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Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science, First Edition. Mark F. Horstemeyer.© 2012 The Minerals, Metals & Materials Society. Published 2012 by John Wiley & Sons, Inc.

CHAPTER 1

AN INTRODUCTION TO INTEGRATED COMPUTATIONAL MATERIALS ENGINEERING ( ICME )

The concept of Integrated Computational Materials Engineering (ICME) arises from the new simulation - based design paradigm that employs a hierar-chical multiscale modeling methodology for optimizing load - bearing struc-tures. The methodology integrates material models, structure – property relationships that are observed from experiments, and simulations starting at the quantum level. At the structural level, heterogeneous microstructures are embedded in the fi nite element analysis. Because these microstructures are included, the paradigm shift from safety factors to predicting failure is fundamental.

ICME ’ s opportunity has emerged because of the recent confl uence of smaller desktop computers with enhanced computing power coupled with the advent of physically based material models. Furthermore, the clear trend in modeling and simulation is to integrate more knowledge into materials processing and product performance. I propose that ICME is the appropriate means to garner the required accuracy for a simulation - based design and manufacturing para-digm, and this book is a means for engineers to realize that goal. This fi rst chapter includes Horstemeyer’s [1] review of the various multiscale method-ologies related to solid materials and the associated experimental infl uences, the various infl uences of multiscale modeling on different disciplines, and some examples of multiscale modeling in design of structural components.

1

COPYRIG

HTED M

ATERIAL

2 INTRODUCTION TO ICME

1.1 BACKGROUND

Although computational multiscale modeling methodologies were developed in the very late 20th century, the fundamental notions of multiscale modeling have been around since da Vinci studied different sizes of ropes. The recent rapid growth in multiscale modeling arose from the confl uence of parallel computing power, experimental capabilities that characterize structure – property relations down to the atomic level, and theories that admit multiple length scales. The ubiquitous research focused on multiscale modeling has since broached different disciplines (solid mechanics, fl uid mechanics, mate rials science, physics, mathematics, biological, and chemistry), different regions of the world (most continents), and different length scales (from atoms to autos).

With the advent of accurate modeling and simulation and signifi cant increases in economical computing power, virtual design and manufacturing provides the means to reduce product development time and cost while improv-ing overall quality and manufacturing effi ciency. However, the quality of the end product depends on the quality of the modeling with respect to the par-ticular conditions involved and the computational effi ciency of the simulations (e.g., plasticity and fracture of specifi c materials under extremely rapid stress conditions). Several case studies are later shown to demonstrate the important benefi ts of such analysis and design. The knowledge gained and the computa-tional tools developed in these illustrations can then be rapidly applied to other product designs for structural components. Design optimization with different quality standards and uncertainty can then be achieved using virtual simula-tions with rapid turnaround — even more important in the context of optimizing the “ system ” with various subsystem and component trade - offs. The foundation for these results is built on the accuracy of the modeling and computational accuracy and effi ciency. Accordingly, ICME offers the ideal venue for the simulation - based design and manufacturing paradigm that will be presented in this book.

1.2 THE APPLICATION OF MULTISCALE MATERIALS MODELING VIA ICME

Although this book is dedicated primarily to the application of ICME prin-ciples to the design and manufacture of structural materials with nonlinear behavior, the integrated circuits industry provides an analogy of the progress achieved in modeling components and systems with different modeling accu-racies and utilizing multilevel simulation tools. Even with the huge advance-ments in parallel computing power, it is simply impractical to model a complete system that accommodates all possible application conditions (e.g., test condi-tions) with the highest - accuracy physics. However, these highly accurate physics models are able to capture the phenomena under various extreme

THE APPLICATION OF MULTISCALE MATERIALS MODELING VIA ICME 3

conditions and thereby provide the basis for more abstract models (e.g., current - voltage model of transistors), which are more computationally effi -cient. These transistor models become the basis for an even higher level of abstraction (e.g., switch level or logic gate level), which in turn becomes the basis for simulating logic blocks or subsystems over the greater range of test conditions with the environmental conditions already being validated by the lower level modeling. Often the systems are simulated using simply input – output logic signals with various degrees of timing delays determined from lower level simulations. Tools have evolved which support concurrent multi-level simulations (e.g., different logic blocks are simulated with different levels of accuracy and with appropriate interfaces between the simulation blocks in order to focus the computing power on particular issues).

These multilevel modeling and simulation tools support rapid virtual design without resorting to the time - consuming physical prototyping until there is suffi cient confi dence in the design. This virtual simulation capability further supports design optimization because it allows far more iterations than can be achieved utilizing physical prototyping and supports the evolution of more sophisticated and “ tuned ” design optimization tools, which become the basis of perhaps “ automatically generated ” subsystem design optimization. In turn, these simulation capabilities become the basis for designing for manufactur-ability and quality (e.g., selecting test vector conditions for physical testing to minimize expensive test time and correlating product test results to manufac-turing variations to assure quality). It simply is not practical to consider design-ing modern electronic systems with several hundred million transistors on a “ chip ” without using these virtual computational tools. For such system com-plexity, a more rapid time - to - market provides a competitive advantage as well as saving costs.

The semiconductor and computer industries have been the leaders in devel-oping these methodologies, representing a signifi cant paradigm shift and pro-viding a means for sustaining their phenomenal growth over so many years, illustrated by Gordon Moore ’ s infamous laws of complexity doubling every 2 years and performance doubling every 18 months while maintaining equiva-lent costs. With such industrial productivity exemplifi ed by the culture of “ Silicon Valley, ” the competition drives new product development with rapid time - to - market being vital.

With the currently available computing power, the question remains: can a similar philosophy of multilevel modeling and simulation be utilized in material - based mechanical systems in order to achieve rapid time - to - market in product design and manufacturing? If so, what is the current state of knowl-edge and practice? What can one expect in the future? Clearly, the primary underlying issue relates to the multilevel material modeling (multiscale) involved in order to achieve the required accuracy and computational effi -ciency under the associated conditions .

Several detailed case studies are presented in this book to help the para-digm shift for employing multiscale modeling methods for structural and


mechanical designed components. In each example case, this book considers an aluminum alloy for instructional purposes and consistency, but the ICME methodology of multiscale modeling has been used for magnesium, steel, and other alloys.

Whether designing an automobile, airplane, building, or any structural system for that matter, large - scale systems tests are both expensive and time - consuming. However, the cost models for using virtual design methodologies that are physics based show signifi cant reduction in the time - to - market and costs. The advantages include the following:

1. ICME can reduce the product development time by alleviating costly trial - and - error physical design iterations (design cycles) and facilitate far more cost - effective virtual design optimization.

2. ICME can reduce product costs through innovations in material, product, and process designs.

3. ICME can reduce the number of costly large systems scale experiments. 4. ICME can increase product quality and performance by providing more

accurate predictions of response to design loads. 5. ICME can help develop new materials. 6. ICME can help medical practice in making diagnostic and prognostic

evaluations related to the human body.

These benefi ts, which are now being realized, are the market drivers for such an explosion of multiscale modeling into various industrial sectors.

1.3 HISTORY OF MULTISCALE MODELING

The recent surge in multiscale modeling, from the smallest scale (atoms) to full system level (e.g., autos) related to solid mechanics, that has now grown into an international multidisciplinary activity, was birthed from an unlikely source. Since the U.S. Department of Energy ( DOE ) national labs (Los Alamos National Laboratory (LANL), Lawrence Livermore National Laboratory (LLNL), Sandia National Laboratories (SNL), and Oak Ridge National Labo-ratory (ORNL)) started to reduce nuclear underground tests in the mid - 1980s, with the last one in 1992, the idea of simulation - based design and analysis concepts emerged. After the Comprehensive Test Ban Treaty of 1996 in which many countries pledged to discontinue all systems level nuclear testing, pro-grams like the Advanced Strategic Computing Initiative (ASCI) were initiated within the U.S. DOE and managed by the labs. The basic premise of ASCI was to provide more accurate and precise simulation-based design and analysis tools. In essence, the numerous, large - scale systems level tests that were previ-

HISTORY OF MULTISCALE MODELING 5

ously used to validate a design were no longer acceptable, thus warranting the tremendous increase in reliance upon simulation results of complex systems for design verifi cation and validation purposes.

Because of the requirement for greater complexity in these simulations, advancing parallel computing and multiscale modeling became top priorities. With this perspective, experimental paradigms shifted from the large - scale, complex tests to multiscale experiments that provided material models with validation at different length scales. If the modeling and simulations were physically based and less empirical, then a predictive capability could be real-ized for other conditions. As such, various multiscale modeling methodologies were independently but concurrently created at the U.S. DOE national labs. In addition, personnel from these national labs encouraged, funded, and managed academic research related to multiscale modeling. Hence, the cre-ation of different methodologies and computational algorithms for parallel environments gave rise to different emphases regarding multiscale modeling and the associated multiscale experiments.

Signifi cant advances in parallel computing capabilities further contributed to the development of multiscale modeling. Since more degrees of freedom could be resolved by parallel computing environments, more accurate and precise algorithmic formulations could be admitted. This thought also drove political leaders to encourage the simulation - based design concepts.

At LANL, LLNL, and ORNL, the multiscale modeling efforts were driven from the materials science and physics communities with a bottom - up approach. Each had different programs that tried to unify computational efforts, mate-rials science information, and applied mechanics algorithms with different levels of success. Multiple scientifi c articles were written, and the multiscale activities took different lives of their own. At SNL, the multiscale modeling effort was an engineering top - down approach starting from a continuum mechanics perspective, which was already rich with a computational paradigm. SNL tried to merge the materials science community into the continuum mechanics community to address the lower length scale issues that could help solve engi-neering problems in practice.

Once this management infrastructure and associated funding were in place at the various U.S. DOE institutions, different academic research projects started, initiating various satellite networks of multiscale modeling research. Technological transfer also arose into other labs within the Department of Defense and industrial research communities.

The growth of multiscale modeling in the industrial sector was primarily due to fi nancial motivations. From the U.S. DOE national labs ’ perspective, the shift from large - scale systems experiments mentality occurred because of the 1996 Nuclear Ban Treaty. Once industry realized that the notions of mul-tiscale modeling and simulation - based design were invariant to the type of product and that effective multiscale simulations could in fact lead to design optimization, a paradigm shift began to occur, in various measures within


different industries, as cost savings and accuracy in product warranty estimates were rationalized.

1.3.1 Bridging between Scales: A Difference of Disciplines

Synergistic systems thinking and interdisciplinary thinking bring about the concept of the whole being greater than the sum of the parts. This relates to the “ integrated ” part of ICME. Multiscale modeling requires that several disciplines interact, which has led to miscommunications and misunderstand-ings between communities, particularly about core multiscale modeling and the bridging methodologies between length scales.

Clearly, a key issue in multiscale modeling is how to handle the bridging. Without offi cially stating the bridging methodology, each discipline has its own methods. Before we discuss each discipline ’ s bridging paradigm, let us consider an analogy of the Brooklyn Bridge in New York versus the Golden Gate Bridge in San Francisco. If one were to just translate the Brooklyn Bridge to San Francisco and call it the Golden Gate Bridge, one would fi nd that the old adage of a “ square peg in a round hole ” applies. The Golden Gate Bridge required a design different from that of the Brooklyn Bridge because of what was required on each side of the bridge and the environments that must be sustained. In other words, the boundary conditions played a major role in the design of the bridge. The same notion needs to be considered when developing bridges for multiscale modeling between different length scales. However, the different research disciplines (materials science, applied mechanics, atmospheric sciences, etc.) tend to focus on the research at each pertinent length scale and not so much on the bridge. In fact, modern compu-tational tools at each length scale were just recently published in Yip ’ s [2] Handbook of Materials Modeling , which provides a thorough review of a wide variety of current tools; however, this work did not really deal with bridging methodologies.

In the next sections, the multiscale modeling methods are presented from the different disciplines ’ perspectives. Clearly, one could argue that overlaps occur, but differences in the multiscale methods arise based on the paradigm from which they originated. For example, the solid mechanics internal state variable (ISV) theory includes mathematics, materials science, and numerical methods. However, it clearly started from a solid mechanics perspective, and the starting points for mathematics, materials science, and numerical methods has led to other different multiscale methods. ICME starts from the materials perspective, but it is worth noting the context of other disciplines in this textbook.

Some general guidelines about length scale bridging include the following list. Keep in mind that terminology is often an issue when bringing different disciplines together. In the context of this writing, the term “ upscaling ” means basically a bottom - up approach in which the simulations are performed at a particular scale and the results are averaged in some sense to pass to the next


higher scale. Alternatively, “ downscaling ” is a top - down approach, like the ISV continuum theory that is defi ned at the continuum level but allows lower length scale features via the ISVs:

1. For both downscaling and upscaling, only use the minimum required degrees of freedom necessary for that particular length scale for the type of structural problem being produced (e.g., the ISV may need to have the grain size in some cases but maybe does not need it in other cases).

2. For both downscaling and upscaling, be consistent with the energy within the pertinent volume between length scales; note that the geometric effects will most likely not be the same between length scales within a volume.

3. For both downscaling and upscaling, verify the numerical model ’ s imple-mentation and usage before starting a sequence of calculations that might otherwise lead to erroneous results.

4. For downscaling, start with downscaling before upscaling to help make clear the fi nal goal, requirements, and constraints at the highest length scale.

5. For downscaling, fi nd the pertinent variable and associated equation(s) to be the repository of the structure – property relationship from subscale information.

6. For upscaling, fi nd the pertinent “ effect ” for the next higher scale. Dif-ferent methods can be used: analysis of variation (ANOVA) methods, computations, experiments, and so on.

7. For upscaling, validate the “ effect ” by an experiment at the particular length scale before using it in the next higher length scale.

8. For upscaling, quantify the uncertainty (error) bands (upper and lower values) of the particular “ effect ” before using it in the next higher length scale and then use those limits to help determine the “ effects ” at the next higher level scale.

