Mechanical Behavior of AA1050

at High Strain Rates

João Miguel Silva Morais

Thesis to obtain the Master of Science Degree in

Mechanical Engineering

Supervisors: Prof. Pedro Alexandre Rodrigues Carvalho Rosa

Prof. Paulo António Firme Martins

Examination Committee

Chairperson: Prof. Rui Manuel dos Santos Oliveira Baptista

Supervisor: Prof. Pedro Alexandre Rodrigues Carvalho Rosa

Members of the Committee: Dr. Carlos Manuel Alves da Silva

November 2014

Agradecimentos

O autor expressa o seu agradecimento ao seu orientador Professor Pedro Alexandre Rodrigues

Rosa e ao co-orientador Professor Paulo António Firme Martins, pelo importante apoio e orientação

durante todo o curso do trabalho.

O autor gostaria também de agradecer o apoio prestado pelo Dr. Carlos Silva, pelo

acompanhamento durante todo este processo, dando sugestões muito valiosas.

Uma especial menção para os meus pais, que me apoiaram durante todo este processo, tanto

a nível técnico como pessoal.

Finalmente um obrigado ao Eng. Carlos Santos, ao Técnico Superior João Paixão e ao

Coordenador Técnico Rogério Pereira do LNEC por me terem ajudado a resolver alguns dos

principais problemas com que me deparei durante esta tese.

I

Resumo

A caracterização do comportamento mecânico dos materiais em condições de elevada

velocidade de deformação é um parâmetro de entrada necessário para a simulação numérica de

vários processos de fabrico (laminagem, maquinagem, etc.). Os resultados de uma simulação

numérica tendo por base exclusivamente o comportamento mecânico do material obtido em

condições quasi-estáticas de deformação verifica-se em geral insuficiente quando comparado com os

resultados experimentais, já que os processos referidos anteriormente não ocorrem em condições de

deformação quasi-estática. Resolver este problema é importante para permitir melhorar a qualidade

das previsões teóricas no projecto das ferramentas e definição dos parâmetros operativos dos

processos de fabrico.

Disto surge o objectivo desta tese, estudar o comportamento de um material (alumínio AA1050)

em condições dinâmicas, que se assemelham aos processos de fabrico referidos anteriormente. Este

estudo foi executado através de um ensaio de compressão, com a aplicação de um modelo

Viscoplástico, seguido de um ensaio de relaxação. O modelo Viscoplástico usado foi desenvolvido

numa tese anterior no IST, sendo que este trabalho serviu para consolidar a aplicabilidade do modelo

em questão. O estudo da relaxação em metais à temperatura ambiente é um parâmetro do material

pouco estudado, mas que tem forte influência tanto na qualidade da simulação dos processos como

na resistência mecânica dos componentes fabricados.

Estes ensaios foram executados usando equipamento desenvolvido no IST, tendo sido

necessário fazer algumas modificações à montagem experimental.

Como principal resultado da tese é proposto uma equação que permite modelar o

comportamento à relaxação Viscoplástica do AA1050 à temperatura ambiente.

Palavras-Chave

Viscoplasticidade, Teste de Relaxação, Alumínio AA1050, Actuador Electromagnético

II

Abstract

The characterization of the mechanical behavior of materials under high speed conditions of

deformation is an input parameter in the numerical simulation process of various manufacturing

processes (machining, hot rolling, etc.). The results obtained from these simulations based exclusively

on the mechanical behavior of the material in quasi-static deformation conditions, are generally

inadequate when compared with experimental data. Solving this problem is important in order to

improve the quality of the theoretical predictions in the design of tools and operation parameter

definition for manufacturing processes.

This is the aim of this thesis, study the behavior of a material (AA1050 aluminum) in dynamic

conditions that resemble the manufacturing processes referred earlier. This study was performed with

a compression test, coupled with the application of a Viscoplastic model, followed by a relaxation test.

The Viscoplastic model used was developed in a previous thesis in IST, being that this work served to

further consolidate the applicability of the model. The study of the relaxation phenomenon in metals at

room temperature represents a material parameter with little study, but which has a strong influence in

both the simulation quality of the processes and on the mechanical strength of manufactured

components.

These tests were performed with equipment developed in IST, having been necessary to first

perform some modifications on the experimental set-up.

As the main result of the thesis, a new model is proposed that allows the modeling of the

Viscoplastic relaxation behavior of AA1050 at room temperature.

Keywords

Viscoplasticity, Relaxation Test, AA1050 Aluminum, Electromagnetic Actuator

III

Table of Contents

1. Introduction............................................................................................................................... 1

2. Bibliographic Review................................................................................................................2

2.1 Dynamic constitutive models.............................................................................................2

2.2 Viscoplastic Models...........................................................................................................4

2.2.1 Johnson-Cook...........................................................................................................5

2.2.2 Zerilli–Armstrong.......................................................................................................6

2.2.3 Hybrid Model.............................................................................................................6

2.3 Dynamic Experimental Tests.............................................................................................7

2.3.1 Split-Hopkinson Pressure Bar Test............................................................................8

3. Material and Experimental Procedures...................................................................................10

3.1 Material Preparations......................................................................................................10

3.2 Experimental Set-Up........................................................................................................11

3.2.1 Structural Parts........................................................................................................12

3.2.2 Compression Tool....................................................................................................12

3.2.3 Electromagnetic Actuator.........................................................................................13

3.2.4 Pneumatic damper..................................................................................................15

3.3 Data Acquisition...............................................................................................................16

3.4 Experimental Plan...........................................................................................................17

4. Results and Discussion..........................................................................................................18

4.1 Experimental results........................................................................................................18

4.2 Data Analysis................................................................................................................... 21

4.2.1 Stiffness Modeling...................................................................................................23

4.3 Model Application............................................................................................................24

4.3.1 Compression Model.................................................................................................25

4.3.2 Relaxation Model.....................................................................................................27

5. Conclusion.............................................................................................................................. 30

6. References............................................................................................................................. 31

7. Annexes.................................................................................................................................. 33

7.1 Electromagnetic Actuator Firing Procedure (in Portuguese)............................................33

7.2 Electrical Circuit Diagram................................................................................................36

7.3 LabView Program............................................................................................................37

7.4 Experimental Equipment Data Sheets.............................................................................38

7.5 Translation Cam Schematic (Logistic).............................................................................39

IV

Figure Index

Figure 2.1 - Example of a stress-strain curve (engineering curve)...............................................3

Figure 2.2 - Influence of strain rate and temperature on the flow stress of most materials...........4

Figure 2.3 - expected curve of the compression test (1) followed by a relaxation test (2) indynamic conditions performed in this thesis (segmented curve is the quasi-staticevolution)..................................................................................................................4

Figure 2.4 - Relaxation testing equipment: compression, traction and on liquids tests (left toright)......................................................................................................................... 8

Figure 2.5 - SHPB diagram...........................................................................................................8

Figure 3.1 - (a) aluminum bar, (b) aluminum rod, (c) test specimen...........................................10

Figure 3.2 - Experimental schematic: 1- pneumatic actuator, 2- Translation Cam, 3-Compression Tool, 4- Displacement Sensor, 5- Load Cell, 6- CompressionFollower, 7- Ram, 8- Electromagnetic Actuator.......................................................11

Figure 3.3 - Experimental set-up used: (a) compression tool and (b) electromagnetic actuator. 12

Figure 3.4 - Displacement and Velocity conversion curves.........................................................12

Figure 3.5 - Electromagnetic actuator: (a) head view and (b) end view......................................13

Figure 3.6 - Electrical Circuit (Capacitor Bank)...........................................................................14

Figure 3.7 - Test energy flow......................................................................................................15

Figure 3.8 - Pneumatic actuator: (a) picture and (b) schematic on the right (extracted from thecylinder data-sheet). The air pocket is created around point 4...............................15

Figure 3.9 - (a) Load Cell and (b) Vishay Amplifier.....................................................................16

Figure 3.10 - Displacement sensor: (a) picture and (b) operation diagram.................................16

Figure 4.1 - Tested Specimens...................................................................................................18

Figure 4.2 - Voltage (related to the force) and the displacement of a discarded test. The mainproblem of this one was the force discontinuity caused by a small dent in thetranslation cam's profile (1)....................................................................................19

Figure 4.3 - Example of a good test after the initial treatment (constant displacement after thecompression phase and beginning of the force decline in the same instant).........19

Figure 4.4 - Graph's origin adjustment: (a) Diagram of the compression plates adjustment to thetest specimen and (b) Force-Displacement curve before correction......................20

Figure 4.5 - (a) Force-Displacement curves and (b) Strain rate evolution with strain progression............................................................................................................................... 20

