Article published in Theoretical and Applied Fracture Mechanics. The final publication is available atwww.sciencedirect.com, doi:10.1016/j.tafmec.2017.05.015

Influence of carbide particles on crack initiationand propagation with competing ductile-brittle

transition in ferritic steel

N.A. Giang ∗, M. Kuna, G. Hutter

Institute of Mechanics and Fluid DynamicsTU Bergakademie Freiberg, Lampadiusstr. 4, 09596 Freiberg, Germany

∗ Ngoc-Anh.Giang@imfd.tu-freiberg.de

Carbide particles play an important role for the fracture toughness of ferritic steels inthe ductile-brittle transition (DBT) region as cracks mainly originate from a broken ordebonded carbide. It is well-known that the size, volume fraction, strength and distributionof carbides are relevant for the competition of ductile and brittle mechanisms of failure. Inthe present study, the influence of the carbide strength and carbide-ferrite interaction on thefracture toughness of ferritic steel in the DBT regime is investigated by a micromechanical,deterministic FEM model. The carbide particles are resolved discretely in the fractureprocess zone at the crack tip. Cleavage of ferrite, of carbides and debonding of the carbide-ferrite interface are modeled by cohesive zones. The simulated fracture toughness in theDBT do compare well with experimental data from literature demonstrating the capabilityof the model to capture the competition of the various microscopic mechanism.Keywords: Ferritic steel, Ductile-brittle transition (DBT), Fracture mechanics, FEM

Nomenclature

A void nucleation pre-factor

A0 radius of boundary layer

∆a effective crack length

cs speed of shear waves

E Young’s modulus

f void volume fraction

fg rate of void growth

fn rate of void nucleation

fN void nucleation constant

Geff effective shear modulus of homogenizedmaterial

Gm shear modulus of matrix phase

Gp shear modulus of particle phase

J J-integral

JIc fracture initiation toughness

Keff effective bulk modulus of homogenized ma-terial

KI far-field stress intensity factor

Km bulk modulus of matrix phase

Kp bulk modulus of particle phase

1

Lx weighting factor of effective crack growth

N hardening exponent

qi GTN parameters

sN void nucleation constant

X0 average distance between carbide particles

Wi current width of ith ligament

W remi remaining width of ith ligament

W0 initial ligament width

δ cohesive separation

δ0 critical separation

εplv rate of volumetric plastic strain

Γ0 cohesive work of separation

Γ0M cohesive work of separation of ferritic ma-trix

Γ0P cohesive work of separation of carbidle par-ticle

Γ0PM cohesive work of separation of paticle-matrix interface

σc cohesive strength

σe von Mises equivalent stress

σm hydrostatic stress

σcM cohesive strength of ferritic matrix

σcP cohesive strength of carbide particle

σcPM cohesive strength of paticle-matrix inter-face

1 Introduction

In the range of room temperature, ferritic steels possess excellent fracture properties. For this reason,these materials are often used for pressure vessels, compressors and steam generators of nuclear powerplants [1].

The reason for the high fracture toughness is the ductile mechanism of nucleation, growth and coales-cence of voids, which requires a large amount of plastic deformations. In contrast, at low temperatures(and/or increased rates of loading), ferritic steels fail by cleavage. This mechanism dissipates consid-erably less energy and the fracture toughness is considerably lower than in the ductile regime, whichis why cleavage is associated with a macroscopically brittle behavior. In the ductile-brittle transitionregime (DBT) the ductile mechanism and cleavage compete.

The DBT is driven by an increasing resistance to dislocation motion with decreasing temperatureresulting in an increasing yield stress. Consequently, plastic deformations require higher yield stresseswhich favor cleavage. Typically, cleavage initiates at local stress concentrators like grain boundariesor defects. For this reasons, in the DBT the microstructure has a strong influence on the macroscopicfracture properties. In particular, it is well-known that carbide particles at grain-boundaries play anessential role for the DBT of ferritic steels, see Fig. 1.

