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Constitutive Modeling of Creep in Leaded and Lead-Free Solder Constitutive Modeling of Creep in Leaded and Lead-Free Solder

Alloys Using Constant Strain Rate Tensile Testing Alloys Using Constant Strain Rate Tensile Testing

Eric Thomas Stang Wright State University

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CONSTITUTIVE MODELING OF CREEP IN LEADED AND LEAD-FREE SOLDER

ALLOYS USING CONSTANT STRAIN-RATE TENSILE TESTING

A thesis submitted in partial fulfillment

Of the requirements for the degree of

Master of Science in Mechanical Engineering

By

ERIC THOMAS STANG

B.S.M.E., Wright State University, 2012

2018

Wright State University

WRIGHT STATE UNIVERSITY

GRADUATE SCHOOL

December 12, 2018

I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY

SUPERVISION BY Eric Thomas Stang ENTITLED Constitutive Modeling of Creep

in Leaded and Lead-Free Solder Alloys Using Constant Strain-Rate Tensile Testing

BE ACCEPTED IN PARTIAL FULLFILLMENT OF THE REQUIRMENT FOR

DEGREE OF Master of Science of Mechanical Engineering.

Committee on final examination:

________________________________

Daniel Young, Ph.D.

________________________________

Joseph C. Slater, Ph.D., P.E.

________________________________

Raghavan Srinivasan, Ph.D., P.E.

________________________________

Barry Milligan, Ph.D.

Interim Dean of the Graduate School

________________________________

Daniel Young, Ph.D.

Thesis Director

________________________________

Joseph C. Slater, Ph.D., P.E.

Chair, Department of Mechanical and

Materials Engineering

iii

ABSTRACT

Stang, Eric Thomas. M.S.M.E., Department of Mechanical and Materials Engineering, Wright

State University, 2018. Constitutive Modeling of Creep in Leaded and Lead-Free Solder Alloys

Using Constant Strain-Rate Tensile Testing.

Environmental and safety concerns have necessitated a phase-out of lead-based alloys, which are

often used in electronics solder applications. In order to properly assess suitable replacement

materials, it is necessary to understand the deformation mechanisms relevant to the application. In

the case of electronics solder, creep is an important mechanism that must be considered in the

design of reliable devices and systems. In this study, Power-Law and Garofalo constitutive creep

models were derived for two medium temperature solder alloys. The first alloy is known by the

commercial name Indalloy 236 and is a quaternary alloy of lead, antimony, tin, and silver. The

lead-free alternative is a binary alloy of tin and antimony known by the trade name Indalloy 264.

Constant strain rate tests were conducted at temperatures from -20 to 175 Celsius using constant

strain rate tensile testing in the range of e-5 s-1 to e-1 s-1. Creep constants were defined for use

in materials selection and design analysis activities.

iv

Table of Contents

Chapter 1 Introduction .................................................................................................................... 1

1.1 Motivation ........................................................................................................................ 1

1.2 Industrial Solder Applications ......................................................................................... 2

1.3 Material Considerations in Electronics Solder Selection................................................. 5

1.4 Definition of Creep as a Deformation Mechanism .......................................................... 6

1.5 Reliability Considerations and Damage Models.............................................................. 7

Chapter 2 Fundamental Evaluations of the Creep Mechanism ..................................................... 11

2.1 Macro Level Observations of Creep Behavior .............................................................. 11

2.2 Microscopic Evaluation of Creep Mechanisms. ............................................................ 14

2.3 Deformation-Mechanism Maps ..................................................................................... 16

2.4 Quantifying and Modeling Creep .................................................................................. 18

2.5 Power-Law and Garofalo Parameter Estimation for CSR Testing ................................ 23

Chapter 3 Experimental Setup and Procedure .............................................................................. 28

3.1 Testing Equipment ......................................................................................................... 28

3.2 Test Samples .................................................................................................................. 30

3.3 Testing Parameters ......................................................................................................... 30

3.4 Test Procedure ............................................................................................................... 33

Chapter 4 Test Results .................................................................................................................. 35

4.1 Creep Testing Results .................................................................................................... 35

v

4.2 Data Analysis – Power-Law .......................................................................................... 45

4.3 Data Analysis – Garofalo ............................................................................................... 52

Chapter 5 Discussion .................................................................................................................... 58

5.1 Microstructure ................................................................................................................ 58

5.2 Post-Test Evaluation and Fractography ......................................................................... 65

5.3 Issues with Elastic Test Data ......................................................................................... 70

5.4 Observations .................................................................................................................. 71

5.5 Comparisons to Published Data ..................................................................................... 72

Chapter 6 Conclusions and Future Work ...................................................................................... 75

6.1 Conclusions .................................................................................................................... 75

6.2 Contributions.................................................................................................................. 75

6.3 Future Work ................................................................................................................... 76

APPENDIX A: Certificate of Analysis for Provided Samples ..................................................... 77

APPENDIX B: Example MATLAB Script .................................................................................. 79

Bibliography ................................................................................................................................. 81

vi

List of Figures

Figure 1.2-1. Microbumps with and without a copper pillar used in "flip-chip" manufacturing

processes [15].................................................................................................................................. 4

Figure 1.2-2. Schematic diagram of a flip chip assembly process [14]. ........................................ 4

Figure 1.2-3. Exploded lid attachment schematic for a hermetically sealed MEMS device [12]. . 5

Figure 1.5-1. Exemplar Design for Reliability process for solder joint characterization and design

[6]. ................................................................................................................................................... 7

Figure 2.1-1. Proposed creep test schematics for: a.) Constant Force and b.) Constant Stress [23].

....................................................................................................................................................... 11

Figure 2.1-2. Creep-Rupture plot showing the time to rupture vs. applied stress [19]. ................ 12

Figure 2.1-3. Typical creep curve divided into three stages [18]. ................................................ 13

Figure 2.1-4. Exemplar Stress-Strain Diagram for a CSR Creep Test. ........................................ 14

Figure 2.2-1. Prevailing activation energy of deformation vs. temperature for single crystal

aluminum [17]. .............................................................................................................................. 15

Figure 2.2-2. Schematic of dislocation migration in a grain under uniaxial tension [27]. ........... 16

Figure 2.3-1. Proposed Deformation-Mechanism Map for lead developed by Ashby [28]. ........ 17

Figure 2.3-2. Nickel Deformation Processing Map with added Grain Boundary Sliding field [29].

....................................................................................................................................................... 18

Figure 2.4-1. Multiple strain vs time curves (a) are used to make isochronous stress-strain curves

(b) [23]. ......................................................................................................................................... 19

Figure 2.4-2. Effect of stress on the energy barriers for vacancy motion in a hypothetical creep

mechanism [17]. ............................................................................................................................ 22

vii

Figure 2.5-1. Strain-rate and stress plots for austenitic stainless steel used for Garofalo Parameter

Estimation. .................................................................................................................................... 26

Figure 3.1-1. Test setup for sub-ambient tensile test. ................................................................... 29

Figure 3.2-1. Generic test specimen diagram. .............................................................................. 30

Figure 3.3-1. Axial stress distribution in a necked specimen [31]................................................ 33

Figure 4.1-1. Indalloy 236 test plots for 10-1 s-1 strain-rate. ......................................................... 37

Figure 4.1-2. Indalloy 236 test plots for 10-3 s-1 strain-rate. ......................................................... 38

Figure 4.1-3. Indalloy 236 test plots for 10-4 s-1 strain-rate. ......................................................... 39

Figure 4.1-4. Indalloy 236 test plots for 10-5 s-1 strain-rate. ......................................................... 40

Figure 4.1-5. Indalloy 264 test plots for 10-1 s-1 strain-rate. ......................................................... 41

Figure 4.1-6. Indalloy 264 test plots for 10-3 s-1 strain-rate. ......................................................... 42

Figure 4.1-7. Indalloy 264 test plots for 10-4 s-1 strain-rate. ......................................................... 43

Figure 4.1-8. Indalloy 264 test plots for 10-5 s-1 strain-rate. ......................................................... 44

Figure 4.2-1. Power-Law parameter determination for Indalloy 236, individual models. ........... 45

Figure 4.2-2. Power-Law parameter determination for Indalloy 264, individual models. ........... 46

Figure 4.2-3. Fit and parameter comparisons for Power-Law, individual modeling. ................... 47

Figure 4.2-4. Power-Law parameter determination for Indalloy 236, combined model. ............. 48

Figure 4.2-5. Power-Law parameter determination for Indalloy 264, combined model. ............. 48

Figure 4.2-6. Fit and parameter comparisons for Power-Law, combined model. ........................ 49

Figure 4.2-7. Fit and parameter comparisons for Power-Law, MATLAB model. ....................... 49

Figure 4.2-8. 3D surface fitting of Power-Law Equation for Indalloy 236. ................................. 50

Figure 4.2-9. 3D surface fitting of Power-Law Equation for Indalloy 264. ................................. 51

Figure 4.3-1. Garofalo parameter determination for Indalloy 236. .............................................. 53

viii

Figure 4.3-2. Garofalo parameter determination for Indalloy 264. .............................................. 54

Figure 4.3-3: Fit and parameter comparisons for Garofalo models from iterative analysis ......... 55

Figure 4.3-4. Fit and parameter comparisons for Garofalo models from MATLAB. .................. 55

Figure 4.3-5. 3D Surface fitting of Garofalo Equation for Indalloy 236. ..................................... 56

Figure 4.3-6. 3D Surface fitting of Garofalo Equation for Indalloy 264. ..................................... 57

Figure 5.1-1. Phase diagrams for Sn-Sb binary system (dashed line for Indalloy 264). Left: [38].

