International Journal of Solids and Structures 51 (2014) 1235–1245

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International Journal of Solids and Structures

journal homepage: www.elsevier .com/locate / i jsols t r

Formulation of a macroscale corrosion damage internal state variablemodel

0020-7683/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijsolstr.2013.12.007

⇑ Corresponding author. Tel.: +1 662 325 5569; fax: +1 662 325 5433.E-mail address: cwalton@cavs.msstate.edu (C.A. Walton).

Christopher A. Walton ⇑, M.F. Horstemeyer, Holly J. Martin, D.K. FrancisCenter for Advanced Vehicular Systems (CAVS), Mississippi State University (MSU), 200 Research Boulevard, Starkville, MS 39759, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 22 August 2013Received in revised form 5 November 2013Available online 12 December 2013

Keywords:DamageCorrosionInternal state variableKinematicsThermodynamics

A new consistent formulation coupling kinematics, thermodynamics, and kinetics with damage using anextended multiplicative decomposition of the deformation gradient that accounts for corrosion effects isproposed. The corrosion model, based upon internal state variable (ISV) theory, captures the effects ofgeneral corrosion, pit nucleation, pit growth, pit coalescence, and intergranular corrosion. The differentgeometrically-affected rate equations are given for each mechanism after the ISV formalism and havea thermodynamic force pair that acts as an internal stress. Pit nucleation is defined as the number densitythat changes as a function of time driven by the local galvanic electrochemical potential between basematrix material and second phase material. Pit growth is defined as pit surface area growth. Pit coales-cence is the interaction of the pits as they grow together and is often characterized by transgranularcorrosion and is mathematically constructed from Coulomb’s Law and the Maxwell stress. General cor-rosion is signified by thickness loss of the material and is characterized by a modified Faraday’s Law.The intergranular corrosion rate is related to the grain boundary effects so that it is characterized bythe misorientation between grains. The total damage (void volume or area fraction) is the addition ofthe general, pitting, and intergranular corrosion. The ability of the model to predict aspects of the corro-sion mechanisms and aging history effects of an engineering material are then illustrated by comparisonwith experimental data of an extruded AZ31 magnesium alloy.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

In many automotive and aerospace structural component appli-cations, ductile polycrystalline materials undergo a combination ofmechanical and environmental loading. Magnesium alloys are cur-rently being investigated for use within these industries because ofits low density, excellent castability, easy machinability, and highmechanical stiffness (Froes et al., 1998; Jambor and Beyer, 1997).When exposed to saltwater, magnesium has a high corrosion rate,relegating its current use to unexposed locations within the vehi-cle (Makar and Kruger, 1993; Shaw, 2003; Song and Atrens, 2003).By altering the microstructure with the addition of variouselements, including aluminum, zinc, manganese, and rare earthelements, the corrosion rate of magnesium alloys can be reduced(Ambat et al., 2000; Lunder et al., 1985; Song and Atrens, 1999).In addition to the alteration of the microstructure, the method ofprocessing magnesium alloys can also affect the corrosion rate.The presence of an as-cast skin often found on castings, which con-sists of very small grains formed during alloy cooling, decreasesthe corrosion resistance of the magnesium alloy (Feliu Jr. et al.,

2011; Song and Xu, 2010). While castings are commonly usedfor parts of a vehicle, such as control arms or engine cradles,extruded metal is also used for bumpers and doors. Therefore, adetailed understanding of magnesium’s response to variouscorrosion conditions from different processing and product envi-ronments, through the modeling of general, pitting, and intergran-ular corrosion is needed.

Over the past few decades, significant research has been pub-lished to formulate constitutive model for ductile materials usingmultiplicative decomposition of the deformation gradient (e.g.Rice, 1971; Murakami, 1988, 1990; Bammann and Aifantis, 1989;Marin and McDowell, 1996; Steinmann and Carol, 1998; Voyiadjisand Park, 1999; Brünig, 2002; Regueiro et al., 2002; Solanki, 2008).The idea of utilizing multiplicative decomposition of the deforma-tion gradient to describe the finite deformation elastic–plastic re-sponse was introduced by Bilby et al. (1957) and independentlyby Kröner (n.d.). For this model, a kinematic approach where thedeformation due to internal defects appears naturally and capableof describing the isotropic behavior will be used. Previousresearchers (Bammann, 2001; Bammann et al., 1996, 1993;Clayton et al., 2005; Davison, 1995; Francis et al., n.d.; Horstemey-er et al., 2000; Solanki, 2008) have taken the same approach ofkinematic decomposition.

1236 C.A. Walton et al. / International Journal of Solids and Structures 51 (2014) 1235–1245

Coleman and Gurtin (1967) were the first lay out a frameworkfor internal state variable (ISV) theory. Horstemeyer andBammann (2010) recently presented a history review of ISV the-ory but did not mention any application to corrosion. In the con-text of the present article, Horstemeyer et al. (2000) modified theBammann ISV Plasticity model (Bammann et al., 1993) to capturethe stress state dependent damage evolution and to include theheterogeneities of microstructure for damage progression andfailure analysis. This model will be used as the basis for our studyand requires additional modifications to account for corrosion-based stress state dependent damage evolution. The rate equa-tions are generally written as objective rates with indifferenceto the continuum frame of reference assuming a Jaumann ratein which the continuum spin equals the elastic spin (Horstemey-er, 2012). The ISV equations are functions of the observablevariables: temperature, stress, and strain. In general, the rateequations of generalized displacements, or thermodynamicsfluxes, describing the rate of change may be written as indepen-dent equations for each ISV or as derivatives of a suitably chosenpotential function arising from the hypothesis of generalized nor-mality (Rice, 1971). The selection of the ISVs may, in principle, besomewhat arbitrary, but the kinematic hardening, isotropic hard-ening, and damage rate equations are physically motivated andstrongly influence the history of the material and can be garneredfrom lower length scale arguments (Horstemeyer, 2012). The ISVmodel accounts for deviatoric inelastic deformation resultingfrom the presence of dislocations in crystallographic material,dilatational deformation, and ensuing failure from damage pro-gression (Horstemeyer, 2012). Damage will reduce the materialstrength, enhance the inelastic flow, and soften the elasticmoduli.

