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MK_mk17799 - 16.4.03/druckhaus köthen L. L. Mishnaevsky Ir. et al.: Micromechanisms and modelling of crack initiation and growth in tool steels Leon L. Mishnaevsky Jr.a, Nils Lippmannb, Siegfried Schmaudera a Staatliche Materialprüfungsanstalt (MPA), University of Stuttgart, Stuttgart, Germany b Robert Bosch GmbH, Stuttgart, Germany Micromechanisms and modelling of crack initiation and growth in tool steels: role of primary carbides Dedicated to Professor Dr. Otmar Vähringer on the occasion of his 65th birthday Micromechanisms of damage initiation and crack growth in high speed and cold work steels are investigated using scan- ning electron microscopy in situ experiments. The role of primary carbides in initiation and growth of cracks in tool steels is clarified. It is shown that initial microcracks in the steels are formed in primary carbides and then join together. A hierarchical finite element model of damage initiation, which included a macroscopic model of the deformation of the specimen under real experimental conditions and a me- somechanical model of damage in real microstructures of steels, was developed. Using the hierarchical model, the conditions of local failure in the steels have been obtained. In order to study the effect of carbide arrangement on frac- ture, numerical simulations of fracture in steels with differ- ent ideal carbide arrangements were carried out and com- pared with each other. It is found that the heterogeneous arrangements of primary carbides can lead to strong devia- tions of a crack from the mode I path and, therefore, to a significant increase of the fracture energy of the steels. Keywords: Mesomechanics; Finite elements; Fracture; Da- mage; Tool steels Mikromechanismen und Modellierung der Rissinitiierung und -ausbreitung in Werkzeugstählen: Rolle von Primärkarbiden Mikromechanismen der Schädigungsinitiierung und -evo- lution in Schnellarbeits- und Kaltarbeitsstählen werden un- ter Verwendung von ln-situ-Rasterelektronenmikroskopie- Biegeversuchen untersucht. Die Rolle der Primärkarbide bei der Schädigungsinitiierung und beim Risswachstum in Werkzeugsstählen wird geklärt. Es wurde gezeigt, dass sich die Anfangsmikrorisse in den Primärkarbiden bilden und dann zusammenwachsen. Ein hierarchisches Finite-Ele- mente- Modell der Schädigungsinitiierung, das auch das reale Mikrogefüge des Stahls berücksichtigt, wurde ent- wickelt. Mit dem Mesomodell wurden die lokalen Bruchbe- dingungen bestimmt. Um den Einfluss der Primärkarbid- anordnungen auf das Bruchverhalten der Stähle zu untersuchen, wurden numerische Simulationen des Bruchs in Stählen mit unterschiedlichen idealen Karbidanordnun- Z. Metallkd. 94 (2003) 6 © earl Hanser Verlag, München gen durchgeführt und die Ergebnisse miteinander vergli- chen. Es wurde gefunden, dass die heterogenen Karbid- verteilungen in den Stählen die starken Risspfad- abweichungen von dem Mode-I-Risspfad verursachen und somit zu signifikant höheren Bruchenergien führen. 1. Introduction The improvement of service properties of tool steels pre- sents an important source of increasing the efficiency of metalworking industry. In order to develop a numerical model of damage or fracture in the steel, which should serve to predict the lifetime, or to improve the properties, one needs to know the mechanisms of damage and fracture in the steels [1-4]. The direct in situ observation of the fracture mechanisms of the steels under a microscope is quite difficult as compared with the case of more ductile . materials, since the material fails abruptly. Then, not only qualitative parameters of fracture (like its mechanisms) but also quantitative ones (like critical damage parameters) are of interest. The purpose of this work was to study the mechanisms and conditions of damage initiation and growth in the tool steels both qualitatively and quantitatively. The work in- cludes the following steps: Scanning electron microscopy (SEM) in situ experi- ments on 3-point bending of specimens with inclined notches. Finite element (FE) simulation of the deformation of the specimens on macro- and mesolevel, taking into ac- count the real microstructure of the steels observed in the SEM -experiments. Numerical analysis of the effect of the arrangement of primary carbides in the tool steels on the fracture beha- VIOr. 2. Micromechanisms of damage initiation in tool steels The mechanisms of local failure and critical values for fail- ure of the constituents of the steel have been determined. The constitutive law and elastic constants of the steel con-
6

