Mechanics and Modeling Issues in Materials Design

Overview

Simulation-Assisted Materials Design for the Concurrent Design of Materials and Products David L. McDowell

Engineering design has historically been taught using the paradigm of selecting materials on the basis of tabulated databases of properties (mechanical, physical, chemical, etc.). Recent trends have moved toward concurrent design of material composition and microstructure together with the component/system level. The goal is to tailor materials to meet specified ranges of performance requirements of the overall system. Often these multiple performance requirements are in conflict in terms of their demands on composition and microstructure. This paper explores the elements of a decision-based robust design framework for concurrent design of materials and products, focusing on enhancing the fraction of decisions supported by modeling and simulation. INTRODUCTION Viewing material as a hierarchical structure having design degrees of freedom subject to composition and process modification opens new vistas for improving products, including sustainability and life cycle management. Within the past decade, several prominent lines of thought have emerged regarding materials design. One has to do with materials design as a materials selection exercise that emphasizes construction and search of databases for properties or characteristics of responses that best suit a set of specified performance indices,1 often using combinatorial search methods.2 Attention is focused on data mining and visualization, and providing convenient and powerful interfaces for the designer to support materials selection. Another class of approaches advocates simulation-based design to exploit computational materials science and physics in accelerating the discovery of new materials, computing 2007 September • JOM

structure, and properties using a bottomup approach. For example, the vision for Materials Cyber-Models for Engineering is a computational materials physics and chemistry perspective3 on using quantum and molecular modeling tools to explore for new materials and compounds, making the link to properties. The author advocates an approach that integrates computational simulation, systems engineering, manufacturing, and design, similar to the ideas behind the current thrust in Integrated Computational Materials Engineering (ICME) being examined by a National Academy of Engineering National Materials Advisory Board study group.4 This view is also consistent with the broad recommendations of the May 2006 Report of the National Science Foundation Blue Ribbon Panel on Simulation-Based Engineering Science.5 In contrast to the bottom-up, reductionist approach, a top-down, goal-means strategy was elucidated by G.B. Olson6 and defined for the academic and research communities at a 1997 National Science Foundationsponsored workshop.7 The conventional approach in science-based modeling of hierarchical processes and systems is a bottom-up, deductive methodology of

modeling the material’s process path, microstructure, and resulting properties, with properties then related to performance requirements (see Figure 1). This overlapping process-structureproperty-performance hierarchy maps into concurrent hierarchical material and product design, as shown in Figure 2. Already established methods of design-for-manufacture of parts, subassemblies, assemblies, and overall systems must be extended with appropriate methods to address the multiple length and time scales of material structure and responses that govern property-performance relations. Hence, the objective of tailoring the material to specific applications (to the left of the vertical bar in Figure 2) is patently distinct from traditional materials selection. The basic challenges revolve around the fact that hierarchical modeling of materials is in its infancy, and systems-based robust design methods have not been widely applied to the region left of the vertical bar in Figure 2. From a reductionist, bottom-up perspective, many would regard the hierarchy of scales from quantum to continuum on the left in Figure 2 as a multiscale modeling problem. The materials design challenge is to

Figure 1. Olson’s hierarchical concept of Materials by Design.6

21

Figure 2. The hierarchy of levels from atomic scale to system level in concurrent materials and product design. Existing systems design methods focus on levels to the right of the vertical bar, treating materials design as a materials selection problem.