1.3.1.1 Solid Mechanics Bridging (Hierarchical Methods). Inherent within the idea of multiscale modeling is the bridging methodology and the associated length scale of the feature that is necessary to gain the accurate physics required for the engineering problem [3] . To decide on the pertinent length scale for the feature of importance, one must consider that in modern solid mechanics, continuum theories are driven by the conservation laws (mass, momentum, and energy); however, there are more unknowns than the number of equations, so constitutive relations (sometimes erroneously called “ laws ” ) are required to solve the set of differential equations for fi nite element or fi nite difference analysis. Most modern solid mechanics tools employ fi nite element analysis. When developing a multiscale modeling methodology for the


constitutive relations, the kinetics, kinematics, and thermodynamics need to be consistent in the formulation. There are also certain classical postulates in continuum theory that guide the development of the constitutive theory (objectivity, physical admissibility, equipresence, and locality). Objectivity means that the equations must operate in a consistent manner no matter what frame of reference is used. Physical admissibility means that what is true to the material ’ s behavior must be considered in the equations. Equipresence means that when using a variable in one equation, it must be used in all of the equations. Locality means that the observable variables, such as stress and strain, are related to each other just at the local point in space. Multiscale modeling has been driven by the physical admissibility postulate, and rightly so, at the expense of the postulates of equipresence and locality. In terms of multiscale modeling, physical admissibility means to identify the physical mechanism or discrete microstructural feature at the particular length scale that is a root source of the phenomenological behavior.

Two different general multiscale methodologies exist starting from the solid mechanics continuum theory paradigm: hierarchical and concurrent. The key difference is the bridging methodology. In concurrent methods, the bridging methodology is numerical or computational in nature, with the different length scale algorithms performed (essentially) concurrently. In the hierarchical methods, numerical techniques are independently run at disparate length scales. Then, a bridging methodology utilizing one of several methods (e.g., statistical analysis methods, homogenization techniques, or optimization methods) can be used to distinguish the pertinent cause – effect relations at the lower scale to determine the relevant effects for the next higher scale.

One effective hierarchical method for multiscale bridging is the use of thermodynamically constrained ISVs that can be physically based on microstructure – property relations. It is a top - down approach, meaning the ISVs exist at the macroscale but reach down to various subscales to receive pertinent information. The ISV theory owes much of its development to the state variable thermodynamics constructed by Helmholtz [4] and Maxwell [5] . The notion of ISV was introduced into thermodynamics by Onsager [6] and was applied to continuum mechanics by Eckart [7 – 8] .

The basic idea behind the ISV theory is that, in order to uniquely defi ne the Helmholtz free energy [4] of a system undergoing an irreversible process, one has to expand the dimensions of the state space of deformation and tem-perature (state variables commonly employed in classical thermodynamics to study elastic materials) by introducing a suffi cient number of additional state variables that are considered essential for the description of the internal struc-ture with the associated length scales of the material in question. The number of these ISVs is related to the material structure as well as to the degree of accuracy with which one wishes to represent the material response.

The ISV formulation is a means to capture the effects of a representative volume element (RVE) and not all of the complex causes at the local level;


hence, an ISV will macroscopically average in some fashion the details of the microscopic arrangement. In essence, the complete microstructure arrange-ment is unnecessary as long as the macroscale ISV representation is complete [9] . Figure 1.1 illustrates that we can then “ average ” or “ homogenize ” the effect of the discrete feature into a continuum that includes a number of ISVs. As a result, the ISV must be based on physically observed behaviors and constrained by the laws of thermodynamics [10] . Rice [11] and Kestin and Rice [12] added the notions of inelastic behavior into the context of ISV theory. The inelastic behaviors of importance for design and analysis of metal struc-tural components are typically plasticity, damage, and failure. For inelastic dissipative materials, the ISVs relate microstructural characteristics to mechan-ical behavior and have been used in various materials, polymers, composites, and ceramics [13 – 18] . However, it may be argued that ISV theory has probably had its greatest impact on metals. In the United States, ISV theories have enjoyed success in solving practical engineering problems. The mechanical threshold stress ( MTS ) model [19 – 21] was developed at LANL and focused on the microstructural details in relation to mechanical properties. Freed [22] at NASA developed an ISV model under the paradigm of unifi ed creep - plasticity modeling. Bammann [23 – 24] at SNL developed an ISV modeling framework that was microstructurally based and fi t into the unifi ed creep - plasticity paradigm. In Europe, Chaboche [25] focused on fatigue, large strain plasticity, and damage in his ISV formulations. Although many other ISV models could be discussed here, the above - mentioned theories have been suc-cessfully used in engineering practice on a routine basis for design and manu-facturing for metal and polymer - based material systems.

Figure 1.1 Homogenization of discretized microstructural features into a continuum medium.

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Not only is ISV theory important from a top - down (downscaling) approach, but the linking between scales is critical as well as the different computational methods that can be employed. Figure 1.2 shows a hierarchical multiscale modeling methodology illustrating the different bridges and analyses required to capture the pertinent plasticity, damage, and failure aspects of metal alloys for use in design of an automotive component. An example will be presented later, which describes these different size scale analyses and their associated bridges.

Before the term multiscale modeling was in vogue, methods of bridging lower length information into a continuum at a higher length scale were being addressed in the solid mechanics community. In an upscaling framework, Eshelby [26 – 27] essentially birthed the modern “ micromechanics ” world of analysis, modeling, and simulation by asserting a “ self - consistent ” theme. Essentially, subscale heterogeneities could be assigned an effect within the continuum, and the aggregates response could be averaged to give a contin-uum result. The continuum could be assigned as an element, a “ representative volume element ” or RVE. As such, the composites, metallurgical and applied mechanics communities used this framework for different applications. Modern mixture theory, mean fi eld theory [28] , crystal plasticity, method of cells, the Mura – Tanaka method [29] , homogenization theory, and the general-ized method of cells were developed from the self - consistent Eshelby formula-tion. Mura [30] , Budiansky [31] , and Nemat - Nasser et al. [32] gave a thorough review of these micromechanical self - consistent methods.

A part of the self - consistent method is determining how the RVE should be and how the boundary conditions work down from higher scales to lower scales. Fairly recently, Gokhale and Yang [33] , in the spirit of ICME, performed a hierarchical multiscale modeling study examining different length scale microstructures with digital imaging methods and fi nite element analysis. The technique brought together effects of features at higher length damage evolu-tion and local fracture processes occurring at lower length scales. Hao et al. [34] also employed hierarchical fracture methods at different scales.

In fi nalizing this section, it is worth noting that aside from the modern - day multiscale modeling using ISV theory and self - consistent theories, determin-ing the length scale effects on mechanics properties has been around a long time and has been summarized by Bazant and Chen [35] . Da Vinci in the 1500s made the statement that a longer rope was weaker than a smaller rope. In the 1600s, Galileo disagreed with da Vinci about the strengths and lengths of ropes as he studied the size effects of bones. In the 1700s, Euler related buckling to a column length. In the 1800s, Cauchy related the stress state to radius of a cylinder. In the early 1900s, Bridgman [36] showed for many metal alloys that the notch radii changed the stress state of the material. In all of these examples, the length scale parameter was related to the geometry of the component, not to anything internal to the component. Certainly, these types of size scale effects have been a part of modern fi nite element analysis. However, the multiscale modeling that we are discussing in this book

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1.3.1.2 Numerical Methods (Concurrent Methods). Concurrent methods typically try to combine different scale algorithms together with matching procedures invoked in some overlapping domain in order to resolve the important multiscale physics. A good review of these concurrent methods and the associated challenges can be found in the books by Phillips [37] and Liu et al. [38] and the review articles of Fish [39] and de Pablo and Curtin [40] . Typical concurrent methods have two different length scales, or at most three different length scales, of interest in the formulations. Too many length scales are cost prohibitive due to the complexity.

The fi rst and most active area of research for concurrent methods has been the coupling of atomistic level simulations to continuum level simulations concurrently with different information being shared between the two. The development of fi ne grid – coarse grid fi nite element methods [41] was the basis for these different concurrent multiscale methods. For multiscale modeling, the fi ne grid method was traded for other lower length scale methods, such as atomistic methods. For example, Kohlhoff et al. [42 – 43] were probably the fi rst to join atomistic level simulation capabilities within a fi nite element code. Another example was the quasicontinuum approach of Tadmor et al. [44] that came about by joining the physics community ’ s embedded atom method ( EAM ) molecular statics code developed by Daw and colleagues [45 – 46] at SNL, and the solid mechanics fi nite element formulations of Ortiz et al. [47] . The notion was to have the atomistic method local to a region where plasticity, fracture, or any activity caused dissipation, and the fi nite element method would apply to the appropriate boundary conditions. Shenoy et al. [48] later applied the quasicontinuum method to interfaces, illustrating that concurrent multiscale modeling can be applied to an application in which single length scale analyses could not be appropriately addressed otherwise. Miller et al. [49 – 50] further extended the concurrent quasicontinuum application space to plasticity and fracture. Shenoy et al. [51] later added an adaptive fi nite element meshing capability to the quasicontinuum method in order to broaden the application space. At this time, Abraham et al. [52] with the help of Plimpton [53] from SNL used an atomistic fi nite element code to examine fracture in brittle materials under dynamic loads. There was another group researching concurrent methods at this time as well. Rudd and Broughton [54] and Brough-ton et al. [55] examined dynamic aspects of concurrent multiscale modeling with atomistics, fi nite elements, and their inherent limitations on particular applications.

Others went on to study various aspects of quasicontinuum concurrent multiscale methods. Lidirokis et al. [56] studied local stress states around Si nanopixels using this method. Bazant [57] argued that these atomistic fi nite


element multiscale methods cannot really capture inelastic behavior like frac-ture because the softening effect requires a regularization of the local region that is not resolved.

By the year 2000, the notion of concurrent multiscale modeling had diffused into the greater solid mechanics community at different length scales besides the atomic continuum level. Shilkrot et al. [58 – 59] introduced a method that essentially combined the quasicontinuum method with the discrete dislocation method. In this same spirit, Shiari et al. [60] and Dewald and Curtin [61] con-nected atomistics with dislocation dynamics simulations in a two - scale meth-odology to focus on grain boundary effects. At a higher length scale, Zbib and Diaz de la Rubia [62] joined discrete dislocation methods with fi nite element methods. Hence, the local geometric conditions and kinetics that controlled the dynamics of dislocations were placed into a continuum fi eld theory code that could solve boundary value problems.

Although the crystal plasticity modeling community focused on several length scales in their simulations, the term concurrent multiscale modeling was not historically used. Since crystal plasticity formulations start at the scale of the grain and volume average up to give polycrystalline results, and the poly-crystalline results can affect the results within the grain, they are, in a sense, a pseudo - concurrent method and really a self - consistent method. They can also bring in lower length scale effects in a hierarchical manner [63] . Crystal plastic-ity models became popular during the 1980s as a tool to study deformation and texture behavior of metals during material processing [64] and shear localization [65 – 66] . The basic elements of the theory comprise (i) kinetics related to slip system hardening laws to refl ect intragranular work hardening, including self and latent hardening components [67] ; (ii) kinematics in which the concept of the plastic spin plays an important role; and (iii) intergranular constraint laws to govern interactions among crystals or grains. The theory is commonly acknowledged for providing realistic prediction/correlation of texture development and stress – strain behavior at large strains as it joins continuum theory with discretized crystal activity.

Similar to crystal plasticity modeling and simulation in which discrete enti-ties such as crystals were placed in a larger continuum domain, Shephard et al. [68] discussed concurrent automatic interacting models at different scales considering geometric representations and discretizations required by microstructure. Still others have focused on the bridging algorithms: the bridg-ing domain method of Xiao and Belytschko [69] , the bridging scale method of Wagner and Liu [70] , and Karpov et al. [71] . Nonrefl ecting boundary condi-tion methods originating from the seminal work of Adelman and Doll [72] led to recent works of concurrent multiscale models of Cai et al. [73] , Huang [74] , Gruttmann and Wagner [75] , Karpov et al. [71] , and Park et al. [76] . Essen-tially, the nonrefl ecting boundary allows molecular dynamics ( MD ) to simu-late a considerably smaller lattice around the local physics of interest, such as a crack tip, while keeping the effects of the eliminated degrees of freedom on


the reduced MD system. Also, in an effort to address the quasistatics problem and the dynamics problems discussed by Horstemeyer et al. [77] , Fish and Chen [78] and Fish and Yuan [79] , who introduced timescale incongruencies along with size scale issues, Klein and Zimmerman [80] developed a con-current method that could be used in one - , two - , and three - dimensional analysis.

Concurrent multiscale methods have also been employed to address fatigue. Oskay and Fish [81] and Fish and Oskay [82] introduced a nonlocal temporal multiscale model for fatigue based on homogenization theory. Although these formulations were focused on metals, Fish and Yu [83] and Gal et al. [84] used a similar concurrent multiscale method for analyzing fatigue of composite materials.

1.3.1.3 Materials Science Bridging. Perhaps the materials science com-munity has had the greatest impact on modern multiscale modeling methods, thus it is befi tting that the term ICME came out of this community ’ s paradigm. When addressing the physical admissibility postulate for continuum theory, the materials science community has provided the data. It is inherent within the materials science community to study the various length scale effects although they have not necessarily focused on multiscale modeling until recently. The terminology typically used in the materials science community highlights the “ structure – property ” relations. Essentially, it is the different structures within the material (grains, particles, defects, inclusions, etc.) that dictate the performance properties of the materials [85 – 86] .