To capture the corrosion behavior in a macroscale continuummodel, the kinematics, thermodynamics, and kinetics of the ISVsmust be coupled together to have a consistent theory that capturesthe entire process. The proposed model will capture the degrada-tion effects as a scalar variable. A scalar damage variable whichrepresents a strength decrease was first introduced by Kachanov(1958). Kachanov postulated that the loss of stiffness and integrityattributed to microcracks can be measured by a deterministic,macroscopic damage parameter and its change may be definedby the evolution of an ISV that depends on the expected value ofthe micro-defect density (Bammann and Solanki, 2010; Kachanov,1958). Essentially, the damage state can be thought of as the voidvolume fraction (or void area fraction).

Although the complex interactions of pit nucleation, pit growth,and pit coalescence as related to surface effects are recognized, nohydrogen bulk effects are addressed. For example, in a deleteriouscorrosive environment, hydrogen ions can diffuse into the magne-sium lattice and induce what is often called hydrogen embrittle-ment (HE) (Kappes et al., 2013; Toh and Baldwin, 1956). Byreviewing prior work (Song et al., 1997), the overall reaction forthe corrosion of pure magnesium is shown as

Mgþ 2H2O!MgðOHÞ2 þH2 ð1Þ

This shows the production of hydrogen gas from the systemwhich, in its ionic state, in-turn can absorb, and desorb, intothe matrix of the magnesium (Kappes et al., 2013). General corro-sion under stress is quite different from stress-free conditions(Walton et al., 2013). The combined action of corrosion and stresscan cause premature failure compared to the failure rate of thetwo loading cases separately. The conceptual interrelationshipof corrosion, stress effects, and HE play a huge role in the under-standing all of the key mechanisms. The most serious practicalsituation is when all three phenomena interact. The prominentdamage mechanism explaining the interaction of Mg alloys undersimultaneous corrosion and stress environments is called stress

corrosion cracking (SCC) (Toh and Baldwin, 1956). SCC is brokendown into two main forms: intergranular stress corrosion crack-ing (IGSCC) and transgranular stress corrosion cracking (TGSCC)(Atrens et al., 2011; Winzer et al., 2008). IGSCC is typically causedby a continuous second phase along grain boundaries whereTGSCC is caused by an interaction of hydrogen with the micro-structure (Atrens et al., 2011). The transformation in physical–mechanical properties and microstructure of a solid under theinfluence of chemical reactions which proceed on specimen sur-faces causing additional dislocation flux is called the chemome-chanical effect (Unigovski et al., 2007). Hence, the influence ofactive and aggressive environment on mechanically inducedstress manifests itself through the influence on the dislocationevolution observed in the form of chemomechanical effect(Unigovski et al., 2007). The kinetics behind this phenomenonwill not be addressed in this study.

The research presented here is the first of its kind in creating aphysically-based macroscale ISV inelasticity model for corrosion.The corrosion ISVs are essentially added to the thermomechanicalplasticity-damage model of Bammann (2001), Bammann et al.(1996, 1993), Horstemeyer (2012) and Horstemeyer et al. (2000)but includes electrochemical environments using conservationlaws and empirical relationships. The kinematics, thermodynam-ics, and kinetics of the surface corrosion progression are introducedin a self-consistent manner. In Horstemeyer and Bammann (2010)a history of the development of internal state variable models aredescribed and the key is the interconnectivity between the ther-modynamics, kinematics, and kinetics. Although many researchersfocus on the kinetics without considering the kinematics and ther-modynamics, they miss the constraints that the kinematics and en-ergy provide on the constitutive equations. As such, they mightappear to be physical but in reality they might violate conservationlaws. Hence, this paper starts with the kinematics and thermody-namics. Quantification work on mass loss, pit nucleation, pitgrowth, pit nearest neighbor distance (NND), and intergranularcorrosion effects similar to prior work (e.g. Alvarez et al., 2010;Martin et al., 2010a,b, 2011, 2012; Walton et al., 2012) are avail-able to validate such a model. The ability of the model to predictaspects of the corrosion process (general, pitting, and intergranularcorrosion) as a history effect are then illustrated by comparisonwith experimental data of an extruded AZ31 magnesium alloy(Walton et al., 2012).

1.1. Notation

Standard notation will be followed in this formulation. In thistext, tensors are denoted by boldface font while scalar value willhave the standard weight. All tensor components are written withrespect to a fixed Cartesian coordinate system. Special care is givento specify configurations throughout the derivation by using accentmarks where the tilde (eB), circumflex (B), and macron (B) representdifferent intermediate configurations. The following definitions areused in the text: AB) ðA � BÞij ¼ AikBkj, a� b) ða� bÞij,A : B ¼ AijBij, trðAÞ ¼ Aii, ðATÞij ¼ Aji, and kAk ¼ ðAijAijÞ1=2. The over-dot denotes the material time derivative.

2. Kinematics for a elastically damage material with corrosion

From standard continuum mechanics, all equations are writtenin the current configuration. Any motion, x, maps a particle from itsinitial position, X, in the reference configuration to its position inthe current configuration can be represented by the deformationgradient, F.

