Viscoelastic Equation Research Articles

Existence and decay estimate of global solution for a viscoelastic wave equation with nonlinear boundary source term

In this paper, we consider a viscoelastic wave equation with nonlinear boundary source term. By the combination of Galerkin approximation and potential well methods, we prove the global existence of solutions. Then, we give an decay rate estimate of the energy by making use of the perturbed energy method.

Global Existence and Uniqueness for Viscoelastic Equations With Nonstandard Growth Conditions

This paper is devoted to the study of generalized viscoelastic nonlinear equations with Dirichlet-Neumann boundary conditions. We establish the local and uniqueness of weak solutions results in Sobolev spaces with variable exponents. Solutions are constructed as a limit of approximate solutions by a method independent of a compactness argument. We also discuss the global existence of solutions in the energy space. Keywords: Viscoelastic equation, global existence, nonlinear dissipation, energy estimates.

General Decay Rates of the Solution Energy in a Viscoelastic Wave Equation With Boundary Feedback and a Nonlinear Source

In a bounded domain, we consider a viscoelastic equation with a nonlinear feedback localized on a part of the boundary and certain initial data. We establish an explicit and general decay rate result, using some properties of the convex functions. Our new results substantially improve several earlier related results in the literature. Keywords: General decay, nonlinear source, viscoelastic, wave equation, relaxation function.

General decay results for a viscoelastic wave equation with the logarithmic nonlinear source and dynamic Wentzell boundary condition

In this work we investigate a viscoelastic wave equation involving a logarithmic nonlinear source and dynamic Wentzell boundary condition. Making some assumptions on the memory kernel function and using convex function theory and Lyapunov method, we establish the general decay estimate of the solutions. Finally we give two examples to illustrate our results.

Exact three-dimensional elasticity analysis for buckling of composite laminated plates resting on viscoelastic foundation

This paper aims to present novel exact solutions for the buckling of a laminated plate resting on the viscoelastic foundation with both normal and shear viscoelastic layers. The governing equations of plate buckling are derived using three-dimensional elasticity theory and state-space formulation. The normal and shear layers of the viscoelastic foundations are modeled using the generalized Maxwell model to represent both the elastic and viscose properties of the foundation. To couple the viscoelastic foundation equation with the buckling equation, Boltzmann’s superposition principle along with the Laplace transform is utilized. Then, the effects of geometry, relaxation modulus of normal and shear layers, viscosity, and time are investigated on the buckling load. The results reveal that the higher viscosity coefficient leads to a slower rate of change in the buckling loads. In addition, the viscoelastic properties have a significant impact on the buckling behavior of the plate. In this regard, the results show that instead of the expected second mode at a constant aspect ratio, the plate experiences the first mode as time passes. The computed results also show that there is a critical threshold. When the foundation stiffness exceeds this threshold, the conventional method of reducing the aspect ratio to prevent buckling not only proves ineffective in reducing the probability of buckling but also, in fact, leads to an increase in buckling occurrences. In addition to the analytical investigation, a finite element (FE) analysis is carried out to study the buckling response of the composite plate. The finite element results also show a reasonably good agreement with those of the analytical method.

On a class of a coupled nonlinear viscoelastic Kirchhoff equations variable-exponents: global existence, blow up, growth and decay of solutions

In this work, we consider a quasilinear system of viscoelastic equations with dispersion, source, and variable exponents. Under suitable assumptions on the initial data and the relaxation functions, we obtained that the solution of the system is global and bounded. Next, the blow-up is proved with negative initial energy. After that, the exponential growth of solutions is showed with positive initial energy, and by using an integral inequality due to Komornik, the general decay result is obtained in the case of absence of the source term.

General energy decay estimates for a variable coefficient viscoelastic semilinear wave equation with logarithmic source term and acoustic boundary conditions

In this paper, we consider a variable coefficient viscoelastic semilinear wave equation with logarithmic source term and acoustic boundary conditions in domains with non‐locally reacting boundary. By potential well method, we get the global existence of solution and through several important lemmas, we prove the general energy decay estimate of the system.

Viscoelastic plate equation with variable exponents: existence and blow-up

Viscoelastic plate equation with variable exponents: existence and blow-up

Blow up, growth, and decay of solutions for a class of coupled nonlinear viscoelastic Kirchhoff equations with distributed delay and variable exponents

In this work, we consider a quasilinear system of viscoelastic equations with dispersion, source, distributed delay, and variable exponents. Under a suitable hypothesis the blow-up and growth of solutions are proved, and by using an integral inequality due to Komornik the general decay result is obtained in the case of absence of the source term f1=f2=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f_{1}=f_{2}=0$\\end{document}.