Due to its relevance in engineering applications, models have been developed to describe the loss oftoughness in the DBT. The most prominent models are the probablistic weakest-link model of Beremin[2] and the deterministic Ritchie-Knott-Rice model (RKR model) [3]. Both approaches have experiencednumerous modifications. For an overview, the reader is referred to the review article [4]. All approachesprovide criteria for fracture initiation based on relatively simple idealizations of the mechanism ofcleavage initiation. In the literature, in particular the Beremin model was combined with models ofductile damage like those of Gurson [5] or Rousselier [6] in order to cover the complete DBT, see e. g.[7–9].

Increasing computing capacities allowed to investigate the interactions of the competing damagemechanisms by more detailed computational models. So, Petti and Dodds [11] resolved the microstruc-ture in the fracture process zone (FPZ) discretely in form of voids to model the ductile mechanism andapplied a modified Beremin model to account for cleavage initiation. However, only a 2D model wasemployed and the Beremin approach does not allow to simulate the propagation of cleavage microcracks.

2

Crack tipInclusion

0.5mm

20 m

20 m 5 m

m

a)

b) c)

d)

Figure 1: Carbide particle arrangement in ferritic steel [10]: a) Cavities nucleated from non-metallicinclusions located ahead of the notch tip, b) Zoom of the ductile crack at notch tip, c)Microvoid nucleation at carbide particles along strain localisation path d) Microvoid nucleationclose to the ductile crack tip

Faleskog, Kroon and co-workers [12–14] proposed a refined model that resolves the microstructure inform of discrete, spherical carbide particles. Cleavage of the ferritic matrix, fracture of the carbideparticle and failure of its interface were modeled by a cohesive zone. In fact, two principal parametersin the cohesive zone model are associated with the physical fracture process: i) the cohesive strengthdenotes the necessary local stress to initiate cleavage (the stress criterion) and ii) the work of separationcorresponds to fracture toughness of the material in case of pure cleavage (energy criterion). Faleskog,Kroon et al. [12, 13] performed cell model simulations under uniform loading. However, at a crack tipthe loading is highly inhomogeneous and strong gradients occur over distances which are comparable tomicrostructural length scales like the distance between carbides. For this reason, in a preliminary work[15], the authors adopted the advantage of the approaches of Petti et al.[11] and of Kroon et al.[12, 13]and resolved discrete spherical voids in the FPZ while modeling cleavage of the ferritic matrix by acohesive zone. For nodular cast iron, where the graphite particles are only loosely bonded to the ma-trix, the idealization of the particles to voids was found to be reasonable. However, for ferritic steels itis well-known that the strength of the carbide particles plays an essential role in the DBT. Thus, thescope of the present paper is to incorporate carbide particles with finite strength in the discrete processzone approach of [15] in order to simulate the effect of carbide particles and their strength on the DBT.

2 Model

2.1 Global model

The global model used in the present study is shown schematically in Fig. 2. As already mentioned,the microstructure characterized by spherical carbide particles with a volume fraction fp, is resolveddiscretely in the FPZ at an initially present macroscopic crack. Potential cleavage of the ferritic matrix aswell as decohesion and breakage of the particle are modeled by cohesive zones. The mechanical behaviorof the ferritic matrix is described by Mises plasticity, whereas the carbide particles are assumed to be

3

Homogenised: GTN

Elastic-plasticity

Cohesive zone model

Crack tip

In process zone

Cohesive zone model

𝐾𝐾𝐼𝐼

𝐾𝐾𝐼𝐼

𝑋𝑋𝑜𝑜

𝑡𝑡

𝐴𝐴𝑜𝑜

Carbide-particle

Figure 2: FE model used in the analysis

Ferrite

Homogeneous GTN model

Particle

Interface cohesive zone

Cohesive zone inside carbide particle

Ferritic matrix

X

Z

X

Y

E2

E3E1

U2

U3U1

R

Lo Lo

Lo

Homogeneous GTN model

Unit cell model

Carbide particle

X

Y

E2

E3

E1

U2

U3

U1

R

Lo Lo

Lo

Z

a)a)

b)

Cohesive zone inner particle

Paticle-ferrite interface

FerriteParticle

X

Y

U2

U3U1

R

Lo Lo

Lo

Z

a) b)

Cohesive zone inner particle

Paticle-ferrite interface

crack font

d=<110>carbide

Lo

Figure 3: Crack growth direction in bcc particle arrangement

purely elastic. The isotropic hardening of the matrix is described by the implicit power law