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Page 1: Micromechanisms and modelling of crack initiation and ... · primary carbides in the tool steels on ... Micromechanisms and modelling of crack initiation and growth ... Micromechanisms

MK_mk17799 - 16.4.03/druckhaus köthen

L. L. Mishnaevsky Ir. et al.: Micromechanisms and modelling of crack initiation and growth in tool steels

Leon L. Mishnaevsky Jr.a, Nils Lippmannb, Siegfried Schmauderaa Staatliche Materialprüfungsanstalt (MPA), University of Stuttgart, Stuttgart, Germanyb Robert Bosch GmbH, Stuttgart, Germany

Micromechanisms and modelling of crackinitiation and growth in tool steels:role of primary carbidesDedicated to Professor Dr. Otmar Vähringer on the occasion of his 65th birthday

Micromechanisms of damage initiation and crack growth inhigh speed and cold work steels are investigated using scan­ning electron microscopy in situ experiments. The role ofprimary carbides in initiation and growth of cracks in toolsteels is clarified. It is shown that initial microcracks in thesteels are formed in primary carbides and then join together.A hierarchical finite element model of damage initiation,which included a macroscopic model of the deformation ofthe specimen under real experimental conditions and a me­somechanical model of damage in real microstructures ofsteels, was developed. Using the hierarchical model, theconditions of local failure in the steels have been obtained.

In order to study the effect of carbide arrangement on frac­ture, numerical simulations of fracture in steels with differ­ent ideal carbide arrangements were carried out and com­pared with each other. It is found that the heterogeneousarrangements of primary carbides can lead to strong devia­tions of a crack from the mode I path and, therefore, to asignificant increase of the fracture energy of the steels.

Keywords: Mesomechanics; Finite elements; Fracture; Da­mage; Tool steels

Mikromechanismen und Modellierung derRissinitiierung und -ausbreitung in Werkzeugstählen:Rolle von Primärkarbiden

Mikromechanismen der Schädigungsinitiierung und -evo­lution in Schnellarbeits- und Kaltarbeitsstählen werden un­ter Verwendung von ln-situ-Rasterelektronenmikroskopie­Biegeversuchen untersucht. Die Rolle der Primärkarbidebei der Schädigungsinitiierung und beim Risswachstum inWerkzeugsstählen wird geklärt. Es wurde gezeigt, dass sichdie Anfangsmikrorisse in den Primärkarbiden bilden unddann zusammenwachsen. Ein hierarchisches Finite-Ele­

mente- Modell der Schädigungsinitiierung, das auch dasreale Mikrogefüge des Stahls berücksichtigt, wurde ent­wickelt. Mit dem Mesomodell wurden die lokalen Bruchbe­dingungen bestimmt. Um den Einfluss der Primärkarbid­anordnungen auf das Bruchverhalten der Stähle zuuntersuchen, wurden numerische Simulationen des Bruchsin Stählen mit unterschiedlichen idealen Karbidanordnun-

Z. Metallkd. 94 (2003) 6 © earl Hanser Verlag, München

gen durchgeführt und die Ergebnisse miteinander vergli­chen. Es wurde gefunden, dass die heterogenen Karbid­verteilungen in den Stählen die starken Risspfad­abweichungen von dem Mode-I-Risspfad verursachen undsomit zu signifikant höheren Bruchenergien führen.

1. Introduction

The improvement of service properties of tool steels pre­sents an important source of increasing the efficiency ofmetalworking industry. In order to develop a numericalmodel of damage or fracture in the steel, which shouldserve to predict the lifetime, or to improve the properties,one needs to know the mechanisms of damage and fracturein the steels [1-4]. The direct in situ observation of thefracture mechanisms of the steels under a microscope isquite difficult as compared with the case of more ductile .materials, since the material fails abruptly. Then, not onlyqualitative parameters of fracture (like its mechanisms) butalso quantitative ones (like critical damage parameters) areof interest.