On calculating glacial isostatic adjustment

Modeling the earth's fluid and elastic response to the melting of the glaciers of the last ice age is the most direct way to infer the earth's radial viscosity profile. Here, we compare two methods for calculating the viscoelastic response to surface loading. In one, the elastic equation of motion is converted to a viscoelastic equation using the Correspondence Principle. In the other, elastic deformation is added to the viscous flow as isostatic adjustment proceeds. The two modeling methods predict adjustment histories that are different enough to potentially impact the interpretation of the observed glacial isostatic adjustment (GIA). The differences arise from buoyancy and whether fluid displacements are subjected to hydrostatic pre-stress. The methods agree if they use the same equations and boundary conditions. The origin of the differences is determined by varying the boundary conditions and pre-stress application.

Fractional-order velocity-dilatation-rotation viscoelastic wave equation and numerical solution based on a constant-Q model

Viscoelastic wave equations based on the constant- Q (CQ) model can accurately describe the amplitude dissipation and phase distortion of waves in anelastic media. However, only three velocity or displacement components can be obtained directly by solving such equations. Starting from the time-domain second-order displacement viscoelastic wave equation, we derived the decoupled P- and S-wave displacement vector viscoelastic wave equation by using the polarization difference of P- and S-wave propagation in isotropic media. The equation can be transformed into the velocity-dilatation-rotation viscoelastic wave equation containing the first-order temporal derivative and fractional Laplacian operators, which can be solved directly by using the staggered-grid finite-difference and pseudospectral methods. We use the low-rank decomposition method to approximate the derived mixed space-wavenumber domain fractional Laplacian operators for modeling wave propagation in heterogeneous attenuating media. We also demonstrated the precision of our equation by comparing the numerical solutions with the analytical solutions. Furthermore, compared with the conventional velocity-stress viscoelastic wave equation, experimental results demonstrate that our equation can separate the pure P and S waves from the mixed wavefield during wavefield continuation. In addition, it can be separated into an equation containing predominantly an amplitude attenuation or a phase distortion term.

On a coupled system of viscoelastic wave equation of infinite memory with acoustic boundary conditions

This work deals with a coupled system of viscoelastic wave equation of infinite memory with mixed Dirichlet-Neumann boundary conditions. The coupling is via the acoustic boundary conditions on a portion of the boundary. The semigroup theory is used to show the well-posedness and regularity of the initial and boundary value problem. Moreover, we investigate exponential stability of the system taking into account Gearhart-Prüss’s theorem. Mathematics Subject Classification (2010): 35A01, 74B05, 93D15. Received 11 June 2021; Accepted 13 October 2021

A Class of Fractional Viscoelastic Kirchhoff Equations Involving Two Nonlinear Source Terms of Different Signs

A class of fractional viscoelastic Kirchhoff equations involving two nonlinear source terms of different signs are studied. Under suitable assumptions on the exponents of nonlinear source terms and the memory kernel, the existence of global solutions in an appropriate functional space is established by a combination of the theory of potential wells and the Galerkin approximations. Furthermore, the asymptotic behavior of global solutions is obtained by a combination of the theory of potential wells and the perturbed energy method.

Blow up at Infinite Time of Solutions for a Plate Equation with Delay Term

In this article, we investigate the viscoelastic plate equation with both delay and source terms. Initially, we give the local-global existence results. Later, we establish the blow-up results at infinite time by utilizing the energy method when E (0) < 0 under suitable conditions. Delays effect generally seems in many practical problems for instance medicine, biological, chemical, physical, thermal, economic phenomena, electrical engineering systems and mechanical applications.

Ocean bottom dual-sensorQ-compensated elastic least-squares reverse time migration based on acoustic and separated-viscoelastic coupling equations

The ocean bottom dual-sensor (OBD) observation system places water detectors (pressure sensors) and land detectors on the irregular seabed interface to receive acoustic pressure fields and elastic wavefields, respectively. These data are usually applied to effectively remove ghost waves and ringing interference in the seawater layer, rather than being jointly used for migration. We propose an ocean bottom dual-sensor viscoacoustic and separated-viscoelastic coupled Q-compensated least-squares reverse time migration (OBD-VSV- QLSRTM) in curvilinear coordinates. In this method, to solve the Q-compensated elastic inverse problem, we jointly use the recorded acoustic pressure and elastic multicomponent seismic data from the OBD observation system. A new misfit function in the sense of least-squares inversion is constructed based on viscoacoustic and separated viscoelastic wavefields. Acoustic and separated viscoelastic equations are used to describe the acoustic wavefield in the acoustic medium and the Q-compensated P and S wavefields in the underlying viscoelastic medium, respectively. On the acoustic-viscoelastic interface, we implement a stable governing equation to ensure continuous and steady transmission of Q-compensated stresses in the subseabed and pressure in the seawater. To address the irregular seabed problem, we mesh the acoustic-viscoelastic models onto curvilinear grids and apply coordinate transformations to map the models and equations to a new curvilinear coordinate system. We derive viscoacoustic and separated-viscoelastic coupled adjoint and demigration operators in the curvilinear coordinates, enabling linear acoustic-viscoelastic modeling and stable receiver-side back-propagated wavefield continuation. Numerical examples conducted on two typical acoustic-viscoelastic models demonstrate that our proposed OBD-VSV- QLSRTM in curvilinear coordinates can produce highly accurate P- and S-wave images efficiently.