F ¼ @x@X

ð2Þ

C.A. Walton et al. / International Journal of Solids and Structures 51 (2014) 1235–1245 1237

The deformation gradient assumes a sufficient continuity,where the local deformation at X is characterized as the gradientof the motion, which is a second order, two-point tensor. Fromhere the Green elastic strain tensor, E, or Lagrangian strain tensor,is defined with respect to the reference coordinates as

E ¼ 12

FT F � I� �

ð3Þ

where I is the identity matrix. The Almansi strain tensor, B, or Eule-rian strain tensor, can also be expressed with respect to currentcoordinates as

B ¼ 12

I � ðFFTÞ�1� �

ð4Þ

For small strains, E ¼ B but will differ once the rotational de-grees of freedom start to influence the strain response. The twostrain tensors give a measure of how the lengths of line elementsand angles between line elements change between configura-tions. When adding inelasticity into the kinematics for largestrains, one can follow closely the original works of Davisonet al. (1977), Bammann and Aifantis (1989) and Horstemeyeret al. (2000). The kinematics of motion combine elastic straining,inelastic flow, and formation and growth of damage, which iscaptured by the multiplicative decomposition of the deformationgradient. The deformation can be decomposed into the elastic, Fe,damage, F/, and plastic, Fp, deformation gradients given as thefollowing,

F ¼ FeF/Fp ð5Þ

Eq. (5) assumes that the motion of the body is described by asmooth displacement function. This precludes the initiation ofdiscrete failure surfaces but still allows a continuum descriptionof damage. The elastic deformation gradient, Fe, represents latticedisplacements from equilibrium. The volumetric inelastic defor-mation gradient, F/; represents a continuous distribution of corro-sion and mechanical damage throughout the specimen. By thisnotion, the damage deformation gradient then can be decomposedby the chemical, F/c

, and mechanical, F/m, deformation gradients as

shown in the following:

F/ ¼ F/cF/m

ð6Þ

The finite strain deformation gradient (Eq. (5)) can then berewritten as

F ¼ FeF/cF/m

Fp ð7Þ

Fig. 1 shows the modified finite strain deformation gradient de-scribed in Eq. (7). The elastic deformation gradient is written firstbecause we want our material to unload elastically through F�1

e

to an inelastically damaged material state. The Jacobian, J, of thedeformation gradient can be written as

J ¼ det F ¼ det Fp det F/mdet F/c

det Fe ð8Þ

Fig. 1. Multiplicative decomposition of the deformation gradient into the plastic,chemical damage, mechanical damage, and elastic parts.

The Jacobian of the volumetric damage deformation gradient isrelated to the change in volume or change in density for constantmass (Horstemeyer et al., 2000) as

J/ ¼ det F/ ¼eVV

ð9Þ

and must be positive. The change in volume from the reference con-figuration (B0) to the intermediate configurations (B and B) or thecurrent configuration (B) is V ¼ V0 þ Vvoids þ Vc . The volume in thereference configuration is given by V0. In transforming the configu-ration from B0 to B, an added volume from the corrosion effects, Vc ,is introduced to the total volume. Damage can be defined as theratio of the change in volume of an element in the elasticallyunloaded state eB from its volume in the initial reference state toits volume in the elastically unloaded state; here we assume thatelastic volumetric deformation is minimal compared to the corro-sion and thus assume that the volume does not change from theintermediate configuration and current configuration,

/ ¼ VceV ð10Þ

From this definition, we get

V0 ¼ ð1� /ÞeV ð11Þ

where now the determinant is determined by the damage parame-ter as

det F/ ¼1

1� /ð12Þ

Consequently, the restriction that damage is assumed toproduce isotropic dilatation gives the volumetric part of thedeformation gradient as

F/ ¼1

ð1� /Þ1=3 I ð13Þ

The total strain can be obtained by pulling the intermediateconfigurations all back to the configuration x

E ¼ Ee þ E/cþ E/m

þ Ep ð14Þ

The corresponding Lagrangian strain tensors may be written as

E ¼ 12ðC � IÞ; EF ¼

12ðCp � IÞ; eE/c

¼ 12ðC/c

� IÞ;

E/m¼ 1

2ðeC/m

� IÞ; Ee ¼12ðCe � IÞ ð15Þ

where the stretch tensors, C, are defined as

C ¼ FT F; Cp ¼ FTpFp; eC/c

¼ FT/c

F/c; C/m

¼ FT/m

F/m;

Ce ¼ FTe Fe ð16Þ

Each Cauchy–Green deformation tensors in Eq. (16) canundergo a spectral decomposition of the form

C ¼X3

i¼1

k2i ni � ni ð17Þ

where ki is each positive eigenvalue which corresponds to eachorthonormal eigenvector, ni. The positive eigenvalues representthe principle stretches. Each deformation gradient has a polardecomposition of the form

F� ¼ R�U� ð18Þ

where R is the rotation tensor, U is the right stretch tensor, and (�)can be any of the configuration (e,/c ,/m,p). The relationshipbetween C and U is

1238 C.A. Walton et al. / International Journal of Solids and Structures 51 (2014) 1235–1245

U ¼ffiffiffiffiCp¼X3

i¼1

kini � ni ð19Þ

The velocity gradient associated with the deformation gradientin the current configuration separated into the elastic, corrosion,mechanical, and plastic parts is given by

l ¼ _FF�1 ¼ le þ l/cþ l/m

þ lp ð20Þ

where

_F ¼ _FeF/cF/m

Fp þ Fe_F/c

F/mFp þ FeF/c

_F/mFp þ FeF/c

F/m_Fp ð21Þ

F�1 ¼ F�1p F�1

/mF�1

/cF�1

e ð22Þ

This leaves

l ¼ _FeF�1e þ Fe

_F/cF�1

/cF�1

e þ FeF/c_F/m

F�1/m

F�1/c

F�1e

þ FeF/cF/m

_FpF�1p F�1

/mF�1

/cF�1

e ð23Þ

With the relationship described in Eq. (20), this means

le ¼ _FeF�1e

l/c¼ Fe

_F/cF�1

/cF�1

e

l/m¼ FeF/c

_F/mF�1

/mF�1

/cF�1

e

lp ¼ FeF/cF/m

_FpF�1p F�1

/mF�1

/cF�1

e

As with the strain tensor, similar additive equation holds for thevelocity gradient with respect to every configuration. By pullingback the above equation through the inverse of the elasticdeformation gradient, the velocity gradient in the intermediateconfiguration results,el ¼ ele þel/c

