Int J Fract (2012) 176:81–96 DOI 10.1007/s10704-012-9728-9
ORIGINAL PAPER
Ductile crack propagation by plastic collapse of the intervoid ligaments Geralf Hütter · Lutz Zybell · Uwe Mühlich · Meinhard Kuna
Received: 28 February 2012 / Accepted: 10 May 2012 / Published online: 6 June 2012 © Springer Science+Business Media B.V. 2012
Abstract In the present study ductile crack initiation and propagation is investigated by means of a micro-mechanical model under small-scale yielding conditions. Voids are resolved discretely in the fracture process zone where steep gradients occur during the loading history and are taken into accounted by a homogenized porous plasticity law elsewhere. The size of the region of discrete voids is not set a priori but is determined consistently. The results show that effective crack growth occurs by plastic collapse, i.e. purely geometric softening of the intervoid ligaments without incorporating material separation. Due to this mechanism a limit load exists coinciding with the maximum fracture toughness. In addition, it turns out that the shielding due to the growth of voids around the crack plane has a considerable influence on the computed R-curves compared to models neglecting this effect. Depending on the void arrangement a diffuse softening zone or even crack branching is observed. A comparison with experimental data from literature indicates that plastic collapse and the formation of diffuse zones of void growth are realistic mechanisms of ductile crack propagation. G. Hütter (B) · L. Zybell · U. Mühlich · M. Kuna TU Bergakademie Freiberg, Institute for Mechanics and Fluid Dynamics, Lampadiusstr. 4, 09596 Freiberg, Germany e-mail:
[email protected] M. Kuna e-mail:
[email protected]
Keywords Ductile fracture · Discrete voids · Finite element analysis · Void growth and coalescence · Plastic collapse · Limit load · Small-scale yielding
1 Introduction In the range of room temperature, typical engineering metals fail by a ductile mechanism. In this process, voids are created from second-phase particles which break or debond from the embedding metallic matrix. These voids grow and finally coalesce to a macroscopic crack, Fig. 1. The void volume fractions of these particles differ and lie considerably below 1 % in steels up to more than 10 % in ductile cast iron. A lot of experimental and theoretical studies contributed to the understanding of this mechanism. An overview can be found e.g. in (Benzerga and Leblond 2010). On the theoretical field, mainly two types of models have been adopted to study the phenomenon of ductile damage. The first branch deals with the behavior of a unit cell. So Gurson (1977) obtained his famous model from a limit-load analysis of a representative volume element. The numerical investigations of Tvergaard (1981, 1982) of a unit cell with a cylindrical, respectively spherical void, a so-called cell model, showed that mesoscopic softening can occur despite stable deformation behavior of the metallic matrix material due to a microscopic plastic collapse, i.e. geometric softening of the intervoid ligaments. This process initiates void coalescence. Cell models were employed
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Fig. 1 Fracture surface after ductile failure in a pressure vessel steel (Seidenfuss et al. 2011)
to study many effects, such as e.g. the influences of secondary voids (Kim et al. 2003; Tvergaard 1988), the spatial void arrangement (Kuna and Sun 1996) or strain-gradient hardening (Niordson and Tvergaard 2006). The second group of studies focuses on numerical simulation of the interaction of voids with a crack tip. In the early investigations (McMeeking 1977; Rice and Johnson 1970) potential void growth was evaluated by post-processing of the solution obtained for compact material. Later the evolution of a single discrete void in front of the crack tip was simulated numerically (Aoki et al. 1984; Aravas and McMeeking 1985a,b; Hom and McMeeking 1989). Dahlberg (2004), Gu (2000), Tvergaard and Niordson (2008) as well as Gao et al. (2005) and Kim et al. (2003) resolved up to six discrete voids in the crack plane in front of the crack tip and found an influence of the shape of the voids, of an additionally added second row of voids, of the crack tip constraint and of strain gradient hardening. Petti and Dodds (2005), Tvergaard (2007) and Mostafavi et al. (2011) incorporated up to 30 discrete voids in a single row in front of the crack tip. Gao et al. (2005), Kim et al. (2003), Petti and Dodds (2005), Tvergaard (2007), Tvergaard and Hutchinson (2002) defined a critical value of the ligament reduction ratio. If this value is reached they assumed an immediate loss of the material cohesion in the corresponding intervoid ligament. Implementing a node-release technique in the FE-model this assumption allowed to trace the crack propagation. Tvergaard and Hutchinson (2002) and Kim et al. (2003) found an influence of the critical ligament reduction ratio on the predicted
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crack growth resistance. In addition, the aforementioned authors observed the transition from a void by void to a multiple void interaction mechanism of growth with increasing initial void volume fraction and with ongoing crack propagation for intermediate void volume fractions. In all studies the zone with discretely resolved voids is embedded in a compact region having the same elastic and isochoric plastic properties like the intervoid matrix material. This is not consistent since the homogenized void-matrix composite is expected to show dilatational plastic flow and a lower elastic stiffness. No studies were performed regarding the necessary number of voids to be resolved discretely although this could be essential especially if a multiple void interaction mechanism occurs. In the present study the voids are resolved discretely within a considerably larger region around the crack tip and a convergence study regarding the necessary dimensions of this process zone is performed. The material of the surrounding compact zone is described consistently by the GTN-model representing the homogenized behavior adequately. No assumptions are made regarding a critical reduction of the intervoid ligaments but the effective crack propagation only due to microscopic plastic collapse is considered.