Right: [39]. .................................................................................................................................... 59

Figure 5.1-2. Equilibrium phase composition for Indalloy 236.................................................... 60

Figure 5.1-3. Equilibrium phase composition for Indalloy 264.................................................... 60

Figure 5.1-4. Planar definitions of metallurgical specimen cross-sections. ................................. 62

Figure 5.1-5. Micrographs of Indalloy 236 and Indalloy 264, as-received. ................................. 63

Figure 5.1-6. Micrographs of Indalloy 236 and Indalloy 264, post-test. ...................................... 64

Figure 5.2-1. Leaded solder specimen fractures after high strain-rate testing. ............................. 66

Figure 5.2-2. Leaded solder specimen fractures after low strain-rate testing. .............................. 67

Figure 5.2-3. Lead-free solder specimen fractures after high strain-rate testing. ......................... 68

Figure 5.2-4. Lead-free solder specimen fractures after low strain-rate testing. .......................... 69

Figure 5.3-1. Repitition of apparent elastic modulus observed during reloading cycles. ............. 71

ix

List of Tables

Table 1.1-1. Known properties of the alloys under evaluation [9, 10] ........................................... 2

Table 3.3-1. Proposed test matrix for tensile testing. ................................................................... 31

Table 4.1-1. Summary of stress data used for creep analysis. ...................................................... 36

Table 5.5-1. Comparison of parameter estimation from Power-Law analysis with literature. ..... 72

Table 5.5-2. Comparison of parameter estimation from Garofalo analysis with literature. ......... 73

x

Acknowledgements

I would be remiss if I didn’t dutifully and humbly acknowledge the support, encouragement,

and advisement provided to me by the very people and groups who also gave me the means and

the reasons for pursuing this opportunity in higher education.

First, I would like to thank ASM Materials Education Program and Thermo-Calc who provided

Wright State University with access to the Thermo-Calc software and solder modules through the

Materials Genome Toolkit Program. This software contributed greatly to the depth of this project.

I would also like to personally thank Dr. Henry Daniel Young of Wright State University,

whose guidance has pressed me into the curious pursuit of knowledge and whose tutelage has

reigned in that curiosity to define a clear research path. As an atypical graduate student, I am

thankful to be able to pursue a research degree because of the opportunity that he has given me.

Many thanks are due to my colleagues and mentors at Emerson. Aditya Sakhalkar has been a

key stakeholder in my pursuit of this degree, who intimately understood its value to my career

trajectory and our team’s collective capabilities. Titus Broek was my initial engineering mentor

and has continued to provide me with guidance and resources, some of which have been included

in this present work. I wish I had space enough to mention every member of the Applied

Mechanics team by name, you personify the spirit of camaraderie and technical excellence.

Lastly, my greatest source of energy and purpose has been my family and friends. My life has

seemed at times to be on the precipice of calamity, but you have helped me to see that the increase

in “human entropy” has all been in the pursuit of an equilibrium state. You have provided me with

more support and care than I can repay in a dozen lifetimes, but I’ll do my best to pay it forward.

“I do not wish to expiate, but to live!” – Ralph Waldo Emerson

1

Chapter 1 Introduction

1.1 Motivation

It seems to be common that the great problems in engineering require a balance to satisfy an

array of competing interests to find a satisfactory solution and then continually innovating and

improving to advance and optimize a technology. In our present era, consumers require smaller,

safer, more reliable, and more affordable electronic devices to fully participate in society and the

workforce [1]. This demand necessitates the investigation of new materials and processes on a

rapid and evolutionary scale [2, 3]. The exponential return on iterative improvement in processing

bandwidth was famously predicted (and proven true over a finite time span) by Gordan Moore in

1965, and became known as Moore’s Law [4]. The shrinking size and increasing capacity of

processors has required a similarly rapid improvement in the electrical connections within the

integrated circuit.

Solder alloys in various forms have been used by humankind for practical and ornamental

purposes for millennia. A particular application in the electrical and electronics industry is to

provide a thermally and electrically conductive bond to join electrical components into a useable

circuit [5]. The alloys used for electronics manufacture require a relatively low melting point and

are classified as soft solders (henceforth these will simply be referred to as solders, as hard solders

are traditionally used in brazing processes). A primary element in these alloys has been lead,

which is abundantly available, possesses beneficial characteristics in this application, and has been

extensively studied. Lead, hexavalent chromium, cadmium, asbestos, and other materials have

been used extensively in the last century during a period of rapid technological advance. However,

recent medical and environmental research has necessitated an ethical and humanitarian shift away

2

from these materials which have been shown to adversely affect biological health in humans and

animals [6, 7, 8].

For this reason, many new lead-free solder alloys are being evaluated. In this study, two

commercially available solders will be mechanically tested to determine the constitutive

deformation models that govern their mechanical behavior. The first alloy is a quarternary alloy

known by the commercial name Indalloy 236. This alloy, composed of four alloying elements,

contains 83% lead by mass. The leaded alloy will act as a point of comparison for a lead-free alloy

with a similar melting temperature range. The lead-free alloy, Indalloy 264, is a binary alloy

composed of tin and antimony. The known properties of these alloys are catalogued in Table

1.1-1. Known properties of the alloys under evaluation Table 1.1-1[9, 10].

Table 1.1-1. Known properties of the alloys under evaluation [9, 10]

1.2 Industrial Solder Applications

Solder alloys can be used in a variety of electronics applications and are highly dependent on

the application temperature and current density requirements. This creates sub-classifications of

solders based upon their melting temperature. Lower temperature solders are used often to surface

mount components at the board level. The melting point of these solder should be less than 240

°C, especially if mounting to an organic substrate [11]. Intermediate level solders are often used

for package assemblies prior to board level assembly. They will not reflow at the board level

mount while remaining at safe temperatures for the package subcomponents. This can be used for

mounting subcomponents to a lid that hermetically seals a MEMS based inertial sensor like an

3

accelerometer or a gyroscope [12, 9]. Lastly, high temperature alloys are used for demanding

applications for power transmission or in die attachment processes. The large current density

required for power transmission is known to create heat in the solder connection [13]. The solder

alloys considered in this paper would be classified as intermediate temperature alloys that melt

just above the cut-off temperature of 240 °C.

In the advent of the semi-conductor era, various electronics packaging schemes have been used

to interconnect and support microchips. Specifically, the “flip-chip” process is used to surface

mount semi-conductor processors and MEMS devices to produce robust electronic packaging with

the capacity for mass production in the digital age [6]. In this process, a semiconductor chip is

attached onto an array of solder bumps that lie on either a printed circuit board (PCB) or, in the

case of intermediate temperature alloys, an electronic package. The former process is commonly

referred to as flip chip on board (FCOB) while the latter is called flip chip in package (FCIP).

Figure 1.2-1 shows an older design iteration known as a C4 bump aside of a smaller diameter

copper-pillar microbump that allows for a greater area density for interconnects. The connection

is typically made by theromocompression bonding or reflow. This relatively simple, but nuanced

approach is schematically represented in Figure 1.2-2 for an FCOB process. Advances in this

design have created space savings that are of great desire for manufacturers of small consumer

electronics [14].

4

Figure 1.2-1. Microbumps with and without a copper pillar used in "flip-chip" manufacturing processes [15].

Figure 1.2-2. Schematic diagram of a flip chip assembly process [14].

Intermediate temperature alloys are also used in the assembly of MEMS sensors. Robust

inertial sensors, such as gyroscopes and accelerometers are hermetically sealed to protect the

sensing element. In such a case the sensing elements and conditioning electronics can be

connected to a lid which seals the MEMS system and provides conductive channels to the terminals

5

of the chip carrier. This is shown schematically in Figure 1.2-3. Exploded lid attachment schematic

for a hermetically sealed MEMS device . Microphones and pressure sensors cannot be hermetically

sealed because they must sense pressure and pressure waves from the environment. They can,

however, still be lid or base mounted using the same category of solder alloys [12].

Figure 1.2-3. Exploded lid attachment schematic for a hermetically sealed MEMS device [12].

1.3 Material Considerations in Electronics Solder Selection

Because of the relatively low melting point of solder alloys used in the electronics industry,

they are often subject to high application temperatures relative to their melting temperature [6].

The internal friction of electrons passing through the solder connection (known as Joule heating)

is itself a direct internal heat source, which contributes to the increased temperature [16]. The

definition of a proper material model for such materials in this application will be generally

relevant to other applications, such as hot-work metal forming processes, turbine engine

applications, and nuclear power generation, which must also deal with high temperatures relative

to the materials melting point. This is made evident by the multiple disciplines represented in the

bibliography of the work.

6

This brings into play strain-rate sensitivity considerations which are not common in traditional

structural engineering problems. The mechanical properties of the materials are evaluated using

either an elastic-plastic-creep or viscoplastic model [6]. In the present work we will focus on the

former, elastic-plastic-creep model, to characterize the deformation of the relevant solder alloys

for future design consideration.

1.4 Definition of Creep as a Deformation Mechanism

Throughout this paper the phenomenon of creep will be evaluated from the macroscopic and

microscopic perspectives. The macroscopic perspective will be used to understand the stress,

strain, deformation, and fracture on a larger scale. These are general principles used in mechanical

engineering design. Materials science and engineering principles will be used to evaluate the

microscopic (and submicroscopic) mechanisms that collectively lead to observations of

macroscopic creep.

