elements of the plane strain type TRIP 6 and size 100 m x $00 m was located in the lower left corner of the model. As boundary conditions the displacements ...
Proceedings of IMECE 2001: International Mechanical Engineering Congress and Exposition November 11-16, New York
COMPUTATIONAL DESIGN OF MULTIPHASE MATERIALS AT THE MESOLEVEL
Leon Mishnaevsky Jr MPA, University of Stuttgart, Pfaffenwaldring 32, 70569 Stuttgart, Germany
Nils Lippmann Robert Bosch GmbH, 70442 Stuttgart, Germany
ABSTRACT A concept of optimal design of multiphase materials on the basis of numerical simulation of damage and fracture growth in real and artificial microstructures of the materials is formulated. The suggested procedure includes the following steps: image analysis of the material structure; determination of properties of constituents of the materials; search for regularities or periodicity in the microstructure; simulation of fracture in real microstructures and comparison of the results of the simulation with experiments; simulation of damage and fracture in typical idealized microstructures of material; determination of the direction of the structure optimization and further iterative simulation/optimization of structure. Numerical simulations of fracture in real microstructures of tool steels and some artificial idealized microstructures are presented. The comparison of microstructures is carried out, and some recommendations to the improvement of tool steels are given. KEYWORDS: mesomechanics, finite elements, fracture, damage, fractals INTRODUCTION The problem of the optimal design of multiphase materials on the basis of computational simulations of their behaviour attracts growing interests of researchers in the last decades [1]. Table 1 reviews some works in the area of the material improvement. Comparing the recommendations for the materials design given in Table 1, the following main conclusion may be drawn: by varying the mutual arrangement of phases in a multiphase material, the materials properties may be improved. The paper seeks to develop a systematic computational approach to optimize the microstructures of materials. This
Siegfried Schmauder MPA, University of Stuttgart, Pfaffenwaldring 32, 70569 Stuttgart, Germany
general approach is applied here to optimize the microstructure of tool steels in order to improve the fracture resistance of the tool steels by varying their microstructure. Since tool steels are widely used in the metal-working industry, the problem of the improvement of the tool steel quality and reliability is very important. The suggested approach includes the following steps: 1. Determination of the mechanical properties, damage mechanisms and failure conditions for the constituents of the material to be optimized (tool steel, in our case), 2. Choice of the appropriate simulation approach: unit cell approach (for regular microstructures) or real structure simulation, cohesive concepts of fracture or element elimination [1, 2], 3. Simulation of fracture in a real microstructures of the material; comparison of the results with the experimental data and correction of the model, if necessary; 4. Simulation of crack growth in ideal artificial microstructures; comparison and recommendations for the improvement of the microstructures of materials, 5. Realization of the recommended microstructures in cooperation with industry, using powder metallurgy technology, etc. NOMENCLATURE EC, EN Young moduli of carbides and matrix, Fmax peak load on force-displacement curve, G nominal specific energy of the formation of unit new surface LRS linear size of the real microstructure
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Pi ui HS KA KTYP
pl y
-
-
force for each loading step, displacement for each loading step, high speed steel HS6-5-2 cold work steel X155CrVMo12-1 the number of Gaussian points in a finite element when it is decided to which phase should be assigned the element plastic strain von Mises equivalent stress,
Table 1. Some recommendations on the improvement of fracture resistance of materials [1] Materials Ceramics; CMC [3, 4] MMC [5]
FGM [6]
Nanocomposite Al ceramics [7]
Tool steels [8, 9, 10]
The real structure simulations are carried out either using (automatic) FE mesh generation of the microstructural images by single phased elements in such a way that the phase boundaries in the microstructure correspond to the “surface” boundaries in the FE mesh [11], or by using multiphase finite elements (MPFE) [1, 2, 12]. The main idea of the MPFE is that the different phase properties are assigned to individual integration points in the element (as differentiated from the common approach, when each element of the FE mesh is attributed to one phase and the phase boundaries are supposed to coincide with the edges of finite elements) (see Figure 1). Therefore, the FE-mesh in this case is independent of the phase arrangement of the material, and relatively simple FE-meshes can be used in order to simulate the deformation in a complex microstructure.
