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1

The mixed FEM for analysis of quantum-dot systems based on gradient theory

Sladek Jan, Sladek Vladimir, Repka Miroslav, Hrcek Slavomir

Computer Methods in Materials Science

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2018

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Vol. 18, No. 3

81--89

EN

The QD nanostructures are analyzed under a thermal load. The dimensions of the QDs are of the same order as the material length scale. Therefore, the gradient elasticity theory should be applied to account for the size-dependent behavior of such nano-sized QDs. Since governing equations contain higher order derivatives than in conventional approaches the C1-elements are required for approximation of primary fields in the FEM. The mixed FEM are developed here, where C0 continuous interpolation is applied independently for displacement and displacement gradients. The kinematic constraints between strains and displacements are satisfied by collocation at some cleverly chosen internal points in elements. A unit cell of Indium Arsenide QD in a finite sized Gallium Arsenide (GaAs) substrate is analysed.

2

Energetyka przepływów jonowych w teorii gradientowej

Kubik J.

Roczniki Inżynierii Budowlanej

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2017

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z. 17

79--84

EN

The energy of ionic flow in the gradient thermomechanics is analysed in the paper taking into consideration changes in electrostatic forces. In particular, taking into account the balances of energy and entropy, the residual inequality for the considered case is formulated. Finally, a set of physical equations of the process is obtained.

3

Więzy termomechaniczne w teorii gradientowej

Kubik J.

Roczniki Inżynierii Budowlanej

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2013

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z. 13

111--114

EN

The theory of thermomechanical constraints is used in the work in order to derive particular forms of equations for the theory of gradient thermodiffusion. The constraints are considered for the material medium in which the gradient components of the equations do not cause additional energy dissipation.

4

Symetrie szczególnych teorii gradientowych

Kubik J.

Roczniki Inżynierii Budowlanej

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2013

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z. 13

37--40

EN

The symmetry of particular forms of equations in the theory of gradient viscoelasticity is analysed in the work. Basing on the considerations the reciprocity principle is derived for this case. The principle enables solving boundary problems as for the simplified from of the theory and full one as well.

5

Termodyfuzja gradientowa w wieloskładnikowym ciele odkształcalnym

Kubik J.

Roczniki Inżynierii Budowlanej

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2008

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z. 8

31--36

EN

In the paper, thermodiffusion viscoelastic equation, which in addition to stress and strain tensors appear in their gradients, are given. This type of theory describes the changes of stress fields in the neighborhood of the coast or interfacial border. Starting point for these considerations are the equations of the mixtures theory, which as a result of border crossing, we get the equation of gradient thermodiffusion.

6

Non-local coupling of viscoplasticity and anisotropic viscodamage for impact problems using the gradient theory

Voyiadjis G. Z., Abu Al-Rub R. K., Palazotto A. N.

Archives of Mechanics

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2003

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Vol. 55, nr 1

39-89

EN

A general framework for the analysis of heterogeneous media that assesses a strong coupling between viscoplasticity and anisotropic viscodamage evolution is formulated for-impact related problems within the framework of thermodynamic laws and nonlinear continuum mechanics. The proposed formulations include thermo-elasto-viscoplastici- ty with anisotropic thermo-elasto-viscodamage, a dynamic yield criterion of a von Mises type and a dynamic viscodamage criterion, the associated flow rules, non-linear strain hardening, strain-rate hardening, and temperature softening. The constitutive equations for the damaged material are written according to the principle of strain energy equivalence between the virgin material and the damaged material. That is, the damaged material is modeled using the constitutive laws of the effective undamaged material in which the nominal stresses are replaced by the effective stresses. The evolution laws are impeded in a finite deformation framework based on the multiplicative decomposition of the deformation gradient into elastic, viscoplastic, and viscodamage parts. Since the material macroscopic thermomechanical response under high-impact loading is governed by different physical mechanisms on the macroscale level, the proposed three-dimensional kinematical model is introduced with manifold structure accounting for discontinuous fields of dislocation interactions (plastic hardening), and crack and void interactions (damage hardening). The non-local theory of viscoplasticity and viscodamage that incorporates macroscale interstate variables and their higher-order gradients is used here to describe the change in the internal structure and in order to investigate the size effect of statistical inhomogeneity of the evolution-related viscoplasticity and viscodamage hardening variables. The gradients are introduced here in the hardening internal state variables and are considered to be independent of their local counterparts. It also incorporates the thermomechanical coupling effects as well as the internal dissipative effects through the rate-type covariance constitutive structure with a finite set of internal state variables. The model presented in this paper can be considered as a framework, which enables one to derive various non-local and gradient viscoplasticity and viscodamage theories by introducing simplifying assumptions.

7

On nonlocal gradient model of inelastic heterogeneous media

Stumpf H., Saczuk J

Journal of Theoretical and Applied Mechanics

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2002

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Vol. 40 nr 1

205-234

EN

The aim of this paper is to investigate the influence of nonlocality on the physical and material field equations of heterogeneous media. Taking into account that plastic deformations in metals or damage in brittle and ductile materials are governed by physical mechanisms observed on levels with different lengthscales, we introduce a 6-dimensional kinematical concept with two locally defined vectors to model the material behaviour on a macro- and meso- or microlevel. Using a variational procedure the physical and material balance laws, boundary and transversality conditions are derived for macrp- and microdeformations of heterogeneous media. The disspation inquality including relaxation terms for transport processes is presented. The constitutive equations are formulated with macro- and microstrain measures, their gradients and time rates, and the anisotropy tensor as arguments, where the latter can be considered as a coupling measure between the deformed macrostates with compatible microstates. The model presented in this paper delivers a framework, which enables one to derive various nonlocal and gradient theories by introducing simplifying assumptions. As the special case a solid-void model is considered.

PL

Celem pracy jest zbadanie wpływu nielokalności na fizyczne i materialne równania pola ośrodków heterogenicznych. Biorąc pod uwagę, że plastyczna deformacja w metalach lub zniszczenie w kruchych i ciągliwych materiałach rządzone są przez fizyczne mechanizmy na różnych poziomach skali, wprowadzono 6-wymiarową strukturę z dwoma lokalnie zdefiniowanymi wektorami do modelowania materialnego zachowania ośrodka na poziomie makro- i mezo- lub mikroskali. Wykorzystując wariacyjną procedurę otrzymano fizyczne i materialne prawa bilansu, warunki brzegowe i transwersalność dla makro- i mikrodeformacji ośrodków heterogenicznych. Przedstawiona nierówność dyssypacyjna zawiera człony relaksacyjne procesów transportu. Sformułowane równania konstytutywne wyrażono przy pomocy miar makro- i mikroodkształcenia, ich gradientów i przyrostów oraz tensora anizotropii, gdzie ostatni argument może być traktowany jako miara sprzężenia pomiędzy odkształconymi makrostanami i kompatybilnymi mikrostanami. Przedstawiony w pracy model dostarcza podstaw, które poprzez wprowadzenie uproszczających założeń umożliwiają otrzymanie różnych postaci nielokalnych i gradientowych teorii. Jako przypadek szczególny rozpatrzono model typu ciało stałe-pustka.