þel/mþelp ð24Þ

whereele ¼ F�1e

_Feel/c¼ _F/c

F�1/cel/m

¼ F/c_F/m

F�1/m

F�1/celp ¼ F/c

F/m_FpF�1

p F�1/m

F�1/c

The velocity gradient then can be decomposed into skew, w,and symmetric, d, parts in any configuration

l ¼ dþw ð25Þ

where

d ¼ symðlÞ ¼ 12ðlþ lTÞ; w ¼ skewðlÞ ¼ 1

2ðl� lTÞ

The volumetric part of the velocity gradient is then given by

el/ ¼ _F/F�1/ ¼

_/3ð1� /Þ I ð26Þ

which defines the volumetric rate of deformation as

d/ ¼_/

3ð1� /Þ I ð27Þ

The trace of the volumetric part, Eq. (27), is given as

trðd/Þ ¼_/

ð1� /Þ ð28Þ

so that the damage parameter directly relates to the volumetric rateof deformation. The elastic rate of deformation relates to the volu-metric rate of deformation by the additive decomposition of thedeformation rates,

de ¼ d� d/ � dp ð29Þ

Similarly, the rotation rates of deformation would be

we ¼ w�w/ �wp ð30Þ

Considering that corrosion will only induce inelastic volumetricchanges not deviatoric inelastic changes, we will assume that noplasticity is occurring. As a result, we can assume for small strains,the anti-symmetric component of the velocity gradient is zero,therefore, the strain rate tensors can be written as:

_e ¼ _ee þ _e/ þ _ep ð31Þ

where _ee, _ep, and _e/ are the elastic, plastic and damage-induced vol-umetric strain, respectively. Based on (Davison et al., 1977; Horste-meyer et al., 2000), we assume that damage-induced deformation isisotropic, and damage-induced strain may be written as

_e/ ¼13ð1� /Þ�1 _/I ð32Þ

3. Thermodynamics for an elastically damage material withcorrosion

The First Law of Thermodynamic in the local form is given byMalvern (1969),

q _u ¼ S : dþ qr �r � q ð33Þ

where u is the internal energy per unit mass, S is the Kirchhoffstress tensor, r is the specific heat generation rate, q is the heat fluxvector, and q is the density. The First Law can be defined in theintermediate configuration by pushing forward the symmetric partof the velocity gradient to the tilde configuration.

eq _eu ¼ eS : FTe dFe þ qr �r � q ð34Þ

For the isothermal deformation being considered, the First Lawcan be simplified to the following:

eq _eu ¼ eS : FTe dFe ð35Þ

Also, using a small strain assumption

_eEe ¼ FTe deFe ¼ FT

e dFe � ed/ � edp � ðelT �elTe ÞeEe � eEeðel �eleÞ ð36Þ

ed ¼ FTe symðelÞFe ¼ FT

e dFe ð37Þ

Eq. (35) can be written as

eq _eu ¼ eS : FTe symðelÞFe ð38Þ

By employing the ISV formulation of Coleman and Gurtin(1967), we can assume an isothermal Helmholtz free energy perunit mass, ew, as the following

ew ¼ eu; _eu ¼ _ew ð39Þ

By rearranging and substituting Eqs. (36) and (39) into Eq. (38),we obtain a modified Clausius–Duhem inequality:

�eq _ew þ eS :_eEe þ ed/ þ edp þ ðelT �elT

e ÞeEe þ eEeðel �eleÞj k

P 0 ð40Þ

The free energy, ew, may be defined as a function of a local statethat may be characterized by observable variables such elasticstrain, eEe, a set of i number of strain-like variables, eAi, with ISVssuch as thermodynamic displacement caused by damage, /. Thepower, n, describes the strength of damage, or how fast the moduliwill degrade.

ew ¼ ew eEeeC�n

/m; eEe

eC�n/c; eAi

� �ð41Þ

Fig. 2. A square bar subjected to uniaxial tension with damage effects frommechanical and chemical interactions.

C.A. Walton et al. / International Journal of Solids and Structures 51 (2014) 1235–1245 1239

eAi ¼ eesetd; ebeC�n/m; ebeC�n

/cð42Þ

where etd ¼ treC /

3 and ees ¼ bffiffiffiffiffiffieqs

p.