Ways to increase fracture resistance of materials 1. Controlled microfracture: brittle particles fail in the stress field of a crack, 2. ductile second phase network, 3. frictional toughening Fracture toughness of MMC with a continuous precipitate network is much higher than that for dispersed particles. 1. Increasing the connectivity of the metal phase, 2. crack arrest function of FGM is improved by metal fiber premixing in the ceramic-rich region. 1. "switching from intergranular cracking to transgranular one" (by nano-particles along the grain boundaries), 2. "fracture surface roughening by zigzag crack path" (by the fluctuated residual stresses from the nano-particles within the grains), and 3. "shielding by clinched rough surfaces near the crack tip". 1. Increasing the width of crack path (in the hardened and low tempered states) by increasing the cell size in netlike structure; 2. increasing matrix ductility, 3. "double dispersion" microstructure of the steels.
NUMERICAL APPROACH Modelling complex microstructures of materials The problem to be solved (optimization of the inclusions arrangement in materials in order to increase the fracture resistance) requires a numerical approach which allows both to take into account the microstructure of the material and to simulate crack growth without prescribing the expected crack. In order to take into account a microstructure of material, either a unit cell approach or real structure simulations are used in most cases (see detailed review in [1]).
Phase boundary
Integration point
FE edges
Figure 1. Scheme: multiphase finite elements [12]. Both approaches have their advantages and disadvantages: whereas the automatic mesh generation using single phase elements leads to the simplification of the phase boundary contours, the multiphase elements do not allow to take into account the local interphase effects in materials. Simulation of crack growth Among the numerical approaches used to simulate the crack growth, the representation of cracking by the separation of element boundaries [13, 14, 15], by element „softening“ and by element elimination (the last two approaches are often confused) [2, 16, 17, 18] may be mentioned. Some other approaches to the modelling of crack growth are given in Table 2. The separation of element boundaries can be done, for instance, if all the elements in the model, or the elements along the expected crack path present contact elements. The element “softening” is done if each element is assumed to be weakened as the local stress or damage parameter exceeds a critical level. All components of stress tensors and all forces in this element are set to zero. Therefore, this element stops to transmit load to neighboring non-eliminated elements. In solving the problem the tangential stiffness matrix has to be corrected after the element elimination. This is done by setting the Young‘s modulus of the eliminated elements equal to
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(almost) zero. In order to avoid numerical problems related to strong local loss of equilibrium, the stresses are set to be equal to zero in several steps (called "relaxation steps"). The Young‘s moduli in the eliminated elements are set to be equal to zero in the last relaxation step. The possibility to model the crack growth by such a model is given by the subroutines UMAT or USDFLD in the ABAQUS FE code and was implemented in LARSTRAN FE code as well [2, 18]. Another way to simulate crack growth is to remove the elements from the model, and then to restart the simulation without the eliminated elements. Table 2. Method of modelling fracture Model
Main ideas
Cohesive zone models (CZM) [19]
Crack path is prescribed, and presented as a thin material layer with its own constitutive relation (traction-separation law).
Computational cell methodology (CCM) [20, 21]
Crack path is given as a layer consisting of cubic cells with a void. When the void volume fraction in a cell in front of the crack tip reaches some critical level, the cell is removed and therefore the crack grows.
Cell model of material [22]
A generalized formulation of the cell models of crack growth. The material is accepted to consist on cells, which is characterized by its size and cohesiondecohesion relation.
Smeared crack A crack is considered as a continuous models [23, 24] degradation (reduction of strength/stiffness) along the process zone. The displacement jump is smeared out over some characteristic distance across the crack, which is correlated with the element size. Embedded crack model (ECM) [25]
Finite elements with an embedded discontinuity line or localization band. The constitutive model of the element with an embedded discontinuity is given by both the traction-separation law and the stress-strain law.
