- Email: [email protected]
Generalized Cohesive Zone Model: incorporating triaxiality dependent failure mechanisms
Computational Materials Science 16 (1999) 267±274
Generalized Cohesive Zone Model: incorporating triaxiality dependent failure mechanisms K. Keller a,*, S. Weihe a, T. Siegmund b, B. Kr oplin a a b
Institute for Statics and Dynamics of Aerospace Structures, University of Stuttgart, Pfaenwaldring 27, D-70569 Stuttgart, Germany School of Mechanical Engineering, Purdue University, 1288 Mechanical Engineering Building, West Lafayette, IN 47907-1288, USA
Abstract The present work has been inspired by a presentation given in the preceding conference of this series (T. Siegmund, W. Brocks, Int. J. Fracture, in print). In this study, a modi®ed Gurson Model has been adopted as a reference solution and the response of classic Cohesive Zone Models (CZM) has been evaluated. It has been shown that the conventional CZM in general is not able to predict the in¯uence of the triaxiality on the failure initiation, and that it is not possible to reproduce the expected reference behaviour with a single set of calibration parameters. In the presented framework of modelling, the feature of mode I failure and its transition to mixed mode failure is incorporated within a Generalized Cohesive Zone Model (GCZM; S. Weihe, Dissertation, 1995). Fracture is initiated by a strength criterion while progressive material degradation is based on energy criteria in analogy to Fracture Mechanics. Complete separation due to fracture is obtained when the critical fracture toughness Gcf has been dissipated, where the actual value of Gcf is dependent on the predicted failure mode. It is shown that the transition to mixed mode failure allows the GCZM to reproduce the varying contributions of modes I and II over the triaxiality regime realistically. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Smeared crack; Cohesive zone; Triaxiality dependence; Mixed mode
1. Introduction and overview Due to the nature of engineering materials and the applied loads, degradation and failure behaviour diers signi®cantly. Since a tendency to mode I failure (e.g., represented by the Rankine criterion) is observed for brittle materials under normal stress domination, the importance of mixed mode failure and slip in the fracture plane increases with increasing ductility.
* Corresponding author. Tel.: +49-0711/685-3612; fax: +490711/685-3706.
For the case of brittle fracture, a Cohesive Zone Model (CZM), ®rst proposed by Barenblatt [1] was established to describe material degradation and separation in a process zone in front of a crack tip. The constitutive description of the surrounding material diers from that within the process zone and was introduced to avoid singular stresses around the crack as compared to linear-elastic fracture mechanics. The application of the CZM to ductile fracture was introduced by Tvergaard and Hutchinson [7]. Recently, a comparative approach to the more physically based Gurson Model was given in [5,6]. The presented Generalized Cohesive Zone Model (GCZM) describes the discontinuity of the
0927-0256/99/$ - see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 9 9 ) 0 0 0 6 9 - 5
268
K. Keller et al. / Computational Materials Science 16 (1999) 267±274
strain or displacement rates in a continuum representation. The displacement jump is homogenized over the crack band width h. The feature of mixed mode failure is incorporated in this environment, and therefore, the term ``generalized'' is introduced. It is also known as an Adaptive Fixed Crack Model. Material parameters should by de®nition be constant over the whole loading history. The objective of this investigation is to observe whether the material parameters remain valid over the triaxiality regime. The solution to which it is compared is in this case given by the modi®ed Gurson relation since the traction separation response of this model diers from Gurson's solution. The models are calibrated to yield the same energy dissipation until a total loss of load carrying capacity, i.e., total decohesion or degradation, is obtained. Cohesive strength, cohesive energy and characteristic length are the interesting parameters for this investigation in the CZM [5]. In the framework of the GCZM, both fracture toughnesses GIf and GII f , for modes I and II failure, have to be calibrated. Consistent interaction of both modes is provided in the GCZM. In the following sections, a brief description of the underlying theory of the GCZM is given, and the study about the CZM [5] is shortly introduced to compare both subsequently. 2. Generalized Cohesive Zone Model Fracture is of physical and geometrical nature. In this model the physical part contains material degradation via reducing the strength at the microlevel, i.e., in the plane of degradation (POD) as de®ned in [8], by a softening mechanism. The geometrical counterpart includes orientational preference of the localized discontinuity which is homogenized over the crack band width, and the relation to the three-dimensional continuum is accounted for by embedding this planar degraded subspace by the geometrical transformation. The crack initiation is controlled by a strength criterion, while the failure propagation is guided by the dissipated energy.
