Final Technical Report Transport Modeling of Hydrogen in Metals for Application to Hydrogen Assisted Cracking of Metals by Dr. James P. Thomas (PI) and Mr. Charles E. Chopin for the Office of Naval Research Grant Number: N00014-93-1-0845 DTIC C ELECTE OCT 1 6 1995 F Department of Aerdj'i-ce and Mechahicai Engineering University of Notre Dame ON • S N A Notre Dame, Indiana 46556 19951012 037 I i TR: 001-4/95
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Final Technical Report
Transport Modeling of Hydrogen in Metals forApplication to Hydrogen Assisted Cracking of
Metalsby
Dr. James P. Thomas (PI) and Mr. Charles E. Chopin
for theOffice of Naval Research
Grant Number: N00014-93-1-0845
DTICC ELECTE
OCT 1 6 1995
F Department of Aerdj'i-ce
and Mechahicai Engineering
University of Notre DameON • S N A Notre Dame, Indiana 46556
19951012 037
I i TR: 001-4/95
Final Technical Report
Transport Modeling of Hydrogen in Metals forApplication to Hydrogen Assisted Cracking of
Metalsby
Dr. James P. Thomas (PI) and Mr. Charles E. Chopin
for theOffice of Naval Research
Grant Number: N00014-93-1-0845
0 (
"TR " 001-4195
Final Technical Report
Transport Modeling of Hydrogen in Metalsfor Application to Hydrogen Assisted Cracking of Metals
submitted by:
Dr. James P. Thomast and Mr. Charles E. ChopintUniversity of Notre Dame
Department of Aerospace and Mechanical EngineeringNotre Dame, Indiana 46556-5637
April 4, 1995
submitted to:
Dr. A. John Sedriks, Code 3312
Office of Naval ResearchBallston Tower One
800 North Quincy StreetArlington, Virginia 22217-5660
Grant Number: N00014-93-1-0845September 1, 1993 to December 31, 1994
Principal Investigator, Assistant Professor.t Graduate Research Assistant
TABLE OF CONTENTS
ONR Contract Inform ation ............................................................................................... iiiExecutive Sum m ary ............................................................................................................... ivFinal Report ............................................................................................................................ 1
Introduction ................................................................................................................ 1Theoretical M odeling ............................................................................................... 3
Finite Elem ent M odeling ......................................................................................... 6Programm ing Sim plifications ...................................................................... 7Formulation of the Finite Element Matrix Equations ................................... 8Interpolation Functions for the Concentration and Displacement ................ 8Isoparam etric Interpolation Functions ........................................................ 10Gauss-Legendre Num erical Integration ...................................................... 11Finite Elem ent Equations ........................................................................... 12N on-Linear Solutions via Newton-Raphson ............................................... 13Re-Ordering the Degrees of Freedom ........................................................ 14AB AQUS User Elem ent Subroutines ........................................................ 15
M odeling Applications ........................................................................................... 16Results ....................................................................................................................... 17
Case 1 and 2 Problem s ............................................................................... 17Case 3 Problem s ......................................................................................... 21
D iscussion ................................................................................................................. 24Sum m ary and Future Research ........................................................................... 26
Appendix A : Analytical Solutions via M aple .................................................................. A-]Appendix B: User Elem ent Subroutine ........................................................................... B-1Appendix C: Sample Input D eck .................................................................................... C-1Appendix D : Sample Output ........................................................................................... D -1Appendix E: Conference Paper ....................................................................................... E-1Appendix F: Summary of Publications/Reports/Presentations During Grant Period .... F-1
ii
ONR CONTRACT INFORMATION
Contract Title: "Transport Modeling of Hydrogen in MetalsFor Application to Hydrogen Assisted Cracking"
Performing Organization: University of Notre Dame
Principal Investigator: Dr. James P. Thomas
Contract Number: NOO 14-93-1-0845
R & T Project Number: cor5247---01
ONR Scientific Officer: Dr. A. John Sedriks, Code 3312
iii
EXECUTIVE SUMMARY
The focus of this research was on the development of a finite element code for solutetransport and trapping in linear elastic mixtures for use in modeling the hydrogen transportprocess in metals undergoing hydrogen assisted cracking. Specific objectives included:
1.) Completion of the development of a solute transport and trapping model withcoupling between the concentration, deformation, and thermal field variables andtrapping at reversible and irreversible trap sites.
2.) Implementation of the above theory in a finite element code.3.) Calculation of the crack tip deformations and chemical state variables for some high
strength steels under a variety of loading, environment, and material conditions.A solute transport model has been developed for linear elastic mixtures with coupling
between the deformation, concentration, and thermal variables and trapping at reversible andirreversible trap sites. A journal publication describing this model is in preparation 1.
Work on a 2-D finite element implementation and its application to the modeling ofhydrogen transport in a cracking metal is ongoing. A 1-D code has been developed and testedon a variety of problems with known analytical solutions. The "code" consists of a Fortran"user element" subroutine for use with the ABAQUS 2 finite element program.Documentation of the 1-D user element subroutine is contained within this report.
Work on objective 3.) could not be started without a 2-D version of the finite elementcode. The 1-D code was used to model a variety of hydrogen transport problems with theobjective of learning more about the fully coupled transport theory. We were able to "invent"a problem with a square-root singular stress (a, - 1/ Vx). This problem was used to gainpreliminary insight into the hydrogen distribution problem in planar crack geometries. Theproblem consisted of subjecting a 4340 steel rod (10 cm x 1 cm 2) to a singular body force (-1/x 3/2 ) resulting in the square root singular stress. The steady-state hydrogen concentrationsand deformations were determined using the fully coupled theory, developed in this work,and classical stress assisted diffusion (SAD) theory.
The fully coupled predictions showed slightly higher hydrogen concentrations, a moresevere singularity in the concentration, larger axial and dilatational strains, and larger axialdisplacements, all of which depended on the extent of hydrogen trapping. The mathematicalsolution for the concentration became multi-valued as the singularity was approached. It wasshown that this "non-physical" result would be eliminated if the product of the density, themolar volume of hydrogen in the mixture, and the bulk modulus decrease linearly or betterwith increasing hydrogen concentration. These findings are documented in the report and in aconference paper included as Appendix E.
Development of a 2-D user element subroutine is ongoing. The 2-D code will initiallybe limited to simple linear elastic mixture behavior and equilibrium trapping. Extensions ofthe model to include non-equilibrium trapping effects and plastic crack tip deformations areplanned. An effort is also being made to interface the ABAQUS code with our user elementroutines to the Patran 3 Solid Geometry Modeling program for more convenient meshing ofcomplex 2-D geometries and for displaying the finite element results.
I J. P. Thomas and P. Matic, "Solute Transport Modeling in Elastic Solids", Int. J. Engnr. Sci., in preparation.2 ABAQUS is a finite element code supported by HKS, Inc., Pawtucket, RI.3 Patran is a solid geometry modeling program supported by PDA Engineering, Costa Mesa, CA.
iv
TRANSPORT MODELING OF HYDROGEN IN METALSFOR APPLICATION TO HYDROGEN ASSISTED CRACKING
INTRODUCTIONTwo fundamental questions naturally arise in modeling the influence of hydrogen on
the crack growth rate of metals. Namely, how does hydrogen "enhance" the crack growth rate(CGR); and what is the relationship between the hydrogen distribution within the materialand the corresponding "enhancement" in the CGR? Quantitative knowledge of the crack tiphydrogen distribution under service or laboratory test conditions is requisite to addressingthese questions and will require: a) a hydrogen transport model that incorporates trapping anddeformation-concentration coupling effects (governing equations); b) knowledge of the timedependent, non-uniform hydrogen distribution along the crack walls (boundary conditions);and c) a mathematical solution technique for the resulting system of non-linear equations.
Stress-assisted diffusion (SAD) theory [1,2] is commonly used to model hydrogentransport in cracking metals systems. This theory extends classical diffusion by adding ahydrostatic stress gradient term as a driving force for diffusive transport. Equilibriumtrapping effects are included through the use of an effective diffusion coefficient.
The influence of hydrogen on the material deformation state is assumed to benegligible, and this uncouples the deformation equations from the concentration variable.Hydrostatic stresses determined from solutions to standard elasticity or plasticity problemsare used with the SAD equation to solve for the resulting concentration. The steady-statehydrogen distribution for Mode I cracks in linear elastic materials with uniform hydrogenconcentration along the boundaries is given by [2]:(K
c coCXp (consta•t x K/r cos0
where co is the boundary concentration, K is the stress intensity factor, and r and 0 are polarcoordinates centered at the crack tip. Note that c --* oc as r -* 0 (the crack tip).
Transient and steady-state hydrogen distributions for a plastically deforming crack iniron have been obtained by Sofronis & McMeeking [3] using finite element methods. Theirresults show large, but finite, concentrations at the crack tip region, primarily in traps near thecrack surface. They conclude that the crack tip hydrogen distribution is primarily determinedby the creation of dislocation traps via plastic straining at the crack tip.
Damage models that attempt to link the crack tip hydrogen distribution to the fractureprocess have been reviewed in [4-6]. Applications to service cracking problems have metwith some success, but the lack of information on the crack wall hydrogen distributions underservice or laboratory conditions has limited the usefulness of these models.
The task of specifying the crack wall hydrogen distribution is difficult because of thecomplex nature of the interacting chemical, mechanical, and metallurgical processesoperative during hydrogen assisted cracking of metals (Fig. 1). In aqueous systems, thehydrogen production process is driven by the rapid and irreversible evolution of thechemically unstable "bare" surface at the crack tip to a more stable equilibrium "filmed"state. An electron flow is induced between the bare and filmed crack flank surfaces; net
anodic (dissolution/filming) reactions take place on the bare surface and net cathodic(hydrogen reduction) reactions take place on the filmed surfaces.
Adsorbed hydrogen, Hads, can be produced on both the bare and filmed cracksurfaces by: (1) the reduction of hydrogen ions in acidic environments; or (2) by thereduction of water in alkaline environments. The MHod, species are then free to be absorbedby the transition reaction (a); or combine to form H 2 gas via: recombination (bl); orelectrochemical desorption (b2). Reactions (a), (bl), and (b2) occur in parallel, but one ofthe two (bl) or (b2) reactions is usually dominant (Fig.2).
Crack Tip RegionBare C-local
Steady State Film Coverage Film Growth Surface
Me
1. Electrochemical Mass Transport '71ocal2. Anodic and Cathodic Surface Reactions3. Hydrogen Absorption Reactions4. Hydrogen Transport and Trapping5. Hydrogen Damage
Figure 1: Schematic of the processes responsible for hydrogen assisted crack growth.
(1) Acidic: M + H+ + C- # MHads(2) Alkaline: M + H 20 + e- < MH~d + OH
(a) Adsorption-Absorption: MHads • MHaibs
(bl) Recombination: MH~d, + MHad• : H2 + 2M
(b2) EC Desorption: MHd + H+ + e6 4= H 2 + M
Figure 2: Summary of the hydrogen producing reactions.
The distribution of MHabs along the crack surface is governed by the surfacecoverage of MHad, and the kinetics of reaction (a) acting in parallel with reaction (bl) or(b2). These factors are influenced, in turn, by: the electrochemical environment at the cracktip region (e.g., the potential, pH, species concentrations, dissolved 02, etc.); the kinetics ofthe bare and filmed surface reactions; and the rate of transport of H~b, from the crack surfaceinto the material.
2
Iyer and Pickering [7] have reviewed and modeled the kinetics of hydrogen evolutionand entry in stress free metallic systems with homogeneous electrochemical conditions at themetal surface. Their model is used to quantify the rate constants associated with reactions (1)or (2) and (a) and (bl) or (b2) via analysis of experimental permeation data. Turnbull [8] hasreviewed electrochemical conditions in cracks with particular emphasis on corrosion fatiguecracks of structural steels in sea water. Similarly, Beck [9] and Newman [10] have examinedexperimental techniques for characterizing bare (and filmed) surface reaction kinetics. Theabove models, data, and techniques, plus information concerning the rate controlling processduring crack growth, will have to be used in an analysis of the mass transport process withinthe crack to develop realistic predictions of the MHabs distribution.
We begin with a description of fully coupled transport and trapping theory. The use ofthe ABAQUS finite element "User Element" subroutines for solving 1-D problems is thenoutlined in full detail. This is followed by a description of three 1-D rod problems that havebeen studied. The results of these studies are reported next, followed by a discussion. Andfinally, some conclusions are drawn, and suggestions are made for further research.
THEORETICAL MODELING
This section begins with a brief description of the theory adopted for modelingcoupled diffusion/trapping processes. Derivation of the finite element equations using themethod of weighted residuals is described, followed by a description of analytical and finiteelement solutions to three simple steady-state hydrogen transport problems. While themotivation for this work is the modeling of hydrogen transport in cracking metal systems, theformulation presented here is generalized to generic solute-solid mixtures.
Three solute species are modeled in the analysis:SL - interstitial or lattice species
SR = weak or moderately (reversibly) trapped speciesSr -_ strongly (irreversibly) trapped species
Balance EquationsBalance equations for the mixture mass, the three solute species masses, mixture
linear momentum and moment of momentum, mixture energy, and mixture entropy can bewritten. In this report, our modeling considerations will be restricted to isothermal linearelastic mixtures so that only the balance equations for the solute species mass and mixturelinear momentum need be explicitly considered. Assuming negligible inertial effects, they aregiven by:
Mass: u+ V a ( = L, R, or0 ) (1)
Linear Momentum: o7ijj + F, = 0 ( .j = y, , or z) (2)
where: ck -_ mass fraction concentrations for Sk [kg/kg].
Jk concentration flux vectors for Sk [kg/kg. rm/s].Sak •mass supply rates for Sk [kg/lkgs].aij - stress tensor [N/rn2 ].
Fj =- the it component of body force vector [N/M 3 ].
3
Trapping AnalysisExpressions for the mass supply rates ak in Eq. (1) are written in accordance with the
trapping model of McNabb & Foster [11]:
Stoichiometry: SL 4* SR and SL = Si (3)
The stoichiometric relations require the supply terms sum to zero (i.e., aR + ai - aL).
Kinetics of Trapping: aR =k k(1 - OR)CL - k'40R (4)
ai = k!f(1 - O0)CL - k'O, ,.z kf (1 -- OI)CLwhere: i4 ,kf, kb , kb4 = forward and backward trap rate constants for SR and S" [1/s].
0 R and 0- cR/CR and cI/ci, respectively, are the fraction of filled trap sites [1].cR and ces saturation mass fraction concentration of SR and S. [kg/kg].
The quantities kf, kf, k , k , c' and cs are related to trap site densities, probability ofcapture, etc. and can be quantified via experimental measurement (see, for example, [12-14]).
Significant simplifications are effected when the rate constants for trapping are muchgreater than those for diffusive transport. Trapping can then be modeled as a steady-stateprocess (i.e., a = aR =- ar = 0). This case is considered below.SS Trapping: CR =cR± KRf CL- R _T = C'(5)
1 + KR C,(k f AEbN
KR j eb-:xp~ /1kR
The total internal solute concentration is simply a linear function of CL (Xi, t):
The three versions of Eq. (1) (one for each species) can be summed to give a singleequation by the following considerations. First, JR = J, = 0 because of the linear elasticmaterial assumption which precludes trap site motion (e.g., dislocation motion during plasticdeformation). Second, from Eq. (5a):
OCft &CL acic0 CR S I-- and -- 0 (7)
at at at
Summation of the three balances results in the following single mass balance equation:
S+ OCL ' (8)at
Constitutive Equations
Mass Flux: JcL CL V/LIL Linear Onsager Relation (9)R'L T
where [IL is a mass based chemical potential [15] defined by:
4
1L = 1, (°L(T) + RTln(cL) - Vske) (10)ML L L kg (10)19L ML
and: DL diffusion coefficient for SL [mr2/s].
RL gas constant for SL = R/ML [J/kg 'K].
ML = molecular mass of SL [kg/mol].T temperature [0K].
S=---(CL, CR, c-; cjj; T) free energy per unit mixture mass [J/kg].It'(T) = reference potential for SL at temperature T [J/rnol].
V, - partial molar volume of solute in the metal [m 3/mol S].
k bulk modulus of elasticity [N/mr2].
e trace of the strain tensor (i.e., e = Eii = ±ll + E22 + E33) [m/m].
The use of a mass based chemical potential simplifies the analysis of fully coupleddeformation-diffusion processes because of the primary role played by mass in thedeformation equations.
The resulting expression for concentration flux is given by:
JL DL Vs D+_L_ kcLVe (11)-DLVCL±IRT
The constitutive equation for the stress consists of Hooke's law combined with adilatational stress contribution due to changes in the total solute concentration:
Stress: ij = P a A e 6ij + 2G6ij - 3k a, (1 + c' KR) ACL 6ij (12)
where: p mass density of the solid [kg/m 3 ].
A Lame' constant [N/mr2].ij- Kronecker delta (6ij = 1 for i j and 0 otherwise).
ui the ith component of the displacement vector Irn].a, - linear expansion coefficient for internal solute = 1 k4A
3 ML r/nA]ACL - CL - cO = change in cLfrom the reference level, co.
(1 + c' KR)ACL -- change in Ctotal from the reference level.
The influence of the irreversibly trapped solute on the deformation of the solid hasbeen ignored in Eq. (12). The assumption of equilibrium trapping implies that ci-- -*c
immediately upon introducing solute into the solid. We have assumed, therefore, thatc, = c', and that all deformations are measured with respect to an initial uniformdeformation arising from the presence of the irreversibly trapped solute species at itssaturation concentration level.
5
The relationship between the chemical potential and stress (Eqs. (10) and (12)) is
dictated by the thermodynamic reciprocity relationship:
a/IL _ o(ij/P) (13)
Dcij DCL
The equations of classical stress assisted diffusion violate this requirement by ignoring theconcentration induced dilatational stresses.
Governing EquationsCombining the mass and momentum balance equations with the constitutive relations
results in the following system of governing equations for transport:
Diffusion Equation:
OCL Deff VCL - VDf k( (CL . e + CLVe) (14)at R-Df 2L ] T -- :
Deformation Equations (i = 1, 2, 3):
(A + G) e + ±GV2u± + F, = 3k ca(l + csKR) CL (15)DxiR(15)
where Doff is an "effective" diffusion coefficient defined by: Deff =- DL/(1 + CsKR).Equation (14) is identical to the SAD equations published in the literature with the
exception of the V2e term which is identically zero when linear elastic material behavior isassumed (no coupling and F2 = 0). In the present formulation it is given by:
V2e = 3k oz(16)A+ 2 G(1 + S KR)VCL (16)
FINITE ELEMENT MODELING
Equations (14) and (15) form a system of non-linearly coupled partial differentialequations that must be solved for CL and ui as functions of the space and time coordinates(xi, t). In developing the finite element equations, it is more convenient to work with thesystem of balance equations joined with appropriate constitutive equations. The 1-D form ofthe equations, with both "plane stress" and "plane strain" constitutive relations, is given by:
A single differential equation for CL, under plane stress conditions, can be obtainedusing Eqs. (17) through (19):
9CL _ [ -2 \OL s f 1 (21)t- -x LDeff 1 MLIRT (+c'KR)CL - ÷ + 3 ffFxCL(
In the absence of body forces, this is a standard diffusion equation with a concentrationdependent diffusion coefficient. Solutions to the steady-state problem, with and without bodyforces, can be obtained straightforwardly by integration; symbolic computation isrecommended (see Appendix A).
Programming Simplifications
The following generalizations are made to the expressions for dilatational strain e andaxial stress Oa to simplify the subsequent finite element coding. They are obtained using Eqs.(19) and (20) and the definition for ACL:
e=Al U + A 2CL - A3 (22)ax-, = Ble - B2CL + B 3
where Ai and Bi (i = 1, 2, or 3) are constants defined in the table below:
Constant Plane Stress Plane Strain
A, (1 - 2u) 1
A2 2(1 + P)a,(1 + c'K2 ) 0
A3 A) x co 0
B1 3k (A + 2G)
B2 9ka,(1 + c`)•,) 3ka, (I + c'Kr)
B 3 B2x cQ B2 x cO
Table 1: Constants for the dilatational strain, c, and axial stress, ax, relations.
7
The elastic constants needed as input to the finite element program are the elasticmodulus, E, and Poisson's ratio, v. Conversions from bulk modulus, k, and A + 2G(A = Lam6 constant and G = shear modulus) are given by:
Ek-= (23)
3(1- 2v)A + 2G = (1 - u)E
(I + v)(1- 2v)
The equations of classical stress assisted diffusion result by setting the constants A2 , A3 , B2 ,and B3 equal to zero.
Formulation of the Finite Element Matrix Equations
Galerkin method of weighted residuals is used to formulate the finite element matrixequations. This procedure uses the interpolation functions as weights in the integralformulation given below [ 16]:
( L+ MJ )GidV= 0 (24)
OU± + Fx HI dVe 0
where: Gi ith' concentration interpolation function, (Gi Gi(x)).Hi ft displacement interpolation function, (H= Hj(x)).
dV- differential element volume = A6 dx.
A, -6 cross-sectional area (A. z Ae(x)).
Integrating the second term of Eq.(24a) and the first term of Eq.(24b) by partsyields the coupled set of equations for a single finite element of length h (0 < x < h):
h C Gj xj -JL {& i(x) A ,d ,7 i( )IIh ( 5_ X1 A, dx - A 6 5J; f (25)
- •' {& +) F• {Hj(x)})A~dx = - A•, {H5(x)} i
The RHS term AJL represents the axial solute influx through the element boundaries andAeu, represents the axial force applied at each end of the element.
