ARL-RP-0625 ● MAY 2018

US Army Research Laboratory

Mesoscale Models of Interface Mechanics in Crystalline Solids: A Review

by JD Clayton

Reprinted from J Mater Sci. 2018;53:5515–5545.

Approved for public release; distribution is unlimited.

NOTICES

Disclaimers The findings in this report are not to be construed as an official Department of the Army position unless so designated by other authorized documents.

Citation of manufacturer’s or trade names does not constitute an official endorsement or approval of the use thereof.

Destroy this report when it is no longer needed. Do not return it to the originator.

ARL-RP-0625 ● MAY 2018

US Army Research Laboratory

Mesoscale Models of Interface Mechanics in Crystalline Solids: A Review by JD Clayton Weapons and Material Research Directorate, ARL Reprinted from J Mater Sci. 2018;53:5515–5545. Approved for public release; distribution is unlimited.

ii

REPORT DOCUMENTATION PAGE Form Approved OMB No. 0704-0188

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing the burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS.

1. REPORT DATE (DD-MM-YYYY)

May 2018 2. REPORT TYPE

Reprint 3. DATES COVERED (From - To)

July 2016–January 2018 4. TITLE AND SUBTITLE

Mesoscale Models of Interface Mechanics in Crystalline Solids: A Review 5a. CONTRACT NUMBER

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S)

JD Clayton 5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

US Army Research Laboratory ATTN: RDRL-WMP-C Aberdeen Proving Ground, MD 21005

8. PERFORMING ORGANIZATION REPORT NUMBER

ARL-RP-0625

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

10. SPONSOR/MONITOR'S ACRONYM(S)

11. SPONSOR/MONITOR'S REPORT NUMBER(S)

12. DISTRIBUTION/AVAILABILITY STATEMENT

Approved for public release; distribution is unlimited.

13. SUPPLEMENTARY NOTES Reprinted from J Mater Sci. 2018;53:5515–5545.

14. ABSTRACT

Theoretical and computational methods for representing mechanical behaviors of crystalline materials in the vicinity of planar interfaces are examined and compared. Emphasis is on continuum-type resolutions of microstructures at the nanometer and micrometer levels, i.e., mesoscale models. Grain boundary interfaces are considered first, with classes of models encompassing sharp interface, continuum defect (i.e., dislocation and disclination), and diffuse interface types. Twin boundaries are reviewed next, considering sharp interface and diffuse interface (e.g., phase field) models as well as pseudo-slip crystal plasticity approaches to deformation twinning. Several classes of models for evolving failure interfaces, i.e., fracture surfaces, in single crystals and polycrystals are then critically summarized, including cohesive zone approaches, continuum damage theories, and diffuse interface models. Important characteristics of compared classes of models for a given physical behavior include complexity, generality/flexibility, and predictive capability versus number of free or calibrated parameters.

15. SUBJECT TERMS

interfaces, crystals, mechanics, grain boundaries, twin boundaries, fracture surfaces

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT

UU

18. NUMBER OF PAGES

36

19a. NAME OF RESPONSIBLE PERSON

JD Clayton a. REPORT

Unclassified b. ABSTRACT

Unclassified c. THIS PAGE

Unclassified 19b. TELEPHONE NUMBER (Include area code)

(410) 278-6146 Standard Form 298 (Rev. 8/98)

Prescribed by ANSI Std. Z39.18

INTERFACE BEHAVIOR

Mesoscale models of interface mechanics in crystalline

solids: a review

J. D. Clayton1,2,3,*

1Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY 10027, USA2 A. James Clark School of Engineering, University of Maryland, College Park, MD 20742, USA3 Impact Physics, US ARL, Aberdeen, MD 21005-5066, USA

Received: 14 June 2017

Accepted: 18 September 2017

Published online:

4 October 2017

� Springer Science+Business

Media, LLC 2017

ABSTRACT

Theoretical and computational methods for representing mechanical behaviors

of crystalline materials in the vicinity of planar interfaces are examined and

compared. Emphasis is on continuum-type resolutions of microstructures at the

nanometer and micrometer levels, i.e., mesoscale models. Grain boundary

interfaces are considered first, with classes of models encompassing sharp

interface, continuum defect (i.e., dislocation and disclination), and diffuse

interface types. Twin boundaries are reviewed next, considering sharp interface

and diffuse interface (e.g., phase field) models as well as pseudo-slip crystal

plasticity approaches to deformation twinning. Several classes of models for

evolving failure interfaces, i.e., fracture surfaces, in single crystals and poly-

crystals are then critically summarized, including cohesive zone approaches,

continuum damage theories, and diffuse interface models. Important charac-

teristics of compared classes of models for a given physical behavior include

complexity, generality/flexibility, and predictive capability versus number of

free or calibrated parameters.

Introduction

Interfaces may have a profound effect on the physical

responses of solid materials subjected to various

external stimuli or environmental conditions. This

article is focused on crystalline solid materials sub-

jected to mechanical loading [1–3]. In particular,

material classes considered herein may be single

crystals or polycrystals, comprised of grains of

metallic, ceramic, molecular, and/or mineralogical

origin. In order to maintain a reasonable scope, not

considered in this article are polymers and polymer

composites, though it is acknowledged that interfa-

cial mechanics is often crucial to the local and global

responses of these classes of materials as well [4, 5].

An interface is defined herein as a planar boundary

between two regions of a solid across which physical

Invited review article for Special Issue of Journal of Materials Science.

Address correspondence to E-mail: jdclayt1@umd.edu; john.d.clayton1.civ@mail.mil

https://doi.org/10.1007/s10853-017-1596-2

J Mater Sci (2018) 53:5515–5545

Interface Behavior

properties and/or response characteristics vary sub-

stantially. Local behaviors of the material in the

immediate vicinity of such an interface may also vary

from that of bulk regions of crystal far from the

boundary on either side. Examples of interface types

that may exist prior to deformation or mechanical

loading include grain [6] and subgrain boundaries,

twin boundaries, andphase boundaries. Those that are

often induced by loading include fracture surfaces [7],

stacking faults, and shear bands [8]. Grain and twin

boundaries may also be deformation induced, as in

cases of grain subdivision during plastic deformation

[9] or deformation twinning [10, 11].

This paper is focused on modeling techniques,

including theory and methods of solution, both ana-

lytical and numerical. Experiments are mentioned at

times in supporting discussion, e.g., in the context of

model validation, but this paper does not critically

examine experimental methods. The scope is limited

to continuum models that resolve microstructural

features, labeled here as mesoscale models. Length

scales resolved span approximately tens of nanome-

ters to millimeters, with micrometers the primary

scale of most representations discussed herein. Not

addressed are atomic scale theories [12, 13], or cou-

pled atomic-continuum models [14–17] which tend to

focus on somewhat finer scales of resolution. General

classes of models include sharp interface theories,

whereby jumps in properties and/or field variables

exist across internal boundaries, and diffuse interface

theories, wherein some smoothing or continuous

interpolation of properties and/or response functions

is invoked across an interface [18]. The overall scope

of this paper is thought to be unique in terms of the

ensemble of topics and their comparisons. An

attempt to sufficiently mention relevant works in

each area is made, but inclusion of even a modest

fraction of the vast number of historical and recent

articles on any broad subject is currently impossible

for a review paper of any reasonably finite length,

especially when one considers prolific publishing

practices in contemporary fundamental sciences.

Later in this work, for each kind of represented

physics—grain boundaries, twin boundaries, failure

boundaries—the corresponding classes of models are

compared with regard to flexibility, complexity, and

predictive capability. Flexibility, i.e., generality, refers

to the ability of a model to represent a breadth of

physical behaviors without adjustment of its funda-

mental governing equations. Complexity of a theory

is rather self-explanatory, with availability of ana-

lytical solutions and ease of numerical implementa-

tion both tending to decrease with increasing

complexity. The viewpoint adopted here is that a

given theory or model is considered more predictive

than another if it equally or better represents

observed physics with fewer calibrated parameters

[19–21].

This paper is organized as follows, noting that as is

the case with nearly any review, the present subject

matter deals with topics of the author’s own experi-

ence. The ‘‘Grain boundaries’’ section addresses grain

boundaries in polycrystals, including sharp interface,

continuum defect, and diffuse interface or phase field

representations. The ‘‘Twinning’’ section covers twin

boundaries, again considering sharp interface and

phase field models as well as pseudo-slip theories for

deformation twinning. The ‘‘Failure and localization’’

section describes fracture and localization models in

single crystals and polycrystals: cohesive zone

methods, continuum damage models, and regular-

ized failure models such as those of phase field type.

Fractures may be transgranular or intergranular, with

spall fractures potentially falling into either category.

The ‘‘Discussion’’ section summarizes major features

of the approaches including their strengths and

weaknesses according to the author’s viewpoint.

Although this review is biased more toward gen-

eral themes than intensive derivations, inclusion of

key mathematical relations for theories and methods

of solution is necessary for the intended rigorous

evaluation. Notation of continuum physics is invoked

in such instances, where vectors and tensors are

written in bold italic font, and scalars and scalar

components in italic font. Occasionally, the index

notation is used for clarity, with summation implied

over repeated indices. Further details regarding

notation will be clear from context.

Grain boundaries

Prior to discussion of specific classes of models in the

context of mechanical response, some universal fea-

tures and requisite notation for a grain boundary

(GB) are introduced. Useful references for further

study are [22, 23].

Superscripts in parentheses are used to denote

grain numbers in a polycrystal, where such a number

runs from 1 to the total number of crystals within the

5516 J Mater Sci (2018) 53:5515–5545

body. Denote by XðgÞ the volume occupied by grain g,

prior to deformation. Let X denote the position vector

of a material point in the body, again prior to defor-

mation induced by loading, and referred to a fixed

Cartesian reference frame. Let oXðgÞ denote the

external boundary of a crystal, which may include

free surfaces and/or grain boundaries (GBs). The

boundary between two grains with numbers g1 and

g2 is denoted by oXðg1;g2Þ, and so forth.

Bravais lattice vectors ai, where i ¼ 1; 2; 3, are

assigned to each single crystal and are assumed

uniform in each crystal volume, i.e.,

aiðXÞ ¼ aðgÞi 8X 2 XðgÞ: ð2:1Þ

By definition, the Bravais lattice is discontinuous

across (misoriented) grain boundaries. Let a0i denote

a particular set of the Bravais lattice vectors referred

to a given coordinate frame. Then, the orientation of

the crystal lattice in any grain is related to these

vectors via the orientation matrix GðgÞ:

aðgÞi ¼ GðgÞa0i : ð2:2Þ

The misorientation matrix for any two crystals with

grain numbers g1 and g2 is defined as

Mðg1;g2Þ ¼ ðGðg1ÞÞ�1Gðg2Þ: ð2:3Þ

Orientation matrices and misorientation matrices are

all proper orthogonal, i.e., each has equal transpose

and inverse, with a determinant of value þ1. Such

matrices can be represented in terms of an angle uand axis of rotation r of unit magnitude. For a generic

misorientation matrix M with entries MIJ

(I; J ¼ 1; 2; 3), the angle and components of the axis

are calculated as follows [23]:

cosu ¼ 1

2ðM11 þM22 þM33 � 1Þ; ð2:4Þ

r1 ¼M23 �M32; r2 ¼M31 �M13; r3 ¼M12 �M21:

ð2:5Þ

Often later in this article, the term misorientation will

be used to refer to the magnitude juj, where it is

understood that a complete description of the geo-

metric relation between orientations of the two lat-

tices of the same crystal structure requires the axis r

as well. Furthermore, it is noted that a complete

description of a planar interface requires five inde-

pendent scalar parameters; the misorientation matrix

or angle/axis pair each supplies only three. For

example, a complete characterization of a GB orien-

tation can be achieved via specification of the direc-

tion cosines of the boundary plane in the coordinate

systems of each neighboring grain along with the

angle of twist of both plane stacks normal to the

boundary plane [23].

The coincident site lattice (CSL) model of GBs

[6, 24] will be used in some later descriptions. For a

certain misorientation across a GB, or between two

interpenetrating lattices, some fraction of lattice sites

will coincide, forming a periodic sublattice. The CSL

parameter R is defined as the reciprocal density of

coinciding sites. Boundaries associated with low

values of R are of high interest since special physical

properties are often a result [23]. The use of the grain

boundary character distribution to design polycrys-

talline materials with optimum physical properties

has been suggested [25]. However, a low R value

does not always correlate with GB strength [26].

The maximum deviation from an exact CSL usually

is taken to correspond to the angular limit for a low-

angle boundary: juj ¼ 15�. Thus, a low-angle

boundary is described as R1. Annealing twin

boundaries are a subset of R3 boundaries. An

empirical relationship between maximum angular

deviation um of an arbitrary GB from any exact CSL

with value R is um ¼ u0R�1=2, where u0 is the afore-

mentioned maximum deviation, typically 15� [6, 23].

The jump in any field variable or material property

AðXÞ, which could be a scalar, vector, or higher-order

tensor, is defined as the difference between its values

at two corresponding locations Xð1Þ and Xð2Þ:

sAðXð1Þ;Xð2ÞÞt ¼ AðXð2ÞÞ � AðXð1ÞÞ: ð2:6Þ

The jump across a (sharp) grain boundary corre-

sponds to the difference in limiting values of A as X

approaches the shared point on boundary from either

side. In this work, attention is restricted to coherent

interfaces in the undeformed reference configuration,

meaning sXt ¼ 0. Later, in the ‘‘Failure and localiza-

tion’’ section, surfaces of discontinuity such as frac-

tured interfaces will be discussed, but these are

assumed to be induced by loading or deformation

from a coherent reference state.

Sharp interface models

In what is termed herein as a sharp interface model of

a GB, properties of crystals in the immediate vicinity

of the boundary are identical to those in the interior

J Mater Sci (2018) 53:5515–5545 5517

of each crystal, far from the interface. Therefore,

according to (2.3) and (2.6), the lattice vectors will

demonstrate a discrete jump across the GB for any

nontrivial misorientation matrix between the two

grains:

sait ¼ aðg2Þi � a

ðg1Þi ¼ ½Gðg2Þ � Gðg1Þ�a0i

¼ Gðg1Þ½Mðg1;g2Þ � 1�a0i on oXðg1;g2Þ:ð2:7Þ

The identity tensor is denoted by 1. The difference in

orientation of the lattice across the boundary can

induce various local physical responses when the

aggregate is subjected to far-field mechanical loading,

temperature change, and so forth. A geometric rep-

resentation of a sharp interface discretization of a

polycrystal is shown on the left side of Fig. 1, where

individual crystals are polyhedral shaped and GBs

are faceted (flat) planar surfaces. Such representa-

tions are characteristic of modern finite element

simulations of crystal elasticity [27, 28] and crystal

plasticity [26, 29].

This work will deal with potentially large strains

and rotations, as may occur during deformation of

ductile metals [3, 26], and even in ceramics and

minerals when loading is predominantly compres-

sive [31, 32]. Governing equations from finite elas-

ticity and plasticity of crystals are now reviewed to

lend context to the discussion of interfacial mechan-

ics. Only essential theoretical relations are provided,

and Cartesian coordinates are implied when index

notation is invoked. For a more comprehensive

treatment of finite crystal elastoplasticity that also

encompasses curvilinear coordinate representations,

see [3]. Two other useful references, albeit primarily

limited to finite anisotropic elasticity, are [33, 34].

