This document discusses numerical experiments in micromechanics of materials. It describes three main topics: 1. How to introduce microstructures into computational models through techniques like unit cell modeling, mesh generation from micrographs, and voxel-based modeling. 2. How to model damage in microstructural models using approaches like property modification with element weakening, smeared cracking models, and embedding discontinuities. 3. How to conduct numerical testing of microstructures to optimize materials, including developing software for automatic microstructure generation and voxel-based reconstruction, as well as using experiments to determine local material properties and failure mechanisms to inform computational models.

1. NUMERICAL EXPERIMENTS
IN MICROMECHANICS OF
MATERIALS
Leon Mishnaevsky Jr.
Risø National Laboratory for Sustainable Energy,
Technical University of Denmark

2. RISØ NATIONAL LABORATORY
FOR SUSTAINABLE ENERGY
Danish National Laboratory, 660
employee, and 8 departments.
founded in 1956 by Niels Bohr,
In 2007, Risø merged with
the Technical University of
Denmark

3. TECHNICAL UNIVERSITY OF
DENMARK
founded in 1829 by Hans Christian
Ørsted (discoverer of electro-magnetism
and aluminium)
DTU ranks as the best Scandinavian
(THES) and 5th best European
university (Leiden ranking).
6000 students (2006), about 700 Ph D
fellows and 580 full and associate
professors

4. Group COMPOSITES at the Materials
Research Department
Development of wind
Sino-Danish project
UPWIND ”3D virtual testing of
energy technologies
in Nepal on the basis
composites for wind
of natural materials
energy applications“
UpWind: Integrated
Wind Turbine Design
funded by EU, 40
participating Funded by Danish Agency
institutions from 40 Sci., Technol. & Innovation, Funded by Royal Ministry
countries; 2005-2011 2009-2010 of Denmark, ~700.000
EURO

5. TODAY’s TALK
Can one optimize microstructures of composites?
How to introduce microstructures into computational
models?
How to model damage in microstructural models?
Computational models of several groups of composites
Particle Reinforced Lightweight Composites
Gradient Composites
Interpenetrating Phase Composites
Fiber Reinforced Composites
Wood as a Hierarchical, Cellular Materials with Layered, Fibril
Reinforced Cell Walls

6. CAN ONE OPTIMIZE
MICROSTRUCTURES OF
MATERIALS?

7. GRADIENT, LAYERED AND
SURFACE COMPOSITES
FGM: Graded, smooth variation Coatings: one of the oldest Surface composites (Singh and
of materials properties allow to technologies to improve the Fitz-Gerald): graded properties of
increase the lifetime of materials reliability and lifetime of materials are achieved by transforming the
under cyclic loading. and components. surface of the bulk material into
truncated cone-like structures using
Graded composites Example: Fatigue life of multiple pulse irradiation
have 4..5 times higher stainless steels coated with technique+ deposition of the
ZrNx increases by surface phase.
service life.
400…1100%.

8. CLUSTERED/ „DOUBLE
DISPERSION“ MICROSTRUCTURES
Fine primary carbides lead to Coarse primary carbides are Double dispersion structure
higher strength and lower wear- brittle and result in the higher ensures high wear-resistance
resistance wear-resistance but lower and high strength
strength
Tool steels with “double dispersion structure” ensures 30%
higher fracture toughness and 8 times higher lifetime, than
a standard tool steel (s. Berns and colleagues, 1998)

9. NETLIKE MICROSTRUCTURES
Inclusion networks can determine the
crack path, and, thus, increase the
fracture toughness of materials
(Broeckmann, 1994, Gross-Weege et al, 1996)

10. HIERARCHICAL MATERIALS
EXAMPLE: Synergy Ceramics Project by the Consortium of Japanese Universities.
The idea is to create a new family of ceramic materials, by tailoring material
properties using the “simultaneous control of different structural elements, such as
shape and size, at plural scale levels”.
Combining of aligned
+ anisotropic grains with the
intragranular dispersion of
nanoparticles in ceramics
= gives high toughness and
high strength (Kanzaki et al,
1999)

11. Properties of materials can be improved
just by varying their microstructures. But
how can we determine the optimal
microstucture?