2.1. Geometrical transformation The conventional tensile and shear strengths do not necessarily coincide with the tractions on the relevant subspace which is examined for crack initiation. The transformation between the two scales is used to determine the initial material strength on the POD from conventional experimental data [2]. The transformation between the global continuum level and the relevant physical subspace of the POD can be followed over static and/or kinematic quantities (cf. Fig. 1). However, only one of these constraints can be ful®lled in a strong sense if a nonlinear constitutive relationship is applied. Static constraint : rij Tim Tjn rmn ; Kinematic constraint : eij Tim Tjn emn :
1 2
Herein the Cauchy transformation is formulated in dependence of the direction cosines Tij of the normal to the POD. The normal (index n) and shear (index t) components of the tractions qi as well as the displacements ui across the POD are given by: stat: : kin: :
T
T
3
T
T
4
r11 ; r12 ; r13 ; qn ; qt2 ; qt3 un ; ut2 ; ut3 he11 ; e12 ; e13 :
The displacements are smeared over the crack band width h. In an elasto-plastic formulation, the total strain is split up into an elastic part due to the undamaged continuum and a nonlinear, plastic part activated during failure. The static constraint is ful®led in a strong sense, since the stress state is a function of the elastic strains only. 2.2. Crack initiation The decisive strength criterion for crack initiation consists of an interaction hypothesis of the applied tractions. Its shape is of hyperbolic form. If the fracture criterion F is violated (i.e., F becomes positive), a crack is initiated and a POD activated. The stress is relaxed until the stress state becomes admissible, i.e., until F 0. The relaxation rule used is nonassociated.
K. Keller et al. / Computational Materials Science 16 (1999) 267±274
269
Fig. 1. Transformation between structural ``continuum'' level and physical subspace.
ÿ
F ÿ qn ÿ qn;f ÿ a
2
q2t a2 ; tan2 /
5
calibrating parameters are the current tensile strength qn;f and the asperity parameter a, which is related to the current shear strength qt;f of the POD and the angle of internal friction / (cf. Fig. 2) q 6 qt;f tan / q2n;f 2aqn;f :
2.3. Material degradation The evolution of damage is energy controlled by the fracture toughnesses for modes I and II, namely GIf and GII f , respectively. The material is degraded in the sense that the material strength of the virgin material is decreased corresponding to the active failure modes. The fracture toughnesses are de®ned as follows:
GIf
Z
GII f
1
ucr n 0
Z
qn ducr n
1 ucr t 0
; qt 0
ducr >0
n ! cr qt ÿ qt;r du t
:
7
qn 0
This nonlinear evolution of the material degradation is realized analogous to classical elasto-plasticity following a nonassociated ¯ow rule [4,8]. The dissipated energy for each failure mode is characterized by two normalized internal state variables n: ( ducr 1=GIf qn ducr n ; n P 0; I dn 0; ducr n < 0; 8 1 ; dnII II jqt j ÿ qt;r ducr t Gf where the residual shear strength qt;r is subtracted to gain only these contributions apart from pure friction, 0; qn P 0; 9 qt;r ÿqn tan /; qn < 0: In consequence, the material strengths and the asperity parameter can be determined in a straightforward way during the whole load history according to the equations depending on the individual state variables j: qn;f 1 ÿ jn qn;f 0 ;
a 1 ÿ ja a0 ;
10
where
ÿ jn min 1; nI nII ; Fig. 2. Failure criterion.
ja nII :
11
The residual forms of the fracture criterion are found in Fig. 2 as dashed curves. For pure mode I
270
K. Keller et al. / Computational Materials Science 16 (1999) 267±274
failure, the hyperbolic shape is simply shifted to the origin, which means that the current tensile strength has decreased to zero. In the opposite case of pure mode II failure, the residual form is de®ned by the Mohr±Coulomb criterion of pure frictional contact. 2.4. Crack orientation The orientation of the most critical plane on which the tractions reach the limit state ®rst is found analytically following Mohr's approach [2,9]. When the Mohr circle becomes tangential to the fracture criterion, the critical POD has been found, and a ®rst crack at an angle acrit is introduced. The angle acrit is de®ned as the orientation of the POD normal to the direction of the major principal stress in the r1 ±r3 -plane, cos 2acrit:
f
r1 r3 p rh 3 : fcrit r1 ÿ r3 reff
13
with the hydrostatic stress rh and the von Mises stress reff . Possible ranges are ÿ1
ÿ1
0
+1
1
Hydrostatic compression
Uniaxial compression r2 0
Pure shear r1 ÿ r3 plane)
Uniaxial tension r2 0
Hydrostatic tension
And the relative shear strength, given by the ratio of material shear to tensile strength, c0 :
qt;f : qn;f
14
ÿ p Af tan2 / ÿ c20 tan2 / 1 tan2 / p ; f c20 tan2 / 1 tan2 / ÿ A 1 tan2 / q ÿ 2 12 A 4c20 ÿ 1 ÿ f2 c20 ÿ tan2 / ;
In the subsequent load steps, new cracks can emerge ± the PODs are degrading independently of each other. The crack orientation remains ®xed during further degradation.
