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A coupled diffusion and cohesive zone modelling approach for numerically assessing hydrogen embrittlement of steel structures

20 Apr 2017-International Journal of Hydrogen Energy (Elsevier)-Vol. 42, Iss: 16, pp 11980-11995

Abstract: Simulation of hydrogen embrittlement requires a coupled approach; on one side, the models describing hydrogen transport must account for local mechanical fields, while on the other side, the effect of hydrogen on the accelerated material damage must be implemented into the model describing crack initiation and growth. The present study presents a review of coupled diffusion and cohesive zone modelling as a method for numerically assessing hydrogen embrittlement of a steel structure. While the model is able to reproduce single experimental results by appropriate fitting of the cohesive parameters, there appears to be limitations in transferring these results to other hydrogen systems. Agreement may be improved by appropriately identifying the required input parameters for the particular system under study.
Topics: Hydrogen embrittlement (69%), Hydrogen (55%), Diffusion (business) (51%)

Summary (4 min read)

1. Introduction

  • Hydrogen induced degradation of mechanical properties, often termed hydrogen embrittlement (HE), is a well recognized threat for structural steels.
  • It manifests as loss in ductility, strength and toughness, which may result in unexpected and premature catastrophic failures.
  • Rather it appears that different mechanisms apply to different systems, and that a combination of mechanisms is more likely in many cases.
  • In recent years, cohesive zone modelling has gained increasing interest as suitable method for modelling hydrogen embrittlement [10, 11, 12, 14, 16], with the possibility of providing increased understanding of the involved process and their interactions combined with reduced time and costs compared to experimental programs.
  • The coupling aspect between hydrogen transport and cohesive zone modelling is discussed and put in conjunction with hydrogen diffusion models in Section 2.

2. Hydrogen transport models

  • The process that results in hydrogen embrittlement includes an important transport stage of hydrogen to the site of degradation.
  • Atomic hydrogen is generally considered to reside either at normal interstitial lattice sites (NILS) or being trapped at microstructural defects like dislocations, carbides, grain boundaries and interfaces.
  • Traps generally reduce the amount of mobile hydrogen, thus decreasing the apparent diffusivity and increasing the local solubility of the system.
  • To date, models of transient hydrogen diffusion generally account for trapping by dislocations and hydrostatic drift.
  • Recent approaches include capturing the effect of multiple trap sites and hydrogen transport by dislocations [13, 21, 22].

2.1. Hydrogen in lattice

  • Given a metal lattice, the hydrogen concentration in NILS, CL, can be expressed by [8] CL = βθLNL (1) with θL being the lattice site occupancy, NL the density of solvent atoms and β the number of NILS per solvent atom, usually assigned to be 6 under the assumption of tetrahedral site occupancy in iron.
  • The density of solvent atoms, NL, can be calculated through NL = NA VM (2) where NA is the Avogadro constant equal to 6.022 · 1023 mol-1, and VM is the molar volume of the host lattice, which for iron is 7.106 ·10-6 m3/mol at room temperature.

2.2. Hydrogen in traps

  • Similarly as for lattice sites, the hydrogen concentra- tion in a specific trapping site can be expressed by [8].
  • The ability of a trap site to hold hydrogen is associated with the trap binding energy, representing the attractive interaction of a trap site compared to a normal lattice site.
  • There have been significant advances in theoretical approaches to capture the effect of traps on hydrogen transport, with models by McNabb and Foster [31] and Oriani [32] describing the process for steel.
  • The dislocation trap density is then expressed as a function of the dislocation density ρ and the lattice parameter a N (d) T = √ 2 ρ a (7) The dislocation density (measured in dislocation line length per cubic meter) is considered to vary linearly with the equivalent plastic strain according to ρ = ρ0 + γεp for εp < 0.51016 for εp ≥ 0.5 (8) where ρ0 = 10 10 line length/m3, denotes the dislocation density at zero plastic strain, and γ = 2.0·1016 line length/m3.
  • Generally, the trapped concentration increases with increasing lattice concentration and increasing trap binding energy, until saturation is reached.