Hybrid fracture/ damage approach (HFDA) [26]
At the crack tip, a "super-element" consisting of a singular crack tip element and several variable-node elements is located. When crack grows, the material stiffness is reduced in each timestep at the crack tip.
Element elimination and element softening approaches [18, 2]
Crack growth is presented as a weakening or removal of finite elements in which the local stress or damage parameters exceed some critical level.
The main advantages of the element elimination and element softening approaches (as compared with other methods given in Table 2) are that both microdamage and crack propagation can be simulated using the same local failure criterion. In our simulation, we shall use the multiphase finite element method and the element softening technique.
DETERMINATION OF INPUT DATA FOR THE SIMULATIONS In this section, the input data for the simulation of the steel behaviour are determined. These include: - mechanisms of local failure, - critical values for failure of the constituents of the steel. The constitutive law and elastic constants of the steel constituents are already known from literature [27, 28, 29, 30]. The tool steels are assumed to consist of primary carbides and a “matrix” (which includes also the small secondary carbides). The analysis of the mechanisms of damage initiation in the tool steels includes SEM-in situ experiments and FE-simulation of the deformation of the specimens on macro- and mesolevel. The SEM in-situ observation of the damage initiation seeks to clarify the micromechanisms of damage initiation, whereas the hierarchical finite element model (macro- and mesomodel) is applied to determine the failure conditions for steel constitutents using the real loading conditions and real microstructures of the steel. SEM in-situ study of micromechanisms of damage initiation In order to clarify the mechanisms of damage initiation and growth in the steels, a series of SEM-in-situ-experiments was carried out. 3-point bending specimens with an inclined notch, as described in [29], were used in these tests. These specimens allow to observe the micro- and mesoprocesses of local deformation and failure of carbides and the matrix of steels during loading of macroscopic specimens in the SEM. The shape and sizes of the specimens are shown schematically in Figure 2. 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-point bending specimens). Therefore, one can observe this place with high magnification during loading and identify the load and the point in time at which the first microcracks form very exactly. Specimens made from the cold work steel X155CrVMo121 (in further text denoted as KA) and the high speed steel HS65-2 (denoted as HS) have been used. In the experiments, the specimens with different orientations of primary carbide layers were studied. Since the tool steels are produced in the form of
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round samples and because they were subject to hot reduction after austenitization and quenching, they are anisotropic: the carbide layers are oriented typically along the axis of the cylinder (this is the direction of hot reduction). Therefore, the following designation of the specimen orientation was used: L – the direction along the carbide layers, R – radial direction in the workpiece, C – the direction along the workpiece axis. In the experiments, specimens with orientations CL (the specimen is oriented along the carbide layers; the observed area is oriented along to the ingot axis), LC (the specimen is oriented along the round ingot axis) and CR (the specimen is oriented along the carbide layers; the observed area is oriented normally to the ingot axis) have been used. The specimens have been subjected to the heat treatment (hardening at 1070oC in vacuum and tempering 2 times at 510oC), and then polished with the use of the diamond pastes of different sizes till the roughness Rz of the surface of the specimens does not exceed 3 region of the specimens was etched with 3 % and 10 % HNO3 until the carbides were seen on the surface.
Generally, the course of failure of the specimens was as follows: (1) formation of a microcrack at some carbide, (2) formation of several microcracks at many carbides in different places of the observed area (in so doing, the microcracks are formed rather at larger carbides at some distance from the boundary of the specimen, than in more strained macroscopically areas in the vicinity of the lower boundary of the specimen; the local fluctuations of stresses caused by the carbides have evidently much more influence on microcracking than the macroscopic stress field), and (3) after the failure of many carbides, the microcracks (or plastic zones in front of the microcracks) begin to grow into the matrix; just after this occurs, the specimens fail. The failure of many carbides was observed just before the specimens failed. The differences between the loads at which the microcracks are formed, and that at which the specimen failed was in most cases very small. Figure 3 shows the SEM micrographs of a typical primary carbide in the notch region of steels before and after its failure.
