Consistent micro–macro transitions at large objective strains in curvilinear convective coordinates

Grytz, Rafael; Meschke, Günther

doi:10.1002/nme.2105

Cited by 15 publications

(12 citation statements)

References 26 publications

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“…According to the previously mentioned connection Fig. 2 Scale-up and scale-down operations for second order tensors [5] between Y i m and Y i M the homogenization of the presented heterogeneous micro-structure in Figure 1 results in a transversal orthotropic material response at the macro-scale with preferred direction Y 1 M . Thus, the orientation of microscopic fibers coincide with the curvilinear coordinate lines…”

mentioning

confidence: 79%

“…We introduce the operations scale-up and scale-down [5], which define the representation of a microscopic tensor variable at the macro-scale and vice versa. They will be used for representation of the scale transitions in absolute tensor notation.…”

Section: Scale-up and Scale-down Operationsmentioning

confidence: 99%

“…The rules introduced above are illustrated in Figure 2. As an example, the scale-up operation defined in (5) can be used to reformulate the homogenization of the stress field in absolute tensor notation for the consideration of different physical spaces at different scales…”

Section: Scale-up and Scale-down Operationsmentioning

confidence: 99%

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Computational micro‐macro transitions at large strains for curvilinear physical directions

Grytz

Meschke

2007

Proc Appl Math and Mech

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Biological tissues such as those involved in the eye, heart, veins or arteries are heterogeneous on one or another spatial scale and can undergo very large elastic strains. Frequently, these tissues are characterized by shell-like structures at the macroscopic scale and the physical material directions follow curvilinear paths. We consider a homogenized macro-continuum formulated in curvilinear convective coordinates with locally attached representative micro-structures. Micro-structures attached to different macroscopic points are assumed to be rotated counterparts according to the curvilinear path of the physical material directions at the macro-scale. The solution of such multi-scale problems according to the computational homogenization scheme [1, 2, 3] would need a different RVE at each macroscopic point. The goal of this paper is to use the same initial RVE at each macroscopic point by generalizing the computational homogenization scheme to a formulation considering different physical spaces at the micro-and macro-scale. The deformation and the reference frame of the micro-structure are assumed to be coupled with the local deformation and the local reference frame at the corresponding point of the macrocontinuum. For a consistent formulation of micro-macro transitions physical reference directions are defined on both scales, where the macroscopic one follows a curvilinear path. To formulate the generalized micro-macro transitions in absolute tensor notation the operations scale-up and scale-down are introduced. Definition of micro-macro transitions at finite strains in curvilinear coordinates Kinematics of micro-and macro-continuum in curvilinear convective coordinates at finite strainsLet B M0 and B m0 be the configurations of a macro-and micro-structure at time t 0 in a 3D Euclidean space R 3 . In shell analyses it is preferable to define the position vector as a function of three independent parameters Θ i representing curvilinear coordinates. Note that for shell-like structures Θ 1 and Θ 2 specify the middle surface of the shell and Θ 3 points in thickness direction. The position vector of a typical point P M0 (P m0 ) of B M0 (B m0 ) is denoted by

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confidence: 79%

Section: Scale-up and Scale-down Operationsmentioning

confidence: 99%

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Computational micro‐macro transitions at large strains for curvilinear physical directions

Grytz

Meschke

2007

Proc Appl Math and Mech

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“…In this contribution, all relations are represented only in Lagrangian description, considering that it is straightforward to formulate the problem in Eulerian description [206,402,408,458]. Furthermore, it is possible to establish the homogenization theory based on Green-Lagrange strain and Piola-Kirchhoff stress as alternatively suitable strain and stress measures [459,460].…”

Section: Theorymentioning

confidence: 99%

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Saeb

Steinmann

Javili

2016

Applied Mechanics Reviews

185

138

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The objective of this contribution is to present a unifying review on strain-driven computational homogenization at finite strains, thereby elaborating on computational aspects of the finite element method. The underlying assumption of computational homogenization is separation of length scales, and hence, computing the material response at the macroscopic scale from averaging the microscopic behavior. In doing so, the energetic equivalence between the two scales, the Hill-Mandel condition, is guaranteed via imposing proper boundary conditions such as linear displacement, periodic displacement and antiperiodic traction, and constant traction boundary conditions. Focus is given on the finite element implementation of these boundary conditions and their influence on the overall response of the material. Computational frameworks for all canonical boundary conditions are briefly formulated in order to demonstrate similarities and differences among the various boundary conditions. Furthermore, we detail on the computational aspects of the classical Reuss' and Voigt's bounds and their extensions to finite strains. A concise and clear formulation for computing the macroscopic tangent necessary for FE 2 calculations is presented. The performances of the proposed schemes are illustrated via a series of two-and three-dimensional numerical examples. The numerical examples provide enough details to serve as benchmarks.

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“…Homogenization procedures are most often formulated using the first Piola–Kirchhoff stress and deformation gradient as measures for stress and strain. Grytz and Meschke show how to obtain results in terms of the second Piola–Kirchhoff stress and Green–Lagrange strain in a consistent manner for strain‐driven homogenization. However, it is not directly obvious how to use alternative finite stress and strain measures in a stress‐driven homogenization.…”

Section: Introductionmentioning

confidence: 99%

Formulation and implementation of stress‐driven and/or strain‐driven computational homogenization for finite strain

Dijk

2016

Numerical Meth Engineering

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SUMMARYIn this paper, we present a homogenization approach that can be used in the geometrically nonlinear regime for stress-driven and strain-driven homogenization and even a combination of both. Special attention is paid to the straightforward implementation in combination with the finite-element method. The formulation follows directly from the principle of virtual work, the periodic boundary conditions, and the Hill-Mandel principle of macro-homogeneity. The periodic boundary conditions are implemented using the Lagrange multiplier method to link macroscopic strain to the boundary displacements of the computational model of a representative volume element. We include the macroscopic strain as a set of additional degrees of freedom in the formulation. Via the Lagrange multipliers, the macroscopic stress naturally arises as the associated 'forces' that are conjugate to the macroscopic strain 'displacements'. In contrast to most homogenization schemes, the second Piola-Kirchhoff stress and Green-Lagrange strain have been chosen for the macroscopic stress and strain measures in this formulation. The usage of other stress and strain measures such as the first Piola-Kirchhoff stress and the deformation gradient is discussed in the Appendix.

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Consistent micro–macro transitions at large objective strains in curvilinear convective coordinates

Cited by 15 publications

References 26 publications

Computational micro‐macro transitions at large strains for curvilinear physical directions

Computational micro‐macro transitions at large strains for curvilinear physical directions

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Formulation and implementation of stress‐driven and/or strain‐driven computational homogenization for finite strain

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