Inverse problems for the dynamic characterization of materials

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DYNAMIC CHARACTERIZATION OF

MATERIALS

by:

CAMILO HERNANDEZ ACEVEDO

Doctoral Thesis Submitted in Partial Fulfilment of the Requirements for the Award of Doctor of Philosophy

Structural Integrity Research Group Mechanical Engineering Department

Universidad de los Andes 2014

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DYNAMIC CHARACTERIZATION OF

MATERIALS

by:

Camilo Hernandez Acevedo

Supervisor:

Dr. Alejandro Maranon

External Advisor: Dr. Ian A. Ashcroft

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This is to certify that I am responsible for the work submitted in this thesis, that the orig-inal work is my own except as specified, and that neither the thesis nor the origorig-inal work therein has been submitted to this or any other institution for a degree.

CAMILO HERNANDEZ ACEVEDO August 6, 2014

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I would like to express my gratitude to all those who make this work possible and in any way contributed to the culmination of this dream. First, I would like to express my sincere appreciation for all the support and guidance of my supervisor Dr Alejandro Maranon. His leadership was fundamental for the research. I am extremely thankful to my parents for all their hard work and sacrifice in life that have allowed me the opportunity to receive the best education. Finally, I want to thank my beloved Alicia for her understanding, pa-tience, love and continued support.

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1 INTRODUCTION 1

1.1 EARLY YEARS OF MECHANICAL CHARACTERIZATION OF

MA-TERIALS . . . 1

1.2 INVERSE PROBLEMS . . . 3

1.3 NUMERICAL SIMULATIONS . . . 7

1.4 OPTIMIZATION . . . 8

1.5 LOADING CONDITIONS . . . 11

1.6 HIGH STRAIN RATES . . . 14

1.7 SUMMARY AND CONCLUSION . . . 17

1.8 PROBLEM STATEMENT . . . 18

1.9 RESEARCH OBJECTIVES . . . 18

1.10 SCOPE OF THE RESEARCH . . . 18

1.11 RELEVANCE OF THE RESEARCH . . . 19

1.12 ROAD MAP TO THIS THESIS . . . 19

1.13 DISSEMINATION AND EXPLOITATION . . . 20

2 MATERIAL CHARACTERIZATION PROCEDURE 23 2.1 MECHANICAL CHARACTERIZATION OF MATERIALS AT HIGH STRAIN RATES . . . 24

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2.2 INVERSE PROBLEM FORMULATION . . . 26

2.3 INPUT: CHARACTERISTIC SIGNAL FROM A HIGH STRAIN RATE TEST . . . 29

2.3.1 Data reduction operator . . . 30

2.3.2 Line Moments . . . 32

2.4 OBJECTIVE FUNCTION . . . 35

2.5 REFERENCE MODEL . . . 38

2.6 OPTIMIZATION OPERATOR: GENETIC ALGORITHMS . . . 39

2.6.1 Initialization . . . 40

2.6.2 Evaluation . . . 42

2.6.3 Selection . . . 42

2.6.4 Crossover . . . 43

2.6.5 Mutation . . . 44

2.6.6 Elitism . . . 44

2.7 COMPUTATIONAL INVERSE CHARACTERIZATION TECHNIQUE . 44 2.8 Output . . . 45

2.9 SUMMARY AND DISCUSSION . . . 47

3 CASE I: INVERSE PROBLEM OF DROP TEST FOR CHARACTERI-ZATION OF SOFT MATERIALS 49 3.1 INTRODUCTION AND LITERATURE SURVEY . . . 50

3.2 DROP TEST . . . 53

3.2.1 Average strain rate of the Drop test . . . 54

3.4 INVERSE PROCEDURE FOR MATERIAL CHARACTERIZATION FROM DROP TEST . . . 55

3.4.1 Objective function . . . 56

3.4.2 Finite element model of Drop Test . . . 58

3.5 VALIDATION OF THE COMPUTATIONAL CHARACTERIZATION TOOL WITH A THEORETICAL EXAMPLE . . . 63

3.5.1 Results of the theoretical validation . . . 64

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3.6.1 Power Law plasticity material model . . . 70

3.6.2 Material homogenization and conditioning . . . 72

3.7 EXPERIMENTAL PROCEDURE . . . 72

3.7.1 Quasi-static compression . . . 72

3.7.2 Drop Tests . . . 73

3.8 RESULTS . . . 75

3.8.1 Quasi-static compression . . . 75

3.8.2 Drop Tests . . . 75

3.9 PARAMETERS DETERMINATION . . . 79

3.9.1 Characterization from quasi-static compression tests . . . 79

3.9.2 Characterization from the proposed inverse method . . . 79

3.10 VALIDATION OF MATERIAL PARAMETERS . . . 83

4 CASE II: INVERSE PROBLEM OF SPLIT HOPKINSON PRESSURE BAR TEST FOR CHARACTERIZATION OF MATERIALS 91 4.1 INTRODUCTION AND LITERATURE SURVEY . . . 92

4.2 SPLIT HOPKINSON PRESSURE BAR TEST . . . 93

4.2.1 Wave interaction analysis and stress-strain curve reconstruction . 94 4.3 INVERSE PROBLEM FORMULATION . . . 100

4.4 INVERSE PROCEDURE FOR MATERIAL CHARACTERIZATION FROM A SINGLE SHPB TEST . . . 100

4.4.2 Finite element model of SHPB . . . 102

4.6 MATERIAL: COPPER ALLOY UNS C83600 . . . 113

4.7.1 Description of SHPB equipment used and implemented in UNIAN-DES . . . 115

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4.9.1 Characterization from curve-fitting method . . . 120

5 CASE III: INVERSE PROBLEM OF TAYLOR TEST FOR CHARAC-TERIZATION OF MATERIALS 133 5.1 INTRODUCTION AND LITERATURE SURVEY . . . 134

5.2 TAYLOR IMPACT TEST . . . 135

5.4 INVERSE PROCEDURE FOR MATERIAL CHARACTERIZATION FROM TAYLOR IMPACT TEST . . . 138

5.4.2 Finite element model of the Taylor test . . . 140

5.6 MATERIAL: LOW CARBON STEEL AISI 1018 . . . 152

5.7.1 Split Hopkinson Pressure Bar Tests . . . 153

5.7.2 Taylor test . . . 153

5.8 EXPERIMENTAL RESULTS . . . 155

5.8.1 Split Hopkinson Pressure Bar Tests . . . 155

5.8.2 Taylor test . . . 155

5.9.1 Characterization from curve-fitting method . . . 155

6 CONCLUSIONS AND FUTURE WORK 167 6.1 CONCLUSIONS . . . 167

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6.2 MAIN CONTRIBUTION . . . 171 6.3 FUTURE WORK . . . 172

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2.1 Material characterization procedure diagram . . . 27

2.3 Characteristic signals used as input from three differen high strain rate test. Drop test (left), Split Hopkinson pressure bar test (center), Taylor impact test (right) . . . 36

2.4 Input characteristic signal measured from a SHPB test . . . 36

2.7 Genetic algorithm flowchart . . . 41

2.8 Illustration of uniform crossover [1] . . . 46

2.9 Inverse computational technique for determining the constitutive model parameters from a high strain rate test . . . 46

