Investigation of Mechanical Properties of Thermoplastics with Implementations of LS-DYNA Material Models.

Investigation of Mechanical Properties of

Thermoplastics with Implementations of

LS-DYNA Material Models

Peter Appelsved

Degree project in

Solid Mechanics

Second level, 30.0 HEC

Stockholm, Sweden 2010

Investigation of Mechanical Properties of

Thermoplastics with Implementations of

LS-DYNA Material Models

Peter Appelsved

Degree project in Solid Mechanics

Second level, 30.0 HEC

Stockholm, Sweden 2012

A

BSTRACT

The increased use of thermoplastics in load carrying components, especially in the automotive industry, drives the needs for a better understanding of its complex mechanical properties. In this thesis work for a master degree in solid mechanics, the mechanical properties of a PA 6/66 resin with and without reinforcement of glass fibers experimentally been investigated. Topics of interest have been the dependency of fiber orientation, residual strains at unloading and compression relative tension properties. The experimental investigation was followed by simulations implementing existing and available constitutive models in the commercial finite element code LS-DYNA.

The experimental findings showed that the orientation of the fibers significantly affects the mechanical properties. The ultimate tensile strength differed approximately 50% between along and cross flow direction and the cross-flow properties are closer to the ones of the unfilled resin, i.e. the matrix material. An elastic-plastic model with Hill’s yield criterion was used to capture the anisotropy in a simulation of the tensile test. Residual strains were measured during strain recovery from different load levels and the experimental findings were implemented in an elastic-plastic damage model to predict the permanent strains after unloading. Compression tests showed that a stiffer response is obtained for strains above 3% in comparison to tension. The increased stiffness in compression is although too small to significantly influence a simulation of a 3 point bend test using a material model dependent of the hydrostatic stress.

Keywords: glass-fiber reinforced thermoplastics, polyamide, Hill’s plasticity criterion, strain recovery, cyclic loading, cyclic softening, compression strength, 3 point bend test, LS-DYNA

ii

Undersökning av mekaniska egenskaper

av termoplaster med implementeringar

av materialmodeller i LS-DYNA

Peter Appelsved

Examensarbete i Hållfasthetslära

Avancerad nivå, 30 hp

Stockholm, Sverige 2012

S

AMMANFATTNING

Termoplaster används i allt högre grad i lastbärande komponenter, framförallt inom bilindustrin, vilket kräver en bättre förståelse av dess komplexa mekaniska egenskaper. I detta examensarbete i hållfasthetslära har de mekaniska egenskaperna hos en PA 6/66 polymer med och utan förstärkning av glasfiber undersökts experimentellt. Fiberorienteringens inverkan på styvhet och styrka, kvarvarande töjningar vid avlastning samt skillnader i tryck respektive drag har varit av intresse. Den experimentella undersökningen följdes upp med simuleringar genomförda med befintliga och tillgängliga konstitutiva modeller i den kommersiella finita element lösaren LS-DYNA.

De experimentella resultaten visade att orienteringen av fibrerna påverkade de mekaniska egenskaperna betydligt. Draghållfastheten varierade ca 50% mellan längs respektive tvärs flödesriktningen och för den tvärsgående riktningen var materialegenskaperna närmre materialet utan glasfiber, det vill säga matrismaterialet. En elastisk-plastisk modell med Hills flytvillkor användes för att beskriva de anisotropiska egenskaperna i en simulering av ett enaxligt dragprov. Resttöjningar mättes under återhämtningen från olika belastningsnivåer och de experimentella resultaten implementerades i en elastisk-plastisk skademodell. Kompressionsprov visade att en styvare respons i förhållande till drag erhålls för töjningar över 3%. Den ökade styvheten i kompression kom dock inte att bidra betydande vid simulering av ett 3 punkts böjprov med en materialmodell beroende av hydrostatiska trycket.

Nyckelord: glasfiberförstärkt termoplast, polyamide, Hills flytvillkor, töjningsåterhämtning vid cyklisk last, cykliskt mjuknande, kompressionsstyrka, 3 punkt böjprov, LS-DYNA

iii

A

CKNOWLEDGEMENTS

I would like to thank all colleagues at the CAE department at Kongsberg Automotive for support and fruitful discussions concerning the work. Special thanks to Magnus Hofwing for guidance and planning as supervisor, Johan Haglind for design of the injection molded plates and Kent Salomonsson for helpful advices concerning the presentation of the material. However, the work would not have been possible without additional help from Marcus DeSalareff and Andreas Lindqvist from the prototype department and Stefan Jakobsson and Henrik Karlsson from the test department in Mullsjö.

And last, I thank Henrik Rudelius for his encouragement and for giving me a position as structural analyst at Kongsberg Automotive and the possibility to continue the work with mechanical properties of thermoplastics.

Peter Appelsved May 2012

iv

C

ONTENTS

Abstract ... i Sammanfattning... ii Acknowledgements ...iii Contents ... iv 1. Introduction ... 1 1.1 Background ... 1 1.2 Thesis objective ... 1 1.3 Restrictions ... 2

2. General Properties of Polymers ... 3

2.1 Molecule and Microstructure ... 3

2.2 Mechanical Properties ... 4

2.2.1 Uniaxial Stress-Strain Properties ... 4

2.2.2 Creep, Relaxation and Recovery ... 5

2.2.3 Determining Irreversible Strains ... 6

2.2.4 Dependency of Hydrostatic Stress ... 7

2.3 Reinforcement of Glass Fibers ... 8

3. Experiments for Material Testing ... 9

3.1 Experimental Setup ... 10

3.1.1 Tensile Test ... 10

3.1.2 Compression ... 10

3.1.3 Three Point Bend Test (3PBT) ... 11

3.2 Experimental Results and Discussion ... 12

3.2.1 Ultimate Tensile Stress... 12

3.2.2 Stress-Strain Curves Cyclic Loading ... 14

3.2.3 Recovery after Unloading ... 16

3.2.4 Remaining Deformation as Function of Load Level ... 18

3.2.5 Uni-Axial Compression... 20

3.2.6 Flexural Stiffness ... 21

4. Material modeling, FEA and Results ... 23

4.1 Account for Fiber Orientations using Hill’s Yield Criterion... 23

4.1.1 Analysis Description Fiber Orientation ... 23

4.1.2 Yield Criterion ... 24

4.1.3 Results and Discussion ... 26

4.2 Simulate Loading/Unloading using Damage Modeling ... 27

v

4.2.2 Damage Parameters ... 27

4.2.3 Results and Discussion of the Loading/Unloading Simulation ... 29

4.3 Simulation of Three Point Bend Test ... 30

4.3.1 Analysis Description ... 30

4.3.2 Definition of Material Model ... 30

4.3.3 Results and Discussion for 3PBT Simulations ... 31

5. Remarks ... 32

6. Conclusion ... 33

6.1 Fiber Orientation ... 33

6.2 Loading/Unloading Behavior ... 33

6.3 Compression Properties ... 33

7. Recommendations and Future Work ... 34

8. References ... 35

Appendix – Experimental Results ... 37

Tensile Tests for Maximum Strength ... 37

Cyclic Loading and Strain Recovery ... 39

1

1. I

NTRODUCTION

In the automotive industry, the use of plastics, i.e. polymers, has increased ever since plastics were introduced in the mid 1960s. Plastics were in the beginning only used for non load carrying applications in the interior of the car, but today one will find plastics in all parts of a modern car. Historically plastics gained a bad reputation in the everyday speech as a cheap and low quality material in comparison to metals. In fact, plastics are in many applications superior to metals in the sense of freedom in design and machineability, cost and environmental benefits, resistance and high stiffness in comparison to weight. Since plastics therefore tend to continuously replacing metals in load carrying and critical components, the need for and also the demands on structural analysis of plastics have increased. The mechanical properties of polymers differs in several aspects from metals, and also highly between different types of polymers, requiring more complex constitutive models accounting for time dependency, rate effects, non-linearities and anisotropy.