Different scale features give different scale properties. At the smallest level, the lattice parameter is a key length scale parameter for atomistic simulations. Since atomic rearrangement is intimately related to various types of disloca-tions, Orowan [87] , Taylor [88] , Polyani [89] , and Nabarro [90] developed a relationship for dislocations that related stress to the inverse of a length scale parameter, the Burgers vector [91] , which laid the foundation for all plasticity theories. Nabarro [90] also showed that the diffusion rate is inversely propor-tional to the grain size, another length scale parameter. Hall [92] and Petch [93] related the work hardening rate to the grain size. Ashby [94] found that the dislocation density increased with decreasing second - phase particle size. Frank [95 – 96] and Read and Brooks [97] showed a relation with a dislocation bowing as a function of spacing distance and size, and Hughes et al. [98 – 99] discovered that geometrically necessary boundary spacing decreases with increasing strain.

Other experimental studies have revealed that material properties change as a function of size. For example, Fleck et al. [100] have shown in torsion of thin polycrystalline copper wires the normalized yield shear strength increases by a factor of 3 as the wire diameter is decreased from 100 to 12.5 μ m. Some have argued that the data by Fleck et al. [100] should be criticized because the large strains would have caused nonhomogeneous deformations and the material was polycrystalline. However, St ö lken and Evans [101] observed a


substantial increase in hardening during the bending of ultrathin beams. In micro - indentation and nano - indentation tests [102 – 107] , the measured inden-tation hardness increased by a factor of 2 as the depth of indentation decreased from 10 to 1 μ m. When Lloyd [108] investigated an aluminum – silicon matrix reinforced by silicon carbide particles, he observed a signifi cant increase in strength when the particle diameter was reduced from 16 to 7.5 μ m while holding the particle volume fraction fi xed at 15%. Hughes et al. [98 – 99] inves-tigated deformation induced from frictional loading and found that the stresses near the surface were much greater than that predicted by the local macroscale continuum theory; that is, a length scale dependence was observed. Elssner et al. [109] measured both the macroscopic fracture toughness and the atomic work associated with the separation of an interface between two dissimilar single crystals. The interface (crack tip) between the two materials remained sharp, even though the materials were ductile and contained a large number of dislocations. The stress level necessary to produce atomic decohesion of a sharp interface is on the order of 10 times the yield stress, while local theories predict that the maximum achievable stress at a crack tip is no larger than four to fi ve times the yield stress.

In terms of damage/fracture, Griffi th [110] found a relation between the crack length and the stress intensity factor. Before this time, Roberts - Austen [111] performed a set of experiments that showed the tensile strength of gold had a strong dependence on the impurity size. Fairly recently, McClintock [112] determined the void growth rates as a function of the void size. Void/crack nucleation was determined by various aspects of the second - phase particle size distribution by Gangalee and Gurland [113] . Horstemeyer et al. [114 – 115] and Potirniche et al. [116 – 119] determined the nearest - neighbor distance as a length scale parameter for void coalescence modeling for differ-ent metals that was experimentally validated by Jones et al. [120] . It is clear that whether damage mechanics or fracture mechanics is employed, the length scale of interest is important to model this type of inelastic behavior.

In terms of fatigue, the materials science community has revealed different length scales of interest as well. Local inclusions or defects such as pores, second - phase particles, and constituents can induce local stress concentrations, and the size of the local inclusion or defect can induce local stress concentra-tions large enough to induce fatigue cracks. Neuber [121] and Peterson [122] clearly showed the relation of notch root sizes on fatigue life. Harkegard [123] illustrated using elastic – plastic fi nite element simulation results that the local stress concentration at pores can induce fatigue cracks. Smith et al. [124 – 125] shortly thereafter performed studies on different notch root radii, showing that fatigue life was directly related to the size of notches. Although these experimental studies included external notches, clearly the signifi cance to multiscale modeling was that the notches could be internal or external to induce fatigue cracks. It was the works of Lankford et al. [126 – 128] that started addressing the difference of length scales related to fatigue crack initia-tion, small crack propagation, and long crack propagation. Couper et al. [129]


and Major [130] experimentally determined that the total fatigue life was directly correlated to the length scale of the pore size in high cycle fatigue of cast aluminum alloys. Davidson [131] and Laz and Hillberry [132] quantifi ed fatigue crack mechanisms from different microstructural features and/or defects. These features confi rmed Major ’ s [130] analysis with pores but also included other intermetallics and second - phase particles. In an automotive aluminum alloy, Gall et al. [133 – 135] quantifi ed the effect of particles, inter-metallics, and particle clusters. Recently, Wang et al. [136] watched via in situ scanning electron microscopy the growth of a small fatigue crack through a magnesium alloy that showed crack resistance arising from different size scale features: pores, dendrites, grains, and intermetallics. Kumar and Curtin [137] lately gave a review of various multiscale modeling issues and experiments related to fatigue cracks and the associated microstructure that affects crack growth.

In terms of cyclic plasticity, Shenoy et al. [138 – 139] , Wang et al. [136] , and McDowell [140] performed hierarchical multiscale modeling of Ni - based superalloys employing ISV theory. Fan et al. [141] performed a hierarchical multiscale modeling strategy for three length scales.

In more recent years since the advent of parallel computing, computational materials science has revealed some various length scale dependencies on yield and plasticity as well. Horstemeyer et al. [63, 142 – 151] found that the yield stress is a function of the volume - per - surface - area length scale parameter that correlates well with dislocation nucleation. Figure 1.3 illustrates that for any kind of size scale experiment related to volume - per - surface - area or fun-damental simulation capability where dislocation nucleation is critical, a clear relationship arises [142] . Potirniche et al. [117] showed at the atomic scale through MD that void growth and coalescence showed length scale differences in the elastic region as the specimen size increased but remained scale invari-ant in the plasticity regime.

Results from parallel computing have also helped to understand local structure – property relations for fatigue. Through fi nite element analysis, Fan et al. [141, 152] helped quantify the driving force resistance of microstructur-ally small fatigue cracks at pores and defects. Potirniche et al. [118, 153] and Johnston et al. [154] employed computational methods to study fatigue crack growth at different length scales: atomic level and crystal level. Gall et al. [155] employed fi nite element simulations to understand the crack incubation pro-cesses that lead to a fatigue crack. After the crack has started but still in the microstructurally small regime, Deshpande et al. [156 – 157] examined the local plasticity arising from dislocations at a crack tip and the associated fracture growth using a dislocation dynamics theory. Morita and Tsuji [158] , Mastora-kos and Zbib [159] , and Groh et al. [160] used dislocation dynamics theory in a computational setting to understand relationships of plasticity and crack sizes regarding fatigue crack growth.

One can summarize that the materials science multiscale frame of refer-ence, whether it be from experiments or computations, has been an upscaling


(bottom - up) approach in trying to fi nd the structure – property relationships. McDowell and Olson [161] gave a comprehensive summary of computational materials with respect to multiscale modeling and an associated educational program. The examples given here are certainly not all - inclusive, but at least they are typical of most studies and substantiate the perspective.

Figure 1.3 Yield stress normalized by the elastic shear modulus plotted against a size scale parameter (volume per surface area) illustrating the 6 orders of magnitude of stress levels and 10 orders of magnitude of size related to plastic behavior of single crystal metals [142] (reprinted from Horstemeyer et al. [142] ). See color insert.

atomistic simulationsFleck et al. (1995)Klepaczko (1974)Hughes and Nix (1988)Horstemeyer (1995)Canova et al. (1982)Michalske and Houston (1998)McElhaney et al. (1997)Schmid and Boas (1950)

Phillips (1962)Jackson and Basinski (1967)Quilici et al. (1998)

Edington (1969)

Horstemeyer et al. (2000)Haasen (1958)Windle and Smith (1967)

Follansbee and Gray (1991)

max crystalline strength

ylei

d st

ress

/ela

stic

mod

ulus

1

0.1

0.01

0.001

0.0001

10–5

10–6

10–10 10–8 10–6 0.0001 0.01 1

volume/surface area (m)


1.3.1.4 Physics Perspective. At the lowest scales, computational physi-cists pursued both hierarchical and concurrent methods for joining the electronics principles scale simulations with the atomic scale simulations. Ramasubrama-niam and Carter [162] summarized some of the current multiscale methods at the lower scales. In a concurrent manner, Lu and Kaxiras [163] and Choly et al. [164] coupled density functional theory ( DFT ) with the EAM potentials at the atomic scale to analyze different dislocation behavior. Lu et al. [165] extended the original Tadmor et al. [44] quasicontinuum method to couple another length scale by implementing quantum mechanical DFT calculations so that three concurrent domains were represented. To accomplish this, Lu et al. [165] needed to include Ercolessi and Adams [166] and Li et al. ’ s [167] force matching algorithms for the three disparate length scales. Still others have studied concurrent methods joining quantum models with DFT [168 – 171] .

A different concurrent multiscale method developed within the physics community was termed by the authors as the “ learn - on - the - fl y ” ( LOTF ) strat-egy [172] . In this scheme, simple forms of potentials are chosen to represent the interatomic forces. Unlike conventional empirical potentials whose param-eters are global and constant, these parameterized potentials in the LOTF scheme are local and variable and can be changed at run time by means of accessory quantum calculations when deemed necessary. The original potential is assumed to be constructed to accurately describe bulk processes such as the elastic deformation. Hence, most atoms in the system will experi-ence elastic deformation much of the time, but when a defect or crack is introduced, the quantum calculations will be invoked. The classical potential “ learns ” and adapts to the local environment “ on the fl y. ” The missing informa-tion is computed using a “ black box ” engine based on a DFT or a tight - binding formalism. The dynamic force matching employed here includes the work of Li et al. [167] .

Probably the most used hierarchical multiscale method that joined elec-tronics principles simulation results to the atomic level came by means of the EAM or MEAM potentials for metals. The EAM and MEAM have been called “ semiempirical ” interatomic potentials because of their determination in a hierarchical manner. Daw et al. [45 – 46] were the fi rst to propose a numeri-cal method for calculating atomic energetics (EAM). The major component of EAM is an embedding energy of an atom determined by the lower - scale local electron density into which that atom is placed. An embedding energy is associated with placing an atom in that electron environment to represent the many - body effect of the neighbors. The formalism is nonlocal in nature, and hence because of the many - body interactions, metal responses could be real-ized. Daw et al. [46] reviewed the basic method and several applications of EAM. The MEAM potential, later proposed by Baskes et al. [173 – 175] , was the fi rst semiempirical atomic potential using a single formalism for face cen-tered cubic (FCC), body centered cubic (BCC), hexagonal closed pack (HCP), diamond - structured materials, and even gaseous elements that produced good agreement with experiments or fi rst principles calculations. The MEAM


extended EAM to include angular forces, which is similar, in a solid mechanics sense, to the bending of an Euler beam to a Timoshenko beam.

The EAM and MEAM potentials once determined from electronics prin-ciples calculations [175 – 176] have been used to reproduce physical properties of many metals, defects, and impurities. For example, EAM molecular statics, MD, and Monte Carlo simulations were performed on hydrogen embrittle-ment effects on dislocation motion and plasticity [45, 173 – 174, 176] . These potentials have been used to analyze plasticity [77, 142 – 143, 146 – 147, 150, 177 – 178] , cracks and fracture [116 – 117, 179 – 181] , and fatigue [118, 153, 180, 182 – 183] .

1.3.1.5 Mathematics Perspective. Although mathematics plays a major role in all of the previous perspectives (solid mechanics, numerical methods, materials science, and physics), these previously discussed perspectives essen-tially use mathematics as a means to an end. There are aspects of the math perspective, however, that start purely from mathematical concepts. Brandt et al. [184] and Weinan and Engquist [185] have suggested various mathemati-cal paradigms to address multiscale modeling. Some of the perspectives include dimensional analysis (relates to self - consistency of the parametric dimensions of physical phenomena), fractal and self - similarity analysis (relates to splitting a geometric shape into parts, each of which is approximately a reduced - size copy of the whole, a property called self - similarity), percolation theory (relates to the behavior of connected clusters in a random graph), and statistical design of experiments (DOE) (for multivariable applications, the systematic approach to separate the interactions or relationships into the primary dependencies and parameters — at least as few as possible with the strongest relationships identifi ed).

A good example of starting from a mathematical perspective is the work of Barenblatt [182] who, using dimensional analysis and physical self - similarity arguments, described the art of scaling relationships. He demonstrated the concepts of intermediate asymptotes and the renormalization group as natural consequences of self - similarity and showed how and when these tools could be used for different length scale analyses. In comparing dimensional analysis and renormalization group theory, Carpinteri et al. [183] examined the ductile to brittle transition in metals in which, by increasing the structural size, the tensile strength and fracture energy changed. In another dimensional analysis study, Cheng and Cheng [186] employed dimensional analysis with fi nite element simulations to quantify the size scale relationship with conical indentation of elastic – plastic work hardening behavior. Using asymptotic expansions, Ansini et al. [187] employed a nonlocal functional to capture behavior for a multiscale analysis and applied this analysis concept to a non-linear elastic spherical shell.

Self - similarity is a mathematical concept in which fractals have been applied to predict or simulate some solid material behavior. A fractal is generally “ a rough or fragmented geometric shape that can be split into parts, each of which


is (at least approximately) a reduced - size copy of the whole ” [188] . A fractal typically has a coarse structure with a subscale fi ne structure that is approxi-mately similar to the course structure. It is typically irregular having a Haus-dorff dimension greater than its topological dimension. Long before Mandelbrot thought about fractals, the notion of self - similarity and the associated math-ematics began in the 17th century when Leibniz considered recursive self - similarity. In Mandelbrot [189] , the fracture surfaces of metals were argued to have fractal characteristics. About the same time, Mark and Aronson [190] suggested that fractals can be related to brittle fracture surfaces of rocks. Several years later, Chelidze and Guguen [191] found experimental evidences of fractal type fractures in different rock types. Mosolov et al. [192] then extended the application of fractal geometries to other brittle materials that fractured as a result of compressive loads.

