Plasticity-Related Microstructure-Property …...Plasticity-Related Microstructure-Property...

Plasticity-Related Microstructure-Property

Relations for Materials Design

David L. McDowella, Hae-Jin Choib, Jitesh Panchalc, Ryan Austind, Janet Allene and Farrokh Mistreef

GWW School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405 USA

[email protected],

[email protected],

[email protected],

[email protected],

[email protected], and

[email protected]

Keywords: Plasticity, Materials Design, Microstructure-Property

Abstract. Design has traditionally involved selecting a suitable material for a given application.

A materials design revolution is underway in which the classical materials selection approach is

replaced by design of material microstructure or mesostructure to achieve certain performance

requirements such as density, strength, ductility, conductivity, and so on. Often these multiple

performance requirements are in conflict in terms of their demands on microstructure.

Computational plasticity models play a key role in evaluating structure-property relations

necessary to support simulation-based design of heterogeneous, multifunctional metals and alloys.

We consider issues related to systems design of several classes of heterogeneous material systems

that is robust against various sources of uncertainty. Randomness of microstructure is one such

source, as is model idealization error and uncertainty of model parameters.

An example is given for design of a four-phase reactive powder metal-metal oxide mixture for

initiation of exothermic reactions under shock wave loading. Material attributes (e.g. volume

fraction of phases) are designed to be robust against uncertainty due to random variation of

microstructure. We close with some challenges to modeling of plasticity in support of design of

deformation and damage-resistant microstructures.

Introduction

Traditionally, materials are selected from a materials database in which properties of materials

are tabulated. However, the paradigm is now evolving into designing materials concurrently with

products/components to meet specific performance requirements. By definition, a multifunctional

material is one for which performance dictates multiple property requirements that are often

conflicting, for example both strength and ductility. In view of the complexity of modeling material

process history and nonequilibrium changes of microstructure during inelastic deformation,

emerging capabilities in computational materials science and constitutive modeling must be

combined with systems engineering methods to realize this goal. McDowell [1,2] recently

discussed challenges to microstructure design of materials in relation to elastic-plastic deformation

and damage evolution. This paper describes a methodology developed in the intervening years to

address, in part, these challenges.

Olson [3] has provided a philosophical underpinning of the goal-oriented approach to materials

design. The conventional approach in mechanics modeling of hierarchical processes and systems is

a “bottom-up”, cause-and-effect, deductive methodology of modeling the material’s process path,

nanostructure and microstructure, resulting properties, and then relating properties to performance,

as shown in Fig. 1. In contrast, we seek “top-down”, goals/means, inductive methods to design the

process route and resulting microstructure and mesostructure of a material if our goal is to tailor a

material to a set of specified performance requirements.

We see that certain mappings (models, codes, and heuristics) are necessary to support materials

design according to Olson’s hierarchy in Fig. 1:

Key Engineering Materials Vols. 340-341 (2007) pp. 21-30online at http://www.scientific.net© (2007) Trans Tech Publications, Switzerland

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Structure

Properties

Performance

Goals/

mea

ns (ind

uctive

)

Cause a

nd effec

t (ded

uctive)

Processing

Structure

Properties

Performance

Goals/

mea

ns (ind

uctive

)

Cause a

nd effec

t (ded

uctive)

Processing

Process-structure (PS) relations: Establish

manufacturing constraints, cost factors,

thermodynamic feasibility, and kinetic feasibility

(feasibility of rates of process, necessary driving

forces, and long term stability of metastable

microstructures)

Structure-property (SP) relations: Relate

composition, phase and morphology information,

expressed using distribution functions or digital

representations amenable to computation, to response

functions that relate to properties of relevance to the

design. This is most often an intrinsically hierarchical

modeling exercise.

Property-performance (PP) relations: Relate feasible

properties to response functions that are relevant to imposed performance requirements, either

through detailed point-by-point computational models or by construction of approximate response

surface or surrogate models, depending on the degree of computational intensity.