3.1 Penetration history over time from Drop Test on Plastilina Roma No.1 . . 57

3.2 Inverse procedure for material characterization from Drop Test . . . 57

3.3 Drop Test finite element model . . . 59

3.4 Comparison of penetration history between numerical simulations of the Drop Test with analytical solution at different configurations. . . 62

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3.6 Computational effort. Quality of solution over time . . . 69

3.7 Boxplots for each parameter of the characterization procedure in a theo-retical example . . . 71

3.8 Roma Plastilina No.1 color Grey-Green from Sculpture House (USA) . . 71

3.9 (a) Drop test experimental assembly (b) Final indentation measurement example . . . 74

3.10 Load-Heigh reduction curve from quasi-static compression tests on Roma Plastilina No. 1 . . . 76

3.11 Flow curve from quasi-static compression tests on Roma Plastilina No. 1 . 76 3.12 Drop test on Roma Plastilina No. 1 - Sphere of 63.5 mm dropped from 2 m height. Frames captured at 1472 fps . . . 77

3.13 Penetration profiles over time from Drop Tests on Roma Plastilina No. 1 . 78 3.14 Drop Test penetration curves from experimental tests and optimization results for sphere of 63.5 mm dropped from 2.0 m. . . 82

3.15 Computational effort. Quality of solution over time . . . 82

3.16 Drop Test penetration curves from analytical and FE models and valida-tion with experimental results for sphere of 44.45 mm dropped from (a) 1.0 m (b) 1.5 m (c) 2.0 m. . . 84

3.19 Drop Test penetration curves from FE model using parameters deter-mined from quasi-static tests and using parameters deterdeter-mined from the inverse procedure. . . 88

4.1 Illustration of SHPB apparatus . . . 96

4.2 Striker impacting the incident bar . . . 96

4.3 Schematic of the specimen interfaces between SHPB bars . . . 99

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4.5 Inverse method for determining material parameters from a single SHPB test . . . 103 4.6 Schematic of the finite element model of the SHPB test (not to scale) . . . 103 4.7 Comparison of strain history between numerical simulations of the SHPB

test with experimental measurements and analytical reference values at different impact velocities (a) 6.40 m/s (b) 16.5 m/second (c) 22.24 m/s (d) 27.11 m/s . . . 106 4.8 Transmitted pulse from theoretical example and results . . . 112

4.9 Computational effort. Quality of solution over time . . . 112 4.10 Boxplots for each parameter of the characterization procedure in a

theo-retical example . . . 114 4.11 Custom-made SHPB equipment implemented in the laboratories of the

Mechanical Engineering Department in Universidad de los Andes . . . . 116 4.12 Strain signals capture on bars during a SHPB test at 2714 s−1 . . . 119 4.13 Stress-strain curves at different strain rates for copper alloy UNS C83600 121 4.14 Determination of parameter from quasi-static stress-strain curve for

cop-per alloy UNS C83600. (Below) Zoom of elastic zone. . . 122 4.15 Transmitted pulse from SHPB test at 2713 s−1 used as input for the

in-verse characterization method . . . 124 4.16 Comparison between the experimental transmitted pulse and the

numer-cial simulations . . . 125 4.17 Computational effort. Quality of solution over time . . . 125 4.18 Transmitted pulses at strain rate of 1322 s−1 and FE simulation using

material parameters determined by curve-fitting and inverse method . . . 129

4.19 Transmitted pulses at strain rate of 2713 s−1 and FE simulation using material parameters determined by curve-fitting and inverse method . . . 129 4.20 Transmitted pulses at strain rate of 4325 s−1 and FE simulation using

material parameters determined by curve-fitting and inverse method . . . 130 4.21 Transmitted pulses at strain rate of 4921 s−1 and FE simulation using

material parameters determined by curve-fitting and inverse method . . . 130 4.22 Stress-strain curves for copper alloy UNS C83600 at different strain rates

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5.1 Schematic of the Taylor impact specimen (a) before and (b) after impact. v0impact velocity,L0 initial length,D0 initial diameter,Lf final length,

Df deformed diameter,Xundeformed length. . . 137

5.2 Extraction of silhouette from Taylor specimen. Photography (left), En-hanced contrast (center), Boundary detection (right) . . . 137 5.3 Silhouette of the Taylor specimen extracted from photography used as

input for the inverse material characterization technique . . . 137 5.4 Inverse computational technique for determining strength model

param-eters from a Taylor impact test . . . 141 5.5 Numerical simulation of the Taylor test implemented in the software

ANSYS/LS-DYNA . . . 141 5.6 Results of numerical model validation of Taylor test. Comparisson

be-tween specimen length ratio of aluminum alloy 6061 from simulations and experimental results, Wilkins and Guinan [2] . . . 144 5.7 Final shape of Taylor specimen of aluminum alloy 6061 from numerical

simulation and experimental result by Wilkins and Guinan [2].V0=275 m/s,

L0=46.94 mm,D=7.62 mm. . . 144

5.8 Silhouette of Taylor specimens from theoretical example and results . . . 148 5.9 Computational effort. Quality of solution over time . . . 150 5.10 Boxplots for each parameter of the characterization procedure in a

theo-retical example . . . 151 5.11 Schematic of the gas gun used in the Taylor test . . . 154 5.12 Stress-strain curves at different strain rates for low carbon steel AISI 1018 157 5.13 Silhouettes of the Taylor specimens after impact . . . 158 5.14 Computed silhouettes of the Taylor specimens using the results of the

inverse procedure . . . 160 5.15 Computational effort. Quality of solution over time . . . 161 5.16 Comparison between experimental Taylor specimen silhouettes at

differ-ent impact velocities and numerical simulations using material parame-ters determined by inverse procedure . . . 163

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2.1 Normalized Central Line Moments computed from the input characteris-tic signal of the SHPB test shown in Figure 2.4 . . . 37

3.1 Properties of each part of the numerical model of the Drop Test . . . 59 3.2 Material parmeters of modeling clay [3] . . . 61

3.3 Characteristics of the Drop Tests configurations used to validate the nu-merical model . . . 61 3.4 Search space limits for theoretical example . . . 66 3.5 Results of the inverse characterization procedure with theoretical

exam-ple. 10 consecutive runs. . . 66 3.6 Quality of results. Deviation of determined parameters from optimal . . . 67 3.7 Precision uncertainty and confidence interval with a confidence level of

95% of the computational characterization procedure with a theoretical example . . . 69 3.8 Configurations of the Drop Test performed on Plastilina Roma No. 1 . . . 74 3.9 Indentation depth from Drop Tests on Roma Plastilina No. 1 . . . 81 3.10 Roma Plastilina No.1 material parametrers from quasi-static tests . . . 81 3.11 Power Law plasticity material parameters for Roma Plastilina No. 1

