Integrated Computational Materials Engineering (ICME) for Metals E-Book
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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
Focuses entirely on demystifying the field and subject of ICME and provides step-by-step guidance on its industrial application via case studies This highly-anticipated follow-up to Mark F. Horstemeyer's pedagogical book on Integrated Computational Materials Engineering (ICME) concepts includes engineering practice case studies related to the analysis, design, and use of structural metal alloys. A welcome supplement to the first book--which includes the theory and methods required for teaching the subject in the classroom--Integrated Computational Materials Engineering (ICME) For Metals: Concepts and Case Studies focuses on engineering applications that have occurred in industries demonstrating the ICME methodologies, and aims to catalyze industrial diffusion of ICME technologies throughout the world. The recent confluence of smaller desktop computers with enhanced computing power coupled with the emergence of physically-based material models has created the clear trend for modeling and simulation in product design, which helped create a need to integrate more knowledge into materials processing and product performance. Integrated Computational Materials Engineering (ICME) For Metals: Case Studies educates those seeking that knowledge with chapters covering: Body Centered Cubic Materials; Designing An Interatomic Potential For Fe-C Alloys; Phase-Field Crystal Modeling; Simulating Dislocation Plasticity in BCC Metals by Integrating Fundamental Concepts with Macroscale Models; Steel Powder Metal Modeling; Hexagonal Close Packed Materials; Multiscale Modeling of Pure Nickel; Predicting Constitutive Equations for Materials Design; and more. * Presents case studies that connect modeling and simulation for different materials' processing methods for metal alloys * Demonstrates several practical engineering problems to encourage industry to employ ICME ideas * Introduces a new simulation-based design paradigm * Provides web access to microstructure-sensitive models and experimental database Integrated Computational Materials Engineering (ICME) For Metals: Case Studies is a must-have book for researchers and industry professionals aiming to comprehend and employ ICME in the design and development of new materials.
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Table of Contents
Cover
Title Page
Copyright
List of Contributors
Foreword
Preface
Chapter 1: Definition of ICME
1.1 What ICME Is NOT
1.2 What ICME Is
1.3 Industrial Perspective
1.4 Summary
References
Section I: Body-Centered Cubic Materials
Chapter 2: From Electrons to Atoms: Designing an Interatomic Potential for Fe–C Alloys
2.1 Introduction
2.2 Methods
2.3 Single-Element Potentials
2.4 Construction of Fe–C Alloy Potential
2.5 Structural and Elastic Properties of Cementite
2.6 Properties of Hypothetical Crystal Structures
2.7 Thermal Properties of Cementite
2.8 Summary and Conclusions
Acknowledgments
References
Chapter 3: Phase-Field Crystal Modeling: Integrating Density Functional Theory, Molecular Dynamics, and Phase-Field Modeling
3.1 Introduction to Phase-Field and Phase-Field Crystal Modeling
3.2 Governing Equations of Phase-Field Crystal (PFC) Models Derived from Density Functional Theory (DFT)
3.3 PFC Model Parameters by Molecular Dynamics Simulations
3.4 Case Study: Solid–Liquid Interface Properties of Fe
3.5 Case Study: Grain Boundary Free Energy of Fe at Its Melting Point
3.6 Summary and Future Directions
References
Chapter 4: Simulating Dislocation Plasticity in BCC Metals by Integrating Fundamental Concepts with Macroscale Models
4.1 Introduction
4.2 Existing BCC Models
4.3 Crystal Plasticity Finite Element Model
4.4 Continuum-Scale Model
4.5 Engineering Scale Applications
4.6 Summary
References
Chapter 5: Heat Treatment and Fatigue of a Carburized and Quench Hardened Steel Part
5.1 Introduction
5.2 Modeling Phase Transformations and Mechanics of Steel Heat Treatment
5.3 Data Required for Modeling Quench Hardening Process
5.4 Heat Treatment Simulation of a Gear
5.5 Summary
References
Chapter 6: Steel Powder Metal Modeling
6.1 Introduction
6.2 Material: Steel Alloy
6.3 ICME Modeling Methodology
6.4 Summary
References
Chapter 7: Microstructure-Sensitive, History-Dependent Internal State Variable Plasticity-Damage Model
7.1 Introduction
7.2 Internal State Variable (ISV) Plasticity-Damage Model
7.3 Simulation Setups
7.4 Results
7.5 Conclusions
References
Section II: Hexagonal Close Packed (HCP) Materials
Chapter 8: Electrons to Phases of Magnesium
8.1 Introduction
8.2 Criteria for the Design of Advanced Mg Alloys
8.3 Fundamentals of the ICME Approach Designing the Advanced Mg Alloys
8.4 Data-Driven Mg Alloy Design – Application of ICME Approach
8.5 Outlook/Future Trends
Acknowledgments
References
Chapter 9: Multiscale Statistical Study of Twinning in HCP Metals
9.1 Introduction
9.2 Crystal Plasticity Modeling of Slip and Twinning
9.3 Introducing Lower Length Scale Statistics in Twin Modeling
9.4 Model Implementation
9.5 The Continuum Scale
9.6 Summary
Acknowledgment
References
Chapter 10: Cast Magnesium Alloy Corvette Engine Cradle
10.1 Introduction
10.2 Modeling Philosophy
10.3 Multiscale Continuum Microstructure-Property Internal State Variable (ISV) Model
10.4 Electronic Structures
10.5 Atomistic Simulations for Magnesium Using the Modified Embedded Atom Method (MEAM) Potential
10.6 Mesomechanics: Void Growth and Coalescence
10.7 Macroscale Modeling and Experiments
10.8 Structural-Scale Corvette Engine Cradle Analysis
10.9 Summary
References
Chapter 11: Using an Internal State Variable (ISV)–Multistage Fatigue (MSF) Sequential Analysis for the Design of a Cast AZ91 Magnesium Alloy Front-End Automotive Component
11.1 Introduction
11.2 Integrated Computational Materials Engineering and Design
11.3 Mechanical and Microstructure Analysis of a Cast AZ91 Mg Alloy Shock Tower
11.4 A Microstructure-Sensitive Internal State Variable (ISV) Plasticity-Damage Model
11.5 Microstructure-Sensitive Multistage Fatigue (MSF) Model for an AZ91 Mg Alloy
11.7 Summary
References
Section III: Face-Centered Cubic (FCC) Materials
Chapter 12: Electronic Structures and Materials Properties Calculations of Ni and Ni-Based Superalloys
12.1 Introduction
12.2 Designing the Next Generation of Ni-Base Superalloys Using the ICME Approach
12.3 Density Functional Theory as the Basis for an ICME Approach to Ni-Base Superalloy Development
12.4 Theoretical Background and Computational Procedure
12.5 Ni-Base Superalloy Design using the ICME Approach
12.6 Conclusions and Future Directions
Acknowledgments
References
Chapter 13: Nickel Powder Metal Modeling Illustrating Atomistic-Continuum Friction Laws
13.1 Introduction
13.2 ICME Modeling Methodology
13.3 Atomistic Studies
13.4 Summary
References
Chapter 14: Multiscale Modeling of Pure Nickel
14.1 Introduction
14.2 Bridge 1: Electronics to Atomistics and Bridge 4: Electronics to the Continuum
14.3 Bridge 2: Atomistics to Dislocation Dynamics and Bridge 5: Atomistics to the Continuum
14.4 Bridge 3: Dislocation Dynamics to Crystal Plasticity and Bridge 6: Dislocation Dynamics to the Continuum
14.5 Bridge 7: Crystal Plasticity to the Continuum
14.6 Bridge 8: Macroscale Calibration to Structural Scale Simulations
14.7 Summary
Acknowledgments
References
Section IV: Design of Materials and Structures
Chapter 15: Predicting Constitutive Equations for Materials Design: A Conceptual Exposition
15.1 Introduction
15.2 Frame of Reference
15.3 Critical Review of the Literature