Figure 2 shows how this

hierarchy is broken down into

mappings of vertical type (PS,

SP, PP) and lateral type, the

latter involving a reduction of

order of the material

representation to affect a

reduction of the degrees of

freedom. Vertical mappings

of SP type are sometimes

referred to as homogenization

relations that associate a

certain microstructure

idealization with a set of

desired property and response

attributes.

Of course, this goal of top-

down materials design has

many challenges:

• The role of extreme value distributions (not just mean field averages) of certain

microstructure features that control properties such as fracture and fatigue must be

addressed.

• The nonlinear, path dependent behavior of metals and alloys limits extent of parametric

study and parallelization of continuum analyses. Moreover, it engenders dependence upon

initial conditions and limits pursuit of inverse problems in a strict sense.

• A wide range of local solutions can be realized for specified objective functions in terms of

property or performance requirements, leading to non-uniqueness and perhaps large families

of possible solutions.

• Representation of microstructure representation presents challenges in terms of how much

information to store and in what format. Moreover, the goal of materials design is to explore

microstructures that do not exist, using computational simulations to estimate properties.

One cannot neglect the fact that process capabilities constrain achievable microstructures,

Figure 1. Olson’s hierarchical concept of

‘Materials by Design’ [3].

Composition,

initial

microstructure

α; To,to

Composition,

actual

microstructure,

A,T,t,Ri

Microstructure

Attributes, A’,T,t,Ri

∗ dist functions

* explicitProperties,

overlay on

A,T,t,Ri,

Properties,

overlay on

A’,T,t,Ri

Performance

Requirements

PS

SP

PP

T – temperature

A – microstructure attributes

t – time

Ri – forcing functions

Composition,

initial

microstructure

α; To,to

Composition,

actual

microstructure,

A,T,t,Ri

Microstructure

Attributes, A’,T,t,Ri

∗ dist functions

* explicitProperties,

overlay on

A,T,t,Ri,

Properties,

overlay on

A’,T,t,Ri

Performance

Requirements

PS

SP

PP

T – temperature

A – microstructure attributes

t – time

Ri – forcing functions

Figure 2. Hierarchy of mappings from process route to

microstructure to properties to performance.

Engineering Plasticity and Its Applications22

and that thermodynamics and kinetics (history) considerations limit the range of accessible

or feasible microstructures. Hand-in-hand with selection of material representation is the

issue of computational tractability.

• Major sources of uncertainty in processing, microstructure, modeling, etc. must be taken into

account, as they can dominate the configuration of the design process and range of

acceptable solutions. Uncertainty in assigning local properties of phases is significant. In

fact, this consideration often demands robust design approaches against such uncertainty

rather than some sort of multi-objective optimization.

• Material models (PS, SP) and design results must be validated, using principles of internal

consistency, statistical realizability, and validation of material response at various length

scales by direct or indirect measurement [4]. Certain microstructure features are quite

amenable to high resolution characterization over substantial fields of view, such as grain

size and orientation information. Others are not, such as fine scale precipitates.

Multiscale Modeling and Materials Design

There are at least two compelling reasons to understand and model plasticity over a range of

length scales [1]:

(i) to design material microstructures to tailor plastic deformation-related properties, and

(ii) to develop consistent, unified theories of the formation and evolution of defects during

plastic flow at various length scales (cf. [5].

The hierarchy of mappings in Fig. 2 is closely related to a hierarchy of material length (and

time) scales that are of utility in materials

design, shown in Fig. 3. Numerous

methods for multiscale modeling have been

developed, including concurrent multiscale

models, domain decomposition for discrete

to continuous transitions, two scale

homogenization, statistical mechanics of

evolutionary systems, “Handshaking”

methods for informing reduced DOF

models based on results of higher DOF

models, variational principle of virtual

velocities (PVV), based on the working rate

of microstructural rearrangements

(dislocation glide, diffusive

rearrangements, relative grain boundary

motions, etc.) (cf. [6]), and a number of other emerging multiscale homogenization approaches.

