Master of Science in Mechanical Engineering June 2021
Material characterization of long-term stress relaxation in a semi-crystalline
polymer material
a semi-crystalline polymer
Jakob Görtz
-An experimental and numerical study
Contact Information:
Author:
Jakob Görtz
Jago15@student.bth.se
University advisor:
Prof. Sharon Kao-Walter
Department of Mechanical Engineering Tetra Pak® advisor:
Dr. Eskil Andreasson
Eskil.Andreasson@TetraPak.com Packaging Solutions AB Tetra Pak®assistant advisor:
Viktor Petersson
Viktor.Petersson@TetraPak.com Packaging Solutions AB
This thesis is submitted to the Faculty of Mechanical Engineering at Blekinge Institute of Technology in partial fulfilment of the requirements for the degree of Master of Science in Mechanical Engineering. The thesis is equivalent to 20 weeks of full time studies.
The author declares that he is the sole author of this thesis and that he has not used any sources other than those listed in the bibliography and identified as references. He further declares that he has not submitted this thesis at any other institution to obtain a degree.
A CKNOWLEDGEMENTS
This thesis would not have been possible without the support of Sharlin Shahid, Viktor Petersson, Sharon Kao-Walter, and Eskil Andreasson.
Sharlin got me started by helping me understand the capabilities of the equipment at BTH and by teaching me how to use them.
Viktor contributed with invaluable insight concerning simulation modelling and offered thoughtful advice throughout the thesis.
Sharon tried her best to prevent me from overworking this thesis; with varying effectiveness. She has shown concern for my well-being and has always been available to give me valuable mentorship.
Eskil has been both a scientific advisor and motivational coach for me. He gave me the freedom to approach this work in a way that suited me. Even if he knew that some of my ideas were flawed, he encouraged me to try them out. This might not be the easiest approach, but I gained so much more knowledge and experience for having done it. I am hugely grateful for his enthusiasm and
encouragement.
A BSTRACT
As the plastic and packaging industry is looking to increase the longevity of plastic products as well as recycling used material, there is a need to understand how material properties respond and change during long periods of mechanical loading. Physical tensile experiments on thin plant-based High-Density Polyethylene (HDPE) are conducted with the intent of capturing relaxation behavior from a short-term (1-3 hours) and long-term (29-56 days) perspective. Experimental tests aiming to capture short-term relaxation behavior prior to necking at various loads are made on a MTS Qtest100 tensile-machine in the laboratory at BTH. Long-term experiments are conducted on a custom-built tensile machine stationed in the author’s apartment.
Data gathered from the experiments are swiftly converted into true stress and strain based on the derived mathematical expressions in preparation for computer simulations, i.e. modeling the behavior using two expressions and the Finite Element Method (FEM) in the general purpose FE-software AbaqusTM R2020. The loading curve, i.e. uniform deformation, prior to geometrical necking, was modeled using the Ramberg-Osgood expression and captured the mechanical non-linear behavior accurately. Two expressions are initially used to capture the stress decay, referred to as relaxation behavior: the first one is Guiu and Pratt and the second one is a data-generated Four Parameter Logistics (4PL) expression. A comparison between the two expressions, show that the 4PL expression captures the entire short-term behavior of the experiments. The 4PL expression could also predict the long-term behavior without further calibration. The Guiu and Pratt expression could not predict the behavior as accurately as the 4PL expression.
Using the converted physical data to calibrate a Parallel Rheological Framework (PRF) model in the MCalibration software proved to be time consuming. A combination of the Ramberg Osgood and 4PL expression is used to re-create the converted physical experiment data which reduces both noise and size of the datasets dramatically. The calibration time was significantly reduced because the datasets were much smaller. With a material model calibrated using the re-created data, simulations could be conducted in Abaqus, creating a virtual twin of the physical experiments. Results from the physical experiments are compared to the results of the virtual simulations proving that the PRF model can capture the relaxation behavior shown in the short-term experiments. The model also works for long- term relaxation behavior and only a slight increase in stress relaxation compared to the physical experiments was observed.
Key words: Abaqus, MCalibration, PRF, HDPE, Stress relaxation, Long-term testing, Theoretical expressions, Material calibration
S AMMANFATTNING
I dagens plast och paketeringsindustri finns ett behov att öka produkters livstid samt att använda återvunnet material. Med detta finns då behovet av att bättre förstå hur plasternas egenskaper förändras under långa lastperioder. Fysiska tester kommer därav genomföras med tunna testbitar gjorda av organiskt HDPE med målet att fånga spännings relaxationen från både ett kort (1-3 timmar) till ett långt (29-56 dagar) tidsperspektiv. Experimentella tester som fångar det korta tidsperspektivet görs med olika lastfall före “necking” och genomförs på en MTS Qtest100 dragprovsmaskin på labbet på campus BTH.
Tester som fångar det långa tidsperspertivet görs på en dragprovsmaskin som är tillverkad för detta syftet och är stationerad i författarens lägenhet.
Datan som är tagen från experimenten är först konverterade till sann spänning och töjning för att sedan modeleras utav två matematiska uttryck och en model i Finita Element Metod programmet AbaqusTM R2020. Det matematiska uttrycket Ramberg-Osgood användes för att modellera pålastningskurvan före
“necking” och gorde detta tillfredställande. Två uttryck jämfördes för att modellera relaxationskurvan, ena var Guiu and Pratt uttrycket och det andra var en data-genererad Fyra Parameter Logistik (4PL) uttryck. Jämförelsen visade att 4PL uttrycket fångade hela kurvaturen ur det korta tidsperspektivet. Det visade sig även att 4PL uttrycket kunde prediktera det långa tidsperspektivet utan att göra några extra kailbreringarna från de korta tidsperspektivets kalibrering. Guiu and Pratts uttryck hade problem i bade de korta och långa tidsperspektivet.
Med den omvandlade datan från de fyska testerna börjades kalibreringen av en “Parallel Rheological Model” (PRF) materialmodel i programmet MCalibration. Detta visade sig kräva mycket tid då datafilerna från de fysiska testera var mycket stora och hade även en del brus. Med detta gjordes valet att använda Ramberg Osgood uttrycket samt det data-genererade 4PL uttrycket för att skapa matematiskt beräknade testdata för att minska mängden datapunkter samt ta bort bruset. Med de nya datafilerna blev kalibreringstiden av materialmodellen mycket mindre och det kunde användas för att skapa en “virtual twin” av dragprovsanordningen. Resultatet från simuleringarna av den virtuella dragprovsbiten visar på att modellen fångar den korta tidsaspeketen väldigt bra. För det långa tisperspektivet fungerade modellen men med lite större stress relaxation jämfört med de fysiska experimenten.