Spatial coordinates of the deformed solid are rela-

ted to material coordinates by the time-dependent

motion

x ¼ xðX; tÞ ¼ X þ uðX; tÞ; ð2:8Þ

with u the displacement vector. The deformation

gradient is, with r0ð�Þ the referential gradient,

F ¼ r0x ¼ FEFP: ð2:9Þ

Thermoelastic and plastic deformations are FE and

FP, respectively. The plastic velocity gradient is the

sum of contributions of slip rates _ca, where the

superscript denotes a slip system for dislocation glide

with direction sa and plane normal ma:

LP ¼ _FPFP�1 ¼X

a

_casa �ma: ð2:10Þ

Here the slip direction and slip plane normal are

orthogonal and of unit length, i.e., are those of the

crystal lattice prior to thermoelastic deformation. The

thermoelastic strain used in standard crystal hyper-

elasticity [33] is the Green strain tensor:

E ¼ 1

2ðFEÞTFE � 1h i

; EIJ ¼1

2ðFEkIFEkJ � dIJÞ:

ð2:11Þ

The thermoelastic volume change is measured by

JE ¼ detFE. Cauchy stress r is related to elastic sec-

ond Piola–Kirchhoff stress S via

Figure 1 Geometric rendering of polycrystal with grain boundaries (GBs) represented as sharp interfaces: without secondary GB phase

(left); with secondary GB phase (right) [30].

5518 J Mater Sci (2018) 53:5515–5545

r ¼ 1

JEFESðFEÞT; rij ¼

1

JEFEiKSKLF

EjL: ð2:12Þ

The thermoelastic stress–strain relation is

SIJ ¼ CIJKLEKL þ1

2!CIJKLMNEKLEMN

þ 1

3!CIJKLMNPQEKLEMNEPQ þ � � � � vIJDT � � � � ;

ð2:13Þ

where CIJKL��� are isothermal elastic constants of sec-

ond and higher orders, DT is temperature change

measured from a reference state, and vIJ are thermal

stress coefficients related to elastic constants and

thermal expansion coefficients via vIJ ¼ CIJKLaKL. A

free energy function per unit volume in the ther-

moelastically unloaded configuration is

w ¼ wðE;T; fngÞ; ð2:14Þ

with fng a set of internal state variables that affect the

energy stored in the crystal, e.g., dislocation density.

The viscoplastic flow rule for slip rates is of the

general form

_ca ¼ _caðsa;T; fngÞ; sa ¼ JEr : ½FEsa � ðFEÞ�Tma�:ð2:15Þ

The resolved shear stress acting on a system is sa. Thelocal balance of energy, in the absence of point heat

sources, can be cast as the following temperature rate

equation:

c _T ¼X

a

sa _ca � owofng � T

o2woTofng

� �f _ng � Tv : _E

þ �r � ðK �rTÞ ¼ bX

a

sa _ca � Tv : _Eþ �r � ðK �rTÞ;

ð2:16Þ

where c is the specific heat at constant thermoelastic

strain, the rightmost term accounts for heat conduc-

tion (see e.g., [3, 21] for details), and b is the fraction

of plastic work converted to heat energy, i.e., the

fraction of stored energy of cold work is 1� b.Let C denote any of the tensors of elastic moduli in

(2.13). These elastic constants, which are the usual

second-order type as well as third- and fourth-order

constants important in high-pressure applications

[31, 33, 35, 36], depend on the orientation of the

crystal lattice in the reference configuration, as do the

slip directors and slip plane normal vectors:

CðXÞ¼C½aiðXÞ�¼C½GðXÞ�; saðXÞ¼sa½aiðXÞ�¼sa½GðXÞ�;maðXÞ¼ma½aiðXÞ�¼ma½GðXÞ�:

ð2:17Þ

Similarly, thermal stress coefficients v and thermal

conductivity K also depend on crystal orientation in

the unloaded state for Lagrangian thermoelasticity.

Functional forms of material coefficients for various

crystal classes are available in [3, 33, 34].

In the present context of fully coherent GBs, and in

the absence of any transgranular separation modes,

continuity of displacement and traction in grain

interiors and/or along GBs is stated mathematically

as

suðX; tÞt ¼ 0 , sxðX; tÞt ¼ 0;

ð8X 2 XðgÞ and 8X on oXðg1;g2ÞÞ;ð2:18Þ

stt ¼ srtn ¼ 0; ð8Xðx; tÞon oXðg1;g2ÞÞ: ð2:19Þ

In the traction continuity equation (2.19), n is normal

to the surface, and it is assumed that shock waves

(i.e., stress jumps in solid dynamics) are absent.

Consider a general boundary value problem for a

polycrystal, where traction and/or displacement are

imposed along the far-field (external) boundary. The

continuity equations in (2.18) and (2.19) impose

restrictions on the solution strain and stress fields E

and r. In the present sharp interface representation,

the jump conditions on the lattice vectors in (2.7) lead

to jumps in elastic constants and slip director/normal

vectors in (2.17). It is these potentially drastic changes

in local properties, taken in combination with the

continuity constraints for coherent sharp GBs, that

may give rise to concentrated stress, strain, and/or

slip activity in the vicinity of GBs in solutions to

general polycrystal boundary value problems [37].

The propensity for statistically more pronounced

stress concentrations at GBs and triple point junctions

(i.e., where three GBs intersect) has been predicted in

finite deformation crystal plasticity simulations

[29, 38].

Lattice mismatch does not necessarily preclude

development of continuous bands, across GBs, of

homogeneous or localized field variables such as

plastic strain, temperature, and/or dislocation den-

sity (when modeled as an internal variable depend-

ing on cumulative slip), as demonstrated in early

[39, 40] and more recent [41] finite element simula-

tions of shear banding in polycrystals. Preferential

lattice orientations may promote localized slip bands

J Mater Sci (2018) 53:5515–5545 5519

in isothermal simulations [39, 40], whereas the com-

bination of temperature rise from (2.16) in conjunc-

tion with increased slip activity due to thermal

activation if included in (2.15) will promote adiabatic

shear banding [8]. As is evident from Fig. 2, correla-

tion between dissipation fraction b, dislocation den-

sity n, and local strain and temperature

concentrations has been predicted in mesoscale

crystal plasticity simulations [41].

The earliest finite element simulations of finite

(poly)crystal plasticity appeared in the early to mid-

1980s [42]. Thereafter, transmission and/or blockage

of slip at special (CSL R9 and R17b) high-angle

boundaries in copper have been studied via compu-

tational crystal plasticity and sharp interface repre-

sentations of GBs [43]; reported results are in general

qualitative agreement with experimental trends

relating dislocation pile ups and suggested pending

fracture mechanisms.

Although typically less prone than most metals to

plastic deformation from dislocation glide, ceramics

and minerals also demonstrate concentrated ther-

moelastic strain and stress concentrations due to

lattice mismatch at GBs. A relatively early numerical

study of stress concentrations in polycrystals with

sharp interface GB representations is [44], which

considers elastic anisotropy and anisotropic thermal

expansion coefficients and concludes that anisotropy

Figure 2 Predicted strain localization in Al polycrystal at applied tensile strain of 8%: plastic strain, temperature, heat dissipation fraction

b, and dislocation density [41].

Figure 3 Tilt boundary (left) comprised of edge dislocations

(center) or partial disclination dipoles (right) [3, 53].

5520 J Mater Sci (2018) 53:5515–5545

of either property may affect statistical distributions

of tractions at GBs. This study will also be referred to

later in the ‘‘Failure and Localization’’ section in the

context of failure models, since it includes results

incorporating cohesive fracture elements at GBs for

some simulations. An important physical feature of

many industrial ceramics is the presence of sec-

ondary, often glassy, phases of impurities at GBs

[30, 45]. In a sharp interface model, such phases can

be introduced via a thin layer of secondary material

between crystals as shown on the right side of Fig. 1.

In summary, sharp interface representations of GBs

allow for explicit modeling of strain and stress con-

centrations of field variables near mismatched inter-

faces. No additional constitutive model parameters

are required beyond those invoked for the response

of the bulk material in grain interiors. The approach

is considered physically realistic for modeling con-

tinuum thermoelasticity since thermoelastic proper-

ties and hardness tend to vary little from points near

the GB to points far in the grain interior [46]. In

contrast, properties associated with dissipative phe-

nomena, including dislocation nucleation and motion

in ductile crystals, may vary strongly in the vicinity

of GBs, as will be discussed more in the ‘‘Continuum

dislocation and disclination models’’ section. Fur-

thermore, since no intrinsic length scale exists in

classical crystal elasticity and plasticity theories, the

sharp interface models that invoke classical bulk

constitutive laws are unable to predict size effects

such as Hall–Petch-type behavior (i.e., strength

increases as grain size decreases).

The preceding treatment is focused on stationary

grain boundaries. Depending on thermomechanical

loading conditions, the mobility of GBs becomes

important, for example under conditions pertinent to

grain growth, recrystallization, or some kinds of

phase transformations. Sharp interface models of

surfaces of strain discontinuity and phase bound-

aries—including geometric, kinematic, thermody-

namic, and kinetic aspects—were advanced in the

late twentieth century in several seminal works

[47–49].

Continuum dislocation and disclinationmodels

Grain boundaries, particularly those of low angle

type (i.e., R1), often contain high densities of dislo-

cations depending on the material and processing

history. As shown on the left side of Fig. 3, a tilt

boundary with misorientation magnitude u can be

represented in terms of a density of edge dislocations

of like sign and Burgers vector magnitude b, with

spacing h given by [50]

h ¼ b=u; ðu.u0 ¼ p=12Þ: ð2:20Þ

To a certain extent, the larger the misorientation, the

more closely spaced the dislocations, leading to an

overall higher density (line number per unit area).

However, the maximum misorientation is limited to

around 15� in this model since high-angle grain

boundaries are not well represented physically by

large dislocation densities. The low-angle boundaries

represented in such cases may be present in the

undeformed (poly)crystal, or may be induced by

severe plastic deformation [9].

Yield behavior has been observed in indentation

experiments to vary with distance from GBs in met-

als, with such dependency varying with orientation

mismatch [46]. Boundaries may act as sources or

sinks for dislocation generation and interactions.

Furthermore, slip transmission or blockage may

ensue depending on alignment of slip systems across

a GB [43]. Early continuum models accounting for

different plastic properties near GBs versus grain

interiors treated each crystal as a composite material

with a boundary layer of properties differing from

those of the bulk [37, 51]. More recent computational

models have incorporated rules for GB slip interac-

tions via specialized interfacial finite elements along

grain or subgrain boundaries [26, 52].

The GB dislocations concentrated at interfaces

between severely deformed crystals, which are

required to maintain compatibility of the total

deformation gradient field, account for the incom-

patibility (i.e., failure of integrability or anholonomic

character [54–57]) of the thermoelastic and plastic

deformations individually in (2.9). The spatial den-

sity tensor of geometrically necessary dislocations

(GNDs) is, in coordinate free tensor notation [3, 57]

�qG ¼ � �C : e ¼ � �T : e ¼ �fFE½rðFE�1Þ�g : e; ð2:21Þ

where e is the permutation tensor, �C is the integrable

crystal connection whose curvature tensor vanishes

identically, and �T is the torsion tensor of this con-

nection. The spatial gradient operator is rð�Þ. Sincethe GND tensor in (2.21) does not account for distri-

butions of dislocations whose net Burgers vector

vanishes, a scalar density of statistically stored

J Mater Sci (2018) 53:5515–5545 5521

dislocations (SSDs), qS, is also introduced. Let qG be

the frame invariant GND tensor mapped to the

elastically unloaded intermediate configuration

[3, 57–59], a tensor which can equivalently be con-

structed from the transformation of (2.21) or the

material gradient of FP. Then, the state variable list

for the crystal plasticity constitutive model entering

the free energy and/or flow rule in (2.14) and (2.15)

is, for the present class of models,

fng ¼ fqG; qSg: ð2:22Þ

The flow rule is supplemented with a kinetic equa-

tion for the rate of change of qS, which generally

tends to increase with cumulative plastic slip. Model

parameters are needed to scale the magnitude of the

contribution of GNDs to stored energy and harden-

ing behaviors. Similarly, parameters are required to

quantify contributions of SSDs to stored energy and

slip resistance. A common practice involves use of an

effective shear modulus and scalar Burgers vector to

provide the correct units for normalization. It is

understood that (2.22) must be extended if slip sys-

tems harden unequally (i.e., different self- and latent

hardening), in which case dislocation densities rele-

vant to each system may become distinct state vari-

ables [52].

Inclusion of the GND density tensor in the consti-

tutive model provides a regularizing effect on

numerical solutions due to the presence of the gra-

dient term [i.e., rðFE�1Þ] in (2.21). Size effects can be

predicted using this class of models if properly cali-

brated; for example, such models may depict

increased strength under indentation with a decrease

in indenter size, or Hall–Petch-type behavior (essen-

tially, smaller ¼ stronger) [26]. In simulations of

polycrystals, GNDs tend to build up in the vicinity of

grain boundaries, thereby quantifying intergranular

incompatibility and physically representing pile ups

and increased hardness in such regions [26, 52].

Boundary conditions supplementing those of classi-

cal crystal plasticity are required for solution of

boundary value problems, e.g., the thermoelastic or

plastic deformation gradient may need to be specified

over some part of the external boundary of the

domain. A recent novel scheme for numerical repre-

sentation of grain boundaries in polycrystals via

smoothed jumps of local lattice rotation in the refer-

ence state is described in [60].

A more sophisticated class of elastic–plastic models

intended to capture physics of initial and/or evolving

boundaries of lattice misorientation in crystals sup-

plements the dislocation (GND and SSD) description

with disclinations. Dislocations are fundamental

translational defects, while disclinations are funda-

mental rotational defects, so the latter perhaps more

naturally should be used to represent rotations of the

lattice across boundaries of misorientation. As shown

on the right side of Fig. 3, the mean spacing l of

defects in a tilt boundary represented by a density of

parallel partial wedge disclination dipoles of strength

x and dipole radius r is

l ¼ 2rx=u: ð2:23Þ

The magnitude of misorientation is u, not necessarilyrestricted to small angles. This disclination descrip-

tion credited to Li [61] is an alternative to the Read–

Shockley model of (2.20). With r and x fundamental

constant properties of the crystal lattice associated

with a particular material (i.e., rotational analogs of

b), (2.23) suggests a generally increasing density of

disclinations per unit area with increasing misorien-

tation in the vicinity of grain boundaries.

In the context of finite deformation continuum field

theories of elastic–plastic crystal mechanics, a spatial

tensor of disclination density �h is defined as [3, 53, 62]

�h ¼ � 1

4ðR : eÞ : e; R ¼ RðC;rCÞ; C ¼ �Cþ Q:

ð2:24Þ

In differential-geometric language [56, 57, 63], R is the

fourth-order curvature tensor that depends on the

connection coefficients of C and spatial gradients of

these coefficients. The latter total connection C con-

sists of the sum of the crystal connection defined in

(2.21) and the micro-rotation variable Q, which

physically represents lattice rotation gradients [62].

When Q vanishes, the curvature tensor and the

disclination density tensor also vanish. The spatial

GND density is constructed as in (2.21), but with �Creplaced with C. The list of state variables entering

the free energy function and flow rule is extended to

include disclination density mapped to the interme-

diate configuration h, i.e.,

fng ¼ fqG; qS; hg: ð2:25Þ

The list may be further augmented to include a scalar

density of statistically stored disclinations with

5522 J Mater Sci (2018) 53:5515–5545

vanishing average Frank vector [3, 53], but such a

description is not essential in the present context.

Continuum dislocation–disclination field theory

has been used to describe a hierarchy of cellular

microstructures in ductile metallic crystals under

severe plastic deformation [9, 53]. In this context,

low-angle misorientations across cell walls at a rela-

tively small length scale, termed incidental disloca-

tion boundaries, are resolved by GNDs. Disclinations

resolve potentially higher angle misorientations

across larger cell blocks separated by geometrically

necessary boundaries. The disclination concept

thereby introduces another length scale for regular-

ization into the theory and in conjunction with GNDs

enables a natural description of microstructure evo-

lution at different scales.