Figure 4.6 - (a) Force-Time curves and (b) zoom on the compression phase............................21

Figure 4.7 - Problems detected in the relaxation phase. The correct graph shape (16 – lowstrain rate) isn’t represented in the second curve (31 – high strain rate). End ofcompression point (A) for Case 16.........................................................................21

Figure 4.8 - Cam Clearance (blue arrow is the closing direction)...............................................22

Figure 4.9 - Displacement problem. The test specimen (31) suffers an additional deformationbut immediately returns to correct displacement value...........................................22

Figure 4.10 - Final corrected curves: recovery of the correct stress and strain values. Quasi-static curve example...............................................................................................23

Figure 4.11 - Diagrams with the elements considered in the stiffness series..............................24

Figure 4.12 - Compression stress-strain curve for the real data and the model.........................26

Figure 4.13 - Viscoplastic compression curve examples: (a) test 16 and (b) test 31..................26

Figure 4.14 - Relaxation curve example with the previously mentioned Final Compression stress(1) and the Relaxation stress (2). This quasi-static curve is just representative, notfrom a real test.......................................................................................................27

Figure 4.15 - Viscoplastic models example................................................................................28

Figure 4.16 - Viscoplastic relaxation curve examples: (a) test 17 and (b) test 31.......................29

V

Table Index

Table 3.1 - test specimens: identification, height and diameter...................................................11

Table 3.2 - Average energy values..............................................................................................15

Table 3.3 - Test Plan................................................................................................................... 17

Table 4.1 - Most representative tests..........................................................................................18

Table 4.2 - Stiffness values.........................................................................................................24

Table 4.3 - Viscoplastic model initial parameters........................................................................25

Table 4.4 - Viscoplastic model new parameters..........................................................................25

Table 4.5 - Viscoplastic model fitting results...............................................................................25

Table 4.6 - Viscoplastic relaxation parameters...........................................................................28

VI

Symbols List

ε True strain

ε̇ Strain rate

σ True Stress

σ y Yield Stress

T Temperature

t Time

E Young Modulus

η Viscosity

L Test Specimen Height

Ø Test Specimen Diameter

v Velocity

Abbreviations List

IST Instituto Superior Técnico

SHPB Split-Hopkinson Pressure Bar

VII

1. INTRODUCTION

The deformation process of a material can be represented by a constitutive equation, which

describes the relationship between stress, strain, strain rate and temperature. This is important in

today's effort of simulating manufacturing processes with the Finite Element Method in order to better

understand and optimize them. Each of these equations uses parameters that depend on the material

in study. These parameters can only be determined with experimental tests, such as traction,

compression or torsion configurations.

The main goal of this thesis is to characterize the AA1050 aluminum (99,5% of Al) and

determine it's parameters for a model used in a previous work from IST (Eurico Kupessa [1]) and

developed by Carlos Silva [10]. The mechanical characterization was achieved by means of

compression tests performed in a cam plastometer test machine [16]. The other important subject of

this thesis is to fill a gap in a very specific field of study which is Viscoplastic models for metals, more

precisely the relaxation phenomenon in metals. The studies in the relaxation phenomenon are mainly

focused in polymers, with fewer studies performed for metals [example in 17]. So a new model for

Viscoplastic stress relaxation was proposed for the AA1050 aluminum. A better understanding of this

phenomenon in metals can also help in the process of designing better manufacturing tools and

processes.

This dissertation is structured into five chapters, including this introduction (Chapter 1): Chapter

2 presents a global literature review about Viscoplastic models together with a small introduction to

basic metal deformation theory for the inexperienced reader, Chapter 3 consists in a description of the

experimental procedure used in this thesis, Chapter 4 presents the results obtained in the tests and

the application of mathematical models to the data obtained from them and finally Chapter 5 presents

conclusions about the tests and future work prospects.

1

2. BIBLIOGRAPHIC REVIEW

Exact modeling of the dynamic mechanical behavior of materials is a prerequisite for an

effective analysis of a specific production process [18], although this kind of studies poses great

complexity. Throughout the years a series of empirical constitutive models have been developed in

order to predict the flow stress with high accuracy for the manufacturing processes. However, most of

these models are based on several assumptions in order to reduce the complexity of the problem. The

success of a particular constitutive model depends on how effective it is at reproducing the actual

conditions of the manufacturing process, as well as it's ability to incorporate all relevant parameters in

the equation.

So in this chapter, a brief synthesis about dynamic constitutive models and metal deformation

theory is presented, followed by a description of the more common constitutive models for

Viscoplasticity (scientific field that studies the strain rate dependent deformation in metals). Also a brief

description of the procedure used in dynamic metal characterization tests is presented, with a special

mention to the Split-Hopkinson Pressure Bar (SHPB) test, since it is relevant for the study performed

in the thesis. Also in this chapter, a brief description is made of the typical experimental equipment

used to perform relaxation and creep tests on polymers, as an example of the apparatus necessary to

perform this type of tests.

2.1 Dynamic constitutive models

It was in the early 19th century that the study of material properties under impact conditions

began with the work of Henri Tresca, Saint Venant and Levy on the Maximum Shear Criterion [2].

Experimental investigations were hampered by the lack of adequate instrumentation, but a consider-

able theoretical progress was still made in understanding pressure wave propagation on finite materi-

als, such as circular rods. An improved plasticity model was presented in 1913 by von Mises which is

now referred to as the von Mises Yield Criterion [3]. As for Viscoplasticity, the development of a

mathematical model heads back to 1910 with the representation of primary creep by Andrade's law [4].

The need for dynamic constitutive models as increased throughout the last century. This

happened because most mechanical properties available in databases are obtained in traction and

compression tests, with very low strain rates, which don´t replicated the real conditions of several

manufacturing processes. So it was necessary to carry out tests in realistic conditions in order to have

a better understanding of them, since strain rates of 100 s−1up to 5000 s−1

are common in many

engineering processes such as forging and machining. The development of new technological

processes, or the optimization of existing ones (in the sense of increasing the cadence of manufacture

or reducing production costs) contributes to the need to have more information about the physical

properties of materials.

In the nineteenth century, some researchers such as Maxwell and Boltzmann began

experimenting with creep and relaxation of glasses, metals, and rubbers [5]. Viscoelasticity (similar to

2

Viscoplasticity, but for polymers) was further examined in the late twentieth century when synthetic

polymers were developed and used in a variety of applications. Several models were developed for

polymers during this time period, with fewer work being directed into models for metallic materials.,

since these types of phenomenons is difficult to observe in metals except in very specific conditions

(high strain rates or high temperature).

Plastic deformation in ductile materials occurs when the initial yield stress is surpassed, either in

quasi-static or dynamic conditions. If a material is strained with an impulse that goes beyond its elastic

limit this will be decomposed into a wave of elastic deformation and a wave of plastic deformation.

This translates into the common stress-strain curve (see figure 2.1) that characterizes the mechanical

behavior of most ductile metals with a linear elastic zone followed by a plastic region. The elastic

stress plus the plastic stress represent the flow stress that the material is sustaining at any given

moment.

The elastic stress is mainly related with material properties, mainly the Young Modulus, while

the plastic stress depends on the material ductility, strain level applied and, contrary to what is usually

taken into consideration, temperature and strain rate used (Viscoplasticity). Variations in these two

parameters from their usual values (standard room temperature and quasi-static speed for

experimental set-ups) modify the shape of the curve for any given material (see figure 2.2) [6], which

can better describe the conditions in how a specific manufacturing process occurs, allowing for a more

accurate simulation of it´s progress.

3

Figure 2.1 - Example of a stress-strain curve (engineering curve)

Str

ess

(Pa)

Yield Stress

Ultimate Tensile Stress

Fracture Point

Young's Modulus

Strain

Also important in Viscoplasticity studies of a given material is to know it's viscosity (related to

the material strain rate dependance). In order to determine this value, a relaxation test can be

performed. The important values in the study of a dynamic relaxation test are the Final Loading stress

(defined by the final strain value imposed on the material for the relaxation test), the Relaxation stress

and the material's viscosity (see figure 2.3). The Final Loading stress sets the starting point for the

relaxation test and the Relaxation stress is the target value, which according to the theory should be

equal to the upper limit of elasticity of the material [7]. For some materials, this upper limit of elasticity

corresponds to the flow stress obtained if the same material was to be strained to the same strain

value but in quasi-static conditions.