There are, of course, practical limitations to concurrent multiscale modeling in addition to

computational issues, such as assignment of initial conditions for all microstructure features and

evolving, embedded finer scale attributes (e.g., dislocation densities for various populations, crack

distributions, etc.), With regard to materials design objectives, concurrent multiscale modeling

schemes or homogenization concepts may not be necessary in many cases, because the goal is not to

accurately predict properties but to understand sensitivity to microstructure and to capture essential

dominant mechanisms and their transitions as a function of forcing functions and responses

applicable to the given design scenario. Although seamless ‘bottom-up’ modeling is scientifically

appealing, it is often too idealized and chained with a compilation of approximations that may

compromise viability for materials design. For purposes of materials design, we seek to employ

models in the range for which they are most appropriate and accurate, and use decision-based (e.g.,

utility theory) protocols for informing models/decisions based on these analyses. The necessity of

seamless scale transitions must be determined by the degree of coupling in the system and the utility

of information gained by modeling that coupling in great detail. In general, a hierarchy of models at

Continuum

Microscale

Atomistic

Quantum

Continuum

Microscale

Atomistic

QuantumFigure 3. Hierarchy of

length scales from atomic to

mesoscopic to continuum.

Key Engineering Materials Vols. 340-341 23

various scales may be employed, with different purposes, at different length and time scales, to

inform decisions.

Uncertainty in Materials Design

One must deal with various sources of uncertainty in designing materials, since it is inherently

an exercise of stochastic character, including [7]:

Natural Uncertainty (system variability)

� Parameterizable: Errors induced by processing, operating conditions, etc. (noise and

control factors)

� Unparameterizable: random microstructure

Model Parameter Uncertainty (parameter uncertainty)

� Incomplete knowledge of model parameters due to insufficient or inaccurate data

Model Structural Uncertainty (model uncertainty)

� Uncertain structure of a model due to insufficient knowledge (approximations and

simplifications) about a system.

Propagated Uncertainty in a Process Chain (process uncertainty)

� Propagation of natural and model uncertainty through a chain of models.

To address these sources of uncertainty in designing materials, there are several possible routes.

One is to improve models and characterization of material structure. Another practical approach is

to quantify the uncertainty to the extent possible and then design to seek solutions that are less

sensitive to variation of microstructure and various sources of uncertainty. To this end, we introduce

the compromise Decision Support Problem (cDSP) methodology [8], in which 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 (cf. [9,10]). Effectively,

we employ this decision-based strategy for linking information from one sort of simulation or

heuristic to advise inputs to another model within an

hierachical structure, as shown in Fig. 4. 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

variation of parameters, well characterized uncertainty

and a single, dominant functional requirement – a

demanding set of considerations! For multiple design

objectives, robustness establishes preference among

candidate solutions; we seek solutions with less

sensitivity to variation of noise and control parameters.

In addition, we seek designs that are robust against

variability associated with process route and initial

microstructure, forcing functions, cost factors, design

goals, etc. Moreover, we introduce new methods to

deal with uncertainty due to microstructure variability

and models [11] as well as chained sequences of

models in a multi-level (multiscale) context [12]. The

designer effectively receives decision support from the

cDSP and must make choices within the flexibility of the design process.

Given n, number of decision variables

p, number of equality constraints

q, number of inequality constraints

m, number of system goals

gi(x), constraint functions

Find x (system variables)

di- ,di

+ (deviation variables)

Satisfy

System constraints:

g(x)=0 i=1,...,p

g(x)<0 i=p+1,...,p+q

System goals:

Ai(x)/Gi + di- - di

+ = 1

Bounds:

Ximin < Xi < Xi

max

di- ,di

+ > 0 and di- . di

+ = 0

Minimize

Z = [f1(di-,di

+), …, fk (di-,di

+)] preemptive

Z = �Wi (di- + di

+) Archimedean

Figure 4. Compromise Decision

Support Problem (cDSP) formulation

for multi-objective design, with

deviations from multiple goals

minimized within constraints.


