Experimental Techniques for Shear Testing of Thin Sheet Metals and Compression Testing at Intermediate Strain Rates

THESIS

Presented in Partial Fulﬁllment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University

Kevin Alexander Gardner, B.S.

Graduate Program in Mechanical Engineering

The Ohio State University

2013

Master’s Examination Committee:

Dr. Amos Gilat, Advisor Dr. Mark Walter c Copyright by

Kevin Alexander Gardner

2013 Abstract

A new specimen geometry for the characterization of thin sheet metals in simple

shear is introduced. The objective of this work is to develop methods to generate

experimental data that populate material models for numerical simulations. The

new simple shear specimen, based on ASTM B831, can be tested in both quasi-

static and dynamic conditions using a servo-hydraulic load frame and tension Kolsky

bar, respectively. Specimens are fabricated from a 0.5in Al2024-T351 plate with their gage sections orientated in various directions. Tests are conducted at shear strain rates ranging from 0.01s−1 to 9000s−1. Traditionally, shear characterization is performed through torsion tests on thin walled tube specimens, which are impossible to fabricate from thin sheet metals. The proposed specimen geometry is evaluated by comparing data obtained using the new specimen to existing torsion data. Three- dimensional Digital Image Correlation (DIC) is used to directly measure deformation on the surface of specimen gage sections for all tests. Stress versus strain curves obtained from tests using both specimen geometries agree, indicating that the new specimen geometry is suitable for use in characterizing thin sheet metals in shear.

Additionally, the new specimen geometry is able to capture anisotropic eﬀects which are averaged in torsion data on thin walled tube specimens. A parallel LS-DYNA simulation is conducted to investigate the strain state within the gage section during

ii a test and compare to experimental data measured with DIC. Results show that a

nearly uniform state of shear strain exists until large strains are developed.

An intermediate strain rate apparatus is used to characterize Al2024-T351 and

Cu-101 in compression at a strain rate of 100s−1. The proposed intermediate strain rate apparatus consists of a linear hydraulic actuator to generate the loading and a long transmitter bar. The specimen is placed on the end of the transmitter bar and loaded directly by the actuator. When the specimen is loaded, a compression wave propagates down the transmitter bar and reflects back towards the specimen when it reaches the end of the bar. A long transmitter bar allows the test to continue until the reflected wave reaches the specimen. Ample time (16ms) is provided to accumulate significant strain at intermediate strain rates without inertial effects (ringing) that are common to other intermediate strain rate testing techniques. Two materials, Al2024-

T351 and Cu-101 are tested. Previous data shows Al2024 does not exhibit strain rate sensitivity below 5000s−1 while Cu-101 does. Specimens from both materials are tested in compression at strain rates ranging from 0.01s−1 to 5000s−1 using a load frame, the proposed intermediate strain rate apparatus, and a compression Kolsky bar. DIC is used to measure deformation on the surface of the specimen for all tests.

Experimental data shows the intermediate strain rate apparatus is able to capture the expected data trends and is not subjected to the ringing observed by other common intermediate strain rate test techniques.

iii This document is dedicated to my family and close friends.

iv Acknowledgments

There have been many people who have helped me reach this point in my life and

I would like to acknowledge them for all of their help and support. First and foremost

I would like to thank my parents Kevin and Anayanci Gardner. Their constant love

and support has helped me in attaining my goals.

My advisor, Professor Amos Gilat, has made my graduate experience all that it is.

He is both inspiring and supportive and it is a great honor to work with such a highly regarded expert in the ﬁeld of mechanics of materials. Recognition is due to Dr.

Jeremy Seidt as well. Jeremy has become a good friend and an esteemed colleague, constantly oﬀering insight and guidance making him invalueable as a second advisor.

I would also like to thank Professor Mark Walter for taking the time to serve as my thesis defense committe member.

I would like to thank Jerry Hoﬀ, Larry Antal, and Ryan Shea of the Chemistry

Dept’s machine shop. They fabricated all of the specimens and ﬁxtures used in this research.

This research was funded by the Federal Aviation Administration. Thanks to Don

Altobelli, Bill Emmerling, and Chip Queitzsch from the FAA for all of the support given and the strong relationship they have developed with our research group. The construction of the intermediate strain rate apparatus was supported by NASA (NRA

Grant NNX08AB50A).

v Thanks also to Steven Whitaker, Jarrod Smith, Tim Liutkus, Jeremiah Hammer,

Tom Matrka, and Bob Lowe. These students and researchers in the Mechanical and

Aerospace Engineering Department at The Ohio State University have been integral in my eﬀorts through technical discussions and more importantly, the friendships they have provided.

vi Vita

2007 ...... Warren Local High School, Vincent, OH 2011 ...... B.S. Mechanical Engineering, The Ohio State Univeristy 2011-2012 ...... Graduate Fellow, Dynamic Mechanics of Materials Laboratory, Department of Mechanical and Aerospace Engineering, The Ohio State University 2012-present ...... Graduate Research Assistant, Dynamic Mechanics of Materials Laboratory, Department of Mechanical and Aerospace Engineering, The Ohio State University Publications:

Gardner, K.A., Seidt, J.D., Isakov, M., Gilat, A., “Characterization of Sheet Metals in Shear over a Wide Range of Strain Rates”, Proceedings of the 2013 SEM Annual Conference and Exposition on Experimental and Applied Mechanics, Lombard, IL, June, 2013

Fields of Study

Major Field: Mechanical Engineering

Specializations: Experimental Mechanics, Dynamic Behavior of Materials, Plastic- ity, Computational Mechanics

vii Table of Contents

Page

Abstract...... ii

Dedication...... iv

Acknowledgments...... v

Vita ...... vii

ListofTables...... xi

ListofFigures ...... xii

1. Introduction...... 1

1.1 MotivationandObjectivesfortheResearch ...... 4

2. ShearTestingofThinSheetMetals ...... 6

2.1 Literature Review - Characterizing Thin Sheet Metals in Shear . . 6 2.1.1 DoubleNotchShearSpecimen ...... 7 2.1.2 Eccentric Notch Shear Specimen ...... 10 2.1.3 Twin Bridge Shear Specimen ...... 12 2.1.4 Shear-CompressionSpecimen ...... 14 2.1.5 Simple Shear Specimen ...... 16 2.2 IntroductionofaShearSpecimen ...... 18

3. Simple Shear Experimental Results and Discussion ...... 20

3.1 ExperimentalTestPlan ...... 21 3.1.1 Simple Shear Specimens ...... 23 3.1.2 TorsionSpecimens ...... 24

viii 3.2 Quasi-StaticTestingTechnique ...... 25 3.3 HighStrainRateTestingTechniques ...... 28 3.3.1 High Strain Rate Tension Apparatus ...... 29 3.3.2 High Strain Rate Torsion Apparatus ...... 38 3.4 DigitalImageCorrelation ...... 43 3.5 ExistingExperimentalData ...... 47 3.6 StrainRateEﬀects...... 51 3.7 AnisotropicEﬀects ...... 52 3.8 Results from Numerical Simulation ...... 53

4. IntermediateStrainRateTesting ...... 59

4.1 Literature Review - Intermediate Strain Rate Testing ...... 59 4.1.1 RepeatedLoadingTechnique ...... 60 4.1.2 ModalAnalysisofLoadFrame ...... 62 4.1.3 LongKolskyBar ...... 64 4.2 Introduction of an Intermediate Strain Rate Apparatus ...... 65

5. Compression Experimental Results and Discussion ...... 67

5.1 ExperimentalTestPlan ...... 67 5.1.1 CompressionSpecimens ...... 69 5.2 Quasi-StaticTestingTechnique ...... 70 5.3 Intermediate Strain Rate Testing Technique ...... 71 5.4 HighStrainRateCompressionApparatus ...... 78 5.5 IntermediateStrainRateData ...... 83 5.6 StrainRateEﬀect ...... 85

6. SummaryandConclusions...... 87

6.1 Simple Shear Experimental Conclusions ...... 87 6.2 Compression Experimental Conclusions ...... 88

Appendices 90

A. Experimental Results from the Simple Shear Test Series ...... 90

A.1 SimpleShearResults...... 91 A.1.1 Rolled Direction ...... 91 A.1.2 TransverseDirection ...... 95 A.1.3 45◦ from Rolled Direction ...... 99 A.2 TorsionResults...... 103

ix B. Experimental Results from the Test Series Evaluating the Intermediate StrainRateApparatus ...... 104

B.1 Results from the 0.5inAl2024-T351Plate ...... 105 B.2 Results from the 0.25inCu-101Rod ...... 109

Bibliography ...... 113

x List of Tables

Table Page

3.1 Test plan to evaluate simple shear geometry ...... 22

3.2 Material parameters used for numerical study ...... 55

5.1 Test plan to evaluate intermediate strain rate apparatus ...... 68

xi List of Figures

Figure Page

1.1 Experimental test techniques for various strain rates ...... 5

2.1 Ti-6Al-4V shear stress vs. shear strain data from torsion tests on spool shapedspecimensatvariousstrainrates ...... 7

2.2 Guo, et al.’s FE model of the SHB clamp assembly ...... 8

2.3 Guo, et al.’s double notch specimen ...... 8

2.4 Guo, et al.’s experimental data for Ti-6Al-4V determined from the double notch specimen, before and after calibration ...... 9

2.5 Peirs, et al.’s eccentric notch shear sample (dimensions in mm) . . . . 10

2.6 DIC data Measured on the surface of the eccentric notch specimen . . 11

2.7 Simulated and experimental data using the eccentric notch specimen . 12

2.8 Brosius, et al.’s twin bridge shear specimen ...... 13

2.9 Equivalent plastic strain from the twin bridge shear specimen . . . . 14

2.10 Various shear-compression samples after testing ...... 15

2.11 Experimental comparison of the shear-compression specimen to a compressionspecimen...... 16

2.12 Shear specimen geometries (a)ASTM B831, (b)modiﬁed sample proposed by Isakov, et al. (dimensions in mm) ...... 17

xii 2.13 Shear specimen deformation (a)without thickness reduction, (b)with thicknessreduction ...... 17

2.14 Experimental data from uniaxial tension tests and simple shear tests 18

2.15 Novel specimen for characterizing thin sheet metals in shear . . ... 19

3.1 Specimen fabrication orientations ...... 23

3.2 Simple shear specimen used in plastic deformation testing (dimensions ininches) ...... 24

3.3 Thin walled tube (spool) specimen used in torsion testing (dimensions ininches) ...... 25

3.4 Instron 1321 bi-axial servo-hydraulic load frame (hydraulic wedge grips shown)...... 27

3.5 Simple shear specimen placed in load frame prior to testing ...... 28

3.6 Schematic representation of the direct tension Kolsky bar ...... 29

3.7 Direct tension (left) and torsion (right) Kolsky bars in the DMML . . 30

3.8 Clamping assembly used on the direct tension Kolsky bar ...... 31

3.9 Simple shear specimen glued to the direct tension Kolsky bar . . . . . 32

3.10 Comparison of the various methods for determining shear strain with thesimplesheargeometry ...... 36

3.11 Typical wave data from a tension Kolsky bar experiment on the simple shearspecimen ...... 37

3.12 Processed wave data from a typical tension Kolsky bar experiment on thesimpleshearspecimen ...... 38

3.13 Schematic representation of the stored torque Kolsky bar ...... 39

3.14 Spool sample glued to the stored torque Kolsky bar ...... 40

xiii 3.15 Typical wave data from a torsion Kolsky bar experiment ...... 42

3.16 Processed wave data from a typical torsion Kolsky bar experiment.. 43

3.17 DIC data from a compression test with Vic-3D (a)2D image of speckled specimen and platens, (b)3D reconstruction of specimen and platens. 45