The theory of [3, 53], thought to be the first ther-

modynamically complete constitutive framework for

continuum dislocation–disclination mechanics at

finite strain, includes a set of microscopic equilibrium

equations, that in combination with kinematic rela-

tions between lattice spin and disclination density,

enable prediction of the evolution of Q in an incre-

mental boundary value problem. Numerical solu-

tions of this formidable set of equations, which

inevitably include entities from Riemannian and non-

Riemannian geometry, remain elusive to date due to

complexity. Furthermore, parameters are required

that specify how free energy and slip resistance may

depend on disclination density. Linear [3, 61] and

nonlinear [64] elastic solutions for discrete defects

offer insight into energies contributed by disclina-

tions in certain arrangements, but unique assignment

of parameters for hardening due to GNDs, SSDs, and

disclinations remains problematic and highly mate-

rial dependent.

In contrast, more recent work has obtained

numerical solutions for dislocation–disclination field

theory applied to grain boundaries and triple junc-

tions in metals [65], albeit with the constitutive theory

restricted to the geometrically linear regime. A finite

deformation description of the kinematics of GBs in

relatively brittle minerals (i.e., low dislocation

mobility) invoking a disclination density tensor has

also been exercised [66].

A final concept is of importance when describing

regions of very large defect density using continuum

mechanical concepts. In such cases, the two term

multiplicative decomposition of (2.9) may be

insufficient when FP is attributed to dislocation slip

processes alone as implied in (2.10). An intermediate

term, denoted here by FI , augments the classical

crystal plasticity decomposition, accounting for

residual lattice deformation due to defects within the

local volume element of crystalline material at X to

which F is ascribed:

F ¼ r0x ¼ FEFIFP: ð2:26Þ

The particular form of FI depends on the class of

defect, defect arrangement, and scale of resolution. A

tensor equation for FI has been derived via homog-

enization (i.e., volume averaging) methods for poly-

crystals [29] and single crystals containing subgrain

boundaries [59]. Elsewhere it has been calculated via

consideration of the linear elastic fields of periodic

arrays of edge dislocations [67, 68]. Solutions also

exist for a volume element containing a single edge or

screw dislocation [69, 70]. Representations in terms of

nonlinear elasticity and anharmonic molecular statics

have also been derived [3, 71], the former invoking

third order elasticity of cubic crystals [72].

Analytical calculations have demonstrated the

importance of inclusion of FI in the constitutive

description when dislocation densities approach the

theoretical maximum, an occurrence possible in

regions of crystal near boundaries induced during

severe plastic deformation or shock loading [70, 73].

In particular, since FP is isochoric when attributed

solely to slip, any residual volume changes in the

crystal are omitted if not captured by FI . Further-

more, lattice rotations induced by disclinations may

be described by rotational part of FI [53], and volume

changes associated with point defects may be inclu-

ded in its determinant [62, 74]. If included in a con-

stitutive model framework, an evolution equation or

one of the aforementioned subscale solutions for FI

must be invoked. Though not essential, the residual

lattice deformation may be included in the list of state

variables fng, in which case additional model

parameters may be needed to relate it to stored

energy and strain hardening kinetics, for example.

In summary, the classical sharp interface descrip-

tion invoked in crystal plasticity simulations, as

presented in the ‘‘Sharp interface models’’ section,

can be augmented to explicitly account for defect

densities in the vicinity of GBs and subgrain bound-

aries. Potential benefits of the model classes covered

here in the ‘‘Continuum dislocation and disclination

J Mater Sci (2018) 53:5515–5545 5523

models’’ section include physical flexibility, where

size effects and microstructure evolution at different

resolutions can be incorporated somewhat naturally.

Regularization associated with higher-order gradient

terms in the free energy and balance or kinetic laws

may facilitate mesh-size independence of numerical

solutions, an issue which is of particular importance

for modeling localization phenomena. A drawback is

that additional parameters must be prescribed and

often calibrated rather than determined from first

principles, especially those relating defect densities to

slip kinetics. Enhanced kinematics and stored energy,

on the other hand, may be reasonably incorporated

without calibrated parameters, via consideration of

mathematical physics of defect densities (i.e., differ-

ential-geometric relations) and (non)linear elasticity

solutions for individual non-interacting defects or

those in idealized yet still sufficiently realistic

arrangements. Regardless of the source of parame-

ters, increased model sophistication results in an

increase in computational expense and enables fewer,

if any, available analytical solutions to boundary

value problems for validation of computational

results.

Diffuse interface models

In the sharp interface classes of models in the ‘‘Sharp

interface models’’ and ‘‘Continuum dislocation and

disclination models’’ sections, lattice orientations

demonstrate jump discontinuities across GBs, as do

material properties such as elastic constants and

thermal expansion coefficients that depend on the

reference orientation of the crystal in Lagrangian

constitutive models of anisotropic media. Even

though regions of finite volume in the vicinity of

boundary interfaces may contain distributions of

defects represented by density tensors that may tend

to smooth the mechanical response over small but

finite distances from GBs, the corresponding models

discussed in the ‘‘Continuum dislocation and discli-

nation models’’ section still treat GB interfaces as

discrete/sharp surfaces with regard to properties

such as elastic coefficients.

Diffuse interface models, perhaps most notably

those termed phase field models, treat GBs as regions

of finite volume over which referential properties

vary continuously with distance from the interior of

one crystal to the interior of its neighbor. Denote a

scalar order parameter associated with a given grain

boundary shared by grains g1 and g2 by g 2 ½0; 1�,such that

gðX; tÞ ¼ 0 8X 2 Xðg1Þ; gðX; tÞ ¼ 1 8X 2 Xðg2Þ;

gðX; tÞ 2 ð0; 1Þ 8X 2 oXðg1;g2Þ;

ð2:27Þ

where now boundary zone oXðg1;g2Þ is of finite vol-

ume. Time is denoted by t. Then, a generic Lagran-

gian property A, which could be a tensor, vector, or

scalar, is interpolated in GB regions from its constant

values Aðg1Þ and Aðg2Þ initially assigned to regions

deep within neighboring crystals as

AðX; tÞ ¼ Aðg1Þ þ /½gðX; tÞ�sAt ¼ Aðg1Þ

þ /½gðX; tÞ�ðAðg2Þ � Aðg1ÞÞ;ð2:28Þ

where / is an interpolation function minimally sat-

isfying the end conditions /ð0Þ ¼ 0 and /ð1Þ ¼ 1.

As discussed in the monograph [18] for example,

an immense literature exists on diffuse interface

models used to describe microstructure generation

and evolution under various thermal and chemical

processes: solidification, grain growth, recrystalliza-

tion, mass transport, diffusion, etc. The scope of the

present discussion in the ‘‘Diffuse interface models’’

section is hereafter limited to classes of models that

treat GBs as diffuse but also address the mechanical

response, specifically stress–strain behavior in crys-

tals, which may be elastic or elastic–plastic.

A relatively recent example of a coupled descrip-

tion of microstructure and mechanics via diffuse

grain boundaries is reported in [75], which is focused

on the electromechanical response of polycrystalline

ferroelectrics. The phase field approach to lattice

orientation assignment and GB representation fol-

lows that of [76], with essential equations of the

method outlined in what follows next. A single order

parameter suffices for description of one GB, e.g., that

in a bicrystal. Diffuse interface models of GBs in

polycrystals with many (n) grains require multiple

order parameters, labeled gðiÞ, where i ¼ 1; 2; . . .; n.

Following [75, 76], regions deep within each grain are

characterized by values

gðiÞðX; tÞ ¼ �1; gðjÞðX; tÞ ¼ 0 8X 2 XðiÞ; ði 6¼ jÞ:ð2:29Þ

For any GB region, for at least one value of i,

5524 J Mater Sci (2018) 53:5515–5545

jgðiÞj 2 ð0; 1Þ: ð2:30Þ

In the absence of mechanical (or thermal, electrical,

etc.) loading, a free energy density per unit reference

volume is prescribed as

fðgðiÞ;r0gðiÞÞ ¼ f0ðgðiÞÞ þ

1

2

X

i

jijr0gðiÞj2; ð2:31Þ

whereji arematerial constants.The smaller thevalue(s)

of ji, the thinner the equilibrium width of a GB inter-

facial zone. The particular function f0 used in [75] is

f0 ¼X

i

� a2ðgðiÞÞ2 þ b

4ðgðiÞÞ4

� �þ c

X

i

X

j 6¼i

ðgðiÞÞ2ðgðjÞÞ2;

ð2:32Þ

with a, b, and c material properties. The function f0

contains 2n wells/minima at ðgð1Þ; gð2Þ; . . .; gðnÞÞ¼ ð1; 0; . . .; 0Þ, ð�1; 0; . . .; 0Þ, ð0; 1; . . .; 0Þ; . . .. Function f0is of a local minimum value within a grain, and

r0gðiÞ ¼ 0 8iwithin a grain, i.e., far from an interface.

The free energy functional for the unloaded body is the

integral over the entire polycrystalline volume X:

F ¼Z

XfdX: ð2:33Þ

Evolution of order parameters follows the Allen and

Cahn [77] formalism, also known as the time-de-

pendent Ginzburg–Landau (TDGL) equation, which

drives the total free energy F of the system to a

minimum. Letting l denote the mobility of the pro-

cess (i.e., a parameter controlling the timescale of GB

kinetics), the local rate equation for each order

parameter in this approach is

_gðiÞ ¼ �ldf

dgðiÞ¼ �l

of0ogðiÞ

� r0 �of

or0gðiÞ

� �

¼ �l agðiÞ � b½gðiÞ�3 � 2cgðiÞX

j6¼i

½gðjÞ�2 þ jir20g

ðiÞ

0@

1A:

ð2:34Þ

The following scalar function that has a value of unity

in each grain and a magnitude less than unity within

each GB region is introduced:

f½gðiÞðX; tÞ� ¼X

i

½gðiÞ�2 2 ð0; 1�: ð2:35Þ

An interpolation function for properties used in [75]

to define the fracture surface energy as GGB ¼ /GC of

GBs is f itself:

/½fðgðiÞðX; tÞÞ� ¼ fðgðiÞðX; tÞÞ 2 ð0; 1�: ð2:36Þ

This approach assigns the critical energy release rate

property to a point X in a GB, denoted by GGB, as

some positive fraction of the constant critical energy

release rate GC assigned to all bulk crystals. Fur-

thermore, in 2D simulations, an orientation angle H is

defined at any point X via interpolation as [75]

HðX; tÞ ¼ 1

fðX; tÞX

i

HðiÞ½gðiÞðX; tÞ�2: ð2:37Þ

Orientation-dependent Lagrangian properties such

as anisotropic elastic constants are then assigned to

GB regions based on the local value of H, where HðiÞ

is the uniform angular orientation of grain i. Notice

that according to this model, lattice orientation in a

given GB region potentially depends on orientations

of (many other) grains not in contact with that

boundary. The diffuse interface framework described

in (2.29)–(2.37) is invoked in [75] to create a poly-

crystalline microstructure and assign orientations

and fracture strengths to all points within the

domain, i.e., GB regions and bulk crystalline regions.

The TDGL equation is solved over a finite domain in

time, beginning with randomly seeded admissible

values of gðiÞ at points X 2 X used as initial condi-

tions, from which grain growth take place. Time

integration of (2.34) must be halted when a realistic

microstructure (e.g., a realistic average grain size) is

obtained; otherwise, a uniform single crystal would

ultimately produce the minimum value of total sys-

tem energy F. The polycrystal is then subjected to

electromechanical loading, with the reference con-

figuration (i.e., microstructure) held fixed in

simulations.

Another example of diffuse interface modeling of

GBs in deformable polycrystals is reported in [78, 79].

Crystal elastic–plastic theory is used to represent the

deformation behavior of aluminum grains in the

aggregate. Accumulated dislocations and associated

stored energy then supply driving forces for GB

motion, e.g., grain growth during recrystallization. A

staggered numerical scheme is implemented to solve

the governing equations of crystal plasticity and

phase field kinetics, where the solution of one set of

physical laws influences that of the other set in suc-

cessive iterations.

Advantages of the diffuse interface models of GBs

include the following. Microstructure evolution, e.g.,

J Mater Sci (2018) 53:5515–5545 5525

GB motion, can be addressed in a more detailed and

realistic way than in phenomenological models with

sharp interfaces. In particular, kinetics of

microstructure evolution are motivated by the fun-

damental principle that a system should seek a con-

figuration for which its total free energy is a

minimum. Regularization associated with gradient

terms in the energy functional introduces length scale

effects and facilitates mesh-size independence of

numerical solutions. Disadvantages of the diffuse

interface models include requisite prescription of

non-unique interpolation functions for property val-

ues within interfacial zones such as / of (2.36), as

well as parameters in the energy functional and

kinetic law such as (ji; a; b; c; l) in the framework of

[75]. In computer simulations, mesh resolution must

be fine enough to resolve field variables and their

gradients within the often very narrow GB zones.

Simultaneous solution of the coupled governing

equations for the microstructure-mechanics problem

may be challenging, especially if vastly different

timescales for GB evolution and stress dynamics (e.g.,

wave propagation) arise.

Twinning

Twins are microstructure features observed in many

kinds of crystals. Twinning may be caused by

mechanical forces, in which case it is termed defor-

mation twinning or mechanical twinning. Twins may

also be induced by other physical stimuli, a

notable example being annealing twins produced via

thermal processing of a material. The present section

focuses mostly on deformation twins, particularly

model descriptions of pseudo-slip and phase field

type in ‘‘Continuum pseudo-slip models’’ and ‘‘Dif-

fuse interface models’’ sections, respectively. Many

aspects of sharp interface models discussed in the

‘‘Sharp interface models’’ section may apply to nearly

any kind of twin, regardless of its origin. Prior to

presentation and evaluation of the aforementioned

classes of twinning models, a few fundamental con-

cepts are reviewed. More complete treatments of

twinning in the context of elastic–plastic continuum

mechanics include [3, 10, 11].

Twinning is a general term that may be used to

describe energy invariant transformations of a crystal

structure with certain characteristics. A twin in a

crystalline solid is usually defined as two regions of a

crystal separated by a coherent planar interface called

a twin boundary. As will be described mathemati-

cally in the ‘‘Sharp interface models’’ section, limiting

values of deformation gradients in each region, on

either side of the twin boundary interface, differ by a

simple shear. Unstressed twinned regions of the

crystal far from boundaries or defects possess the

same strain energy density as the unstressed parent

(i.e., the same energy density as the original crystal

prior to twinning), such that twinning shears are said

to be energy invariant [80, 81].

In the context of ductile solids, deformation twin-

ning is most often associated with thermodynami-

cally irreversible shape deformation in

correspondence with collective motion of partial

dislocations and formation of stacking faults [11, 82].

Deformation twinning is preferred over slip in cases

wherein resistances to dislocation glide are very large

in certain directions, often in crystal systems of low,

e.g., non-cubic, symmetry. In addition to their emer-

gence in ductile metals, deformation twins may also

appear in ceramics, minerals, and molecular crystals

[83], though complex low-symmetry crystal struc-

tures do not ensure their occurrence [84]. Twinning is

often preferable to slip at lower temperatures or at

very high strain rates, though exceptions are not

unusual, depending on material. Mechanical work

done during deformation twinning is dissipative

when resulting from defect motion associated with

shearing. Any stored energy is associated only with

defects left behind in the crystal, for example those

comprising the twin boundary. From the standpoint

of continuum thermodynamics, the driving force for

twin propagation is the resolved shear stress on the

habit plane in the direction of twinning shear, as will

be exploited in the context of pseudo-slip models in

the ‘‘Continuum pseudo-slip models’’ section.

