1. Micromechanical motivations of generalised continuum models
Stefanos-Aldo Papanicolopulos
The University of Edinburgh
Aussois, 28 September 2015
The session is titled “From discrete methods to continuum methods: Applications, Problems and Solutions”.

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Acknowledgements
This research effort is funded from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/ ) under REA grant agreement nº 
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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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5. 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 response
Grain breakage within a shear band (Karatza et al., 2015)
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6. Generalised continuum models
Classical continuum:
​Traction on a face is linear in the normal
Space and body are Euclidean, connected
No volume/surface couples
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
Point 1 of the classical continuum is “Cauchy’s postulate”
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7. Mathematical origin of generalised continuum models
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8. 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: 𝜎=𝑓 𝜀,𝛻𝜀,…
Using 𝛻𝜀 necessarily introduces a new “double stress” that is energy conjugate to 𝛻𝜀 {𝜎,𝜇}=𝑓 𝜀,𝛻𝜀,…
Using 𝛻𝜀 also introduces a material length ℓ
One could say that the classical case is a simplification of the generalised, instead of saying that the generalised is an extension of the classical.
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Cosserat models
The material point has both displacement 𝑢 and rotation 𝜓
General form 𝜎,𝜇 =𝑓 𝛾 𝑖𝑗 , 𝜅 𝑖𝑗
Relative deformation 𝛾 𝑖𝑗 = 𝑢 𝑗,𝑖 − 𝑒 𝑖𝑗𝑘 𝜓 𝜅
Distortion 𝜅 𝑖𝑗 = 𝜓 𝑗,𝑖
The Cosserat material point has the same dofs as a rigid body…
…so it could model granular materials
The (continuum) Cosserat rotation 𝜓 is not the (discrete) rotation of the grains
Symmetric part of the relative deformation is the strain, the antisymmetric is the relative rotation
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Micromorphic models
The material point is deformable
E.g. microdeformation as a new (tensor) dof 𝜓 𝑖𝑗 = 𝑢 𝑗,𝑖 ′
Stress 𝜏 𝑖𝑗 , relative stress 𝜎 𝑖𝑗 and double stress 𝜇 𝑖𝑗𝑘 : 𝜏 𝑖𝑗 , 𝜎 𝑖𝑗 , 𝜇 𝑖𝑗𝑘 = 𝑓 (𝑢 𝑖,𝑗 , 𝑢 𝑗,𝑖 − 𝜓 𝑖𝑗 , 𝜓 𝑖𝑗,𝑘 )
Mindlin (1964)
The kinematic quantities are: strain, relative deformation and micro-deformation gradient
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11. 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
Point 1: Loss of ellipticity; Point 5: …or a GCM basis for physical observations
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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)