3.18 Strain measurement comparison between DIC and the load frame mea- surementfortestsonTi-6Al-4V ...... 47

3.19 Comparison of previous Al2024-T351 torsion data to current results . 48

3.20 Comparison of simple shear and torsion data ...... 50

3.21 DIC strain measurement location for a simple shear test ...... 50

3.22 Simple shear data for Al2024-T351 in the rolled direction at various shearstrainrates ...... 51

3.23 Simple shear data from diﬀerent specimen orientations at a shear strain rate of 0.01s−1 ...... 53

3.24 FEmodelusedfornumericalstudy ...... 54

3.25 FEmodelaftermeshing ...... 55

3.26 Shear strain distribution measured with DIC during a simple shear test 56

3.27 Shear strain distribution from LS-DYNA simulation ...... 56

3.28 Vertical strain (ǫyy) distribution measured with DIC during a simple sheartest ...... 57

3.29 Vertical strain (ǫyy) distribution from LS-DYNA simulation ...... 58

3.30 Average shear strain and vertical strain (ǫyy) measured with DIC dur- ingaquasi-staticsimplesheartest ...... 58

4.1 Comparison of single step and incremental loading at quasi-static conditions...... 61

xiv 4.2 Experimental data from an incremental test at 340s−1 ...... 61

4.3 Intermediate strain rate data exhibiting ringing ...... 63

4.4 Experimental record from a test on PMDI foam using a long Kolsky bar 65

4.5 Schematic representation of the intermediate strain rate apparatus. . 66

5.1 Specimen fabrication orientations: (a) 0.5in thick Al2024-T351 plate, (b) 0.25indiameterCu-101rod ...... 69

5.2 Compression specimen used in plastic deformation testing(dimensions ininches) ...... 70

5.3 Steel pushrods with tungsten carbide inserts ...... 71

5.4 Intermediate strainrateapparatusat theDMML ...... 72

5.5 Intermediate strain rate apparatus hydraulic actuator ...... 73

5.6 Load cell glued to transmitter bar of the intermediate strain rate apparatus...... 74

5.7 Force history from a typical test on Cu-101 with the intermediate strain rateapparatus...... 77

5.8 Processed data from a typical intermediate strain rate test on Cu-101, average strain rate 70s−1 ...... 77

5.9 CompressionKolskybarusedbytheDMML ...... 79

5.10 Schematic representation of the compression Kolsky bar ...... 79

5.11 Typical wave data from a compression Kolsky bar experiment on Al2024- T351...... 82

5.12 Processed wave data from a typical compression Kolsky bar experiment onAl2024-T351...... 82

5.13 DIC measurement locations during a compression test ...... 83

xv 5.14 Comparison of Al204-T351 data between load frame and intermediate strainrateapparatus ...... 84

5.15 True stress versus true strain for Cu-101 over a range of strainrates . 86

A.1 Shear test data spread: rolled direction,γ ˙ =0.01s−1 ...... 91

A.2 Shear test data spread: rolled direction,γ ˙ = 2000s−1 ...... 92

A.3 Shear test data spread: rolled direction,γ ˙ = 4500s−1 ...... 93

A.4 Shear test data spread: rolled direction,γ ˙ = 9000s−1 ...... 94

A.5 Shear test data spread: transverse direction,γ ˙ =0.01s−1 ...... 95

A.6 Shear test data spread: transverse direction,γ ˙ = 2000s−1 ...... 96

A.7 Shear test data spread: transverse direction,γ ˙ = 4500s−1 ...... 97

A.8 Shear test data spread: transverse direction,γ ˙ = 9000s−1 ...... 98

A.9 Shear test data spread: 45◦ from rolled direction,γ ˙ =0.01s−1 .... 99

A.10 Shear test data spread: 45◦ from rolled direction,γ ˙ = 2000s−1 .... 100

A.11 Shear test data spread: 45◦ from rolled direction,γ ˙ = 4500s−1 .... 101

A.12 Shear test data spread: 45◦ from rolled direction,γ ˙ = 9000s−1 .... 102

A.13 Torsion test data with spool specimen,γ ˙ = 2000s−1 ...... 103

B.1 Compression test data spread: Al2024-T351, rolled direction,ǫ ˙ =0.01s−1 105

B.2 Compression test data spread: Al2024-T351, rolled direction,ǫ ˙ =1.0s−1 106

B.3 Compression test data spread: Al2024-T351, rolled direction,ǫ ˙ = 100s−1 107

B.4 Compression test data spread: Al2024-T351, rolled direction,ǫ ˙ = 5000s−1 ...... 108

B.5 Compression test data spread: Cu-101,ǫ ˙ =0.01s−1 ...... 109

xvi B.6 Compression test data spread: Cu-101,ǫ ˙ =1.0s−1 ...... 110

B.7 Compression test data spread: Cu-101,ǫ ˙ = 100s−1 ...... 111

B.8 Compression test data spread: Cu-101,ǫ ˙ = 5000s−1 ...... 112

xvii Chapter 1: Introduction

Numerical methods, such as the ﬁnite element method, have had an ever increasing

presence in the engineering design process. Finite element analyses have become a cost

eﬃcient method to analyze and optimize complex components during the engineering

design process, even before the fabrication of an initial prototype. Despite the fact

that a majority of these analyses are concerned with only the elastic response of the

mechanical system, it is becoming increasingly important to have accurate material

models that capture a material’s plastic response. One example of this need can

be given by automotive crashes. Engineers now have the ability to run numerical

simulations of crash events and identify components that are likely to fail, potentially

causing injury to the passengers, as well as improving the crash characteristics of the

complete vehicle.

Prior to yielding, the elastic response of engineering materials is easily understood through the linear relationship between stress and strain within this regime and the material properties, such as Young’s modulus and Poisson’s ratio, are easily available.

When compared to the plastic behavior, the linear elastic response is typically less inﬂuenced by the strain rate, temperature, and anisotropy of the material. Before yielding, the material’s molecular bonds stretch and no permanent damage results since they can spring back into their original conﬁgurations. However, after yielding,

1 the material no longer behaves linearly and more complex material models are needed

to describe the material behavior that arises from the tearing and reconﬁguration of

the molecular bonds, which results in permanent damage [1]. Plastic deformation

is signiﬁcantly more complex and is known to depend more heavily on the strain

rate, temperature, material anisotropy, residual stresses, and other factors such as

the details of the microstructure [2].

While it has become increasingly clear that numerical methods, such as the com- mercially available numerical codes LS-DYNA [3] and ABAQUS [4], have the ability to generate accurate results to complex problems, the level of accuracy attainable is dependent upon the validity of the material models used. Components are subjected to very complex stress states and in order to have conﬁdence in the simulation results, the material models used to describe the deformation and failure behavior of the various components must be based on and calibrated with experimental data.

Many products, such as automobiles and aircraft are comprised largely of thin sheet metal, therefore it is important to determine the shear response of the materials used, since it has been shown that the eﬀective stress vs equivalent plastic strain curves generated from tension and shear tests do not always coincide [5, 6]. Similarly, material failure is known to depend on the stress state [7, 8, 9], therefore it is necessary to characterize the materials used under shear loading conditions.

Typically, the shear response of ductile metals is determined by conducting torsion tests on thin walled tube samples. The samples used in a torsion test are spool shaped with a thin walled tube gage section in the middle and ﬂanges on either side that are used to attach the specimen to the test apparatus. While this sample geometry is easily fabricated from round stock or thicker plates, it is impossible to fabricate this

2 geometry from thin sheet metals, dictating the need for a new sample geometry to

be used in characterizing thin sheet metals in shear.

It is well known that the mechanical response of many engineering materials is dependent upon the applied strain rate. Hammer [10] performed tests on Ti-6Al-4V plate stock in tension, compression, and shear (with torsion tests) over a range of strain rates and has shown that Ti-6Al-4V exhibits an increase in yield stress as well as ﬂow stress in conjuction with an increase in strain rate. Cheng [11] and Vural, et al. [12] have independently shown that AISI 1018 cold-rolled steel exhibits strain hardening in a shear loading through strain rates up to 5000s−1, after which strain softening is observed. This illustrates the need for experimental data over the full gamut of strain rates, from quasi-static through high strain rates.

Recent advances in computational mechanics and computer hardware have made it possible to simulate a complete automotive crash event. Therefore, there is an increasing need for reliable material data at intermediate strain rates in the range of

101s−1 to 103s−1 [13]. Traditionally, tests at high strain rates from 103s−1 to 104s−1 are

performed using the Kolsky bar technique [14] and servo-hyraulic load frames are used

for quasi-static tests in the range of 10−5s−1 to 1s−1. Tests at intermediate strain rates

are diﬃcult to perform because the strain rates are too low to accumulate signiﬁcant

amounts of strain in the specimen within the test duration when using the Kolsky bar

technique. Similarly, the strain rates are too high to be performed using hydraulic

machines because inertia eﬀects inﬂuence the data. There is a need to develop an

experimental technique to accurately determine material behavior at intermediate

strain rates. Naturally, this technique must be immune from the shortcomings of

conventional techniques when applied to this strain rate regime.

3 1.1 Motivation and Objectives for the Research

Developments in numerical methods have improved our ability to simulate complex events such as automobile collisions. There is an increasing need to develop reliable techniques to determine the shear characteristics of thin sheet metals and to test engineering materials at intermediate strain rates from 101s−1 to 103s−1. This

research, therefore, consists of two main goals:

First, to develop a testing technique that can be used to reliably determine the

shear characteristics of thin sheet metals. There have been many sample geometries

proposed by previous researchers, however, many of these are actually subjected to

combined stress states and not pure shear. Pure shear tests can be conducted by

applying torque to thin walled tube specimens. Despite the fact that the thin walled

tube specimen is in pure shear, one major drawback to this sample geometry is that it

is impossible to fabricate from thin plate stock. Additionally, all anisotropic eﬀects are

averaged on the plane of the gage section so there is no ability to study variances from

diﬀerent specimen orientations. Conversely, most of the sample geometries proposed

for thin sheet metals do allow anisotropic eﬀects to be studied. Thus, the goal of this

work is to introduce a simple sample geometry for thin sheet metals that produces a

stress state approaching pure shear.

Second, to introduce an apparatus designed to test materials at intermediate strain rates in compression. Many methodologies have been introduced to ﬁll the gap between the Kolsky bar technique and the use of servo-hydraulic load frames, illustrated by Figure 1.1, however many of these techniques yield unsatisfactory results, are tedious and time consuming, or require arduous calculations. Thus, another goal of this work is to introduce an apparatus that is suitable for characterizing materials in

4 compression within the intermediate strain rate regime while producing data that is free from inertial eﬀects.

Figure 1.1: Experimental test techniques for various strain rates

5 Chapter 2: Shear Testing of Thin Sheet Metals

2.1 Literature Review - Characterizing Thin Sheet Metals in Shear

Many researchers have introduced various sample geometries to characterize thin sheet metals in shear. Several of the proposed sample geometries are discussed in the following sections. Most of the proposed samples are loaded in tension so that testing can be performed easily with common test apparatuses such as load frames and tension Kolsky bars. The goal is to produce a state of pure shear, however, pure shear cannot be obtained in a tensile loaded specimen because it is impossible to load the specimen without a bending component [15]. Therefore, the accuracy of the experimental data obtained from a certain specimen design depends on the uniformity of the shear deformation in the intended shear zone. Ideally, the specimen design should provide experimental data that is comparable to that obtained through torsion tests on spool shaped specimens. Typical data from a torsion test series using spool shaped geometry is shown in Figure 2.1 [10]. Hammer [10] performed these tests on Ti-6Al-4V.

6 Figure 2.1: Ti-6Al-4V shear stress vs. shear strain data from torsion tests on spool shaped specimens at various strain rates

2.1.1 Double Notch Shear Specimen

Guo, et al. [16] introduced a dynamic shear test technique and used it to evaluate the shear properties of Ti-6Al-4V. Tests were performed using a modiﬁed Kolsky compression bar that had a clamping assembly attached as illustrated by Figure 2.2

[16]. The clamp was used to constrain the double notch specimen, shown in Figure

2.3 [16]. Finite element simulations were performed on the test ﬁxture to optimize the design of the clamping assembly as well as the design of the double notch specimen.

The clamping assembly is needed in order to improve the shear strain distribution in the sample. Without the clamp, large errors arise from the bending moment that is introduced between the loading site and the specimen support sites. However, the clampling assembly renders the transmitted wave data unusable because of the elastic wave reﬂections that are caused by its large size and complex geometry. Because of this, the loading condition of the sample was determined from the incident and reﬂected waves (see Section 5.4).

