Dynamic Stability of Orthotropic Viscoelastic Rectangular Plate of an Arbitrarily Varying Thickness

The research object of this work is an orthotropic viscoelastic plate with an arbitrarily varying thickness. The plate was subjected to dynamic periodic load. Within the Kirchhoff–Love hypothesis framework, a mathematical model was built in a geometrically nonlinear formulation, taking into account the tangential forces of inertia. The Bubnov–Galerkin method, based on a polynomial approximation of the deflection and displacement, was used. The problem was reduced to solving systems of nonlinear integrodifferential equations. The solution of the system was obtained for an arbitrarily varying thickness of the plate. With a weakly singular Koltunov–Rzhanitsyn kernel with variable coefficients, the resulting system was solved by a numerical method based on quadrature formulas. The computational algorithm was developed and implemented in the Delphi algorithmic language. The plate’s dynamic stability was investigated depending on the plate’s geometric parameters and viscoelastic and inhomogeneous material properties. It was found that the results of the viscoelastic problem obtained using the exponential relaxation kernel almost coincide with the results of the elastic problem. Using the Koltunov–Rzhanitsyn kernel, the differences between elastic and viscoelastic problems are significant and amount to more than 40%. The proposed method can be used for various viscoelastic thin-walled structures such as plates, panels, and shells of variable thickness.

Dynamic analysis of an orthotropic viscoelastic cylindrical panel of variable thickness

The intensive development of the modern industry is associated with the emergence of a variety of new composite materials. Plates, panels, and shells of variable thickness made of such materials are widely used in engineering and machine building. Modern technology for the manufacture of thin-walled structures of any configuration makes it possible to obtain structures with a given thickness variation law. Such thin-walled structures are subjected to various loads, including periodic ones. Nonlinear parametric vibrations of an orthotropic viscoelastic cylindrical panel of variable thickness are investigated without considering the elastic wave propagation. To derive a mathematical model of the problem, the Kirchhoff-Love theory is used in a geometrically nonlinear setting. The viscoelastic properties of a cylindrical panel are described by the hereditary Boltzmann-Volterra theory with a three-parameter Koltunov-Rzhanitsyn relaxation kernel. The problem is solved by the Bubnov-Galerkin method in combination with the numerical method. For the numerical implementation of the problem, an algorithm and a computer program were developed in the Delphi algorithmic language. Nonlinear parametric vibrations of orthotropic viscoelastic cylindrical panels under external periodic load were investigated. The influence of various physical, mechanical, and geometric parameters on the panel behavior, such as the thickness, viscoelastic and inhomogeneous properties of the material, external periodic load, were studied.

Vibrations of a geometrically nonlinear viscoelastic shallow shell with concentrated masses

Shell structures are widely used in various fields of technology and construction. Often, they play the role of a bearing surface with assemblies, overlays, and aggregates installed on them. At the same time, in solving various problems, such attached elements are considered as the elements concentrated at the points and rigidly connected. Vibrations of an orthotropic viscoelastic shallow shell with concentrated masses in a geometrically nonlinear setting are considered. The equation of motion for a shallow shell is derived based on the Kirchhoff-Love theory. The traditional Boltzmann-Volterra theory is used to describe the viscoelastic properties of a shallow shell. The effect of concentrated masses is taken into account using the Dirac delta function. Using the polynomial approximation of the deflections of the Bubnov-Galerkin method, the problem is reduced to solving a system of ordinary nonlinear integro-differential equations with variable coefficients. In the calculations, the three-parameter Koltunov-Rzhanitsyn kernel was used as a weakly singular relaxation kernel. A numerical method was used to solve the resulting system that eliminates the singularity in the relaxation kernel. The problem of nonlinear vibrations of an orthotropic viscoelastic shallow shell with concentrated masses is solved. The influence of concentrated masses and location, properties of the shell material, and other parameters on the amplitude-frequency response of the shallow shell vibrations is investigated.

Theoretical and numerical stability results for a viscoelastic swelling porous-elastic system with past history

<abstract><p>The purpose of this paper is to establish a general stability result for a one-dimensional linear swelling porous-elastic system with past history, irrespective of the wave speeds of the system. First, we establish an explicit and general decay result under a wider class of the relaxation (kernel) functions. The kernel in our memory term is more general and of a broader class. Further, we get a better decay rate without imposing some assumptions on the boundedness of the history data considered in many earlier results in the literature. We also perform several numerical tests to illustrate our theoretical results. Our output extends and improves some of the available results on swelling porous media in the literature.</p></abstract>

Internal resonances and relaxation memory kernels in composites

In heterogeneous composite materials, the behaviour of the medium on larger scales is determined by the microgeometry and properties of the constituents on finer scales. To model the influence of the microlevel processes in composite materials, they are described as materials with memory in which the constitutive relations between stress and strain are given as time-domain convolutions with some relaxation kernel. The paper reveals the relationship between the viscoelastic relaxation kernel and the spectral measure in the Stieltjes integral representation of the effective properties of composites. This spectral measure contains all information about the microgeometry of the material, thus providing a link between the relaxation kernel and the microstructure of the composite. We show that the internal resonances of the microstructure determine the characteristic relaxation times of the fading memory kernel and can be used to introduce a set of internal variables that captures dissipation at the microscale. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.