1.1

B

ACKGROUND

Kongsberg Automotive is a global provider of engineering, design and manufacturer of seat comfort, driver and gear shifter systems within the vehicle industry. Many of the components are manufactured in polymeric material and practically every new project requires new designs. The customers, i.e. automotive manufacturers, constantly increase the demands concerning stiffness, strength, robustness and weight. Finite element (FE) simulations are used with purpose to optimize the design and to ensure that the customer’s demands will be fulfilled before the components are physically tested and later implemented in the production.

Figure 1.1 Examples of products developed by Kongsberg; gear shifters and headrestraints.

1.2

T

HESIS

O

BJECTIVE

The often used design criterions are different static abuse loads, representing accidentally violence to the component and therefore a worst case design. In the evaluation, both the risk for fracture and remaining deformation of the component must then be revised.

Polymers seems troublesome from a design and computational point of view, since it does not exist any obvious yield point as for metals where one clearly can see the deviation from linearity. An obvious criterion for classifying critical stresses is therefore missing regarding remaining deformation.

In order to cost effectively increase stiffness and strength, polymer resins are often reinforced with short glass fibers (GF). However, the fibers tend to orient with the flow when the components are injection molded, resulting in anisotropic properties.

2

The aim of this master thesis is therefore to investigate remaining deformation and anisotropy for a common engineering thermoplastic by performing material testing and apply constitutive models that are implemented and available in the finite element code LS-DYNA.

Specifically, the following topics are addressed in the work

 Comparison of stiffness and strength in different directions of the flow, i.e. the fiber orientation, and for the corresponding non reinforced material.

 Unloading behavior and degree of strain recovery after unloading in order to evaluate the residual strains from different load levels as function of time.

 Possible differences in strength and stiffness in compression relative to tension.

1.3

R

ESTRICTIONS

One common polymer resin at Kongsberg Automotive, a polyamide 6/66 compound (PA6/66), was chosen for material testing and evaluation with the finite element method (FEM). The resin was tested both as reinforced with glass fibers and as non reinforced (unfilled) resin. Several aspects of thermoplastics behavior were tried to be captured in order to enlighten the complexity rather than secure statistical confidence. Testing procedures were based on international standards as well as methods presented in technical papers within the research area.

FE simulations were restricted to already existing and implemented models in LS-DYNA, both due to the complexity to implement user defined models and to obtain useful results in the daily engineering work at Kongsberg Automotive.

3

2. G

ENERAL

P

ROPERTIES OF

P

OLYMERS

Polymers are usually grouped into thermosets and thermoplastics, which differs both in characheteristic mechanical properties and in the manner of forming. Thermoplastics stand although for 90% of all worldwide produced plastics and are in absolute majority in the automotive industry due to low cost and easily formability [1].

2.1

M

OLECULE AND

M

ICROSTRUCTURE

Polymers are in general synthetic compounds with basically a carbon-carbon structure modified with an organic side group. Characteristic for polymers are its structure, where small molecular units, monomers, are covalent bonded together into long molecular chains. The process is called polymerization and result into a material with significantly high molecular weight [2].

The intermolecular bonds between the chains themselves differ although between thermoplastics and thermosets. Thermoplastics are linked together only by weak intermolecular bonds, i.e. Van der Waals or hydrogen bonds. Thermosets on the other hand, has strong covalent bonds between the chains and therefore naturally stiffer. In order to receive a plastic resin useful in engineering, additives as stabilizers and flame retardants, are necessary to add to the polymer base.

Thermoplastics are further divided into semi-crystalline (often just referred to as crystalline) and amorphous based on their degree of ordered microstructure. As understood from the notation, semi-crystalline thermoplastics have a microstructure consisting of small regions with ordered structure, in contrast to amorphous polymers which is entirely randomly ordered. A fully ordered structure is not possible due to the significant length and lack of symmetry in the molecular chains comparing to other groups of material [3].

Amorphous thermoplastics are in general stiffer and more brittle than semi-crystalline plastics, but have a more uniform and quantitatively lower shrinkage during processing. Amorphous thermoplastics can be made transparent and often referred to as glassy thermoplastics.

All thermoplastics are strongly temperature dependent, which is easily seen by plotting the stiffness against temperature. For amorphous thermoplastics, a suddenly drop in stiffness will be obtained when all intermolecular bonds breaks. The specific temperature is called the glass transition temperature, 𝑇𝑔, and amorphous thermoplastics could therefore only be used in temperatures below 𝑇𝑔. Semi-crystalline thermoplastics are also affected by 𝑇𝑔 and the amorphous regions cause a first drop in stiffness. The crystalline regions are although more resistance to increased temperature, so a final drop in stiffness is obtained at the melt temperature,𝑇𝑚. Semi-crystalline thermoplastics are therefore preferably used between 𝑇𝑔 and 𝑇𝑚, where its ductile and impact resistance properties are attractive. Note although, that 𝑇𝑔 and 𝑇𝑚 could vary significantly between amorphous and semi-crystalline thermoplastics, and also between different resins, i.e. 𝑇𝑔for an amorphous resin could equal 𝑇𝑚 for a semi-crystalline. In Figure 2.1, the characteristics of the temperature dependent stiffness of thermoplastics can be seen. Thermoplastics are formed after heated to high temperature, 𝑇𝑔 respectively 𝑇𝑚, where the solid polymer turns into viscous fluid and could be molded and dyed. Thermoplastics are therefore recyclable.

4

Figure 2.1 The characteristic temperature dependency for thermoplastics showing the glass and melt temperature. Left: Amorphous. Right: Semi-crystalline

2.2

M

ECHANICAL

P

ROPERTIES

The special microstructure of polymers with long molecular chains results in time dependent mechanical properties often denoted as viscous, which refers to the behavior of fluids. Viscous properties include the phenomenon creep, relaxation, strain recovery and rate dependency. The viscous behavior of glass fiber reinforced polymers will be less pronounced, due to the glass fibers more or less linear elastic response. The viscous properties complicate designing and dimensioning in the engineering work, since the highly non-linear behavior affects both stiffness and strength. In addition, long term influence (ageing) from air, sunlight and chemicals often results in a more brittle behavior. Basic mechanical parameters as the elastic modulus and yield point are therefore not as easy to define as for steel and will not remain constant if the loading conditions are varied.

2.2.1

U

NIAXIAL

S

TRESS

-S

TRAIN

P

ROPERTIES

A thermoplastic without any reinforcement has a typical stress-strain curve as seen in Figure 2.2. The curve is characterized by a local stress maxima followed by a softening behavior and finally re-hardening before rupture.

Figure 2.2 Typical stress-strain curves for non reinforced thermoplastic.

The behavior originates in the molecular structure, proposed by the pioneering work of Harward and Thackaray in 1968 [4]. According to Figure 2.2 (left), phase A is dominating up to the stress maxima and the deformation are mainly generated from movement of the molecular chains relatively each other. Eventually, the weak intermolecular bonds rupture resulting in the strain softening explaining the local stress maxima. In phase B, the molecule chain itself is straightened, resulting in re-hardening at large

𝐸 𝑇 𝐸 𝑇 𝑇𝑔 _𝑇 𝑔 𝑇𝑚

5

strains. The alignment at large strains cause transverse isotropy, i.e. that pure isotropic behavior could no longer be assumed. The straightening of the molecular chain makes the material stronger than the non- straightened, resulting in a different necking behavior in contrast to metals at uniaxial tensile tests. In metals, the specimen will be weakened in the necking region due to reduced cross-sectional area. In polymers, the necking region will instead grow due to that the necking region has become stronger than its surroundings, as seen in Figure 2.2 (right).

The structure of long molecular chains leads to a different behavior in compression, where the chains instead tend to orient in a plane perpendicular to the load direction resulting in higher strength. For thermoplastics, the maximum strength for compression could be up to 30% higher than in tension [5]. Important when it comes to polymers is that the yield point, 𝜎𝑌, not necessarily equals onset of plastic deformation like in metals. According to international standard ISO-EN 527 [6], the yield stress is defined as

𝜎𝑌=𝜕𝜎

𝜕𝜀= 0; 𝜎𝑌> 0 Eq. 2.1

i.e. the local stress maxima seen in Figure 2.2 (right). Not all types of thermoplastics show the characteristic stress maxima and other yield criterions are suggested and also used in the literature [7]. Glass-fiber reinforced thermoplastics have in general practically no necking due to brittle failure, and no yield criterion is used for these materials.