New Types of Robust Design for Materials Design

Robust design, proposed by Taguchi [13] is a method for improving the quality of a product by

minimizing the effect of uncertainty on product performance. From Taguchi’s perspective, design

performance should be on-target with low variability. This philosophy leads to designs that are very

different from traditionally preferred optimum solutions. ‘Optimal’ solutions offer performance that

may be on-target nominally but often deteriorates significantly when conditions or assumptions

change. Herbert Simon proposed a philosophy similar to Taguchi when he introduced the notion of

“satisficing” and contrasted it with optimizing: “The decision that is optimal in the simplified

model will seldom be optimal in the real world. The decision maker has a choice between an

optimal decision from an imaginary simplified world, or decisions that are ‘good enough’, that

satisfice, for a world approximating the complex real one more closely.” [14]

Our philosophy in designing materials is grounded in the philosophies of Simon and Taguchi.

Because material systems are complex and prone to many of the sources of uncertainty discussed in

the previous section, we seek materials design solutions that are satisficing and robust.

There are several categories of robust design, associated with different types of uncertainty. Type I

robust design, originally proposed by Taguchi, centers on achieving insensitivity in performance

with regard to noise factors—parameters that designers cannot control in a system. Relevant

examples of noise factors are variation of ambient temperature, morphology changes, etc. Type II

robust design, proposed by Chen and coauthors, 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, namely, the Robust Concept Exploration Method [8,9] has

been proposed. The Robust Concept Exploration Method (RCEM) is a domain-independent

approach for generating robust, multidisciplinary design solutions. Robust solutions to multi-

functional design problems are preferable trade-offs between expected performance and sensitivity

of performance due to deviations in design or uncontrollable variables. These solutions may not be

absolute optima within the design space.

The types of robust design have been extended to include Type III. Type III robust design

considers sensitivity to uncertainty embedded within a model (i.e., model parameter/structure

uncertainty). Model parameter/structure uncertainty is typically different from the uncertainty

associated with noise and

control factors, because it

could exist in the parameters

or structure of constraints,

meta-models, engineering

equations, and associated

simulation or analysis

models. The approach for

Type III robust design is

focused on incorporating the

Error Margin Indices [11]

within the RCEM (RCEM-

EMI), and it is based on

simultaneous incorporation

of Type I, II, and III robust

design techniques. Type III

robust design is typically

required to manage (a)

y = f(x,z)x= Control Factors

z= Noise Factors

y= Responses

TYPE III: Uncertainty in function relationship between

control/noise and response

• Natural system randomness due to unparameterizable

variability (natural uncertainty)

• Simplifying assumptions (model structure uncertainty)

• Limited data (model parameter uncertainty)

TYPE III: Uncertainty in function relationship between

control/noise and response

• Natural system randomness due to unparameterizable

variability (natural uncertainty)

• Simplifying assumptions (model structure uncertainty)

• Limited data (model parameter uncertainty)

TYPE I: Natural Uncertainty in system

variables that designer cannot

control

• Variability in boundary conditions

• Operating conditions

TYPE I: Natural Uncertainty in system

variables that designer cannot

control

• Variability in boundary conditions

• Operating conditions

TYPE II: Natural uncertainty in

design variables

• Material specifications (e.g.

particle sizes, volume fractions

of mixture)

• Geometric specifications

• Material processing parameters

TYPE II: Natural uncertainty in

design variables

• Material specifications (e.g.

particle sizes, volume fractions

of mixture)

• Geometric specifications

• Material processing parameters

Figure 5. Three types of robust design, including insensitivity

to uncertainty in noise factors (Type I) and control factors

(design variables) (Type II), as well as uncerainty in models

(PS and SP) and microstructure (Type III) [15].


inherent variability that is

difficult or impossible to

parameterize, such as

stochastic microstructure,

(b) limited data, and (c)

limited knowledge in new

domains such as a new

class of microstructures, as

shown in Fig. 5.

Figure 6 clarifies the

application of Types I-III

robust design, showing that

while application of

traditional Types I-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 randomness 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 (cf. [11]). This combination of “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 address realistic

considerations in materials design.