The purpose of this work was to study the mechanismsand conditions of damage initiation and growth in the toolsteels both qualitatively and quantitatively. The work in­cludes the following steps:• Scanning electron microscopy (SEM) in situ experi­

ments on 3-point bending of specimens with inclinednotches.

• Finite element (FE) simulation of the deformation of thespecimens on macro- and mesolevel, taking into ac­count the real microstructure of the steels observed inthe SEM -experiments.

• Numerical analysis of the effect of the arrangement ofprimary carbides in the tool steels on the fracture beha­VIOr.

2. Micromechanisms of damage initiation in toolsteels

The mechanisms of local failure and critical values for fail­ure of the constituents of the steel have been determined.The constitutive law and elastic constants of the steel con-

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Fig. 1. 3-point bending specimen: (a) scheme and (b) sideview of loading device.

stituents are already known from literature and from ourprevious investigations [3, 4, 6]. The analysis of the me­chanisms of damage initiation in the tool steels includesSEM in situ experiments and FE simulation of the deforma­tion of the specimens on macro- and meso level. The SEMin situ observation of the damage initiation seeks to c1arifythe micromechanisms of damage initiation, whereas thehierarchical finite element model (macro- and mesomodel)is applied to determine the failure conditions for steel con­stituents using the real loading conditions and real micro­structures of the steel.

In order to c1arify the mechanisms of damage initiationand growth in the steels, aseries of SEM in situ experimentswas carried out. 3-point bending specimens with an inclinednotch, as described in [5], were used in these tests. Thesespecimens allow to observe the micro- and mesoprocessesof local deformation and failure of carbides and the matrixof steels during loading of macroscopic specimens in theSEM. The shape of the specimens is shown schematicallyin Fig. 1a. A photograph of the specimen under loading isgiven in Fig. 1b. The advantage of the specimen with the in­clined notch is that the most probable location of first mi­crocrack initiation in the specimen notch can be simply pre­dicted (which is not the case in the conventional 3-pointbending specimens). Therefore, one can observe this loca­tion with high magnification during loading and identifyvery exactly the load and the point in time at which the firstmicrocracks form.

Specimens made from the cold work steelX155CrVMo12-1 (in further text denoted as KA) and thehigh speed steel HS6-5-2 (denoted as HS) have been used.In the experiments, the specimens with different orienta­tions of primary carbide layers were studied. Since the toolsteels are produced in the form of round sampIes and be­cause they were subject to hot reduction after austenitiza­tion and quenching, they are anisotropic: the carbide layersare oriented typically along the axis of the cylinder (this isthe direction of hot reduction). Therefore, the followingdesignation of the specimen orientation was used: L - thedirection along the carbide layers, R - radial direction inthe workpiece, C - the direction along the workpiece axis.In the experiments, specimens with orientations CL (thespecimen is oriented along the carbide layers; the observedarea is oriented along to the ingot axis), LC (the specimenis oriented along the round ingot axis) and CR (the speci­men is oriented along the carbide layers; the observed areais oriented normally to the ingot axis) have been used. Thespecimens have been subjected to the heat treatment (hard­ening at 1070°C in vacuum and tempering 2 h at 510 0c),and then polished with the use of the diamond pastes tillthe roughness Rz of the surface of the specimens does notexceed 311m. The notch region of the specimens was etched

2

Table 1. Critical forces in the tests.

Type of Force at which a firstForce at which thethe

microcrack was observedspecimen failed (N)specimen

in the specimen (N)

KALC

95,52,37.5155,85,160KACR

50,55,37.595,95,70HSCR

45,50,5095,80,95HSLC

50,72.52, 127200, 190, 195

with 3 and 10 % HN03 until the carbides were c1early seenon the surface.