applied load
a)
observed area
Figure 2. 3-point bending specimens The force-displacement curves were recorded during the tests. The loading was carried out in small steps, with a rate of loading of about 1 mm/s. The places in the specimen notch where microcrack initiation was expected have been observed through scanning electron microscopy (SEM) during the tests. It was observed that the first microcracks formed only in the primary carbides, and not in the “matrix” of the steel. Also, no microcrack along 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 3. Table 3. Critical Forces in the Tests Type of the Force at which a first Force at which specimen microcrack was the specimen observed in the failed, specimen, N N KALC 95, 52, 37.5 155, 85,160 KACR 50, 55, 37.5 95, 95, 70 HSCR 45, 50, 50 95, 80, 95 HSLC 50, 72.52, 127 200, 190, 195
b)
Figure 3. A carbide grain before (a) and after failure (b). (Area size 40x100 Failure Condition of Primary Carbides To simulate the deformation of 3-point bending specimens with inclined notch, a three-dimensional FE model of the specimen was developed. The forces measured in the tests described above were applied in the simulations. The displacements from the boundary nodes of elements which are located in the vicinity of the symmetry plane and at the lower notch boundary are used as boundary conditions in the mesomechanical simulation of carbide failure. Then, the 2D mesomechanical simulations of carbide failure have been carried out for each microstructure and each load, measured in the experiments. A 2D-model was created, which represents the cluster of finite elements at the notch region of the specimen. The real structure region with 5000 elements of the plane strain type TRIP 6 and size 100
"!$#% &" '()+*-,'. (/#-% ("01#(23'4&"% 05,(06% 27'-.(89% ) ("#-:;
boundary conditions the displacements from the model of deformation of 3-point bending specimen were taken. Since the
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mesh density in the 2D case is higher, the calculated displacements have been linearly interpolated between the points which were available in the 3D simulation. The mesomechanical simulation was performed with the use of the multiphase element method [2]. The micrograph of the carbide, obtained in SEM-in-situ experiments was digitized and then automatically imposed on the region of the real structure. The micrographs to be digitized were chosen in such a way that they were representative enough for the given materials. Due to the inclined notch surface, the micrographs in Figure 3 have different scales in X- and Y-directions. To take that into account, the micrographs were scaled with the use of the image analysis software XView accordingly to their scales in both directions. The properties of carbide and matrix are as follows [27, 28, 29]: (cold work steel) EC=276 GPa, EM=232 GPa, constitutive law of the matrix: y = 1195 +1390 [1-exp (pl/0.0099)]; (high speed steels) EC=286 GPa, EM=231 GPa, constitutive law of the matrix: y = 1500 + 471 [1-exp(pl/0.0073)], Poisson's ratio - 0.19 (carbides) and 0.3 (matrix).
simulation, it means that we can use the multiphase element method. Then, the main input data for the simulation (i.e. the carbide failure condition) was determined with the use of the combined SEM in-situ and FEM approach.
CRACK GROWTH IN A REAL MICROSTRUCTURE As a result of the analysis above, we know the mechanical properties of the constituents of the steel. Then, we chose the method of modelling: the multiphase finite elements and the element elimination technique (since no interface fracture was observed in the experiments, the MPFE may be used in the simulations). Figure 5 gives the metallographic micrograph (digitized) of the high speed steel HS6-5-2 which was used in the simulations. In this section, the simulation of crack growth in the real microstructure of the steels is carried out.
Table 4. Calculated failure stresses of primary carbides in tool steels Type of the steel KALC KACR HSCR HSLC Failure stress of 1826 1840 1604 2520 carbides, MPa Figure 4 gives the distribution of von Mises stress in the real microstructure of the cold work steel at the loads at which the carbide failed. Supposing that failure of the carbides is determined by the action of maximal normal stresses, one obtains the failure stresses of carbides for different steels and orientations (see Table 4).