2 Model 2.1 Problem formulation Mode I crack propagation in a material with pre-existing cylindrical voids is investigated under plane-strain conditions. The limit case of small-scale yielding is considered in order to exclude possible effects of a specimen geometry. For this purpose in a so-called boundary layer approach the linear elastic K I -solution is prescribed as displacement boundary condition at a circumference whose radius A0 is large compared to the maximum size of the plastic zone. The matrix material between the voids is described by an isotropic hypoelastic-plastic formulation for large strains with Mises yield surface and isotropic hardening. Within the process zone the voids are introduced as a number of cylindrical holes arranged in several rows in front of the crack tip. The material behavior around the discretely resolved zone is described in a homogenized way by means of the Gurson-model in the modification
Ductile crack propagation by plastic collapse
83
KI homogenised: GTN
matrix: Mises-plasticity
X0 2R0 A0
Fig. 2 Semi-infinite crack with resolved process zone
of Tvergaard and Needleman (GTN-model). The whole model is shown schematically in Fig. 2. A one-parametric power law of hardening is utilized for the matrix material, so that the uniaxial response between true stress and logarithmic strain is given by σ/E σ < σ0 . (1) ε= 1/N σ0 /E (σ/σ0 ) else The symbols E, σ0 and N denote Young’s modulus, initial yield stress and the hardening exponent. These values and the Poisson ratio ν are set to E = 333 σ0 , N = 0.1 and ν = 0.3 corresponding to the behavior of typical medium strength engineering metals. The GTN yield condition reads σeq 2 3q2 σm − (1 + q3 f 2 ) ≤ 0 + 2q1 f cosh σM 2 σM (2) in which σeq , σm and σM denote the Mises and the hydrostatic stress, respectively the effective yield stress of the matrix material. It is taken into account that no accelerated void growth occurs in the region where the GTN-model is applied so that the actual porosity f is equal to the effective one. The evolution equations and associative flow rule are well known and are not repeated here. The heuristic parameters q1 , q2 and q3 have been introduced by Tvergaard (1981) to fit the model to the results of computations on cell models with cylindrical voids. Within that study the value of the void volume fraction of the cylindrical pores (equal to the area fraction of the circles in the plane) was used for the porosity f giving the fitted values q1 = 1.5, q2 = 1 and q3 = q12 . The present study adopts this definition and employs these values. In principle the yield curve σM = f (¯εM ) of the matrix in the yield condition Eq.
(2) should be the one corresponding to Eq. (1) (subtracting the elastic part). However, as it will be shown in Sect. 3 a slight calibration is necessary to improve the fit of the GTN-model to the behavior of the unit cells in the relevant range of strains. The symbol ε¯ M denotes the equivalent plastic strain of the matrix. The effective elastic properties E eff and νeff of a material with cylindrical voids under plane strain can be found in text books (e.g. Gross and Seelig 2006). The values used in the present study are summarized in Table 1. The change of the elastic properties with ongoing void growth is not taken into account. With these values the far-field energy release rate is obtained as 2 K I2 1 − νeff . (3) J= E eff Several regular arrangements of voids relative to each other and to the crack plane are investigated. The cubic primitive and hexagonal basic patterns are employed. A sketch including the used abbreviations and symbols is shown in Fig. 3. Correspondingly, the initial porosity is computed as Table 1 Effective elastic properties for a Poisson ratio of the matrix ν = 0.3 f0
0.00087
0.0035
0.014
0.056
E eff /E
0.997
0.990
0.96
0.86
νeff
0.2999
0.2995
0.293
0.290
W0
cp-0° 45°
X0
cp-45°
(a) W0
hex-0°
60° 30° X0
hex-30°
(b) Fig. 3 Void arrangements: a cubic primitive and b hexagonal
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χ 1
Δa ligaments i 0 x/X0
Fig. 4 Smeared measure of crack extension
cubic: f 0 = π
R0 X0
2
, hexagonal:
2π f0 = √ 3
R0 X0
2
initial crack tip .
Fig. 5 Finite element mesh near the crack tip
(4) 2.3 Numerical implementation A number of 40–140 discrete voids is resolved in the crack plane in the model depending on the amount of crack growth to be simulated. Section 4.2 deals with the number of layers of discrete voids to be incorporated.
2.2 Effective crack length The presented model does not incorporate material decohesion in the intervoid ligaments so that the current crack tip cannot be defined by the end of the zone of fully separated material. Nevertheless, at appropriate loading the ligaments soften geometrically so that the active softening zone and thus the active plastic zone move macroscopically corresponding to an effective crack growth. As there is no sharp crack tip, the latter is defined in a smeared sense as the center of the currently active damage zone as sketched in Fig. 4. Thereby, the relative width χi = Wi /W0 of each intervoid ligament in the crack plane is taken as weight and the effective crack growth is computed as a = X 0
n lig
(1 − χi ) .
(5)
i=1
Thereby, W0 and Wi denote the width of the ligament i in the initial and in the current configuration, respectively. For the cp-45° and hex-30° configurations Eq. (5) cannot be adopted easily. Necessary modifications are pointed out at appropriate positions.