In view of the hierarchy of length scales corresponding to microstructure and mechanisms that

affect PS and SP relations, another essential ingredient of robust design is designing a system that is

insensitive to the propagated

uncertainty in the design

process. The basic idea of our

approach for finding solutions

that are robust against the

propagated uncertainty in an

multi-level model chain is

passing down the feasible

solution range in an inverse

manner, from given final

performance range to design

space, instead of a bottom-up

manner, passing mean and

variance (or deviation) of a

output to the next model. This

approach allows for a designer

to find not just a single robust

solution, but also ranged sets

of robust solutions among

which a designer expresses

preference for a solution.

Figure 6. Illustration of Types I and II robust design solutions

relative to optimal solution based on extremum of objective

function. Type III robust design minimizes deviation from flat

region associated with model and microstructure uncertainty

[15].

Y

XType I, II, III

Robust

Solution

Upper Limit

Lower Limit

Deviation or

Objective

Function

Deviation

at Optimal

Solution

Deviation

at Type I, II

Robust Solution

Deviation

at Type I, II, III

Robust Solution

Design

VariableType I, II

Robust

Solution

Optimal

Solution

Y

XType I, II, III

Robust

Solution

Upper Limit

Lower Limit

Upper Limit

Lower Limit

Deviation or

Objective

Function

Deviation

at Optimal

Solution

Deviation

at Type I, II

Robust Solution

Deviation

at Type I, II, III

Robust Solution

Design

VariableType I, II

Robust

Solution

Optimal

Solution

y = f(x,z)x= Control

Factors

z= Noise Factors

y= Responses

y = f(x,z)x= Control

Factors

z= Noise Factors

y= Responses

y2 = g(y,z)

z2= Noise Factors

y2= Responses

y2 = g(y,z)

z2= Noise Factors

y2= Responses

Design 1

Design 2

Constraint boundary

: Feasible region : Infeasible region

x space y space y2 space

Design 1

Design 2

Constraint boundary

: Feasible region : Infeasible region

x space y space y2 space

Figure 7. Multi-level robust design in which (top) responses

y from model 1 serve as inputs to model 2, with output y2;

(bottom) designs 1 & 2 have different model projections

(e.g., PS) on the intermediate space, but design 1 is further

from the constraint and is therefore preferred [15].


Figure 7 illustrates our concept for selecting the most effective robust solution in multi-level

robust design, applicable to multiscale modeling and simulation. If we assume that we have two

candidate solutions, Designs 1 and 2, of which final performances are identical, then which design

should we select among the two? Since the projected range of Design 2 in the intermediate space of

Y is farther from the constraint boundary than Design 1, we believe that Design 2 is better than

Design 1; Design 2 is more reliable with regard to potential errors (unquantified uncertainty) in the

mapping functions, f. Current robust design method cannot address this issue since the method only

focus on the final performance range. With this approach, we believe, the better solution may be

obtained considering model uncertainty that is hard to quantify as well as quantified propagated

uncertainty.

Materials Design as a Process

The foregoing framework of

multi-objective decision support for

designing materials can be readily

extended to incorporate the design

of the material (composition,

morphology, etc.) as part of a larger

overall systems design process.

Results of models at multiple length

scales and time scales can be

analyzed as necessary to provide

decision support for selection of

hierarchical morphology and

process path to deliver a required

set of multifunctional, often

conflicting properties. Moreover,

the same framework can embody the

hierarchy of process-structure-

properties-performance set forth by Olson [3], shown in Fig. 1. The inductive goals/means

engineering approach is relevant to design, and contrasts with the usual approach taken in

application of the scientific method, which focuses on deductive cause-and-effect (bottom-up).

Figure 8 shows a flow diagram of the design process, with design requirements as inputs and robust

sets of design specifications on design variables as output.

We can map Olson’s inductive goals/means concept of materials design directly to the systems-

based approach adopted in the present work, as indicated in Fig. 8. Effectively, process-structure-

property relations inform the designer and map directly into the design process that facilitates

transformation of overall design requirements into a set of robust specifications for the material

system of interest.