Sharp interface models

A sharp interface model of a twin boundary (TB) is

similar to a sharp interface model of GB. In the latter,

as discussed in the ‘‘Sharp interface models’’ section,

a planar interface oXðg1;g2Þ separates two distinct

crystals g1 and g2 with different referential lattice

orientations, and Lagrangian material properties are

discontinuous across the GB. In the former (TB), a

planar interface oXðp;t1Þ separates the original crystal

(parent) p and the twinned crystal t1, or more

5526 J Mater Sci (2018) 53:5515–5545

generally, two different twins if p is replaced with t2.

Lattice orientations demonstrate a jump discontinuity

across the TB, as do associated anisotropic Lagran-

gian properties such as elastic moduli. However, the

misorientation across a TB is restricted by the crystal

structure and type of twin, while that across a GB is

relatively unrestricted. Furthermore, a simple shear-

ing process describes the transformation from the

original lattice to the twinned lattice, whereas no

such process generally exists for an arbitrary GB.

Unlike general GBs that have a finite radius of cur-

vature, fully formed TBs tend to be flat, though

exceptions are common for growing or receding

twins or those induced by concentrated forces [10]

such as those encountered in (nano)indentation.

The present focus is restricted to coherent TBs, for

which continuity of displacement and traction hold

analogously to (2.18) and (2.19):

suðX; tÞt ¼ 0 , sxðX; tÞt ¼ 0;

ð8X 2 XðpÞ;Xðt1Þand 8X on oXðp;t1ÞÞ;ð3:1Þ

stt ¼ srtn ¼ 0; ð8Xðx; tÞ on oXðp;t1ÞÞ: ð3:2Þ

Kinematics of twinning can be described, in part,

by invoking geometrically nonlinear elasticity theory.

Let two regions of the crystal, which are labeled as

parent p and twin t1, be separated by a surface oXðp;t1Þ

across which displacements of the material are con-

tinuous as in (3.1), but across which gradients of

displacement are not. This surface of composition

[80], which need not be planar, corresponds to the

habit plane in the traditional description of mechan-

ical twinning [82]. Let

FðpÞ ¼ r0xðpÞ; Fðt1Þ ¼ r0x

ðt1Þ ð3:3Þ

denote constant limiting values of deformation gra-

dient FðX; tÞ in each region in the vicinity of the TB,

where X are reference coordinates of the original

crystal prior to twinning. Since volumes and masses

remain positive, detFðpÞ [ 0 and detFðt1Þ [ 0. Let m0

be a unit normal vector to oXðp;t1Þ, pointing from

parent side to the twinned side. The compatibility

requirement that the interface be coherent (i.e., con-

tinuous coordinates x along the surface of composi-

tion or TB) necessitates that Hadamard’s jump

conditions apply [1, 80, 85]:

sFt ¼ Fðt1Þ � FðpÞ ¼ a�m ¼ c0s�m0; ð3:4Þ

where c0 [ 0 is a scalar magnitude of the twinning

deformation (eigen-shear) and s is a spatial unit

vector. Let WðFÞ denote the strain energy density per

unit reference volume of the crystal. Energy invari-

ance of twinning demands that

W ½Fðt1ÞðXÞ� ¼ W ½Q0FðpÞðXÞH�; ð3:5Þ

where Q0 is a proper orthogonal tensor

(Q�10 ¼ QT

0 ; detQ0 ¼ 1) and H is an energy invariant

transformation of the crystal, not necessarily orthog-

onal, that depends on the material’s intrinsic struc-

ture/symmetry. In a single global Cartesian

coordinate system, assuming that H does not induce

volume changes since the converse assertion would

permit the total strain energy of a sample of fixed

mass to remain constant as the volume of the sample

is increased without bound, and noting that

detH ¼ 1, at the same limiting point X 2 oXðp;t1Þ,

detFðt1Þ ¼ detFðpÞ ¼ det½FðpÞ þ a�m0�¼ detFðpÞ½1þ c0s0 �m0�:

ð3:6Þ

It follows that the pullback of s, i.e., s0 ¼ ½FðpÞ��1s,

must be orthogonal to unit normal m0:

s0 �m0 ¼ 0: ð3:7Þ

When the parent is taken as a perfect reference lattice,

then

FðpÞ ¼ 1 ) Fðt1Þ ¼ 1þ c0s0 �m0; ð3:8Þ

demonstrating that Fðt1Þ is indeed a simple shear. In

that case, the equivalent product Q0H is also a simple

shear, possibly of large magnitude, that shifts the

perfect crystal to another minimum energy configu-

ration, with the strain energy density of this config-

uration equivalent to that of the parent. In this

context, the strain energy function W can be inter-

preted as a multi-well potential, with global minima

corresponding to conditions Wð1Þ ¼ WðQ0HÞ ¼ 0, a

characteristic feature that will be exploited later in the

‘‘Diffuse interface models’’ section in the context of

diffuse interface theories of twinning. Notice that the

above description does not account for any (surface)

energy associated with defects along the boundary of

the twin, which can be reflected in continuum theo-

ries via augmentation of the free energy function with

internal state variables, as will be demonstrated in the

‘‘Continuum pseudo-slip models’’ section.

J Mater Sci (2018) 53:5515–5545 5527

The preceding treatment addresses kinematics and

strain energy density for sharp interface model rep-

resentations of twin boundaries. Elements of such a

treatment can be invoked to analyze and predict

occurrence of various preferred microstructures. For

example, laminated twin arrangements (i.e., her-

ringbone patterns) and other characteristic features of

martensite [1, 85] can be predicted from considera-

tion of compatibility constraints and free energy

minimization for certain crystal structures. No

material parameters generally need to be calibrated

in such analyses, which rely on fundamental prop-

erties such as symmetry operations—and transfor-

mation strains if solid–solid phase changes are

involved—associated with crystal structure.

The above treatment does not enable explicit pre-

diction of time-dependent motion of twin bound-

aries, e.g., twin growth and dynamic interactions

with other twins or other crystals in a polycrystal. For

such predictions, a kinetic law for twin boundary

dynamics must supplement the kinematic descrip-

tion, and a (numerical) scheme must be invoked to

track the position of interface(s) as deformation pro-

ceeds in time. Derivation of the corresponding

equations is beyond the present scope, but one such

example of this class of sharp interface model for

twinning dynamics is the 2-D theory and level-set

numerical method of [86]. Therein, a stored energy

function is non-convex with multiple wells. Evolu-

tion of twin interfaces is governed by a kinetic rela-

tion for the twin boundary velocity as a function of

the local driving traction and boundary orientation.

A regularized version of the theory is constructed via

the level-set method which removes the requirement

of treatment of explicit jump conditions. Numerical

finite difference results in [86] compare favorably

with observed phenomena in martensite: cusp for-

mation, needle growth, spontaneous tip splitting, and

microstructure refinement. Unlike purely (nonlinear)

elastic treatments, however, sharp interface models of

TB motion require kinetic laws and material param-

eter(s) that relate driving forces to interface velocities,

for example.

Continuum pseudo-slip models

In what is termed here as a pseudo-slip class of

model, a local volume element of a crystal consists of

fractions of the parent and one or more deformation

twins. Volume or mass fractions of twins evolve

according to a kinetic law, with the driving force for

twinning typically a resolved shear stress acting on

the habit plane of the twin system, in the direction of

twinning shear for that system. Twin boundary

interfaces are not resolved explicitly within each

volume element. However, the boundary between a

fully twinned domain and the parent or a domain

containing twins of other twin systems is captured in

a homogenized or smoothed sense, whereby neigh-

boring coordinates X may support different volume

fractions of each twin variant.

The pseudo-slip approach was apparently first

introduced in [87, 88] where it was used for crystal-

lographic texture predictions. Finite element imple-

mentations of a purely mechanical theory accounting

for elasticity, slip, and twinning were perhaps first

reported in [89, 90], where pseudo-slip laws were

invoked for ductile metals. The first complete ther-

momechanical frameworks accounting for such

deformation phenomena—both exercised to describe

shock or high-pressure phenomena for which higher-

order thermoelasticity is essential—are described in

[73, 83]. The first to also include GNDs in a gradient

theory, merging the nonlinear thermoelasticity, crys-

tal plasticity, and pseudo-slip twinning descriptions

with concepts described in ‘‘Continuum dislocation

and disclination models’’ section, was presented in

[91]. Results of these works address ceramics and

minerals [73, 91] or molecular energetic crystals [83].

The forthcoming presentation summarizes the

theory developed in [3, 73, 91]. The deformation

gradient is decomposed into a product of three terms:

F ¼ r0x ¼ FEFgFP; ð3:9Þ

where thermoelastic deformation FE and deforma-

tion from plastic slip FP have the same meanings as

in (2.9) of the ‘‘Continuum dislocation and disclina-

tion models’’ section. The contribution of twinning

shear to the total deformation gradient for a volume

element at point X and time t is denoted here by

FgðX; tÞ. A term akin to detFI of (2.26) is also used in

the full kinematic framework of [3, 73, 91] to account

for possible volume changes associated with lattice

defects (including dislocation cores, stacking faults,

and TBs) but is omitted here in the interest of brevity.

Let gbðX; tÞ denote the volume fraction of twin

variant b of the material at point X and time t, where

b ¼ 1; 2; . . .; q, with q the number of twin systems. Let

cb0 denote the stress free twinning shear associated

5528 J Mater Sci (2018) 53:5515–5545

with variant b, which has shearing direction sb0 and

habit plane normal mb0. Then, the rate of deformation

and spin from twinning is computed via

Lg ¼ _FgFg�1 ¼X

b

_gbcb0sb0 �mb

0 : ð3:10Þ

The plastic velocity gradient of (2.10) is replaced with

an augmented equation that accounts for the change

in lattice orientation of twinned domains:

LP ¼ Fg _FPFP�1Fg�1

¼ ð1� gTÞX

a

_casa �ma þX

b

gbX

a

_cabsab �ma

b

!:

ð3:11Þ

Here, gT ¼P

gb 2 ½0; 1� is the total twinned volume

fraction, and the second (double) sum is over twin-

ned domains with slip rates _cab and rotated or reflec-

ted director vectors sab and mab. The rotation or

reflection matrices depend on the crystal structure

and twin type, e.g., a type I or type II twin [3, 11].

Equations for thermoelasticity such as (2.11)–(2.13)

still hold, but with the anisotropic thermoelastic

moduli updated according to the weighted rotation/

reflection matrices to account for the twinning

transformation. The free energy function is of the

same generic form as in (2.14), but the twinning

shears must be included in the list of internal state

variables fng to enable description of the effect of

twinning on the anisotropic thermoelastic coeffi-

cients. In the theory of [91], for example,

fng ¼ fgb; qG; qSg: ð3:12Þ

The evolution of the twin variants is dictated by a

pseudo-slip law:

_gb ¼ _gbðsb;T; fngÞ; ð3:13Þ

where the twinning direction (sign of resolved shear

stress sb acting on the variant’s habit plane) must be

respected to account for increased deformation

resistance in the anti-twinning sense. The local

energy balance (i.e., temperature rate equation)

accounts for dissipation from twinning shear in

addition to that from plastic work, extending the

elastic–plastic representation in (2.16).

The pseudo-slip-based class of models to which the

above theory and those developed in [83, 89, 90] belong

enables reasonably accurate predictions of the onset

and evolution of bulk twinningbehavior andassociated

crystallographic texture changes. Such models can be

implemented in existing crystal plasticity simulation

frameworks with modest additional effort. However,

specific kinetic equations and parameters must be

assigned to (3.13), and effects of twinning on slip must

be included in the flow rule for the slip rates via aug-

mentation of (2.15). Calibration and validation of such

features are often problematic, with unique property

selection difficult, if not impossible, due to the immense

number of possible slip–slip, slip–twin, and twin–twin

system interactions, each of which may most generally

demonstrate different physical behaviors [11, 89]. Even

more complexity is introduced if de-twinning is incor-

porated. Phenomenological expressions may be

assigned to describe an evolving thickness to each local

twin variant as in [91], but shapes of each twin variant

are not predicted within a volume element at X by this

class of models. If the gradient aspect of the theory (i.e.,

qG) is omitted as in [73, 83, 89, 90], themodel contains no

intrinsic length for regularization; thus, those models’

predictions do not depend on the absolute size of the

domain. Twin boundary migration has recently been

explicitly incorporated in a computational crystal

plasticity framework applied to nano-twinned metals

[92].

Diffuse interface models

The diffuse interface representation of a twin

boundary (TB) has many similarities to the diffuse

interface modeling scheme of GBs outlined in the

‘‘Diffuse interface models’’ section. One or more order

parameter(s) are introduced that delineate the parent

crystal from one or more twin variant(s). Deep within

the parent and deep within each twin, order param-

eters are homogeneous, typically with numerical

values of zero or unity depending on details of the

model formulation. Twin boundaries are represented

by finite volumes within which spatial gradients of

order parameter(s) do not vanish. Physical properties

that depend on lattice orientation (e.g., anisotropic

Lagrangian elastic constants) are interpolated

between parent and twin(s) across the boundary

regions. Unlike the GB models in which deformation

mechanics are not often addressed, in diffuse inter-

face models of deformation twinning, accounting for

the kinematics is paramount. Specifically important

are the kinematics of the formation of the interface

and the transformation (i.e., stress free shearing) of

the twin variant(s).

J Mater Sci (2018) 53:5515–5545 5529

The first nonlinear phase field theory for twinning

in crystals, incorporating both nonlinear anisotropic

elasticity and geometric nonlinearity, appears to be

that of [93]. Around the same time [94, 95] appeared,

albeit limited to small deformations, e.g., linear

elasticity. In general applications, incorporation of

nonlinear elasticity is deemed crucial, since different

predictions arise in analytical [96] and numerical

results [97, 98], and since shears, rotations, and/or

reflections inherent to the twinning process all tend

to be large, exceeding the usual limits of continuum

linear elastic constitutive models. Supporting the

asserted general necessity of nonlinear theory, Fig. 4

demonstrates different results for twinning in calcite

single crystals modeled via the nonlinear theory of

[93, 97] and its linearization. The lamellar features

observed in many instances (e.g., in martensite) are

present in the nonlinear result but absent in the linear

result, speculatively due to some greater departure

from convexity of the total potential energy in the

former. Another nonlinear theory appearing soon

after [93, 97] is applied to martensite in [99].

In what follows next, key features of the varia-

tional, finite deformation, phase field theory for

deformation twinning developed and refined in

[93, 97, 100] are reviewed. Attention here is limited to

a single crystal with a single twin variant; extension

to multiple twin systems is conceptually straightfor-

ward and is described in an appendix of [93]. This

particular theory has not been used in conjunction

with a model component for plastic slip (i.e., glide of

dislocations distinct from twinning partials), mean-

ing the response is limited to combined elastic

deformation and twinning deformation. However, an

example of a coupled phase field-crystal plasticity

model for metals that undergo simultaneous dislo-

cation slip and twinning has been reported [101].

Thermal effects are omitted.

Let the reference volume of a crystal X (which may

be contained within a polycrystal) be divided into

parent XðpÞ, twin Xðt1Þ, and boundary oXðp;t1Þ regions.

The order parameter associated with twinning is

denoted by g and obeys

gðX; tÞ ¼ 0 8X 2 XðpÞ; gðX; tÞ ¼ 1 8X 2 Xðt1Þ;

gðX; tÞ 2 ð0; 1Þ 8X 2 oXðp;t1Þ:

ð3:14Þ

Deformation and displacement are defined as in (2.8),

and the continuity requirements in (3.1) and (3.2) still

apply. However, unlike the sharp interface treatment

of the ‘‘Sharp interface models’’ section, the defor-

mation gradient now is presumed continuous

everywhere, including points in oXðp;t1Þ.

The total deformation gradient obeys the

decomposition

F ¼ r0x ¼ FEFg; ð3:15Þ

where FE is the elastic part and Fg accounts for

shearing due to mechanical twinning. Let s0 and m0

denote constant orthogonal unit vectors in the

direction of twinning shear and normal to the habit

plane, respectively. Let c0 [ 0 denote the magnitude

of stress free shear for a fully transformed domain.

Then,

FgðgÞ ¼ 1þ ½/ðgÞc0�s0 �m0: ð3:16Þ

The interpolation function /ðgÞ obeys

/ðgÞ 2 ½0; 1�; /ð0Þ ¼ 0; /ð1Þ ¼ 1;

d/dg

ð0Þ ¼ d/dg

ð1Þ ¼ 0:ð3:17Þ

Let w denote the free energy density per unit refer-

ence volume of the following form:

wðF; g;r0gÞ ¼ W ½FEðF; gÞ� þ fðg;r0gÞ; ð3:18Þ

with W the elastic strain energy density. The function

f, which is nonzero only in TB regions, consists of the

sum of a double-well potential f0 and a gradient

contribution:

Figure 4 Phase field order parameter g representing twinning in

nano-indentation of calcite single crystal with 120� wedge:

nonlinear theory (left) and linear theory (right) [97].

5530 J Mater Sci (2018) 53:5515–5545

fðg;r0gÞ ¼ f0ðgÞ þ j : r0g�r0g

¼ Ag2ð1� gÞ2 þ j : r0g�r0g:ð3:19Þ

Here, constant A quantifies the depth of the energy

wells, and second-order tensor j penalizes sharp

interfaces. When j ¼ j1, with j a scalar, the TB

energy is isotropic. In this case, the equilibrium

thickness l and equilibrium surface energy per unit

reference area C are related to parameters in (3.19) by

[93]

A ¼ 12C=l; j ¼ 3Cl=4: ð3:20Þ

Let the total free energy functional be denoted by

W, and let t0 denote the mechanical traction vector per

unit reference area and h a conjugate force to the

order parameter acting on global external boundary

oX that has unit outward normal vector n. The fol-

lowing variational principle is applied:

dW ¼ dZ

XwdX ¼

I

oXt0 � dudoXþ

I

oXhdgdoX:

ð3:21Þ

Application of standard mathematical techniques for

analysis of continuous media then results in local

equilibrium equations in X and natural boundary

conditions on oX:

r0 � P ¼ r0 �oW

oF¼ 0;

df0dg

þ oW

og¼ 2r0 � jr0g;

ð3:22Þ

t0 ¼ P � n; h ¼ 2j : r0g� n: ð3:23Þ

The first Piola–Kirchhoff stress is P ¼ ðdetFÞrF�T.

The constitutive model is complete upon specifi-

cation of the strain energy function W and the inter-

polation function /. Regarding the former, any

hyperelastic potential suitable for crystalline media

can be used, but the elasticity tensor(s) C (if aniso-

tropic) must be interpolated in the TB regions, e.g.,

C½gðX; tÞ;X� ¼ CðpÞðXÞ þ /½gðX; tÞ�fCðt1ÞðXÞ � CðpÞðXÞg:ð3:24Þ

Two interpolation functions have used with success

to date, a cubic polynomial and a Fermi–Dirac

exponential [98]:

/ ¼ ð3� 2gÞg2; / ¼ 1

1þ exp½�30ðg� 1=2Þ� :

ð3:25Þ

The latter function in (3.25) yields a rather steep

change in / near g ¼ 1=2.

Although this model is variational and quasi-static,

evolution of twin morphology during deformation

paths is predicted by sequential energy minimization

(i.e., minimization of W subject to any essential

boundary constraints) as loads are incrementally

applied. The predictive capability of the theory

described above has been validated or favorably

compared with results from experiments, analysis, or

atomic simulations for a number of crystalline

materials—calcite, sapphire, and magnesium—for

problems involving twin nucleation and/or growth

from a seedling [93], indentation [97], and a notch or

crack tip [98]. A generic twinning criteria based on

analytical solutions to the phase field theory for

localized versus diffuse transformation has also been

validated [100].

The primary purpose of the diffuse interface

approach to modeling twins and TBs is prediction of

detailed, fine-scale twin morphology. Balance laws

(or kinetic laws if a TDGL or Allen-Cahn [77] equa-

tion is invoked as in [94, 95]) are derived from fun-

damental principles of energy minimization rather

than user-prescribed phenomenology as is typical in

pseudo-slip models of the ‘‘Continuum pseudo-slip

models’’ section. Furthermore, few, if any, material

parameters must be calibrated. This advantage par-

tially disappears if continuum slip laws requiring

parameterization are added to the theory, as in [101],

for example. The length scale of regularization l is

controlled by the modeler and dictates the equilib-

rium width of TB zones. The regularization process

renders numerical simulations mesh-size indepen-

dent so long as the discretization is fine enough to

resolve order parameter gradients in such zones. This

is often a drawback in simulations involving

twin(s) of size much larger than their boundary

zones, since very fine meshes must be used to resolve

boundaries that encompass a relatively small overall

fraction of the entire problem domain.

The motion of twin or crystallographic phase

boundaries—be it via dynamic extension, thickening,

or migration—may crucially influence the mechanical

response of certain crystals, particularly nano-twin-

ned metals [102], shape memory alloys, and

martensite [85, 103]. A novel theory and comple-

mentary computational method were recently

developed that includes distinct prescriptions of

J Mater Sci (2018) 53:5515–5545 5531

nucleation through the source term of the phase field

conservation law and kinetics through a distinct

interfacial velocity field [104, 105]. This approach

alleviates potential obscurity of these physical phe-

nomena suffered by classical phase field

implementations.

Failure and localization

Mesoscopic continuum models of damage processes

in polycrystals are now addressed. These processes

tend to be irreversible and dissipative. Examples

include fracture, void formation and growth, pore

collapse, and shear localization. In crystalline mate-

rials, fractures may occur along GBs, i.e., intergran-

ular modes, and/or along cleavage planes within a

grain, i.e., transgranular modes. Regarding inter-

granular fracture, some kinds of crystals demonstrate

preferred planes, typically of low surface energy,

while others do not, in which case fractures tend to be

conchoidal [106]. Void mechanisms arise in ductile as

opposed to brittle solids, wherein plastic deformation

of surrounding matrix material enables volumetric

growth of damage rather than planar fractures. In

materials with initial porosity, including many min-

erals and their composites (e.g., concrete [107]) pores

may be irreversibly compressed out of the material

due to mechanical pressure, leading to an increase in

compressive bulk modulus. As discussed already in

the ‘‘Sharp interface models’’ section, shear bands

may arise due to lattice orientation effects [39], but

are perhaps most common in high rate deformation

processes wherein localization is promoted by

(nearly) adiabatic heating and thermal softening [8].

Tensile wave interactions also can induce damage

mechanisms, specifically spall failure [108], which

may entail dynamic brittle fracture or ductile failure

via void coalescence, depending on ductility of the

polycrystal.

Cohesive fracture models

Cohesive fracture models are a kind of sharp inter-

face representation of failure at the mesoscale. The

theoretical concept of such models, most often

attributed to [109, 110], is that along surfaces near a

crack tip, the degraded material supports a nonzero

traction vector over some finite distance, called the

cohesive zone length. The traction tends to decrease

in magnitude as the crack opening displacement

increases. Traction generally includes both normal

and shear components, in association with mode I

and mode II/III crack opening dispacements [7]. The

earliest numerical implementation of a cohesive fail-

ure model in a finite element context appears to be

documented in [111]. Subsequently, cohesive failure

elements have been invoked in quasi-static crystal

plasticity simulations [112, 113], thermoelastic simu-

lations of ceramics [44], and elastodynamic simula-

tions of crack branching [114] and spall [45]. The first

dynamic simulations coupling finite crystal ther-

moelasto-plasticity with cohesive failure along GBs

seem to be those reported in [115, 116], with a follow-

up study of spall in [117]. Representative simulation

results for cohesive failure modeling of dynamic

fracture in ceramic and metallic polycrystalline

microstructures are shown in Figs. 5 and 6, respec-

tively. In all such simulations, the bulk material

response within continuum finite elements is mod-

eled via standard thermoelasticity or crystal elasto-

plasticity, e.g, models discussed in the ‘‘Grain

boundaries’’ section of this work. The cohesive zone

model is invoked for failure/separation behavior

along element boundaries.

A few key equations for a basic cohesive zone

model of fracture are reviewed next. An immense

literature on cohesive fracture modeling has emerged

over the previous two decades, with more sophisti-

cated theoretical and numerical formulations

accounting for various subscale physical mechanisms

and thermodynamic aspects now available. The pre-

sentation below is the minimum deemed necessary to

illustrate the fundamental mechanical concepts.

Let the crack opening displacement vector across

two crack faces initially coincident at point X be

defined as the displacement jump

dðX; tÞ ¼ suðX; tÞt; ð4:1Þ

where now obviously the continuity constraint in

(2.18) is violated across the crack surfaces. Denote by

nðXÞ the unit normal to the surface of impending

fracture, referred to the reference coordinate system.

Let t0 denote the traction vector per unit reference

area:

t0 ¼ P � n; ð4:2Þ

with P the first Piola–Kirchhoff stress tensor. In the

cohesive zone, a general form of traction–separation

law is prescribed:

5532 J Mater Sci (2018) 53:5515–5545

t0 ¼ t0ðd; fvgÞ: ð4:3Þ

Potential history effects are captured by state vari-

able(s) fvg. Typically, a magnitude of displacement

dC is assigned as a material property or parameter,

beyond which traction vanishes and the crack sur-

faces become free surfaces. For quasi-static imple-

mentations, a stiffness matrix is usually needed, in

which case the traction function must have a con-

tinuous derivative with respect to opening displace-

ment. For simulations invoking explicit numerical

integration, on the other hand, this differentiability

restriction does not hold. The work done during

separation can be related to a surface energy of

fracture C, equal to half the critical strain energy

release rate in the context of linear elastic fracture

mechanics:

C ¼ 1

2

Z dC

0

t0 � dd; ð4:4Þ

where the path of integration ends when the critical

separation magnitude is attained. Perhaps the sim-

plest realistic model is the triangular cohesive law

jt0j ¼ rCð1� jdj=dCÞ; ð4:5Þ

which can be invoked separately for magnitudes j � jof normal and shear components. Here, rC is the

strength required to initiate fracture, i.e., the resolved

scalar stress component at which the cohesive zone

starts to open. Only two of the three parameters

(C; rC; dC) need be prescribed since (4.4) enables one

of these to be eliminated algebraically. More sophis-

ticated models accounting for mode mixity are typi-

cal [114, 118], but these often need additional

calibration or parameters. The length of the cohesive

zone in the context of isotropic linear elastic fracture

mechanics is [7, 118]

Figure 5 Shear stress r in AlON ceramic under simulated shear and compression with nonlinear elastic grains and cohesive finite

elements at GBs: external view (left) and internal view (right) [27].

Figure 6 Particle velocity up in shock wave prior to spall (left, arrows are velocity vectors; colors indicate velocity magnitude) and

effective stress at spall zones (right) in dynamic crystal plasticity-cohesive finite element simulations [117].

J Mater Sci (2018) 53:5515–5545 5533

lC � 2EC

pr2Cð1� mÞ2; ð4:6Þ

where E and m are the elastic modulus and Poisson’s

ratio.

Cohesive failure models often enable realistic pre-

dictions of brittle or ductile failure in microstructures.

Distinct fracture entities, i.e., crack sizes and shapes,

are fully resolved, and interactions among entities are

naturally addressed. Advantages of this class of

model include relatively few parameters [minimally

two, e.g., dC and rC in the context of (4.5)] and distinct

behavior of interfaces and bulk material, meaning

that traditional solid continuum elements can be used

for representing the latter. However, real polycrys-

talline solids demonstrate a distribution of strengths

and surface energies among potential failure sites

(e.g., various kinds of GBs, different families of

cleavage planes, and initial defect structures that

affect toughness), a characteristic often ignored in

deterministic computer simulations. Numerical

implementation of the basic model is rather

straightforward, though additional nodal degrees of

freedom increase the computational expense where

duplicate nodes are inserted along failure planes. A

potentially severe drawback is that fracture paths are

constrained to follow element boundaries in most

computer implementations. Therefore, meshes must

be constructed such that realistic crack paths are

possible, often requiring some a priori knowledge of

failure morphologies. This drawback is alleviated

when fractures are restricted to preexisting interfaces

such as GBs, which makes the approach ideal for

representing intergranular failure or fracture between

phases of heterogeneous crystals such as reported in

[115–117]. Cleavage fracture on specific planes is

more difficult to address via cohesive element mod-

eling, but examples of success have been reported

[119]. Finally, cohesive finite element sizes must be

small enough to resolve behavior over the cohesive

length, e.g., must be smaller than lC of (4.6). Extre-

mely fine meshes are often necessary for representing

fracture zones of materials with high strength and

low surface energy, e.g., many strong yet brittle

ceramics, as modeled, for example, in [120].

Continuum damage mechanics

In continuum damage mechanics models, individual

failure entities such as discrete cracks and voids are

not resolved explicitly. Instead, one or more state

variable(s) is introduced as a function of time t and

material coordinates X that accounts for degradation

of the material at a local material point and a given

time instant. Such state variables, which may be

scalar-, vector-, or tensor-valued, quantify the effects

of multiple damage entities contained within a local

volume element of material at that point. Research

books on the subject include [121, 122]. The funda-

mental concept, in the context of a scalar damage

mechanics theory, is often attributed to [123].

The forthcoming presentation considers a

(poly)crystalline solid whose damage is represented

by a scalar state variable DðX; tÞ 2 ½0; 1�. Generaliza-

tion to a vector, tensor, or multiple scalars is in

principle straightforward but notationally more

cumbersome. The solid is assumed to be hyperelastic

and may undergo plastic slip, following the finite

deformation continuum formalism discussed in the

‘‘Sharp interface models’’ and ‘‘Continuum disloca-

tion and disclination models’’ sections. The defor-

mation gradient is decomposed multiplicatively as

F ¼ r0x ¼ FEFDFP; ð4:7Þ

where FE includes thermoelastic deformation of the

crystal lattice as well as mechanically reversible

changes in damage (e.g., elastic crack closure on load

release), FP accounts for plastic slip from dislocations

as described in the ‘‘Sharp interface models’’ section,

and FD accounts for mechanically irreversible dam-

age mechanisms such as cracks and voids that remain

in the solid upon elastic unloading. A three term

decomposition of this general form was proposed in

[124]. Other forms of the deformation gradient that

reflect residual damage modes include additive

[115, 125, 126] and hybrid additive–multiplicative

[113, 127] decompositions, often derived or moti-

vated from homogenization of discrete displacement

jumps associated with cracks within a volume ele-

ment of material. Transformations between additive

and multiplicative descriptions have also been

derived [125, 128]. The volume fraction of damage N

(i.e., porosity) is related to the determinant of FD as

[3, 129]

JD ¼ detFD ¼ ð1� NÞ�1=3: ð4:8Þ

Besides its use for solids with voids [124, 129] or

pores [21, 107], a multiplicative damage term has

been introduced for cleavage cracking in crystals

5534 J Mater Sci (2018) 53:5515–5545

driven by resolved normal and shearing tractions in

pseudo-slip-type models [32, 128, 130].