Interpolation Functions for the Concentration and DisplacementThe concentration is represented using a linear interpolation function, and the
displacement is represented using a quadratic interpolation function:
where L'"J indicates a row matrix and {...} a column matrix. Substitution of these intoEqs. (22a,b) and (17b) for the dilatational strain, stress, and mass flux, respectively, give:
e A1 d Hfx) {uj(t)} + A2 LGi(x)J{ci(t)} - A3 (28)crx •A d~ dx) 1!
=BIA, ~dIjjx) {uj(t)}+(BIA 2 - B 2) LGi(x)J{ci (t)} + (B 3 - B 1A3 )
Dc- D di(x) K, d {c((t)} ± Df D{(t) I
Substituting Eqs.(28a,b,c) into Eq.(25a,b) and collecting terms in {c(t)b}, {uM(t)}, {1(t) },and {zij(t)} yields the following matrix equation:
[[d E01 f I+ [KcI [K.] ] {c} {R}{ (29)101 101 f 7ijI[K. jI J 1
{!R•} - A, f ({Hj(x)}Fx - (B3 - B1A3) dHj(x) dx + Aex{Hj(x)} h• dx ýo • H()o
and where the cross-sectional area, Ae, is assumed to be constant within the element.The fact that [K0 ,] is zero might appear to imply that there is no coupling between
concentration and deformation, but this is not the case. The [K,] contains the first derivativeof the hydrostatic strain, ac/Ox, which contains, in turn, the second derivative in thedisplacement, u (from Eq. (22a)):
Oe 02u A2 OCL (34)OX 'ax-2 aX
The presence of the second order derivative in u would normally require the use ofC' continuous elements in order for u to satisfy element interface compatibility requirements[16]. To avoid this complication, the values of u and CL from the previous time step are usedto approximate ae/ax for the current time calculations.
The integral expressions in Eqs. (31)through(33) use interpolation functionsexpressed in terms of the global coordinate, x. The code will be implemented usingisoparametric coordinates 4 which requires replacement of the functions Gi(x) and Hj(x) byan appropriate set of isoparametric interpolation functions.
Isoparametric Interpolation Functions
Standard linear and quadratic isoparametric interpolation functions, expressed interms of the local coordinate r (- 1 < r < 1) are given by [17]:
SThe transformation from global to local coordinates is isoparametric in displacement and superparametric inconcentration. The term isoparametric will be used here to refer to the local coordinate parametrization of theelement geoimetry which extends from -1 at the left-hand end of the element to +1 at the right-hand end.
10
For an element with the global node locations, xi, the correspondence between the globalcoordinate, x, and the local coordinate, r, is given by the transformation:
x = [hj(r)J{xj} (36)
Equations (31) through (33) can now be expressed in terms of the local r coordinates via thefollowing substitutions:
is the "scalar" Jacobian transformation matrix, J, with determinant J J, and inverse[J-l] = 1. Using Eqs.(36) through (38) in Eqs. (29), (30), and (33) yields the followingrelations:
[Cc] - A {gi(r)} Lg (r)J J dr (39)
[K,,1- [0] (40)
[K,]- Ae fB 1 A1 dhj(r) dhj(r) I - drfl-i dr' JL dr IJ J'
[K,,,]A, 1 A A~2 - B2) d r) Lgi(r)j dr-1 dr" ý
/+1 ({dgi (r`) dgi (r`) iKldgi(r) D e d1[K,]- A+f D~f f d r K rýO gl)
+r1
{Rc} A, 17 {gjQr)} (41)
[±l-Ae+ B3113 1 dhr)+1JR, A (I h- (B3 - l dh3 r Jdr +Acuzfhj(r)} E
and Oe/Ox in [K,] has been replaced with (Oe/&r)/J via use of the chain-rule ofdifferentiation.
Gauss-Legendre Numerical IntegrationThe above integrals are evaluated using a Gauss-Legendre numerical integration
scheme. The variable r occurs as a 3rd degree quantity in the expression for [IV,]. A two-
11
point integration using Gauss-point locations, rk + 1/V3, and Gauss weights, Wk = 1.0
is therefore selected for use in the numerical calculations [16,17]:
2
[Co-- L_,AeWk J(rk){gi(rk)} [gi (rk)J (42)k=1
[K,,] [0] (43)AWk BA1 { dhj(rk) dhj(rk)
[ -K J(rk) dr dr
[K.] - -Ae.Wk(BiA2 - B2) dr Lgi(N)Ik=1
[K,] E AWk Deff dgi(rk) dgi(rk) K, de i(rk)} Lgi(rk)Jk=1 J(rk) d'r dr
+1{R} =- Ae• {gi(r) -1 (44)
{fRlL•= AeWk J(rk)jhj(rk)}-(B3 - BA 3) dr) + Aec7x{hj(r)} _1k=1 dr
Finite Element EquationsThe matrix finite element equations given by Eq. (29) are rewritten, following
reference [16], in the form:
[C(vo)]{fi+}o + [K(vo) ]{v}0 = {R(to)} (45)
where {v} is the vector of nodal degrees of freedom and 0 (0 < 0 < 1) parameterizes thetime integration scheme via the following definitions:
to = t + OAt{v}o (1 - 0){v}f0 + O{V}f 1+ (46)
f {)10_ VIn"i - M"~At
For 0 = 0 and a lumped capacitance matrix, the algorithm is explicit; for 0 = 1, thealgorithm is Crank-Nicolson; for 0 - 2 the algorithm is Galerkin; and for 0 = 1, the
3algorithm is the backward difference.
The coupled temperature-displacement solver routine in ABAQUS is used to solvethe fully coupled system of transport equations represented by Eq. (45). This particularABAQUS routine uses the backward difference algorithm for time integration. Substituting0 = I into Eqs. (46) gives:
12
t8=1 = t.+1 = tn + At (47){1v}e=l1 {v}n+l
I i) 10=1 = I i) 1+l 1Vn+-- {v1}n
At
which, when substituted into Eq.(45) yields:
[[K(v+)] [C(vn+l)] vn+l-AA {V} -- {IR(tn+i)} (48)
If [C] and [/C] are independent of the degrees of freedom {v}, then the problem is linear, andthe equations may be solved directly for {v} 1+, via standard linear systems solver routines.On the other hand, if [C] or [1C] is a function of v, then Eq. (48) is non-linear and must besolved using more specialized techniques. One of the techniques used by ABAQUS is theNewton-Raphson method. It is described below.
Non-Linear Solutions via Newton-RaphsonAssume that {v}j, is known and we wish to determine {v},+1 . First, define the
"residual" vector {f (vn+±)}:
{f(vn+l)} +- AP+A] t 01 -){v}n+l (49)
[C(v+i )1)- /At {t }-{R-(tn+l)}
where {f(v,+l)} = {0} for a {v} n+ which is a solution to Eq. (48). Defining {v}n'l as the
ith iterated approximation to the actual solution, {v} n+, leads to the following definition for
the (i + 1)"t iterated approximation:{v}ij3 {} + {Av}i+ (50)
i+1 A i+1
where jAvj'+ 1 is a correction vector. This correction vector {Avjn+1 can be determined byexpanding {f(v±+l)} in a Taylor's series approximation about the point {v}l'+l, retainingonly the zeroth and first order terms, and then setting the resulting expression equal to zero:
A simplification used in the numerical computations is to assume that [/I7'] and [AC']are approximately zero. This results in significant computational savings but reduces the
convergence speed for the interative solution. Eq. (55) defines the 5 x 5 "Jacobian" matrixdenoted in ABAQUS by "A MATtRX".-
Re-ordering the Degrees of Freedom
The ordering of the degrees of freedom (do) in the system matrix equation, Eq. (29)
or (45), violates the ABAQUS convention which groups the degrees of freedom by node[181. The current layout of [(C] in Eq. (45) is given by.
=[Ki K K• K,• K[2 K(V3 (56)
[hen: LUJj '+j I~ +~ KAt K 2
a non-sparse block matrix. Rearranging the order of the dofs to conform with the ABAQUS
convention requires:
14
This reordering gives the new element stiffness matrix layout:K1' 0 0 Ki 2 0
K 11 K11 K12 K12 K13[ KE] K2 KU21 K22 K 22 K2 (58)
K,21 0 0 KC2 0
The capacitance matrix must also be altered to reflect the new ordering given by Eq. (57).
ABAOUS User Element Subroutines
ABAQUS executes a Fortran subroutine named UEL for each "user defined" finiteelement in the model. Current values for the: 1. material properties; 2. total and incrementalnodal degrees of freedom; 3. time incrementation parameters; and 4. user-defined statevariables (i.e., the dilatational strain and axial stress at the Gauss integration points;extrapolated axial stress at the element boundaries; the mass flux at the center of the element;and the Gauss point locations) are passed from the main ABAQUS code into the UEL.Depending on the exact stage in the time increment, the subroutine UEL must return somecombination of the: "Jacobian" matrix (AMATRX); right-hand side vector (RHS); andupdated values of the state variables.
From Eq.(5 1), we have:a If (vi+ - {If (V +l)} (59)
n I ^ fvinl
or AMATRX { Avl+'= - RHS (60)
1
where AMATRX,• 1 [C(vin+)] + [±(Vn+l)] (61)At
and RHS = {ýR} - [iC]{ [C{Av}+ (62)
On return from subroutine UEL, ABAQUS assembles the AMATRX and RHSfrom each element into the global matrix equation:
[AMATRX]M{Av}K+11 {RHS}G (63)
and adds contributions from concentrated loads into the RHS array. ABAQUS solves Eq.(63) for the incremental correction {Avy+1 using standard solution techniques. The solutionprocess is repeated until the residual, that is, the {RHS}G vector, is smaller than somespecified tolerance, at which point the solution is accepted as correct. The tolerances usedwith this UEL subroutine must typically be set in the ABAQUS input deck, since theresiduals for the displacement and concentration variables generally show many orders ofmagnitude difference.
15
MODELING APPLICATIONSA variety of one-dimensional hydrogen transport problems have been used to explore
the differences between the fully coupled transport theory, described in this paper, andclassical SAD theory. Analytical and finite element solutions for the steady-state distributionsof: hydrogen, axial stress, axial and dilatational strains, and the axial displacement have beenobtained for a (10 cm x 1.0 cm 2) cylindrical rod of 4340 steel subjected to various boundaryconditions and applied body forces. Plane stress conditions are assumed for all problems.
The boundary conditions consist of concentration or mass flux and displacement orload (stresses) at each end of the rod. The analytical solutions are obtained using the Maplesymbolic computation program, and the numerical solutions are obtained using ABAQUSwith our custom Fortran user element subroutine. The problems examined both analyticallyand numerically are summarized in Table 2. Other problems that have been examinedanalytically are documented in Appendix A.
Case # Deformation Boundary Conditions Diffusion Boundary Conditions Body Force # Elementsla u(LHS) = 0.0 P (RHS) 0.0 cL(LHS) = 0.0 CL(RHS) = 10-1 0 102a u(LHS) = 0.0 u(RHS) = 0.0 cL(LHS) = 0.0 CL(RHS) = 10-6 0 10
Table 2: Displacements, u, are specified in [m]; loads, P•, in [N]; concentrations, CL, in[gH/gFe]; mass flux, JL, in [m/s]; and body force, Fx, in [N/r 3 ]. Case 3a (LT)and (HT) correspond to low and high trapping (1 + clKR = 20 and 500).
Case 1 and 2 problems were used to gain insight into the differences between the fullycoupled theory, SAD theory, and classical diffusion theory. They also proved useful asbenchmark problems for debugging and verifying the user element subroutine.
Case 3 was posed in order to better assess the influence of the hydrogen induceddeformations, which are not accounted for in classical SAD theory, at a crack-like stresssingularity. A square-root singular stress was introduced in the rod by subjecting it to thefictitious body force:
15 x 106 (64)VI(I - (64
The variable e is a small constant (2.0 x 1 0 -1S) included to prevent inadvertent Fortranerrors at x = 0. Substituting this into Eq. (18) and integrating results in the square-rootsingular stress:
30 x 106(65)
The numerator of Eq. (65) is equivalent to a stress-field intensity factor, K, which means thatwe have adopted a stress-field K value of 30 [MPa/in] for these Case 3 problems.
The introduction of this singularity into the 1-D rod problem resulted in someinteresting, but unexpected, behavior in the mathematical solution for the concentration as a
16
function of x. We found that the concentration becomes a multi-valued function of x as thesingularity at x = 0 is approached. The exact point at which the concentration becomesmulti-valued is predictable and depends only on material constants. Various features of theanalytical solution will be discussed in the following Results and Discussion sections, but asa consequence, we have had to modify the rod geometry in our Case 3 numerical analysis sothat the left hand boundary starts at x = 4.0 x 10-5[m] rather than x = 0. This avoids thenumerical problems associated with multi-valued concentrations.
The material properties used in the analytical and numerical analyses are given belowin Table 3:
Property ValueMass Density, p 7.8 [g/cm 3]Temperature, T 293 [K]
Lattice Diffusion Coefficient1 , DL 1 x 10-5 [cm 2/s]
Partial Molar Volume of Hydrogen 2, VH 2.0 [cm 3r/mol]Saturation Trapping Concentration 3, cR 2 x 1012 to 10-' [g H/g Fe]Reversible Trap Binding Enthalpy3 , HB 3.3 to 30 [kJ/mol]
KR = exp(-T) 4 to 2.2 x 105
Trapping Factor, (1 + c'KR) 20 and 500Molecular Weight of Hydrogen, .ML 1.00797 [g/mol]
Modulus of Elasticity, E 200 [Gpa]Poisson's Ratio, v 0.3
Reference Concentration, co 0 [g H/g Fe]
Table 3: Material property values used in the Maple and finite element analyses.
RESULTSThe results are presented in two subsections. The first subsection corresponds with
the Case la and 2a problems, and the second subsection corresponds with the Case 3problem. Each subsection is further divided into parts describing the analytical and finiteelement solution procedures and description of results.
Case 1 & 2 ProblemsAnalytical solutions to the 1-D problem can be obtained by solving Eq. (21) for
concentration. For the convenience of the reader, Eq. (21) is repeated below:
OCL =9 0l V, f; 1 I sKR CL +V, Def fFCL (1OCt O [Df i MLRT(1-c•KR)CL)± +3{fFxcL (21)
I Approximate value taken for a-Fe from Figure 12.4 of reference [19].2 Approximate value taken from Section III.A.2 of reference [6].
3 Range of values obtained from Table I of Section III.B. I of reference [6] with HB < 30 [kJ/mol].
17
This is a non-linear diffusion equation with a concentration dependent diffusion coefficient.Under steady state conditions the equation simplifies to:
( + c' KR ) dcL +Vs FCL = Constant (66)1 A4LRT i •R )c ---x 31R--T
where the constant is proportional to the mass flux that obtains under steady-state conditions.For Cases 1 & 2, the body force, Fz, is also taken to be equal to zero, so that the
governing differential equation simplifies to:
(I M=c'TKR) d Constant (67)pV(1k R ) d-L
This equation can be integrated to give cL(x), and once CL(x) is known, Eq. (18) and (22b)can be combined to give:
de _ B 2 dCL(X) (68)dx B1 dx
This can be integrated using the known function cL(x) to give the dilatational strain, e, andthe axial strain and axial stress can then be obtained directly from Eqs. (22a,b):
du _ e(x) _ A 2 A 3dx A1 A, A(
rx = Bie(x) - B2CL(X) + B 3
Finally, the displacement, u, can be obtained by integrating Eq. (69a). The constants ofintegration in the expressions for: CL(x), JL, e(x), cx(x), g,(x), and u(x) can be determinedusing the given boundary data.
The above steps were performed, in this work, using the Maple V symboliccomputation program on a Sun SPARC1O workstation. The general results are given below:
CLW = CX + C I - C 2X (70)
U(x)=C 3 x+(C 4 -C 5 X)!1- C2 x-C 4 (71)
(x) = C6 - C 7 \ - C 2X (72)
-7 Cs (73)
JL = C9 (74)
where the Q, i c {1, ... , 9} are constants that depend on the boundary conditions andmaterial property values. Documentation of the Maple analyses is given in Appendix A.
18
For the ABAQUS finite element analyses, the 10 [cm] rod is discretized using tenelements of length 1 [cm]. The initial concentration and displacement are specified as zerothroughout the rod, and the boundary conditions are as indicated in Table 2, and in thefigures below. A "coupled temperature-displacement; steady-state" ABAQUS analysis isperformed with an initial time increment of 5 x 10-3 [s], a total step time of 1.0 [s], and amaximum allowable increment of 0.5 [s]. The residual magnitude for the concentration dof issignificantly smaller than the residual for displacement dof. A separate convergence criterionfor the concentration (temperature in ABAQUS) dof is therefore adopted. The initial time-average flux for convergence of the concentration dof is set at 5.0 x 10-21 (see the AUM[II;9.6.2-1]). The input decks for the Case la and 2a analyses are given in Appendix C.
Case la and 2a rod geometries and boundary conditions are shown in Figures 3 and 4.Figures 5 through 11 show plots of: CL (X); the difference between the linear concentrationdistribution that obtains from classical SAD and diffusion analysis and cL(x); e(x); u(x);and orx(x). Keep in mind while examining these plots that e(x), cx(x), u(x), and o-x(x) areidentically zero for the classical SAD or diffusion analyses.
Figure 3: The rod geometry and boundary Figure 4: The rod geometry and boundaryconditions for Case la. conditions for Case 2a.
1 .0 6.010-
0.8 - ABAQUSAnalytical
0.6
0.4 0
2.0
o 0.2
U
0.0 0.0'0 2 4 6 8 10 0 2 4 6 8 10
Distance, x [cm] Distance, x [cm]
Figure 5: Analytical and finite element Figure 6: Difference between classicalconcentration predictions for Cases la diffusion or SAD and the fully coupledand 2a using the fully coupled theory. concentrations (i.e., 10- 7 x - CL(X) ).
Figures 5 and 6 show the concentration and concentration difference, which areidentical for the Case la and 2a problems, as a function of position along the rod. Theconcentration difference is defined as: 10-7x - CL(X) where 10-7X is the concentrationdistribution that obtains for the SAD and classical diffusion theories. The difference ismaximum at the center of the rod, but is very small at • 5.5 parts per billion (ppb). Thedifference between the analytical and finite element predictions is less than 0.05 ppb.
19
0.6:,0.3 " ABAQUS T /S-- Analytical l sAB QU
X ABAQUSW 0.4 Analytical= 0.2 x
0.1o E 0.2Cd
0.0 0.0
0.0 2.0 4.0 6.0 8.0 10.0 0 2 4 6 8 10
Distance, x [cm] Distance, x [cml
Figure 7: Fully coupled dilatational strain Figure 8: Fully coupled displacementprediction for Case la. The FE predictions prediction for Case la. The FE predictionsare given at Gauss integration points, are given at the nodes.
Figures 7 and 8 show the dilatational strain and the axial displacement experienced bythe rod for the case la conditions. The rod is traction free, and there are no applied bodyforces, so by Eq. (18), the stress in the rod must be zero. By way of verification, we note thatthe FE analysis (not shown) also predicted a zero stress throughout the rod. The maximumdilatational strain and axial displacement occur at the free end of the rod with magnitudes of; 310 [/um/m] and ; 5.2[/,m] respectively.
0.000.3 - ABAQUS
a. ABAQUSX 02, -0.04 Analytical
0.2
- -0.08- 0.1 C
Ca aa -0.12
0.0 2.0 4.0 6.0 8.0 10.0 0 2 4 6 8 10
Distance, x [cm] Distance, x [cm]
Figure 9: Fully coupled dilatational strain Figure 10: Fully coupled displacementprediction for Case 2a. The FE predictions prediction for Case 2a. The FE predictionsare given at the Gauss integration points, are given at the nodes.
Figures 9 and 10 show the dilatational strain and axial displacement for Case 2a. Therod, stress free in the absence of hydrogen, tries to expand when the hydrogen is introduced.A uniform axial compressive stress of : 10.4 [Ivf Pa] (see Figure 11) is introduced becauseof the zero displacement boundary restraints at the walls. The dilatational strain is positiveand maximum at the RH boundary of the rod, and small, but negative at the LH boundary.The axial displacement is negative throughout the rod indicating a leftward movement of therod material. The maximum displacement of z - 1.3 [t.nm] occurs at the center of the rod.Recalling the definition of axial strain, , = du/dx, we see that it is negative in the left halfof the bar, positive in the right half, and zero in the middle. The maximum strain ( ± 50[fu]) occurs at each end of the rod (see the figure on page A-2a-14 of Appendix A).
20
0.0
(a1
S-5.0 * ABAQUS- Analytical
S-10.0
-15.00.0 2.0 4.0 6.0 8.0 10.0
Distance, x [cm]
Figure 11: Fully coupled axial stress prediction for Case 2a. The FE predictionsare given at the Gauss integration points.
Case 3 ProblemsThe governing differential equation for the Case 3 problem is obtained from Eq. (66)
by substituting in for the applied body force, Fx:
Analytical solutions to this equation are obviously more difficult to obtain than in theprevious two cases.