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3.12 Precision uncertainty and confidence interval with a confidence level of 95% of the computational characterization procedure . . . 90 3.13 Power Law plasticity material parameters for Roma Plastilina No. 1

de-termined by inverse method with a confidence of 95% . . . 90

4.1 Properties of each part of the numerical model of the SHPB test . . . 103 4.2 Comparison of characteristic properties of the incident pulse between

nu-merical simulations of the SHPB test with experimental measurements and analytical reference values at different impact velocities. . . 107 4.3 Material parameters and properties for 1018 steel [4] . . . 107 4.4 Search space limits for theoretical example . . . 107 4.5 Results of the inverse characterization procedure with theoretical

exam-ple. 10 consecutive runs. . . 111 4.6 Quality of results. Deviation of determined parameters from optimal . . . 111 4.7 Precision uncertainty and confidence interval with a confidence level of

95% of the computational characterization procedure with a theoretical example . . . 114 4.8 Chemical composition of copper alloy sample and UNS C83600

chemi-cal requirements [5] . . . 114 4.9 Specimen dimensions and impact speed for SHPB tests . . . 116 4.10 Geometric dimentions of the SHPB bars . . . 119 4.11 Cowper-Symonds material parameters for copper alloy UNS C83600

de-termined by curve-fitting method . . . 121 4.12 Cowper-Symonds material parameters for copper alloy UNS C83600

de-termined by inverse method . . . 124 4.13 Precision uncertainty and confidence interval with a confidence level of

95% of the computational characterization procedure . . . 128 4.14 Cowper-Symonds material parameters for copper alloy UNS C83600

de-termined by inverse method with a confidence of 95% . . . 128 4.15 Comparisson of the transmitted pulse peak value. Experimental and

nu-merical measurements . . . 128

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5.2 Material parmeters of aluminum alloy 6061 [6, 7] . . . 143

5.3 Comparison of length ratio (Lf/L0) of aluminum alloy 6061 specimens from numerical model and experimental measurements by Wilkins and Guinan [2] . . . 143

5.4 Search space limits for theoretical example . . . 148

5.5 Results of the inverse characterization procedure with theoretical exam-ple. 10 consecutive runs. . . 148

5.6 Quality of results. Deviation of determined parameters from optimal . . . 150

5.7 Precision uncertainty and confidence interval with a confidence level of 95% of the computational characterization procedure with a theoretical example . . . 151

5.8 Chemical composition of low carbon steel AISI 1018 . . . 154

5.9 Specimen dimensions and impact speed for SHPB tests . . . 154

5.10 Taylor impact specimens . . . 154

5.11 Steel AISI 1018 strength material parameters from the material charac-terization using the customary curve-fitting method . . . 157

5.12 Cowper-Symonds material parameters for low carbon steel AISI 1018 determined by inverse method . . . 160

5.13 Precision uncertainty and confidence interval with a confidence level of 95% of the computational characterization procedure . . . 160

5.14 Cowper-Symonds material parameters for low carbon steel AISI 1018 determined by inverse method with a confidence of 95% . . . 161

5.15 Comparisson of the Taylor specimen final length. Experimental and nu-merical measurements . . . 166

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INTRODUCTION

Mechanical characterization of materials, in terms of determining the constant parame-ters that defines the mechanical behavior of materials by a given constitutive model, is a topic in research and engineering that has gained importance over the last decades. This high level of importance has been driven by the use of the finite element analysis (FEA), as a primary tool for modern engineering design.

History of mechanical characterization has been intimately related to the development of numerical simulations, and the need for new computational models of materials that can accurately describe their mechanical behavior under high strain loading. In following sections, the development of mechanical characterization procedures is reviewed. Its main trends and the techniques used to characterize materials at high strain rates are explored.

1.1 EARLY YEARS OF MECHANICAL CHARACTERIZATION OF

MA-TERIALS

The industrialization boom, which followed the Second World War, relied on both the understanding and perfectioning of manufacturing processes that included plastic de-formation of metals. For this purpose, early research efforts were made to formulate

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constitutive models to model plasticity at different strain rates; for example the Power Law plasticity model [8], and the Cowper-Symonds material model [9]. These material models consist of phenomenological stress-strain relationships that describe the plastic-ity of metals using material parameters to scale its mechanical response. In these studies, through the use of quasi-static tension and flexion tests the relations among stress, strain, and in some cases strain rate, were determined by curve fitting. Currently, these material models are still used in fields such as metal forming and impact analysis.

From the decade of 1980’s, with the rising of the computational power and machine computing, numerical models and computer simulations found an optimal environment to grow. Along with the fast growing of numerical simulation methods, the models to describe the mechanical behavior of materials found a favorable atmosphere that en-hanced the development and proposal of new constitutive models. Some of the most used material models presented during this period are the basis of the current numerical simulations. For example, Steinberg et al. [10] presented a constitutive model for metals to be used in computer hydrocodes for simulating impact events. Material parameters needed to implement the model were determined for several metals using a large number of experiments.

Probably, the currently most used constitutive model was proposed by Johnson and Cook [11, 12]. This material model, devised to implement numerical models of metals subjected to plastic deformation, is a phenomenological relation among stress, strain, strain rate and temperature. It consists of five material parameters that are determined for various material by uncoupling the model and curve fitting them from tension, torsion and dynamic compression tests over a range of temperatures.

Another approach was presented by Zerilli and Armstrong [13]. In this study the effects of strain hardening, strain-rate hardening, and thermal softening were modeled using a physical concept based on dislocation mechanics combined with phenomenolog-ical observations. The material parameters required for this model were also determined by curve fitting from diverse experimental tests.

More complex material constitutive models were also proposed by some authors try-ing to describe the behavior of different types of materials; Barlat and Lian [14] pro-posed a model to describe the behavior of certain metals in forming processes. This material model can describe the orthotropic behavior exhibit by metal sheets with planar

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anisotropy and subjected to plane stress conditions. Johnson and Holmquist [15, 16] pre-sented a model to represent the mechanical behavior of brittle materials such as ceramics and concrete.

All of the above mentioned studies have a common feature: the constitutive model formulation is accompanied by the characterization of some materials to demonstrate its performance. In all the cases, the characterization was performed following direct methods involving the uncoupling of the material model and determining the material parameters individually by curve fitting from stress-strain curves obtained from diverse tests, in most cases quasi-static tests. This line of working is still used nowadays; in fact, direct procedures to identify material properties are the customary methods proposed in a number of standards [17, 18].

Characterization procedures with similar direct methods, but adding new features, were proposed from this approach by some authors. For example, Müller and Hartmann [19] described an optimization method based on biologic evolution to ease the identi-fication of material parameters and shorten the computation time. Bruhns and Anding [20] devised a method to determine the material parameters simultaneously. The pre-sented method determines the material model parameters from uniaxial experiments, and although curve fitting methods based on a least-squares criterion were used, the parame-ters were identified simultaneously without uncoupling the model.

Nowadays, numerical tools based on direct methods and curve fitting are being pro-posed for automated identification of constitutive model parameters, such as the one pre-sented by Mohmann [21], among others. With these kind of computational tools material parameters are adjusted with respect to experimental data by fitting the constitutive model with an optimization method. For example, in the case of the methodology presented by Mohmann the selected optimization method was based on a simple gradient method.

1.2 INVERSE PROBLEMS

Although direct curve fitting methods proved to be convenient, novel characterization procedures arose from different fields seeking to provide reliability to material charac-terization methodologies. New requirements from more complex finite element simula-tions, applied to miscellaneous engineering fields, involving different loading conditions

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and demanding more accurate material models, drove the researchers to develop novel characterization methods able to identify the materials from new sources of experimen-tation and trustworthy procedures. Hence, researchers turned their attention to inverse problems as a method for this task.