2.3. Hydrogen diffusion

  • The main mechanism for hydrogen diffusion in steel is lattice diffusion by interstitial jumps, where the hydrogen atom occupy interstitial sites and move by jumping from one interstitial site to a neighbouring one [38].
  • Chemical potential gradients constitute the main driving force for hydrogen diffusion in steel; hydrogen will diffuse from regions where the chemical potential is high to regions where it is low, and the process ceases once the chemical potentials of all atoms are everywhere the same and the system is in equilibrium [39].
  • Assuming that the diffusion flux is proportional to the concentration gradient, which often is the case, Fick’s laws are the governing equations describing the processes.
  • These laws represent a continuum description and are purely phenomenological.
  • The large scatter observed for ferritic steels is generally considered to be associated with trapping [41].

2.4. Implications of the hydrogen transport model

  • In the following section, the effect of varying the hydrogen solubility, the trap binding energy and the trap density on the total hydrogen distribution, as controlled by the hydrogen transport model in Equation (14), is illustrated.
  • Displacements are enforced on the circular boundary, controlled by the stress intensity factor KI .
  • Figure 6a-d displays the resulting hydrogen profiles in front of the notch tip at the end of loading, for trap binding energies in the range 20-60 kJ/mol and two initial hydrogen concentrations of 0.00034 wppm and 1 wppm, representative of the theoretical solubility of hydrogen in ferrite and a 3 % NaCl aqueous solution [11], respectively.
  • In the plots, CL is therefore included as one single line, representing all the cases considered.
  • For the low trap density model, the maximum attainable trapped concentration is 0.033 wppm, 100 times an initial lattice concentration of 0.00034 wppm.

3. A cohesive zone modelling approach to hydro-

  • Gen embrittlement Cohesive models were first formulated by Barenblatt [44] and Dugdale [45], who introduced finite non-linear cohesive tractions in front of an existing crack, as a mean to overcome the crack tip stress singularity.
  • To date, the cohesive model is extensively applied for crack propagation analysis using the finite element method.
  • Among the various approaches available, it is appealing in that it requires few parameters and in its universality of applicability [46].

3.1. The cohesive model

  • The cohesive theory of fracture is a purely phenomenological continuum framework, not representative of any physical material.
  • Common to most cohesive laws is that they can be described by two independent parameters out of the following three: the cohesive strength σC , the critical separation δC and the cohesive energy ΓC .
  • The area embedded by the curve represents the cohesive energy.
  • Alvaro et al. [15] points out the importance of this in relation to modelling hydrogen embrittlement.
  • Yu et al. [57] have applied the viscosity term by Gao and Bower [56] in a three step, un-coupled, hydrogen in- formed cohesive zone model under constant displacement, and found the viscous regularization to be effective in solving the convergence problem with good accuracy.

3.2. Implementing hydrogen influence

  • Most known attempts of implementing hydrogen influence into the cohesive model is through the HEDE principle [11, 15, 16, 58, 59, 60]; hydrogen reduction of the cohesive energy at fracture.
  • The data fit is illustrated by the red line in Figure 8b.
  • Using the coupling between hydrogen coverage and bulk concentration as supplied by the Langmuir-McLean isotherm, Serebrinsky et al. [11] suggested ∆G0b = 30 kJ/mol, which represents the trapping energy of hydrogen at a Fe grain boundary, yielding a threshold concentration of about 0.001 wppm and an embrittlement saturation level of about 5 wppm.
  • Figure 10b illustrates hydrogen influence, in terms of hydrogen coverage, according to Equation (20), on both the cohesive strength and on the critical separation, for the polynomial cohesive law by Needleman [48].
  • 64], no quantification of any effect of hydrogen on the critical separation is found to date.