with the load type parameter or stress triaxiality f and the relative shear strength c0 (Fig. 3). The load type parameter or stress triaxiality is introduced as
2.5. Fracture mode Failure under mode I is given by acrit 0 . Even in the above-mentioned case of brittle materials and crack initiation under mode I, mixed mode failure can appear during the relaxation process and stress redistribution during the following load history. Shear stresses can occur and increase. The possible crack orientations are given by 0 6 a 6 12 p2 ÿ / :
Fig. 3. Analytical determination of the orientation of the introduced POD.
15
If the relative shear strength c0 is smaller for another material or experiment, the probability of mode II failure increases and a tendency to more ductile behaviour can be observed (cf. Fig. 4). Under pure compressive loads, no failure is assumed within this framework. If the load-type parameter or stress triaxiality is higher, the mode I failure becomes more dominant and the impor-
K. Keller et al. / Computational Materials Science 16 (1999) 267±274
271
Fig. 4. Correlation between fracture mode, stress state and material parameters.
tance of mixed mode failure increases with decreasing triaxiality.
3. Cohesive Zone Model More recently, a lot of applications and modi®cations have been introduced in the CZM [5]. A comparative approach to a more physically based model, the modi®ed Gurson Model [3], which in the constitutive equations refers to initial void volume fraction, void growth, nucleation and coalescence, was given in [5]. In the investigation of Siegmund and Brocks [5], a CZM was applied to the prediction of ductile crack growth. The parameters cohesive energy C, cohesive strength rmax and characteristic length d were observed over a triaxiality regime for 1:7 6 f 6 8:7. In general, the cohesive zone material parameters are introduced to be independent of all other speci®cations and processes. With respect to ductile fracture, the triaxiality at the crack tip is of signi®cant interest. The cohesive energy C Z usep xx rxx dux 16 C 0
is dissipated during the failure process until complete loss of load carrying capacity, i.e., total separation, is obtained. The cohesive strength is the maximum stress reached over this decohesion process, while the characteristic length of the CZM
represents the crack tip opening displacement (CTOD) at which the main portion of the cohesive energy is already spent. Only two of these three parameters are independent of each other, since C c rmax d
17
with c as integration constant. The material separation occurs under pure mode I and the crack path is prede®ned. The results have been compared to the modi®ed Gurson Model to observe whether the variation of the material parameters are `true' parameters, i.e., stay constant. It was concluded that the material parameters remain constant only over a small range of triaxiality. It has been concluded in [5] that improved CZMs should incorporate a nonlocal approach in order to possibly overcome this problem. For more detailed information the interested reader is referred to [5].
4. Comparison of the GCZM to the CZM Incorporating the described theory, this investigation is made on the material point level. At this material point, multiple simultaneously active cracks at dierent inclinations can be introduced or be selectively deactivated by the relaxation process between the elastic predictor and plastic corrector.
272
K. Keller et al. / Computational Materials Science 16 (1999) 267±274
The applied stress triaxiality regime is 1:7 6 f 6 8:7, equivalent to that in the comparative study [5].
gime. In the late softening range, a kink appears conjugated to void coalescence. There is a suplementary traction-separation curve plotted for uniaxial straining (uyy 0; y L).