Nyckelord: ABAQUS, MCalibration, PRF, HDPE, spännings relaxation, långtids mätningar, teoretiska uttryck, Kalibrering av materialmodel
N OMENCLATURE
Abbreviation Term
4PL Four Parameter Logistic curve
CAD Computer Aided Design
CAM Computer Aided Manufacturing CNC Computer Numerical Control DIC Digital Image Correlation
FE Finite Element
FEA Finite Element Analysis
FEM Finite Element Method
HDPE High Density Polyethylene
SMHI Swedish Meteorological and Hydrological Institute
Term Description
Arduino yún Arduino board with wifi access and webhosting capabilities Long-term 29-56 days
Mach3 Computer Numerical Control software used to drive stepper motors MCalibration Software used to calibrate material models
MTS Qtest100 Professional tensile machine at BTH
Net-flow Movement in a fluid or viscoelastic material PolymerFEM Developer of the MCalibration software
Quasi-static Analysis type required for materials with time-dependent properties Short-term 1-3 hours
Specimen Dog-bone shaped material test piece Stepper motor Motor used for high accuracy movements
Strain-to-failure Experiment that extends a specimen at a constant velocity until it breaks TestWorks 4 Software used to operate the MTS Qtest100
Virtual-twins Method of attempting to virtually replicate real experiments and systems
Variable Definition
𝐴 Actual cross-section area
𝐴0 Initial cross-section area
𝛼 Ramberg-Osgood parameter
𝐵 Guiu and Pratt parameter
𝛽𝜎 Linear shifts
𝐶 Abbreviated constant
𝑐 Guiu and Pratt parameter
∆𝐻 Potential energy barriers
∆𝐿 Change in clamp distance
𝐸 Young’s modulus
𝜀 Strain
𝜀𝐸 Engineering strain
𝜀𝑒 Elastic strain
𝜀𝑇 True strain
𝜀𝑣 Eyring/Viscous flow strain
𝐹 Load
𝐾 Generated expression parameter
𝑘 Boltzmann’s constant
𝐿 Actual clamp distance
𝐿0 Initial clamp distance, 60mm
𝑚 Generated expression parameter
𝑛 Ramberg-Osgood parameter
𝑟 Generated expression parameter
𝜎𝐸 Engineering stress
𝜎𝑇 True stress
𝜎𝑡 Stress at time 𝑡
𝜎𝑢𝑡𝑠 Ultimate tensile strength
𝜎0 Stress at time 𝑡 = 0
𝑇 Absolute temperature
𝑡 Time
𝑉 Volume
𝑉𝑎 Activation volume of a molecular event
𝑣 Chain-sliding frequency
C ONTENTS
ACKNOWLEDGEMENTS ... IV ABSTRACT ... V NOMENCLATURE ... VII CONTENTS ... IX
1 INTRODUCTION ... 1
1.1 Background ... 1
1.2 Problem statement ... 3
1.3 Objective ... 4
1.4 Outline ... 4
1.5 Limitations ... 6
2 PHYSICAL STRESS RELAXATION EXPERIMENTS ... 7
EXPERIMENT METHOD ... 9
2.1 Manufacturing of additional equipment ... 9
2.2 Specimen preparation ... 11
2.3 Tensile machine operations ... 12
EXPERIMENT RESULTS ... 13
2.4 Strain-to-fail experiment ... 13
2.5 Short-term experiments ... 14
2.6 Long-term experiments ... 16
2.7 Defined regions of stress relaxation ... 17
2.8 Necking experiments ... 18
3 THEORETICAL EXPRESSIONS ... 19
THEORETICAL ANALYSIS METHOD ... 22
3.1 Converting physical data to strain and stress ... 22
3.2 Ramberg-Osgood formula ... 23
3.3 Relaxation expressions ... 23
3.4 Re-creating experiments using mathematical expressions ... 23
THEORETICAL RESULTS BASED ON EXPERIMENTAL DATA ... 24
3.5 Converted data to true strain/stress ... 24
3.6 Parameter study based on Ramberg Osgood formula ... 25
3.7 Parameter study based on relaxation expression ... 25
3.8 Combined expression results ... 29
4 MATERIAL MODEL CALIBRATION ... 31
CALIBRATION METHOD ... 32
4.1 Calibrating the material model ... 32
4.2 Simulating the physical experiments ... 32
CALIBRATION RESULTS ... 34
4.3 Material model ... 34
4.4 Abaqus simulation model of virtual relaxation experiments ... 34
4.5 Simulated and physical data comparison ... 36
5 DISCUSSION ... 38
5.1 Generated expression ... 38
5.2 Issues with the custom-built tensile machine ... 39
5.3 Increased load in the simulation experiments ... 39
5.4 Fluctuations in the long-term experiments ... 41
6 CONCLUSION AND FURTHER WORK ... 43
6.1 Conclusion ... 43
7 REFERENCES ... 46
APPENDIX A ... 48
APPENDIX B ... 52
APPENDIX C ... 53
1 I NTRODUCTION 1.1 Background
Plastics have gradually, over several decades, become more frequently incorporated in manufactured products. Plastics are currently in everything from single-use utensils to integral parts of vehicles. One industry that commonly uses plastics is the packaging industry. Companies like Tetra Pak® for example regularly incorporate thermoplastic in their packaging solutions. Figure 1.1 shows a packaging solution from Tetra Pak® with the top section entirely made from injection moulded High Density Polyethylene, also known as HDPE.
Figure 1.1: Packaging solution containing injection moulded HDPE [1]
The increase in use of plastics has resulted in an unacceptable increase plastic waste. In response to this environmental impact, the European Commission [2] introduced stricter rules regarding single-use plastics in 2019. As a result, companies are working on solutions that fulfill the new requirements and on finding long-term solutions that reduce the environmental impact of plastics. Tetra Pak® for example offers its customers the option to use a plant-based polymer in their packaging solutions [3] or certified recycled polymers [4].
One European Commission goal is that by 2030 all plastic caps will have to be permanently attached to the packaging unit to reduce the number of caps that end up in the ocean [2]. The development of a cap that remains attached to the package and retains its sealing function over time is a challenge. Virtual models that simulate the mechanical properties of potential solutions are used to efficiently evaluate viable options [5] [6].
1.1.1 Thermoplastics, an overview
Thermoplastics are commonly used for packaging solutions because they are stable, flexible and suitable for injection moulding. HDPE, a type of thermoplastic, is a semi-crystalline material. This means that during its solid state it has a part crystalline and part amorphous structure. The crystalline structure is built up by several polymers packed together that form an orthorhombic unit cell. The amorphous solid is comprised by entangled polymers that deform when a load is applied. Once the load is removed, only the elastic portion of the polymers will return [7]. A visual representation of the different structures can be seen in Figure 1.2.
Figure 1.2: Material structure (adapted from [7])
When a constant tensile load is applied to a semi-crystaline structure, some bindings will break and the polymer will deform. With small loads the deformation is limited to a tangential value whereas for larger loads the deformation will continue to deform non-linearly until failure occurs [8]. Another way of observing the behaviour is called stress relaxation. It describes how the material loses some of its internal tensile strength if held at a constant deformation. A visualization of this can be seen in Figure 1.3. The initial curve illustrates loading of the material and the deformation. The second part, “Strain held constant”, is where stress relaxation is observed.