15.4 Crystal Plasticity-Based Virtual Experiment Model
15.5 Hierarchical Strategy for Developing a Constitutive EQuation (CEQ) Expansion Model
15.6 Closing Remarks
Nomenclature
Acknowledgments
References
Chapter 16: A Computational Method for the Design of Materials Accounting for the Process–Structure–Property– Performance (PSPP) Relationship
16.1 Introduction
16.2 Frame of Reference
16.3 Integrated Multiscale Robust Design (IMRD)
16.4 Roll Pass Design
16.5 Microstructure Evolution Model
16.6 Exploring the Feasible Solution Space
16.7 Results and Discussion
16.8 Closing Remarks
Acknowledgments
Nomenclature
References
Section V: Education
Chapter 17: An Engineering Virtual Organization for CyberDesign (EVOCD): A Cyberinfrastructure for Integrated Computational Materials Engineering (ICME)
17.1 Introduction
17.2 Engineering Virtual Organization for CyberDesign
17.3 Functionality of EVOCD
17.4 Protection of Intellectual Property
17.5 Cyberinfrastructure for EVOCD
17.6 Conclusions
References
Chapter 18: Integrated Computational Materials Engineering (ICME) Pedagogy
18.1 Introduction
18.2 Methodology
18.3 Course Curriculum
18.4 Assessment
18.5 Benefits or Relevance of the Learning Methodology
18.6 Conclusions and Future Directions
Acknowledgments
References
Chapter 19: Summary
19.1 Introduction
19.2 Chapter 1 ICME Definition: Takeaway Point
19.3 Chapter 2: Takeaway Point
19.4 Chapter 3: Takeaway Point
19.5 Chapter 4: Takeaway Point
19.6 Chapter 5: Takeaway Point
19.7 Chapter 6: Takeaway Point
19.8 Chapter 7: Takeaway Point
19.9 Chapter 8: Takeaway Point
19.10 Chapter 9: Takeaway Point
19.11 Chapter 10: Takeaway Point
19.12 Chapter 11: Takeaway Point
19.13 Chapter 12: Takeaway Point
19.14 Chapter 13: Takeaway Point
19.15 Chapter 14: Takeaway Point
19.16 Chapter 15: Takeaway Point
19.17 Chapter 16: Takeaway Point
19.18 Chapter 17: Takeaway Point
19.19 Chapter 18: Takeaway Point
19.20 ICME Future
19.21 Summary
References
Index
End User License Agreement
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Guide
Cover
Table of Contents
Foreword
Preface
Begin Reading
List of Illustrations
Chapter 1: Definition of ICME
Figure 1.1 Schematic illustration of solving the inverse problem where the performance requirements are examined first and then the creation of new materials is backed out at the end (Horstemeyer, 2012).
Figure 1.2 In order to capture the Cradle-to-Grave history, robust models must be able to capture various manufacturing and in-service design scenarios (Horstemeyer and Wang, 2003). This example shows that integration of information passage is required for both the “Horizontal ICME” sequence and the “Vertical ICME” sequence.
Figure 1.3 “Vertical ICME” bridging between two different length scales of simulations showing the sequential steps of the ICME methodology.
Figure 1.4 “Horizontal ICME” bridging between two different steps in the process–performance sequence of simulations showing the sequential steps of the ICME methodology.
Figure 1.5 Uncertainty analysis is useful in bringing robustness to an industrial usage of ICME. Here, the modeling and simulations need to be validated by examination of an uncertainty analysis.
Figure 1.6 The ICME cyberinfrastructure houses repositories for models and codes, materials characterization data, experimental stress–strain data, and different calibration tools. Examples of running different codes are also included in a tutorial fashion.
Figure 1.7 ICME as the value creation at the junction point of design, materials, and manufacturing through a computational framework.
Figure 1.8 Methodology adoption curve in an industry.
Figure 1.9 (a) Significant value creation by co-adoption of new design, materials and manufacturing techniques resulting in breakthrough products. (b) For this specific opportunity, the technology readiness level is higher and the gap is to identify business opportunities.
Figure 1.10 Schematic illustrating the co-adoption of multiscale models that were experimentally validated within a finite element method (FEM) coupled with a cost analysis, uncertainty analysis, and multiobjective design optimization analysis can help design new materials, new structures, and new manufacturing processes.
Chapter 2: From Electrons to Atoms: Designing an Interatomic Potential for Fe–C Alloys
Figure 2.1 Schematic showing the sequence of steps in vertical bridging between the smallest length scale (electrons) and the next higher length scale (atoms) for examining the formation of cementite.
Figure 2.2 Energy versus volume curves for Fe in bcc, fcc, and hcp crystal structures. The solid curve is constructed from experimental values in Table 2.2. For ease of comparison, the DFT curves are shifted vertically by a constant amount equal to the difference between experimental and DFT cohesive energies of Fe in bcc at equilibrium volumes.
Figure 2.3 Energy versus nearest neighbor distance curves for C in diamond and graphite. The solid curve is constructed from experimental values in Table 2.2. For comparison, the DFT curve is shifted vertically to the experimental cohesive energy at the equilibrium nearest neighbor distance.
Figure 2.4 Cohesive energy of graphite as a function of the
c
/
a
ratio. Energy at zero is set to the minimum energy predicted by the MEAM potential.
Figure 2.5 Sensitivity (change in target versus range of sensitivity) of selected properties of the Fe–C system: Heat of formation (HF) of Fe–C in the B
1
(B
1
HF) and L
12
structure (L
12
HF), HF of cementite (Fe
3
C HF), interstitial energies of C in bcc Fe in tetrahedral (
E
int-tet
) and octahedral positions (
E
int-oct
).
Figure 2.6 Heat of formation of cementite as a function of (a) density scaling factor
ρ
0
, (b) additional cubic term in the universal energy equation
a
3
, (c) heat of formation of the reference structure Δ (eV), and angular screening parameters, (d)
C
min
(Fe–Fe–C) and (e)
C
min
(C–C–Fe).
Figure 2.7 Comparison of energy versus volume curves for cementite. The dashed-line curve is constructed from experimental values of the cohesive energy, equilibrium volume and polycrystalline bulk modulus, of cementite. For comparison, the DFT curve is shifted vertically to the experimental cohesive energy at the equilibrium volume.
Figure 2.8 Comparison of the energy versus volume curves of Fe–C alloy system in the B
1
structure. The DFT curve is shifted vertically to the MEAM-predicted cohesive energy at the equilibrium nearest neighbor distance to aid the comparison with the MEAM curve.
Figure 2.9 Comparison of the energy versus volume curves of Fe–C alloy system in the L
12
structure. The DFT curve is shifted vertically to the MEAM-predicted cohesive energy at the equilibrium nearest neighbor distance aid the comparison with the MEAM curve.
Figure 2.10 Snapshots of the two-phase MD simulation in the NPT ensemble with
T
= 1420 K and
P
= 0. Red spheres are Fe atoms and blue spheres are C atoms. (a) Initial state of the simulation box, which contains both liquid and solid phases of cementite. (b) Intermediate state of the simulation box at 16 ns, as the solid phase propagates to the liquid phase. (c) Final state of the simulation box at 32 ns, when the entire system has turned into a solid phase.
Figure 2.11 Snapshots of the two-phase MD simulation in the NPT ensemble with
T
= 1430 K and
P
= 0. Red spheres are Fe atoms and blue spheres are C atoms. (a) Initial state of the simulation box, which contains both liquid and solid phases of cementite. (b) Intermediate state of the simulation box at 20 ns, as the liquid phase propagates to the solid phase. (c) Final state of the simulation box at 30 ns, when the entire system has turned into a liquid phase.