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The described boundary value problem is solved numerically with the commercial FE-code Abaqus/ Standard. A boundary layer of radius A0 = (1, 500 . . . 4, 500) X 0 is spatially discretized. Only a half-model needs to be taken into account due to the symmetry. Since large plastic deformations occur in the intervoid ligaments and no remeshing is performed during the simulation, the mesh needs to be sufficiently fine. Nevertheless, the elements at the surface of the voids experience several hundred percent of straining and distort at some time. However, this does not occur until the respective intervoid ligament deforms mainly in a uniaxial mode which can still be represented adequately. A typical mesh near the crack tip used for the simulations is shown in Fig. 5. Quadrilateral elements with quadratic shape functions and reduced integration are used. No special averaging techniques are employed to couple the discretely resolved and the homogenized regions. Due to the large number of discretely resolved voids steady-state crack propagation is expected to occur during the simulations where the crack propagates at practically constant loading. At this point there is no unique correlation between the far-field loading and the crack growth near the crack tip anymore. This behavior cannot be handled by a standard load application scheme, but either a generalized arc length method or a dynamic computation needs to be applied. In this study, a dynamic simulation under quasi-static loading with implicit time integration (Euler backward) is performed. It has to be ensured that the time
Ductile crack propagation by plastic collapse
85
Σyy
lateral contraction
Σxx /Σyy = 0.1
Σyy /σ0
2
Σxx
longitudinal strain
Σxx /Σyy = 0.3
1
cell model GTN GTN fit
y 0 -0.02
x
3 Cell model for calibration In the present model the uniformly distributed voids are resolved discretely in the fracture process zone and are taken into account in a homogenized way by means of the GTN-model outside this zone. Thus, the parameters of the GTN-model need to be calibrated so that the behavior of the porous material is captured in the relevant range of straining. In the simulations of the fracture process with the boundary layer model the strains outside the discretely resolved process zone are in the range of 1–2 % at intermediate biaxialities. So, the second in-plane principal stress reaches a level of 10–30 % of the maximum principal stress. For the calibration simple cell model computations have been performed as depicted in Fig. 6. Due to the mirror symmetry of the configuration the condition of plane boundaries in the cell model coincides with periodic boundary conditions. The obtained stress-strain curves are plotted in Fig. 7 in comparison to those of the GTN-model. Firstly, the parameters given by Tvergaard (1981) were employed (denoted “GTN” in Fig. 7) which were obtained as fit to
0
Exx
Fig. 6 Cell model for f 0 = 0.014
elastic waves need to proceed through the process zone is small compared to the time scale of loading τL . For this purpose the mass density is specified so that values of X 0 /c = 1/650, 000 τL are obtained with c denoting the velocity of the faster longitudinal waves. This choice assures sufficiently small inertia forces during stable crack propagation but does not lead to too small time increments necessary to drive the crack in the steady-state stage.
-0.01
0.01
0.02
Eyy
Fig. 7 Stress-strain curves of cell model computations and GTN-model ( f 0 = 0.014) Table 2 Fitted initial matrix yield stress σM0 of the GTN-model f0
0.00087
0.0035
0.014
0.056
σM0 /σ0
0.996
0.985
0.95
0.90
exactly the same cell model as investigated here. However, Tvergaard (1981) focused on the later stages of void growth. The results in Fig. 7 show that behavior of the cell model is captured quite well with this parameter set but that the stress level is slightly overestimated in the relevant range of straining. The relative difference of the stresses between cell model and GTN-model with original parameters is practically independent of the levels of strain and biaxiality. That is why it is not reasonable the tune to GTN parameters q1 and q2 . Instead, the matrix yield curve σM = f (¯εM ) of the GTN-model was calibrated. So, not the “true” yield curve of the matrix material in the discrete void region is taken for σM = f (¯εM ) but the nominal matrix initial yield stress σM0 in the GTN model is slightly scaled down compared to the actual value σ0 . With this calibration a very good agreement with the results of the cell model is obtained as Fig. 7 shows. The fitted values of the GTN initial matrix yield stress σM0 are given in Table 2 for all investigated values of the initial porosity f 0 . Changes of the hardening exponent N are not necessary. Another point is the anisotropy introduced by the regular void arrangement. This anisotropy with respect the elastic and plastic behavior was investigated by further cell model computations not shown here. So in
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Fig. 8 Deformed configuration (a) in the initial region J = 0.26 σ0 X 0 and (b) in a later stage J = 0.98 σ0 X 0 ( f 0 = 0.014, cp-0°)
(a)
(b)
(a)
1
0.8
rel. void width W / X0
addition to the situation above where the principal axes of loading are aligned parallel to the lattice axes ([1 0] and [0 1]), computations with 45° rotated axes of loading ([1 1] and [1 −1]) were performed. The sections of the unit cell in the numerical model were chosen appropriately. The relative difference between the results for parallel and rotated principal axes of loading lies below one per mill and can thus be neglected. This holds for the difference between cubic and hexagonal arrangement, too.