develop methods that employ bottomup modeling and simulation, calibrated and validated by characterization and measurement to the extent possible, facilitated by top-down, requirementsdriven exploration of the hierarchy of material length scales shown in Figure 1. It is a multilevel design problem. To address the problem of systemsbased design of materials, one must first rectify the notions of hierarchy in Figures 1 and 2. Figure 3 provides the common ground, whereby the composition and process path give rise to structure/microstructure, in turn affecting properties that are subsequently related to performance. Three things deserve notice in Figure 3: first, the hierarchy is decomposed as a set of multilevel mappings (process-structure [PS] relations, structure-property [SP] relations, and property-performance [PP] relations); second, lateral movement can occur at each level by reducing the order of the model or the conceptualization; and third, the arrows connect through the hierarchy from the bottom up. In view of the hierarchy of material length scales in Figure 2, it is also apparent that these various scales may each reside within each level of hierarchy shown in Figure 3. It is also noted that the shaded area at the upper right in Figure 3 represents the materials selection problem, which occurs at just one or two levels of the hierarchy; it involves selection based on tabulated data from models or experiments and 22

may be approached using informatics (e.g., data mining, combinatorics, and so forth).1–3 The PS relations must abide by processing and manufacturing constraints, cost factors, thermodynamic accessibility, and kinetic feasibility (rates of process, necessary driving forces, and long-term stability of metastable microstructures). Given the cost of modeling and simulation above the PS level in the hierarchy, it makes little

sense to devote effort to broad searches of structure-property relations for otherwise inaccessible or unstable material structures. Hence, in many respects the PS relations constrain the processes of design exploration at higher levels of the hierarchy. However, one can pursue SP relations in parallel, guided increasingly by accessible regions of the PS mappings as they are identified. Moreover, SP searches can guide the exploration of process routes to achieve structures that satisfy required property sets. Lateral mappings or transformations at each level of the hierarchy in Figure 3 represent a reduction of order of the description. For example, when representing microstructure with low order moments of the spatial correlations of phases and other microstructure attributes (e.g., volume fractions), we concede accuracy of the representation in favor of efficiency, both in terms of storage and in terms of computational idealization. This is a common and necessary feature of design exploration for microstructures that may satisfy property/response requirements; they must be performed as parametric studies unless the set of possible microstructures is discrete, in which case combinatorial search is warranted (as in designer molecules for nanoscale functionality, such as chemical sensors and virus

Figure 3. The hierarchy of mappings in multi-level materials design.

JOM • September 2007

Figure 4. A schematic of steps 1 and 2 in IDEM. Step 1 involves bottom-up simulations or experiments, typically conducted in parallel fashion, to map composition into structure and then into properties, with regions in yellow showing the feasible ranged set of points from these mappings. Step 2 involves top-down evaluation of points from the ranged set of specified performance requirements that overlap feasible regions established by bottom-up simulations in Step 1.

blockers). In general, microstructures that are reconstructed for purposes of SP simulations do not possess accuracy at higher order moments of the spatial distribution of the attributes, which may have implications for modeling responses that depend on extreme value characteristics such as fracture, fatigue, and strength. The reduction of order mappings introduce irreversibility of information flow in materials design, which is one source of difficulty in the top-down inversion of structure-property relations. However, there are other even more potent sources. The bottom-up direction of arrows in Figure 3 indicates the flow of model execution and input from experiments. As a practical matter, very few SP relations can be uniquely inverted or inverted in any sense, owing to nonlinear, non-equilibrium characteristics, dependence on initial conditions, and non-uniqueness of solutions. The same is true for PS relations, often to an even greater degree. Clearly, an iterative approach of bottom-up information flow (simulations, experiments) combined with top-down guidance from applications and performance requirements is essential. Anything other than combinatorial searches at the molecular level is complicated by the multiple levels of hierarchy that are evident in microstructures and products/applications. To this end, an approach has been developed called the inductive design exploration method (IDEM).8–10 The method has two major objectives: to explore top-down, requirements-driven design, guiding 2007 September • JOM

bottom-up modeling and simulation, and to manage uncertainty in model chains. Figure 4 illustrates the concept of IDEM. Consider multiple spaces of different dimension (DOF) between which models, simulations, or experiments serve as mappings that project points in a given space (e.g., composition) to the next (e.g., microstructure) and to the final (e.g., properties/responses). First, the design system must be configured, including information flow between specific databases and modeling and simulation tools, and a ranged set of performance requirements specified. Then, steps in IDEM, shown in Figure