Figure 1.3: Relaxation test behavior of stress and strain during strain-gauge experiments.
This phenomenon can be described phenomenologically using a system composed of dampers and springs. A model from the 1980’s is shown in the Figure 1.4. The figure illustrates the effect of loading and un-loading [8].
Figure 1.4: Mechanical description of plastics behavior (adapted from [8])
Abaqus R2020 [9] enables the use of the Parallel Rheological Framework (PRF) model which is a modern approach to this type of system. PRF consists of several springs and dampers and a slider in a parallel system. PRF and other similar models are used to create more accurate models of plastic’s behavior in Finite Element Method/Analysis (FEM/FEA) programs such as Abaqus. A visualization of this type of model can be seen in Figure 1.5.
Figure 1.5: A PRF model. The number of dampers can vary but 2-3 is most common.
1.2 Problem statement
Much of today’s plastic is used in disposable products, which have a large negative effect on the environment if not recycled or otherwise suitably disposed of. There are many ways of combating this issue, two of which are using organic based polymers and increasing the products lifetime. This thesis will focus on increasing the knowledge of long-term stress relaxation of a sugarcane-based plastic with properties close to HDPE and creating material models that include the relaxation. Increased knowledge in this area will simplify modeling and developing reusable products with longer lifetimes.
Previous work in the relaxation area involves short-term testing e.g., seconds or minutes. Tests done during 1-8 hours have been viewed as long-term tests. This thesis includes longer tests that evaluate the mechanical behavior during stress relaxation. With results from physical tests, a material model will be created using a three-system model.
Virtual twins, a virtual replica corresponding to the experimental tests performed in this thesis, will then be used to assess the model. The term virtual twins means a computer replicated model of an a physical object. The accuracy of the model can be compared to the actual tests by virtually creating the specimen and recreating the physical tests.
1.3 Objective
Research questions:
• How many relaxation tests are needed to create an accurate understanding of the material properties?
• Identify and define the different stages of the stress relaxation curve ranging from seconds, minutes, hours and weeks.
• How can mechanical material behavior be accurately generated using different mathematical expressions?
• What is the minimum relaxation time needed during physical testing to accurately determine a theoretical expression that replicates the material properties?
• Is it possible to create a numerical material model using data from the physical experiments to simulate the material behavior during the tensile tests?
1.4 Outline
The research is presented in 3 sections; each with its own method and results. This approach is used as sections 2 and 3 are heavily dependent on the result of the previous section. Figure 1.6 is a workflow chart that describes the approach.
- Physical stress relaxation testing
Preparation of strain gauge machines, additional equipment, and specimens and results of the physical experiments.
- Theoretical expressions
Using the results of the physical tests, the data is analyzed to create expressions matching the physical data.
- Creating and Calibrating a material model
Figure 1.6: workflow chart.
The discussion includes all three research sections and concludes with conclusions and recommendations for further work.
1.5 Limitations
• Only one manufacturing process setting and one type of HDPE material is used for all experiments.
• Duration of the relaxation phase and the length increase of the loading are the only parameters that change in all experiments.
• Only the material flow direction, referred as MD direction in this work, is considered in the experiments.
• Expressions and the material model only include before-neck mechanical behavior.
• Temperature was not controlled or monitored during the physical experiments. It can be assumed that temperature varied more during the experiments on the custom-built equipment which took place in an apartment with large south-facing windows than during the experiments performed on the MTS Qtest100, located in BTH’s climate-controlled laboratory. Experiments on the MTS Qtest100 only lasted for up to three hours, whereas the experiments on the custom- build tensile machine lasted for up to 56 days, during which daily temperature fluctuations would have occured.
• The number of elements per model is limited in the student version of Abaqus CAE.
• Access to the tensile machine MTS Qtest100 at BTH was limited due to the Covid-19 pandemic.
A small custom-built tensile machine was therefore constructed which consequently enabled long-term testing.
1.5.1 Limitations specific to custom-built tensile machine:
• Data gathering is limited to 5 datasets per second.
• The tensile machine was calibrated against the MTS Qtest100.
• Pre-load data collected by the software is deleted from the data set in order to standardize the starting point of the data sets. This data handling could potentially introduce error.
• The experiments were only filmed during the first hour of long-term testing. Any potential issues after the first hour cannot therefore be investigated with video material.
2 P HYSICAL STRESS RELAXATION EXPERIMENTS
One of the processing methods for thermoplastics is injection moulding. Injection moulding involves pushing molten plastic material into a mould. One method uses a single-screw injection-moulding machine as illustrated in Figure 2.1 [10]. This method was used to create the material plates used in this thesis. Figure 2.2 shows the injection-moulded plate with its dimensions.
Figure 2.1: Single-screw injection-moulding machine.
Figure 2.2: Injection-moulded plate with dimensions 79x90mm and thickness 0.6mm.
There are different ways of testing a material’s properties. The method used for this thesis is tensile testing. This experimental test is done by fastening a material specimen and extending it until failure while simultaneously recording displacement and force data [11]. During the extension of materials like HDPE, a stable neck occurs. Göktepe and Serdar [12] describe’s how glassy polymers during the cold- drawing process undergo necking. If a glasssy polymer is extended at a constant velocity, necking will eventually occur. The necking phenomenon consists of the initiation, propagation, and stable phases. A depiction of the before neck, initiation, and stable neck phase is shown in Figures 2.3 and 2.4. Figure 2.3 illustrates the data gathered during a test while Figure 2.4 shows the specimen’s geometry.
Figure 2.3: The three stages of a tensile test that result in a stable neck
Figure 2.4: Illustration of the tensile test stages.
This thesis focuses primarily on stress relaxation that occurs before necking, i.e. prior to the maximum load obtained and hence during the first stage of the deformation. Some tests are however specifically performed on the neck to obtain supplementary information. The tests are conducted to measure the stress relaxation that occurs in the material. Stress relaxation is when a material is loaded and then held at a constant strain. The material loses stress as soon as the loading stops because the chains in the material shift [11]. It is difficult to extract conclusive data during the stable neck phase as the material behaves differently in the neck than it does in the rest of the specimen; the effects of the varying behaviors cannot be distinguished in the collected data [12].
Experiment method
2.1 Manufacturing of additional equipment 2.1.1 Adjustable camera holder and floating table
Physical experiments are filmed to enable Digital Image Correlation (DIC) and to better understand the results. A stable and adjustable holder for the camera is essential to ensure the quality of the video. A platform was designed and built by the author for this purpose using plywood, bolts, and foam. The holder was designed using Computer Assisted Design (CAD) in Autodesk Fusion 360. The integrated Computer Assisted Manufacturing (CAM) was used to generate toolpaths to cut the plywood on a Computer Numerical Control (CNC) machine.
