Measurements and simulations on the (dynamic) properties ... · PDF fileMeasurements and simulations on the (dynamic) properties of aluminium alloy AA6060 M. Beusink MT 11.25 June

Measurements and simulations on the(dynamic) properties of aluminium alloyAA6060

M. BeusinkMT 11.25

June 2011

Supervisors:prof.dr.ir. M.G.D. Geersprof.dr.ing. A.H. Clausenprof.dr.ing. T. Børvikprof.dr.ing. O.S. Hopperstad

Faculty of Mechanical EngineeringEindhoven University of Technology

SIMLab, Department of Structural EngineeringNorwegian University of Science and Technology

Abstract

During this project, carried out at the Structural Impact Laboratory, aluminium alloy AA6060 is inves-tigated under several testing conditions: specimens are machined from an extrusion billet obtained fromHydro Aluminium and are subjected to a heat treatment, before they are used for testing on either asplit-Hopkinson tension bar or a hydraulic tensile testing machine. During the tests temperatures will bevaried from room temperature up to 723 [K] and strain rates will vary from 0.01 to 1000 [s−1].

The heat treatment is applied to the specimens because an earlier study on the same material showedthat under the combination of high temperatures and strain rates, the specimens would fracture brittly. Itis suspected that this is caused by local melting of precipitates present in the alloy, although no metallurgicstudy has been carried out to confirm this suspicion.

After performing the tests, it is observed that the heat treatment has not prevented brittle fracture.Furthermore, due to several problems with the testing equipment the quantity of usable test results isfound to be insufficient for further analysis. The decision is made to continue with results obtained inearlier studies.

After obtaining the test results from earlier tests, three material models are calibrated to describethe material: the modified Johnson-Cook model with Ludwik’s hardening relation, the modified Johnson-Cook model with Voce’s hardening relation and the Zerilli-Armstrong model. The modified Johnson-Cookmodel with Ludwik’s hardening relation is found to give the best results. Simulation are carried outusing an axisymmetric numerical model, which reveals that this calibrated model performs less at highertemperatures, especially for quasi-static tests. The reason for this is believed to be the thermal softeningduring the quasi-static tests.

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Contents

1 Preface 7

2 Theoretical introduction 9

2.1 The material: aluminium alloy AA6060 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Split-Hopkinson tensile bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Hydraulic tensile testing machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Material models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.1 The modified Johnson-Cook model . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.2 The modified Johnson-Cook model with Voce hardening . . . . . . . . . . . . . . . 16

2.4.3 The Zerilli-Armstrong model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Experimental procedures and test results 19

3.1 The specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Measurement procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.1 Heating system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.2 Tests on the SHTB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.3 Tests on the hydraulic tensile testing machine . . . . . . . . . . . . . . . . . . . . . 22

3.3 Measurement results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Calibration and comparison of material models 27

4.1 Calibration of the material models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1.1 Calibration of the modified Johnson-Cook material models . . . . . . . . . . . . . 27

4.1.2 Calibration of the modified Zerilli-Armstrong material model . . . . . . . . . . . . 28

4.2 Calibration results and comparison of material models . . . . . . . . . . . . . . . . . . . . 28

5 Numerical model and simulations 31

5.1 Numerical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6 Conclusion and discussion 33

Bibliography 35

A Measurement procedures 37

A.1 Measurement procedure for the split-Hopkinson tensile bar . . . . . . . . . . . . . . . . . 38

A.2 Measurement procedure for the hydraulic testing machine . . . . . . . . . . . . . . . . . . 39

B Test results series 2 41

C Test results series 1 57

3

D Comparison test results series 1 and 2 59

E Data fitting results 63

F Simulation results 67

4

5

6

Chapter 1

Preface

Figure 1.1: An aluminium space frame of a car.

The Structural Impact Laboratory (SIMLab) in Trondheim is hosted by the Department of StructuralEngineering at the Norwegian University of Science and Technology (NTNU) in cooperation with theDepartment of Materials Technology, NTNU and SINTEF Materials and Chemistry. The main objectiveof the center is to develop safe and cost-effective solutions for aluminium, high strength steel and poly-mer structures. Applications are found in automotive and offshore industry, infrastructural safety andprotective structure for international peacekeeping operations.

Aluminium alloy AA6060 is used in multiple industries and for numerous products because of it’srelatively low density, good formability, corrosion resistance, etcetera. The reason to perform this projectis to attempt to accurately describe the material behavior of aluminium alloy AA6060 under wide rangesof strain rates and temperatures which are commonly encountered in hot working production processessuch as extrusion.

To investigate the material behavior, tests are performed on two testing machines: the split-Hopkinsontension bar and a hydraulic screw-driven tensile testing machine. The first is used to perform tests athigh strain rates, the latter is used for tests at quasi-static testing conditions. After the data has beenprocessed, three material models described in section 2.4 will be calibrated using the least squares methodto fit the data obtained by the measurements.

After calibrating the material models, they will be used in a numerical model to try to reproduce thedata which was obtained from tests. Although the numerical model did contain some critical instabilitiesat first, a working model is created. The simulation results show that the calibrated model is not able tocapture the full ranges of the strain rate and temperature.

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8

Chapter 2

Theoretical introduction

Before the tests are run, some theory involving the material, the measurements and the equipment usedis discussed. After this the models selected during this project are introduced.

2.1 The material: aluminium alloy AA6060

The metal investigated during the project is the aluminium alloy AA6060. Some typical applications ofthis alloy are architectural members (i.e. glazing bars and window frames), windscreen sections and inroad transport. The alloy is mainly used because of it’s good formability, corrosion resistance and it’ssuitability for surface treatments. Tables 2.1 and 2.2 show the chemical composition and some materialparameters of the alloy, respectively.

Figure 2.1 shows the face centered cubic crystallographic structure. In this unit cell the {111}-familyof planes are the four most densely packed planes and combined with the most densely packed directions,the 〈110〉-family, this crystallographic structure has a total of 12 slip systems. These slip systems playan important role during plastic deformation: the larger the number of slip systems, the more freelydislocations can move through the structure. High temperatures also have a positive effect on dislocationmobility, which explains why crystalline materials are more ductile when the temperature rises. Thereare however also limiting factors. Grain boundaries, foreign particles, precipitates and other dislocationsare all possible slip barriers which reduce the mobility of dislocations and thus reduce the ductility of thematerial.

Figure 2.1: Schematic representation of the arrangement of atoms in a FCC crystallographic structure.

The aluminium alloy under examination during this project is a polycrystalline material, meaning thatit is built up of a great amount of grains. All these grains have the fcc crystallographic structure shown inFigure 2.1, but each one has a different orientation. In a uniaxial tensile test, the maximum shear stresshas an angular offset of 45o to the applied load [1]. The slip systems that are aligned with the directionof this shear stress will be activated first and after further (plastic) deformation other slip systems will

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Table 2.1: Chemical composition of aluminium alloy AA6060 [2]

Al Si Fe Cu Mn Mg Zn Ti CrOther elementsEach Total

Min. [wt%]Balance

0.40 0.18 - - 0.45 - - - - -Max. [wt%] 0.45 0.22 0.02 0.03 0.50 0.02 0.02 0.02 0.02 0.10

Table 2.2: Material parameters for aluminium alloy AA6060 [2]

Density [kg/m3] 2700-2710

Young’s modulus [MPa] 69·103

Shear modulus [MPa] 26·103

Linear expansion coefficient [K−1] 293-373 [K] 23·10−6

Thermal conductivity [W/(m·K)] 293 [K] 200

Specific heat capacity [J/(kg·K)] 273-373 [K] 880-900

Melting range [K] 873-928

follow. During this process the dislocations will run into the earlier mentioned slip barriers, obstructingfurther movement of the dislocations unless the load is increased. This phenomenon is observed as workhardening.

2.2 Split-Hopkinson tensile bar

In this section the split-Hopkinson tensile bar (SHTB) will be described. Because the focus of thisproject is on analyzing the material under high strain rates and temperatures, the split-Hopkinson barwill be discussed in more detail than the other testing machines used during this project. A schematicrepresentation of this apparatus, of which the origin dates back to the 20th century [3], is given in figure2.2. The diameter of the bar used during the project is 10 [mm].

Figure 2.2: Schematic representation of the split-Hopkinson tensile bar. All measurements are in [mm].

The SHTB is operated by clamping the incident bar (section AC) at point B with a clamp (see figure2.3) and applying a tensile strain to section AB of the bar by means of a force N0 which can be set byusing a hydraulic jack. This force is calibrated with strain gauge 1. The steel used for the bars has a yieldstrength of approximately 900 [MPa] and therefore the force N0 must not exceed 70 [kN] to ensure thatthe deformation in the bar is purely elastic. The friction coefficient between the clamp at point B andthe bar is assumed to be in the order of 0.2−0.3 [4] and thus the clamping force should be approximately5 ·N0 to ensure that the bar will not move during the pretensioning. After removing the clamping forceat point B, a tensile strain pulse, referred to as the incident strain εI , is created in the incident bar.