7 Figure 2.2: Guo, et al.’s FE model of the SHB clamp assembly

Figure 2.3: Guo, et al.’s double notch specimen

8 This technique was used to conduct tests on Ti-6Al-4V at shear strain rates of up to 29000s−1. The experimental data, shown in Figure 2.4 [16], requires a coupled

experimental-numerical data processing procedure to obtain the true shear response

rendering it unsatisfactory when compared to data optained from torsion testing.

Figure 2.4 [16] shows the inﬂuence of this numerical calibration procedure. The

calibration consists of two parameters, a shear stress calibration coeﬃcient and a shear

strain calibration coeﬃcient that were determined through ﬁnite element simulations

and then used to modify the experimental data. The calibration coeﬃcients address

the issue of non-uniform strain in the gage sections and the presence of deformation

in the zones surrounding the gage sections.

Figure 2.4: Guo, et al.’s experimental data for Ti-6Al-4V determined from the double notch specimen, before and after calibration

9 2.1.2 Eccentric Notch Shear Specimen

Peirs, et al. [17] introduced the sample geometry shown in Figure 2.5 [17]. The specimen was designed for shear testing of sheet metals over a wide range of strain rates. The sample is meant to be tested using existing tension testing apparatuses with little to no modiﬁcation. For dynamic testing the sample is glued into slits on a tensile Kolsky bar. The eccentric notches in the sample are a result of numerical studies performed with the goal of optimizing the strain state in the intended gage section of the sample. The optimal notches result in a nearly uniform state of shear strain within the gage section.

Figure 2.5: Peirs, et al.’s eccentric notch shear sample (dimensions in mm)

Static and dynamic shear tests were performed on a 0.6mm Ti-6Al-4V sheet using this specimen geometry. Samples were machined in two diﬀerent orientations, one set with the axis of the sample aligned with the roll direction (RD), and another in the transverse direction (TD). Quasi-static tests were performed at a shear strain rate of 0.005s−1 and dynamic tests were performed at a rate of 750s−1. Digital Image

Correlation, see Section 3.4, was used to measure the strains directly on the surface

10 of the specimens during the tests. Figure 2.6 [17] shows the typical contour plot of

ǫ12 during a test indicating that much of the deformation is limited to within the intended gage section and a nearly uniform state of shear strain has been achieved with the proposed geometry.

Figure 2.6: DIC data Measured on the surface of the eccentric notch specimen

The sample geometry proposed by Peirs, et al. [17] generates a nearly uniform

state of stress and limits much of the deformation to within the intended gage section.

Figure 2.7 [17], which shows simulated and experimental data for shear stress and

shear strain vs displacement, exhibits oscillations in the stress data that occurs with

the dynamic tests. It is unknown whether this can be attributed to elastic wave

reﬂections from the specimen geometry or if it is simply an eﬀect of the test apparatus

being used for the dynamic tests.

11 Figure 2.7: Simulated and experimental data using the eccentric notch specimen

2.1.3 Twin Bridge Shear Specimen

Brosius, et al. [18] introduced a specimen geometry called the twin bridge shear specimen. This geometry consists of two concentric circular disks that are connected by shear bridges, which serve as the gage sections. This geometry is shown in Figure

2.8 [18]. The specimen is loaded by rotating the two clamped regions, denoted by hatching, relative to each other thus transferring a plane torsional moment through the specimen. The geometry of the specimen causes strain localization at the two shear bridges. The shear bridges are arranged symmetrically in order to avoid any unintended resultant forces or moments. Any number of shear bridges can be used in the specimen design, however the use of only two allows anisotropic material behavior to be investigated without any averaging eﬀects. Rotating the clamps relative to each other results in both bridges being sheared symmetrically.

12 Figure 2.8: Brosius, et al.’s twin bridge shear specimen

This specimen geometry was evaluated in a purely numerical study. Figure 2.9

[18] shows the distribution of equivalent plastic strain in one of the shear bridge gage sections. As illustrated, the deformation is not limited to within the gage section. A considerable amount of deformation occurs outside the gage section. Also, the strain distribution is not uniform due to the stress concentrations that arise where the shear bridge joins the inner and outer rings. This sample was only evaluated numerically, so the experimental performance of the sample geometry is unknown. However, it can be deduced that the sample would be diﬃcult to use in a practical situation.

The required loading configuration could pose complications and require a modified torsion Kolsky bar for dynamic tests and a rotational servo-hydraulic load frame even for static tests. The geometry of the specimen would also make it difficult, although not impossible, to utilize Digital Image Correlation during a test.

13 Figure 2.9: Equivalent plastic strain from the twin bridge shear specimen

2.1.4 Shear-Compression Specimen

Rittel, et al. [19] introduced the shear-compression specimen that has been de-

veloped for large strain testing of materials. While this specimen geometry would

be diﬃcult to fabricate from very thin sheet metals, it requires less thickness than

required to fabricate the spool shaped torsion specimens. It also provides the ability

to study anisotropic eﬀects. The shear-compression specimen, shown in Figure 2.10

[19], consists of a cylinder in which two diametrically opposed slots are machined at

45◦ with respect to the longitudinal axis, thereby creating a deﬁned gage section. The

specimen is easily loaded along its axis by existing compressive testing apparatuses

and also allows very high shear strain rates to be obtained in the range of 47 × 104s−1

[12].

Numerical studies were performed using ABAQUS [4] to determine the state of

strain within the gage section. Results indicated that the state of strain is nearly

uniform across the entire gage section and that shear is the dominant mode of de-

formation. The results were also used to develop simple relationships that express

14 Figure 2.10: Various shear-compression samples after testing

the equivalent stress and strain in terms of the geometrical parameters of the shear-

compression specimen and the applied load and displacement. These equations were

then used to compare data extracted from the shear-compression specimen to that

obtained from compression specimens at an eﬀective strain rate of 9000s−1, shown in

Figure 2.11 [19]. The data measured with the shear-compression specimen agrees well with data from a cylindrical sample. The initial overshoot in the shear-compression sample is attributed to the friction at the interface of the specimen and platens, since the cylindrical ends of the specimen must translate relative to each other during the shear deformation as can be observed from Figure 2.10 [19]. The experimental data shows the viability of the sample geometry and improved methods of reducing friction at the specimen-platen interface can improve the use of this geometry.

15 Figure 2.11: Experimental comparison of the shear-compression specimen to a compression specimen

2.1.5 Simple Shear Specimen

Isakov, et al. [20] used a modiﬁed specimen geometry, based on ASTM B831

[21], to characterize ferritic stainless sheet steel in shear. ASTM B831 presents a specimen geometry intended to determine the ultimate shear strength of wrought or cast alluminum alloys. The specimen geometry, shown in Figure 2.12(a) [21], consists of a planar specimen with a gage section created by two opposing 45◦ slots through

the specimen thickness. This sample geometry was modiﬁed for use in a tension

Kolsky bar and initial experimental data at quasi-static conditions dictated the need

for a thickness reduction of the gage section as shown in Figure 2.12(b) [20].

Digital Image Correlation was used to measure surface strains of a scaled version of

the geometry proposed in B831. The results of this test dictated the need for reducing

the thickness of the gage section. Figure 2.13 [20] shows the deformation of the shear

specimen with(b) and without(a) the thickness reduction of the gage section. The

specimen without the thickness reduction exhibits severe distortion outside of the

16 (a) (b)

Figure 2.12: Shear specimen geometries (a)ASTM B831, (b)modiﬁed sample proposed by Isakov, et al. (dimensions in mm)

intended gage section. Failure also occurrs outside the gage section. The specimen with the thickness reduction has a well deﬁned gage section and DIC data from tests conﬁrm that the deformation was limited within the gage section.

Figure 2.13: Shear specimen deformation (a)without thickness reduction, (b)with thickness reduction

17 Isakov used specimens with thickness reduction to characterize the stainless steel

sheet at quasi-static and high strain rates. Uniaxial tension tests were also performed

to compare to the shear data. The data, shown in Figure 2.14 [20], illustrates an

agreement between the tension and the simple shear tests. The specimen geometry

provides accurate data at a wide range of strain rates and is easily implemented since

it can be tested with existing tensile testing apparatuses.

Figure 2.14: Experimental data from uniaxial tension tests and simple shear tests

2.2 Introduction of a Shear Specimen

The shear response of ductile metals is generally investigated through torsion tests on spool shaped specimens with thin walled tube gage sections. It is the goal of this work to create a specimen geometry for characterizing thin sheet metals in shear that produces data comparable to that of torsion tests. A 0.5in Al2024-T351 plate was chosen to allow both sample geometries to be fabricated. Initially, simple shear specimens with the geometry proposed by Isakov, see Figure 2.12(b) [20], were

18 fabricated. Tests were performed at both low and high shear strain rates. While the

data at low strain rates was satisfactory, the high strain data was unsatisfactory due

to the long duration required to achieve force equilibrium in the specimen. This was

due to the sample geometry being too long (because of the 45◦ slots). In order to address this issue, a simpler geometry is introduced for the characterization of thin, ductile sheet metals in shear.

The specimen geometry, shown in Figure 2.15, is similar to the design previously proposed by Isakov, et al. [20], however the gage section is constructed through cuts that are made perpendicular to the loading direction as opposed to 45◦ cuts. This change signiﬁcantly reduces the overall exposed length of the specimen between the grips of the test apparatus used, thus reducing the duration required to obtain force equilibrium in the specimen during dynamic tests. This sample geometry is easily tested using existing apparatuses for tension testing. The specimen dimensions used in this work are given in Section 3.1.1.

Figure 2.15: Novel specimen for characterizing thin sheet metals in shear

19 Chapter 3: Simple Shear Experimental Results and Discussion

This chapter presents the experimental procedure that was used to evaluate the

new simple shear sample. The methods used for quasi-static and dynamic testing

and the associated calculations are also presented and discussed. Several apparatuses

were used to conduct the tests discussed in this chapter. Many of the testing pro-

cedures described were originally developed by the Dynamic Mechanics of Materials

Laboratory (DMML).

Experimental results generated during the test series evaluating the simple shear geometry, described in Table 3.1, are presented. Quasi-static and high strain rate data are presented. The simple shear geometry allows the anisotropic eﬀects of the

Al2024-T351 plate used to be studied. Data from specimens orientated in several directions within the plate are presented. Results from a numerical simulation of the simple shear geometry are also presented and compared with the experimental data.

The data from the simple shear geometry shows good agreement with data recorded using conventional thin walled tube specimens. Data from different orienatations of the simple shear geometry shows different plastic deformation and failure characteristics. Data spread for each individual test configuration can be seen in Appendix A.

Representative curves from the test series are selected and presented here.

20 Digital Image Correlation (DIC) was used to measure full-field specimen strains for all tests presented here. DIC, described in greater detail by Sutton, et al. [22] is a non-contact optical method that allows full field strain to be measured directly on the surface of a deforming specimen. DIC allows further insight into the evolution of strain in the specimen during a test. Effects such as strain localization can be directly observed, eliminating the need for estimation and greatly increasing the amount of data that can be gleaned from the testing of materials. This optical measurement technique is briefly discussed in this chapter.

3.1 Experimental Test Plan

Performance of the simple shear specimen is evaluated by completing a simple shear test series with the new specimen geometry and torsion tests on thin walled tube specimens. Specimens were cut from the same material stock: a 0.5in Al2024-T351 plate. Seidt [23] previously characterized the same exact plate in shear with torsion tests on thin walled tube specimens over a range of strain rates while researching the plastic deformation behavior of this plate stock. Due to the amount of previous data available for this stock the shear tests using the thin walled tube specimens did not need to be repeated, however a single test was performed to confirm the measurements previously taken. The new simple shear geometry allows the anisotropic effects of the plate to be determined which is not possible with the thin walled tube geometry since it effectively averages all in-plane shear stresses. This plate stock is also known to be insensitive to strain rate below 5000s−1 [24]. The simple shear test program is outlined in Table 3.1.