In using differential self - similarity concepts, Dyskin [193] determined the effective characteristics and associated stress concentrations of pores, cracks, and rigid inclusions. He proposed a sequence of continua at increasing size scales with each RVE determined by an associated microstructure. Power laws with exponents representing the scaling of the microstructure are not neces-sarily tied to the material fractal dimension. This formalism was based on the earlier microstructurally based RVE defi nition by Graham and Yang [194] and Lee and Rao [195] , who discussed the pertinence of enlarging the size of an RVE because of the multitude of different microstructural features that require admission into an RVE. To numerically address these issues of length of the RVE with respect to the microstructures and their associated gradients and subsequent effects on the periodicity assumption, Fish and Wagiman [196] , Fish and Yuan [79] , and Nuggehally et al. [197] proposed enhancements to the homogenized solutions.

Contrary to self - similar concepts, He et al. [198] and Wang et al. [199] argued that once fracture occurs at a certain scale, the discontinuity invalidates any self - similar notion. In other words, the fracture introduces a singularity within the self - similar material at that scale. A scale lower would then not be self - similar.

Percolation theory has been another mathematical tool to address scaling issues. One of the earliest works on percolation theory related to size scale effects was that of Essam [200] who linked clusters of entities with different correlation functions. Greenspoon [201] employed an asymptotic analysis with a scaling parameter that quantifi ed correlation lengths. Later Kestin [202] presented an overview of percolation theory applications to different size scale issues. Percolation theory has been used for different multiscale aspects of materials. Otsubo [203] used the percolation theory to determine particle sizes in suspensions of polymer systems. Leclerc and Olson [204] pro-posed a percolation model for lignin degradation such that low values for the cluster size exponent obtained during wood degradation were in agreement with diffusion aggregation processes. In a different application, Fu et al. [205]


employed a percolation model to describe porosity distributions in a ceramic. Ostoja - Starzewski [206] combined a self - consistent method with a percolation model to describe size effects of an inelastic material with random granular microstructure. In particular, the author found a decrease in scatter with regard to strength as the specimen size decreased.

In terms of different scale bridging, statistical methods can play a key role in determining which particular subscale features are most important. Bai et al. [207] gave a nice review of using statistical methods for multiscale model-ing. Another example that summarizes statistical methods bridging many length scales was that discussed by Reichl [208] . However, the key is to deter-mine the appropriate microstructural characteristics related to the pertinent statistics. For example, a dilemma that arises is that the size of the RVE needs to be large enough to capture the statistics of the represented microstructures and defects but must be small enough to represent a continuum element in fi nite element analysis. As discussed earlier, materials scientists may assert that dendrite cell size is a predominant feature on the fatigue life of a cast speci-men because they perform structure – property experiments to quantify the effect. Even though there may be a direct correlation between the size scale feature called dendrite cell size and fatigue life, the dendrite cell size is not the cause of the fatigue failure. For example, as a casting solidifi es, the den-drites push the porosity arising from hydrogen within the melt. As such, the dendrite cell size correlates with the porosity and scales with the pore size. Major [130] , Zhang et al. [209] , Horstemeyer et al. [115] , and Horstemeyer and Wang [210] showed that it was the pore size and associated statistics related to pore nearest - neighbor distance and pore volume fraction that were the critical cause – effect parameters related to fatigue life and not the dendrite cell size. However, it took experiments and statistical methods coupled with fi nite element analysis to help determine this relationship for a hierarchical multi-scale method for fatigue.

In statistics, analysis of variance (ANOVA) is a collection of statistical models and their associated procedures in which the observed variance is partitioned into components due to different explanatory variables. The initial techniques of the ANOVA were developed by the statistician and geneticist R.A. Fisher in the 1930s and are sometimes known as Fisher ’ s ANOVA or Fisher ’ s analysis of variance due to the use of Fisher ’ s F - distribution as part of the test of statistical signifi cance.

These techniques by Fisher [211, 212] can help sort out the parametric effects effi ciently for the multiscale analysis. The DOE approach, popularized by Taguchi [213, 214] in the fi eld of quality engineering, has recently been utilized in various contexts of mechanics problems and design by Horstemeyer and Gokhale [145] , Horstemeyer and Ramaswamy [114] , and Horstemeyer et al. [215] . The DOE methodology enables an investigator to select levels for each subscale parameter and then conduct numerical experiments in order to evaluate the effect of each parameter in an effi cient manner. Any number of


parameters and levels for each parameter can be placed in an orthogonal array, which lends itself to optimal determination of parametric effects. Here, orthogonality refers to the requirement that the parameters be statistically independent. The basic terminology of orthogonal arrays L a ( b c ) is as follows: a denotes the number of calculations, b denotes the number of levels for each parameter, and c denotes the number of parameters. As such, many parame-ters can be evaluated with others at that particular scale, and the most infl u-ential ones can be determined to be and then used for the next higher - scale analysis.

1.4 ICME FOR DESIGN

A signifi cant opportunity exists for ICME in the next wave of design para-digms for components, subsystems, or large - scale systems by accounting for the structure – property relations that have a real impact on measured system response to design loads. As multiscale modeling is refi ned and enhanced, the degree of model accuracy and precision is directly and benefi cially related to the level of product design optimization that can be achieved. When combin-ing ICME into simulation - based design concepts, several challenges include the following: lack of validated techniques for bridging the various time and length scales, management of models with the associated uncertainties, man-agement of a huge amount and variety of information, and development of simpler, user - friendly methods for effi cient decision making.

Panchal et al. [216] discussed how multiscale modeling deals with effi cient integration of information from multiscale models to gain a holistic under-standing of the system, whereas multiscale design deals with effi cient utiliza-tion of information to satisfy design objectives. In order to address the challenges associated with multiscale design, Panchal et al. [216] proposed a domain - independent strategy based on generic interaction patterns between multiscale models.

In a different vein, Olson [85, 217] has described the processing – structure – performance relationship as a function of multiscale, multilevel materials attri-butes that has lent itself to multiscale modeling and simulation. In the U.S. Defense Advanced Research Projects Agency (DARPA) Accelerated Inser-tion of Materials (AIM) project, for which Dr. Olson was a lead researcher, a Ni - based superalloy employing a hierarchical multiscale method was used to design a component in a gas turbine engine with the objective of increasing burst speed and decreasing the disk weight. The focus on materials design was exemplifi ed in Olson [218] in which martensite was examined through a multiscale method.

In joining the Olson and Panchal et al. strategies, the Mistree [219] research group at Georgia Tech has focused on using hierarchical multiscale modeling to optimize materials design. The classical materials selection approach is being replaced by the design of material structure and processing paths based

ICME FOR DESIGN 23

on a hierarchy of length scales for multifunctional performance requirements. In Seepersad et al. [220, 221] , they optimized the mechanical properties for cellular foam with voids.

1.4.1 Design Optimization

As illustrated in Figure 1.4 , in the absence of using ICME tools, the conven-tional design method has a person start from a preconceived design, then alter it based on his or her experience, and fi nally build a prototype to test the design under conditions anticipated in the real world. One uses ad hoc analysis tools, and engineers then make judgments with all of this information. However, there are problems with employing this current practice method:

• The current design method depends on the designer ’ s intuition, experi-ence, and skill.

• If the experienced designer leaves, the experience leaves with the person. • The current design method is a trial - and - error method and as such is

costly. • The current design method is not easy to apply to a complex system

because of the limitations of a human.

Figure 1.4 Conventional design method currently used in engineering practice.

Initial design

Analyze the system

Is designsatisfactory?

No

Change design based onexperience


• The current design method does not always lead to the best possible design, although it can lead to a working design.

• The current design method is a qualitative and not quantitative design.

To realize a more optimized design, one must understand the different opti-mization algorithms. Design optimization is simply fi nding the maxima and minima of mathematical functions related to the engineering application. Recall from calculus that this means fi nding the derivatives of the function. It is much more complex than that, however, because although the concept is simple, the design variables, constraints, and objectives of a particular engi-neering problem complicate the process quickly. Let us start with some defi nitions:

Design Variables (d) : A design variable is a specifi cation that is controllable by the designer (e.g., thickness, material, etc. ) and is often bounded by maximum and minimum values. Sometimes these bounds can be treated as constraints .

Constraints (g, h) : A constraint is a condition that must be satisfi ed for the design to be feasible . Examples include physical laws ; constraints can refl ect resource limitations , user requirements , or bounds on the validity of the analysis models. Constraints can be used explicitly by the solution algorithm or can be incorporated into the objective using Lagrange multipliers.

Objectives (f) : An objective is a numerical value or function that is to be maximized or minimized . For example, a designer may wish to maximize profi t or minimize weight. Many solution methods work only with single objectives. When using these methods, the designer normally weights the various objectives and sums them to form a single objective. Other methods, such as the calculation of a Pareto front, allow multi - objective optimization (MOO).

Pareto Frontier : Once there exists more than one objective like in multi - objective design optimization problems, a Pareto curve is required because nonunique solutions will arise. Hence, the Pareto frontier (or front) is a curve that gives the solution space for several, if not many solutions. The Pareto front comes from an economy development in which preferred options and solutions exist on the curve, while nonopti-mum solutions do not.

Models : The designer must also choose models to relate the constraints and the objectives to the design variables. They may include fi nite element analysis and reduced order metamodels.

Reliability : The probability of a component to perform its required func-tions under stated conditions for a specifi ed period of time.

Different kinds of optimization algorithms exist since there are no unique solutions. These can be divided into four categories: gradient - based methods,

ICME FOR DESIGN 25

gradient - free methods, population - based methods, and other methods (there are always researchers introducing something outside of the norm or combin-ing some of the aforementioned methods):

1. Gradient - based methods • Adjoint equation • Newton ’ s method • Steepest descent • Conjugate gradient • Sequential quadratic programming

2. Gradient - free methods • Hooke – Jeeves pattern search • Nelder – Mead method

3. Population - based methods • Genetic algorithm • Memetic algorithm • Particle swarm optimization • Ant colony • Harmony search

4. Other methods • Random search • Grid search • Simulated annealing • Direct search • Indirect optimization based on self - organization (IOSO)

When employing a single objective optimization method in the design process as illustrated in Figure 1.5 , one can use any of the approaches mentioned above, but the key is the constraint on the objective, which still requires human input. Hence, a designer is still required, although a unique solution can arise. A problem with a single - objective constrained design is that most engineering components, particularly structural components, serve more than one purpose and as such have multiple objectives.

When more than one objective is required in the design, which is most of the time, then one must employ what is called MOO methods. Sometimes they are called multicriteria or multi - attribute methods. The idea is that two simultaneous and often confl icting objectives are required in the design. Each of the objectives can also have certain constraints on them that can also be confl icting. In this case, one unique solution does not exist. One cannot just run a battery of simulations employing the ICME tools and a solution will arise. In this case, the solution space is multiple and each individual opti-mized solution will be comprised because of the presence of the other. For an


excellent tutorial, one can go to the Internet ( http://en.wikipedia.org/wiki/Multi - objective_optimization ).

1.4.2 Metamodeling Approaches

If one were to use the ICME tools for a particular design, the analysis tools would require a lot of time to run all of the pertinent simulations demanded by the boundary conditions from the downscaling requirements. Clearly, simulation - based design tools have been in use the past 10 – 15 years, with a good example being the Boeing 777, which was, fi rst designed completely by simulation before a single component was procured [222] . For design optimiza-tion using fi nite element analysis, the number of required simulation runs can be on the order of tens to hundreds of simulations depending on the problem and the optimization algorithms. Additionally, these simulations are typically run in serial, so it would take even a longer time if parallel computing was not available. Using expensive fi nite element simulations directly in a MOO design is still prohibitive today, even in a parallel computing environment (cf. [177, 223] , for crashworthiness simulations).

One way of reducing the amount of time necessary for the large - scale simu-lations is to employ metamodels, which are sometimes called surrogate models or response surface functions. These approximate models (functional relation-

Figure 1.5 Constrained single - objective design optimization process.

Starting point

Analysis

Optimizer

Converge?

Convergencecriteria?

Updated X

X

F(X), g(X)

NoNo

Change design usingoptimization technique

Analyze the system

Initial design

Identify :(1) Design variables (X)(2) Objective functions to be minimized (F)

(3) Constraints that must be satisfied (g)

ICME FOR DESIGN 27

ships between input and output variables) can reduce the simulation times by one to two orders of magnitude. Figure 1.6 summarizes the use of metamodels.

In order to reduce the computational time further, one can also employ DOE methods before the large - scale simulations are conducted. As Figure 1.7 illustrates, the DOE technique is used to help determine which high - fi delity simulations should be run and then, after the simulations are run, their results are used to develop the metamodel.

1.4.3 Design with Uncertainty Analysis

In the context of validation and verifi cation (often called V & V), uncertainty plays a large role in the modeling and experiments [224] . Validation is “ doing the right, ” meaning checking the modeling with the experiments. Verifi cation is “ doing things right, ” meaning the codes and algorithms need to be checked with other codes and algorithms even though the same model might be employed. Hence, to be validated, a model ’ s uncertainty must be inside of the range of the experimental data ’ s uncertainty. If this is true, then the model is said to be validated.

In terms of the ICME tools, uncertainties can come into play from both the experimental and modeling perspectives. Figure 1.8 illustrates the different

Figure 1.6 Metamodeling - based optimization (MBO) process in the context of the optimization problem.

Define optimizationproblem

Minf(x1, x2....xn)s.t. g(x) ≤ 0

h(x) = 0

Evaluate f(x), g(x), h(x)

x1

x2

...xn

fgh

Identify most and leastimportant variables

Perform optimiza-tion

Validation?Yes

No

Stop

Refine design spaceAdd more data points

(not necessary but advisable)

Metamodel


Figure 1.7 Using the design of experiments (DOE) to defi ne the large - scale simula-tions to be used for the analysis, the analysis results are then used to develop the metamodel, which in turn is used to optimize the design.