A linear algorithm for the identification of a weakly singular relaxation kernel using two boundary measurements

We consider a distributed system of a type which is encountered in the study of diffusion processes with memory and in viscoelasticity. The key feature of such a system is the persistence in the future of the past actions due the memory described via a certain relaxation kernel; see below. The parameters of the kernel have to be inferred from experimental measurements. Our main result in this paper is that by using two boundary measurements, the identification of a relaxation kernel which is a linear combination of Abel kernels (as often assumed in applications) can be reduced to the solution of a (linear) deconvolution problem.

Intrinsic decay rates for the energy of a nonlinear viscoelastic equation modeling the vibrations of thin rods with variable density

We consider the long-time behavior of a nonlinear PDE with a memory term which can be recast in the abstract form$\frac{d}{dt}\rho(u_{t})+Au_{tt}+\gamma A^{\theta}u_{t}+Au-\int_{0}^{t}g(s)Au(t% -s)=0,$where A is a self-adjoint, positive definite operator acting on a Hilbert space H, ${\rho(s)}$ is a continuous, monotone increasing function, and the relaxation kernel ${g(s)}$ is a continuous, decreasing function in ${L_{1}(\mathbb{R}_{+})}$ with ${g(0)>0}$. Of particular interest is the case when ${A=-\Delta}$ with appropriate boundary conditions and ${\rho(s)=|s|^{\rho}s}$. This model arises in the context of solid mechanics accounting for variable density of the material. While finite energy solutions of the underlying PDE solutions exhibit exponential decay rates when strong damping is active (${\gamma>0,\theta=1}$), this uniform decay is no longer valid (by spectral analysis arguments) for dynamics subjected to frictional damping only, say, ${\theta=0}$ and ${g=0}$. In the absence of mechanical damping (${\gamma=0}$), the linearized version of the model reduces to a Volterra equation generated by bounded generators and, hence, it is exponentially stable for exponentially decaying kernels. The aim of the paper is to study intrinsic decays for the energy of the nonlinear model accounting for large classes of relaxation kernels described by the inequality ${g^{\prime}+H(g)\leq 0}$ with H convex and subject to the assumptions specified in (1.13) (a general framework introduced first in [1] in the context of linear second-order evolution equations with memory). In the context of frictional damping, such a framework was introduced earlier in [15], where it was shown that the decay rates of second-order evolution equations with frictional damping can be described by solutions of an ODE driven by a suitable convex function H which captures the behavior at the origin of the dissipation. The present paper extends this analysis to nonlinear equations with viscoelasticity. It is shown that the decay rates of the energy are intrinsically described by the solution of the dissipative ODE${S_{t}+c_{1}H(c_{2}S)=0}$with given intrinsic constants ${c_{1},c_{2}>0}$. The results obtained are sharp and they improve (by introducing a novel methodology) previous results in the literature (see [20, 19, 21, 6]) with respect to (i) the criticality of the nonlinear exponent ρ and (ii) the generality of the relaxation kernel.

Solutions of Cattaneo-Hristov model of elastic heat diffusion with Caputo-Fabrizio and Atangana-Baleanu fractional derivatives

Recently Hristov using the concept of a relaxation kernel with no singularity developed a new model of elastic heat diffusion equation based on the Caputo-Fabrizio fractional derivative as an extended version of Cattaneo model of heat diffusion equation. In the present article, we solve exactly the Cattaneo-Hristov model and extend it by the concept of a derivative with non-local and non-singular kernel by using the new Atangana-Baleanu derivative. The Cattaneo-Hristov model with the extended derivative is solved analytically with the Laplace transform, and numerically using the Crank-Nicholson scheme.

Fractional Calculus Approach to Relaxation Behavior of Amorphous Shape Memory Polymers

To select an appropriate relaxation kernel function is significant for shape memory polymers (SMPs) in their thermomechanical constitutive models. The relaxation modulus of SMPs are described by fractional-order viscoelastic (FOV) kernel and three other kinds of viscoelastic kernel, that is, Prony series, Kohlrausch-Williams-Watts (KWW) kernel and Cole-Cole Model (CCM) kernel. The data fitting result shows FOV kernel is a valuable tool to describe the relaxation response of SMPs. Compared with Prony series, KWW kernel and CCM kernel, the FOV kernel can give a comparative description of relaxation modulus of SMPs with fewer material parameters.