Figure 5. Real (digitized) microstructure of high speed steel HS6-5-2. (Area size 100x100 =>? Figure 6 gives the schema of the model. The real microstructure of high speed steel was placed in an area 100 @ x 100 ACBD"EFHG-IDBJ G-K"IML-BMG-IDNA9J O D"P-QSRI DNTUL-V"DJ WHGIDAJ O DPXL-T 500 Y[Z]\-^"_` abc de f f Y[Z5b^g`b ac5hiHb^kj glman_o"^4p ^aq ^"^_7a-b^ right lower corner and the notch tip amounted 300 rts5uvwxy-z{ x 150 |}5~~ 5
-~~ " 5 "5-" ~ |9 "- of the steel, and the real microstructure of the steel, the crack initiation and growth in the steel is simulated.
Figure 4. Von Mises stress distribution in the real microstructure of the steel in the notch of the specimen.
y
Therefore, it was shown that the initial microcracks in the steels are formed in primary carbides (i.e. not along the carbide/matrix interface and not in the matrix). For our further
Figure 6. Loading scheme.
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As criteria for the element elimination, both the value KTYP and the critical values of failure stress (for carbides) and plastic strain (for matrix) were used. The value KTYP means the number of Gaussian points in an element when it is decided whether this element is to be assigned to the matrix or to a carbide. This value was 3; i.e., if 4 (or more) Gaussian points of an element (which contains 6 points) are lying in the matrix, the element was supposed to be eliminated as a matrix element. Otherwise, it was considered as an element in a carbide. The criterion of element elimination in the matrix was determined on the basis of available knowledge about 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-behaviour of the matrix. Yet, during SEM-in-situ-experiments, some plastic 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 follows from SEM-in-situ-experiments described above, the critical plastic strain for the matrix of the steels should be very low.
have carried out the simulation of crack initiation with the carbide failure criterion obtained above and different criteria of crack initiation in the matrix (Rice-Tracey damage criterion, critical stress, different values of the critical plastic strain, etc.), until the above described crack behaviour was obtained in the simulations. As expected, the most appropriate criteria of the crack initiation in the matrix was the critical plastic strain, and the critical level of this value was pl, c = 0.1 %. At this level of critical plastic strain, the small crack increment in the matrix is followed by failure of a carbide in the vicinity of the crack tip, as observed, while e.g. for pl, c = 0.05 % delayed crack growth occurs. Figure 7 shows crack growth in the real (band-like) structure of the high speed steel. The real microstructure area contained 5000 finite elements. We used 6-node triangular elements, with full integration. In Figure 7 it can be seen that the crack grows initially in the matrix almost straightforward. The crack began to grow after the displacement reached 0.0006 mm. (In the experiments presented in [30], the crack in the short rod specimens from high speed steels began to grow when the external displacement reached the value 0.0006 mm, too). The calculation of the maximal stress distribution in the first load step (the displacement 0.0002 mm from the notch) shows a higher stress concentration in the vicinity of the crack tip. The straightforward crack path is stopped due to the carbide bands and the crack begins to grow along the carbide band. Then, the straightforward crack growth in the matrix continued. At a displacement of 0.0008 mm, crack branching caused by the availability of a carbide row in the crack path occurs. Without further increment of load, the full area of the microstructure breaks. So, the direction of crack growth and the structure of the crack are strongly influenced by the carbide rich regions. CRACK PROPAGATION IN IDEALIZED MICROSTRUCTURES OF STEELS
Figure 7. Crack path in the real structure of the high speed steel HS6-5-2. The critical plastic strain value was determined with the use of the numerical experiment technique (this type of modeling is also referred to as inverse modeling) on the basis of the qualitative information about the crack behaviour in tool steels and the above determined carbide failure stress. It is known [8, 9] that a crack in ledeburitic chromium steels is initiated in carbides, if they are available in the vicinity of the notch tip, grows straightforward in the matrix, and kinks into the carbide rich regions and then follows them to a small part, then jumps to the next carbide band, grows in the carbide band furtherly, and so on. As was noted by Berns et al. [8], ”...running crack must follow carbide bands...The width of the crack is restricted to jumps between adjacent carbide bands, but most of the crack surface is produced by cleavage of carbides in one band”. We
TYPICAL
