Viscoelastic Orthotropic Plate Research Articles

Impact of Non-Homogeneity on Vibration Dynamics of Orthotropic Visco-Elastic Plates with Exponential Thickness Variation

Impact of Non-Homogeneity on Vibration Dynamics of Orthotropic Visco-Elastic Plates with Exponential Thickness Variation

Characterization of the full complex-valued stiffness tensor of orthotropic viscoelastic plates using 3D guided wavefield data

A two-stage inversion scheme is proposed to determine the full set of 9 complex-valued stiffness properties of orthotropic viscoelastic plates using their 3D surface velocity response to a broadband vibrational excitation. A combination of the hybrid TLS-ESPRIT and IWC method is employed to the extract complex-valued wavenumber–frequency pairs corresponding to relevant Lamb wave and shear horizontal plate waves from the data from surface velocity response. Particle swarm optimization is chosen to inversely determine the plate’s orthotropic viscoelastic properties using the semi-analytical finite element method as a forward model to compute the real and imaginary parts of the wavenumbers. The inversion procedure first identifies the elastic parameters by matching the real wavenumber values, followed by the characterization of the viscoelastic properties which are merely linked to the imaginary wavenumber values. To validate the accuracy of the proposed method, a series of numerical simulations of 3D wavefield data using a finite element model (COMSOL) is conducted, as well as broadband experimental measurements obtained by means of a 3D Infrared Scanning Laser Doppler Vibrometer. It is shown that the inverted orthotropic viscoelastic properties based on the virtual wavefield data are in close agreement with the target values, showing a mean relative error of less than 2% and 5% error on the elastic and viscous properties, respectively. Additionally, the median absolute deviations are negligible, showing the robust convergence of the inversion procedure to a global minimum, and giving confidence to apply the method on experimental wavefield data in order to reveal the full (unknown) C-tensor for material characterization. The proposed characterization method can be used for various orthotropic viscoelastic materials, including metal sheets, carbon/epoxy laminates, and wooden plates.

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.

Determining the stress concentration change with time in a viscoelastic orthotropic solid

The procedure for solving the plane problem of the linear theory of viscoelasticity by the finite element method is described. Based on the virtual work principle and the assumption of the constancy of the strain rate at small intervals of time, the matrix form of the equilibrium equations of the finite-element approximation of a body is written. The solution procedure is described for the constitutive relations in the Boltzmann—Volterra integral form. This integral is transformed into an incremental form on a time mesh, at each interval of which the problem is solved by the finite element method with unknown increments of displacements. The numerical procedure is constructed by ununiformly dividing the time interval, at which the study is conducted. In this case, the stiffness matrix requires recalculation at each time step. The relaxation functions of the moduli of a viscoelastic orthotropic material are described in the form of the Proni—Dirichlet series. The solution to the problem of determining the change over time of the stress concentration in a body with a round hole in a viscoelastic orthotropic plate is presented. To construct a numerical solution, the three moduli of orthotropic material were written using one exponent with the same relaxation time. For these initial data, an analytic expression for the viscoelastic components of the stiffness matrix of an orthotropic plate under plain stress conditions is constructed. Numerical examples are presented for several ratios of the hole radius to the size of the plate. These results are compared with the solution obtained for an infinite plate by inverse transformation by a numerical method of the well-known analytic elastic solution.

Parametric oscillations of viscoelastic orthotropic plates of variable thickness

Viscoelastic orthotropic rectangular plates of variable thickness are considered in the paper under the effect of periodic load. It is believed that under periodic load, the plates allow displacements commensurate with their thickness. Based on the Kirchhoff-Love hypothesis, a mathematical model of the problem of parametric oscillations of a viscoelastic orthotropic rectangular plate of variable thickness is constructed in a geometrically nonlinear statement. A method for solving the problem under consideration is proposed, based on the application of the Bubnov-Galerkin method with polynomial approximation of displacements and deflection, and on a numerical method based on the use of quadrature formulas. In calculations, the three-parameter Koltunov-Rzhanitsyn kernel is used as a weakly singular kernel. Based on the algorithm for solving the problem, a program was developed in the Delphi algorithmic language to solve the problem of parametric oscillations of viscoelastic orthotropic rectangular plates of variable thickness under the effect of an external periodic load. The effect of geometrical nonlinearity, viscoelastic properties of the material, physico-mechanical and geometrical parameters of a viscoelastic orthotropic plate on the areas of dynamic instability was investigated. The results obtained are in good agreement with the results of other authors.