4.1. Traction-separation response The traction-separation response of the GCZM is dierent from that predicted by the modi®ed Gurson Model as shown in Fig. 5. This may also be due to the dierent criteria for crack initiation. According to the Gurson model, crack initiation occurs when the integrated internal variable for the void volume fraction f reaches the critical value fc . The criterion for crack initiation in the GCZM is based on the actual stress state. The two ®gures illustrate the traction-separation response of the models used. On the left, the CZM with the traction-separation response de®ned by the Gurson relation and on the right for the GCZM. The quantities on the abscissa are equivalent, in the case of the CZM the crack tip opening displacement uxx is made dimensionless by the element length D. The increasing ductility with decreased triaxiality behaviour of the GCZM results from increasing dissipation during the strain softening which is a consequence of an increased activation of mode II failure. The modi®ed Gurson traction-separation relation for the CZM provides ®rst a hardening re-
5. Comparison: behaviour over the triaxiality regime For this study (cf. [5]), the Gurson Model has been accepted as the reference solution. Fig. 6 displays the material parameters of the CZM and the GCZM that are necessary in order to dissipate the same energy during fracture than Gurson's Model. It is seen that the fracture toughness GII f for mode II failure remains constant over the whole triaxiality regime, whereas GIf must be adopted slightly to dissipate the correct amount of cohesive energy. In comparison, the cohesive energy of the CZM has a variation which is four times as high as the one for the GCZM. Referring to Fig. 7, the reason for this superior behaviour of the GCZM over the CZM is explained. Fig. 7 shows the contribution in percent of the energy dissipated by mode II failure in relation to the total dissipated energy through the whole degradation process. As the triaxiality is decreasing, mode II failure gets activated, and thus, the capability of representing mixed mode failure in the GCZM is essential.
Fig. 5. Cohesive (left) and Generalized Cohesive Zone Model (right).
K. Keller et al. / Computational Materials Science 16 (1999) 267±274
Fig. 6. Parameter variation for the CZM and GCZM.
273
gated. In this study it was focussed on the fracture toughnesses GIf and GII f , for modes I and II failure correspondingly. The results have been compared to the results of the CZM, given in [5], in which attention was paid to the cohesive energy C, cohesive strength rmax and characteristic length d as cohesive zone material parameters. The transition from brittle mode I failure to a `quasi-ductile' mode II frictional slip is one of the key enhancements of the GCZM when compared to the classic CZM. For a single, optimized set of both fracture toughnesses GIf and GII f , it is demonstrated that the varying contributions of the dierent failure modes can be predicted realistically and closely to the predictions of Gurson's Model. Acknowledgements The presented work is funded by the German Research Council within the Graduate Collegium ``Internal Interfaces in Crystalline Materials'', which is gratefully acknowledged. References
Fig. 7. Mode II energy contribution.
6. Summary and conclusion The GCZM [8] has been shortly outlined. Crack initiation controlled by a strength criterion in the form of an interacting tractions hypothesis and material degradation depending on the independent failure modes controlled by the dissipated energy has been introduced. The crack orientation is determined by Mohr's approach. The investigation of Siegmund and Brocks [5], which compared the predictions of a CZM to these of the modi®ed Gurson Model, was brie¯y described. The dependence of the material parameters on the triaxiality of the stress state has been investi-
[1] G.I. Barenblatt, The mathematical theory of equilibrium cracks in brittle fracture, Adv. Appl. Mech. 7 (1962) 55± 129. [2] B. Kr oplin, S. Weihe. Constitutive and geometrical aspects of fracture induced anisotropy, in: Owen et al. (Ed.), Int. Conf. on Computational Plasticity, COMPLAS 5, Barcelona (E), 1997, pp. 255±279. [3] A. Needleman, V. Tvergaard, An analysis of ductile rupture in notched bars, J. Mech. Phys. Solids 32 (1984) 461±490. [4] F. Ohmenh auser, S. Weihe, B. Kr oplin, Algorithmic implementation of a Generalized Cohesive Crack Model, Comput. Mater. Sci., 1998. [5] T. Siegmund, W. Brocks, Prediction of the work of separation in ductile fracture and implications to the modeling of ductile fracture, Int. J. Fracture, in print. [6] T. Siegmund, W. Brocks, The role of cohesive strength and separation energy for modeling of ductile fracture, in: K.L. Jerina, P.C. Paris (Eds.), Fatigue and Fracture Mechanics, vol. 30, in print. [7] V. Tvergaard, J.W. Hutchinson, The relation between crack growth resistance and fracture process parameters in elastic±plastic solids, J. Mech. Phys. Solids 40 (6) (1992) 1377±1397.
274
K. Keller et al. / Computational Materials Science 16 (1999) 267±274
[8] S. Weihe, Modelle der ®ktiven Riûbildung zur Berechnung der Initiierung und Ausbreitung von Rissen: Ein Ansatz zur Klassi®zierung, Dissertation, Universit at Stuttgart (D), 1995.
[9] S. Weihe, B. Kr oplin, R. de Borst, Classi®cation of smeared crack models based on material and structural properties, Int. J. Solids Structures 35 (23) (1997) 1289±1308.
