MODELING OF MATERIALS - INTRODUCTION

MODELING OF MATERIALS - INTRODUCTION

INTRODUCTION TO COMPUTATIONAL MECHANICS OF MATERIALS Civil Engineering, 1st cycle studies, 7th semester

elective subject academic year 2014/2015

Institute L-5, Faculty of Civil Engineering, Cracow University of Technology

Jerzy Pamin

Adam Wosatko

Lecture contents

About modelling of materials

From materials science to mechanics of materials

Notion of continuum mechanics

Homogenization - multi-scale analysis

Classification of structural and material models Simplifications and finite elements

Materials used in civil engineering

From materials science to mechanics of materials

(Taken from www.mate.tue.nl)

Levels of observation e.g. for metals

Classification of materials:

I metals and alloys

I ceramics and glass

I polymers

I semiconductors

I composites Applications:

I structural (e.g. wood)

I plastics (e.g. polyvinyl chloride PVC)

I composites (e.g. laminates)

I electronic (e.g. isolators)

I magnetic (e.g. ferromagnetics)

I optical (e.g. LED)

Classification of materials – atomic structure

Distinction based on atom ordering:

I no order (e.g. argon - monoatomic gas)

I short-range order (atom linked only with nearest neighbours, e.g. silicate glass) - amorphous materials

I long-range order (special atomic arrangement over more than 100 nm, e.g.

metals, aloys, semiconductors) - crystalline materials (single crystals, e.g.

silicon, or polycrystals, e.g. aluminium alloys) Remarks:

I Periodic atom structure is preferred, since such configuration is optimal from the viewpoint of thermodynamic stability

I Imperfections (defects) in crystal structure have significant influence on its properties; one distinguishes point defects (vacancies, impurities, dopants - deliberately added compounds), line defects (dislocations), surface defects (grain boundaries)

I Dislocation motions (slips in crystal lattice) decide about plasticity (permanent

deformations) and material ductility

Computational mechanics

Physical scale

I Nanomechanics (particle physics, chemistry)

I Micomechanics (physics of crystals, microstructures)

I Continuum mechanics (assumption of continuous fields, homogenization, phenomenological models)

I Mechanical structures (airplanes, bridges, robots, engines, . . .) Continuum mechanics:

I solids and solid structures

I fluids (CFD)

I coupled problems (multiphysics)

Fire Dynamics Simulator and Smokeview (FDS-SMV)

Fire simulation (M. Kwapisz)

Concept of continuum in mechanics

Continuum

Y Z

X

V

Continuous medium (continuum) is characterized by continuous distribution of matter (mass) in space.

The density function is well-defined at every point:

ρ = dM dV = lim

V n →0

M n

V n

,

where infinitesimally small volume V _n is large enough to contain an infinitely large number of particles (discrete structure of matter is neglected). Continuum is a mathematical idealization of the real world.

Phenomenological models are based on macroscopic observations. To

consider micro-characteristics an improved continuum theory (micropolar

or nonlocal) must be considered.

Numerical homogenization – macro-micro description

Taken from:

V. Kouznetsova, Computational homogenization for the multi-scale analysis of multi-phase

materials, Eindhoven University of Technology, 2002.

Mathematical and physical models

Variation in time:

I stationary problems – independent of time (statics)

I non-stationary problems – dependent on time (dynamics) Possible simplifications based on certain hypotheses:

I kinematic/geometrical, e.g. dominant dimensions, cross section type

I static/dynamic, e.g. slow/fast loads, specific load direction Mathematical models are:

I linear (small deformations – displacements and strains, and Hooke’s law) → superposition principle holds

I nonlinear

Classification of structural models

and finite elements (FEs)

Reduction of dimensionality:

I solids (three-dimensional 3D)

I panel (mambrane), plate and shell structures (two-dimensional 2D)

I bar (frame) structures (one-dimensional 1D)

Finite elements for mechanics:

I 1D – truss element

I 1.5D – beam/frame elements

I 2D – elements for plane stress (panel), plane strain and axial symmetry

I 2.5D – plate/slab, shell elements

I 3D – volume (brick) elements

Classification of material models

Phenomenological (macroscopic)

Material models (constitutive relations):

I elastic – body returns to initial configuration upon unloading, no energy dissipation

I plastic – body exhibits permanent deformation upon unloading

I damage mechanics – body undergoes material degradation upon loading, continuum representation of defects

I fracture mechanics – body undergoes discrete cracking upon

loading, crack propagation is traced

Features of constitutive models

Materials can be:

I homogeneous – properties are the same at all points, e.g. steel

I heterogeneous – properties depend on point, e.g. composites

I isotropic – material properties at a point are similar in all directions, e.g. steel

I anisotropic – different properties in different directions, e.g. wood

I isonomic – properties in certain direction do not depend on sense, e.g. steel

I anisonomic – properties in certain direction depend on sense (e.g.

wood or concrete have different properties in tension and

compression)

Computational models of materials

Options:

I discrete (discrete element method, lattice models)

I continuous (classical FEM)

I discontinuous (interface/crack modelling)

Materials used in civil engineering

I Steel – material assumed to be isonomic, isotropic, ductile (low carbon steel), exhibits permanent strains → plasticity theory

I Concrete – multicomponent material, material model usually homogeneous (aggregate grains and cement matrix not distinguished), anisonomic, isotropic, quasi-brittle, cracking → damage mechanics, macro-crack modelling (e.g. XFEM)

I Timber – material is anisonomic, anisotropic → nonlinear constitutive models for anisotropic materials

I Glass – material is isonomic, isotropic, brittle → fracture mechanics

(linear elastic)

Material properties – stiffness and ductility

Taken from:

D.R. Askeland, P.P. Phul´ e.The Science and Engineering of Materials, Fifth Edition, Thomson,

2006.

Material properties - temperature dependence

Taken from:

D.R. Askeland, P.P. Phul´ e.The Science and Engineering of Materials, Fifth Edition, Thomson,

2006.

Material behaviour - stress and strain response

Taken from:

D.R. Askeland, P.P. Phul´ e.The Science and Engineering of Materials, Fifth Edition, Thomson,

2006.

Material behaviour - stress and strain response

Taken from:

D.R. Askeland, P.P. Phul´ e.The Science and Engineering of Materials, Fifth Edition, Thomson,

2006.