21 Table 3.1: Test plan to evaluate simple shear geometry Test Specimen Testing Specimen Shear Strain Rate No. Geometry Apparatus Orientation (1/s) 1 Instron Load Frame 0.01 2 2000 Rolled 3 Tension Kolsky Bar 4500 4 9000 5 Instron Load Frame 0.01 6 2000 Simple Shear Transverse 7 Tension Kolsky Bar 4500 8 9000 9 Instron Load Frame 0.01 10 ◦ 2000 45 From Rolled 11 Tension Kolsky Bar 4500 12 9000 13 Spool Torsion Kolsky Bar N/A 2000

Simple shear specimens are fabricated in three diﬀerent orientations in order to

study the anisotropic eﬀects of the plate. Specimens are fabricated with their gage sec-

tions aligned with the rolled direction, at 45◦ to the rolled direction, and in the transverse direction. Specimens with the spool shaped geometry are fabricated through the plate thickness. Specimen orientations are illustrated by Figure 3.1.

22 Figure 3.1: Specimen fabrication orientations

3.1.1 Simple Shear Specimens

All tests are conducted on the specimen shown in Figure 3.2. The gage section of the specimen is 0.220in in width, 0.040in in height, and 0.040in in thickness. The specimen is 0.080in thick and 0.020in slots are milled from either side to create a well deﬁned gage section with half the thickness of the surrounding material. The height of the gage section was chosen to achieve high strain rates and allow enough area to reliably measure strains in the notch with DIC. DIC was used to measure shear strains directly on the gage section of the specimen during tests. Shear specimens are fabricated with conventional machining techniques on mills. Necessary specimen dimensions are measured and recorded prior to testing.

23 Figure 3.2: Simple shear specimen used in plastic deformation testing (dimensions in inches)

3.1.2 Torsion Specimens

Torsion tests are conducted on specimens with thin walled tube (spool) geometry.

The shear strain rate of this specimen geometry can also be adjusted by varying the length of the gage section. All torsion tests were conducted on specimens with the geometry shown in Figure 3.3. The length of the gage section (0.100in) was designed to achieve a high strain rate while maintaining the ability to measure specimen strains with DIC. The spool specimens are fabricated with convential machining techniques, their proﬁles are cut from the plate and then they are turned down on a lathe.

Necessary dimensions are measured and recorded prior to testing. The thickness of the gage section is measured in four locations to obtain an average value.

24 Figure 3.3: Thin walled tube (spool) specimen used in torsion testing (dimensions in inches)

3.2 Quasi-Static Testing Technique

An Instron 1321 biaxial servo-hydraulic load frame, shown in ﬁgure 3.4 [23], is used to conduct both the simple shear and compression test series at strain rates at or below

1s−1. This machine has the capability to move axially ±61.2mm and rotate ±45◦.

Therefore, this one test apparatus can be used to conduct tension tests, compression tests, torsion tests, or any combination of these loading conditions. Measurements can be taken with two diﬀerent load cells, depending on the anticipated load. If the anticipated force magnitude is small, under approximately 2000 pounds, an Interface

1216CEW-2k load cell with a maximum axial capacity of 8.9kN and a maximum torsional capacity of 110Nm can be used. For larger forces, a Lebow 6467-107 load cell with an axial capacity of 89kN and a torsional capacity of 1100Nm is used. This machine is controlled by an external computer that works in conjuction with an

25 MTS FlexTest SE controller. Data is acquired by the controller through MTS 493.25

Digital Universal Conditioners that have a maximum data acquisition rate of 100kHz.

The axial motion of the load frame is controlled by a Linear Variable Diﬀerential

Transformer (LVDT) and the rotational motion is controlled by a Rotational Variable

Diﬀerential Transformer (RVDT). Measurements from these instruments are recorded

during a test, however, strains computed from these records are subject to machine

and grip compliance errors.

Many specimen grips can be used with the Instron 1321 load frame to test various specimen geometries. The simple shear geometry (Figure 3.2) is loaded in the same manner as a tension specimen. Figure 3.5 shows a simple shear specimen placed in the Instron load frame prior to testing. These tests are performed using a pair of MTS647.02B-22 hydraulic wedge grips (Figure 3.4) that have a maximum axial load capacity of 31kN. These grips can accomodate several diﬀerent designs of wedge grips. Test specimens are gripped using ﬂat knurled wedges suitable for the specimen thicknesses used in this work (0.080in). Alignment pegs on the wedges are used to

vertically align the specimens during placement.

26 Figure 3.4: Instron 1321 bi-axial servo-hydraulic load frame (hydraulic wedge grips shown)

27 Figure 3.5: Simple shear specimen placed in load frame prior to testing

3.3 High Strain Rate Testing Techniques

The high rate material behavior presented here was experimentally determined using Kolsky Bar or split-Hopkinson Bar tests. This method, ﬁrst proposed by Kolsky

[14], uses the propogation of elastic stress waves in two long bars to determine the dynamic plastic deformation behavior of a specimen that is placed between the bars.

This technique has become a standard method for dynamic material testing and has been modiﬁed to perform both tension and torsion experiments. Typically, strain rates ranging from 500s−1 to 8000s−1 can be obtained, depending on the strength and dimensions of the sample and the loading condtions of the apparatus. The following section describes the tension and torsion Kolsky bars at the DMML and their respective data processing techniques.

28 3.3.1 High Strain Rate Tension Apparatus

The DMML utilizes a direct tension Kolsky bar to conduct high strain rate tension tests. This apparatus was used to test simple shear specimens at high strain rates.

This apparatus is similar to the compression Kolsky bar, however there are several key differences in the machine design and operation. The most notable difference in the direct tension Kolsky bar is the method used to generate the loading pulse in the incident bar. The design of the direct tension bar is different from that of the compression bar as illustrated in Figures 3.6 [23] and 3.7 [23]. There are many methods used to generate an incident tensile wave, however the method used by this apparatus, described in further detail by Staab and Gilat [25], utilizes a long incident bar that is loaded with an initial tensile preload, which is released by fracturing a pin, thus generating the loading wave.

Figure 3.6: Schematic representation of the direct tension Kolsky bar

In order to load the incident bar with the initial preload the bar is clamped near

the middle with the clamping assembly shown in Figure 3.8. This clamping assembly

29 Figure 3.7: Direct tension (left) and torsion (right) Kolsky bars in the DMML

30 holds the bar against a loading block so that a tensile load can be applied from the

end of the bar. The tensile load, P , is applied to the end of the incident bar using a pulley system with a hydraulic actuator. The initial preload, measured using a full

Wheatstone bridge that is located between the pulley and the clamping assembly, is proportional to the desired strain rate in the sample. The top of the clamping assembly is joined by a notched pin while the bottom of the assembly is held together by the hydraulic actuator. As the load in the hydraulic actuator increases so does the loading on the pin, eventually the pin is broken allowing the tensile incident wave to propagate towards the specimen. The magnitude of the incident wave that reaches the specimen is half of the initial preload since equal amplitude waves are generated in both directions from the clamp.

Figure 3.8: Clamping assembly used on the direct tension Kolsky bar

The incident and transmitter bars of the direct tension Kolsky bar are fabricated

from 12.7mm diameter 7075-T651 aluminum. The incident bar is 3.68m long and the transmitter bar is 1.83m long. There have been many methods developed to attach the

31 specimen to the bars. Mechanical connections tend to be subject to relative motion

which causes an overestimation of the sample strain. Mechanical connections can also

introduce wave reﬂections due to impedence mismatches which result in oscillatory

waves and dynamic force imbalance. In this work, specimens are glued into adapters

which are then glued between the two bars using either Hysol Tra-Bond 2106T epoxy

or JB Kwik steel reinforced epoxy as shown in Figure 3.9.

Figure 3.9: Simple shear specimen glued to the direct tension Kolsky bar

It is important to avoid an impedence mismatch between the bars and the specimen adapters to prevent reﬂections of the elastic wave. The adapters used are custom designed for each sample geometry in order to maintain a constant impedence, which can be determined in general by equation 3.1. The interface between the adapters and specimen has two diﬀerent materials in parallel with one another. The following relationship, equation 3.2, must be maintained so that the impedence of the specimen-adapter interface matches that of the bar.

Z = ρAc (3.1)

Zb = Za + Zs = ρaAaca + ρsAscs = ρbAbcb (3.2)

32 where the subscripts a, s, and b represent the adapters, specimen, and bars respec-

tively. The adapter can be designed to maintain constant impedence if the material

properties of all three components are known. In this work, the bars, adapters, and

specimen are all fabricated from aluminum, thus the adapter design is straight for-

ward and only requires designing the adapters to leave a minimal gap at the adapter-

specimen interface based on the dimensions of the specimen.

The strains in both bars are measured using full Wheatstone bridges made from

four Micro-Measurements ED-DY-075AM-10C 1000Ω strain gages. Gage A is located

1825mm from the specimen on the incident bar. Gage B is located 365mm from

the specimen on the incident bar and Gage C is located 365mm from the specimen

on the transmitter bar. 15.0V excitation is provided by a pair of Hewlett Packard

6200B power supplies. Each of the strain gage signals is conditioned by a Tektronix

ADA400A differential amplifier with a low pass filter set to 100kHz. The signals are measured using a four channel 350MHz, 8 bit Tektronix TDS5034B digital phospor oscilloscope with a sampling rate of 2.5 × 106Hz.

The strain rate of a tension specimen,ǫ ˙, in the direct tension Kolsky bar can be determined by u˙ − u˙ ǫ˙ = i t (3.3) lg

where lg is the initial gage length of the specimen.u ˙i andu ˙t are the incident and transmitter bar velocities which are given by

1 l l l l u˙ = F t − A + F t − A +2 B − F t + B (3.4) i ρA c A c A c c B c b b b b b b and 1 l u˙ = F t + C (3.5) t ρA c C c b b b 33 where, ρ is the density of the bars, Ab is the cross sectional area of the bars, and

cb is the elastic wave speed of the bars. lA, lB, and lC are the distances from the

specimen to the strain gages A, B, and C respectively. FA, FB, and FC are the forces

measured by their respective strain gages. The stress history of the specimen can be

determined by F l σ(t)= C t + C (3.6) A c s b where As is the cross sectional area of the specimen.

When a specimen with the introduced simple shear geometry is tested on the direct tension Kolsky bar, the initial calculations are the same. The only diﬀerence arises from the calculations of the shear strain and stress. For the calculations a state of simple shear deformation is assumed and the small angle approximation is used.

Thus, the shear strain rate can be determined by

u˙ − u˙ γ˙ = i t (3.7) h

where h is the height of the gage section. Once the shear strain rate history of the

specimen is known it is integrated to determine the shear strain history of the sample.

The average shear stress in the sample can be determined from

F l τ(t)= C t + C (3.8) A c shear b where Ashear is the cross sectional area of the gage section (width x thickness).

The use of DIC, see Section 3.4, during tests with the simple shear geometry allows several strain measurements to be used. The strain can be measured as an average value over the gage section or can be calculated from the displacements between points on either side of the gage section, similar to the caculation from the recorded waves.

Figure 3.10 shows the time history of shear strain during a test calculalated using

34 the three possible techniques. The shear strain from the waves is calculated using

the equations presented above. The shear strain based on DIC points was calculated

using the displacements of two points, one above and one below the gage section. For

this calculation the shear strain was calculated using the trigonometric deﬁnition of

shear strain that can be determined by

k γ = arctan( ) (3.9) h where k is the relative horizontal displacement between the points above and below

the gage section. The shear strain from the rectangle average returns the average

Lagrange shear strain on the gage section as determined with DIC. As illustrated,

both methods involving DIC data give similar results while the strain calculated from

the elastic waves gives a much larger value. Some of this diﬀerence can be attributed

to not using the trigonometric deﬁnition of shear strain for the wave calculation since

the approximation used is only valid for small shear strains. However, it cannot be

used because shear strain rate, as opposed to shear strain, is calculated from the wave

data. While comparable results can be obtained from the rectangle average and two

point methods using DIC, the average shear strain obtained directly from the gage

section is the most accurate because it is a direct measurement on the surface of the

deforming specimen. Other methods give approximate values that are based on the

gage section deforming ideally, however, as mentioned previously the simple shear

specimen cannot be loaded in tension without generating a bending moment across

the gage section as well. Thus, the most accurate method for calculating shear strain

is to measure it directly on the surface of the gage section.

