Micromechanical motivations of generalised continuum models Stefanos-Aldo Papanicolopulos The University of Edinburgh Aussois, 28 September 2015.

Micromechanical motivations of generalised

continuum modelsStefanos-Aldo Papanicolopulos

The University of Edinburgh

Aussois, 28 September 2015

26ᵗʰ ALERT Geomaterials Workshop 2

Acknowledgements

• This research effort is funded from thePeople Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007-2013) under REA grant agreement nº 618096.

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Outline

• Introduction (Micromechanics & GCMs)• Mathematical origin of GCMs• Theoretical micromechanical arguments• A simple example• Experimental data• Numerical multiscale methods• Conclusions

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Introduction

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Micromechanics

• The study of the mechanical behaviour of materials with microstructure…• …when microstructure

plays an important role.• Considers (at least) two

different length scales• Microstructure size may

affect macro responseGrain breakage within a shear

band (Karatza et al., 2015)

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Generalised continuum models

Classical continuum:1. Traction on a face is

linear in the normal2. Space and body are

Euclidean, connected3. No volume/surface

couples4. No “microstructure”

described by additional degrees of freedom

(Maugin, 2010)

• Generalised continuum models (GCMs) relax (at least) one of the classical assumptions• Examples: Cosserat,

strain-gradient, micromorphic, non-local• Introduce a material

length scale

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Mathematical origin of generalised continuum models

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Strain gradient models

• Cauchy (1851):“[Stresses] can generally be considered as linear functions of the displacements and their derivatives of different orders.”• Second gradient models

Toupin (1962), Mindlin (1964), Mindlin & Eshel (1968)

• Mathematically:

• Using necessarily introduces a new “double stress” that is energy conjugate to

• Using also introduces a material length

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Cosserat models

• The material point has both displacement and rotation • General form

• Relative deformation

• Distortion

• The Cosserat material point has the same dofs as a rigid body…• …so it could model

granular materials• The (continuum)

Cosserat rotation is not the (discrete) rotation of the grains

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Micromorphic models

• The material point is deformable• E.g. microdeformation as a

new (tensor) dof

• Stress , relative stress and double stress :

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Mindlin (1964)

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GCMs as a mathematical tool

• Material behaviours like strain softening lead to ill-posedness of the governing equations for classical continua• Generalised continua

regularise the problem removing ill-posedness• Also introduce a length

scale dimensionally needed for size effect

• GCMs can be (and often are) seen as simply mathematical tools (e.g. Matsushima et al., 2002)• Many attempts to

provide a physical(i.e. micromechanical) basis for GCMs

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“Side effects”

• Additional kinematic and static quantities• Additional boundary

conditions• New material models

introducing additional parameters

• Additional BCs may have a physical meaning• Bolted layer example

(Vardoulakis, 2004)

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Edge tractions in gradient elasticity

• Strain gradient theory has edge tractions associated with corners/ edges• Uniaxial loading in plane

strain, with “rough” BC

• Corners affect the result even without applied edge traction

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Papanicolopulos & Zervos (2010)

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Theoretical micromechanical arguments

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Discrete to continuum

• Extensive work in discrete to continuum transformations• Different definitions/

computations of continuum quantities are possible• Bagi (2006): “…an

overview on 10 different microstructural strain definitions…”

• Different definitions have significant theoretical interest, but not always great practical differences

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Strains

• Strains based on equivalent continuum or on best fit• Particle rotations often

ignored (but e.g Kruyt 2003, Kruyt et al. 2014)• Averaging volume needs

to be defined• Similar discussions

about stressesBagi (1996, 2006)

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Applications to GCMs?

• Discrete to continuum transformations in literature often applied to DEM simulations• Some transformations

use a generalised continuum framework…• But few applications to

obtaining generalised continuum constitutive models

Couple stresses(Ehlers et al., 2003)

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A simple examplePapanicolopulos and Veveakis (2011)

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Grain contact mechanics

• Two grains of radius with sliding or rolling contact

• Interaction through forces (sliding) and couples (rolling)

• Power of forces/couples:

• Relative velocity at contact:

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Continuum embedment [1]

• Embed kinematic quantities in continuous field and linearise:

• Also: • the (Cosserat) rate of

relative deformation and the rate of distortion

• Generate forces from stress field and couples from couple stress field

• Powers:

with

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Continuum embedment [2]

• Referring quantities to the centre of particle 1:

• Powers:

• Same form for but different form for ,

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Fabric averaging

• Continuum embedment gives quantities that depend on the contact normal • Eliminate dependency by computing a fabric average

considering probability distribution of • For uniform distribution

• For (i.e. ) we get the power of Cosserat continuum

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Stress invariants [1]

• 3 independent invariants for symmetric stress • 39 independent invariants

for general and , of which 8 of order 1 or 2, need physical justification• Decompose and in

spherical and deviatoric

• Introduce stress and couple stress vectors

• …and decompose into normal and tangential components, e.g.:

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Stress invariants [2]

• Stresses

• Couple stresses

• Should use contact quantities, so:

• Simple, -type plasticity models should use , , and

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2D Cosserat Mohr-Coulomb plasticity

• Stresses: • Couple stresses?

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Experimental data

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Macroscale

• Many experimental results showing size effect (e.g. shear banding, indentation, torsion)

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Papamichos et al. (2006)

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Microscale

• Gauthier & Jashman (1975): A quest for micropolar elastic constants.• Oda et al. (1982)

experiments on photoelastic oval-section rods (effect of particle rolling)

• Many other experimental results on model materials

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Sibille and Froiio (2007)

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X-ray computed tomography

• Digital image correlation • Particle tracking

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Besuelle et al. (2006) Ando et al. (2012)

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Working with microscale data

• Usually only kinematics• Contact information more difficult to assess• Possibly erroneous or missing measurements• New homogenisation procedures required…• …especially for GCMs• Particle rotations are half of the available information

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Numerical multiscale methods

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FEMxFEM (FE2)

• FEM analysis of an REV for each Gauss point of the macro FEM analysis• No need for a macro-

scale continuum constitutive model• Increased

computational cost Computational homogenization scheme

(Kouznetsova et al., 2002)

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FEMxDEM

• DEM analysis of an REV for each Gauss point of the macro FEM analysis• As in FE2, no need for

macro- model but(even more) increased computational cost

Nguyen et al. (2014)

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Multiscale GCMs

• Multiscale approaches eliminate the macro-scale constitutive model• Must still choose macro-

scale continuum type• A classical continuum

still can’t represent microstructural size• Generalised continua

used for regularisation

• E.g. strain gradient FE2 (Kouznetsova et al., 2002), Cosserat FE2 (Feyel, 2003), strain-gradient FEMxDEM (Desrues et al., 2015)• Clearly micromechanical

analysis…• …but micromechanical

motivation?

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Conclusions

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Conclusions

• Generalised continua an accepted solution for modelling microstructure-size-dependent phenomena• Often seen as a regularisation, but significant interest in

micromechanical motivation• Discrete-to-continuum transformations can provide important

insight• Even simple micromechanical motivations can inform

modelling choices• Experimental particle tracking and numerical multiscale

methods are interesting new applications• Increasing use of generalised continua (in research) but less

emphasis on generalised continuum constitutive models

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