Surrogate-BasedOptimization(Metamodelingor Response SurfaceMethods)

Benefit: quicker answersIssue: less accuracy (maybe)

Objective–Develop metamodels aslow-cost surrogates for expensivehigh-fidelity simulations

DOE

Selected X

AnalysisF(X), g(X)

Metamodel Approximation

F (X) ª S ak · fk (X)L

k=1

Optimizer

Figure 1.8 Uncertainties are prevalent in both the modeling/simulation space and the experimental space. One must quantify both to “ validate ” a model.

SUMMARY 29

uncertainties that one must consider in the validation and in the usage of uncertainty in the design context.

Once a model has been “ validated, ” it can be used for simulation - based design. Now there are a number of methods to employ “ optimization under uncertainty ” techniques, but essentially carrying along the uncertainties of the model and experiments into the simulation space and the design space now gives different parameters to consider for the design. Diwekar [225] and McDowell et al. [226] gave great reviews and discussions on employing the concepts of uncertainty into the design optimization process.

1.5 ICME FOR MANUFACTURING

Besides design, another great opportunity for ICME is the venue of materials processing. Rapid prototyping and other materials processing techniques can facilitate tailoring microstructural topology with high levels of detail. This industrial framework makes it amenable for multiscale materials modeling to affect a paradigm shift with U.S. manufacturing. Economists Grimes and Glazer of the Examination Bureau of Labor Statistics stated that the United States has lost 18% of its manufacturing jobs from 1990 to 2003. Although higher labor costs in the United States are a big factor, U.S. industry has lacked improvements in innovative design. An upturn in the manufacturing industry clearly requires a competitive advantage through better design tools based on information technology and high - fi delity multiscale materials modeling, simulations, and design optimization. Horstemeyer and Wang [210] argued that improved design/analysis capabilities based on recent scientifi c and engineering research advances will provide a substantial benefi t to the U.S. manufacturing industry, but to do so, multiscale modeling is needed in the mix. This applies to the general manufacturing industry, specifi c “ made - to - order ” manufacturing segments, and those involved with cutting - edge tech-nologies. Signifi cant breakthroughs in the knowledge base, software integration, length/time scale bridging, and educational enterprise are required to reverse the negative manufacturing trend in the United States. Right now, multiscale modeling can be used to optimize the process – property – cost for processes such as forging, forming, casting, extrusion, rolling, stamping, and welding/joining.

1.6 SUMMARY

Glimm and Sharp [227] proposed a challenge to the 21st century researcher to consider multiscale materials modeling as a new paradigm in order to realize more accurate predictive capabilities. The context is to predict the macroscale/structural scale behavior without disregarding the important


smaller - scale features. Clearly, the ICME paradigm fi ts into the solution of this proposed problem. ICME is a key for more accurate simulations for solid materials because of the heterogeneities arising from these subscale microstructures. As such, the notion of selecting or disregarding the subscale information requires interdisciplinary methods to make proper judg-ments. When selecting the appropriate subscale features, both upscaling (bottom - up) and downscaling (top - down) methods are needed. As a conse-quence, the terminology and focus in recent years has helped to gain more accurate component and systems level answers. As demonstrated by several examples, multiscale modeling has the capability to ameliorate the conun-drum, trading off higher - fi delity physics versus computational cost and revolutionizing simulation - based design and manufacturing. In the end, higher - quality, more optimized designs are admissible today because of the confl u-ence of computing power, experimental validation techniques, and availability of advanced numerical algorithms (e.g., exploiting parallel processing). For multiscale modeling to have even greater impact in manufacturing and design, computational effi ciency with greater accuracy requires that computational times of the concurrent multiscale methods should be on the order of the computation required for the more accurate treatment using the hierarchical methods. Nothing is really gained from concurrent multiscale methods if this is not true, since hierarchical multiscale methods could be run independently with lesser computational times. This limits the usage of concurrent methods usually to two or three length scales, whereas hierarchical methods are not limited to just two length scales.

Although multiscale modeling has experienced some success with metals, the usage for polymer material systems has been lacking. In particular, the impact on biological tissue has yet to be realized. Furthermore, the common usage of multiscale modeling for design and manufacturing requires a para-digm shift for the industry. Kassner et al. [228] summarized the future needs of multiscale modeling for metallic and polymeric systems, essentially request-ing research related to multiscale modeling that deals with discontinuities/defect/microstructures.

Not mentioned in this introduction but certainly important to ICME are topics such as self - assemblies, thin fi lms, thermal barrier coatings, patterning, phase transformations, nanomaterials design, and semiconductors. All have an economic motivation for study. Studies related to these types of materials and structures require multiphysics formulations to understand the appropriate thermodynamics, kinetics, and kinematics.

Finally, a website that contains information, materials models, codes, and experimental databases to help your personal ICME education is located at ftp://ftp.wiley.com/public/sci_tech_med/icme_metals . This website is wiki - based and was originally funded under the U.S. DOE where the ICME label was fi rst coined. The exercises and projects related to a course taught in context with this textbook can be found at this website, and the development of this website came from the work of Haupt et al. [229 – 231] .

REFERENCES 31

REFERENCES

1 M. F. Horstemeyer , Multiscale modeling: a review , in: J. Leszczynski and M. K. Shukla, eds., Practical Aspects of Computational Chemistry , Springer Science + Business Media, 2009 , pp. 87 – 135 .

2 S. Yip , Handbook of Materials Modeling , Springer , Dordrecht , 2005 . 3 E. B. Tadmor , R. Phillips , and M. Ortiz , Hierarchical modeling in the mechanics

of materials , International Journal of Solids and Structures , vol. 37 , no. 1 – 2 , pp. 379 – 389 , 2000 .

4 H. Helmholtz , Ü ber die Erhaltung der Kraft (lecture presented on July 23, 1847 at the Physika), 1847 .

5 J. C. Maxwell , On the dynamical evidence of molecular constitution of matter , Nature , vol. 11 , pp. 357 – 359 , 1875 .

6 L. Onsager , Reciprocal relations in irreversible processes, Physical Review , vol. 37 , pp. 405 – 426 ; vol. 38 , p. 2265 , 1931 .

7 C. Eckart , The thermodynamics of irreversible processes. I. The simple fl uid , Physical Review , vol. 58 , pp. 267 – 269 , 1940 .

8 C. Eckart , The thermodynamics of irreversible processes. IV. The theory of elastic-ity and anelasticity , Physical Review , vol. 73 , no. 4 , p. 373 , 1948 .

9 E. Kr ö ner , How the internal state of a physically deformed body is to be described in a continuum theory , 1960 .

10 B. D. Coleman and M. E. Gurtin , Thermodynamics with internal state variables , The Journal of Chemical Physics , vol. 47 , p. 597 , 1967 .

11 J. R. Rice , Inelastic constitutive relations for solids: an internal - variable theory and its application to metal plasticity , Journal of the Mechanics and Physics of Solids , vol. 19 , no. 6 , pp. 433 – 455 , 1971 .

12 J. Kestin and J. Rice , Paradoxes in the application of thermodynamics to strained solids (thermodynamic application to strained solids, considering paradoxes, irre-versible processes and continuum mechanics) , 1970 .

13 R. Talreja , Continuum modelling of damage in ceramic matrix composites , Mechanics of Materials , vol. 12 , no. 2 , pp. 165 – 180 , 1991 .

14 B. F. S ø rensen and R. Talreja , Effects of nonuniformity of fi ber distribution on thermally - induced residual stresses and cracking in ceramic matrix composites , Mechanics of Materials , vol. 16 , no. 4 , pp. 351 – 363 , 1993 .

15 V. N. Bulsara , R. Talreja , and J. Qu , Damage in ceramic matrix composites with arbitrary microstructures , Composites and Functionally Graded Materials , vol. 80 , pp. 441 – 446 , 1997 .

16 O. A. Hasan and M. C. Boyce , A constitutive model for the nonlinear viscoelastic viscoplastic behavior of glassy polymers , Polymer Engineering and Science , vol. 35 , no. 4 , pp. 331 – 344 , 1995 .

17 H. D. Espinosa , H. C. Lu , P. D. Zavattieri , and S. Dwivedi , A 3 - D fi nite deforma-tion anisotropic visco - plasticity model for fi ber composites , Journal of Composite Materials , vol. 35 , no. 5 , p. 369 , 2001 .

18 B. A. Gailly and H. D. Espinosa , Modelling of failure mode transition in ballistic penetration with a continuum model describing microcracking and fl ow of pulverized media , International Journal for Numerical Methods in Engineering , vol. 54 , no. 3 , pp. 365 – 398 , 2002 .


19 U. F. Kocks , The relation between polycrystal deformation and single - crystal deformation , Metallurgical and Materials Transactions B , vol. 1 , no. 5 , pp. 1121 – 1143 , 1970 .

20 P. Follansbee , U. Kocks , and G. Regazzoni , The mechanical threshold of dynami-cally deformed copper and nitronic 40 , 1985 .

21 P. S. Follansbee and U. F. Kocks , A constitutive description of the deformation of copper based on the use of the mechanical threshold stress as an internal state variable , Acta Metallurgica , vol. 36 , no. 1 , pp. 81 – 93 , 1988 .

22 A. Freed , Thermoviscoplastic model with application to copper , 1988 . 23 D. J. Bammann , An internal variable model of viscoplasticity , International Journal

of Engineering Science , vol. 22 , no. 8 – 10 , pp. 1041 – 1053 , 1984 . 24 D. J. Bammann , Modeling temperature and strain rate dependent large deforma-

tions of metals , Applied Mechanics Reviews , vol. 43 , p. S312 , 1990 . 25 J. L. Chaboche , Viscoplastic constitutive equations for the description of cyclic

and anisotropic behaviour of metals , Academie Polonaise des Sciences, Bulletin, Serie des Sciences Techniques , vol. 25 , no. 1 , p. 33 , 1977 .

26 J. D. Eshelby , The determination of the elastic fi eld of an ellipsoidal inclusion, and related problems , Proceedings of the Royal Society of London. Series A, Mathe-matical and Physical Sciences , vol. 241 , no. 1226 , p. 376 , 1957 .

27 J. D. Eshelby , The elastic fi eld outside an ellipsoidal inclusion , Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences , vol. 252 , no. 1271 , pp. 561 – 569 , 1959 .

28 R. Hill , A self - consistent mechanics of composite materials , Journal of the Mechanics and Physics of Solids , vol. 13 , no. 4 , pp. 213 – 222 , 1965 .

29 T. Mori and K. Tanaka , Average stress in matrix and average elastic energy of materials with misfi tting inclusions , Acta Metallurgica , vol. 21 , no. 5 , pp. 571 – 574 , 1973 .

30 T. Mura , Micromechanics of Defects in Solids , Martinus Nijhoff , the Hague, Netherlands , 1982 .

31 B. Budiansky , Micromechanics , Computers & Structures , vol. 16 , no. 1 – 4 , pp. 3 – 12 , 1983 .

32 S. Nemat - Nasser , M. Lori , and S. K. Datta , Micromechanics: overall properties of heterogeneous materials , Journal of Applied Mechanics , vol. 63 , p. 561 , 1996 .

33 A. M. Gokhale and S. Yang , Application of image processing for simulation of mechanical response of multi - length scale microstructures of engineering alloys , Metallurgical and Materials Transactions A , vol. 30 , no. 9 , pp. 2369 – 2381 , 1999 .

34 S. Hao , B. Moran , W. K. Liu , and G. B. Olson , A hierarchical multi - physics model for design of high toughness steels , Journal of Computer - Aided Materials Design , vol. 10 , no. 2 , pp. 99 – 142 , 2003 .

35 Z. P. Bazant and E. P. Chen , Scaling of structural failure , Applied Mechanics Reviews , vol. 50 , pp. 593 – 627 , 1997 .

36 P. W. Bridgman , The compressibility of 30 metals as a function of temperature and pressure , in Proceedings of the American Academy of Arts and Sciences , vol. 58 , pp. 165 – 242 , 1923 .

37 R. B. Phillips , Crystals, Defects and Microstructures: Modeling Across Scales , Cam-bridge University Press , Cambridge, UK , 2001 .

REFERENCES 33

38 W. K. Liu , E. G. Karpov , and H. S. Park , Nano Mechanics and Materials: Theory, Multiscale Methods and Applications , Wiley , West Sussex, UK , 2006 .

39 J. Fish , Bridging the scales in nano engineering and science , Journal of Nanopar-ticle Research , vol. 8 , no. 5 , pp. 577 – 594 , 2006 .

40 J. J. de Pablo and W. A. Curtin , Multiscale modeling in advanced materials research: challenges, novel methods, and emerging applications , MRS Bulletin - Materials Research Society , vol. 32 , no. 11 , pp. 905 – 912 , 2007 .

41 W. Hackbush , Multigrid Methods and Applications , Springer , New York , 1985 . 42 S. Kohlhoff , P. Gumbsch , and H. F. Fischmeister , Crack propagation in bcc crystals

studied with a combined fi nite - element and atomistic model , Philosophical Mag-azine A , vol. 64 , no. 4 , pp. 851 – 878 , 1991 .

43 S. Kohlhoff and S. Schmauder , in: V. Vitek and D. J. Srolovitz , eds., Atomistic Simulation of Materials: Beyond Pair Potentials , Plenum , New York , 1989 , pp. 411 – 418 .

44 E. B. Tadmor , M. Ortiz , and R. Phillips , Quasicontinuum analysis of defects in solids , Philosophical Magazine A , vol. 73 , no. 6 , pp. 1529 – 1563 , 1996 .

45 M. S. Daw and M. I. Baskes , Embedded - atom method: derivation and application to impurities, surfaces, and other defects in metals , Physical Review B , vol. 29 , no. 12 , p. 6443 , 1984 .