12. NUMERICAL EXPERIMENTS in the
Mesomechanics of Materials
Development Computational Testing
Numerical of Microstructures
Tools
Acquiring
Experimental Determination of
Data Optimal Micro-
structures and
their Realization

13. CHALLENGES
Numerical experiments require a large number of
complex numerical models. How can they be generated?
How to determine local properties of materials?
How to introduce the complex
microstructures into the
numerical models?

14. HOW TO INTRODUCE
MICROSTRUCTURES
INTO COMPUTATIONAL
MODELS?

15. HOMOGENIZATION
Pierre Suquet (1987): In order to determine
the constitutive equations for the averaged
properties of a heterogeneous material
One defines of a volume element, which is
statistically representative for the whole
microstructure.
Localization (macro-micro transition):
microscopic boundary conditions are
determined on the basis of the macroscopic
strain tensor.
Homogenization (micro-macro
transition): macroscopic properties of the
equivalent homogeneous medium are
determined on the basis of the analysis of
the microscopic behavior of the RVE.

16. 1. Determination of RVE
(Representative Volume
Element). Unit cell models

17. RVE DESIGN: Classification of methods according to
Professor Helmuth BÖHM (TU Vienna)
Periodic Microfield Approaches (PMA) or Unit Cell (UC)
Methods: assuming the periodic phase arrangement, one
analyses a repeating unit cell in the microstructure
Embedded Cell Approach: the materials is represented as
a cut-out (unit cell) with a real microstructure, embedded
into a region of the material with averaged properties
“Windowing approach”: microstructure samples, chosen
using “mesoscale test windows”, randomly placed in a
heterogeneous material, are subject to homogeneous
boundary conditions. By averaging the results for several
“windows”, one can obtain bounds for the overall behavior
of the material (Nemat-Nasser and Hori,1993)
Modeling the full microstructure of a sample
From: H.J. Böhm: Short Introduction to Basic Aspects of Continuum Micromechanics of Materials.Galway, Ireand; Juli 1998.

18. UNIT CELL FOR FIBER
REINFORCED COMPOSITES
(Li, 1999)

19. DESIGN OF MINIMAL UNIT
CELLS
Selection of UC, using
symmetry analysis
(after Li, 1999)
Axisymmetric UC for particle reinforced composite:

20. 3D UNIT CELLS FOR PARTICLE
REINFORCED COMPOSITES
(Bao et al, 1991)

21. AXISYMMETRIC UNIT CELLS
WITH DAMAGE
Debonding
Crack in the
particle
Void
(Mozhev and Kozhevnikova, 1997, Steglich and
Brocks, 1997)

22. UC WITH AND WITHOUT
EMBEDDING

23. 2. Microstructure-based
finite element model
generation

24. MICROSTRUCTURE BASED MESH
GENERATION
Geometry- Voxel-based
based mesh modelling:
generation: ragged phase
from digitized boundaries
photos to FE
mesh

25. VOXEL-BASED GENERATION OF
REAL MICROSTRUCTURES

26. VORONOI CELL FEM
Voronoi-cell finite element method: a microstructure is divided
into Voronoi polygons (a), which are then used as hybrid finite elements (b)
(after Moorthy and Ghosh, 1998)
Dirichlet
tesselation
Prescribed
traction
boundary
Interelement
boundary
Prescribed
displace-ment
boundary

27. MULTIPHASE FINITE ELEMENTS:
Multiphase finite elements:
Interface may run across FE elements.
phase boundary
Integration points of one and the same
element can be assigned to different
phases.
integration FE edges
points
Reconstruction of microstructures from serial sections, and generation of a
microstructural model:

28. HIERARCHICAL MODEL: Example
After Planckensteiner
et al:
the microstructure of a
high speed steel with
the carbide strings is
modeled as a layered
material at the
mesolevel and as a
statistically
homogeneous two-
phase material inside
the strings

29. HOW TO INCLUDE DAMAGE
IN CONTINUUM
MICROMECHANICAL
MODELS?
Challenge:
evolving physical discontinuities with infinitely
small tips are incorporated into continuum
mechanical, discretized problem.

