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Effective Properties of Micro-HeterogeneousMaterials
Regular Course in Summer Term 2011
Lectures:Dr.-Ing. Daniel Balzani (V15 S06 D21)Dates:Tuesday 9:45 - 12:00, V15 S04 C57 (rotation with exercise)
Exercises:Dr.-Ing. Daniel BalzaniDates:Tuesday 9:45 - 12:00, V15 S04 D22 (rotation with lecture)
Examinations:seminar paper written with Latex (critical review on scientificpaper) and oral examination
Course material: lecture notes and additional literature
Moodle: all announcements (room changes, news, etc.) and all files required forthe exercises as well as the lecture notes are provided at moodle
c Dr.-Ing. Daniel Balzani, Institut fur Mechanik, Universitat Duisburg-Essen
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Preliminaries
Who attended the modules...?
Finite-Element-Method 1
Continuum Mechanics
Finite-Element-Method 2
At which level are your programming skills?
Matlab is required for the exercises, so please make sure before the first exercisethat you know about the basics of matlab
Do you know Latex?
Latex is a document language for the professional editing of scientific books,papers or reports.The seminar paper has to be written in Latex, thus, please make yourself familiarwith the basic structural elements and commands.
c Dr.-Ing. Daniel Balzani, Institut fur Mechanik, Universitat Duisburg-Essen
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Homogeneous Materials?
Macroscale Mesoscale (Microscale) Microscale
Usual procedure:
1. Assume homogeneous material
2. Perform (macroscopic) experiments to determine macroscopic material behavior3. Construct suitable constitutive law, e.g. Hooke ( =E)
4. Calculate (macroscopic) structural problems
Limitation: Usage of simple constitutive laws is not possible for all materialclasses ( anisotropy, damage evolution, plasticity, microscopic eigenstresses)
c Dr.-Ing. Daniel Balzani, Institut fur Mechanik, Universitat Duisburg-Essen
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The Problem of Scales
Macroscale Mesoscale Microscale Nanoscale
Lengthscale:
Timescale: days 106 sec 1014 sec
meters m nmmm
sec
Systems material
matrix-/Structuresinclusions
grains
crystals
molecules
atoms
Focus in this course: Different lengthscales rather than timescales
c Dr.-Ing. Daniel Balzani, Institut fur Mechanik, Universitat Duisburg-Essen
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Nanoscale Problems
c Dr.-Ing. Daniel Balzani, Institut fur Mechanik, Universitat Duisburg-Essen
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Multiphase Steels
Field of applications:
light-weight constructions, bridge structures
enhancement of crash safety
Stonecutters Bridge Hong Kong
Material behavior governed by complexcomposition of microstructure
several phases: inclusion phases + matrix phase
Ferritic-/perlitic steel Ferritic-/perlitic-/martensitic steel Ferritic-/martensitic steel (DP-steel)
c Dr.-Ing. Daniel Balzani, Institut fur Mechanik, Universitat Duisburg-Essen
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Analysis of a DP-steel/max[M]
l/l0
Martensite (M)
Volfrac-computation ()
DP-steelFerrite (F)
1. Laboratory generation of pure ferritic and pure martensitic steel
2. Mechanical testing (uniaxial tension) of individual phases
3. Adjustment of a simple material law to experiments
4. Identification of martensitic volume fraction (VM 0.2) and computation ofvolumetric average =VMM+ VFF (volfrac computation)
Oversimplified volfrac-computation does not fit experiment of DP-steelc Dr.-Ing. Daniel Balzani, Institut fur Mechanik, Universitat Duisburg-Essen
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Polycrystalline materials
Ferritic steel Damascus steel
Micro-heterogeneities need not to be necessarily defined by the existence ofdifferent materials
Polycrystals (as e.g. ferritic steels): different grains are characterized by differentcrystal orientations
No matrix-inclusion microstructure
Higher stiffness at grain boundaries leads to a grain size dependency of thematerial behavior
c Dr.-Ing. Daniel Balzani, Institut fur Mechanik, Universitat Duisburg-Essen
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Natural Material: Wood
Composition:
c Dr.-Ing. Daniel Balzani, Institut fur Mechanik, Universitat Duisburg-Essen
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Human Bodies are Multiscale Systems
c Dr.-Ing. Daniel Balzani, Institut fur Mechanik, Universitat Duisburg-Essen
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Hierarchical Structure of Bones
c Dr.-Ing. Daniel Balzani, Institut fur Mechanik, Universitat Duisburg-Essen
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Histology of Arterial Walls
Healthy elastic artery (Junqueira[1991]:
Adventitia
IntimaMedia
Membrana
elastica internaEndothel
Collagen fibers:
Differentiate two types of arteries:
Elastic arteries (large diameter, located close to heart, e.g. aorta) Muscular arteries (located at periphery, e.g. cerebral arteries)
Composition:
Ground-substance
Embedded fibers (collagen-/elastin fibers, smooth muscle cells)c Dr.-Ing. Daniel Balzani, Institut fur Mechanik, Universitat Duisburg-Essen
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Reinforcing Collagen Fibers are Substructured
(10 % of the length of Tropocollagen;
columns overlaps
microfibril
Collagenfibril Collagenfiber
Collagenfibers
300 nm
Tropocollagen
bundle ofoverlaps
light stripe)
With columns(dark stripe,
35 nm)
67 nm
c Dr.-Ing. Daniel Balzani, Institut fur Mechanik, Universitat Duisburg-Essen
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Further Multiscale Engineering Materials
Polymers:
Glass-filled polymer Single polymer chains
Engineering Composite Materials:
Carbon-fiber tube Sandwich construction
c Dr.-Ing. Daniel Balzani, Institut fur Mechanik, Universitat Duisburg-Essen
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Simple Example for Scale-Bridging
Attempt to explain linear elasticity based on the description of the atomic bonding
Macroscopic observation:
E
At the macroscale many materials undergoing small deformations can bedescribed by Hookes law:
=E
c Dr.-Ing. Daniel Balzani, Institut fur Mechanik, Universitat Duisburg-Essen
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Simple Model for Atomic Bindings
Material constants (depending on kind of atoms): A and B
The stiffness of the binding is then computed by
S=dF
dr
Evaluation of stiffness at reference state leads to
S0=dF
dr
r0
c Dr.-Ing. Daniel Balzani, Institut fur Mechanik, Universitat Duisburg-Essen
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Elasticity of a Crystal Lattice
Body-Centered-Cubic (BCC)
Face-Centered-Cubic (FCC)
Hexagonal-Closest-Packing (HCP)
Mechanical stress is computed by
=N F =N S0(r r0)
With N bindings/unit area
N 1
r20and by defining the strain
=r r0
r0
we obtain the stresses
1
r
2
0
S0 r0 =S0
r0
and the elasticity modulus yields
ES0
r0, attention: r0=f()
Crystal materials behave linear elastic in the small strain domain!
c Dr.-Ing. Daniel Balzani, Institut fur Mechanik, Universitat Duisburg-Essen
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Micro-Structural AnalysisMicroscale
Macroscale
Real micro-heterogeneous
material behavior
Approximative homogeneous
behavior at the macroscale
Is it possible to find the properties of a homogeneous material that approximatesthe behavior of the original micro-heterogeneous case based on micro-mechanical
considerations?c Dr.-Ing. Daniel Balzani, Institut fur Mechanik, Universitat Duisburg-Essen
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