Inverse problems consist of finding the values of the unknowns in a mathematical or numerical model such as the calculated response matches with the observed and mea-sured data. These kinds of problems closely resemble the nature of characterization of materials. New features such as the simultaneous estimation of parameters, the use of different experimental tests and more accurate solutions were added with the use of in-verse problems in the material characterization process. However, since most inin-verse problems are ill-posed and the uniqueness of the solution is not guaranteed, the solving process tends to be more difficult and requires well established procedures [22].

Probably one of the first studies involving the estimation of material parameters in solid mechanics was presented by Schnur and Zabaras [23]. In this study, they proposed an inverse method to determine the elastic properties and geometric parameters of defects in materials, specifically a circular inclusion. The inverse problem consisted in measuring the traction and displacement of a specimen and by matching a finite element model the material and geometric parameters were identified. Later, Zabaras and Badrinarayanan [24] extended the work to other parameter identification challenges, classified some of the problems of material processing that can be formulated as inverse problems, and discussed the ways of tackling these problems. Using this inverse approach and applying different experimental tests to diverse materials, a string of studies were published in the second half of the decade of the 1990’s.

Banks and Smith [25] presented a model for coupled torsion and bending in a beam to describe vibrations. The model was implemented in an inverse problem fashion for identifying the beam parameters, including material and geometric properties. Using ac-celerometer data, material properties such as elastic modulus, stiffness, torsional rigidity and density were estimated. Gelin and Ghouati [26, 27] applied the inverse method to determine viscoplastic material parameters of aluminum alloys from plane strain com-pression tests and axisymmetric comcom-pression test. Likewise, Moulton et al. [28] used a similar method in the field of biomedical engineering. They determined myocardial ma-terial properties by solving an inverse boundary value problem of the difference between

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model-predicted and measured strains obtained in the intact heart by magnetic resonance imaging radiofrequency tissue tagging. Chenot et al. [29] also used the inverse procedure to optimize the shape and materials parameters for the simulation of forming processes; one of the fields that pushed the topic of material characterization. Other experimental tests, such as torsion and compression, were also used in inverse problems to identify material parameters. Lam et al. [30] used these two experimental tests as input data in an inverse procedure to determine the constitutive parameters of metals.

Finally, it is worth mentioning the work by Gavrus et al. [31]. They proposed an identification method for the automatic computation of the rheological parameters of a material undergoing a large plastic deformation. The material consistency, the strain rate sensitivity with respect to the generalized plastic strain and the temperature softening were simultaneously estimated using the proposed method.

Lately, inverse problems devised to identify the material parameters has widespread to exploit the modern computational hardware. Inverse methods have been adapted to use modern finite element simulations of mechanical tests jointly with a wide range of opti-mization methods. This offers the possibility to characterize different types of materials in diverse loading conditions and using complex constitutive models. Some examples of characterization procedures based on inverse problems proposed in the last years that take advantage of the currently available tools are reviewed in the following paragrahs.

Zhou et al. [32] combined an optimization technique with the finite element method in an inverse problem to determine the material parameters of a constitutive model. Axisym-metric compression tests at high temperatures were used as input data to characterize a ceramic composite. Another inverse approach for the determination of constitutive model parameters was proposed by Cooreman et al. [33]. The inverse method was based on the comparison of strain measurements after the necking in experimental tensile tests with finite element simulations. The optimization of the material parameters was aided by the calculation of a sensitivity matrix, which express the sensitivities of the strains with respect to the material parameters. Using this approach, less iterations of the numerical model were required. Andrade-Campos et al. [34] determined the material parameters for a thermoelastic-viscoplastic constitutive model of an aluminum alloy by proposing an inverse problem in which the experimental results from uniaxial tension and simple shear at different temperatures were compared with the computation of a mathematical model.

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They compared two optimization methods, a gradient based method and an evolutionary algorithm, to determine the best suited material parameter set, concluding that the evolu-tionary algorithm is a better optimization method due to the starting point independence and the higher probability to find the global minimum.

In the last few years, the use of this kind of characterization based on inverse meth-ods has multiplied and the line of work is directed to the improvement of resources and the reduction of computational time. Eggertsen and Mattiasson [35] based their char-acterization procedure on an inverse problem. This procedure consisted of fitting finite element simulations to the moment-curvature curve at the middle of the sheet strip during a cyclic three-point bending test. This procedure offered advantages in terms of comput-ing time since the finite element model is solved less number of times. Chen et al. [36] presented an approach for identifying the material parameters of an orthotropic medium based on an inverse technique. The displacement measurements during a uniaxial tension test were used to optimize the difference with numerical simulations of the test. By using the scaled boundary finite element method, computation time was reduced. Bäker and Shrot [37] proposed a method to determine the material parameters using and inverse problem. They varied the standard inverse methods by including an auxiliary strategy in terms of physical descriptors of the experimental test in order to pre-estimate the ma-terial constitutive parameters. With the proposed strategy a reduction of the number of numerical simulations needed to determine the material parameters was achieved. How-ever, understanding of the physics of the problem was required to find the appropriate descriptor.

As reviewed, inverse problems offer a valuable methodology to decipher material char-acterization problems. However, to exploit the advantages provided by the inverse prob-lems it is essential to implement computational algorithms that ease the parameter iden-tification. This computational routines usually combine the performance of two main tools: a numerical simulation and an optimization method. The interaction of these two allows the identification of material parameters accurately, simultaneously, from diverse experimental tests using diverse loading conditions. In the following sections, studies concerning these two main topics are reviewed.

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1.3 NUMERICAL SIMULATIONS

With the emergence of inverse problems applied to the characterization of materials, the parameter identification methodologies evolved dramatically. Customary direct char-acterization procedures that determined the material parameters by curve fitting from stress-strain curves were replaced by procedures in which numerical simulations of ex-perimental tests are adjusted to exex-perimental measurements by varying the material pa-rameters. These procedures enabled the identification of more accurate parameters from a great variety of experimental tests. Additionally, the material parameters determined by this method, have the advantage to be ready to be used and virtually calibrated to use in other numerical simulations.

Many characterization procedures were proposed using the combined approach of in-verse problems with experimentation and numerical simulations. For instance, Meuwis-sen et al. [38] proposed a mixed numerical - experimental method to determine the con-stitutive model parameters. From uniaxial tensile tests instrumented with retro-reflective markers, the parameters were determined by optimizing the difference with a finite ele-ment model of test.

Another example was proposed by Huang et al. [39] for the characterization of com-posite materials. They proposed an identification algorithm to determine the properties of a two-dimensional orthotropic material. The identification algorithm was based on the minimization of the difference between measured and calculated displacements by a boundary element model. Numerical examples using uniform tension tests were used to validate the proposed algorithm. They noticed that using boundary element models, the computation is more efficient in terms of processing time.

A similar study to determine the properties of composite materials was presented by Anghileri et al. [40]. In this case, they proposed an inverse technique to determine the constitutive parameters of an ortho-tropic constitutive material model under dynamic loading. The inverse technique used an inverse mixed numerical-experimental approach to determine the parameters by correlating the numerical simulation with experimental results of dynamic axial crushing tests of carbon reinforced composites.

