Optimization of the Microstructure of Ledeburitic Tool Steels: a Fractal Approach

L. Mishnaevsky Jr

13.01.2000

Werkstoffkolloquium (MPA, University of Stuttgart)

 


Simple Geometrical Model for Determining a Relationship between the Fractal Dimension of a Crack and Surface Roughness

Appearance of a Peak on the Fracture Surface

Determination of the Fractal Dimension of a Crack: by equating the real length L of this part of the surface profile, which is calculated with the use of the classical integral equation for the length of a curve, to that, calculated with the use of the fractal model, one obtains:

D = 1 + ln (k)/[ln Sm – ln d]

where
D - fractal dimension of the crack, k - parameter of the peak shape, k2= Rz/Sm-1, d - average carbide size, Rz and Sm - surface roughness parameters; height and width of peaks of the profile of surface roughness, respectively.

The fractal dimension of a fracture surface increases with increasing the roughness of fracture surface.

Surface Roughness of Fracture Surface, Fractal Dimension of Crack and Fracture Toughness: Interrelations.
(Analytical Results)

G = G0 (d/L0)1-D,

where
d - yardstick length (here - carbide size), L0 - nominal crack size, G0 - value of G for ordinary, non-fractal crack.

KIc ~ (Rz/Sm+1)0.5 B ln(d/Lo)

where B - material parameter.



=> Fracture toughness of steels increases with increasing the width of crack path (or with increasing the height of the fracture surface).

- Experimental equation obtained by Lippmann (Ph.D. Dissertation, 1995) for high speed steel S6-5-2

 

KIc ~ ln (Rz)

- Results of Berns et al., Eng. Fract. Mech., Vol. 58, No. 4, 1997

 

"...a high crack deflection corresponds to a high fracture toughness..."


Probabilistic Model of Crack Growth in Netlike Structures of Steels

  • Mechanism of crack growth: crack follows carbide-rich regions, and can change its direction on every junction of carbide bands.

  • The probability of a given height (Rz) of a peak of the surface profile is calculated as the probability that the crack does not change its direction on M=Rz/C junctions of carbide bands:

Prob{Rz} ~ 0.5Rz/C

where C - diameter of unit cell in the netlike structure.

  • The probability of higher peaks on the fracture surface increases with increasing the cell size C.

  • Results of other authors (for comparison):
"the bigger the metal cells the higher the crack deflection" and "High crack deflection causes ... a high demand of energy of crack propagation" (Broeckmann, Proc. 4th Int. Conf. Tooling, Bochum, 1996)

Crack Growth in Bandlike Structures of Steels

 

1st case: Crack parallel to bands







2nd case:Crack at an angle with bands

  • Mechanism of crack growth: crack follows only carbide-rich regions, if the load is applied normally to the bands (1st case), and grows in both matrix and bands otherwise (2nd case).

  • Bandlike structure can be taken as a limiting case of the netlike structure, with cell size approaching infinity.

  • Fractal dimension of the fracture surface:

    • 1st case - D~ 1;
    • 2nd case – the fractality of the fracture surface is determined by both parts of the crack path: the part in the carbide bands (D ~ 1), and in the matrix.

Conclusions from the Analysis of the Fractality of Fracture in Different Steel Structures
  • Fracture toughness of steels increases with increasing the width of the crack path (or with increasing surface roughness).

  • On the basis of a probabilistic analysis of crack kinking it was demonstrated that the fracture toughness of tool steels with a netlike structure increases with increasing the size of the cell of a netlike structure of steels:

  • The fractal dimension of a crack characterizes the width of the crack path (i.e., the height of the surface roughness Rz)

    Micromechanical Simulation of Crack Growth in Real Structures of Steels
    1. Macro-micro transition
    2. Microscopical Simulation using
      • the failure stress of carbides obtained experimentally
      • the stress-strain curve for the matrix taken from literature
      • and the digitized micrographs of the real structures of ledeburitic steels provided by Boehler Edelstahl
      in order to study the effect of material structure on the crack path numerically


    Crack Growth in Real Microstructure of High Speed Steel
    Yellow/red areas - primary carbides, blue areas - matrx of the steel

    Input Data for the Simulation of Fracture in Real and Idealized Structures

    1. Failure stress of carbides: obtained by MPA in the SEM in-situ Experiments and mesomechanical simulations

    2. Young Moduli of Carbides and Matrix: obtained by MPA with the use of the Microindentation Technique:

      3. (for cold work steel)

      E Matrix = 232 GPa,

      E Carbide = 276 GPa.

    3. Stress-strain curves of the matrix: approximated on the basis of the experimental data by CEIT:

      4.

      Cold work steel

      		 sigmay = 1195 +1390 [1-exp (-epsilonpl/0.0099)]

       

      High speed steel

      		 sigmay = 1500 +1101 [1-exp(-epsilonpl/0.00369)]

    4. Critical plastic strain of the matrix: obtained by MPA by the comparison of experimental observations and numerical simulations of crack growth in real structures:

    Plastic strain ~ 0.1 %

    Transition from Macromodel of Short-Rod Specimen to the Micromodel

    • Displacement distribution on the boundaries of small area (300 x 500 mcm) near the notch of short-rod-specimen was

      • determined from the Macromodel (with homogeneous material properties, and realistic loading conditions),

      • approximated by a linear function of the coordinates of a point

      • approximated by a linear function of loading step in the macromodel,

      • transmitted to the Micromodel (which contains the Real Microstructure of the steel, size 100 x 100 mcm).