A typical experimental record from the simple shear geometry is shown in Figure

3.11. The incident wave begins at approximately 60µs and increases to a value of

35 Figure 3.10: Comparison of the various methods for determining shear strain with the simple shear geometry

36 about 10kN. The incident wave reaches gage B at approximately 356µs. The incident wave appears to have a much shorter duration at gage B, however, this is due to the reﬂected portion of the wave unloading the bar. The portion of the wave that travels through the specimen is the transmitted wave, measured by gage C, and is proportional to the stress in the specimen. The test ends when the transmitted wave drops abruptly, indicating failure of the specimen. The strain rate of the sample can be adjusted by varying the magnitude of the incident wave by adjusting the preload or by changing the gage length, or in the case of the simple shear sample the height of the gage section. Processed experimental data from the recorded waves can be seen in Figure 3.12, which shows the stress, strain rate, and strain history of a specimen from a typical test of the simple shear geometry on the direct tension Kolsky bar.

Figure 3.11: Typical wave data from a tension Kolsky bar experiment on the simple shear specimen

37 Figure 3.12: Processed wave data from a typical tension Kolsky bar experiment on the simple shear specimen

3.3.2 High Strain Rate Torsion Apparatus

The DMML utilizes a stored torque Kolsky bar, shown in Figure 3.7, similar in design to the direct tension Kolsky bar as illustrated by Figure 3.13 [23]. The technique is similar, with the only diﬀerence being the method used to generate the incident wave. The incident bar is clamped near the center, see Figure 3.8, and a pre-torque is applied to the clamped section using a pulley system with a hydraulic actuator. The incident wave is released by fracturing a pin, similar to the direct tension Kolsky bar. This technique is described in further detail by Gilat [26]. The incident wave amplitude is equal to half of the stored torque since the wave propagates in both directions from the clamp.

38 Figure 3.13: Schematic representation of the stored torque Kolsky bar

The stored torque Kolsky bar consists of two 22.23mm diameter 7075-T651 aluminum bars. The incident bar has an overall length of 3510mm, while the clamped portion has a length of 1227mm. The transmitter bar is 2026mm long. Strains in the bars are measured using full Wheatstone bridges with each being comprised of four

Micro-Measurements 1000Ω strain gages, orientated at 45◦, at the locations shown in

Figure 3.6 [23]. Gage A is located 1715mm from the specimen on the incident bar.

Similarly gage B is located 385mm from the specimen on the incident bar and gage

C is located 385mm from the specimen on the transmitter bar. 20.0V excitation is provided to each bridge from a pair of HP3611A power supplies. The stored torque

Kolsky bar uses the same signal conditioning and recording devices as the direct tension Kolsky bar.

The specimen geometry (Figure 3.3) used for the stored torque Kolsky bar consists of a thin-walled tube gage section that has ﬂanges on either end that allow it to be glued to the bars as shown in Figure 3.14. Specimens are glued to the bars using either

Hysol Tra-Bond 2106T epoxy or JB Kwik steel reinforced epoxy. The geometry of

39 the specimen can be changed in order to vary the strain rate, shortening the gage

length of the thin walled tube section between the ﬂanges increases the strain rate.

However, limitations arise because a smaller gage length makes it increasingly diﬃcult

to measure strain data on the surface of the gage section using DIC, see Section 3.4.

Additionally, there is also a stress concentration where the gage section meets the

ﬂanges that is accentuated by decreasing the gage length.

Figure 3.14: Spool sample glued to the stored torque Kolsky bar

Data processing is similar to that of the direct tension Kolsky bar. The amplitude of the incident shear strain wave is

Trb γi = (3.10) 2GbJb

where T is the magnitude of the torque preload, rb is the radius of the bar, Gb is the shear modulus of the bar, and Jb is the polar moment of inertia of the bar. As the incident wave reaches the specimen, a portion of the wave passes through the specimen, loading it and generating the transmitted wave. The remaining portion of

40 the incident wave reﬂects back (reﬂected wave) and is proportional to the shear strain

rate. The angular velocity of the incident bar is determined by

1 θ˙i(t)= [TA (t − tA)+ TA (t − tA +2tB) − TB (t + tB)] (3.11) ρbJbct where ρb is the density of the bar material. TA and TB are torques measured at gages

A and B respectively. tA, tB, and tC are the time required for the elastic wave to travel the distance between the specimen and gages A, B, and C respectively. ct is

the transverse elastic wave speed which can be determined from

Gb ct = (3.12) s ρb

The angular velocity of the transmitter bar, θ˙t(t), is

1 θ˙t(t)= [TC (t − tC )] (3.13) ρbJbct where TC is the torque measured by gage C on the transmitter bar. The shear strain rate can be determined by

rm θ˙i(t) − θ˙t(t) γ˙ (t)= (3.14) h ls i

where rm is the mean gage radius of the specimen and ls is the gage length of the specimen. Once the strain rate history is known, it can be integrated to determine the strain history of the specimen. The shear stress is determined by

TC (t − tC ) τ(t)= 2 (3.15) 2πrmls

The experimental record from a typical test performed on the stored torque Kolsky

bar is presented in Figure 3.15. The incident wave reaches gage A at approximately

70µs and reaches a value of approximately 100Nm. The incident wave reaches gage B

41 at approximately 500µs having the same amplitude until the reﬂected wave reaches gage B. The portion of the incident wave that travels through the specimen is the transmitted wave, measured by gage C, and is proportional to the shear stress in the sample. The test is completed when the transmitted wave suddenly drops in magnitude indicating specimen failure. Processed experimental data from the torsion

Kolsky bar is shown in Figure 3.16. The ﬁgure shows the shear stress, shear strain rate, and shear strain histories of the spool sample. It can be observed from the shear strain rate history that the rise time of the transverse wave is longer than those of the axial waves measured with the compression and direct tension Kolsky bars.

Figure 3.15: Typical wave data from a torsion Kolsky bar experiment

42 Figure 3.16: Processed wave data from a typical torsion Kolsky bar experiment

3.4 Digital Image Correlation

Digital Image Correlation (DIC) is an optical technique that allows non-contact measurements of deformation and strain. DIC systems allow researchers to measure full ﬁeld displacements and strains directly on the surface of a specimen during a test. An overview of the technique is provided here and an in depth discussion of the technique is provided by Sutton, Orteu, and Schreier [22]. Systems are available for both two-dimensional and three-dimensional measurements. The DMML uses three-dimensional systems. The three dimensional DIC system utilizes two cameras, arranged in a stereo array, to track displacements on the surface of the specimen.

Several commercial systems are available, however Correlation Solutions VIC-3D 2010

[27] is used by the DMML.

43 Setup involves first arranging the cameras so that their fields of view are centered on approximately the same spot on the surface of the sample. Depending on the sample geometry and orientation of the cameras this may not be feasible, but it is preferable to have the cameras aligned on at least one axis to reduce calculation errors. The three-dimensional DIC system is calibrated by photographing a panel with a dotted grid of known spacing. The panel is photographed in several different orientations providing data points for the calibration algorithm. This calibration procedure establishes the global coordinate system.

During a test, the cameras take two synchronized images, at a set framerate, of the specimen that has been coated with a random speckle pattern as shown in Figure

3.17(a). The algorithm used by the software then discretizes the images into user deﬁned subsets of n × n pixels. Gray scale values within these subsets are used to identify the subsets in subsequent, deformed images. Subset displacements between the current frame and the reference frame are calculated. Once these displacements are known the software can calculate the strains on the surface of the specimen utilizing one of the strain tensor deﬁnitons available. Since a stereo camera array is used, the three dimensional DIC system is also able to generate a 3D representaion of the sample as shown in Figure 3.17(b).

The software contains many available options for post-processing the data. Data can be extracted at a single point, over an area and averaged, or through virtual extensometers. Point data is useful for tracking the strain history at localization points and failure locations. Average data within a box is useful for evaluating the general strain state of a specimen. Virtual extensometers are also extremely useful

44 (a) (b)

Figure 3.17: DIC data from a compression test with Vic-3D (a)2D image of speckled specimen and platens, (b)3D reconstruction of specimen and platens

in that they can be used to compare DIC data to data acquired with conventional

mechanical extensometers.

The DMML has several camera setups available with a wide range of possible framerates for low rate and dynamic testing. Typically, for strain rates less than

1.0 × 10−2s−1, a pair of Point Grey Research GRAS-20S4M-C cameras are used.

These cameras can record images at framerates up to 19fps. They have a constant resolution of 1624 × 1224 pixels. These cameras can be ﬁtted with a variety of lenses

but typically Schneider 35mm lenses are used. Schneider 17mm and Sigma 50mm

lenses are also available for larger and smaller ﬁelds of view, respectively. At strain

rates of 1.0 × 10−1s−1 to 1.0s−1 a pair of Photron MC2 cameras are used. These

cameras are capable of 2000fps at their maximum resolution of 512 × 512 pixels and

higher frame rates can be achieved with lower resolutions. The maximum framerate

of 10, 000fps can be achieved with a resolution of 512 × 96pixels. The Photron MC2

45 cameras are typically used with the same lenses as the Point Grey cameras. High

strain rate tests utilize a pair of Photron SA1.1 cameras. These cameras have a

maximum resolution of 1024 × 1024 pixels that can be used at framerates up to

5, 400fps. Higher framerates can be achieved at reduced resolutions. The maximum framerate of 675, 000fps provides a reduced resolution of 64×16 pixels. Most tests are performed with a framerate of either 100, 000fps or 125, 000fps to provide suﬃcient number of frames during a test and reasonable resolution. Typical resolutions are

192 × 256 pixels. The Photron SA1.1 cameras are typically used with either 90mm or 180mm Tamron lenses.

DIC has proven to be a useful tool that provides many advantages over traditional

measurement techniques such as mechanical extensometers and strain gages. Strain

gages are difficult to install and the proficiency of the installer has a large effect on

the quality of data measured since they are easily damaged. Additionally, they are

typically limited to small strains of less than 8%. Mechanical extensometers are easier

to use, however, they are unable to capture localizations and only provide an average

strain measurement.

DIC also eliminates error that arises due to the compliance of the test devices used.

For example, a typical load frame has an LVDT that measures the movement of the

actuator used to load the specimen, however, this measurement is taken far away from

the specimen and when high loads are applied the compliance of the actuator between

the specimen and LVDT introduces large errors into strain calculations. Since DIC

measures strain directly on the surface of the sample, it is not subject to this source of

error and thus diﬀerences in stress strain curves for samples requiring large loadings,

such as those performed on Ti-6Al-4V by Hammer [10], can arise as shown in Figure

46 3.18 [10]. The eﬀect of compliance is proportional to the loading applied by the test

apparatus and aﬀects all tests to some degree.

2000

1500

1000

Tension (DIC) Stress (MPa) Tension (Machine LVDT) 500 Compression (DIC) Compression (Machine LVDT) Torsion (DIC) Torsion (Machine RVDT) 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Strain Figure 3.18: Strain measurement comparison between DIC and the load frame measurement for tests on Ti-6Al-4V

3.5 Existing Experimental Data

The 0.5in Al2024-T351 plate used for the simple shear test series has been exten- sively characterized in shear (torsion) over a range of strain rates from 1 × 10−4s−1 to 5000s−1 by Seidt [23]. Accordingly, the torsion tests using thin walled tube geometry do not need to be repeated, however a single test is performed at a shear strain rate of 2000s−1 to conﬁrm the previous experimental data. A comparison of previous experimental data at various shear strain rates with the additional test performed is provided in Figure 3.19. As illustrated, the previous experimental data and the additional test performed agree well.

