Eective Properties of Micro-Heterogeneous Materials

8/10/2019 E ective Properties of Micro-Heterogeneous Materials

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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.



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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)



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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



5/18

Nanoscale Problems



6/18

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)



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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



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Natural Material: Wood

Composition:



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Human Bodies are Multiscale Systems



11/18

Hierarchical Structure of Bones



12/18

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



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Further Multiscale Engineering Materials

Polymers:

Glass-filled polymer Single polymer chains

Engineering Composite Materials:

Carbon-fiber tube Sandwich construction



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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



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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



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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!



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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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Eective Properties of Micro-Heterogeneous Materials