So

ew ¼ ew eEeeC�n

/m; eEe

eC�n/c; eesetd; ebeC�n

/m; ebeC�n

/c

� �ð43Þ

Taking the time derivative of ew to get the following,

_ew ¼ @ew@ eEe

eC�n/m

� � :_eEeeC�n

/mþ eEe �neC/m

� �ð�n�1Þ @ew@ eEe

eC�n/m

� �: eEe

_eC�n/mþ @ew@ eEe

eC�n/c

� � :_eEeeC�n

/cþ eEe �neC/c

� �ð�n�1Þ @ew@ eEe

eC�n/c

� �: eEe

_eC�n/cþ @ew@eesetd

: _eesetd þ@ew@eesetd

: ees_etd þ

@ew@ebeC�n

/m

:_ebeC�n

/mþ eb �neC/m

� �ð�n�1Þ @ew@ebeC�n

/m

: eb _eC�n/mþ @ew@ebeC�n

/c

:_ebeC�n

/cþ eb �neC/c

� �ð�n�1Þ @ew@ebeC�n

/c

: eb _eC�n/c

ð44Þ

By substituting Eq. (44) into Eq. (40), we get

eS� eq @ew@ eEe

eC�n/m

� � eC�n/m

� �T� eq @ew

@ eEeeC�n

/c

� � eC�n/c

� �T

24 35 :_eEe

þ eS :

ed/þ edpþ elT �elTe

� �eEe

þeEeel�ele

� �264

375� eq eEe �neC/m

� �ð�n�1Þ @ew@ eEe

eC�n/m

� � : eEe_eC�n

/m

24 35� eq eEe �neC/c

� �ð�n�1Þ @ew@ eEe

eC�n/c

� � : eEe_eC�n

/c

24 35� eq @ew@eesetd

: _eesetd� eq @ew@eesetd

: ees_etd

� eq @ew@ebeC/m

:_ebeC/m

� eq eb �neC/m

� �ð�n�1Þ @ew@ebeC�n

/m

: eb _eC�n/m

" #

� eq @ew@ebeC/c

:_ebeC/c

� eq eb �neC/c

� �ð�n�1Þ @ew@ebeC�n

/c

: eb _eC�n/c

" #P 0 ð45Þ

The constitutive law for eS is obtained as

eS ¼ eq @ew@ eEe

eC�n/m

� � eC�n/m

� �Tþ eq @ew

@ eEeeC�n

/c

� � eC�n/c

� �Tð46Þ

Thus, the dissipation can be simplified as:

eS :

ed/þ edpþ elT �elTe

� �eEe

þeEeel�ele

� �264

375� eq eEe �neC /m

� �ð�n�1Þ @ew@ eEe

eC�n/m

� � : eEe_eC�n

/m

24 35� eq eEeð�neC/c

Þð�n�1Þ @ew

@ eEeeC�n

/c

� � : eEe_eC�n

/c

24 35� eq @ew@eesetd

: _eesetd

� eq @ew@eesetd

: ees_etd� eq @ew

@ebeC/m

:_ebeC/m

� eq ebð�neC/mÞð�n�1Þ @ew

@ebeC�n/m

: eb _eC�n/m

" #

� eq @ew@ebeC /c

:_ebeC/c

� eq eb �neC/c

� �ð�n�1Þ @ew@ebeC�n

/c

: eb _eC�n/c

" #P 0 ð47Þ

The first term is the external work from deviatoric plastic defor-mation; the second term represents external work from damage-induced volumetric expansion, and the third term is the dissipationfrom internal work from the dislocations. The third term is relatedto ISVs.

4. Kinetics

4.1. Moduli degradation

The development of the continuum damage mechanics beganwith the introduction of a scalar damage variable (Kachanov,1958) that represents the degradation of strength in a one-dimen-sional tensile bar due to creep. Bammann et al. (1993) extendedthe idea to the so-called Unified Creep-Plasticity (UCP) framework.Similar to Bammann and Solanki (2010), we can consider a uniformbar subjected to a uniaxial tensile stress, r, as shown in Fig. 2. Theuniaxial case can easily be expressed using the formula

r ¼ PA

ð48Þ

where P is the applied load and A is the cross sectional area. In orderto use the principles of continuum damage mechanics, the follow-ing expression for the effective uniaxial stress (Kachanov, 1958;Rabotnov, 1969) is derived such that:

rD ¼P

Að1� /Þ ð49Þ

where damage is defined as the loss of cross-sectional area or a de-crease of load carrying capacity. For our case, the total damage isbased off an additive decomposition of mechanical, /m, and corro-sion, /c , interactions and is defined as the following:

/ ¼ /m þ /c ð50Þ

The effective stress tensor, rD, and the nominal stress tensor, r,for the isotropic damage case can then be rewritten as follows:

rD ¼r

ð1� /m � /cÞð51Þ

For Hookean elasticity, we get

rD ¼ Eð1� /m � /cÞee ð52Þ

Fig. 3. Macroscopic observation examples of the (a) general, (b) pitting, and (c) intergranular damage mechanisms on specimen surfaces (Alvarez et al., 2010). For the generalcorrosion micrograph (a), the bubbles are form across the surface indicating a uniform attack. The micrograph for a pit (b) shows a perspective of the in-plane surface area. Forthe intergranular corrosion micrograph (c), one can observe the corrosion attack at the grain boundaries.

1240 C.A. Walton et al. / International Journal of Solids and Structures 51 (2014) 1235–1245

where damage tends to degrade the elastic moduli. A detailed threedimensional approximation was derived by Budiansky and O’Con-nell (1976) using self-consistent techniques. They found the elasticmoduli were degraded according to

rD ¼ Kð/Þ � 23lð/Þ

� �I þ 2lð/Þee ð53Þ

where the bulk modulus coefficient Kð/Þ ¼ K0 1� 3ð1�v0Þ2ð1�2v0Þ

/h i

and

the shear modulus coefficient lð/Þ ¼ l0 1� 15ð1�v0Þ7�5v0

/h i

as discussed

in Bammann and Solanki (2010).