For thermoplastics, an increased strain rate results in general in an increased stiffness, i.e. that different stress-strain response is obtained depending on how fast the strain has been applied as illustrated in Figure 2.3. In contradiction to metals, the strain rate is of importance at all rates including rates that traditionally is referred to as quasi-static [3, 7, 8,].

Figure 2.3 Principle of rate dependency.

2.2.2

C

REEP

,

R

ELAXATION AND

R

ECOVERY

The viscous behavior is causing the well known phenomenon creep and relaxation, which are easily visualized in Figure 2.4 – 2.5. At creep the strain continuously increases, although the applied stress is constant. On the other hand, if the strain is held constant, the stress will continuously decrease resulting in relaxation.

The term recovery refers to a phenomenon occurring after unloading to zero stress, where the resulting strain will continue to decrease if it is unconstrained, i.e. that the material recovers. Creep and relaxation behavior of thermoplastics with varying load paths has been studied among several authors [8, 9, 10].

𝜀 𝑡 𝜀 𝜎 𝜀1 𝜀1 𝜀2 𝜀3 𝜀2 𝜀3

6

Figure 2.4 Principle of creep; increasing strain at constant stress.

Figure 2.5 Principle of relaxation; decreasing stress at constant strain.

2.2.3

D

ETERMINING

I

RREVERSIBLE

S

TRAINS

Since the onset of irreversible strains could not be concluded from the stress-strain curve, additional loading/unloading experiments have to be done monitoring the resulting irreversible strains from different load levels. A good example is the study by Brusselle-Dupend et.al. [11], focusing on the uniaxial behavior before necking on polypropylene (PP). Loading/unloading to different stress levels are followed by recovery (zero stress) until the residual strains has stabilized and could be concluded as permanent. The experimental study highlights the complex hysteresis unloading behavior of semicrystalline polymers including elastic, viscoelastic and plastic parts. Similar experimental setup for classifying residual strains after unloading in recoverable and permanent are found in the literature [7, 12] and illustrated in Figure 2.6.

Figure 2.6 Loading/unloading behavior with recovery 𝜀 𝑡 𝑡 𝜎 𝜎 𝑡 𝑡 𝜀 Recovery

𝜀

𝜀

𝜎𝑌

𝜎

Possible onset of irreversible strains

𝑡

𝜀𝑝 𝜀𝑝 _𝜀0 𝜀0

7

2.2.4 D

EPENDENCY OF

H

YDROSTATIC

S

TRESS

In the most common yield criteria, like the von Mises or Tresca, the hydrostatic stress is not included. For metals, where plastic flow usually is referred to as shearing of dislocation planes, this has been showed to be a satisfying description. Yielding of polymers and other materials, like soil, rocks and concrete, have on the other hand shown dependency of hydrostatic stress, or mean stress, defined as

𝜎𝑕= −p =

𝜎𝑥+ 𝜎𝑦+ 𝜎𝑧

3 Eq. 2.2

Several experimental studies [13, 14, 15] have been performed in order to investigate how a superimposed hydrostatic pressure influence the yielding of non reinforced polymers. Pae [13] investigated the yield surface of POM and PP by immerse test specimens in hydro-static pressure and superimpose tension, compression and shear. Common for both resins are the increasing yield strength with increasing pressure, i.e. negative (compressive) hydrostatic stress. In the most structural engineering applications one should therefore be aware of that when one has loading situations causing hydrostatic tension; the yield strength will decrease for superimposed tension. The Drucker-Prager yield criterion takes the hydrostatic pressure into account as the comparison with von Mises in Figure 2.7 show.

Figure 2.7 Illustration of Drucker-Prager yield criterion including the hydrostatic stress 𝑝 in comparison

to the common von Mises criterion [100].

Several investigations have been performed [16, 17] where the yield surface has been determined by uniaxial tensile, compression and shear tests together with biaxial tests. Often a yield surface as seen in Figure 2.8 is found, which does not coincide with the von Mises. A softening in biaxial loading is seen together with an increased strength in shear and compression. The yield surface for a specific polymer resin will although vary.

Figure 2.8 Left; Possible yield surface of a thermoplastic in comparison to von Mises. Right; resulting displacement-force curve for thermoplastics with an higher stiffness in compression.

𝑑 𝐹 𝜎1 𝜎2 Tension Compression von Mises Experimentally

8

2.3

R

EINFORCEMENT OF

G

LASS

F

IBERS

Reinforcement of glass fibers are a cost effective solution to increase stiffness and strength and still enables injection molding. The lengths could vary from tenths of a millimeter (referred as short) up to ten millimeters (long) and usually 20 to 50 wt%. Strength could be increased several hundred percentages compared to the base polymer, but the behavior will become more brittle and notch sensitive.

Mechanical properties will vary significantly in the different regions of an injection molded component when using glass fiber reinforcement with both in-plane and through thickness variations. Experiments have shown that short fibers tend to orient parallel to the flow near the walls of the mold tool, but randomly or even cross flow oriented in the core (skin-core-skin morphology) [2, 3, 18], as seen in Figure 2.9. The thickness will affect the size of the total amount of skin morphology, as it varies from around 90% for 2 mm thickness to 75% for 6.4 mm thickness [3]. Everywhere in the component where one could expect irregular flow, like around sharp corners or narrow gaps, the fiber orientations will be less pronounced.

Figure 2.9 Skin-core-skin morphology with less oriented fibers in the core.

The point wise orientation is usually determined by an averaged second order orientation tensor [17, 19] defined as

𝑎𝑘𝑖 = 𝑝𝑘𝑝𝑖𝜓 𝒑 𝒑

𝑑𝒑 Eq. 2.3

where 𝒑 is the axial orientation of an individual fiber oriented by two spherical angles with respect to a fix Cartesian system. 𝜓 𝒑 is the normalized orientation distribution function, i.e. the probability to find oriented fibers between 𝒑 and 𝒑 + 𝑑𝒑. The tensor 𝒂 is then written as

𝒂 = 𝑎𝑥𝑥 𝑎𝑦𝑥 𝑎𝑧𝑥 𝑎𝑥𝑦 𝑎𝑦𝑦 𝑎𝑧𝑦 𝑎𝑥𝑧 𝑎𝑦𝑧 𝑎𝑧𝑧 _𝒆 𝒙,𝒆𝒚,𝒆𝒛 Eq. 2.4

From the definition of 𝒂 follows 𝑎𝑥𝑥+ 𝑎𝑦𝑦 + 𝑎𝑧𝑧 = 1. A fully alignment in the x-direction according to Figure 2.9 would then imply that 𝑎𝑥𝑥 = 1.0 and a fully random orientation that 𝑎𝑥𝑥 = 𝑎𝑦𝑦 = 𝑎𝑧𝑧 = 1/3. Representative values for PA +30w.t% GF is 𝑎𝑥𝑥 = 0.8 in the skin and as low as 𝑎𝑥𝑥 = 0.2 − 0.4 in the core where the core region increases with thickness [19].

Stress-strain curves provided according to ISO527 is based on specimens directly injection molded into its shape and therefore representing results along the flow orientation. When reviewing the manufacture’s data sheets, one should be aware of that it is the best possible strength and stiffness which is reported, i.e. non conservative since the same ideally circumstances seldom are fulfilled in most injection molded components. skin skin core 𝑧 x z 𝑎𝑥𝑥 skin skin core

9

3. E

XPERIMENTS FOR

M

ATERIAL

T

ESTING

The experimental testing has been performed to investigate the specific properties of the chosen resin and to give input and reference results for finite element simulations. The following tests and their expected output was performed

 Uni-axial tensile test until break for determination of stiffness and maximum strength for the unfilled and reinforced resins. For the reinforced resin, specimens were prepared in 0°, 30°, 60° and 90° in relation to the flow direction in order to capture the flow dependency.