voids no. 1,2,4,7,10,20,30,40
0.6
0.4
0.2
0
4 Results
Jlim 0
0.2
0.4
0.6
0.8
1
1.2
1.4
J/ ( σ0 X0 )
4.1 Crack growth by plastic collapse
(b)
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Jlim = 1.1σ0 X0
1.2 1
J/ ( σ0 X0 )
The deformed state of the voids in the process zone for cubic-primitive arrangement cp-0° is shown in Fig. 8 for different stages of loading. The evolution of the width of some voids in the crack plane with increasing loading J is plotted in Fig. 9a. It shows that the size of the first voids increases initially with a certain offset until all voids grow successively when reaching the limit load Jlim . This limit load corresponds to the fracture toughness. According to the definition of crack growth Eq. (5) as a measure of accumulated void growth the effective crack growth resistance curve in Fig. 9b is obtained. The distribution of the current void sizes with ongoing deformation is depicted in Fig. 10. It can be seen that after the initiation a damage profile forms which is finally shifted congruently along the crack plane when a stationary stage of crack growth is reached. In the employed numerical model this process is limited by the width of the region with discretely resolved voids. Figure 10 shows that the damage profile becomes flatter
0.8 0.6 0.4 0.2 0
0
10
20
30
40
50
Δa / X0 Fig. 9 Evolution of the width of the voids in the crack plane (a) and crack growth resistance curve (b) ( f 0 = 0.014, cp-0°)
with the transition from the initial to the steady state as already found by Tvergaard (2007). In the stationary regime the active damage zone encompasses about 15
Ductile crack propagation by plastic collapse
87
1
0.8
Jlim = 1.1 σ0 X0
1− χ
0.6
J = 0.90 σ0 X0
0.4
0.2
J = 0.65 σ0X0 0
0
5
10
15
20
25
30
35
40
intervoid ligaments Fig. 10 Evolution of the void size distribution in the ligament ( f 0 = 0.014, cp-0°)
voids. This is the minimum number of discrete voids to be incorporated in the model along the crack plane in order to observe the plastic collapse mechanism. The ligaments in the wake behind the current crack tip experience unlimited straining if the active damage zone moves sufficiently far away. Unavoidably, the elements distort and the numerical accuracy decreases in this region. Finally, the field quantities computed at the strongly distorted elements become even completely worthless. However, this numerical inaccuracy does not affect the global behavior if it is ensured that the elements provide sufficiently accurate results until the active softening zone passed away. The ligaments in the wake behind the active zone are contracted so strong that deviations in the computed true stress do not affect the nominal stress anymore. The latter vanishes practically and so do the contributions to the nodal forces. For the present configuration strains of several hundred percent have to be resolved accurately corresponding to the left tails of the steady-state void growth profiles in Fig. 10. The employed meshes as shown in Fig. 5 fulfill this requirement.
4.2 Size of the region of discrete voids In the present model the uniformly distributed voids are accounted for in a homogenized way where possible and resolved discretely only where necessary. In order to determine the size of the region where discrete voids have to be incorporated, the number of layers of
discrete voids has been varied. According to Sect. 4.1 enough voids are incorporated along the direction of crack growth, see also Sect. 2.1. It is expected that convergence of the fracture behavior with respect the number of layers of voids occurs since the homogenized constitutive approach should describe the material behavior adequately in sufficiently large distance to the fracture process zone, at the latest in the elastic region. The obtained effective crack growth resistance curves are plotted in Fig. 11a. The results show that by incorporating more layers of discrete voids, the tearing region of the R-curve becomes considerably wider and that the limit load toughness Jlim increases. Convergence is not reached until ten layers of voids (in the half model) are resolved discretely corresponding to almost one third of the height of the plastic zone in the stationary state. Obviously, there is a strong shielding effect by the voids in the plastic zone. The observation that so many rows of discrete voids need to be included is remarkable as the GTN material parameters were fitted to the behavior of a unit cell in the relevant range of straining. Apparently, the simple deformation state in the cell model does not meet the situation prevailing in large parts of the plastic zone. In a further simulation not shown here we verified that the large number of necessary layer is not attributed to some features of the employed coupling between homogeneous and discrete void region. So the boundary of the last cells at the connection to the homogenous region was fixed to remain planar as in the cell model. This did not lead to any noteworthy change. Figure 11b shows the R-curve computed with ten layers of voids and the standard GTN parameters, i.e. without the fit performed in Sect. 3. Comparing this curve with the already converged curve obtained with fitted parameters and the same number of layers indicates that a change of a few per cent in the GTN matrix yield stress σM0 leads to differences of about 10 % in the limit load toughness Jlim . More layers of voids are necessary with the standard GTN parameters in order to reach a converged state at which the model is consistent. Furthermore, a R-curve computed for the “classical” case of a single layer of voids with identical Mises plasticity behavior in the intervoid ligaments and in the homogeneous, compact region is incorporated in Fig. 11b. This model does not include the shielding and thus underestimates the tearing modulus and the
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(a)
G. Hütter et al. Table 3 Number of layers of voids in the half model
1.4
1,4,7,10,16 layer(s)
1.2
J/ ( σ0 X0 )
1 0.8
0.00087
0.0035
0.014
0.056
cp-0°
16
16
10
10
hex-0°
−
−
14
14
cp-45°& hex-30°
−
17
17
17
0.6 0.4 0.2 0
0
10
20
30
40
50
60
Δa/ X0
(b)
f0
1.4 1.2
the end of the discretely resolved region. The results given in the following exclude regions of such inconsistent interactions. The numbers of layers of discrete voids used in the following computations are given in Table 3. These numbers are oriented at the section (about one third) of the plastic zone to be modeled with discrete voids as found for the example above. The consistency of this choice was verified for some configurations with a second simulation with one or two additional layers.