The force-displacement curves were recorded during thetests. The loading was carried out in small steps, with a rateof loading of about 1 mmJs. The places in the specimennotch where microcrack initiation was expected have beenobserved through SEM during the tests. It was observed thatthe first microcracks formed only in the primary carbides,and not in the "matrix" of the steel. Also, no microcrackalong the carbide/matrix interface was observed in the tests.The forces at which the failure of primary carbides was ob­served in each specimen are given in Table 1. Fig. 2 showsSEM micrographs of a typical primary carbide in the notchregion of steels before and after failure. Since the picturewere taken frontal and the specimens used were with theinclined notch, the magnifications of micrographs in x­and y-directions in Fig. 2 are different. In some carbides,multiple cracking was observed as weil (see Fig. 3).

E5-o

40pm

Fig.2. Carbide grains before (a, c) and after failure (b, d).

Z. Metallkd. 94 (2003) 6

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L. L. Mishnaevsky Ir. et al.: Micromechanisms and modelling of crack initiation and growth in tool steels

Ea-o~

40l1mFig. 3. Multiple cracks in a primary carbide.

Fig.4. Microcracks (shown by arrows) in primary carbides.

Generally, the course of failure of the specimens was asfolIows:1. Formation of a microcrack at some carbide.

2. Formation of several microcracks at many carbides atdifferent locations of the observed area (in so doing,the microcracks are formed rather at larger carbides atsome distance from the boundary of the specimen, thanin more strained macroscopically areas in the vicinityof the lower boundary of the specimen; the local fIuc­tuations of stresses caused by the carbides have evi­dently much more infIuence on microcracking than themacroscopic stress field).

3. After the failure of many carbides, the microcracks (orplastic zones in front of the microcracks) begin to growinto the matrix; just after this occurs, the specimens fai!.

The failure of many carbides was observed just before thespecimens failed. Fig. 4 shows a segment of the loaded areawith many failed carbides.

3. Condition of failure of primary carbides in toolsteels

To simulate the deformation of 3-point bending specimenswith inclined notch, a three-dimensional (3D) FE model of

Z. Metallkd. 94 (2003) 6

the specimen was developed. The forces measured in thetests described above were applied in the simulations. Thedisplacements from the boundary nodes of elements whichare located in the vicinity of the symmetry plane and at thelower notch boundary (Fig. 1) are used as boundary condi­tions in the micromechanical simulation of carbide fail­ure.

Then, the two-dimensional (2D) rnicromechanical simu­lations of carbide failure have been carried out for each mi­crostructure and each load, measured in the experiments: a2D model was created, which represents the cut-out at thenotch region of the specimen. The real structure region ofthe micromodel contains 5000 elements of the plane straintype TRIP 6 and size 100 l!m x 100 11mand is placed in thelower left corner of the macroscopic 3-point bending mod­el, where the carbide was observed experimentally. Asboundary conditions the displacements from the model ofdeformation of 3-point bending specimen were taken. Sincethe mesh density in the 2D case is higher, the calculated dis­placements have been linearly interpolated between thepoints which were available in the 3D simulation. The mi­cromechanical simulation was performed with the use ofthe multiphase element method [8-11]. The micrographof the carbide, obtained in SEM in situ experiments was di­gitized and then automatically imposed on the region of thereal structure. The rnicrographs to be digitized were chosenin such a way that they were representative enough for thegiven materials. Due to the inclined notch surface, the mi­crographs in Fig. 2 have different scales in x- and y-direc­tions. To take that into account, the micrographs werescaled with the use of the image analysis software XViewaccordingly to their scales in both directions. The propertiesof carbide and matrix are as follows [3 - 6] : (cold worksteel) Young's modulus Ec = 276 GPa, EM = 232 GPa,constitutive law of the matrix: Sy = 1195 + 1390 [l-exp(-epl/0.0099)]; (high speed steels) Ec = 286 GPa, EM =231 GPa, constitutive law of the matrix: Sy = 1500 + 471[l-exp(-ep1/O.0073)], Poisson's ratio - 0.19 (carbides)and 0.3 (matrix).

827.7

806. !785.7

76'U;

7436

7221)

7016

680.&

&596

638.6

617ß

5SS.!>

575.6

564.6

533.6

Fig. 5. Von Mises stress distribution in the real microstructure of thesteel in the notch of the specimen (the boundary conditions in the 2Dmicromodel were taken from the 3D macromodel of the 3-point bend­ing specimen).

