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

ABSTRACT 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

SAMMANFATTNING 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

ACKNOWLEDGEMENTS 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

CONTENTS 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

4.2.1 Analysis Description Unloading ...................................................................................... 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

Three Point Bend Test ........................................................................................................................... 41

1

1. INTRODUCTION 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 BACKGROUND

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 THESIS OBJECTIVE

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 RESTRICTIONS

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. GENERAL PROPERTIES OF POLYMERS 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 MOLECULE AND MICROSTRUCTURE

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 MECHANICAL PROPERTIES

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 UNIAXIAL STRESS-STRAIN PROPERTIES 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 𝑇𝑔 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 CREEP, RELAXATION AND RECOVERY

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 DETERMINING IRREVERSIBLE STRAINS 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 DEPENDENCY OF HYDROSTATIC STRESS 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 REINFORCEMENT OF GLASS FIBERS

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 𝑎𝑧𝑥𝑎𝑥𝑦 𝑎𝑦𝑦 𝑎𝑧𝑦𝑎𝑥𝑧 𝑎𝑦𝑧 𝑎𝑧𝑧

𝒆𝒙,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. EXPERIMENTS FOR MATERIAL TESTING 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 EXPERIMENTAL SETUP

3.1.1 TENSILE TEST

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 COMPRESSION 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 THREE POINT BEND TEST (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 =𝜀𝑚𝑎𝑥𝜀 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 EXPERIMENTAL RESULTS AND DISCUSSION

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 ULTIMATE TENSILE STRESS 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

𝜎𝑌𝑑𝑟𝑦 ,2

= 70 𝑀𝑃𝑎, 𝜀𝑌𝑑𝑟𝑦 ,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 1600

10

20

30

40

50

60

70

Eng. Strain [%]

Eng.

Str

ess [

MP

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 9 100

20

40

60

80

100

120

140

Eng. strain [%]

Eng.

str

ess [

MP

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 STRESS-STRAIN CURVES CYCLIC LOADING 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 60 900

10

20

30

40

50

60

70

80

90

100

110Stress as function of offset angle

Offset angle relativly flow direction []

Eng.

str

ess [

MP

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 50

10

20

30

40

50

60

70

80

90

100

110

Eng. strain [%]

Eng.

str

ess [

MP

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 RECOVERY AFTER UNLOADING 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 90

10

20

30

40

50

60

Eng. strain [%]

Eng.

str

ess [

MP

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 90

5

10

15

20

25

30

35

40

45

Eng. strain [%]

Eng.

str

ess [

MP

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]

Eng.

str

ain

[%

]Strain recovery, reinforced 0

20 MPa

40 MPa

60 MPa

80 MPa

100 MPa

0,2% strain

0 500 1000 1500 2000 2500 3000 35000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [s]

Eng.

str

ain

[%

]

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 REMAINING DEFORMATION AS FUNCTION OF LOAD LEVEL 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 35000

0.5

1

1.5

2

2.5

3

Time [s]

Eng.

str

ain

[%

]

Strain recovery, non reinforced

20 MPa

30 MPa

40 MPa

0,2% strain

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

Applied max load [MPa]

Rem

ain

ing s

train

[%

]

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 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Applied max load [MPa]

Rem

ain

ing s

train

[%

]

Remaining strains, reinforced 90

No relaxation

5 min relaxation

1 h relaxation

0,2% strain

0 10 20 30 400

0.5

1

1.5

2

2.5

3

Applied max load [MPa]

Rem

ain

ing s

train

[%

]

Remaining strains, non reinforced

No relaxation

5 min relaxation

1 h relaxation

0,2% strain

20

3.2.5 UNI-AXIAL COMPRESSION 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 60

20

40

60

80

100

120

140

Strain [%]

Str

ess [

MP

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 FLEXURAL STIFFNESS 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 9 100

5

10

15

20

25

30

35

40

45

50

Strain [%]

Str

ess [

MP

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 20 250

50

100

150

200

250

300

350

400

450

500

Displacement [mm]

Forc

e [

N]