Boundary conditions must be specified, material models must be defined, parameters must be evaluated
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13. Edge tractions in gradient elasticity
Strain gradient theory has edge tractions associated with corners/ edges
Uniaxial loading in plane strain, with “rough” BC 𝜕 𝑢 𝑦 𝜕𝑦 =0 for 𝑦=±𝐻
Corners affect the result even without applied edge traction
Papanicolopulos & Zervos (2010)
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14. 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 stresses
Bagi (1996, 2006)
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17. 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 example
Based on an idea by I. Vardoulakis
Papanicolopulos and Veveakis (2011)
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19. Grain contact mechanics
Two grains of radius 𝑅 𝑔 with sliding or rolling contact ℓ 𝑖 = 𝑅 𝑔 𝑛 𝑘
Interaction through forces (sliding) and couples (rolling)
Power of forces/couples: 𝑃 (𝑛𝑠) = 1 𝑉 𝑓 𝑖 2,1 𝑐 𝑣 𝑖 2,1 𝑐 𝑃 (𝑛𝑠) = 1 𝑉 𝑚 𝑖 2,1 𝑐 𝑤 𝑖 2,1 𝑐
Relative velocity at contact: 𝑣 𝑖 2,1 𝑐 = 𝑣 𝑖 2 − 𝑣 𝑖 − 𝑒 𝑖𝑗𝑘 𝑤 𝑗 𝑤 𝑗 ℓ 𝑘
Reference volume V not specified, 𝑣 𝑖 2,1 𝑐 is the relative velocity of grain 2 with respect to grain 1 at the contact, 𝑓 𝑖 2,1 𝑐 is the force acting on grain 2 by grain 1, ℓ 𝑘 = 𝑅 𝑔 𝑛 𝑘 is “half” of the branch vector
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20. Continuum embedment [1]
​Embed kinematic quantities in continuous field and linearise: 𝑣 𝑖 (1) = 𝑣 𝑖 𝑐 − ℓ 𝑚 𝑣 𝑖,𝑚 𝑐 𝑣 𝑖 (2) = 𝑣 𝑖 𝑐 − ℓ 𝑚 𝑣 𝑖,𝑚 𝑐 so that 𝑣 𝑖 2,1 𝑐 =2 ℓ 𝑚 Γ 𝑚𝑖 𝑐
Also: 𝑤 𝑖 2,1 𝑐 =2 ℓ 𝑚 K 𝑚𝑖 𝑐
Γ 𝑚𝑖 𝑐 the (Cosserat) rate of relative deformation and K 𝑚𝑖 𝑐 the rate of distortion
Generate forces from stress field and couples from couple stress field 𝑓 𝑖 2,1 𝑐 = 𝜎 𝑘𝑖 𝑐 𝑛 𝑘 𝑆 𝑚 𝑖 2,1 𝑐 = 𝜇 𝑘𝑖 𝑐 𝑛 𝑘 𝑆
Powers: 𝑃 (𝑛𝑠) =𝛼 𝑛 𝑘 𝑛 𝑚 𝜎 𝑘𝑖 𝑐 Γ 𝑚𝑖 𝑐 𝑃 (𝑛𝑟) =𝛼 𝑛 𝑘 𝑛 𝑚 𝜇 𝑘𝑖 𝑐 𝐾 𝑚𝑖 𝑐 𝑃 (𝑛) =𝛼 𝑛 𝑘 𝑛 𝑚 ( 𝜎 𝑘𝑖 𝑐 Γ 𝑚𝑖 𝑐 +𝜇 𝑘𝑖 𝑐 𝐾 𝑚𝑖 𝑐 )
with 𝛼= 2 𝑅 𝑔 𝑆/𝑉
The surface 𝑆 is not specified, as we did before for the volume 𝑉.
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21. Continuum embedment [2]
Referring quantities to the centre of particle 1: Γ 𝑖𝑗 𝑐 = Γ 𝑖𝑗 (1) + 𝑒 𝑗𝑘𝑙 ℓ 𝑘 𝐾 𝑖𝑙 (1) 𝜎 𝑖𝑗 𝑐 = 𝜎 𝑖𝑗 (1) 𝐾 𝑖𝑗 𝑐 = 𝐾 𝑖𝑗 (1) 𝜇 𝑖𝑗 𝑐 = 𝜇 𝑖𝑗 (1) + 𝑒 𝑗𝑘𝑙 ℓ 𝑘 𝜎 𝑖𝑙 (1)
Powers: 𝑃 (𝑛𝑠) =𝛼 𝑛 𝑘 𝑛 𝑚 𝜎 𝑘𝑖 𝑐 Γ 𝑚𝑖 𝑒 𝑖𝑗𝑙 ℓ 𝑗 𝐾 𝑚𝑙 𝑃 (𝑛𝑟) =𝛼 𝑛 𝑘 𝑛 𝑚 𝜇 𝑘𝑖 𝑒 𝑖𝑗𝑙 ℓ 𝑗 𝜎 𝑘𝑙 𝐾 𝑚𝑖 𝑐 𝑃 (𝑛) =𝛼 𝑛 𝑘 𝑛 𝑚 𝜎 𝑘𝑖 1 Γ 𝑚𝑖 1 +𝜇 𝑘𝑖 1 𝐾 𝑚𝑖 1
Same form for 𝑃 (𝑛) but different form for 𝑃 (𝑛𝑠) , 𝑃 (𝑛𝑟)
The contact quantities are “transferred” quantities
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Fabric averaging
Continuum embedment gives quantities that depend on the contact normal 𝑛 𝑖
Eliminate dependency by computing a fabric average ∙ considering probability distribution of 𝑛 𝑖
