Nonlinear Law Research Articles

NUMERICAL IMPLEMENTATION OF THE TSCHOEGL–CHANG–BLOCH NONLINEAR VISCOELASTIC CONSTITUTIVE LAW FOR UNVULCANIZED CARBON BLACK–FILLED RUBBERS

ABSTRACT Rubber engineering design analysis requires a fundamental understanding of the mechanical behavior of polymers, especially in their unvulcanized state. It necessitates the establishment of a three-dimensional constitutive material to account for the observed mechanical behavior. This law is required to produce realistic descriptions of the viscoelastic performance in a mathematically simple form that is easy to implement in engineering applications. This article describes the theory of the Tschoegl–Chang–Bloch time-dependent nonlinear viscoelastic constitutive law. The experimental verification of this law is provided under different deformation fields and multiple load steps. Laboratory test procedures to obtain the parameters required to describe the material under consideration are provided in detail. A recursive form of the constitutive law, suitable for finite element application, is derived and coded in the finite element commercial code Abaqus via the user subroutine UMAT. Comparisons between the experimental observations, the theoretical results, and the numerical data are drawn for simple test models examined under creep or shear relaxation conditions. The excellent agreements observed indicate the suitability of the governing law in analyzing viscoelastic problems of unvulcanized carbon black–filled rubbers.

Physics-based neural network as constitutive law for finite element analysis of sintering

In the sintering of advanced ceramics, a digital twin consisted of a finite element model can predict the final shape and microstructure of a sintered ceramic. However, to minimise uncertainties, it is essential to continually update the mechanical properties in the digital twin using data collected from the manufacturing process. One promising approach for achieving this is through artificial neural networks (ANN). This study introduces a machine learning strategy to update the constitutive behaviour of advanced ceramics in finite element analysis of sintering deformation. A major challenge in implementing machine learning in material processing is the huge amount of data required by the training and validation of an ANN, which are often unavailable or incomplete in a real manufacturing process. This study demonstrates that the data requirement can be reduced by employing a two-step training technique. Firstly, the ANN is trained using a nonlinear constitutive law, which describes a general relationship between the strain rates and stresses. Subsequently, the weights and biases of the ANN are transferred for the retraining using limited experimental data for an actual ceramic. It is shown that such approach can successfully capture the nonlinear constitutive behaviour of fine-grained alumina without demanding for large amount of experimental data. A case study is provided, highlighting the feasibility of implementing the ANN in a commercial finite element package, to replace the constitutive law and predict the shrinkage and distortion of a sintering part. In particular, the sintered dumb-bell shape part, simulated using the retrained ANN, showed grain size and relative density that markedly different from those using the nonlinear constitutive law. It is important to note that the proposed methodology is generic and can be used to create ANNs to replace constitutive laws in finite element analysis in digital twins for a wide range of other engineering processes.

Investigation of thermomagnetic convective flow in vertical layers between water and kerosene based magnetic fluids

In this article, the convective flow in a vertical layer subjected to a consistent magnetic field with nonzero gravity condition is investigated. One side of the vertical walls is heated, while the opposite side is cooling. The magnetic field strength, field orientation angle, magnetization effect, and thermal deviation of vertical walls play important roles as controlling parameters on the flow stabilization. The goal of this investigation is to analyze the flow characteristics and find out the significant distinction between water and kerosene based magnetic fluids under the variation of thermal and magnetic effects. The numerical results are obtained by the pseudo-spectral Chebyshev expansion method. The properties of all instability modes caused by three major mechanisms, namely; thermomagnetic, thermogravitational and magneto-gravitational are analyzed. In the normal magnetic field, the wave speed responds faster, and it is recognized by a relatively small wave number in water based fluid than that in kerosene based fluid. In oblique magnetic field, the waves propagate faster in both kerosene and water based fluids with the field inclination angle increases, but they still propagate slower in kerosene based fluid comparatively in water based fluid. According to the linear or non-linear magnetization law, both upward and downward propagating waves in kerosene based fluid propagate slower, and they are recognized by greater wave numbers than that in water based fluid. It is found that the basic flow in water based fluid is much more stable than in kerosene based fluid.