30. CLASSIFICATION OF METHODS
according to Professor Tony Ingraffea (Cornell)
Non-geometrical representation:
properties modification approaches: methods, based on the local
reduction of element stiffness used to represent the crack path
(“constitutive methods”, as computational cells, smeared crack,
element elimination), and
kinematic methods (xFEM, enriched elements),
Geometrical representation:
mesh modification approaches: constrained shape (i.e., if the crack
path is prescribed by the faces of existing elements or by some
theory-based assumptions) and
arbitrary shape methods (meshfree, adaptive FEM/BEM, lattice
methods, etc.).

31. MESH MODIFICATION APPROACHES
Element elimination:
The element is removed
Nodal decoupling (often, followed
by remeshing):

32. PROPERTY MODIFICATION
APPROACHES
Element weakening:
Stiffness of an element is Smeared crack model: The
reduced displacement jump is smeared out over
some characteristic distance across the
crack, which is correlated with the
element size. The degradation of
individual failure planes is described by
the constitutive law.

33. UNIT CELLS PLACED ALONG THE
CRACK PATH
Computational cell model
(after Xia and Shih):
L
Cell model of material
(Broberg)

34. SPECIAL FINITE ELEMENT
FORMULATIONS
FE with special constitutive behavior: cohesive
elements, described by traction-separation law. CE placed in the mesh in sites of
potential damage initiation.
FE with embedded discontinuities (Belytchko, Jirasek) Crack is
simulated using the corresponding choice of the kinematic representation of localized fracture.
The discontinuity, which crosses the element and divides it into two parts, is represented by
additional degrees of freedom, corresponding to the normal and tangential components of the
displacement jump.
Generalized FEM (Babuška) combines the advantages of meshless methods and
Nodal decoupling (often, followed
the standard FEM. Taking into account that the nodal shape functions sum up to unity in the
by remeshing):
modeled area, they suggested to enrich the element shape functions by assumed local
functions.
eXtended FEM (XFEM): The displacement fields is presented as a sum of the
regular displacement field (for the case without any discontinuities), and the enriched
displacement field. Discontinuous enrichment functions are added to take into account the
cracks and singular enrichment functions are added to account for the crack tips.
Meshfree, other connectivity-free, adaptive methods.
Galerkin (EFG) method for the discretization of structures, which are described by gradient-dependent damage models. Since the shape functions in the EFG method are formulated on the basis of the moving least squares principle, not on the basis of element connectivity, one can easily obtain higher order continuity shape functions.
Askes et al. (2000) applied the element free

35. NUMERICAL TESTING OF
MICROSTRUCTURES
How to optimize microstructures of
different groups of materials?

36. 1. Some software
developed in our group
Automatic 3D model generation

37. Automatic Generation of FE Models
of 3D Microstructures
1. Free meshing + Geometry-based modelling
for Particle Reinforced Composites
Exact geometry. Best for testing artificial
microstructures.
2. Voxel-based 3D reconstruction of
microstructures
Geometry is approximated by discrete voxels.
3. Fiber Reinforced Composites with
Damageable elements and Interface Layer

38. 1st Code:
FREE MESHING + 3D GEOMETRY-
BASED MODEL
Input data:
(3, 3, 7)
(6, 6, 2)
volume content of particles, (2, 7, 2)
....
type of particle arrangement
(graded, cluster, random
uniform, regular),
particle shape (sphere,
ellipsoid),average radius of
particles, and standard
deviation of radii.
Output data:
MSC/PATRAN Database
statistical analysis of
generated micro-
structures

39. Automatic Generation and Meshing of
Artificial 3D and 2D Microstructures
Some Designed Microstructures: Localized Structures: graded Embedded
Randomly arranged particle, with and clustered Cell FE
constant and random sizes Model:
Varied particle orientations:
Varied gradient degrees: aligned, random, staggered

40. 2nd Code:
VOXEL-BASED GENERATION OF 3D
MICROSTRUCTURE MODELS
Input data:
[1,0,1, 0,0,0,0 1, 1
0,0,1, 0, 0, 1, 1, 1
voxel array, or
........................]
statistical parameters of
microstructure
Output data:
3D microstructural FE
model (MSC/PATRAN
Database)
The program carries out the
percolation analysis of the
microstructure.