2.2 Viscoplastic Models

Viscoplasticity is a theory that describes the strain rate dependance of solids. Strain rate

dependence means that the deformation of the material depends on the rate at which the load is

applied. The inelastic behavior described in Viscoplasticity pertains to plastic deformation which

translates into an unrecoverable deformation suffered by the materiel when a certain load level is

4

Figure 2.3 - expected curve of the compression test (1) followed by a relaxation test (2) in dynamic conditions

performed in this thesis (segmented curve is the quasi-static evolution)

Str

ess

(Pa)

Time (s)

2

Final Loading Stress

Relaxation Stress

1

Figure 2.2 - Influence of strain rate and temperature on the flow stress of most materials

ε̇2

ε̇1

ε̇3

ε̇1> ε̇2> ε̇3 T1>T 2>T3

T3

T2

T1

Str

ess

(Pa)

Str

ess

(Pa)

Strain Strain

achieved. Rate-dependent plasticity is important for transient plasticity calculations, especially when

studying manufacturing processes. The main difference between rate-independent plastic and

Viscoplastic material models is that the latter exhibit not only permanent deformations after the

application of loads but continue to undergo a creep phenomenon as a function of time under the

influence of the applied load [7].

For metals and alloys, Viscoplasticity is a macroscopic phenomenon caused by the movement

of dislocations inside the grain structure, with additional effects of inter-crystalline movement. This

phenomenon usually becomes dominant at temperatures greater than approximately one third of the

absolute melting temperature, in quasi-static conditions.

In the constant effort to estimate and simulate today's manufacturing processes to improve our

knowledge of them, engineers have been using the Finite Element Method as a very powerful tool to

perform this work. To apply this technique, it is essential to use empirical models based on the

traditional traction or compression tests in quasi-static and dynamic conditions in order to accurately

estimate the flow stress in every operation.

Various dynamic hardening models have been suggested to represent the effect of the strain,

the strain rate, and the temperature on the hardening characteristics of metallic materials. In this

section, two well known models are presented: the Johnson–Cook model and the Zerilli–Armstrong

model [8]. Then a special reference is made to the new Hybrid model developed in IST, which is the

one used in this work. A drawback of these models is that they can´t model some of the more complex

Viscoplastic phenomenons, like relaxation or creep in metals. Fixing this problem is one of the

objectives of this thesis, which can have an important impact in the manufacturing industry [18].

2.2.1 Johnson-Cook

Johnson and Cook (1983) [9] proposed a dynamic hardening model that is capable of

representing the effects of the strain, the strain rate, and the temperature on the flow stress of metallic

materials.

σ=[A+B(εeffp)N] [1+C ln(ε̇ )][1−(T H)

M] (1)

ε̇=εeff

p

ε̇0

(2)

T H=T−T R

TM−T R

(3)

εeffp

is the effective plastic strain, TH is the Homologous temperature, TM is the Melt

temperature of the material in study and finally ε̇0 and T R are the strain rate and temperature values

used to determine the remaining constants.

5

The expression in the first term of brackets provides the stress as a function of strain for ε0 and

T=T R . The expressions in the second and third brackets represent the effects of the strain rate and

the temperature, respectively.

This model works well with copper alloys, and is the most widely used today because of its

simplicity and convenience, but it has some problems in representing the hardening characteristics of

other types of materials. The strain rate hardening term in the second bracket is expressed as the

logarithm of the strain rate. This evolution of the flow stress isn´t true for some materials who instead

exhibit a more complex dependence on the strain rate, exemplifying one of the shortcomings of this

standard model (there are some modified models for more specific situations that deal with this

problem).

2.2.2 Zerilli–Armstrong

Zerilli and Armstrong (1987) [8] suggested two different types of models for FCC and BCC

crystalline configurations based on the dislocation dynamics for each group:

σ=C0+[C1+C2 √ε ]exp [−C3T ln(ε̇ )]+C5 εn (4 )

The C's are the material constants that also set the equation either in FCC or BCC mode. For

the FCC configuration C1=C5=0 and for the BCC equation C2=0 .

In the FCC equation, the main consideration is that the temperature softening and strain rate

hardening dependencies of the flow stress are greater with increased strain hardening. In the BCC

case, the strain hardening factor is uncoupled from the strain rate hardening and the thermal softening

terms.

From the expressions of the two Zerilli–Armstrong models, a shortcoming of each model can be

perceived. In the FCC case, C0 is independent of the strain rate and the temperature. Therefore, for

FCC model the yield stress is constant with changes in strain rate and/or temperature, which doesn't

correspond to what happens in a real situation, were the yield stress varies with these parameters. In

the BCC case, the strain hardening factor of C5 εn

is uncoupled from the strain rate hardening and the

thermal softening terms, making the model unable to show the hardening change with variations in the

strain rate and temperature.

2.2.3 Hybrid Model

The application of the previous models doesn't always reproduce the real behavior of materials

according to their specific characteristics and operation parameters, because of the simplifications and

hypothesis used in them, so there is a need to work with an endless number of models in order to

accommodate all these potential variables. Faced with this problem, a new constitutive model was

developed by Carlos Silva [10], that can describe a high number of material behaviors and even

reproduce some of the traditional models with only one empirical equation (hence the name Hybrid).

6

This new model takes into account the combined effects of strain and strain rate variation in the

flow stress, allowing for a better estimation of the material's behavior in a wide range of strain rates.

The only drawback of this model is that it doesn't take into account the temperature influence in the

flow stress, reducing it's applicability to manufacturing processes were temperature isn’t a

predominant factor.

σ=[A+exp (mε)εn] [B+Cln(D+ ε̇)] (5)

The material constants A, B, C, D, m and n are closely dependent on the conditions under which

the experimental tests are performed. Looking at the mathematical expression, you can separate the

content of the two brackets that are involved in the equation: the first directly related to the influence of

strain and strain hardening in the flow stress and the second with strain rate.

2.3 Dynamic Experimental Tests

There are two generic experimental procedures used to determine the viscosity of the material

to be studied, which can be important in the followup material characterization tests in dynamic

conditions [8]:

• Creep tests, which are used to determine the evolution of strain as a function of time in a

material subjected to uniaxial stress at a constant temperature. Creep is the tendency of a

solid material to slowly move or deform permanently under constant load and is closely related

to it's viscosity value.

• Relaxation tests, where the stress response due to a constant strain during a set period of

time can be determined. In viscous materials, relaxation tests demonstrate the stress

relaxation in uniaxial loading at a constant strain, complementing the information obtained in

the creep test.

Having this information, a strain hardening test can be performed in order to obtain the stress-

strain curve for the material in dynamic conditions, whose shape usually isn't very different from the

quasi-static one. Still, three important differences can be perceived:

• at the same strain, the higher the strain rate the higher the stress;

• a change in the strain rate during the test results in an immediate change in the stress–strain

curve;

• the concept of a plastic yield limit is no longer strictly applicable.

There are multiple examples of experimental procedures that can be used to perform dynamic

mechanical characterization tests, one being the SHPB which is one of the more popular. Other

example are high speed hydraulic presses and drop hammers. For relaxation tests in metals, a typical

procedure includes using a screw-driven press in displacement mode with fixtures that allows the

experiments to be done at room temperature or surrounded by a furnace for high temperature tests.

7

Just as a comparison, stress relaxation tests for polymers should follow the ISO DIS 3384 1997

[12], which describes two methods A and B (see figure 2.4). The test specimens are either a cylindrical

disc or a ring with very specific dimensions. In method A, the compression is applied and all counter

force measurements are made at the test temperature. In method B, the compression is applied and

all counter force measurements are made at standard laboratory temperature (23 °C).

In this work, the relaxation test and the strain hardening test were performed in the same

operation in order to save time and test specimens, but also because the experimental set-up that was

available didn't allowed for the relaxation test to be performed separately.

2.3.1 Split-Hopkinson Pressure Bar Test

The Hopkinson Pressure Bar was first suggested by Bertram Hopkinson in 1914 [13] as a way

to measure stress pulse propagation in a metal bar. Later, in 1949 H. Kolsky refined Hopkinson's

technique by using two Hopkinson bars in series, now known as the Split-Hopkinson bar, to measure

stress and strain in the test specimen. Later modifications have allowed for tensile, compression, and

torsion testing.

Although there are various setups and techniques currently in use for the Split-Hopkinson

pressure bar, the underlying principles for the test and measurement are the same. The specimen is

placed between the ends of two straight bars, called the incident bar and the transmitted bar (see

figure 2.5).