A separate floating table is also needed to support the camera in its holder because the MTS Qtest100 vibrates. After taking measurements of the tensile machine, a table was designed so the table surface could be placed as close to the specimen as possible without touching the machine. The table was built in BTHs wooden machine workshop. A picture of the table and camera holder can be seen in Figure 2.5.
Figure 2.5: Camera holder standing on the floating table. (Photographer: Pratik Rajesh
Powar)2.1.2 Specimen cutting die
Tetra Pak® supplied a limited number of pre-cut test specimens together with the offcuts of the injection moulded plates. To ensure that there were sufficient test specimens, additional specimens for informal testing were made using the offcuts. A cutter specifically for this purpose was made using BTH’s equipment.
The cutting profile of the bottom die was designed based on the test specimens supplied by Tetra Pak® and surrounded by a milled slot for the cutter. The top die has the same shape with an additional 0.02mm clearance around the edges to fit over the bottom die. The dies were designed in the CAD software Autodesk Inventor and machined in the machine laboratory’s HAAS CNC milling machine. The cutting edge of the top die was then filed and sanded to shape by hand as no appropriate tooling for this operation was available. The guide rods were made using the machine laboratory’s lathe and was fit into place first using a hydraulics press to insert the rods in the bottom die and then adjusted by chiseling around the rods. The finished cutter can be seen in Figure 2.6. The dies are made from construction steel and are not hardened. As a result, the edge needs to be sharpened often to ensure a clean cut.
Figure 2.6: Specimen cutting die.
2.1.3 Designing and manufacturing long-term testing machine
An additional tensile-testing machine was built because access to the MTS Qtest100 at BTH’s campus was limited due to the Covid-19 pandemic. Access to an additional tensile testing machine also enabled long-term testing that would not have been feasible on the MTS Qtest100. The objective was to build a machine that functioned as similarly as possible to the MTS Qtest100 at BTH with the capability to conduct unsupervised long-term tests. The design requirements are:
• Located and operational at the user’s home.
• The loadsensor only measures the force on the specimen.
• The data from the machine is accessible at all times.
• Specimen loading is exactly 30mm/min and stops at a defined distance.
• Programing and control of the machine is relatively simple.
• Collected data remains intact after a poweroutage and unforseen re-start.
• The specimen clamps are in line with each other, same as the MTS Qtest100 at BTH.
The resulting machine can be seen in the Figure 2.7. It is constructed of aluminium profiles intended for constructing machines and custom plates that were made from sheets of aluminium. The loadsensor is driven by an Arduino Yún [13] that has two separate processors: one for running C++ and interacting with the available pins and a second to run a local website. The prossesor that runs C++ code takes measurements from the loadsensor, keeps track of time, and deposits data at set intervals into a text file accesible to the second processor. The second processor hosts a simple website that only contains a link to download the text file with the data stored on it. Pictures of the tensile machine’s components can be found in Appendix A.
Loading of the tensile machine is done separately using a software called mach3 [14] that is mainly used to run CNC machines. By setting the stepper motor’s jog speed to 30mm/min with almost instant acceleration, it is possible to insert a single G-code line to control the total displacement of the machine with high accuracy.
To calibrate the machine, two tests were conducted on both the MTS Qtest100 and and on the custom- made machine. To match the data from the two machines, it was necessary to multiply the load with a single factor and shift the time axis data because the processor on the Arduino board has a delay that had not been considered. The calibration can be seen in Appendix A.
Figure 2.7: The custom-made tensile machine for long-term testing.
2.2 Specimen preparation
Injection-moulded plates and specimens were produced and supplied by Tetra Pak® using their punching equipment to ensure consistency throughout testing. All specimens are cut in the direction of the material flow in the center of the plate as other angles on the plate have different properties [15]. The placement of the specimen is important as the two circular extrusions seen in Figure 2.2 cause turbulence during the injection moulding process, which means the structure of the material in the plate is not uniform. All experiments are therefore conducted using specimens cut from the middle of the plate in line with the injection-moulding flow.
Before cutting the specimen, the bottom of the plate is marked to indicate the direction of material flow.
The plate is then cut by Tetra Pak® with the machine shown in Appendix A.
The specimen is marked with lines 60mm apart indicating the clamping positions in the strain-gauge machines. A DIC pattern is added between the clamping lines as a backup if any measurements require confirmation. Figure 2.8 illustrates the process of cutting and preparing the specimen.
Figure 2.8: The stages of specimen preparation.
Additional specimens for informal testing were made using the offcuts seen in Figure 2.8 B, using the tool in Figure 2.6. These specimens were prepared in the same way without the addition of a DIC pattern.
2.3 Tensile machine operations
Tensile testing is performed on two systems. All tests conducted on the MTS Qtest100 system located at BTH have a relaxation duration of 3 hours or less. Tests conducted using the custom-built machine last for weeks.
The MTS Qtest100 system is run by the software TestWorks 4 with a limited license; due to this, the machine can only conduct experiments with pre-defined programs. One of these programs is suitable to conduct simple relaxation tests. The program initially extends the specimen to a pre-set load or extension with the speed 30mm/min. The machine then holds the specimen at that location for a set amount of time. The program is set to record 200 data points during the relaxation phase. After the relaxation phase, the load on the specimen is brought to zero, the program is shut down, the specimen is removed, and the data is exported. Appendix A contains a picture of the MTS Qtest100 system.
Tests conducted on the custom-built tensile machine are performed with the same loading speed, 30mm/min as the MTS Qtest100 system. With the custom-built tensile machine, it is necessary to use extension distance to determine the start of the relaxation phase because data gathering and loading is controlled separately. A benefit of this separation is that the data gathering program does not need to be adjusted between tests. Given the long duration of these tests, fewer data points are captured over time because less relaxation is expected; 2000 data points at 0.2-second intervals to start with, followed by 500 data points at 0.5-second intervals, then 500 data points at 5-second intervals, and finally every 30 minutes until the test is stopped.
Once data gathering is started and confirmed by accessing the output file, a line of G-code is sent to Mach3 to pull the specimen a set number of millimeters at the given 30mm/min speed. After 10 minutes, the Mach3 software and stepper motor driving the displacement shuts down, leaving the specimen at the desired relaxation distance. The machine is then left alone gathering data until it is stopped. Appendix A contains a picture of the custom-built tensile machine in use with a specimen.
The specimen is fastened with the clamps at a measured 59mm distance. Once the specimen is securely in place, it is stretched taught without the sensor registering over 5N. This brings the specimen close to the clamping distance of 60mm. The camera holder in Figure 2.5 is used for all tests to capture the loading and initial part of the relaxation phase. For the custom-built tensile machine, shutdown of the loading system is also filmed to capture any potential movements. All videos are viewed to check for any issues during the test as an additional control.