 Uni-axial tensile test with cyclic loading/unloading with a stepwise increased maximum load and recovery following each unloading, i.e. with a scheme like loading up to 20% of the tensile strength - unloading – recovery – loading up to 40% of the tensile strength - unloading – recovery and so on. The residual strains could then be categorized in recoverable and irreversible as function of load levels and allowed recovery time. The unfilled and reinforced resins with fibers in 0° and 90° were tested.

 Compression tests for determination of stiffness and strength for comparison to results obtained in tension. The unfilled and reinforced resins with fibers in 0° were tested.

 Three point bend test (3PBT) in order to evaluate the flexural stiffness in comparison to the tensile and compressive stiffness. The unfilled and reinforced resins with fibers in 0° and 90° were tested.

All specimens were cut out from injection molded plates designed in order to have a directed flow resulting in a preferred orientation of the glass fibers. The thickness of the plate was chosen to 3 mm, which is a typical thickness in injection molded structural components. In order to secure proper flow in the plate, a filling simulation was run using the commercial software Moldflow. The average fiber orientation according to Eq. 2.4 along the flow, i.e. y axis in Figure 3.1, was 0.8 in the skin and 0.6 in the core.

Figure 3.1 Injection molded plate for specimen preparation designed for even flow with oriented fibers. The chosen resin polyamide is sensitive for moisture absorption which will act like a plasticizer affecting both strength and stiffness. Material properties are therefore presented in dry respectively conditioned state. In the automotive industry in general, conditioned state is used which has been the focus in this work. However, some measurements in dry state have been included to point out the effect.

Moisture’s strong influence on the mechanical properties is found in the molecular structure of polyamides since the intermolecular hydrogen bonds between amide chains are interrupted and replaced with water bridges [20]. The entanglement and bonding between the molecule chains are then reduced resulting in decreased stiffness and strength and increased energy absorption, i.e. that moisture acts like a plasticizer.

10

Conditioned state specimens were conditioned according to ISO-EN 1110 [21] with a temperature of 70℃ and a relative humidity (r.h.) of 62% for approximatly 160 hours in order to receive conditioned state. After the accelerated conditioning, the specimens were kept in a controlled environment of 23℃ and 50% r.h. to preserve the conditioning.

Tensile test were performed at Jönköping University using a uni-axial electro-mechanical testing machine, Zwick Model E with 120𝑘𝑁 load capacity, together with a clip-gauge extensometer. The compression and the three point bend test were done at Kongsberg Automotive’s test facilities with a

Lloyd Instruments testing machine with a load capacity of 10𝑘𝑁 with strains evaluated from cross-head

displacement.

3.1

E

XPERIMENTAL

S

ETUP

3.1.1 T

ENSILE

T

EST

The tensile tests were displacement controlled with a cross-head speed of 0.5 𝑚𝑚/𝑚𝑖𝑛 in all tests, i.e. the ultimate tensile stress and cyclic measurements. Equivalent strain rate, based on the free length of 115 𝑚𝑚 between the grips, was then 7.2 ∙ 10−5 _𝑠−1 _{and the low strain rate made it possible to have a} displacement controlled unloading. After each load level in the cyclic loading scheme, the unloading was followed by a period of recovery. The load was removed by releasing one of the grips so the specimen solely was hanging free. The recovery time were depending on load level and material.

Specimens were CNC-milled from the plates in the different flow directions with dimensions according to ISO-EN 527 [6] as seen in Figure 3.2 and listed in Table 1.

Figure 3.2 Dimensions of the tensile specimen according to ISO-EN 527 (type 1B specimen).

3.1.2 C

OMPRESSION

No international standard was followed when the compression test was designed since the available plates limited the opportunities to design a short thick cylindrical specimens commonly used for these tests [19]. A similar setup were instead used as described in the experiments by Kolling et.al [16] and Becker et.al [22], which have performed tests in compression with flat specimens.

A fixture, as seen in Figure 3.3 with specimen dimensions listed in Table 2, was designed to give support and prevent buckling. No strain measurements were possible so the strain had to be derived from the cross-head displacement. Rectangular specimens were therefore cut from the injection molded plates.

𝐿𝑡𝑜𝑡 𝐿1 𝐿

𝐷

W

Table 1. Dimensions of tensile test specimen acc. to ISO527 Total length including grips 𝐿𝑡𝑜𝑡 ≥ 150 𝑚𝑚

Distance between grips 𝐿 = 115 ± 1

Effective test length 𝐿₁= 60 ± 0,5

Depth 𝐷 = 3 𝑚𝑚

Width 𝑊 = 20 ± 0,2

11

Each specimen was machined to a thickness that smoothly slides in the fixture. The same cross-head speed of 0.5 𝑚𝑚/𝑚𝑖𝑛 as for the tensile specimen was used.

Figure 3.3 Design of the fixture for compression tests and definition of specimen dimensions.

Table 2. Dimensions of compression test specimen Effective test length 𝐿1= 100 𝑚𝑚

Depth 𝐷 = 3 𝑚𝑚

Width 𝑊 = 20 𝑚𝑚

3.1.3 T

HREE

P

OINT

B

END

T

EST

(3PBT)

A simple 3 point bend test setup was used, as seen in Figure 3.4, with rectangular specimens with dimensions listed in Table 3. The span between the supports was 60 𝑚𝑚.

Figure 3.4 Setup for the 3 point bend test. Table 3. Dimensions of 3PBT specimen

Effective test length 𝐿1= 130 𝑚𝑚

Depth 𝐷 = 3 𝑚𝑚 Width 𝑊 = 20 𝑚𝑚 𝐷 𝐿1 𝑊 𝐷 𝐿1 𝑊

12

Test speed was determined so that approximately the same strain rate was obtained in the region of maximum strain as in the tensile tests. A finite element analyze, FEA, was therefore made in advance to simulate the test. At a certain deflection, 7 mm was chosen in this case, the 3 point bend simulation resulted in a maximum strain of 𝜀𝑚𝑎𝑥 = 0.031. The same strain rate as in the uni-axial tensile test was then given by 𝑡𝑖𝑚𝑒3𝑝𝑜𝑖𝑛𝑡 ,7𝑚𝑚 = 𝜀𝑚𝑎𝑥 𝜀 𝑡𝑒𝑛𝑠𝑖𝑙𝑒 = 0.031 7.25 ∙ 10−5 = 428 𝑠 = 7.13 𝑚𝑖𝑛 Eq. 3.1 𝑑 3𝑝𝑜𝑖𝑛𝑡 = 7 7.13 = 0.98 𝑚𝑚 𝑚𝑖𝑛 Eq. 3.2

A test speed of 1.0 𝑚𝑚/𝑚𝑖𝑛 was therefore used in the 3 point bend test in order to have a similar strain rate as in the tensile tests.

3.2

E

XPERIMENTAL

R

ESULTS AND

D

ISCUSSION

The below presented results are categorized after test outcome and show representative median curves or mean values. All obtained data could although be seen in the Appendix.

3.2.1 U

LTIMATE

T

ENSILE

S

TRESS

The stress-strain curves seen in Figure 3.5 for the unfilled resin and Figure 3.6 for the reinforced were obtained from the uni-axial tensile tests.

Both dry and conditioned states are included for the unfilled resin showing different behaviors. In the dry state, a stiffer response is obtained and the maximum stress is approximately 35% higher compared to the conditioned state. Some different characteristics could be noticed between the both curves. The dry state exhibit a plateau followed by a re-hardening up to the maximum stress of 70 𝑀𝑃𝑎 which is assumed to be associated with collapsing intermolecular bonds. A neck then develops with decreasing engineering stress and final rupture at approximately 58% strain. The conditioned state shows on the other hand a local stress maximum at 38% strain followed by softening and re-hardening until failure at approximately 150% strain. In the conditioned state, the inter-molecular hydrogen bonds are assumed to already be dissolved by the moisture and a smooth response is obtained up to the local stress maxima.