σ

σ0=

(σ

σ0+E

σ0εp

)N

(1)

in which σ0, εp, N and E are the initial yield stress, the effective plastic strain, the hardeningexponent and Young’s modulus, respectively. Outside of the fracture process zone, the behavior of the“composite” of matrix and carbides is described in a homogenized way by the Gurson model in themodification of Tvergaard and Needleman (GTN-model). For the particle distribution in the processzone, a regular body-centered cubic (bcc) arrangement of the carbide particles is considered1 with crackpropagation along a (110) in direction 〈110〉, see Fig. 3. In a bcc arrangement with an edge length L0 ofthe unit cell, the mean distance of particles is X0 = L0/

3√

2 [16]. The regular bcc arrangement allows toexploit the symmetry in thickness direction (001), which is why only a “slice” with a thickness t = L0/2of half the length L0 of a bcc unit cell needs to be implemented in the FEM model. Macroscopic plain-strain conditions and ideal small-scale yielding (SSY) under mode-I loading are assumed ensuring a highlevel of crack-tip constraint. SSY conditions can easily implemented in a FEM model by prescribingthe KI solution as boundary condition in sufficient distance A0 to the initial crack tip, which is oftenreferred to as a boundary layer (BL) model. Under SSY, the J-integral in the far-field corresponds to

1Furthermore, compared to a cubic-primitive arrangement, the anisotropy of fracture toughness in the ductile regimewas found to be considerably lower [16].

4

the energy release rate and is related to the stress intensity factor KI by

J =(1− ν2

eff)

EeffK2

I , (2)

whereby Eeff and νeff are the homogenized Young’s modulus and Poisson ratio, respectively.

2.2 Homogenized material model and constitutive parameters

Outside the process zone, material behavior is described in a homogenized way by the GTN model. Itsyield surface reads (σeq

σ

)2

+ 2q1f cosh

(3q2

2

σm

σ

)− 1− (q1f)2 = 0 (3)

where σeq, σm and σ are von-Mises stress, hydrostatic stress and effective yield stress of the ferriticmatrix, respectively. Void nucleation is assumed to originate from debonded or broken carbide particles.Thus, there is no initial void volume fraction in the material (f0 = 0). The GTN parameters q1 and q2

as well as σ0 are determined by fitting the numerical simulation results between GTN model and cellmodel simulations, which is performed in section 2.6. Void growth rate results from the nucleation ofnew voids and the growth of existing voids

f = fn + fg (4)

wherefg = (1− f)εp

v (5)

fg is the growth rate of existing voids, expressed in terms of the volumetric part εpv of plastic rate of

deformation.According to [17, 18], a plastic strain-controlled void nucleation law

fn = A ˙εp

(6)

is adequate if the void nucleates from small particles. The function A(εp) of effective plastic strain ischosen according to Chu and Needleman [19]

A =fN

sN

√2π

exp

[−1

2

(εp − εN

sN

)2]

(7)

in which fN = fp is the volume fraction of nucleable carbides. Furthermore, εN is the strain at which thevoid nucleation rate achieves its maximum and sN is the standard deviation of the nucleation function.In practice, sN is chosen ≈ εN/5 [20]. The values εN and sN are estimated in section 2.6.

The employed material properties as well as the resulting homogenized values are given in Tab. 1.The homogenized elastic properties, Young’s modulus and Poisson ratio Eeff and νeff , are calculated bythe Mori-Tanaka method as given in the appendix A. The particle volume fraction is chosen as fp=1.4%in ferritic steel (the volume fraction of MnS is quite small [21] and neglected in this study).