The LHS of Eq. (75) is directly proportional to the mass flux JL; the mass flux istherefore constant throughout the rod. The Case 3 problem under study imposes a zero massflux boundary condition on the LH boundary. The constant in Eq. (75) can therefore be setequal to zero, leading to the following governing equation:
(I k( 1 + C' KR)CL dCL v (15 x 10L6)LLRT - 3 / (76)
This must be solved for CL as a function of x (i.e., for CL(X)).An interesting observation regarding Eq. (76) can be made. As CL(X) increases from
some vanishingly small value, the coefficient of the derivative term will go from positive tonegative, passing through zero on the way. When the coefficient equals zero, the derivativeterm drops out and we are left with the equation:
V8 (15 X 10L6)
3 R T 7x 3
This equation cannot be satisfied because CL is not zero, x is finite-valued, and the rest of thevariables are either non-zero constants or material properties. A mathematical solution to theproblem can only be obtained by letting dcL/dx -- z: as:
21
,LX)A4LRT (78)CLI~X) --' -2p Vs k(1+ KR)
Allowing this results in multi-valued concentration solutions to Eq. (76); Eq. (78) defines thepoint at which the solution becomes multi-valued. Since multi-valued concentrations are notphysically realistic, the assumptions made in the derivation of the fully coupled transportequations must be re-examined. This is done in the following Discussion section.
The solution to Eq. (76) is obtained by integration using Maple (see Appendix A):
eL(X) = -- C10w CIO (79)
W(x) is Lambert's W function 4 , and the Ci, i C {10, 11, 12} are constants related to thematerial properties and boundary conditions. This equation is multi-valued in x, but it can beinverted to give x as a single-valued function of eL:
X(CL) (C12)2 (80)01-0 -- in(CL)- C11
The stress field is obtained by integrating Eq. (18) with the applied body forcesubstituted in for Fz, yielding:
30 x 106ax(X)= V/•(81)
The dilatational strain can be written in terms of a, and CL using Eq. (22b):
e (X) =_ 1(GuX(x) + B2 CL (X) - B 3 ) (82)B1
Similar operations can be performed on Eq. (22a) to determine du/dx which can then beintegrated (analytical integration may not be possible) to obtain the displacement, u(x).
The ABAQUS analysis for Case 3 requires significantly smaller elements than theCase 1 and 2 analyses, particularly near the singularity. The rod is discretized into 200elements of length 1i = 0.104/0.94i-1 [trn] where the first and smallest element, 11 = 0.104[trm], is placed at the left hand end of the bar, x = 4 x 10- 3 [cm]. The 0.94 factor in thedenominator is known as the element "bias" in ABAQUS. A "coupled temperature-displacement; steady-state" analysis is used with an initial time increment of 1.0 x 10- 4 [s],a total time of 1.0 [s], and a maximum time increment of 5.0 x 10-3 [s]. The initialdisplacements in the rod are taken as zero, but an initial uniform concentration equal to theRH boundary concentration of 1 x 10-7 [g H/g Fe] is assumed. To match this initialcondition in the analysis, the RH boundary concentration is applied as a step rather than a
4 Lambert's W function satisfies the equation: W(x) x exp (W(x)) = x. Additional information can be foundusing the interactive Maple V help program.
22
ramp function. Providing an initial concentration resulted in a more rapid convergence of thesolution to the steady-state values. The residual magnitude for the concentration dof issignificantly smaller than the residual for displacement dof. A separate convergence criteriafor the concentration (temperature in ABAQUS) dof was therefore adopted. The startingtime-average flux for convergence of the concentration dof is set at 5.0 x 10-21 (see theAUM [II;9.6.2-1]). The input deck for the Case 3 analysis is given in Appendix C.
The Case 3 rod geometry and boundary conditions are shown in Figure 12. Figures 13through 17 show: cL(x); ln(cL(X)) - constant which illustrates the different concentrationsingularities; e(x); u(x); and ax(x). Plots are given for the fully coupled theory under lowand high trapping conditions, for SAD theory, and in many cases, for Fx 0. The finiteelement results are plotted for every 20th element.
Figure 12: The rod geometry, boundary Figure 13: Concentration predictions forconditions, and applied body force for the Case 3. The SAD results lie under the LTCase 3 problem. (20) curve of the fully coupled theory.
Figure 14: Concentration singularities. Figure 15: Dilatational strain for Case 3The SAD & LT-20 curves are coincident, using the fully coupled and SAD theories.
Figures 13 and 14 show the predicted concentrations for the Case 3 problems usingfully coupled and SAD theories. The SAD predictions are made using the classic expression:
CLW =c CXP(V, I 30X- 10) (83)
23
The fully coupled model predicts slightly larger concentrations than the SAD model, withincreasing differences as the singularity is approached and as the degree of trappingincreases. Figure 14 shows that the singularity for the fully coupled concentration is moresevere than the exp (I/Jf) singularity of the SAD model, at least for the high trapping case.
10.0 10,- Fully Coupled w/ F.
,S 8.0 FullyCoupledw/oF. 2.48-0.50log()C OUncoupled with F. 500 2 5
S6.0
"I • • 103S4.0S+
. 2.0 .-
0. ý 101 ,-0 2 4 6 8 10 10-3 10-2 10-1 100 10'
Distance, x [cm] Distance, x [cm]
Figure 16: Nodal displacement curves for Figure 17: Gauss point stresses with athe FE analysis of Case 3 for the fully least squares line fit showing the requiredcoupled and SAD theories, straight line behavior with a - 1/2 slope.
Figures 15 and 16 illustrate the dilatational strains and nodal displacement curves. Aswith Cases 1 and 2, the fully coupled model predicts larger displacements and dilatationalstrains throughout the rod. Again, the difference between the fully coupled and SAD modelpredictions increase with the degree of trapping. The zero-body force strains anddisplacements are also shown for comparison with the zero valued strains and displacementsthat obtain for the SAD and classical diffusion model. Figure 17 shows the FE calculatedGauss point stresses; these match the expected behavior.
DISCUSSIONThe above three rod problems illustrate the differences between the fully coupled and
classical SAD theory under steady-state conditions (additional results can be found inAppendix A). The results show that the concentration differences are generally small, butgrow in the region very near to the singularity. The deformation differences are a bit morepronounced, especially in Cases 1 and 2 where the boundary and body force loadings areabsent. All of the fully coupled results are dependent on the degree of trapping.
The results also showed that accurate finite element solutions are possible for thefully coupled transport theory using ABAQUS with custom user element subroutines. Theexperience gained during the development and application of this 1 -D user element routine isproving to be very useful in the ongoing development of the 2-D user element subroutine.
The dependence of the concentration and deformation distributions on the trappedhydrogen may be useful in the development of new "Gorsky effect" experiments for trappingcharacterization. Perhaps the time dependent displacement at the free end of a cantileveredrod can be related to some transport or trapping parameter of interest. Transient Case 1 typeproblems are probably be most useful in this regard.
24
Regarding the multi-valued concentrations that appear in the mathematical analysis ofthe Case 3 problem, the values of CL satisfying Eq. (78) for the 4340 steel considered are:2.3 x 10-5 and 9.3 x 10-7 [g H/g Fe] for the low (LT) and high (HT) trapping conditions,respectively. The total internal hydrogen, in mass fraction concentration units5, is given byEq. (6): ctotal - 4.7 x 10-4 + c' [gH/gFe]. Now, this is a very large concentrationcompared with the solubility of hydrogen in pure iron at room temperature and 1 [atm]pressure (Ctotal - 2 x 10-' [gH/gFe] or 1 x IO7 [H/Fe]). It leads us to question theassumptions made regarding the magnitude of the concentration in the development of thefully coupled model. The three major assumptions include: 1.) "infinite" dilution of the totalhydrogen in the mixture; 2.) ideal behavior; and 3.) material properties that are independentof the hydrogen concentration level.
The magnitude of the lattice concentrations at the critical point are still much lessthan one. It is unlikely, therefore, that the assumption of infinite dilution is playing any rolein this particular situation. Extension of the model to include finite concentrations, while stillretaining the ideal solution assumption, can be made by adopting a "reduced" chemicalpotential [15], which takes into account the blocking of interstitial sites in the neighborhoodof a hydrogen atom 6 . Reduced chemical potentials will be adopted in a future version of thetheory assuming that the multi-valued solutions for the concentration can be eliminated.
If we maintain the infinite dilution assumption, then non-ideality cannot play a rolebecause an infinitely dilute mixture is by definition, ideal. Non-ideal effects are only possiblein finite dilution mixtures [15], and the use of non-ideal expressions in classical SADmodeling is almost non-existent. Kirchheim and Hirth [20] have proposed a first orderextension which accounts for H-H interactions. The work by Fukai [21] and coworkers onpredicting hydrogen solubility in metals under very high pressures may also be of use inextending our model to include non-ideal effects. The question still remains; will theadoption of a non-ideal chemical potential result in governing equations with single-valuedconcentrations? We do not presently have an answer for this question; it remains to beinvestigated in the future.
The last possibility involves the assumption of constant valued material parameters. Ifthe RHS of Eq. (78) were to increase at least linearly with CL, through some concentrationdependence of the material parameters, then the multi-valued solutions will not occur. Alinear decrease in p, or k, or a square-root decrease in Vs, with increasing CL levels wouldsatisfy this requirement. Experimental evidence for these decreases in 4340 steel do notappear to be available in the literature. There is evidence, though, for decreases in V, at largeCL values for various other alloys (see, for example, the article by Peisl in [19; pp. 53-74]; orFukai [21; pp. 95-100]. This possibility will also have to be investigated in the future.
There remains one final aspect of this multi-valued solution dilemma that needs to bediscussed. That is the validity of square-root stress singularities in the fully coupled theory.Remember that this singular body force was artificially introduced in order to get the square-root singular stress. We have not yet determined whether square-root singular stresses willnaturally occur at the tip of a crack in the fully coupled theory. The 2-D finite element modelunder development should shed some light on this issue.
5 Multiplication of ctota! by AMFC/MAI ! 55 gives the mole fraction concentration, xtotai = 2.6 x 10-2 + 55 cs6 Reduced chemical potentials are also referred to as the "Fermi-Dirac" potentials [20-22].
25
SUMMARY AND FUTURE RESEARCHThe focus of this research was on the development of a finite element code for
coupled hydrogen transport and trapping in linear elastic metals for use in modeling hydrogenassisted cracking processes. A fully coupled solute transport model was developed; a 1-Dversion of the model was implemented in a finite element code via a custom Fortran "userelement" subroutine for use with the ABAQUS finite element program. A series of threesimple 1-D problems were posed to develop an understanding of the fully coupled transporttheory, and for verifying the accuracy and coding of the l-D user element subroutine. Steady-state solutions for the fully coupled theory, and for classical diffusion and stress-assisteddiffusion theories, were obtained analytically and numerically using the Maple symboliccomputation program and ABAQUS. Differences in the predicted concentrations anddeformations between the various theories were observed.
One of the problems incorporated a square-root singular stress as a "l-D analog" ofthe hydrogen transport problem in planar crack geometries. The fully coupled predictionsshowed slightly higher hydrogen concentrations, a more severe singularity in theconcentration, larger axial and dilatational strains, and larger axial displacements, all ofwhich depended on the extent of hydrogen trapping. The results indicated that the hydrogeninduced deformations, present only in the fully coupled theory, were more influential withregard to the displacements and strains.
Development of a 2-D user element subroutine is ongoing. The initial versions will belimited in scope to simple linear elastic mixture behavior and equilibrium trapping atreversible and irreversible trap sites. Rectilinear isoparametric 8-node displacementl4-nodeconcentration interpolation functions will be adopted. Extensions of the model to includenon-equilibrium trapping effects and plastic crack tip deformations are planned. An effort isalso being made to interface the ABAQUS code with our user element routines to the PatranSolid Geometry Modeling program. This will provide us with a convenient means ofmeshing complex 2-D geometries and manipulating and displaying the finite element results.
After the 2-D code has been developed, it will be used to calculate deformations andconcentrations in the crack tip region of a metal with uniform hydrogen concentrationsimposed along the crack walls. Steady-state SAD solutions for this idealized problem areavailable for comparison. The results will provide definitive answers on the importance ofthe hydrogen induced deformations in crack tip modeling.
A longer term goal is the use of the 2-D code to help establish an energy-basedparameter that characterizes the driving force for crack growth in the presence of adeleterious environment. This parameter, the.free energy release rate F, will generalize theclassical strain energy release rate g to include the hydrogen induced deformation energy andthe "free" chemical energy. It is defined by:
A(Externally Supplied Work - Specific Free Energy of the Hydrogen-Metal Mixture)A(Crack Length)
The use of F as a driving force for crack growth in environmentally assisted crackingappears to be a novel extension of the classical fracture mechanics concept. The advantage ofF over the stress intensity factor K (or AK) approach will be its ability to account forloading, environment, and material effects on the driving force for crack growth in terms of asingle variable. The concept will be applicable many material-environment crack systems of
26
technological interest including: hydrogen assisted cracking of metals, high temperatureoxidation cracking of superalloys, and moisture induced cracking of organic composites.
The magnitude of F in any environmental cracking situation will have to bedetermined by an analysis of the deformation and diffusion/trapping processes operative inthe crack tip region. More specifically, the deformation state (stresses and strains) andchemical state (potentials and concentrations for each solute species) of the mixture will haveto be determined as a function of the loading (e.g., a or Au, R, f, waveform, etc.),environment (e.g., phase, species, concentrations, T, pH, potentials, partial pressures, etc.)and material (e.g., elastic moduli, trapping parameters, solubility, diffusion coefficient, etc.)parameters. This information will be obtained using: a.) the fully coupled theory to model thedeformation-diffusion-trapping processes occurring in the crack tip region (governingequations); b.) an experimentally determined distribution of absorbed species along the crackwalls (boundary conditions); and c.) a method for solving the resulting mathematicalequations (the FE code).
Our initial efforts will be focused on the application of this concept to hydrogenassisted cracking of high strength metals. This system is ideal in the sense that it shows largeenvironmental effects, is important in many applications, and is supported by an extensivedata base. It also avoids the complications of large scale crack tip plasticity. Validation of theconcept will require demonstration of a unique correlation between the crack growth rate and.F (or A.F) for a variety of materials subject to a variety of loading, environment, andmaterial conditions.
ACKNOWLEDGMENTSThe first author's discussions with Dr. Peter Matic of the Naval Research Laboratory
concerning various aspects of the theoretical model, and both author's discussions with Dr.David Kirkner of the University of Notre Dame concerning the finite element modeling aregratefully acknowledged. The support of this work by Dr. John Sedriks of the Office of NavalResearch under Grant No. N00014-93-1-0845 is also gratefully acknowledged.
27
REFERENCES1. J. C. M. Li, R. A. Oriani, and L. S. Darken, (1966), "The Thermodynamics of Stressed
Solids", Z. Physik. Chem., Vol. 49, pp. 271-290.2. H. W. Lui, (1970), "Stress Corrosion Cracking and the Interaction Between Crack-Tip
Stress Fields and Solute Atoms", Trans. ASME-J. Basic Engng., Vol. 92, pp. 633-638.
3. P. Sofronis, and R. M. McMeeking, (1989), "Numerical Analysis of Hydrogen TransportNear a Blunting Crack Tip", J. Mech. Phys. Solids, Vol. 37, pp. 317-350.
4. H. K. Birnbaum, (1990), "Mechanisms of Hydrogen-Related Fracture of Metals", inEnvironment-Induced Cracking of Metals, Gangloff, R. P., and M. B. Ives, Eds., NACE-10, National Association of Corrosion Engineers, Houston, TX.
5. Hydrogen Degradation of Ferrous Alloys, (1985), R. A. Oriani, J. P. Hirth, and M.Smialowski, Eds., Noyes Publications, Park Ridge, NJ.
6. J. P. Hirth, (1980), "Effects of Hydrogen on the Properties of Iron and Steel", Metall.Trans. A, Vol. 11A, pp. 861-890.
7. R. N. Iyer and H. W. Pickering, (1990), Annu. Rev. Mater. Sci., Vol. 20, pp. 299-338.8. A. Turnbull, (1982), "Review of the Electrochemical Conditions in Cracks with
Particular Reference to Corrosion Fatigue of Structural Steels in Sea Water", Reviews inCoatings and Corrosion, Vol. 5, pp. 43-171.
9. T. R. Beck, (1977), "Techniques for Studying Initial Film Formation on Newly GeneratedSurfaces of Passive Metals", in Electrochemical Techniques in Corrosion, R. Baboian,Nat. Assoc. of Corrosion Engineers, Houston, TX, pp. 27-34.
10. Newman, R. C., (1984), "Measurement and Interpretation of Electrochemical Kinetics onBare Metal Surfaces" in Corrosion Chemistry Within Pits, Crevices, and Cracks, A.Turnbull, Ed., Her Majesty's Stationery Office, London, UK, pp. 317-356.
11. A. McNabb and P. K. Foster, (1963), "A New Analysis of the Diffusion of Hydrogen inIron and Ferritic Steels", Trans. TMS-AIME, Vol. 227, pp. 618-627.
12. B. G. Pound, (1989), "The Application of a Diffusion/Trapping Model for HydrogenIngress in High-Strength Alloys", CORROSION, Vol. 45, pp. 18-25.
13. H. H. Johnson, (1988), "Hydrogen in Iron", Metall. Trans. A, Vol. 19A, pp. 2371-2387.14. R. Oriani, (1970), "The Diffusion and Trapping of Hydrogen in Steel", Acta Metall.,Vol.
18, pp. 147-157.15. J. P. Thomas, (1993), "A Classical Representation for a Mass based Chemical Potential",
Int. J. Engng. Sci., Vol. 31, pp. 1279-1294.16. K. H. Huebner and E. A. Thorton, (1982), The Finite Element Method for Engineers, 2nd
ed., Wiley, NY.17. R. D. Cook, D. S. Malkus, and M. E. Plesha, (1989), Concepts and Applications of Finite
Element Analysis, 3rd ed., Wiley, NY.!8. ABAQUS Theory and User's Manuals, (1993), Standard Version 5.3-1, Hibbitt, Karlson,
& Sorensen, Inc., Pawtucket, RI.
28
19. J. V611d and G. Alefeld, (1978), "Diffusion of Hydrogen in Metals", in Hydrogen inMetals I, Topics in Applied Physics Vol. 28, G. Alefeld and J. V61kl, Eds., Springer, NY,pp. 321-348.
20. R. Kirchheim and J. P. Hirth, (1986), "Hydrogen Absorption at Cracks in Fe, Nb, andPd", in Prospectives in Hydrogen in Metals, M. F. Ashby and J. P. Hirth, Eds., PergamonPress, NY, pp. 95-98.
21. J. P. Hirth and B. Carnahan, (1978), "Hydrogen Absorption at Dislocations and Cracks",'Acta Metallurgica, Vol. 26, pp. 1795-1803.
22. Y. Fukai, (1993), The Metal Hydrogen System, Springer-Verlag, NY, pp. 53-57 and 95-100.
29
APPENDIX A: Analytical Solutions Using Maple
This Appendix contains the Maple analyses of the three steady-state rod transportproblems studied in this work. All analyses were performed using Maple V, Release 3, on aSun SPARC 10 workstation. The specific cases analyzed are summarized in the tables below:
Case # Deformation Boundary Conditions Diffusion Boundary Conditions c'KR Page
Table A-i: Case summary for rod problem #1. Displacements, u, are specified in [m]; loads,P, in [N]; concentrations, CL, in [kg H/kg Fe]; and mass flux, JL, in [m/s].
Table A-3: Summary for rod problem #3. The rod length in used in this exact analysis is 10
[cm], in distinction with the rod length used in the finite element analysis. Displacements, u,are specified in [m]; loads, P, in [N]; concentrations, CLin [kg H/kg Fe]; mass flux, JL,
in [m/s]; and body force, F•, in [N/m 3].
A-1
1. MAPLE PROGRAM DIRECTORY AND FILE NAME:/afs/nd.edu/user4/jthomas5/Maple/Transport/lDSSsol#la.ms
2. ORIGINATOR:Dr. James P. ThomasUniversity of Notre DameDepartment of Aerospace and Mechanical Engineering374 Fitzpatrick HallNotre Dame, IN 46556-5637(219) 631-9371
4. PROGRAM DESCRIPTION:One-dimensional, steady-state analysis of the fully coupled solute transport and trappingequations using "plane stress" constitutive equations for stress. The particular problemsolved is shown below:
1. dJ/dx=O (steady-state diffusion equation)2. J=Ds*(Kl*c*de/dx-dc/dx) (mass flux)3. dS/dx=O (steady-state deformation equation with zero body forces)4. S=Bl*e-B2*c+B3 (axial stress)5. de/dx-K2*dc/dx=O (dilational strain gradient)6. e=Al*du/dx+A2*c-A3 (dilatational strain)
The constants used in the plane stress analysis are defined below:
Ds:=lattice diffusivity [m^2/sec]
Kl:=(Vs*K)/(R*T)Vs:= partial molar volume of solute [m^3/mol solute]K:= bulk modulus=Em/(3*(l-2*nu)) [N/m^2]Em:= Modulus of Elasticity [N/m^2]nu:= Poisson's ratio [1]R:= universal gas constant=8.31432 [J/mol-K]T:= temperature [K]
K2:=3*alpha*(l+traps)alpha:= (rho*Vs) / (3*MWs)=solute concentration expansion coefficient [m/m/Delc]rho:= mass density of the solid [kg/m^3]MWs:- molecular weight of the solute [kg/mol]Delc:= c-cO [1]
+ B2 l + 2 KI K2 Cl x + 2 K1K2 _Cl C2 + B3 KlK2)( KlK2)
Note #3: At this point, we will introduce expressions for the constants andthen simplify the results before quantifying the constants anddetermining the constants of integration.