3.3. Coupling of diffusion and mechanical models

  • The Langmuir-McLean isotherm defines the necessary coupling between the hydrogen diffusion model in Section 2.3 and the hydrogen-dependent cohesive law described in the previous section.
  • Choosing a ∆G0b level in the lower range may be justified, conforming to the findings by Novak et al. [13] and Ayas et al. [37] that the only possible trap sites associated with hydrogen embrittlement are low-binding energy traps.
  • Olden et al. [12, 15] and Moriconi et al. [16] have applied the latter approach.
  • (Equation (14)) to account for hydrogen transport by dislocations.
  • Brocks et al. [14, 68] have developed a model of hydrogen induced cracking, which in addition to the coupled interactions of hydrogen diffusion and reduced cohesive strength, also includes the effect of surface kinetics on hydrogen absorption and hydrogen induced softening of the local yield strength (HELP mechanism).

5. Conclusion

  • A coupled mass transport and cohesive zone modelling approach for simulating hydrogen induced cracking is described and discussed.
  • Based on calculations, the main findings are summarized as follows: •.
  • These levels have again significant influence on hydrogen induced reduction of the cohesive strength.
  • New developments within modelling of mass transport may improve the agreement.
  • Further, transferability may be improved by appropriately identifying the required input parameters for the particular system under study.

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A coupled diffusion and cohesive zone modelling approach for numerically assessing
hydrogen embrittlement of steel structures
L. Jemblie
a,
, V. Olden
b
, O. M. Akselsen
a,b
a
Department of Engineering Design and Materials, NTNU, 7456 Trondheim, Norway
b
SINTEF Materials and Chemistry, 7456 Trondheim, Norway
Abstract
Simulation of hydrogen embrittlement requires a coupled approach; on one side, the models describing hydrogen transport
must account for local mechanical fields, while on the other side, the effect of hydrogen on the accelerated material
damage must be implemented into the model describing crack initiation and growth. The present study presents a
review of coupled diffusion and cohesive zone modelling as a method for numerically assessing hydrogen embrittlement
of a steel structure. While the model is able to reproduce single experimental results by appropriate fitting of the cohesive
parameters, there appears to be limitations in transferring these results to other hydrogen systems. Agreement may be
improved by appropriately identifying the required input parameters for the particular system under study.
Keywords: Hydrogen embrittlement, Hydrogen transport, Cohesive zone modelling
1. Introduction
Hydrogen induced degradation of mechanical proper-
ties, often termed hydrogen embrittlement (HE), is a well
recognized threat for structural steels. It manifests as loss
in ductility, strength and toughness, which may result in
unexpected and premature catastrophic failures. The phe-
nomenon was first reported by Johnson in 1874 [1], and
has later been extensively researched both experimentally
[2, 3, 4, 5, 6, 7] and numerically [8, 9, 10, 11, 12, 13, 14, 15,
16], yielding a number of models accounting for the phe-
nomenon. However, no consensus about the basic mech-
anisms responsible for hydrogen embrittlement is reached
yet. Two theories have advanced as the more accepted
ones for the case of hydrogen degradation in steel: Hy-
drogen Enhanced Decohesion (HEDE), in which intersti-
tial atomic hydrogen reduces the bond strength and thus
the necessary energy to fracture [17, 18]; and Hydrogen
Corresponding author
Email address: lise.jemblie@ntnu.no (L. Jemblie)
Enhanced Localized Plasticity (HELP), in which atomic
hydrogen accelerates dislocation mobility through an elas-
tic shielding effect which locally reduces the shear stress
[19, 20]. Today it is seemingly recognized that no single
mechanism can comprehensively explain all the phenom-
ena associated with hydrogen embrittlement. Rather it
appears that different mechanisms apply to different sys-
tems, and that a combination of mechanisms is more likely
in many cases.
In recent years, cohesive zone modelling has gained
increasing interest as suitable method for modelling hy-
drogen embrittlement [10, 11, 12, 14, 16], with the pos-
sibility of providing increased understanding of the in-
volved process and their interactions combined with re-
duced time and costs compared to experimental programs.
The damage process is classically described by interface el-
ements, which constitutive relation is defined by a cohesive
law (traction separation law). Simulation of hydrogen in-
duced degradation requires a coupled approach, including
modelling of transient mass transport, plastic deformation,
Preprint submitted to International Journal of Hydrogen Energy November 28, 2016

