Two-scale modelling of micromorphic continua

Jänicke, Ralf; Diebels, Stefan; Sehlhorst, Hans-Georg; Düster, Alexander

doi:10.1007/s00161-009-0114-4

Cited by 65 publications

(34 citation statements)

References 21 publications

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“…With this, condition (ii) in (48) requires the periodicity of ϕ at the boundary. Both conditions fulfill the well-established HillMandel microheterogeneity condition which plays a key role in many numerical homogenisation schemes (see, e.g., [45][46][47]). Moreover, conditions (48) obviously fulfill restriction (5) 2 as well.…”

Section: Finite Element Discretisationmentioning

confidence: 83%

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Partially relaxed energy potentials for the modelling of microstructures – application to shape memory alloys

Bartel

Menzel

2012

GAMM-Mitteilungen

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Energy relaxation is well-established by several researchers -especially in the field of the modelling of solid-solid phase transformations. Nevertheless, critics still counter this concept by considering it as a purely mathematical tool with poor physical significance. In this contribution we aim at emphasising the significance of energy relaxation methods for the modelling of dissipative solids and especially microstructure formation and its further evolution. In particular, we shall point out aspects and advantages of this concept which are not straight forward to achieve within alternative schemes.

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Section: Finite Element Discretisationmentioning

confidence: 83%

“…One key point of the model is exemplified by the phase distriutions given in points 3 and 8 of Figure 5. Mesoscale: x ∈ B mes ⊂ B mac localisation procedure according to (45), (46).…”

Section: A (Quasi) Three-scale Mesoscopic Modelmentioning

confidence: 99%

Partially relaxed energy potentials for the modelling of microstructures – application to shape memory alloys

Bartel

Menzel

2012

GAMM-Mitteilungen

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“…However, the micromorphic approach reduces to the second gradient concept, if the micro deformation is considered to equal the first displacement gradient. In the sequel, only homogenisation rules for the micromorphic substitute medium will be discussed, which have been initially proposed by Forest et al [5][6][7][8][9]. The kinematic averaging rules for a quadratic unit cell of the size V m = l 2 read…”

Section: Second Order Homogenisationmentioning

confidence: 99%

“…By consequence it is not possible to prescribe Dirichlet type conditions on the boundary ∂V m [6]. In literature, several approaches are to be found [5,7] dealing with a cubic polynomial for the microscopic displacement field. However, the displacement field has to be prescribed on the entire micro volume V m .…”

Section: Second Order Homogenisationmentioning

confidence: 99%

Relaxed loading conditions for higher order homogenisation approaches

Jänicke

Uribe

Ruíz

et al. 2011

Proc Appl Math and Mech

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The present paper deals with the formulation of minimal loading conditions for the application of numerical homogenisation techniques, namely the FE 2 methodology. Based on the set of volume averaging rules connecting the heterogeneous micro and the homogeneous macro scale, the minimal constraints on the deformation of a micro volume are derived for a classical Cauchy continuum as well as for a micromorphic continuum theory. For both cases, numerical studies are included highlighting the main aspects of the proposed procedure within the context of small deformations.

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“…In the last 15 years, it has been proposed to resort to the mechanics of generalized continua which incorporate additional degrees of freedom or higher order strain gradient terms, see for instance [10,13,16,17,18,19,21]. Higher order homogenization schemes have been proposed to replace the composite material by a homogeneous substitution generalized medium.…”

Section: Introductionmentioning

confidence: 99%

Generalized continuum overall modelling of periodic composite structures

Trinh

Forest

2011

Vietnam J. Mech.

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Abstract. Classical homogenization methods fail to reproduce the overall response of composite structures when macroscopic strain gradients become significant. Generalized continuum models like Cosserat, strain gradient and micromorphic media, can be used to enhance the overall description of heterogeneous materials when the hypothesis of scale separation is not fulfilled. We show in the present work how the higher order elasticity moduli can be identified from suitable loading conditions applied to the unit cell of a periodic composite. The obtained homogeneous substitution generalized continuum is used then to predict the response of a composite structure subjected to various loading conditions. Reference finite element computations are performed on the structure taking all the heterogeneities into account. The overall substitution medium is shown to provide improved predictions compared to standard homogenization. In particular the additional boundary conditions required by generalized continua makes it possible to better represent the clamping conditions on the real structure.

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Two-scale modelling of micromorphic continua

Cited by 65 publications

References 21 publications

Partially relaxed energy potentials for the modelling of microstructures – application to shape memory alloys

Partially relaxed energy potentials for the modelling of microstructures – application to shape memory alloys

Relaxed loading conditions for higher order homogenisation approaches

Generalized continuum overall modelling of periodic composite structures

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