Using the above approach, one can carry out computational experiments to study the effects of the material structure on the fracture behaviour. The next step in the numerical optimization of a material is the simulation of deformation and fracture in artificial quasi-real microstructures. By testing some typical idealized microstructures of a considered material in such numerical experiments, one can determine the directions of the material optimization and preferable microstructures of materials under given service conditions. Such simulations should be carried out for the same loading conditions and material, as the real structure simulations which proved to reflect adequately the material behaviour. Among the types of idealized microstructures, as a first approximation random and periodic microstructures are usually taken. In studying cast and deformed metals, it is advisable to consider also the net-like (typical for the as-cast state) and band-like microstructures (hot formed steels). To investigate the
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structures of hard alloys and other ceramic materials, one should take into account the degree of clustering of hard (or in some cases, ductile) particles and vary such parameters as the connectivity, degree of clustering, etc. as well as the distributions of particle shapes and sizes. To determine the optimal microstructure of the steel, bandlike, net-like and random microstructures were considered. Two types of each microstructure were taken: a fine one with carbide size of 2.5 " "5U -¡N"5¢ - U-£" ¤¦¥ §©¨ 9§
The crack path in the band-like structure (Figure 10) shows markable crack deflection at the carbide bands. In the matrix, the crack grows rectilinearly. In the fine net-like microstructure (Figure 9), the crack is instantly directed to the carbide network, and then follows exclusively the carbide network (this mechanism of crack growth was considered theoretically in [31]). It is of interest that the force-displacement curve for this microstructure gives the highest value of the peak force. As differentiated from the fine net-like microstructure, the crack in the coarse net-like microstructure was initiated in the matrix. This influenced the further crack growth sufficiently: the carbide layers are passed by the crack, and lead only to relatively slight crack deflections. This behavior of the crack in the coarse net-like microstructure is similar to that in the bandlike structure.
Figure 8. Simulated crack path in a microstructure with random carbide distribution (coarse microstructure). The artificial net-like, band-like and fine random microstructures with 200 round particles of a given radius [such that the surface content (in this case, volume content) of the particles is about 10 %] were created with the use of the graphics software XFIG. In the case of the random microstructures, the particles were randomly distributed thereafter. The particles in the band- and net-like structure were distributed in such a manner that the distance between bands or the cell size (respectively) was about 20 times the particle diameter for the coarse structures, and 10 times the particle diameter for the fine microstructures. The coarse random structure possesses only 100 particles, the size of which was selected in such a way that the volume content of carbides amounts to 10 %. The simulations for these artificial microstructures were carried out in each case with the same boundary conditions as for the real structure. Figures 8-10 show the crack path in the artificial microstructures of the steels. For all the simulations, the forcedisplacement curves were determined numerically. COMPARISON OF THE MICROSTRUCTURES Let us compare the structure of the crack path, as well as qualitative and quantitative parameters of fracture for different ideal arrangements of carbides.
Figure 9. Simulated crack path in a microstructure with net-like continuous carbide distribution (fine microstructure).
Figure 10. Simulated crack path in a microstructure with band-like carbide distribution (coarse microstructure).
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Table 5 gives some main quantitative characteristics of the crack growth in the different structures. The value Fmax characterizes the critical load at which the crack begins to propagate. The value G (nominal specific energy of the formation of unit new surface) for each microstructure characterizes the fracture resistance of each of the structures. The physical meaning of the averaged force Fav is close to that of G: this value characterizes the fracture resistance of each of the microstructures, yet, not in relation to the unit of nominal new surface, but in relation to the applied displacement. The difference between these two values (Fav and G) is caused by the fact that cracks in each case passed the microstructure area at different applied displacements The nominal specific energy of the formation of unit new surface for each microstructure was calculated as follows: G=ª
i
(Pi ui)/ LRS
(1)
where Pi – force for each loading step, ui – displacement for each loading step, LRS - linear size of the real microstructure, the summation is carried out for all loading steps until the crack passes the real microstructure. Figure 11 gives the specific energies of new surface formation for all the microstructures. Table 5. Quantitative parameters of fracture behaviour of the artificial structures
Fmax, N Fav, N Rmax,
«
Net-like coarse fine 44.9 52.3 13
36
Band-like coarse fine 43.8 43.6 24
Random coarse fine 47.0 50.2
14
18
12
800
200 100 0
¬ ® ¯ ° ± ²³ ® ´
Band-like Coarse
300
Net-like Coarse
400
Random Fine
500
Band-like Fine
600
Random Coarse
700
Net-like Fine
Specific fracture energy G, N/m
900
Figure 11. Specific energy of new surface formation for each of the microstructures.