8

Figure 2.4 - Relaxation testing equipment: compression,

traction and on liquids tests (left to right)

Figure 2.5 - SHPB diagram

Striker Bar Incident Bar TransmittedBar

Test Specimen

V V1 V2

At the end of the incident bar (some distance away from the specimen, typically at the far end),

a stress wave is created which propagates through the bar toward the specimen. This wave is referred

to as the incident wave, and upon reaching the specimen, splits into two smaller waves. One of which,

the transmitted wave, travels through the specimen and into the transmitted bar, causing plastic

deformation in the specimen. The other wave, called the reflected wave, is reflected away from the

specimen and travels back down the incident bar. Most modern setups use strain gauges on the bars

to measure strains caused by the waves. Assuming deformation in the specimen is uniform, the stress

and strain can be calculated from the amplitudes of the incident, transmitted, and reflected waves

using mathematical equations.

For compression testing, two symmetrical bars are situated in series, with the sample in

between. The incident bar is struck by a striker bar during testing. The striker bar is usually fired from a

gas gun. The transmitted bar collides with a momentum trap (typically a block of soft metal). Strain

gauges are mounted on both the incident and transmitted bars.

A new methodology using the SHPB for relaxation tests is being develop in recent years [20],

that represents an alternate procedure to the one used in this work. The major drawback is still the

need to use mathematical equations and models to extrapolate the data, instead of obtaining it

directly.

9

3. MATERIAL AND EXPERIMENTAL PROCEDURES

Given the high standards required in material characterization tests in dynamic conditions, it's

important to develop an experimental methodology that allows the characterization of the material

itself and at the same time gives the user control over the deformation speed of the test specimen and

the value of displacement to be applied. In this context, this chapter presents the experimental

procedure developed in this work along with a brief description of the equipment used, from the

preparation of the AA1050 test specimens to the test plan implemented. Also the full maintenance

procedure applied to the equipment by the author is described for each component, which was highly

required after the extensive use the equipment underwent in previous theses.

As previously explained, the objective of this work was to perform a compression test in

Aluminum AA1050 specimens followed by a relaxation test. Both of these tests require very specific

conditions to be met, mainly tool stability and durability for the first and the ability to apply the same

displacement to all the specimens and hold it during a short time period for the latter.

3.1 Material Preparations

Since the AA1050 was supplied in a 5mm thick plate and the objective was to produce 6×6 mm

cylinders as the test specimens, there was a need for an unusual manufacturing procedure to be

developed, since metal casting was discarded due to it's inherent difficulty and time consumption.

The first step to obtain the specimens required for the tests from the AA1050 plate was to cut off

smaller 100×25 mm strips from the plate. Then using a hydraulic press, the strips were compressed

into 100 mm long bars with an almost quadrangular section (see figure 3.1a).

Next these bars were machined with a lathe into circular rods with 8 to 9 mm of diameter (see

figure 3.1b). Then, still using the lathe, the specimens were machined out from the rods one by one

into the test specimen's final shape, 6×6 mm cylinders(see figure 3.1c).

The final step was to perform an annealing procedure on the specimens. They were placed in

the oven at 350 ºC for 3 hours and then cooled down at room temperature.

In the end, 40 specimens were made using this process (see table 3.1).

10

Figure 3.1 - (a) aluminum bar, (b) aluminum rod, (c) test specimen

(a) (b) (c )

3.2 Experimental Set-Up

The equipment selected to perform the material characterization test with high strain rates and

the subsequent relaxation test (for a full schematic, see figure 3.2) was a compression test using a

special equipment design by Vasco Ezequiel [14], Gonçalo Vargas [19] and Carlos Silva (hereby

refereed as compression tool (see figure 3.3a). This tool was coupled with another unique equipment

also design and built in IST [16], an electromagnetic actuator, as the source of the force required to

perform the tests together with the capability of applying that force with the required speed to perform

them.

Aside from the fact that this equipment was “homemade” in IST, making it unique, the main

property of this system is that it greatly reduces the level of vibrations that occur in these tests from

interfering with the test procedure. This happens mainly because the impulse direction is

perpendicular to the compression direction (as showed by the blue arrows on figure 3.2), so most of

the vibration waves generated can't change direction and go through the cam to the compression

follower and then to the specimen. This will be further described in the following topics.

11

Table 3.1 - test specimens: identification, height and diameterNumber Height (L) Diameter (Ø) Number Height (L) Diameter (Ø) Number Height (L) Diameter (Ø)

1 6,03 6,03 15 6,05 5,94 28 6,05 6,12 5,92 6,08 16 5,96 5,76 29 5,95 5,983 6,02 6,03 17 6 5,94 30 5,98 6,094 6,02 6 18 5,88 5,95 31 6,05 6,075 6,06 5,97 19 5,93 6,06 32 6,07 5,876 5,98 6,03 20 6,08 5,88 33 6,07 6,027 5,97 6,07 21 6,01 6,07 34 6 5,918 6,08 6,08 22 5,97 5,98 35 6 69 6,07 6,09 23 6,09 6,06 36 5,92 5,9310 6,06 6,03 24 5,96 5,87 37 5,92 5,9711 6,01 6,05 25 5,6 6,03 38 5,92 6,1612 6,03 5,92 26 5,84 6,06 39 5,99 5,9313 5,97 5,99 27 5,95 6,09 40 6,03 614 6,03 6,06

Figure 3.2 - Experimental schematic: 1- pneumatic actuator, 2- Translation Cam, 3- Compression Tool, 4-

Displacement Sensor, 5- Load Cell, 6- Compression Follower, 7- Ram, 8- Electromagnetic Actuator

3.2.1 Structural Parts

This section refers to structural parts for the test equipment namely the I beam and the support

frames (painted in blue). The main objective of this structure is to provide the necessary stability to the

tests while providing housing for the compression tool and the electromagnetic actuator.

3.2.2 Compression Tool

Important mechanical parts in this set-up comprise the compression tool which includes: a fixed

housing containing two flat compression plates, a translating cam and a follower to pass the kinetic

impulse generated in the actuator to the plates without the mechanical vibration associated with this

type of tests [16].

The surface contour of the cam (cam profile, see Annex 7.5) was designed with the intent of

transmitting the movement from the actuator to the compression plates as smoothly as possible. The

follower traces the cam profile and converts horizontal movement of the ram to vertical displacement

of the lower flat compression plate (as seen in figure 3.2). This solution allows the same displacement

be applied in every test and due to the shape of cam profile in the final section, that position can be

maintained as long as necessary.

The conversion of movement is schematically illustrated in figure 3.4, one of which present the

vertical displacement Y of the follower as a function of the horizontal displacement X of the cam.

12

Figure 3.3 - Experimental set-up used: (a) compression tool and (b) electromagnetic actuator

(a)(b)

Figure 3.4 - Displacement and Velocity conversion curves

The velocity v y of the follower is directly related to the first derivative of the displacement curve

because the velocity of the ram v x is approximately constant in the working region of the cam follower.

The acceleration a y of the cam follower is computed from the variation in the velocity v y .

v y=dydt

=vxdydx

a y=dv y

dt(6)(7)

The small values of acceleration at the entry dwell of the logistic cam profile combined with the

fact that pressure angles θ of the follower are kept below 30° (θmax=22.5 °) , helps keeping inertia

forces at a small level and justify the reason why the proposed testing equipment worked smoothly

without shocks and vibrations while performing material testing at high rates of loading.

During the initial tests, some problems were detected in the data collected that would be later

related to small deformations and dents in the translation cam. So the compression tool was

disassembled, cleaned of any residues, repaired with sandpaper and a metallic file and reassembled.

3.2.3 Electromagnetic Actuator

The impulse source used in the compression of the test specimens was an electromagnetic

actuator develop and periodically improved in the past few years in IST (see figure 3.5) [16].

The electromagnetic actuator consists of electrical circuits for charging and firing 5 banks of

energy-storage capacitors (20 capacitors total, each with 6 mF) and a series of coils that generate the

pressure to accelerate the ram linked to the translating cam. The actuator combines the reluctance

and induction electromagnetic features of the assembly between coils and ram.

Typical coils a total length of 9 mm, an external diameter of 160 mm and an internal diameter of

90 mm, were utilized. The ram consists of a long (1,5 m) and heavy (5 kg) bimetallic bar made of AA

6082-T651 Aluminum with slotted hollow half-ring surface inserts (to reduce eddy current losses) of

DIN St52.3 Steel (see Annex from 15).

The reluctance of the electromagnetic actuator derives from the very large positive magnetic

susceptibility and attractive ferromagnetic properties of the slotted half-ring inserts of Steel placed

13

Figure 3.5 - Electromagnetic actuator: (a) head view and (b) end view(a) (b)

along the surface of the Aluminum ram. The induction results from the repulsive forces originated by

the eddy currents that are induced in the surface of the ram when the coils are fired. It is worth a

mention that Aluminum is paramagnetic and its magnetic susceptibility is much smaller than that of

Steel.

