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Key Engineering Materials
ISSN: 1662-9795, Vol. 754, pp 153-156
© 2017 Trans Tech Publications, Switzerland
Online: 2017-09-05
A Phase Field Approach to Fracture with Mass Transport Extension for
the Simulation of Environmentally-Assisted Cracking
Falkenberg Rainer a
BAM Bundesanstalt für Materialforschung und -prüfung, Department 5.2, Unter den Eichen 87,
12205 Berlin, Federal Republic of Germany
Keywords: phase field, crack propagation, mass transport, environmentally-assisted cracking, hydrogen embrittlement
Abstract. The fracture mechanics assessment of materials exposed to harmful environments requires
the understanding of the interaction between the soluted species and the affected mechanical behaviour. With the introduction of a mass transport mechanism the entire problem is subjected to a
time frame that dictates the time-dependent action of soluted species on mechanical properties. A numerical framework within the phase field approach is presented with an embrittlement-based coupling
mechanism showing the influence on crack patterns and fracture toughness. Within the phase field approach the modeling of sharp crack discontinuities is replaced by a diffusive crack model facilitating
crack initiation and complex crack topologies such as curvilinear crack patterns, without the requirement of a predefined crack path. The isotropic hardening of the elasto-plastic deformation model and
the local fracture criterion are affected by the species concentration. This allows for embrittlement
and leads to accelerated crack propagation. An extended mass transport equation for hydrogen embrittlement, accounting for mechanical stresses and deformations, is implemented. For stabilisation
purposes a staggered scheme is applied to solve the system of partial differential equations. The simulation of showcases demonstrates crack initiation and crack propagation aiming for the determination
of stress-intensity factors and crack-resistance curves.
Stress corrosion cracking (SCC) is the generally accepted term for describing subcritical cracking of
materials under sustained loads (residual or applied) in most liquid and some gaseous environments
[1]. Therefore the simulation of SCC must involve the interaction of the soluted species, provided
by the environment, with the material. For hydrogen-assisted cracking, several mechanisms can be
addressed, whereupon the most important ones are adsorption-induced hydrogen dislocation emission
(AIDE) [2], hydrogen-enhanced localised plasticity (HELP) [3] and hydrogen-enhanced decohesion
(HEDE) [4]. According to the AIDE mechanism, dislocations are emitted by the absorption of hydrogen that weakens interatomic bonding forces. The HELP mechanism describes the shielding of
dislocations by soluted hydrogen yielding an enhanced dislocation mobility. With the HEDE mechanism, the soluted hydrogen degrades the interatomic bonding strength and leads to premature decohesion. Finite-element calculations take those mechanisms into account by mass transport simulations
and coupling of the local hydrogen concentration to material strength and toughness properties, e.g.
a hydrogen-affected cohesive zone model was combined with a phenomenological constitutive relation, in which the local stress is a decreasing function of the local hydrogen concentration, to model
hydrogen-induced crack propagation [5]. Surface kinetics and hydrogen mass diffusion were added
in [6] in conjunction with concentration dependent cohesive zone elements and J2 plasticity under
isotropic hardening. A major drawback of cohesive zone elements is the necessity of a predefined
crack path, so that complex crack patterns are not assessable. To overcome this drawback, the application of a crack phase field model is taken into account. The phase field model is based on a variational
approach [7] and the crack discontinuity regularisation by a scalar variable [8]. A thermodynamically
consistent framework was introduced afterwards for brittle fracture [9] and for ductile fracture [10].
All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans
Tech Publications, (#103374919, University of Auckland, Auckland, New Zealand-12/11/17,12:52:33)
Advances in Fracture and Damage Mechanics XVI
The application of a crack phase field model towards environmentally-assisted cracking of brittle materials was conducted by means of a mass transport coupling for concrete [11] and for a high strength
steel [12]. In this paper an extension of this approach is presented, incorporating ductile material behaviour in terms of J2 plasticity under isotropic hardening and an extended mass diffusion equation
accounting for mechanical interactions on mass transport.