4.2. Damage evolution

The damage due to the mechanical loading effects follows theHorstemeyer and Gokhale (1999) and Horstemeyer et al. (2000)model where

/m ¼ gvc ð54Þ

In an effort to create a consistent, physically-based modeldescribing the corrosion of a material, we have employed conser-vation laws and empirical relationships for the following proposedmodel. The total corrosion damage is as follows:

/c ¼ /gc þ /pc þ /ic ð55Þ

where /gc is the damage from general corrosion, /pc is the damagefrom pitting corrosion, and /ic is the damage from intergranularcorrosion. Fig. 3 shows macroscopic observation examples of thegeneral (Fig. 3a), pitting (Fig. 3b), and intergranular (Fig. 3c) damagemechanisms on specimen surfaces (Alvarez et al., 2010). To calcu-late the corrosion due to pitting, a similar model to Horstemeyeret al. (2000) mechanically induced defects was developed and canbe expressed as the following:

/pc ¼ gpvpcp ð56Þ

where gp is the pit number density related to nucleation of pits(number per unit area), vp is the area of pit growth related togrowth of the pits, and cp is a function of the NND related to the coa-lescence of the pits. The generalized rate form for the total damagethen can be written as the following:

_/c ¼ _/gc þ _/pc þ _/ic ð57Þ

The assumption here is that the corrosion damage rate at anygiven time due to these mechanisms is only a surface effect.

4.2.1. General corrosion rateGeneral corrosion will be referred to corrosion dominated by

uniform thinning that proceeds without appreciable localizedattack (Davis et al., 1987). For the general corrosion rate, _/gc , amodified Faraday’s Law of Electrolysis to summarize mass loss

based on the total electric charge of the substance is presented.Faraday’s Law (Ehl and Ihde, 1954) is expressed as the following:

m ¼ QMFz

ð58Þ

where m is the mass of the substance liberated at an electrode ingrams, Q is the total electric charge that passed through the sub-stance, F is the Faraday constant = 96;485 C=mol, M is the molarmass of the substance, and z is the valence number of ions of thesubstance. For mass loss, a linear relationship with respect to timewas assumed, m ¼ C1t. Rearranging Eq. (58), we arrive at thefollowing relationship for general corrosion

_/gc ¼ _QðtÞ ¼ C1FztM

ð59Þ

where C1 is a material constant. The relationship ensures that incombination with the chemical balance equations that conservationof mass is realized.

4.2.2. Pitting corrosion rateIn the magnesium literature ‘‘localized corrosion’’ is commonly

used as pitting (Song and Atrens, 2003). Since we are using a mag-nesium alloy for our application at the end of this paper, we relatelocalized corrosion to pitting. Also, pitting corrosion can occur ineither a passive or a nonpassive manner (Davis et al., 1987). Thegeneralized pitting corrosion rate can be expressed as thefollowing:

_/pc ¼ _gpvpcp þ gp _vpcp þ gpvp _cp ð60Þ

The damage evolution of the pit nucleation mechanism followsa piecewise power and exponential trend. The relationship can beexpressed as the following,

_gpðtÞ ¼C2tC3 if t < ty

gsat þ C4tC5 if t P ty

(ð61Þ

where ty is the transition time between nucleation and coalescencedominated damage; gsat is the saturation level of the number den-sity; and C2, C3, C4, and C5 are material constants. The transitionoccurs when pits begin to coalescence at a faster rate than theynucleate across the surface. This causes the relationship to initiallyincrease when pit nucleation is dominant, and then decreasetowards the saturation level when coalescence is dominant.

The pit growth rate, _vp, can be described by the following powerlaw equation,

_vpðtÞ ¼ C6tC7 ð62Þ

where C6 and C7 are material constants. This signifies the areafraction at which pits are growing across the surface. Physicalmeasurements of the nucleation density are a number count per

C.A. Walton et al. / International Journal of Solids and Structures 51 (2014) 1235–1245 1241

unit area, so the void growth area must be used. One could employ avoid volume measurement instead of the void area measurement,but then the number density in Eq. (61) would need to be per unitvolume and not per unit area. Either choice can work; they just haveto be consistent.

The coalescence rate, _c, is assumed to relate to Coulomb’s Law(Amis, 1951) and Maxwell’s stress. Both represent an electrostaticinteraction between two charged particles with respect to the forceon the body. In our study, we focus on the electrochemical interac-tion between two pits (with some electric charge) on the surface,so we assume an averaged charged value in the region as a contin-uum point; hence, for coalescence we assume two continuumpoints. The scalar form of Coulomb’s Law is the following,

F ¼ keq1q2

r2 ð63Þ

where r is the separation distance, ke is the proportionality con-stant = 8:987� 109 Nm2=C2, q1 and q2 are point charges, and F isthe force interaction between the points. For a foundation to thismodel, the point charges will be equivalent to the elementarycharge (e = 1:602� 10�19 C) of an electron (Hull and Williams,1925; Roberge, 2008). The separation distance, r, will be definedas the nearest neighbor distance. As the NND approaches zero, theforce of interaction reaches its maximum potential. In combinationwith Maxwell’s stress, if we take the relationship and rearrange toget a rate form of the coalescence, we arrive at the following rateequation,

_cpðtÞ ¼keq1q2

pe0ðN _NDðtÞÞ4ð64Þ

Fig. 4. Theoretical corrosion scenarios in which the respective (a) general corrosion, (bmechanism dominates. The dotted line shows a baseline reference to help visualize and

where e0 is the electric constant = 8:854� 10�12 F=m and

N _NDðtÞ ¼ C8tC9 if t < ty

C10tC11 if t P ty

�with material constants of C8, C9,

C10, and C11.