2.3 Cohesive zone model

The cohesive zone model with exponential law

t = σcδ

δ0exp

(1− δ

δ0

)(8)

between traction t and separation δ was established by Xu and Needleman [23]. Therein, σc is thecohesive strength (the maximum transferable traction) and δ0 refers to the characteristic separation

5

Table 1: Material characteristic [15, 22]

Material Eeff/σ0 νeff E/σ0 ν fp [%] X0[µm] N

Homogenized GTN model 336 0.298 - - - - 0.1

Ferritic matrix - - 333 0.3 - - 0.1

Carbide - - 666 0.3 1.4 20 -

Δ𝑎𝑎

REAL (in process zone)

crack

Traction-separation

Brocken fiber

Crack initiation Cohesive traction

crack

Separation of cohesive

Mode I

Interface elements

FEM ELEMENTS COHESIVE LAWS

Exponential law

Brittle

Ductile

Trapezoidal law

Bilinear

Dugdale’s

Dissipated energy

Γ𝑜𝑜

Γ0 Γ𝑜𝑜

𝛿𝛿𝑜𝑜

T(δ)

δ

𝛿𝛿𝑐𝑐

𝛿𝛿𝑐𝑐

𝜎𝜎𝑐𝑐

𝜎𝜎𝑐𝑐

𝜎𝜎c

𝑇𝑇

𝑇𝑇 𝑇𝑇

𝑇𝑇

𝑇𝑇

δ

δ

𝜎𝜎𝑐𝑐 Ductile+cleavage

Cleavage Ductile damage

(Chakraborty & Biner)

δ0

Figure 4: Traction-separation law

when σc is reached. The cohesive work of separation Γ0 = exp(1)σcδ0 corresponds to the area belowthe cohesive traction-separation curve. Roth et al.[24] extended this cohesive law for general loadingproblems as shown in Fig. 4. This cohesive law is implemented in this study for modeling of particlecracking, debonding of the particle-ferrite interface and cleavage in the ferritic matrix. Under mode-Iloading, the direction of maximum principal stress is perpendicular to the crack plane and coincideswith the traction t of the cohesive zone model. Therefore, the initiation of softening after reachingthe cohesive strength σc corresponds to the maximum principal stress criterion as being established forcleavage initiation [15].

In order to evaluate the fracture toughness of the material at the macro-scale under the influence ofmicrostructural features, several authors [25–29] used the relationship between the fracture toughnessJIc and micro structural characteristic length scale X0 (average void or particle spacing) as JIc =F · X0σ0. Therein, F is a function of microstrutural features of the material such as the initial voidvolume fraction f0, the hardening exponent N , the normalized cohesive strength σc/σ0 and the ratioσ0/E. To determine F in the present study, firstly cohesive work of separation Γ0 is chosen as lowbound value of fracture toughness JIc in which pure cleavage dominates. According to Knott [30], thelower-bound fracture toughness of ferrite corresponds to the work of separation Γ0 and lies in the range5 ÷ 7 kJ/m2. Here, Γ0 is chosen to be 5 kJ/m2. Therefore, the work of separation of ferritic matrixΓ0M=5 kJ/m2. Then, the factor F is estimated by the normalized work of separation to X0σ0. Inferritic steel, the average distance of carbide particles is in the range of X0 = 30÷130 µm and the yieldstress amounts to σ0 = (250 ÷ 300) MPa [15, 31], which provides a normalized work of separation inthe range of Γ0/X0σ0 = 0.1÷ 0.6. Here, the value of Γ0/X0σ0 = 0.25 is chosen as in [15].

Regarding the temperature dependence, it is well-known that the cleavage stress of mild steel is hardlydependent on temperature, i. e. the cohesive strength of ferritic matrix σcM is constant. In contrast,the yield stress σ0 of ferrite increases with decreasing temperature [22]. Consequently, the ratio σc/σ0

increases with increasing temperature and vice versa. In the lower-shelf regime, when σcM/σ0 is low,the crack grows only by softening of cohesive zone without any plastic deformation. On the contrary,when σcM/σ0 is high, a large amount of plastic deformations will be necessary before the local stresses

6

𝑢𝑢𝑦𝑦 𝑢𝑢𝑥𝑥

𝐴𝐴0 (MBL radius) 𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌 (𝑈𝑈2 = 0)

𝑈𝑈3 = 0

Process zone

c) Back surface

b) Front surface

Cohesive zone element

Y

X Y

X

a) Boundary layer 𝑟𝑟𝑡𝑡 𝑙𝑙𝑒𝑒

Figure 5: Geometry model and applied load on boundary layer: a) Boundary layer; b) Front face of theprocess zone; c) Back face of the process zone

reach σcM due to the work-hardening. The occurring plastic deformations firstly dissipate much energyand secondly drive the ductile mechanism of void growth and coalescence. In order to assess the effectof the temperature, a parameter study is performed with respect to σcM/σ0 .