Figure 5. A Compromise Decision Support Problem (cDSP) formulation for multi-objective design, with deviations from multiple goals minimized within constraints.12

4, are as follows: v Perform parallel discrete points evaluation at each level of design process (bottom-up simulations and experiments) v Perform inductive (top-down) feasible space exploration based on metamodels from step 2 v Obtain a robust solution with respect to model uncertainty Projected points need not comprise simply connected regions in each space, but they are plotted as such for simplicity. There is also provision for reconfiguring the design system in case there is no overlap between performance requirements and feasible ranges of the bottom-up pro-

Figure 6. An illustration of Types I and II robust design solutions relative to optimal solution based on an extremum of objective function. Type III robust design minimizes deviation of the objective function from the flat region associated with model and microstructure uncertainty.8,16

23

jections and/or to mitigate uncertainty. The mitigation of uncertainty is a second primary feature of IDEM. If two candidate solutions exist, say designs 1 and 2, for which final performance is identical, then which design should be selected? In general, we select solutions that lie the furthest from the constraint boundaries in each space, including the boundaries of feasible regions established in step 1 of IDEM (yellow regions in Figure 4). Current robust design methods cannot address this issue since these methods only focus on the final performance range. Ultimately, design is a decisionmaking process, whether one is designing materials, systems, or both in concurrent fashion. A realistic goal of simulation-assisted, systems-based materials design is to enhance the fraction of design decisions that are supported by simulations. Hence, a decision-support framework is essential to the vision of concurrent materials and product design. Such a framework recognizes that decisionbased robust design is necessary to mitigate various sources of uncertainty, including:11 v Parameterizable (errors induced by processing, operating conditions, etc.) and unparameterizable (e.g., random microstructure) natural variability v Incomplete knowledge of model parameters due to insufficient or inaccurate data v Uncertain structure of a model due to insufficient knowledge (approximations and simplifications) about a system v Propagation of natural and model uncertainty through a chain of models A practical approach is to quantify the uncertainty to the extent possible and then seek robust solutions that are less sensitive to variation of microstructure and various other sources of uncertainty. To this end, the compromise Decision Support Problem (cDSP) protocol,12 shown in Figure 5, is introduced as the primary decision support tool in IDEM. In the cDSP, multiple design objectives are set as targets, with deviations from these goals minimized subject to user preferences to select from among a family of solutions, subject to a set of constraints (see, for example, References 24

13 and 14). Materials design typically involves multiple objectives mapping into several property domains. For example, gas turbine engine blades involve thermal, chemical, mechanical, and thermomechanical properties. The notion of optimization, typically applied to a single objective, is sensible only for limited sensitivity to the variation of parameters, well characterized uncertainty, and a single, dominant functional requirement. For multiple design objectives, robustness establishes preference among candidate solutions;15 we seek solutions with less sensitivity to variation of noise and control parameters. In addition, designs are needed that are robust against variability associated with process route and initial microstructure, forcing functions, cost factors, design goals, etc. The authors’ collaborative efforts at the Georgia Institute of Technology have yielded new methods to deal with uncertainty due to microstructure variability and models 8,16 as well as chained sequences of models in a multilevel (multiscale) context.17 There are several categories of robust design that deal with different types of uncertainty. Type I robust design, originally proposed by Taguchi,15 focuses on achieving insensitivity of performance with respect to noise factors—parameters that designers cannot control in a system. Type II robust design, proposed by Chen and coauthors,13 relates to insensitivity of a design to variability or uncertainty associated with design variables—parameters that a designer can control in a system. A method for Types I and II robust design has been proposed, namely the robust concept exploration method.12,13 These types of robust design have recently been extended to include Type III,8 which considers sensitivity to uncertainty embedded within a model (i.e., model parameter/structure uncertainty). Figure 6 clarifies the application of Types I–III robust design, showing that while application of traditional Types I and II robust design methods seek solutions that are insensitive to variations in control or noise parameters, Type III robust design additionally seeks solutions that have minimum distance between upper and lower uncertainty bounds on the response function(s) of interest associated with material random-