A strain-to-fail test is the first experiment and is conducted to decide which points are used for relaxation testing. The strain-to-fail test consists of extending a specimen at the velocity 30mm/min until failure
Experiment results
The first experiment was a strain-to-failure and was conducted by extending a specimen at 30mm per minute. From the data gathered in this experiment, points are selected to where a relaxation experiment will be conducted, shown in this chapter. Frames from the video during the strain-to-failure experiment are shown in Figure 2.9, with the initial clamping distance displayed during the different stages of the experiment. All specimens that were used for experiments in the MTS Qtest100 are shown in Appendix A.
Figure 2.9: Pictures on how the different stages looked during the tensile experiments on the MTS Qtest100 system.
2.4 Strain-to-fail experiment
Figure 2.10 shows the results of the strain-to-failure experiment. The experiment ends before failure because a safety switch prevented the machine from extending beyond a set distance. As necking occurs before the safety switch activates and post-necking behavior is not in scope for this thesis, additional testing is not necessary.
Figure 2.10: Strain to failure experiment. Failure does not occur due to a pre-set
limitation on the machine.
2.5 Short-term experiments
30N, 35N, and 40N were chosen for the first relaxation experiments based on the strain-to-failure results and these were selected at around half the maximum load. Additional experiments at 60N and 69N were performed thereafter to cover a wider range of data below the point of neck inititation; the 60N test was conducted three times as a control. The results of these tests are shown in table 2.1.
Table 2.1: Relaxation tests over 1h before necking on the MTS QTest100 Initial relaxation load [N] Relaxation extension [mm] Relaxation time [hours]
30 0,655 3
35 0,813 3
40 0,998 3
60 2,097 3
69 5,002 1
60 1,983 1
60 2,162 1
The first 100 seconds of the relaxation experiments were repeated for 30N, 35N, and 40N to collect more data to accurately describe the relaxation curve. The data collection program was adjusted to record up to 20 000 data points instead of 200 during the relaxation phase as defined by the MTS Qtest100 system’s standard program. All experiments from this point on were performed with increased data capture during the relaxation.
Results of both 30N experiments can be seen in Figure 2.11.
Figure 2.11: 30N relaxation experiments, one with 3 hours duration and one with 100 seconds, conducted on MTS Qtest100; data plotted over the first 350 seconds.
It is evident from the relaxation curves in Figure 2.11 that data collection during the first minute was insufficient in the 3-hour experiment. Relaxation data of 30N, 35N, and 40N are presented hereon in as a hybrid of the 3-hour and 100-second experiments. Relaxation data for all loads is presented in figures 2.12-2.15.
Figure 2.12: Relaxation experiments before necking and the strain-to-failure experiment, conducted on MTS Qtest100; data plotted with load over extension.
It appears that initiation of necking occurs in the 69N curve as the measured load begins to drop before the loading phase ends, although this is not visible in the video material.
Figure 2.13: Relaxation experiments before necking and the strain-to-failure experiment, conducted on MTS Qtest100; data plotted with extension over the first 60 seconds.
Figure 2.14: Relaxation experiments before necking and the strain-to-failure experiment,
conducted on MTS Qtest100; data plotted with load over the first 350 seconds.
Figure 2.15: 40N, 60N, 69N relaxation experiments before necking and the strain-to- failure experiment, conducted on MTS Qtest100; data plotted with load over the full duration.
2.6 Long-term experiments
A load of 65N was chosen for the long-term experiments in order to extend the specimen as far possible without risking necking as was suspected at 69N. The experiment was performed twice to demonstrate that the results could be reproduced. Figure 2.16 shows the first minutes of the experiments’ results obtained with the custom-built tensile machine together with the strain-to-fail curve obtained with the MTS Qtest100. Figure 2.17 contains the data from the entire 56 days of the experiments.
Figure 2.16: Relaxation experiments before necking on the custom-built tensile machine and the strain-to-failure experiment from MTS Qtest100; data plotted load over the first 350
seconds.
2.7 Defined regions of stress relaxation
Using the data from the 65N long-term experiments that lasted 29-56 days, three regions are used to depict the relaxation behavior: initial, short-term, and long-term. The initial region contains the first 20 seconds and contains most of the relaxation. The short-term region lasts one hour, based on the long- term and short-term experiments. The long-term region lasts two weeks; any relaxation after this was too small to isolate from noise. Figure 2.18 and 2.19 illustrate the different regions and how much of the load has decayed using the longest of the two 65N experiments.
Figure 2.18: initial and short-term stress relaxation regions.
Figure 2.19: Long-term relaxation region.
2.8 Necking experiments
The experiments that include the initiation and stable stages of necking are shown in a load over displacement graph in Figure 2.20. The figures 2.21 and 2.22 show the load over time behavior of the necked relaxation experiments. Note that the relaxation curve seems to be close to the same measured load for all experiments.
Figure 2.20: Relaxation experiments during initiation and stable neck with the strain-to- failure experiment, conducted on MTS Qtest100; data plotted with load over extension.
Figure 2.21: Relaxation experiments during initiation and stable neck with the strain-to- failure experiment, conducted on MTS Qtest100; data plotted with load over the first 350
seconds.
Figure 2.22: Relaxation experiments during initiation and stable neck with the strain-to-
failure experiment, conducted on MTS Qtest100; data plotted with load over the full duration.
3 T HEORETICAL EXPRESSIONS Strain and stress types
To better understand the mechanical material response and use it for other geometries, the data obtained in the physical tests must be converted into strain and stress. Data gathered is load (N), extension (mm), and time (sec). To use this data, extension is converted into the unitless dimension strain and load is converted into stress (MPa). There are two ways of calculating strain and stress using the dimensions illustrated in Figure 3.1: Engineering strain/stress and True strain/stress.
Figure 3.1: Elongation of a specimen with original area A
0to the left and the deformed specimen with the true area A to the right.
The Engineering strain (3.1) 𝜖𝐸 is given by the change in length ∆𝐿 over the initial length 𝐿0 of a specimen. The Engineering stress (3.2) 𝜎𝐸 is given by the load 𝐹 divided by the initial cross-section area 𝐴0.
𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑠𝑡𝑟𝑎𝑖𝑛: 𝜖𝐸 =∆𝐿
𝐿0 (3.1)
𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑆𝑡𝑟𝑒𝑠𝑠: 𝜎𝐸= 𝐹
𝐴0 (3.2)
A limitation of the Engineering approach is that the cross-section area is assumed to be constant throughout loading. The True strain 𝜖𝑇 and True stress 𝜎𝑇 take the change in cross-section area into consideration. The True strain (3.3) 𝜖𝑇 is given by the logarithmic value of the total length 𝐿 divided by the initial length 𝐿0. The True stress (3.4) 𝜎𝑇 is given by the load 𝐹 divided by the cross-section area 𝐴.
𝑇𝑟𝑢𝑒 𝑠𝑡𝑟𝑎𝑖𝑛: 𝜖𝑇 = ln (𝐿
𝐿0) (3.3)
𝑇𝑟𝑢𝑒 𝑠𝑡𝑟𝑒𝑠𝑠: 𝜎𝑇 =𝐹
𝐴 (3.4) If volume 𝑉 is assumed to be constant, True strain can be calculated using the change in cross-section area (3.5). [16]
𝑇𝑟𝑢𝑒 𝑠𝑡𝑟𝑎𝑖𝑛: 𝜖𝑇 = ln (
𝑉 𝐴𝑉 𝐴0
) = ln (𝐴𝐴0) (3.5)
It is possible to relate the two different stress/strain measures to each other by the formulas (3.6) [6].