NN‐mCRE: A modified constitutive relation error framework for unsupervised learning of nonlinear state laws with physics‐augmented neural networks

AbstractThis article proposes a new approach to train physics‐augmented neural networks with observable data to represent mechanical constitutive laws. To train the neural network and learn thermodynamics potentials, the proposed method does not rely on strain‐stress or strain‐free energy pairs but needs only partial strain or displacement measurements inside the structure. The neural network is trained thanks to an unsupervised procedure in which the modified constitutive relation error (mCRE) is minimized. The mCRE functional provides a bias‐aware data assimilation framework with a rich physical sense as the constitutive relation error (CRE) part can be interpreted as a modeling error continuously defined over the structure, and can be used as a prediction quality in the inference phase. This article also extends previous works on the mCRE by introducing a new minimization procedure in the case of nonlinear state laws. As typical structural health monitoring applications may require that the neural networks should be trained online, an important focus is thus made on automatic and adaptive tuning of sensitive hyperparameters (learning rate, weighting between losses, number of epochs and initialization). It is shown that when the training database is rich enough with respect to the loading cases, the proposed method achieves remarkable performance regarding the quality of the learned model, noise robustness, and low sensitivity to user‐defined hyperparameters. The method is evaluated on two test cases: a non‐quadratic potential in the small strain regime with synthetic optic fiber measurements, and a Mooney–Rivlin model in the hyperelastic case with synthetic digital image correlation observations.

Orbital instability of periodic waves for scalar viscous balance laws

The purpose of this paper is to prove that, for a large class of nonlinear evolution equations known as scalar viscous balance laws, the spectral (linear) instability condition of periodic traveling wave solutions implies their orbital (nonlinear) instability in appropriate periodic Sobolev spaces. The analysis is based on the well-posedness theory, the smoothness of the data-solution map, and an abstract result of instability of equilibria under nonlinear iterations. The resulting instability criterion is applied to two families of periodic waves. The first family consists of small amplitude waves with finite fundamental period which emerge from a local Hopf bifurcation around a critical value of the velocity. The second family comprises arbitrarily large period waves which arise from a homoclinic (global) bifurcation and tend to a limiting traveling pulse when their fundamental period tends to infinity. In the case of both families, the criterion is applied to conclude their orbital instability under the flow of the nonlinear viscous balance law in periodic Sobolev spaces with same period as the fundamental period of the wave.

A Convergence Criterion for a Class of Stationary Inclusions in Hilbert Spaces

Here, we consider a stationary inclusion in a real Hilbert space X, governed by a set of constraints K, a nonlinear operator A, and an element f∈X. Under appropriate assumptions on the data, the inclusion has a unique solution, denoted by u. We state and prove a covergence criterion, i.e., we provide necessary and sufficient conditions on a sequence {un}⊂X, which guarantee its convergence to the solution u. We then present several applications that provide the continuous dependence of the solution with respect to the data K, A and f on the one hand, and the convergence of an associate penalty problem on the other hand. We use these abstract results in the study of a frictional contact problem with elastic materials that, in a weak formulation, leads to a stationary inclusion for the deformation field. Finally, we apply the abstract penalty method in the analysis of two nonlinear elastic constitutive laws.

Designing metainterfaces with specified friction laws.

Many devices, including touchscreens and robotic hands, involve frictional contacts. Optimizing these devices requires fine control of the interface's friction law. We lack systematic methods to create dry contact interfaces whose frictional behavior satisfies preset specifications. We propose a generic surface design strategy to prepare dry rough interfaces that have predefined relationships between normal and friction forces. Such metainterfaces circumvent the usual multiscale challenge of tribology by considering simplified surface topographies as assemblies of spherical asperities. Optimizing the individual asperities' heights enables specific friction laws to be targeted. Through various centimeter-scaled elastomer-glass metainterfaces, we illustrate three types of achievable friction laws, including linear laws with a specified friction coefficient and unusual nonlinear laws. This design strategy represents a scale- and material-independent, chemical-free pathway toward energy-saving and adaptable smart interfaces.

Research on nonlinear corrosion law of epoxy coating on steel bridges based on image processing

This paper adopted the accelerated corrosion test method to study the corrosion law of epoxy coatings on steel bridges. An image processing algorithm based on MATLAB was used to perform grayscale conversion, normalized histogram drawing, and binary image processing on the surface morphology images of coated specimens. Four coated specimens were prepared and placed in the cyclic accelerated corrosion environment. The coated specimens were subjected to the accelerated corrosion test for 90 days, observed changes in coating thickness and corrosion area rate and analyzing the corrosion laws of epoxy coatings in different corrosion environments. Using image processing technology to process images of coating corrosion and analyze the corrosion mechanism of coatings. The research results indicate that the corrosion rate of epoxy coatings exhibits a non-linear corrosion law of slow corrosion in the early stage and accelerated corrosion in the middle and later stages. The corrosion rate of epoxy coatings in saltwater corrosive environments is faster than in freshwater corrosive environments. Based on the corrosion laws and mechanisms of coatings, a nonlinear corrosion function model for epoxy coatings on steel bridges under different accelerated corrosion environment was established, which has a high correlation with experimental data