To instantly achieve a constant strain rate during the measurements, it is desirable to produce aperfectly rectangular strain pulse. This is however physically impossible and therefore the clamping

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system shown in figure 2.3 is used to achieve a close approximation. Component f is loaded by applyingthe clamping force to component a by means of a pneumatic jack. After pretensioning the incident bar,the force applied to component a is increased until component f fails. This failure should be instantaneousto make the rise time of the strain pulse as short as possible and therefore component f should be madefrom a brittle material. A notched bolt made of hardened steel is selected to serve as component f [4].

Figure 2.3: Schematic representation of the clamping system used for the split-Hopkinson tension bar.

The created tensile strain pulse will partially propagate through the sample and into the transmissionbar (section DE in figure 2.2). The remaining part of the strain wave, referred to as the reflected strainεR, will reflect back into the bar as a compressive stress wave. The incident strain and reflected strainare measured by strain gauge 2, while the transmitted strain (εT ) is measured by strain gauge 3. Straingauge 2 and 3 are positioned at 600 [mm] from point C and D indicated in figure 2.2, respectively.

In figure 2.4 the recorded signals of strain gauge 2 and 3 during a typical measurement are shown.The incident strain is determined from the strain pulse recorded by strain gauge 2, while the reflectedstrain is calculated by subtracting ε2 from εI . The transmitted strain is equal to the strain recorded bystrain gauge 3. Figure 2.4 shows that duration of a test is in the order of 1 [ms].

As mentioned earlier, the incident, reflective and transmitted strain are measured approximately600 [mm] from the sample. To use these measurements to calculate the engineering stress, engineeringstrain and engineering strain rate in the sample, it has to be assumed that the stress wave is one-dimensional and no dispersion of the stress wave occurs. In this case the strains measured by the straingauges will be the same as in the sample. After making this assumption, the engineering stresses at bothends of the sample can then be calculated with equations (2.1) and (2.2).

σ1 =Ab

AsEb(εI + εR), (2.1)

σ2 =Ab

AsEbεT , (2.2)

where A refers to the cross sectional areas and the subscripts b and s refer to the bar and the specimen,

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

0

1

2

3

4

5

6

7

8

9x 10

−4

Time t [ms]

Str

ain

ε [−

]

Example strain signals

Strain gauge 2 (ε2)

Strain gauge 3 (ε3)

εR = ε

I − ε

2

εT = ε

3

εI

Figure 2.4: Recorded strain signals during a split-Hopkinson tensile bar test.

Figure 2.5: Schematic representation of the specimen in the split-Hopkinson tensile bar.

respectively. It is assumed that the specimen is in stress equilibrium, so:

σ1 = σ2, (2.3)

and thus:

εI + εR = εT . (2.4)

The velocities v1 and v2 of the ends of the bars that are connected to the specimen, depicted in figure2.5, can be related to the strains with one dimensional wave theory in the following manner:

v1 = cb(εI − εR), (2.5)

v2 = cbεT , (2.6)

where cb refers to the speed of the stress wave through the bars:

cb =√Eb/ρb. (2.7)

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In this project Eb = 210 · 103 [MPa] and ρb = 7850 [kg/m3] and thus cb ≈ 5100 [m/s]. The engineeringstrain rate is expressed as:

e =v1 − v2Ls

=cbLs

(εI − εT − εR). (2.8)

It is important to note that Ls is the parallel section of the specimen (as indicated in figure 2.5). Theengineering strain is now given by:

e =

∫t

cbLs

(εI − εT − εR)dτ, (2.9)

where t represents the duration of the test. If equation (2.22) is combined with equations (2.1) or (2.2),(2.8) and (2.9), the final expressions for obtaining the engineering stress, engineering strain rate andengineering strain in the specimen are given as:

ss =Ab

AsEbεT , (2.10)

es = −2cbLsεR, (2.11)

es = −2cbLs

t∫0

εR dτ. (2.12)

During the experiment an important assumption is made about the place where deformation occurs:the deformation is assumed to take place in the gauge section of the specimen. After yielding takes placesomewhere in the gauge section the deformation will indeed localize here and the assumption is valid,but before yielding some elastic deformation will also take place in transitional regions of the samplewhere the diameter decreases (see Figure 2.5). Consequently, when the signals are processed with theformulas presented in this section an error will exist in the experimentally determined Young’s modulus.A correction on the measured strain values can be applied [6] by using the correct Young’s modulus (alsosee Figure 2.6):

ε = ε′ − σ · E − E′

E · E′. (2.13)

This correction has been applied and checked on the SHTB used during this project with good results [4].

Figure 2.6: Correction of stress-strain curve.

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2.3 Hydraulic tensile testing machine

The tests at low strain rates are performed on a Zwick/Roell hydraulic testing machine with a loadcapacity of 30 [kN]. The machine is controlled by a computer on which the velocity of deformation ucan be assigned in [mm/s]. When the test is started, the computer will record the time t, force F anddisplacement u. As a function of these parameters, the engineering stress-strain curves can be calculatedwith the following equations:

es =u

Ls, (2.14)

ss =F

As. (2.15)

The engineering strain rate is calculated with the displacement per unit of time which is entered inthe computer:

es =u

Ls. (2.16)

2.4 Material models

The material AA6060 is assumed to be a elastic-viscoplastic material. The corresponding constitutiveequations, using the Von Mises yield criterion, are given in the following paragraphs and sections. Thestrain tensor can be decomposed into two contributing terms, namely the elastic and viscoplastic parts:

ε = εe + εp. (2.17)

Using the generalized Hooke’s law the stress tensor is then defined by:

σ = 4C : εe, (2.18)

in which 4C is the fourth order elasticity tensor which is given as:

4C = KII + 2G(4IS − 1

3II). (2.19)

K and G are the bulk and shear modulus respectively. The viscoplastic strain rate is expressed as:

εp = κ3σ′

2σ. (2.20)

with the deviatoric stress σ′, the equivalent stress σ and the equivalent strain rate κ defined as:

σ = σ′ − 1

3trσI, (2.21)

σ =

√3

2σ′ : σ′, (2.22)

κ =

√2

3εp : εp, (2.23)

respectively. In uniaxial tension equation (2.22) simplifies to σ = σ and equation (2.23) becomes κ = εp.Next a loading function f is presented:

f(σ, κ, κ, T ) = σ(σ)− σf (κ, κ, T ), (2.24)

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where an elastic limit surface is defined as f(σ, κ, κ, T ) = 0. If the material is loaded and the elastic limitsurface is reached, plastic deformation will start to occur if further deformation is applied and the elasticlimit surface will start to evolve. This evolution can mathematically be represented with the Kuhn-Tuckerconditions:

f(σ, κ, κ, T ) ≤ 0, κf(σ, κ, κ, T ) = 0, κ ≥ 0. (2.25)

In equation (2.24) σf represents the flow stress, which takes work hardening, strain rate sensitivityand temperature into account. This flow stress can be described by multiple models. For the numericalmodel in the LS DYNA FEM software a selection of these material models are considered. The mostsuitable of these models is chosen by determining the quality of the fits of each model on the availablemeasurement data. The models which are considered are:

• The modified Johnson-Cook model with Ludwik’s hardening relation (referred to as the modifiedJohnson-Cook model in the rest of the report).

• The modified Johnson-Cook model with Voce’s hardening relation.

• The Zerilli-Armstrong model.

These three models are selected because they are expected to give the best results of the models thatare available within the LS DYNA software package. The fitting procedures for each model are discussedin Section 4.1.

To account for large strains, a hypoelastic formulation using the objective Jaumann stress rate is usedby the FEM software. An additive decomposition of the rate-of-deformation tensor D is adopted [5]:

D = De + Dp + Dt, (2.26)

in which De is the elastic part, Dp is the plastic part and Dt is the thermal part. The hypoelasticformulation in terms of the Jaumann stress rate is expressed as:

σOJ = K tr(De)I + 2GDe′, (2.27)

where

De′ = De − 1

3tr(De)I (2.28)

is the deviatoric part of the elastic rate-of-deformation tensor. The objective Jaumann stress rate is givenas:

σOJ = σ −W · σ − σ ·WT (2.29)

In which σ is the time derivative of the Cauchy stress tensor σ and W is the spin tensor. If equation(2.27) and (2.29) are combined the final expression is obtained:

σ = W · σ + σ ·WT +K tr(De)I + 2GDe′. (2.30)

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2.4.1 The modified Johnson-Cook model

The empirical Johnson-Cook model dates back to 1983 and it’s aim is to combine the effects of largestrains, high strain rates and high temperatures in one constitutive relation [7]. The relation is given by

σf = (A+Bκn)(1 + C ln(κ∗))(1− T ∗m), (2.31)

where

κ∗ =κ

κ0(2.32)

represents the dimensionless strain rate and

T ∗ =T − TrTm − Tr

(2.33)

is the homologous temperature. The parameter κ0 is a user defined reference strain rate, while Tr andTm are defined as the room and melting temperature, respectively. The first term in the model describesthe strain hardening and consists of parameters A, B and n, the second term describes the dependenceof the strain rate and consists of parameter C and the third and last term represents the temperaturesoftening behavior and is described with parameter m. The second and third term essentially shift thehardening curve up or down, depending on the strain rate and temperature conditions.