If the definition of yield strength, according to the ISO-EN 527 defined in Eq. 2.1, should be applied, two possible yield points are possible for the dry state due to the plateau and the following stress maximum. For the plateau one obtain

𝜎𝑌𝑑𝑟𝑦 ,1= 63 𝑀𝑃𝑎, 𝜀𝑌𝑑𝑟𝑦 ,1= 4%

and for the stress maximum

𝜎_𝑌𝑑𝑟𝑦 ,2= 70 𝑀𝑃𝑎, 𝜀_𝑌𝑑𝑟𝑦 ,2 = 28%

For the conditioned state the plateau is missing and the definition is straightforward 𝜎𝑌𝑐𝑜𝑛𝑑 _{= 52 𝑀𝑃𝑎, 𝜀𝑌}𝑐𝑜𝑛𝑑 _{= 38%}

The determined yield points should in the next chapter be seen in relation to the onset of irreversible strains.

13

Figure 3.5 Stress-strain curves for the unfilled resin in dry respectively conditioned state.

Figure 3.6 Stress-strain curves from different flow directions for the glass fiber reinforced resin showing both dry and conditioned state.

The glass fiber reinforced resins all had brittle failure without necking and the presented curves in Figure 3.6 is shown up to the maximum stress which correspond to the yield point defined in ISO-EN 527. Dry state data are included for the resins along and across the flow direction and it could be observed that the stress for the 0° and 90° curves in the conditioned state relates to the dry state by a factor of 1.3 at corresponding strain for both flow directions.

The tendency of decreasing stiffness and strength with increasing offset angle to the flow direction is remarkable clear and in Figure 3.7 is the corresponding stress for 1, 2, 3 and 4% strain plotted as function

0 20 40 60 80 100 120 140 160 0 10 20 30 40 50 60 70 Eng. Strain [%] E n g . S tr e s s [ M P a ]

Strength non reinforced resin



Yield cond.



Yield dry 1



Yield dry 2 Conditioned state Dry state 0 1 2 3 4 5 6 7 8 9 10 0 20 40 60 80 100 120 140 Eng. strain [%] E n g . s tr e s s [ M P a ]

Strength reinforced resin

0 (cond.) 0 (dry.) 30 (cond) 60 (cond) 90 (cond) 90 (dry) Increasing angle

14

of the offset angle to the flow direction. Up to 60° is an almost linear fit possible, but then a smaller difference is seen between 60° and 90°. Similar behavior with smaller difference between the 60° and 90° results could be found in the book by Trantina and Nimmer [2] for 30% glass fiber filled Polyethylene (PE). In contradiction, results has been shown by Andriyana et al. [8] with an equally difference between 30° to 60° and 60° to 90° offset angle. However, for both references, the largest difference in strength is obtained when going from 0° to 30°, i.e. that the stiffness and strength are sensitive for rather small offset angles to the flow direction, but this is not seen in the measurements performed in this work. The referenced authors, although, report a higher average degree of orientation with the flow, which then is assumed to affect the proportion of the stress-strain curves in the different flow directions.

Several other sources are reported to influence the results. Liang et al. [18] has extensively investigated different molding settings such as fill times and properties of the specimens like thickness and from which position of the plate it has been machined out. In addition, Liang et al. conclude that the cross flow measurement shows larger variation regarding stiffness than the along flow measurements which also is found in this work.

Figure 3.7 Stress at different strains as function of the offset angle relatively flow direction (conditioned state)

3.2.2 S

TRESS

-S

TRAIN

C

URVES

C

YCLIC

L

OADING

The load levels in the cyclic loading tests were based on the results obtained in the ultimate tensile stress results, i.e. that a number of load levels were evenly distributed along the stress-strain curve. Each unloading was followed by a period of recovery. The lower grip was released at zero stress in order to obtain a non restricted strain recovery. Unfortunately, the cyclic loading was not possible to perform with continuous measuring of the strain. The extensometer was therefore set to zero before every new load cycle.

In Figure 3.8 –3.10, the results are shown for the different resins with rather similar characteristics. The non-linear unloading and hysteresis for the unloading is as expected and in agreement with similar investigations [3, 8, 11, 23]. Noticeable is how a weaker response is obtained for every additional load cycle. It is important to keep in mind that the figures does not show the accumulated strain, i.e. that zero

0 30 60 90 0 10 20 30 40 50 60 70 80 90 100 110

Stress as function of offset angle

Offset angle relativly flow direction []

E n g . s tr e s s [ M P a ] 1% strain 2% strain 3% strain 4% strain 5% strain

15

strain has been defined for every new load cycle. The phenomenon is referred to as cyclic softening and has been shown by Launay et al. [3, 23] when experimentally testing a glass-fiber reinforced PA 66 resin for deriving a constitutive model for cyclic loading. The stiffness loss may be a mixture of several physical sources such as transformation of semi-crystalline matrix structure, fibre/matrix debonding or void formation [10, 24]. Launay take the cyclic softening in consideration in a proposed constitutive model by letting the stiffness decrease with an increasing inelastic energy, i.e. the difference between total mechanical energy and instantaneous elastic energy. Further investigations are although proposed by Launuay [23] in order to asses if the cyclic softening is an irreversible process or if a long-term recovery of the stiffness is possible.

Figure 3.8 Cyclic stress-strain curves for the reinforced resin loaded along the flow direction 0° .

Notice that the same specimen is used except in the reference curve (dashed) from Figure 3.6 and that the strain measurement has been put to zero before every new cycle.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 10 20 30 40 50 60 70 80 90 100 110 Eng. strain [%] E n g . s tr e s s [ M P a ]

Reinforced 0, cyclic loading

20 MPa 40 MPa 60 MPa 80 MPa 100 MPa Max load cyclic Max load single

16

Figure 3.9 Cyclic stress-strain curves for the reinforced resin loaded across the flow direction (90°).

Notice that the same specimen is used except in the reference curve (dashed) from Figure 3.7 and that the strain measurement has been put to zero before every new cycle.

Figure 3.10 Cyclic stress-strain curves for the unfilled resin loaded along the flow direction. Notice that the same specimen is used except in the reference curve (dashed) from Figure 3.5 and that the strain

measurement has been put to zero before every new cycle.

3.2.3 R

ECOVERY AFTER

U

NLOADING

Long term recovery times were not able to be performed due to limited available time for completing all tensile tests. A few tests with recovery times above one hour were although performed, see Appendix. Extrapolation of the presented mean results by fitting exponential functions has instead been done for all

0 1 2 3 4 5 6 7 8 9 0 10 20 30 40 50 60 Eng. strain [%] E n g . s tr e s s [ M P a ]

Reinforced 90, cyclic loading

20 MPa 30 MPa 40 MPa 50 MPa Max load cyclic Max load single

0 1 2 3 4 5 6 7 8 9 0 5 10 15 20 25 30 35 40 45 Eng. strain [%] E n g . s tr e s s [ M P a ]

Non reinforced, cyclic loading

20 MPa 30 MPa 40 MPa Max load cyclic Max load single

17

curves and agrees well with the few long term measurements. However, extrapolated material data will always raise some uncertainties regarding the reliability and therefore marked in the figures with dashed lines. As a reference, the common engineering assumption of using 0,2% of remaining strains as yield criteria for metals has been included in Figure 3.11 – 3.13 where the relaxation results are shown.

Figure 3.11 Relaxation for the reinforced resin loaded along the flow direction (0°) for different load

levels. Dashed lines indicate that extrapolation of the measured values by fitting an exponential function.

Figure 3.12 Relaxation for the reinforced resin loaded across the flow direction (90°) for different load

levels. Dashed lines indicate that extrapolation of the measured values by fitting an exponential function.

0 500 1000 1500 2000 2500 3000 3500 0 0.1 0.2 0.3 0.4 0.5 0.6 Time [s] E n g . s tr a in [ % ]

Strain recovery, reinforced 0

20 MPa 40 MPa 60 MPa 80 MPa 100 MPa 0,2% strain 0 500 1000 1500 2000 2500 3000 3500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time [s] E n g . s tr a in [ % ]

Strain recovery, reinforced 90

20 MPa 30 MPa 40 MPa 50 MPa 0,2% strain

18

Figure 3.13 Relaxation for the unfilled resin for different load levels. Dashed lines indicate that extrapolation of the measured values by fitting an exponential function.