Creep, in its most concise definition, is the permanent deformation of a material subjected to a

stress field that is time or rate dependent. Elastic and plastic deformation mechanisms require a

change in the stress-state to produce a change in strain. Thus, they are generally evaluated without

consideration of the application rate or duration of the loading. Creep deformation, however, will

progress without the necessity of a change in the stress state [17]. Betten distinguishes creep from

elasticity and plasticity based upon the “rate-dependent” behavior of the creep deformation [18].

Creep deformation and damage at high stress occurs by way of the same dislocation

mechanisms that cause plastic deformation, namely slip and climb. At low stress and high

temperature creep deformation is governed by vacancy diffusion along grain boundaries or through

the crystal lattice. In contrast to plastic deformation, creep occurs at high temperatures, relative to

the material’s melting or liquidus temperature, due to the increase in kinetic vibrations at the

7

atomic level. This increase in internal energy allows for the movement of vacancies at lower stress

levels which may or may not be static [19, 17].

1.5 Reliability Considerations and Damage Models

Once the deformation mechanisms are modeled, this information can be implemented into the

design validation and structural reliability studies. The iterative process of choosing a material,

defining the material behavior, generating the design, and evaluating the reliability is commonly

referred to as the Design-For-Reliability method. A process map for an exemplar “design for

reliability” methodology is shown in Figure 1.5-1 [6].

Figure 1.5-1. Exemplar Design for Reliability process for solder joint characterization and design [6].

Aspects of the solder design process which need to be emphasized are frequently determined

using a Failure Modes Effect and Analysis (FMEA) process. This methodology involves a joint

8

assessment of the probability of occurrence, probability of timely detection, and overall impact of

all potential failure modes. This is done with particular consideration of the application loads,

application environment, and design life by consulting with an experienced team. The output tends

to abide by the Pareto Principle, whereby a subset of potential causes is determined to represent

the most likely root causes of impactful field issues [20]. This assessment will direct the practical

engineer in determining which types of damage and what damage models to consider when

evaluating the design. In the present document we will only seek to define the types of damage

which can occur due to the creep mechanisms in crystalline materials.

Creep rupture or fracture is the complete separation of a material subjected to creep

deformation. Creep rupture testing is used to define the amount of time that a specimen can endure

a constant load or constant stress before it fractures. This is the simplest type of creep test and the

creep rupture failure criterion is generally considered to be relevant to situations where shorter

design lives are being evaluated [19].

Excessive deformation is another simple failure criterion considered in the design of materials

subject to creep deformation. It is easy to see how this could present an issue in the turbine power

generation industry, where tight tolerances are required to ensure maximum efficiency. Over time,

the blades are subjected to a variety of loads, including centrifugal loads, and high temperature.

Creep deformation can lead to a loss of efficiency, a loss of power output, and potentially

dangerous failure. Therefore, the effect of creep over long time periods is assessed and compared

against a maximum allowable strain or deformation in the application [17, 19].

Fatigue is another failure consideration for the design of solder joints that must consider creep

behavior. Fatigue damage occurs from repetitive plastic strain cycles. At low stress levels the

deformation is localized and can be approximated using stress analysis. This is commonly referred

9

to as high cycle fatigue. At high stress levels macro level plasticity must be considered which

causes low cycle fatigue. The combined effect of high and low cycle fatigue is evaluated using

the Manson-Coffin relationship, which is shown in Equation 1.5-1. Similarly, Knecht and Fox

have proposed a fatigue damage model for cyclic creep damage by considering creep strain cycles

according to Equation 1.5-2. These models can be combined using the linear damage

accumulation model attributed to Palmgren and Miner, Equation 1.5-3 [6]. Alternatively,

Schubert and coworkers have proposed fatigue damage models which considers either the cyclic

equivalent creep strain or viscoplastic strain energy dissipation which abide by the Basquin

relationship employed by Manson and Coffin [21].

∆𝜀

2=

∆𝜀𝑒

2+

∆𝜀𝑝

2=

𝜎′𝑓

𝐸′(2𝑁𝑓)

𝑏+ 𝜀′𝑓(2𝑁𝑓)

𝑐 Equation 1.5-1 [22]

𝑁𝑓 – Number of Cycles to Fracture

∆𝜀𝑒 – Elastic Strain; ∆𝜀𝑝 – Plastic Strain 𝜎′𝑓

𝐸′ – High Cycle Fatigue Intercept; 𝜀′𝑓 – Low Cycle Fatigue Intercept

𝑏 – Fatigue Strength Exponent; 𝑐 – Fatigue Ductility Exponent

𝑁𝑓 =𝐶

∆𝛾𝑚𝑐 Equation 1.5-2 [6]

𝑁𝑓 – Number of Cycles to Fracture

𝐶 – Constant Related to Failure Criterion

∆𝛾𝑚𝑐 – Strain Range Due to Matrix Creep

1

𝑁𝑓=

1

𝑁𝑓,𝑓𝑎𝑡𝑖𝑔𝑢𝑒+

1

𝑁𝑓,𝑐𝑟𝑒𝑒𝑝

Equation 1.5-3 [6]

The first two failure criteria, or damage models, can be considered as stress problems. These

types of problems can frequently be analyzed using hand calculations. This is because the

deformation occurs in the bulk material and can be approximated using a similar technique to

elastic-plastic analysis. Excessive deformation can be evaluated by applying what are called

isochronous stress-strain curves, while creep fracture can be evaluated with rupture life data [23].

10

Fatigue damage models must account for permanent deformation on a highly localized level

which lead to fracture initiation and progression. Sophisticated finite element software is often

employed to perform fatigue analysis. Elastic modeling with Neuber correction or elastic-plastic

modeling is sufficient to provide the strain data used to perform Manson-Coffin fatigue analysis

[24]. However, elastic-plastic-creep or viscoplastic models must be developed to account for

cyclic creep damage [25].

In either case, the creep behavior needs to be understood in order to perform engineering

analysis according to the design for reliability methodology shown in Figure 1.5-1. Therefore, the

creep mechanism must be experimentally evaluated to create a valid material model to input into

either the hand calculations or advanced finite element analysis.

11

Chapter 2 Fundamental Evaluations of the Creep Mechanism

2.1 Macro Level Observations of Creep Behavior

Without understanding the mechanisms of creep on the microscopic and atomic scale, the

phenomenon can be observed on the macroscopic scale by way of a constant load or constant stress

creep test. This traditional creep test is performed on a tensile specimen which is subjected to

stress from a hanging mass. Examples of setups for constant load and constant stress testing are

shown in Figure 2.1-1 [23]. This test is performed at elevated temperature to determine either the

creep rupture life or the minimum strain rate. Creep rupture time is generally plotted against the

stress at various temperatures as in Figure 2.1-2 [19]. Additional instrumentation to measure strain

can be used to create isochronous stress-strain curves or determine the minimum strain rate.

Figure 2.1-1. Proposed creep test schematics for: a.) Constant Force and b.) Constant Stress [23].

12

Figure 2.1-2. Creep-Rupture plot showing the time to rupture vs. applied stress [19].

An exemplar strain vs. logarithmic time plot for a constant load creep test is shown in Figure

2.1-3 [18]. As illustrated, the creep phenomenon occurs in three stages. The first stage, primary

creep, is characterized by a decreasing rate of creep strain which follows the initial elastic-plastic

strain state, ε0, caused by the applied load. The secondary stage of creep is characterized by a

constant strain rate, which is the minimum strain rate, for most of the specimen life. The final

stage is characterized by an increase in strain rate until specimen fracture. The second stage of

creep is dominant for most of the specimen life and is therefore of most concern to the practical

engineer [23].

13

Figure 2.1-3. Typical creep curve divided into three stages [18].

The decrease in strain rate during the primary stageg is indicative of strain hardening which

occurs due to an increase in dislocation density. The increase in dislocations will tend to retard

dislocation motion and is thought to be involved in subgrain formation at high temperature. In the

secondary stage the recovery processes, due to cross-slip and edge dislocation climb, are

dynamically balanced with the strain hardening effects. In the final stage prior to rupture the

hardening and recovery process again become imbalanced. This observed increase in strain rate

is due to the combined effects of metallurgical instabilities and microstructural changes. These

include necking, corrosion, intercrystalline fracture, microvoid formation, brittle particle

precipitation, and recrystallization [19].

An alternative graphical representation is used in the special case of creep testing performed

at a constant strain rate. The curve in Figure 2.1-4 is plotted in stress-strain space with the regions

of elasticity, primary creep, secondary creep, and tertiary creep identified. This exemplar stress-

strain curve is particularly valid when the true strain rate is controlled as a constant, where the

14

theoretical steady state stress in region two is constant [26]. Stage I is characterized by the

phenomena of strain hardening outlined above while Stage III is dominated by metallurgical

instabilities and necking. Creep analysis for constitutive modeling will focus on the behavior in

Stage II where the deformation is stabilized and the stress state is relatively constant [26].

Figure 2.1-4. Exemplar Stress-Strain Diagram for a CSR Creep Test.

