Continua with Microstructure Part II: Second-gradient Theory

Continua with Microstructure Part II: Second-gradient Theory Stefanos-Aldo Papanicolopulos 1 Antonis Zervos 2 1 Department of Mechanics, National Technical University of Athens 2 School of Civil Engineering and the Environment, University of Southampton ALERT Doctoral Shool Thursday, October 7, 2010 S.-A. Papanicolopulos, A. Zervos Second-gradient theory 1 / 52

Acknowledgement The research leading to these results has received funding from the European Research Council under the European Community s Seventh Framework Programme (FP7/2007 2013) / ERC grant agreement n o 228051. S.-A. Papanicolopulos, A. Zervos Second-gradient theory 2 / 52

Outline Introduction Gradient Elasticity Other gradient theories Example problems Numerical implementation S.-A. Papanicolopulos, A. Zervos Second-gradient theory 3 / 52

Outline Introduction Continua with microstructure Introducing second-gradient models Gradient Elasticity Other gradient theories Example problems Numerical implementation S.-A. Papanicolopulos, A. Zervos Second-gradient theory 4 / 52

Continua with microstructure Microstructure of materials Materials always have a microstructure...... possibly at different length scales Overall mechanical behaviour depends on mechanical behaviour of microstructure Usually whole is less than sum of parts (continuum averaging) Grain of expanded perlite (Georgopoulos, 2006) S.-A. Papanicolopulos, A. Zervos Second-gradient theory 5 / 52

Continua with microstructure Classical Continuum Discrete microstructure to continuum model Representative elementary volume (REV) associated to material point REV has finite size Material point in classical continuum is dimensionless (no microstructure) REV S.-A. Papanicolopulos, A. Zervos Second-gradient theory 6 / 52

Continua with microstructure Continuum with microstructure Material point has microstructure, finite dimensions Introduction of internal length as material parameter Comparison between problem size and material size Interaction between different length scales REV S.-A. Papanicolopulos, A. Zervos Second-gradient theory 7 / 52

Continua with microstructure Generalised continuum theories Different theories: Cosserat, second-gradient, non-local, micromorphic,... Mechanics of generalised continua is... the mechanics of matter for circumstances in which the conventional continuum mechanics cannot offer a promising approach (Kröner, 1968) Clear qualitative difference from classical continuum S.-A. Papanicolopulos, A. Zervos Second-gradient theory 8 / 52

Continua with microstructure Comparison with classical continuum What is being generalised? A generalised continuum may: 1. Violate Cauchy s postulate (traction on a face is linear in the normal to the face) 2. Refer to non-euclidean or not-connected space or body 3. Have applied point couples in volume or surface 4. Have additional internal degrees of freedom describing microstructure See G. Maugin, Generalized Continuum Mechanics: What Do We Mean by That? (2010) S.-A. Papanicolopulos, A. Zervos Second-gradient theory 9 / 52

Introducing second-gradient models Second-gradient theory The basic concept Additional dependence of mechanical behaviour on the second spatial derivative of displacement Can be seen as dependence on the strain and the (first) strain gradient Also called first strain gradient, strain gradient, second grade, grade two or gradient theory Used in problems of elastic deformation, fracture, strain localisation S.-A. Papanicolopulos, A. Zervos Second-gradient theory 10 / 52

Introducing second-gradient models Cauchy s higher-gradient concept «Note sur l équilibre et les mouvements vibratoires des corps solides», 1851:... les composants A, F, E; F, B, D; E, D, C des pressions supportées au point P par trois plans parallèles aux plans coordonnés des yz, des zx et des xy, pourront être généralement considérées comme des fonctions linéaires des déplacements ξ, η, ζ et des leurs dérivées des divers ordres. The stresses are considered as linear functions of the displacements and their derivatives of various orders. S.-A. Papanicolopulos, A. Zervos Second-gradient theory 11 / 52

Introducing second-gradient models Mindlin s elasticity with microstructure Micro-structure in linear elasticity (1964) A more general theory Each material point has a deformable micro-volume Two scales: micro-medium & macro-medium Each scale has it s own deformation Second-gradient theory seen as limit case for zero relative deformation x 2 x 2 x 1 x 1 S.-A. Papanicolopulos, A. Zervos Second-gradient theory 12 / 52