The list of internal state variables of (2.22) is

extended to include damage in addition to

dislocations:

fng ¼ fqG; qS;Dg: ð4:9Þ

Damage variable D varies from zero to unity as the

material at the corresponding point loses integrity.

The free energy density w therefore depends on

damage in addition to elastic strain E, temperature T,

and dislocation densities. The elastic second Piola–

Kirchhoff stress and conjugate force to damage are

S ¼ owoE

; f ¼ � owoD

: ð4:10Þ

An equation similar in form to (2.13) applies for the

stress, but now elastic moduli depend on damage.

The simplest degradation model of the moduli is

linear in D:

C½DðX; tÞ;X� ¼ ½1�DðX; tÞ�C0ðXÞ; ð4:11Þ

with C0ðXÞ ¼ Cð0;XÞ the tensor of elastic moduli for

the undamaged crystal at the corresponding material

point. For an isotropic solid, (4.11) leads to a constant

Poisson’s ratio. More sophisticated degradation laws

are required to more realistically capture physics of

arbitrary loading paths, such as damage induced

anisotropy and differences in tensile versus com-

pressive degradation, where generally the former is

more severe.

The work done by the appropriate stress tensor

acting on the rate of FD contributes to local dissipa-

tion, as does the product f _D. Kinetic equations must

be supplied for time rates of FD and D, e.g., of general

forms

_FD ¼ _FDðE; fng;T;FDÞ; _D ¼ _DðE; fng;T;FDÞ:ð4:12Þ

Dependence on elastic strain and internal state vari-

ables is often conveniently replaced with physically

more transparent dependence on stress and other

thermodynamic driving forces. Kinetic equations for

plasticity and dislocation density are likewise affec-

ted by nonzero damage [124, 130]; e.g., stress con-

centrations may increase the tendency for plastic

flow.

Capabilities and caveats of the continuum damage

classes of material models can be summarized as

follows. The present class of models to which the

above theory, and typical of those discussed in

[121, 122], enables reasonably accurate predictions of

the onset and evolution of bulk damage behavior and

associated changes in elastic stiffness. These models

can often be implemented in existing continuum

mechanics simulation frameworks, including those

accounting for crystal plasticity, with modest over-

head. Unlike cohesive finite element approaches, no

special interfacial elements or node duplications are

necessary. Unfortunately, specific kinetic equations

and parameters must be assigned to (4.12), and cou-

pled effects of D must be included in the flow rule for

the slip rates via augmentation of (2.15). Calibration

and validation of these aspects of the model are often

difficult, and unique specification of all parameters

cannot usually be ensured from available test data for

a particular material. Additional complexity is injec-

ted when more realistic formulations involving

multiple damage variables or vector- or tensor-val-

ued damage variables are employed for anisotropic

media. Since damage is homogenized at each point X,

this class of models is unable to resolve solution

fields of discrete cracks or voids and their interac-

tions. The damage components of this class of model

contain no intrinsic length for regularization, and

problems that involve the usual material softening

with increasing D often suffer from mesh-size-de-

pendent numerical solutions. Such problems can be

alleviated when gradient terms are considered, e.g.,

qG if GNDs are addressed as in (4.9) or [130]. Gra-

dients of the damage variable can also be introduced

into the balance equations and/or kinetics, which

renders the continuum damage theory nonlocal and

hence non-classical. This type of regularization is a

characteristic advantage of diffuse interface or phase

field approaches discussed next in ‘‘Diffuse interface

models’’ section.

Diffuse interface models

Diffuse interface classes of models for failure mech-

anisms, which include phase field representations of

fracture, do not resolve discrete jumps in the dis-

placement field in contrast to sharp interface-type

(e.g., cohesive zone) models. Instead, fracture sur-

faces (or other failure surfaces) are described by one

or more order parameter(s). In the usual case, an

order parameter is assigned a uniform value (e.g.,

zero or unity) in regions of material where no dam-

age exists or where complete failure has occurred.

J Mater Sci (2018) 53:5515–5545 5535

The change from initial or undamaged state to fully

failed state can be thought of as a transition from a

perfect crystal to a liquid state or a vacuum,

depending on the strength/stiffness properties

attributed to the fully failed state. Boundaries

between fully failed and perfectly intact material are

represented by order parameter gradients, and in

such boundary regions the order parameter takes on

a value between its two extremes. Lagrangian elastic

properties depend on the order parameter, with a

decrease in local tangent stiffness correlating to an

increase in local damage.

Two of the earliest diffuse interface models of

fracture are reported in [131, 132]. These early models

invoked linearity in terms of deformation kinematics

and elastic response. A nonlinear diffuse interface

model for isotropic solids is presented in [133] in

concordance with 2-D simulations. The first nonlinear

phase field model for fracture in anisotropic crystals

is described in [134]. In that work, 3-D simulations for

mode I and mode II loading are validated versus

analytical solutions, and a problem associated with

crack bridging is investigated. Specifically, the ten-

dency for a cleavage fracture to propagate around a

spherical inclusion or cut through the inclusion is

quantified in terms of stiffness and strength proper-

ties of the matrix and the sphere. A particular result

in which the crack is deflected around the inclusion is

shown in Fig. 7. Mesoscale phase field simulations of

simultaneous intergranular and transgranular frac-

tures in anisotropic polycrystals, with and without

secondary GB phases, are reported in [30], with

characteristic results shown in Fig. 8.

The forthcoming discussion presents key aspects of

the nonlinear anisotropic phase field model for frac-

ture of [134]. As noted above, displacement and

traction are continuous fields, so constraints (2.18)

and (2.19) hold. Plastic deformation, twinning, and

thermal effects are omitted in what follows, noting

that a coupled nonlinear phase field theory for

simultaneous fracture and twinning mechanisms has

been developed and analyzed in [100], with numeri-

cal simulation results for polycrystals given in [135].

Let the reference volume of a crystal X be divided

into perfect XðpÞ, fully failed XðfÞ, and boundary oXðp;fÞ

regions. The theory considers a single order param-

eter gðX; tÞ, where now

gðX; tÞ ¼ 0 8X 2 XðpÞ; gðX; tÞ ¼ 1 8X 2 XðfÞ;

gðX; tÞ 2 ð0; 1Þ 8X 2 oXðp;fÞ:

ð4:13Þ

Deformation and displacement are defined as in (2.8),

with the deformation gradient and its determinant

associated with volume change given by the usual

equations

F ¼ r0x; J ¼ detF[ 0: ð4:14Þ

The free energy density per unit reference volume is

of the following form:

wðF; g;r0gÞ ¼ WðF; gÞ þ fðg;r0gÞ; ð4:15Þ

with W the elastic strain energy density. The function

f, which vanishes identically in the perfect/pristine

regions of the crystal(s), consists of the sum of a

quadratic potential f0 and a quadratic gradient

contribution:

fðg;r0gÞ ¼ f0ðgÞ þ j : r0g�r0g

¼ Clg2 þ fCl½1þ bð1�m�mÞ�g : r0g�r0g:

ð4:16Þ

Here, C is the surface energy of fracture, l is the

equilibrium width of the diffuse fracture zone, m is a

unit normal vector to a preferred cleavage plane in

the reference configuration, and b is a penalty factor

that should be prescribed as a large number to restrict

Figure 7 Phase field prediction of crack deflection around a

strong second-phase inclusion for far-field mode I loading with

failed material (g[ 0:7) removed to visualize crack propagation

[134].

5536 J Mater Sci (2018) 53:5515–5545

fractures to take place along m. Isotropic fracture

corresponds to b ¼ 0, in which case m is not needed.

The strain energy density W may be any nonlinear

elastic potential suitably modified to account for

degradation of stiffness with increasing values of g.For isotropic neo-Hookean elasticity considered in

some simulations [134], the shear modulus ldegrades from its initial value l0 as

lðgÞ ¼ l0½fþ ð1� fÞð1� gÞ2�; ð0 f 1Þ:ð4:17Þ

Parameter f, if nonzero, provides for finite stiffness infully fractured zones. The bulk modulus degrades in

tension but not in compression via a criterion based

on local values of volume ratio J. Similar treatments

for degradation of anisotropic elastic constants for

crystals are described in [30].

Derivation of the Euler–Lagrange equations paral-

lels that of the ‘‘Diffuse interface models’’ section. The

total free energy functional is W, t0 denotes the

mechanical traction vector per unit reference area,

and h is a conjugate force to the order parameter

acting on boundary oX that has unit outward normal

vector n. The first Piola–Kirchhoff stress tensor is P. A

variational principle is applied:

dW ¼ dZ

XwdX ¼

I

oXt0 � dudoXþ

I

oXhdgdoX;

ð4:18Þ

which results in local equilibrium equations and

natural boundary conditions:

r0 � P ¼ r0 �oW

oF¼ 0; 2

Cglþ oW

og¼ 2r0 � jr0g;

ð4:19Þ

t0 ¼ P � n; h ¼ 2j : r0g� n: ð4:20Þ

Diffuse interface models of fracture or material

failure exhibit some important positive characteris-

tics: structural transformations occur naturally via

energy minimization as in the approach outlined

Figure 8 Phase field predictions of inter- and transgranular

fracture in anisotropic Zn polycrystal of edge length L under axial

tensile strain e with failed material (g[ 0:7) removed to visualize

crack propagation [30]: a L ¼ 100 lm, e ¼ 0:15% b L ¼ 100lm,

e ¼ 0:20% c L ¼ 100 nm, e ¼ 4:4% d L ¼ 100 nm, e ¼ 8:0%.

J Mater Sci (2018) 53:5515–5545 5537

above, or Ginzburg–Landau-type kinetics [131],

without recourse to phenomenology associated with

often ad hoc, user-prescribed rate equations for

damage evolution. Correspondingly, relatively few

material parameters or properties are usually needed.

For example, in the theory of [30, 134] described

above, the only required parameters are the elastic

constants, fracture surface energy C, and the length

constant l. Regarding the latter, numerical solutions

are regularized and thus are rendered mesh-size

independent by l, which is associated with interfacial

width. Regularized variational models invoking the

form of modulus degradation in (4.17) may also

demonstrate so-called gamma convergence toward

the classical Griffith theory of fracture mechanics as

the regularization zone shrinks to a singular surface.

Detailed damage morphologies can be predicted by

numerical simulations invoking phase field fracture

theory, including crack nucleation and branching and

interactions among multiple cracks [136]. Standard

continuum finite element technologies and meshing

strategies can be used since fields are continuous, but

additional degree(s) of freedom [i.e., order parame-

ter(s)] must be updated at each node as the calcula-

tion proceeds incrementally. A disadvantage of the

phase field approach to modeling fracture is the

requirement of resolution of damage surfaces/

boundaries. A very fine mesh is needed to resolve

order parameter gradients across narrow zones

where crack surfaces exist. Dynamic remeshing

strategies have been invoked in some cases to deal

with this issue [137]. Problems associated with

proper modeling of large crack velocities via phase

field models in the setting of elastodynamics have

also been reported [138].

Although the preceding discussion deals with

physics of fracture, separation of material in con-

junction with shear localization can be addressed via

similar principles. A combined phase field model for

plastic deformation, adiabatic shear localization, and

ductile fracture in metals is described in [139]. Shear

band thickness may evolve, which can complicate

mesoscale mechanics of plasticity at their interfaces,

for example a realistic prescription of a regularization

length. Insight into the width and structure of shear

bands in ductile metals may be found in analysis and

numerical solutions reported in [8, 140, 141].

Another recent diffuse interface approach is

applied to shear localization or shear fracture in

magnesium [142] and boron carbide [143, 144], with

the latter material known to undergo stress-induced

amorphization promoted by compression and shear

[145]. The novelty of the recent theory presented in

[142, 144]—with initial developments first reported in

[146]—is use of Finsler differential geometry to for-

mulate the governing kinematic and equilibrium

equations. Essentially, the generalized theory of

Finsler-geometric continuum mechanics permits the

metric tensor and corresponding local volume ele-

ments to depend on the order parameter(s) compris-

ing the state vector of pseudo-Finsler space, enabling

a natural coupling between dilatation or volume

collapse and shearing modes, without introduction of

spurious fitting equations or calibrated parameters.

Though not yet undertaken, the same approach could

be used to address dilatation in the vicinity of

stacking faults or twin boundaries in crystals

[147–149]. The first analytical solutions for Finsler-

geometric continuum theory are derived in [146];

subsequent solutions are also validated versus

experimental observations for the above noted crys-

talline materials in [142, 144, 150, 151]. An alternative

Finsler-based thermomechanical theory was applied

in numerical simulations of shear localization in

metals in [152, 153].

Discussion

Capabilities and notable advantages and disadvan-

tages of modeling techniques—sharp interface, con-

tinuum defect, and diffuse interface

representations—are summarized next. The author’s

viewpoint is that all such models have been suitably

validated versus experiment or atomic simulation for

their intended applications, as has been noted

already with supporting references in corresponding

sections of the main text of this article and references

cited therein.

First consider sharp interface models. Sharp inter-

face representations of grain boundaries (GBs) permit

explicit modeling of strain and stress concentrations

of field variables near mismatched interfaces. No

additional constitutive model parameters are

required beyond those for the bulk material. The

approach is physically realistic for thermoelasticity

since properties and hardness tend to vary little from

points near the GB to points far in the grain interior.

The approach is also considered physically accurate

in the sense that atomically sharp boundaries are

5538 J Mater Sci (2018) 53:5515–5545

modeled by jumps in properties, with no artificial

smoothing. Similar statements apply for sharp inter-

face models of twin boundaries (TBs). No spurious

material parameters generally need to be calibrated

to address problems in elastostatics. To address

interface motion, however, a kinetic law for GB or TB

dynamics must supplement the kinematic descrip-

tion, and a (numerical) scheme must be invoked to

track the position of interface(s) as deformation pro-

ceeds in time. Implementation of explicit front-

tracking often proves challenging. Cohesive failure

models are categorized as sharp interface models for

material separation. Distinct crack morphologies are

suitably captured, as are interactions among cracks

and other heterogeneities. Advantages include rela-

tively few parameters for basic crack separation laws

and distinct behavior of interfaces and bulk material

permitting use of traditional solid continuum finite

elements for the bulk. Variations in initial failure

properties, which may be extreme in brittle solids,

can unfortunately increase the number of parameters

needed for realistic predictions. Numerical imple-

mentation is usually straightforward. A disadvantage

is that fracture paths are often constrained to follow

element boundaries. Cohesive finite element sizes

must be small enough to resolve behavior over the

cohesive length, an issue that may necessitate very

fine meshes at correspondingly high computational

expense.

Next consider continuum defect descriptions of

interfaces in crystalline solids. Crystal plasticity can

be augmented to explicitly account for defect (e.g.,

dislocation and disclination) densities in the vicinity

of grain and subgrain boundaries. Benefits include

physical flexibility, with size effects and microstruc-

ture evolution at different resolutions naturally

addressed. Regularization associated with higher-

order gradient terms may enable mesh-size inde-

pendence of numerical solutions. Additional param-

eters must be prescribed and often calibrated rather

than determined from first principles. Furthermore,

increased model sophistication contributes compu-

tational expense and limits availability of analytical

solutions for model verification. Pseudo-slip-type

twinning models enable reasonably accurate predic-

tions of the onset and evolution of bulk twinning

behavior and texture changes. These can usually be

implemented in crystal plasticity frameworks with

modest effort. Kinetic equations and parameters must

be assigned, and effects of twinning on slip must be

included in the flow rule. Calibration and validation

of such features is often problematic. Shapes of each

twin variant within a local material volume element

are not predicted. Continuum damage models can

reasonably predict onset and evolution of bulk

damage behavior and changes in stiffness. These

models can often be implemented in existing simu-

lation frameworks with modest overhead, and no

special interfacial elements or node duplications are

needed. Kinetic equations and parameters must be

assigned for damage and its effects on plastic flow.