fracture and their interactions. On one side, the models
describing hydrogen diffusion must account for local me-
chanical field quantities; i.e. hydrostatic stress and plastic
strain. On the other side, the effect of hydrogen on the ac-
celerated material damage must be implemented into the
cohesive law.
The present work presents a review of coupled diffu-
sion and cohesive zone modelling as a method for numer-
ically assessing the hydrogen embrittlement susceptibility
of a steel structure. In Section 2, established and re-
cent models for hydrogen transport are summarised and
discussed. Section 3 gives an overview of cohesive zone
modelling in general and approaches for implementing hy-
drogen influence. The coupling aspect between hydrogen
transport and cohesive zone modelling is discussed and put
in conjunction with hydrogen diffusion models in Section
2. Section 4 discusses some practical applications of the
presented model.
2. Hydrogen transport models
The process that results in hydrogen embrittlement in-
cludes an important transport stage of hydrogen to the site
of degradation. In order to predict the degrading effect of
hydrogen on the mechanical properties, it is of fundamen-
tal importance to correctly assess the hydrogen distribu-
tion in the material.
Atomic hydrogen is generally considered to reside ei-
ther at normal interstitial lattice sites (NILS) or being
trapped at microstructural defects like dislocations, car-
bides, grain boundaries and interfaces. Traps generally
reduce the amount of mobile hydrogen, thus decreasing
the apparent diffusivity and increasing the local solubility
of the system. To date, models of transient hydrogen dif-
fusion generally account for trapping by dislocations and
hydrostatic drift. Recent approaches include capturing the
effect of multiple trap sites and hydrogen transport by dis-
locations [13, 21, 22].
2.1. Hydrogen in lattice
Given a metal lattice, the hydrogen concentration in
NILS, C
L
, can be expressed by [8]
C
L
= βθ
L
N
L
(1)
with θ
L
being the lattice site occupancy, N
L
the density
of solvent atoms and β the number of NILS per solvent
atom, usually assigned to be 6 under the assumption of
tetrahedral site occupancy in iron. The density of solvent
atoms, N
L
, can be calculated through
N
L
=
N
A
V
M
(2)
where N
A
is the Avogadro constant equal to 6.022 · 10
23
mol
-1
, and V
M
is the molar volume of the host lattice,
which for iron is 7.106 ·10
-6
m
3
/mol at room temperature.
According to Equation (1), this gives a lattice occupancy
to NILS concentration ratio of θ
L
/C
L
= 10
5
wppm
1
,
and thus θ
L
<< 1 for most practical purposes.
2.2. Hydrogen in traps
Similarly as for lattice sites, the hydrogen concentra-
tion in a specific trapping site can be expressed by [8]
C
T
= αθ
T
N
T
(3)
where θ
T
is the occupancy, N
T
is the density of the specific
trap site (dislocation, carbide etc.) and α is the number
of sites per trap.
The ability of a trap site to hold hydrogen is associated
with the trap binding energy, representing the attractive
interaction of a trap site compared to a normal lattice
site. Trap sites and trap binding energies can be estab-
lished experimentally for a microstructure using varies ap-
proaches like electrochemical permeation or thermal des-
orption spectroscopy (TDS), with TDS considered best
suited to provide detailed trap characteristics [5, 13, 23].
A considerable amount of data is reported in literature
for various steels. Selected data on trap binding energies
are summarized in Table 1, where the large discrepancy
2