Furthermore, values σcP, σcPM, Γ0P and Γ0PM of cohesive strength and work of separation for breakageof the carbide particle and for the particle-matrix interface, respectively, need to be specified. Actually,these values are quite difficult to be assessed experimentally. Thus, the values Γ0P=Γ0PM = Γ0M areadopted from [12–14, 32, 33] in the present study. The cohesive strength is selected as σcP = σcPM ≤ σcM

[33].

2.4 Numerical implementation

Due to symmetrical loading in mode I and symmetrical distribution of carbide particles in the bccconfiguration (see Fig. 2), only a semi-circular slice of the boundary layer (Fig. 5) needs to be meshedin the FEM analysis. Both, at the front and back surface of the BL, out-of-plane displacement aresuppressed (uz = 0) in order to ensure macroscopic plane strain conditions. These boundary conditionscorrespond to the mirror symmetry of microstructure. In [15] it was pointed out that a certain numberof layers of discrete layers is necessary due to potential shielding of the process zone by void growtharound it. Therefore, 5 layers of discrete particles together with 16 unit cells in the crack growthdirection are generated in the process zone. The process zone is connected with the homogenized regionby a kinematic constraint.

During the simulation, in particular when cleavage is dominant, the crack growth could becomepartially unstable. In order to handle local unstable crack growth, a dynamic analysis is performedwith quasi-static loading. This means that the time of elastic waves traveling from the outer boundaryto the center of the boundary layer has to be smaller compared to time period of loading τL. Accordingto [15], the time period can be chosen to be τL = 0.02X0/cs wherein cs is the speed of shear waves,cs =

√0.5ρEeff/(1 + νeff) and ρ denotes the mass density.

7

𝑊𝑊𝑖𝑖

W

𝑊𝑊0

ligament i

∆𝑎𝑎

Initial crack tip

𝑊𝑊0 𝑊𝑊0

𝑊𝑊1𝑟𝑟𝑟𝑟𝑟𝑟 𝑊𝑊2

𝑟𝑟𝑟𝑟𝑟𝑟

ligament i

∆𝑎𝑎 𝐿𝐿𝑥𝑥

𝑊𝑊𝑖𝑖

𝑊𝑊0 𝐿𝐿𝑥𝑥 𝑊𝑊0

𝑊𝑊3

x

Initial crack tip

carbides

z

y

𝑊𝑊2𝑟𝑟𝑟𝑟𝑟𝑟 𝑊𝑊1

𝑟𝑟𝑟𝑟𝑟𝑟

Figure 6: Definition of effective crack growth [15]

Simulations are performed with the commercial software Abaqus/Standard. Hexahedral elementswith quadratic shape functions and reduced integration (C3D20R) are used for both FPZ and ho-mogenized region. Cohesive elements with quadratic shape functions are implemented by UEL sub-routine [24]. The length of cohesive elements and the adjacent continuum elements are chosen to bele = (0.03÷0.06)X0 [16]. An initial crack tip with small radius rt is designed in order to avoid numericalproblems during the initial crack tip blunting, rt = L0/10. The radius A0 of the boundary layer modelis chosen large enough to ensure small scale yielding conditions: A0 = (700÷ 7500)X0.

2.5 Effective crack growth and fracture initiation

The present model is designed to simulate the competition between brittle and ductile mechanismson the microscale. Both mechanisms will lead to an active softening zone of finite size. For cleavage,this is the region where the cohesive zone is in the softening regime of the traction-separation law. Inthe ductile mechanism, the geometric softening of the ligaments between debonded particles cannotbe compensated by the plastic strain hardening and internal necking will initiate. If both mechanismsare active at the same time, it is not trivial to define a suitable measure of crack growth ∆a at themicroscale. Here, we adopt the approach from [34] and define the amount of effective crack growth asthe distance from the initial crack tip to the center of currently active softening zone as sketched inFig. 6. In particular, the relative reduction (1−Wi/W0) of the ith ligament is taken as the weightingfactor for each unit cell in the crack plane. Therein, Wi and W0 are the current and initial ligamentwidth, respectively, which are measured on the crack plane in the direction of crack growth. Thus, thecrack growth is calculated by

∆a = Lx

nlig∑i=1

(1− Wi

W0

1− W remi

W0

)(9)

Furthermore, W remi is the remaining width of ith ligament in the wake behind the active softening zone

and Lx is the distance between two neighbouring particles on the crack plane. According to Fig. 3,Lx = 25/6 ·X0.