J/( σ0 X0 )
1 0.8
4.3 Initial porosity
0.6 0.4
GTN σM0 fitted, 10 layers GTN σM0 = σ0 , 10 layers Mises, single layer
0.2 0
0
10
20
30
40
50
60
Δa/ X0 Fig. 11 Crack resistance curves (a) for several number of layers of discretely resolved voids and (b) for fitted and non-fitted GTN parameters ( f 0 = 0.014, cp-0°)
limit load toughness considerably (This holds as well for smaller values of the initial porosity, see also Fig. 17 below). This large difference of the crack growth resistance for the classical single layer model and the consistent one with uniformly distributed voids exceeds the deviation of the Young’s modulus of about 5 % between both approaches by far. In addition, the R-curve of the model with a single layer of voids increases again when the active plastic zone reaches the end of the zone of discrete voids. Then, the last discrete void blunts only and the model is not valid anymore. The latter applies as well if the GTN-model is employed for the homogenized material. In this case the deformation localizes within one row of elements when the active process zone reaches
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The effective crack growth resistance curves for several values of the initial porosity f 0 are depicted in Fig. 12. As expected, the limit load toughness Jlim decreases with increasing f 0 . In addition, the tearing region becomes wider for higher values of the initial porosity corresponding to the transition from a void by void growth to a multiple void mechanism (Tvergaard and Hutchinson 2002). For lower porosities f 0 ≤ 0.0035 the crack problem becomes dynamic after having reached the limit load Jlim and the crack accelerates up to one third of the Rayleigh wave speed. This indicates that in these cases the limit load Jlim forms a local maximum of the static R-curve lying above the steady-state toughness. A local maximum Jlim in the R-curve was observed by Tvergaard (2007), too. The evolution of the relative width χ of the ligament between the initial crack tip and the first void is plotted in Fig. 13. It shows that for low porosities there is a linear initial region corresponding to crack tip blunting followed by a transition to a second nearly linear section. Gu (2000) defined this transition point as crack initiation. The second region is also observed for higher porosities whereas there is no distinct transition from the initial stage anymore. Again, this is attributed to the multiple void mechanism. In this context it has to be recalled that within the present simulations the initial radius of the crack tip is set equal to the
Ductile crack propagation by plastic collapse
89 1
2
1− χ
0.8
J/ ( σ0 X0 )
1.5
1
0.4
f0 = f0 = f0 = f0 =
0.2 0 -15
0.5
0
10
20
30
40
50
Δa/ X0 Fig. 12 Crack resistance curves for several values of the initial porosity (cp-0°)
f0 = f0 = f0 = f0 =
1
0.8
0.00087 0.0035 0.014 0.056
χ
0.6
0.4
0.2
0 0
0.2
0.4
0.6
0.8
1
1.2
-10
0.00087 0.0035 0.014 0.056 -5
0
5
10
15
(x − Δa)/ X0
f0 = 0.014 f0 = 0.056
f0 = 0.00087 f0 = 0.0035 0
0.6
1.4
1.6
J/( σ0 X0 ) Fig. 13 Relative width of the ligament between the initial crack tip and the first void (cp-0°)
initial void radius which is larger for higher porosities. In the numerical model the elements at the surface of the respective ligament distort so that χ can hardly be evaluated reliably anymore in the final stages χ 0.1. Figure 14 shows the reduction of the relative width χ of the ligaments in the crack plane around the current crack tip at steady-state crack propagation. In contrast to the initial stage the influence of the initial porosity f 0 is only moderate. A distinct void by void mechanism cannot be identified anymore even for small porosities as already noted by Tvergaard (2007).
Fig. 14 Profile of the relative reduction of the width of the intervoid ligaments around the current crack tip in the steadystate (cp-0°)
4.4 Void arrangement Deformed configurations for other void arrangements than cubic cp-0° are depicted in Fig. 15. If the void next to the initial crack plane lies 30° inclined to the latter a diffuse damage zone is observed whose width decreases with ongoing deformation, Fig. 15b. In the steady-state mainly the voids in the crack plane and the nearest ones next to the crack plane grow, leaving a diffuse zone of grown voids and remanent bridges between the crack plane voids in the wake behind the currently active softening zone. Due to the left bridges of width W rem definition (5) of effective crack growth cannot be applied reasonably but is modified for the hex-30° configuration as √ n lig 1 − χi 3 X0 . (6) a = 2 1 − Wirem /W0 i=1
Thereby, ligaments i are counted √ between the voids in the crack plane and the factor 3/2 accounts for the geometry in the hex-30° arrangement. For the hex-0° arrangement mainly the voids in the crack plane grow as with cp-0° and a deformed configuration is not shown here. The obtained effective crack growth resistance curves are depicted in Fig. 16. The R-curves computed for cp-0° and hex-0° are similar and the reached values of the limit load toughness differ by about 20 %. This plausible difference can partly be attributed to the slightly wider initial width of the intervoid ligaments for the hexagonal configuration due to the higher packing density for the same porosity. For the hex-30°-arrangement a considerable higher toughness is obtained than for cp-0° and hex-0° due to
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Fig. 15 Deformed configuration for (a) cp-45°, J = 3.5 σ0 X 0 and (b) for hex-30° in the wake behind the current crack tip ( f 0 = 1.4 %)
(a)
W1rem
W2rem . . .