Table 1: Selected trap binding energies for hydrogen in steel.
Trap site E
b
[kJ/mol] Steel Ref.
Dislocation 24 - 26.8 Ferritic [3, 24]
10-20 Austenitic [25, 26]
Elastic dislocation 0-20.2 - [13, 27]
Screw dislocation core 20-30 - [27]
Mixed dislocation core 59.9 - 61 - [2, 5]
Grain boundary 17.2 - 59 - [3, 13, 24, 27, 28]
Austenite-ferrite interface 52 Duplex [29]
Carbide interface 67-94 - [5, 13, 24, 27, 28]
in binding energies for similar trapping sites reflects varia-
tions in microstructural features and experimental details.
Traps with a binding energy above 60-70 kJ/mol are
typically denoted irreversible traps [30], characterized by
a high binding energy not possible to overcome by nor-
mal tempering procedures. These hydrogen atoms may
be regarded as permanently removed from the diffusion
process.
There have been significant advances in theoretical ap-
proaches to capture the effect of traps on hydrogen trans-
port, with models by McNabb and Foster [31] and Oriani
[32] describing the process for steel. Oriani [32] proposed
that hydrogen in NILS and hydrogen in reversible traps
are always in local equilibrium, an approach which is valid
in the domain of rapid trap filling and escape kinetics, such
that
θ
T
1 θ
T
=
θ
L
1 θ
L
exp
E
B
RT
(4)
with E
B
being the trap binding energy. Considering that
θ
L
<< 1 for most purposes, C
L
and C
T
relates through
C
T
=
K
αN
T
βN
L
C
L
1 +
K
βN
L
C
L
(5)
where K is the equilibrium constant, as defined by the
exponential term in Equation (4). A consequence of this
equilibrium condition is that the trap site occupancy be-
comes independent of the number of traps. By making
use of Equation (2) and (4), contours of the trap bind-
ing energy as a function of the trap occupancy and the
NILS concentration are plotted in Figure 1. The theoret-
ical solubility of hydrogen in steel at normal temperature
Figure 1: Relationship between NILS concentration, trap occupancy
and trap binding energy, as proposed by Oriani [32]. The binding
energy E
B
is given in units of kJ/mol.
and pressure, measured in wppm, is on the order of 10
4
for α-iron [27] and 1 for austenitic stainless steels [33]. It
is noticeable that for binding energies greater than about
60 kJ/mol, all traps will be completely saturated, θ
T
1
(unless C
L
is small, less than 10
4
wppm). In contrast,
traps with binding energies below 10-15 kJ/mol will be
drained of hydrogen, θ
T
0 (unless C
L
has a high value
in excess of 10 wppm).
For microstructural defects like carbides and grain bound-
aries, the trap densities are often assumed constant through-
out the material. For dislocations, however, the trap den-
sity varies point-wise dependent on the local plastic strain.
Kumnick and Johnson [2] have studied hydrogen trapping
3

in zone refined deformed iron by performing permeation
transient measurements. By rolling at room temperature,
0 - 60 % cold work were obtained. The resulting trap bind-
ing energies and trap densities were inferred from time lag
measurements and interpreted in terms of the McNabb-
Foster [31] approach. The trap density was found to in-
crease sharply with deformation at low levels and more
gradually with further deformation. Based on these ob-
servations, Sofronis and McMeeking [8] proposed the fol-
lowing relationship between the dislocation trap density,
N
(d)
T
, and the equivalent plastic strain, ε
p
, for iron:
log N
(d)
T
= 23.26 2.33 exp(5.5ε
p
) (6)
Similar relationships can be determined experimentally for
the applicable steel by performing hydrogen permeation
measurements at various levels of plastic deformation.
An alternative theoretical approach has been proposed
by Sofronis et al. [34, 35], assuming one trap site per
atomic plane threaded by a dislocation, maintaining that
this is consistent with the experimental work of Thomas
[25]. The dislocation trap density is then expressed as a
function of the dislocation density ρ and the lattice pa-
rameter a
N
(d)
T
=
2
ρ
a
(7)
The dislocation density (measured in dislocation line length
per cubic meter) is considered to vary linearly with the
equivalent plastic strain according to
ρ =
ρ
0
+ γε
p
for ε
p
< 0.5
10
16
for ε
p
0.5
(8)
where ρ
0
= 10
10
line length/m
3
, denotes the dislocation
density at zero plastic strain, and γ = 2.0·10
16
line length/m
3
.
Using the lattice parameter of BCC iron a = 2.86
˚
A, the
trap densities according to the data from Kumnick and
Johnson [2] and the model by Sofronis et al. [34, 35] are
compared in Figure 2. It can be concluded that the model
by Sofronis et al. yields a dislocation trap density about
Figure 2: Dislocation trap densities according to the work by Kum-
nick and Johnson [2] and the model by Sofronis et al. [34, 35]. In
calculating C
T
, it is assumed αθ
T
= 1, which accordingly gives the
maximum possible hydrogen concentration trapped at dislocations.
three orders of magnitude larger than the data by Kum-
nick and Johnson. The maximum trapped concentration
as predicted by the Sofronis model is in line with mea-
sured hydrogen concentrations in ferritic steel (1.5 - 2.5
wppm [36]), while the data from Kumnick and Johnson is
more comparable to the theoretical equilibrium solubility
of hydrogen in iron.
Using Equation (1) - (5), the dislocation trapped hy-
drogen concentration, C
T
, is calculated as a function of the
lattice hydrogen concentration, C
L
, in terms of the trap-
ping models by Kumnick and Johnson [2] and Sofronis et
al. [34, 35], assuming V
M
= 7.106 · 10
6
m
3
/mol, β = 6,
α = 1 and room temperature. The results are displayed in
Figure 3 for two levels of equivalent plastic strain, 0 and
0.8, and three levels of trap binding energy, 20, 40 and 60
kJ/mol.
Generally, the trapped concentration increases with in-
creasing lattice concentration and increasing trap binding
energy, until saturation is reached. With a higher lattice
concentration, more hydrogen is available for trapping.
With a higher binding energy, the attractive interaction
4