41. 3rd Code:
3D MODEL OF FIBER REINFORCED
COMPOSITES
Input data:
volume content
and amount of
fibers
Output:
MSC/PATRAN
Database

42. ABAQUS Subroutine for Damage
Simulation in Multiphase Materials
ABAQUS Subroutine USDFLD :
damage in element is simulated as local reduction
of stiffness (element weakening),
applies different damage criteria for different
phases of the material:
⌧critical principal stress for brittle phases (particles),
⌧Lemaitre damage or critical strain failure condition (for matrix)
stiffness of an
element is reduced

43. 2. Experiments
Before we simulate, we must know
(a) local properties and
(b) damage mechanisms

44. WHICH MICROMECHANISMS CONTROL
DAMAGE AND FRACTURE OF MATERIALS?
SEM in-situ Investigations of Micromechanisms of Deformation and
Damage in AlSi cast Alloys and Tool Steels
AlSi cast alloys 3-point bending specimen Tool steels
load
observed
area
How the micro- Primary carbides: before and after failure
structure influen-
ces the strength
of the alloys?
* Cracks are initiated by failure of Si
particles caused by dislocation pile-
ups. * Coalescence of cracks follows
the shear bands.
* Al cast alloys with globular microstructures have
much higher failure strain than the alloys with
lamellar microstructure.
L. Mishnaevsky Jr et al., Eng. Fract. Mech. 63/ 4, 1999, pp. 395-411, Zeitschrift f. Metallkunde, 94, 2003, 6, pp. 676-681

45. HOW TO DETERMINE LOCAL PROPERTIES
OF CONSTITUENTS AND PHASES?
Hierarchical and inverse modelling
Hierarchical (macro-micro) FE model of
carbide failure in tool steels:
micromodel includes the real microstructure
macromodel reproduces the specimen
Primary carbides:
before & after failure
applied load
Type of the Cold High speed steel, High speed steel,
steel work normal to the bands along the bands
Failure
stress of 1826 1604 2520
observed carbides
area (MPa)
L. Mishnaevsky Jr et al., Zeitschrift f. Metallkunde, 94, 2003, 6, pp. 676-681

46. Experimental-Numerical Methode:
MAIN POINTS:
SEM in-situ experiments: whereas the damage process is
observed and recorded under SEM, the macroscopical F-U
curve is recorded as well.
Macro-micro simulation: the macroscopical model of the full
specimen with a submodel (microstructural model) of the zone
where the damage is observed.
Inverse modeling: the strength and failure conditions of
phases (local) is determined by comparing the micro-macro FE
model with micro-macro observations in the experiments.

47. 3. Numerical testing of
different microstructures
Lightweight metal matrix (Al)
composites reinforced by ceramics
particles (SiC)

48. 3D Numerical Testing of
Microstructures of Al/SiC Composites
Distributions of Plastic
Strains: on box Stress-strain curves & fraction of failed
boundary particles vs. strain
On particle/matrix
interface
Failure strain of composite:
Failure strain of compo-
sites increases in the
In a vertical section following order:
clustered < regular <
random < gradient
microstructure.