2.2 Microscopic Evaluation of Creep Mechanisms.

Creep is generally differentiated into two different prevailing mechanisms, dislocation and

diffusion. Dislocation creep tends to dominate at lower temperature and greater stress. Diffusional

creep is enabled by thermal energy at the atomic level, however, and is dominant at higher

temperatures under lower stress states. Creep processes are usually modeled with an Arrhenius

thermal activation energy term, which will be discussed later. Figure 2.2-1 shows the activation

energy of the dominant deformation mechanism plotted against the test temperature for aluminum

with controlled processing [17]. Dislocation and diffusion creep processes can be further

subdivided as discussed below.

15

Figure 2.2-1. Prevailing activation energy of deformation vs. temperature for single crystal aluminum [17].

Dislocation creep is controlled by the same mechanisms as plastic flow. However, the

increased thermal energy allows for easier motion of dislocations by slip and climb. In general,

slip is a lower energy transport mechanism, but which may require iterations of climb to overcome

obstacles to motion. For this reason, slip dominates in low temperature, high stress creep.

Alternatively, the dislocation climb mechanism can occur independently at higher temperature.

This distinction is sometimes used to differentiate between low and high temperature dislocation

creep, both of which tend to occur at higher stress levels.

Diffusion creep occurs when vacancies diffuse along the grain boundaries or across the grains

themselves. The governing mechanism is controlled based upon temperature and grain size.

Diffusion along grain boundaries, called Coble Creep, requires a lower thermal activation energy

and therefore dominates at lower temperature levels. Additionally, grain boundary diffusion is

preferential in fine grain materials where grain boundaries are abundantly available. At higher

16

temperatures and in coarse grain materials, Herring-Nebarro creep occurs as vacancies traverse

through the body of the grain [19].

In diffusional creep, the material voids tend to migrate from grain boundaries under tensile

stress to grain boundaries under a net compressive stress. A corresponding mass of atoms must

then migrate in the opposite direction. This phenomenon is schematically shown in Figure 2.2-2,

and the differentiation can occur due to a multiaxial stress state or simply due to the Poisson Effect

[27]. This migration causes the grains to elongate and promotes separation. Separation is

prevented as the elongation is accommodated by the effect of grain boundary sliding [19].

Figure 2.2-2. Schematic of dislocation migration in a grain under uniaxial tension [27].

2.3 Deformation-Mechanism Maps

In 1972 Ashby proposed a lucid graphical method for determining the dominaεting mechanism

for deformation based upon the temperature and applied stress. The so-called deformation

mechanism map is segmented into fields which indicate the primary mechanism for deformation

of a material in normalized temperature-stress space. The process map for lead, originally

published in this seminal article, is shown in Figure 2.3-1 [28]. Above the theoretical shear stress

17

is “defect-less flow”, which is the shear strength of a perfect crystal. This is only mathematically

valid as real-world materials possess defects which result in a much lower observed strength [17].

The field labelled “Dislocation Glide” represents primary plastic flow which is controlled by

dislocation glide with climbs to avoid obstacles. This field is also commonly referred to as

“Power-Law Breakdown”. The dislocation creep and diffusional flow fields are of particular

importance to this study, as they represent the two primary creep mechanisms [28].

Figure 2.3-1. Proposed Deformation-Mechanism Map for lead developed by Ashby [28].

Sometimes called process maps, these plots have been improved and updated to include

additional sub-classifications of deformation for completeness, as originally proposed by Ashby

18

[28]. These additional sub-classifications of the deformation map include core diffusion, grain

boundary sliding, and dynamic recrystallization. Sometimes they are represented as individual

fields, as in Figure 2.3-2 or they are shown as shaded areas which exists within the previously

defined fields [29]. Some experimentalists have added iso-lines to represent a constant creep strain

rate throughout the map [19].

Figure 2.3-2. Nickel Deformation Processing Map with added Grain Boundary Sliding field [29].

2.4 Quantifying and Modeling Creep

In order to properly assess the effects of creep it must be quantitatively defined. This can be

done empirically with data tables and curves or mathematically by determining a constitutive

equation. Hand calculations, such as those used for creep-rupture and excessive deformation

evaluations, can be assessed using a rupture-life curve or an isochronous stress-strain curve,

respectively. The methodology of deriving an isochronous stress-strain curve is shown in Figure

19

2.4-1 [23]. Whenever complex geometry or fatigue are considered a constitutive equation must

be derived for use in computational modeling as was done by Lai and Wang and Tseng et al. [25,

30].

Figure 2.4-1. Multiple strain vs time curves (a) are used to make isochronous stress-strain curves (b) [23].

A great many constitutive equations for quantifying creep have been developed in the last half

century. Some have limited applicability and some have purely empirical derivations. Two of the

most commonly applied creep models are the Power-Law and Garofalo creep models [19, 17].

Power-Law is preferred when it is applicable because of the simplicity of the model, but the

robustness of the Garofalo model is necessary to accommodate a wider range of temperatures and

stresses.

Weertman originally derived the Power-Law model to explain diffusional creep at high

temperatures and low stresses [17]. This equation includes a power law relationship to stress and

an Arrhenius activation energy term. It is shown, for uniaxial tension, in Equation 2.4-1. In the

equation the coefficient A, stress exponent, n, and activation energy, Q, are constants to be

experimentally determined. It is believed that these constants derive from combinations of the

following: crystal structure, material diffusivity, deformation mechanism, grain size, elastic

20

modulus, and presence of intermetallic or secondary phases [19]. At temperatures near the melting

point and with low stresses the stress exponent n can be approximated to unity [19].

𝜀̇ = 𝐴𝜎𝑛𝑒−𝑄𝑅𝑇

Equation 2.4-1 [17]

The power law relationship holds for intermediate to high stresses. However, at higher stresses

Shoeck has shown that the strain rate is exponentially proportional to the stress. He used this to

empirically define Equation 2.4-2. This was experimentally shown in Aluminum tested at high

temperatures and high stress levels [17]. Weertman has theorized that the exponential creep at

high stress is due to accelerated diffusion caused by excessive vacancy concentration brought on

by interactions between dislocations [19].

𝜀̇ = 𝐴𝑒𝛼𝜎𝑒−𝑄𝑅𝑇

Equation 2.4-2 [17]

With observations of both Power-Law and exponential creep, Garofalo proposed a combined

equation that simplifies to each relationship at its extreme ends. The equation which utilizes a

hyperbolic sine function is shown in Equation 2.4-3. At lower stress levels the function simplifies

to Power-Law creep, while at higher levels it simplifies to exponential creep. It has been proposed

that these transitions occur when the argument, ασ, is less than 0.8 or greater than 1.2 [19].

𝜀̇ = 𝐴[𝑠𝑖𝑛ℎ 𝛼𝜎]𝑛𝑒−𝑄𝑅𝑇 Equation 2.4-3 [19, 31]

A hypothetical process was modeled and described by Reed-Hill that seeks to explain,

mathematically, why the strain-rate has a hyperbolic sine relationship to stress and an exponential

relationship with temperature. The equilibrium state of vacancy formation and frequency of

vacancy motion have been defined by the work required to form or transport a vacancy. The energy

barrier to motion, q, must be overcome for a vacancy to move, and the frequency of motion is

defined by an Arrhenius relationship in Equation 2.4-4 [17].

21

𝑓𝑣 = 𝐴𝑒−𝑞

𝑘𝑇

𝑓𝑣 – Frequency of vacancy motion (moves/second)

𝐴 – A constant; material and mechanism specific

𝑞 – The energy barrier to motion (Joules)

𝑘 – Boltzmann’s Constant (Joules/Kelvin)

𝑇 – Absolute Temperature (Kelvin)

Equation 2.4-4 [17]

While this is true for random vacancy motion, it was shown in Figure 2.2-2 that an applied

stress will cause directional motion of the vacancies. To explain this, a hypothetical creep process

has been proposed and explained in Figure 2.4-2. In the first image there is no applied stress and

the energy barrier from state A to state C is the same as it is from C to A. The second image shows

the as stressed state which shows a directional change in the energy barrier which is then extended

in the final image. In this case the frequency of vacancy motion from A to C will be different from

the frequency of motion from C to A as defined in Equation 2.4-7 [17]. The frequency of

preferential, net motion will be the difference between the frequencies as shown in Equation 2.4-7.

The portion of the energy barrier defined by 𝑤 can be considered proportional to the applied

stress and the bracketed terms in Equation 2.4-7 can be reduced into a hyperbolic sine function.

This process is happening simultaneously throughout the material and not just at a single vacancy,

so Avogadro’s Number can be applied to consider a mole of vacancies. Since a corresponding

mass of atoms will move in the opposite direction, the net effect is creep strain that is proportional

to the net frequency of motion. By combining all of the proportionality constants and simplifying

we arrive at Equation 2.4-7, which is quite similar to the Garofalo equation defined above.

22

Figure 2.4-2. Effect of stress on the energy barriers for vacancy motion in a hypothetical creep mechanism [17].

𝑓𝐴−𝐶 = 𝐴𝑒−(𝑞−𝑤)

𝑘𝑇

𝑓𝐶−𝐴 = 𝐴𝑒−(𝑞+𝑤)

𝑘𝑇

𝑓𝑛𝑒𝑡 = 𝑓𝐴−𝐶 − 𝑓𝐶−𝐴 = 𝐴𝑒−𝑞

𝑘𝑇 [𝑒𝑤

𝑘𝑇 − 𝑒−𝑤

𝑘𝑇 ]

𝜀̇ = 𝐴 sinh (𝛼𝜎

𝑅𝑇) 𝑒

−𝑄

𝑅𝑇

Equation 2.4-5 [17]

Equation 2.4-6 [17]

Equation 2.4-7

23

Diffusional creep deformation has a greater activation energy requirement than dislocation

creep. Because the Arrhenius term is only dependent upon the activation energy and temperature,

the coefficient for diffusional creep must be orders of magnitude greater. When evaluating a

material across multiple modes of creep it is expected that the activation energies and coefficients

should increase for higher temperatures.

