Micropolar theory is a weakly non-local higher-order continuum theory based on the inclusion of independent (micro-)rotational degrees of freedom. Subsequent introduction of couple-stresses and an internal length scale mean the micropolar continuum is therefore capable of modelling size effects. This paper proposes a non-linear Finite Element Method based on the spatial micropolar equilibrium equations, but using the classical linear micropolar constitutive laws defined in the reference configuration. The method is verified rigorously with the Method of Manufactured Solutions, and quadratic Newton-Raphson convergence of the minimised residuals is demonstrated

Augarde, Charles E.

Coombs, William M.

Gourgiotis, Panos A.

O'Hare, Ted J.

English

Durham Research Online

UKACM 2023 Conference, April 19-21, 2023, The University of Warwick, Coventry, UKA GEOMETRICALLY-EXACT FINITE ELEMENT METHOD FORMICROPOLAR CONTINUA WITH FINITE DEFORMATIONSTed J. O’Hare1∗, Panos A. Gourgiotis2, William M. Coombs1 and Charles E. Augarde11 Department of Engineering, Durham University, DH1 3LN, UK.ted.o’hare@durham.ac.uk2 Department of Civil Engineering, University of Thessaly, Pedion Areos, GR-38334, Volos, Greece.Abstract. Micropolar theory is a weakly non-local higher-order continuum theory based on the inclusionof independent (micro-)rotational degrees of freedom. Subsequent introduction of couple-stresses andan internal length scale mean the micropolar continuum is therefore capable of modelling size effects.This paper proposes a non-linear Finite Element Method based on the spatial micropolar equilibriumequations, but using the classical linear micropolar constitutive laws defined in the reference configu-ration. The method is verified rigorously with the Method of Manufactured Solutions, and quadraticNewton-Raphson convergence of the minimised residuals is demonstrated.Key words: Micropolar elasticity; Non-linear FEM; Updated Lagrangian; Method of ManufacturedSolutions1 IntroductionThe modelling of materials with microstructures has become an important research topic in the field ofsolid mechanics. The use of micropolar continua, first proposed by the Cosserat brothers in 1909 [1],has been proven to be an effective approach for capturing the complex behaviour of such materials,especially granular media. The theory introduces an additional kinematic field of rigid-body properorthogonal rotations to the conventional continuum formulation, which occur in the microstructure (e.g.soil grains) independently of the macro-deformation. Micropolar continua therefore have six degrees offreedom: three for the translations and three for the rotations. A kinematic measure of the gradient ofthese rotations is included in the thermodynamic formulation, which is conjugate to couple-stress (torqueper unit area). As a result, the theory is characterised as weakly non-local and a characteristic length isintroduced, allowing the observation of size effects and the natural evolution of strain localisation.To date, few numerical methods have been developed to model micropolar continua in the finite strain andmicro-rotation (meaning occurring in the microstructure, not of the order 10−6) regime. Although othershave focused on both material and geometric non-linearity [2,3], the only Finite Element Method (FEM)implementations dealing with purely geometric non-linearity appear to be [4] and [5]. However, theconstitutive model used in [4] is based on a small strain assumption, and [5] provides a total Lagrangianformulation. The present contribution is therefore the first fully geometrically-exact updated Lagrangianformulation which still makes rigorous use of the linear micropolar constitutive equations.2 Micropolar theory2.1 KinematicsLet a micropolar continuum occupy a volume Ω in its current (deformed) configuration. The translationvector ui emanates from the Cartesian reference position Xi of each point in the undeformed volume Ω0 to1Ted J. O’Hare, Panos A. Gourgiotis, William M. Coombs and Charles E. Augardeits current position xi in Ω, and the deformation gradient tensor Fiθ = ∂xi∂Xθ provides the fundamental linkbetween reference and current coordinates. At every point in the micro-continuum there exists a rigidbody, attached to which is a set of axes that are free to rotate independently of deformation occurring atthe continuum scale. Each rotated axis ki in the current configuration is related to its counterpart Wψ inthe reference configuration via ki = QiψWψ, where Qiψ ∈ SO(3) is a proper orthogonal tensor termed themicro-rotation tensor. The rotation may also be parameterised as a vector ϕk of Euler angles around thereference axes; alternatively ϕk may be identified as the axis of rotation with the angle its magnitude. Askew-symmetric tensor Φi j =−ei jkϕk (where ei jk is the third-order Levi-Civita, or permutation, tensor)is then used to compute the micro-rotation tensor using the (Euler-)Rodrigues formulaQiψ = δiψ +sin |ϕ||ϕ|Φiψ +1− cos |ϕ||ϕ|2Φi jΦ jψ, (1)where δiψ denotes the Kronecker delta and |ϕ| is the magnitude of ϕk. Two Lagrangian measures are usedto quantify micropolar deformation: a strain tensor, and a measure of the rotation gradient (curvature),named the wryness tensor which endows the theory with its non-local propertyEγπ = QiγFiπ −δγπ and Γγπ =−12eγτηQpτ∂Qpη∂Xπ. (2)2.2 Constitutive and balance equationsTo preserve objectivity the constitutive laws are defined only in the reference frame. Biot-like stress Bαβand couple-stress Sαβ are obtained directly from the Lagrangian strain and wryness measures asBαβ = λδαβEγγ +(µ+ν)Eαβ +(µ−ν)Eβα = DαβγπEγπ (3)Sαβ = αδαβΓγγ +(β+ γ)Γαβ +(β− γ)Γβα = D̂αβγπΓγπ (4)where Dαβγπ and D̂αβγπ are constitutive tensors which include the Lamé parameters λ and µ and microp-olar constants ν, α, β and γ based on information about the material’s characteristic length scale [6]. Theinverse Piola transformation leads to the Cauchy stress and couple-stressσi j = J−1QiαBαβFjβ and mi j = J−1QiαSαβFjβ (5)respectively, where J = det(F) is the volume ratio between the original and deformed states. The spatialforms of linear and angular momentum balance in the quasi-static case read∂σi j∂x j+ pi = 0 and∂mi j∂x j− ei jkσ jk +qi = 0, (6)where pi and qi are the body force and body couple respectively. Note that the presence of couples in theangular momentum balance equation, which is satisfied trivially in classical continua through equality ofcomplementary shear stresses, means the stress tensor is not required to be symmetric.3 Numerical formulationDiscretisation and application of Galerkin’s method produces weak forms of (6) at a node I,∫Ω∂NI∂x jσi j dΩ =∫ΩNI pi dΩ and∫Ω(∂NI∂x jmi j +NIei jkσ jk)dΩ =∫ΩNIqi dΩ , (7)2Ted J. O’Hare, Panos A. Gourgiotis, William M. Coombs and Charles E. Augardewhere NI is the shape function associated with node I. Solution of the discretised boundary-value prob-lem for a fixed external load with a Newton-Raphson scheme requires an iterative sequence of lineari-sation and incrementation of the internal force pintIi and couple qintIi expressions (the LHS of (7)1 and(7)2 respectively) until a convergence criterion is met. The linearisation procedure produces a tangentstiffness matrix KIiJ j which essentially relates an increment in deformation to an increment in force andcouple, such that∆pintIi = KpuIiJ j∆uJ j +KpwIiJ j∆wJ j and ∆qintIi = KquIiJ j∆uJ j +KqwIiJ j∆wJ j (8)for deformations at all nodes J, where ∆wJ j denotes an incremental rotation at J around axis k j as orientedat the beginning of the current iteration. Once the deformation increments are obtained, the kinematicfield is updated for the (k+1)th iteration from that in the kth withxk+1i = xki +∆ui and Qk+1iψ = (∆Qi j)Qkjψ (9)where ∆Qi j is computed by substituting ∆wk for ϕk in (1). As noted in [4], the complete consistentlinearisation of pintIi and qintIi with respect to nodal displacement and rotation increments is lengthy andarduous, and although it has been completed and used in this work, its inclusion lies beyond the scope ofthis contribution.4 VerificationThe formulation’s accuracy and convergence properties are assessed by means of the Method of Manu-factured Solutions (MMS). In the MMS, a