𝜎𝑇 = 𝜎𝐸(1 + 𝜖𝐸), 𝜖𝑇= ln (1 + 𝜖𝐸) (3.6)
Ramberg Osgood
The strain/stress curve of a non-linear material can be depicted with a single analytical expression. The Ramberg Osgood expression (3.7) [17], for example, can be used to approximate a material with a smooth non-linear response curve. This expression is mostly used for paper and aluminum but can also be used to mimic the behavior of other material [18]. The expression calculates the strain of a material using the Young’s modulus 𝐸, ultimate tensile strength 𝜎𝑢𝑡𝑠, and two dimensionless parameters 𝛼 and 𝑛.
𝜀𝑇 =𝜎
𝐸(1 + 𝛼 ( 𝜎
𝜎𝑢𝑡𝑠)𝑛−1) (3.7)
Relaxation theory
In materials with linear-viscoelasticity, one can observe creep or stress relaxation. For materials that have non-linear behavior, many of the properties associated with linear theory do not apply. Modeling this behavior is therefore done with expressions that are supported by physical experiments. Ward and Sweeney [19] describe three different approaches to creating expressions for this purpose: Engineering, Rheological, and Molecular.
The Engineering approach predicts behavior for a specific scenario with as few experiments as possible.
The Rheological approach has existed for some time with solutions ranging from generalized linear theory to more complex approaches with multiple integrals, but few solutions remain in use today.
One Molecular approach created by Eyring et al. [20] assumes that deformations of polymer material are the result of thermally activated processes, caused by the movement of molecular chains in the material over potential barriers. Ward and Sweeney [19] present equation (3.8) as a description of the frequency of molecular events where ∆𝐻 is the potential energy barrier, 𝑣 is for the chain-sliding frequency of the polymers, 𝑘 is Boltzmann’s constant, and 𝑇 is the absolute temperature.
𝑣 = 𝑣0exp (−∆𝐻
𝑘𝑇) (3.8) Material flow in the direction of an applied stress can be expressed using equations (3.9) and (3.10) given that linear shifts 𝛽𝜎 of the barriers ∆𝐻 occurs when stress 𝜎 is applied.
𝑣1 = 𝑣0exp (−(∆𝐻−𝛽𝜎)
𝑘𝑇 ) (3.9)
𝑣 = 𝑣1− 𝑣2= 𝑣0exp (−∆𝐻
𝑘𝑇) [𝑒𝑥𝑝 (𝛽𝜎
𝑘𝑇) − 𝑒𝑥𝑝 (−𝛽𝜎
𝑘𝑇)] (3.11)
Assuming the net-flow direction is in the same direction as the applied stress and that the flow direction is directly related to the changing rate of strain, the equation (3.12) is given, where 𝜀̇ is the constant pre- exponential factor, 𝑉𝑎 replaces 𝛽 and is termed the activation volume of a molecular event.
𝑑𝜀
𝑑𝑡 = 𝜀̇ = 𝜀̇0exp (−∆𝐻
𝑘𝑇) sinh (𝑉𝑎𝜎
𝑘𝑇) (3.12)
Guiu and Pratt use the Eyring approach by assuming that total strain consists of part elastic strain 𝜀𝑒 and part Eyring/viscous flow 𝜀𝑣. The total strain is then given by equation (3.13)
𝜀 = 𝜀𝑒− 𝜀𝑣 (3.13)
Adding time gives the equation (3.14)
𝜀̇ = 𝜀̇𝑒− 𝜀̇𝑣 (3.14) Replacing 𝜀̇𝑣 with the stress adaptation from Eyring in equation (3.12) gives (3.15); 𝐶 and 𝐵 being abbreviated constants.
𝜀̇𝑣 = 𝜀̇0exp (−∆𝐻
𝑘𝑇)1
2exp (𝑉𝑎𝜎
𝑘𝑇) = 𝐶𝑒𝑥𝑝(𝐵𝜎) (3.15) Assuming linear elasticity, (3.15) can be expressed as (3.16).
𝜀̇ =𝜎̇𝐸+ 𝐶𝑒𝑥𝑝(𝐵𝜎) (3.16) As strain is constant during the relaxation period, stress relaxation can be expressed as (3.17).
0 =𝜎̇𝐸+ 𝐶𝑒𝑥𝑝(𝐵𝜎) (3.17) Solving this equation with variable separation will give the Guiu and Pratt expression (3.18) with constants 𝐵, 𝑐, and stress 𝜎0 at the start of relaxation, 𝑡 = 0.
𝜎0− 𝜎 =1
𝐵𝑙𝑛 (1 +𝑡
𝑐) (3.18) The Guiu and Pratt equation is presented in detail by Sweeney et al. [21] with illustrations and issues concerning the short-term inconsistencies of the expression in comparison to experiment data. Sweeney et al. then present a complex two-system model that greatly reduces the short-term inconsistency in the Guiu and Pratt equation.
Theoretical analysis method
3.1 Converting physical data to strain and stress
Data from five physical experiments were converted into True strain and stress. The four experiments from the MTS Qtest100 and one from the custom-built tensile machine that were converted are shown in Table 3.1.
Table 3.1: Experiments converted to True strain and stress.
MTS Qtest100 Strain-to-failure 3-hour 40N relaxation 3-hour 60N relaxation 1-hour 69N relaxation Custom-built tensile machine
56-day 65N relaxation
The physical data is converted using the equations (3.4) and (3.5). Before the data is converted the geometry is simplified by removing the radiused corners of the specimen, leaving a rectangular 60x5x0.6mm block. This simplification is illustrated in Figure 3.2.
Figure 3.2: Specimen showing the true and the simplified geometry for strain and stress calculations.
Using geometric simplification of the specimen and assuming volume 𝑉 is unchanged throughout
Using Excel, the cross section was calculated using the known parameters 𝑉 and 𝐿0 with the measured extension of the specimen ∆𝐿. This was then used in equations (3.4) and (3.5) together with the measured load F to calculate the True stress and True strain.
3.2 Ramberg-Osgood formula
To find the Ramberg-Osgood curve (3.7) that best fit the material used in the physical experiments, a spreadsheet was created using graphs as a visual aid when calibrating Young’s modulus 𝐸 and the constants 𝛼 and 𝑛. Young’s modulus was initially calculated from the converted strain-to-failure data and the two unknown constants were given the value 1. These values are calibrated by changing them until the Ramberg-Osgood curve adequately matched the converted strain-to-failure data. The ultimate tensile strength 𝜎𝑢𝑡𝑠 was calculated from the strain-to-failure data and was used throughout the calibration.