2.5 Power-Law and Garofalo Parameter Estimation for CSR Testing

Much of the historical testing for creep modeling has been done using constant load tensile

testing due to the simplicity of the apparatus, as shown above. Alternative loading schemes, such

as indentation, have become more common for estimation of the stress exponent and activation

energy [32, 33, 34, 35]. Most recently, with advances in machine controllers, strain-rate controlled

testing has been implemented in mechanical creep testing. Constant load testing requires the

experimentalist to wait through the primary creep stage until steady state creep is reached and

confirmed. Constant strain rate (CSR) testing has advantages because the minimum creep rate is

achieved by test design. This presents a considerable time savings when time-to-rupture data is

not required and can allow for multiple valid tests of the same specimen with greater consistency

[33].

The methodologies presented in this paper are valid for constant strain rate tensile testing. To

complete this evaluation, it is necessary to test the material at multiple strain rates and

temperatures. The test parameters should be determined so that they will be applicable to the

design environment for the application. The data acquisition must acquire load and strain data

throughout the test as it will be used for evaluation. The constant stress portion of the curve or

maximum stress is used to determine the creep flow stress which, combined with the test

temperature and strain-rate, will be used for parametric evaluation of the constitutive model.

24

Power-Law parameter estimation is to be done in three steps. First, the Power-Law equation

is linearized by taking the natural logarithm of both sides. It is first rearranged into slope intercept

form for the estimation of the stress exponent, as shown in Equation 2.5-1. The data sets are split

by the test temperature and plotted individually. Linear regression is used to find the slope, n, for

each curve. A similar rearrangement can be done to solve for the activation energy, Q (Equation

2.5-2). In this case the data sets are plotted for each strain rate tested, and the slope is again

computed. These plots, used to calculate the activation energy, are commonly referred to as

Arrhenius plots. Once n and Q are known, the linear regression from Equation 2.5-1 can be used

to calculate A based upon the intercept of the regression model.

Equation 2.5-1

Equation 2.5-2

The process of linearization is also used for Garofalo parameter estimation. However, in the

case of the Garofalo equation, the extreme cases of exponential and Power-Law creep are required

to accurately estimate the parameters n and α. The mathematical derivation for Power-Law creep

is based on the Taylor series for the hyperbolic sine function. In Equation 2.5-3 the higher order

terms are ignored because they approximate zero for small values of ασ. This process is repeated

to simplify the equation to exponential creep at high stress. The negative exponential term in

Equation 2.5-4 will approach zero for large values of ασ. These approximations will be used with

linear regression to estimate the parameters n and α.

25

Equation 2.5-3

Equation 2.5-4

As with the Power-Law, Equation 2.4-3, Equation 2.5-3, and Equation 2.5-4 will be linearized

using the natural logarithm and rearranged into slope-intercept form. In order to calculate the

exponent, n, we will use the linearized power law equation, which is shown in Equation 2.5-5.

Secondly, nα is calculated using Equation 2.5-6, which is the linearization of the exponential

region of the curve. A lucid interpretation of this process is shown in Figure 2.5-1Figure 1.2-1

[31]. For strain rate testing, Pernis has suggested using the high temperature data to estimate n

and the low temperature data to estimate nα since the temperature and stress vary inversely [36].

Equation 2.5-5

Equation 2.5-6

26

Figure 2.5-1. Strain-rate and stress plots for austenitic stainless steel used for Garofalo Parameter Estimation.

Arrhenius plots are generated for the linearization of the Garofalo equation as shown Equation

2.5-7. The Zener-Hollomon parameter, Z, is introduced for to facilitate the calculation of the

coefficient, A. Also called the temperature corrected strain rate, Z is shown for the Garofalo

model in Equation 2.5-8 and in linearized form in Equation 2.5-9. Linear regression models are

then used to find the slope and intercept of these equations to determine the activation energy and

coefficient terms [31, 36].

Equation 2.5-7

Equation 2.5-8

Equation 2.5-9

Alternatively, advanced computational algorithms have been developed for surface modeling

of custom input equations. For example, the Curve Fitting Toolbox in MATLAB can be used to

estimate the parameters of the power-law and Garofalo equations by modeling a 3D surface in

27

stress-time-strain space given the coordinates of the test points and the form of the constitutive

equation. This method, which relies on numeric optimization (by maximizing the coefficient of

determination), will be used as a point of comparison to the traditional, iterative methodologies

outlined above.

28

Chapter 3 Experimental Setup and Procedure

3.1 Testing Equipment

Testing was conducted at Wright State University using an Instron Universal Test Machine

(system ID 4505C0502). An Instron 2630-109 extensometer was used to monitor strain and an

Instron 5 kN load cell was used to measure the force on the specimen. The extensometer was

calibrated and verified using DoALL A+ grade gauge blocks (Set 86-R, Serial 7701) through its

full range. This calibration was performed within the Bluehill test software program using the

Calibration Wizard.

Temperatures above ambient were controlled and monitored using an Astro Industries 12K25

temperature controller using a type K thermocouple and clamshell furnace. Temperatures below

ambient were controlled and monitored using an Omega CSC32 temperature controller using a

type T thermocouple, a liquid nitrogen valve, and a pressurized nitrogen dewar. The clamshell

furnace was used for sub-ambient testing for insulation with none of the heating elements

energized. To allow for the inclusion of the extensometer, wedges of fire brick and loose fiberglass

insulation were included to wedge open the furnace and insulate the test environment. This setup

for sub-ambient testing is shown in Figure 3.1-1.

29

Figure 3.1-1. Test setup for sub-ambient tensile test.

Particular evaluation of the extensometer measurement was performed because the specimen

was subjected to temperatures in the range of -20 to 175 Celsius. A rod and tube extensometer

was initially evaluated, to protect the measurement device. This setup presented particular

difficulties due to the softness of the solder alloy specimens. Ultimately, it was discovered that

the 2630-100 extensometer was rated for operation through the entire temperature range with

negligible effects on the gauge length or sensitivity [37].

30

3.2 Test Samples

Rods of diameter 0.500” with a length of 12” of Indalloy 236 (83Pb 10SB 5SN 2AG, Lot:

BF9014) and Indalloy 264 (91.5SN 8.5SB, Lot: BF9184) were used to generate tensile test

specimens which would be compatible with the Instron test machine. Specimens were machined

into round tensile specimens with a 1” gauge length with a 0.25” diameter. The grip section was

machined with ½-13 UNC threads, and the fillet from the grip section to the reduced section was

specified to a minimum of radius of 5/16”. A generic specimen diagram is shown in Figure 1.

Surface roughness (Ra) measurements on the reduced section of every eighth specimen were taken

which resulted in a mean roughness of 10 µin and a standard deviation of 3 µin on a sample size

of 10.

Figure 3.2-1. Generic test specimen diagram.

3.3 Testing Parameters

Testing was conducted at three temperatures and four strain rates. Table 3.3-1 details the test

matrix of required test conditions. All tests were to be conducted with three replicates at each test

point (with some exceptions due to test machine and data acquisition errors). This resulted in a

projected total sample size of 72 specimens (two materials, three replicates, three temperatures,

and four strain rates). Testing at high strain-rate was to be used to evaluate the elastic-plastic

31

behavior of the material. The high strain-rate testing is used to minimize the effect of creep and

recovery processes, which require time to manifest. Lower strain-rate tests would illuminate the

creep behavior by maintaining the strain-rate and evaluating the relatively constant stress over

time.

Table 3.3-1. Proposed test matrix for tensile testing.

Testing was initially conducted until specimen fracture. However, it was decided that tests

would be terminated at 50% extension or fracture, whichever came first. Because the creep and

relaxation behavior are time dependent, it was not practical to remove the extensometer and

continue testing. Also, the extensometer was physically limited at the ends of its range to prevent

damage to the instrument. Therefore, without removing the extensometer, the knife edges used to

grip the specimen would drag across the specimen surface. Since the extensometer was not capable

of measuring strains in excess of 50% the strain in excess of 50% would need to be correlated

using cross-head extension assuming all strain occurred in the reduced section. The required

assumptions to make the correlation would not produce any additional insight into the stage II

creep model, and thus the correlation was not quantitatively evaluated.

Data was collected using the Instron Universal Test System through the Bluehill software suite.

Data collected at a minimum frequency of 1 kHz for high strain rate and a frequency no less than

32

two orders of magnitude greater than the strain rate (the strain rate in (in/in)/s reduces to s-1 which

is consistent with the unit Hertz). The engineering stress was calculated by dividing the measured

force by the specimen initial cross-sectional area and the extensometer directly measured the

engineering strain. Bluehill software calculated the true stress and strain according to Equation

3.3-1.

𝜖 = ln(1 + 𝜀), 𝜎𝑇 = 𝜎(1 + 𝜀) Equation 3.3-1

The approximations for true stress and true strain are not valid in the tertiary stage of the curve

shown in Figure 2.1-4 because of the onset of necking. The Poisson effect has a profound effect

on the stress state of the necked portion and creates a triaxial stress state. Details of the axial stress

state are shown in Figure 3.3-1, and the stress state is a function of the local radius of the neck

[31]. An accurate approximation of the stress can be found using the Bridgman relation [31, 19].