Introducing second-gradient models Mindlin s elasticity with microstructure Micro-structure in linear elasticity (1964) A more general theory Each material point has a deformable micro-volume Two scales: micro-medium & macro-medium Each scale has it s own deformation Second-gradient theory seen as limit case for zero relative deformation x 2 x 1 S.-A. Papanicolopulos, A. Zervos Second-gradient theory 12 / 52

Introducing second-gradient models A simple 1D second-gradient model Average y of field y = f (x) over length L (REV) y = 1 L L/2 f (x + ξ) dξ L/2 Express f (x + ξ) as Taylor series around ξ = 0 f (x + ξ) = f (x) + f (x)ξ + 1 2 f (x)ξ 2 + 1 6 f (x)ξ 3 + O(ξ 4 ) Case 1: local curvature is negligible y y locally homogeneous or local continuum S.-A. Papanicolopulos, A. Zervos Second-gradient theory 13 / 52

Introducing second-gradient models A simple 1D second-gradient model Average y of field y = f (x) over length L (REV) y = 1 L L/2 f (x + ξ) dξ L/2 Express f (x + ξ) as Taylor series around ξ = 0 f (x + ξ) = f (x) + f (x)ξ + 1 2 f (x)ξ 2 + 1 6 f (x)ξ 3 + O(ξ 4 ) Case 2: local curvature is not negligible y y L2 24 d 2 y dx 2 locally inhomogeneous or non-local continuum S.-A. Papanicolopulos, A. Zervos Second-gradient theory 13 / 52

Introducing second-gradient models A simple 1D second-gradient model Average y of field y = f (x) over length L (REV) y = 1 L L/2 f (x + ξ) dξ L/2 Express f (x + ξ) as Taylor series around ξ = 0 f (x + ξ) = f (x) + f (x)ξ + 1 2 f (x)ξ 2 + 1 6 f (x)ξ 3 + O(ξ 4 ) Case 2: local curvature is not negligible y y L2 d 2 y Second-gradient theory with an 24 dx 2 internal length locally inhomogeneous or non-local continuum S.-A. Papanicolopulos, A. Zervos Second-gradient theory 13 / 52

Outline Introduction Gradient Elasticity Introductory concepts Variational formulation Linear and isotropic gradient elasticity Other gradient theories Example problems Numerical implementation S.-A. Papanicolopulos, A. Zervos Second-gradient theory 14 / 52

Introductory concepts Gradient elasticity Work by Toupin (1962), Mindlin (1964), Mindlin & Eshel (1968) Simplest non-trivial second gradient model Shows basic properties of second-gradient theories Allows analytical treatment of simple problems Rigorous mathematical framework Here we assume: (Quasi-) Static case Small deformations Cartesian coordinates S.-A. Papanicolopulos, A. Zervos Second-gradient theory 15 / 52

Introductory concepts Indicial notation for tensors Tensors are indicated using indices i, j, k,... x i, u i, τ ij, κ ijk,... Indices from 1 to 3 x i = {x 1, x 2, x 3 } Repeated indices indicate summation τ ii = τ 11 + τ 22 + τ 33 A comma indicates spatial differentiation u i,j = u i / x j Kronecker delta 1 if i = j δ ij = 0 if i = j Permutation symbol 1 123,231,312 e ijk = 1 321,213,132 0 otherwise S.-A. Papanicolopulos, A. Zervos Second-gradient theory 16 / 52

Introductory concepts Grouping the displacement derivatives Classical theory Classical theory is first-gradient Mechanical behaviour depends only on u i,j Actually only on the symmetric part (strain) ε ij = 1 2 (u i,j + u j,i ) 6 independent components out of 9 We can decompose into spherical and deviatoric components ε ij = 1 3 ε kkδ ij + ε ij Convenient in some applications Equivalent to using just the strain S.-A. Papanicolopulos, A. Zervos Second-gradient theory 17 / 52