The ultimate tensile strength was used when creating the stress data for the Ramberg Osgood equation;
200 datapoints spaced evenly from zero to the ultimate tensile strength. These points are used in the Ramberg Osgood equation to calculate strain, using the previously calculated Young’s modulus and the two constants. Both the Ramberg Osgood and the converted strain-to-failure data are plotted together and the values 𝐸, 𝛼, and 𝑛 are gradually calibrated. A picture of the spreadsheet with the initial values from the converted strain-to-failure data and Ramberg Osgood curve is found in Appendix B.
3.3 Relaxation expressions
To better understand the results from the physical relaxation experiments, the last load value from the relaxation period was divided by the initial value at the start of the relaxation. These were plotted together to give an overview of how the material reacted to the applied load. A small experiment was then conducted using data from the 60N relaxation experiment. Six data points were chosen closest to the times 0, 1, 10, 102, 103, and 104 seconds. The load measured at these points was then divided with the initial load to give a normalized curve starting at 1. This data is then entered into several data- generated functions to obtain an overview of the available mathematical expressions that might fit the datapoints.
The existing expression Guiu and Pratt (3.18) was suitable to model and predict the material properties measured in the physical experiments. A spreadsheet is created to help calibrate the expression to the converted relaxation data, by changing the Guiu and Pratt parameters until the expression adequately matched the converted physical experiment data. Using the three first hours of the relaxation period to calibrate the expression, it is used to generate the full relaxation period of the tests. This is used to assess the expression’s ability to predict the initial seconds of the test as well as the long-term behavior observed by the 65N experiments.
3.4 Re-creating experiments using mathematical expressions
Using the Ramberg Osgood and relaxation expression, an attempt is made to mathematically re-create the converted physical experiment data. The point chosen to graft the two expressions together is chosen from the converted experiment’s strain value, as the load applied to the specimen was strain driven during the experiments. These curves are then used to generate a FEM material model and to look for potential ways of generating experimental data in the gaps between experiments.
Theoretical results based on experimental data
3.5 Converted data to true strain/stress
The before-neck strain-to-failure data converted using the True strain and True stress method, is shown in Figure 3.3.
Figure 3.3: Converted train-to-failure data plotted true stress over true strain.
The same method to convert the strain-to-failure experiment is used to convert the 40N, 60N, and 69N experiments. These are shown plotted stress over strain in Figure 3.4.
Figure 3.4: Converted strain-to-failure and 40N, 60N, and 69N relaxation data. Data plotted true stress over true strain.
The long-term 65N experiment is not included as the extension of the specimen was not measured during the experiment. To convert this curve into strain and stress, the data from the strain-to-failure experiment was used to estimate the strain at the measured load 65N. This was then used as the constant strain during the relaxation of the 65N experiment. Strain during loading was assumed to be linear to estimate the loading curve. The four converted relaxation tests are illustrated in Figure 3.5 plotting stress over time.
Figure 3.5: Converted relaxation experiments. Data plotted stress over the initial 60sec, 3 hours.
3.6 Parameter study based on Ramberg Osgood formula
The fitted Ramberg Osgood curve did not exactly fit the converted strain-to-failure data, however, for the purpose of this thesis the difference is negligible. The formula with the calibrated parameters is expressed in equation (3.20) and is compared to the data from the converted strain-to-failure test in Figure 3.6.
𝜀 =1050𝜎 (1 + 1,97 (24,7𝜎 )5,2−1) , 0 ≤ 𝜎 ≤ 24,7 , 𝑆𝑡𝑟𝑒𝑠𝑠 𝑔𝑖𝑣𝑒𝑛 𝑖𝑛 𝑀𝑃𝑎 (3.20)
Figure 3.6: Converted strain-to-failure test data compared to the fitted Ramberg Osgood curve.
3.7 Parameter study based on relaxation expression
The plot of the residual stress after three hours of stress relaxation indicates that as the initial stress of the relaxation increases, the amount of residual stress on the specimen increases as well. A plot of the three-hour tests on the MTS Qtest100 showing the residual stress is shown in figure 3.7.
Figure 3.7: Residual stress from the three-hour experiments conducted on the MTS Qtest100.
The 60N data was chosen for the short experiment with the curve-generated expressions because a large amount of data points were collected during relaxation and the data was verified by two additional experiments. The data used is seen in Table 3.2 and was used on the website MyCurveFit [22] that calibrates expressions to submitted data.
Table 3.2: 3-hour 60N data used to generate an expression.
t [sec] 0 0,84 9,64 100,04 1000,04 10 000,04
𝜎(𝑡)/𝜎0 1 0,883 0,702 0,529 0,406 0,343
From the website, the non-linear for Four Parameter Logistic (4PL) method, was chosen as it matched the data geometry. The expression with values is shown in equation (3.21).
𝑦 = 0,3 + ( 1−0,3
1+(𝑥
19)0,433) (3.21)
𝑦 is replaced with 𝜎(𝑡)/𝜎0 and 𝑥 with 𝑡. The values 0,3, 19, and 0,433 are set as parameters 𝑚, 𝐾 and 𝑟. With these changes the expression became (3.22).
𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑒𝑑 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛: 𝜎(𝑡) = 𝜎0{𝑚 + ( 1−𝑚
1+(𝑡
𝐾)𝑟)} (3.22) From the existing methods of predicting stress relaxation, a rearranged version of the Guiu and Pratt expression (3.23) was chosen. As the re-defined generated expression (3.22) had an additional parameter compared to Guiu and Pratt, the expression was included in the modeling and prediction of the converted physical data. The spreadsheet used for calibrating the Guiu and Pratt expression was expanded to include both expressions.
𝐺𝑢𝑖𝑢 𝑎𝑛𝑑 𝑃𝑟𝑎𝑡𝑡: 𝜎𝑡= 𝜎0− {1
𝐵 𝑙𝑛 (1 +𝑡
𝑐)} (3.23)
Table 3.3: Guiu and Pratt expression values Tests on the MTS Qtest100
Test B c
40N 1,9 0,00009
60N 1,88 0,000004
69N 1,5 0,000002
Test on the custom-built machine
65N 1,68 0,000002
Table 3.4: Generated expression values Tests on the MTS Qtest100
Test m K r
40N 0,23 19 0,433
60N 0,3 19 0,433
69N 0,35 19 0,433
Test on the custom-built machine
65N 0,384 19 0,433
3.7.1 Expression fitting physical data
Figures 3.8-3.10 show the converted 40N, 60N, and 69N experiment data with the Guiu and Pratt and generated expression calibrated to best match the physical data.
Figure 3.8: Converted 40N data with Guiu and Pratt and Generated expressions. Data plotted stress over the full test duration.
Figure 3.9: Converted 60N data with Guiu and Pratt and Generated expression. Data
Figure 3.10: Converted 69N data with Guiu and Pratt and Generated expressions. Data plotted stress over the full test duration.