Calibration is often difficult, and unique specification

of all parameters cannot usually be ensured. Com-

plexity increases as multiple scalar damage variables

or vector- or tensor-valued damage variables are

employed for anisotropic media. Solution fields of

discrete cracks or voids and their interactions are not

well described. Usually, such models contain no

intrinsic regularization length.

Finally, consider diffuse interface representations

which notably encompass phase field descriptions.

Microstructure evolution can often be addressed in a

natural way, whereby kinetics of microstructure

evolution is motivated by the fundamental principle

that a system should seek a minimum energy state.

Regularization is obtained as a by-product of gradi-

ent terms in the energy functional. Disadvantages

include necessary choices of non-unique interpola-

tion functions for property values within interfacial

zones and parameters in the energy functional and

kinetic laws that may lack obvious physical inter-

pretation. Mesh resolution must be fine enough to

resolve field variables and their gradients. Disparate

timescales for microstructure evolution and stress

dynamics may complicate numerical solution proce-

dures. The diffuse interface approach to modeling

twins and twin boundaries enables prediction of

detailed, fine-scale twin morphologies for equilib-

rium states. Diffuse interface models of fracture

likewise naturally enable prediction of failure

behavior via energy minimization, without recourse

to phenomenology usually associated with contin-

uum damage mechanics theories. Relatively, few

material parameters or properties are needed. Com-

plex crack morphologies can be predicted by

numerical simulations invoking phase field fracture

theory. Standard continuum finite elements and their

meshes can be used since fields are continuous, but

additional degree(s) of freedom are needed to track

order parameter(s). A disadvantage is that very fine

J Mater Sci (2018) 53:5515–5545 5539

meshes needed to resolve order parameter gradients

across narrow zones (i.e., regularization lengths)

where crack surfaces exist. The regularization length

is thus often artificially increased in a compromise of

physical realism with computational cost.

General trends in capabilities are summarized in

Table 1, representative of many of the theories

developed or implemented in works cited through-

out the text. Exceptions to these trends are possible.

This article, as implied by its title, has focused on

interfacial physics from the perspective of (contin-

uum or mesoscale) mechanics. Applications of sharp

and diffuse interface models to broad classes of

materials science problems, including those not

addressed herein such as solidification, for example,

have been discussed in books and other review arti-

cles on these subjects [18, 154–156].

Conclusions

Models of interfaces in crystalline solids have been

categorized, summarized, and evaluated. Emphasis

has been given to finite deformation descriptions at

the mesoscale, with attention limited to continuum

mechanical frameworks, as opposed to molecular

statics/dynamics. Particular physical boundaries

considered have been grain and subgrain boundaries,

twin boundaries, and failure surfaces (e.g., fracture

planes). For each type of physical boundary, corre-

sponding models have been grouped into one of

three general categories: sharp interface models,

continuum defect models, and diffuse interface (e.g.,

phase field) models. Though moderately lengthy, the

present review of course does not claim to cover all

important works, with any notable exclusions unin-

tended. It is hoped that the present article will be a

useful aid to researchers beginning studies on the

subject and a useful reference for those more expe-

rienced, pointing out some key historic and more

modern, perhaps overlooked, references wherein

further details can be found.

Acknowledgements

Much of this paper was written while the author

served as a visiting research fellow at Columbia

University in the Department of Civil Engineering

and Engineering Mechanics of the Fu Foundation

School of Engineering and Applied Science in New

York, NY, USA. The author acknowledges the cour-

tesy of Dr. WaiChing (Steve) Sun for hosting his

sabbatical visit at Columbia University in 2016.

References

[1] Phillips R (2001) Crystals, defects and microstructures:

modeling across scales. Cambridge University Press,

Cambridge

[2] Rohrer GS (2001) Structure and bonding in crystalline

materials. Cambridge University Press, Cambridge

[3] Clayton JD (2011a) Nonlinear mechanics of crystals.

Springer, Dordrecht

[4] Yadav S, Ravichandran G (2003) Penetration resistance of

laminated ceramic/polymer structures. Int J Impact Eng

28:557–574

Table 1 Classes of mesoscopic interfacial models, examples, and general characteristics

Sharp interface Continuum defect Diffuse interface

Examples

(Grain boundary) Discrete GB models

(property jumps)

Distributed dislocation/disclination Phase field models of GBs

(Twinning) Discontinuous

deformation gradient

Pseudo-slip twinning models Phase field models of twinning

(Failure) Cohesive fracture models Continuum damage mechanics Phase field models of fracture

Complexity

(usual governing equations)

Low High Moderate

Generality

(usual physics addressed)

Low High Moderate

Phenomenology

(typical calibration and parameters)

Low High Moderate

5540 J Mater Sci (2018) 53:5515–5545

[5] Clayton JD (2015) Modeling and simulation of ballistic

penetration of ceramic-polymer-metal layered systems.

Math Probl Eng 2015:709498

[6] Brandon DG (1966) The structure of high-angle grain

boundaries. Acta Metall 14:1479–1484

[7] Rice JR (1968) Mathematical analysis in the mechanics of

fracture. In: Liebowitz H (ed) Fracture: an advanced trea-

tise. Academic Press, New York, pp 191–311

[8] Wright TW (2002) The physics and mathematics of adia-

batic shear bands. Cambridge University Press, Cambridge

[9] Hughes DA, Hansen N, Bammann DJ (2003) Geometri-

cally necessary boundaries, incidental dislocation bound-

aries and geometrically necessary dislocations. Scr Mater

48:147–153

[10] Boiko VS, Garber RI, Kosevich AM (1994) Reversible

crystal plasticity. AIP Press, New York

[11] Christian JW, Mahajan S (1995) Deformation twinning.

Prog Mater Sci 39:1–157

[12] Dongare AM, LaMattina B, Irving DL, Rajendran AM,

Zikry MA, Brenner DW (2012) An angular-dependent

embedded atom method (A-EAM) interatomic potential to

model thermodynamic and mechanical behavior of Al/Si

composite materials. Modelling Simul Mater Sci Eng

20:035007

[13] Zhigilei LV, Volkov AN, Dongare AM (2012) Computa-

tional study of nanomaterials: from large-scale atomistic

simulations to mesoscopic modeling. In: Bhushan B (ed)

Encyclopedia of nanotechnology. Springer, Berlin,

pp 470–480

[14] Abraham FF, Broughton JQ, Bernstein N, Kaxiras E (1998)

Spanning the continuum to quantum length scales in a

dynamic simulation of brittle fracture. Europhys Lett

44:783–787

[15] Knap J, Ortiz M (2001) An analysis of the quasicontinuum

method. J Mech Phys Solids 49:1899–1923

[16] Clayton JD, Chung PW (2006) An atomistic-to-continuum

framework for nonlinear crystal mechanics based on

asymptotic homogenization. J Mech Phys Solids

54:1604–1639

[17] Chung PW, Clayton JD (2007) Multiscale modeling of

point and line defects in cubic crystals. Int J Multiscale

Comput Eng 5:203–226

[18] Emmerich H (2003) The diffuse interface approach in

materials science: thermodynamic concepts and applica-

tions of phase-field models. Springer, Berlin

[19] Schoenfeld SE, Wright TW (2003) A failure criterion based

on material instability. Int J Solids Struct 40:3021–3037

[20] Wallace DC (2003) Statistical physics of crystals and liq-

uids: a guide to highly accurate equations of state. World

Scientific, Singapore

[21] Clayton JD, Tonge A (2015) A nonlinear anisotropic

elastic–inelastic constitutive model for polycrystalline

ceramics and minerals with application to boron carbide. Int

J Solids Struct 64–65:191–207

[22] Bunge H-J (1982) Texture analysis in materials science:

mathematical methods. Butterworths, London

[23] Randle V, Engler O (2000) Introduction to texture analysis:

macrotexture, microtexture, and orientation mapping.

Gordon and Breach, Amsterdam

[24] Grimmer H, Bollmann W, Warrington DH (1974) Coinci-

dent-site lattices and complete pattern-shift lattices in cubic

crystals. Acta Crystallogr A 30:197–207

[25] Watanabe T (1984) An approach to grain boundary design

for strong and ductile polycrystals. Res Mech 11:47–84

[26] Roters F, Eisenlohr P, Hantcherli L, Tjahjanto DD, Bieler

TR, Raabe D (2010) Overview of constitutive laws, kine-

matics, homogenization and multiscale methods in crystal

plasticity finite-element modeling: theory, experiments,

applications. Acta Mater 58:1152–1211

[27] Clayton JD, Kraft RH, Leavy RB (2012) Mesoscale mod-

eling of nonlinear elasticity and fracture in ceramic poly-

crystals under dynamic shear and compression. Int J Solids

Struct 49:2686–2702

[28] Clayton JD (2013a) Mesoscale modeling of dynamic

compression of boron carbide polycrystals. Mech Res

Commun 49:57–64

[29] Clayton JD, McDowell DL (2003a) A multiscale multi-

plicative decomposition for elastoplasticity of polycrystals.

Int J Plast 19:1401–1444

[30] Clayton JD, Knap J (2015a) Phase field modeling of

directional fracture in anisotropic polycrystals. Comput

Mater Sci 98:158–169

[31] Clayton JD (2013b) Nonlinear Eulerian thermoelasticity for

anisotropic crystals. J Mech Phys Solids 61:1983–2014

[32] Clayton JD (2014a) Analysis of shock compression of

strong single crystals with logarithmic thermoelastic–plas-

tic theory. Int J Eng Sci 79:1–20

[33] Thurston RN (1974) Waves in solids. In: Truesdell C (ed)

Handbuch der Physik, vol 4. Springer, Berlin, pp 109–308

[34] Teodosiu C (1982) Elastic models of crystal defects.

Springer, Berlin

[35] Clayton JD (2014b) Shock compression of metal crystals: a

comparison of Eulerian and Lagrangian elastic–plastic

theories. Int J Appl Mech 6:1450048

[36] Clayton JD (2015b) Crystal thermoelasticity at extreme

loading rates and pressures: analysis of higher-order energy

potentials. Mech Lett 3:113–122

[37] Meyers MA, Ashworth E (1982) A model for the effect of

grain size on the yield stress of metals. Philos Mag A

46:737–759

J Mater Sci (2018) 53:5515–5545 5541

[38] Clayton JD, Schroeter BM, Graham S, McDowell DL

(2002) Distributions of stretch and rotation in OFHC Cu.

J Eng Mater Technol 124:302–313

[39] Harren SV, Deve HE, Asaro RJ (1988) Shear band for-

mation in plane strain compression. Acta Metall

36:2435–2480

[40] Harren SV, Asaro RJ (1989) Nonuniform deformations in

polycrystals and aspects of the validity of the Taylor model.

J Mech Phys Solids 37(2):191–232

[41] Clayton JD (2009a) Modeling effects of crystalline

microstructure, energy storage mechanisms, and residual

volume changes on penetration resistance of precipitate-

hardened aluminum alloys. Compos B Eng 40:443–450

[42] Needleman A, Asaro RJ, Lemonds J, Peirce D (1985) Finite

element analysis of crystalline solids. Comput Methods

Appl Mech Eng 52:689–708

[43] Zikry MA, Kao M (1996) Inelastic microstructural failure

mechanisms in crystalline materials with high angle grain

boundaries. J Mech Phys Solids 44:1765–1798

[44] Ortiz M, Suresh S (1993) Statistical properties of residual

stresses and intergranular fracture in ceramic materials.

J Appl Mech 60:77–84

[45] Espinos HD, Zavattieri PD (2003a) A grain level model for

the study of failure initiation and evolution in polycrys-

talline brittle materials. Part I theory and numerical

implementation. Mech Mater 35:333–364

[46] Pathak S, Michler J, Wasmer K, Kalidindi SR (2012)

Studying grain boundary regions in polycrystalline materials

using spherical nano-indentation and orientation imaging

microscopy. J Mater Sci 47:815–823. doi: 10.1007/s10853-

011-5859-z

[47] Grinfeld M (1991) Thermodynamic methods in the theory

of heterogeneous systems. Longman Scientific and Tech-

nical, Sussex

[48] Abeyaratne R, Knowles JK (1990) On the driving traction

acting on a surface of strain discontinuity in a continuum.

J Mech Phys Solids 38:345–360

[49] Abeyaratne R, Knowles JK (1991) Kinetic relations and the

propagation of phase boundaries in solids. Arch Ration

Mech Anal 114:119–154

[50] Read WT, Shockley W (1950) Dislocation models of

crystal grain boundaries. Phys Rev 78:275–289

[51] Mughrabi H (1983) Dislocation wall and cell structures and

long-range internal stresses in deformed metal crystals.

Acta Metall 31:1367–1379

[52] Rezvanian O, Zikry MA, Rajendran AM (2007) Statisti-

cally stored, geometrically necessary and grain boundary

dislocation densities: microstructural representation and

modelling. Proc R Soc Lond A 463:2833–2853

[53] Clayton JD, McDowell DL, Bammann DJ (2006) Modeling

dislocations and disclinations with finite micropolar

elastoplasticity. Int J Plast 22:210–256

[54] Clayton JD, Bammann DJ, McDowell DL (2004a)

Anholonomic configuration spaces and metric tensors in

finite strain elastoplasticity. Int J Non Linear Mech

39:1039–1049

[55] Clayton JD (2012a) On anholonomic deformation, geom-

etry, and differentiation. Math Mech Solids 17:702–735

[56] Clayton JD (2014c) Differential geometry and kinematics

of continua. World Scientific, Singapore

[57] Steinmann P (2015) Geometrical foundations of continuum

mechanics. Springer, Berlin

[58] Regueiro RA, Bammann DJ, Marin EB, Garikipati K

(2002) A nonlocal phenomenological anisotropic finite

deformation plasticity model accounting for dislocation

defects. J Eng Mater Technol 124:380–387

[59] Clayton JD, McDowell DL, Bammann DJ (2004b) A

multiscale gradient theory for elastoviscoplasticity of single

crystals. Int J Eng Sci 42:427–457

[60] Admal NC, Po G, Marian J (2017) Diffuse-interface poly-

crystal plasticity: expressing grain boundaries as geomet-

rically necessary dislocations. Mater Theory 1:1–16

[61] Li JCM (1972) Disclination model of high angle grain

boundaries. Surf Sci 31:12–26

[62] Clayton JD, Bammann DJ, McDowell DL (2005) A geo-

metric framework for the kinematics of crystals with

defects. Philos Mag 85:3983–4010

[63] Steinmann P (2013) On the roots of continuum mechanics

in differential geometry. In: Altenbach H, Eremeyev VA

(eds) Generalized continua-from the theory to engineering

applications. Springer, Udine, pp 1–64

[64] Clayton JD (2015c) Defects in nonlinear elastic crystals:

differential geometry, finite kinematics, and second-order

analytical solutions. Z Angew Math Mech ZAMM)

95:476–510

[65] Upadhyay M, Capolungo L, Taupin V, Fressengeas C

(2011) Grain boundary and triple junction energies in

crystalline media: a disclination based approach. Int J

Solids Struct 48:3176–3193

[66] Sun XY, Cordier P, Taupin V, Fressengeas C, Jahn S (2016)

Continuous description of a grain boundary in forsterite

from atomic scale simulations: the role of disclinations.

Philo Mag 96:1757–1772

[67] Gerken JM, Dawson PR (2008) A crystal plasticity model

that incorporates stresses and strains due to slip gradients.

