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The purpose of this paper is to test the capability to properly analyze the electrical circuits of a novel constitutive relation of capacitor.

For ceteris paribus, the constitutive relations of the resistor and inductor have been reformulated by following the novel constitutive relation of capacitor. The responses of RL, RC, LC and RLC circuits defined on the fractal set described by these definitions have been derived by means of the fractal calculus and fractal Laplace transformation. A comparative Hamiltonian formalism-based analysis has been performed where the circuits described by the conventional and the formerly proposed revisited constitutive relations have also been considered.

This study has found that the novel constitutive relations give unreasonable results unlike the conventional ones. Like such previous revisited constitutive relations, an odd Hamiltonian has been obtained. On the other hand, the conventional constitutive relations give a reasonable Hamiltonian.

To the best of the author’s knowledge, for the first time, the analysis of fractal set defined electrical circuits by means of unconventional constitutive relations has been performed where the deficiency of the tested capacitive constitutive relation has been pointed out.

As a powerful mathematical analysis tool, the local fractional calculus has attracted wide attention in the field of fractal circuits. The purpose of this paper is to derive a new ℑ-order non-differentiable (ND) R-C zero state-response circuit (ZSRC) by using the local fractional derivative on the Cantor set for the first time.

A new ℑ-order ND R-C ZSRC within the local fractional derivative on the Cantor set is derived for the first time in this work. By defining the ND lumped elements via the local fractional derivative, the ℑ-order Kirchhoff voltage laws equation is established, and the corresponding solutions in the form of the Mittag-Leffler decay defined on the Cantor sets are derived by applying the local fractional Laplace transform and inverse local fractional Laplace transform.

The characteristics of the ℑ-order R-C ZSRC on the Cantor sets are analyzed and presented through the 2-D curves. It is found that the ℑ-order R-C ZSRC becomes the classic one when ℑ = 1. The comparative results between the ℑ-order R-C ZSRC and the classic one show that the proposed method is correct and effective and is expected to shed light on the theory study of the fractal electrical systems.

To the best of the authors’ knowledge, this paper, for the first time ever, proposes the ℑ-order ND R-C ZSRC within the local fractional derivative on the Cantor sets. The results of this paper are expected to give some new enlightenment to the development of the fractal circuits.

The purpose of this paper is to derive a new fractal active low-pass filter (LPF) within the local fractional derivative (LFD) calculus on the Cantor set (CS).

To the best of the author’s knowledge, a new fractal active LPF within the LFD on the CS is proposed for the first time in this work. By defining the nondifferentiable (ND) lumped elements on the fractal set, the author successfully extracted its ND transfer function by applying the local fractional Laplace transform. The properties of the ND transfer function on the CS are elaborated in detail.

The comparative results between the fractal active LPF (for γ = ln2/ln3) and the classic one (for γ = 1) on the amplitude–frequency and phase–frequency characteristics show that the proposed method is correct and effective, and is expected to shed light on the theory study of the fractal electrical systems.

To the best of the author’s knowledge, the fractal active LPF within the LFD calculus on the CS is proposed for the first time in this study. The proposed method can be used to study the other problems in the fractal electrical systems, and is expected to shed a light on the theory study of the fractal electrical systems.

The fractal and fractional calculus have obtained considerable attention in the electrical and electronic engineering since they can model many complex phenomena that the traditional integer-order calculus cannot. The purpose of this paper is to develop a new fractional pulse narrowing nonlinear transmission lines model within the local fractional calculus for the first time and derive a novel method, namely, the direct mapping method, to seek for the nondifferentiable (ND) exact solutions.

By defining some special functions via the Mittag–Leffler function on the Cantor sets, a novel approach, namely, the direct mapping method is derived via constructing a group of the nonlinear local fractional ordinary differential equations. With the aid of the direct mapping method, four groups of the ND exact solutions are obtained in just one step. The dynamic behaviors of the ND exact solutions on the Cantor sets are also described through the 3D graphical illustration.

It is found that the proposed method is simple but effective and can construct four sets of the ND exact solutions in just one step. In addition, one of the ND exact solutions becomes the exact solution of the classic pulse narrowing nonlinear transmission lines model for the special case 9 = 1, which strongly proves the correctness and effectiveness of the method. The ideas in the paper can be used to study the other fractal partial differential equations (PDEs) within the local fractional derivative (LFD) arising in electrical and electronic engineering.

The fractional pulse narrowing nonlinear transmission lines model within the LFD is proposed for the first time in this paper. The proposed method in the work can be used to study the other fractal PDEs arising in electrical and electronic engineering. The findings in this work are expected to shed a light on the study of the fractal PDEs arising in electrical and electronic engineering.

This paper aims to present a novel scheme for imposing periodic boundary conditions with downscaled macroscopic strain measures of gradient Cosserat continuum on the representative volume element (RVE) of discrete particle assembly in the frame of the second-order computational homogenization methods for granular materials.

The proposed scheme is based on the generalized Hill’s lemma of gradient Cosserat continuum and the incremental non-linear constitutive relation condensed to the peripheral particles of the RVE of discrete particle assembly. The generalized Hill’s lemma conducts to downscale the macroscopic strain or stress measures and to impose the periodic boundary conditions on the RVE boundary so that the Hill-Mandel energy equivalence condition is ensured. Because of the incremental non-linear constitutive relation condensed to the peripheral particles of the RVE, the periodic boundary displacement and traction constraints together with the downscaled macroscopic strains and strain gradients, micro-rotations and curvatures are imposed in the point-wise sense without the need of introducing the Lagrange multipliers for enforcing the periodic boundary displacement and traction constraints in a weak sense.