46 M. S. Daw , S. M. Foiles , and M. I. Baskes , The embedded - atom method: a review of theory and applications , Materials Science Reports , vol. 9 , no. 7 – 8 , pp. 251 – 310 , 1993 .

47 M. Ortiz , Y. Leroy , and A. Needleman , A fi nite element method for localized failure analysis , Computer Methods in Applied Mechanics and Engineering , vol. 61 , no. 2 , pp. 189 – 214 , 1987 .

48 V. B. Shenoy , R. Miller , E. B. Tadmor , R. Phillips , and M. Ortiz , Quasicontinuum models of interfacial structure and deformation , Physical Review Letters , vol. 80 , no. 4 , pp. 742 – 745 , 1998 .

49 R. Miller , M. Ortiz , R. Phillips , V. Shenoy , and E. B. Tadmor , Quasicontinuum models of fracture and plasticity , Engineering Fracture Mechanics , vol. 61 , no. 3 – 4 , pp. 427 – 444 , 1998 .

50 R. Miller , E. B. Tadmor , R. Phillips , and M. Ortiz , Quasicontinuum simulation of fracture at the atomic scale , Modelling and Simulation in Materials Science and Engineering , vol. 6 , p. 607 , 1998 .

51 V. B. Shenoy , R. Miller , E. B. Tadmor , D. Rodney , R. Phillips , and M. Ortiz , An adaptive fi nite element approach to atomic - scale mechanics - the quasicontinuum method , Arxiv preprint cond - mat/9710027 , 1997 .

52 F. F. Abraham , J. Q. Broughton , N. Bernstein , and E. Kaxiras , Spanning the con-tinuum to quantum length scales in a dynamic simulation of brittle fracture , Europhysics Letters , vol. 44 , p. 783 , 1998 .

53 S. Plimpton , Fast parallel algorithms for short - range molecular dynamics , Journal of Computational Physics , vol. 117 , no. 1 , pp. 1 – 19 , 1995 .

54 R. E. Rudd and J. Q. Broughton , Coarse - grained molecular dynamics and the atomic limit of fi nite elements , Physical Review B , vol. 58 , no. 10 , pp. 5893 – 5896 , 1998 .

55 J. Q. Broughton , F. F. Abraham , N. Bernstein , and E. Kaxiras , Concurrent coupling of length scales: methodology and application , Physical review B , vol. 60 , no. 4 , p. 2391 , 1999 .


56 E. Lidorikis , M. E. Bachlechner , R. K. Kalia , A. Nakano , P. Vashishta , and G. Z. Voyiadjis , Condensed matter: structure, etc. – – coupling length scales for multi-scale atomistic - continuum simulations: atomistically induced stress distributions in Si/Si 3 N 4 nanopixels , Physical Review Letters , vol. 87 , no. 8 , pp. 86104 – 86400 , 2001 .

57 Z. P. Bazant , Can multiscale - multiphysics methods predict softening damage and structural failure? International Journal of Multiscale Computational Engineering , vol. 8 , no. 1 , pp. 61 – 67 , 2010 .

58 L. E. Shilkrot , R. E. Miller , and W. A. Curtin , Coupled atomistic and discrete dislocation plasticity , Physical Review Letters , vol. 89 , no. 2 , p. 25501 , 2002 .

59 L. E. Shilkrot , R. E. Miller , and W. A. Curtin , Multiscale plasticity modeling: coupled atomistics and discrete dislocation mechanics , Journal of the Mechanics and Physics of Solids , vol. 52 , no. 4 , pp. 755 – 787 , 2004 .

60 B. Shiari , R. E. Miller , and W. A. Curtin , Coupled atomistic/discrete dislocation simulations of nanoindentation at fi nite temperature , Journal of Engineering Materials and Technology , vol. 127 , p. 358 , 2005 .

61 M. P. Dewald and W. A. Curtin , Multiscale modelling of dislocation/grain bound-ary interactions. II. Screw dislocations impinging on tilt boundaries in Al , Philo-sophical Magazine , vol. 87 , no. 30 , pp. 4615 – 4641 , 2007 .

62 H. M. Zbib and T. Diaz de la Rubia , A multiscale model of plasticity , International Journal of Plasticity , vol. 18 , no. 9 , pp. 1133 – 1163 , 2002 .

63 M. F. Horstemeyer , D. L. McDowell , and R. D. McGinty , Design of experiments for constitutive model selection: application to polycrystal elastoviscoplasticity , Modelling and Simulation in Materials Science and Engineering , vol. 7 , p. 253 , 1999 .

64 P. R. Dawson , On modeling of mechanical property changes during fl at rolling of aluminum , International Journal of Solids and Structures , vol. 23 , no. 7 , pp. 947 – 968 , 1987 .

65 D. Peirce , R. J. Asaro , and A. Needleman , An analysis of nonuniform and localized deformation in ductile single crystals , Acta Metallurgica , vol. 30 , no. 6 , pp. 1087 – 1119 , 1982 .

66 M. M. Rashid and S. Nemat - Nasser , Modeling very large plastic fl ows at very large strain rates for large - scale computation , Computers & Structures , vol. 37 , no. 2 , pp. 119 – 132 , 1990 .

67 U. F. Kocks , C. N. Tom é , and H. R. Wenk , Texture and Anisotropy: Preferred Ori-entations in Polycrystals and Their Effect on Materials Properties , Cambridge University Press , Cambridge, UK , 2000 .

68 M. S. Shephard , M. W. Beall , R. Garimella , and R. Wentorf , Automatic construc-tion of 3 - D models in multiple scale analysis , Computational Mechanics , vol. 17 , no. 3 , pp. 196 – 207 , 1995 .

69 S. P. Xiao and T. Belytschko , A bridging domain method for coupling continua with molecular dynamics , Computer Methods in Applied Mechanics and Engi-neering , vol. 193 , no. 17 – 20 , pp. 1645 – 1669 , 2004 .

70 G. J. Wagner and W. K. Liu , Coupling of atomistic and continuum simulations using a bridging scale decomposition , Journal of Computational Physics , vol. 190 , no. 1 , pp. 249 – 274 , 2003 .

REFERENCES 35

71 E. G. Karpov , G. J. Wagner , and W. K. Liu , A Green ’ s function approach to deriv-ing non - refl ecting boundary conditions in molecular dynamics simulations , International Journal for Numerical Methods in Engineering , vol. 62 , no. 9 , pp. 1250 – 1262 , 2005 .

72 S. A. Adelman and J. D. Doll , Generalized Langevin equation approach for atom/solid - surface scattering: general formulation for classical scattering off harmonic solids , The Journal of Chemical Physics , vol. 64 , p. 2375 , 1976 .

73 W. Cai , M. de Koning , V. V. Bulatov , and S. Yip , Minimizing boundary refl ections in coupled - domain simulations , Physical Review Letters , vol. 85 , no. 15 , pp. 3213 – 3216 , 2000 .

74 Z. Huang , A dynamic atomistic - continuum method for the simulation of crystalline materials , Arxiv preprint cond - mat/0112173, 2001 .

75 F. Gruttmann and W. Wagner , A linear quadrilateral shell element with fast stiff-ness computation , Computer Methods in Applied Mechanics and Engineering , vol. 194 , no. 39 – 41 , pp. 4279 – 4300 , 2005 .

76 H. S. Park , E. G. Karpov , W. K. Liu , and P. A. Klein , The bridging scale for two - dimensional atomistic/continuum coupling , Philosophical Magazine , vol. 85 , no. 1 , pp. 79 – 113 , 2005 .

77 M. F. Horstemeyer , M. I. Baskes , V. C. Prantil , J. Philliber , and S. Vonderheide , A multiscale analysis of fi xed - end simple shear using molecular dynamics, crystal plasticity, and a macroscopic internal state variable theory , Modelling and Simula-tion in Materials Science and Engineering , vol. 11 , p. 265 , 2003 .

78 J. Fish and W. Chen , Discrete - to - continuum bridging based on multigrid princi-ples , Computer Methods in Applied Mechanics and Engineering , vol. 193 , no. 17 – 20 , pp. 1693 – 1711 , 2004 .

79 J. Fish and Z. Yuan , Multiscale enrichment based on partition of unity , Interna-tional Journal for Numerical Methods in Engineering , vol. 62 , no. 10 , pp. 1341 – 1359 , 2005 .

80 P. A. Klein and J. A. Zimmerman , Coupled atomistic - continuum simulations using arbitrary overlapping domains , Journal of Computational Physics , vol. 213 , no. 1 , pp. 86 – 116 , 2006 .

81 C. Oskay and J. Fish , Fatigue life prediction using 2 - scale temporal asymptotic homogenization , International Journal for Numerical Methods in Engineering , vol. 61 , no. 3 , pp. 329 – 359 , 2004 .

82 J. Fish and C. Oskay , A nonlocal multiscale fatigue model , Mechanics of Advanced Materials and Structures , vol. 12 , no. 6 , pp. 485 – 500 , 2005 .

83 J. Fish and Q. Yu , Computational mechanics of fatigue and life predictions for composite materials and structures , Computer Methods in Applied Mechanics and Engineering , vol. 191 , no. 43 , pp. 4827 – 4849 , 2002 .

84 E. Gal , Z. Yuan , W. Wu , and J. Fish , A multiscale design system for fatigue life prediction , International Journal for Multiscale Computational Engineering , vol. 5 , no. 6 , pp. 435 – 446 , 2007 .

85 G. B. Olson , New age of design , Journal of Computer - Aided Materials Design , vol. 7 , pp. 143 – 144 , 2000 .

86 G. B. Olson , Systems design of hierarchically structured materials: advanced steels , Journal of Computer - Aided Materials Design , vol. 4 , no. 3 , pp. 143 – 156 , 1998 .


87 E. Orowan , Zur Kristallplastizit ä t. I , Zeitschrift f ü r Physik A Hadrons and Nuclei , vol. 89 , no. 9 , pp. 605 – 613 , 1934 .

88 G. I. Taylor , The mechanism of plastic deformation of crystals. Part I. Theoretical , Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character , vol. 145 , no. 855 , pp. 362 – 387 , 1934 .

89 M. Z. Polyani , Uber eine Art Gilterstorung die einen Kristall plastisch machen konnte , Zeitschrift f ü r Physikalische , vol. 89 , pp. 660 – 664 , 1934 .

90 F. R. N. Nabarro , Mathematical theory of stationary dislocations , Advances in Physics , vol. 1 , pp. 269 – 394 , 1952 .

91 J. M. Burgers , Some Considerations on the Fields of Stress Connected with Disloca-tions in a Regular Crystal Lattice. I , Koninklijke Nederlandse Akademie van Wetenschappen , 1939 .

92 E. O. Hall , The deformation and ageing of mild steel: III discussion of results , Proceedings of the Physical Society. Section B , vol. 64 , p. 747 , 1951 .

93 N. J. Petch , The cleavage strength of polycrystals , The Journal of the Iron and Steel Institute , vol. 174 , no. 1 , pp. 25 – 28 , 1953 .

94 M. F. Ashby , in: A. Kelly and R. B. Nicholson , eds., Strengthening Methods in Crystals , 1971 , p. 137 .

95 F. C. Frank , The infl uence of dislocations on crystal growth , Discussions of the Faraday Society , vol. 5 , pp. 48 – 54 , 1949 .

96 F. C. Frank , Crystal dislocations. Elementary concepts and defi nitions , Philosophi-cal Magazine , vol. 42 , no. 331 , pp. 809 – 819 , 1951 .

97 W. T. Read , Jr. and H. Brooks , Dislocations in crystals , Physics Today , vol. 8 , p. 17 , 1955 .

98 D. A. Hughes , D. B. Dawson , J. S. Korellls , and L. I. Weingarten , Near surface microstructures developing under large sliding loads , Journal of Materials Engi-neering and Performance , vol. 3 , no. 4 , pp. 459 – 475 , 1994 .

99 D. A. Hughes and W. D. Nix , The absence of steady - state fl ow during large strain plastic deformation of some FCC metals at low and intermediate temperatures , Metallurgical and Materials Transactions A , vol. 19 , no. 12 , pp. 3013 – 3024 , 1988 .

100 N. A. Fleck , G. M. Muller , M. F. Ashby , and J. W. Hutchinson , Strain gradient plasticity: theory and experiment , Acta Metallurgica et Materialia , vol. 42 , no. 2 , pp. 475 – 487 , 1994 .

101 J. S. St ö lken and A. G. Evans , A microbend test method for measuring the plastic-ity length scale , Acta Materialia , vol. 46 , no. 14 , pp. 5109 – 5115 , 1998 .

102 W. D. Nix , Mechanical properties of thin fi lms , Metallurgical and Materials Trans-actions A , vol. 20 , no. 11 , pp. 2217 – 2245 , 1989 .

103 M. S. De Guzman , G. Neubauer , P. Flinn , and W. D. Nix , The role of indentation depth on the measured hardness of materials , in Thin Films: Stresses and Mechan-ical Properties IV , 1993 , pp. 613 – 618 .

104 N. A. Stelmashenko , M. G. Walls , L. M. Brown , and Y. V. Milman , Microindenta-tions on W and Mo oriented single crystals: an STM study , Acta Metallurgica et Materialia , vol. 41 , no. 10 , pp. 2855 – 2865 , 1993 .

105 Q. Ma and D. R. Clarke , Size dependent hardness of silver single crystals , Journal of Materials Research , vol. 10 , pp. 853 – 863 , 1995 .

REFERENCES 37

106 W. J. Poole , M. F. Ashby , and N. A. Fleck , Micro - hardness of annealed and work - hardened copper polycrystals , Scripta Materialia , vol. 34 , no. 4 , pp. 559 – 564 , 1996 .

107 K. W. McElhaney , J. J. Vlassak , and W. D. Nix , Determination of indenter tip geometry and indentation contact area for depth - sensing indentation experi-ments , Journal of Materials Research , vol. 13 , no. 5 , pp. 1300 – 1306 , 1998 .