synthetic solution field is designed and the corresponding bodyforce/couple and boundary conditions generated via the governing equations. Numerical accuracy is thenobserved by comparing the numerically-approximated solution of the problem with the pre-determinedanalytical solution. To that end, the arbitrary displacement-rotation fieldu1 = · · ·= ϕ3 = asin(2πX1)sin(2πX2)sin(2πX3), (10)was chosen, where Xi ∈ [0,1] defines a unit cube domain and a = 1100 ensures displacement is sufficientlysmall to ensure numerical stability. This particular trigonometric function was selected as it is contin-uously and infinitely differentiable and cannot be captured exactly by Lagrange interpolation. In thisstudy, the problem is simulated using a three-dimensional FE implementation of the formulation, witha discretisation of tri-linear hexahedral elements. The Euclidean norms of the displacement error andCauchy stress error (normalised by the analytical norm) are then integrated over the domain; the result-ing convergence graphs for elements of decreasing dimension h are shown on log-log scales in Figures1(a) and (b). As linear elements are used, quadratic displacement and (approaching) linear stress errordecay is observed. Asymptotically quadratic Newton-Raphson convergence is demonstrated in Figure1(c) for a separate problem involving very large displacements and rotations. The force residual is com-puted by taking its Euclidean norm normalised by the external load, and the energy residual is the scalarproduct of the force residual and the incremental deformations.5 ConclusionsA geometrically-exact FEM for modelling the behaviour of micropolar continua under finite strains andmicro-rotations has been presented. The governing equations are solved in the spatial frame, making the3Ted J. O’Hare, Panos A. Gourgiotis, William M. Coombs and Charles E. Augarde(a) (b)10-10 100 1010previous residual100residual2energyforce(c)Figure 1: Convergence with mesh refinement of (a) translations/rotations and (b) Cauchy stress/couple-stress. Newton-Raphson convergence through a single loadstep is depicted in (c).method updated Lagrangian, however the constitutive parameters used are defined in the reference frameand therefore all have physical meaning. Additionally, the method is computationally efficient (in termsof its convergence rate) and can be used to model complex problems accurately.AcknowledgementThis work was supported by the Engineering and Physical Sciences Research Council [grant numberEP/T518001/1]. For the purpose of open access, the authors have applied a Creative Commons Attribu-tion (CC BY) licence to any Author Accepted Manuscript version arising.REFERENCES[1] Cosserat, E. and Cosserat, F. Théorie des corps déformables. A. Hermann et fils, Paris (1909).[2] Bauer, S., Dettmer, W.G., Perić, D. and Schäfer, M. Micropolar hyperelasticity: constitutive model,consistent linearization and simulation of 3D scale effects. Computational Mechanics (2012)50:383-396.[3] Bauer, S., Dettmer, W.G., Perić, D. and Schäfer, M. Micropolar hyperelasto-plasticity: constitu-tive model, consistent linearization, and simulation of 3D scale effects. International Journal forNumerical Methods in Engineering (2012) 91:39-66.[4] Bauer, S., Schäfer, M. Grammenoudis, P. and Tsakmakis, Ch. Three-dimensional finite elements forlarge deformation micropolar elasticity. Computer Methods in Applied Mechanics and Engineering(2010) 199:2643-2654.[5] Erdelj, S.G. and Jelenić, G. and Ibrahimbegović, A. Geometrically non-linear 3D finite-elementanalysis of micropolar continuum. International Journal of Solids and Structures (2020) 202:745-764.[6] Lakes, R.S. Physical meaning of elastic constants in Cosserat, void, and microstretch elasticity.Journal of Mechanics of Materials and Structures (2016) 11:217-229.4

A geometrically-exact Finite Element Method for micropolar continua with finite deformations

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A geometrically-exact Finite Element Method for micropolar continua with finite deformations

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