The capacitors are charged by means of single-phase alternating current supplied with 230 V

that is converted to higher-voltage direct constant current by means of a charging circuit consisting of

a variable-voltage transformer and a constant current rectifier system. The variable-voltage

transformer is made of a variable-voltage regulator which can change the incoming line voltage from

0% to 100% and a transformer capable of producing 3.6 times the input voltage. This means that the

overall rating of the variable-voltage transformer system is approximately 1,000 V for a single-phase

input line voltage of 230 V.

Once the capacitors are charged, the charging circuits are closed and the thyristor switches

located in the discharging circuits are activated to simultaneously fire each capacitor into its

associated coil. The resulting current pulse will only last for a few milliseconds but the amount of time

will be enough for the ram to achieve large velocity and kinetic energy. The firing procedure is fully

described in Annex 7.1.

In case of a single-phase 230 V directly inputted from the supply line without any intervening

transformers, the current pulse will accelerate the ram to levels of velocity up to 18 m/s. However, the

proposed equipment is capable of operating across a broad range of testing conditions because

individual coils of the electromagnetic actuator can be easily changed or temporarily switched off

during testing and because kinetic energy of the ram is capable of rising as the operating voltage of

the charging and discharging circuits is increased.

As for the maintenance applied to this equipment, the entire electrical circuit was tagged and a

new electrical circuit diagram was drawn with this new information (see Annex 7.2 and figure 3.6).

14

Figure 3.6 - Electrical Circuit (Capacitor Bank)

3.2.4 Pneumatic damper

During this work, after performing some preliminary tests with the equipment, the need to

dissipate the remaining kinetic energy of the kinematic set (ram+cam) at it's end instead of at it's head

(rubber damper against the Electromagnetic Actuator's body) became apparent with the constant

deterioration of the set's screw connections after each test (see figure 3.7 and table 3.2).

Some ideas were considered, but the one used was to implement a pneumatic actuator

(Pneumatic cylinder CNOMO NFE 49-001 from Rexroth) attached to the end of the kinematic set (see

figure 3.2 and 3.8). This solution could provide the necessary damping effect required to preserve the

equipment using the air pocket created inside the actuator near the end course as a damper (the

actuator can dissipated around 750 J in this configuration) and also work as an actual actuator to reset

the kinematic set to it´s starting position using the room's pressurized air line.

15

Figure 3.8 - Pneumatic actuator: (a) picture and (b) schematic on the right (extracted from the cylinder data-

sheet). The air pocket is created around point 4

Figure 3.7 - Test energy flow

Table 3.2 - Average energy valuesTensão (V) Energia nos Condensadores (J) Energia na Came (J)

120 860 215200 2400 600280 4700 1175360 7770 1942,5

Energia na Barra de Compressão (J) Energia a amortecer (J)43 172120 480235 940

388,5 1554

(a) (b)

3.3 Data Acquisition

The data extracted from the compression tests was the usual pair: force applied on the test

specimen and the displacement imposed by the compression plates. In order to obtain this data, a

force transducer and a displacement sensor were installed in the compression tool.

The force transducer is a commercial, 50 kN load cell from HBM (type C9B), with nominal

sensitivity 1 mV/V and accuracy class 0.5 that allows measuring static and dynamic compressive

forces (see figure 3.9). Since the output from the load cell is too weak to be accurately read by the

data acquisition system (DAQ) used, a Vishay signal amplifier (model 2100) was used to increase the

signal by 1000 (transforming mV to V). So the output voltage value should be multiplied by 10000

(nominal factor) in order to obtain the force in the specimen.

The displacement reading equipment consists of a probe, a target (metallic disc) and the sensor

itself that operates through reading the variation of Eddy currents produced in the probe against the

target (see figure 3.10). In more detail, this equipment works as following: the probe is fixed and

generating a magnetic field and the target is a magnetic material coupled follower from the

compression tool. When the compression starts, the target moves away from the probe, interfering

with the magnetic field generated by the probe, causing an output voltage signal proportional to the

distance between the probe and the target which is interpreted by the sensor into a measurement of

the displacement that occurred.

All this data was acquired by a DAQ NI-USB-6251 terminal with 16 analogue inputs (16-bit),

1.25 MS/s single-channel (1 MS/s aggregate) and processed by a LabView program made by the

author (see Annex 7.3) were the data is compiled into a text file to be later analyzed. The data is

16

Figure 3.10 - Displacement sensor: (a) picture and (b) operation diagram

Figure 3.9 - (a) Load Cell and (b) Vishay Amplifier

(a) (b)

(a) (b)

acquired at 200k Hz during 2 seconds in order to accurately register the compression test and the

relaxation test.

This part of the set-up required extensive maintenance, ranging from damaged cables to

recalibrations:

• the force transducer's cable was damaged near the plug, with broken wires and no electrical

noise isolation, so a new plug was properly installed to solve these issues;

• the displacement sensor probe also had a damaged cable, from a possible door accident,

which was also repaired;

• both sensors underwent a recalibration procedure in order to obtain their actual gain factor,

which in both cases were still very close to original values.

3.4 Experimental Plan

As previously mentioned, the experimental tests were divided into 4 sets, each with different

energy levels. Since the easiest way of monitoring the energy stored in all the capacitors was reading

their electrical voltage, the 4 sets used were as follow: 120 V, 200 V, 280 V and 360 V. The choice of

these values was due to several reasons which were:

• The minimum voltage necessary to move the kinematic set is around 100 V, so in

order to have a proper test, the first set was done using 120 V.

• Since the maximum voltage allowed by the capacitors used was 450 V, the highest

voltage considered was 400 V to keep the capacitors in top condition, avoiding further

damage.

• In order to have visible differences between the sets, a minimum offset in the range of

50 to 100 V was required.

So having all these requirements, a 4 set plan was created using the previous values (see

table 3.3). The data was obtained at 200 kHz during 2 seconds, resulting in 400000 data lines with

values for time, force and displacement.

To obtain the necessary quasi-static properties of the material and the corresponding

engineering curve, an additional set was performed using the compression tool and part of the

pneumatic actuator in a “manual mode”, which consisted in applying the compression force with a

wrench against the screwed rod of the pneumatic cylinder. The results obtained with this procedure

weren’t of the highest quality, but good enough to continue the tests.

17

Table 3.3 - Test PlanTest Type Voltage (V) Average Cam Speed (mm/s)

Quasi-static - 0,2

Dynamic

120 3500200 5500280 7000360 8000

4. RESULTS AND DISCUSSION

This chapter is divided in three parts. First the initial treatment and triage applied to the data

obtain from the experimental tests. Then a description, step by step, of the analysis applied to the

data. The last part comprises the application of the Viscoplastic models to the data and subsequent

results.

4.1 Experimental results

Excluding all the tests done during the maintenance process applied to the equipment (over 20)

in order to solve several issues that the compression tool and data acquisition equipment acquired

over the previous years of use (e.g.: broken cables, re-calibrations, small deformations and dents in

the translation cam, electrical and mechanical noise in the readings, etc), all of the 40 initial test

specimens were used in actual tests (see figure 4.1).

Of the 40 tested specimens, only 16 were usable in the end (see table 4.1), due to the difficulty

in obtaining repeatable results in these tests, consequence of the compression tool and

electromagnetic actuator inherent unreliability (see figure 4.2). Causes for this unreliability go from

18

Table 4.1 - Most representative tests

Case 38 Case 39 Case 40 Case 15 Case 16 Case 17 Case 19 Case 20 Case 215,92 5,99 6,03 6,05 5,96 6 5,93 6,08 6,016,16 5,93 6 5,94 5,76 5,94 6,06 5,88 6,071,62 1,624 1,668 1,618 1,59 1,598 1,622 1,604 1,609

Case 22 Case 27 Case 28 Case 30 Case 31 Case 32 Case 335,97 5,95 6,05 5,98 6,05 6,07 6,075,98 6,09 6,1 6,09 6,07 5,87 6,021,592 1,567 1,578 1,58 1,574 1,549 1,559

Quasi-Static 3500 mm/s 5500 mm/s

Initial Height (mm)Initial Diameter (mm)Final Heigth (mm)

7000 mm/s 8000 mm/s

Initial Height (mm)Initial Diameter (mm)Final Heigth (mm)

Figure 4.1 - Tested Specimens

misfires with the computer (synchronizing the data acquisition with the shot) to excessive vibrations in

the system due to a loosened component or a poorly placed specimen.

Taking a closer look at the data obtained from the 16 tests, at a glance the force data showed

no problems, but the displacement data displayed a constant small amplitude oscillation throughout

the entire test (visible in figure 4.2, zone 2). This was supposedly due to an internal behavior of the

displacement sensor and was solved by doing an average of the values every two data lines (see

figure 4.3).