108 D. J. Lloyd , Particle reinforced aluminium and magnesium matrix composites , International Materials Reviews , vol. 39 , no. 1 , pp. 1 – 23 , 1994 .

109 G. Elssner , D. Korn , and M. Ruehle , The infl uence of interface impurities on fracture energy of UHV diffusion bonded metal - ceramic bicrystals , Scripta Metal-lurgica et Materialia (United States) , vol. 31 , no. 8 , pp. 1037 – 1042 , 1994 .

110 A. A. Griffi th , The phenomena of rupture and fl ow in solids , Philosophical Trans-actions of the Royal Society of London, V , vol. 221 , p. 21 , 1921 .

111 W. C. Roberts - Austen , On certain mechanical properties of metals considered in relation to the periodic law , Philosophical Transactions of the Royal Society of London. A , vol. 179 , pp. 339 – 349 , 1888 .

112 F. A. McClintock , A criterion for ductile fracture by the growth of holes , Journal of Applied Mechanics, Transactions ASME , vol. 35 , pp. 363 – 371 , 1968 .

113 A. Gangalee and J. Gurland , On the fracture of silicon particles in aluminum - silicon alloys , Transactions of the Metallurgical Society of AIME , vol. 239 , no. 2 , pp. 269 – 272 , 1967 .

114 M. F. Horstemeyer and S. Ramaswamy , On factors affecting localization and void growth in ductile metals: a parametric study , International Journal of Damage Mechanics , vol. 9 , no. 1 , p. 5 , 2000 .

115 M. F. Horstemeyer , M. M. Matalanis , A. M. Sieber , and M. L. Botos , Microme-chanical fi nite element calculations of temperature and void confi guration effects on void growth and coalescence , International Journal of Plasticity , vol. 16 , no. 7 – 8 , pp. 979 – 1015 , 2000 .

116 G. P. Potirniche , J. L. Hearndon , M. F. Horstemeyer , and X. W. Ling , Lattice ori-entation effects on void growth and coalescence in FCC single crystals , Interna-tional Journal of Plasticity , vol. 22 , no. 5 , pp. 921 – 942 , 2006 .

117 G. P. Potirniche , M. F. Horstemeyer , G. J. Wagner , and P. M. Gullett , A molecular dynamics study of void growth and coalescence in single crystal nickel , Interna-tional Journal of Plasticity , vol. 22 , no. 2 , pp. 257 – 278 , 2006 .

118 G. P. Potirniche , M. F. Horstemeyer , P. M. Gullett , and B. Jelinek , Atomistic model-ling of fatigue crack growth and dislocation structuring in FCC crystals , Proceed-ings of the Royal Society A: Mathematical, Physical and Engineering Science , vol. 462 , no. 2076 , p. 3707 , 2006 .

119 G. P. Potirniche , M. F. Horstemeyer , and X. W. Ling , An internal state variable damage model in crystal plasticity , Mechanics of Materials , vol. 39 , no. 10 , pp. 941 – 952 , 2007 .

120 M. K. Jones , M. F. Horstemeyer , and A. D. Belvin , A multiscale analysis of void coalescence in nickel , Journal of Engineering Materials and Technology , vol. 129 , p. 94 , 2007 .

121 H. Neuber , Notch stress theory , DTIC Document, 1965 . 122 R. E. Peterson , in: G. Sines and J. L. Waisman , eds., Notch Sensitivity, Metal Fatigue ,

McGraw - Hill, New York, 1959 , pp. 293 – 306 .


123 G. Harkegard , Application of the fi nite element method to cyclic loading of elastic - plastic structures containing effects , International Journal of Fracture , vol. 9 , p. 322 , 1973 .

124 R. A. Smith , K. Jerram , and K. J. Miller , Experimental and theoretical fatigue - crack propagation lives of variously notched plates , The Journal of Strain Analysis for Engineering Design , vol. 9 , no. 2 , pp. 61 – 66 , 1974 .

125 R. A. Smith and K. J. Miller , Fatigue cracks at notches , International Journal of Mechanical Sciences , vol. 19 , no. 1 , pp. 11 – 22 , 1977 .

126 J. Lankford , Inclusion - matrix debonding and fatigue crack initiation in low alloy steel , International Journal of Fracture , vol. 12 , no. 1 , pp. 155 – 157 , 1976 .

127 J. Lankford , D. L. Davidson , and K. S. Chan , The infl uence of crack tip plasticity in the growth of small fatigue cracks , Metallurgical and Materials Transactions A , vol. 15 , no. 8 , pp. 1579 – 1588 , 1984 .

128 J. Lankford and F. N. Kusenberger , Initiation of fatigue cracks in 4340 steel , Metal-lurgical and Materials Transactions B , vol. 4 , no. 2 , pp. 553 – 559 , 1973 .

129 M. J. Couper , A. E. Neeson , and J. R. Griffi ths , Casting defects and the fatigue behaviour of an aluminium casting alloy , Fatigue & Fracture of Engineering Mate-rials & Structures , vol. 13 , no. 3 , pp. 213 – 227 , 1990 .

130 J. F. Major , Porosity control and fatigue behavior in A356 - T61 aluminum alloy , Transactions - American Foundrymens Society , vol. 102 , pp. 901 – 906 , 1998 .

131 D. L. Davidson , The effects on fracture toughness of ductile - phase composition and morphology in Nb - Cr - Ti and Nb - Si in situ composites , Metallurgical and Materials Transactions A – – Physical Metallurgy and Materials Science , vol. 27 , pp. 2540 – 2556 , 1996 .

132 P. J. Laz and B. M. Hillberry , Fatigue life prediction from inclusion initiated cracks , International Journal of Fatigue , vol. 20 , no. 4 , pp. 263 – 270 , 1998 .

133 K. Gall , M. Horstemeyer , D. L. McDowell , and J. Fan , Finite element analysis of the stress distributions near damaged Si particle clusters in cast Al - Si alloys , Mechanics of Materials , vol. 32 , no. 5 , pp. 277 – 301 , 2000 .

134 K. Gall , M. F. Horstemeyer , M. Van Schilfgaarde , and M. I. Baskes , Atomistic simulations on the tensile debonding of an aluminum - silicon interface , Journal of the Mechanics and Physics of Solids , vol. 48 , no. 10 , pp. 2183 – 2212 , 2000 .

135 K. Gall , N. Yang , M. Horstemeyer , D. L. McDowell , and J. Fan , The debonding and fracture of Si particles during the fatigue of a cast Al - Si alloy , Metallurgical and Materials Transactions A , vol. 30 , no. 12 , pp. 3079 – 3088 , 1999 .

136 X. S. Wang , F. Liang , J. H. Fan , and F. H. Zhang , Low - cycle fatigue small crack initiation and propagation behaviour of cast magnesium alloys based on in - situ SEM observations , Philosophical Magazine , vol. 86 , no. 11 , pp. 1581 – 1596 , 2006 .

137 S. Kumar and W. A. Curtin , Crack interaction with microstructure , Materials Today , vol. 10 , no. 9 , pp. 34 – 44 , 2007 .

138 M. M. Shenoy , R. S. Kumar , and D. L. McDowell , Modeling effects of nonmetallic inclusions on LCF in DS nickel - base superalloys , International Journal of Fatigue , vol. 27 , no. 2 , pp. 113 – 127 , 2005 .

139 M. Shenoy , J. Zhang , and D. L. McDowell , Estimating fatigue sensitivity to poly-crystalline Ni - base superalloy microstructures using a computational approach ,

REFERENCES 39

Fatigue & Fracture of Engineering Materials & Structures , vol. 30 , no. 10 , pp. 889 – 904 , 2007 .

140 D. L. McDowell , Simulation - assisted materials design for the concurrent design of materials and products , Journal of the Minerals, Metals and Materials Society , vol. 59 , no. 9 , pp. 21 – 25 , 2007 .

141 J. Fan , Z. Gao , and X. Zeng , Cyclic plasticity across micro/meso/macroscopic scales , Proceedings of the Royal Society of London. Series A, Mathematical, Physi-cal and Engineering Sciences , vol. 460 , no. 2045 , p. 1477 , 2004 .

142 M. F. Horstemeyer , M. I. Baskes , and S. J. Plimpton , Computational nanoscale plasticity simulations using embedded atom potentials , in: G. Sih , ed., Prospects in Mesomechanics , Theoretical and Applied Fracture Mechanics , vol. 37 , nos. 1 – 3, pp. 49 – 98 , 2001 .

143 W. W. Gerberich , N. I. Tymiak , J. C. Grunlan , M. F. Horstemeyer , and M. I. Baskes , Interpretations of indentation size effects , Journal of Applied Mechanics , vol. 69 , p. 433 , 2002 .

144 M. F. Horstemeyer , Damage infl uence on Bauschinger effect of a cast A356 alu-minum alloy , Scripta Materialia , vol. 39 , no. 11 , pp. 1491 – 1495 , 1998 .

145 M. F. Horstemeyer and A. M. Gokhale , A void - crack nucleation model for ductile metals , International Journal of Solids and Structures , vol. 36 , no. 33 , pp. 5029 – 5055 , 1999 .

146 M. F. Horstemeyer and M. I. Baskes , Atomistic fi nite deformation simulations: a discussion on length scale effects in relation to mechanical stresses , Journal of Engineering Materials and Technology , vol. 121 , p. 114 , 1999 .

147 M. F. Horstemeyer , M. I. Baskes , and S. J. Plimpton , Length scale and time scale effects on the plastic fl ow of FCC metals , Acta Materialia , vol. 49 , no. 20 , pp. 4363 – 4374 , 2001 .

148 M. F. Horstemeyer , Mapping failure by microstructure - property modeling , Journal of the Minerals, Metals and Materials Society , vol. 53 , no. 9 , pp. 24 – 27 , 2001 .

149 M. F. Horstemeyer , J. Fan , and D. L. McDowell , From atoms to autos: part 2 fatigue modeling , Sandia National Laboratories Report, SAND2001 - 8,661, vol. 4, 2001 .

150 M. F. Horstemeyer , M. I. Baskes , A. Godfrey , and D. A. Hughes , A large deforma-tion atomistic study examining crystal orientation effects on the stress - strain relationship , International Journal of Plasticity , vol. 18 , no. 2 , pp. 203 – 229 , 2002 .

151 M. F. Horstemeyer , et al., Torsion/simple shear of single crystal copper , Journal of Engineering Materials and Technology , vol. 124 , p. 322 , 2002 .

152 J. Fan , D. L. McDowell , M. F. Horstemeyer , and K. Gall , Cyclic plasticity at pores and inclusions in cast Al - Si alloys , Engineering Fracture Mechanics , vol. 70 , no. 10 , pp. 1281 – 1302 , 2003 .

153 G. P. Potirniche and M. F. Horstemeyer , On the growth of nanoscale fatigue cracks , Philosophical Magazine Letters , vol. 86 , no. 3 , pp. 185 – 193 , 2006 .

154 S. R. Johnston , G. P. Potirniche , S. R. Daniewicz , and M. F. Horstemeyer , Three - dimensional fi nite element simulations of microstructurally small fatigue crack growth in 7075 aluminium alloy , Fatigue & Fracture of Engineering Materials & Structures , vol. 29 , no. 8 , pp. 597 – 605 , 2006 .


155 K. Gall , M. F. Horstemeyer , B. W. Degner , D. L. McDowell , and J. Fan , On the driving force for fatigue crack formation from inclusions and voids in a cast A356 aluminum alloy , International Journal of Fracture , vol. 108 , no. 3 , pp. 207 – 233 , 2001 .

156 V. S. Deshpande , A. Needleman , and E. Van Der Giessen , Scaling of discrete dislocation predictions for near - threshold fatigue crack growth , Acta Materialia , vol. 51 , no. 15 , pp. 4637 – 4651 , 2003 .

157 V. S. Deshpande , A. Needleman , and E. Van Der Giessen , Discrete dislocation plasticity modeling of short cracks in single crystals , Acta Materialia , vol. 51 , no. 1 , pp. 1 – 15 , 2003 .

158 T. Morita and H. Z. Tsuji , Dislocation dynamics simulation of fatigue crack initia-tion , Journal of the Society of Materials Science (Japan) , vol. 53 , pp. 647 – 653 , 2004 .

159 I. N. Mastorakos and H. M. Zbib, DD simulations of dislocation - crack interaction during fatigue , Journal of ASTM International , vol. 4 , pp. 1 – 9 , 2007 .

160 S. Groh , S. Olarnrithinun , W. A. Curtin , A. Needleman , V. S. Deshpande , and E. V. Giessen , Fatigue crack growth from a cracked elastic particle into a ductile matrix , Philosophical Magazine , vol. 88 , no. 30 , pp. 3565 – 3583 , 2008 .

161 D. L. McDowell and G. B. Olson , Concurrent design of hierarchical materials and structures , Journal of Scientifi c Modeling and Simulation , vol. 15 , pp. 207 – 240 , 2008 .

162 A. Ramasubramaniam and E. A. Carter , Coupled quantum – atomistic and quantum – continuum mechanics methods in materials research , MRS Bulletin , vol. 32 , no. 11 , pp. 913 – 918 , 2007 .

163 G. Lu and E. Kaxiras , Handbook of Theoretical and Computational Nanotechnol-ogy , American Scientifi c , Stevenson Ranch, CA , 2004 .

164 N. Choly , G. Lu , E. Weinan , and E. Kaxiras , Multiscale simulations in simple metals: a density - functional - based methodology , Physical Review B , vol. 71 , no. 9 , p. 094101 , 2005 .

165 G. Lu , E. B. Tadmor , and E. Kaxiras , From electrons to fi nite elements: a concur-rent multiscale approach for metals , Physical Review B , vol. 73 , no. 2 , p. 024108 , 2006 .