For uniform distribution 𝑃 (𝑛) = 1 3 𝛼 𝜎 𝑖𝑗 Γ 𝑖𝑗 + 𝜇 𝑖𝑗 𝐾 𝑖𝑗
For 𝛼=3 (i.e. 2 𝑅 𝑔 𝑆=3𝑉) we get the power of Cosserat continuum
Note that if S and V are the surface and volume of a sphere of radius Rg, then 𝛼=6
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Stress invariants [1]
3 independent invariants for symmetric stress 𝜎 𝑖𝑗
39 independent invariants for general 𝜎 𝑖𝑗 and 𝜇 𝑖𝑗 , of which 8 of order 1 or 2, need physical justification
Decompose 𝜎 𝑖𝑗 and 𝜇 𝑖𝑗 in spherical and deviatoric 𝜎 𝑖𝑗 = 𝑠 𝑖𝑗 +𝑝 𝛿 𝑖𝑗 𝜇 𝑖𝑗 = 𝑐 𝑖𝑗 + 𝜇 𝜏 𝛿 𝑖𝑗
Introduce stress and couple stress vectors 𝑡 𝑖 = 𝜎 𝑘𝑖 𝑛 𝑘 𝑚 𝑖 = 𝜇 𝑘𝑖 𝑛 𝑘
…and decompose into normal and tangential components, e.g.: 𝑡 (𝑛) = 𝑡 𝑖 𝑛 𝑖 𝑡 𝑖 (𝑡) = 𝑡 𝑖 − 𝑡 (n) n i
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Stress invariants [2]
Stresses 𝑡 (𝑛) =𝑝 T 2 = 𝑡 𝑘 (𝑡) 𝑡 𝑘 (𝑡) = 𝑠 𝑖𝑗 𝑠 𝑖𝑗 − 𝑠 𝑖𝑗 𝑠 𝑗𝑖
Couple stresses 𝑚 (𝑛) = 𝜇 𝜏 M 2 = 𝑚 𝑘 (𝑡) 𝑚 𝑘 (𝑡) = c 𝑖𝑗 c 𝑖𝑗 − c 𝑖𝑗 c 𝑗𝑖
Should use contact quantities, so: 𝑚 𝑛 𝑐 = 𝑚 (𝑛) = 𝜇 𝜏 𝑀 𝑐 2 = 𝑚 𝑘 𝑡 c 𝑚 𝑘 𝑡 c = 𝑀 2 + 𝑅 𝑔 𝑇 2
Simple, 𝐽 2 -type plasticity models should use 𝑝, 𝑇, 𝜇 𝜏 and 𝑀 𝑐
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25. 2D Cosserat Mohr-Coulomb plasticity
Stresses: 𝑡 (𝑡) ≤− 𝑡 (𝑛) tan 𝜙
Couple stresses? 𝑚 𝑡 𝑐 ≤− 𝑅 𝑔 𝑡 (𝑛) tan 𝜙 𝑟
We can consider two yield mechanisms (two yield surfaces); the y.s. for the stresses will depend only on the stresses, while for the couple stresses on both couple stresses and stresses. Note that for the couple stresses we use the transported invariant. Discuss DEM use of rolling friction 𝜙 𝑟 . AM is the asymmetric part of the stress; note that stress asymmetry brings us closer to failure!
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Experimental data
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Macroscale
Many experimental results showing size effect (e.g. shear banding, indentation, torsion)
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
No elastic good results (Maraganti and Sharma also don’t find something with numerical simulations)
Sibille and Froiio (2007)
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29. X-ray computed tomography
Digital image correlation
Particle tracking
Besuelle et al. (2006)
Ando et al. (2012)
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30. 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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31. Numerical multiscale methods
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32. 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?
Number of Gauss points important
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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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