Generalized Cohesive Zone Model: incorporating triaxiality dependent failure mechanisms K. Keller a,*, S. Weihe a, T. Siegmund b, B. Kr oplin a a b
Institute for Statics and Dynamics of Aerospace Structures, University of Stuttgart, Pfaenwaldring 27, D-70569 Stuttgart, Germany School of Mechanical Engineering, Purdue University, 1288 Mechanical Engineering Building, West Lafayette, IN 47907-1288, USA
Abstract The present work has been inspired by a presentation given in the preceding conference of this series (T. Siegmund, W. Brocks, Int. J. Fracture, in print). In this study, a modi®ed Gurson Model has been adopted as a reference solution and the response of classic Cohesive Zone Models (CZM) has been evaluated. It has been shown that the conventional CZM in general is not able to predict the in¯uence of the triaxiality on the failure initiation, and that it is not possible to reproduce the expected reference behaviour with a single set of calibration parameters. In the presented framework of modelling, the feature of mode I failure and its transition to mixed mode failure is incorporated within a Generalized Cohesive Zone Model (GCZM; S. Weihe, Dissertation, 1995). Fracture is initiated by a strength criterion while progressive material degradation is based on energy criteria in analogy to Fracture Mechanics. Complete separation due to fracture is obtained when the critical fracture toughness Gcf has been dissipated, where the actual value of Gcf is dependent on the predicted failure mode. It is shown that the transition to mixed mode failure allows the GCZM to reproduce the varying contributions of modes I and II over the triaxiality regime realistically. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Smeared crack; Cohesive zone; Triaxiality dependence; Mixed mode
1. Introduction and overview Due to the nature of engineering materials and the applied loads, degradation and failure behaviour diers signi®cantly. Since a tendency to mode I failure (e.g., represented by the Rankine criterion) is observed for brittle materials under normal stress domination, the importance of mixed mode failure and slip in the fracture plane increases with increasing ductility.
* Corresponding author. Tel.: +49-0711/685-3612; fax: +490711/685-3706.
For the case of brittle fracture, a Cohesive Zone Model (CZM), ®rst proposed by Barenblatt [1] was established to describe material degradation and separation in a process zone in front of a crack tip. The constitutive description of the surrounding material diers from that within the process zone and was introduced to avoid singular stresses around the crack as compared to linear-elastic fracture mechanics. The application of the CZM to ductile fracture was introduced by Tvergaard and Hutchinson [7]. Recently, a comparative approach to the more physically based Gurson Model was given in [5,6]. The presented Generalized Cohesive Zone Model (GCZM) describes the discontinuity of the
0927-0256/99/$ - see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 9 9 ) 0 0 0 6 9 - 5
268
K. Keller et al. / Computational Materials Science 16 (1999) 267±274
strain or displacement rates in a continuum representation. The displacement jump is homogenized over the crack band width h. The feature of mixed mode failure is incorporated in this environment, and therefore, the term ``generalized'' is introduced. It is also known as an Adaptive Fixed Crack Model. Material parameters should by de®nition be constant over the whole loading history. The objective of this investigation is to observe whether the material parameters remain valid over the triaxiality regime. The solution to which it is compared is in this case given by the modi®ed Gurson relation since the traction separation response of this model diers from Gurson's solution. The models are calibrated to yield the same energy dissipation until a total loss of load carrying capacity, i.e., total decohesion or degradation, is obtained. Cohesive strength, cohesive energy and characteristic length are the interesting parameters for this investigation in the CZM [5]. In the framework of the GCZM, both fracture toughnesses GIf and GII f , for modes I and II failure, have to be calibrated. Consistent interaction of both modes is provided in the GCZM. In the following sections, a brief description of the underlying theory of the GCZM is given, and the study about the CZM [5] is shortly introduced to compare both subsequently. 2. Generalized Cohesive Zone Model Fracture is of physical and geometrical nature. In this model the physical part contains material degradation via reducing the strength at the microlevel, i.e., in the plane of degradation (POD) as de®ned in [8], by a softening mechanism. The geometrical counterpart includes orientational preference of the localized discontinuity which is homogenized over the crack band width, and the relation to the three-dimensional continuum is accounted for by embedding this planar degraded subspace by the geometrical transformation. The crack initiation is controlled by a strength criterion, while the failure propagation is guided by the dissipated energy.