(b)
cp-45°
the currently active softening zone in the 45° inclined branch projected onto the crack plane as in (Needleman and Tvergaard 1987). The dimensions of the branching zone scale approximately with the size of the plastic zone. This linear characteristic is indicated as an arrow in Fig. 16.
hex-30°
J/ ( σ0 X0 )
2
hex-0° 1
cp-0°
5 Discussion 0
0
5
10
15
20
25
30
5.1 Micro-mechanical models of ductile crack initiation and propagation
Δa/ X0 Fig. 16 Crack resistance curves for several void arrangements ( f 0 = 0.014)
the large plastic deformations in the wider and diffuse damage zone. In addition, the formation of the diffuse damage zone leads to a more distinct and almost linear tearing region. With the cp-45° configuration the crack branches completely, see Fig. 15a. In this case the FE-mesh had to be refined around affected voids, too. (This applies to the second row of voids with hex-30°as well). For the cp-45°-arrangement none of the definitions (5) or (6) of the current crack length could be applied. The current crack tip was extracted manually as the center of
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Models with discrete voids have been employed in numerous investigations documented in the literature. Most of them focus on a point of damage initiation in the first ligament. In many cases (Aravas and McMeeking 1985a; Baaser and Gross 2003; Dahlberg 2004; Gao et al. 2005; Hom and McMeeking 1989; Mostafavi et al. 2011) a post-processing criterion was applied to obtain this point defined as critical. Tvergaard and Hutchinson (Tvergaard 2007; Tvergaard and Hutchinson 2002), Kim et al. (2003) and Petti and Dodds (2005) employed a node-release technique to trace the crack growth after having fulfilled such a criterion. Aravas and McMeeking (1985b) and Aoki et al. (1984) described the matrix material with the GTN
Ductile crack propagation by plastic collapse
model to compute the damage by a second population of voids directly but pointed out the inability of a classical material model to describe final failure. Tian et al. (2010) modeled the secondary void growth by means of a micromorphic damage approach. However, the present results show that material degradation or decohesion is not inevitably necessary for crack propagation. Merely, the plastic collapse, i.e. the geometrical softening of the intervoid ligaments, is sufficient to induce a shifting of the active plastic and softening zones effectively corresponding to crack propagation. In this context the effective crack tip was defined as center of the currently active softening zone. Gu (2000) and Chew et al. (2007) employed models with discrete voids in front of the crack tip and defined the effective crack tip as a particular point of the active softening zone as well. If enough discrete voids along the crack plane are incorporated a steady-state is reached where the softening zone shifts congruently and the exact definition of the current effective crack tip has no significance anymore. In the present study it was assumed that voids are present not only at the crack tip but everywhere in the material. This is reasonable and consistent if voids are distributed uniformly in the material. This concerns the number of discrete voids to be incorporated in front of the crack tip and around the crack plane. It was found that considerable more voids have to be resolved in front of the crack tip than done in most preceding studies (Aoki et al. 1984; Aravas and McMeeking 1985a; Hom and McMeeking 1989; Kim et al. 2003; Tvergaard and Hutchinson 2002). If enough voids are included along the crack plane, a limit load exists corresponding to a maximum fracture toughness. The additional voids around the crack plane are responsible for a strong shielding effect. It was already found by Gao et al. (2005) that a second layer (in the half-model) of voids decelerates the growth of the voids in the crack plane. Pardoen and Hutchinson (2003) employed a damage cell model and report on a higher computed crack growth resistance if void growth around the crack plane is taken into account. Another point concerns the material degradation in the intervoid ligaments. Of course, the assumption made in the present study that the intervoid matrix material can undergo arbitrary large deformations is not realistic. Decohesion or other secondary damage mechanisms emerge in the intervoid ligaments of real materials. However, the limit load toughness computed
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in the present study should be the upper limit of all models with the same void arrangement, pre-damage plastic properties and consistently enough voids in front of the crack tip. This limit can be attained if the degradation of the matrix does not occur until the zone of active geometrical softening already migrated further. In this case it is hardly reasonable to identify the point of crack initiation with beginning decohesion. The extent of crack growth and thus the corresponding R-curve defined in this way was very sensitive to small disturbances in the decohesion behavior whereas such disturbances do not affect the effective crack growth and the effective R-curve. Another problem occurs in models with voids in front of the crack tip due to effective crack growth preceding decohesion. In (Kim et al. 2003; Petti and Dodds 2005; Tvergaard 2007; Tvergaard and Hutchinson 2002) the nodes of one micro-ligament in the crack plane were released when the relative ligament width χ reached a critical value χc . Then, the released ligament was evaluated as a step of crack growth. As discussed above if this value χc is small enough and enough discrete voids along the crack plane are incorporated the computed steady-state toughness Jss should coincide with the limit load toughness Jlim of the plastic collapse solution. In the latter the width of the first ligament and thus χc decreases asymptotically towards zero. The problem is that the number of discretely resolved voids along the crack plane is limited in the numerical implementation of the model. Inevitably, the active softening zone finally piles up at the end of the region of discretely resolved voids as described in Sect. 4.2. At this point the first micro-ligament has a finite width depending on the number of discrete voids. So if the critical relative width χc is chosen below the value at pile-up it happens that, although enough discrete voids may have been incorporated to observe an effective crack growth by plastic collapse, the computed toughness is already in the pile-up region. Therefore, this toughness value has to be considered as a numerical artifact and is too high. In Fig. 17 the limit load toughness values Jlim computed in the present study are plotted against the ratio of initial void radius R0 to distance X 0 (adopted from McMeeking 1977) together with results from literature. If a crack growth resistance curve reaching a steadystate was given in the literature then the steady-state crack growth resistance Jss is plotted whereas in other cases the published initiation toughness Jc is depicted.