The graphs in Figures 3.8-3.10 illustrate that the Guiu and Pratt expression does not accurately predict the initial seconds of the material behaviour. Sweeney et al. [21] show how this can be solved by adding an additional system, however this would greatly increase the complexity of the calibration. The results show that the generated expression is sufficiently accurate and has the benefit of fewer parameters than a two-system Guiu and Pratt model.
With the two expressions fitted to the converted data from the first three hours of the 65N experiment, they were used to predict the entirety of the experiment without changing any parameters. Figure 3.11 shows the first 30 seconds and the initial three-hour window that the expressions were calibrated to.
Figure 3.12 shows the full duration of the experiment together with the extended experiments to evaluate their ability to predict long-term aspect.
Figure 3.11: Converted 69N data with Guiu and Pratt and Generated expressions. Data
plotted stress over the initial three hours.
As seen in figure 3.12, the Guiu and Pratt expression predicts a lower remaining stress then the converted 65N data. The generated expression has a slightly higher remaining stress but is much closer to the converted 65N data than the Guiu and Pratt expression. As the generated expression more accurately predicts the initial few seconds and deviated less in the long-term prediction, it was chosen to attempt to model the converted physical experiments together with the Ramberg Osgood formula.
3.7.2 Calibration result using one hour relaxation data
Using the 65N experiment, the calibration was reduced to only include the first hour of the experiment.
This was then used to calibrate new parameters and to evaluate the shorter duration’s ability to capture the long-term aspect. The results show that one hour is sufficient for the generated expression, however, the Guiu and Pratt expression ends with a lower relaxation compared to the three-hour data calibration.
The results are shown in Figures 3.13 and 3.14.
Figure 3.13: Converted 65N data with Guiu and Pratt and Generated expression.
Expressions calibrated to the first hour and then extended to the full duration.
Figure 3.14: Converted 69N data with Guiu and Pratt and Generated expression.
Expressions calibrated to the first hour and then extended to the full duration.
3.8 Combined expression results
The combined-expression predicted-relaxation experiments used to re-create the converted physical experiment data are compared together with true stress over true strain in Figure 3.15.
Figure 3.15: Expression predicted strain and stress data compared to the converted physical experiment data.
Figure 3.15 illustrates how the 69N expression-predicted data goes to a larger strain than the Ramberg Osgood curve is calibrated to (0,063), shown in Figure 3.6. This causes the calculated stress at this point to be higher than the results from the converted physical data. Because of this, the expression-predicted curve for 69N is not used. Figures 3.16 and 3.17 illustrate the curves created by the expressions compared to the converted physical data that is used for the FEM computer modeling. The expression- predicted data for the 65N experiment incudes data for all 56 days of the physical experiment.
Figure 3.16: Expression-predicted and converted physical data for 40N, 60N, and 65N relaxation experiments. Plotted stress over the initial 350 seconds.
Figure 3.17 Expression-predicted and converted physical data for 40N, 60N, and 65N
relaxation experiments. Plotted stress over the initial 3 hours.
4 M ATERIAL MODEL CALIBRATION
The concept of virtual twins is used to validify the material model. This concept originates from the time of NASA’s Apollo program, creating physical “twins” to simulate a vehicle’s response under specific flight conditions. This concept has since been further developed in virtual simulation environments and is currently used to simulate entire manufacturing systems and product lifecycles [23].
This thesis will use the virtual-twins concept to simulate physical experimental tensile tests to assess a material model’s ability to accurately predict the response of a physical material.
Finite Element Method
To create and execute the simulation needed for the physical specimen’s virtual twin, FE-software AbaqusTM 2020 Student Edition is used. For the simulation of the long relaxation periods of the physical experiments, one of the two dynamic methods, Explicit and Implicit, is used. In addition to these methods, Quasi-static analysis is required because the material has responses such as creep, viscoelasticity, and relaxation [24].
The Explicit method is a conditionally stable method, requiring small time durations between calculated increments. This is necessary because the method starts calculating the load region and includes additional regions as the load spreads in a dynamic pattern. This is often used for high impact loads because it reduces the number of regions the program solves per increment but can become unstable if time between the increments is increased.
The Implicit method is always stable, because the size of each increment does not affect the outcome because it calculates the whole model in every increment. For high-impact loads, this method takes more time to solve compared to the Explicit method. On the other hand, it is preferable for modeling long durations of relaxation testing as the program can be defined with longer intervals between increments which requires far less calculations than the Explicit method [25].
Material modeling
A viscoelastic material’s response to time-dependent loading can be captured by a three-network representation. The three-network model used by Bergström and Bischoff [26] is comprised of three molecular networks acting in parallel. The Parallel Rheological Framework illustrated in Figure 1.5 [27]
gives a visual representation of the molecular networks. Two of the networks are modeled with energy activation mechanisms aimed to capture the amorphous and crystalline structure of the material. The last network is for large strain responses that are controlled by entropic resistance.
Shahin et al. [28] used the three-network model to simulate the temperature- and time-dependent behavior of HDPE with physical tests lasting up to 8 hours. They used the PolyUMod [29] software to define the three-network model’s parameters and results show the model has the capability to capture most of the material’s behavior.
Calibration method
4.1 Calibrating the material model
Calibration of the three-network model of the material was done in the computer software Mcalibration developed by PolymerFEM [30]. The model is calibrated by incrementally adding more data manually to the software in order to avoid any local maximums. The model is initially calibrated to the converted strain-to-failure data, with the intention to capture the viscoelastic behavior of the loading phase. Once the three-network model accurately matches the converted strain-to-failure data, the converted 40N and 60N experiments are added. These are added with the intention to further calibrate the model and include the plant-based HDPE material’s relaxation properties during the three hours of the physical experiments. Lastly, the entire converted 65N experiment is added with the intention to include the long- term relaxation behavior of the material. Once the software calibrates a three-network model to fit the converted physical data adequately accurate, it is exported as a PRF AbaqusTM material model. The same method is used to calibrate a material model using the data created by the expression-predicted data.
4.2 Simulating the physical experiments
Simulations are made using the student version of the software AbaqusTM CAE. The 3D geometry of the specimen is created using measurements taken from one of the specimens before it is used for formal testing. The geometry only includes the region between the clamps. As a substitute for the clamped material, two reference points are created 20mm away from the ends of the specimen geometry and connectors added from each point to respective end surface. This removes the need for calculating the clamped geometry and makes it possible to increase the elements on the loaded part of the specimen.
Another benefit is the ability to gather specific load and extension data from the reference point giving the same data as with both tensile test machines.
The material model is first created using the Abaqus user interface, Aabaqus CAE, to define the viscoelastic and density properties taken from the exported MCalibration file. The two energy activated networks of the exported material model are copied and placed into the keyword file of the Abaqus model because the user interface does not support the type of model created by the three-network model.
The material model is defined in the option Model>Edit Keywords according to Figure 4.1 and with the keywords edited, a section with the material model is created from the specimen geometry. The specimen is then meshed as finely as possible with the student version of Abaqus.