The entire process of mechanical deformation, crack growth and mass transport is governed by the
total free energy functional
Ψ(ui , ω, α, c) = Ψm (ui , ω, α, c) + Ψd (c) + Ψc (ω, c),
consisting of the functionals of the mechanical free energy Ψm , diffusion free energy Ψd and crack
free energy Ψc , that depend on the displacement ui , the scalar-valued crack phase field ω, the total
mass concentration c and the hardening variable α. The mechanical free energy functional is affected
by the crack phase field
] 0
Ψm (ui , ω, α, c) =
(1 − ω)2 + k ψm,e
(ui ) + ψm,p
(α, c) dΩ,
where the mechanical free energy density consists of an elastic part ψm,e
and a plastic part ψm,p
. The
material degradation takes effect by the factor [(1 − ω) + k] acting on the elastic part, where ω = 1
defines a fully degradated material. The conditioning parameter k > 0 ensures the well-posedness
of the problem at full degradation by an artificial residual energy. Additional material degradation is
superimposed by the local concentration acting on the plastic properties of ψm,p
. The evaluation of the
diffusion free energy functional Ψd leads to the extended mass diffusion equation [13]
cL + cT (1 − θT ) ∂cL
dNT ∂εp
− ∇ · (DL ∇cL ) + ∇ ·
∇σh + θT
= 0,
dεp ∂t
cL (cT ) is the hydrogen concentration in lattice (trap) sites, the total hydrogen concentration is c = cL +
cT , V H is the partial molar volume of hydrogen, σh is the hydrostatic stress, DL is the concentration
independent lattice diffusivity, θL (θT ) is the occupancy of lattice (trap) sites, NL (NT ) is the number
of lattice (trap) sites per unit volume, R is the universal gas constant, T is the absolute temperature,
εp is the equivalent plastic strain. The crack free energy functional Ψc covers the energy with respect
to crack propagation and is defined for the domain Ω as
Ψc (ω, c) = Ω [GC (c)γ(ω, ∇ω)] dΩ,
GC is the concentration-dependent Griffith energy release rate and γ(ω, ∂i ω) = (1/2l)ω 2 +(l/2)(∇ω)2
is the crack surface density [14]. The function γ regularises the discrete sharp crack by a continuous
transition between the crack zone and the surrounding defect-free continuum by introducing the internal length l.
Finite-element simulations of a C(T)-40 fracture mechanics specimen, s. fig. 1(a), subjected to displacement and mass flux boundary conditions under different deformation rates were conducted. As
the mass transport process introduces a time frame, the entire simulation becomes rate-dependent.
The mechanical properties were set to E = 210GP a and ν = 0.3, the critical energy release rate was
assumed as GC = 10.83kJ/m2 . The J2 plasticity model with linear isotropic hardening was adjusted
to FeE690T high-strength steel. The diffusion coefficient was assumed to be D = 7.5 · 10−8 m2 /s
Key Engineering Materials Vol. 754
according to the diffusion of H in bcc-F e [15]. The internal length l is set twice the characteristical
element length h. The fatigue pre-crack a0 = 1mm was simulated as initial boundary condition by
setting the corresponding nodes to ω = 1. A normalised concentration was applied from the start at
the specimen edges to model the species absorption. Since by crack growth, evolving new surfaces
are usually prone to absorb species, an adaptive concentration boundary condition was assumed to be
active at the current crack tip. A monotonuous linear increasing deflection u(t) was applied in slowstrain-rate regimes of u̇1 = 100µm/h, u̇2 = 10µm/h, u̇3 = 1µm/h. The hydrogen distribution at
the onset of crack initiation is shown in fig. 1(b) and for the end of the simulation in 1(c). The contribution of the species absorbed by the specimen edges is almost vanishing whereas the concentration
introduced across the crack tip surface dominates the species distribution.
(a) Model
(b) Concentration field at crack initiation,
red: max. total concentration
F [N ]
vll [mm]
(c) Concentration field at end of
simulation, red: max. total
(d) Force-deflection curves undergoing
different deformation rates with
environmental effects
Fig. 1: Simulation results with a C(T)-40-fracture mechanics specimen
The crack evolves from the fatigue crack tip in fig. 1(b) and propagates in a straight line through
the ligament, s. fig. 1(c), as expected. The failed elements facing ω = 1 were removed and the plotted
deformation was scaled by a factor of 20. The force-deflection force curves under different deformation rates show the detrimental influence of the species, s. fig. 1(d). The high deformation rate u̇1
shows a minor influence by the soluted species on the drop of the maximum force value, with an increasing effect at the lower deformation rates u̇2 and u̇3 . This shows that at high deformation rates the
mechanical processes, such as deformation and crack propagation, are occuring faster than the mass
transport of the species. This situation is inverted at low deformation rates, where the mass transport
receives more time to reach higher concentrations and cover longer distances in the material.
Advances in Fracture and Damage Mechanics XVI
Environmentally-assisted crack growth could be simulated by a phase field model with a superimposed
time-dependent mass transport field. The local concentration affects predominantly the local critical
energy release rate that controls the crack process. Hence, the HEDE mechanism, leading to decohesion by the weakening of the interatomic bonding strength, is mainly addressed in this approach.
The detrimental effect of soluted hydrogen on the material could be simulated and shown by means
of force-deflection curves. Furthermore the deterioration of ductility as a consequence of hydrogen
embrittlement could be pointed out by CTOD crack resistance r-curves. At high deformation rates,
the material degradation takes less effect because of lower hydrogen distribution leading to steep rcurves that represent high fracture resistance. On the contrary at low deformation rates the hydrogen
distribution is high and leads to a significant loss of ductility causing flat r-curves that represent brittle
fracture behaviour.
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