Then the next step was removing all the data from each test that concerned the time period

previous to the compression phase (just fractions of a second that translate into several thousand data

lines). This was a two phase task: first to find the beginning of the compression process (done by

19

Figure 4.2 - Voltage (related to the force) and the displacement of a discarded test. The main problem of this one

was the force discontinuity caused by a small dent in the translation cam's profile (1)

Voltage

Displacement

Figure 4.3 - Example of a good test after the initial treatment (constant displacement after the compression phase

and beginning of the force decline in the same instant)

Voltage

Displacement

(1)

(2)

monitoring when the displacement started to change) and then remove part of the displacement

concerning the adjustment of the compression plates to the test specimen (see figure 4.4). This

adjustment refers to alignment problems between the compression plates and the specimen and also

that the interacting surfaces from the specimen and the plates aren't perfectly parallel.

Having performed these steps, it was necessary to readjust the data so that the initial section of

the force-displacement curve would pass through the graphs origin. As it will be discussed later, this

section's slope represents the test stiffness, which isn´t equal to the Elastic Modulus of the test

specimen, as expected in compression tests.

So after this initial data treatment process, here are the more representative force-time curves

and force-displacement curves of these tests, which excluding the oscillation in the first micro-seconds

due to the impact, these graphs have the expected shape similar to a regular forging process

(compression phase) followed by the relaxation phase. The typical strain rate evolution is also

presented, which is in accordance with what was expected [16] (see figures 4.5 and 4.6).

20

Figure 4.4 - Graph's origin adjustment: (a) Diagram of the compression plates adjustment to the test

specimen and (b) Force-Displacement curve before correction

Figure 4.5 - (a) Force-Displacement curves and (b) Strain rate evolution with strain progression

Case 16

Case 31

(a)

(b)

Case 16

Case 16

4.2 Data Analysis

After the preliminary data treatment, the data analysis process could start. True stress and

strain were calculated for each data set, using the initial measurements for each specimen and the

Volume Conservation Principle applied to metal deformation. After this initial step, some problems

were detected in data, mainly in the tests performed with higher strain rates (those with strain rates of

7000 mm/s and 8000 mm/s).

The main problems that can be observed in figure 4.7 from case 31 are: the intended final strain

value for test 31 was surpassed (should stop around point A) and then instead of it remaining constant

like on the low strain rate tests (vertical evolution like case 16), the test specimen retracted to the

desired strain value abruptly (B); additional vibrations were detected after the compression phase

(zone B); the stress decrease in the relaxation phase presents a discontinuity instead of being a

continuous process like in case 16 (∆σ ) .

Prior on finding a solution suited to these problems, it was necessary to identify the cause.

Further analysis of the test data revealed an unusual phenomenon in the displacement after the

21

Figure 4.7 - Problems detected in the relaxation phase. The correct graph shape (16 – low strain rate) isn’t

represented in the second curve (31 – high strain rate). End of compression point (A) for Case 16

Figure 4.6 - (a) Force-Time curves and (b) zoom on the compression phase

Case 16

Case 31

Case 16

Case 31

(A)

(B)

(∆σ )

Case 16

Case 31

(zoom)

compression phase (see figure 4.9), at around 3 ms, which indicates that the compression tool was

adding further displacement to the specimen. This was most likely caused by the larger amount of

kinematic energy in the system, that closed some of the natural clearances of the kinematic

mechanism in use (see figure 4.8). This wouldn't happen in a lower strain rate situation because the

resultant kinematic force to be dissipated is lower than the natural attrition force between the

contacting surfaces.

This additional displacement was around 100 micrometers, which resulted in two major

problems, aside from some additional vibration: as the test specimen was deformed beyond the

desired mark, the resulting stress (Final Compression stress) became higher than what it should be in

a normal test; after this additional deformation, with the system still under load from the elastic

recovery component of the specimen, the previous clearances were regained, allowing the specimen

the recover this additional displacement, reducing the residual stress levels in it (this could potentially

interfere with the relaxation test results).

So after further analysis of these problems, a set of corrections were applied to the data in order

to recover them to workable conditions. These corrections included the following steps, with some

hypothesis taken into consideration: assuming that the (∆σ ) from these tests had not changed, but

had just suffered an offset in relation to the correct values, this offset value was calculated and added

22

Figure 4.9 - Displacement problem. The test specimen (31) suffers an additional deformation but immediately

returns to correct displacement value

Case 16

Case 31

Figure 4.8 - Cam Clearance (blue arrow is the closing direction)

to all the stress values after Final Compression stress in order to recover the correct relaxation data;

application of a limit value for the strain and stress data in order to remove the extra caused by this

problem. The limit for the strain was extracted from the strain value at the end of the relaxation

process (long after this brief elastic recovery phase, to assure that this value was no longer influenced

by the problem) and the stress limit value corresponds to the stress of the specimen before this extra

deformation (determined with the strain limit value).

After the application of these corrections, all 16 data sets could be used in further analysis (see

figure 4.10).

After all this analysis process on the dynamic results, the next focus was on the quasi-static

results. These show a higher than expected flow stress values since for the final strain value, the

corresponding stress is around 210 MPa when it should be close to 140 MPa [10]. But as it can

perceived in figure 4.10, the quasi-static evolution is in accordance with the dynamic curves, even

verifying the relaxation theory in study. Causes for this difference can be: the material supplied was an

aluminum alloy with a metallurgical constitution a little different from the theoretical AA1050 or the

normal metallurgical condition of the test specimens wasn't completely recovered with the applied

annealing process. Either way, these results verify one of the important aspects of this thesis about the

relaxation phenomenon.

4.2.1 Stiffness Modeling

In an attempt to justify the final height differences between the test specimens (table 4.1) and

upon realizing that the stiffness value of the specimen's elastic deformation wasn't related to the

Young's Modulus of the AA1050 aluminum (70 GPa), a new challenge was proposed to find an

explanation to the new stiffness value being observed. If the compression tool elements were all

completely rigid, the final height of the specimens should all be the same, but if the system suffered

23

Figure 4.10 - Final corrected curves: recovery of the correct stress and strain values. Quasi-static curve example

Case 16

Case 31

Quasi-static

some minor elastic deformation (see figure 4.11), that could explain these differences. In order to

prove this, the stiffness value observable in the data must be lower than that solely of the test

specimen.

After considering several possibilities, the solution that showed the best results was a model

which included the following elements in a stiffness series (see figure 4.11): test specimen, load cell

measuring membrane, paper spacer between the load cell and it's external housing and the 4 screws

fixing this housing to the rest of the compression tool (aside from a small influence from the

compression plates).

As illustrated in table 4.2, the model stiffness value and the value extracted from the data are

similar, but still lower than that of the test specimen proving the validity of this hypothesis. The

difference between the model and the data values can have several explanations, such as:

measurement and extraction errors for the data value or a missing element in the stiffness series for

the theoretical value.

4.3 Model Application

The final objective of this work was to apply two models to this data. First, use the Hybrid model

previously mentioned in order to further study it's applicability to this type of situations. Second,

develop a new Viscoplastic model for metals with the intent of describing the relaxation process visible

in these tests.

24

Figure 4.11 - Diagrams with the elements considered in the stiffness series

Table 4.2 - Stiffness valuesTest Specimen Model Data

Stiffness Values329194 57470 47172

(N/mm)

4.3.1 Compression Model

Continuing the work done with this model from previous theses, the starting information for this

part was the model's equation and the parameters to be applied with the pair (material+tool) used (see

table 4.3, extracted from [10]).

σ=[A+exp (mε)εn] [B+Cln(D+ ε̇)] (5)

After some initial tests with this model on the data, it became apparent that the parameters

needed to be fine tuned to this case, either because the material wasn’t in the same metallurgical

condition as from previous tests or the tool was further worn out from previous works. So a new fitting

operation was in order. Using the spreadsheet file with the data and applying the Least Square Root

method with the program's internal solver, the new set of parameters was determined (see table 4.4).

The parameter D wasn't used from this point on, because the fitting process used couldn't work with it

(it would result in incorrect values that would not have the necessary performance throughout all the

tests, only fitting correctly one data set), but since the this parameter was only for small adjustments to

the strain rate influence in the flow stress, this was seen as only a small, discardable problem. In this

fitting process, both the strain and strain rate were used implicitly in the analysis.