47 Figure 3.19: Comparison of previous Al2024-T351 torsion data to current results

48 The simple shear geometry allows the characterization of thin sheet metals in shear since spool shaped specimens cannot be fabricated from thin plates. Specimens were fabricated from the 0.5in Al2024-T351 plate so that data from simple shear tests

can be compared with torsion data from spool shaped specimens. Figure 3.20 shows

data from the simple shear geometry in each of the three speicmen orientations and

the additional torsion test with the spool shaped geometry. All test data shown is

from experiments with a shear strain rate of 2000s−1. The experimental data from

the simple shear geometry agrees very well with the torsion data. This correlation

indicates that the simple shear geometry can be eﬀectively used to characterize thin

sheet metals in shear. In addition, the eﬀects of anisotropy can also be determined.

Data from tests performed at high strain rates are free from oscillations and wave

reﬂections observed with the various other sample geometries proposed, see Section

2.1. Strain data presented is measured with DIC by averaging the Lagrange shear

strain in a rectangular area placed on the gage section, see Figure 3.21. Failure

strains are signiﬁcantly higher with the simple shear geometry. This is the subject of

continued investigation, however, an initial hypothesis is that the torsion specimens

fail prematurely because of the stress concentration where the tube meets the ﬂange.

49 Figure 3.20: Comparison of simple shear and torsion data

Figure 3.21: DIC strain measurement location for a simple shear test

50 3.6 Strain Rate Eﬀects

The 0.5in Al2024-T351 plate used to fabricate the simple shear specimens is known to be insensitive to strain rate at equivalent strain rates below 5000s−1 [24]. Thus, it is expected that this would also be observed with the simple shear tests. Figure

3.22 shows data from specimens machined in the rolled direction at shear strain rates ranging from 0.01s−1 to 9000s−1. As indicated, there is no observable strain rate eﬀect over the large range of strain rates tested with the simple shear geometry conﬁrming the previous results. Failure strain increases as the shear strain rate increases, however this can be attributed to the heat generation in the specimen increasing the ductility of the material.

Figure 3.22: Simple shear data for Al2024-T351 in the rolled direction at various shear strain rates

51 3.7 Anisotropic Eﬀects

Anisotropic effects can be studied by conducting experiments on simple shear samples in various orientations. Anisotropic effects are not measurable with torsion tests because the data is the average shear response of the plate being tested in the plane of the gage section. The simple shear specimen allows anisotropy to be studied at any angle to the rolled direction with thin plates, and with thicker plates specimens could be fabricated with any orientation desired thus increasing the amount of material response data that can be determined. Specimens in this study are fabricated in three different orientations through the plate thickness, in the rolled direction(OR1), perpendicular to the rolled direction(OR2), and at 45◦ to the rolled direction(OR3), as shown in Figure 3.1, to investigate the shear response anisotropy of the 0.5in

Al2024-T351 plate. Figure 3.23 shows experimental data from low rate shear tests on specimens orientated in the three different directions. The data from specimens orientated in the rolled direction and 45◦ to the rolled direction give nearly identical results, however, the specimens orientated transverse to the rolled direction exhibit higher failure strains and lower plastic flow stresses at shear strains up to 30%. The simple shear geometry can be used to measure anisotropic effects that cannot be measured with torsion tests.

52 Figure 3.23: Simple shear data from diﬀerent specimen orientations at a shear strain rate of 0.01s−1

3.8 Results from Numerical Simulation

LS-DYNA [3] is used to simulate the simple shear test. The numerical results are compared to the experimental data. For the simulation it is assumed that during a test the portion of the sample held by the hydraulic grips on the load frame or in the adapters during a high rate test would not aﬀect the gage section and thus was omitted from the model. The geometry used in the numerical study is shown in Figure 3.24.

Hypermesh 11.0 is used to mesh the ﬁnite element model with 8-node quadilateral elements. The elements all have an aspect ratio of 1. Elements are sized so that the gage section consists of 110 elements across its width, and 20 elements in both height and thickness as illustrated by Figure 3.25. Elements on the left boundary are

53 constrained in all directions and elements on the right boundary are given a velocity

of 2.0m/s that corresponds to an average shear strain rate of approximately 2000s−1.

The simulation in LS-DYNA was performed using a simpliﬁed Johnson-Cook model

(MAT098) [28] and constant stress elements. The material parameters used for the model were deterimined from the previous characterization of the 0.5in Al2024-T351 plate and are presented by Seidt and Gilat [24]. The parameters were determined from the combined response of the material determined by compression, tension, and torsion testing. The parameters used are presented in Table 3.2.

Figure 3.24: FE model used for numerical study

Particularly of interest is the strain state of the specimen during a test. Figure 3.26

shows DIC shear strain data in the gage section of the specimen during a test. Data is

54 Figure 3.25: FE model after meshing

Table 3.2: Material parameters used for numerical study ρ(g/c3) E(GP a) ν A(MP a) B(MP a) n C 2.78 73.1 0.33 299 471 0.406 0

not available at the edges of the gage section because of the correlation algorithm used

by the software and the size of the DIC subsets. However, the DIC data indicates a

nearly uniform state of shear strain, with variations only arising near the free edges of

the gage section. Similarly, Figure 3.27 shows the shear strain distribution obtained

from the LS-DYNA simulation which also indicates a nearly uniform state of shear

strain in the gage section with deviations at the free edges.

Simple shear specimen geometries that are loaded axially cannot obtain a state of pure shear because of the emergence of a coupled bending moment across the gage section. Figure 3.28 shows DIC measurements of strain in the vertical direction during a test. These values are obtained when the average shear strain in the specimen

55 Figure 3.26: Shear strain distribution measured with DIC during a simple shear test

Figure 3.27: Shear strain distribution from LS-DYNA simulation

56 gage section has an approximate value of 0.20. Similarly, Figure 3.29 shows strain in the vertical direction obtained from the simulation when the shear strain in the gage section has an approximate value of 0.20. The experimental data and the simulation results show similar distributions with both showing a concentration of tensile strain near the edges of the gage section. Additionally, the simulation results indicate compressive strains along the boundary of the gage section and the rest of the specimen.

These strains are not captured by the DIC because the algorithm used is not able to extract data at the edges of the speciﬁed area of interest. Nevertheless, the strains in the vertical direction within the majority of the gage section remain insigniﬁcant until large values of shear strain are obtained as illustrated by Figure 3.30, which shows the average values of shear strain and vertical strain (ǫyy) in the gage section measured with DIC during a quasi-static simple shear test. Failure ultimately initiates in tension near the edges of the specimen.

Figure 3.28: Vertical strain (ǫyy) distribution measured with DIC during a simple shear test

57 Figure 3.29: Vertical strain (ǫyy) distribution from LS-DYNA simulation

Figure 3.30: Average shear strain and vertical strain (ǫyy) measured with DIC during a quasi-static simple shear test

58 Chapter 4: Intermediate Strain Rate Testing

4.1 Literature Review - Intermediate Strain Rate Testing

Many techniques have been developed to bridge the strain rate gap between quasi-

static and high strain rate testing and provide a method for intermediate strain rate

testing of engineering materials. Several of the proposed methods are discussed in

this section. Researchers have tried to perform intermediate strain rate tests using

standard hydraulic load frames, however the results from these tests are unsatisfactory

due to the fact that the load frame is not in equillibrium. Intermediate strain rate

tests occur at a timescale that is similar to the time it takes for stress waves to travel

through the machine and for the components of the load frame to accelerate. A direct

result of this is a large amplitude oscillatory response superposed on the load history

data. This is commonly referred to as ringing. When a load frame is used for low

rate testing the specimen and load frame are in a state of static equilibrium, however,

when these machines are used at intermediate strain rates the system is not in a state

of equilibrium as these inertia eﬀects arise. Ideally, the design of an intermediate

strain rate apparatus should eliminate these inertia eﬀects.

Intermediate strain rate tests are also diﬃcult to perform using apparatuses designed for dynamic testing because of the longer test durations required to accumulate

59 signiﬁcant strains in the specimen at intermediate rates. The duration of the loading

pulse with the Kolsky bar technique is proportional to the length of the striker used.

Typically, Kolsky bars are designed to give loading durations in the range of 0.25ms to 0.5ms. These durations do not allow signiﬁcant strain to be accumulated in the specimen at intermediate strain rates.

4.1.1 Repeated Loading Technique

A technique for intermediate strain rate testing was introduced by Gupta, et al. [13] that is based on repeated loadings of a sample using a conventional Kolsky bar apparatus designed for high strain rate testing. This method was applied to compressive testing where the specimen was repeatedly loaded at an intermediate strain rate until desired strains were achieved. The stress strain curves from each test were simply combined to generate the complete stress strain plot for a given strain rate. This method was used to characterize Mg-Al-Zn alloy.

The repeated loading method was ﬁrst evaluated with quasi-static tests performed on a load frame. Samples were tested in a typical single step compression test, and a second set of samples were tested incrementally. Test data from both methods, shown in Figure 4.1 [13], shows that a sample loaded incrementally and a sample loaded in a single step produce similar stress strain curves at quasi-static conditions. Once evaluated at quasi-static conditions, this method was then used to test specimens at a strain rate of 340s−1. Incremental experimental data for this intermediate strain rate test is shown in Figure 4.2 [13].

The incremental technique does not provide the same level of accuracy at intermediate strain rates. Combining the individual components of the stress strain plot

60 Figure 4.1: Comparison of single step and incremental loading at quasi-static conditions

Figure 4.2: Experimental data from an incremental test at 340s−1

61 results in an irregular master curve. The main issue facing this technique is that of temperature effects. Increases in specimen temperature at low strain rates are negligi- ble because deformation happens slowly and generated heat is removed by conduction with the platens. At intermediate and high strain rates, however, these temperature effects cannot be ignored. Typically, the strain rate effect that is observed in materials is a combination of strain hardening due to the plastic deformation and thermal softening due to the increase in specimen temperature. The temperature rise in a specimen can be significant and since the test duration is very short the specimen is considered to be adiabatic. This incremental loading method allows the sample to cool between each loading period, effectively changing the effect of thermal softening throughout the test. Thus, data obtained from this method will not match data obtained from a single step loading method at intermediate and higher strain rates. This method is also very tedious and time consuming. Any mistakes during one of the incremental steps would require starting over from the beginning with a new specimen.

4.1.2 Modal Analysis of Load Frame

A common method for intermediate strain rate testing involves performing the tests on a standard load frame, despite ringing as illustrated by Figure 4.3 [29], and then using numerical studies of the entire test apparatus with an assumed specimen constitutive model. If the assumed specimen response simulates ringing similar to that measured experimentally,the assumed response curve is assumed to be the true response curve of the material. A complimentary technique, outlined by Zhu, et al.

62 [29], involves determining the modal response of the load frame experimentally so

that it may be removed from the test data.

Figure 4.3: Intermediate strain rate data exhibiting ringing

Zhu outfitted an MTS load frame with a pair of Kistler 8702B500 accelerometers in four different configurations to characterize its dynamic response. For each of the different configurations at least five tests were performed where the vibration of the machine was recorded after an impact was generated between the grips. A frequency response function was determined from the measured response of the load frame which was used to filter experimental data.

Both techniques require tedious calculations and in the first case, relies on numerical studies to modify experimental data, which should be avoided. Additionally, these methods require studies to be performed on every load frame configuration used since modifications to the setup, such as using different specimen grips, will change the frequency response of the system.

63 4.1.3 Long Kolsky Bar

A technique employed by several researchers [30, 31] for intermediate rate compression testing simply involves scaling up the dimensions of the Kolsky bar. The duration of the loading pulse generated by a striker bar is directly proportional to double the length of the striker bar. When the striker bar impacts the end of the incident bar a compression waves propogate into both the incident and striker bars.

When the compression wave reaches the end of the striker bar, it reﬂects as a tensile wave. When the tensile wave reaches the impact interface the striker separates from the incident bar, setting the duration of the loading pulse. Thus, by scaling up the length of the striker, a longer duration loading pulse is obtained. A longer duration pulse is needed at intermediate strain rates because the amount of strain accumulated in the sample during the test is determined from the strain rate and duration of the pulse. Tests at lower strain rates require a longer pulse to accumulate enough plastic strain in the sample. The downside to simply scaling the dimensions of the Kolsky bar is that wave dispersion becomes an increasingly important issue as the bars are lengthened.