3

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Table 2. Calculated failure stresses of primary carbides in toolsteels.

Type of the steel KALCKACRHSCRHSLC

Failure stress of carbides

1826184016042520

(MPa)

Fig. 5 gives the distribution of von Mises stress in the realmicrostructure of the cold work steel at the loads at which

the carbide failed. Supposing that failure of the carbides isdetermined by the action of maximal normal stresses, oneobtains the failure stresses of carbides for different steelsand orientations (see Table 2).

The initial microcracks in the steels are formed withinprimary carbides (i. e., not along the carbide/matrix inter­face and not in the matrix). For our further simulation, thismeans that we can use the multi phase element method.The main input data for the simulation (i. e., the carbidefailure condition) was determined with the use of the com­bined SEM in situ and FE model approach.

4. Effect of the arrangement of primary carbides onthe fracture resistance of tool steels

As a result of the analysis above, we know the mechanicalproperties of the constituents of the steel. Then, we seek tostudy the effect of the arrangement of the primary carbidesin tool steels on the fracture behavior of the steels. To do

this, we carry out aseries of finite element simulations ofcrack propagation in tool steels with the mechanical prop­erties determined above. Different arrangements of the pri­mary carbides are designed using graphics software, andincluded into mesomechanical FE models (with carbidesas "black" regions and "matrix" as white area). In the si­mulations, two approaches were used both to take into ac­count the microstructure and to simulate the crack propa­gation: multi phase finite elements (with "fuzzy" interfacebetween carbides and matrix) (for "simple" microstruc­tures, see Figs.6a-d) and traditional single-phase ele­ments (the interface corresponds to the boundaries of thebodies of FE model and to the element boundaries) for"complex" microstructures, (see Figs. 6-f), as well as theelement elimination and softening techniques. As demon­strated in [10], the methods are fully compatible and givethe same results with respect to the crack path, fracture en­ergy and crack roughness, when applied in equivalentcases.

Fig. 7 gives the scheme of the model of a short rod frac­ture specimen. The area with carbides (100 J-lmx 100 J-lm)was placed near the notch in the model, and contained5000 finite elements (6-node triangular elements, with fullintegration). The size of the model is 500 J-lm(length) x300 J-lm(height). The distance between the right lower cor­ner and the notch tip amounted 300 J-lm(length) x 150 J-lm.We consider here the cold work steel and use the propertiesfor the steel as obtained above.

As criteria for the element elimination, the critical valuesof failure stress (for carbides) and plastic strain (for matrix)were used. The criterion of element elimination in the ma­trix was determined on the basis of available knowledgeabout the micromechanisms of fracture of steel matrix.Any damage criteria based on void growth seemed inapplic­able in this case due to the mainly brittle macro-behaviourof the matrix. Yet, during SEM in situ experiments, someplastic deformation has been observed at the microlevel(which, however, is quickly followed by the failure of spe­cimens). Thus, we chose the critical plastic strain as a criter­ion of the element elimination in the matrix. As followsfrom the SEM in situ experiments described above, the cri­tical plastic strain in the matrix of the steels should be verylow. The critical plastic strain value was determined withthe use of the numerical experiment technique [8, 10]. Atthe level of critical plastic strain Gphc = 0.1 %, the smallcrack increment in the matrix is followed by failure of a car­bide in the vicinity of the crack tip, as observed, while e. g.for Gp"c = 0.05 % delayed crack growth occurs.

