a rig to test the strength of steel wires by dropping a weight.
In 1914, Bertram Hopkinson (John Hopkinson’s son) developed a system to measure the pressure produced by high explosives or bullet impacts. In the past, the results were disperse and untrustworthy resulting of inaccurate measurement equipment. It was hard
to conduct the same test with the same parameters repeatedly (Chen and Song,2011).
In 1948, Davis discussed the dispersion of stress waves when they propagate in a long rod. Kolsky extended Hopkinson’s work to measure stress-strain response of various ma-terials under impact loading conditions. Kolsky bar had the same principle of Davis’ but was made up of two elastic rods, one on each side of the specimen. One of the rods would then be struck by an explosive blast.
The typical workings of a SHPB is as follows: sudden force is generated via rapid gas expansion, which then triggers a series of impacts on a set of bars and, among them, the specimen; it initially accelerates a striker that impacts an incident bar; this incident bar compresses the specimen against a transmission bar; A compressive wave goes through both incident and transmitter bars, thus generating transmission and reflected responses which differ, due to having the specimen placed in between; resorting to Kraft’s strain gauges (developed in 1954) these responses are recorded and compared against, evidencing the specimen’s own response. Kolsky apparatus differs from others in that its bars are not significantly stiffer than the specimen.
20 The Split-Hopkinson Pressure Bar
3.2
SHPB constitutive parts
As referred in Section 3.1, the basic apparatus components are an incident bar, a
trans-mission bar and a striker device, as well as a stable support. According toChen and Song
(2011), they can be divided in three basic areas in the apparatus. These are the loading
device, the bar components and the data acquisition and recording system.
Firstly, the striker hits the incident bar. For the purposes of proper testing, the launch of a striker must be repeatable. A gas gun is arguably the best choice as it allows a appreciable degree of control. By varying the pressure in its gas chamber, accounting for striker weight and other systemic circumstances, one can predict and set the speed at which the striker hits the incident bar. Strike length, i.e. the width of the compressive wave, depends on striker geometry and material. The set up used in the tests for this
thesis is shown in Figure 3.2. Sharing high yield strength, and the designed ability to
function in the elastic domain, incident and transmitter bars are usually alike, except for their length. To avoid overlapping responses, incident bars are typically longer than their counterparts. On the accuracy side, all bars have particularly low straightness and cylindricity tolerances, and the alignment system must ensure exceptional coaxiality. This ensures one-dimensional wave propagation. A momentum trap is placed at the end of the bar train.
The data acquisition system is composed of sensors, signal conditioners, and a recording device. To account for minor buckling phenomena, a pair of strain gauges are placed on opposite sides and each attached to a half Wheatstone Bridge. A dummy half-bridge strain gauge is placed for the sake of temperature compensation. Further information about this
matter will be carried out in Section3.7.1. A typical computer-operated recording device
stores the amplified signal. The schematics of all three parts are portrayed in Figure 3.1.
3.3 Workings of an SHPB 21
Ideally, according to Chen and Song (2011), the response frequency from from all the
data-acquisition components should not be less than 100 kHz.
(a) (b)
Figure 3.2: Actual SHPB used in the tests located in LOME.
3.3
Workings of an SHPB
Once activated by the gas gun, the striker generates a compressive wave that transforms into a tension wave which travels through the input bar after impact. If the impedance of the specimen is less than that of the bars, an elastic tensile wave is reflected into the incident bar and an elastic compression wave is transmitted into the out put bar as was
mentioned in Section3.1. Figure3.3illustrates this operation in an intuitive way.
The striking time T is dependent of the length of the rod L and cst, the latter being the
elastic wave speed of the striker material. As such, it is possible to derive from Equation (3.1).
T = 2L
cst (3.1)
The stress, strain and strain rate in the specimen can be obtained from the recorded transmitted and reflected strains by the following equations:
σspe=AbAEb spe t (3.2) spe= 2cb Hs Z r(t).dt (3.3)
22 The Split-Hopkinson Pressure Bar
Figure 3.3: Wave signal transmission and reflection representation (Chen and Song,2011).