Comparison 3PBT

0 reinforced

90 reinforced

Non reinforced

23

4. MATERIAL MODELING, FEA AND RESULTS 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 ACCOUNT FOR FIBER ORIENTATIONS USING HILL’S YIELD

CRITERION

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 ANALYSIS DESCRIPTION FIBER ORIENTATION

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 YIELD CRITERION

In MAT103 the orthotropic material is defined by Hill’s yield criterion given as

𝐹 𝜎22 − 𝜎33 2 + 𝐺 𝜎33 − 𝜎11

2 + 𝐻 𝜎11 − 𝜎22 2 + 2𝐿𝜎23

2 + 2𝑀𝜎312 + 2𝑁𝜎12

2 = 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:

𝐺𝜎11

2 + 𝐻𝜎112 = 𝜎𝑠

2 Eq. 4.5

2 direction:

𝐹𝜎22

2 + 𝐻𝜎222 = 𝜎𝑠

2 Eq. 4.6

3 direction: 𝐺𝜎332 + 𝐹𝜎33

2 = 𝜎𝑠2 Eq. 4.7

𝐹 =1

2 𝜎𝑠

2

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 𝜎𝑠

2

𝜎112 −

𝜎𝑠2

𝜎222 +

𝜎𝑠2

𝜎332 =

1

2

1

1−

1

0.482+

1

0.482 =

1

2 Eq. 4.9

𝐻 =1

2 𝜎𝑠

2

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′ = 𝜎1 cos2 𝜑 + 𝜎2 sin

2 𝜑 + 2τ12 sin𝜑 cos𝜑 Eq. 4.11

σ2′ = 𝜎1 cos2 𝜑 + 𝜎2 sin

2 𝜑 + 2τ12 sin𝜑 cos𝜑 Eq. 4.12

τ12′ =𝜎2 − σ1

2 sin 2𝜑 + τ12 cos 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′ = 𝜎1 cos2 45° + 𝜎2 sin

2 45° = 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 + Gσ11

2 + H σ11 − σ22 2 = 𝜏12′

2 𝐹 + 5𝐺 = 𝜎𝑠2 → 𝜏12′

2 =𝜎𝑠

2

𝐹 + 5𝐺 = 𝜏𝑠

2 Eq. 4.19

𝜎1 𝜎1

𝜎2

𝜎2

𝜏12′

1

2

1′ 2′

𝜑 𝜏12′

𝜏12′

𝜏12′

26

The constant 𝑁 was then determined to

2𝑁𝜎122 = 𝜎𝑠

2 = F + 5G τs2 → 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 ∙ 3.17𝜎122 = 𝜎𝐻𝑖𝑙𝑙

2 Eq. 4.23

4.1.3 RESULTS AND DISCUSSION 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 9 100

20

40

60

80

100

120

140

True strain [%]

Tru

e st

ress

[M

Pa]

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 SIMULATE LOADING/UNLOADING USING DAMAGE MODELING

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 ANALYSIS DESCRIPTION UNLOADING 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 DAMAGE PARAMETERS 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 −𝐸𝑑𝐸

Eq. 4.24

𝜎𝑌,𝑒𝑓𝑓 =𝜎𝑌

1 − 𝑑 Eq. 4.25

𝜀𝑝 = 𝜀 −𝜎𝑌,𝑒𝑓𝑓

𝐸= 𝜀 −

𝜎𝑌𝐸𝑑

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 0.090

10

20

30

40

50

60

70

80

True strain

Tru

e s

tress [

MP

a]

Determination of input parameters MAT187

Measured

Fitted curve

Y

(p) MAT024

Y,eff

(p) MAT187

29

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

4.2.3 RESULTS AND DISCUSSION OF THE LOADING/UNLOADING SIMULATION 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 0.040

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Plastic strain p

Dam

age p

ara

mete

r d

Damage paramter MAT187

Calculated di

Interpolated d(p)

0 1 2 3 4 5 6 7 80

10

20

30

40

50

60

True strain [%]

Tru

e s

tress [

MP

a]