For both net-like and band-like microstructures the fracture resistance of the steels is much higher for the fine than for the coarse versions of the structures. Although this effect was not observed for the random microstructure, we assume that it is caused only by the different number of particles in the fine and coarse random microstructures, and that the general tendency remains the same for the random microstructures as well. The fracture energy G is higher for the net-like than for the bandlike microstructures. It is of interest that the averaged force Fav hardly increases with increasing the peak load. If one considers only the coarse microstructures, one can see that Fav even decreases with increasing the peak load Fmax. So, increasing the resistance of the materials to crack initiation does not mean necessarily the increase of the energy consumption in crack growth (i.e. fracture toughness). This suggests that an approach which includes both the optimal design of parts (tools) and the optimal design of the tool material should be used for the design of tools from these steels. Namely, in the parts of the cutting tool, in which the tensile stresses are maximal and the crack initiation is therefore most probable, microstructures which ensure maximum Fmax should be used (first of all, the cutter face, especially in the vicinity of the cutter edge is meant). In the rest of the tool material, a microstructure which ensures the maximum energy consumption in crack propagation should be taken. Therefore, it can be useful to consider the possibility of using gradient microstructures to optimize the tool steels. Let us look at the form and structure of the crack path in different microstructures. As parameters of the crack path form, we considered the maximal height of the roughness peak Rmax, the amount of eliminated elements not connected to the main crack as well as branches of the crack, and the fractal dimension of the crack. The maximal height of the roughness peak Rmax was calculated from the crack profile as the distance between highest and lowest points of the crack path measured along the perpendicular to the initial crack direction (horizontal). The difference in the structure of crack path for different microstructures of steels is rather obvious: whereas a crack grows almost straightforward in the matrix or between carbide layers, it kinks at the carbide layers or even follows the carbiderich region if this does not require strong deflection of the crack path from its initial direction. A general tendency is that the maximal height of the fracture surface peak is much higher for the coarse than for the fine microstructure (this was not observed for the net-like microstructure only since the full change of the mechanism of crack growth occured in our simulation – the crack in the coarse net-like microstructure grew like in the band-like microstructures, i.e. it did not follow the carbide network, but just deflected at the carbide layers). Microcracking at random sites in the microstructure (not in front of the growing crack) was observed in the band-like and random microstructures, but not in the net-like microstructures. Intensive crack branching took place also in the band-like and
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random microstructures. This can be explained by comparison with the results by Berns et al. [8], who showed that when carbides at some distance from the crack tip fail, that leads to the change of the direction of crack growth. Therefore, high intensity of microcracking in random sites of the material causes a high intensity of crack branching. FRACTALITY OF SIMULATED CRACK PATH Consider now the fractality of a growing crack in the real materials. The very often observed mechanism of crack growth, when a crack grows by joining microcracks which are formed in front of the crack tip (but not necessarily in the plane of crack propagation) and which are much smaller than the crack [32, 33] causes random variations of the direction of crack growth (see Figure 12). These variations determine the fractal dimension D of a fracture surface [33]. The growth of a crack by joining of microcracks is similar to the formation of a fractal cluster from randomly moving particles which may join the cluster [32]. For a fractal cluster which grows by such a mechanism, the fractal dimension of a cluster is given by the formula: i ~ L D,
(2)
where i – the amount of particles (in our case, the number of unit steps L of the crack growth, or eliminated finite elements), L – projected crack length. Table 6. Fractal dimensions for the simulated fracture surfaces
D
Net-like Band-like coarse fine coarse fine 1.285 1.593 1.442 1.40
Random coarse fine 1.372 1.446.