Minimal port-Hamiltonian modeling of voice production: choices of fluid flow hypotheses, resulting structure and comparison

Voice production results from the interaction between expelled airflow and soft tissues in the larynx and the vocal tract. Among the large literature on this topic over the last fifty years, a few nonlinear fluid-structure interaction models have been proposed in the port-Hamiltonian framework for passivity purposes. In this paper, we examine, compare and discuss two lumped-element port-Hamiltonian models from the literature, both derived from distributed parameter descriptions and simplifying assumptions chosen to integrate the minimal relevant phenomena involved in the larynx (for self-oscillations) or the vocal tract (during articulation). These models are recalled and reformulated using common terminology and notations. This leads to equivalent circuit representations, the components and the structure of which allow direct comparison (about causality, dimension, nonlinear laws, coupling) and physical interpretation. These results highlight important properties to consider and provide guidelines for future modelling improvement to be used in simulation.

Tracking control for a class of fractional order uncertain systems with time-delay based on composite nonlinear feedback control

<abstract><p>In this paper, we dealt with the tracking control problem of a class of fractional-order uncertain systems with time delays. In order to handle the effects brought by the uncertainties, external disturbances, time-delay terms, and to overcome the obstacles caused by inputs saturation, the tracking controller, which consisted of linear control law, nonlinear law, and robust control law proposed in this paper, was designed by combining the composite nonlinear feedback control method and the properties of fractional order operators. Furthermore, the validation of this tracking controller was proved.</p></abstract>

Quiescent Bright Optical Solitons for Radhakrishnan�Kundu�Lakshmanan Equation with Nonlinear Chromatic Dispersion and Power�Law of Self�Phase Modulation by Lie Symmetry

Quiescent Bright Optical Solitons for Radhakrishnan�Kundu�Lakshmanan Equation with Nonlinear Chromatic Dispersion and Power�Law of Self�Phase Modulation by Lie Symmetry

Nonlinear Guidance Laws for Anti-tank Guided Missile to Intercept Maneuvering Tank Targets Using Optimal Error Dynamics and Relative Virtual Model

Nonlinear Guidance Laws for Anti-tank Guided Missile to Intercept Maneuvering Tank Targets Using Optimal Error Dynamics and Relative Virtual Model

Nonlinear Seepage Mechanism and Evolution Law of CO2 Enhancing Coalbed Methane Recovery

China’s coal seam permeability is low, and the original coal seam gas extraction is difficult. CO2 displacement of coal seam CH4 technology is an effective gas extraction technology. CO2 is injected into the coal seam under pressure, and competitive adsorption occurs with CH4 in the pores. The gas composition is nonuniformly distributed, and its viscosity μ is the dynamic parameter. As the gas is compressible, the pressure drops, and migration distance does not satisfy a linear relationship. Therefore, the gas transport does not conform to Darcy’s law. The mass transfer process and a multicomponent gas competitive adsorption were investigated theoretically and experimentally. The adsorption characteristics and gas compressibility determine the distribution of the gas components in pores and change the gas dynamic viscosity in different regions. The change in the gas dynamic viscosity in the channel is the direct reason for the nonlinear pressure gradient and gas flow curve. The permeability and gas component affect the degree of nonlinear deviation of the gas flow and pressure gradient curve. This affects the nonlinear deviation degree of the curve by changing the gas dynamic viscosity in the pore channel during displacement. The reasonable displacement pressure is the critical pressure (PO) through experimental and theoretical analysis.

Physically Admissible and Inadmissible Exact Localized Solutions in Problems of Nonlinear Wave Dynamics of Cylindrical Shells

It is shown that, when studying nonlinear longitudinal deformation waves in cylindrical shells, it is possible to obtain physically admissible solitary wave solutions using refined shell models. In the article, a physically admissible exact localized solution based on the Flügge – Lurie – Byrne model is constructed. An analysis of the influence of the external nonlinear elastic medium on the exact solutions obtained is carried out. It is established that the use of quadratic and cubic nonlinear deformation laws leads to nonintegrable equations with exact soliton-like solutions. However, the amplitudes of the exact solutions exceed the values of permissible displacements corresponding to the maximum points on the curves of the deformation laws of the external medium, which leads to the physical inadmissibility of these solutions.

