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”.

26ᵗʰ ALERT Geomaterials Workshop Acknowledgements This research effort is funded from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007-2013) under REA grant agreement nº 618096. 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

26ᵗʰ ALERT Geomaterials Workshop Outline Introduction (Micromechanics & GCMs) Mathematical origin of GCMs Theoretical micromechanical arguments A simple example Experimental data Numerical multiscale methods Conclusions 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

26ᵗʰ ALERT Geomaterials Workshop Introduction 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

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) 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

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” 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

Mathematical origin of generalised continuum models 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

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. 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

26ᵗʰ ALERT Geomaterials Workshop 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 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

26ᵗʰ ALERT Geomaterials Workshop 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 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

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 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

26ᵗʰ ALERT Geomaterials Workshop “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 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

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) 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

Theoretical micromechanical arguments 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

26ᵗʰ ALERT Geomaterials Workshop 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 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

26ᵗʰ ALERT Geomaterials Workshop 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) 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

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) 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

26ᵗʰ ALERT Geomaterials Workshop A simple example Based on an idea by I. Vardoulakis Papanicolopulos and Veveakis (2011) 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

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 1 − 𝑒 𝑖𝑗𝑘 𝑤 𝑗 2 + 𝑤 𝑗 1 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 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

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. 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

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 1 + 𝑒 𝑖𝑗𝑙 ℓ 𝑗 𝐾 𝑚𝑙 1 𝑃 (𝑛𝑟) =𝛼 𝑛 𝑘 𝑛 𝑚 𝜇 𝑘𝑖 1 + 𝑒 𝑖𝑗𝑙 ℓ 𝑗 𝜎 𝑘𝑙 1 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 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

26ᵗʰ ALERT Geomaterials Workshop 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 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

26ᵗʰ ALERT Geomaterials Workshop 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 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

26ᵗʰ ALERT Geomaterials Workshop Stress invariants [2] Stresses 𝑡 (𝑛) =𝑝 T 2 = 𝑡 𝑘 (𝑡) 𝑡 𝑘 (𝑡) = 4 15 𝑠 𝑖𝑗 𝑠 𝑖𝑗 − 1 15 𝑠 𝑖𝑗 𝑠 𝑗𝑖 Couple stresses 𝑚 (𝑛) = 𝜇 𝜏 M 2 = 𝑚 𝑘 (𝑡) 𝑚 𝑘 (𝑡) = 4 15 c 𝑖𝑗 c 𝑖𝑗 − 1 15 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 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

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! 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

26ᵗʰ ALERT Geomaterials Workshop Experimental data 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

26ᵗʰ ALERT Geomaterials Workshop Macroscale Many experimental results showing size effect (e.g. shear banding, indentation, torsion) Papamichos et al. (2006) 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

26ᵗʰ ALERT Geomaterials Workshop 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) 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

X-ray computed tomography Digital image correlation Particle tracking Besuelle et al. (2006) Ando et al. (2012) 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

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 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

Numerical multiscale methods 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

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) 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

26ᵗʰ ALERT Geomaterials Workshop 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) 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

26ᵗʰ ALERT Geomaterials Workshop 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 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

26ᵗʰ ALERT Geomaterials Workshop Conclusions 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop

26ᵗʰ ALERT Geomaterials Workshop 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 28/09/2015 26ᵗʰ ALERT Geomaterials Workshop