A long Kolsky bar with incident and transmitter bars both measuring 11m and a 2.5m long striker bar was developed by Song, et al. [30]. The apparatus generates a 1ms incident wave. This duration was increased by the use of a foam pulse shaper that slowed the momentum transfer between the striker and incident bar increasing the duration of the incident wave to 3ms. Tests were conducted on PMDI foam, which had an ultimate stress of 9MPa, at strain rates on the order of 10s−1. Experimental data from a test is given in Figure 4.4 [30]. This technique, although conceptually

64 simple, becomes unwieldy due to the large dimensions of the components as longer

pulse durations are needed for the accumulation of larger strains.

Figure 4.4: Experimental record from a test on PMDI foam using a long Kolsky bar

4.2 Introduction of an Intermediate Strain Rate Apparatus

A novel apparatus for compression testing at intermediate strain rates, introduced by Gilat and Matrka [32], is discussed. The apparatus, shown schematically in Figure

4.5 [32], is comprised of a hydraulic actuator and a long transmitter bar, 40m in length.

The specimen is placed between the end of the transmitter bar and the actuator. The specimen is loaded directly by the actuator. The custom designed hydraulic actuator can generate a force up to 22kN at speeds up to 2m/s. The actuator has a stroke of

90mm. As the sample is loaded, a compression wave propagates down the transmitter bar. The wave is measured with a Wheatstone bridge located near the specimen on the transmitter bar. The experiment can continue until the wave propagates down the

65 transmitter bar, reﬂects back and reaches Wheatstone bridge again, approximately

16ms with the current conﬁguration. All strain measurements are made using Digital

Image Correlation, see Section 3.4.

Figure 4.5: Schematic representation of the intermediate strain rate apparatus

This intermediate strain rate apparatus alleviates the shortcomings associated with other methods for testing at this strain rate. Ringing is not an issue due to the long transmitter bar and the test is able to continue for a considerably long duration, allowing large strains to be accumulated at low strain rates. The force measured by the transmitter bar can be used to determine the response of the material without the arduous numerical calculations or experimental calibrations necessary when traditional load frames are used at this strain rate. Similarly, this design is able to generate signiﬁcantly longer loading pulses than a similarly sized compression kolsky bar. A kolsky bar of immense length would be needed to generate a loading pulse of

16ms and at this scale dispersion would complicate matters. The use of a hydraulic actuator eliminates the need for a striker and an incident bar and the location of the

Wheatstone bridge eliminates the eﬀects of dispersion in the transmitter bar. The abilities of the test apparatus are limited only by the speciﬁcations of the hydraulic actuator used, namely its force capacity, velocity limit, and stroke range.

66 Chapter 5: Compression Experimental Results and Discussion

This chapter presents the experimental program that was used in evaluating the

proposed intermediate strain rate apparatus. The methods used for quasi-static and

dynamic testing as well as the associated calculations are also presented and discussed.

Experimental results from the intermediate strain rate apparatus test series, de-

scribed in Table 5.1, are presented. Quasi-static, intermediate, and high strain rate

data are presented. Data from the intermediate strain rate apparatus shows good

agreement with data recorded using existing test apparatuses. Data spread for each

test conﬁguration can be seen in Appendix B. Representative curves from the test

series are selected and presented here to illustrate the data trends.

5.1 Experimental Test Plan

The performance of the intermediate strain rate apparatus is evaluated by completing a strain rate test series on two materials in compression. First, specimens were fabricated from the same 0.5in Al2024-T351 plate used in the shear test series.

Specimens were cut such that their axes of symmetry were parallel to the rolled direction of the plate. Al2024-T351 does not exhibit strain rate sensitivity at or below

5000s−1 [24]. Second, samples were fabricated from a 0.25in diameter rod of Cu-101.

67 This material is known to exhibit strain rate sensitivity [33]. A test plan was created

to test both materials over a range of strain rates from 0.01s−1 to 5000s−1. Tests were conducted with existing compression apparatuses and the intermediate strain rate apparatus. The test plan for this test series is outlined by Table 5.1. Specimen orientations are illustrated by Figure 5.1. Results from these tests are presented at the end of this chapter.

Table 5.1: Test plan to evaluate intermediate strain rate apparatus Test Specimen Testing Strain Rate No. Material Apparatus (1/s) 1 0.01 Load Frame 2 1.0 Al2024-T351 3 Intermediate Strain Rate Apparatus 100 4 Compression Kolsky Bar 5000 5 0.01 Load Frame 6 1.0 Cu-101 7 Intermediate Strain Rate Apparatus 100 8 Compression Kolsky Bar 5000

68 Figure 5.1: Specimen fabrication orientations: (a) 0.5in thick Al2024-T351 plate, (b) 0.25in diameter Cu-101 rod

5.1.1 Compression Specimens

Compression tests are conducted using solid cylindrical specimens with the geometry shown in Figure 5.2. The length to diameter ratio of the specimens is one, signiﬁcantly higher ratios increase the chance for specimen buckling which would result in a complex stress state that is no longer uniaxial. Specimens are fabricated with conventional machining techniques and dimensions are recorded prior to testing.

69 Figure 5.2: Compression specimen used in plastic deformation testing(dimensions in inches)

5.2 Quasi-Static Testing Technique

Quasi-static compression tests were performed using the Instron model 1321 load frame discussed previously in Section 3.2. Compression tests are conducted using a pair of short steel pushrods loaded into collets shown in Figure 5.3. Masking tape is used to prevent glare from external lighting that would be detrimental to the Digital

Image Correlation system used for all tests, see Section 3.4. Tungsten Carbide inserts are used as platens during compression testing to avoid fixture deformation. The pushrods are aligned through the use of a pin that connects both rods together until the collets are tightened. After the collets are tightened the platens are installed and checked to ensure parallelism. Interfaces between the specimen and the platens are lubricated with Molybendum Disulfide (MoS2) grease to minimize the effects of friction and reduce specimen barreling, which results in a non-uniaxial stress state. Prior

70 to testing, specimens are centered on the platens and a small preload (approximately

5lb) is applied to keep the surfaces in contact.

Figure 5.3: Steel pushrods with tungsten carbide inserts

5.3 Intermediate Strain Rate Testing Technique

Intermediate strain rate tests are performed using the intermediate strain rate testing apparatus introduced in Section 4.2. The apparatus, shown in Figure 5.4 was used to conduct compression tests on Cu and Al samples at a strain rate of 100s−1.

The apparatus consists of a custom hydraulic actuator designed with TestResources

71 [34], shown in Figure 5.5. The actuator is mounted to a counterweight, and is capable

of generating 22kN of force while moving at velocities up to 2m/s with a stroke of

90mm. A long transmitter bar, 40m in length, is used to measure force. A spring

and damper at the end of the transmitter bar brings it to rest after a test. Diﬀerent

transmitter bar materials and diameters are available depending on the strength of

the sample. The apparatus can be ﬁtted with either a 12.7mm Ti-6Al-4V bar, a

22.23mm 304L stainless steel bar, or a 38.1mm 304L stainless steel bar. Bars with higher impedences have a lower particle velocity, however, the force sensitivity also decreases. All bars have a cumulative length of 40m and are comprised of individual

10 to 12 foot long bars joined with threaded couplings. This test program employed the 22.23mm 304L stainless steel transmitter bar.

Figure 5.4: Intermediate strain rate apparatus at the DMML

Although the design eliminates ringing with the use of a long transmitter bar, dispersion eﬀects can still disrupt the strains measured by the Wheatstone bridge

72 Figure 5.5: Intermediate strain rate apparatus hydraulic actuator

near the specimen for determining the loading if a specimen with a diameter much smaller than that of the bar is used. The specimens being tested have a diameter of 3.05mm, and due to the high forces needed for large deformation the 22.23mm transmitter bar was needed. A Futek load cell, shown in Figure 5.6, is glued directly on the end of bar and serves as a redundant force measurement. Force measurements were recorded with both the Futek load cell and instrumented transmitter bar during tests.

Force is measured by strain gages in a full Wheatstone bridge located 365mm from the end of the transmitter bar to determine the force in the specimen. The bridge is construced from four Micro-Measurements ED-DY-125AC-10C 1000Ω strain gages. 15.0V excitation is provided by an Agilent 3611A power supply. The signal is conditioned by a LeCroy DA1855A differential amplifier with a gain of 10 and a low pass filter set to 100kHz. The force transmitted by the sample is also measured with a Futek LCA300 load cell that has a capacity of 22.3kN. 10.0V excitation and

73 Figure 5.6: Load cell glued to transmitter bar of the intermediate strain rate apparatus

signal conditioning are both provided by a Vishay Measurements Model 2311 signal

conditioning ampliﬁer with a gain of 10 and a low pass ﬁlter set to 10.0kHz. Both signals are recorded using a four channel 350MHz, 8 bit Tektronix TDS5034B digital phosphor oscilloscope.

The hydraulic actuator is controlled by a TestResources 2350-02 controller oper- ated by a laptop via USB. The controller allows diﬀerent displacement curves to be programmed for the actuator. For this test series a sinusoidal displacement curve was used. The frequency of the sinusoidal curve can be adjusted to provide diﬀerent actuator velocities during the test. The location of the transmitter bar is adjusted so impact occurs just before constant velocity is achieved. The location of the actuator is measured by the controller with a Biss Bi-LV-3/19-025 LVDT. The data from the

74 LVDT is also output from the controller through a digital to analog converter as a

±10.0V signal that is also recorded by the oscilloscope during testing.

During a test the sample is impacted directly by the hydraulic actuator generating a compression wave that propagates down the transmitter bar. This wave is measured by both the Wheatstone bridge and the load cell. Molybendum DiSulﬁde grease is used to lubricate the interfaces with the specimen to reduce the eﬀects of friction and maintain a state of uniaxial compression. The engineering stress in the sample is proportional to the magnitude of the wave, ǫt, and can be determined from the

Wheatstone bridge measurement by

E A ǫ σ(t)= b b t (5.1) As

where Eb is the modulus of the bar material, Ab is the cross sectional area of the bar,

and As is the initial cross sectional area of the specimen. The engineering stress can also be determined from the measurement obtained by the load cell from

F σ(t)= c (5.2) As

where Fc is the force measured by the load cell.

All specimen strain data is recorded using Digital Image Correlation during tests.

DIC provides strain data directly on the surface of the specimen and also the relative

positions of the platens. Force data from the Wheatstone bridge and the Futek load

cell recorded during a typical test on Cu-101 is shown in Figure 5.7. The specimen

is subjected to a peak load of roughly 7kN. The loading duration was approximately

8ms, after which the sample deformation slowed, indicating the end of the test. Mea-

surements from both the Wheatstone bridge and Futek load cell agree. All data

presented is based on the Wheatstone bridge measurement. Processed experimental

75 data can be seen in Figure 5.8, which shows the stress, strain rate, and strain history

of a specimen from a typical test on the intermediate strain rate apparatus. The

strain history of the specimen was determined from DIC data. The derivative of the

strain history was then calculated to determine the strain rate history. The strain rate

data was then ﬁltered using a moving average with a window of 100 points to reduce

noise. As illustrated on the strain rate history, the strain rate of the specimen during

the test is not constant. The average strain rate of the specimen for the data shown

is 70s−1. The hydraulic actuator used is tunable in that the speed of the actuator can be adjusted during the test duration. Further reﬁnement of the test technique could allow a more constant strain rate to be achieved. Additionally, by adjusting the speed of the actuator to compensate for the increase in force as the specimen is deformed, the decrease in strain rate as the test progresses could be lessened.