At high temperatures and strain-rates a diffuse neck of large radius can occur which is emblematic

of superplasticity, which is observed in some materials. This detailed analysis was not carried out

as the mechanics of monotonic material rupture were not being evaluated.

33

Figure 3.3-1. Axial stress distribution in a necked specimen [31].

3.4 Test Procedure

The test specimen was inserted into the threaded hole on the upper tensile fixture bar which

was pinned to the load cell. The load on the test frame was zeroed after the installation of the test

specimen. The lower fixture bar was then threaded into the lower portion of the test specimen.

The crosshead was manually raised until the lower fixture bar could be pinned to the crosshead.

Upon pinning the specimen and fixture bars, the extensometer was inserted onto the specimen

gauge section at a gauge length of one inch. Zero extension was set using the mechanical zeroing

feature on the extensometer. The extensometer used knife edges and a spring clip to grip the

specimen.

Upon completion of the specimen installation, the oven was closed against the fire brick

wedges. All cavities were insulated with loose fiberglass to reduce heat transfer between the test

chamber and ambient environment. At this point, the temperature control system was energized.

Sufficient time for specimen and test chamber equilibrium was around forty-five minutes for the

34

initial test and fifteen minutes for subsequent tests. Test temperature at the surface of the grip was

monitored with a second digital temperature meter utilizing a K-type thermocouple to ensure

equilibrium. The baseline of zero strain was reestablished upon reaching thermal equilibrium to

account for any thermal expansion.

Bluehill software was used to communicate with the controller for the conduction of the test

and the acquisition of discrete data. For the duration of the test, the software relied upon PID

feedback control to match the measured time signal of strain to the expected strain signal. The test

continued until specimen fracture or 50% strain, whichever came first.

The PID control loop parameters were tuned on the onset of testing to ensure that they would

be sufficient for control at all strain rates. Upon observation of multiple unloading and reloading

cycles at low strain rate, the potential causes and effects were weighed against the results. It was

concluded that the discrete nature of the measurement and control systems would inherently cause

the signals to deviate from a purely linear signal. Data would be accepted if there was sufficient

time of near constant stress at the strain rate, which tended toward linear over longer time intervals.

35

Chapter 4 Test Results

4.1 Creep Testing Results

To perform proper constitutive modeling the important parameters of the data points were the

true stress, true strain-rate, and absolute specimen temperature. The strain-rate and absolute

specimen temperature were achieved by test design as detailed in the test matrix in Table 3.3-1

The applicable true stress was determined from the stress-strain curve. For low strain-rate testing

(≤0.001 s-1) the average stress during the steady state period was recorded for data analysis. At

high strain-rate testing (0.1 s-1) there was no prolonged steady state stress. Therefore, the

maximum stress was recorded for data analysis, which is consistent with the method used by

Rudolf Pernis [36]. The sample data used for the detailed analysis is catalogued in Table 4.1-1.

Cells corresponding to omitted data include a brief explanation detailing the reason for the

omission. Some cells are blank due to the specimen shortage which was caused by prior test faults.

Data at low strain-rate was often observed to show cyclic loading behavior in the steady state range

due to the discrete PID controller. Samples which exhibited large levels of variation were omitted

because the oscillations in the strain rate were large relative to the average. The time range used

to average the stress is indicated above the stress value for low-strain rate samples.

36

Table 4.1-1. Summary of stress data used for creep analysis.

The important data points can be observed in both stress vs. strain and stress vs. time plots,

from which the data in Table 4.1-1 was extracted. The stress vs. strain and stress vs. time plots for

Indalloy 236, the leaded alloy, are shown in Figures 4.1-1 through 4.1-4. Similarly, Figures 4.1-5

through 4.1-8 show these curves for Indalloy 264, the lead-free alternative. The figures are

arranged by decreasing strain rate and each summarizes the testing at the specified strain rate.

These plots were generated in Microsoft Excel, which was also used for iterative modeling of the

linearized constitutive equations for parameter estimation.

37

Figure 4.1-1. Indalloy 236 test plots for 10-1 s-1 strain-rate.

38

Figure 4.1-2. Indalloy 236 test plots for 10-3 s-1 strain-rate.

39

Figure 4.1-3. Indalloy 236 test plots for 10-4 s-1 strain-rate.

40

Figure 4.1-4. Indalloy 236 test plots for 10-5 s-1 strain-rate.

41

Figure 4.1-5. Indalloy 264 test plots for 10-1 s-1 strain-rate.

42

Figure 4.1-6. Indalloy 264 test plots for 10-3 s-1 strain-rate.

43

Figure 4.1-7. Indalloy 264 test plots for 10-4 s-1 strain-rate.

44

Figure 4.1-8. Indalloy 264 test plots for 10-5 s-1 strain-rate.

45

4.2 Data Analysis – Power-Law

Power-law creep was first assessed individually at each test temperature. This analysis is used

to determine a specific stress exponent, n, and intercept, A, derived for each alloy at a given test

temperature. The iterative process of determining power law creep parameters outlined above is

graphically summarized in Figure 4.2-1 and Figure 4.2-2 for Indalloy 236 (leaded) and Indalloy

264 (lead-free), respectively. The final complete curve fits in stress and strain-rate space are shown

in Figure 4.2-3. The model parameters and coefficients of determination of the curve fits are also

summarized.

Figure 4.2-1. Power-Law parameter determination for Indalloy 236, individual models.

46

Figure 4.2-2. Power-Law parameter determination for Indalloy 264, individual models.

47

Figure 4.2-3. Fit and parameter comparisons for Power-Law, individual modeling.

Subsequently, the Power-Law model was fitted as a common model for each material,

applicable for all temperatures. The iterative process of determining Power-Law creep parameters

was repeated, but the slope and intercept were averaged. The process is summarized in Figure

4.2-4 and Figure 4.2-2 for Indalloy 236 (leaded) and Indalloy 264 (lead-free), respectively. The

final complete curve fits in stress and strain-rate space are shown in Figure 4.2-6 with the

applicable curve fitting parameters. The Curve-Fitting Toolbox in MATLAB was used to surface

model the Power-Law equation. The output was another singular model, which was determined

using optimization tools. The summary of these curve fits, in the same manner, can be seen in

Figure 4.2-7. An alternative visualization shows the comparison of the data points (blue) against

the surface plot of the constitutive model plotted in MATLAB. This is shown in for the Indalloy

236 in Figure 4.2-8 and Indalloy 264 in Figure 4.2-9.

48

Figure 4.2-4. Power-Law parameter determination for Indalloy 236, combined model.

Figure 4.2-5. Power-Law parameter determination for Indalloy 264, combined model.

49

Figure 4.2-6. Fit and parameter comparisons for Power-Law, combined model.

Figure 4.2-7. Fit and parameter comparisons for Power-Law, MATLAB model.

50

Figure 4.2-8. 3D surface fitting of Power-Law Equation for Indalloy 236.

51

Figure 4.2-9. 3D surface fitting of Power-Law Equation for Indalloy 264.

52

4.3 Data Analysis – Garofalo

The same exercise of curve fitting performed for the combined Power-Law model can be completed for the

Garofalo estimation using the linearization methods outlined in Section 2.5. For iterative modeling the process

detailed by Rudolf Pernis was used to determine n and α. The data at maximum temperature is used to represent the

limiting case of low stress (Equation 2.5-5), and the data at minimum temperature is used to represent the limiting

case of high stress (Equation 2.5-6). Therefore, the slope of the logarithm of strain rate vs the logarithm of stress at

maximum temperature represents n. Similarly, the slope of the logarithm of strain rate vs. the stress at minimum

temperature represents nα. Slopes of the Arrhenius plots were used to determine the activation energies and Zener-

Hollomon plots were used to determine the coefficients. This process is detailed in

Figure 4.3-1 and Figure 4.3-2 for Indalloy 236 (leaded) and Indalloy 264 (lead-free),

respectively. These curve fits in stress and strain-rate space are detailed in Figure 4.3-3. Similar

fitting from MATLAB is detailed in Figure 4.3-4. The 3D visualizations for the leaded and lead-

free solder are shown in Figure 4.3-5 and Figure 4.3-6, respectively.

53

Figure 4.3-1. Garofalo parameter determination for Indalloy 236.

54

Figure 4.3-2. Garofalo parameter determination for Indalloy 264.

55

Figure 4.3-3: Fit and parameter comparisons for Garofalo models from iterative analysis

Figure 4.3-4. Fit and parameter comparisons for Garofalo models from MATLAB.

56

Figure 4.3-5. 3D Surface fitting of Garofalo Equation for Indalloy 236.

57

Figure 4.3-6. 3D Surface fitting of Garofalo Equation for Indalloy 264.

58

Chapter 5 Discussion

5.1 Microstructure

Before performing physical microstructure analysis, a study began into the phase composition

and formation for these solder alloys. The quaternary alloy contains lead, tin, antimony, and silver

and is most lucidly explained using graphs of the equilibrium phase composition plotted against

temperature. Because the tin and antimony alloy is binary, it could be evaluated either with the

with equilibrium phase composition graphs or using the binary phase diagram.