49. Mechanical Behavior of Polymer
Composites (Polypropylene + Glass)
Steps: Here: D0 is the instantaneous
Homogenisation of the ε( t ) = g 0 (σe )D0 Se : σ( t ) + compliance, g0; g1; g2 and a are
composite (nonlinear scalar functions of an equivalent
t ⎛ t dt ' ⎞
viscoelastic matrix +elastic + g1 (σe ) ∫0 ΔD⎜ ∫τ
⎜ a (σ ) ⎟
⎟ stress σ; ΔD(t) is a linear
⎝ e ⎠ viscoelastic creep compliance;
( )dτ
particles), using
tensors Se and Sc are 4th order
affine formulation (tangent d g 2 (σe )Sc : σ(τ)
tensors containing the elastic and
linearisation of each phase) and
Mori-Tanaka scheme dτ creep Poisson’s ratios.
Implementation of 3D
Shapery law into UMAT
Comparison of the
theoretical model with the
numerical 3D model
This work has been carried out together with Dr. M. Levesque and Prof. D. Baptiste (ENSAM,
France). (see M. Levesque, et al., Composites Part A: Appl. Sci & Manuf, 35, 2004, 905-913)

50. 4. Fiber reinforced
plolymer composites
Competing damage mechanisms: fiber
cracking, matrix fracture and interface
damage
L. Mishnaevsky Jr., P. Brøndsted, Composites Science and Technology, Vol. 69, No. 7-8, 2009, pp 1036-1044 Vol. 69, No3-4, 2009, pp. 477-484, Computational
Materials Science, Vol. 44, No 4, 2009, pp 1351-1359

51. MODELLING OF DAMAGE IN FRC
Fiber cracking: in Matrix cracking: in
potential damageable ”interface layer”
planes
F M
Fiber bridging:
F I M
Matrix crack growth from a fiber crack

52. MODELLING OF DAMAGE IN FRC
Overloaded fibers
near a failed fiber: Effect of variability of fiber properties:
400
350
300
Stress, MPa
250
200
Random (Weibull)
150 fiber strenths,
viscoelastic matrix
100
Constant fiber
50 strength
0
0 0,005 0,01 0,015 0,02 0,025 0,03
Strain
3 Competing Damage Modes:
Damage Evolution: Strong Matrix
The interface crack is formed in the vicinity of a fiber crack
Fiber cracking causes interface damage, and then leads to and the matrix crack is formed far away. Weak interface
interface damage at neighbouring fiber delays matrix cracking!

53. 5. Functionally gradient
composites
What is the effect of microstructural
gradient on strength and damage?
L. Mishnaevsky Jr., Composites Sci. & Technology, 2006, Vol 66/11-12 pp 1873-1887

54. Numerical Testing of Generic
Gradient Microstructures
Design of Artificial Graded Von Mises Stress Distribution
Gradient 3 microstructure
Microstructures:
Disp=1. Disp=5. Disp=15.
Damage in particles and in matrix
Varying the dispersion of the
distribution, we can obtain highly
gradient, as well as almost non-
gradient particle distributions.

55. Effect of the Degree of Gradient on
the Strength and Damage Evolution
Fraction of failed particles vs. strain for
different gradient degrees:
Flow stress and stiffness of
composites decrease, and
the failure stress increases
with increasing the gradient
degree.
Degree of homogeneity =1/Gradient degree

56. 6. Crack growth in tool
steels
Which arrangement of primary carbides
ensures maximum toughness?
L. Mishnaevsky Jr et al., Int. J. Fracture, Vol. 120, Nr. 4, 2003, pp. 581-600, Int. J. Fracture, 125: 33-50, 2004

57. NUMERICAL TESTING OF TOOL STEELS (1)
Effect of Microstructure on the Fracture Toughness of Tool Steels
FE Simulation of Crack Growth FE Simulation of Crack Growth in
in the Real Microstructure Artificial Microstructures

58. NUMERICAL TESTING OF TOOL STEELS (2)
Effect of Microstructure on the Fracture Toughness of Tool Steels
60
50 Band-like fine
Net-like fine
40
Random fine
30
20
10
0
0 0,001 0,002 0,003 0,004 0,005
Displacement, mm
Fracture energy is calcu- Fracture resistance of steels with layered & clustered
lated as: G=Σ Pi ui/L, microstructures are higher than those with simple micro-
where Pi - force and ui - dis-
placement at increment i, L- structures. Net-like fine microstructure shows an exception to this rule. However,
length of the microstructure such a mechanism of toughening (crack follows the carbide network) is unstable.
area.