In order to avoid unwanted effects when ε∗eq < 1, the strain rate sensitivity term in the Johnson-Cookmodel was adjusted [8]. This led to the modified Johnson-Cook model [9], given by:

σf = (A+Bκn)(1 + κ∗)C(1− T ∗m). (2.34)

2.4.2 The modified Johnson-Cook model with Voce hardening

In this version of the Johnson-Cook model the term which is accountable for the strain hardening isreplaced by two terms consisting of the Voce hardening law. The model is now expressed as:

σf = (A+

2∑i=1

Qi(1− exp(−Ciκ)))(1 + κ∗)C(1− T ∗m). (2.35)

Due to the two Voce terms, this model should improve the accommodation for different hardeningregions in the plastic stress-strain curve and thus might provide a better prediction of the materialbehavior. A disadvantage of this model however, is the additional number of parameters to be fittedcompared to the modified Johnson-Cook model. A third difference is that the modified Johnson-Cookmodel with Ludwik’s hardening relation describes a monotonously increasing stress as the plastic strainincreases, while with Voce’s hardening relation there is a horizontal asymptote defined by Q1 and Q2.

2.4.3 The Zerilli-Armstrong model

The Zerilli-Armstrong material model is a physically based model for metals [10] and, unlike the (modified)Johnson-Cook model, provides for coupling between thermal softening and strain rate sensitivity. Themodel is given in equation (2.36).

σf = ∆σ′G + c2κ1/2 exp(−c3T + c4T ln(κ)) + kl−1/2. (2.36)

In this relation, ∆σ′G is a term that accounts for the the effect of solute and initial dislocation density onthe yield stress. The term kl−1/2 considers the effect of the raise in polycrystal flow stress at relatively lowtemperatures, due to the requirement of stress concentrations at grain boundaries to enable transmission

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of plastic deformation between the grains. The parameter k represents the microstructural stress intensityand l is the average grain diameter. The constants c2, c3 and c4 are to be calibrated. Because the modelis developed with respect to dislocation theory, it consequently is different for each crystallographicmicrostructure. The model given above is for fcc structures.

In 1990 the Zerilli-Armstrong model was revised [11], this time combining ∆σ′G and kl−1/2 into oneconstant σA. The model is rewritten and now obtains the following form:

σf = σA +Aκ1/2 exp(−(α0 − α1 ln(κ))T ), (2.37)

where σA, A, α0 and α1 are the parameters to be fitted.

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Chapter 3

Experimental procedures and testresults

The performed tests were carried out under varying conditions concerning the temperature (room tem-perature up to approximately 723 [K]) and the strain rates (0.01 [s−1] up to 1000 [s−1]). Two testingmachines were used to obtain data at different strain rates:

• A hydraulic tensile testing machine (0.01 [s−1] up to 1 [s−1]).

• A split-Hopkinson tensile bar (100 [s−1] up to 1000 [s−1]).

In the following sections more will be explained about the specimens and measurement procedures.

3.1 The specimens

A schematic representation of the specimens used on the split-Hopkinson bar and the hydraulic testingmachine is given in Figure 3.1. The small specimen size was required due to the high strain rate testson the SHTB and to prevent possible size effects, the same specimen was used on the hydraulic testingmachine.

Figure 3.1: Schematic of the specimens used during the tests. All dimensions are in [mm].

The threads are required to fix the specimen in both machines. The specimens are machined to theshown specifications from the center of an extrusion ingot made of aluminium alloy AA6060 to avoid skineffects as a consequence of the casting of the ingot. After machining the specimens they are subjected to aheat treatment shown in Figure 3.2. The heat treatment is carried out in a Nabertherm N60/85HA oven,which can be programmed to perform the heating steps automatically, and is divided into a number ofsteps. First the specimens are placed in the oven, which is preheated to 353 [K], and they are kept at thistemperature for 2 minutes. Then, for 44.5 minutes, the specimens are heated at a rate of 10 [K/minute].The next stage in the heat treatment consists of keeping the temperature constant at 798 [K] and after13 minutes the specimens are quenched in water which is at room temperature. The specimens can then

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be stored for a maximum of 5 days in the freezer at 255 [K] to minimize natural aging of the heat treatedspecimens.

0 500 1000 1500 2000 2500 3000 3500 4000200

300

400

500

600

700

800

900Heat treatment samples

Time t [s]

Tem

pera

ture

T [K

]

Figure 3.2: The applied heat treatment to the specimens.

The heat treatment is applied to the specimens due to results from earlier tests [12], named series1. In those tests a loss of ductility was observed in the specimens at high temperatures and strainrates. Similar observations are reported for aluminium alloy AA5383 [13]. For this alloy (AA5383) twotypes of embrittlement were observed by the authors of the article. At several combinations of hightemperatures and/or strain rates a mix of ductile fracture, brittle intergranular fracture and cleavagefracture was noted. When temperature and strain rate are increased even further, the fracture modespresent are brittle intergranular fracture and cleavage fracture. In this work an important observation ismade: the intergranular fracture surface is extremely smooth and brittle, which suggests the presence ofa liquid film at the grain boundary prior to fracture, possibly caused by melting of deep eutectics. It issuspected that the brittle fracture in the earlier tests with aluminium alloy AA6060 is caused by similareffects, namely local melting of the Mg2Si precipitates which are responsible for the age hardening ofthis material [12]. The heat treatment is applied in order to try to reduce the precipitation and thus toprevent embrittlement.

3.2 Measurement procedures

The steps followed during the measurements and the problems encountered with equipment and duringtesting are discussed in the following sections.

3.2.1 Heating system

The heating system used to increase the temperature of the specimen to the desired value is an inductionheater with a remote heat station from MSI Automation Incorporated. The heat station is shown in

20

Figure 3.3a, the coil, set up on the SHTB, is shown in Figure 3.3b.

(a) (b)

Figure 3.3: The heating system used during the tests on both the SHTB and the hydraulic tensile testingmachine.

An Eddy current is induced in the specimen by adjusting the frequency of the alternating currentflowing through the coil, which, according to Joule’s first law (equation (3.1), where Q is the heat, Ithe current, R the resistance and t the time), is transformed into heat. The amount of heat producedis dependent on the amount and the frequency of the alternating current, the geometry of the specimenand the ohmic resistance of the material which is heated. A disadvantage of this method is the possibleoccurrence of the skin effect: the alternating current induced in the specimen will mainly flow throughthe “skin” of the specimen and thus the heating will not be uniform throughout the specimen. Becausethe specimen dimensions are relatively small and the the aluminium alloy has a high thermal conductivitythe temperature is assumed to be homogeneous within the entire specimen.

Q = I2Rt (3.1)

3.2.2 Tests on the SHTB

Figure 3.4: The split-Hopkinson tension bar.

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Procedures followed during the tests on the SHTB are given in Appendix A. Due to the large number ofacts performed, these tests had to be carried out with three people. Setting the strain rate proved to bedifficult during the tests, since it was not possible to enter or set a discrete value. The strain rate couldonly be set by adjusting the pretensioning force in the incident bar, which dropped during the additionalloading to release the clamping system. As a consequence the strain rate will vary around the values thatwere selected for testing. The temperature was measured with a probe which was fixed into position toensure constant contact with the sample during heating. Some problems were encountered during testing.On some occasions the pretensioned incident bar slipped through the clamping system, deforming thespecimens in the process and therefore rendering them inadequate for further testing. In such a case,the bolt in the locking mechanism and the specimen were replaced. Increasing the clamping force provedto be difficult since the bolts were already loaded within quite a small margin until failure, which wasexpressed by component f failing to early at some occasions. In this case a new test was started with anew sample.

3.2.3 Tests on the hydraulic tensile testing machine

Figure 3.5: The hydraulic tensile testing machine.

Procedures followed during the tests on the hydraulic tensile testing machine are given in Appendix A.During the measurements on the hydraulic testing machine the temperature was again measured witha probe, which was fixed into position with a number of clamps to ensure that the probe would stayin contact with the surface of the specimen. After a number of tests were carried out, some problemsoccurred with the heating system. The system leaked some water which was used to cool the coil and thecontrol system for the frequency of the alternating current broke down, making it impossible to controlthe temperature in the specimen with a good accuracy. Fluctuations in the temperature of up to 75 [K]were observed. The effects of these fluctuations on the stress-strain curves are shown in the followingsection.