Amplitude-preserving P-/S-wavefield separation with the discontinuous Galerkin method on unstructured meshes

The decoupling of P and S waves is an essential prerequisite for elastic reverse time migration, which effectively reduces crosstalk artifacts, but most wavefield separation algorithms are implemented on uniform rectangular grids. We develop an amplitude- and phase-preserving P- and S-wavefield separation approach on unstructured meshes, which can effectively decompose the original elastic wavefield into P and S wavefields. The isotropic case is considered. With the aid of viscoelastic theory, we choose to attenuate P or S waves and preserve the other wave mode, so as to achieve wavefield decomposition. Viscoelastic wave equations are first reformulated as decoupling wave equations with a selective strong attenuation. We then use the discontinuous Galerkin (DG) method to simulate decoupling P or S wavefield propagation on triangular and tetrahedral meshes. We adopt a quadrature-free DG approach and the arbitrary mesh is mapped into the reference mesh for numerical calculation, where no additional volume and surface integrations are involved. The amplitude and phase information of this vector decomposition agrees with that of the original elastic data. Four numerical examples are used to demonstrate the superior performance of this vector decomposition algorithm. The isotropic example indicates the applicability and correctness of our scheme and the second example displays superiority in handling strong velocity contrasts. The third example exhibits mesh flexibility in dealing with complex structures, such as caves, faults, and undulating surfaces. The last example indicates the effectiveness of our developed algorithm extended to a 3D case.

Effect of the magnetic field on the thermomechanical flexural wave propagation of embedded sandwich nanobeams

This work examines thermo-mechanical bending wave propagation in a sandwich nanobeam using advanced sandwich nanobeam and nanolocal strain gradient elasticity theories. The sandwich nanobeam is a unique structure with biocompatible ceramic ZrO2 and metal Ti6Al4V on the top and bottom sides. Sandwich nanobeam cores have functionally graded materials. This combination gives the nanobeam distinctive qualities and opens up many uses in diverse industries. The wave propagation equation is computed by applying the Navier method to the medium’s thermal, Lorentz, and viscoelastic equations of motion. The sandwich nanobeam is analyzed using four distinct models, taking into account its composition of ceramic and metal materials. The various factors that affect sandwich nanobeam bending wave propagation have been extensively studied. In the scenario where the magnetic field intensity is Hm = 0, an increase in temperature difference causes the wave frequency of all models (except Model 2) to decrease to zero, resulting in buckling. In Model 2, the sandwich nanobeam exhibits a phase velocity of 0.43 Km/s at ΔT = 0, which subsequently decreases by ∼9% to 0.39 km/s at ΔT = 500. These factors include the strength of the magnetic field, the impact of thermal loads, the nonlocal effect, the dimensions of the sandwich nanobeam, and the foundation’s influence. The findings of this research will help build nanosensor systems that can work in aerospace applications under extreme temperatures. These findings will contribute to the optimization of the design process, ensuring the reliability and functionality of the nanosensors under severe thermal conditions.

On the numerical resolution of the second-order viscoacoustic and viscoelastic anisotropic wave equations using the recursive convolution method

Accurate and efficient seismic wave modeling is fundamental for the high-resolution seismic full-waveform inversion and correct interpretation of seismic data. Based on the viscoelastic mechanism of the Generalized Standard Linear Solid and the constitutive relationship between stress and strain, we demonstrate a generalized recursive formula using the Taylor series expansion to directly compute the temporal convolution in the constitutive equation, rather than solving auxiliary differential equations of the memory variables in the conventional method. New sets of the second order viscoacoustic and viscoelastic anisotropic wave equations are established with the new recursive temporal convolutions, which completely replaces the traditional auxiliary differential equations of the memory variables. The error analysis shows the proposed method can offer better accuracy than the existing recursive methods, and is very competitive with the widely used memory variable method. The numerical results comparing the different methods show the accuracy and feasibility of the generalized recursive convolution method to simulate the viscoacoustic and viscoelastic anisotropic waves.

Blow-up of Solutions for a Problem with Balakrishnan-Taylor Damping and Nonlocal Singular Viscoelastic Equations

In this paper, we study the nonlinear one-dimensional viscoelastic nonlocal problem with Balakrishnan-Taylor damping terms and nonlinear source of polynomial type. We demonstrate that the nonlinear source of polynomial type is able to force solutions to blow up infinite time even in presence of stronger damping with non positive initial energy combined with a positive initial energy.

Finite element approximation and analysis of damped viscoelastic hyperbolic integrodifferential equations with [formula omitted] kernel

In this paper, we propose a finite element method for solving viscoelastic hyperbolic integrodifferential equations with L1 kernel, involving a nonlocal and nonlinear damped coefficient. Firstly, we discuss and deduce that the L1 kernel is of positive type in two cases. Subsequently, based on the continuous Galerkin technique and the energy argument, we prove the global existence and uniqueness of the discrete solution. Finally, using the basic properties of the positive-type kernel, we derive the error estimates of the finite element solutions.