Introductory concepts Grouping the displacement derivatives Second-gradient theory We define: κ ijk = u k,ij ˆκ ijk = ε jk,i κ ij = 1 2 e jlmu m,li κ ijk = 1 3 (u k,ij + u j,ki + u i,jk ) Dependence on u k,ij u k,ij symmetric in i, j 18 independent components out of 27 Different groupings (decompositions) u k,ij = κ ijk (I) = ˆκ ijk + ˆκ jki ˆκ kij (II) = κ ijk 2 3 e ilk κ jl 2 3 e jlk κ il (III) Three different Forms (other possible) All equivalent We ll use Form II S.-A. Papanicolopulos, A. Zervos Second-gradient theory 18 / 52

Variational formulation Variational formulation Potential energy density W = W(ε ij, κ ijk ) where κ ijk = ε jk,i Variation of W δw = W ε ij δε ij + W κ ijk δκ ijk = τ ij δε ij + μ ijk δκ ijk introduces stresses and double-stresses Total potential energy W = V W dv Equilibrium requires δw = δp where P the work of external actions and δw = (τ ij δε ij +μ ijk δκ ijk ) dv V S.-A. Papanicolopulos, A. Zervos Second-gradient theory 19 / 52

Variational formulation Variation of potential energy Start with (V is volume) δw = (τ ij δε ij + μ ijk δκ ijk ) dv V Calculate (S is surface) δw = + n j (τ jk μ ijk,i )δu k ds S (τ jk μ ijk,i ),j δu k dv V n i μ ijk δu k,j ds S On S, δu k,j is not independent of δu k Introduce normal derivative operator D n j x j and surface-gradient operator D j n j D x j S.-A. Papanicolopulos, A. Zervos Second-gradient theory 20 / 52

Variational formulation Introducing external actions After calculations (C are edges, [[ ]] is jump ) δw = (τ jk μ ijk,i ),j δu k dv V + nj (τ jk μ ijk,i ) D j (n i μ ijk ) + (D l n l )n j n i μ ijk δuk ds + S n i n j μ ijk Dδu k ds + S s m e mlj [[n l n i μ ijk ]]δu k dc C This suggests (F k body forces, P k surface tractions, R k surface double-tractions, E k edge tractions) δp = V F k δu k dv+ P k δu k ds+ S R k Dδu k ds+ S E k δu k dc C S.-A. Papanicolopulos, A. Zervos Second-gradient theory 21 / 52

Variational formulation Equilibrium equations Since δw = δp we obtain [strong form] (τ jk μ ijk,i ),j + F k = 0 in V P k = n j (τ jk μ ijk,i ) D j (n i μ ijk ) + (D l n l )n j n i μ ijk R k = n i n j μ ijk E k = s m e mlj [[n l n i μ ijk ]] on S on S on C but also [weak form / virtual work equation] V (τ ij δε ij + μ ijk δκ ijk ) dv = V F k δu k dv+ P k δu k ds+ S R k Dδu k ds+ S E k δu k dc C S.-A. Papanicolopulos, A. Zervos Second-gradient theory 22 / 52

Variational formulation Remarks on equilibrium equations Additional boundary tractions w.r.t. classical continuum (R k, E k ) P k not according to Cauchy s postulate Weak form simpler than strong form* You cannot ignore R k and E k in either case Setting σ jk = τ jk μ ijk,i yields σ jk,j + F k = 0 Some authors use σ jk as a Cauchy or equilibrium stress However, P k = n j σ jk S.-A. Papanicolopulos, A. Zervos Second-gradient theory 23 / 52

Variational formulation Remarks on equilibrium equations Additional boundary tractions w.r.t. classical continuum (R k, E k ) P k not according to Cauchy s postulate Weak form simpler than strong form* You cannot ignore R k and E k in either case Setting σ jk = τ jk μ ijk,i yields σ jk,j + F k = 0 Some authors use σ jk as a Cauchy or equilibrium Cannot stress However, ignore R k, E k P k One = n j σstress, jk energy conj. to strain S.-A. Papanicolopulos, A. Zervos Second-gradient theory 23 / 52

Linear and isotropic gradient elasticity Linear gradient elasticity Up to now, only W = W(ε ij, κ ijk ) was assumed A quadratic form of W W = 1 2 c ijpqε ij ε pq + f ijkpq κ ijk ε pq + 1 2 a ijkpqrκ ijk κ pqr results in linear gradient elasticity τ ij = c ijpq ε pq + f pqrij κ pqr μ ijk = f ijkpq ε pq + a ijkpqr κ pqr c ijpq, f ijkpq and a ijkpqr are material parameter tensors with 21, 108 and 171 independent parameters (due to various symmetries) Total of 300 independent parameters for general anisotropic linear case S.-A. Papanicolopulos, A. Zervos Second-gradient theory 24 / 52