    • Whereas the displacement Uy in the Macromodel presents a concentrated load and varies from 0 to 1 mm, the displacement applied to the boundary of the Micromodel is distributed along its upper and lower boundaries, and varies as follows:

    Uymicro = 0.0002 * (Number of increments) * (-1.96 X +1),

    where

    X = x-coordinate of the point.


    Conclusions from the Micromechanical FE-Simulation of Crack Growth in Real Structure
    • The numerically obtained crack path corresponds qualitatively to the typical crack trajectory in these steels:

      • the crack is initiated in carbides,

      • follows the carbide rich regions to a small part and then

      • moves to the next carbide band, and so on.

    • Results of other authors (for comparison):

    "...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" (Berns et al., Eng. Fract. Mech., Vol. 58, No. 4, 1997)

    Micromechanical Simulation of Crack Growth in Idealized Artificial Microstructures of the Steels

    Main points:
    1. Macro-Meso-Simulation: the displacement on the boundaries of a small area containing the real structure are obtained from macro-simulation (full short-rod specimen is simulated) and transmitted to the micromodel.

    2. Advanced FE techniques:

      • Multiphase Finite Elements and

      • Element Elimination Technique

    3. Input Data for Simulations are obtained experimentally and tested in the simulations of real structures.

    4. Typical Idealized Microstructures of steels are used to study the fracture propagation, namely:

      • net-like, band-like and random carbide distributions, with coarse and fine microstructure


    Idealized Quasi-Realistic Microstructures of Steels used in the Simulations

    - Band-like fine and coarse, net-like fine and coarse, random fine and coarse microstructures:

    Crack Growth in Netlike Fine Microstructure

    Crack Growth in Random Fine Microstructure

    Qualitative Parameters of the Crack Behaviour in Quasi-Real Microstructures

    Parame-ter

    Sym-bol

    Method of determination

    Physical Meaning

    Peak load of the force-displacement curve, N

    Fmax

    maximal force on the force-displacement curve

    critical load at which the crack begins to propagate

    Nominal specific energy of the formation of unit new surface, J/m2

    G

    by the formula:

    G = Sumi (Pi ui)/ LRS,

    where Pi – force for each loading step, ui – displacement for each loading step, LRS = 100 mcm - linear size of the real microstructure, the summation is carried out for all loading steps until the crack passes the real microstructure.

    resistance of the material to the crack growth (in relation to the unit of nominal new surface)

    Averaged loading force, N

    Fav

    by the formula:

    Fav =Sumi Pi ui/ um,

    where um=0.0026 mm, ui – displacement for each loading step, the summation is carried out until u= um.

    the resistance of the material to crack growth (in relation to the applied displacement)

    Maximal height of the roughness peak mcm

     

    Rmax

    the distance between highest and lowest points of the crack path measured along the perpendicular to the initial (horizontal) crack direction

    the degree of variations of crack path



    Development of the Method of Calculation of Energetic and Fractal Characteristic of Fracture from FE-Simulations

    • By definition of the Fractal Dimension,

    i ~ L0D,

         where i - amount of unit steps of crack growth (or the amount of "yardstick lengths" in the real crack length – in terms of the fractal theory), L0 - projected crack length, D - fractal dimension of the fracture surface. In our FE model, i = amount of eliminated elements.

  • Method of numerical determination of the fractal dimension of crack includes the following steps:

    1. Amount of eliminated elements in each increment of the FE-simulations is plotted vs. the length of the as-formed crack,

    2. The power in the relationship between these two values presents the fractal dimension D.

  • Modeling of the crack growth with the use of the Element Elimination Technique makes it possible to analyze the fractality of the crack. Using any model of crack growth with prescribed crack path excludes the possibility to determine the fractality of the crack numerically.

Comparison of Analytical and Numerical Results
Results of the Analytical and Probabilistic Models FE-Simulation
G is a power function of D. The plot of G versus D looks like a power function (with deflections)
D is a linear function of log (Rz). The plot of Rmax versus D looks as an exponential function
Net-like microstructure (assuming the crack follows the carbide network) ensures maximum fracture resistance, and highest surface roughness Fine net-like microstructure (in which the crack follows the carbide network) gave the highest G and Rmax values
BUT:
The fracture energy and surface roughness increase with increasing the network cell size in the net-like microstructure 		When the network cell size in the net-like microstructure exceeds some threshold, the crack stops to follow the carbide network

where G - specific fracture energy, D - fractal dimension of fracture surface, Rmax - height of fracture roughness peaks.

Conclusions

  • Fracture toughness of steels increases with increasing the width of the crack path (or with surface roughness) and the fractal dimension of fracture surface

    • In all microstructures, the crack path is strongly influenced by the carbide distributions. In the matrix, the crack path is usually rectilinear. In the band-like and net-like microstructures, rather large crack deflections on the carbide-rich regions were observed

      • Different mechanisms of the toughening of tool steels are identified:
          • crack deflection by the carbide layers oriented perpendicularly to the initial crack path (net-like coarse and band-like microstructures),
          • crack follows the carbide network (net-like fine microstructure), and
          • damage formation in random sites of the steels + crack branching (random microstructures).
        • The analytical and numerical investigations of fracture of tool steels give similar results.


        Send your comments to: impgmish@mpa.uni-stuttgart.de (Dr. L. Mishnaevsky)