Example: Reaction Initiation with Dynamic Plasticity in an Energetic Material

Multiphase energetic materials (thermite mixtures) are mixtures of micron scale (or finer) metal

and metal oxide (sometimes intermetallic) powders with a binder and limited porosity. Hence, these

are complex, three or four phase mixtures. Shock compression of these materials can result in the

initiation of exothermal chemical reaction that may or may not propagate in self-sustained manner at

scales well above the mean particle size. Shock compression of spatially-resolved particle systems

must be modeled to understand the effects of dynamic plastic deformation, particle interactions and

pore collapse at the mesoscale on the reaction initiation behavior to design the mixture for specified

reaction initiation conditions (shock strength). The simulations involve propagating shock waves

through aluminum-iron oxide thermite systems (Al+Fe2O3) composed of micron-size particles

suspended in a polymer binder [16]. As shown in Fig. 9, a fully thermomechanically coupled

Simulation InfrastructureSimulation Infrastructure

Input and Output

Design Process

Analysis Program

Input and Output

Design Process

Analysis Program

Compromise DSP

FindDesign Variables

SatisfyConstraintsGoalsBounds

MinimizeObjective Function

Analysis Programs,

Databases, etc.

Surrogate /

Approx Model

Construction

Robust Design

Specifications

Uncertainty

Analysis

Search /

Optimization

Algorithms

Design Space

StructureDesign Variables

BehaviorPerformance

RequirementsConstraintsGoals

Overall Design

Requirements

Design Process

Performance Property Structure Process

Process-Structure-Property Relations

Figure 8. Systems-based materials design approach.


Eulerian code RAVEN [17 ] is used to model the progression of a shock wave induced by imposed

high particle velocity on a statistic volume element (SVE) of microstructure. The simulated

(reconstructed) microstructures are based on particle size distributions, volume fractions, and levels

of porosity measured experimentally for as-processed materials. The measured nearest neighbor

distribution of the Al particles (largest particles in Fig. 9, approximately 1-2 microns in diameter) is

met through simulated annealing, followed by constrained Poisson point placement of the Fe2O3

particles and pores. The epoxy binder then constitutes the remainder. Post-processing enables

determiation of the shock velocity in the mixture (Hugoniot behavior), the pressure profile throught

the SVE as a function of wavefront position, and the detailed temperature rise throughout the SVE.

Appropriate rate- and temperature-dependent constitutive equations are used for the constituents

(Klepazcko model [18] for 1100 Al, Hasan-Boyce model [19] for Epon 828, and an athermal model

for elastic-plastic behavior of Fe2O3).

The probability of shock-induced reaction initiation is then evaluated from the temperature field

at hot spots using the Merzhanov instability criterion for thermal explosion [20], which essentially

considers whether or not heat generated at hot spots by the reaction can be transferred by conduction

to the surrounding material at a rate high enough to quench the reaction. Because the SVE is not

large enough relative to the particle size and spacing, it cannot serve as a representative volume

element (RVE) for the desired response (number density of sites for reaction initiation) and it is

necessary to build up the statistics for number density of sites from a set of instantiations or

realizations of SVEs for the same microstructure subjected to the same shock loading condition. It

is noted that the RVE size is not based simply on spatial statistics of geometric features, as is

commonly implied, but must come from analyses of responses; certain responses, such as localized

plasticity, or in this case reaction initiation, depend on extrema of microstructure features and

therefore require much larger (too large

for computational practicality) volumes

of material to be analyzed. From these

simulations, statistics are compiled and

the mean response (fo) and 99%

prediction interval for upper (f1) and

lower (f2) uncertainty bound functions

represented using an iterative re-

weighted generalized linear model, with

variance function estimation by

maximizing the Pseudolikelihood

estimator. In this case, variance is

due largely to microstructure

stochasticity. For given Al and Fe2O3

particle size distributions, for

example, Fig. 10 shows the number

density of reaction initiation sites as a

function of mean void size and

volume fraction. This facilitates either optimization (based on mean) or Types I-III robust design

outlined earlier of a mixture with specified reaction initiation requirements.