Figure 2.12 Variation of properties of the two-phase system over the temperature. (a) Total energy of the system, (b) Volume of the system. The curve with solid symbols is first-principles data by Dick
et al.
[Dick, Kormann, Hickel, and Neugebauer, 2011], (c) Specific heat of the system. The curve with solid symbols is the first-principles data, and (d) dV/dT of the system.
Chapter 3: Phase-Field Crystal Modeling: Integrating Density Functional Theory, Molecular Dynamics, and Phase-Field Modeling
Figure 3.1 “Vertical ICME” bridging between Phase-Field Crystal (PFC) modeling and molecular dynamics (MD) simulations.
Figure 3.2 Schematics of the solid–liquid equilibrating structure using (a) MEAM-MD simulations and (b) PFC simulations (density plot).
Source:
Adapted from Asadi
et al
. (2014).
Figure 3.3 The PFC simulations of symmetric grain boundaries with misorientation angles: (a) and (b) .
Figure 3.4 Plot of the grain boundary free energy
γ
GB
for Fe versus a misorientation angle
θ
.
Chapter 4: Simulating Dislocation Plasticity in BCC Metals by Integrating Fundamental Concepts with Macroscale Models
Figure 4.1 “Vertical ICME” bridging connecting different length scales of simulations showing the sequential steps of the ICME methodology for describing dislocation plasticity in BCC metals.
Figure 4.2 Dislocations in BCC metals exhibit a strong lattice friction that requires thermal activation to overcome the barriers. The Peierls potential has a washboard shape, as illustrated here, and the dislocation must overcome this barrier. One dislocation shows a bulge in the line (on the left), described by the line tension model; and the second dislocation (right) has well-formed kinks, described by the elastic interaction model. The valleys and peaks are separated by the spacing
h
and the height of the Peierls potential is Δ
U
.
Figure 4.3 (a) The Peierls potential predicted by DFT compared against analytical models. DFT simulations demonstrate that the single peak potential is an accurate description, which can be used to predict the energetics of kink-pair nucleation. (b) The activation enthalpy plotted versus stress for different models analytical Peierls potential models.
Figure 4.4 The thermal part of the flow stresses extracted from experiments for four different BCC metals: (a) Mo, (b) Ta, (c) W, and (d) Nb. The fits of the elastic interaction model (EI, Regime I) and the line tension model (LT, Regime II) are shown in solid lines (Lim
et al
., 2015a).
Figure 4.5 (a) The activation enthalpy computed from atomistics under different applied stress conditions and (b) the stress normalized using the expression discussed in the text, which collapses the data onto a single universal activation energy curve.
Figure 4.6 (a) Measured and calibrated temperature-dependent flow stresses of iron single crystals for orientation “A” using non-Schmid and Schmid models. The experiments shown are tensile data only and the Schmid-based model predicts equal tensile and compressive strengths. The non-Schmid model calibrated to the tensile data and predicted in compression, exhibiting tension–compression asymmetry. (b) Measured and predicted temperature-dependent flow stresses of iron single crystals for orientation “B” using non-Schmid and Schmid models (Lim
et al
., 2015b).
Figure 4.7 Schematic view of typical texture analysis and crystal plasticity finite element method (CP-FEM) models.
Figure 4.8 (a) Measured and predicted temperature-dependent stress–strain response of single-crystal iron and (b) predicted stress–strain responses of polycrystalline iron. A total of 125,000 hexahedral finite elements were used to create 139 grains in polycrystal simulation.
Figure 4.9 (a) Temperature-dependent biaxial yield surfaces for iron single crystals. Here,
x
-axis in [001] direction, while
y
-axis is in [] direction. (b) Predicted (data points) and fitted (solid lines) polycrystalline biaxial yield surfaces at different temperatures (Lim
et al
., 2015b).
Figure 4.10 Inverse pole Figure of the tensile axis in single-crystal iron as it rotates in the standard unit triangle during plastic deformation at 100, 200, and 300 K for tension and compression.
Figure 4.11 Effects of
α
1
and
α
2
on yield surface in Eq. (4.21). Here,
α
1
represents contributions from the pressure, while
α
2
determines the shape of the yield surface.
Figure 4.12 A plot of yield stresses of tantalum fit to CP-FEM simulations using the JC, ZA, and MTS models for different (a) temperatures and (b) strain rates (Lim
et al
., 2015a).
Figure 4.13 Specimen shapes and equivalent plastic strain (EQPS) maps predicted by CTH simulations of impact of a two-dimensional, axisymmetric tantalum specimen into a rigid wall. The tantalum material was described using parameters from Johnson–Cook yield (a) model 1 and (b) model 2, Mechanical Threshold Stress model with yield parameters described by fits to (c) crystal plasticity finite element and (d) kink-pair results, and Zerilli–Armstrong (e) model 1 and (f) model 2 (see Tables 4.3–4.5). The images were captured after 200 µs of simulation time. The dimensions on the axes are in centimeters. Regions colored in gray have plastic strain values less than 10
−3
. Images are mirrored once about the axis of symmetry (
x
= 0) for clarity.
Figure 4.14 Specimen shapes and equivalent plastic strain maps predicted by ALEGRA simulations of impact of a three-dimensional quarter-symmetric tantalum specimen into a 4340 steel plate. The tantalum material was described using parameters from Johnson–Cook yield (a) model 1 and (b) model 2, Mechanical Threshold Stress model with yield parameters described by fits to (c) crystal plasticity finite element and (d) kink pair results, and Zerilli–Armstrong (e) model 1 and (f) model 2 (see Tables 4.3–4.5). The images were captured after the projectile reflected off the steel plate such that the center point of the bottom of the (deformed) specimen reached its initial position of approximately 0.1 mm above the target surface. Images are mirrored thrice about the planes of symmetry (
x–z
planes) for clarity.
Figure 4.15 The JC, ZA, and MTS fits from Figure 4.12, extrapolated to higher strain rates.
Figure 4.16 Projectile shape profiles (not to scale) predicted by all six ALEGRA simulations, with results reported from earlier Taylor cylinder impact experiments (Maudline
et al
., 1999) (using the measurements from the minor axis). Open circles are extrapolations of the final measured specimen height to its center axis.
Chapter 5: Heat Treatment and Fatigue of a Carburized and Quench Hardened Steel Part
Figure 5.1 Dilatometric data for AISI 1050 steel continuously cooled from 840 °C at the indicated rates.
Figure 5.2 Dilatometric data for AISI 8620 steel isothermally transformed.
Figure 5.3 Schematic of tensile properties by phase.
Figure 5.4 Thermal strain curves in terms of temperature during cooling of Pyrowear 53 with base carbon (0.1%C), 0.3%C, 0.5%C, and 0.8%C.
Figure 5.5 Respective heat transfer regimes for quenching in liquids that boil.
Figure 5.6 Surface heat transfer coefficients for still and agitated salt and oil immersion quenching showing variation of heat transfer coefficient with part surface temperature.
Figure 5.7 Comparison of cooling curves for some common quenchants.
Figure 5.8 Test gear: (a) CAD model and (b) dimensions of cross section.
Figure 5.9 Single tooth finite element meshing: (a) overall view and (b) zoomed in view of the tooth section.
Figure 5.10 Carbon distribution after gas carburization process: (a) overall view and (b) zoomed in view of the tooth section. Unit is weight fraction.
Figure 5.11 Carbon weight percentage as a function of depth at the flank and root fillet locations.
Figure 5.12 Phase distributions after carburization and cooling to room temperature: (a) retained austenite, (b) martensite and (c) tempered martensite.
Figure 5.13 Residual stress distributions after carburization and cooling to room temperature: (a) minimum principal stress and (b) maximum principal stress.
Figure 5.14 Stress relaxation during reaustenitization process: (a) temperature, (b) austenite and (c) minimum principal stress.
Figure 5.15 Carbon distribution after reaustenitization process: (a) overall view and (b) zoomed in view of the tooth section. Unit is weight fraction.