3.3 Measurement results

After performing the measurements on both the testing machines, the data is processed into the graphsshown in Appendix B. The engineering stress-strain curves are obtained from the measurement databy using the methods discussed in Chapter 2. It should be mentioned that the strain is based on thedisplacement of the ends of the specimen, not on the elongation of the gauge length Ls. This means that

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H

Table 3.1: Series 2 test results.

Test ε T σ0 Q C H σ4 σ8 σ12 σu εpu εf[s−1] [K] [MPa] [MPa] [-] [MPa] [MPa] [MPa] [MPa] [MPa] [-] [-]

s2t1 400 289 74.4 35.8 16.2 69.6 94.3 106.0 113.4 127.8 0.259 1.616

s2t2 320 289 47.5 47.6 14.3 235.2 77.6 98.8 114.8 169.2 0.317 1.578

s2t3 950 289 62.5 138.2 3.2 57.0 81.4 98.3 113.4 179.9 0.369 1.936

s2t4 300 458 45.9 38.8 26.5 161.6 77.7 93.0 102.5 114.8 0.188 1.689

s2t6 1000 478 56.8 87.7 4.3 0.0 70.7 82.3 92.2 123.6 0.330 1.867

s2t7 250 628 46.2 11.1 28.2 77.7 56.8 62.4 66.2 75.8 0.239 2.383

s2t9 900 653 42.6 37.2 6.7 0.0 51.3 58.0 63.2 70.7 0.211 2.958

s2t11 300 693 35.0 13.4 24.7 63.5 46.0 51.6 55.3 67.6 0.302 3.148

s2t13 0.01 294 48.8 67.7 7.8 192.3 74.6 95.6 113.0 186.0 0.380 1.126

s2t14 1 294 56.0 58.3 7.4 180.5 76.4 95.4 111.2 174.2 0.355 1.455

s2t15 1 473 54.2 31.1 21.2 145.7 77.1 91.0 100.3 124.3 0.268 1.603

s2t16 0.01 473 54.4 36.1 16.7 156.2 78.2 93.5 104.4 130.5 0.259 1.545

s2t20 1 573 - - - - - - - 38.1 0.057 4.240

s2t21 1 723 - - - - - - - 12.0 0.083 7.371

the strain is overestimated since there will also be some deformation in the remaining parts of the sample.This error should however be the same in all the tests, i.e. it should not depend on the temperature andstrain rate, and should not influence the measured strain rate and temperature sensitivity of the material.The true stress-strain curves are then calculated with the equations (3.2) and (3.3), in which σ and ε arethe true stress and strain and s and e are the engineering stress and strain.

σ = s(1 + e) (3.2)

ε = ln(1 + e) (3.3)

These equations are only valid up to necking. After calculating the true stress-strain curves the trueplastic strain is determined by using Hooke’s law:

εp = ε− σ

E. (3.4)

The resulting data, especially those from the dynamic tests, still contain some noise. Using a constitu-tive relation which combines Voce hardening and linear hardening (equation (3.5)) to fit on the data, it ispossible to obtain smooth, filtered curves which accurately describe the experimental data. The resultingcurves and parameters can also be observed in Appendix B as well as in Table 3.1.

σ = σ0 +Q · (1− exp(−Cεp)) +Hεp (3.5)

Table 3.1 also contains the temperatures (T ), strain rates (ε), stress values at plastic strains of 0 [-](σ0), 0.04 [-] (σ4), 0.08 [-] (σ8), 0.012 [-] (σ12), the maximum stress before necking (σu) together with thecorresponding plastic strain (εpu) and the fracture strain (εf ). The fracture is calculated by measuring theaverage diameters of the fracture surfaces with an optical microscope and using equation (3.6).

εf = 2 ln(D0

Df) (3.6)

23

First the results of tests s2t20 and s2t21 should be mentioned. Due to the combination of a hightemperature and a slow strain rate in these tests, the material shows no hardening and therefore relation(3.5) was not fitted to this data. Secondly, some test results are not shown in Appendix B and in the table.This could be due to a number of reasons, consisting of unreliable data due to temperature fluctuationsduring the tests, failure and slipping of the clamping system during the SHTB measurements and brittlefracture of the specimen. Some of the problems encountered will be explained in the next paragraphs.

During the measurements at low strain rates, the regulation of the induction heater broke down.Controlling the temperature accurately became impossible due to the faulty equipment, causing largefluctuations in the temperature (up to several dozens of degrees Kelvin). Figure B.15 in Appendix Billustrates the effect of these temperature fluctuations: the stress-strain curve shows peaks and falls whichcorrespond to the changes in temperature. The data generated during this break down of equipment isconsidered unreliable and therefore must be discarded from further analysis. Unfortunately, at the timeof performing the measurements the heating system could not be repaired in time for a rerun of theconcerning tests. The number of available specimens was also limited.

As described in Section 3.1, the specimens were subjected to a heat treatment. The goal of applyingthis treatment is to prevent brittle fracture of the specimens during measurements at high temperaturesand strain rates, which was observed in earlier tests [12]. Up to a temperature of 693 [K] no problemswere experienced and indeed an increase in ductility is observed during the tests(comparative test resultsfrom the series 1 experiments are shown in Appendix C). The heat treatment also lowered the yield pointof the material. Figures 3.6, D.1, D.3 and D.2 show plots of the Voce data of some comparative tests atroom temperature from series 1 and 2. Both the (slight) increase in ductility and the lower yield pointsare visible. At room temperature, the work hardening rate seems to be unaffected by the heat treatment.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

50

100

150

200

250

300

True plastic strain εp [−]

Tru

e st

ress

σ [M

Pa]

Test s2t13Test s1t46Test s1t47

Figure 3.6: A comparison of series 1 and series 2 tests at room temperature to illustrate the effect of theapplied heat treatment. The strain rate for the tests illustrated here is ∼0.01 [s−1]

It appears that the heat treatment has had some effect, however when tests were performed at hightemperatures (> 733 [K]), again a sudden loss of ductility was observed in the specimens. Figure 3.7shows a sample which has failed by brittle fracture. The specimens which endured brittle fracture all

24

failed in the transition area where the diameter changes from 5 [mm] to 3 [mm] or in the area where thediameter is equal to 5 [mm].

Figure 3.7: Test specimen from test s2t10, performed on the split-Hopkinson tensile bar at T = 733 [K].

Due to the large number of tests that were deemed unsuitable for further analysis for a variety ofreasons explained above and the fact that brittle failure was not prevented by the heat treatment, thedecision was made to continue the project working with the series 1 data listed in Table C.1 [12]. A shortsummary of the observations made in the study leading to the series 1 data will be given in the followingparagraph.

At low temperatures the strain rate sensitivity is rather low. A change in strain rate in the order oftwo decades gives no considerable change in the observed true plastic stress-strain curve. This changeshowever for high temperatures: the strain rate sensitivity seems to increase considerably, even for slightchanges in strain rate. Another observation made is the decrease in work hardening for quasi-static testsat high temperatures, where almost no work hardening takes place at all. At high strain rates and hightemperatures however, significant hardening takes place.

25

26

Chapter 4

Calibration and comparison of materialmodels

In this chapter the calibration of the material models will be presented. After calibration the parametersare given and the quality of the calibrated models is evaluated.

4.1 Calibration of the material models

In order to make use of the material models available in the FEM software, the parameters of each materialmodel should be fitted to the experimentally obtained stress-strain curves. Each material model requiresa certain number of steps to be completed in order to get an accurate fit. These steps will be explainedin the following subsections.

4.1.1 Calibration of the modified Johnson-Cook material models

Because the modified Johnson-Cook relations discussed in Sections 2.4.1 and 2.4.2 and their fitting proce-dure are so similar, the calibration methods for both models are discussed in this section. The calibrationsof the terms for strain rate sensitivity and thermal softening are exactly the same.

As explained the modified Johnson-Cook models consist of three terms which describe strain harden-ing, strain rate sensitivity and thermal sensitivity. Each of these three terms is fitted on tests in whichonly the variables which govern the concerning term will vary. The first term to be fitted consists of theparameters A, B and n and is fitted under the conditions T ∗m = 0 and κ∗ = 1:

σf = (A+Bκn)2C . (4.1)

The strain rate sensitivity at room temperature for this alloy is known to be very low [12], from whichwe assume that C � 1. This means that equation (4.1) can be approximated by:

σf ≈ A+Bκn. (4.2)

The user defined reference strain rate is chosen to be κ0 = 0.01 [s−1] and the parameters A, B and nare fitted on tests that fulfill the above mentioned conditions using the method of least squares. The sameconditions and approximations apply for the modified Johnson-Cook relation with Voce strain hardening,however the parameters fitted for this model are A, Q1, C1, Q2 and C2 and the term to be fitted in thefirst step is:

σf ≈ A+

2∑i=1

Qi(1− exp(−Ciκ)). (4.3)

27

The next step in the fitting process is to fit parameter C. For this step only the condition T ∗m = 0 isapplied and using the parameters which are fitted in the first step, the part of the model which is to befitted in this step is given by:

σf = (A+Bκn)(1 + κ∗)C . (4.4)

The last step consists of calibrating m. Again using the parameters obtained in the first step, the partof the model which will be calibrated is now represented as:

σf ≈ (A+Bκn)(1− T ∗m). (4.5)

The conditions applied are κ∗ = 1 and C is again assumed to be small. To fulfill these conditions, thisterm is fitted on quasi-static test results. These tests are assumed to be isothermal.