The following arrangements of carbides were consid­ered: random, net-like (typical for the as-cast state), band­Iike (typical for the hot formed steels), layered (one half ofthe microstructure area is filled with randomly distributedsmall carbides and another half is filled with large carbides)and clustered arrangements (see Fig. 6). In the cases of net­Iike, band-like, clustered and random arrangements, twotypes of each microstructure were taken: a fine one withcarbide size of 2.5 11m(200 carbides in the area) and acoarse one with carbide size of 3.6 J-lm(100 carbides in thearea). In the case of layered arrangement of carbides, bothsmall and large carbides were combined in one and thesame microstructure. The surface area content (in this case,volume content) of the primary carbides was taken to be10 %. Three types of layered arrangements were consideredin the simulations, which differ by their orientation relativeto the assumed mode-I crack path: (1) Large-small case (orL->S) - a crack propagates first through the part with largecarbides, goes through the "interface" (perpendicular tothe "interface") and propagates further through the part

Fig. 6. AITangernents of primat'y carbides indesigned "simple" (a-d) and "complex" (e­h) microstructures: (a) band-like, (b) randorn,(c), net-like, large grains, (d) net-Iike, smallgrains, (e-g) layered, (h) clustered. Three or­ientations of the layered structure are consid­ered: (e) small-large (or S->L) orientation, (f)large/small (or L/S) orientation, and (g) large­small (or L->S) orientation.

(e)

(h)

r···.'::' •••·•••~.... •••••~'. " ~

'J:::):~~:::(:;·\r··J.:~l;.•·t.....;....;.

(g)

(a)

4 Z. Metallkd. 94 (2003) 6

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Cwss-scction ofshmt md

sp ...~.•.'itrlt~n Wil,h

t1\'cmged Sll',eI

propen.ics

Fig.7. Loading scheme.

with small carbides, (2) reverse case (S->L): first, smallparticle area, then large particle area, and (3) mode-I crackpropagates along the "interface" (L/S). For all the simula­tions, the force-displacement curves were determined nu­merically. Table 3 gives the nominal specific energy G ofthe formation of a unit of new surface for the different ar­rangements of the primary carbides. This value charac­terizes the fracture resistance of each of the structures, andwas caIculated on the basis of the force-displacementcurves as folIows: G = ~i (Pju;)/LRS, where Pi is the forcefor each loading step, Jli is the displacement for each load­ing step, and LRS is the size of the real microstructure area(100 Ilm). The summation is carried out for allioading stepsuntil the crack passes the real microstructure [10]. Fig. 8gives two examples of the simulated crack path: a curved/zigzagged, highly energy-consuming crack path in the net­like arrangement of small carbides, and a straightforward,low energy consuming crack path in the band-like micro­structure. As parameters of the deviations from the mode-Icrack path, we considered the maximum height of theroughness peak Rmax. Rmax was caIculated from the crackprofile as the distance between highest and lowest pointsof the crack path measured perpendicularly to the initial

crack direction (horizontal) [13]. Table 3 gives the valuesof Rmax for the considered ideal microstructures.

It can be seen from Table 3 that the microstructureswhich lead to the strong crack deviations (i. e., higher RmaJensure rather high fracture resistance. The strong crack pathdeviations from its initial direction increase the real cracklength (i. e., energy consumption in fracture) without in­creasing the stress intensity factor K[ [13-14]. One maynote that more complex microstructures (like clustered orlayered) ensure always higher fracture resistance than thesimple symmetrie microstructures. The one exception tothis rule is the net-like fine microstructure, which forcesthe crack to follow the carbide network, ensuring the high­est fracture resistance, even higher than all the complexnllcrostructures. However, such a mechanism of tougheningis unstable: the net-like coarse microstructure (with largercells) gives very low fracture toughness, since the crackdoes not follow the carbide network, but just goes throughit.

From the investigations one can conclude that the com­plex (cluster and gradient) microstructures have a great po­tential for improving the fracture toughness of tool steels.

5. ConcIusions

The mechanisms of damage initiation and growth in toolsteels were investigated and the role of primary carbides indamage and fracture of the steels was clarified.

The experimentally observed course of damage evolutionin the steels was as folIows: One microcrack appears in acarbide, then several microcracks appear in other carbidesat different locations of the observed area. In so doing, themicrocracks are formed rather at larger carbides at somedistance from the boundary of the specimen, than in themacroscopically more strained areas in the vicinity of the

Fig. 8. Examples of the simulated crack path:(a) Curvedizigzagged path in a net-like ar­rangement of sm all carbides, and (b) straight­forward crack path in a band-like microstruc­ture. The corresponding microstructures areshown in the right upper corners.