Figure 5.16 Effect of temperature and phase transformation on stress evolution at 13.4 s during oil quench process: (a) temperature, (b) martensite and (c) minimum principal stress.
Figure 5.17 Effect of temperature and phase transformation on stress evolution at 23.5 s during oil quench process: (a) temperature, (b) martensite and (c) minimum principal stress.
Figure 5.18 Retained austenite and residual stress distribution at the end of oil quench and cooling to room temperature: (a) retained austenite, (b) minimum principal stress and (c) maximum principal stress.
Figure 5.19 Retained austenite and residual stress distribution after deep freeze treatment: (a) retained austenite, (b) minimum principal stress and (c) maximum principal stress.
Figure 5.20 Retained austenite and residual stress distribution after low temperature tempering: (a) tempered martensite, (b) minimum principal stress and (c) maximum principal stress.
Figure 5.21 Response of gear root fillet during quenching: (a) surface and inner points selected and (b) relation between temperature and martensite transformation at the two selected points.
Figure 5.22 Response of gear root fillet during quenching: (a) relation between temperature and tangential stress and (b) relation between temperature and martensite transformation at the two selected points.
Figure 5.23 Comparison of model predicted circumferential stress and stresses determined from XRD measurements at the root center as a function of depth from the surface.
Figure 5.24 Schematic bending fatigue loading test setup.
Figure 5.25 Pyrowear 53 Gear Tooth Bending Fatigue Data.
Figure 5.26 Maximum principal stress plot for tooth bending model with a stress free initial state. Unit is MPa.
Figure 5.27 Maximum principal stress plot for tooth bending model accounting for initial residual compressive surface stress state. Unit is MPa.
Figure 5.28 (a) Maximum principal stress and (b) minimum principal stress versus depth from the root fillet surface for finite element models with and without residual compressive surface stress due to carburizing and quench hardening.
Chapter 6: Steel Powder Metal Modeling
Figure 6.1 Modeling sequence used in this project for a powder metal FC-0205 steel automotive bearing cap.
Figure 6.2 Schematic illustrating different notions of failure locations. The maximum stress was found in location (B), the maximum porosity was found in location (F); as the heterogeneously distributed porosity simulation results show that location (F) was the failure spot, because damage accumulation was driven by the very high initial porosity.
Figure 6.3 Modified ISV Cap Model: yield surfaces in the |
s
−
α|
−
p
plane.
Figure 6.4 Comparison of the finite element analysis and the three different experimental results (Archimedes: immersion and Metaldyne), image analysis, and X-ray computed tomography.
Figure 6.5 Comparison of proposed internal state variable compaction model (CAVS model) with (a) experiment, (b) Gurson model, (c) Oyama model, (d) Fleck–Gurson using tuned low stress, (e) Cam clay, and (f) Drucker–Prager/cap for an aluminum 6061 alloy.
Figure 6.6 Sensitivity of (a) compressibility curve, (b) interparticle friction (tan β), (c) material cohesion, (d) cap eccentricity (
R
), and (e) elastic modulus (
E
) for FC-0205 with 0.6% wax.
Figure 6.7 Uncertainty in the compression curve for FC-0205 with 0.6% wax.
Figure 6.8 The compaction simulation results with its associated uncertainties are given as inputs for the sintering simulations.
Figure 6.9 Schematic diagram of two-particle sintering and vector plot for displacement of tungsten atoms during isothermal sintering simulation at 2000 K.
Figure 6.10 Densification behavior during isothermal sintering at different temperatures; (a) shrinkage and (b) densification.
Figure 6.11 Calculated master sintering curve (DiAntonio
et al
., 2005).
Figure 6.12 Two different forms of grain growth plots for 17–4 PH stainless steel. (a) Time. (b) Master sintering.
Figure 6.13 Representation of the sigmoid function and its different shape parameters.
Figure 6.14 Overall algorithm for construction of master sintering curve.
Figure 6.15 Master sintering curve for the FC-0205 powder.
Figure 6.16 Comparison of the axial shrinkage curves measured in dilatometer and predicted by FEM model for FC-0205 powder compacts (60% dense) sintered in nitrogen at 10 °C/min to 1120 °C with 30-min holding.
Figure 6.17 Density distribution and shrinkage of the MBC after sintering with a deformation scale factor of 100 (mesh of green part is shown as transparent).
Figure 6.18 The sintering simulation results with its associated uncertainties are given as inputs for the performance simulations.
Figure 6.19 Internal state variable plasticity-damage model calibration for mean monotonic stress–strain behavior under different stress states and temperatures with an initial (a) low porosity and (b) high porosity.
Figure 6.20 Compression experiment-model comparison at (a) 293 K and (b) 573 K.
Figure 6.21 Tension experiment-model comparison at (a) 293 K and (b) 573 K.
Figure 6.22 Load–displacement comparison between the experimental results and the finite element model (FEA) for notch tensile tests with R60 and R150 specimens.
Figure 6.23 Plot of load versus displacement for the monotonic MBCs.
Figure 6.24 Comparison of (a) finite element model with (b) experimental results indicating the crack initiation point, and (c) the regions of maximum Von Mises with the tabulated results shown in (d).
Figure 6.25 Comparison of the experimental data with the finite element model at two different locations of bearing cap (where the strain gage is located).
Figure 6.26 MultiStage Fatigue (MSF) model calibration with low- and high-porosity fatigue experimental data.
Figure 6.27 Partial Arc Failure in a failed main bearing cap (Ilia
et al
., 2003).
Figure 6.28 Effective strain and shear strain for a shaft load of 1000 lbs.
Figure 6.29 Effective strain and shear strain amplitudes for a shaft load of 20,000 lbs.
Figure 6.30 Different Young's modulus distributions based on density distributions.
Figure 6.31 Modeling results of failure location in accordance with fatigue testing results.
Figure 6.32 The top view of bearing cap with design variables (
R
1
,
R
2
,
R
3
, and
T
) for the optimization.
Figure 6.33 The finite element simulation modeling approach showing (a) the initial density distribution of the powder in the die prior to compaction, (b) the density distribution solution after springback, (c) the optimized performance model solution, and (d) the optimized fatigue model solution.
Figure 6.34 Comparison of the performance of the optimized MBC design and the current MBC design.
Chapter 7: Microstructure-Sensitive, History-Dependent Internal State Variable Plasticity-Damage Model
Figure 7.1 Seven processing steps for the AISI 1010 steel alloy that started as a steel sheet and ended up as a tube. Before and after each step, the structure–property relationships were quantified and used to validate the internal state variable (ISV) plasticity-damage model.
T
and
L
represent the temperature and mechanical load in the material at each step of the process. A difference in the subscript of
T
and
L
represents a temperature or load change in that step of the process.
Figure 7.2 Material constants calibration using DMG fit routine. Marks are experimental data from tension tests at 298 K and 623 K, and dark line is a calibrated model for data at 298 K while light gray line is for higher temperature at 623 K.
Figure 7.3 Steel 1010's stress–strain curves acquired from the tension test at each stage.
Figure 7.4 Tube forming simulation model showing the sheet steel at entrance and 14 sets of rollers.
Figure 7.5 State 1 (the raw sheet material) stress–strain verification of ISV Plasticity-Damage model in FE tension simulation. It is compared with experimental data.
Figure 7.6 Tube forming simulation showing the von Mises stress (in psi) progression from sheet to tube. (a) Flat sheet state. (b) Plate at the sixth roller. (c) Plate at the 11th roller. (d) Tube shape at final.
Figure 7.7 Tube forming simulation showing the plastic equivalent strain progression from sheet to tube. (a) Flat sheet state. (b) Plate at the sixth roller. (c) Plate at the 11th roller. (d) Tube shape at final.