Furthermore, the use of finite element simulations in the process of material charac-terization was evaluated and compared with other techniques. De-Carvalho et al. [41]

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compared the performance of inverse problems to determine the material constitutive pa-rameters from two point of view. They evaluated the use of a single point method, in which a single infinitesimal point is used to compare the stress value, and the use of finite element analysis to fit the material parameters. They concluded that, although the single point method is considered more efficient in terms of computational cost, the finite ele-ment analysis allows knowing the entire deformation story and making a more accurate prediction of the material parameters.

On the other hand, numerical simulations have been used in material characterization procedures to validate the determined parameters. Hartmann et al. [42] implemented a direct search method complemented with a finite element model to determine the material parameters of a constitutive plasticity model. Although finite element simulations were successfully integrated to direct search methods, the advantages of numerical simulations are not fully exploited. The numerical simulations are merely used as a validation method for the parameters found.

To sum up, the integration of numerical simulation of experimental tests has become a dominant tool in inverse material characterization procedures. The performance of numerical simulations as a reference model to determine the material constitutive pa-rameters has been compared with other material characterization routines. Nevertheless, inverse problems in connection with numerical simulations have shown better and more accurate results than customary curve fitting procedures.

However, the coupled method of inverse problems with numerical simulations requires an extra tool consisting of an optimization procedure. Due to the complex search space derived from this kind of problems, diverse authors have studied the optimization tech-niques suitable for the specific application.

1.4 OPTIMIZATION

The optimization process during the determination of material constitutive parameters is a key aspect that defines the success or the failure of the inverse material characteriza-tion procedure. The optimizacharacteriza-tion process on inverse characterizacharacteriza-tion problems consist of minimizing the difference between numerical simulations and experimental measured data. However, the minimization process is complicated in most material

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characteriza-tion procedures, requiring the implementacharacteriza-tion of optimizacharacteriza-tion methods with features that allow the solution of complex search spaces with multiple local minima.

As a result of the complex search spaces derived from inverse characterization prob-lems, authors evaluated the possible optimization methods suitable for the task. Tradi-tional gradient-based optimization methods were first evaluated and poor results were obtained due to the incapability of gradient-based methods to find global minima in com-plex search spaces. This circumstance led the implementation of new optimization meth-ods based on evolutionary algorithms.

Szeliga et al. [43] compared the performance of different optimization methods in their work. Three optimization methods were evaluated for application in inverse problems to the identification of material model parameters. Genetic algorithms, derivative-free methods and gradient methods were used to identify parameters of diverse material mod-els. Advantages and disadvantages of the three optimization strategies were described, highlighting genetic algorithms as the most efficient optimization tool for the particular application regardless of the large computational cost. Another work was presented by Li et al. [44]. They evaluated three algorithms based on evolutionary programming to solve the optimization problem yielded in determination of material constants. They found that the objective functions to be optimized during material characterization procedures con-tained narrow ridges and large plateaus that make the optimization particularly difficult to classical gradient-based optimization techniques. The results of the study showed that the evolutionary algorithms are ideally suited to the problem. Following this recommen-dations, Franulovic et al. [45] and Dusunceli et al. [46] among many others, implemented characterization procedures with genetic algorithms as optimization method. In the case of Franulovic and coworkers, an inverse procedure from low-cycle fatigue tests was de-veloped using genetic algorithms and finite elements; and in the case of Dusunceli, an optimization procedure based on genetic algorithms was implemented to determine the material parameters for a viscoplastic model. The procedure minimizes the difference between experimental and simulation results of loading-unloading tensile tests.

On the other hand some authors combined the advantages of the two optimization techniques into hybrid methods. The ability to search wide and complex spaces of the evolutionary techniques and the agility to find the optimum in convex spaces of the gra-dient based methods were merged in hybrid methods. Chaparro et al. [47] evaluated two

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types of optimization techniques used in material parameters identification problems. A gradient-based strategy and an evolutionary algorithm were used to identify the constitu-tive model parameters of an aluminum alloy from tensile and shear test. They found that both optimization methods were capable to fit the experimental behavior, each one with particular problems. In the case of evolutionary methods, large calculation times were encountered; while in the case of gradient-based techniques, locking in local minima ocurred. To improve the optimization of material parameters a hybrid method combining both methods was proposed: a genetic algorithm was implemented to reach the vicin-ity of the global minimum and then, a derivative-based algorithm was used to reach the optimal set of parameters. A similar approach was used by Furukawa et al. [48]. They proposed an automated optimization system to determine the material parameters of in-elastic constitutive models by means of a hybrid optimization method that incorporated the advantages of gradient-based and evolutionary optimization techniques. Likewise, de-Carvallo et al. [49] evaluated the performance of two optimization strategies when used to determine the material parameters. An inverse problem to characterize materi-als from tensile tests was implemented and two optimization strategies based on cascade and parallel techniques were evaluated. They proposed a hybrid method, combining both strategies, to determine the material parameters.

Complementary studies on the optimization techniques were performed by other au-thors. These studies were focused on the objective functions and the nature of the exper-imental data used as input for the characterization procedures. For example, Andrade-Campos et al. [50] proposed variations of objective functions in constitutive model pa-rameter identification problems, in order to ease the optimization of the difference be-tween experimental and numerical results. The proposed objective function based on the addition of two main features: (i) all the experimental data points on the curve and all experimental curves should have equal opportunity to be optimized ; and (ii) differ-ent units and/or the number of curves in each sub-objective should not affect the overall performance of the fitting.

An alternative approach was considered by Seiberg et al. [51] in which an optimiza-tion procedure was proposed to identify the material parameters of inelastic constitutive models using a stochastic scheme. A stochastic model for generating artificial data was used in order to increase the sample size and enhance the quality of the material

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param-eter fitting. Creep and tensile tests were used as experimental data. Hart et al. [52] also used an stochastic model to generate artificial data to improve the confidence in the fit of the identification of material parameters. Experimental creep, tensile and cyclic tension-compression tests were used in the implementation of the method in combination with an optimization process. The objective of this characterization procedure was to reduce the number of experimental tests without losing substantial information and accuracy in the parameter identification. A similar method was used by Norenberg and Mahnken [53] to characterize the properties of adhesive materials from tension tests.

1.5 LOADING CONDITIONS

Besides the accuracy and the multi-objective determination of the material parameters gained with the implementation of inverse problems in the topic of material characteri-zation; one of the most valuable features added with the inverse problem technique is the ability to determine material parameters from different experimental tests. A broad range of experimental tests can be used as input in the inverse characterization procedures de-termining the material constitutive parameters at different loading conditions for diverse applications.