"> K:=(Em/(3*(1-2*nu)));
Em
3 -6v"> lambda:=(Em*nu)/((1+nu)*(1-2*nu));
Em v
(1 +v)(1 -2v)"> mu:=Em/(2*(1+nu));
Em2+2v
"> aIpha:=(rho*Vs)/(3*MWs);
A-i a-5
1 p Vs3 MWs
" K :=(Vs*K)/(R*T);
Vs EmK1:=
(3 -6v)RT"> K2:=3*alpha*(1 +traps);
p Vs ( 1 + traps)K2 :MWs
"> Kl*K2;
Vs2 Em p (1 + traps)
(3 - 6 v) R TMWs"> A1 :=(1-2*nu);
A]:= 1 -2 v"> A2:=2*(l +nu)*alpha*(l +traps);
2 (1 +v)p Vs(l +traps)A2 : =3 MWs
"> A3:=A2*cO;
2 (1+v)pVs(1+traps)cOA3 := 3 MWs
"> B1:=3*K;
EmB] :=3-
3 -6v"> B2:=9*K*alpha*(l+traps);
B2: :3 Em p Vs ( 1 + traps)(3 - 6 v) MWs
"> B3:=B2*cO;
Em p Vs ( 1 + traps) cOB3 :=3(3 - 6 v) MWs
A-la-6
"> cl (X);
1+, 1+2 Vs 2 Emp(1+traps) xClx(+2 Vs2 Emp(l+traps) _Cl C2 (3 6v)RT1 (3 -6 v) RTMWs (3 - 6v) RTMWs )MWs/(Vs2 Em p (1 + traps))
"> c2(x);
Vs2 Emp(1 +trapsClx 2 Vs 2 Emp(1+traps) -l (36v)RT1 (3 - 6 v) R TMWs (3 - 6 v) R TMWs (3 v
MWs/(Vs2 Em p (1 + traps))
"> el(x);
Vs2 Emp(I+traps) -Clx Vs2 Emp(I+traps)-C1-C2 C3 Vs EmK1+2-Cl +2
Vs2 Em p (1 + traps) -C1 x Vs2 Em p (1 + traps) C1 C2%1 : 1 +2 +2 -
(3 - 6v) R TMWs (3- 6v) R TMWs
"> Sl(x);
- (MWs _C3 Vs Em - 3 R TMWs + 6 R TMWs v +Em p Vs2 cO + Em p Vs2 cO traps)/(
(-1 + 2 v) Vs MWs)"> S2(x);
- (MWs _C3 Vs Em- 3 R TMWs + 6 R TMWs v +Em p Vs2 cO + Em p Vs2 cO traps)/(
(-1 + 2 v) Vs MWs)
"> S1 (x)-S2(x);
0
Note #4: The stresses are same for each solution root, and do not depend onthe position x! The stresses are therefore constant throughout thebody. We will solve for C3 in terms of the boundary stress, SO.
"> _C3:=simplify(solve(S1 (x)=SO,_C3));
_C3:= - (-3 R TMWs + 6 R TMWs v + Em p Vs 2 cO + Em p Vs2 cO traps - SO Vs MWs
+ 2 SO Vs MWs v)/(MWs Vs Em)
A-ia-8
"> S1(x);
So"> S2(x);
so"> Sl:=x->S0;
Si :=x - SO"> S2:=x->SO;
82 :=x-->SO
Note #5: Now we will introduce numerical values for the material properties.The values selected are typical for internal hydrogen as asolute in AISI 4340 steel.
"> SO:=0.0;
SO:= 0> Ds:=l e-9;
Ds:= .110-"> Vs:=2.02e-6;
Vs :=.202 10-'"> traps:=19;
traps := 19"> cO:=0.0;
cO:= 0
"> MWs:=0.00100797;
MWs :=.00100797"> Em:=200e9;
Em :=.200 1012
"> nu:=0.3;
V :=.3
"> rho:=7800;
p := 7800> T:=293;
T:= 293> R:=8.31432;
R:= 8.31432> K;
.166666666666667 1012
"> lambda;
.115384615384616 1012
"> mu;
.769230769230769 1011"> alpha;
A-ia-9
5.21047253390476"> Rh:=R/MWs;
Rh := 8248.57882675080"> Ki;
138.199274509089"> K2;
312.628352034286"> KI*K2;
43205.0114421104"> 1/(Kl*K2);
.0000231454631447068"> Al;
.4"> A2;
270.944571763048"> A3;
0>BI;
.500000000000001 1012
"> B2;
.156314176017143 1015
"> B3;
0> 03;
.00723592800000000"> SI(x);
0"> S2(x);
0
Note #6: Now let's solve for the integration constants: _Cl, _C2, and C4, using theboundary data.
"> cl (X);
.0000231454631447068 + .0000231454631447068
SIl + 86410.0228842208 _Cl x + 86410.0228842208 Cl _C2"> c2(x);
.0000231454631447068 - .0000231454631447068
I + 86410.0228842208 _Cl x + 86410.0228842208 _Cl _C2"> evalf(cl (x),5);
Note #7: The solution cl(x) is not capable of satisfying the boundaryconditions, as evidenced by the lack of solution for the givenboundary conditions. The concentration function c2(x) is thecorrect solution in this case!
-.0072359 F1. - .84542 x + .0072359"> evalf(ex(x),5);
1 x.0024120-.00080399 +.00067972 .0016080 fl. - .845 4 3 x
1.-.84543 x , 1.-.84543 x
"> J2(x);
-.978397494278943 10-14
A-la-14
1. MAPLE PROGRAM DIRECTORY AND FILE NAME:/afs/nd.edu/user4/j thomas5/Maple/Transport/lDSSsol#lb.ms
2. ORIGINATOR:Dr. James P. ThomasUniversity of Notre DameDepartment of AerospaCe and Mechanical Engineering374 Fitzpatrick HallNotre Dame, IN 46556-5637(2i9) 631-9371
4. PROGRAM DESCRIPTION:One-dimensional, steady-state analysis of the fully coupled solute transport and trappingequations using "plane stress" constitutive equations for stress. The particular problemsolved is shown below:
1. dJ/dx=0 (steady-state diffusion equation)2. J=Ds*(Kl*c*de/dx-dc/dx) (mass flux)3. dS/dx=G (steady-,state deformation equation with zero body forces)4. S=Bl*e-B2*c+B3 (axial stress)5. de/dx-K2*dc/dxý0 (d•iational strain gradient)6. e=Al*du/dx+A2*c.-A.i (dilatational strain)
The constants used in (ie plane stress analysis are defined below:
Ds:=lattice diffusivily Im^2/sec]
KI:=(Vs*K)/(R*T)Vs:= partial molar voihlime of solute [m^3/mol solute]K:= bulk modulus=Em/(3*(l-2*nu)) [N/m^2]Em:= Modulus of Elast icity [N/m^2]nu:= Poisson's ratio [I]R:= universal gas consant=8.31432 [J/mol-K]T:ý temperature [K)
K2:=3*alpha*(l+traps)alpha:= (rho*Vs)/)3WMWs)=solute concentration expansion coefficient [m/m/Delc]rho:ý mass density of the solid [kg/m^3]MWs:= molecular weight of the solute (kg/mol]Delc:= c-co [1)
Vs2 Em p (1 + traps) Cl x Vs2 Em p (1 + traps) _C C2%1:=1+2 +2 _
(3 - 6 v) R TMWs (3 - 6 v) R TMWs
"> S1(x);
- (MWs _C3 Vs Em - 3 R TMWs + 6 R TMWs v +Em p Vs 2 cO + Em p Vs2 cO traps)/(
(-1 +2v) VsMVLs)"> S2(x);
- (MWs _C3 Vs Em - 3 R TMWs + 6 R TMWs v +Em p Vs2 cO +Em p Vs2 cO traps)/(
(-1 + 2 v) Vs MWs)
"> SI(x)-S2(x);
0
Note #4: The stresses are same for each solution root, and do not depend onthe position x! The stresses are therefore constant throughout thebody. We will solve for C3 in terms of the boundary stress, SO.
"> C3:=simplify(soIve(S1 (x)=SO,_C3));
_C3:= - (-3 R T MWs + 6 R T MWs v + Em p Vs 2 cO + Em p Vs2 cO traps - SO Vs MWs
+ 2 SO Vs MWs v)/(Em MWs Vs)"> S1 (x);
SO"> S2(x);
A-lb-8
So"> Si :=x->SO;
Si := x SO"> S2:=x->SO;
S2 :=x- SO
Note #5: Now we will introduce numerical values for the material properties.The values selected are typical for internal hydrogen as asolute in AISI 4340 steel.
"> So:=0.0;
SO:= 0"> Ds:=le-9;
Ds:= .1 10-8
> Vs:=2.02e-6;
Vs :=.202 10-5"> traps:=19;
traps := 19"> cO:=0.O;
cO:= 0"> MWs:=0.00100797;
MWs :=.00100797
> Em:=200e9;
Em :=.200 1012
"> nu:=0.3;
V :=.3
"> rho:=7800;
p := 7800
"> T:=293;
T:= 293
"> R:=8.31432;
R:= 8.31432
> K;
.166666666666667 1012
"> lambda;
.115384615384616 1012
"> mu;
.769230769230769 1011
"> alpha;
5.21047253390476
"> Rh:=R/MWs;
A-lb-9
Rh:= 8248.57882675080"> K1;
138.199274509089"> K2;
312.628352034286"> KI*K2;
43205.0114421104"> 1/(Kl*K2);
.0000231454631447068"> Al;
.4
"> A2;
270.944571763048"> A3;
0"> B1;
.500000000000001 1012
"> B2;
.156314176017143 1015"> B3;
0> 03;
.00723592800000000"> Sl(x);
0"> S2(x);
0
Note #6: Now let's solve for the integration constants: _C, _C2, and _C4, using theboundary data.
"> ci(X);
.0000231454631447068 + .0000231454631447068
j1 + 86410.0228842208 _C] x + 86410.0228842208 Cl _C2"> c2(x);
.0000231454631447068 - .0000231454631447068
J + 86410.0228842208 _C] x + 86410.0228842208 _CI _C2"> evalf(cl(x),5);
.000023146 + .000023146 j 1. + 86408. _C] x + 86408. CI _C2
Note #7: The solution cl(x) is not capable of satisfying the boundaryconditions, as evidenced by the lack of solution for the givenboundary conditions. The concentration function c2 (x) is thecorrect solution in this case!
-.0072359 F1. - .086222 x + .0072359"> evalf(ex(x),5);
1 x.0024120 - .00080399 + .000069323
ý1. -. 086223x 1.-.086223x
- .0016080 1 1. - .086223x
"> J2(x);
-.997839749427893 10-15
A-lb-14
1. MAPLE PROGRAM DIRECTORY AND FILE NAME:/afs/nd.edu/user4/jthomas5/Maple/Transport/lDSSsol#lc.ms
2. ORIGINATOR:Dr. James P. ThomasUniversity of Notre DameDepartment of Aerospace and Mechanical Engineering374 Fitzpatrick HallNotre Dame, IN 46556-5637(219) 631-9371
4. PROGRAM DESCRIPTION:One-dimensional, steady-state analysis of the fully coupled solute transport and trappingequations using "plane stress" constitutive equations for stress. The particular problemsolved is shown below:
1. dJ/dx=O (steady-state diffusion equation)2. J=Ds*(Kl*c*de/dx-dc/dx) (mass flux)3. dS/dx=O (steady-state deformation equation with zero body forces)4. S=Bl*e-B2*c+B3 (axial stress)5. de/dx-K2*dc/dx=O (dilational strain gradient)6. e=Al*du/dx+A2*c-A3 (dilatational strain)
The constants used in the plane stress analysis are defined below:
Ds:=lattice diffusivity [m^2/sec]
Kl:=(Vs*K)/(R*T)Vs:= partial molar volume of solute [m^3/mol solute]K:- bulk modulus=Em/(3*(l-2*nu)) [N/m^2]Em:= Modulus of Elasticity [N/m^2]nu:= Poisson's ratio [1]R:= universal gas constant=8.31432 [J/mol-K]T:= temperature [K]
K2 :3*alpha*(l+traps)alpha:= (rho*Vs) / (3* MWs)=solute concentration expansion coefficient [m/m/Delc]rho:- mass density of the solid [kg/m^3]MWs:= molecular weight of the solute [kg/mol]Delc:= c-cO [1]
- B2 J 1 + 2 K1 K2 _C] x + 2 K1 K2 _C] _C2- B3 K K2)/(K] K2)
Note #3: At this point, we will introduce expressions for the constants andthen simplify the results before quantifying the constants anddetermining the constants of integration.
"> K:=(Em/(3*(1-2*n u)));
EmK:= 3-6v
"> Iambda:=(Em*nu)/((l+nu)*(1-2*nu));
Em v
(1+v)(1 -2v)"> mu:=Em/(2*(l+nu));
Em2+2v
"> alpha:=(rho*Vs)/(3*MWs);
A-Ic-5
I p Vs
3 MWs"> K1 :=(Vs*K)/(R*T);
Vs EmK1.-
(3 -6v)RT"> K2:=3*alpha*(1 +traps);
p Vs ( 1 + traps)K2 :MWs
"> KI*K2;
Vs2 Em p (1 + traps)
(3 - 6 v) R TMWs"> A1:=(1-2*nu);
Al:= 1 -2 v"> A2:=2*(l +nu)*alpha*(l +traps);
2 (1 +v)p Vs(1 +traps)A2:=3 MWs
"> A3:=A2*cO;
2 (1 +v) p Vs (1 +traps) cOA3 =-3 MWs
"> B1 :=3*K;
Em3-6v
"> B2:=9*K*alpha*(1 +traps);
Em p Vs (1 + traps)B2 :=3(3 - 6 v) MWs
"> B3:=B2*cO;
Em p Vs (1 + traps) cOB3:=3(3 - 6 v) MWs
A-ic-6
"> cl (X);
(1+ 1+2 Vs2 Emp(1+traps) Clx Vs + 2 Emp(I+traps) Cl C2 (3 6v)RTJ (3 -6v) RTMWs (3 - 6v) RTMWsMWs/(Vs2 Em p (1 + traps))
"> c2(x);S / Vs2 Em p(1+ traps) -Cl x Vs2 Em p( 1+ traps) -C1 C21
-f1+j1+2 Vs +2 -- (-vR1 (3 - 6 v) R TMWs (3 - 6 v) R TMWs
MWs/(Vs2 Em p (1 + traps))
"> el(x);
r Vs2 Em l(+traps) Clx Vs2 Emp(1+traps) _C1 C2 C3VsEmJ1+2 (3-6v)RTMWs (3-6v)RTMWs (3-6v)RTT
Vs2 Em p (1 + traps) -C1 x Vs2 Em p (1 + traps) -C1 C2%1 := 1 +2 +2
(3 - 6 v) R TMWs (3 - 6 v) R TMWs
"> S1(x);
- (MWs _C3 Vs Em- 3 R TMWs + 6 R TMWs v + Em p Vs2 cO +Em p Vs2 cO traps)/(
(-1 +2v) VsMWs)"> S2(x);
- (MWs _C3 Vs Em - 3 R TMWs + 6 R TMWs v + Em p Vs2 cO +Em p Vs2cO traps)/(
(-1 + 2 v) Vs MWs)
"> Sl(x)-S2(x);
0
Note #4: The stresses are same for each solution root, and do not depend onthe position x! The stresses are therefore constant throughout thebody. We will solve for _C3 in terms of the boundary stress, SO.
> C3:=simplify(solve(Sl(x)=SO,_C3));
_C3:= - (-3 R TMWs + 6 R TMWs v + Em p Vs 2 cO + Em p Vs2 cO traps - SO Vs MWs
+ 2 SO Vs MWs v)/(Em Vs MWs)
A-Ic-8
"> Sl(x);
so"> S2(x);
so"> S1:=x->S0;
Si :=x - SO"> S2:=x->SO;
52 := x-- SO
Note #5: Now we will introduce numerical values for the material properties.The values selected are typical for internal hydrogen as asolute in AISI 4340 steel.
"> SO:=O.O;
SO:= 0"> Ds:=le-9;
Ds:= .1 10.8"> Vs:=2.02e-6;
Vs := .202 10-5
"> traps:=499;
traps:= 499"> cO:=O.O;
cO:=0" MWs:=O.O0100797;
MWs :=.00100797"> Em:=200e9;
Em :=.200 1012
"> nu:=0.3;
V :=.3
"> rho:=7800;
p := 7800"> T:=293;
T:= 293"> R:=8.31432;
R:= 8.31432> K;
.166666666666667 1012
"> lambda;
.115384615384616 1012
"> mu;
.769230769230769 1011"> alpha;
A-ic-9
5.21047253390476"> Rh:=R/MWs;
Rh:= 8248.57882675080"> K1;
138.199274509089"> K2;
7815.70880085715"> K1*K2;
.108012528605276 107
"> 1/(Kl*K2);
.925818525788271 10-6
"> Al;
.4"> A2;
6773.61429407620"> A3;
0"> B1;
.500000000000001 1012
"> B2;
.390785440042860 1016
"> B3;
0"> C3;
.00723592800000000"> S2(x);
0"> S2(x);
0
Note #6: Now let's solve for the integration constants: _Cl, C2, and _C4, using theboundary data.
> cl (X);
.925818525788271 10-6+ .925818525788271 10-6
J1 +1.216025057210552 107 _C] x + .216025057210552 107 _C] _C2
Note #7: The solution cl(x) is not capable of satisfying the boundaryconditions, as evidenced by the lack of solution for the givenboundary conditions. The concentration function c2 (x) is thecorrect solution in this case!
1. MAPLE PROGRAM DIRECTORY AND FILE NAME:/afs/nd.edu/user4/jthomas5/Maple/Transport/lDSSsol#ld.ms
2. ORIGINATOR:Dr. James P. ThomasUniversity of Notre DameDepartment of Aerospace and Mechanical Engineering374 Fitzpatrick HallNotre Dame, IN 46556-5637(219) 631-9371
4. PROGRAM DESCRIPTION:One-dimensional, steady-state analysis of the fully coupled solute transport and trappingequations using "plane stress" constitutive equations for stress. The particular problemsolved is shown below:
1. dJ/dx=O (steady-state diffusion equation)2. J=Ds*(Kl*c*de/dx-dc/dx) (mass flux)3. dS/dx=O (steady-state deformation equation with zero body forces)4. S=Bl*e-B2*c+B3 (axial stress)5. de/dx-K2*dc/dx=O (dilational strain gradient)6. e=Al*du/dx+A2*c-A3 (dilatational strain)
The constants used in the plane stress analysis are defined below:
Ds:=lattice diffusivity [m^2/sec]
Kl:=(Vs*K)/(R*T)Vs:= partial molar volume of solute [m^3/mol solute]K:= bulk modulus=Em/(3*(l-2*nu)) [N/m^2]Em:= Modulus of Elasticity [N/m^2]nu:= Poisson's ratio [1)R:= universal gas constant=8.31432 [J/mol-K]T:= temperature [K]
K2:=3*alpha*(l+traps)alpha:= (rho*Vs)/(3*MWs)=solute concentration expansion coefficient [m/m/Delc]rho:= mass density of the solid [kg/m^3]MWs:= molecular weight of the solute [kg/mol]Delc:= c-c0 [I1
A-id-i
Al: =l-2*nuA2:=2*(l+nu)*alpha*(l+traps)A3 :=A2*cO
5. REQUIRED INPUTS:
Ds:=lattice diffusivity [m^2/sec]Vs:: partial molar volume of solute [m^3/mol solute]traps:= Csr*Kr=trapping constant [1]cO:= reference solute concentration [1]MWs':: molecular weight of the solute [kg/mol]
Em:= Modulus of Elasticity [Pa]nu:= Poisson's ratio [1]rho:= mass density of the solid [kg/m^3]
T:= temperature [K]
6. SYMBOLIC ANALYSIS:
"> restart;
"> Digits:=trunc(evalhf(Digits));
Digits:= 15
"> J:=Ds*(Kl*K2*c(x)-l)*diff(c(x),x);
J:= Ds (K1 K2 c(x) - 1) c(x)
"> deqn:=J=O;
deqn :=Ds (K1 K2 c(x) -1) c(x) 0
"> csoln:=dsolve(deqn,c(x),explicit);
1csoln := c(x) - , c(x) = C]K] K2
Note #1: There are apparently two "roots" to the solution for c(x) . Thefirst solution is obviously incorrect in that the concentrationthroughout the rod is defined in terms of the constants K1 and K2.The second solution is adopted as correct.
Note #3: At this point, we will introduce expressions for the constants andthen simplify the results before quantifying the constants anddetermining the constants of integration.
Em(--C2 MWs + p Vs _C] + p Vs _C1 traps -p Vs cO - p Vs cO traps)
(-1 + 2 v) MWs
A-ld-4
Note #4: Now we will introduce numerical values for the material properties.The values selected are typical for internal hydrogen as asolute in AISI 4340 steel.