NUMERICAL IMPLEMENTATION OF THE TSCHOEGL–CHANG–BLOCH NONLINEAR VISCOELASTIC CONSTITUTIVE LAW FOR UNVULCANIZED CARBON BLACK–FILLED RUBBERS

ABSTRACT Rubber engineering design analysis requires a fundamental understanding of the mechanical behavior of polymers, especially in their unvulcanized state. It necessitates the establishment of a three-dimensional constitutive material to account for the observed mechanical behavior. This law is required to produce realistic descriptions of the viscoelastic performance in a mathematically simple form that is easy to implement in engineering applications. This article describes the theory of the Tschoegl–Chang–Bloch time-dependent nonlinear viscoelastic constitutive law. The experimental verification of this law is provided under different deformation fields and multiple load steps. Laboratory test procedures to obtain the parameters required to describe the material under consideration are provided in detail. A recursive form of the constitutive law, suitable for finite element application, is derived and coded in the finite element commercial code Abaqus via the user subroutine UMAT. Comparisons between the experimental observations, the theoretical results, and the numerical data are drawn for simple test models examined under creep or shear relaxation conditions. The excellent agreements observed indicate the suitability of the governing law in analyzing viscoelastic problems of unvulcanized carbon black–filled rubbers.

A Novel Adaptive Nonlinear Reaching Law for DC-link Voltage Control of DC-biased Vernier Reluctance Generator

A Novel Adaptive Nonlinear Reaching Law for DC-link Voltage Control of DC-biased Vernier Reluctance Generator

On determining and breaking the gauge class in inverse problems for reaction-diffusion equations

Abstract We investigate an inverse boundary value problem of determination of a nonlinear law for reaction-diffusion processes, which are modeled by general form semilinear parabolic equations. We do not assume that any solutions to these equations are known a priori, in which case the problem has a well-known gauge symmetry. We determine, under additional assumptions, the semilinear term up to this symmetry in a time-dependent anisotropic case modeled on Riemannian manifolds, and for partial data measurements on ${\mathbb R}^n$ . Moreover, we present cases where it is possible to exploit the nonlinear interaction to break the gauge symmetry. This leads to full determination results of the nonlinear term. As an application, we show that it is possible to give a full resolution to classes of inverse source problems of determining a source term and nonlinear terms simultaneously. This is in strict contrast to inverse source problems for corresponding linear equations, which always have the gauge symmetry. We also consider a Carleman estimate with boundary terms based on intrinsic properties of parabolic equations.

Пространственная фильтрационная модель гидратосодержащих сред с неклассическим законом движения

The work provides a description of the spatial filtration problem in a three-phase hydrate equilibrium zone. A mathematical model is presented for studying two-dimensional fluid flows taking into account the solid hydrate phase and the irregular structure of formations. A non-classical form of the motion law is used, applicable at low permeability and low pressure drops. Efficient computational algorithms based on the support operator method are proposed that make it possible to separate the hyperbolic and dissipative subsystems of the problem. The algorithms are implemented on meshes of irregular structure to model the two-dimensional multiphase processes of gas hydrate dissociation. Testing is carried out on model piezoconductive processes with saturation transfer, where it is shown that depression processes are less expressed when using a nonlinear law of motion.

Optical solitons in birefringent fibers for perturbed complex Ginzburg–Landau equation with polynomial law of nonlinearity

Optical solitons in birefringent fibers for perturbed complex Ginzburg–Landau equation with polynomial law of nonlinearity

Physics-informed Neural Network to predict kinetics of biodiesel production in microwave reactors

Microwaves are a process intensification (PI) method to deliver energy to reactive systems. Microwaves act directly on molecules’ dipolar moment, generating volumetric heating that allows temperature to rise rapidly, which directly impacts the overall rate of a reaction. Because reaction rates adhere to non-linear rate laws, predicting them is challenging. We use a Physics-informed Neural Network (PINN), a physics-driven model, to identify the reaction kinetics of a biodiesel production process. PINNs perform a regression on very few experimental data points and try to fit the physics at hand. We use a microwave reactor with a constant power input to perform the transesterification reaction, measure the infrared temperature and analyze the concentration of glycerides using GC-FID at different reaction times. We train the PINN to predict the reaction rates with respect to the Arrhenius equation. Results show that the PINN successfully identifies the rate constants, including their temperature dependency. Furthermore, the PINN can extrapolate its predictions to other power inputs without ever seeing the concentration data, generating a digital twin of the microwave-assisted reaction.