Most likely, one of the first fields that demanded the need to characterize materials under loading conditions different than the customary tension or compression tests was the metal forming. Ghouati and Gelin [54] presented an inverse characterization scheme in which the material parameters are determined directly from a metal forming tests. The proposed scheme was based on the coupling of the finite element method and an optimization technique to determine the material parameters under the loads established during the forming process. Furthermore, Ghouati and Gelin [55] extended their work to other loading conditions. A coupled finite element method with an optimization tech-nique was used to identify the material parameters under different loading conditions. Experiments such as the tensile tests, the deep drawing test and the bulge test were used with the proposed method to determine plasticity parameters. They concluded the method was more efficient than classical approaches and require fewer assumptions leading to more accurate material parameters. Similarly, Ponthon and Kleinermann [56], motivated by modeling the metal forming process, proposed the identification of material

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param-eters of constitutive models by minimizing the difference between experimental results and finite element simulations. Non-linear gradient-based optimization methods were used to solve the inverse problem. The punch test was used as experimental technique by Li and He [57] to determine the parameters of metals subjected to loading conditions similar to sheet forming. They proposed an identification procedure to determine the material parameters from the punch stretch test and an identification strategy that com-bined an optimization algorithm based on genetic algorithms and a finite element model of the test. Anisotropic behavior of materials described during sheet metal forming was characterized successfully using this technique. Dorogoy and Rittel [58] determined the Johnson-Cook material parameters with an inverse procedure in which the experimental results from numerical simulations of shear compression specimens were adjusted to ex-perimental tests. Quasi-static and dynamic tests at room and high temperature were used to determine the material parameters of metals used in metal forming processes.

On the other hand, novel measuring devices and experimental techniques that yield full-field measurements were also implemented in inverse characterization problems. For instance, Kajberg and Lindkvist [59] presented a method for the characterization of materials subjected to large strains using digital speckle photography. The character-ization method consisted of an inverse model of a uniaxial tensile test in which strain measurements are captured using digital speckle photography. The material constitutive parameters are determined by minimizing the difference between the experimental and finite element calculated displacement and strain fields. Pagnacco et al. [60] proposed a strategy to determine the elastic and viscoelastic material parameters using as experi-mental data full-field measurements. They used a single experiexperi-mental test to determine the parameters based on the gap minimization between experimental measurements and displacements given by the finite element model. Experimental measurements using high speed photography were also used as input data. Avril et al. [61] proposed a method for identifying the elasto-visco-plastic material parameters from tensile loading experiments on double-notched specimens to observe non-uniform strain rate conditions. The identi-fication was performed by comparing the full-field deformation measurements acquired with a high speed camera and compared with a virtual field method solution. Results from the proposed technique were compared with results from standard tensile tests, noticing discrepancies in the values of material parameters.

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Other group of experimental tests also used frequently for characterizing materials is the indentation test. From instrumented spherical indentation tests, Cao and Lu [62] or Kucharski and Mroz [63] proposed a characterization method based on inverse problems. The material parameters were determined from indentation tests using the loading curve in the case of Cao or the unloading curve in the case of Kucharski. Finite element compu-tations of the indentation test were used to compare with experimental results. A similar approach was used by Rauchs and Bardon [64]. In this study nano-indentation tests were used to determine material parameters. Fatigue properties were also determined using inverse characterization techniques. Franulovic et al. [65] published a study that presents an inverse procedure for characterizing the material parameters for a low-cycle fatigue model. An automated system based on a genetic algorithm was developed to determine the material parameters from cyclic uniaxial tension-compression tests. Equally, Eggert-sen and Matiasson [66] determined the material parameters for a constitutive model using an inverse procedure using two experimental tests: cyclic three-point bending and cyclic tensile/compression tests. The inverse determination of parameters yielded different set of hardening parameters with both experimental tests; demonstrating the importance of the selection of the adequate loading test conditions for the desired application.

Even composite materials with orthotropic properties were characterized using inverse procedures. Lecompte et al. [67] presented a mixed numerical-experimental technique to the identification of the elastic in-plane engineering constants for orthotropic composite material based on surface measurements. Biaxial tensile loading on a cruciform specimen were used and by means of digital image correlation surface displacements measured, the strains were calculated. The difference between measured strain and numerical strains calculated from a finite element model were minimized to determine the engineering pa-rameters. A Newton-Raphson algorithm was used as optimization routine. Another study on composite materials was presented by Rikards et al. [68]. They presented a method of material parameter identification based on experiment design and inverse problems. Mechanical properties of laminated composites were determined from measured eigen-frequencies of plates obtained with vibration tests. The proposed a method that mini-mized the difference of experimental results with finite element simulations, which were performed only in reference points of the structures to reduce the computational time.

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1.6 HIGH STRAIN RATES

Nowadays, a topic with accelerated growth in the field of numerical simulations is the modeling of dynamic events. Such simulations usually involve impacts, crashes and collisions where materials deform rapidly. Hence, the implementation of an accurate material model that describes truthfully the mechanical behavior is essential for the suc-cess of the numerical simulations. However, the material models required for this kind of applications can be complex and their material parameters should be determined under loading conditions that reflect the high strain rate condition of the simulated event. As result, over the years many authors have presented diverse characterization procedures to determine the material parameters using high strain rate tests.

One of the first high strain rate tests used to determine the material parameters from constitutive plasticity models was the Taylor impact test. This simple test, that consists of striking a small specimen against a rigid wall was used by Rule [69] in a characterization scheme. The author proposed a method for determining the material constants of a plas-ticity model from the Taylor impact test. The deformed geometry of a Taylor impact test was measured and used as input for the determination of the material constants. Appling an inverse problem, an analytical model of the deforming Taylor specimen was numer-ically integrated and used as reference. The material constants of the Johnson–Cook model were adjusted using an optimizer so that experimentally measured deformed ge-ometries were matched with those calculated by the analytical model. Likewise, Allen et al. [70] developed a methodology to allow a single Taylor impact test to be used as a simple and cost efficient means for obtaining and refining constants for dynamic material strength models. The methodology presented obtains the constants by minimizing the difference between the displacement results of a Taylor impact test and a hydrocode sim-ulation of the event. The material parameters were determined by minimizing the volume difference between a deformed Taylor test cylinder and the finite element simulation of the test.

Although the Taylor impact tests stands out due to its simplicity and economy, the split Hopkinson pressure bar test have become the most used dynamic tests worldwide. The capability of the test to deliver data in direct manner at high strain rates has made the Hopkinson devise the favorite dynamic experiment of numerous authors. Many

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dy-namic characterization procedures found in open literature are based on the use of split Hopkinson pressure bar tests as input experiments.

Kajberg and Wikman [71] used an inverse approach to determine the material con-stitutive parameters by minimizing the difference between the experimental and finite element calculated displacement and strain fields at high strain rates. They used a split Hopkinson pressure bar device to tests materials and by means of high speed photography and digital speckle photography the nonhomogeneous states of deformation were obtain and used in the material optimization process with the finite element analysis. Sasso et al. [72] also used dynamic tensile tests on a split Hopkinson pressure bar device to identify the material parameters for plasticity constitutive models. A post-process proce-dure based on the optimization of the difference between finite element simulations and the experimental nominal curves was used to obtain the material coefficients. A similar approach was implemented by Sedighi et al. [73]. They determined the optimal set of material parameters of some constitutive models by minimizing the standard deviation of the numerically obtained stress-strain curve from the experimental data and computed constitutive models.

A different method was proposed by Milani et al. [74]. The authors, in this oppor-tunity, proposed a method for determining the constitutive material parameters using a weighted multi-objective identification strategy. The characterization method used as in-put experimental tests from uniaxial tensile tests and dynamic compression tests from a split Hopkinson pressure bar device. The main advantage of the proposed method is that the material parameters are determined simultaneously.