"> SO:=0.0;
SO:= 0"> Ds:=Ie-9;
Ds:= .1 108
"> Vs:=2.02e-6;
Vs .202 10-5
"> traps:=19;
traps := 19"> cO:=0.0;
cO:=0"> MWs:=0.00100797;
MWs :=.00100797"> Em:=200e9;
Em :=.200 1012
"> nu:=0.3;
v :=.3"> rho:=7800;
p := 7800"> T:=293;
T:= 293"> R:=8.31696;
R := 8.31696> K;
.166666666666667 1012
"> lambda;
.115384615384616 1012
"> mu;
.769230769230769 1011"> alpha;
5.21047253390476"> Rh:=R/MWs;
Rh := 8251.19795232001"> K1;
138.155406787625"> K2;
312.628352034286"> KI*K2;
A-ld-5
43191.2971486416"> 1/(K1*K2);
.0000231528123954816"> Al;
.4"> A2;
270.944571763048"> A3;
0">B1;
.500000000000001 1012
"> B2;
.156314176017143 1015"> B3;
0
Note #6: Now let's solve for the integration constants: _Cl, -C2, and -C3, using theboundary data.
"> c2(x);
_C!"> C1:=le-7;
_CI := .1 10-6"> c2(x);
.1 10-6
"> u2(x);
-.0000677361429407620 x + 2.50000000000000 x _C2 + 1.00000000000000 _C3
1. MAPLE PROGRAM DIRECTORY AND FILE NAME:/afs/nd.edu/user4/jthomas5/Maple/Transport/lDSSsol#le.ms
2. ORIGINATOR:Dr. James P. ThomasUniversity of Notre DameDepartment of Aerospace and Mechanical Engineering374 Fitzpatrick HallNotre Dame, IN 46556-5637(219) 631-9371
4. PROGRAM DESCRIPTION:One-dimensional, steady-state analysis of the fully coupled solute transport and trappingequations using "plane stress" constitutive equations for stress. The particular problemsolved is shown below:
1. dJ/dx=O (steady-state diffusion equation)2. J=Ds*(Kl*c*de/dx-dc/dx) (mass flux)3. dS/dx=O (steady-state deformation equation with zero body forces)4. S=Bl*e-B2*c+B3 (axial stress)5. de/dx-K2*dc/dx=O (dilational strain gradient)6. e=Al*du/dx+A2*c-A3 (dilatational strain)
The constants used in the plane stress analysis are defined below:
Ds:=lattice diffusivity [m^2/sec]
Kl::(Vs*K)/(R*T)Vs:= partial molar volume of solute [m^3/mol solute]K:= bulk modulus=Em/(3*(l-2*nu)) [N/m^2]Em:= Modulus of Elasticity [N/m^2]nu:= Poisson's ratio [1]R:= universal gas constant=8.31432 [J/mol-K]T:= temperature [K]
K2:=3*alpha*(l+traps)alpha:= (rho*Vs)/(3*MWs)=solute concentration expansion coefficient [m/m/Delc]rho:= mass density of the solid [kg/m^3]MWs:= molecular weight of the solute (kg/mol]Delc:= c-cO [11
Ds::lattice diffusivity [m^2/sec]Vs:= partial molar volume of solute [m^3/mol solute]traps:= Csr*Kr=trapping constant [1]cO:= reference solute concentration [1]MWs!= molecular weight of the solute [kg/moll
Em:= Modulus of Elasticity [Pa]nu:= Poisson's ratio [1]rho:= mass density of the solid [kg/m^3]
T:= temperature [K]
6. SYMBOLIC ANALYSIS:
"> Digits:=tr unc(evalhf(Digits));
Digits:= 15"> J:=Ds*(Kl*K2*c(x)-l)*diff(c(x),x);
J:=Ds(K] K2c(x)- 1) c(x)
"> deqn:=J=O;
deqn :=Ds(K1K2c(x)- 1) -c(x) = 0
"> csoln:=dsolve(deqnc(x),explicit);
1csoln c(x) - , c(x) = C]
Note #1: There are apparently two "roots" to the solution for c(x) . Thefirst solution is obviously incorrect in that the concentrationthroughout the rod is defined in terms of the constants Kl and K2.The second solution is adopted as correct.
Note #3: At this point, we will introduce expressions for the constants andthen simplify the results hefore quantifying the constants anddeternining the constants of integration.
EmK.=
"> Iambda:=(Em*nu)/((1+nu)*(1-2*nu));
Em v
(1 +v) (I - 2v)"> mu:=EmIC2*(1+nu));
Em
2±+2 v"> alpha:=(rho*Vs)I(3*MWs);
I 1p Vs
3 MWs"> Ki :=(Vs*K)I(R*T);
Vs EmnK]:
(3 -6 v) R T"> K2:=3*alpha*(l +traps);
A-Ie-3
K2 := p Vs ( 1 + traps)
MWs
"> Kl*K2;
Vs 2 Em p (1 + traps)
(3- 6v) R TMWs"> A1 :=(1-2*nu);
A]:= 1 -2 v"> A2:=2*(l +nu)*alpha*(l +traps);
2 (1 +v)p Vs(l +traps)A2:=3 MWs
"> A3:=A2*cO;
2 (1+v)pVs(I+traps)cOA3 :=- 3 MWs
"> B1:=3*K;
EmB]:= 3 -
3 -6v"> B2:=9*K*alpha*(l+traps);
Em p Vs ( 1 + traps)B2 :3 .(3 - 6 v) MWs
"> B3:=B2*cO;
Em p Vs ( 1 + traps) cOB3 3(3 - 6 v) MWs
"> c2(x);
-C]
" e2(x);
_C2
"> collect(u2(x),x);
2(1+ v) pVs(1+traps) C 2(1 +v)p Vs(I+traps)cO C(+vp MWs 3 MWs -C x
33 + C3
1 -2v
"> S2(x);
Em (--C2 MWs + p Vs _Cl + p Vs _C] traps - p Vs cO - p Vs cO traps)
(-1 + 2 v) MWs
A-le-4
Note #4: Now we will introduce numerical values for the material properties.The values selected are typical for internal hydrogen as asolute in AISI 4340 steel.
"> Ds:=le-9;
Ds: .1 10-1"> Vs:=2.02e-6;
Vs :=.202 10'"> traps:=499;
traps := 499
"> cO:=O.O;
cO:= 0"> MWs:=O.00100797;
MWs :=.00100797
"> Em:=200e9;
Em :=.200 1012
"> nu:=0.3;
v :=.3"> rho:=7800;
p := 7800
"> T:=293;
T:= 293
"> R:=8.31432;
R:= 8.31432
> K;
.166666666666667 1012
"> lambda;
.115384615384616 1012
"> mu;
.769230769230769 1011"> alpha;
5.21047253390476"> Rh:=R/MWs;
Rh := 8248.57882675080"> K1;
138.199274509089"> K2;
7815.70880085715"> KI*K2;
.108012528605276 107
A-le-5
"> 1/(K1*K2);
.925818525788271 10-6
"> Al;.4
"> A2;
6773.61429407620"> A3;
0"> Bi;
.500000000000001 1012
"> B2;
.390785440042860 1016
"> B3;
0
Note #6: Now let's solve for the integration constants: CI, _C2, and C3, using theboundary data.
"> c2(x);
C]"> _Cl:=l e-7;
C] := .1 10-6
"> c2(x);
.1 10.6
"> u2(x);
-.00169340357351905 x + 2.50000000000000 x _C2 + 1.00000000000000 _C3"> S2(x);
.000260523626695245 x + 1.00000000000000 _C3"> C3:=solve(u2(O.O)=O,-C3);
C3 :=0"> u2(x);
.000260523626695245 x"> e2(x);
.000781570880085718
A-le-6
1. MAPLE PROGRAM DIRECTORY AND FILE NAME:/afs/nd.edu/user4/jthomas5/Maple/Transport/lDSSsol#2a.ms
2. ORIGINATOR:Dr. James P. ThomasUniversity of Notre DameDepartment of Aerospace and Mechanical Engineering374 Fitzpatrick HallNotre Dame, IN 46556-5637(219) 631-9371
4. PROGRAM DESCRIPTION:One-dimensional, steady-state analysis of the fully coupled solute transport and trappingequations using "plane stress" constitutive equations for stress. The particular problemsolved is shown below:
1. dJ/dx:O (steady-state diffusion equation)2. J=Ds*(Kl*c*de/dx-dc/dx) (mass flux)3. dS/dx=O (steady-state deformation equation with zero body forces)4. S=Bl*e-B2*c+B3 (axial stress)5. de/dx-K2*dc/dx=O (dilational strain gradient)6. e=Al*du/dx+A2*c-A3 (dilatational strain)
The constants used in the plane stress analysis are defined below:
Ds:=lattice diffusivity [m^2/sec3
Kl:=(Vs*K)/(R*T)Vs:= partial molar volume of solute [m^3/mol solute]K:= bulk modulus=Em/(3*(l-2*nu)) [N/m^2]Em:= Modulus of Elasticity IN/m^2]nu:= Poisson's ratio [1]R:= universal gas constant=8.31432 [J/mol-K]T:= temperature [K]
K2:=3*alpha*(l+traps)alpha:= (rho*Vs)/(3*MWs)=solute concentration expansion coefficient [m/m/Delc]rho:= mass density of the solid [kg/m^3]MWs:= molecular weight of the solute [kg/mol]Delc:= c-c0 [1]
A-2a-i
Al: =1-2 *nu
A2:=2*(l+nu)*alpha*(l+traps)A3 : =A2 *cO
5. REQUIRED INPUTS:
Ds:=lattice diffusivity [m^2/sec]Vs:: partial molar volume of solute [m^3/mol solute]traps:= Csr*Kr=trapping constant [1]cO:= reference solute concentration [1]MWs:= molecular weight of the solute [kg/mol]
Em:= Modulus of Elasticity [Pa]nu:= Poisson's ratio [1]rho:= mass density of the solid [kg/m^3]
Note #3: At this point, we will introduce expressions for the constants andthen simplify the results before quantifying the constants anddetermining the constants of integration.
Vs2 Em p( 1 +traps) C1 x Vs2 Em p( 1 +traps)jC1 _C2%1 := 1 +2 +2
(3- 6v) R TMWs (3 - 6 v) R TMWs
"> S1 (x);
- (MWs _C3 Vs Em - 3 R TMWs + 6 R TMWs v + Em p Vs 2 cO + Em p Vs 2 cO traps)/(
(-1 + 2 v) Vs MWs)
"> S2(x);
- (MWs _C3 Vs Em - 3 R TMWs + 6 R TMWs v + Em p Vs2 cO + Em p VS2 cO traps)/(
(-1 +2v) VsMWs)"> S1(x)-S2(x);
0
Note #4: The stresses are same for each solution root, and do not depend onthe position x! The stresses are therefore constant throughout thebody. We define a new stress function, Sx.
"> Sx::S2(x);
Sx:= - (MWs _C3 Vs Em - 3 R TMWs + 6 R TMWs v + Em p Vs2 cO +Em p Vs2 cO traps)l(
(-1 +2v) VsMWs)
A-2a-9
Note #5: Now we will introduce numerical values for the material properties.The values selected are typical for internal hydrogen as asolute in AISI 4340 steel.
"> Ds:=le-9;
Ds:= .1 10-8"> Vs:=2.02e-6;
Vs :=.202 10-"> traps:=19;
traps := 19
"> cO:=0.0;
cO:= 0"> MWs:=0.00100797;
MWs :=.00100797"> Em:=200e9;
Em := .200 1012
"> nu:=0.3;
v := .3"> rho:=7800;
p := 7800"> T:=293;
T:= 293"> R:=8.31432;
R := 8.31432
> K;
.166666666666667 1012
"> lambda;
.115384615384616 1012
"> mu;
.769230769230769 1011"> alpha;
5.21047253390476"> Rh:=R/MWs;
Rh:= 8248.57882675080"> KI;
138.199274509089"> K2;
312.628352034286"> K1*K2;
43205.0114421104
A-2a-10
"> 1/(K1*K2);
.0000231454631447068"> Al;
.4"> A2;
270.944571763048"> A3;
0"> Bi;
.500000000000001 1012
"> B2;
.156314176017143 1015
"> B3;
0
Note #6: Now let's solve for the integration constants: _CI, _C2, _C3, and C4,using the boundary data.
"> cl (x);
.0000231454631447068 + .0000231454631447068
S_1 + 86410.0228842208 _C] x + 86410.0228842208 _Cl _C2"> c2(x);
.0000231454631447068 - .0000231454631447068
SI + 86410.0228842208 _CI x + 86410.0228842208 _Cl _C2"> evalf(cl (x),5);
Note #7: The solution cl(x) is not capable of satisfying the boundaryconditions, as evidenced by the lack of solution for the givenboundary conditions. The concentration function c2(x) is thecorrect solution in this case!
1 x.0023603-.00080399 +.00067972 -. 0016080 1. - .84543 x
,1.- .84543 x J1.-.84543 x
A-2a-15
1. MAPLE PROGRAM DIRECTORY AND FILE NAME:/afs/nd.edu/user4/jthomas5/Maple/Transport/lDSSsol#2b.ms
2. ORIGINATOR:Dr. James P. ThomasUniversity of Notre DameDepartment of Aerospace and Mechanical Engineering374 Fitzpatrick HallNotre Dame, IN 46556-5637(219) 631-9371
4. PROGRAM DESCRIPTION:One-dimensional, steady-state analysis of the fully coupled solute transport and trappingequations using "plane stress" constitutive equations for stress. The particular problemsolved is shown below:
1. dJ/dx=O (steady-state diffusion equation)2. J=Ds*(Kl*c*de/dx-dc/dx) (mass flux)3. dS/dx=O (steady-state deformation equation with zero body forces)4. S=Bl*e-B2*c+B3 (axial stress)5. de/dx-K2*dc/dx=O (dilational strain gradient)6. e=Al*du/dx+A2*c-A3 (dilatational strain)
The constants used in the plane stress analysis are defined below:
Ds::lattice diffusivity [m^2/sec]
Kl::(Vs*K)/(R*T)Vs:= partial molar volume of solute [m^3/mol solute]K:= bulk modulus=Em/(3*(l-2*nu)) [N/m^2]Em:= Modulus of Elasticity [N/m^2]nu:= Poisson's ratio [1]R:- universal gas constant=8.31432 [J/mol-K]T:= temperature [K]
K2::3*alpha*(l+traps)alpha:= (rho*Vs) / (3*MWs)=solute concentration expansion coefficient [m/m/Delc]rho:= mass density of the solid [kg/m^3]MWs:= molecular weight of the solute [kg/mol]Delc:= c-cO [1]
Note #3: At this point, we will introduce expressions for the constants andthen simplify the results before quantifying the constants anddetermining the constants of integration.
"> K:=(Em/(3*(1-2*nu)));
EmK:=
3 -6v"> lambda:=(Em*nu)/((1+nu)*(1-2*nu));
Em v
(1+ v)(1 -2v)
A-2b-5
"> mu:=Em/(2*(l+nu));
Em2+2v
"> alpha:=(rho*Vs)/(3*MWs);
1 p Vs
3 MWs
"> Ki :=(Vs*K)/(R*T);
Vs EmK1:=
(3-6v)RT"> K2:=3*alpha*(l +traps);
p Vs (I + traps)K2 :MWs
"> KI*K2;
Vs2 Em p ( 1 + traps)
(3 - 6 v) R T MWs"> A1 :=(]-2*nu);
A :=1 -2v
"> A2:=2*(l+nu)*alpha*(l+traps);
2 (1 +v)p Vs(l +traps)A2 :3 MWs
"> A3:=A2*cO;
2 (1 +v) p Vs (1 + traps)cOA3 :=- 3 MWs
> BI:=3*K;
EmB/ := 3 -3-6v
"> B2:=9*K*alpha*(l+traps);
Em p Vs ( 1 + traps)B2 :=3(3 - 6 v) MWs
"> B3:=B2*cO;
B3:=3 Em Vs (1 + traps) cO
(3 - 6 v) MWs
A-2b-6
> cl (x);
+I Vs m Xp(+1traps) l x Vs2 m mp(1+traps) l C22(3 -(3-6v)RTMWs (3-6v)RTMWs
MWs/( Vs2 Em p (1 + traps))"> c2(x);
. /1+2 VsEm- p(1+traps)+Clx Vs2 Em p (1+traps)C C2 6v)-1+ 1+2 (3-6v)RTMWs (3-6v)RTMWs (3
- (MWs _C3 Vs Em - 3 R TMWs + 6 R TMWs v + Em p Vs2 cO + Em p Vs2 cO traps)/(
(-1 + 2 v) Vs MWs)"> S2(x);
- (MWs _C3 Vs Em - 3 R TMWs + 6 R TMWs v + Em p Vs2 cO +Em p Vs2 cO traps)/(
(-1 + 2 v) Vs MWs)
"> Sl(x)-S2(x);
0
Note #4: The stresses are same for each solution root, and do not depend onthe position x! The stresses are therefore constant throughout thebody. We define a new stress function, Sx.
"> Sx:=S2(x);
Sx:: - (MWs _C3 Vs Em - 3 R TMWs + 6 R TMWs v +Em p Vs2 cO + Em p Vs2 cO traps)/(
(-1 +2v) VsMWs)
A-2b-9
Note #5: Now we will introduce numerical values for the material properties.The values selected are typical for internal hydrogen as asolute in AISI 4340 steel.
"> Ds:=le-9;
Ds :=.1 10'"> Vs:=2.02e-6;
Vs :=.202 10.'"> traps:'=19;
traps := 19"> cO:=0.0;
cO:= 0
"> MWs:=0.00100797;
MWs :=.00100797"> Em:=200e9;
Em :=.200 1012
"> nu:=0.3;
v := .3"> rho:=7800;
p := 7800"> T:=293;
T:= 293"> R:=8.31432;
R:= 8.31432
> K;
,166666666666667 1012
"> lambda;
.115384615384616 1012
"> mu;
.769230769230769 1011"> alpha;
5.21047253390476"> Rh:=R/MWs;
Rh := 8248.57882675080"> K1;
138.199274509089"> K2;
312.628352034286"> KI*K2;
43205.0114421104
A-2b-1O0
"> 1/(K1*K2);
.0000231454631447068"> Al;
.4"> A2;
270.944571763048"> A3;
0">B1;
.500000000000001 1012
"> B2;
.156314176017143 1015
"> B3;
0
Note #6: Now let's solve for the integration constants: _Cl, _C2, _C3, and _C4,using the boundary data.
Note #7: The solution cl(x) is not capable of satisfying the boundaryconditions, as evidenced by the lack of solution for the givenboundary conditions. The concentration function c2(x) is thecorrect solution in this case!
-.0072359 11. - .086222 x + .0072338"> evalf(ex(x),5);
1 x.0024068 -. 00080399 + .000069323
.-. 086223x 1. -. 086223 x
- .0016080 J1. -. 086223 x
A-2b-15
1. MAPLE PROGRAM DIRECTORY AND FILE NAME:/afs/nd.edu/user4/jthomas5/Maple/Transport/lDSSsol#2c.ms
2. ORIGINATOR:Dr. James P. ThomasUniversity of Notre DameDepartment of Aerospace and Mechanical Engineering374 Fitzpatrick HallNotre Dame, IN 46556-5637(219) 631-9371
4. PROGRAM DESCRIPTION:One-dimensional, steady-state analysis of the fully coupled solute transport and trappingequations using "plane stress" constitutive equations for stress. The particular problemsolved is shown below:
1. dJ/dx=O (steady-state diffusion equation)2. J=Ds*(Kl*c*de/dx-dc/dx) (mass flux)3. dS/dx=0 (steady-state deformation equation with zero body forces)4. S=Bl*e-B2*c+B3 (axial stress)5. de/dx-K2*dc/dx=O (dilational strain gradient)6. e=Al*du/dx+A2*c-A3 (dilatational strain)
The constants used in the plane stress analysis are defined below:
Ds:=lattice diffusivity [m^2/sec]
Kl:=(Vs*K)/(R*T)Vs:= partial molar volume of solute [m^3/mol solute]K:= bulk modulus=Em/(3*(l-2*nu)) [N/m^2]Em:= Modulus of Elasticity [N/m^2]nu:= Poisson's ratio [1]R:= universal gas constant=8.31432 [J/mol-K]T:= temperature [K]
K2:=3*alpha*(l+traps)alpha:- (rho*Vs) / (3*MWs)=solute concentration expansion coefficient [m/m/Delc]rho:= mass density of the solid [kg/m^3]MWs:- molecular weight of the solute [kg/mol)Delc:= c-cO [1]
Note #3: At this point, we will introduce expressions for the constants andthen simplify the results before quantifying the constants anddetermining the constants of integration.
Vs2 Em p (1 + traps) -C1 x Vs2 Em p (1 + traps) -C1 C2%1: 1 +2 +2
(3 - 6 v) R TMWs (3 - 6v) R TMWs
"> S1 (x);
- (MWs _C3 Vs Em - 3 R TMWs + 6 R TMWs v + Em p Vs2 cO +Em p VS2 cO traps)/(
(-1 + 2 v) Vs MWs)
"> S2(x);
- (MWs _C3 Vs Em - 3 R TMWs + 6 R TMWs v +Em p Vs2 cO + Em p Vs2 cO traps)!(
(-1 +2v) VsMWs)"> S 1(x)-S2(x);
0
Note #4: The stresses are same for each solution root, and do not depend onthe position x! The stresses are therefore constant throughout thebody. We define a new stress function, Sx.