of the trap site increases and, correspondingly, more hy-
drogen atoms will reside in traps. This is consistent with
Figure 1, resulting in an increased trap occupancy level.
A higher level of plastic strain increases the trap density
and, thus, the overall trapped concentration of the system.
The trap occupancy is maintained, which by definition is
independent of the trap site density.
For E
B
= 60 kJ/mol, all traps are saturated, and the
trapped concentration is independent of the lattice con-
centration. Novak et al. [13] found that high-binding
energy traps cannot account for the loss in strength ob-
served on hydrogen charged steel, because these traps re-
main saturated with hydrogen regardless of loading condi-
tions and/or hydrogen exposure conditions. Similar find-
ings have been reported by Ayas et al. [37]. Rather, it is
the lattice sites and low-binding energy trap sites which
holds a critical role. Novak et al. [13] postulated that low-
binding energy dislocation traps are the governing con-
tribution promoting hydrogen induced fracture. On the
other hand, Ayas et al. [37] reported that the presence
of lattice hydrogen is the critical event with low energy
trapped hydrogen only having a negligible effect.
It is noticeable from Figure 3, that in most cases, either
C
L
or C
T
yield the dominating influence on the total hy-
drogen concentration. Considering now only trapped con-
centration levels below saturation, θ
T
1. According to
the trapping model by Kumnick and Johnson [2], assum-
ing ε
p
0.8 (maximum), trapping yields the dominating
influence when E
B
37 kJ/mol. Similarly, according to
the model by Sofronis et al. [34, 35], trapping yields the
dominating influence when E
B
23 kJ/mol. Conform-
ing to the findings from Novak et al. [13] and Ayas et al.
[37], assuming the only possible trap sites associated with
hydrogen induced fracture are low-binding energy disloca-
tions, it can be concluded that C
L
will be the dominat-
ing influence on the total hydrogen concentration for most
practical purposes.
2.3. Hydrogen diffusion
The main mechanism for hydrogen diffusion in steel is
lattice diffusion by interstitial jumps, where the hydrogen
atom occupy interstitial sites and move by jumping from
one interstitial site to a neighbouring one [38].
Chemical potential gradients constitute the main driv-
ing force for hydrogen diffusion in steel; hydrogen will dif-
fuse from regions where the chemical potential is high to
regions where it is low, and the process ceases once the
chemical potentials of all atoms are everywhere the same
and the system is in equilibrium [39]. Assuming that the
diffusion flux is proportional to the concentration gradi-
ent, which often is the case, Fick’s laws are the governing
equations describing the processes. These laws represent
a continuum description and are purely phenomenological.
Fick’s first law gives the flux of diffusing particles, which
for an isotropic medium is given by [40]
J = D
C
L
x
(9)
whit D being the diffusion coefficient. The transient diffu-
sion process is described by Fick’s second law, also denoted
the fundamental differential equation for diffusion [40]
C
L
t
= D
2
C
L
x
2
+
2
C
L
y
2
+
2
C
L
z
2
(10)
It can be derived from Equation (9) by considering a con-
trolled volume element. The diffusion coefficient holds an
Arrhenius-type dependence on temperature [33]
D = D
0
exp
E
a
RT
(11)
where D
0
is a pre-exponential factor independent of tem-
perature and E
a
is the activation energy (energy barrier)
for hydrogen jumping between interstitial sites. Figure 4
displays a summary of reported diffusion coefficients for
hydrogen in iron and steel. The substantially higher diffu-
sivity in ferrite compared to austenite is due to the lower
packing density of bcc metals, reducing the potential en-
ergy barrier for jumps. In contrast, the larger interstice of
fcc metals yields a higher hydrogen solubility in austenite.
5