Combination of dynamic tests were also used by Majzoobi et al. [75, 76]. In this inves-tigation authors determined the material properties of materials at high strain rates using dynamic experimental tests. Dynamic tensile tests on a flying wedge device and dynamic compression tests on a Hopkinson bar apparatus were performed. The parameters for plasticity constitutive material models were determined by a computational method that combined experimental, numerical and optimization approaches. Later, the study was improved by the same authors [77], implementing a new computational optimization procedure based on genetic algorithms. With this improvement, there was no need for experimental stress-strain curve which is always accompanied by restricting phenomenon such as necking in tension and bulging in compression. Instead of stress-strain curve, the

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difference between the post-deformation profiles of specimens obtained from experiment and the numerical simulations was adopted as the objective function for optimization pur-poses.

On the other hand, ad hoc dynamic experimental tests were used in different studies to characterize materials at high strain rate but under specific loading conditions. For ex-ample, Kajberg et al. [78] employed a method based on inverse problems to characterize materials at high strain rates under impact conditions. The authors proposed an experi-mental device to measure the strains in a small specimen impacted by a hammer using high speed photography equipped with a microscope lens. This method although demon-strated good potential for characterizing materials at high strain rates required a lot of improvement due to the complexity of the experimental set-up. Likewise, Notta-Cuvier et al. [79] used photography to measure strain fields. They proposed a method for the identification material parameters of a constitutive model using the virtual fields method and full-fields strain measurements by digital image correlation. The experimental data was acquired from notched flat specimens subject to quasi-static tensile tests and from dynamic tensile tests.

Another approach was proposed by Liu et al. [80] with a hybrid method to determine the material parameters. The method used as experimental data the lateral compression test on thin-walled tubes. The characterization methodology employed finite element simulations and a genetic algorithm to determine the material parameters. In comparison with classical characterization methods, the proposed method demonstrated to be more efficient and accurate in the parameter identification of structures composed by tubes and subject to crashing. An analogous study was also published by Markiewicz et al. [81]. They developed an inverse method to determine the properties of thin-walled square tubes from quasi-static and dynamic axial crushing. Using a mathematical optimizer and nu-merical simulations the material parameters from a constitutive model were determined. The parameters from the constitutive model were determined by correlating the results of the numerical simulation with the global variables obtained by the experimental tests. The identified parameters were used to predict crash behavior of structures.

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1.7 SUMMARY AND CONCLUSION

The use of numerical models to predict the mechanical response have expanded to many applications during the past years. However, to obtain accurate results from numerical simulations, a truthful mechanical characterization of the materials involved should be performed, particularly when the simulations undergo high strain rates.

Although much research have been done in characterization techniques and method-ologies to determine the constitutive parameters of a given material, from literature it is possible to identify the following:

1. Material parameters are determined individually and sequentially by uncoupling the constitutive material model, dismissing the interaction among material param-eters.

2. Several experimental tests are required to complete the characterization process.

3. Input data for the characterization techniques are often limited to stress-strain flow curves, restricting the use of diverse experimental high strain rate tests that pro-vides data in other forms.

Additionally, previous studies have produced important lessons that need to be incor-porated in future research, as follows:

1. Inverse problems in combinations with optimization algorithms showed better re-liability to determine material parameters due to the flexibility to use diverse ex-perimental tests.

2. Simultaneous identification of material parameters allows implementing better com-puter simulations due to the interaction among material parameters.

3. Genetic algorithms showed better performance than other tested optimization tech-niques due to the ease of handling complex and large search spaces present in the determination of material parameters.

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1.8 PROBLEM STATEMENT

Considering the aforementioned limitations identified in the currently available material characterization techniques at high strain rates, the following statement condenses the research problem that is investigated in this thesis:

GIVENboth a material and a mathematical constitutive model, with unknown constant parameters,DEVELOPan inverse technique to determine the parameters from a high

strain rate experiment.

1.9 RESEARCH OBJECTIVES

Considering the aforementioned challenges and to overcome the existing limitations of material characterization techniques, the main objectives in this research are:

1. Develop an inverse characterization procedure to determine the material constant parameters associated to a given mathematical constitutive model from a high strain experiment.

2. Implement the proposed characterization technique on a flexible and efficient com-putational procedure.

3. Verify the performance of the proposed computational procedure by applying it to case studies.

1.10 SCOPE OF THE RESEARCH

This thesis concentrates on the determination of material constitutive parameters of mod-eling clay Roma Plastilina No. 1, copper alloy UNS C83600 and low carbon steel AISI 1018. Experimental mechanical high strain rate tests such as Drop Test, Split Hopkinson Pressure Bar and Taylor impact test are used as input data. The material parameters are determined given constitutive material models such as the Power Law plasticity model and the Cowper-Symonds constitutive model. The selection of the constitutive model suitable for each material is assumed to be known and accurate. As well, issues associ-ated with the performance of the used experimental techniques are beyond the scope of this thesis.

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1.11 RELEVANCE OF THE RESEARCH

This research provides useful benefits to other researchers, laboratories and companies in the field of mechanical characterization of material at high strain rates; since the inverse characterization procedure proposed in this study delivers a quick and efficient tool to determine constitute material parameters using diverse experimental tests.

Additionally, professionals in the field of numerical simulation of events involving dynamic loading and impacts are benefited due to the potential of the characterization technique to implement numerical models of material at high strain rate.

1.12 ROAD MAP TO THIS THESIS

1. PART I: "INTRODUCTION AND PROBLEM DESCRIPTION"

• Chapter 2, "Literature Review". This chapter presents other research pub-lications relevant to the work reported in this thesis.

2. PART II: "MATERIAL CHARACTERIZATION PROCEDURE"

• Chapter 3, "Material characterization procedure". In this chapter a novel computational inverse technique to characterize materials at high strain rates is proposed.

3. PART III: "APPLICATIONS AND CASE STUDIES"

• Chapter 4, "Inverse problem of drop test for characterization of soft materials". A case study to characterize soft materials is presented to test the performance of the proposed characterization technique.

• Chapter 5, "Inverse problem of split Hopkinson pressure bar test for characterization of materials". The proposed inverse procedure is tested with the characterization at high strain rates of a metal using as experimental tool a split Hopkinson pressure bar test.

• Chapter 6, "Inverse problem of Taylor test for characterization of mate-rials". A Taylor impact tests is used as input for the inverse characterization procedure to tests its performance.

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4. PART IV: "CONCLUSIONS"

• Chapter 7, "Conclusions and future work". Conclusions and research findings are summarized on this chapter, and finally, possible future work is discussed.

1.13 DISSEMINATION AND EXPLOITATION

Selected portions of the work have been published in international conferences and jour-nals, as listed below:

• Hernandez, C., Maranon, A., Ashcroft, I. A. and Casas-Rodriguez, J.P. An inverse problem for the mechanical characterization of steels from the Taylor test.In Pro-ceedings of the ASME 2010 International Mechanical Engineering Congress & Exposition.November 12-18, 2010, Vancouver, Canada.