Linear and isotropic gradient elasticity Isotropic linear gradient elasticity For isotropic material parameter tensors (invariant w.r.t. rotation) W = 1 2 λε iiε jj + με ij ε ij + a 1 κ iik κ kjj + a 2 κ ijj κ ikk + a 3 κ iik κ jjk + a 4 κ ijk κ ijk + a 5 κ ijk κ kji τ ij = λδ ij ε pp + 2με ij μ ijk = 1 2 a 1(δ ij κ kpp + 2δ jk κ ppi + δ ik κ jpp ) + 2a 2 δ jk κ ipp + a 3 (δ ij κ ppk + δ ik κ ppj ) + 2a 4 κ ijk + a 5 (κ kji + κ jki ) Five additional material parameters a 1,..., a 5 f ijkpq = 0 so τ ij depends only on ε ij and μ ijk on κ ijk S.-A. Papanicolopulos, A. Zervos Second-gradient theory 25 / 52

Linear and isotropic gradient elasticity Equilibrium equation for displacements Introducing two lengths l 2 1 = 2(a 1 + a 2 + a 3 + a 4 + a 5 ) λ + 2μ > 0 l 2 2 = a 3 + 2a 4 + a 5 2μ > 0... gives ( 2 ( ) = ( ),qq ) (λ+2μ)(1 l 2 1 2 )u p,pk μ(1 l 2 2 2 )(u p,pk u k,pp )+F k = 0 Simplified model: a 1 = a 3 = a 5 = 0, a 2 = λl 2 /2, a 4 = μl 2 so that l 1 = l 2 = l (1 l 2 2 ) λu p,pk + μ(u p,pk + u k,pp ) + F k = 0 S.-A. Papanicolopulos, A. Zervos Second-gradient theory 26 / 52

Outline Introduction Gradient Elasticity Other gradient theories Gradient plasticity and damage Example problems Numerical implementation S.-A. Papanicolopulos, A. Zervos Second-gradient theory 27 / 52

Gradient plasticity and damage Strain-gradient plasticity and damage Second-gradient theory is a general framework Various classical models may be extended to the second-gradient case Gradient plasticity has been proposed and used to regularise problems of strain localisation Also some work on gradient damage theories S.-A. Papanicolopulos, A. Zervos Second-gradient theory 28 / 52

Gradient plasticity and damage Theories with gradients of internal variables Plasticity and damage are theories with internal variables Introducing dependence on gradients of internal variables introduces an internal length The resulting models regularise localisation problems These gradient plasticity or gradient damage models are not second-gradient (of displacement) models S.-A. Papanicolopulos, A. Zervos Second-gradient theory 29 / 52

Outline Introduction Gradient Elasticity Other gradient theories Example problems Thick hollow cylinder under external normal traction Infinite layer under shear Infinite layer with bolts Plane strain uniaxial loading Mode I crack Numerical implementation S.-A. Papanicolopulos, A. Zervos Second-gradient theory 30 / 52

Thick hollow cylinder under external normal traction Thick hollow cylinder: description r a p b r b Simplified isotropic linear second gradient elasticity, material length l, plane strain (all examples) Only classical BCs r b /r a = 3, ν = 1/4, different r a /l All results will be given in normalised (dimensionless) form S.-A. Papanicolopulos, A. Zervos Second-gradient theory 31 / 52

Thick hollow cylinder under external normal traction Thick hollow cylinder: radial strain radial strain εrr/(pb/μ) 0.2 0.1 0.0-0.1-0.2 l = 0 r a /l = 32 r a /l = 16 r a /l = 8 r a /l = 4-0.3 1.0 1.5 2.0 2.5 3.0 radial position r/r a Classical solution has strong strain gradient near inner boundary... so difference of strain gradient solution larger Difference increases as r a /l gets smaller (i.e. the hole gets smaller) S.-A. Papanicolopulos, A. Zervos Second-gradient theory 32 / 52