Simulation loading/unloading

MAT187 - no recovery

MAT187 - 5min recovery

MAT024

Measured - no recovery

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 SIMULATION OF THREE POINT BEND TEST

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 ANALYSIS DESCRIPTION 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 DEFINITION OF MATERIAL MODEL 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 + 𝑝𝑡𝜎𝑌 = 𝑠𝑐𝑎𝑙𝑒 ∙ 𝑓𝑡 𝑝 + 1 − 𝑠𝑐𝑎𝑙𝑒 ∙ 𝑓𝑐 𝑝

Eq. 4.27

Indentor

Specimen Support

31

The reinforced resin in the flow direction is simulated with a hardening curve given as input based on the

experimental results obtained and showed in Figure 3.6 and 3.17. As a reference and comparison also

MAT024 is included based on tension data.

4.3.3 RESULTS AND DISCUSSION FOR 3PBT SIMULATIONS The force-displacement curves are shown in Figure 4.10 together with the experimental curves. In the

initial linear region the simulated results agree with the measured results, but start to deviate at a

displacement of about 7 𝑚𝑚 . At this point, the specimen starts to slide against the supports in the

simulations and the interface energy increase, which not is seen in the same extent for the measured

result. However, it can be seen how MAT124 gives a stiffer response than MAT024 and MAT124 is also

closer to the measured result.

Figure 4.10 Comparison of the simulation and measured results for the three point bend test

Even if it could be discussed whether the simulation gives the thru behavior above a displacement of

about 7 𝑚𝑚 due to the increased sliding, it could be seen that the major part of the displacement- force

response is captured by MAT024. As seen in this example, the influence of the hydrostatic pressure is

limited at a typical thickness of 3 𝑚𝑚 at bending.

However, the difference between measured and simulated results at higher strains complicates evaluation

of the accuracy in the compression tests showed in chapter 3.2.5 and which the MAT124 results are based

upon. A more suitable setup for the three point bend test had been to have a rolling contact between the

specimen and supports and thereby exclude the friction as a source of error.

0 2 4 6 8 10 120

50

100

150

200

250

300

350

400

450

500Simulation 3PBT

Displacement [mm]

Forc

e [

N]

Simulated MAT024

Simulated MAT124

Experimentally measured

32

5. REMARKS In the performed work several topics have been enlighten and investigated regarding the mechanical

properties of polyamide and thermoplastics in general. Discussions have been addressed for each topic

when presenting results and below follow additional remarks for the work in a wider perspective

concerning sources of error.

Only a few repetitions of each measurement were possible to perform. There is then a risk for misleading

result influenced from fluctuations of specific specimens. No notice were taken to how the specimen were

prepared regarding the specific position on the plate or from which plate they were machined from.

Variations between the different plates are therefore unknown. Different degree of fiber orientation is

most likely found depending on position on the plate. In future work it is recommended to keep track of

from which specific plate the specimens are machined from.

The injection molded plates varied in thickness and so also the specific specimens. A thickness could then

only be specified as an average for a specific specimen, which will add uncertainties in the measurements.

Less variation in the plate thickness would simplify repetitive results.

All results showed in this work will only be valid for the specific conditions for which the results are

obtained. Alternative strain rates will result in different results due to the viscous properties of the

material which should be taken in concern if implementing the experimental finding.

33

6. CONCLUSION The performed work has showed some important aspect regarding the behavior of thermoplastics in

general and polyamide resins especially.

6.1 FIBER ORIENTATION

Both stiffness and strength are highly dependent of the orientation of the fibers in the tested

glass-fiber reinforced polyamide when injection molded with a thickness of 3 𝑚𝑚 . The

maximum strength cross the flow direction was approximately 50% of the strength along the

flow direction and closer the strength of the corresponding non-reinforced resin.

The fiber orientation could be simulated with the anisotropic yield criterion by Hill and

implemented in LS-DYNA as material model MAT103. This requires that the flow must be

determined in each structural element and the possible element formulations for MAT103 are

restricted to hexahedral shell or solids.