3.2.4 R

EMAINING

D

EFORMATION AS

F

UNCTION OF

L

OAD

L

EVEL

The same data used in 3.2.3 Recovery after unloading has been manipulated to show the remaining deformation as a function of applied load level in Figure 3.14 – 3.16 with the purpose to serve as a reference if evaluating remaining deformation based on stress level. Continuous curves have been obtained using piecewise continuous interpolation of cubical splines.

Figure 3.14 Residual strains as function of applied load for different relaxation times for the reinforced resin loaded along the flow direction.

0 500 1000 1500 2000 2500 3000 3500 0 0.5 1 1.5 2 2.5 3 Time [s] E n g . s tr a in [ % ]

Strain recovery, non reinforced

20 MPa 30 MPa 40 MPa 0,2% strain 0 20 40 60 80 100 0 0.1 0.2 0.3 0.4 0.5 0.6

Applied max load [MPa]

R e m a in in g s tr a in [ % ]

Remaining strains, reinforced 0 No relaxation

5 min relaxation 1 h relaxation 0,2% strain

19

Figure 3.15 Residual strains as function of applied load for different relaxation times for the reinforced resin loaded across the flow direction.

Figure 3.16 Residual strains as function of applied load for different relaxation times for the unfilled resin.

Comparing with the maximum obtained stresses in Figure 3.5 and Figure 3.6, it could be observed that 80% and 70% of the maximum stress could be applied with 0,2% residual strains when allowing an recovery of 5 min for the reinforced resins along respectively cross flow direction. For the unfilled resin, only 50% of the maximum stress is possible to apply if as low as 0.2% residual strains are acceptable with 5 min of recovery.

0 10 20 30 40 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Applied max load [MPa]

R e m a in in g s tr a in [ % ]

Remaining strains, reinforced 90 No relaxation 5 min relaxation 1 h relaxation 0,2% strain 0 10 20 30 40 0 0.5 1 1.5 2 2.5 3

Applied max load [MPa]

R e m a in in g s tr a in [ % ]

Remaining strains, non reinforced No relaxation

5 min relaxation 1 h relaxation 0,2% strain

20

3.2.5 U

NI

-A

XIAL

C

OMPRESSION

No device to measure strain was available when performing the compression tests so the cross-head displacement of the testing machine was monitored instead. Weakness in the machine setup and specimen fixture was taken into account by measuring the deflection when compressing without a specimen. A polynomial was then least square fitted so the fixture deflection was given as a function of applied force and possible to extract from the compression measurement of the specimen.

The reliability of the curves presented here could be questioned, but it seemed that the specimen’s thickness in relation to the gate in the fixture where it was supposed to slide had a major impact on the results. If the thickness of the specimen were too thick, it had limited possibilities to slide when compressed due to the Poisson’s effect. On the other hand, if the specimen was to thin, it allowed more deflection resulting in a weaker response and lower collapsing buckling force. Presented curves in Figure 3.17 – 3.18 are therefore the test that showed the best deformation response, but in the same time correlated best with the corresponding tensile curves showed in Figure 3.5 and Figure 3.6.

In the presented results, it could be observed that at small strains below 2.5%, practically no difference is seen between tension and compression for both the reinforced and the unfilled resin. Above 2.5% strain, the compression curve started to deviate from the tension curve. For the reinforced resin a buckling collapse was obtained at 3.6% strain and the curve is extrapolated from this level by an exponential fit in order to be used in later FE implementations. No failure due to buckling was obtained for the unfilled resin, instead the fixture was preventing further compression, since the upper grip compressing the specimen came in contact with the fixture. Therefore, also the curve for the unfilled resin was extrapolated for use in FE implementations.

Similar trend with an increased deviation of the compression stiffness at increasing strain is seen in the literature, for example by Ghorbel [17] presenting tension and compression data for PA12.

Figure 3.17 Stress-strain curve in compression compared to tension for the reinforced resin loaded along the flow direction. Dashed lines indicate extrapolated values.

0 1 2 3 4 5 6 0 20 40 60 80 100 120 140 Strain [%] S tr e s s [ M P a ]

Compression vs tension, reinforced 0

Compression Tension

21

Figure 3.18 Stress-strain curve in compression compared to tension for the unfilled resin. Dashed lines indicate extrapolated values.

3.2.6 F

LEXURAL

S

TIFFNESS

The three point bend test show how the flow direction will affect the response in a perhaps more common load case compared to uni-axial loading. In Figure 3.19, it is seen that a completely different behavior is obtained between the specimens with the bending stresses parallel to the flow and the ones with the stresses directed cross the flow. Specimens loaded along the flow break at approximately half the prescribed displacement of 25 𝑚𝑚, while the specimen cross the flow does not and shows a behavior more similar to the unfilled specimen. Although failure is not obtained in the cross flow specimen, the load carrying capacity in the flow directed specimen is at least a factor of 2 higher.

The local stress maximum for the cross flow and unfilled specimen is a result of the specimens sliding at the supports. Unfortunately, the cross-section area of the 3 point bend (3PB) specimen was varying ±3,8% and the thickness ±5,7% itself. Bernoulli beam theory was used to compare the initial elastic modulus so the result for the median specimen regarding stiffness could be presented in Figure 3.19 for the different resins. In Appendix all the 3PB measurements are found including the median specimens shown below. 0 1 2 3 4 5 6 7 8 9 10 0 5 10 15 20 25 30 35 40 45 50 Strain [%] S tr e s s [ M P a ]

Compression vs tension, non reinforced

Compression Tension

22

Figure 3.19 Force-displacement response for the 3 point bend test for the reinforced resin along and across flow direction and the unfilled resin. The cross-section areas of the specimens differed although as

followed; 0° = 3.25 ∙ 25.07 𝑚𝑚, 90° = 3.16 ∙ 25.85 𝑚𝑚, unfilled = 3.30 ∙ 25.44 𝑚𝑚 0 5 10 15 20 25 0 50 100 150 200 250 300 350 400 450 500 Displacement [mm] F o rc e [ N ] Comparison 3PBT 0 reinforced 90 reinforced Non reinforced

23

4. M

ATERIAL MODELING

,

FEA

AND

R

ESULTS

It is important to be aware of the difficulties in capturing the true behavior of thermoplastics in material models when performing finite element analysis (FEA). Three different types of simulations have been performed in this work aimed to capture and highlight some of the complexity regarding fiber orientation, unloading and hydrostatic pressure dependency.

4.1

A

CCOUNT FOR

F

IBER

O

RIENTATIONS USING

H

ILL

’

S

Y

IELD

C

RITERION

The anisotropy introduced in a component made of injection molded thermoplastic is complex to handle in FEA. In order to obtain accurate properties for each individual element, flow orientations from a filling simulation must be mapped to the structural mesh. Even if the correct orientation could be obtained in each element, a fully anisotropic, non-linear constitutive formulation is as well costly. A cost effective solution is to make use of anisotropic yield criteria, for example Hill’s yield criterion implemented in LS-DYNA MAT103 Anisotropic viscoelasticity. The model has been used to simulate the measured fiber orientations with input data obtained and correlated to the 0° and 90° measurements.

4.1.1 A

NALYSIS

D

ESCRIPTION

F

IBER

O

RIENTATION

The experimental tensile test specimen is modeled and constraints applied similar as the grips in physical testing, see Figure 4.1.

Figure 4.1 Mesh density and boundary conditions for simulation of uniaxial tensile test.