So here is an example of the curve obtained from the application of this model to the data and

the resulting fitting parameters (R2) (see figure 4.12 and table 4.5):

25

Table 4.3 - Viscoplastic model initial parametersA B C D m n

0,22 134 12,5 0,8 -0,34 0,36

Table 4.5 - Viscoplastic model fitting results

15 16 17 19 20 210,911 0,7671 0,929 0,589 0,78 0,6590,988 0,974 0,995 0,866 0,949 0,825

22 27 28 30 31 32 330,44 0,422 0,649 0,488 0,639 0,542 0,6480,511 0,551 0,691 0,645 0,694 0,482 0,692

3500 mm/s 5500 mm/s

Model R 2̂Trend Line R 2̂

7000 mm/s 8000 mm/s

Model R 2̂Trend Line R 2̂

Table 4.4 - Viscoplastic model new parametersA B C m n

0,29 195,19 0,35 -0,12 0,47

The new parameters are still similar to the previous group, with some differences most likely

related to the vibration problem and the issues previously mentioned.

The low R2values from the model for the higher strain rate tests can be justified by the

additional vibration in this tests caused by the higher impact force, instead of inherent problems in the

model or the parameters used. This correlation can be further illustrated by applying a Quadratic Trend

Line (since the graphic has a small curvature with no inflection, which can be accurately fitted by a

quadratic equation) to each data batch and monitor it's R2 evolution throughout the tests. If the Trend

Line R2is low, then the data must have some problems (in this case the mechanical vibration), which

is related to a low R2for the model. When there is low vibration, both R2

are high, which is the case

for the low energy level tests (120 V) (see figure 4.13).

As expected, the flow stress is higher throughout the test when higher strain rate values were

used (higher energy). Also the duration of compression part in the test became shorter with increase in

the compression speed (more energy used).

26

Figure 4.13 - Viscoplastic compression curve examples: (a) test 16 and (b) test 31

Case 16

Model

Case 31

Model

(a) (b)

Figure 4.12 - Compression stress-strain curve for the real data and the model

Case 17

Polinomial TL

Model

The unexpected result was that the Final Compression stress should have increased with the

strain rate, and even though it did, the evolution was smaller than expected. This is most likely a

characteristic of this material and not a defect of the test procedure. This can be explained by a

saturation of the strain rate influence in the flow stress, which was likely achieved in these tests.

Regarding the model, it's application was successful, further illustrating the influence of the

strain rate in the aluminum's flow stress, while being able to follow the material's stress-strain

response accurately in all tests, all with the same set of parameters.

4.3.2 Relaxation Model

As previously stated, there aren't Viscoplastic models specifically for metals that describe the

relaxation phenomenon. So in this work a new model is proposed to describe the relaxation process

metals undergo if submitted to a constant strain for a set time period.

The first step in this analysis was to understand and determine the type of equation that could fit

into this curve (see figure 4.14).

As it can be observed, it has a decaying behavior characteristic of fractional or logarithmic

equations. So, after experimenting with several models, here are the two more promising (see

figure 4.15):

- Logistic Model σ=A+B

1+[(t−t 0) /C ]D

(8)

- Exponential Model σ=A∗exp(−B∗t)+C (9)

27

Figure 4.14 - Relaxation curve example with the previously mentioned Final Compression stress (1) and the

Relaxation stress (2). This quasi-static curve is just representative, not from a real test

(1)

(2)

Case 16

Quasi-static

While both can fit the relaxation curve, the second is more adequate because it can adapt to

both zones with it's two curvature parameters (C and D). The value t 0 is the time when the relaxation

process begins.

Having found the model to use, it was necessary to perform a fitting process to each data batch

in order to obtain the equation parameters for each one. Using a methodology similar to the

Viscoplastic model, the parameters were determined and later analyzed (see figure 4.16 and

table 4.6).

28

Figure 4.15 - Viscoplastic models example

Table 4.6 - Viscoplastic relaxation parameters

15 16 17 19 20 21R 2̂ 0,98 0,98 0,97 0,99 0,98 0,99A 211,4 219,7 218,2 220,9 207,6 217,2B 34,9 32,4 27,5 51,1 40,7 37,4C 0,011 0,008 0,011 0,006 0,005 0,006D 0,774 0,728 0,672 0,837 0,690 0,676

A+B 246,3 252,1 245,7 272,1 248,2 254,6

22 27 28 30 31 32 33R 2̂ 1,0 1,0 1,0 1,0 1,0 1,0 1,0A 199,6 210,4 206,0 229,2 226,0 219,8 221,2B 29,5 20,0 27,2 20,5 26,4 20,7 21,3C 0,005 0,011 0,004 0,008 0,005 0,006 0,011D 0,710 0,662 0,532 0,685 0,749 0,559 0,800

A+B 229,1 230,3 233,2 249,7 252,4 240,4 242,5

A B C D A+B215,9 30,0 0,007 0,698 245,98,4 9,2 0,003 0,086 11,5

3500 mm/s 5500 mm/s

7000 mm/s 8000 mm/s

AverageSt. Deviation

Case 31

Exponential Model

Logitc Model

In accordance with the Viscoplastic theory accepted at the time of this work, the flow stress after

the relaxation process should be close to the flow stress of the same material strained to the same

mark in quasi-static conditions (at room temperature) if given enough time. This was exemplified in the

tests and further accurately reproduced by the model.

Since each test has it's parameter set instead of a parameter group for the pair (material + tool)

in use like in the previous Viscoplastic model, the problem of estimating these parameters in order to

apply the model in other tests became evident. The parameters A and B can be estimated with a

Viscoplastic model (like the Hybrid Model used) by introducing the desired conditions for the relaxation

test (strain, strain rate and temperature) in the model and extracting the corresponding flow stress

values: A+B (Final Compression stress) can be obtained from the Viscoplastic Model in dynamic

conditions with the initial conditions for the test; B (Relaxation stress) can be extracted from the

Viscoplastic Model in quasi-static conditions for the final strain value; A can be calculated by

subtracting the Relaxation stress from the Final Compression stress [A=(A+B)– B] .

Meanwhile the fact that the C and D parameters are likely related with the material's viscosity

means that they primarily depend on the type of material used (according to the data obtained, the

viscosity dependance on the strain rate is very small while the study of the influence of other

parameters wasn´t the focus of this work). To have values for other material types would require an

extensive work building a parameter's data base for different materials, which was clearly out of the

scope of this work.

All this information pertaining the parameters is true because the C and D remain almost

constant throughout all the tests (related to the material viscosity) while the A and B differ from test to

test, as illustrated in table 4.6.

29

Figure 4.16 - Viscoplastic relaxation curve examples: (a) test 17 and (b) test 31

Case 17

Model

Case 31

Model

(a) (b)

5. CONCLUSION

The main objective of this work was to further study the dynamic characteristics of AA1050

aluminum, with a special focus on Viscoplastic relaxation properties. A new model was developed for

the relaxation curve that can describe this behavior, but the author was unable to verify it's reliability

on this or other material types which would require further tests and analysis that couldn’t be

performed in this thesis time frame. Also the application of the Hybrid model to a new batch of tests

further demonstrated it's capability to accurately describe this type of compression tests, increasing

our understanding of metal deformation in dynamic conditions.

Secondary objectives of this thesis were to perform an upgrade and some improvements to the

original experimental set-up. This was achieved with the addition of a pneumatic damper to the

kinematic set which helped improve it's durability and longevity.

Concerning possible future work done to further this thesis contributions to the Viscoplastic field

for metallic materials or just improve the work done here, there are several suggestions: re-test the

specimens used in this work to verify if their yield stress matches the final stress observed in data

proving the theory behind the relaxation Viscoplastic model; further study the stiffness of the

experimental set-up to improve the model developed in this thesis in order to enhance the use of the

equipment; some improvements could also be made to the compression tool, such as a mechanism

that would allow the user to adjust the initial space between the compression plates as to reduce the

initial impact when the compression process starts; study the influence of other parameters in the

AA1050 aluminum’s viscosity (different strain values for the relaxation test or different temperatures),

keeping the new experimental conditions as close as possible to those used in these tests; model the

whole test in a Finite Element software and implement both models in order to further verify their

applicability to real cases (this one is the more important of the group since it's results could be directly

applied to an industrial production process); related to the previous suggestion, in order to better

simulate today's manufacturing processes, more materials should be tested using this procedure with

the objective of assembling a parameter data base with all the engineering material in use.

As a final remark, the relaxation model developed in this work can potentially be extremely

useful in the industry sector, for those situation were there is a need to estimate the surface conditions

of a given material after it was submitted to the manufacturing process (ex: estimate the residual

stress level in the new surface after a machining process [18 and 21]

30

6. REFERENCES

[1] Kupessa, Eurico (2010), Caracterização do Comportamento Mecânico da Liga de

AlumínioAA1050. Master Degree in Mechanical Engineering, Instituto Superior Técnico,

UniversidadeTécnica de Lisboa.