2.1. Geometrical transformation The conventional tensile and shear strengths do not necessarily coincide with the tractions on the relevant subspace which is examined for crack initiation. The transformation between the two scales is used to determine the initial material strength on the POD from conventional experimental data [2]. The transformation between the global continuum level and the relevant physical subspace of the POD can be followed over static and/or kinematic quantities (cf. Fig. 1). However, only one of these constraints can be ful®lled in a strong sense if a nonlinear constitutive relationship is applied. Static constraint : rij Tim Tjn rmn ; Kinematic constraint : eij Tim Tjn emn :
1 2
Herein the Cauchy transformation is formulated in dependence of the direction cosines Tij of the normal to the POD. The normal (index n) and shear (index t) components of the tractions qi as well as the displacements ui across the POD are given by: stat: : kin: :
T
T
3
T
T
4
r11 ; r12 ; r13 ; qn ; qt2 ; qt3 un ; ut2 ; ut3 he11 ; e12 ; e13 :
The displacements are smeared over the crack band width h. In an elasto-plastic formulation, the total strain is split up into an elastic part due to the undamaged continuum and a nonlinear, plastic part activated during failure. The static constraint is ful®led in a strong sense, since the stress state is a function of the elastic strains only. 2.2. Crack initiation The decisive strength criterion for crack initiation consists of an interaction hypothesis of the applied tractions. Its shape is of hyperbolic form. If the fracture criterion F is violated (i.e., F becomes positive), a crack is initiated and a POD activated. The stress is relaxed until the stress state becomes admissible, i.e., until F 0. The relaxation rule used is nonassociated.
K. Keller et al. / Computational Materials Science 16 (1999) 267±274
269
Fig. 1. Transformation between structural ``continuum'' level and physical subspace.
ÿ
F ÿ qn ÿ qn;f ÿ a
2
q2t a2 ; tan2 /
5
calibrating parameters are the current tensile strength qn;f and the asperity parameter a, which is related to the current shear strength qt;f of the POD and the angle of internal friction / (cf. Fig. 2) q 6 qt;f tan / q2n;f 2aqn;f :
2.3. Material degradation The evolution of damage is energy controlled by the fracture toughnesses for modes I and II, namely GIf and GII f , respectively. The material is degraded in the sense that the material strength of the virgin material is decreased corresponding to the active failure modes. The fracture toughnesses are de®ned as follows:
GIf
Z
GII f
1
ucr n 0
Z
qn ducr n
1 ucr t 0
; qt 0
ducr >0
n ! cr qt ÿ qt;r du t
:
7
qn 0
This nonlinear evolution of the material degradation is realized analogous to classical elasto-plasticity following a nonassociated ¯ow rule [4,8]. The dissipated energy for each failure mode is characterized by two normalized internal state variables n: ( ducr 1=GIf qn ducr n ; n P 0; I dn 0; ducr n < 0; 8 1 ; dnII II jqt j ÿ qt;r ducr t Gf where the residual shear strength qt;r is subtracted to gain only these contributions apart from pure friction, 0; qn P 0; 9 qt;r ÿqn tan /; qn < 0: In consequence, the material strengths and the asperity parameter can be determined in a straightforward way during the whole load history according to the equations depending on the individual state variables j: qn;f 1 ÿ jn qn;f 0 ;
a 1 ÿ ja a0 ;
10
where
ÿ jn min 1; nI nII ; Fig. 2. Failure criterion.
ja nII :
11
The residual forms of the fracture criterion are found in Fig. 2 as dashed curves. For pure mode I
270
K. Keller et al. / Computational Materials Science 16 (1999) 267±274
failure, the hyperbolic shape is simply shifted to the origin, which means that the current tensile strength has decreased to zero. In the opposite case of pure mode II failure, the residual form is de®ned by the Mohr±Coulomb criterion of pure frictional contact. 2.4. Crack orientation The orientation of the most critical plane on which the tractions reach the limit state ®rst is found analytically following Mohr's approach [2,9]. When the Mohr circle becomes tangential to the fracture criterion, the critical POD has been found, and a ®rst crack at an angle acrit is introduced. The angle acrit is de®ned as the orientation of the POD normal to the direction of the major principal stress in the r1 ±r3 -plane, cos 2acrit:
f
r1 r3 p rh 3 : fcrit r1 ÿ r3 reff
13
with the hydrostatic stress rh and the von Mises stress reff . Possible ranges are ÿ1
ÿ1
0
+1
1
Hydrostatic compression
Uniaxial compression r2 0
Pure shear r1 ÿ r3 plane)
Uniaxial tension r2 0
Hydrostatic tension
And the relative shear strength, given by the ratio of material shear to tensile strength, c0 :
qt;f : qn;f
14
ÿ p Af tan2 / ÿ c20 tan2 / 1 tan2 / p ; f c20 tan2 / 1 tan2 / ÿ A 1 tan2 / q ÿ 2 12 A 4c20 ÿ 1 ÿ f2 c20 ÿ tan2 / ;
In the subsequent load steps, new cracks can emerge ± the PODs are degrading independently of each other. The crack orientation remains ®xed during further degradation.