6.2 LOADING/UNLOADING BEHAVIOR

Non-linear hysteresis response was seen in the loading and unloading paths for both the unfilled

and glass-fiber reinforced resins, although less pronounced for the reinforced resin.

The recovery of the residual strains after unloading was significant and approximately 45% of

the residual strains were recovered after 15 minutes of zero stress.

Simulation of the cyclic loading was made by the damage model implemented in MAT187,

where the elastic modulus continuously was lowered thru the loading sequence to match the

residual strains determined experimentally. The common elasto-plastic model MAT024 was

shown to highly over predict the permanent strains at unloading if the non-linear stress-strain

curve should be followed.

6.3 COMPRESSION PROPERTIES

A tendency of higher strength in compression than in tension was shown by uni-axial

compression test. However, no significant difference in stiffness was seen for moderate strains

below 3%.

The influence of the higher stiffness and strength in compression was in addition shown by

simulating a three point bend test and comparing it to the physical measurements. The simulation

where the material model MAT124 was used, which accounts for the hydrostatic stress, was

closer to the measured result than a simulation with MAT024 with a von Mises symmetric yield

criteria. The difference was although too small for recommending MAT124 in general

engineering applications.

34

7. RECOMMENDATIONS AND FUTURE WORK As seen in the experimental results, the spread in stiffness and strength is largely due to the flow

direction. Simulating glass-fiber reinforced resins with isotropic models will then raise some uncertainties

in how to evaluate results concerning critical stresses, i.e. failure criterion, and stiffness predictions. In

general, tensile data from material suppliers is given in the flow directional properties without any further

information regarding the fiber dependency. Safety factors must then be used in order to compensate for

the uncertainties which follows by reinforce plastics with short glass fibers. Future work is recommended

for evaluating the possibilities to mapping anisotropic data from flow filling simulations to structural

simulations in order to describe the material more correct and reduce the need of safety factors.

This work has showed that it is possible to some extent simulate the permanent strains after unloading by

an elasto-plastic model with a simple damage function. However, the method requires extensively testing

of material properties and calibration. Further work and evaluation of visco-elastic models in LS-DYNA

is recommended for simulation of the non-linear loading/unloading behavior, creep and relaxation.

Several subjects have not been included into this thesis work, but are recommended for future work. The

strain rate will affect all the measured data presented in this work so the extend of its influence must be

evaluated.

In the automotive industry, fatigue issues must always be considered. In the performed work, a softening

behavior was already seen for the materials at only a few repetitive loading cycles. Further investigations

regarding softening in relation to fatigue failure are therefore recommended. In addition, complex

industrial components are often subjected to multi-axial loading conditions. Tests to investigate possible

softening compared to uni-axial loading would therefore also be recommended.

35

8. REFERENCES

[1] N. G. McGrum, C. P. Buckley, C. B. Bucknall, “Principles of Polymer Engineering”, Oxford

University Press, Second edition, New York (1997).

[2] Trantina G., Nimmer R., “Structural Analysis of Thermoplastic Components”, McGraw-

Hill,Inc, New York (1994).

[3] Vincent M., Giroud T., Clarke A., Eberhardt C., “Description and Modeling of Fiber Orientation

in Injection Molding of Fiber Reinforced Thermoplastics”, Polymer 46, pp 6719-6725 (2005).

[4] Harward R., Thackaray G., “The use of a Mathematical Model to describe Isothermal Stress-

Strain Curves in Glassy Thermoplastics”, Procedings Royal Society A 302, pp 453-472 (1968).

[5] Raghava R., Cadell R., Yeh G., “The Macroscopic Yield Behavior of Polymers”, Journal of

Materials Science 8, pp225-232 (1973).

[6] International Standard ISO Plastics, “Determination of Tensile Properties – Part 2: Testing

Conditions for Moulding and Extrusion Plastics”, Reference number ISO 527-2:1993(E)

[7] Shan G.-F., Yang W., Yang M.-B., Xie B.-H., Qiang F., Mai Y.-W., ”Investigat