Infinite layer under shear Shear layer: description H y τ Classical solution has no strain gradient We impose kinematic second-gradient BC x u x / y = 0 for y = ±H H τ ( rough boundary BC) Different H/ l Symmetry wrt x axis, consider only upper half S.-A. Papanicolopulos, A. Zervos Second-gradient theory 33 / 52

Infinite layer under shear Shear layer: shear strain 1.0 0.8 position y/ H 0.6 0.4 0.2 H/l = 64 H/l = 32 H/l = 16 H/l = 8 H/l = 4 0.0 0.0 0.2 0.4 0.6 0.8 1.0 shear strain γ xy /(τ/μ) S.-A. Papanicolopulos, A. Zervos Second-gradient theory 34 / 52

Infinite layer under shear Shear layer: shear strain 1.0 position y/ H 0.8 0.6 0.4 0.2 H/l = 64 H/l = 32 H/l = 16 H/l = 8 H/l = 4 0.0 0.0 0.2 0.4 0.6 0.8 1.0 shear strain γ xy /(τ/μ) Classical solution cannot accomodate rough boundary Gradient solution creates boundary layer Size of boundary layer scales with l For l comparable to H, boundary layers merge S.-A. Papanicolopulos, A. Zervos Second-gradient theory 34 / 52

Infinite layer with bolts Bolted layer: description R y We impose static second-gradient BC H y R y = 0 for y = ±H H R y x R y is self-equilibrating Vardoulakis proposed R y < 0 to simulate effect of rock bolts Symmetry wrt x axis, consider only upper half S.-A. Papanicolopulos, A. Zervos Second-gradient theory 35 / 52

Infinite layer with bolts Bolted layer: strain 1.0 0.8 position y/ H 0.6 0.4 0.2 H/l = 64 H/l = 32 H/l = 16 H/l = 8 H/l = 4 0.0 0.0 0.2 0.4 0.6 0.8 1.0 strain ε yy /(R y /l/(λ + 2μ)) S.-A. Papanicolopulos, A. Zervos Second-gradient theory 36 / 52

Infinite layer with bolts Bolted layer: strain 1.0 position y/ H 0.8 0.6 0.4 0.2 H/l = 64 H/l = 32 H/l = 16 H/l = 8 H/l = 4 0.0 0.0 0.2 0.4 0.6 0.8 1.0 strain ε yy /(R y /l/(λ + 2μ)) Classical elasticity cannot accomodate BC for R y Surface effect of R y (self-equilibrating), creates boundary layer Size of boundary layer scales with l S.-A. Papanicolopulos, A. Zervos Second-gradient theory 36 / 52

Plane strain uniaxial loading Uniaxial loading: description H H H y u y0 x H Uniaxial loading in plane strain Classical solution has no strain gradient Rough boundary BC u y / y = 0 for y = ±H Different H/l, ν = 0.35 u y0 S.-A. Papanicolopulos, A. Zervos Second-gradient theory 37 / 52

Plane strain uniaxial loading Uniaxial loading: description H H H y u y0 x H Uniaxial loading in plane strain Classical solution has no strain gradient Rough boundary BC u y / y = 0 for y = ±H Different H/l, ν = 0.35 Calculate displacements of top half of right edge u y0 S.-A. Papanicolopulos, A. Zervos Second-gradient theory 37 / 52

Plane strain uniaxial loading Uniaxial loading: description H H H y u y0 u y0 x H Uniaxial loading in plane strain Classical solution has no strain gradient Rough boundary BC u y / y = 0 for y = ±H Different H/l, ν = 0.35 Calculate displacements of top half of right edge There are corners (edges in 3D) S.-A. Papanicolopulos, A. Zervos Second-gradient theory 37 / 52

Plane strain uniaxial loading Uniaxial loading: edge displacement 1.0 0.8 position y/ H 0.6 0.4 0.2 classical H/l = 64 H/l = 32 H/l = 16 H/l = 8 H/l = 4 0.0-0.7-0.6-0.5-0.4-0.3-0.2 horizontal displacement u x /u y0 S.-A. Papanicolopulos, A. Zervos Second-gradient theory 38 / 52