4.1.2 Calibration of the modified Zerilli-Armstrong material model

The Zerilli-Armstrong relation is calibrated in two consecutive steps [14]. The term fitted first is givenby equation (4.6).

σf = σA +A∗κ1/2 (4.6)

This term is fitted on quasi-static test results (κ = 0.01 [s−1]) obtained at room temperature. Afterobtaining values for σA and A∗ the parameters A, α0 and α1 are fitted by using the complete model andfitting it on all the available data, under the constraint that:

A∗ = A exp(−(α0 − α1 ln(κ))T ), (4.7)

when κ = 0.01 [s−1] and T = Tr. Because the dynamic tests, which are assumed to be adiabatic, areincluded in this step, the rise in temperature during the tests has to be calculated. Equation (4.8) is usedto calculate the temperature due to adiabatic plastic deformation:

T = T0 +χ

ρCp

∫σdεp, (4.8)

where ρ is the density, Cp is the heat capacity and χ is the Tailor-Quinney coefficient, which determinesthe fraction of plastic work that is transformed into heat. In this case χ = 0.9 [-].

4.2 Calibration results and comparison of material models

The parameters obtained by using the previously described fitting procedures are given in Tables 4.1a,4.1b and 4.1c for the modified Johnson-Cook model (JC), modified Johnson-Cook model with Voce’shardening relation (JCwV) and the Zerilli-Armstrong model (ZA) respectively.

In all the models, the strain rate sensitivity appears to be low. This observation seems to coincidewith earlier conclusions drawn about the series 1 test results [12]. To further evaluate the calibration ofthe models, Tables 4.2, E.1 and E.2 are presented. In these tables the mean errors of all the stress-straincurves produced with the calibrated models with respect to the series 1 tests are shown in percentageversus the temperature range and strain rate with which they correspond. The colors green, yellow,orange and red represent the error intervals 0− 10 [%], 11− 20 [%], 21− 30 [%] and > 30 [%] respectively.

28

Table 4.1: Fitted parameters for the three selected material models

(a)

JC

A [MPa] 70.28B [MPa] 302.5n [-] 0.46C [-] 1 · 10−4

m [-] 0.70

(b)

JCwV

A [MPa] 85.50Q1 [MPa] 200.0C1 [-] 1.12Q2 [MPa] 99.85C2 [-] 12.50C [-] 1 · 10−4

m [-] 0.70

(c)

ZA

σA [MPa] 77.88A [MPa] 299.2α0 [-] 2.6 · 10−6

α1 [-] 1.2 · 10−7

Table 4.2: Fit of the Johnson-Cook model: mean errors in percentage per strain rate and temperaturerange.

Strain rates [s−1]0.01 1 40 250 300 350 400 550 600 650 700 900 1000

Tem

per

atu

res

[K]

290 < T < 300 4 5 3 7 5 6 3 2 6410 < T < 420 19420 < T < 430 13 20430 < T < 440 16550 < T < 560 12 15560 < T < 570 8570 < T < 580 69 33 8640 < T < 650 8660 < T < 670 6680 < T < 690 9720 < T < 730 260 45740 < T < 750 32750 < T < 760 32

Table 4.2 shows that the modified Johnson-Cook model gives good results around room temperature,with the mean errors not exceeding 7 [%]. At intermediate temperatures the model gives less good results,with mean errors between 10 [%] and 20 [%]. At temperatures between 560 [K] and 690 [K] a divisioncan be observed in the performance of the calibrated model. For quasi-static tests (strain rates up to1 [s−1]) this calibrated model shows mean errors up to values as high as 69 [%]. At higher strain rateshowever, the model seems to give fairly good results. At temperatures higher than 690 [K] this model’sperformance seems to drop. The model’s performance agrees with earlier observations: the quality of thecalibrated model appears to be less sensitive to the strain rate (with exception of the quasi-static tests)and is affected more by the temperature.

The modified Johnson-cook model with Voce’s hardening relation gives very similar results as themodified Johnson-Cook model, not only in terms of mean errors, but also in terms of sensitivity to strainrate and temperature. At room temperature the mean errors are low, but for higher temperatures thiscalibrated model seems to give less accurate results. In comparison with the modified Johnson-Cookmodel, this model seems to perform marginally better at room temperature, but produces slightly worseresults at higher temperatures. Judging from these results, the usage of Voce’s hardening relation in steadof Ludwik’s hardening relation in the Johnson-Cook model has no real advantages.

The calibration of the Zerilli-Armstrong model appears to give very poor results. At room temperaturethe errors do not exceed 10 [%], with the exception of the mean error at 350 [s−1]. At higher temperatures

29

however the errors rise dramatically. The cause for this bad performance seems to be the calibration stepsfollowed. Because the term σA is fitted on tests at room temperature it is rather high, having a value of77.88 [MPa]. When the temperature increases, this parameter is too high and this cannot be compensatedby the sum with the second term A exp(−(α0 − α1 ln(κ))T ), which can only be positive. A differentcalibration method, where all parameters are fitted on all the tests at the same time, is used. This yieldsmuch better results, as can be seen in Table E.3). At room temperature the fit is good, however in thetemperature range 410 [K] and 730 [K] the fit shows the same trends as the other two models, althoughin general the mean errors are slightly higher. At high temperatures (> 730 [K]) this calibration of themodel seems to supply better results.

Although the Zerilli-Armstrong model seems to give slightly better results at high temperatures,the overall performance of the modified Johnson-Cook models appears to be better. For the numericalsimulations the modified Johnson-Cook model with Ludwik’s hardening relation is used, since this modelseems to give the best overall performance out of the three models selected.

30

Chapter 5

Numerical model and simulations

After the most suitable material model is selected, the obtained parameters can be used in the FEMsoftware. To carry out the simulations in order to reproduce the experimental data, first a simplifiednumerical model is created. Next the boundary conditions are defined and applied and the results areextracted for analysis during post processing.

5.1 Numerical model

The axisymmetric specimens were modeled in LS DYNA using the 2D mesh geometry shown in Figure5.1. Because the tests were performed on different apparatuses, only the specimen is modeled. For the 2Dnumerical model four node axisymmetric shell elements with one integration point were used. In addition,due to the single integration point per element, stiffness-based hourglass control [15] was applied. In theradial direction 20 elements are used, leading to an approximate element size of 0.075×0.075 [mm2]. Thetotal model consists of 3400 elements and 3591 nodes. According to earlier research [16], this is consideredto give a reasonable balance between numerical accuracy and computational efficiency.

Figure 5.1: The axisymmetric mesh geometry used in the simulations.

The load is applied by assigning a velocity to one end of the sample, while putting a constraint on theother end. To prevent possible oscillations due to stress wave propagation [9], the velocity was given ashort rise time to reach the desired value. Because the applied deformation velocities are low in the quasi-static tests, mass scaling was applied to reduce computation time. The (change in) kinetic energy waschecked in order to determine if the effects of the mass scaling on the dynamic response were negligible.

The initial intention was to solely use the modified Johnson-Cook material card available in LS DYNAto describe both the material parameters as well as the applied temperature. The software uses equation(4.8), where χ = 0 represents isothermal conditions and χ = 0.9 represents adiabatic conditions) tocalculate the temperature rise due to plastic deformation and takes this into account while calculatingthe specimen’s response. Unfortunately, the simulations calculated with this method proved unstable,showing large oscillations in the stress-strain curve. Considerable time was spent in order to find possiblemistakes in the input of the numerical model before contacting the developer of the FEM software. Itappeared that the cause for the instability was a bug in the implementation of the modified Johnson-Cookmaterial card in the software. Unable to fix this bug due to limited time available for this project, thedecision was made to include a thermal analysis into the computations in order to be able to simulate atdifferent temperatures.

31

5.2 Simulation results

In Figure F.11 a comparison between the simulation and experimental results for test s1t16 are shown.In Appendix F more comparisons for some selected tests are shown. Because of the time available, it wasdecided not to simulate all the tests, a selection is made which covers both the range of temperatures andstrain rates of the experiments.