Linear hexahedral elements (LS-DYNA parameter ELFORM 2) were used with an average element length of 0.85 mm, resulting in 6 500 elements with 4 elements thru the thickness. In order to use MAT103 in LS-DYNA, a local coordinate system had to be defined for each element used for the principal material orientations 1,2 and 3. The compatible LS-DYNA preprocessor, LS-Prepost, was used for the purpose. In general for anisotropic models in LS-DYNA, the principal material orientations are defined by a user defined element coordinate system a-b-c [25]. For solids, with the parameter AOPT equal to zero, see Figure 4.2 (left), are the vectors a and d defined and c and b given by the cross-products 𝒄 = 𝒂 𝐱 𝒅 and 𝒃 = 𝒄 𝐱 𝒂. In the current model, the vectors have been defined by the global coordinate system so the 𝑐 axis coincide with the global 𝑧 direction according to Figure 4.2 (right). The coordinate system was then rotated around its 𝑐 axis with the parameter BETA in order to obtain the different flow directions. The 1,2 and 3 directions in the Hill criterion then relates to the LS-DYNA local element system as 1 = 𝑎, 2 = 𝑑 and 3 = 𝑐.

Constrained in all DOFs

Prescribed displacement

24

e

Figure 4.2 Left: Definition of material principal axes in LS-DYNA. Right: Global coordinate system of the specimen.

4.1.2 Y

IELD

C

RITERION

In MAT103 the orthotropic material is defined by Hill’s yield criterion given as 𝐹 𝜎22− 𝜎33 2_{+ 𝐺 𝜎33− 𝜎11} 2_{+ 𝐻 𝜎}

11− 𝜎22 2+ 2𝐿𝜎232 + 2𝑀𝜎312 + 2𝑁𝜎122 = 𝜎𝐻𝑖𝑙𝑙2 Eq. 4.1 By introducing plasticity in the very beginning of the load curve it is possible to capture the anisotropic behavior thru the entire loading sequence. The hardening curve in the 0° direction was given as tabulated input. The difference to the usually used von Mises yield criterion is, depending on how the constants F,G,H,L,M and N are defined, that the criterion will be scaled depending on the direction of the stress. The constants F,G,H,L,M and N had to be determined by measuring yield stresses from uniaxial tensile tests in the 1,2 and 3 directions and yield stresses from shear in 12, 13 and 23 directions. The 1 direction was determined to be the flow direction and by assuming a transversely isotropic material gives

𝜎11= 𝜎0°= 𝜎𝑠= σHill Eq. 4.2

𝜎22= 𝜎33= 𝜎90° Eq. 4.3

The relation between 𝜎0° and 𝜎90° on the average thru out the loading sequence in the experimental results was

𝜎90°= 0.48𝜎0° Eq. 4.4

The constants were possible to be determined explicitly by assuming uni-axial loading in each principal material direction. 1 direction: 𝐺𝜎112 + 𝐻𝜎₁₁2 = 𝜎𝑠2 Eq. 4.5 2 direction: 𝐹𝜎₂₂2 + 𝐻𝜎₂₂2 = 𝜎𝑠2 Eq. 4.6 3 direction: 𝐺𝜎332 + 𝐹𝜎332 = 𝜎𝑠2 Eq. 4.7 𝐹 =1 2 𝜎𝑠2 𝜎₂₂2 − 𝜎𝑠2 𝜎₁₁2 + 𝜎𝑠2 𝜎₃₃2 = 1 2 1 0.482− 1 + 1 0.482 = 3.84 Eq. 4.8 y x 1 2

25 𝐺 =1 2 𝜎𝑠2 𝜎₁₁2 − 𝜎𝑠2 𝜎₂₂2 + 𝜎𝑠2 𝜎₃₃2 = 1 2 1 1− 1 0.482+ 1 0.482 = 1 2 Eq. 4.9 𝐻 =1 2 𝜎𝑠2 𝜎112 + 𝜎𝑠2 𝜎222 − 𝜎𝑠2 𝜎332 = 1 2 1 1+ 1 0.482− 1 0.482 = 1 2 Eq. 4.10

In the outlined work, no shear stress measurements were possible to perform. Therefore, a pure shear stress state were expressed from a combination of loads in the principal material directions, i.e. 𝜎1, 𝜎2 and 𝜎3.

Figure 4.3 Transformation in the 1-2 plane for the pure shear stress state where the

applied tension/compression 𝜎1/𝜎2 equals the shear 𝜏12′ in a rotated coordinate system.

A pure shear stress state is found as shown in Figure 4.3 and the transformation of the axes gives σ1′ = 𝜎₁cos2𝜑 + 𝜎₂sin2𝜑 + 2τ₁₂sin 𝜑 cos 𝜑 Eq. 4.11

σ2′ = 𝜎1cos2𝜑 + 𝜎2sin2𝜑 + 2τ12sin 𝜑 cos 𝜑 Eq. 4.12

τ12′ =𝜎2− σ1

2 sin 2𝜑 + τ12cos 2𝜑 Eq. 4.13

In the initial system, with an applied tensile/compression stress, the following is assumed

𝜎1, 𝜎2≠ 0, τ12= 0 Eq. 4.14

In the transformed system, a pure shear state prevail

𝜎1′ = 𝜎2′ = 0; 𝜏12′ ≠ 0 → 𝜎1≠ 𝜎2 Eq. 4.15

Eq. 4.11-4.13 together with Eq. 4.14-4.15 gives the rotation angle

σ1 1 − tan2_{𝜑 = σ2 1 − tan}2_{𝜑 → 𝜑 = 45° Eq. 4.16}

Using the found angle from Eq. 4.16 in Eq. 4.11 and 4.13 then gives

𝜎1′ = 𝜎₁cos245° + 𝜎₂sin245° = 0 → 𝜎₁= −𝜎₂ Eq. 4.17

𝜏12′ =

𝜎2− σ1

2 sin 2 ∙ 45° = 𝜎2= −𝜎1 Eq. 4.18

Inserting eq. 4.18 into eq. 4.1 results in

Fσ222 + Gσ112 + H σ11− σ22 2= 𝜏12′2 𝐹 + 5𝐺 = 𝜎𝑠2→ 𝜏₁₂′2= 𝜎𝑠 2 𝐹 + 5𝐺 = 𝜏𝑠2 Eq. 4.19

𝜎

1

𝜎

1

𝜎

2

𝜎

2

𝜏

₁₂′

1

2

1

′ 2′

𝜑

𝜏

₁₂′

𝜏

₁₂′

𝜏

₁₂′

26 The constant 𝑁 was then determined to

2𝑁𝜎122 _{= 𝜎𝑠}2_{= F + 5G τs}2 _{→ 𝑁 =}F + 5G

2 = 3.17 Eq. 4.20

In the same manner was 𝑀 and 𝐿 determined to

M = N = 3.17 Eq. 4.21

L =1

2 4F + 2H = 8.18 Eq. 4.22

The yield criteria in the model was then finally defined as

0.50 𝜎22− 𝜎33 2+ 0.50 𝜎33− 𝜎11 2+ 3.84 𝜎11− 𝜎22 2

+2 ∙ 8.18𝜎232 _{+ 2 ∙ 3.17𝜎31}2 _{+ 2 ∙ 3.17𝜎12}2 _{= 𝜎𝐻𝑖𝑙𝑙}2 _{Eq. 4.23}

4.1.3 R

ESULTS AND

D

ISCUSSION

The simulation was compared in Figure 4.4 to the physical testing by plotting the stress-strain response in the global 𝑥 direction.

Figure 4.4 Simulation o measured fiber orientations using MAT103

As could be expected, the simulation matches the measured results in the 0° direction, simply because the strain-stress curve for this direction was given as input to the model. In the other directions, the given curve was scaled by the specified factors, which are determined from the relation between 𝜎0 and 𝜎90. Since the relation differ thru out the loading scheme was an average value used as specified in Eq. 4.4. The result for the 30°, 60° and 90°directions are therefore dependent on how the relation 𝜎90/𝜎0 are specified, since it will result in different factors 𝐹, 𝐺, 𝐻, 𝑀, 𝑁, 𝐿. For presented set of parameters, the maximum difference between measured and simulated result is 17% at 8% strain for the 90° direction.

