HIGHER-ORDER ASYMPTOTIC HOMOGENIZATION OF PERIODIC MATERIALS WITH LOW SCALE SEPARATION

2016 ◽  
Author(s):  
M. Mohammed Ameen ◽  
R.H.J. Peerlings ◽  
M.G.D. Geers
Keyword(s):  
Higher Order ◽  
Asymptotic Homogenization ◽  
Scale Separation ◽  
Periodic Materials

Higher order asymptotic homogenization and wave propagation in periodic composite materials

2008 ◽  
Vol 464 (2093) ◽  
pp. 1181-1201 ◽  
Author(s):  
Igor V Andrianov ◽  
Vladimir I Bolshakov ◽  
Vladyslav V Danishevs'kyy ◽  
Dieter Weichert
Keyword(s):  
Composite Materials ◽  
Exact Analytical Solution ◽  
Higher Order ◽  
Square Lattice ◽  
Infinite Plane ◽  
Dimensional Problem ◽  
Asymptotic Homogenization ◽  
Plane Shear ◽  
Frequency Bands ◽  
Periodic Composite

We present an application of the higher order asymptotic homogenization method (AHM) to the study of wave dispersion in periodic composite materials. When the wavelength of a travelling signal becomes comparable with the size of heterogeneities, successive reflections and refractions of the waves at the component interfaces lead to the formation of a complicated sequence of the pass and stop frequency bands. Application of the AHM provides a long-wave approximation valid in the low-frequency range. Solution for the high frequencies is obtained on the basis of the Floquet–Bloch approach by expanding spatially varying properties of a composite medium in a Fourier series and representing unknown displacement fields by infinite plane-wave expansions. Steady-state elastic longitudinal waves in a composite rod (one-dimensional problem allowing the exact analytical solution) and transverse anti-plane shear waves in a fibre-reinforced composite with a square lattice of cylindrical inclusions (two-dimensional problem) are considered. The dispersion curves are obtained, the pass and stop frequency bands are identified.

Generalized FVDAM Theory for Periodic Materials Undergoing Finite Deformations—Part II: Results

2013 ◽  
Vol 81 (2) ◽  
Author(s):  
Marcio A. A. Cavalcante ◽  
Marek-Jerzy Pindera
Keyword(s):  
Displacement Field ◽  
Computer Code ◽  
Biological Tissues ◽  
Order Theory ◽  
Substantial Improvement ◽  
Higher Order ◽  
Finite Deformations ◽  
Stress Distributions ◽  
Field Representation ◽  
Periodic Materials

In Part I, a generalized finite-volume direct averaging micromechanics (FVDAM) theory was constructed for periodic materials with complex microstructures undergoing finite deformations. The generalization involves the use of a higher-order displacement field representation within individual subvolumes of a discretized analysis domain whose coefficients were expressed in terms of surface-averaged kinematic variables required to be continuous across adjacent subvolume faces. In Part II of this contribution we demonstrate that the higher-order displacement representation leads to a substantial improvement in subvolume interfacial conformability and smoother stress distributions relative to the original theory based on a quadratic displacement field representation, herein called the 0th order theory. This improvement is particularly important in the finite-deformation domain wherein large differences in adjacent subvolume face rotations may lead to the loss of mesh integrity. The advantages of the generalized theory are illustrated through examples based on a known analytical solution and finite-element results generated with a computer code that mimics the generalized theory's framework. An application of the generalized FVDAM theory involving the response of wavy multilayers confirms previously generated results with the 0th order theory that revealed microstructural effects in this class of materials which are important in bio-inspired material architectures that mimic certain biological tissues.

scholarly journals Study of wave dispersion in periodic composites by higher-order asymptotic homogenization method

PAMM ◽  
2007 ◽  
Vol 7 (1) ◽  
pp. 1090403-1090404
Author(s):  
Igor V. Andrianov ◽  
Vladimir I. Bolshakov ◽  
Vladyslav V. Danishevs'kyy ◽  
Dieter Weichert
Keyword(s):  
Wave Dispersion ◽  
Homogenization Method ◽  
Higher Order ◽  
Asymptotic Homogenization ◽  
Periodic Composites ◽  
Asymptotic Homogenization Method

Finite Volume-Based Asymptotic Homogenization of Periodic Materials Under In-Plane Loading

2020 ◽  
Vol 87 (12) ◽  
Author(s):  
Zhelong He ◽  
Marek-Jerzy Pindera
Keyword(s):  
Finite Volume ◽  
Local Equilibrium ◽  
Shear Loading ◽  
Local Fields ◽  
Homogenization Theory ◽  
Asymptotic Homogenization ◽  
Plane Shear ◽  
Strain Gradients ◽  
Combination Approach ◽  
Periodic Materials

Abstract The previously developed finite volume-based asymptotic homogenization theory (FVBAHT) for anti-plane shear loading (He, Z., and Pindera, M.-J., “Finite-Volume Based Asymptotic Homogenization Theory for Periodic Materials Under Anti-Plane Shear,” Eur. J. Mech. A Solids (in revision)) is further extended to in-plane loading of unidirectional fiber reinforced periodic structures. Like the anti-plane FVBAHT, the present extension builds upon the previously developed finite volume direct averaging micromechanics theory applicable under uniform strain fields and further accounts for strain gradients and non-vanishing microstructural scale relative to structural dimensions, albeit with multidimensional in-plane loadings incorporated. The unit cell problems at different orders of the asymptotic field expansion are solved by satisfying local equilibrium equations and displacement and traction continuity in a surface-averaged sense which is unique among the existing asymptotic homogenization schemes, leading to microfluctuation functions that yield homogenized stiffness tensors at each order for use in macroscale problems. The newly extended multiscale theory is employed in the analysis of a structural boundary-value problem under in-plane loading, illustrating pronounced boundary effects. A combination approach proposed in the literature is subsequently employed to mitigate the boundary layer effects by explicitly accounting for the microstructural details in the boundary region. This combination approach produces accurate recovery of the local fields in both regions. The extension to in-plane problem marks FVBAHT as an alternative, self-contained asymptotic homogenization tool, with documented advantages relative to current numerical techniques, for the analysis of periodic materials in the presence of strain gradients produced by three-dimensional loading regardless of microstructural scale.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

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