Figure 5.1-1 shows two renderings of the Sn-Sb phase diagram from different authors which

are not consistent. The diagram on the left predicts Sb2Sn3 over a broad range of composition and

temperature and does not predict the formation of SbSn [38]. The diagram on the right predicts

Sb2Sn3 in a narrower region with SbSn existing in a large range [39]. The diagrams were formed

using different assumptions and both authors attempted to address the variation. The Sb2Sn3

structure is not completely defined, as no crystal structure was proposed by either author.

While it is possible to use tie-lines to evaluate the binary alloy, this will not be done for two

reasons. First, as discussed above, there is not a consensus on the stoichiometry of the intermetallic

phases and their positions in the tin-antimony phase-diagram. The second reason is that the

quaternary alloy cannot be evaluated in the same way. To accurately compare the microstructure

predictions both Indalloy 236 and Indalloy 264 should be compared using the same type of

analysis.

59

Figure 5.1-1. Phase diagrams for Sn-Sb binary system (dashed line for Indalloy 264). Left: [38]. Right: [39].

To predict the composition of the alloys with a consistent method, Thermo-Calc software was

used. The chosen modules in the software calculated the phase composition at thermodynamic

equilibrium and did not consider reaction kinetics when predicting phases. The software was used

to approximate the volume and mass of each phase that should be present in the alloys and to

evaluate the elemental composition of the present phases. The phase composition vs temperature

graphs for the two alloys are shown in Figure 5.1-2 and Figure 5.1-3. These graphs show the

calculated phase composition at thermodynamic equilibrium for one mole of the alloys within a

relevant temperature range. The Thermo-Calc predictions are discussed below.

60

Figure 5.1-2. Equilibrium phase composition for Indalloy 236.

Figure 5.1-3. Equilibrium phase composition for Indalloy 264.

61

For Indalloy 236 the first phase that begins to solidify is Ag3Sn followed by the FCC phase

and, subsequently, the SbSn and Rhombohedral phase. The Ag3Sn phase also includes Sb, which

diminishes with decreasing temperature. The FCC phase is almost entirely Pb with traces of all

three other elements. The SbSn phase is primarily composed of SbSn in the same structure as

sodium chloride, but also contains dissolved Pb. Lastly, the rhombohedral phase is almost entirely

composed of Sb with dissolved Pb, Sn, and Ag atoms. Here the primary matrix is FCC Pb which

will contain intermetallic phases Ag3Sn, SnSb, and rhombohedral Sb. The intermetallics are

predicted to represent 25 - 30% of the volume of the alloy after solidification.

Thermo-Calc software predicts that Indalloy 264 will first solidify as BCT Sn (β-Sn) with

dissolved Sb atoms. As the temperature decreases the SbSn phase begins to form as an

intermetallic, with a varying volume concentration of 2-19%. The software predicts that the

primary matrix will suffer from “tin-sickness” and transform into gray tin (diamond lattice

structure) at about 15.5 Celsius. It has been shown that the presence of the alloying element, Sb,

can inhibit this formation with concentrations as low as .015% by mass [40]. Therefore, it is

unlikely that any appreciable amount of gray tin will form even at long hold times at low

temperature.

Noting the inclusion of SbSn and the absence of Sb2Sn3, the results are comparable to what

would be predicted using the second phase diagram from Figure 5.1-1. As it relates to gray tin,

the phase diagram does not include temperatures low enough to compare with the Thermo-Calc

prediction. The other phase diagram in Figure 5.1-1 varies greatly from the software predictions.

Micrographs were taken on specimens which were etched with 5% nital. The as-received

material was evaluated in the base plane, normal to the axis of the cylinder, and an orthogonal

plane which is parallel to the cylinder’s axis. This is shown graphically in Figure 5.1-4. Post-test

62

specimens were only evaluated in the base plane in the reduced section near the neck. Specimens

were cut with a hacksaw and polished. Polish included 320 grit silicon-carbide sand paper,

diamond suspension (six, three, and one µm diameter), and 0.05 µm alumina powder in a colloidal

suspension.

Figure 5.1-4. Planar definitions of metallurgical specimen cross-sections.

As received specimens of the leaded and lead-free alloy were evaluated in orthogonal

orientations to evaluate any possible effects of drawing or extruding, which may have been part of

the material processing. In addition, it was decided that important specimens should be chosen for

each material to evaluated any microstructural changes that occurred due to testing. These

important specimens were chosen to be the highest strain-rate and lowest temperature specimens

as well as the lowest strain-rate and highest temperature specimens. The high strain-rate, low

temperature specimens were expected to have the least variation compared to the baseline.

Conversely, the low-strain rate, high temperature specimens would remain at elevated temperature

through the test which should allow for greater metallurgical changes.

The micrographs for the as received material are shown in Figure 5.1-5. The dark field is

believed to be a matrix of the primary alloying element (FCC Lead or BCT β-Sn, depending on

the alloy) while the brighter fields are believed to be the intermetallic compounds. Intermetallics

63

in the leaded alloy appear to be flakey or needle shaped while they are irregular-spheroidal in the

lead-free alloy. While the particles do not appear to show any elongation in the cross-sections

there is some preferential streaking of particles in the axial direction for both specimens. This

supports the hypothesis that the alloys were extruded or drawn into the 0.5-inch diameter bars that

were provided.

Figure 5.1-5. Micrographs of Indalloy 236 and Indalloy 264, as-received.

Subsequent micrographs were taken of the etched specimens after testing. These micrographs

appear in Figure 5.1-6 alongside the baseline for comparison. The most discernable change is

observed in the high-temperature specimens where the particles appear to have refined. This is

64

more apparent for irregular SbSn particles in the binary alloy than for the dendritic particles found

in the quaternary lead alloy.

Figure 5.1-6. Micrographs of Indalloy 236 and Indalloy 264, post-test.

65

5.2 Post-Test Evaluation and Fractography

Photographs were taken of specimens which had fractured at high and low strain rate. Leaded

specimens are shown in their post-test fractured state in Figure 5.2-1 and Figure 5.2-2 for high and

low strain-rate, respectively. Similarly, lead-free alloys are shown in Figure 5.2-3 and Figure

5.2-4. The bent specimen in Figure 5.2-2 was due to buckling and bending after fracture. This

occurred because of test control instability caused by stationary strain feedback.

The tin-antimony alloy exhibited more ductility at fracture in general, which is most noticeable

at low temperature. This also coincided with an orange-peel like surface morphology. At elevated

temperature and high strain rate the diffuse nature of the neck and surface morphology is more

pronounced.

It is worth noting that the soft specimens were susceptible to fracture at the knife edge

attachment points of the extensometer. This is likely induced by the transverse compression from

the spring-loaded grips. This biaxial stress state increases the shear stress locally as the cross-

section is in tension. This effect was more pronounced at low strain rate and high temperature

when the material strength is at its lowest.

66

Figure 5.2-1. Leaded solder specimen fractures after high strain-rate testing.

67

Figure 5.2-2. Leaded solder specimen fractures after low strain-rate testing.

68

Figure 5.2-3. Lead-free solder specimen fractures after high strain-rate testing.

69

Figure 5.2-4. Lead-free solder specimen fractures after low strain-rate testing.

70

5.3 Issues with Elastic Test Data

During testing it was observed that the apparent elastic modulus of the materials determined

from testing was not consistent within the test range and often showed a large degree of variation

at individual test conditions. Below is a list of some of the proposed causes of the discrepancies:

• Indentation of the extensometer knife-edge into the soft specimen will introduce some error

when the arms of the extensometer are not perpendicular to the specimen. This effect will

be more pronounced in the limited elastic range than in the expanded full strain range.

• The early onset of primary creep could be occurring before defined plasticity. This is

clearly a concern at high temperature where the observed stress is low and the propensity

for diffusional creep is increased.

• The strain-rate dependence of the elastic modulus for solder alloys has been proposed by

some authors [6]. The elastic range is small relative to the full creep curve and the

controller has not yet stabilized during its measurement. This would lead to variation

within and across measurements.

It was observed that the apparent elastic modulus at low strain-rates repeated during the cyclic

loading brought on by the settling of the PID controller. A zoomed view of the stress-strain curve

for one specimen is shown in Figure 5.3-1. This observation lends credence to the latter two

explanations, but the first should be increasingly more pronounced as the strain increases.

71

Figure 5.3-1. Repitition of apparent elastic modulus observed during reloading cycles.

5.4 Observations

One of the primary observations, prior to complete data analysis, is that the lead-free tin-

antimony alloy is more resistant to creep than the quaternary lead alloy in the test range. This is

apparent as the flow stress at all temperatures and strain-rates was equivalent or greater for the

lead-free alloy.

Interpolation within the test range should be valid assuming the rate-controlling mechanism is

the same, however extrapolation should be performed cautiously as the deformation mechanism

could change. Furthermore, the data range for strain and temperature is valid for many

applications. Extremely high strain-rates would only be relevant for similarly short design life and

a sufficiently low strain-rate can be defined for “infinite life” designs.

72

5.5 Comparisons to Published Data

The variables determined from the iterative analysis were compared against published data for

various materials. Comparison data spans strain-rate and load controlled tensile tests as well as

indentation and stress-relaxation testing. Power-Law data is widely available for solder alloys, but

many authors only evaluated individual parameters such as the stress exponent, n, or activation

energy, Q. Garofalo fits were compared against undoped and doped SAC solder as well as

aluminum and brass alloy. Comparisons for Power-Law curve fits are shown in Table 5.5-1, while

Garofalo data is summarized in Table 5.5-2.

Table 5.5-1. Comparison of parameter estimation from Power-Law analysis with literature.