Figure 7.8 Post tube forming (State 2) stress–strain behavior: comparison between the finite element model results and experimental data. The thick error bars with light gray are uncertainty of experimental data while the thin error bars are uncertainty propagated from the simulation.
Figure 7.9 Progression of isotropic (
κ
) and kinematic (
α
) hardening history variable on the middle elements of the tube in forming.
Figure 7.10 Sizing simulation showing the von Mises stress (in psi) at different instants. (a) Tube at the first roller. (b) Tube at the third roller. (c) Tube at the sixth roller. (d) Final state.
Figure 7.11 Sizing simulation showing the plastic equivalent strain at different instants. (a) Tube at the first roller. (b) Tube at the third roller. (c) Tube at the sixth roller. (d) Final state.
Figure 7.12 Post sizing (State 3) stress–strain behavior: comparison of the finite element model results and experimental data.
Figure 7.13 Progression of isotropic (
κ
) and kinematic (
α
) hardening history variable on the middle elements of the tube in sizing.
Figure 7.14 Calibrated grain size evolution curves with 10 combinations between soak time and temperatures. Note that the grains enlarged as temperature and time increased. The boundary condition of 1700 °F with 10 min was used in the annealing model.
Figure 7.15 The first annealing simulation showing temperature (in kelvin) progression. (SDV8 is temperature in ISV's output). (a) Initial state. (b) Heating at 1200 K (after 10 min). (c) Cooling at atmosphere (after 20 min). (d) Final state (after 30 min).
Figure 7.16 Post first anneal (State 4) stress–strain behavior: comparison of the finite element model results and experimental data.
Figure 7.17 Progression of isotropic (
κ
) and kinematic (
α
) hardening history variables on the middle elements of the tube in the first annealing.
Figure 7.18 First drawing simulation showing von Mises stress (left) and the equivalent plastic strain (SDV19, right) progressions. (a) Initial state when drawing starts. (b) Elongation of the tube by die and mandrel.
Figure 7.20 Post first and second drawing (States 5 and 6) stress–strain behaviors: comparison of the finite element model results and experimental data.
Figure 7.19 Second drawing simulation presenting von Mises stress (left, in psi) and the equivalent plastic strain (SDV19, right) progressions. At this process, tube was significantly deformed due to small radius of the die. (a) Initial state when drawing starts. (b) Elongation of the tube by die.
Figure 7.21 Tube specimen microscopy at 200× in pulling direction after first drawing (a) and second drawing (b) showing significant grain elongation after second drawing.
Figure 7.22 Progression of isotropic (a) and kinematic (b) hardening history variables on the middle elements of the tube in first drawing.
Figure 7.23 Progression of isotropic (a) and kinematic (b) hardening history variables on the middle elements of the tube in the second drawing.
Figure 7.24 (a) Post second annealing stress–strain behavior showing excellent agreement between model result and experiment, (b) Progression of isotropic (
κ
) hardening history variable on the middle elements of the tube in the second annealing. The variable was completely softened during the annealing process.
Figure 7.25 Progression of isotropic and norm value of kinematic hardening through the tubing sequences. Stages indicates 1–2 stage is the tube forming, 2–3 the sizing, 3–4 the first annealing, 4–6 the drawings, and 6–7 the second anealing.
Chapter 8: Electrons to Phases of Magnesium
Figure 8.1 The schematic diagram showing the data-driven development of advanced Mg alloys via the ICME approach, which integrates the first-principles calculation and CALPHAD modeling.
Figure 8.2 The CALPHAD approach (Saunders and Miodownik, 1998). (Paul Mason is acknowledged for the figure.)
Figure 8.3 Eight different jump frequencies in an hcp lattice for migration within a basal plane (a) and between two adjacent basal planes (b).
Figure 8.4 The comparison of deformation electron density between non-fault planes in an extrinsic fault and those in a perfect Mg together with the previous theoretical work by Blaha
et al
., (1988).
Figure 8.5 (010)
s. c.
Plane contour plots of
Δρ
of Mg (a) I1, (b) I2, and (c) EF with 0.0005 intervals, generated using VESTA (Momma and Izumi, 2008, 2011) with light gray for and dark gray for (Wang
et al
., 2012).
Figure 8.6 The electron density isosurface of I1 at different levels. The points in light gray are the charge distribution region with , while the points in dark gray are the charge distribution region with .
Figure 8.7 0.5
Δρ
max Isosurface of (100) plane view, (a–g): I1, I2, 6
H
, 10
H
, 14
H
, 18
R
, and 24
R
, generated using VESTA, with letters in light grey denoting fault layers (Wang
et al
., 2014a).
Figure 8.8 Comparison of the calculated enthalpies of formation for the binary compounds in the Mg−X binary systems with experimental measured values. The solid line shows unity , while the dashed lines represent an error range of . The region inside the dotted lines in (a) is enlarged in (b) (Zhang
et al
., 2009).
Figure 8.9 Calculated phonon dispersion curves for Al
12
Mg
17
pertaining to the equilibrium lattice parameters at 298 K (left figure). Thermodynamic properties: entropy (S) and Gibbs energy (G) for Al
12
Mg
17
. The solid lines represent the calculated results with only phonon contribution (Ph), and the dotted lines represent results with phonon and TECs (Zhang
et al
., 2010).
Figure 8.10 The predicted fractions of solid (hcp), liquid, and Al
12
Mg
17
phases in AZ61 alloy by equilibrium (solid line) and Scheil (dashed line) models (a). The predicted distributions of Al and Zn in solid (hcp), liquid, and Al
12
Mg
17
phases as a function temperature in AZ61 alloy by equilibrium (solid line) and Scheil (dashed line) models (b) (Shang
et al
., 2008).
Figure 8.11 Ternary phase diagrams of Mg–Sn–Ca system (a) and Mg–Sn–Sr system (b) at 298 K.
Figure 8.12 Calculated self-diffusion and dilute solute tracer diffusion coefficients in hcp Mg along (a) basal plane and (b) normal to the basal plane . The original data of diffusion coefficients are based on the first-principles calculations by Ganeshan
et al
. (2010, 2011) and Wang
et al
. (2015).
Figure 8.13 Predicted variation tendency
B
/
G
ratio affected by solute atoms in Mg
95
X referring to (a) concentration of solute atoms and (b) volume of solute atoms in HCP. The original volume data of solute atoms are based on the first-principles calculations by Shang
et al
. Various symbols are used to identify the crystal structures of each individual solute atom at room temperature, such as FCC, HCP, BCC, and other complex structures.
Figure 8.14 Predicted stable fault energy and ideal shear strengths affected by solute atom in Mg
95
X (a) GSF (I2) and (b) twin fault (T2) based on the pure alias shear deformations. The original data are based on the first-principles calculations by Shang
et al
.. Various symbols are used to identify the crystal structures of each individual solute atom at room temperature, such as FCC, HCP, BCC, and other complex structures.
Figure 8.15 Twinnability (
Λ
FCC
) of Mg
95
X affected by solute atoms (a) variation tendency of
Λ
FCC
and (b) the change of
Λ
FCC
as a function of volume of FCC solute atom. The original data of twinnability and volume of FCC solute atoms are from Shang
et al
., respectively.
Figure 8.16 Isosurfaces of the deformation electron density of Mg
95
X. These differential charge densities are parallel to the {0001} planes and close to the alloying elements X in Mg
95
X. The dark gray circles highlight the reduced
Δρ
of Mg in the first nearest neighbor of X (Shang
et al
., 2014).
Figure 8.17 Contributions of lattice distortion on the bond structure and the force field of 10
H
LPSO of Mg–10Gd (wt%). (a) isosurface of (100) plane in the positive and negative mode (a-1) and the positive mode (a-2); (b) local force field caused by the segregation of Gd in the fault layers of 10H LPSO; and (c) contour plots of the local force (eV/Å) along
V
1 ,
V
2 , and
V
3 (⟨001⟩
s. c.