J Mech Phys Solids 56:1651–1672

[68] Luscher DJ, Mayeur JR, Mourad HM, Hunter A, Kena-

mond MA (2016) Coupling continuum dislocation

5542 J Mater Sci (2018) 53:5515–5545

transport with crystal plasticity for application to shock

loading conditions. Int J Plast 76:111–129

[69] Clayton JD, Hartley CS, McDowell DL (2014) The missing

term in the decomposition of finite deformation. Int J Plast

52:51–76

[70] Clayton JD (2014d) An alternative three-term decomposi-

tion for single crystal deformation motivated by non-linear

elastic dislocation solutions. Q J Mech Appl Math

67:127–158

[71] Clayton JD, Bammann DJ (2009) Finite deformations and

internal forces in elastic–plastic crystals: interpretations

from nonlinear elasticity and anharmonic lattice statics.

J Eng Mater Technol 131:041201

[72] Toupin RA, Rivlin RS (1960) Dimensional changes in

crystals caused by dislocations. J Math Phys 1:8–15

[73] Clayton JD (2009b) A continuum description of nonlinear

elasticity, slip and twinning, with application to sapphire.

Proc R Soc Lond A 465:307–334

[74] Clayton JD (2009c) A non-linear model for elastic dielec-

tric crystals with mobile vacancies. Int J Non Linear Mech

44:675–688

[75] Abdollahi A, Arias I (2012) Numerical simulation of

intergranular and transgranular crack propagation in ferro-

electric polycrystals. Int J Fract 174:3–15

[76] Fan D, Chen L-Q (1997) Computer simulation of grain

growth using a continuum field model. Acta Mater

45:611–622

[77] Allen SM, Cahn JW (1979) A microscopic theory for

antiphase boundary motion and its application to antiphase

domain coarsening. Acta Metall 27:1085–1095

[78] Abrivard G, Busso EP, Forest S, Appolaire B (2012a) Phase

field modelling of grain boundary motion driven by cur-

vature and stored energy gradients. Part I: theory and

numerical implementation. Philos Mag 92:3618–3642

[79] Abrivard G, Busso EP, Forest B, Appolaire S (2012b)

Phase field modelling of grain boundary motion driven by

curvature and stored energy gradients. Part II: application to

recrystallisation. Philos Mag 92:3643–3664

[80] James RD (1981) Finite deformation by mechanical twin-

ning. Arch Ration Mech Anal 77:143–176

[81] Zanzotto G (1996) The Cauchy–Born hypothesis, nonlinear

elasticity and mechanical twinning in crystals. Acta Crys-

tallogr A 52:839–849

[82] Bilby BA, Crocker AG (1965) The theory of the crystal-

lography of deformation twinning. Proc R Soc Lond A

288:240–255

[83] Barton NR, Winter NW, Reaugh JE (2009) Defect evolu-

tion and pore collapse in crystalline energetic materials.

Modelling Simul Mater Sci Eng 17:035003

[84] Clayton JD, Becker R (2012) Elastic–plastic behavior of

cyclotrimethylene trinitramine single crystals under spher-

ical indentation: modeling and simulation. J Appl Phys

111:063512

[85] Bhattacharya K (2003) Microstructure of martensite: why it

forms and how it gives rise to the shape-memory effect.

Oxford University Press, New York

[86] Hou TY, Rosakis P, LeFloch P (1999) A level-set approach

to the computation of twinning and phase-transition

dynamics. J Comput Phys 150:302–331

[87] Chin GY, Hosford WF, Mendorf DR (1969) Accommoda-

tion of constrained deformation in FCC metals by slip and

twinning. Proc R Soc Lond A 309:433–456

[88] Van Houtte P (1978) Simulation of the rolling and shear

texture of brass by the Taylor theory adapted for mechan-

ical twinning. Acta Metall 26:591–604

[89] Kalidindi SR (1998) Incorporation of deformation twinning

in crystal plasticity models. J Mech Phys Solids

46:267–290

[90] Staroselsky A, Anand L (1998) Inelastic deformation of

polycrystalline face centered cubic materials by slip and

twinning. J Mech Phys Solids 46:671–696

[91] Clayton JD (2010a) Modeling finite deformations in trig-

onal ceramic crystals with lattice defects. Int J Plast

26:1357–1386

[92] Mirkhani H, Joshi SP (2014) Mechanism-based crystal

plasticity modeling of twin boundary migration in nan-

otwinned face-centered-cubic metals. J Mech Phys Solids

68:107–133

[93] Clayton JD, Knap J (2011a) A phase field model of

deformation twinning: nonlinear theory and numerical

simulations. Physica D 240:841–858

[94] Hu SY, Henager CH, Chen L-Q (2010) Simulations of

stress-induced twinning and de-twinning: a phase field

model. Acta Mater 58:6554–6564

[95] Heo TW, Wang Y, Bhattacharya S, Sun X, Hu S, Chen L-Q

(2011) A phase-field model for deformation twinning.

Philos Mag Lett 91:110–121

[96] Bhattacharya K (1993) Comparison of the geometrically

nonlinear and linear theories of martensitic transformation.

Contin Mech Thermodyn 5:205–242

[97] Clayton JD, Knap J (2011b) Phase field modeling of

twinning in indentation of transparent single crystals.

Modelling Simul Mater Sci Eng 19:085005

[98] Clayton JD, Knap J (2013) Phase field analysis of fracture

induced twinning in single crystals. Acta Mater

61:5341–5353

[99] Hildebrand FE, Miehe C (2012) A phase field model for the

formation and evolution of martensitic laminate

microstructure at finite strains. Philos Mag 92:4250–4290

J Mater Sci (2018) 53:5515–5545 5543

[100] Clayton JD, Knap J (2015b) Nonlinear phase field theory

for fracture and twinning with analysis of simple shear.

Philos Mag 95:2661–2696

[101] Kondo R, Tadano Y, Shizawa K (2014) A phase-field

model of twinning and detwinning coupled with disloca-

tion-based crystal plasticity for HCP metals. Comput Mater

Sci 95:672–683

[102] Li X, Wei Y, Lu L, Lu K, Gao H (2010) Dislocation

nucleation governed softening and maximum strength in

nano-twinned metals. Nature 464:877–880

[103] Yang L, Dayal K (2010) Formulation of phase-field ener-

gies for microstructure in complex crystal structures. Appl

Phys Lett 96:081916

[104] Agrawal V, Dayal K (2015a) A dynamic phase-field model

for structural transformations and twinning: regularized

interfaces with transparent prescription of complex kinetics

and nucleation. part I: formulation and one-dimensional

characterization. J Mech Phys Solids 85:270–290

[105] Agrawal V, Dayal K (2015b) A dynamic phase-field model

for structural transformations and twinning: regularized

interfaces with transparent prescription of complex kinetics

and nucleation. part II: two-dimensional characterization

and boundary kinetics. J Mech Phys Solids 85:291–307

[106] Schultz MC, Jensen RA, Bradt RC (1994) Single crystal

cleavage of brittle materials. Int J Fract 65:291–312

[107] Clayton JD (2008) A model for deformation and frag-

mentation in crushable brittle solids. Int J Impact Eng

35:269–289

[108] Antoun T (2003) Spall fracture. Springer, New York

[109] Dugdale DS (1960) Yielding of steel sheets containing slits.

J Mech Phys Solids 8:100–104

[110] Barenblatt GI (1962) The mathematical theory of equilib-

rium cracks in brittle fracture. Adv Appl Mech 7:55–129

[111] Needleman A (1987) A continuum model for void nucle-

ation by inclusion debonding. J Appl Mech 54:525–531

[112] Xu X-P, Needleman A (1993) Void nucleation by inclusion

debonding in a crystal matrix. Modelling Simul Mater Sci

Eng 1:111–132

[113] Clayton JD, McDowell DL (2004) Homogenized finite

elastoplasticity and damage: theory and computations.

Mech Mater 36:799–824

[114] Xu X-P, Needleman A (1994) Numerical simulations of fast

crack growth in brittle solids. J Mech Phys Solids

42:1397–1434

[115] Clayton JD (2005a) Dynamic plasticity and fracture in high

density polycrystals: constitutive modeling and numerical

simulation. J Mech Phys Solids 53:261–301

[116] Clayton JD (2005b) Modeling dynamic plasticity and spall

fracture in high density polycrystalline alloys. Int J Solids

Struct 42:4613–4640

[117] Vogler TJ, Clayton JD (2008) Heterogeneous deformation

and spall of an extruded tungsten alloy: plate impact

experiments and crystal plasticity modeling. J Mech Phys

Solids 56:297–335

[118] Espinosa HD, Zavattieri PD (2003b) A grain level model

for the study of failure initiation and evolution in poly-

crystalline brittle materials. Part II: numerical examples.

Mech Mater 35:365–394

[119] Kraft RH, Molinari JF (2008) A statistical investigation of

the effects of grain boundary properties on transgranular

fracture. Acta Mater 56:4739–4749

[120] Foulk JW, Vogler TJ (2010) A grain-scale study of spall in

brittle materials. Int J Fract 163:225–242

[121] Krajcinovic D (1996) Damage mechanics. Elsevier,

Amsterdam

[122] Voyiadjis GZ, Kattan PI (2005) Damage mechanics. CRC

Press, Boca Raton

[123] Kachanov LM (1958) Time of the rupture process under

creep conditions. Isv Akad Nauk SSR Otd Tekh Nauk

8:26–31

[124] Bammann DJ, Solanki KN (2010) On kinematic, thermo-

dynamic, and kinetic coupling of a damage theory for

polycrystalline material. Int J Plast 26:775–793

[125] Del Piero G, Owen DR (1993) Structured deformations of

continua. Arch Ration Mech Anal 124:99–155

[126] Clayton JD (2006) Continuum multiscale modeling of finite

deformation plasticity and anisotropic damage in poly-

crystals. Theor Appl Fract Mech 45:163–185

[127] Clayton JD, McDowell DL (2003b) Finite polycrystalline

elastoplasticity and damage: multiscale kinematics. Int

Solids Struct 40:5669–5688

[128] Clayton JD (2010b) Deformation, fracture, and fragmen-

tation in brittle geologic solids. Int J Fract 163:151–172

[129] Bammann DJ, Aifantis EC (1989) A damage model for

ductile metals. Nucl Eng Des 116:355–362

[130] Aslan O, Cordero NM, Gaubert A, Forest S (2011)

Micromorphic approach to single crystal plasticity and

damage. Int J Eng Sci 49:1311–1325

[131] Jin YM, Wang YU, Khachaturyan AG (2001) Three-di-

mensional phase field microelasticity theory and modeling

of multiple cracks and voids. Appl Phys Lett 79:3071–3073

[132] Karma A, Kessler DA, Levine H (2001) Phase-field model

of mode III dynamic fracture. Phys Rev Lett 87:045501

[133] Del Piero G, Lancioni G, March R (2007) A variational

model for fracture mechanics: numerical experiments.

J Mech Phys Solids 55:2513–2537

[134] Clayton JD, Knap J (2014) A geometrically nonlinear phase

field theory of brittle fracture. Int J Fract 189:139–148

5544 J Mater Sci (2018) 53:5515–5545

[135] Clayton JD, Knap J (2016) Phase field modeling of coupled

fracture and twinning in single crystals and polycrystals.

Comput Methods Appl Mech Eng 312:447–467

[136] Spatschek R, Brener E, Karma A (2011) Phase field mod-

eling of crack propagation. Philos Mag 91:75–95

[137] Borden MJ, Verhoosel CV, Scott MA, Hughes TJR, Landis

CM (2012) A phase-field description of dynamic brittle

fracture. Comput Methods Appl Mech Eng 217:77–95

[138] Agrawal V, Dayal K (2017) Dependence of equilibrium

Griffith surface energy on crack speed in phase-field

models for fracture coupled to elastodynamics. Int J Fract

207:243–249

[139] McAuliffe C, Waisman H (2015) A unified model for metal

failure capturing shear banding and fracture. Int J Plast

65:131–151

[140] Wright TW, Ockendon H (1992) A model for fully formed

shear bands. J Mech Phys Solids 40:1217–1226

[141] Wright TW, Walter JW (1996) The asymptotic structure of

an adiabatic shear band in antiplane motion. J Mech Phys

Solids 44:77–97

[142] Clayton JD (2017a) Finsler geometry of nonlinear elastic

solids with internal structure. J Geom Phys 112:118–146

[143] Clayton JD (2014e) Phase field theory and analysis of

pressure-shear induced amorphization and failure in boron

carbide ceramic. AIMS Mater Sci 1:143–158

[144] Clayton JD (2017b) Generalized finsler geometric contin-

uum physics with applications in fracture and phase

transformations. Z Angew Math Phys (ZAMP) 68:9

[145] Clayton JD (2012b) Towards a nonlinear elastic represen-

tation of finite compression and instability of boron carbide

ceramic. Philos Mag 92:2860–2893

[146] Clayton JD (2016a) Finsler-geometric continuum mechan-

ics. Technical Report ARL-TR-7694, US Army Research

Laboratory, Aberdeen Proving Ground MD

[147] Kenway PR (1993) Calculated stacking-fault energies in a-

Al2O3. Philos Mag B 68:171–183

[148] Clayton JD (2010c) Modeling nonlinear electromechanical

behavior of shocked silicon carbide. J Appl Phys

107:013520

[149] Clayton JD (2011b) A nonlinear thermomechanical model

of spinel ceramics applied to aluminum oxynitride (AlON).

J Appl Mech 78:011013

[150] Clayton JD (2016b) Finsler-geometric continuum

mechanics and the micromechanics of fracture in crystals.

J Micromech Mol Phys 1:164003

[151] Clayton JD (2017) Finsler-geometric continuum dynamics

and shock compression. Int J Fract. doi:10.1007/s10704-

017-0211-5

[152] Saczuk J (1996) Finslerian foundations of solid mechanics.

Polskiej Akademii Nauk, Gdansk

[153] Stumpf H, Saczuk J (2000) A generalized model of oriented

continuum with defects. Z Angew Math Mech (ZAMM)

80:147–169

[154] Sethian JA (1999) Level set methods and fast marching

methods: evolving interfaces in computational geometry,

fluid mechanics, computer vision, and materials science.

Cambridge University Press, Cambridge

[155] Steinbach I (2009) Phase-field models in materials science.

Modelling Simul Mater Sci Eng 17:073001

[156] Chen L-Q (2002) Phase-field models for microstructure

evolution. Ann Rev Mater Res 32:113–140

J Mater Sci (2018) 53:5515–5545 5545

Approved for public release; distribution is unlimited. 32

1 DEFENSE TECHNICAL (PDF) INFORMATION CTR DTIC OCA 2 DIR ARL (PDF) IMAL HRA RECORDS MGMT RDRL DCL TECH LIB 1 GOVT PRINTG OFC (PDF) A MALHOTRA 48 ARL (PDF) RDRL CIH C J KNAP RDRL D M TSCHOPP RDRL DP T BJERKE RDRL WM B FORCH S KARNA A RAWLETT S SCHOENFELD J ZABINSKI J MCCAULEY RDRL WML B I BATYREV R PESCE-RODRIGUEZ B RICE D TAYLOR N WEINGARTEN RDRL WML H D MALLICK C MEYER B SCHUSTER RDRL WMM J BEATTY RDRL WMM B G GAZONAS D HOPKINS B LOVE B POWERS T SANO R WILDMAN RDRL WMM E J LASALVIA J SWAB RDRL WMM G J ANDZELM RDRL WMP A S BILYK RDRL WMP B C HOPPEL J MCDONALD

M SCHEIDLER T WEERISOORIYA RDRL WMP C R BECKER D CASEM J CLAYTON M FERMEN-COKER M GREENFIELD R LEAVY J LLOYD C MEREDITH S SATAPATHY S SEGLETES A SOKOLOW A TONGE C WILLIAMS RDRL WMP D R DONEY C RANDOW RDRL WMP E B AYDELOTTE