0 0.02 0.04 0.06 0.08 0.1 0.120

10

20

30

40

50

60

70

80


Tru

e st

ress

σ [M

Pa]

Comparison of simulation and experimental data for test s1t16

modified Johnson−Cook (numerical)Experimental data

Figure 5.2: A comparison of experimental data and simulation results at 600 [s−1] and 571 [K].

The simulations show very similar results as Table 4.2: the results at room temperature are in fairlygood accordance, however when the temperature rises, some deviations occur. In general the curvefrom the simulations is shifted downwards in comparison with the experimental data, although the workhardening rates seem to agree rather well. For test s1t58 however, the stress-strain curve is shiftedup. The work hardening rate obtained from the simulation of this test is too high compared to theexperimental one. It seems likely that the fit of the last term of the modified Johnson-Cook model, whichdescribes the thermal softening, in combination with the low work hardening rate for quasi-static tests athigh temperatures, is the cause for this trend. The modified Johnson-Cook model doesn’t accommodatefor full coupling between thermal softening, strain rate sensitivity and work hardening, while formerreports [12] do show a relation between the strain rate sensitivity and thermal softening for this alloy.

During the simulations it is assumed that tests performed at ε ≥ 40 are adiabatic. After studyingliterature, it appears that this assumption is uncorrect. Earlier research [17] shows that the transitionrange from isothermal to fully adiabatic conditions for aluminium alloys is approximately 10-1000 [s−1].The test data available are exactly in this strain rate range.

For tests that were assumed to be adiabatic (ε ≥ 40), it is striking that there are some minor fluctua-tions present in the stress-strain curve from the simulations. For the quasi-static tests, which are assumedisothermal and thus χ = 0, these fluctuations are not present. Apparently the thermal coupling causessome type of instabilities, although it is unsure where they exactly originate from.

32

Chapter 6

Conclusion and discussion

Uniaxial tests were performed on specimens made from the aluminium alloy AA6060. A heat treatmentwas applied to the specimens in order to prevent brittle fracture, which was observed in earlier tests athigh strain rates and temperatures. Although the heat treatment did have effect on the ductility and thestress-strain curve, it did not prevent brittle fracture. In further work a metallurgic study of the failuresurface of the brittley fractured specimens might point out the reason why this phenomenon is observed.A possible method to avoid brittle fracture could be to perform compression in stead of tension tests.

During the testing the heating equipment, which was controlled manually, broke down on two occa-sions. First water used to cool the coil leaked out of the system and second the controls malfunctioned,making it difficult to perform reliable measurements. The effect of the temperature fluctuations resultingfrom this faulty equipment is illustrated by stress-strain curve obtained during test s2t22 (Figure B.15).Also the heating of the incident and transmission bar in the SHTB due to a phenomenon called hysteresisloss, and it’s effects on the specimen, are not yet measured and are therefore not fully known. Possiblythis could as well be a cause for the brittle fracture, since brittle fracture is almost always observed in asection of the specimen near the bars. It is suggested that in further research a different heating systemis used, e.g. a furnace, which allows for a more reliable, automated control of the temperature. Duringthis study, data up to the point of necking was used for analysis. Additional data after necking couldnot be used since the applied assumptions are not met under these conditions. If a laser or digital imagecorrelation is used, the diameter reduction could be measured and the data after necking is also usablefor further processing, possibly improving the calibration of the material model.

After obtaining the series 2 test results, the decision was made to continue working with the series1 results. The amount of usable data obtained from the series 2 tests was insufficient to calibrate thematerial models selected. For each material model selected a calibration procedure was used to obtain therequired parameter values. The initial calibration method for the Zerilli-Armstrong model, even though itwas similar to the method used in literature, gave bad results and therefore all the parameters were fittedat once. This improved the fit of the model, although it still was not as good as the modified Johnson-Cookmodels. The difference between the modified Johnson-Cook model with Ludwik’s hardening relation andthe modified Johnson-Cook model with Voce’s hardening relation was minor, with the first performingslightly better.

After a material model was selected, the specimen was modeled in the LS DYNA software. Someinitial instabilities were encountered, which were caused by an error in the implementation of the thermaleffects in the material model in the software. To avoid these instabilities, the thermal effects were switchedoff in the material card and separate thermal coupling was used. Simulations were carried out with thenumerical model, which showed similar results as Table 4.2: at low temperatures the model performedwell, but as the temperature increases the model performs precarious. For quasi-static tests the model doesnot capture the lack of work hardening which is shown by the experimental results at high temperatures.Because these tests are used in the calibration of the thermal term of the model, they affect the fit quitedrastically. For higher strain rates and at intermediate temperatures the calibrated model shows errors

33

up to 20 percent, however between 560 and 690 [K] (for the dynamic tests) the errors are below 10percent, which is quite good. An additional possible source for the errors is the assumption of adiabatictesting conditions at ε ≥ 40. A literature study revealed that the transition from isothermal to adiabaticconditions for aluminium alloys is approximately in the range 10-1000 [s−1]. To illustrate, Figure 6.1shows the temperature as a function of true plastic strain for test s1t2. The rise in temperature underfull adiabatic conditions is in the order of 10 [K]. If the mechanical work is only partially transformedinto heat, this could make a difference in the response of the material.

0 0.05 0.1 0.15 0.2

294

296

298

300

302

304

306

Temperature under adiabatic conditions for test s1t2


Tem

pera

ture

T [K

]

Figure 6.1: The rise in temperature under the assumed adiabatic conditions.

For further work a reduction of the ranges for strain rates and temperatures should be consideredin order to calibrate a new model to improve the performance. The temperature should be monitoredand logged if possible, so they can be used to obtain better fits. In addition to the material modelsmentioned in this report the Sellars-Tegart model [18] (equation (6.1)) and slightly modified version ofthe Sellars-Tegart model (equation (6.2)) proposed in literature [19] could be considered as well. Earlierresearch [12] shows that these models perform well for AA6060 when calibrated at a strain rate intervalof 0.01-1000 [s−1] and a temperature interval of 473-723 [K]. A disadvantage of these models is thatthey don’t have a strain dependence. An implementation of calibrated models to several strain values incombination with interpolation in the LS DYNA FEM software could present an acceptable solution.

σf =1

αarcsinh((

κ

Aexp(

QZH

RT))1/n) (6.1)

σf =1

αarcsinh((

κ+ ∆k

Aexp(

QZH

RT))1/n) (6.2)

In these models T is the absolute temperature, R is the universal gas constant, QZH is an activationenergy and α, A and n are material constants. The parameter ∆k is temperature dependent, and includesan elastic domain in the formulation.

34

Bibliography

[1] Ashby M.F. & Jones D.R.H. (1980). “Engineering Materials 1, An introduction to their Propertiesand Applications.” Pergamon Press, Oxford.

[2] “Technical Datasheet: AlMgSi alloy 606035.” Hydro Aluminium Product Handling Center, Havik.

[3] Chen W.W. & Song B. (2011). “Split Hopkinson (Kolsky) Bar: Design, Testing and Applications”Springer Science+Business Media, Boston.

[4] Clausen A.H. & Auestad T. (2002). “Split-Hopkinson Tension Bar: Experimental Set-up and Theo-retical Considerations.” Technical report R-16-02, SIMLab, NTNU, Department of Structural Engi-neering, Trondheim, Norway.

[5] Khan A.S. & Huang S. (1995). “Continuum Theory of Plasticity.” John Wiley & Sons, Inc., NewYork.

[6] Albertini C. & Montagnani M. (1977). “Dynamic material properties of several steels for fast breederreactor safety analysis.” Technical report No. EUR 5787 EN, Applied Mechanics Division, JointResearch Centre, Ispra.

[7] Johnson G.R. & Cook W.H.. (1983). “A constitutive model and data for metals subjected to largestrains, high strain rates and high temperatures.” Proceedings of the 7th International Symposiumon Ballistics, Hague, 541-547.

[8] Camacho G.T. & Ortiz M. (1997). “Adaptive Lagrangian modelling of ballistic penetration of metallictargets.” Int. J. Comput. Methods Appl. Mech. Engrg, 142, 269-301.

[9] Børvik T., Hopperstad O.S., Berstad T. & Langseth M. (2001). “A computational model of viscoplas-ticity and ductile damage for impact and penetration.” Eur. J. Mech. A/Solids, 20, 685-712.

[10] Zerilli F.J. & Armstrong R.W. (1987). “Dislocation-mechanics-based constitutive relations for mate-rial dynamics calculations.” J. Appl. Phys., 61 (5), 1816-1825.

[11] Zerilli F.J. & Armstrong R.W. (1990). “Description of tantalum deformation behavior by dislocationmechanics based constitutive relations.” J. Appl. Phys., 68 (4), 1580-1591.

[12] Hopperstad O.S., Clausen A.H. & Børvik T. (2010). “Behaviour and modelling of the aluminiumalloy AA6060 for a wide range of strain rates and temperatures.” SIMLab, NTNU, Department ofStructural Engineering, Trondheim, Norway.