4.2.3. Intergranular corrosion rateThe definition of intergranular corrosion is localized corrosion

that occurs at the grain boundaries caused by precipitates and seg-regation leading to the formation of microgalvanic cells (Daviset al., 1987). For the intergranular corrosion rate, _/ic , we postulatethe following equation,

_/ic ¼ IC _AFðtÞðMOMO0Þ

zic

ð65Þ

where IC _AFðtÞ ¼ C12tC13 if t < tc

C14tC15 if t P tc

�and tc is the transition time at

which intergranular corrosion reaches its maximum rate. C12, C13,C14, and C15 are material constants. ðMO=MO0Þzic is a misorientationfactor that represents galvanic cells formed. A fundamental aspectof the corrosion of magnesium alloys is the role of second phaseparticles. The particles cause the galvanic cells (Song and Atrens,2003) that induce a faster corrosion rate at the grain boundary thanwithin the matrix material, and Eq. (65) accounts for thatphenomenon.

5. Corrosion cases

The following section describes different corrosion scenarios inwhich each mechanism presented above is the dominant mecha-nism. Fig. 4 illustrates five cases in comparison with a baselinereference (dotted line) to help visualize the differences.

) pit nucleation, (c) pit growth, (d) pit coalescence, and (e) intergranular corrosioncompare the different cases. The solid black line represents the predicted response.

0

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003

0 10 20 30 40 50 60 70

Pit

Num

ber

Den

sity

(1/

µm2 )

Time (hr)

ModelExperiment

Fig. 6. Comparison of model and experimental data for the pit number density of anextruded AZ31 magnesium alloy immersed in a 3.5 wt% NaCl solution at roomtemperature.

1242 C.A. Walton et al. / International Journal of Solids and Structures 51 (2014) 1235–1245

5.1. Case 1: general corrosion is dominant

When general corrosion is dominant (Fig. 4a), the rate of gen-eral corrosion, _/gc , will be greater than the other rates ( _/pc and_/ic). In this case, the thickness reduces faster than pit nucleation,pit growth, pit coalescence, and intergranular corrosion rates. Sincegeneral corrosion removes material that otherwise would accountfor the pit nucleation, growth, and coalescence, one might notexperimentally see many pits. This does not mean that pitting cor-rosion would not occur, but only that the general corrosion occursso much faster that that pitting is minimized.

5.2. Case 2: pit nucleation is dominant

When pit nucleation is dominant, the pit nucleation rate, _gp, isgreater so more pits will be prevalent as opposed to more pitsgrowing. Also, the thickness reduction will be lesser allowing morepits to form, and the pits will nucleate around particles faster thanbetween the grain boundaries. Again, this does not mean that pitgrowth, pit coalescence, general corrosion, or intergranular corro-sion are not occurring, but just that they are occurring at a lesserrate than the pit nucleation rate.

5.3. Case 3: pit growth is dominant

When pit growth is dominant (Fig. 4c), the in-plane or depth pitgrowth rate, _vp, is prevalent than general corrosion, pit nucleation,and pit coalescence. Pit growth can occur by growing deep into thematerial, by spreading across the surface, or both. If this were tooccur, then one would expect to observe the greatest pit growthover time compared to the other cases. Because the general corro-sion rate would be lesser than the pit growth rate, not much thick-ness loss would occur comparatively. When comparing Cases 2 to3, one would expect that the alloying elements are less diffuselydistributed for Case 3 to admit greater pit growth. Alternatively,the alloying elements would be more involved in the intermetallicsor second phase particles for Case 2 to be more prevalent (pitnucleation being more dominant than pit growth).

5.4. Case 4: pit coalescence is dominant

When pit coalescence is dominant (Fig. 4d), the pit coalescencerate, _cp, between pits is more prevalent than general corrosion, pitnucleation, and pit growth. Pit coalescence occurs as pits join to-gether (decrease in NND) and is characterized many times by

-0.125

-0.1

-0.075

-0.05

-0.025

00 10 20 30 40 50 60 70

Mas

s L

oss

(gra

ms)

Time (hr)

ModelExperiment

Fig. 5. Comparison of model and experimental data for the mass loss of an extrudedAZ31 magnesium alloy immersed in a 3.5 wt% NaCl solution at room temperature.

transgranular corrosion. If this were to occur, then one wouldexpect to see the highest values over time compared to the othercases.

5.5. Case 5: intergranular corrosion is dominant

When intergranular corrosion is dominant (Fig. 4e), the rate ofintergranular corrosion, _/ic , is greater than the other rates ( _/pc

and _/gc). As such, the corrosion along grain boundaries occurs ata faster rate than pit nucleation, pit growth, pit coalescence, andgeneral corrosion. With a rapid deterioration of the grain bound-aries, one might also expect to see an influence on generalcorrosion as well.

6. Application of model to a structural magnesium alloy

In order to show the practical engineering use of the aforemen-tioned ISV corrosion model kinetics (Eqs. (55)–(65)), we now applythe model to an extruded AZ31 (Mg–3Al–1Zn) alloy exposed to a3.5 wt% NaCl immersion environment from a previous study(Walton et al., 2012). In this case, the environment dependentparameters were determined (shown in the Appendix A) and thenthe material constants were calibrated to the experimental data,and a comparison of the model with the experimental data isshown in Figs. 5–9 and the associated material constant valueslisted in the Appendix A.

Fig. 5 shows a comparison of the model to mass loss data fromthe immersion tests. This curve reflects the general corrosionmechanism and was calibrated by optimizing the C1 parametergiven in the Appendix A. The environmental dependence is due al-most entirely to the specimen contact with the aggressive environ-ment. This effect is fully detailed by Walton et al. (2012). Tosummarize, the continuous presence of water in the immersionenvironment allowed the corrosion by-products to dissolve, asMg(OH)2 is water soluble, and also kept the salt suspended. Theinability of the pit debris and salt residuals to remain on thesurface meant that material was removed from the Mg surfaces,leading to a decrease in weight (Walton et al., 2012). In additionto the Mg(OH)2 species, magnesium carbonate in the corrosionlayer can influence the corrosion rate (Feliu Jr. et al., 2009; Liuet al., 2012; Wang et al., 2010).