Dynamic stability of viscoelastic rectangular plates with concentrated masses

Thin-walled constructions such as plates and shells, with installed units, devices and assemblies, are widely used in engineering and construction. In calculations, such attached elements are considered as concentrated at points and rigidly fixed elements. The influence of concentrated masses is taken into account in the equation of motion using the Dirac delta function. Recently, more and more attention has been paid to the nonlinear and inhomogeneous properties of a structure. Dynamic stability of viscoelastic orthotropic rectangular plates with concentrated masses in a geometrically nonlinear statement is considered in the paper. Using the Bubnov-Galerkin method, based on a polynomial approximation of deflections, the problem is reduced to solving a system of ordinary nonlinear integro-differential equations. The results of the problem are obtained by the proposed numerical method based on the use of quadrature formulas. Dynamic stability of viscoelastic rectangular plates with concentrated masses under various boundary conditions was studied over a wide range of changes in physico-mechanical and geometrical parameters of the plate.

Dynamic calculation of nonlinear oscillations of viscoelastic orthotropic plate with a concentrated mass

Plates, panels and shells made of composite material with fixed objects in the form of an additional mass have found a wide use due to their viscoelastic and strength properties. An analysis of their dynamic behavior indicates a significant effect of inhomogeneity of an associated mass type on their strength. The problem of oscillations of a viscoelastic orthotropic rectangular plate with an associated mass is considered according to the Kirchhoff-Love hypothesis in a geometrically nonlinear statement. This problem is reduced to solving the systems of nonlinear integro-differential equations with singular relaxation kernels, solved by the Bubnov-Galerkin method in combination with a numerical method based on the use of quadrature formulas. The numerical values of the approximate solution have been calculated in the Delphi programming environment. At wide range of changes in physicomechanical and geometrical parameters, the behavior of the plate has been studied. The effect of viscoelastic and inhomogeneous material properties, concentrated mass and their location on the oscillatory process of a rectangular plate is shown.

To Calculation of Rectangular Plates on Periodic Oscillations

Geometrically nonlinear mathematical model of the problem of parametric oscillations of a viscoelastic orthotropic plate of variable thickness is developed using the classical Kirchhoff-Love hypothesis. The technique of the nonlinear problem solution by applying the Bubnov-Galerkin method at polynomial approximation of displacements (and deflection) and a numerical method that uses quadrature formula are proposed. The Koltunov-Rzhanitsyn kernel with three different rheological parameters is chosen as a weakly singular kernel. Parametric oscillations of viscoelastic orthotropic plates of variable thickness under the effect of an external load are investigated. The effect on the domain of dynamic instability of geometric nonlinearity, viscoelastic properties of material, as well as other physical-mechanical and geometric parameters and factors are taken into account. The results obtained are in good agreement with the results and data of other authors.

Damping sources interactions in impact of viscoelastic composite plates with damping treated SMA wires, using a hyperbolic plate theory

In the present paper, a comprehensive study is made on effects of the viscoelastic and the phase-transformation-based dissipations and their interactions on impact responses of viscoelastic composite plates with damping treated (structural hierarchy) shape memory alloy (SMA) wires, for the first time. In contrast to almost all of the available researches, a high-order hyperbolic plate theory that includes not only odd but also even functions of the transverse coordinate, is proposed and employed here. While a hierarchical viscoelastic constitutive law is employed for both the orthotropic and SMA materials, Brinson's constitutive law is refined to include the loading fluctuations and structural hierarchy of the SMA wire, simultaneously. The traditional Hertz and Yang-Sun contact laws are modified accordingly. The resulting highly nonlinear piecewise-defined integro-differential finite element governing equations are solved by an iterative algorithm within each time step. The presented discussions show that in contrast to the common belief, the zero-shear traction condition on the top and bottom surfaces of the viscoelastic orthotropic plate cannot be satisfied by the available plate theories, even for the symmetric lamination schemes. Results show that the viscoelasticity and phase-transformation effects on the resulting dynamic responses are more pronounced for the low and high energy impacts, respectively.