Similarly, it is not trivial with the present micromechanical model to define the event of fractureinitiation and thus the fracture initiation toughness JIc uniquely. Here, the definition of JIc as firstkink in the plot crack tip opening displacement CTOD vs. J is adopted from [15, 35–37]. Initially, theCTOD-J is linear for pure crack tip blunting and a deviation from this linear regime indicates a stopof blunting and thus fracture initiation.

2.6 Calibration of GTN model

In order to ensure consistency of the mechanical behavior from the outer, homogenized region to thefracture process zone, cell model simulations are performed to identify the parameters of the GTN

8

Ferrite

Homogeneous GTN model

Particle

Interface cohesive zone

Cohesive zone inside carbide particle

Ferritic matrix

X

Z

X

Y

E2

E3E1

U2

U3U1

R

Lo Lo

Lo

Homogeneous GTN model

Unit cell model

Carbide particle

X

Y

E2

E3

E1

U2

U3

U1

R

Lo Lo

Lo

Z

a)a)

b)

Cohesive zone inner particle

Paticle-ferrite interface

FerriteParticle

X

Y

U2

U3U1

R

Lo Lo

Lo

Z

a) b)

Cohesive zone inner particle

Paticle-ferrite interface

crack font

d=<110>carbide

Lo

Figure 7: Unit cell model for bcc arrangement: a) bcc unit cell, b) loading and boundary condition onthe unit cell

Table 2: Parameters for homogenized GTN model with void nucleation corresponding to σcPM

σcPM/σ0 fN q1 q2 εN sN = εN/5 σ0/σ0

0.1 0.014 1.3 0.98 0.005 0.001 0.98

1.0 0.014 1.3 0.98 0.005 0.001 0.98

3.0 0.014 1.3 0.98 0.0035 0.0007 1.02

6.0 0.014 1.3 0.98 0.0035 0.0007 1.04

model. Due to the symmetries of the bcc configuration, only 1/8 of the cubic unit cell needs to beincorporated in the FEM model, see Fig. 7. Outer surfaces of unit cells are kept plane during theloading history by using kinematic constraints. The stress state parameter T is controlled by usingRiks algorithm in Abaqus/Standard. For details on the cell model approach (like the exact definitionof the macroscopic equivalent strain Eeq), the reader is referred e. g. to [38–40].

For materials with pre-existing voids, the GTN parameters q1 = 1.3, q2 = 0.98 were already deter-mined in [39, 40] and it remains to identify the nucleation parameters εN, sN and fN. Furthermore, itturned out that the fit between GTN model and cell model can be improved by a slight calibration of theequivalent initial yield stress σ0 of the GTN model [15]. According to Saje [17], the mean strain value ofvoid nucleation εN is determined at a point where the void nucleation rate achieves its maximum value.The standard deviation of the normal distribution sN is estimated by 20% of εN [20]. For example, inFig. 8 the relationship between the rate of the void volume fraction and the effective strain is plotted forσcP/σ0 = σcPM/σ0 = 3.0. From this diagram the mean strain value of the void nucleation εN = 0.0035is obtained and consequently, the standard deviation becomes sN = εN/5 = 0.0007. It is clear that thevoid nucleation and growth strongly depend on the cohesive strength of particle-ferrite interface. TheGTN parameters were calibrated for various values of the cohesive strength of particle-ferrite interfaceσcPM/σ0 assuming uni-axial tension with T = 1/3. Results are given in Tab. 2.