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present study Jlim (cyl., cp-0°) present study Jlim (cyl., hex-30°) present study Jlim (cyl., 1 layer, Mises) Tvergaard&Hutchinson (2002) Jss (cyl.) Tvergaard (2007) Jss (cyl.) Petti&Dodds (2005) Jss (cyl.) Aoki et al. (1984) Jc (cyl.) Rice&Johnson (1970) Jc (sph.) Kim et al. (2003) Jc (sph.) Gu (2000) Jc (cyl.)
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Fig. 17 Limit load toughness Jlim computed in the present study in comparison to steady-state toughness Jss and crack initiation toughness Jc values from literature for models with cylindrical (cyl.) and spherical (sph.) voids in front of the crack tip
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Our results for a single layer of 24 up to 40 voids embedded in isochorically yielding material (Misesplasticity) are shown, too. The toughness values given in (Aravas and McMeeking 1985a; Hom and McMeeking 1989) were obtained by extrapolation of the initial stage of void growth and are not included here. In many studies (Gao et al. 2005; Gu 2000; Kim et al. 2003; Tvergaard and Hutchinson 2002) not enough discrete voids were resolved so that the results are influenced by pile-up effects. Correspondingly, the computed values of the crack growth, respectively initiation toughness are too high. Petti and Dodds (2005) incorporated 30 cylindrical voids but slightly different material parameters and the trend of their results coincides with that of the present study. The semianalytical model of Rice and Johnson (1970) for growth of a single spherical void leads to a similar trend as well. Tvergaard (2007) resolved 15 voids which could allow for effective crack propagation by plastic collapse with the corresponding limit load toughness. However, Tvergaard (2007) employed a node-release per ligament after a comparatively low critical relative ligament width χc was reached, i.e. after the ligaments underwent large straining. As discussed above, this criterion might be too strict and reached not until the (almost) steady-state active softening zone piles up at the end of the zone of discrete voids. This presumption is substantiated by comparing with own results for a single row of voids. In this context it has to be recalled
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that in the present study the initial radius was set equal to the initial radius of the voids whereas in most of the cited studies a smaller value was chosen. However, this difference should have an effect only in the initial stage of void growth but not on the steady-state regime. Comparing the collapse results for a single row of voids embedded in Mises-material to those of the consistent model (enough voids, GTN around) in Fig. 17 shows that the relative difference becomes smaller for decreasing values of the initial porosity. This is clear as the difference between isochoric and the consistently homogenized dilatational plastic behavior reduces with decreasing porosity. The results of the semi-analytical model of Rice and Johnson (1970) for a spherical void show the same tendency as the present results. Regarding the void arrangement a large difference is observed between the toughness for the cp-0° (and similar values for hex-0°) and the hex-30° configuration. Even highly dissipative complete crack branching was observed for the cubic primitive arrangement with principal axes inclined 45° to the crack plane. In more realistic models with stochastic void arrangements such a branching would of coarse be limited. Nevertheless, the crack propagation toughness would presumably be considerably higher than for the outstanding cp-0° and hex-0° models. Possibly, the hex-30° configuration gives a better approximation of the behavior of models with stochastically distributed voids.
Ductile crack propagation by plastic collapse
Effects of a diffuse damage zone and crack bifurcation have been already reported by Tvergaard and Needleman (2006) for a model with different arrangements of islands of nucleating voids around the crack tip. 5.2 Comparison with experiments In the present study it is assumed that the intervoid material can undergo arbitrarily large deformations. This allows for an effective crack growth by plastic collapse of the intervoid ligaments. As discussed in the introduction, the sustainable deformation of real materials is limited due to secondary damage mechanism such as nucleation and growth of secondary voids (see Fig. 1) or the formation of shear bands. Thus, the question arises whether plastic collapse of the intervoid ligaments is the relevant mechanism for the ductile fracture of real materials. Secondary damage mechanisms lead to a localization zone within the intervoid ligaments whose size is related to a characteristic length such as the size of secondary particles or the width of shear bands. That means, if such a mechanism were relevant, the crack growth resistance would not be proportional to the void spacing X 0 as obtained in the present study but it would depend on the mentioned characteristic length as well.1 This possible size effect is checked for ductile cast iron now. In this material the void volume fraction of the nearly spherical graphite particles can be determined accurately. In addition, these particles debond from the matrix after relatively small deformations and thus can be modeled as initial voids reasonably. For the ductile cast iron EN-GJS-400 with a graphite volume fraction of 12–13 % a lot of data are available in the literature for a relatively broad range of graphite spacings (although the raw data of the R-curves are published seldom). Following Salzbrenner (1987) we define the mean void spacing as 1 X0 = √ (7) 2 NA where N A is the mean number of voids per area in an arbitrary cross section. In order to exclude effects of the 1
The R-curve would be proportionally to X 0 as well if the secondary damage mechanism were relevant but its characteristic length were negligible compared to X 0 . But this is not the case in typical engineering metals where secondary particles or shear bands have dimensions in the range of a few microns.