[13] Deschamps A., Pron S., Brchet Y., Ehrstrm J.C. & Poizat L. (2002). “High temperature, high strainrate embrittlement of Al-Mg-Mn alloy: evidence of cleavage of an fcc alloy.” Materials Science andTechnology, 18, 1085-1091.

[14] Dey S. (2004). “High strength steel plates subjected to projectile impact: An experimental and nu-merical study.” Dr. ing. thesis, SIMLab, NTNU, Department of Structural Engineering, Trondheim,Norway.

35

[15] Hallquist J.O. (2003). “LS-DYNA keyword users manual: Version 970.” Livermore Software Tech-nology Corporation, California.

[16] Børvik T., Hopperstad O.S. & Berstad T. (2003). “On the influence of stress triaxiality and strainrate on the behaviour of a structural steel. Part II. Numerical study” Eur. J. Mech. A/Solids, 22,15-32.

[17] Lindholm U.S. & Johnson G.R. (1982). “Strain-rate effects in metals at large shear strains.” Pro-ceedings of the 29th Sagamore Army Material Research Conference: Material behaviour under highstress and ultrahigh loading rates, New York, 61-79.

[18] Sellars C.M. & Tegart W.J. (1966). “On the mechanism of hot defromation” Acta Metallurgica, 14,1136-1138.

[19] Lof J. (2001). “Elasto-viscoplastic FEM simulations of the aluminium flow in the bearing area forextrusion of thin-walled sections.” Journal of Materials Processing Technology, 114, 174-183.

36

Appendix A

Measurement procedures

37

A.1 Measurement procedure for the split-Hopkinson tensile bar

A short summary of the acts which were performed during the measurements on the split-Hopkinson bar:

1. The sample diameter is measured with a micrometer with an accuracy of 0.01 [mm].

2. The notched bolt (component f in Figure 2.3) is attached to components d and e.

3. The components b to i in Figure 2.3 are put in place and a shield is placed over the locking mechanismto stop flying fragments of the bolt during its failure.

4. The sample is first screwed into the incident bar after which the transmission bar is screwed ontothe sample.

5. An initial load is applied to the locking mechanism with a hydraulic jack to clamp the bar. Theapplied force is measured with a pressure gauge.

6. The sample is manually heated with the induction heater at an approximate rate of 10 [K/s]. Thetemperature is monitored with a temperature probe.

7. The incident bar is pretensioned with a hydraulic jack. The applied force is measured with straingauge 1 (see Figure 2.2).

8. The logging program is started and the induction heater is turned off to reduce noise levels in thesignals of strain gauge 2 and 3.

9. The applied force on the locking mechanism is increased until the notched bolt fails.

10. At the moment of failure the temperature af the sample is noted.

11. The diameter of the fracture surface was measured with an optical microscope in order to determinethe fracture strain.

38

A.2 Measurement procedure for the hydraulic testing machine

A short summary of the acts which were performed during the measurements on the hydraulic testingmachine:

1. The sample diameter is measured with a micrometer with an accuracy of 0.01 [mm].

2. The specimen was screwed in place in clamps of the testing machine.

3. The specimen was heated up to the desired temperature.

4. The desired deformation velocity was set.

5. The logging system and the measurement were started.

6. During the measurement the temperature was monitored and controlled manually with the inductionheater.

7. The diameter of the fracture surface was measured with an optical microscope in order to determinethe fracture strain.

39

40

Appendix B

Test results series 2

41

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

20

40

60

80

100

120

140

Engineering strain e [−]

Eng

inee

ring

stre

ss s

[MP

a]

(a) Engineering stress-strain curve.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.3570

80

90

100

110

120

130

140


Tru

e st

ress

σ [M

Pa]

Experimental testVoce model

Fitted parameters:σ

0 = 74.4 [MPa]

Q = 35.8 [MPa]C = 16.2 [−]H = 69.6 [MPa]

(b) True stress-strain curve and fitted Voce curve.

(c) Test specimen after testing.

Figure B.1: Test s2t1, performed at ε = 385 [s−1] and T = 289 [K].

42

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

140


Eng

inee

ring

stre

ss s

[MP

a]


0 0.05 0.1 0.15 0.2 0.25 0.3 0.3540

60

80

100

120

140

160

180


Tru

e st

ress

σ [M

Pa]



0 = 47.5 [MPa]

Q = 47.6 [MPa]C = 14.3 [−]H = 235.2 [MPa]




43

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

140


Eng

inee

ring

stre

ss s

[MP

a]


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.460

80

100

120

140

160

180

200


Tru

e st

ress

σ [M

Pa]



0 = 62.5 [MPa]

Q = 138.2 [MPa]C = 3.2 [−]H = 57.0 [MPa]




44

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

10

20

30

40

50

60

70

80

90

100


Eng

inee

ring

stre

ss s

[MP

a]


0 0.05 0.1 0.15 0.240

50

60

70

80

90

100

110

120

130


Tru

e st

ress

σ [M

Pa]



0 = 45.9 [MPa]

Q = 38.8 [MPa]C = 26.5 [−]H = 161.6 [MPa]




45

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

10

20

30

40

50

60

70

80

90

100


Eng

inee

ring

stre

ss s

[MP

a]


0 0.05 0.1 0.15 0.2 0.25 0.3 0.3550

60

70

80

90

100

110

120

130

140


Tru

e st

ress

σ [M

Pa]



0 = 56.8 [MPa]

Q = 87.7 [MPa]C = 4.3 [−]H = 0.0 [MPa]




46

0 0.2 0.4 0.6 0.8 1 1.2 1.40

10

20

30

40

50

60

70


Eng

inee

ring

stre

ss s

[MP

a]


0 0.05 0.1 0.15 0.2 0.2545

50

55

60

65

70

75

80

85


Tru

e st

ress

σ [M

Pa]



0 = 46.2 [MPa]

Q = 11.1 [MPa]C = 28.2 [−]H = 77.7 [MPa]




47

0 0.2 0.4 0.6 0.8 1 1.2 1.40

10

20

30

40

50

60


Eng

inee

ring

stre

ss s

[MP

a]


0 0.05 0.1 0.15 0.2 0.2535

40

45

50

55

60

65

70

75


Tru

e st

ress

σ [M

Pa]



0 = 42.6 [MPa]

Q = 37.2 [MPa]C = 6.7 [−]H = 0.0 [MPa]




48

0 0.2 0.4 0.6 0.8 1 1.2 1.40

10

20

30

40

50

60


Eng

inee

ring

stre

ss s

[MP

a]


0 0.05 0.1 0.15 0.2 0.25 0.3 0.3530

35

40

45

50

55

60

65

70

75


Tru

e st

ress

σ [M

Pa]



0 = 35.0 [MPa]

Q = 13.4 [MPa]C = 24.7 [−]H = 63.5 [MPa]




49

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

20

40

60

80

100

120

140


Eng

inee

ring

stre

ss s

[MP

a]


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.440

60

80

100

120

140

160

180

200


Tru

e st

ress

σ [M

Pa]



0 = 48.8 [MPa]

Q = 67.7 [MPa]C = 7.8 [−]H = 192.3 [MPa]



Figure B.9: Test s2t13, performed at ε = 0.01 [s−1] and T = 294 [K].

50

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

20

40

60

80

100

120

140


Eng

inee

ring

stre

ss s

[MP

a]


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.440

60

80

100

120

140

160

180


Tru

e st

ress

σ [M

Pa]



0 = 56.0 [MPa]

Q = 58.3 [MPa]C = 7.4 [−]H = 180.5 [MPa]




51

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

10

20

30

40

50

60

70

80

90

100


Eng

inee

ring

stre

ss s

[MP

a]


0 0.05 0.1 0.15 0.2 0.25 0.3 0.3550

60

70

80

90

100

110

120

130


Tru

e st

ress

σ [M

Pa]



0 = 54.2 [MPa]

Q = 31.1 [MPa]C = 21.2 [−]H = 145.7 [MPa]




52

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

20

40

60

80

100

120


Eng

inee

ring

stre

ss s

[MP

a]


0 0.05 0.1 0.15 0.2 0.25 0.3 0.3550

60

70

80

90

100

110

120

130

140


Tru

e st

ress

σ [M

Pa]



0 = 54.4 [MPa]

Q = 36.1 [MPa]C = 16.7 [−]H = 156.2 [MPa]




53

0 0.2 0.4 0.6 0.8 1 1.2 1.40

5

10

15

20

25

30

35

40


Eng

inee

ring

stre

ss s

[MP

a]


(b) Test specimen after testing.


54

0 0.5 1 1.50

2

4

6

8

10

12


Eng

inee

ring

stre

ss s

[MP

a]




55

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

1

2

3

4

5

6

7

8

9


Eng

inee

ring

stre

ss s

[MP

a]




56

Appendix C

Test results series 1

57

Table C.1: Series 1 test results.

