On a discrete-to-continuum convergence result for a two dimensional brittle material in the small displacement regime

AbstractA celebrated result of Halász describes the asymptotic behavior of the arithmetic mean of an arbitrary multiplicative function with values on the unit disc. We extend this result to multilinear averages of multiplicative functions providing similar asymptotics, thus verifying a two dimensional variant of a conjecture of Elliott. As a consequence, we get several convergence results for such multilinear expressions, one of which generalises a well known convergence result of Wirsing. The key ingredients are a recent structural result for multiplicative functions with values on the unit disc proved by the authors and the mean value theorem of Halász.

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Homogenization in BV of a model for layered composites in finite crystal plasticity

Advances in Calculus of Variations ◽

10.1515/acv-2019-0011 ◽

2019 ◽

Vol 0 (0) ◽

Author(s):

Elisa Davoli ◽

Rita Ferreira ◽

Carolin Kreisbeck

Keyword(s):

Crystal Plasticity ◽

Integral Form ◽

Convergence Result ◽

Variational Model ◽

Functions Of Bounded Variation ◽

Potential Candidate ◽

Gamma Convergence ◽

Two Dimensional ◽

Rigid Component

Abstract In this work, we study the effective behavior of a two-dimensional variational model within finite crystal plasticity for high-contrast bilayered composites. Precisely, we consider materials arranged into periodically alternating thin horizontal strips of an elastically rigid component and a softer one with one active slip system. The energies arising from these modeling assumptions are of integral form, featuring linear growth and non-convex differential constraints. We approach this non-standard homogenization problem via Gamma-convergence. A crucial first step in the asymptotic analysis is the characterization of rigidity properties of limits of admissible deformations in the space BV of functions of bounded variation. In particular, we prove that, under suitable assumptions, the two-dimensional body may split horizontally into finitely many pieces, each of which undergoes shear deformation and global rotation. This allows us to identify a potential candidate for the homogenized limit energy, which we show to be a lower bound on the Gamma-limit. In the framework of non-simple materials, we present a complete Gamma-convergence result, including an explicit homogenization formula, for a regularized model with an anisotropic penalization in the layer direction.

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Numerical Solution of Parabolic Equation on a 2-D Infinite Strip with Composite Hermite-Legendre Spectral Method

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.647.875 ◽

2013 ◽

Vol 647 ◽

pp. 875-879

Author(s):

Ting Jing Zhao

Keyword(s):

Differential Equation ◽

Parabolic Equation ◽

Model Problem ◽

Convergence Result ◽

Parabolic Differential Equation ◽

Infinite Strip ◽

Two Dimensional ◽

Numerical Tests ◽

Legendre Spectral Method ◽

Legendre Galerkin Method

For solving numerically parabolic differential equation on a two-dimensional infinite strip, composite Hermite-Legendre Galerkin method is proposed in this article. By making use of stabilised scaled factor, the proposed method achieves stability. We also establish the convergence result for the proposed method. Numerical tests conduct for the model problem. It is shown that the proposed method is efficient.

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Geometric linearization of theories for incompressible elastic materials and applications

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202521500202 ◽

2021 ◽

pp. 1-32

Author(s):

Martin Jesenko ◽

Bernd Schmidt

Keyword(s):

Small Displacement ◽

Gamma Convergence ◽

Stored Energy ◽

Energy Functionals ◽

Small Length Scale ◽

Nematic Elastomers ◽

Incompressible Materials ◽

Geometric Linearization ◽

Displacement Regime ◽

Nonlinear Energy

We derive geometrically linearized theories for incompressible materials from nonlinear elasticity theory in the small displacement regime. Our nonlinear stored energy densities may vary on the same (small) length scale as the typical displacements. This allows for applications to multiwell energies as, e.g. encountered in martensitic phases of shape memory alloys and models for nematic elastomers. Under natural assumptions on the asymptotic behavior of such densities we prove Gamma-convergence of the properly rescaled nonlinear energy functionals to the relaxation of an effective model. The resulting limiting theory is geometrically linearized in the sense that it acts on infinitesimal displacements rather than finite deformations, but will in general still have a limiting stored energy density that depends in a nonlinear way on the infinitesimal strains. Our results, in particular, establish a rigorous link of existing finite and infinitesimal theories for incompressible nematic elastomers.

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Two-Dimensional Isotropic Damage Elastoplastic Model for Quasi-Brittle Material

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.606.103 ◽

2014 ◽

Vol 606 ◽

pp. 103-106

Author(s):

Pavel Beneš ◽

Daniel Vavřík

Keyword(s):

Brittle Material ◽

Mechanical Model ◽

Two Dimensions ◽

Damage Variable ◽

Two Dimensional ◽

Elastoplastic Model ◽

Isotropic Damage

Micro-mechanical model for isotropic damage of quasi-brittle material including frictionis presented. Damage is assumed to be isotropic and scalar damage variable is employed . Operatorsplitting method is applied. The article contains derived expressions for derivations necessary forcomputation of coefficients in two dimensions for strain and damage normality rules.