Figures (15)
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DOI
12 Sep 2005
Abstract: The Defense Waste Processing Facility (DWPF) is about to process High Level Waste (HLW) Sludge Batch 4 (SB4). This sludge batch is high in alumina and nepheline can crystallize readily depending on the glass composition. Large concentrations of crystallized nepheline can have an adverse effect on HLW glass durability. Several studies have been performed to study the potential for nepheline formation in SB4. The Phase 3 Nepheline Formation study of SB4 glasses examined sixteen different glasses made with four different frits. Melt rate experiments were performed by the Process Science and Engineering Section (PS&E) of the Savannah River National Laboratory (SRNL) using the four frits from the Phase 3 work, plus additional high B2O3/high Fe2O3 frits. Preliminary results from these tests showed the potential for significant improvements in melt rate for SB4 glasses using a higher B2O3-containing frit, particularly Frit 503. The main objective of this study was to investigate the durability of SB4 glasses produced with a high B2O3 frit likely to be recommended for SB4 processing. In addition, a range of waste loadings (WLs) was selected to continue to assess the effectiveness of a nepheline discriminator in predicting concentrations of nepheline crystallization that would be sufficient to influencemore » the durability response of the glass. Five glasses were selected for this study, covering a WL range of 30 to 50 wt% in 5 wt% increments. The Frit 503 glasses were batched and melted. Specimens of each glass were heat-treated to simulate cooling along the centerline of a DWPF-type canister (ccc) to gauge the effects of thermal history on product performance. Visual observations on both quenched and ccc glasses were documented. A representative sample from each glass was submitted to the SRNL Process Science Analytical Laboratory (PSAL) for chemical analysis to confirm that the as-fabricated glasses corresponded to the defined target compositions. The Product Consistency Test (PCT, ASTM C1285) was performed in triplicate on each Frit 503 quenched and ccc glass to assess chemical durability. The experimental test matrix also included the Environmental Assessment (EA) glass and the Approved Reference Material (ARM-1) glass. Representative samples of all the ccc glasses were examined for homogeneity visually and by X-ray diffraction (XRD) analysis. Chemical composition measurements indicated that the experimental glasses were close to their target compositions. PCT results showed that all of the Fit 503 quenched glasses had an acceptable durability compared to the EA benchmark glass. The durability of one of the ccc glasses, NEPHB-04, was statistically greater than its quenched counterpart. However, this was shown to be of little practical significance, as the durability of the NEPHB-04 ccc glass was acceptable when compared to the durability of the EA benchmark glass. Visual observations and PCT results indicated that all of the Frit 503 quenched glasses were free of any crystallization that impacts durability. For the ccc glasses, XRD results indicated that the lower WL glasses (30 to 40 wt%) were amorphous, which was consistent with visual observations and PCT responses. The higher WL glasses (45 and 50 wt%) were shown by XRD to contain spinel (trevorite, NiFe2O4). It is possible that some of the other high WL glasses also contained some nepheline, but that the amount of nepheline crystallization was below the detection limit (0.5 vol%) associated with XRD. The results indicate that Frit 503 is a good candidate for SB4 processing, based on chemical durability of homogeneous and devitrified glasses over a WL range of 30 - 50%. It should be noted that the higher WL glasses would not be fit for processing in DWPF as they exceed other process related criteria (such as liquidus temperature). However, this is only one of many factors influencing the frit selection. Melt rate and the final SB4 composition are also important factors in frit selection. Additional melt rate studies are currently underway, and the final composition projection for SB4 is expected shortly.« less