0 1 2 3 4 5 6 7 8 9 10 0 20 40 60 80 100 120 140 True strain [%] Tr ue s tr es s [M P a]

Comparison simulation to experimental measurments

0 measured 0 simulated 30 measured 30 simulated 60 measured 60 simulated 90 measured 90 simulated

27

As seen in Figure 4.4, MAT103 with Hill’s yield criterion can capture the influence from the fiber orientations in the solution. The use in more complex geometries and loading conditions require although the possibility to assign shifting principal material axes in individual elements based on flow simulation results. Nutini et.al [26] has used MAT103 for 4 node shells and created a mapping algorithm in order to include the orientations given from the mold filling simulation software Moldflow.

4.2

S

IMULATE

L

OADING

/U

NLOADING USING

D

AMAGE

M

ODELING

In most cases, perfectly linear elasticity is not accurate enough to represent the non-linear response in thermoplastics. By introducing non-linear plasticity models, the correct stiffness and stress response is captured and robust models using the von Mises evolution law are available in commercial FE codes. The most common elastic-plastic model in LS-DYNA is MAT024, Piecewise Linear Plasticity, where the hardening curve (true stress verses plastic strain) directly could be tabulated as input and therefore no parameter fitting is needed. However, one should be aware of that irreversible strains not necessary are introduced in the material just because the strain-stress curve starts to deviate from linearity as been stated in chapter 2.2.3 and found in the experiments. The simulation of cyclic loading/unloading using a simple damage model, implemented in MAT187, is aimed to present an alternative approach to the common elastic-plastic models.

4.2.1 A

NALYSIS

D

ESCRIPTION

U

NLOADING

The same mesh and boundary conditions was used as in the simulation of the fiber orientations, see Figure 4.1, except that no material principle coordinate system had to be defined. At the moment, MAT187 is only implemented for the explicit solvers of LS-DYNA so a quasi-static simulation had to be performed. The kinetic energy was held at a minimum so it became negligible in comparison to the total energy, i.e. that a sufficiently long time span was used when applying the load.

In contradiction to the experiment, the load was applied with a prescribed force so the unloading scheme could be simulated. Smooth loading curves (sinusoidal) had to be used for numerical stability.

4.2.2 D

AMAGE

P

ARAMETERS

Haufe et al [27] has developed and recently implemented the model SAMP-1 – A semianalytical model for polymers and referenced as MAT187 in LS-DYNA. In the model, it is a possibility to use damage modeling to get a representative unloading behavior by gradually decreases the elastic modulus thru the loading sequence by a damage parameter d, Eq. 4.24. The decreased elastic modulus, 𝐸𝑑, is in the same time compensated in the plasticity load curve, i.e. by decreased values of plastic strain, 𝜀𝑝, against an increased true stress, 𝜎𝑌,𝑒𝑓𝑓, Eq. 4.25 and 4.26. The principle of determining the damage is shown in Figure 4.5. 𝑑 = 1 −𝐸𝑑 𝐸 Eq. 4.24 𝜎𝑌,𝑒𝑓𝑓 = 𝜎𝑌 1 − 𝑑 Eq. 4.25 𝜀𝑝= 𝜀 − 𝜎𝑌,𝑒𝑓𝑓 𝐸 = 𝜀 − 𝜎𝑌 𝐸𝑑 Eq. 4.26

28

Figure 4.5 Determination of damage as function of plastic strain [31]

The model is used to capture the loading/unloading response of the non reinforced material as seen in Figure 4.6. A single load curve (dashed blue) was fitted to the measured loading/unloading curves (black). A linear approximation was done of the unloading from which the effective elastic modulus was defined, 𝐸𝑖,𝑑. The damage function was then calculated from the effective modulus and a piecewise cubic spline interpolation was used to receive the continuous function shown in Figure 4.7.

The red dashed curve in Figure 4.6 is the load curve compensated for the decreased elastic modulus,

𝜎𝑌,𝑒𝑓𝑓(𝜀𝑝𝑙), and given as input curve to MAT187 together with the continuous damage function 𝑑(𝜀𝑝𝑙).

As a comparison the load curve for MAT024 is plotted (continuous red curve) with plastic strains as usually based on the Young’s Modulus.

Figure 4.6 Determination of input data for MAT187.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 10 20 30 40 50 60 70 80 True strain T ru e s tr e s s [ M P a ]

Determination of input parameters MAT187 Measured Fitted curve

_Y(_p) MAT024

29

Figure 4.7 Damage parameter d as a function of plastic strain.

4.2.3 R

ESULTS AND

D

ISCUSSION OF THE

L

OADING

/U

NLOADING

S

IMULATION The result is shown in Figure 4.8 for the stress-strain components along the specimen. By correlating MAT187 to experimental results, it can be seen that the correct plastic strains, for a given recovery time, could be simulated using the simple damage function presented.

Figure 4.8 Comparison between experimental and simulated cyclic loading of the non reinforced resin. The simulation with MAT187 show the possibility to simulate unloading behavior, but in order to obtain the results some additional work with the input data must be done in comparison MAT024. Data for unloading is seldom provided by the material distributor and therefore additional testing must be performed. 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Plastic strain _p D a m a g e p a ra m e te r d

Damage paramter MAT187

Calculated d_i Interpolated d(_p) 0 1 2 3 4 5 6 7 8 0 10 20 30 40 50 60 True strain [%] T ru e s tr e s s [ M P a ] Simulation loading/unloading MAT187 - no recovery MAT187 - 5min recovery MAT024

30

In the end it could be discussed if the non-linear viscous behavior even should be approached with elastic-plastic models. The presented curves will only be valid for the specific loading condition which prevailed during the experimental measurements and cannot describe situations including creep or relaxation.

4.3

S

IMULATION OF

T

HREE

P

OINT

B

END

T

EST

The aim of simulating the three point bend test is to investigate how the stiffness and strength response differ if including the hydrostatic pressure in the material model. MAT124, Plasticity Compression Tension, is an elastic-plastic model with the possibility to define different hardening curves for compression and tension, i.e. include dependency of the hydrostatic pressure.

4.3.1 A

NALYSIS

D

ESCRIPTION

Since the simulation results were compared to the experimental results by comparison of displacement-force curves, the dimensions of the specific specimens had to be defined in the FE model. Like in the previous analyses, fully integrated linear hexahedral elements were used to model the specimen (LS-DYNA parameter ELFORM 2). The supports are modeled with quadratic shell elements and the indentor by first order tetrahedral elements. The meshed geometry is shown in Figure 4.9 where an element length of 0.5 mm were used in the specimen resulting in 70 000 elements in total and 7 elements thru the thickness.

Figure 4.9 Mesh densities for the three point bend test (Note that the complete model not is shown). The contact between the specimen and indentor were defined with a segment based contact definition (LSDYNA keyword AUTOMATIC_SURFACE_TO_SURFACE_MORTAR). The friction between the support and the specimen was measured to 0.08 using a in house Coloumb friction testing device. Same coefficient for static as for dynamic friction was used in order to obtain a stabile solution. The simulation was performed with a quasi-static implicit solution scheme.

4.3.2 D

EFINITION OF

M

ATERIAL

M

ODEL

By calculating the sign of the hydrostatic stress, either the compression or tension load curve is used in the defined plastic regime of MAT124 [25]. An isochoric flow rule is used to follow either the hardening curve for tension respectively compression. Which curve depends on the sign of the hydrostatic pressure (mean stress). In addition, a specific value of the hydrostatic pressure in compression and tension, 𝑝𝑐 and 𝑝𝑡 respectively, could be defined for when a pure compression, 𝑓 𝑝 , or tension, 𝑓(𝑝), load curve should be used. Between these specified values a weighted average will be used for numerical stability defined as

𝑖𝑓 − 𝑝𝑡 ≤ 𝑝 ≤ 𝑝𝑐 = 𝑠𝑐𝑎𝑙𝑒 = 𝑝𝑐− 𝑝 𝑝𝑐+ 𝑝𝑡 𝜎𝑌= 𝑠𝑐𝑎𝑙𝑒 ∙ 𝑓𝑡 𝑝 + 1 − 𝑠𝑐𝑎𝑙𝑒 ∙ 𝑓𝑐 𝑝 _{Eq. 4.27} Indentor Specimen Support