Test ε T σ0 Q C H σ4 σ8 σ12 σu εpu εf[s−1] [K] [MPa] [MPa] [-] [MPa] [MPa] [MPa] [MPa] [MPa] [-] [-]

s1t2 600 293 105.8 59.3 9.2 219.1 132.9 154.3 171.8 211.0 0.239 1.301

s1t3 1000 293 91.9 53.4 12.3 252.2 122.7 145.4 163.3 213.7 0.278 1.355

s1t4 300 293 78.9 56.6 19.2 281.3 120.5 145.8 163.6 223.4 0.313 1.146

s1t6 1000 293 75.9 58.9 23.2 324.4 124.5 151.6 170.1 231.6 0.298 1.244

s1t7 550 293 95.8 107.0 6.2 113.9 123.7 146.5 165.3 212.6 0.267 1.444

s1t10 400 557 55.7 33.5 16.8 43.1 73.8 83.9 89.9 93.0 0.150 2.117

s1t12 900 552 70.4 6.9 29.9 116.1 79.9 86.0 91.1 95.4 0.156 2.190

s1t13 350 293 86.0 54.0 21.0 281.5 128.0 152.5 169.5 212.9 0.260 1.080

s1t14 700 293 97.2 78.6 10.1 194.2 131.0 156.2 175.6 221.2 0.263 1.093

s1t16 600 571 53.4 14.2 29.1 96.4 67.0 73.9 78.7 79.2 0.124 2.300

s1t17 400 563 41.1 26.9 19.6 88.1 59.2 69.4 76.0 91.8 0.272 2.368

s1t19 400 682 35.3 13.8 20.6 55.1 45.2 50.8 54.5 58.7 0.182 3.225

s1t20 400 646 42.7 21.1 12.4 50.9 53.0 60.0 65.1 78.2 0.294 2.925

s1t21 400 439 59.4 37.6 77.3 323.2 108.1 122.7 135.7 148.0 0.158 1.325

s1t22 400 418 86.0 43.5 19.3 197.8 117.4 136.1 149.0 163.8 0.180 1.402

s1t23 550 752 35.5 17.8 10.5 11.6 42.0 46.5 49.6 53.8 0.211 3.627

s1t24 650 748 38.6 26.0 6.2 0.0 44.3 48.7 52.2 59.0 0.249 3.346

s1t25 350 295 88.2 47.8 13.5 325.1 121.2 145.8 165.6 250.3 0.353 -

s1t28 250 663 32.4 18.3 39.9 72.7 49.9 55.7 59.2 71.8 0.291 2.882

s1t32 40 295 92.2 58.0 18.7 274.0 133.7 159.1 176.9 213.2 0.233 1.063

s1t34 40 295 100.9 38.5 29.8 344.7 141.5 163.4 179.7 237.6 0.285 0.956

s1t46 0.01 295 93.0 84.6 14.2 215.9 138.3 167.8 188.2 246.6 0.323 0.769

s1t47 0.01 295 80.0 88.3 16.8 250.2 133.2 165.3 186.6 250.2 0.329 0.735

s1t48 0.01 424 76.0 53.5 26.3 155.5 117.0 135.4 145.9 176.4 0.301 1.093

s1t49 0.01 425 75.8 53.9 21.9 124.0 112.2 130.2 140.7 167.0 0.302 1.101

s1t51 0.01 573 33.8 2.4 49.4 55.3 38.1 40.6 42.8 45.3 0.173 4.017

s1t52 0.01 723 7.8 1.7 44.7 14.4 9.8 10.6 11.2 13.7 0.235 4.673

s1t53 1 295 79.4 82.1 21.5 292.9 138.4 170.2 190.4 253.8 0.315 0.811

s1t55 1 295 81.0 80.9 23.6 281.3 141.7 172.2 190.9 251.5 0.319 0.735

s1t56 1 426 87.0 56.5 28.3 136.6 130.7 148.5 158.0 177.4 0.249 0.948

s1t57 1 424 62.5 65.7 39.7 180.0 121.9 139.8 149.2 175.8 0.265 1.239

s1t58 1 574 35.4 11.2 130.0 62.3 49.0 51.6 54.1 54.9 0.152 3.386

s1t59 1 576 37.4 8.8 104.0 53.7 48.2 50.5 52.6 53.5 0.151 3.686

s1t60 1 723 19.8 1.2 162.0 16.7 21.6 22.3 23.0 22.7 0.100 8.005

s1t61 1 726 19.4 2.7 154.0 17.7 22.8 23.5 24.2 24.3 0.107 8.091

58

Appendix D

Comparison test results series 1 and 2

59

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.450

100

150

200

250

300


Tru

e st

ress

σ [M

Pa]


Figure D.1: A comparison of series 1 and series 2 tests at room temperature to illustrate the effect of theapplied heat treatment. The strain rate for the tests illustrated here is ∼1 [s1]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.460

80

100

120

140

160

180

200

220

240


Tru

e st

ress

σ [M

Pa]



60

0 0.05 0.1 0.15 0.2 0.25 0.3 0.3540

60

80

100

120

140

160

180

200

220

240


Tru

e st

ress

σ [M

Pa]

Test s2t2Test s1t4


61

62

Appendix E

Data fitting results

63

Table E.1: Fit of the Johnson-Cook model with Voce’s hardening relation: mean errors in percentage perstrain rate and temperature range.

Strain rates [s−1]0.01 1 40 250 300 350 400 550 600 650 700 900 1000

Tem

per

atu

res

[K]

290 < T < 300 4 5 4 6 4 5 4 2 5410 < T < 420 20420 < T < 430 14 21430 < T < 440 17550 < T < 560 14 17560 < T < 570 8570 < T < 580 68 31 11640 < T < 650 9660 < T < 670 7680 < T < 690 11720 < T < 730 259 41740 < T < 750 33750 < T < 760 32

64

Table E.2: Fit of the Zerilli-Armstrong model: mean errors in percentage per strain rate and temperaturerange.

Strain rates [s−1]0.01 1 40 250 300 350 400 550 600 650 700 900 1000

Tem

per

atu

res

[K]

290 < T < 300 2 4 3 12 8 9 6 4 9410 < T < 420 20420 < T < 430 27 19430 < T < 440 31550 < T < 560 92 83560 < T < 570 141570 < T < 580 291 207 110640 < T < 650 180660 < T < 670 207680 < T < 690 219720 < T < 730 1462 533740 < T < 750 241750 < T < 760 254

Table E.3: Fit of the Zerilli-Armstrong model (all parameters at once): mean errors in percentage perstrain rate and temperature range.

Strain rates [s−1]0.01 1 40 250 300 350 400 550 600 650 700 900 1000

Tem

per

atu

res

[K]

290 < T < 300 7 8 7 9 7 9 8 7 8410 < T < 420 24420 < T < 430 18 25430 < T < 440 22550 < T < 560 12 15560 < T < 570 8570 < T < 580 70 35 6640 < T < 650 6660 < T < 670 12680 < T < 690 17720 < T < 730 416 128740 < T < 750 10750 < T < 760 16

65

66

Appendix F

Simulation results

67

0 0.05 0.1 0.15 0.20

20

40

60

80

100

120

140

160

180

200

220


Tru

e st

ress

σ [M

Pa]



Figure F.1: A comparison of experimental data and simulation results at 600 [s−1] and 293 [K].

0 0.05 0.1 0.15 0.2 0.250

20

40

60

80

100

120

140

160

180

200

220


Tru

e st

ress

σ [M

Pa]




68

0 0.05 0.1 0.150

10

20

30

40

50

60

70

80

90

100


Tru

e st

ress

σ [M

Pa]




0 0.05 0.1 0.150

10

20

30

40

50

60

70

80

90

100


Tru

e st

ress

σ [M

Pa]




69

0 0.05 0.1 0.15 0.2 0.250

10

20

30

40

50

60

70

80


Tru

e st

ress

σ [M

Pa]




0 0.05 0.1 0.150

50

100

150


Tru

e st

ress

σ [M

Pa]




70

0 0.05 0.1 0.15 0.20

10

20

30

40

50

60


Tru

e st

ress

σ [M

Pa]




0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

50

100

150

200

250


Tru

e st

ress

σ [M

Pa]




71

0 0.05 0.1 0.15 0.2 0.25 0.30

50

100

150

200

250


Tru

e st

ress

σ [M

Pa]




0 0.05 0.1 0.15 0.2 0.250

20

40

60

80

100

120

140

160

180


Tru

e st

ress

σ [M

Pa]




72

0 0.05 0.1 0.150

10

20

30

40

50

60

70

80

90


Tru

e st

ress

σ [M

Pa]




73

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Measurements and simulations on the (dynamic) properties ... · PDF fileMeasurements and simulations on the (dynamic) properties of aluminium alloy AA6060 M. Beusink MT 11.25 June