Numerical results demonstrate that the applicability and effectiveness of the proposed scheme in imposing the periodic boundary conditions on the RVE. The results of the RVE subjected to the periodic boundary conditions together with the displacement boundary conditions in the second-order computational homogenization for granular materials provide the desired estimations, which lie between the upper and the lower bounds provided by the displacement and the traction boundary conditions imposed on the RVE respectively.

Each grain in the particulate system under consideration is assumed to be rigid and circular.

The proposed scheme for imposing periodic boundary conditions on the RVE can be adopted solely for estimating the effective mechanical properties of granular materials and/or integrated into the frame of the second-order computational homogenization method with a nested finite element method-discrete element method solution procedure for granular materials. It will tend to provide, at least theoretically, more reasonable results for effective material properties and solutions of a macroscopic boundary value problem simulated by the computational homogenization method.

This paper presents a novel scheme for imposing periodic boundary conditions with downscaled macroscopic strain measures of gradient Cosserat continuum on the RVE of discrete particle assembly for granular materials without need of introducing Lagrange multipliers for enforcing periodic boundary conditions in a weak (integration) sense.

We have developed a computer oxidation modeling program, named NOVEL, which has been integrated into our process simulator FINDPRO. It combines the modified Deal‐Grove growth rate model with a nonlinear viscoclastic deformation model to predict both the oxide shape and stress. Modeling the thermal oxidation of silicon presents several numerical challenges. First, the oxide region expands and deforms extensively during the process which has to be modeled as a moving boundary, large deformation problem. Second, the SiO2 mechanical property changes from clastic to viscoclastic to viscous as the processing temperature is changed from a value below the the glass transition temperature (960°C) to one above it. The viscoclastic deformation model which is adequate over the entire temperature range of interest has an intrinsic numerical singularity when the oxide viscosity (divided by time) becomes relatively lower than the elastic modulus at high temperatures. These must be handled appropriately to ensure that the modeled results are correct. In this paper, we present details of how NOVEL solves the above mentioned problems. We show examples of low temperature/high pressure oxidation of a LOCOS structure, trench isolation structure, and the technique by which the finite element program NOVEL interfaces with the finite difference process simulator FINDPRO.

The purpose of this paper is to perform experimental investigation and constitutive modeling of the viscoelastic and viscoplastic behavior of metallocene catalyzed polypropylene (mPP) with application to lifetime assessment under conditions of creep rupture.

Three series of experiments are conducted where the mechanical response of mPP is analyzed in tensile tests with various strain rates, relaxation tests with various strains, and creep tests with various stresses at room temperature. A constitutive model is derived for semicrystalline polymers under an arbitrary three‐dimensional deformation with small strains, and its parameters are found fitting the observations.

Crystalline structure and molecular architecture of polypropylene strongly affect its time‐ and rate‐dependent behavior. In particular, time‐to‐failure of metallocene catalyzed polypropylene under tensile creep noticeably exceeds that of isotactic polypropylene produced by the conventional Ziegler‐Natta catalysis.

Novel stress‐strain relations are developed in viscoelastoplasticity of semi‐crystalline polymers and applied to predict their mechanical behavior in long‐term creep tests.

The purpose of this paper is to compare mechanical response of polypropylene in multi‐cycle tensile tests with strain‐controlled and mixed deformation programs and to develop constitutive equations that describe quantitatively the experimental data.

Multi‐cycle tensile tests are performed on isotactic polypropylene with strain‐controlled (oscillations between fixed maximum and minimum strains) and mixed (oscillations between a fixed maximum strain and the zero minimum stress) programs. A constitutive model is derived in cyclic viscoelasticity and viscoplasticity of semicrystalline polymers, and its parameters are found by fitting observations. The effect of damage accumulation of material parameters is analyzed numerically.

The model predicts accurately mechanical behavior of polypropylene in tests with numbers of cycles strongly exceeding those used to determine its parameters. In the regime of developed damage, material constants in the stress‐strain relations are independent of deformation program.

A novel constitutive model is derived in cyclic viscoelastoplasticity of semicrystalline polymers and comparison of its adjustable parameters is performed for different deformation programs.

A general first-invariant constitutive model has been derived in literature for incompressible, isotropic hyperelastic materials, known as Marlow model, which reproduces test data exactly without the need of curve-fitting procedures. This paper aims to describe how to extend Marlow’s constitutive model to the more general case of compressible hyperelastic materials.

The isotropic constitutive model is based on a strain energy function, whose isochoric part is solely dependent on the first modified strain invariant. Based on Marlow’s idea, a principle of energetically equivalent deformation states is derived for the compressible case, which is used to determine the underlying strain energy function directly from measured test data. No particular functional of the strain energy function is assumed. It is shown how to calibrate the volumetric and isochoric strain energy functions uniquely with uniaxial or biaxial test data only. The constitutive model is implemented into a finite element program to demonstrate its applicability.

The model is well suited for use in finite element analysis. Only one set of test data is required for calibration without any need for curve-fitting procedures. These test data are reproduced exactly, and the model prediction is reasonable for other deformation modes.

Marlow’s basic concept is extended to the compressible case and applied to both the volumetric and isochoric part of the compressible strain energy function. Moreover, a novel approach is described on how both compressive and tensile test data can be used simultaneously to calibrate the model.