with the load type parameter or stress triaxiality f and the relative shear strength c0 (Fig. 3). The load type parameter or stress triaxiality is introduced as
2.5. Fracture mode Failure under mode I is given by acrit 0 . Even in the above-mentioned case of brittle materials and crack initiation under mode I, mixed mode failure can appear during the relaxation process and stress redistribution during the following load history. Shear stresses can occur and increase. The possible crack orientations are given by 0 6 a 6 12 p2 ÿ / :
Fig. 3. Analytical determination of the orientation of the introduced POD.
15
If the relative shear strength c0 is smaller for another material or experiment, the probability of mode II failure increases and a tendency to more ductile behaviour can be observed (cf. Fig. 4). Under pure compressive loads, no failure is assumed within this framework. If the load-type parameter or stress triaxiality is higher, the mode I failure becomes more dominant and the impor-
K. Keller et al. / Computational Materials Science 16 (1999) 267±274
271
Fig. 4. Correlation between fracture mode, stress state and material parameters.
tance of mixed mode failure increases with decreasing triaxiality.
3. Cohesive Zone Model More recently, a lot of applications and modi®cations have been introduced in the CZM [5]. A comparative approach to a more physically based model, the modi®ed Gurson Model [3], which in the constitutive equations refers to initial void volume fraction, void growth, nucleation and coalescence, was given in [5]. In the investigation of Siegmund and Brocks [5], a CZM was applied to the prediction of ductile crack growth. The parameters cohesive energy C, cohesive strength rmax and characteristic length d were observed over a triaxiality regime for 1:7 6 f 6 8:7. In general, the cohesive zone material parameters are introduced to be independent of all other speci®cations and processes. With respect to ductile fracture, the triaxiality at the crack tip is of signi®cant interest. The cohesive energy C Z usep xx rxx dux 16 C 0
is dissipated during the failure process until complete loss of load carrying capacity, i.e., total separation, is obtained. The cohesive strength is the maximum stress reached over this decohesion process, while the characteristic length of the CZM
represents the crack tip opening displacement (CTOD) at which the main portion of the cohesive energy is already spent. Only two of these three parameters are independent of each other, since C c rmax d
17
with c as integration constant. The material separation occurs under pure mode I and the crack path is prede®ned. The results have been compared to the modi®ed Gurson Model to observe whether the variation of the material parameters are `true' parameters, i.e., stay constant. It was concluded that the material parameters remain constant only over a small range of triaxiality. It has been concluded in [5] that improved CZMs should incorporate a nonlocal approach in order to possibly overcome this problem. For more detailed information the interested reader is referred to [5].
4. Comparison of the GCZM to the CZM Incorporating the described theory, this investigation is made on the material point level. At this material point, multiple simultaneously active cracks at dierent inclinations can be introduced or be selectively deactivated by the relaxation process between the elastic predictor and plastic corrector.
272
K. Keller et al. / Computational Materials Science 16 (1999) 267±274
The applied stress triaxiality regime is 1:7 6 f 6 8:7, equivalent to that in the comparative study [5].
gime. In the late softening range, a kink appears conjugated to void coalescence. There is a suplementary traction-separation curve plotted for uniaxial straining (uyy 0; y L).