• Hernandez, C., Maranon, A., Ashcroft, I. A. and Casas-Rodriguez, J.P. Mechanical characterization of oil-based modeling clay from drop-test using inverse problems. In Proceedings of the 19th DYMAT Technical Meeting, Dynamic mechanical be-haviour of polymers and composites, December 1-3, 2010, Strasbourg, France.

• Hernandez, C., Maranon, A., Mateus, L., Ramirez, A. Simulación Numérica de Ensayos Dinámicos en el Dispositivo de Barra Hopkinson.In Proceedings of Ter-cer Simposio en Mecánica de Materiales y Estructuras Continuas - SMEC 2011, October 18-20, 2011, Cartagena de Indias, Colombia.

• Hernandez, C., Maranon, A., Ashcroft, I. A. and Casas-Rodriguez, J.P. An inverse problem for the characterization of dynamic material model parameters from a single SHPB test.Procedia Engineering, 2011.10: p. 1607-1612.

• Hernandez, C., Maranon, A., Ashcroft, I. A. and Casas-Rodriguez, J.P. Quasi-static and dynamic characterization of oil-based modeling clay and numerical simulation of drop-impact test. In Proceedings of the ASME 2011 International Mechanical Engineering Congress & Exposition. November 11-17, 2011, Denver, Colorado, USA.

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• Hernandez, C., Maranon, A., Ashcroft, I. A. and Casas-Rodriguez, J.P. Inverse Methods for the Mechanical Characterization of Materials At High Strain Rates. In Proceedings of the 10th International Conference on the Mechanical and Physical Behaviour of Materials Under Dynamic Loading DYMAT 2012. September 2-7, 2012, Freiburg, Germany.

• Hernandez, C., Maranon, A. Caracterización Dinámica de Plastilina Roma Medi-ante Ensayo de Caída Libre y Problemas Inversos. In Proceedings of the VI Con-greso Internacional de Ingeniería Mecánica, IV ConCon-greso de Ingeniería Mecatrónica, IV Congreso Internacional de Materiales, Energía y Medio Ambiente -CIMM 2013.May 2-4, 2013, Barranquilla, Colombia.

• Hernandez, C., Maranon, A., Ashcroft, I. A. and Casas-Rodriguez, J.P. A com-putational determination of the Cowper-Symonds parameters from a single Taylor test.Applied Mathematical Modelling, 2013.37(7): p. 4698-4708.

• Hernandez, C., Buchely, M.F., Maranon, A., and Casas-Rodriguez, J.P. Material Characterization and Modeling of Drop-Tests on Plasticine.Submitted to..., 2013

• Hernandez, C., Buchely, M.F., Maranon, and A., Ashcroft, I. A. Inverse Method for the Mechanical Characterization og Soft Malleable Materials from Drop Test. Submitted to..., 2013

• Hernandez, C., Maranon, A., and Ashcroft, I.A. Inverse Procedure for the Me-chanical Characterization of Materials froms a Single SHPB Test. Submitted to..., 2013

• Hernandez, C., Maranon, A., and Ashcroft, I.A. Determination of strength model parameters from a Taylor test using an inverse technique.Submitted to..., 2013

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MATERIAL CHARACTERIZATION PROCEDURE

Abstract

This chapter presents a novel dynamic material characterization procedure to

deter-mine the material constitutive parameters associated with a given plasticity

constitu-tive model at high strain rates and using as input a single experimental high strain

rate test.

The proposed material characterization procedure, which is designed to overcome

identified limitations of customary characterization procedures, is based on the

formu-lation and solution of an inverse problem from high strain rate experimental tests. The

solution of the characterization problem is based on the reformulation of the inverse

problem as an optimization scheme in which the difference of the results of a reference

model against the experimental measurements is minimized. A six-step computational

procedure built on three operators, which includes the use of techniques such as Line

Moments, Numerical Simulations and Genetic Algorithms, is presented to determine

the material parameters that defines the mechanical response of the material at high

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2.1 MECHANICAL CHARACTERIZATION OF MATERIALS AT HIGH

STRAIN RATES

The mechanical characterization of materials, motivated by the implementation of ma-terial models to apply in numerical simulations, is defined as the determination of the material constitutive parameters associated with a given plasticity constitutive model. This engineering activity, fundamental to produce accurate numerical simulations, is an active research field that requires the processing of measurements from mechanical ex-perimental tests in order to determine the fittest set of material parameters that defines the mechanical response of the material. This process of determining the material con-stitutive parameters is performed by a material characterization procedure.

Conventional characterization procedures typically use as experimental measurements force-displacement curves obtained from uniaxial tests at quasi-static strain rates to de-termine the material constitutive parameters. However, in order to implement numerical simulations of dynamic events that involve plastic deformation of materials, quasi-static experimental tests and force-displacement curves at quasi-static speeds do not return enough information to complete an accurate mechanical characterization of many ma-terials. Situation that is aggravated by the mechanical behavior of some materials, often influenced by the strain rate. This particular behavior that exhibits some materials makes the characterization process more complicated, and in some cases requires the imple-mentation of several dynamic experimental tests at different strain rates. High strain rate tests that usually produce measurements in forms different to load-displacement curves. In these cases, the characterization procedure plays an important role to process the ex-perimental measurements and determine the optimum set of material parameters at high strain rate.

In the literature review, from the available material characterization procedures at high strain rates to determine material parameters, it was identified that the characterization procedures often submit at least one of three identified limitations. The material char-acterization procedures often requires as input measurements obtained from a specific test, generally in the form of stress-strain flow curves. This limitation restricts the use of several experimental tests that produces data in other forms. Additionally, the character-ization procedures at high strain rates require the use of numerous tests to complete the

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characterization process, making the characterization of materials expensive and compli-cated. And finally, the material parameters are identified individually and sequentially by uncoupling the constitutive material model, dismissing the interactions among parame-ters.

In conclusion, for the characterization of materials at high strain rates it is necessary a characterization procedure that is capable of using as input the measurements obtained from a high strain rate mechanical test, independently of either the nature of the test or the form in which the measurements are yielded. Then, an identification technique de-termines the materials parameters that define the mechanical behavior at high strain rates of the material. The sketch of the material characterization procedure is shown in Figure 2.1. In this figure the input for the material characterization process is represented by a signal from a high strain rate experiment called "characteristic signal". This characteris-tic signal represents the measurements obtained from a given experimental test according to the desired strain rate and loading conditions. The features of the input characteris-tic signal are described in section 2.3. After the characterization procedures performs the identification process, the output parameters, represented by a vector⃗z, is given by the optimum set of constitutive material parameters that defines the mechanical behav-ior associated to a given constitutive model. The output of the material characterization procedure is described in section 2.8.

The objective of this work is to propose a material characterization procedure at high strain rates that completes the process shown in Figure 2.1 with three fundamental fea-tures that overcome the limitations identified during the literature review. These main features are:

1. Use of a single experimental test to profit easiness, simplicity and un-expensiveness.

2. Simultaneous estimation of material parameters that allows observing the interac-tion among parameters.

3. Ability to use diverse experimental high strain rate tests to determine the material parameters.

In the following sections, the material characterization procedure proposed to deter-mine the constitutive parameters from high strain rates is presented. The proposed