For pitting corrosion, the proposed model is multiplicativelydecomposed into the nucleation, growth, and coalescence mecha-nisms which have been correlated to quantifiable data presented

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 10 20 30 40 50 60 70

Pit A

rea

Frac

tion

Time (hr)

ModelExperiment

Fig. 7. Comparison of model and experimental data for the pit area fraction of anextruded AZ31 magnesium alloy immersed in a 3.5 wt% NaCl solution at roomtemperature.

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60 70

Nea

rest

Nei

ghbo

r D

ista

nce

(µm

)

Time (hr)

ModelExperiment

Fig. 8. Comparison of model and experimental data for the pit nearest neighbordistance of an extruded AZ31 magnesium alloy immersed in a 3.5 wt% NaCl solutionat room temperature.

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0 10 20 30 40 50 60 70

Inte

rgra

nula

r C

orro

sion

Are

a F

ract

ion

Time (hr)

ModelExperiment

Fig. 9. Comparison of model and experimental data for the intergranular corrosionarea fraction (ICAF) of an extruded AZ31 magnesium alloy immersed in a 3.5 wt%NaCl solution at room temperature.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 10 20 30 40 50 60 70

Cor

rosi

on D

amag

e -φc

Time (hr)

Fig. 10. Total corrosion damage of an extruded AZ31 magnesium alloy immersed ina 3.5 wt% NaCl solution at room temperature.

C.A. Walton et al. / International Journal of Solids and Structures 51 (2014) 1235–1245 1243

in the literature. Here, the nucleation term was defined as the pitnumber density. Fig. 6 shows a comparison of the model to thepit number density data. Pit growth has been defined as pit surfacearea growth. For comparison, the pit area fraction data not pre-sented by Walton et al. (2012) has been used for model calibration.Fig. 7 shows a comparison of the model to the pit area fractiondata. The pit coalescence term is a function of the NND. Backingout the NND from the model, we are able to compare the modelto the data (Fig. 8). These three pitting mechanisms have been pre-dicted and agree well with the experimental data. An importantaforementioned aspect of the pitting corrosion behavior is shownin the pit number density and NND plots. After four hours of expo-sure time in the immersion environment, the AZ31 magnesium al-loy shows the nucleation (Case 2) to coalescence (Case 4) transitiondominated behaviors as describe above and illustrated in Fig. 4.

Fig. 9 illustrates the intergranular corrosion area fraction pre-diction for the immersion environment. The trend shows thatintergranular corrosion takes place at an increasing rate, but atsome time the rate begins to decrease towards saturation.

With the five damage mechanisms previously described, the to-tal damage with respect to time was found. Fig. 10 shows the totalcorrosion damage, /c , described in Eq. (55). Notice for this case, thecorrosion rate (slope at any point on the curve) continues toincrease with time.

7. Conclusion

In this paper, we present a corrosion constitutive model that isinternally consistent with the thermodynamics, kinematics, andkinetics based upon the original works of Coleman and Gurtin(1967), Bammann and Aifantis (1989), Bammann (2001),Bammann et al. (1996, 1993) and Horstemeyer et al. (2000) whocreated a model in a thermomechanical damage framework. Thiswill enable the prediction of coupled mechanical and corrosion his-tory effects often called stress corrosion cracking. The modeldistinguishes between the functional and rate forms of each mech-anism and is physically motivated by empirically observed andquantified relationships. This theoretical framework presentedhere is easily extendable to the addition of other damage mecha-nisms (not shown here), and can be generalized to the develop-ment of consistent coupled equations.

A tensor ISV is introduced into the model that will affect theevolution of geometrically necessary dislocations and statisticallystored dislocations due to the chemomechanical fundamentalmechanism.

1244 C.A. Walton et al. / International Journal of Solids and Structures 51 (2014) 1235–1245

Finally, the model has been validated with the quantificationwork of an extruded AZ31 magnesium alloy. The model showsgood agreement with the mass loss, pit nucleation, pit growth,pit NND, and intergranular corrosion data (Walton et al., 2012).Although the application was to only one magnesium alloy, themodel is expected to general in nature so is expected to apply toother metal alloys as well.

Acknowledgements

Financial support from the Center for Advanced VehicularSystems at Mississippi State University is gratefully acknowledged.This work was also supported by the Department of Energy and theNational Energy Technology under Award Number No. DE-FC26-02OR22910 and Department of Defense under Award NumberNo. WD41-360413. This report was prepared as an account of worksponsored by an agency of the United States Government. Neitherthe United States Government nor any agency thereof, nor any oftheir employees, makes any warranty, expressed or implied, or as-sumes any legal liability or responsibility for the accuracy, com-pleteness, or usefulness of any information, apparatus, product,or process disclosed, or represents that its use would not infringeprivately owned rights. Reference herein to any specific commer-cial product, process, or service by trade name, trademark, manu-facturer, or otherwise does not necessarily constitute or imply itsendorsement, recommendation, or favouring by the United StatesGovernment or any agency thereof. The views and opinions ofauthors expressed herein do not necessarily state or reflect thoseof the United States Government or any agency thereof. Such sup-port does not constitute an endorsement by the Department of En-ergy nor the Department of Defense of the work or the viewsexpressed herein.

Appendix A.

Material constants used for the extruded AZ31 magnesium alloycorrosion damage model.

C1

1.44E�03 C2 1.85E�04 C3 5.00E�03 C4 3.41E�04 C5 �5.80E�01 C6 2.00E�01 C7 2.40E�01 C8 1.60E+00 C9 4.15E�02 C10 5.07E+00 C11 �8.18E�01 C12 2.10E�01 C13 6.00E�02 C14 3.36E�01 C15 �3.00E�01

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