32 citations


Journal ArticleDOI
Abstract: The effect of cathodic polarization conditions on hydrogen degradation of X2CrNiMoCuN25-6-3 super duplex stainless steel welded joints, obtained using flux cored arc and submerged arc welding methods, was evaluated. Slow strain rate tensile tests of base material and welded specimens, ferrite content measurements, scanning electron microscopy observations, and statistical analysis were performed. It was found that hydrogenation of super duplex steel welded joints under the conditions of cathodic protection in artificial seawater environment leads to hydrogen embrittlement and that the weld area shows the highest degree of degradation. Welded joints made with higher heat input and under higher current density tend to cause an increase in the degradation of mechanical properties. A series of models to analyze the relationship between conditions and properties of material under extreme environmental conditions were successfully elaborated and evaluated.

27 citations


References
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01 Jan 1956
TL;DR: Though it incorporates much new material, this new edition preserves the general character of the book in providing a collection of solutions of the equations of diffusion and describing how these solutions may be obtained.
Abstract: Though it incorporates much new material, this new edition preserves the general character of the book in providing a collection of solutions of the equations of diffusion and describing how these solutions may be obtained

20,482 citations


Journal ArticleDOI
Abstract: Y ielding at the end of a slit in a sheet is investigated, and a relation is obtained between extent of plastic yielding and external load applied. To verify this relation, panels containing internal and edge slits were loaded in tension and lengths of plastic zones were measured.

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"A coupled diffusion and cohesive zo..." refers methods in this paper

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Abstract: A method is presented in which fracture mechanics is introduced into finite element analysis by means of a model where stresses are assumed to act across a crack as long as it is narrowly opened. This assumption may be regarded as a way of expressing the energy adsorption GC in the energy balance approach, but it is also in agreement with results of tension tests. As a demonstration the method has been applied to the bending of an unreinforced beam, which has led to an explanation of the difference between bending strength and tensile strength, and of the variation in bending strength with beam depth.

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Journal ArticleDOI
Abstract: A method is presented in which fracture mechanics is introduced into finite element analysis by means of a model where stresses are assumed to act across a crack as long as it is narrowly opened. This assumption may be regarded as a way of expressing the energy adsorption GC in the energy balance approach, but it is also in agreement with results of tension tests. As a demonstration the method has been applied to the bending of an unreinforced beam, which has led to an explanation of the difference between bending strength and tensile strength, and of the variation in bending strength with beam depth.

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Grigory Isaakovich Barenblatt1Institutions (1)
Abstract: Publisher Summary In recent years, the interest in the problem of brittle fracture and, in particular, in the theory of cracks has grown appreciably in connection with various technical applications. Numerous investigations have been carried out, enlarging in essential points the classical concepts of cracks and methods of analysis. The qualitative features of the problems of cracks, associated with their peculiar nonlinearity as revealed in these investigations, makes the theory of cracks stand out distinctly from the whole range of problems in terms of the theory of elasticity. The chapter presents a unified view of the way basic problems in the theory of equilibrium cracks are formulated and discusses the results obtained thereby. The object of the theory of equilibrium cracks is the study of the equilibrium of solids in the presence of cracks. However, there exists a fundamental distinction between these two problems, The form of a cavity undergoes only slight changes even under a considerable variation in the load acting on a body, while the cracks whose surface also constitutes a part of the body boundary can expand even with small increase of the load to which the body is subjected.

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"A coupled diffusion and cohesive zo..." refers methods in this paper

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    [...]

  • ...Cohesive models were first formulated by Barenblatt [44] and Dugdale [45], who introduced finite non-linear cohesive tractions in front of an existing crack, as a mean to overcome the crack tip stress singularity....

    [...]


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