Plane strain uniaxial loading Uniaxial loading: edge displacement position y/ H 1.0 0.8 0.6 0.4 0.2 classical H/l = 64 H/l = 32 H/l = 16 H/l = 8 H/l = 4 0.0-0.7-0.6-0.5-0.4-0.3-0.2 horiz. displacement u x /u y0 Classical solution leaves sides straight ( uniform solution) Gradient solution with rough BC distorts sides Uniform gradient solution impossible because of BC at corners E i = 0 Effect of corners (edges) even without edge tractions S.-A. Papanicolopulos, A. Zervos Second-gradient theory 38 / 52

Mode I crack Mode I crack: description P 0 y 2a x Classical solution has infinite stress at crack tip Different a/l, ν = 0.2 Semi-analytical gradient solution exists Numerical results presented here P 0 S.-A. Papanicolopulos, A. Zervos Second-gradient theory 39 / 52

Mode I crack Mode I crack: crack opening 1.0 crack opening uy/ u0 0.8 0.6 0.4 0.2 classical a/l = 50 a/l = 20 a/l = 10 a/l = 5 0.0-1.0-0.8-0.6-0.4-0.2 0.0 distance from crack tip x/a S.-A. Papanicolopulos, A. Zervos Second-gradient theory 40 / 52

Mode I crack Mode I crack: crack opening 1.0 crack opening uy/ u0 0.8 0.6 0.4 classical a/l = 50 a/l = 20 0.2 a/l = 10 a/l = 5 0.0-1.0-0.8-0.6-0.4-0.2 0.0 distance from crack tip x/a Results for 1/4 of problem Smaller cracks become stiffer Cusped shape of crack ( u y / x = 0 at tip) Strains and stresses are important S.-A. Papanicolopulos, A. Zervos Second-gradient theory 40 / 52

Mode I crack Mode I crack: strains and stresses strain and stress 3 2 1 0 ε xx ε yy ε xy τ xx τ yy -1-1 -0.5 0 0.5 1 distance from crack tip x/a S.-A. Papanicolopulos, A. Zervos Second-gradient theory 41 / 52

Mode I crack Mode I crack: strains and stresses strain and stress 3 2 1 0-1 ε xx ε yy ε xy τ xx τ yy -1-0.5 0 0.5 1 distance from crack tip x/a Results for a/ l = 10 Strains and stresses at crack tip are Finite Continuous Non-smooth We can use a maximum-stress criterion On crack face τ yy = 0 Attention if other stresses are used S.-A. Papanicolopulos, A. Zervos Second-gradient theory 41 / 52

Mode I crack Mode I crack: strain gradients strain gradient 40 20 0-20 ε xx,x ε yy,x ε xy,x -40-0.1-0.05 0 0.05 0.1 distance from crack tip x/a S.-A. Papanicolopulos, A. Zervos Second-gradient theory 42 / 52

Mode I crack Mode I crack: strain gradients strain gradient 40 20 0-20 -40 ε xx,x ε yy,x ε xy,x -0.1-0.05 0 0.05 0.1 distance from crack tip x/a Strain gradients and double-stresses at crack tip are not continuous and infinite Numerical solution affected by this S.-A. Papanicolopulos, A. Zervos Second-gradient theory 42 / 52

Outline Introduction Gradient Elasticity Other gradient theories Example problems Numerical implementation Basics of the Finite Element Method C 1 discretisation of second-gradient models Alternative numerical formulations S.-A. Papanicolopulos, A. Zervos Second-gradient theory 43 / 52

Basics of the Finite Element Method Very brief review of FEM Use bold notation Must calculate u(x) (displacement field) Interpolate using shape functions & DOFs u(x) = N(x)u N Using matrix notation ε T τ = τ ij ε ij ε(x) = B(x)u N B(x) contains first derivatives of N(x) Elasticity: τ = Dε Substituting in virtual work equation V V B T DBdV N T FdV + or simply Ku N = f u N = N T PdS S S.-A. Papanicolopulos, A. Zervos Second-gradient theory 44 / 52