ness and model structure/parameter uncertainty. These bounds are determined from the statistics obtained from application of models over a parametric range of feasible microstructures and process conditions relevant to the simulations necessary to support design decisions (see References 8 and 16). This combination of seeking “flat regions” of the objective function and tight bounds of variability associated with uncertainty of the functional relationship between control/noise and response is a new concept introduced to more realistically address uncertainty in modeling and simulation to support materials design. APPLICATIONS OF DECISION-BASED ROBUST DESIGN OF MATERIALS The foregoing framework of multiobjective decision support can be readily applied to robust design of materials as part of the design of an overall system. To this end, it is necessary to consider materials design as a process rather than as a bottom-up, multiscale modeling exercise. Applications to date have included design of extruded prismatic metals for multifunctional structural and thermal applications18–20 and design of four phase powder metal oxide mixtures (energetic materials) for combined strength and shock-assisted exothermic chemical reaction,16,21 among others. The challenge is to extend these robust design concepts to tailor microstructures that deliver required multiphysics performance requirements in a wide range of problems, for example: v Phase morphologies, precipitate/ dispersoid distributions, texture, and grain boundary networks in alloy systems for multifunctional performance in terms of strength, ductility, fracture, fatigue, corrosion resistance, etc. v Process path and in-service evolution of microstructure (e.g., plasticity, phase transformation, diffusion, etc.) v Resistance to or preference for shear banding v Formable materials that employ transformation- or twinning-induced plasticity v Fine and coarse scale porosity control in castings v Surface treatment, heat treatment, JOM • September 2007

and residual stresses in alloys with primary inclusions CONCLUSIONS The approach described in this paper is characterized by a confluence of engineering science, mechanics of materials, materials science/physics, informatics, and systems engineering. Potential benefits include more efficient, concurrent design of material and components to meet specified performance requirements, the capability to prioritize models and computational methods in terms of degree of utility in design, prioritizing mechanics and materials science/chemistry/physics phenomena to be modeled, and tools for conducting feasibility studies to establish probable return on investment of new material systems. ACKNOWLEDGEMENTS The author especially wishes to thank his many Georgia Institute of Technology (Georgia Tech) colleagues (SRL faculty F. Mistree and J.K. Allen, and former graduate students C. Seepersad, H.-J. Choi, and J. Panchal) in collaborating to develop many of the systems design concepts and tools outlined here. Early stages of this work were sponsored by the Defense Sciences Office of the Defense Advanced Research Projects Agency (DARPA, N00014-99-1-1016) and Office of Naval Research (ONR, N0014-99-1-0852) on Synthetic Multifunctional Materials, monitored by Dr. L. Christodoulou of DARPA and Dr. S. Fishman of ONR. Support of an Air Force Office of Scientific Research Multi-University Research Initiative (1606U81) on Design of Multifunctional Energetic Structural Materials (Dr. C.S. Hartley,