//⟨0001⟩
10
H
).
Chapter 9: Multiscale Statistical Study of Twinning in HCP Metals
Figure 9.1 Mg AZ31 deformed at 300 K and 10
−3
s
−1
. (a) Basal pole distribution of the initial rolling texture showing texture component around the normal direction (ND) to the plate; (b) stress–strain behavior from deformation by in-plane tension (IPT), in-plane compression (IPC), and through thickness compression (TTC); (c) electron backscatter diffraction (EBSD) micrograph showing {10−12} tensile deformation twins.
Figure 9.2 Flow chart showing schematically how the different length scales and experimental and computational results to be discussed in what follows are linked in order to develop a multiscale model of the mechanical response of HCP materials.
Figure 9.3 Typical shear modes active in HCP materials. (a) basal slip; (b) prismatic slip; (c) pyramidal slip and tensile twinning. The latter typically reorients the
c
-axis by about 90°, depending on
c
/
a
ratio.
Figure 9.4 (a) EBSD showing tensile twins in Zr; (b) a composite grain (CG) representation of parent and twins showing the predominant twin system and the mean free path of dislocations inside parent and twin domain. In the original CG model, volume fraction is transferred from parent to twins during twin growth, and some stress and strain rate components are enforced to be continuous across the parent–twin interface; (c) an alternative model, where continuity conditions across interfaces are eliminated and twins and parents are treated as uncoupled ellipsoids, except for the volume fraction transfer.
Figure 9.5 Measured and predicted stress–strain behavior of rolled Mg AZ31 loaded to 5% strain by in-plane compression and reloaded a further 10% strain by through-thickness compression. Simulation results are obtained with the composite grain (CG) model and with the uncoupled twinning–detwinning–twinning (TDT) model. Predicted basal pole Figure show the detwinning effect at the end of the TTC reload.
Figure 9.6 Flow response of rolled zirconium tested at a rate of 10
−3
s
−1
at 76, 150, 300, and 450 K. (a) in-plane tension (IPT), (b) in-plane compression (IPC), (c) through-thickness compression (TTC). Measured responses are represented by symbols and predicted responses by solid lines. Results obtained using parameters reported by Beyerlein and Tomé (2008) instead of the ones reported in Tables 9.1 and 9.2. The dashed lines correspond to assuming that the propagation stress is constant for compressive {11−22} twins () (Beyerlein and Tomé, 2008).
Figure 9.7 Comparison between IPC experimental stress–strain curves for high-purity Zr and a constitutive model that assumes activation of slip and twinning (a) in a deterministic way (dashed line) and (b) in a probabilistic way (solid line). The evolution of flow stress and hardening is more consistent with experiments in the latter case (Beyerlein and Tomé, 2010).
Figure 9.8 Excess potential energy of grain boundaries as a function of tilt angle
θ
. The regions R-1 to R-7 mark ranges of
θ
in which STGBs can be viewed as having similar atomic structure. R-1 and R-7: arrays of GBDs; R-2: random configuration STGBs; R-3 to R-5: twin boundaries plus GBDs; R-6: {−2021} STGB plus GBDs. Interaction between a STGBs and a four-dislocation pileup: (b) twin nucleation at a 17.35° STGB; (c) migration of a 68.20° STGB twin boundary. Dashed lines in (b) and (c) outline the boundaries.
Figure 9.9 Twin statistics based on EBSD data of rolled Mg compressed 3% along the In Plane direction. The frequency of twin variants as a function of their Schmid Factor (SF).
V
1 is the variant with the highest SF in a grain, and
V
6 the one with the lowest SF.
Figure 9.10 Mg deformed 3% by in plane compression. (a) Number of twins per grain plotted versus grain diameter; (b) twinned volume fraction per grain plotted versus grain diameter; (c) twin thickness plotted versus Schmid factor.
Figure 9.11 Illustration of a grain in a polycrystal with the neighboring grains not shown. As shown, the boundary contains a network of facets (
a
*), triple lines, and quadruple points.
Figure 9.12 FFT simulations for one RVE containing 522 grains and 565,437 grain boundary voxels. (a) Grain structure of the RVE. Grains were generated by Voronoi tessellation and assigned an orientation by sampling from the experimentally measured initial texture. (b) Map of von Mises effective stress in the RVE under applied compression. (c) Map showing the deviation in effective stress from the grain average. The color scale has been truncated for better visual contrast. The actual deviation in von Mises stress could be as high as 800 MPa.
Figure 9.13 Distribution of fluctuations of normal stress component
σ
11
as a function of macroscopic applied strain. The distributions are well approximated by a Gaussian except for the larger values at the distribution tails.
Figure 9.14 (a) Grain size distribution, (b) predicted distribution for the number of neighbor grains, and (c) predicted distribution of grain facet areas for the Zr sample modeled here.
Figure 9.15 Comparison of model predictions with experimental stress–strain curves for: (a) in-plane compression at 76, 150, and 300 K for Zr (Niezgoda
et al
., 2014); (b) same as (a) for the 76 K case comparing predictions of a deterministic model and the current probabilistic twin model (Beyerlein
et al.
, 2011b); (c) in-plane compression of Mg at room temperature.
Figure 9.16 Comparison of predicted and measured basal pole Figure at different amounts of strain for in-plane compression along the TD. Zr was deformed at 76 K (Beyerlein and Tomé, 2010) and Mg at 300 K (Beyerlein
et al
., 2011a). Contour lines are 0.5, 1, 2, 3, 4, 5, 6, 8 m.r.o. Notice the large twin-induced reorientation of the
c
-axis.
Figure 9.17 Comparison of twin volume fraction predicted by the model with measurements made by EBSD on high-purity Zr (Niezgoda
et al
., 2014).
Figure 9.18 In-plane compression of rolled Zr at 76 K. (a), (c), and (e): calculated distribution frequency of twins versus geometric Schmid factor for 5%, 10%, and 20% strain. (c) also reports 10% measured statistics (Capolungo
et al
., 2009b). (b), (d), and (f): Calculated volume distribution of geometric Schmid factors. Histograms should sum to unity as they are normalized by the total volume of all twins.
Figure 9.19 Number of twins per grain for (a) Zr after 10% IPC strain and (b) Mg after 3% IPC strain, (c) and (d) show twin fraction as a function of grain size.
Figure 9.20 Initial texture of clock-rolled Zr. True stress–strain during through thickness compression, in plane compression, and in plane tension axial tests at 300 K (Segurado
et al
., 2012).
Figure 9.21 Four-point bending of textured Zr bars, up to approximately 20% strain in the outer fibers. (a) Predicted final 3-D shapes; (b) predicted final lateral sections, showing the distribution of strain component along the fiber direction. TT vertical and TT horizontal refer to basal texture component contained in the bending plane or perpendicular to it, respectively; and (c) four-point beam testing jig and deformed test specimen for the TT vertical test.
Figure 9.22 Four-point bending of textured Zr bars at 300 K. (a) Predicted final cross sections showing the distribution of the stress component along the fiber direction (in MPa); (b) picture of final cross sections (Kaschner
et al
., 2001); white arrows represent the predominant direction of the basal texture component and dots correspond to the FE-VPSC predicted profile.
Figure 9.23 Four-point bending of textured Zr bars at 300 K. Predicted stress component parallel to the axis of the beam (MPa) along the vertical direction of the beam section: (a) TT vertical and (b) TT horizontal. Stresses are shown evolving as a function of time (strain). The final stress profile associated with an isotropic (von Mises) and ideally plastic material (
σ
yield
= 100 MPa) is superimposed for comparison purposes.
Figure 9.24 Four-point bending of textured Zr bars at 300 K. Measured and predicted strain components along the axis of the bar (
E
yy
), along the bottom-to-top direction (
E
zz
), and normal to the bending plane (
E
xx
). (a) TT vertical and (b) TT horizontal.