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Refined convergence for the Boolean model

Advances in Applied Probability ◽

10.1239/aap/1261669579 ◽

2009 ◽

Vol 41 (4) ◽

pp. 940-957 ◽

Cited By ~ 1

Author(s):

Pierre Calka ◽

Julien Michel ◽

Katy Paroux

Keyword(s):

Probability Space ◽

Order Term ◽

Convergence Result ◽

Boolean Model ◽

Random Sets ◽

Two Dimensional ◽

Convergence In Distribution ◽

Line Process ◽

The Difference ◽

Poisson Line Process

In Michel and Paroux (2003) the authors proposed a new proof of a well-known convergence result for the scaled elementary connected vacant component in the high intensity Boolean model towards the Crofton cell of the Poisson hyperplane process (see, e.g. Hall (1985)). In this paper we investigate the second-order term in this convergence when the two-dimensional Boolean model and the Poisson line process are coupled on the same probability space. We consider the particular case where the grains are discs with random radii. A precise coupling between the Boolean model and the Poisson line process is first established. A result of directional convergence in distribution for the difference of the two sets involved is then derived. Eventually, we show the convergence of the process, measuring the difference between the two random sets, once rescaled, as a function of the direction.

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Refined convergence for the Boolean model

Advances in Applied Probability ◽

10.1017/s0001867800003670 ◽

2009 ◽

Vol 41 (04) ◽

pp. 940-957 ◽

Cited By ~ 1

Author(s):

Pierre Calka ◽

Julien Michel ◽

Katy Paroux

Keyword(s):

Probability Space ◽

Order Term ◽

Convergence Result ◽

Boolean Model ◽

Random Sets ◽

Two Dimensional ◽

Convergence In Distribution ◽

Line Process ◽

The Difference ◽

Poisson Line Process

In Michel and Paroux (2003) the authors proposed a new proof of a well-known convergence result for the scaled elementary connected vacant component in the high intensity Boolean model towards the Crofton cell of the Poisson hyperplane process (see, e.g. Hall (1985)). In this paper we investigate the second-order term in this convergence when the two-dimensional Boolean model and the Poisson line process are coupled on the same probability space. We consider the particular case where the grains are discs with random radii. A precise coupling between the Boolean model and the Poisson line process is first established. A result of directional convergence in distribution for the difference of the two sets involved is then derived. Eventually, we show the convergence of the process, measuring the difference between the two random sets, once rescaled, as a function of the direction.

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Validation of the Small Boundary Displacement Model of Fluid-Structure Coupling for Predicting Fluid-Elastic Lock-In

Volume 4: Fluid Structure Interaction ◽

10.1115/pvp2005-71570 ◽

2005 ◽

Author(s):

David J. Manko ◽

M. M. Sussman

Keyword(s):

Coupling Model ◽

Elastic Response ◽

Small Displacement ◽

Two Dimensional ◽

Fluid Structure ◽

Boundary Displacement ◽

Displacement Theory ◽

Lock In ◽

Displacement Model ◽

Structural Motion

This paper describes the validation of the Small Boundary Displacement Model (SBDM) of fluid-structure coupling for predicting fluid-elastic lock-in response of a D-shaped cylinder in crossflow. This coupling model extends structural small displacement theory to fluid-structure interfaces, eliminating the need for temporally changing meshes when structural motion is small compared with problem dimensions. The SBDM algorithm accurately predicts the range and characteristics of lock-in behavior when compared to an independent two-dimensional numerical solution. Further validation of the SBDM simulations is provided by comparisons to fluid-only solutions at the limits of lock-in where the cylinder boundary is forced to oscillate at the same amplitudes as the corresponding coupled simulations. The SBDM predicted fluid-elastic response exceeds the 20 dB limit commonly used for experimentally identifying lock-in behavior.

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Spectrophotometric measurements of objective-prism spectra

Symposium - International Astronomical Union ◽

10.1017/s007418090005333x ◽

1966 ◽

Vol 24 ◽

pp. 118-119

Author(s):

Th. Schmidt-Kaler

Keyword(s):

Progress Report ◽

Two Dimensional ◽

Objective Prism ◽

Made In

I should like to give you a very condensed progress report on some spectrophotometric measurements of objective-prism spectra made in collaboration with H. Leicher at Bonn. The procedure used is almost completely automatic. The measurements are made with the help of a semi-automatic fully digitized registering microphotometer constructed by Hög-Hamburg. The reductions are carried out with the aid of a number of interconnected programmes written for the computer IBM 7090, beginning with the output of the photometer in the form of punched cards and ending with the printing-out of the final two-dimensional classifications.

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