Indalloy

236 Indalloy

264 SnAgCu

[40] SnAgCuCe

[40] Pure Sn

[41] Sn-2.9Sb

[41]

Temp (°C) -20, 80, 175 25, 50, 75, 100 23 - 175 23 - 175

Strain Range (in/in)/s

e-5 to e-1 e-5 to e-1 e-9 to e-3 e-9 to e-3 e-9 to e-2 e-9 to e-2

A(1/s) 1.00e-3 2.1 e-8 1.5 e-9 6.2 e-11 - -

n 10.692 9.647 5.3 6.1 8.6 4.7-11.2

Q (kJ/mol) 95.850 60.601 71 81 73 101-117

Sn-5.8Sb [41]

Sn-8.1Sb [41]

Sn-10Sb [42]

Sn-5Sb [43]

Sn-5Sb-3.5Ag [43]

Sn-5Sb-1.5.5Au [43]

Temp (°C) 23 - 175 23 - 175 23 25 - 130 25 - 130 25 - 130

Strain Range (in/in)/s

e-9 to e-2 e-9 to e-2 e-5 to e-2 e-6 to e-3 e-6 to e-3 e-6 to e-3

A(1/s) - - - - - -

n 5.3-11.6 5.6-9.8 5.9-10 4.0-7.0 4.5-7.7 6.4-11.0

Q (kJ/mol) 85-95 49-61 - 47.02 69.45 127.47

Sn-3.5Ag [44]

Sn-3.5Ag-0.5Cu [44]

Sn-3.5Ag-Cu

[44]

Sn-37Pb [45]

Sn-37Pb-Ag [45]

Sn-37Pb [46]

Temp (°C) 100 100 100 25 - 125 25 - 125 25 - 125

Strain Range (in/in)/s

e-9 to e-5 e-9 to e-5 e-9 to e-5 e-10 to e-4 e-10 to e-4 Relaxation

A(1/s) - - - - - 0.0138 and 268

n 4.2 6.0 3.9 – 7.6 3.39-4.95 3.32-3.75 3.2 and 6.2

Q (kJ/mol) - - - 45-57 63-67 68.8

73

Table 5.5-2. Comparison of parameter estimation from Garofalo analysis with literature.

Indalloy 236 Indalloy

264 SnAgCu

[40] SnAgCuCe

[40] Brass CuZn30

[36] 2014 Al +

Zr [41]

Temp (°C) -20, 80, 175 25, 50, 75, 100 650 - 850 300 - 500

Strain Range (in/in)/s

e-5 to e-1 e-9 to e-3 0.5 to 25 e-3 to 5

A(1/s) 1.08 e+7 8.13 e+5 3.25 e+5 2.84 e+5 3.22 e+10 -

α (1/MPa) 0.06728 0.03062 0.05217 0.02432 0.00927 0.05

n 4.459 5.891 5.3 6.1 4.37 avg 3

Q (kJ/mol) 83.603 57.582 48.221 53.21 180 143.4

Sn3.9Ag0.6Cu [50]

Sn37Pb [50]

SnAgCu (Various) [51]

Sn3.8Ag0.7Cu [52]

Temp (°C) 25, 75, 125 -50 to 150

Strain Range (in/in)/s

e-8 to e-4 e-8 to e-1 e-8 to e-3

A(1/s) 248.4 1999.4 2.780 e+5 3.2 e+4

α (1/MPa) 0.188 0.2 0.02447 0.037

n 3.788 2.108 6.41 5.3

Q (kJ/mol) 62.917 54.137 54.032 54.246

It is apparent that the values obtained in this study are consistent with the range of what was

found in literature. One primary observation is that introduction of elements that form

intermetallic compounds in solders as well as doping with small amounts of rare earth metals can

cause a significant improvement in the creep resistance of the solder alloys. A complete literature

review and combined plots should be used to determine the best alloys for the application.

74

The refinement of intermetallic particles observed in the microstructure of high temperature

specimens is consistent with what was observed by Beshai, et al. They observed that even when

testing at a single temperature, namely room temperature, the particle refinement introduced by

annealing correlated with a decrease in the Power-Law stress exponent [42]. This may indicate

that annealing could lead to more consistent creep behavior. Also, if elevated temperatures are

observed in service, the microstructure and potentially the creep behavior could change

permanently.

75

Chapter 6 Conclusions and Future Work

6.1 Conclusions

Comparing these solder alloys is relevant because of their similar melting temperature

properties, which makes them suitable for similar applications. From a material selection stand-

point the tin-antimony alloy appears to be a proper replacement for the leaded alloy when

considering creep-resistance, ductility, and engineering ethics. Further analysis of costs as well as

thermal and electrical properties is necessary for any practical conclusion in application.

Also, it is apparent that optimization algorithms tend to fit the data more closely than the

iterative process. However, the iterative process allows the practical engineer to evaluate the

results independently for each variable and better understand their contributions to the combined

model. Also, since the iterative process is dependent upon and defined by the physics of the

problem, it allows the engineer evaluate the veracity of the calculated values. When aided with

this knowledge, the realistic interpretation of the optimized solution can be made.

6.2 Contributions

The expansion of the evaluation and implementation of CSR test methodologies are unique

contributions from this work. Several comparisons were made to data derived from CSR, constant

load, and stress-relaxation testing. This work also expands the test envelope to include sub-

ambient temperatures in creep analysis, which was not commonly seen in the literature search. In

aerospace applications at high altitude, electronics systems are likely to be exposed to sub-ambient

temperatures. This information regarding the materials’ behavior at low temperature is likely

applicable to the stress analysis and reliability modeling of such avionics systems. Further, an in-

depth comparison of Power-Law and Garofalo creep was conducted. This information will aid

76

practical users who must balance the trade-offs in complexity and accuracy for creep testing and

modeling.

6.3 Future Work

The issues with variability in the elastic modulus were not isolated to this study. While the

measured variation at individual temperatures was stark, it has been proposed that the apparent

elastic modulus will vary with temperature and strain-rate for solder materials. Other methods

may be useful in determination of the elastic-plastic behavior of the material. For example, Pang

recommends using the Split-Hopkins pressure bar test (high strain-rate) to evaluate the relationship

between yield stress and strain-rate [6]. This test can also be used to determine the elastic modulus

in response to dynamic stress [40]. This test data could supplement the creep data acquired in this

study in order to form the full elastic-plastic-creep model for analysis.

Also, more controlled tests or post-test evaluation could be used to evaluate the exact

deformation mechanisms and elucidate the micromechanics governing the deformation. As an

example, the activation energy in Figure 2.2-1 was determined using controlled specimens where

slip was limited to two octahedral planes on single-crystal, pure aluminum [17]. Also, post-test

evaluation with Scanning Electron Microscopy could be used to evaluate the specimens after

testing. Evidence of grain boundary sliding or cavitation can be observed in such micrographs.

77

APPENDIX A: Certificate of Analysis for Provided Samples

78

79

APPENDIX B: Example MATLAB Script

% Garofalo Curve Fitting for Lead-Free Solder Alloy

clear all

close all

clc

z20=[0.00001;0.00001;0.0001;0.0001;0.0001;0.001;0.001;0.001;0.1;0.1;0.1];

x20=[52.01;47.96;59.92;58.83;58.96;75.07;76.46;67.45;97.28;101.49;96.93];

z80=[0.00001;0.00001;0.0001;0.0001;0.001;0.001;0.001;0.1;0.1;0.1];

x80=[20.13;22.77;30.78;27.39;36.54;37.89;41.32;59.94;59.24;60.89];

z175=[0.00001;0.00001;0.00001;0.0001;0.0001;0.0001;0.001;0.001;0.001;0.1;0.1;0.1];

x175=[7.55;7.42;8.41;13.15;12.35;15.11;17.57;19.48;18.53;38.53;35.00;39.02];

z20l=log(z20);

z80l=log(z80);

z175l=log(z175);

xall=[x20;x80;x175];

zall_l=[z20l;z80l;z175l];

Temp_20=zeros(length(x20),1);

Temp_80=zeros(length(x80),1);

Temp_175=zeros(length(x175),1);

Temp_20=Temp_20+253.15;

Temp_80=Temp_80+353.15;

Temp_175=Temp_175+448.15;

Temp=[Temp_20;Temp_80;Temp_175];

80

%% Wide Open Garofalo

f_matlab=fit([xall,Temp],zall_l,'A+B*log(sinh(C*x))-

D/(8.314*y)','Lower',[1,1,.0001,1000],'Upper',[50,20,.4,800000])

% figure

% plot(f_matlab,[xall,Temp],zall_l)

%% The Following values for Z_matlab variables was added after analysis

[X,Y]=meshgrid(7:3:100,253.15:3:448.15);

Z_matlab=7.797+4.496.*log(sinh(0.04321.*X))+(-54530./(8.314.*Y));

figure

surf(X,Y,Z_matlab,'EdgeColor','black');

hold

scatter3(xall,Temp,zall_l,'MarkerEdgeColor','white','MarkerFaceColor','blue');

[X1,Y1]=meshgrid(7:3:100,253.15:3:448.15);

Z_excel=11.30650382+5.891.*log(sinh(0.03062.*X1))+(-57582./(8.314.*Y1));

figure

surf(X1,Y1,Z_excel,'EdgeColor','black');

hold

scatter3(xall,Temp,zall_l,'MarkerEdgeColor','white','MarkerFaceColor','blue');

81

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