"> Sx:=S2(x);
Sx:= - (MWs _C3 Vs Em- 3 R T MWs + 6 R T MWs v +Em p Vs2 cO + Em p Vs2 cO traps)/(
(-1 +2v) VsMWs)
Note #5: Now we will introduce numerical values for the material properties.The values selected are typical for internal hydrogen as asolute in AISI 4340 steel.
A-2c-8
"> Ds:=le-9;
Ds:=.1 10-8"> Vs:=2.02e-6;
Vs :=.202 10i'"> traps:=499;
traps:= 499"> cO:=O.O;
cO:= 0"> MWs:=O.00100797;
MWs :=.00100797"> Em:=200e9;
Em :=.200 1012
"> nu:=0.3;
v :=.3"> rho:=7800;
p := 7800"> T:=293;
T:= 293"> R:=8.31432;
R:= 8.31432
> K;
.166666666666667 1012
"> lambda;
.115384615384616 1012
"> mu;
.769230769230769 1011"> alpha;
5.21047253390476"> Rh:=R/MWs;
Rh:= 8248.57882675080"> K1;
138.199274509089"> K2;
7815.70880085715"> KI*K2;
.108012528605276 107
"> 1/(K1*K2);
.925818525788271 10-6
A-2c-9
"> Al;
.4"> A2;
6773.61429407620"> A3;
0> B1;,
.500000000000001 1012
"> B2;
.390785440042860 1016
"> B3;
0
Note #6: Now let's solve for the integration constants: _CI, _C2, C3, and _C4,using the boundary data.
"> cl(x);
.925818525788271 10-6+ .925818525788271 10-6
J1 +_.216025057210552 107 _C1 x + .216025057210552 107 _C] _C2
"> c2(x);
.925818525788271 10-6 - .925818525788271 10-6
Ji +_.216025057210552 107 _C] x +.216025057210552 107 _C1 _C2
Note #7: The solution cl(x) is not capable of satisfying the boundaryconditions, as evidenced by the lack of solution for the givenboundary conditions. The concentration function c2(x) is thecorrect solution in this case!
-.0072359 1. - 2.0436 x + .0071848"> evalf(ex(x),5);
1 x.0022842-.00080399 +.0016430 - .0016080 l. -2.0436 x
S17- 2.0436 x 1.- 2.0436 x
A-2c-14
1. MAPLE PROGRAM DIRECTORY AND FILE NAME:/afs/nd.edu/user4/jthomas5/Maple/Transport/iDSSsol#2d.ms
2. ORIGINATOR:Dr. James P. ThomasUniversity of Notre DameDepartment of Aerospace and Mechanical Engineering374 Fitzpatrick HallNotre Dame, IN 46556-5637(219) 631-9371
4. PROGRAM DESCRIPTION:One-dimensional, steady-state analysis of the fully coupled solute transport and trappingequations using "plane stress" constitutive equations for stress. The particular problemsolved is shown below:
1. dJ/dx=O (steady-state diffusion equation)2. J=Ds*(Kl*c*de/dx-dc/dx) (mass flux)3. dS/dx=O (steady-state deformation equation with zero body forces)4. S=Bl*e-B2*c+B3 (axial stress)5. de/dx-K2*dc/dx=O (dilational strain gradient)6. e=Al*du/dx+A2*c-A3 (dilatational strain)
The constants used in the plane stress analysis are defined below:
Ds:=lattice diffusivity [m^2/sec]
Kl:=(Vs*K)/(R*T)Vs:= partial molar volume of solute [m^3/mol solute]K:= bulk modulus=Em/(3*(l-2*nu)) [N/m^2]Em:= Modulus of Elasticity [N/m^2]nu:= Poisson's ratio [1)R:= universal gas constant=8.31432 [J/mol-K)T:= temperature [K)
K2:=3*alpha*(l+traps)alpha:= (rho*Vs)/(3*MWs)=solute concentration expansion coefficient [m/m/Delc]rho:= mass density of the solid [kg/m^3]MWs:= molecular weight of the solute [kg/mol]Delc:= c-cO [L]
Ds:=lattice diffusivity [m^2/sec]Vs:= partial molar volume of solute [m^3/mol solute]traps:= Csr*Kr=trapping constant [1]cO:= reference solute concentration [1]MWs:= molecular weight of the solute [kg/mol]
Em:= Modulus of Elasticity [Pa]nu:= Poisson's ratio [1]rho:= mass density of the solid [kg/m^3]
T:: temperature [K]
6. SYMBOLIC ANALYSIS:
"> Digits:=trunc(evalhf(Digits));
Digits:= 15
"> J:=Ds*(KI*K2*c(x)-I )*diff(c(x),x);
J:= Ds (K] K2 c(x) - 1) c(x)
"> deqn:=J=O;
deqn:= Ds (K1 K2 c(x) - 1) - (x) 0
"> csoln:=dsolve(deqn,c(x),explicit);
1csoln :=c(x) ,c(x) = _C1
K] K2
Note #1: There are apparently two "roots" to the solution for c(x). Thefirst solution is obviously incorrect in that the concentrationthroughout the rod is defined in terms of the constants Ki and K2.The second solution is adopted as correct.
Note #4: At this point, we will introduce expressions for the constants andthen simplify the results before quantifying the constants anddetermining the constants of integration.
"> K:=(Em/(3*(1-2*nu)));
EmK:=
3 -6v"> Iambda:=(Em*nu)I((1+nu)*(1-2*nu));
Em v
(1+ v) (1 -2v)"> mu:=Em/(2*(l+nu));
Em2+2v
"> alpha:=(rho*Vs)/(3*MWs);
1 p Vs
3 MWs"> K1 :=(Vs*K)I(R*T);
Vs EmK]
(3 -6v)RT
A-2d-3
"> K2:=3*alpha*(l+traps);
p Vs ( 1 + traps)K2 :MWs
"> KI*K2;
Vs 2 Em p (1 + traps)
(3 - 6 v) R TMWs"> A1 :=(1-2*nu);
A]:= 1 -2 v"> A2:=2*(l+nu)*alpha*(l +traps);
2 (1 +v) p Vs (1 + traps)A2:=3 MWs
"> A3:=A2*cO;
2 (1 +v) p Vs (1 + traps) cOA3.-3 MWs
"> B1:=3*K;
EmB1 := 3 -3 -6v
"> B2:=9*K*alpha*(l +traps);
B2 =3 Em p Vs (1 + traps)
(3 - 6 v) MWs"> B3:=B2*cO;
B3:= 3 Em p Vs ( 1 + traps) cO
(3 - 6 v) MWs
"> c2(x);
_C]
"> e2(x);
_C2
"> collect(u2(x),x);
(2(1 +±v) p Vs(1I +traps) C] 2 (1±+V) p Vs (1±+traps) cO _ C2 x3MWs 3 MWs + ) C
2 v+_C3
1 -2v
> S2(x);
Em(-_C2 MWs + p Vs _C] + p Vs _C] traps - p Vs cO - p Vs cO traps)
(-1 + 2 v) MWs
A-2d-4
Note #5: Now we will introduce numerical values for the material properties.The values selected are typical for internal hydrogen as asolute in AISI 4340 steel.
"> Ds:=le-9;
Ds :=.1 10-1"> Vs:=2.02e-6;
Vs :=.202 10'"> traps:=19;
traps := 19"> cO:=0.0;
cO:= 0"> MWs:=0.00100797;
MWs :=.00100797"> Em:=200e9;
Em :=.200 1012
"> nu:=0.3;
v :=.3"> rho:=7800;
p := 7800"> T:=293;
T:= 293"> R:=8.31432;
R:= 8.31432
> K;
.166666666666667 1012
"> lambda;
.115384615384616 1012
"> mu;
.769230769230769 1011"> alpha;
5.21047253390476"> Rh:=RIMWs;
Rh := 8248.57882675080"> K1;
138.199274509089"> K2;
312.628352034286"> K1*K2;
43205.0114421104
A-2d-5
> 1/(Kl*KZ);
.0000231454631447068"> Al;
.4"> A2;
270.944571763048"> A3;
0"> B1;
.500000000000001 1012
"> B2;
.156314176017143 1015
"> B3;
0
Note #6: Now let's solve for the integration constants: _Cl, C2, _C3, and _C4,using the boundary data.
"> c2(x);
_C]"> _Cl :=1e-7;
CI :=. 1 10.6"> c2(x);
.1 10-6
"> u2(x);
-.0000677361429407620 x + 2.50000000000000 x _C2 + 1.00000000000000 _C3"> eqnsetl :={O.O~u2(O.O), 0.0=u2(0.10)};
1. MAPLE PROGRAM DIRECTORY AND FILE NAME:/afs/nd.edu/user4/jthomas5/Maple/Transport/lDSSsol#2e.ms
2. ORIGINATOR:Dr. James P. ThomasUniversity of Notre DameDepartment of Aerospace and Mechanical Engineering374 Fitzpatrick HallNotre Dame, IN 46556-5637(219) 631-9371
3. HISTORY:
Written: June-1994Latest Revision: 13-Feb-1995
4. PROGRAM DESCRIPTION:One-dimensional, steady-state analysis of the fully coupled solute transport and trappingequations using "plane stress" constitutive equations for stress. The particular problemsolved is shown below:
1. dJ/dx=O (steady-state diffusion equation)2. J=Ds*(Kl*c*de/dx-dc/dx) (mass flux)3. dS/dx=O (steady-state deformation equation with zero body forces)4. S=Bl*e-B2*c+B3 (axial stress)5. de/dx-K2*dc/dx=O (dilational strain gradient)6. e=Al*du/dx+A2*c-A3 (dilatational strain)
The constants used in the plane stress analysis are defined below:
Ds:=lattice diffusivity [m^2/sec]
Kl:=(Vs*K)/(R*T)Vs:= partial molar volume of solute [m^3/mol solute]K:= bulk modulus=Em/(3*(l-2*nu)) [N/m^2]Em:= Modulus of Elasticity [N/m^2]nu:- Poisson's ratio [1]R:- universal gas constant=8.31432 [J/mol-K]T:- temperature [K]
K2:=3*aloha*(l+traps)alpha:= (rho*Vs) / (3*MWs)=solute concentration expansion coefficient [m/m/Delc]rho:= mass density of the solid [kg/m^3]MWs:= molecular weight of the solute [kg/mollDelc:= c-cO [1]
Ds:=lattice diffusivity [m^2/sec]Vs:= partial molar volume of solute [m^3/mol solute]traps:= Csr*Kr=trapping constant [1]cO:= reference solute concentration [1]MWs:z molecular weight of the solute [kg/mol]
Em:= Modulus of Elasticity [Pa]nu:= Poisson's ratio [1]rho:-- mass density of the solid [kg/m^3]
T:= temperature [K]
6. SYMBOLIC ANALYSIS:
"> Digits:=trunc(evalhf(Digits));
Digits:= 15
"> J:=Ds*(Kl*K2*c(x)-l)*diff(c(x),x);
J:=Ds(K]K2c(x)-l) C(X)
"> deqn:=J=O;
deqn:=Ds(K]K2c(x)-1) )-C(X) =0
"> csoln:=dsolve(deqn,c(x),explicit);
1csoln c(x) c(x) = -C]
K] K2
Note #1: There are apparently two "roots" to the solution for c(x) . Thefirst solution is obviously incorrect in that the concentrationthroughout the rod is defined in terms of the constants KI and K2.The second solution is adopted as correct.
Note #4: At this point, we will introduce expressions for the constants andthen simplify the results before quantifying the constants anddetermining the constants of integration.
"> K:=(Em/(3*(1-2*nu)));
EmK:=-
3 -6v"> lambda:=(Em*nu)/((1+nu)*(1-2*nu));
Em v
(l+v)(1 -2v)"> mu:=Em/(2*(1+nu));
Em2+2v
"> alpha:=(rho*Vs)/(3*MWs);
1 p VsOU.--
3 MWs"> Ki :=(Vs*K)/(R*T);
Vs EmK]
(3-6v)RT
A-2e-3
"> K2:=3*alpha*(l+traps);
K2 p Vs ( 1 + traps)MWs
"> Kl*K2;
Vs 2 Em p (1 + traps)
(3 - 6 v) R TMWs"> A1 :=(1-2*nu);
A] := 1 -2v
"> A2:=2*(l +nu)*alpha*(l +traps);
2 (1 +v)p Vs(1 +traps)A2:
3 MWs
"> A3:=A2*cO;
2 (l+v)pVs(l+traps)cOA3 :3 MWs
"> B1:=3*K;
EmB:= 3
3 -6v"> B2:=9*K*alpha*(l+traps);
132 :=3 Em p Vs (1 + traps)
(3 - 6 v) MWs"> B3:=B2*cO;
Em p Vs ( 1 + traps) cO
(3 - 6 v) MWs
"> c2(x);
C]
"> e2(x);
C2"> collect(u2(x),x);
2 (1 +v)p Vs(1+ traps) C1 2 (1 +v)p Vs(1 +traps) cO + -C2 x
3 MWs 3 MWs -+ _C3
1 -2v
"> S2(x);
Em (-C2 MWs - p Vs _C] - p Vs _C1 traps + p Vs cO + p Vs cO traps)
(-1 + 2v)MWs
A-2e-4
Note #5: Now we will introduce numerical values for the material properties.The values selected are typical for internal hydrogen as asolute in AISI 4340 steel.
"> Ds:=le-9;
Ds :.1 10-8"> Vs:=2.02e-6;
Vs :=.202 10-'"> traps:=499;
traps := 499"> cO:=0.0;
cO:= 0"> MWs:=O.00100797;
MWs :=.00100797"> Em:=200e9;
Em :=.200 1012
"> nu:=0.3;
V :=.3"> rho:=7800;
p := 7800"> T:=293;
T:= 293"> R:=8.31432;
R := 8.31432> K;
.166666666666667 1012
"> lambda;
.115384615384616 1012
"> mu;
.769230769230769 1011"> alpha;
5.21047253390476"> Rh:=R/MWs;
Rh := 8248.57882675080"> K1;
138.199274509089> K2;
7815.70880085715"> K1*K2;
.108012528605276 107
A-2e-5
"> 1/(K1*K2);
.925818525788271 10-6
"> Al;
.4"> A2;
6773.61429407620"> A3;
0"> B1;
.500000000000001 1012
"> B2;
.390785440042860 1016
"> B3;
0
Note #6: Now let's solve for the integration constants: _Cl, C2, _C3, and C4,using the boundary data.
"> c2(x);
_CI"> C1:=le-7;
C1 := .1 10-6
"> c2(x);
.1 10-6
"> u2(x);
-.00169340357351905 x + 2.50000000000000 x _C2 + 1.00000000000000 _C3"> eqnsetl :={O.O=u2(0.0), 0.0=u2(0.10)};
1. MAPLE PROGRAM DIRECTORY AND FILE NAME:/afs/nd.edu/user4/jthomas5/Maple/Transport/lDSSsol#3a.ms
2. ORIGINATOR:Dr. James P. ThomasUniversity of Notre DameDepartment of Aerospace and Mechanical Engineering374 Fitzpatrick HallNotre Dame, IN 46556-5637(219) 631-9371
4. PROGRAM DESCRIPTION:One-dimensional, steady-state analysis of the fully coupled solute transport and trappingequations using "plane stress" constitutive equations for stress. The particular problem tobe solved is shown below. It is the 1-D SS transport problem, but with a (1/x)^(3/2)singular body force applied. This is meant to "simulate" the stress occurring at the tip ofa crack:
The constants used in the plane stress analysis are defined below:
Ds:=lattice diffusivity [m^2/sec]
Kl:=(Vs*K)/(R*T)Vs:= partial molar volume of solute [m^3/mol solute]K:= bulk modulus=Em/(3*(l-2*nu)) [N/m^2)Em:= Modulus of Elasticity [N/m^2]nu:= Poisson's ratio [1lR:= universal gas constant=8.31432 [J/mol-K]T:= temperature [K]
K2 : :3*alpha* (l+traps)alpha:= (rho*Vs)/(3*MWs)=solute concentration expansion coefficient [m/m/Delc]rho:= mass density of the solid [kg/m^3]MWs:= molecular weight of the solute [kg/mol]Delc:= c-cO [1]
Note #1: The governing differential equation in this case is J=constantbecause dJ/dx=O, and we are specifying the value of J(O)=0.Setting J(O)=constant produces a DE that is not easily solved.
Note #4a: dx/dc->O as c->l/KI*K2, or conversely, dc/dx->infinity asc->I/KI*K2! This is apparently a critical point in themathematical solution for the concentration. The concentrationbecomes multivalued at this point.
Note #4b: We will now determine the value of the constant _Cl in terms ofthe concentration at the houndary c(01)=cl.
"> Cl ::solve(x(cI,A,K2,Cl)=0.1,Cl);
CI:= -.500000000000000
-2. B]2 ci K12 K22 + 2. B] 2 In(ci) K1 K2 + 12.6491106406735 B] K1 2 K2 A
B1 2 K12 K22
.500000000000000
-2. B1 2 cI K] 2 K22 + 2. B] 2 ln(ci) K] K2 - 12.6491106406735 B] K]2 K2 A
ln(cl) A1.00000000000000 ci - 1.00000000000000 - 6.32455532033675 A
K] K2 B] K2"> Clb(cI,A,K2);
ln(ci) A1.00000000000000 ci - 1.00000000000000 -- + 6.32455532033675 A
K1 K2 BI K2
Note #5: The quadratic nature of the solution for the constant _Cl yieldstwo solutions, Cla & Clb, each of which results in a differentsolution for x(c).
"> xa:=(c,A,K2,cl)->x(c,A,K2,Cla(cI,A,K2));
xa:= (c, A, K2, cl) -- >x(c, A, K2, Cla(cl, A, K2))"> xa(c,A,K2,cI);
+ 1.00000000000000 ln(cl) B] - 6.32455532033675 A K] 2
Note #6: Next we will define the constants, and simplify the twoconcentration expressions for the fully coupled theory andclassical stress assisted diffusion theory.
xa (upper) &xb (lower) -vs-c (High Trapping), c(L)=le-7
10,I
1,!
Ie-07 i1le-07 1,2e-07 1.3e-07 1,4e-07 1,5e-07C
Note #11: The above plots show that the Clb parameter is clearly correctfor c(L)=le-7. The xa(c) solution predicts a singularityat the minimum concentration level on the rod, which is of nophysical use. We must now determine, however, whether the xb(c)solution is valid for all c if c(L)=le-7.
"> logplot(xbl(c,15e6,1K2,1e-7),"> c=le-7..2e-4,title='xb -vs- c (Low Trapping), c(L)=l e-7');
"> logplot(xbh(c,15e6,hK2,1 e-7),"> c=le-7..1e-5,title='xb -vs- c (High Trapping), c(L)=le-7');
A-3a-10
xb -vs-c (High Trapping), c(L)=le-7
0.1
0.01
0.001
0.0001
le-05
2e-06 4e-06 6e-06 8e-06 le-05C
Note #12a: THE ABOVE PLOTS DO NOT HAVE THE SAME CONCENTRATION AXES. Whilethe second plot appears to show that the singularity shiftstoward a higher concentration level, this is only caused by achange in the axis scaling -- the singularity shift is towardlower concentration levels.
Note #12b: Both xa(c) & xb(c) become multi-valued at some critical value of cwhich is dependent on the level of trapping present in the model,but independent of the left-hand concentration. We must determinethe value of x where this critical value of concentration occurs as x
Note #13: The limit concentration is independent of c(L), but the value ofx(c) which corresponds to this limit concentration is obviouslyhighly dependent on c(L). We now examine the nature of thisdependence.
> Iogplot(xah(c,15e6,hK2,1 e-6),c=1 e-6..1.5e-6,title='xa -vs- c (High Trapping), c(L)=1e-6');
xa -vs-c (High Trapping), c(L)=le-60.1" .
0.05'
0.01'
le-06 1.1e-06 1.2e-06 1.3e-06 1.4e-06 1.5e-06c
Note #15a: The solution is divided, then, into two regimes. The first,withc(L)<c(L)critical, requires the xb solution, while the second,with c(L)>c(L)critical, requries the xa solution. The exactvalue of c(L)critical is dependent on the level of trappingpresent in the problem.
Note #15b: To directly compare the concentration predictions generated byvariations in trapping and c(L), we define new concentrationfunctions.
"> cb:=(x,A,K2,cl)->c(x,A,K2,Clb(cI,A,K2));
cb := (x, A, K2, cI) -> c(x, A, K2, Clb(cl, A, K2))"> cb(x,AIK2,cl);
Note #16: The above plots indicate that the difference between SAD andFully Coupled hydrogen concentration is strongly dependent onthe degree of trapping.
Note #18: Strain is calculated from the constitutive relationship, Eq. (4).Without displacements due to concentration, strain may becalculated from the standard linear-elastic constitutiverelationship for comparison.
+ 500 cc int(c(x, .15 108, hK2, Clb(.1 10-6,.15 108, hK2)), x 0 .. 'x')"> ux(x);
.000300000000000000 x - .000474342000000000 x + 2605.23626695238
0.925818525788272 10-6
Q "200903303205813 1l-14 "808962789010000 10"• x - .4127337051323 1013" x ))dx
W(-.108012528605276 107 e Xdx
A-3a-19
APPENDIX B: User Element Subroutine
This Appendix contains a copy of the Fortran user element subroutine for solvingfully coupled deformation-diffusion problems in one-dimension. A listing of the subroutine isgiven first, and is followed by an annotated listing.