9

𝜀𝜀N

Figure 8: Void growth vs. effective plastic strain

(a) 𝜎𝜎cM/𝜎𝜎0 = 3.25 (b) 𝜎𝜎cM/𝜎𝜎0 = 3.75 (c) 𝜎𝜎cM/𝜎𝜎0 = 3.85

Figure 9: Crack initiation and propagation with weak particles σcPM/σ0 = 1.0

3 Results

Fig. 9 visualizes (by incorporating the exploited symmetries of the model) the simulated deformationsduring initial crack propagation for weak particles σcP/σ0 = σcPM/σ0 = 1.0, whereby three differentvalues of the matrix strength ratio σcM/σ0 were assumed, which corresponds to different temperatures.In Fig. 9a) at low temperature, corresponding to low strength ratio σcM/σ0 = 3.25, the carbide particlesbreak firstly before cleavage occurs in the matrix on the crack plane. In this case, no plastic deformationsare observed but the matrix remains completely elastic. Fig. 9b) shows results for a cohesive strengthratio σcM/σ0 = 3.75 as a representative for the intermediate DBT region. Obviously, voids have grownand a stretch zone has formed at the initial crack tip but the matrix finally fails by cleavage, i. e. ductileand brittle failure occur simultaneously. Remarkably, the first carbide particles debond from the matrixwhereas those in a certain distance to the initial crack tip are broken. For only a slightly higher strengthratio σcM/σ0 = 3.85, purely ductile crack propagation is observed in Fig. 9c), characterized by particledebonding and subsequently void growth.

Fig. 10 shows corresponding deformation plots for hard particles σcP/σ0 = σcPM/σ0 = 3.0. Notethat for this parameter set, the carbide particles debond in the complete temperature regime, i. e.for all investigated values of the matrix strength ratio σcM/σ0. For example, a matrix strength ratioσcM/σ0 = 3.75 in Fig. 10c), more severe plastic deformation around the particle in front of the cracktip is observed, which is the main factor to debond particle from the matrix.

Extracted crack growth-resistance curves (R-curves), normalized by the initial yield stress σ0 and themean distance of particles X0, are shown in Fig. 11 for different σcM/σ0.

All curves start from the origin due to the employed definition (9) of the effective crack growth

10

(a) 𝜎𝜎cM/𝜎𝜎0 = 3.25 (b) 𝜎𝜎cM/𝜎𝜎0 = 3.5 (c) 𝜎𝜎cM/𝜎𝜎0 = 3.75

Figure 10: Crack initiation and propagation with hard particles σcPM/σ0 = 3.0

Figure 11: Crack growth resistance (σcPM/σ0=1.0)

11

Figure 12: Initial fracture toughness of ferritic steel in the temperature transition region

∆a. The trend towards higher crack growth resistance with increasing σcM/σ0, i. e. with increasingtemperature, is obvious. In the purely brittle regime σcM/σ0 . 3.0, the load cannot be increasedafter crack initiation and the crack propagation becomes dynamic. When the temperature increases tothe DBT region, i. e. with higher σcM/σ0, the contribution of plastic dissipation to the crack growthresistance increases. The corresponding crack tip blunting is manifested in the initially linear regime ofthe corresponding R-curves. Furthermore, the further increase of the R-curve after fracture initiation,the tearing, becomes more pronounced with increasing plastic contributions and thus with increasingfracture initiation toughness. The qualitative features observed in the simulations , i.e. dynamic crackpropagation for pure cleavage and increasing blunting and tearing at higher temperatures, are well-known from experiments.

The influence of the strength ratio σcM/σ0 on the computed fracture initiation toughness JIc is shownin Fig. 12 for three cases including very weak (σcPM/σ0=0.1), weak (σcPM/σ0=1.0) and hard particles(σcPM/σ0=3.0). The S-shape of the curves is typical for the transition from the brittle to the ductileregime. In the DBT, the simulations with hard particles yield slightly higher fracture iniation valuesthan those with weak particles. Recall, that in this regime the fracture mechanism was breakage ofweak particles and debonding of strong particles, compare Figs. 9 and 10. An intuitive expectationcould be, that this trend would persist for all temperatures [18]. However, when the saturation valuesin the ductile regime are considered, i. e. at right-hand side of Fig. 12, the computed fracture toughnessis about from 20% to 25% higher corresponding to weak and very weak particles compared to hardones. In this regime, particle debonding was observed for these parameter sets. The weak and veryweak particles debond earlier from the matrix thereby nucleating voids which grow and coalescence byplastic deformations. In contrast, for hard particles, the matrix material remains longer bonded to theelastic particle. However, during that time the plastic deformations of the ferritic matrix are hindered,reducing their contributions to the crack growth resistance. Consequently, a bonding between matrixand carbide particle which is too strong is found to be unfavorable.