59. 7.Interpenetrating Phase
Composites
L. Mishnaevsky Jr., Materials Science & Engineering A, Vol. 407, No. 1-2, 2005, pp.11-23

60. OVERVIEW
3D cubic model by triangular prism unit cell model
Daehn et al (1996) by Wegner and Gibson (2000).
sphere
interstiti
2-phase and 3-phase models by al matrix
Feng et al. (2003, 2004)
matricity model by Lessle, Dong
and Schmauder

61. OUR APPROACH
Unit cell model of interpenetrating phase composite
Isotropic
Gradient

62. EFFECT OF THE CONTIGUITY OF
INTERPENETRATING PHASES
Stress-strain curves & critical strain plotted versus vol. content of pcles
No M percolation
P & M percolation
No P percolation
Peak stress plotted versus maxim.
cluster size
Stiffness of composites increases almost linearly with
increasing the maximum size of particle cluster up to
the percolation threshold.
A composite (ductile matrix + brittle inclusions)
where the inclusions form a percolation cluster
behaves as a brittle material.

63. GRADIENT INTERPENETRATING
PHASE COMPOSITES
Modeling of sharp/smooth
graded interfaces Examples of the unit cells Stress-strain curves
(examples):
2vc0
vc( y ) =
1 + e g − 2 gy / L
Peak stress vs. sharpness
of the interface
Stiffness of graded
composites increases,
when the graded
interface becomes
smoother.

64. 8. Multiscale model of
wood
Wood as an hierarchical cellular material
with layered cell walls and fibril reinforced
wall sublayers
H. Qing, L. Mishnaevsky Jr., Mechanics of Materials (in press), Comput. Matls Science (in press), Comput. Matls Science, Vol.44, 2, 2008, pp.363-370

65. HIERARCHICAL MODEL OF WOOD
θ θ θ
Halpin-Tsai model
Multiscale model of wood:
Mesolevel: the layered honeycomb like microstructure of
cells is modelled as a 3D unit cell with layered walls.
Submicrolevel: Each of the layers forming the cell walls was
considered as an unidirectional, fibril reinforced
composite.

66. HIERARCHICAL MODEL OF WOOD
COMPUTATIONAL STUDIES SOME OBSERVATIONS
Effect of Micrifibril Angle in S2 Layer The thickest and strong S2
on elastic properties sublayer is responsible for the
shear strength, while strong and
stiff “interphase” layers S1 and S3
are important to ensure the
integrity of wood under XZ loading
Microfibril angles in different
sublayers of the cell wall control
different properties of the wood
Generally, different parameters of
multiscale microstructure are
Effect of cell shape on elastic responsible for different loading
properties strengths.

67. 9. Wear of diamond
grinding wheels

68. MESOMECHANICAL ANALYSIS OF WEAR
OF DIAMOND GRINDING WHEELS
Von Mises strain distribution
on grinding wheel suface
FE model of a cutout of Fraction of failed elements in
wheel surface
diamond grains versus force
Mesomechanics approach is
applicable to the analysis of the
grinding and grinding wheel wear.

69. CONCLUSIONS
Strength and damage resistance of materials can be improved
by varying the microstructures of materials.
The optimal microstructure of materials can be determined by
using numerical experiments.
A number of new numerical tools for the microstructural
computational testing of materials have been developed and
eployed for the numerical testing of microstructures: programs
for the geometry-based and voxel-based generation of 3D microstructural model
of composites, subroutines and programs for damage simulation, etc.

70. References:
L. Mishnaevsky Jr, Computational Mesomechanics of Composites,
Wiley, 2007, 290 pp.
S. Schmauder, L. Mishnaevsky Jr, Micromechanics and
Nanosimulation of Metals and Composites, Springer, 2008, 420 pp.
L. Mishnaevsky Jr, Damage and Fracture of Heterogeneous Materials,
Balkema, Rotterdam, 1998, 230 pp.
Some papers are available
on:
http://risoe-
staged.risoe.dk/
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