DISCLAIMER: This ABAQUS "user element" subroutine is provided free of charge as acourtesy to the technical community. The authors and the University of Notre Dame acceptno responsibility for any decisions based upon the use of this subroutine. Each user isadvised that the subroutine has been tested against a limited number of steady-statedeformation-diffusion problems, and has been found to give accurate answers to theseparticular test problems. The accuracy of this subroutine cannot be guaranteed in anyparticular future analysis.
Developer Contact:
Dr. James P. ThomasUniversity of Notre Dame
Department of Aerospace and Mechanical Engineering374 Fitzpatrick Hall
Notre Dame, IN 46556-5637
Phone: (219) 631- 9371Fax: (219) 631-8341
e-Mail: James.P.Thomas.66 @ nd.edu
B-1
ABAQUS User Element Subroutine for Coupled Deformation-DiffusionProblems: Fortran 77 Code
C * SVARS(1)= DILATATIONAL STRAIN AT INTEGRATION POINT GR(l) [1] *C " SVARS(2)= DILATATIONAL STRAIN AT INTEGRATION POINT GR(2) [1] *C * SVARS(3)= AXIAL STRESS AT INTEGRATION POINT GR(1) [Pa) *C * SVARS(4)= AXIAL STRESS AT INTEGRATION POINT GR(2) [Pa] *C * SVARS(5)= AXIAL STRESS AT LEFT END OF ELEMENT [Pa] *C SVARS(6)= AXIAL STRESS AT RIGHT END OF ELEMENT [Pa] *C * SVARS(7)= MASS FLUX AT THE CENTER OF EACH ELEMENT [m/s] *C * SVARS(8)= X LOCATION OF INTEGRATION POINT GR(1) [m] *C SVARS(9)= X LOCATION OF INTEGRATION POINT GR(2) [a]CC *
C * BASED ON THEORY AND EQUATIONS BY DR. JAMES P. THOMAS *C * CODED BY CHARLES E. P. CHOPIN, 25 AUGUST 1994 *C * LAST MODIFIED BY CHARLES E. P. CHOPIN, 13 MARCH 1995 *C DEPARTMENT OF AEROSPACE AND MECHANICAL ENGINEERING *C * UNIVERSITY OF NOTRE DAME, NOTRE DAME, INDIANAC * *
C DIRECT INQUIRIES REGARDING THIS USER ELEMENT *C * SUBROUTINE TO: *C * [email protected] *
CC DEGREES OF FREEDOM FOR MODEL IN ORDER:C {C1,UI,U2,C2,U3)'CC C = NODAL MASS FRACTION CONCENTRATION OFC SOLUTE [kgSOLUTE/kgSOLID]C U = NODAL DISPLACEMENT Em]C KFLAG CASE FLAG FOR PLANE STRESS OR PLANE STRAIN [I]C AREA CROSS-SECTIONAL AREA OF THE ROD [m-2]
B-2
C E = YOUNG'S MODULUS FOR THE SOLID [Pa]C XNU POISSON'S RATIO FOR THE SOLID [I]C D = DIFFUSION COEFFICIENT FOR SOLUTE IN SOLID [m^2/s]C ALPHA = SOLUTE EXPANSION COEFFICIENT [I]C =(l/3)*(RHO*(PARTIAL MOLAR VOLUME OF SOLUTE))/XMWC CREF REFERENCE CONCENTRATION [kgSOLUTE/kgSOLID]C RHO = DENSITY OF SOLID MIXTURE [kg/m^3]C XMW = MOLECULAR WEIGHT FOR SOLUTE SPECIES [kg/mol]C TEMP = AMBIENT TEMPERATURE (ASSUMED CONSTANT) [K]C TRAPS = TRAP SITES CONSTANT (CSRKR IN (1+CSRKR) TERM) [11C
C 2-POINT GAUSSIAN INTEGRATION LOCATIONS AND WEIGHTSC
data GR/-0.577350269189626D0,0.577350269189626D0/,1 GW/ 1.000000000000000D0,1.000000000000000D0/
CC RS = GAS CONSTANT FOR SOLUTE SPECIESC BULK = BULK MODULUS FOR THE SOLIDC XKON1 = NON-DIMENSIONAL COEFFICIENT FOR (C*DE/DX) TERM OFC MASS FLUX EQUATIONC (J=-DEFF*DC/DX + DEFF*XKONI*C*DE/DX)C DEFF EFFECTIVE DIFFUSION COEFFICIENTC
iff((LFLAGS(3).eq.l).or.(LFLAGS(3).eq.5)) thenCCC <<<<< CALCULATE {RHS} = (R)-[K] (U) >>>>>>>>>C NB: THIS USER ELEMENT ASSUMES THAT ALL OF THEC NEUMANN BOUNDARY DATA ARE HOMOGENEOUS (i.e., ZERO).C INHOMOGENEOUS (NON-ZERO) NEUM4ANN BOUNDARY DATA SUCHC AS NON-ZERO BOUNDARY TRACTIONS AND/OR SOLUTE FLUXESC ARE AUTOMATICALLY INCORPORATED INTO THE COMPUTATIONSC BY ABAQUS. INHOMOGENEOUS DATA SHOULD BE ENTERED INC THE INPUT DECK AS CONCENTRATED NODAL FORCES.CC THE FOLLOWING SECTION CALCULATES THE TOTAL USER SPECIFIEDC DISTRIBUTED LOADS (e.g. BODY FORCES) AT THE GAUSS POINTSC ON THE ROD FOR USE IN FINDING THE RHS VECTOR.C
if((LFLAGS(3).eq.1).or.(LFLAGS(3).eq.5)) thenCC <<<<< CALCULATE {RHS} = (R}-[K){U}-([C]/DTIME){DU}»»»>>CC THE FOLLOWING SECTION CALCULATES THE TOTAL USER SPECIFIEDC DISTRIBUTED LOADS (e.g. BODY FORCES) AT THE GAUSS POINTSC ON THE ROD FOR USE IN FINDING THE RHS VECTOR.C
double precision A,EPSC THIS LOAD ROUTINE CALCULATES THE BODY FORCE AT THEC GAUSS POINTS FOR USE IN FINDING THE RHS VECTOR
IF(JDLTYP.EQ.-l) THENCC LOAD TYPE 1 INDICATES A DISTRIBU-TED BODY FORCE OF THE FORMC F(X)= A/(X-iEPS)V3/2C GRAVITATIONAL BODY FORCE FOR 43-'0 STEEL IS -80.003 [N/m'3]C 'A' SHOULD BE SPECIFIED IN UNIITS OF [FORCE/LENGTH^3/2)
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A. A. C 0A+.0A 0A0 . 0.). 0 A.0 AH) 00 0) +0 0)0 A. A. 00-- 24 0 -r - 0- 0 -A . -- '. ' . ' . ' .
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0 C 0 PQ 0 w
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13 2
APPENDIX C: Sample Input Deck
This Appendix contains a copies of the ABAQUS input decks that were used to solvethe Case la, 2a, and 3a problems. A section-by-section annotated listing is also included.
C-1
ABAQUS Input Deck: Case la
*HEADING, UNSYMMSTATIC DIFFUSION -- STRONGLY COUPLED Standard Case 1
1,0.ODO21,0.10DO*NGEN,NSET=ROD1,21,1*USER ELEMENT, NODES=3,UNSYMM, COORDINATES=1, PROPERTIES=11,VARIABLES=9,TYPE=U1** THE FIRST AND THIRD NODES HAVE DEGREES OF FREEDOM IN CONCENTRATION AND** DISPLACMENT, WHILE THE SECOND NODE ONLY HAS DISPLACEMENT DEGREES OF** FREEDOM. NOTE THAT CONCENTRATIONS ARE TREATED AS TEMPERATURE D.O.F.i1,i
2,13,11,1*ELEMENT,TYPErUl,ELSET=SCID1,1,2,3*ELGEN, ELSET=SC1D1,10,2,1,1,1,1,1*UEL PROPERTY, ELSET=SCID** PROPERTIES ARE IN THE ORDER:** CFLAG,AREA,E,NU,D,ALPHA,CREF,RHO,** MW,TEMP,TRAPS1.0D0,1.0D-4,200.0D9,0.3D0,1.0D-9,5.2D0,0.ODO,7.8D3,1.00797D-3,293.ODO,19.ODO*MATERIAL,NAME=A4340*DEPVAR
9*BOUNDARY** ZERO DISPLACEMENT AT LHS1,1* ZERO CONCENTRATION AT LHS
1,11*INITIAL CONDITIONS, TYPE:TEMPERATURE** 0.0 INITIAL CONCENTRATION OVER ROD LENGTHROD, 0.0D0*WAVEFRONT MINIMIZATION, SUPPRESS*STEP,INC=I000*COUPLED TEMPERATURE-DISPLACEMENT, STEADY STATE0.005,1.0, ,0.05
*CONTROLS, PARANMETERS=FIELD, FIELD=TEMPEPATURE
5.OD-21
*NODE PRINT, SUiMARY=NO,FREQUENCY=100
COORI,Ul,NT*EL PRINT, SUM4-ARY=NO,FREQUENCY=l100
SDV*BOUNDARY** SPECIFY RIGHT HAND BOUNDARY CONCENTRATION AS 1S0E-621,11,11,1.OD-6*END STEP
C-2
ABAQUS Input Deck: Case 2a
*HEADING, UNSYMM
STATIC DIFFUSION -- STRONGLY COUPLED Standard Case 2
1,0.ODO21,0.10DO*NGEN,NSET=ROD1,21,1*USER ELEMENT, NODES=3 ,UNSYMM, COORDINATES=1, PROPERTIES=11,VARIABLES=9,TYPE=U1** THE FIRST AND THIRD NODES HAVE DEGREES OF FREEDOM IN CONCENTRATION AND** DISPLACMENT, WHILE THE SECOND NODE ONLY HAS DISPLACEMENT DEGREES OF** FREEDOM. NOTE THAT CONCENTRATIONS ARE TREATED AS TEMPERATURE D.O.F.i1,1
2,13,11,1"*ELEMENTTYPE=U1,ELSET=SCID1,1,2,3*ELGEN, ELSET=SC1D1,10,2,1,1,1,1,1*UEL PROPERTY, ELSET=SC1D** PROPERTIES ARE IN THE ORDER:** CFLAG,AREA,E,NU,D,ALPHA,CREF,RHO** MW,TEMP,TRAPS1. ODO, 1. OD-4, 200. OD9, 0. 3D0, 1. OD-9, 5. 2D0, 0.0D0,7. 8D3,1.00797D-3,293.ODO,19.ODO"*MATERIAL,NAME=A4340*DEPVAR
9"HBOUNDARY** ZERO DISPLACEMENT AT LHS1,1
** ZERO DISPLACEMENT AT RHS21,1** ZERO CONCENTRATION AT LHSi,11*INITIAL CONDITIONS, TYPE=TEMPERATURE** 0.0 INITIAL CONCENTRATION OVER ROD LENGTH
ROD, 0.ODO"*WAVEFRONT MINIMIZATION, SUPPRESS"*STEPINC=1000"*COUPLED TEMPERATURE-DISPLACEMENT, STEADY STATE
** THE FIRST AND THIRD NODES RAVE DEGREES OF FREEDOM IN CONCENTRATION AND** DISPLACMENT, WHILE THE SECOND NODE ONLY HAS DISPLACEMENT DEGREES OF** FREEDOM. NOTE THAT CONCENTRATIONS ARE TREATED AS TEMPERATURE D.O.F.11,i2,13,11,1
*ELEMENT,TYPE=Ul,ELSET=SCID
1,1,2,3*ELGEN, ELSET=SC1D
1,200,2,1,1,1,1,1*UEL PROPERTY, ELSET=SCID** PROPERTIES ARE IN THE ORDER:
** THE FIRST AND THIRD NODES HAVE DEGREES OF FREEDOM IN CONCENTRATION AND** DISPLACMENT, WHILE THE SECOND NODE ONLY HAS DISPLACEMENT DEGREES OF** FREEDOM. NOTE THAT CONCENTRATIONS ARE TREATED AS TEMPERATURE D.O.F.11,12,13,11,1
** PROPERTIES ARE IN THE ORDER:** CFLAG,AREA,E,NU,D,ALPHA,CREF,RHO** MW,TEMP,TRAPS1.0D0,1 . 0D-4,200.0D9,0. 3D0, i. OD-9, 5. 2D0, 0. 0D0,7.8D31.00797D-3,293.0D0,499.0D0*MATERIALNAME=A4340*DEPVAR
9*BOUNDARY
** ZERO DISPLACEMENT AT LHS1,1
*INITIAL CONDITIONS, TYPE=TEMPERATURE** 1.OE-7 INITIAL CONCENTRATION OVER ROD LENGTHROD, 1.OD-7
*WAVEFRONT MINIMIZATION, SUPPRESS*STEP,INC=5000,AMPLITUDE=STEP*COUPLED TEMPERATURE-DISPLACEMENT, STEADY STATE
1.OD-4,1.0, ,5.OD-3
*CONTROLS, PARAMETERS=FIELD,FIELD=TEMPERATURE
,5.OD-21
*DLOAD,OP=NEW
SC1D,UINU
*NODE PRINT, SDNMARY=NO, FREQUENCY=100
COORI,UT,NT*EL PRINT, SUMMARY=NO, FREQUENCY=100
SDV
*BOUNDARY** SPECIFY RIGHT HAND BOUNDARY CONCENTRATION AS 1.OE-7401,11,11,1.0D-7*END STEP
C-5
CIO
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C-1
APPENDIX D: Sample Output
This Appendix contains a copy of an ABAQUS output listing generated in thesolution of a 1-D fully coupled deformation-diffusion problem.
D-1
ulU)U
U)-mU) w E
QH m U) UH n U
wFU
w ý ý ý H nHHýElH
A A Av H I IH
HI IH H H -
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CY0 Y A Al A AA A A A W lW
CY 0 - - - -- 11
II 00 V VI I EV V H
01 0 01I I01~ 01 0 A IA A A AAHIII
0 01 I I U VV O
010 A AI A A AAA I U
01011010010 v I I------------HIV II V V V ( H
AA A A AA A AAH0
A A A A A AAA(n
-- ------- (nHm QCQ vv I I v vv Hco
m o1 A AA A A A AH X
AA AA PQ
I IAAcoA AA
I H H -
H H H VV I D-V2
00
0-I;?; 0*
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H l M Z~ HO HH 4H 0
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HHH 0 H HD it HO (WW 0 E 4 10ýI
HO HO HH 0 H OHHHO H 0 H 0tiE
00 C)HOE
tO H H)ý 0 Q 0
HO 0 OHH 14 2 Nx EHw OC H : ; U> OH; OHm DHH H1 H-Hý u CMHH > HE E H
El~ O H 004 (n-E00:0 0 >,M 1
H0 H 0 : :OInO a 00 a *H 0 H 0
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'C I onC In 0 0 E0 In 'I I 1 InI
HC I II In .2 .2 In[4D '
In o >1I u) H In 0 In WD En E1 In Q 0041r0~~~~~P IfF0)DC CD -' 0 nP! E I "I Hl I> toI C 0Z 0P- Io 020 F X0~~~~~~~~1 01 -n H In I-4C C 0 ' CI~
'C H O mD ' C o W W 'nIn H Cm InI H H- II o~t 'In CD C . IH'~~ ;E; CD41 0' ' I 04 O
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To appear in Proceedings of the 5th International Conference on Hydrogen Effects on Material Behavior, JacksonLake Lodge, Wyoming, Sept. 1994.
MODELING OF HYDROGEN TRANSPORT
IN CRACKING METAL SYSTEMS
J. P. Thomas and C. E. ChopinUniversity of Notre Dame
Department of Aerospace and Mechanical Engineering374 Fitzpatrick Hall
Notre Dame, IN 46556-5637
Abstract
An extended stress-assisted diffusion theory with concentration induced deformations andequilibrium trapping effects was developed for modeling hydrogen transport in cracking metalsystems. Special Fortran "user element" subroutines were written for the ABAQUS finiteelement code to solve the transport equations in 1-D geometries. A simple rod problem with asquare-root singular stress was analyzed to assess the influence of the hydrogen induceddeformation on the concentration and deformation field variables. Significant differencesbetween the classical and extended stress-assisted diffusion theory were demonstrated and theimplications for modeling crack tip hydrogen distributions discussed.
IntroductionTwo fundamental questions naturally arise in modeling the influence of hydrogen on the
crack growth rate of metals. Namely, how does hydrogen "enhance" the crack growth rate(CGR); and what is the relationship between the hydrogen distribution within the material andthe corresponding "enhancement" in the CGR? Quantitative knowledge of the crack tip hydrogendistribution under service or laboratory test conditions is requisite to addressing these questionsand will require: a) a hydrogen transport model that incorporates trapping and deformation-concentration coupling effects (governing equations); b) knowledge of the time dependent, non-uniform hydrogen distribution along the crack walls (boundary conditions); and c) amathematical solution technique for the resulting system of non-linear equations.
Stress-assisted diffusion (SAD) theory [1] is commonly used to model hydrogen transportin cracking metals systems. This theory represents an extension of classical diffusion with ahydrostatic stress gradient term added as a driving force for diffusive transport. Equilibriumtrapping effects are included through the use of an effective diffusion coefficient. The influenceof hydrogen on the material deformation state is assumed to be negligible, and this uncouples thedeformation equations from the concentration variable. Hydrostatic stresses determined fromsolutions to standard elasticity or plasticity problems are used with the SAD equation to solve forthe resulting concentration. The steady-state hydrogen distribution for Mode I cracks in linearelastic materials with uniform hydrogen concentration along the boundaries is given by [2]:
c coexp (constantt X /r cos0)
where co is the boundary concentration, K is the stress intensity factor, and r and 0 are polarcoordinates centered at the crack tip. An infinite hydrogen concentration is predicted at the cracktip. Transient and steady-state hydrogen distributions for a plastically deforming crack in ironhave been obtained by Sofronis & McMeeking [3] using finite element methods. Their resultsshow large, but finite, concentrations at the crack tip region, primarily in traps near the cracksurface. They conclude that the crack tip hydrogen distribution is primarily determined by thecreation of dislocation traps via plastic straining at the crack tip.
Damage models that attempt to link the crack tip hydrogen distribution to the fractureprocess have been reviewed in [4-6]. Applications to service cracking problems have met withsome success, but the lack of information on the crack wall hydrogen distributions under serviceor laboratory conditions has limited the usefulness of these models.
The task of specifying the crack wall hydrogen distribution is difficult because of thecomplex nature of the interacting chemical, mechanical, and metallurgical processes operativeduring hydrogen assisted cracking of metals (Figure 1). In aqueous systems, the hydrogenproduction process is driven by the rapid and irreversible evolution of the chemically unstable"bare" surface at the crack tip to a more stable equilibrium "filmed" state. An electron flow isinduced between the bare and filmed crack flank surfaces; net anodic (dissolution/filming)reactions take place on the bare surface and net cathodic (hydrogen reduction) reactions takeplace on the filmed surfaces.
Adsorbed hydrogen, Hads, can be produced on both the bare and filmed crack surfacesby: (1) the reduction of hydrogen ions in acidic environments; or (2) by the reduction of water inalkaline environments. The MHads species are then free to be absorbed by the transition reaction(a); or combine to form H 2 gas via: recombination (bl); or electrochemical desorption (b2).Reactions (a), (bl), and (b2) occur in parallel, but one of the two (bl) or (b2) reactions is usuallydominant (Figure 2).
Crack Tip RegionBare alocal
Steady State Film Coverage Film Growth Surface
MeT (Mie" . IHI, C1, Naet , .. )menH
ý - -
1. Electrochemical Mass Transport2. Anodic and Cathodic Surface Reactions local3. Hydrogen Absorption Reactions4. Hydrogen Transport and Trapping5. Hydrogen Damage
Figure 1: Schematic of the processes responsible for hydrogen assisted crack growth.
2
(1)Acidic: M + H+ + e- € MHads
(2)Alkaline: M + H 2 0 + e- < MHads ± 0H
(a) Adsorption-Absorption: MHads 4 MHabs
(bl) Recombination: MHads + MHads • H 2 + 2M
(b2) EC Desorption: MMHads + H+ + e- €:ý H 2 + M
Figure 2: Summary of the hydrogen producing reactions.
The distribution of MHabs along the crack surface is governed by the surface coverage ofMHlds and the kinetics of reaction (a) acting in parallel with reaction (bl) or (b2). These factorsare influenced, in turn, by: the electrochemical environment at the crack tip region (e.g., thepotential, pH, species concentrations, dissolved 02, etc.); the kinetics of the bare and filmedsurface reactions; and the rate of transport of Hab, from the crack surface into the material.
Iyer and Pickering [7] review and model the mechanisms and kinetics of hydrogenevolution and entry in stress free metallic systems with homogeneous electrochemical conditionsat the metal surface. Their model is used to quantify the rate constants associated with reactions(1) or (2) and (a) and (bl) or (b2) via analysis of experimental permeation data. Turnbull [8] hasreviewed electrochemical conditions in cracks with particular emphasis on corrosion fatiguecracks of structural steels in sea water. Similarly, Beck [9] and Newman [10] have examinedexperimental techniques for characterizing bare (and filmed) surface reaction kinetics. It seemsthat the above models, data, and techniques, plus information concerning the rate controllingprocess during crack growth, will have to be used in an analysis of the mass transport processwithin the crack to develop realistic predictions of the MHads distribution.