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particular evaluation method we determined N A from the published micrographs. The normalized crack-growth resistance curves for the data from literature at room temperature are plotted in Fig. 18. The 0.2 % offset yield point is taken as initial yield stress σ0 . Baer and Pusch (Baer 1996; Baer and Pusch 1995) published fitted R-curves which are plotted only within the range between the exclusion lines used for fitting. Firstly, the data in Fig. 18 lie in a band indicating that secondary damage mechanisms play a minor role and that the assumptions underlying the present study seem to be reasonable for this material at room temperature. Dong et al. (1997) focused on later stages of crack growth so that their results fluctuate considerably around the band formed by the other data. In the initiation region a 4 X 0 there is a slight trend towards higher normalized toughness values for lower mean distances X 0 . However, the particular distributions of dimensions, spacing and shape parameters of the graphite particles were not evaluated and it is unknown to which extent these distributions influence the crack growth resistance. Thus, no substantiated statement about the origin of the slight deviations for lower mean distances X 0 of the voids can be established. Possibly, strain gradient hardening effects already occur in this region. If the experimental R-curves shall be related to those computed in the present study it has to be noticed first that the current crack length is determined by partial unloading or potential drop techniques in the experiments giving an effective measure of crack length comparable to the one employed in the present study. Qualitatively, the experimental and computed R-curves in Fig. 18, respectively, Figs. 12 and 16 exhibit a similar tearing behavior. However, the measured curves do not show a steady-state region. All experiments except those of Dong et al. (1997) were performed on specimens with a ligament W − a ≈ 10 mm so that this region is already influenced by interactions of the plastic zone with the specimen geometry which are not accounted for in the present study. Although the order of magnitude is comparable, the computed R-curves lie at a lower level than the experimental ones. However, it could not be expected that a model with cylindrical voids leads to quantitatively realistic predictions. But it is well known that spherical voids lead to a higher toughness compared to models with cylindrical voids for the same
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Fig. 18 Normalized crack growth resistance curves of ductile cast iron EN-GJS-400 at room temperature
void volume fraction (Hom and McMeeking 1989; Kim et al. 2003). In addition, it is an open point how the mean void spacing X 0 defined by Eq. (7) of a stochastic void distribution compares directly to the spacing X 0 in the regular arrangement used in the model. Furthermore, the hardening exponent of ductile cast iron EN-GJS-400 is at least 50 % higher than used in the present study. If this stronger hardening were accounted for in the model, the predicted crack growth resistance would be higher again. In Sect. 4.4 a diffuse damage zone or even crack branching was observed for some void arrangements. Comparing the micrograph in Fig. 19 with Fig. 15 this seems to be a realistic behavior for a material with higher porosity and a stochastic void arrangement such as the considered ductile cast iron.
6 Summary and conclusions In the present paper a micro-mechanical finite element model for crack propagation in ductile materials is presented. A variety of regularly arranged cylindrical discrete voids is included at the crack tip within an elastic-plastic matrix material. Consistently, the Gurson-law is employed for the homogenized, dilatational material behavior outside the region of discrete voids.
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In this region no softening occurs such that the boundary-value problem is well-posed. The limit case of small-scale yielding is addressed by means of a boundary layer model in order to exclude possible influences of a specimen geometry. No damage mechanism of the matrix material is incorporated in the presented model. It turns out that the plastic collapse, i.e. the geometrical softening, of the intervoid ligaments is sufficient to induce the movement of the currently active plastic zone corresponding to effective crack growth. A measure for the position of the effective crack tip is introduced allowing to extract crack growth resistance curves. It is found that a limit load exists if enough voids are resolved discretely ahead of the crack tip. This limit load corresponds to the fracture toughness of the material. The limit load toughness should be the upper limit of fracture toughness values of all models incorporating further damage mechanisms of the intervoid ligaments such as secondary void growth or decohesion (if void arrangement and pre-damage plastic properties are the same). If secondary damage effects are taken into account, effective crack propagation can still precede physical material separation so that the latter need not always be a useful criterion of crack initiation. A convergence study is performed with respect to the number of layers of discrete voids to be incorporated. It reveals that the void interaction has a consid-
Ductile crack propagation by plastic collapse
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Fig. 19 Deformations around the crack tip in EN-GJS-400 (ref. [34] in Zybell et al. 2012)
erable influence within a large part of the plastic zone and that the homogenized material model captures the material behavior adequately only in the outer region of the plastic zone. The shielding of the voids directly at the crack tip by the surrounding voids leads to a considerably higher fracture toughness compared to models neglecting this effect. Several regular void arrangements are investigated. The formation of diffuse damage zones or even crack branching is observed for some configurations. A comparison with experimental data from literature indicates that plastic collapse of the intervoid ligaments can be the main mechanism of ductile crack propagation. Also the occurrence of diffuse damage zones or crack branching is in agreement with experimental observations. In future work more realistic spherical and stochastically aligned voids should be used in a consistent model to investigate whether a quantitative agreement between experiments and predictions is possible. Further microscopic effects such as dislocation interactions or secondary damage mechanisms may be relevant for some materials and can be included in the model as in (Tian et al. 2010; Tvergaard and Niordson 2008) in order to study their competition with the mechanism of plastic collapse of the intervoid ligaments. Acknowledgments The financial support of this investigation by the Deutsche Forschungsgemeinschaft (German Science Foundation) under contracts KU 929/13-2 and KU 929/14-1 is gratefully acknowledged. Furthermore, the authors thank the student A. Burgold for his commitment in performing finite element computations for the present study.
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