Furthermore, Fig. 12 contains the experimental data of Coates and Roberts [41] and Wallin et al.[42] for a quantitative comparison. The macroscopic yield stress of ferritic steels varies in the range

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from 518 MPa (at -110 ◦C) to 412 MPa (at 60 ◦C) [41, 43]. In our study, the yield stress is expressedas a function of temperature adopted from [43]. The parameters σcM and Γ0 of the cohesive zone areessential for modeling cleavage in the ferritic matrix. As aforementioned, the work of separation Γ0

is chosen as the lower-bound fracture toughness of the ferritic steel. According to [22], the fracturetoughness of ferritic steel amounts to Γ0M= 5 kJ/m2 at the temperature -125 ◦C. At this temperaturethe yield stress is calculated to be σ0 = 600 MPa based on [43]. Consequently, the average distanceof carbides is chosen as X0 = Γ0M/σ0 = Γ0/σ0 ≈ 26µm for normalizing experimental data. Thecohesive strength σcM corresponding to local cleavage fracture stress of the ferrite matrix is assumedto be temperature-independent. However, according to Wang and Chen [44] σcM actually depends onboth carbide particle size and grain size of the ferritic matrix and lies in the range of (1300 ÷ 1750)MPa. Here, the cleavage fracture stress is chosen as σcM = 1550 MPa corresponding to coarse carbides(dp ≥ 0.6µm) and a pre-cracked specimen (insensitive to grain size). Thus, the strength ratio σcM/σ0

correlates to the temperature as follows: σcM/σ0 = 2.99 at -110oC, 3.05 at -100 ◦C, 3.17 at -80 ◦C,3.20 at -75 ◦C, 3.34 at -50 ◦C, 3.43 at -30 ◦C, 3.56 at 0 ◦C) and 3.76 at 60 ◦C. Fig. 12 shows, that thesimulation results of the present study agree quite well with the experimental data.

4 Summary

In the present study, a micromechanical model was used for simulating the crack initiation and propa-gation in ferritic steel in the ductile-brittle transition region. In particular, the carbide particles, whichplay an essential role for the ductile-brittle transition, were resolved discretely in the fracture processzone. The competing mechanism of breakage or debonding of carbide particles or cleavage of the ferritematrix were modeled by cohesive zones. Outside the process zone, the material behavior was describedin a homogenized way by means of the GTN-model to keep the computational costs at a manageablelevel.

The results of the simulations show that the model predicts qualitatively correct the most importanttrends and features in the DBT, which are known from experiments . Also quantitatively, the presentresults comply well with experimental data from literature.

A parameter study on the effect of the strength of the particles on the fracture toughness has shownthat the strength of the particles can have a different influence in the DBT than in the purely ductileregime. These first achievements give an optimistic out look future research, where the predictionsof the developed model can be advantageously used to optimize the microstructure of ferritic steelstowards certain application requirements.

Acknowledgements

The financial support by the Deutsche Forschungsgemeinschaft (German Research Foundation) undercontracts HU 2279 (Giang) and INST 267/81-1 FUGG (computing facilities) is gratefully acknowledged.The authors thank Stephan Roth for providing the implementation of the cohesive element for Abaqus.

Appendix A

The macroscopic values of bulk and shear modulus, respectively, can be computed with the Mori-Tanakamethod [45] as

Keff = Km + fp(Kp −Km)Km

Km + α(1− fp)(Kp −Km)(A.1)

Geff = Gm + fp(Gp −Gm)Gm

Gm + β(1− fp)(Gp −Gm)

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wherein Kp,Km, Gp and Gm are the respective values of matrix and particle at the microscale and α =3Km/(3Km+4Gm) and β = (6Km+2Gm)/(5(3Km+4Gm)) refer to the constraint parameters in the Es-helby tensor. The macroscopic moduli Keff and Geff are related to the engineering constants of Young’smodulus and Poisson ratio by Eeff = 9KeffGeff/(3Keff +Geff) and νeff = (3Keff − 2Geff)/(6Keff + 2Geff).

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