4.1. Traction-separation response The traction-separation response of the GCZM is dierent from that predicted by the modi®ed Gurson Model as shown in Fig. 5. This may also be due to the dierent criteria for crack initiation. According to the Gurson model, crack initiation occurs when the integrated internal variable for the void volume fraction f reaches the critical value fc . The criterion for crack initiation in the GCZM is based on the actual stress state. The two ®gures illustrate the traction-separation response of the models used. On the left, the CZM with the traction-separation response de®ned by the Gurson relation and on the right for the GCZM. The quantities on the abscissa are equivalent, in the case of the CZM the crack tip opening displacement uxx is made dimensionless by the element length D. The increasing ductility with decreased triaxiality behaviour of the GCZM results from increasing dissipation during the strain softening which is a consequence of an increased activation of mode II failure. The modi®ed Gurson traction-separation relation for the CZM provides ®rst a hardening re-
5. Comparison: behaviour over the triaxiality regime For this study (cf. [5]), the Gurson Model has been accepted as the reference solution. Fig. 6 displays the material parameters of the CZM and the GCZM that are necessary in order to dissipate the same energy during fracture than Gurson's Model. It is seen that the fracture toughness GII f for mode II failure remains constant over the whole triaxiality regime, whereas GIf must be adopted slightly to dissipate the correct amount of cohesive energy. In comparison, the cohesive energy of the CZM has a variation which is four times as high as the one for the GCZM. Referring to Fig. 7, the reason for this superior behaviour of the GCZM over the CZM is explained. Fig. 7 shows the contribution in percent of the energy dissipated by mode II failure in relation to the total dissipated energy through the whole degradation process. As the triaxiality is decreasing, mode II failure gets activated, and thus, the capability of representing mixed mode failure in the GCZM is essential.
Fig. 5. Cohesive (left) and Generalized Cohesive Zone Model (right).
K. Keller et al. / Computational Materials Science 16 (1999) 267±274
Fig. 6. Parameter variation for the CZM and GCZM.
273
gated. In this study it was focussed on the fracture toughnesses GIf and GII f , for modes I and II failure correspondingly. The results have been compared to the results of the CZM, given in [5], in which attention was paid to the cohesive energy C, cohesive strength rmax and characteristic length d as cohesive zone material parameters. The transition from brittle mode I failure to a `quasi-ductile' mode II frictional slip is one of the key enhancements of the GCZM when compared to the classic CZM. For a single, optimized set of both fracture toughnesses GIf and GII f , it is demonstrated that the varying contributions of the dierent failure modes can be predicted realistically and closely to the predictions of Gurson's Model. Acknowledgements The presented work is funded by the German Research Council within the Graduate Collegium ``Internal Interfaces in Crystalline Materials'', which is gratefully acknowledged. References
Fig. 7. Mode II energy contribution.
6. Summary and conclusion The GCZM [8] has been shortly outlined. Crack initiation controlled by a strength criterion in the form of an interacting tractions hypothesis and material degradation depending on the independent failure modes controlled by the dissipated energy has been introduced. The crack orientation is determined by Mohr's approach. The investigation of Siegmund and Brocks [5], which compared the predictions of a CZM to these of the modi®ed Gurson Model, was brie¯y described. The dependence of the material parameters on the triaxiality of the stress state has been investi-
[1] G.I. Barenblatt, The mathematical theory of equilibrium cracks in brittle fracture, Adv. Appl. Mech. 7 (1962) 55± 129. [2] B. Kr oplin, S. Weihe. Constitutive and geometrical aspects of fracture induced anisotropy, in: Owen et al. (Ed.), Int. Conf. on Computational Plasticity, COMPLAS 5, Barcelona (E), 1997, pp. 255±279. [3] A. Needleman, V. Tvergaard, An analysis of ductile rupture in notched bars, J. Mech. Phys. Solids 32 (1984) 461±490. [4] F. Ohmenh auser, S. Weihe, B. Kr oplin, Algorithmic implementation of a Generalized Cohesive Crack Model, Comput. Mater. Sci., 1998. [5] T. Siegmund, W. Brocks, Prediction of the work of separation in ductile fracture and implications to the modeling of ductile fracture, Int. J. Fracture, in print. [6] T. Siegmund, W. Brocks, The role of cohesive strength and separation energy for modeling of ductile fracture, in: K.L. Jerina, P.C. Paris (Eds.), Fatigue and Fracture Mechanics, vol. 30, in print. [7] V. Tvergaard, J.W. Hutchinson, The relation between crack growth resistance and fracture process parameters in elastic±plastic solids, J. Mech. Phys. Solids 40 (6) (1992) 1377±1397.
274
K. Keller et al. / Computational Materials Science 16 (1999) 267±274
[8] S. Weihe, Modelle der ®ktiven Riûbildung zur Berechnung der Initiierung und Ausbreitung von Rissen: Ein Ansatz zur Klassi®zierung, Dissertation, Universit at Stuttgart (D), 1995.
[9] S. Weihe, B. Kr oplin, R. de Borst, Classi®cation of smeared crack models based on material and structural properties, Int. J. Solids Structures 35 (23) (1997) 1289±1308.