Chapter 10: Cast Magnesium Alloy Corvette Engine Cradle
Figure 10.1 The various length scales for analysis of a Mg alloy structural component.
Figure 10.2 Schematic illustrating different predictions of failure locations using three approaches. The maximum stress was found in location (D), the maximum porosity found in location (C); however, location (E) was the failure spot because damage accumulation represented by a damage internal state variable (ISV) was dependent upon both entities.
Figure 10.3 A summary of the bridging required between the electronic and atomistic scales.
Figure 10.4 Energy–volume dependence of (a) Al and (b) Mg in FCC, HCP, BCC, and simple cubic crystal structures.
Figure 10.5 Al–Mg in NaCl, ZnS, and CsCl crystal structures.
Figure 10.7 Stress–strain behavior of macroscopic AM60 alloy (6% of Al in Mg) showing the tension–compression asymmetry. The experiments were performed at a quasistatic strain rate at room temperature.
Figure 10.6 Summary of the bridge between the mesoscale and the atomistic scale.
Figure 10.8 Atomistic stress–strain behavior comparing compression and tension for single-crystal magnesium simulations using the modified embedded atom method (MEAM) potential at a strain rate of 10
8
/s and associated centrosymmetry plots illustrating dislocation nucleation, twinning, and fracture.
Figure 10.9 True stress–strain behavior illustrated at the atomistic scale using the modified embedded atom method (MEAM) potential for single-crystal magnesium oriented for basal plane loading showing that as the applied strain rate increases, the yield stress increases.
Figure 10.10 True stress–strain tensile behavior illustrated at the atomistic scale using the modified embedded atom method (MEAM) potential for single-crystal magnesium oriented for basal plane loading showing that as the vacancy concentration (local atomistic porosity) increases, the yield stress and the elastic modulus decrease.
Figure 10.11 True stress–strain tensile behavior illustrated at the atomistic scale using the modified embedded atom method (MEAM) potential for single-crystal magnesium oriented for basal plane loading showing as the percentage of substitutional aluminum atoms increases, the yield stress decreases and the elastic modulus increases.
Figure 10.12 Summary of the bridge between the mesoscale and the macroscale modeling.
Figure 10.13 Contour plot of stress triaxiality (SDV12) for AM60 with (a) two cylindrical voids with a two-void diameter spacing at a 15% remote strain in a one-eighth space analysis and (b) two spherical voids with a two-void diameter spacing at a 15% remote strain in a one-eighth space analysis. The loading direction was in the vertical direction.
Figure 10.14 Normalized void fraction versus engineering strain comparing the growth of cylindrical and spherical voids in AM60.
Figure 10.15 Comparison of experimental AM30 magnesium tension data with plasticity model parameters. Quasistatic experiments were performed at room temperature.
Figure 10.20 Comparison of experimental AE44 magnesium tension and compression data at different temperatures with plasticity model parameters. The experiments were performed at a quasistatic strain rate.
Figure 10.21 Boundary conditions for single-element monotonic simulations.
Figure 10.25 Comparing internal state variable (ISV) damage model implemented into the finite element method (FEM) and experimental failure elongation versus temperature for AM50.
Figure 10.26 Damage as a function of strain for different levels of initial porosity arising from the manufacturing process for AE44.
Figure 10.28 Damage as a function of strain for different levels of initial porosity arising from the manufacturing process for AM60.
Figure 10.29 Computed tomography (X-ray) porosity state after manufacturing and before mechanical testing on the left are compared to the initial porosity levels (vvf) in finite element mesh on the right.
Figure 10.30 Computed tomography X-ray of notch fractured AM60B Mg alloy specimen and predicted damage from the finite element simulation. The black regions in the simulation represent element failures due to damage evolution. Note that regions A, B, and C are shown to give similar geometric trends.
Figure 10.31 Boundary conditions for cradle load-to-failure analyse showing the cradle, mounting legs, swivels, and beams.
Figure 10.32 Image of cradle showing marked areas where sections were taken and porosity levels were measured.
Figure 10.33 Predicted failure locations from cradle load-to-failure analysis when
homogeneous
porosity distributions of 0.48% and 1.50% were used.
Figure 10.34 Predicted failure location from cradle load-to-failure analysis using measured porosity levels mapped onto cradle mesh.
Figure 10.35 If only stress or initial porosity were considered independently the failure location would not be predicted correctly because damage is dependent upon both stress and inclusions.
Figure 10.36 Comparison between simulation prediction and experiments showing good correlation.
Chapter 11: Using an Internal State Variable (ISV)–Multistage Fatigue (MSF) Sequential Analysis for the Design of a Cast AZ91 Magnesium Alloy Front-End Automotive Component
Figure 11.1 Shock tower and connecting parts.
Figure 11.2 (a) Processing–structure–property–performance relationships typically miss the macroscale material modeling and simulations and (b) hierarchical top-down and bottom-up approaches in design (Olson 1997).
Figure 11.3 Capturing all the PSPD relationships in the design of a component requires the incorporation of different length scales (Allison, 2006).
Figure 11.4 Multiscale modeling approach (Horstemeyer, 2012).
Figure 11.5 Solving inverse problem approach for the evaluation of cyclic performance of the shock tower.
Figure 11.6 Integrated computational materials engineering (ICME) perspective through an internal state variable (ISV)–multistage fatigue (MSF) sequential modeling for design life analysis of the structural shock tower.
Figure 11.7 Shock tower measured porosity locations (image) and volume fraction results (see Table 11.1).
Figure 11.8 (a) Single optical mosaic image from location L3 and (b) X-ray scanned image of location L3 showing porosity.
Figure 11.9 Representative micrographs from L1 (a) and L2 (b) for as-cast AZ91 (Rettberg
et al
., 2012).
Figure 11.10 Quasistatic (0.001/s) compression and tension tests results from shock tower material.
Figure 11.11 Fractured surface of a quasistatic tension specimen, at four different magnifications, from an AZ91 shock tower.
Figure 11.12 (a) Locations L1 and L2 from where samples were taken. (b) Strain-life curve for the as-cast AZ91 Mg alloy for locations L1 and L2. (Rettberg
et al
., 2012).
Figure 11.13 Crack initiation sites observed for the fatigued fracture surface specimens of the AZ91 Mg alloy (Rettberg
et al
., 2012).
Figure 11.14 Number of cycles to failure against the equivalent initiation pore size for AZ91, in the as-cast condition and at a strain amplitude of 0.2%. Points encircled correspond to initiation sites at surface pores.
Figure 11.15 Comparison of the strain-life experimental results for locations L1 and L2 to the mean, upper, and lower bound estimations of the multistage fatigue model.
Figure 11.16 Load and boundary conditions on the AZ91 Mg alloy shock tower.
Figure 11.17 The four strain metrics investigated to test the usefulness in the MultiStage Fatigue model were the (a) effective strain calculated by integrating the scalar product of the deviatoric strain tensor increment, (b) effective strain calculated by the scalar product of the deviatoric strain tensor, (c) maximum principal strain , and (d) the plastic equivalent strain . is the deviatoric strain rate tensor. The results showed the effective strain tensor best predicted the number of cycles to failure.
Figure 11.18 (a) Solidification time contours for the AZ91 Mg alloy. Maps for distribution of (b) porosity, (c) pore size, and (d) dendrite cell size.
Figure 11.19 Internal state variable (ISV)—Multistage Fatigue (MSF) simulation results for (a) FEA #1, (b) FEA # 3, (c) FEA#5, and (d) FEA #2. Failure location also is indicated.
Figure 11.20 Illustration of test setup (a) and actual test setup (b) for fatigue tests.
Figure 11.21 Load–displacement response for the shock tower fatigue tests.
Figure 11.22 Failure location for fatigue shock tower test 1.