76 Figure 5.7: Force history from a typical test on Cu-101 with the intermediate strain rate apparatus

Figure 5.8: Processed data from a typical intermediate strain rate test on Cu-101, average strain rate 70s−1

77 5.4 High Strain Rate Compression Apparatus

The compression Kolsky Bar at the DMML, see Figures 5.9 [23] and 5.10 [23], consists of three 12.7mm diameter Ti-6Al-4V rods. The incident and transmitter bars are both 1880mm long. A 610mm long striker bar is fired into the incident bar at velocities up to 40m/s. This impact generates the loading wave which has a nominal duration of approximately 250µs. A tube located at the end of the transmitter bar utilizes an energy damping material to gently bring the bars to rest after a test. Strains are measured in the incident and transmitter bars using strain gages arranged in a full Wheatstone bridge configuration. 15.0V excitation is provided by a pair of Hewlett Packard E3611A power supplies. The Wheatstone bridges are each constructed from four Micro-Measurements ED-DY-125AC-10C 1000Ω strain gages located in the center of both the incident and transmitter bars. Signal conditioning is performed by a pair of Standford Research Systems, model SR560, preamplifiers with a low pass filter set to 100kHz. The signals are measured using a four channel 350MHz, 8 bit Tektronix TDS5034B digital phosphor oscilloscope at a rate of

2.5 × 106Hz. The interface between the bars and the specimen is lubricated with

Molybendum Disulﬁde grease in order to reduce the eﬀect of friction and maintain a state of uniaxial compression within the specimen.

An overview of the test methodology and the data analysis necessary for the compression Kolsky bar technique is given by Gray [35]. With this technique a gas gun is used to ﬁre a striker bar which then impacts the end of the incident bar, creating an elastic strain wave in the incident bar. This strain wave, ǫi, travels through the incident bar toward the specimen. The amplitude of the incident wave is dependent

78 Figure 5.9: Compression Kolsky bar used by the DMML

Figure 5.10: Schematic representation of the compression Kolsky bar

79 on the velocity of the striker, v, and is given by

v ǫi = (5.3) 2cb

where cb is the longitudinal elastic wave velocity in the bar that is given by

Eb cb = (5.4) s ρb

where Eb and ρb are the elastic modulus and density, respectively, of the bars used in the test apparatus. When the incident wave reaches the specimen, the specimen begins to deform plastically. A portion of the wave travels through the specimen and into the transmitter bar, which is known as the transmitted wave ǫt. The remaining portion of the incident wave reﬂects back into the incident bar as a tensile wave,

ǫr, unloading the bar. The engineering stress in the specimen is proportional to the transmitted pulse and can be determined by

E A ǫ σ(t)= b b t (5.5) As

where Ab is the cross sectional area of the bars used and As is the initial cross sectional area of the specimen being tested. The strain rate in the sample is proportional to the relative velocity between the ends of the incident, ui, and transmitter, ut, bars at

the interface with the specimen. The strain rate in the sample can be determined by

u˙ − u˙ 2c ǫ ǫ˙ = i t = b r (5.6) ls ls assuming

ǫi = ǫt + ǫr (5.7)

where ls is the initial gage length of the specimen. Once the time dependent strain rate has been determined it is integrated to determine the strain history of the specimen.

80 The forces in the two ends of the specimen can be found from.

Fi = AbEb (ǫi + ǫr) (5.8) and

Ft = AbEbǫt (5.9)

Recorded wave data from a typical test on Al2024-T351, shown in Figure 5.11, shows the incident wave beginning at approximately 35µs and increasing to a value of approximately 28kN. The reﬂected wave begins at approximately 415µs and the transmitted wave begins at approximately 430µs. In this example, the test is considered complete at approximately 550µs when there is a sudden drop in the transmitted force, indicating failure of the specimen. Processed experimental data from the recorded waves can be seen in Figure 5.12, which shows the stress, strain rate, and strain history of a specimen from a typical test on the compression Kolsky bar.

81 Figure 5.11: Typical wave data from a compression Kolsky bar experiment on Al2024- T351

Figure 5.12: Processed wave data from a typical compression Kolsky bar experiment on Al2024-T351

82 5.5 Intermediate Strain Rate Data

Presented strain data are measured with DIC. The compressive strain in Figure

5.14 and in all plots in Appendix B is a hybrid of sample surface strain and strain calculated from platen motion. Initially, the hybrid strain is the average Hencky strain in a rectangular area on the surface of the specimen. After yielding, DIC is used to determine the motion of the platens which is used to determine the engineering strain in the specimen. These measurement locations are illustrated by Figure 5.13.

Figure 5.13: DIC measurement locations during a compression test

The intermediate strain rate apparatus is designed to eliminate ringing during a test. The 0.5in Al2024 plate tested is known to be insensitive to strain rate. Figure

5.14 shows data from Al2024 specimens tested at a low strain rate, 0.01s−1, on an

83 Instron load frame and at an intermediate strain rate, 100s−1, using the proposed apparatus. As illustrated, the intermediate data correlates well with the load frame data and also does not exhibit ringing that is common to other intermediate rate test methodologies.

Figure 5.14: Comparison of Al204-T351 data between load frame and intermediate strain rate apparatus

84 5.6 Strain Rate Eﬀect

Two materials (Al2024-T351 and Cu-101) are tested in compression at various strain rates. Al2024-T351 specimens are fabricated from a previously tested plate known to not exhibit strain rate sensitivity [24]. Conversely, the Cu-101 should exhibit strain rate sensitivity [33]. The experimental data from the intermediate strain rate apparatus should reﬂect these trends as well. Figure 5.15 shows true stress versus true strain data for the Cu-101 specimens at various strain rates. Tests at a strain rate of

0.01s−1 and 1.0s−1 were conducted on an Instron load frame, tests at 100s−1 on the intermediate strain rate apparatus, and tests at 5000s−1 on a compression Kolsky bar.

As illustrated, the intermediate strain rate apparatus captures the strain rate eﬀect of Cu-101 as expected. The agreement of experimental data with expected results illustrates the eﬀectiveness of the intermediate strain rate apparatus and corroborates the test methodology.

85 Figure 5.15: True stress versus true strain for Cu-101 over a range of strain rates

86 Chapter 6: Summary and Conclusions

A new specimen geometry for the characterization of thin sheet metals in sim-

ple shear and an intermediate strain rate apparatus for compression are introduced.

Experimental test plans are created to evaluate the simple shear geometry and inter-

mediate strain rate apparatus. Data spread from the two test series can be seen in

Appendix A and Appendix B respectively.

6.1 Simple Shear Experimental Conclusions

Detailed results from the simple shear test series are presented in Chapter 3.

Experimental methods used in the testing are also described in Chapter 3. A 0.5in

Al2024-T351 plate is tested due to the previously available shear data over a wide range of strain rates from torsion tests with thin walled tube geometry. Simple shear specimens are fabricated and the experimental data is compared to torsion data from specimens with thin walled tube geometry. Tests are performed according to the test plan outlined in Table 3.1. Simple shear specimens are fabricated with their gage sections orientated in various directions. A test is also performed using the thin walled tube specimen geometry to conﬁrm the existing experimental data.

Simple shear specimens are tested in quasi-static conditions with an Instron 1321 load frame and in dynamic conditions with a direct tension Kolsky bar. The test on

87 the thin walled tube geometry is conducted with a stored torsion Kolsky bar. Three-

dimensional Digital Image Correlation is used during all tests to measure deformation

and strains directly on the surface of specimen being tested.

Tests with the simple shear specimen are conducted at shear strain rates ranging from 0.01s−1 to 9000s−1. Stress versus strain curves from both the simple shear and thin walled tube geometries agree indicating that the proposed specimen geometry is suitable for characterizing thin sheet metals in shear. The simple shear geometry is able to measure anisotropic effects which are averaged in torsion tests on thin walled tube specimens. Experimental results indicate that different specimen orientations can result in differences in failure strains and plastic flow stresses. A numerical study is performed using LS-DYNA to determine the state of strain within the gage section and compare results to experimental measurements taken with DIC. The results from the simulation and DIC indicate that most of the gage section is under a uniform state of shear strain and that nearly pure shear is achieved until large shear strains are accumulated in the specimen. Results also indicate that plastic deformation is limited to the gage section of the specimen.

6.2 Compression Experimental Conclusions

Results from the experimental test series to evaluate the proposed intermediate strain rate apparatus are presented in Chapter 5. Experimental methods are also presented in Chapter 5. Two materials are chosen for the experimental testing: a strain rate insensitive material, Al2024-T351, and a strain rate sensitive material, Cu-101.

Tests are performed following the experimental outline given in Table 5.1. Compres- sion tests are performed using cylindrical specimens over a wide range of strain rates

88 ranging from 0.01s−1 to 5000s−1. Quasi-static tests are performed using an Instron

1321 load frame, intermediate rate tests are performed using the proposed intermediate strain rate apparatus, and high rate tests are performed using a compression

Kolsky bar. Three-dimensional DIC is used during all tests to measure deformations and strains.

Experimental data illustrates that the intermediate strain rate apparatus is able to capture the expected data trends for both of the materials, indicating that the proposed apparatus is capable of testing engineering materials at intermediate strain rates. The design allows signiﬁcant strains be accumulated during the test duration at intermediate strain rates. The experimental data also shows that the apparatus is not aﬀected by ringing, which is common with other methods for intermediate strain rate testing.

89 Appendix A: Experimental Results from the Simple Shear Test Series

The experimental results from each of the test sets used to evaluate the simple shear geometry are presented here to show the amount of data spread in each set.

Data are presented in shear stress versus shear strain. Results and discussion are presented in Chapter 3.

90 A.1 Simple Shear Results

A.1.1 Rolled Direction

Figure A.1: Shear test data spread: rolled direction,γ ˙ =0.01s−1

91 Figure A.2: Shear test data spread: rolled direction,γ ˙ = 2000s−1

92 Figure A.3: Shear test data spread: rolled direction,γ ˙ = 4500s−1

93 Figure A.4: Shear test data spread: rolled direction,γ ˙ = 9000s−1

94 A.1.2 Transverse Direction

Figure A.5: Shear test data spread: transverse direction,γ ˙ =0.01s−1

95 Figure A.6: Shear test data spread: transverse direction,γ ˙ = 2000s−1

96 Figure A.7: Shear test data spread: transverse direction,γ ˙ = 4500s−1

97 Figure A.8: Shear test data spread: transverse direction,γ ˙ = 9000s−1

98 A.1.3 45◦ from Rolled Direction

Figure A.9: Shear test data spread: 45◦ from rolled direction,γ ˙ =0.01s−1

99 Figure A.10: Shear test data spread: 45◦ from rolled direction,γ ˙ = 2000s−1

100 Figure A.11: Shear test data spread: 45◦ from rolled direction,γ ˙ = 4500s−1

101 Figure A.12: Shear test data spread: 45◦ from rolled direction,γ ˙ = 9000s−1

102 A.2 Torsion Results

Figure A.13: Torsion test data with spool specimen,γ ˙ = 2000s−1

103 Appendix B: Experimental Results from the Test Series Evaluating the Intermediate Strain Rate Apparatus

Experimental results from each of the tests used to evaluate the intermediate strain rate apparatus are presented here. Data is presented in true stress versus true strain. Note that the sign of the data has been ﬂipped for easier interpretation. The data presented has been clipped to true strain values of 0.36 because at large strains barreling occurs and the stress state is no longer uniaxial. Results and discussion are presented in Chapter 5.

104 B.1 Results from the 0.5in Al2024-T351 Plate

Figure B.1: Compression test data spread: Al2024-T351, rolled direction,ǫ ˙ =0.01s−1

105 Figure B.2: Compression test data spread: Al2024-T351, rolled direction,ǫ ˙ =1.0s−1

106 Figure B.3: Compression test data spread: Al2024-T351, rolled direction,ǫ ˙ = 100s−1

107 Figure B.4: Compression test data spread: Al2024-T351, rolled direction,ǫ ˙ = 5000s−1

108 B.2 Results from the 0.25in Cu-101 Rod

Figure B.5: Compression test data spread: Cu-101,ǫ ˙ =0.01s−1

109 Figure B.6: Compression test data spread: Cu-101,ǫ ˙ =1.0s−1

110 Figure B.7: Compression test data spread: Cu-101,ǫ ˙ = 100s−1

111 Figure B.8: Compression test data spread: Cu-101,ǫ ˙ = 5000s−1

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