The objective of this paper is to address the above items a) and c). A hydrogen transportmodel for isothermal linear elastic materials with mutual deformation-concentration coupling andequilibrium trapping is developed using continuum mixture theory. Computational solutions for1-D geometries and arbitrary boundary conditions are obtained using the finite element codeABAQUSI supplemented with Fortran "user element" subroutines. The code is used to analyzehydrogen transport in a 4340 steel rod subjected to a singular body force( --, A/x 3 /2 ) whichproduces a square root singular stress. The concentrations, displacements, and dilatational strainspredicted by the fully coupled theory are somewhat larger than those predicted by classical SADtheory, depending on the extent of trapping. Further examination of the fully coupled theory inthe context of hydrogen transport in 2-D crack systems is currently underway.
The paper begins with a description of the strongly coupled hydrogen transport model.The use of ABAQUS with its "user element" subroutines for solving 1-D problems is describednext, followed by an analysis and discussion of hydrogen transport in a rod with a singular bodyforce. Extension of the finite element model to 2-D geometries and its application to crackingmetal systems is briefly discussed, followed by suggestions for future work.
I ABAQUS is a commercial finite element code supported by Hibbitt, Karlsson & Sorensen, Inc., Pawtucket, RI.
3
Modeling the Hydrogen Transport and Trapping ProcessThree species of hydrogen are modeled in the analysis:
HL -- interstitial or lattice hydrogenHR = weak or moderately (reversibly) trapped hydrogen (e.g., AEb < 30)
Balance EquationsBalance equations for the mixture mass, the three hydrogen species masses, the mixture
linear momentum and moment of momentum, mixture energy, and mixture entropy can bewritten. Since our modeling considerations are restricted to isothermal linear elastic materials,only the balance equations for the hydrogen species mass and the mixture linear momentum(assuming negligible inertial effects) need be explicitly considered. They are given by:
Mass: 0---+ V -Jk = ak (k=L, R, orI) (1)a9t
Linear Momentum: uij + F = 0 (i,j x, y, or z) (2)
where: Ck - mass fraction concentrations for Hk [kg/kg].
Jk concentration flux vectors for Hk [kg/kg r m/s ].ak mass supply rates for Hk [kg/kg/s].
caij stress tensor [N/m 2 ].
F- the ith component of body force vector [N/M 3].
Trapping AnalysisExpressions for the mass supply rates ak in Eq. (1) are written in accordance with the
trapping model of McNabb & Foster [11]:
Stoichiometry: HL €# HR and HL =- H1 (3)
The stoichiometric relations require the supply terms sum to zero (i.e., aR + al - aL).Kinetics of Trapping: aR = k~f(1- OR)CL Rk•O (4)
k{(1 - O,)CL - k I01 k -(1 O,)CL
where: k{f, kf, k b, k b = forward and backward trap rate constants for HR and H1 [1/s].0 R and 0, - cR/c 8 and c_/cs, respectively, are the fraction of filled trap sites [1].cS and cs - saturation mass fraction concentration of HR and H1 [kg/kg].
The quantities kf, kf, kb, kb, cs and cs are related to trap site densities, probability of capture,etc. and can be determined from experimental measurements (e.g., [ 12-14]).
Significant simplifications are effected when the rate constants for trapping are muchgreater than those for diffusive transport. Trapping can then be modeled as a steady-state process(i.e., aL = aR = al = 0). This case is considered below.
4
C8 KR CLSS Trapping: CR -- 1± KR CL & CI = c4 (5)
kf - xKR R= xp AEb)
so that the total internal hydrogen concentration is simply a linear function of CL(Xi, t):
The three versions of Eq. (1) can be summed to give a single equation by the followingconsiderations. First, JR = JI = 0 because of the linear elastic material assumption whichprecludes trap site motion (e.g., dislocation motion during plastic deformation). And second,from Eq. (5a):
9CR DCL OCIF;•ý CsKR and t 0 (7)at R at at
The resulting mass balance equation is given by:DgCL
where UL is the mass based chemical potential [15] defined by:
PL 1 (p°L(T) +RTln(CL) -VHke) [J] (10)
DCL ML L jand: DL diffusion coefficient for HL [m2/s].
RL gas constant for HL = R/ML [J/kg °'K].
ML =- molecular mass of hydrogen [kg/rnoll.
T - temperature ['K].
0 - V)(CL, CR, c; cij ; T) free energy per unit mixture mass [J/kg].
LP'(T) = reference potential for HL at temperature T [J/kg].
VH = partial molar volume of hydrogen in the metal [m3 /mol H].
k bulk modulus of elasticity [N/m 2].
e trace of the strain tensor (i.e., e = ii = Ell + E22 + E33) [M/m].
The use of mass based chemical potentials simplifies the analysis of fully coupled deformation-diffusion processes because of the primary role played by mass in the deformation equations.
The resulting expression for concentration flux is given by:
JL DL VH DL kCJ76 (11)
5
The constitutive equation for the stress consists of Hooke's law combined with adilatational stress contribution due to changes in the total hydrogen concentration:
Stress: piJ ao A e 6ij + 2Gcij - 3 k ceH (1 + c' KR)A cL 6ij (12)
where p mass density of the solid [kg/M 3].A Lame' constant [N/mr2].6ij Kronecker delta (•ij = I for i z-j and 0 otherwise).G shear modulus [N/mr2].6ij i1 (ui,j + uj,j) infinitesimal strain tensor [mrn/m].
ui --the ith component of the displacement vector [m].iH- linear expansion coefficient for internal hydrogen K VH [M/n/AcH].
(1 + c' KR)ACL -= change in Ctotal from some reference level.
The relationship between the chemical potential and stress (Eqs. (10) and (12)) is dictated by thethermodynamic reciprocity relationship:
D -/L _ (u7i/P) (13)acij DCL
Governing EquationsCombining the mass and momentum balance equations with the constitutive relations
results in the following system of governing equations for transport:Diffusion Equation:
OCL Deff V 2c- VH Deff LV2 (14)
Deformation Equations (i = 1, 2, 3):
(A + G) De +G7U+ ,=3 l(+c'KR) OC (15)x R (15±
where Doff is an "effective" diffusion coefficient defined by: Doff _ DL/(1 + c'KR).Equation (14) is identical to the stress-assisted diffusion equations published in the
literature with the exception of the V2e term which is identically zero when linear elasticmaterial behavior is assumed (and Fj = 0). In the present formulation it is given by:
3•2 3kcrH (1 + c' KR) V 2CL (16)A + 2G
Computational ModelingEquations (14) and (15) form a system of non-linearly coupled partial differential
equations that must be solved for CL and ui as functions of the space and time coordinates (xi,; t).This is accomplished for 1-D geometries by writing Fortran "user element" subroutines for theABAQUS finite element code to effect a computational solution. The 1-D "plane stress" form ofthe governing equations 2 (aor =Y Cz7 = 0) is given by
2 "Plane strain" forms of the 1-D equations can be obtained by a simple change in constants.
6
&CL OJL _0(7Diffusion: (1 + c' KR) -O 0 (17)R at ax
S OCL VH DL OJL ~DL + VHDLkae
a9x IRT ax
Deformation: O + F 0 (18)
Ux 3k(e- 30H (1 + c'KR)A CL) = (a (1E K) )
A single differential equation for CL can be obtained by combining Eqs. (17) and (18):
OcL a F 1 (I +cOK\)CL VH DffLat x [ff MAL R T -R ) ax± 3RT FxcLJ (19)
which, in the absence of body forces, is a standard diffusion equation with a concentrationdependent diffusion coefficient. Solutions to the steady-state problem, with and without bodyforces, can be obtained by integration (symbolic computation is highly recommended).
A coupled set of equations for a single finite element of length h and cross-sectional areaA can be written from Eqs. (17-18) using the method of weighted residuals. The result, afterintegration by parts, is given by:
oh (( + C KR) { x -- Gi(x) d=- h±~R) at fi.x~ J axx)}dx= -J{Gj()} 0 (20){G JJ(xL){} x}-0 (20
-ux o ax _F {H3 (X)} dx= -ux f{Hj(x)}
2
where: CL LGi(x)ci(t) LG (x) {ci(t)} Z-- 1,2 (21)i=1
3
u L7Hi (x)uj =LA J f j VI j 1, 2, 3j=l
are 2 and 3 node finite element representations for the lattice concentration and displacement,Gi(x) and Hj(x) are the respective linear and quadratic interpolation functions, and '"indicates a row matrix while {... } indicates a column matrix.
Substituting Eqs. (21) into (17b) and (18b) and converting to isoparametric coordinates,results in finite element equations of the form:
[ [C]1 [0]1 V{iI} [Kc] [K,.] {ci} {'c([0] [0] {'d I L [K,] I{R} (22)
The capacitance, stiffness, and RHS matrices, [C], [K], and {R}, are functions of CL and u,which necessitates the use of nonlinear solution techniques. The integrations required for thecomputation of the [C] and [K] matrices are accomplished using two-point Gauss quadrature.
The term Oe/Ox in JL includes the second derivative in u (from Eq. (18b):
'7
_±e 1- 2U 02(23)ox 2v)- + 2(1 + v)aH ( + R OCL(x
This would normally require the use of Ci continuous elements in order for u to satisfy theelement interface compatibility requirement [16]. To avoid this complication, we use the valuesof u and CL from the previous time step to approximate Oe/Ox in the current time calculations.
The deformation equations are analogous to those of linear thermoelasticity. This allowsthe use of ABAQUS' coupled temperature-displacement routine to solve the problem. Timeintegration is performed using an implicit backwards difference scheme. The computations andsetup required of the user element subroutine include [17]: calculation of the [C], [K] and {R}matrices; calculation of the "Jacobian" and {right-hand-side} matrices for the Newton-Raphsonnon-linear solver routine; and updating the solution dependent state variables: JL, o, e, and&e/Ox. The subroutine is written in double precision Fortran, and the code is run on a SunSPARC 10 workstation.
Applications
Steady-state hydrogen transport in a 4340 steel rod subjected to a body force selected togive a 1/xV singular stress (Figure 3) was studied using the ABAQUS code. The rod wasdiscretized using 200 elements of length 1i = 0.104/0.94'-' [Mm], where the first and smallestelement (11 = 0.104 [/[m]) was placed at the left-hand end of the bar (x = 4 x 10-3 [cm]). Theparameter values used in the analysis were: p =7.8 [g/cm 3]; T=293 [K]; DL = 1 x 10-5[cm 2/s]; VH = 2.0 [cm 3/mol]; ML 1.008 [g/mol]; (1 + CeKR) = 20 (low trapping) & 500(high trapping); E = 200 [GPa]; LI 0.3; F, 15 x 106 [N/M 3 ], and (CL)ref = 0.
Body Force, F.
]5 x 10' 104
2.48 - 0.50 log(xr)
S101
10cm c
JL(4.]0 x 10 -5)0 L(.0) -7
1(4.0 x 10- 5 )=0 G(O. 10)=O 10 10-2 10 100 10'
Distance, x [cm]
Figure 3: The rod geometry with the applied Figure 4: Gauss integration point stressesbody force and boundary conditions. with a least squares line fit through the data.
ResultsAn analytical solution to this problem 3 was obtained by direct integration of Eq. (19). The
singular nature of the body force results in a mathematical solution for the concentration that is amulti-valued function of x. In order to avoid the multiple concentration values, we must restrict
3 To be described in more detail in a future publication.
our considerations to the region x > 4 x 10-3 [cm]. The difficulty arises when the sign of theOCLIOX coefficient term in Eq. (19) changes from positive to negative. That is, when CL exceeds
some "critical" value given by: ML IR T/p V2 k(1 + c' KR).
Figures 4 through 8 show the steady-state analytical and finite element (FE) results (every20th point) for the high and low trapping conditions. Figure 4 shows the FE stresses at the Gaussintegration points with a least squares fit line through the FE data. The stress in this problem isnot influenced by the hydrogen; it is simply related to the integral of the body force per Eq. (18a).The required square root singularity in the stress is clearly predicted by the FE code as indicatedby the slope of the least squares line fit.
Figures 5 & 6 show the concentration distribution for the fully coupled and stress-assisteddiffusion (SAD) theories with uHYD = 10 x 106/ ,x. The fully coupled model predicts largerconcentrations than the SAD model as the singularity is approached, and the difference increasesas the degree of trapping increases (i.e., 1 + c'KRt = 500 -vs.- 20). Figure 6 shows that the fullycoupled concentration singularity is more severe than the exponential square root singularity ofthe SAD model for the higher trapping case.
5.0 - S.A.D.S4.0 - Analytical 100
53003.0- 20--- 20lx"
S2.0 .= '°x "•. •S.A.D.
O F 1 - - Analyticalo 10-1 0 F.E.M.
1.0 - :- ' I ' = -
10-3 10-2 10-1 100 10-3 10-2 10-' 100
Distance, x [cm] Distance, x [cm]
Figure 5: Concentration predictions for the Figure 6: Plot to illustrate the difference infully coupled and SAD theory under high and singular behavior between the fully coupledlow trapping conditions. and SAD model concentration predictions.
10.0 - Fully Coupled w/ F 10.0 Analytical
2 8.0 -- '- Fully Coupled w/o F F.E.M.D - Uncoupled with F. 20
; 6.0 5.0 . .78500
E 4.0co 20
2.0 - -- 031- - o
0.0 - 0.00 2 4 6 8 10 10-3 10-2 10-' 100 101
Distance, x [cm] Distance, x [cm]
Figure 7: Nodal displacement curves for a Figure 8: Analytical and FE dilatationalvariety of conditions. strains for a variety of conditions.
9
Figures 7 & 8 show nodal displacement curves and Gauss point dilatational strains,respectively. The fully coupled theory predicts larger displacements and dilatational strainsthroughout the rod. Again, the differences increase as the degree of trapping increases. Thedisplacements and strains for the zero body force case are also shown.
Discussion
This simple rod problem with its square root singular stress clearly illustrates thedifferences between the fully coupled and classical SAD models. The concentration differencesare relatively small but dependent on the trapping constants (i.e., the amount of trappedhydrogen). The non-physical nature of the multi-valued concentration in the fully coupled theorymay be an indication that singularities, like the inverse square root singularity found at 2-D cracktips in classical elasticity, do not naturally arise in the fully coupled theory. This could haveimportant implications in modeling the hydrogen damage at 2-D crack tips.
The influence of the concentration coupling to the deformation state, and its role in thehydrogen damage process, is also an important consideration. The fully coupled theory exhibitslarger dilatational strains than those predicted by SAD theory (classical elasticity), but identicalstresses. On the other hand, a bar fixed between two walls and subject to a change inconcentration results in differences between both strain and stress. This is also likely to be true intwo and three-dimensional problems where the deformations in different directions are coupledthrough the constitutive equations.
The results suggest the need for an examination of the coupled theory in the context ofcrack tip hydrogen predictions. We are currently developing 2-D isoparametric rectilinear (8-node displacement; 4-node concentration) and axisymmetric user element routines for use in thistask. Extension of the model to include the effects of non-equilibrium trapping, crack tipplasticity, and hydride formation will be considered in the future.
AcknowledgmentsThe discussions with Dr. D. Kirkner concerning the finite element aspects of this work,
and the support of this work by the Office of Naval Research under Contract No. N00014-93-1-0845, are both gratefully acknowledged.
References1. J. C. M. Li, R. A. Oriani, & L. S. Darken, Z. Physik. Chein., 1966, Vol. 49, pp. 271-290.2. H. W. Lui, Trans. ASME-J. Basic Engng., 1970, Vol. 92, pp. 633-638.
3. P. Sofronis, and R. M. McMeeking, I. Mech. Phys. Solids, 1989, Vol. 37, pp. 317-350.4. H. K. Birnbaum, in Environment-Induced Cracking of Metals, R. Gangloff and M. B.
Ives, Eds., National Association of Corrosion Engineers, Houston, TX, 1990, pp. 21-29.
5. Hydrogen Degradation of Ferrous Alloys, R. A. Oriani, J. P. Hirth, and M. Smialowski,Eds., Noyes Publications, Park Ridge, NJ, 1985.
6. J. P. Hirth, Metall. Trans. A, 1980, Vol. llA. pp. 861-890.
7. R. N. Iyer and H. W. Pickering, Annu. Rev. Mater. Sci., 1990, Vol. 20, pp. 299-338.8. A. Turnbull, Reviews in Coatings and Corrosion, 1982, Vol. 5, pp. 43-171.
9. Beck, T. R., (1977), in Electrochemical Techniques in Corrosion, R. Baboian, Nat.Assoc. of Corrosion Engineers, Houston, TX, 27-34.
10
10. Newman, R. C., (1984), in Corrosion Chemistry Within Pits, Crevices, and Cracks, A.Turnbull, Ed., Her Majesty's Stationery Office, London, UK, 317-356.
11. A. McNabb and P. K. Foster, Trans. TMS-AIME, 1963, Vol. 227, pp. 618-627.
12. B. G. Pound, Acta Metall. Mater., 1991, Vol. 39, pp. 2099-2105.
13. H. H. Johnson, Metall. Trans. A, 1988, Vol. 19A, pp. 2371-2387.
14. R.Oriani, Acta Metall., 1970, Vol. 18, pp. 147-157.
15. J. P. Thomas, Int. J. Engng. Sci., 1993, Vol. 31, pp. 1279-1294.
16. K. H. Huebner and E. A. Thorton, The Finite Element Method for Engineers, 2nd ed., J.Wiley & Sons, NY, 1982, pp. 79-85.
17. ABAQUS Theory and User's Manuals, Standard Version 5.3-1, Hibbitt, Karlson, &Sorensen, Inc., Pawtucket, RI, 1993.
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APPENDIX F: Summary Of The Publications/Reports/Presentations
1. Papers Published in Refereed Journals:1. J. P. Thomas and R. P.Wei, "Standard Error Estimates for Rates of Change From
Indirect Measurements", TECHNOMETRICS, (in review).2. J. P. Thomas and P. Matic, "Solute Transport Modeling in Elastic Solids",
International Journal of Engineering Science, in preparation.
2. Non-Refereed Publications and Technical Reports:
1. J. P. Thomas and C. E. Chopin, "Modeling of Hydrogen Transport in Cracking MetalSystems", Proceedings of the 5th International Conference on Hydrogen Effects onMaterial Behavior, A. W. Thompson and N. R. Moody, Eds., The Materials Society(TMS), Jackson Lake Lodge, Wyoming, Sept. 1994 (to appear).
2. J. P. Thomas and C. E. Chopin, "An ABAQUS User Element Routine for One-Dimensional Coupled Deformation-Diffusion Problems", Department of Aerospaceand Mechanical Engineering Report, University of Notre Dame, Department ofAerospace and Mechanical Engineering, Notre Dame, IN, (in preparation).
3. Presentationsa. Invited:
1. J. P. Thomas, April 1994, "Modeling the Influence of Hydrogen on the CrackGrowth Rate of Metals", Lehigh University, Department of MechanicalEngineering and Mechanics, Bethlehem, PA.
2. J. P. Thomas, July 1994, "Environmental Effects in Fatigue Crack Growth",Ladish Corporation, Inc., Cudahy, Wisconsin.
3. J. P. Thomas, October 1994, Invited Lecture, "Solute Transport in ElasticSolids", D. G. B. Edelen Symposium, 31st Annual Technical Meeting of theSociety of Engineering Science (SES), College Station, Texas.
4. J. P. Thomas, October 1994, "Modeling the Influence of Internal Hydrogen onthe Crack Growth Rate of Metals", University of Kentucky, Department ofEngineering Mechanics, Lexington, Kentucky.
5. J. P. Thomas, January 1995, "Modeling the Influence of Internal Hydrogen onthe Crack Growth Rate of Metals", Westinghouse Electric Corporation, BettisAtomic Power Laboratory, West Mifflin, Pennsylvania.
b. Contributed:1. J. P. Thomas, September 1994, "Modeling Crack Tip Hydrogen
Distributions", Poster Session, 5 th International Conference on HydrogenEffects on Material Behavior, The Materials Society (TMS), Jackson LakeLodge, Wyoming.
2. J. P. Thomas and C. E. Chopin, October 1994, "Finite Element Modeling ofHydrogen Transport in Metals", 3 1st Annual Technical Meeting of the Societyof Engineering Science (SES), College Station, Texas (planned).
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4. Books: none
5. List Of Honors/Awards:
Name of PersonReceiving Award Recipient's Institution NameP Sponsor and
Mr. Charles E. Chopin, Department of Aerospace Purpose of the AwardGraduate Student and Mechanical Engineering SES 31st Annual Technical
University of Notre Dame Meeting Student Stipend
Notre Dame, IN National ScienceFoundation & Office of
Naval Research
To support studentparticipation in the SES
Annual Meeting.
6. Participants And Their Status
Principal Investigator:1. Dr. James P. Thomas, Assistant Professor, University of Notre Dame, Department ofAerospace and Mechanical Engineering, Notre Dame, IN.
Graduate Student:1. Mr. Charles E. Chopin, 2nd year Doctoral Student, University of Notre Dame, Departmentof Aerospace and Mechanical Engineering, Notre Dame, IN.