Figure 11.23 Finite element model and failure locations.
Figure 11.24 Finite element model and fatigue failure location (compared to the experimental failure in Figure 11.22).
Chapter 12: Electronic Structures and Materials Properties Calculations of Ni and Ni-Based Superalloys
Figure 12.1 A schematic diagram showing the data-driven approach to ranking the effectiveness of alloying elements used in future Ni-base superalloy development.
Figure 12.2 Twenty-six alloying elements and their atomic number studied in this work in their approximate location on the periodic table.
Figure 12.3 Illustration showing DFT calculated enthalpy of (a) Ni and (b) NiAl as a function of temperature with and without the thermal electronic contribution, solid and dashed lines, respectively, using linear response theory for the vibrational contribution to the free energy. The calculated results are compared to well-evaluated experimental thermodynamic properties from the literature with details in the reference.
Figure 12.4 A schematic Figure representing the CALPHAD approach (Saunders and Miodownik, 1998).
Figure 12.5 Five-frequency model as developed by Leclaire and Lidiard (1956) and Lidiard (1955) as depicted by Mehrer (2007) showing the five jump frequencies needed to calculate the dilute diffusion coefficient, where represents the solute/impurity atom, represents a vacancy, and represents the host/solvent atom (fcc Ni in the present work).
Figure 12.6 Diagram showing DFT calculated enthalpy of (a) Ni and (b) Ni
3
Al as a function of temperature with the thermal electronic contribution, using the phonon supercell approach (light gray dashed line), the Debye–Grüneisen model (black dotted line) and the Debye–Wang model (solid gray line) for calculating the vibrational contribution to the Helmholtz energy.
Source
: Reproduced from Shang
et al
. (2010b).
Figure 12.7 (a) Entropy, (b) enthalpy, and (c) heat capacity of ReTi as a function of temperature for the B2 structure from the phonon supercell approach (solid line) and the Debye model (dotted line) and for the MoTi structure (dotted-dashed line).
Figure 12.8 Calculated phase diagram of the Re–Ti system (a) without the ReTi phase and (b) with the ReTi phase, with experimental data from Savitskii and Tylkina (1959) showing the melting temperature of Re
24
Ti
5
(), the solidus (), bcc-Ti single-phase region (), and hcp-Ti + bcc-Ti two-phase region (), Savitskii
et al
. (1969) showing hcp-Re single-phase region (), and hcp-Re + Re
24
Ti
5
two-phase region ().
Figure 12.9 Calculated shear moduli and Young's moduli of Ni
31
X alloys at their equilibrium volumes based on the Voigt approach (see Eqs (12.4) and (12.5)) at 0 K and without the zero-point vibrational energies.
Figure 12.10 Predicted properties at 300 K for Ni
31
X, (a) the shear modulus, (b) the length of Burges vector, and (c) stacking fault energy
γ
SF
. Here, the white (low)–light gray (middle)–dark gray (high) color scale is used to mark the change in data based on alloying element.
Figure 12.11 Impurity diffusion coefficient at
T
= 1000 K plotted as a function of increasing atomic number along different transition element rows (3d elements: circles, 4d elements: squares, 5d elements: triangles, other elements: diamonds) in the periodic table.
Figure 12.12 Log of diffusivity of the 26 Ni
31
X systems where X = 3d elements (circles), 4d elements, (squares), 5d elements (triangles), and others (diamonds) plotted as a function of (a) increasing experimental ionic radius (Å) and (b) increasing calculated compressibility, (1/GPa) for all elements.
Figure 12.13 Convex hull of the enthalpy of formation in the Re–Y system. The enthalpy of formation from the CALPHAD modeling is represented by the solid lines and the enthalpy of formation of Re
2
Y calculated via first-principles calculations is represented by .
Figure 12.14 (a) Isothermal section of the Ni–Re–Y system from 1000 K, and (b) the liquidus projection of the Ni–Re–Y showing phases formed during primary solidification.
Figure 12.15 Relative creep rate ratio plotted vs increasing atomic number for 3d elements (circles), 4d elements, (squares), 5d elements (triangles), and others (diamonds) at (a)
T
= 600 K and (b)
T
= 1200 K.
Chapter 13: Nickel Powder Metal Modeling Illustrating Atomistic-Continuum Friction Laws
Figure 13.1 Evolution of the failure and cap yield surfaces of the modified Drucker/Prager Cap model.
Figure 13.2 Representation of the double yield surface for dense powder aggregate.
Figure 13.3 Schematic of two-particle model showing active nickel atoms as white circles and fixed boundary atoms as gray. The arrows indicate the loading direction. The model setup includes contact angles of γ = 0°, 30°, or 60°, diameter D = 3.52, 7.04, 10, or 14.08 nm, and velocity of 0.22 nm/ps up to 20% strain. The orientation of the crystal is shown on the right side.
Figure 13.4 Log–log shear yield stress normalized by the shear modulus versus volume-per-surface area for nickel, gold, and copper for various experiments and MD simulations.
Figure 13.5 Evolution of dislocation structures and plastic deformation between 3 and 5% strain from MD Simulations for 14 nm particle model with a 60 ° contact angle. Dislocation nucleation and glide along discrete slip planes through the centers and along the interface of two contacting particles is demonstrated along with formation of complex dislocation structures attributing to plastic deformation.
Figure 13.6 EAM MD simulation results for tangential force versus normal force for (a) 30 ° and (b) 60 ° contact angle.
Figure 13.7 Correlated friction evolution equation from MD simulation data for 60 ° contact angle between the spherical particles.
Figure 13.8 Friction evolution model for 60 ° contact angle between two spherical particles applied to particles in the micrometer range.
Figure 13.9 Comparison of saturated coefficient of friction values between the model prediction and experimental and MD simulation results based on volume per surface area.
Chapter 14: Multiscale Modeling of Pure Nickel
Figure 14.1 Schematic detailing the upscaling/downscaling relationships. Downscaling requirements are driven from the boundary value problem or highest length scale that needs to be solved. That information is then used from one length scale to the next to determine the upscaling requirements for solving the problem. Uncertainty is quantified at each length scale and directly compared to propagated uncertainty acquired from lower length scales (when applicable). (
See color plate section for the color representation of this figure
.)
Figure 14.2 Typical sources of uncertainty for experimental and model-based analysis of material behavior.
Figure 14.3 Schematic of Bridge 1 between the electron (density functional theory, DFT) and atom (molecular dynamics, MD) length scales and Bridge 4 between the electron (DFT) and the continuum (internal state variable, ISV) element associated with the elastic moduli results for nickel (Ni).
Figure 14.4 (a) Normalized interatomic energy versus volume curves for nickel in BCC, FCC, and HCP structures as calculated using the electronics structures code VASP. Data from VASP were then used to calibrate LAMMPS modified embedded atom method (MEAM) parameters for atomic mobility simulations. LAMMPS MEAM calibration for nickel demonstrates good agreement with DFT data for the available range of DFT data. (b) Generalized stacking fault energy (GSFE) curves for nickel. Density functional theory (DFT) and the modified embedded atom method (MEAM) potential data were gathered for pure nickel and analyzed to produce a composite mean and a 95% confidence interval that is one standard error from the mean. Numerical DFT simulation results were used to calibrate MEAM LAMMPS parameters for use in atomic mobility calculations. The MEAM LAMMPS calibration shows good agreement with the numerical data from DFT calculations.
Figure 14.5 Schematic of Bridge 2 between atoms (molecular dynamics, MD) and dislocations (dislocation dynamics, DD) length scales and Bridge 5 associated with temperature dependence between atoms and the continuum for nickel (Ni).
Figure 14.6 Dislocation core position versus time for an applied shear stress of 100 MPa illustrating the steady-state motion of the dislocation core in single crystal nickel as it moves across periodic boundaries.