Table 3. Quantitative parameters of fracture behavior of the steels for different carbide arrangements.

Complex structuresSimple structures

Layered

ClusterNet-IikeBand-likeRandom

fine L->SL/SS->L (coarselfine)

Nominal specific energy of

668.9786.1733.9635.7436.15/827.0341.28/676.75699.18/557.02

new surface (J/m2)Rmax (Ilm)

44464435 13/36 14/2418/12

Z. Metallkd. 94 (2003) 6 5

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lower boundary of the specimen. The local fluctuations ofstresses caused by the carbides have evidently much moreinfluence on the microcracking than the macroscopic stressfield). Finally, after the failure of many carbides, the micro­cracks (as weil as plastic zones in front of the microcracks)begin to grow into the matrix; just after this occurs, the spe­cimens fai!. The initial microcracks in the steels are formedin primary carbides (i. e., not along the carbide/matrix inter­face and not in the matrix). The investigations of the role ofthe primary carbide arrangements in crack growth allow todraw the conclusions that microstructures which lead tostrong deviations from the mode I crack path (for "com­plex" arrangements of the carbides, like clustered andlayered ones) ensure rather high fracture resistances.

One author (L. L. M. Jr.) is grateful to the Deutsche Forschungsge­meinschaft for its support via the Heisenberg Fellowship. The authorsgratefully acknowledge the financial support by the Deutsche For­schungsgemeinschaft via the project DFG Schm-746/30-1.

References

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[2] H. Berns, C. Broeckrnann, D. Weichert: Key Eng. Mater. 118­119 (1996) 163.

[3] A. Gross- Weege, D. Weichert, C. Broeckmann: Comp. Mater.Sei. 5 (1996) 126.

[4] A. Lehmann: Diploma Thesis, TU Bergakademie Frei berg (1995).[5] N. Lippmann, A. Lehmann, Th. Steinkopff, H.-J. Spies: Comp.

Mater. Sei. 7 (1996)123.

6

[6] N. Lippmann: Dissertation, Freiberg (1995).[7] Chr. Le Calvez, A. Ponsot, G .. Lichtenegger, L. Mishnaevsky Jr,

S. Schmauder, I. Iturriza, M. Rodriguez Ibabe: Final Report RTDEU Project, Influence of Micromechanical Mechanisms ofStrength and Deformation of Tool Steels under Static and CyclicLoad, Creusot Loire, France (2001).

[8] L. Mishnaevsky Jr, N. Lippmann, S. Schmauder, in: Proc. 10thIn!. Conf. Fracture, Honolulu, HA, CD-ROM (2001).

[9] L. Mishnaevsky Jr., S. Schmauder: Appl. Mech. Rev. 54 (2001)49.

[10] L. Mishnaevsky Jr., M. Dong, S. Hoenle, S. Schmauder: Comp.Mater. Sei. 16 (1999) 133.

[11] L. Mishnaevsky Jr., U. Weber, N. Lippmann, S. Schmauder: in:M. Cross, J.W. Evans, C. Bailey (Eds.), Computational Modelingof Materials, Minerals, and Metals Processing, Proc. TMS Con­ference (2001) 673.

[12] L. Mishnaevsky Jr., N. Lippmann, S. Schmauder: in: ASME In!.Mechanical Engineering Congress and Exposition, New York,CD, Vol. 2 (2001).

[13] L. Mishnaevsky Jr.: Damage and Fracture of Heterogeneous Ma­terials, Balkema, Rolterdam (1998).

[14] H. Berns, A. Melander, D. Weichert, N. Asnafi, C. Broeckmann,A. Gross- Weege: Comp. Mater. Sei. II (1998) 166.

(Received October 11, 2002)

Correspondence address

Dr. Leon MishnaevskyStaatliche Materialprüfungsanstalt (MPA)Pfaffenwaldring 32, D-70569 Stuttgart, GermanyTel.: +49711 6852635Fax: +497116852635

E-mail: [email protected]

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