[2] Kojic, M. and Bathe, K-J., (2006). Inelastic Analysis of Solids and Structures. Elsevier.

[3] von Mises, R. (1913). "Mechanik der festen Korper im plastisch deformablen Zustand.".

Gottinger Nachr, math-phys Kl 1913:582–592.

[4] McCrum, Buckley, and Bucknell (2003). "Principles of Polymer Engineering," 117-176.

[5] Betten, J., (2005). Creep Mechanics. 2nd Ed., Springer.

[6] J.L. Chaboche (1996). Thermodynamic Formulation of Constitutive Equations and

Application to the Viscoplasticity and Viscoelasticity of Metals and Polymers. Int. J. Solids

Structures Vol. 34, No. 18, pp. 2239-2254. 1997.

[7] J.L. Chaboche (2008). A review of some plasticity and viscoplasticity constitutive theories.

International Journal of Plasticity.

[8] Hoon Huh, Kwanghyun Ahn, Ji Ho Lim, Hyung Won Kim, Lee Ju Park (2014). Evaluation of

dynamic hardening models for BCC, FCC, and HCP metals at a wide range of strain rates.

Journal of Materials Processing Technology.

[9] Len Schwer (2007). Optional Strain-Rate Forms for the Johnson Cook Constitutive Model

and the Role of the Parameter Epsilon_0. 6th European LS-DYNA Users’ Conference.

[10] Silva, Carlos (2013), Caracterização Mecânica e à fractura de materiais aplicada a

processos de deformação plástica e corte. PhD in Mechanical Engineering, Instituto

Superior Técnico, Universidade Técnica de Lisboa.

[11] David Roylance (2001). Engineering Viscoelasticity. Department of Materials Science and

Engineering.

[12] Göran Spetz (1999). Stress Relaxation Tests – Technical Report 98/1. 2nd Edition

Elastocon AB Sweden.

[13] B. Hopkinson, “A Method of Measuring the Pressure Produced in the Detonation of High

Explosives or by the Impact of Bullets,” Philos. Trans. R. Soc. (London) A, 213, pp. 437-

456, 1914.

[14] Ezequiel, Vasco (2010), Influência da Velocidade de Deformação e da Tensão Normal na

Tenacidade à Fractura da Liga de Alumínio AA1050. Master Degree in Mechanical

Engineering, Instituto Superior Técnico, Universidade Técnica de Lisboa.

31

[15] Pinto, José (2009), Avaliação do Comportamento Mecânico de Blindagens Balísticas.

Master Degree in Mechanical Engineering, Instituto Superior Técnico, Universidade

Técnica de Lisboa.

[16] C.M.A. Silva, P.A.R. Rosa, P.A.F. Martins (2011). Electromagnetic Cam Driven

Compression Testing Equipment. Society for Experimental Mechanics

[17] Mokolobate, Kenosi Norman (2011), Stress Relaxation Tests on a Low Alloy Steel. Master

Degree in Mechanical Engineering, Faculty of Engineering and Built Environment,

University of the Witwatersrand.

[18] T. D. Marusich, M. Ortiz (1995). Modelling and Simulation of High-Speed Machining.

Division of Engineering, Brown University, Providence, USA.

[19] Vargas, Gonçalo (2009), Contribuição para uma nova Abordagem na Caracterização do

Comportamento Mecânico de Materiais. Master Degree in Mechanical Engineering,

Instituto Superior Técnico, Universidade Técnica de Lisboa.

[20] Sia Nemat-Nasser, Jon B. Isaacs, John E. Starrett (1991). Hopkinson Techniques for

Dynamic Recovery Experiments. Proceedings: Mathematical and Physical Sciences,

Volume 435, Issue 1894

[21] J.C. Outeiro, F. Rossi, G. Fromentin, G. Poulachon, G. Germain, A.C. Batista (2013).

Process Mechanics and Surface Integrity Induced by Dry and Cryogenic Machining of

AZ31B-O Magnesium Alloy. 14th CIRP Conference on Modeling of Machining Operations

(CIRP CMMO)

32

7. ANNEXES

7.1 Electromagnetic Actuator Firing Procedure (in Portuguese)

Armário e Actuador:

• Verificar que o transformador regulável está na posição de 0 V

• Ligar os voltímetros e ligar o actuador à rede eléctrica

• Ligar o Geral. Verificar as ligações das extensões na zona dos condensadores

• Ligar o Carga para iniciar o processo de carga dos condensadores

◦ Ligar os disjuntores dos bancos de condensadores a carregar

◦ Regular o transformador, aumentado a tensão sem que a corrente geral passa 5 A, até

que os condensadores atinjam o valor de carga desejado

◦ Desligar os disjuntores

◦ Desligar o Carga

• Ligar o Descarga

◦ Fazer o disparo com o computador. Colocar a gravar e logo depois carregar no Trigger

◦ Desligar o Descarga

• Desligar o Geral e os voltímetros

• Desligar o actuador da rede.

Ferramenta e Provete:

• Verificar a mobilidade da barra de impacto

◦ Colocar na posição de disparo (face da Came no plano da ferramenta)

◦ Verificar a posição do batente de borracha

• Verificar a camada de óleo em toda a ferramenta

• Colocar o provete entre os pratos o mais centrado possível

• Verificar as ligações aparafusadas nas pastes móveis do equipamento

◦ Ligação da barra de impacto à Came

◦ Ligação da barra de seguimento ao alvo do sensor de deslocamento

◦ Ligação do prato superior à barra de seguimento

33

• Verificar o aperto da tampa da ferramenta

Móvel do Computador:

• Verificar a ligação da extensão exterior à rede eléctrica. Verificar a extensão interior

• Ligar a extensão interna no botão

◦ Ligar o computador

◦ Ligar a placa de aquisição

◦ Ligar o amplificador (deixar aquecer durante 30 minutos)

Sensores (no início de cada período de ensaios):

• Verificar a excitação e amplificação da célula de carga no amplificador (usar o multímetro)

◦ Ver tensão de excitação nas saídas do módulo da fonte de energia do amplificador

◦ Verificar o ganho de 1000 do amplificador com o multímetro (ocasionalmente) *

• Verificar o sensor de deslocamento

◦ Verificar as condições do cabo da sonda

◦ Verificar o ganho do sensor

• Verificar a taxa de aquisição da placa (adquirir 400k pontos a uma taxa de 200k >>> 2s)

• Se o sinal obtido tiver ruído em excesso, verificar as ligações dos cabos dos sensores

◦ À placa de aquisição e ao amplificador

◦ À terra

◦ Entre a malha do cabo da célula de carga e o amplificador

Cilindro Pneumático

• Função de amortecedor no ensaio:

◦ Colocar o batente da haste na posição de disparo (encostado à Came)

◦ Verificar que não há pressão no cilindro **

◦ Verificar que a válvula de saída está aberta

◦ Regular a válvula de escape do cilindro para a posição de amortecimento desejado ***

◦ Verificar que a tampa da ferramenta está bem apertada ***

• Função de actuador pneumático:

34

◦ Encostar o batente da haste à face da Came

◦ Abrir a válvula de admissão de ar a ¼ do curso, com a válvula de escape aberta

◦ Regular a velocidade da haste com a válvula de escape, podendo desapertar a tampa

da ferramenta ou aumentar o caudal de ar para facilitar a tarefa

* F(N)=10000xV(saída)

** Assumindo que o posicionamento do cilindro é para esta situação.

*** Caso contrário o cilindro funciona como uma mola

Nota: Montagem do Cilindro Pneumático na configuração usada nesta tese

Montar o cilindro pneumático de modo a que a ponta do batente da haste, quando esta

estiver deslocada para fora 62 mm desde a posição recolhida, ficar a tocar na face da Came quando

esta estiver na posição de disparo.

35

7.2 Electrical Circuit Diagram

36

7.3 LabView Program

Front Panel

Back Panel (part 1)

37

Back Panel (part 2)

7.4 Experimental Equipment Data Sheets

Here is a list of the equipment's data sheets used in this work.

• Signal Conditioning Amplifier 2100 (2011), Vishay Precision Group.

• C9B Force Transducers 50 kN, Hottinger Baldwin Messtechnik GmbH.

• ECL100 Series Eddy-Current Displacement Sensors, Lion Precision.

• Power supply unit - MINI-PS-100-240AC/10-15DC/2 - 2938756, Phoenix Contact.

• NI 6251 Specifications (2007), National Instruments Corporation.

38

7.5 Translation Cam Schematic (Logistic)

39