To determine the fractal dimension D of the crack path from the above simulations, we determined the number of eliminated elements and the projected length of the crack after each loading step, and determined the power in the i-Lrelationship for each of the structures. The fractal dimensions for each microstructure are given in Table 6. One can see that the maximal fractal dimension is found in the band-like fine microstructure. Comparing results of the Tables 5 and 6, it can be seen that the height of the peaks of the fracture surface increases with increasing the fractal dimension as well. It is of interest to compare the last conclusion with the analytical results from [31]. It was shown in [31] on the basis of a probabilistic analysis of the distribution of peak heights on the fracture surface that the height of the peaks increases with increasing D; yet, the result was obtained only for the net-like microstructures of steels, while our present calculations show this trend for the other microstructures as well.
L
µ
L
Figure 12. Scheme: Crack path variations due to the microcracks
One should note here that the numerical determination of the fractal dimension of a crack is possible only with the use of the above techniques: the element elimination technique (it is evident that prescribing crack path in simulations of the crack growth excludes any possibility to consider the fractality of the fracture surface), and the real microstructure simulation (it is also evident that a crack which grows only according to the fracture and continuum mechanics laws, without taking into account “real world” will be not fractal). FUTURE WORKS As further directions of the work, investigations in the following directions may be carried out. First, it is of interest to compare the two numerical methods: MPE and the single-phase finite elements microstructure simulations (this approach assumes the mesh generation on the basis of the microstructural images). It is expected that the comparison of the methods and their comparison with experiments help to improve the numerical approach further. Then, the next step includes investigations of the crack behaviour in some other types of microstructures which by our opinion or according to the results of other groups can be very promising [10]. Among them are the cluster and gradient distributions of the carbides. Another interesting direction of work would be to optimize numerically the materials’ microstructure taking into account not only the fracture resistance, but also the wear-resistance [34]. Such an approach could be well realized in the framework of the element removal simulation/restart techniques [17], rather than the element softening and MPE. CONCLUSIONS Comparing the fracture behaviour of different microstructures of steels, the following conclusions can be drawn. The fracture resistance of the steels is much higher for the fine than for coarse version of the same type of microstructures. The resistance of the steels to crack initiation, characterized by the peak load on the force-displacement curve, is lowest for the
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band-like structures, and is again sufficiently higher for the fine than for coarse microstructures (especially in the cases of the random and band-like fine microstructures). The roughness of the fracture surface is higher for the coarse than for the fine microstructures. Microcracking at random sites in the microstructure (not in front of the growing crack) as well as intensive crack branching, were observed in the band-like and random microstructures but hardly in the netlike microstructures. Generally, the fracture resistance of steels increases in the following order: band-like random net-like microstructure. Furthermore, one may note some interrelations between the geometrical and energy parameters of fracture: the fracture toughness increases with increasing the fractal dimension and height of the roughness profile of the fracture surface. The interrelations between the specific fracture energy, fractal dimension and the roughness height, obtained in our numerical experiments, correspond to the analytical results from [31, 33]. On the basis of the above study, one may speculate about possible directions of the optimization of steels. The following effects which increase the toughness of the steels were observed in the considered structures: • crack deflection by the carbide layers oriented perpendicularly to the initial crack path (net-like coarse microstructure, band-like microstructures), •
the crack follows the carbide network (net-like fine microstructure), and
•
damage formation at random sites of the steels and following crack branching (random microstructures).
[3]
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[10] It may be noted that all above effects increase the ratio of the summary area of as-formed surface to the size of the failed specimen (projected crack length). ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support by the Deutsche Forschungsgemeinschaft via the project DFG Schm-746/30-1 “Optimization of High Speed Steels by Varying their Microstructures on the basis of the Fractal Analysis of Interrelations between the Microstructure and Fracture”.
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