166 F. Ercolessi and J. B. Adams , Interatomic potentials from fi rst - principles calcula-tions: the force - matching method , Europhysics Letters , vol. 26 , p. 583 , 1994 .

167 Y. Li , D. J. Siegel , J. B. Adams , and X. Y. Liu , Embedded - atom - method tantalum potential developed by the force - matching method , Physical Review B , vol. 67 , no. 12 , p. 125101 , 2003 .

168 N. Govind , Y. A. Wang , A. J. R. Da Silva , and E. A. Carter , Accurate ab initio energetics of extended systems via explicit correlation embedded in a density functional environment , Chemical Physics Letters , vol. 295 , no. 1 – 2 , pp. 129 – 134 , 1998 .

169 T. Kl ü ner , N. Govind , Y. A. Wang , and E. A. Carter , Prediction of electronic excited states of adsorbates on metal surfaces from fi rst principles , Physical Review Letters , vol. 86 , no. 26 , pp. 5954 – 5957 , 2001 .

170 Y. A. Wang , N. Govind , and E. A. Carter , Orbital - free kinetic - energy functionals for the nearly free electron gas , Physical Review B , vol. 58 , no. 13 , p. 465 , 1998 .

171 Y. A. Wang , N. Govind , and E. A. Carter , Orbital - free kinetic - energy density functionals with a density - dependent kernel , Physical Review B , vol. 60 , no. 24 , p. 16350 , 1999 .

REFERENCES 41

172 G. Cs á nyi , T. Albaret , M. C. Payne , and A. De Vita , “ Learn on the fl y ” : a hybrid classical and quantum - mechanical molecular dynamics simulation , Physical Review Letters , vol. 93 , no. 17 , p. 175503 , 2004 .

173 M. I. Baskes , Application of the embedded - atom method to covalent materials: a semiempirical potential for silicon , Physical Review Letters , vol. 59 , no. 23 , pp. 2666 – 2669 , 1987 .

174 M. I. Baskes , J. S. Nelson , and A. F. Wright , Semiempirical modifi ed embedded - atom potentials for silicon and germanium , Physical Review B , vol. 40 , no. 9 , p. 6085 , 1989 .

175 M. I. Baskes and R. A. Johnson , Modifi ed embedded atom potentials for HCP metals , Modelling and Simulation in Materials Science and Engineering , vol. 2 , p. 147 , 1994 .

176 M. I. Baskes , J. E. Angelo , and C. L. Bisson , Atomistic calculations of composite interfaces , Modelling and Simulation in Materials Science and Engineering , vol. 2 , p. 505 , 1994 .

177 H. Fang , M. Rais - Rohani , Z. Liu , and M. F. Horstemeyer , A comparative study of metamodeling methods for multiobjective crashworthiness optimization , Com-puters & Structures , vol. 83 , no. 25 – 26 , pp. 2121 – 2136 , 2005 .

178 K. Solanki , M. F. Horstemeyer , M. I. Baskes , and H. Fang , Multiscale study of dynamic void collapse in single crystals , Mechanics of Materials , vol. 37 , no. 2 – 3 , pp. 317 – 330 , 2005 .

179 D. Farkas , Atomistic studies of intrinsic crack - tip plasticity , MRS Bulletin , vol. 25 , no. 5 , pp. 35 – 39 , 2000 .

180 D. Farkas , M. Willemann , and B. Hyde , Atomistic mechanisms of fatigue in nano-crystalline metals , Physical Review Letters , vol. 94 , no. 16 , p. 165502 , 2005 .

181 A. Luque , et al., Molecular dynamics simulation of crack tip blunting in opposing directions along a symmetrical tilt grain boundary of copper bicrystal , Fatigue & Fracture of Engineering Materials & Structures , vol. 30 , no. 11 , pp. 1008 – 1015 , 2007 .

182 G. I. Barenblatt , Scaling , vol. 34 , Cambridge University Press , Cambridge, UK , 2003 .

183 A. Carpinteri , P. Cornetti , F. Barpi , and S. Valente , Cohesive crack model descrip-tion of ductile to brittle size - scale transition: dimensional analysis vs. renormaliza-tion group theory , Engineering Fracture Mechanics , vol. 70 , no. 14 , pp. 1809 – 1839 , 2003 .

184 A. Brandt , J. Bernholc , and K. Binder , Multiscale Computational Methods in Chemistry and Physics , Vol. 177 , IOS Press , Heidelberg , 2001 .

185 E. Weinan and B. Engquist , The heterogeneous multiscale methods , Communica-tions in Mathematical Sciences , vol. 1 , no. 1 , pp. 87 – 132 , 2003 .

186 Y. T. Cheng and C. M. Cheng , Fundamentals of nanoindentation and nanotribol-ogy , Materials Research Society Symposium Proceedings , vol. 522 , pp. 139 – 144 , 1998 .

187 N. Ansini , et al., Multiscale analysis by gamma - convergence of a one - dimensional nonlocal functional related to a shell - membrane transition , SIAM Journal on Mathematical Analysis , vol. 38 , pp. 944 – 976 , 2006 .

188 B. B. Mandelbrot , D. E. Passoja , and A. J. Paullay , Fractal character of fracture surfaces of metals , 1984 .


189 B. B. Mandelbrot , The Fractal Geometry of Nature , W.H. Freeman , New York , 1983 . 190 D. M. Mark and P. B. Aronson , Scale - dependent fractal dimensions of topographic

surfaces: an empirical investigation, with applications in geomorphology and computer mapping , Mathematical Geology , vol. 16 , no. 7 , pp. 671 – 683 , 1984 .

191 T. Chelidze and Y. Guguen , Evidence of fractal fracture = Mise en é vidence de la fracture fractale , International Journal of Rock Mechanics and Mining Sciences , vol. 27 , no. 3 , pp. 223 – 225 , 1990 .

192 A. B. Mosolov , F. M. Borodich , and A. Maciulaitis , Fractal fracture of brittle bodies during compression , Soviet Physics. Doklady , vol. 37 , pp. 263 – 265 , 1992 .

193 A. V. Dyskin , Effective characteristics and stress concentrations in materials with self - similar microstructure , International Journal of Solids and Structures , vol. 42 , no. 2 , pp. 477 – 502 , 2005 .

194 S. Graham and N. Yang , Representative volumes of materials based on micro-structural statistics , Scripta Materialia , vol. 48 , no. 3 , pp. 269 – 274 , 2003 .

195 S. Lee and R. Rao , Scale - based formulations of statistical self - similarity in images , in 2004 International Conference on Image Processing, 2004. ICIP ’ 04 , vol. 4 , 2004 , pp. 2323 – 2326 .

196 J. Fish and A. Wagiman , Adaptive, multilevel, and hierarchical computational strategies , in Winter Annual Meeting of the American Society of Mechanical Engi-neers , vol. 157 , pp. 95 – 117 , 1992 .

197 M. A. Nuggehally , M. S. Shephard , C. Picu , and J. Fish , Adaptive model selection procedure for concurrent multiscale problems , International Journal for Multi-scale Computational Engineering , vol. 5 , no. 5 , pp. 369 – 386 , 2007 .

198 G. W. He et al., Multiscale coupling: challenges and opportunities , Prog. in Nat. Sci. , vol. 14 , pp. 463 – 466 , 2004 .

199 H. Y. Wang et al., Multiscale coupling in complex mechanical systems , Chem. Eng. Sci. , vol. 59 , pp. 1677 – 1686 , 2004 .

200 J. W. Essam , Phase transitions and critical phenomena , Phase Transitions and Critical Phenomena , vol. 2 , pp. 1583 – 1585 , 1972 .

201 S. Greenspoon , Finite - size effects in one - dimensional percolation: a verifi cation of scaling theory , Canadian Journal of Physics , vol. 57 , p. 550 , 1979 .

202 H. Kestin , Percolation Theory for Mathematicians , Proc. Cambridge Phil. Soc., Boston , 1982 .

203 Y. Otsubo , Elastic percolation in suspensions fl occulated by polymer bridging , Langmuir , vol. 6 , no. 1 , pp. 114 – 118 , 1990 .

204 D. F. Leclerc and J. A. Olson , A percolation - theory model of lignin degradation , Macromolecules , vol. 25 , no. 6 , pp. 1667 – 1675 , 1992 .

205 R. Fu , Q. Y. Long , and C. W. Lung , Interpretation of porosity effect on strength of highly porous ceramics , Scripta Metallurgica et Materialia , vol. 25 , no. 7 , pp. 1583 – 1585 , 1991 .

206 M. Ostoja - Starzewski , Mechanics of damage in a random granular microstruc-ture: percolation of inelastic phases , International Journal of Engineering Science , vol. 27 , no. 3 , pp. 315 – 326 , 1989 .

207 Y. L. Bai , H. Y. Wang , M. F. Xia , and F. J. Ke , Statistical mesomechanics of solid, linking coupled multiple space and time scales , Applied Mechanics Reviews , vol. 58 , p. 372 , 2005 .

REFERENCES 43

208 L. E. Reichl , A Modern Course in Statistical Physics , University of Texas Press, Austin , 2009 .

209 B. Zhang , D. R. Poirier , and W. Chen , Microstructural effects on high - cycle fatigue - crack initiation in A356. 2 casting alloy , Metallurgical and Materials Trans-actions A , vol. 30 , no. 10 , pp. 2659 – 2666 , 1999 .

210 M. F. Horstemeyer and P. Wang , Cradle - to - grave simulation - based design incor-porating multiscale microstructure - property modeling: reinvigorating design with science , Journal of Computer - Aided Materials Design , vol. 10 , no. 1 , pp. 13 – 34 , 2003 .

211 S. R. A. Fisher , Statistical Methods for Research Workers , Oliver and Boyd, Edin-burgh , 1932 .

212 R. A. Fisher , The Design of Experiments , 1935 . 213 G. Taguchi , Reports of statistical application research , JUSE , vol. 6 , pp. 1 – 52 , 1960 . 214 G. Taguchi , System of Experimental Design , vols. 1 and 2 , American Supplier

Institute Inc., Center for Taguchi Methods , Dearborne, MI , 1987 . 215 M. F. Horstemeyer , S. Ramaswamy , and M. Negrete , Using a micromechanical

fi nite element parametric study to motivate a phenomenological macroscale model for void/crack nucleation in aluminum with a hard second phase , Mechan-ics of Materials , vol. 35 , no. 7 , pp. 675 – 687 , 2003 .

216 J. H. Panchal , H. J. Choi , J. Shepherd , J. K. Allen , D. L. McDowell , and F. Mistree , A strategy for simulation - based design of multiscale, multi - functional products and associated design processes , in Proceedings of the Design Engineering Techni-cal Conferences (DETC ’ 05) , 2005 .

217 G. B. Olson , Computational design of hierarchically structured materials , Science , vol. 277 , no. 5330 , p. 1237 , 1997 .

218 G. B. Olson , Advances in theory: martensite by design , Materials Science and Engineering: A , vol. 438 , pp. 48 – 54 , 2006 .

219 F. Mistree , C. C. Seepersad , B. M. Dempsey , D. L. McDowell , and J. K. Allen , Robust concept exploration methods in materials design , in 9th AIAA/ISSMO Symposium and Exhibit on Multidisciplinary Analysis and Optimization , Atlanta, GA, 2002 .

220 C. C. Seepersad , J. K. Allen , D. L. McDowell , and F. Mistree , Robust design of cellular materials with topological and dimensional imperfections , Journal of Mechanical Design , vol. 128 , p. 1285 , 2006 .

221 C. C. Seepersad , B. M. Dempsey , J. K. Allen , F. Mistree , and D. L. McDowell , Design of multifunctional honeycomb materials , AIAA Journal , vol. 42 , no. 5 , pp. 1025 – 1033 , 2004 .

222 P. E. Eden , Civil Aircraft Today: The World ’ s Most Successful Commercial Aircraft , Amber Books , London , 2008 .

223 H. Fang , K. Solanki , and M. F. Horstemeyer , Numerical simulations of multiple vehicle crashes and multidisciplinary crashworthiness optimization , International Journal of Crashworthiness , vol. 10 , no. 2 , pp. 161 – 172 , 2005 .

224 H. W. Coleman and W. G. Steele , Experimentation and Uncertainty Analysis for Engineers , Wiley - Interscience , New York , 1999 .

225 U. M. Diwekar , Introduction to Applied Optimization , Vol. 22 , Springer - Verlag , New York , 2008 .


226 D. L. McDowell , J. H. Panchal , H. J. Choi , C. C. Seepersad , J. K. Allen , and F. Mistree , Integrated Design of Multiscale, Multifunctional Materials and Products , Elsevier, Butterworth - Heinemann , Oxford, UK , 2010 .

227 J. Glimm and D. H. Sharp , Multiscale science: a challenge for the twenty - fi rst century , Advances in Mechanics , vol. 30 , p. 04 , 1998 .

228 M. E. Kassner , et al., New directions in mechanics , Mechanics of Materials , vol. 37 , no. 2 – 3 , pp. 231 – 259 , 2005 .

229 T. Haupt , Cyberinfrastructure support for integrated materials engineering , in: J. Allison , P. Collins , and G. Spanos , eds., The Proceedings of the 1st World Congress on Integrated Computational Materials Engineering (ICME) , Wiley , 2011 , pp. 229 – 234 .

230 T. Haupt , N. Sukhija , and M. F. Horstemeyer , Cyberinfrastructure Support for Engineering Virtual Organization for CyberDesign , in 2nd Workshop on Scalable Computing in Distributed Systems (SCoDiS ’ 11) , Torun, Poland, September 11 – 14, 2011 .

231 T. Haupt , N. Sukhija , and I. Zhuk , Autonomic execution of computational work-fl ows , in Federated Conference on Computer Science and Information Systems (FedCSIS’11) , Szczecin, Poland, September 18 – 12, 2011 .

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