To facilitate higher length scale calculations, it is necessary to use the discrete particle

simulations to inform extended irreversible thermodynamics continuum internal state variable

models with additional flux terms to model delay of macroscopic reaction propagation at

component length scales [21]. Multi-level robust design concepts, as shown in Fig. 7, come into

play in facilitating such hierarchical relations.

Microstructural

Descriptors

Microstructural

Descriptors

Simulated

Microstructures

Actual Microstructure

Explicit Hydrocode

Simulations

of Shock Wave Propagation:

Parametric Studies and

Design

of Microstructures

Figure 9. Linkage of actual microstructure to

reconstruction algorithm to support parametric Eulerian

hydrocode studies of shock wave propagation through

energetic materials and characterization of reaction

initiation probability [11,15].


{ }0

1 ,1 / 2

2 ,1 / 2

ˆ( )=exp 2

ˆˆ( )=exp exp 2

2

ˆˆ( )=exp exp 2

2

converged

converged

converged N P

converged

converged N P

f

f t

f t

α

α

− −

− −

⋅ −

⋅ ⋅ + ⋅ −

⋅ ⋅ − ⋅ −

0 0

'

0

0 0

'

0

0 0

x β x

θ xx β x

θ xx β x0.02

0.040.06

0.080.1

0.5e-3

1e-30

5

10

15

20

25

30

Mean Radius of Voids (mm)

At Mean Ra of Al = 0.75e-3 and Fe2O3 = 0.4e-3 (mm)

Vf of Voids

Num

of

Rxn S

ites

Upper limit of the

prediction interval

The mean model

Lower limit of the

prediction interval

Samples

0.020.04

0.060.08

0.1

0.5e-3

1e-30

5

10

15

20

25

30

Mean Radius of Voids (mm)

At Mean Ra of Al = 0.75e-3 and Fe2O3 = 0.4e-3 (mm)

Vf of Voids

Num

of

Rxn S

ites

Upper limit of the

prediction interval

The mean model

Lower limit of the

prediction interval

Samples

The challenge is to extend these robust design concepts to tailor microstructures that deliver

required performance requirements with consideration of multi-physics in problems involving, for

example:

• Phase morphologies of alloy systems for multifunctionality

• Fracture resistant microstructures without a priori specification of fracture paths

• Evolution of microstructure (e.g. plasticity, phase transformation, diffusion, etc.)

• Texture and grain boundary networks for fracture and fatigue resistance

• Precipitate distributions and morphologies for fatigue resistance

• Microstructures with resistance to or preference for shear banding

• Formable materials with transformation- or twinning-induced plasticity

• Processing with targeted porosity control

• Surface treatments, heat treatments, inclusions, and residual stresses

Closure

A systems-based approach has been developed for designing material microstructures to meet

multiple performance/property requirements. The approach embodies robust design rather than

single objective optimization owing to the prevalence of uncertainty in process route, stochasticity

of microstructure, and nonequilibrium, path dependent nature of inelastic deformation and

associated constitutive models. Challenges for design of practical alloy systems and inelastic

deformation and damage mechanisms are outlined. We suggest a hierarchical materials design

framework in which mechanistic models are exercised to inform designers regarding decisions that

contribute to the goal of robust design. The approach is characterized by a confluence of

engineering science and mechanics, materials science/physics, 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 conducting feasibility studies to establish probable return on

investment of new material systems.

Figure 10. Mean response surface model fit to shock simulation results of mixture for number

of reaction sites in SVE as a function of volume fraction and mean radius of voids for a

prescribed size distribution of Al and Fe2O3 particles, along with upper and lower uncertainty

bounds on response [11]. This enables combined Types I-III robust design of the material

[15].


Acknowledgments

The authors acknowledge support from an AFOSR Multi-University Research Initiative

(1606U81) on Design of Multifunctional Energetic Structural Materials.

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