2007 September • JOM

J. Tiley, and B. Connor) is also acknowledged. The systems design approach outlined here is being further developed and extended with support of the Center for Computational Materials Design (CCMD), a National Science Foundation Industry/University Cooperative Research Centers program jointly founded by Pennsylvania State University and Georgia Tech (DLM, co-director), www.ccmd.psu.edu/. References 1. M.F. Ashby, Materials Selection in Mechanical Design, 2nd Edition (Oxford, UK; Butterworth-Heinemann, 1999). 2. C. Shu et al., “Combinatorial Materials Design Through Database Science,” Materials Research Society Symposium—Proceedings, v 804, Combinatorial and Artificial Intelligence Methods in Materials Science II (Warrendale, PA: Materials Research Society, 2003), pp. 333–341. 3. S.J.E. Billinge, K. Rajan, and S.B. Sinnot, From Cyberinfrastructure to Cyberdiscovery in Materials Science: Enhancing Outcomes in Materials Research, Education and Outreach (Report from NSF-sponsored workshop held in Arlington, Virginia, 3–5 August 2006), www.mcc.uiuc.edu/nsf/ciw_2006/. 4. T.M. Pollock and J. Allison, Committee on Integrated Computational Materials Engineering: Developing a Roadmap for a Grand Challenge in Materials (Washington, D.C.: National Materials Advisory Board, National Academy of Engineering, 2007), http://www7. nationalacademies.org/nmab/CICME_home_page.html. 5. J.T. Oden et al., Simulation-Based Engineering Science: Revolutionizing Engineering Science through Simulation (Report of NSF Blue Ribbon Panel on Simulation-Based Engineering Science, May 2006), www.nsf.gov/pubs/reports/sbes_final_report.pdf. 6. G.B. Olson, “Computational Design of Hierarchically Structured Materials,” Science, 277 (5330) (1997), pp. 1237–1242. 7. D.L. McDowell and T.L. Story, New Directions in Materials Design Science and Engineering (Report of NSF DMR-sponsored workshop held in Atlanta, GA, 19–21 October 1998). 8. H.-J. Choi, “A Robust Design Method for Model and Propagated Uncertainty” (Ph.D. Dissertation, G.W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, 2005). 9. J.H. Panchal et al., “A Strategy for Simulation-Based Multiscale, Multifunctional Design of Products and Design Processes” (Presentation at the ASME Design

Automation Conference, Long Beach, CA, 2005), paper number DETC2005-85316. 10. H.-J. Choi et al., “An Inductive Design Exploration Method for the Integrated Design of Multiscale Materials and Products” (Presentation at the ASME Design Automation Conference, Long Beach, CA, 2005), paper number DETC2005-85335. 11. S.S. Isukapalli, A. Roy, and P.G. Georgopoulos, Risk Analysis, 18 (3) (1998), p. 351. 12. F. Mistree, O.F. Hughes, and B.A. Bras, Structural Optimization: Status and Promise, ed. M.P. Kamat (Washington, D.C.: AIAA, 1993), p. 247. 13. W. Chen, “A Robust Concept Exploration Method for Configuring Complex Systems” (Ph.D. Dissertation, G.W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, 1995). 14. C.C. Seepersad et al., “Foundations for a SystemsBased Approach for Materials Design” (Presentation at the 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Albany, NY, 2004), pp. AIAA2004-4300. 15. G. Taguchi, Taguchi on Robust Technology Development: Bringing Quality Engineering Upstream (New York; ASME Press, 1993). 16. H.-J. Choi et al., “An Approach for Robust Design of Reactive Powder Metal Mixtures Based on Nondeterministic Micro-scale Shock Simulation,” Journal of Computer-Aided Materials Design, 12 (1) (2005), pp. 57–85. 17. J.H. Panchal, “A Framework for Simulation-Based Integrated Design of Multiscale Products and Design Processes” (Ph.D. Dissertation, G.W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, 2005). 18. C.C. Seepersad, “A Robust Topological Preliminary Design Exploration Method with Materials Design Applications” (Ph.D. Dissertation, G.W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, 2004). 19. C.C. Seepersad et al., “Multifunctional Design of Prismatic Cellular Materials,” Journal of Computer-Aided Materials Design, 11 (2-3) (2005), pp. 163–181. 20. C.C. Seepersad et al., “Design of Multifunctional Honeycomb Materials,” AIAA Journal, 42 (5) (2004), pp. 1025–1033. 21. J.H. Panchal et al., “Designing Design Processes for Integrated Materials and Products Realization: A Multifunctional Energetic Structural Material Example” (Presentation at the 2006 ASME Design Automation Conference, Philadelphia, PA, 2006), paper no. DETC2006-99449. David L. McDowell is with the Woodruff School of Mechanical Engineering, School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405 USA; (404) 894-5128; e-mail [email protected].

25