˙spe =
2cb
Hs
r(t).dt (3.4)
where E represents the Young’s modulus, cb is the elastic wave speed of the bar and
Aspe , b is the cross-sectional area of the specimen and bars.
Figure 3.4: Testing section detail in an SHPB (Chen and Song,2011).
3.4
FEM modelling of SHPB tests
Finite Element Analysis is used as a system and process optimization tool at not only the product design stages but also at failure investigation, among others. Its growing adoption stems from allowing users an unprecedented level of prediction of complex systems behavior, now-a-days in any field of engineering. FEA users must be aware of its garbage-in, garbage-out nature. Careful model construction and critical thinking is paramount at all stages, from FEA-friendly part modelling and proper meshing, to how systems are built and components interact. More importantly, seasoned knowledge of how to read and interpret results is key. While FEA material models may follow a semblance of Pareto’s principle,
3.5 Specimen geometry effect in testing 23
where the basic setup is an excellent starting point, trustworthy results in mission-critical applications requires diving deeper - and safer. Material model calibration is a fairly simple process, in which real-world tests are modelled in the FEA environment, and model setup and material properties are tuned. Its goal is to replicate real results, and obtain reliable FEA material parameters.
However, a lot of care must be taken when building an FEM model because the accuracy of the model depends largely on the input data. If the wrong contact properties are used, wrong stress/strain curve inserted, bad elements in the mesh, geometry, loads, boundary conditions etc., the results are not as they should be and it might be, at first, undetectable.
3.5
Specimen geometry effect in testing
The axial wave propagation one-dimensional simplification problem benefits from cylindri-cal specimen usage. On the other hand, a square cross-sectional specimen can present a few advantages such as allowing for a easier application of the DIC technique, due to its flat sides, directly extracting strain from the specimen. In spite of that, DIC will still be used in one of the tests to accurately uncover the strain-rate of a PC specimen.
A critical assumption in the SHPB technique is that of stress equilibrium. Pankow
et al. (2009) uncovered that, after testing various shapes of specimen (cylinder, cube and ellipse) with different length to diameter ratios, or L/D ratios, it would be advantageous to use a smaller ratio (usually not over 0,5) since not only do the specimens reach equilibrium faster but they also do it with a better pulse signal. However, in order to obtain the strain rate dependency of the Young’s modulus, a thicker specimen must be used because it allows a prolonged S-S curve, allowing constant strain-rate to be achieved in the linear elastic regime of the material.
As such, to cover all the desired characteristics of the different geometries, two types of specimen will be used - Type A and B. Type A specimen will have a thickness or height
of espe = 10mm and a diameter φspe = 10 mm. Type B will have the same diameter but
will have espe= 5 mm as can be seen from Figure3.5.
3.5.1 Impact in the presence of oil film
The existence of an oil film in the ends of the specimen is a prompt way of confirming its coaxial positioning in reference to the bars. When a test is carried out, if a radial
ejection of oil (as seen in Figure 3.6a) is verified, the correct position of the specimen is
confirmed.Adrian et al.(2007) found not only this but also that the oil film helps prevent
24 The Split-Hopkinson Pressure Bar
Figure 3.5: Technical drawing of both Types A and B Specimens that will be used.
(a) Detail of film with thick-ness of 0,1 mm.
(b) Stage of impact and ejec-tion simulaejec-tion.
(c) Uniform radial ejection on impact - high speed camera.
Figure 3.6: Simulation and test of impact in the using an oil film (Adrian et al.,2007).
3.6
The Pulse Shaping technique
The Pulse Shaping technique was first used extensively by Chen and Ravichandran in 1997 in the study of brittle materials in the SHPB. It was then introduced by Chen et al. in 1999 in the study of softer materials.
Pulse shaping consist in placing solid (or even fluid) material between the striker and incident bars. As the pulse develops, changes in impedance will cause changes in the pulse’s shape. This technique’s goal is to help minimize response signal noise and flatten the strain-time curve, thus achieving dynamic stress equilibrium early, when specimens are
still elastic. Furthermore, this helps testing specimen at constant strain rates. Frew et al.
(2005) states that a slowly rising incident pulse is preferred to a steeply rising pulse so
that the specimen can achieve dynamic stress equilibrium prior to occurrence of massive damage on the sample.