C 1 discretisation of second-gradient models The second-gradient case Still Ku N = f with K = V BT DBdV and τ = Dε, but: 1. Vector of external actions f = N T FdV + N T P + D(N T )R ds + N T EdC V S C 2. Additional second-gradient terms in τ, ε, D ε T τ = τ ij ε ij + μ ijk κ ijk 3. B(x) has both first and second derivatives of N(x) Due to (3), C 1 continuity (continuous first derivative) required for interpolation of u(x) Usual finite elements cannot be used with second-gradient theories S.-A. Papanicolopulos, A. Zervos Second-gradient theory 45 / 52

C 1 discretisation of second-gradient models The second-gradient case Still Ku N = f with K = V BT DBdV and τ = Dε, but: 1. Vector of external actions f = N T FdV + N T P + D(N T )R ds + N T EdC V S C 2. Additional second-gradient terms in τ, ε, D ε T τ = τ ij ε ij + μ ijk κ ijk 3. B(x) has both first and second derivatives Usual of N(x) finite Due to (3), C 1 continuity (continuous first elements derivative) required for interpolation of u(x) cannot be used with Usual finite elements cannot be used with 2 second-gradient theories nd -gradient theories S.-A. Papanicolopulos, A. Zervos Second-gradient theory 45 / 52

C 1 discretisation of second-gradient models C 1 continuity of interpolation Polynomial interpolation within element is C 1 Possible problem at boundary between elements 1D: single node [easy, use Hermite elements] 2D: edges & nodes [use plate-bending elements] 3D: faces, edges & nodes [more elements needed] S.-A. Papanicolopulos, A. Zervos Second-gradient theory 46 / 52

C 1 discretisation of second-gradient models Note about gradients of internal variables The C 1 continuity requirement applies to second-gradient (strain-gradient) theories Plasticity/damage theories with gradients of internal variables do not generally require C 1 continuity Other difficulties are introduced We must calculate the gradient of a value evaluated at discrete (integration) points S.-A. Papanicolopulos, A. Zervos Second-gradient theory 47 / 52

Alternative numerical formulations Alternative numerical formulations C 1 requirement poses some limitations Few elements available Excessive continuity at nodes Alternative C 0 formulations are sought (simpler, cheaper, more flexible) Various mixed formulations have been proposed (discretise other fields besides displacement) Other possible methods Meshless finite element methods Boundary element method S.-A. Papanicolopulos, A. Zervos Second-gradient theory 48 / 52

Alternative numerical formulations Example of mixed formulation [1] Target is to avoid C 1 requirement...... created by second gradients ( e.g. in B) Discretise three fields: 1. u = Nu N displacements 2. v = Mv N relaxed displacement gradients 3. λ = Λλ N Lagrange multipliers Here v is not defined as u Constraint λ : ( u v)dv = 0 enforces u v V DOFs u N, v N, λ N are not necessarily evaluated at the same positions (generally different interpolations N, M, Λ are used) S.-A. Papanicolopulos, A. Zervos Second-gradient theory 49 / 52

Alternative numerical formulations Example of mixed formulation [2] Introducing the groupings û = {u, v, λ} û N = {u N, v N, λ N } ˆε = { u, v, v, λ} ˆτ T = {τ λ, μ, λ, u v} N C F I N M ˆN = M B = M D = F A I Λ Λ I I We obtain the familiar relations û = ˆNû N, ˆε = Bû N, ˆτ = Dˆε, ˆε T ˆτ = τ ij ε ij + μ ijk κ ijk Now B has up to first derivatives of shape functions D non-symmetric, has some zero diagonal elements S.-A. Papanicolopulos, A. Zervos Second-gradient theory 50 / 52

Alternative numerical formulations Notes on C 1 vs mixed formulations Usual finite elements cannot be used All appropriate elements introduce shortcomings (but at least they work!) Mixed methods do offer greater flexibility (in the choice of elements) but also introduce one additional approximation Computational cost is an important factor C 1 elements generally have more DOFs (more expensive) but also higher interpolation (richer) When there is a C 1 interpolation, it is optimum Mixed elements discretise more fields than strictly necessary S.-A. Papanicolopulos, A. Zervos Second-gradient theory 51 / 52

Summary Continua with microstructure Second-gradient elasticity Other second-gradient models Example problems in second-gradient elasticity Numerical implementation issues S.-A. Papanicolopulos, A. Zervos Second-gradient theory 52 / 52