Investigation of guided waves propagation in orthotropic viscoelastic carbon–epoxy plate by Legendre polynomial method

The effect of viscoelasticity on the guided waves propagation in viscoelastic plate has been investigated according to multi-aspect. To this purpose, an extension of the Legendre polynomial method is proposed to formulate the guided waves equation in orthotropic viscoelastic plate composed of carbon–epoxy. The validity of the proposed Legendre polynomial method is illustrated by comparison with available data. The convergence of the method is discussed through a numerical example. The hysteretic and Kelvin–Voigt viscoelastic models are used to integrate the imaginary part of the complex stiffness matrix associated with the viscoelastic plate in this study. Accordingly, both viscoelastic models do not affect on the dispersion curves results. However, appreciable effects are seen in the attenuation curves. Also, the sensitivity of the guided waves propagation caused by variations of elastic and viscoelastic modulus has been studied in detail. Finally, the advantages of the Legendre polynomial method are described.

Evolution of Stresses in the Bolted Connection Node of Viscoelastic Composite Plates

Orthotropic viscoelastic plates’ connection with made open bolt has been studied. Ends of plates are loaded with constant equally spaced tension. The problem has been solved in three-dimensional statement in the frames of linear theory of viscoelasticity. Solution has been done by the method of quasi constant operators with finite-element realization. The picture of stress state connection evolution at time-constant external load has been obtained. The estimation of connection limiting stress has been done in the frames of A.A.Ilyushin’s limiting stress theory. The possibility of joint immobility disturbance in the course of time has been defined. Unsafe stress as well as the values of constant external action and characteristic time of strength failure occurrence has been determined.

Mode I Crack Initiation in Orthotropic Viscoelastic Plates Under Biaxial Loading

The subcritical propagation of a crack in orthotropic viscoelastic plate under time-constant biaxial external loading is investigated on the basis of generalization of the Leonov–Panasyuk–Dugdale crack model to the case of orthotropic materials satisfying a strength condition of arbitrary form. The crack is directed along one of the anisotropy axes with external loads applied parallel and perpendicularly to this axis. To find the rheological characteristics of the composite material, we apply the method of operator continued fractions. The relationships used to determine the duration of incubation and transition periods of crack propagation are deduced on the basis of the Volterra principle and the solution of the corresponding elastic problem. The influence of the biaxiality of external loading on the safe loading and subcritical crack growth is analyzed within the framework of the critical crack opening displacement criterion.

Vibration of Visco-elastic Orthotropic Parallelogram Plate with Linear Thickness Variation in Both Directions

A simple model presented here for vibration of a visco-elastic, orthotropic parallelogram plate with linear thickness variation in both directions. Using the separation of variables method, the governing differential equation has been solved for the vibration of a visco-elastic, orthotropic parallelogram plate having clamped boundary conditions on all four of the edges. An approximate but convenient frequency equation is derived by using the Rayleigh-Ritz technique with a two-term deflection function. The time period and deflection function at different points for the first two modes of vibration are calculated for various values of taper constants, aspect ratio and skew angle.

Моделирование колебательных процессов вязкоупругих ортотропных пластин с переменной жёсткостью.

Моделирование колебательных процессов вязкоупругих ортотропных пластин с переменной жёсткостью.

Photoelastic modeling of the fracture of viscoelastic orthotropic plates with a crack

The well-known equations of photoelasticity of linear viscoelastic bodies are used to describe the photoelastic behavior of a viscoelastic orthotropic plate with a crack. Expressions for the stress intensity factors (SIFs) at the crack tip are obtained using photoelastic measurements. The time dependence of the SIFs is analyzed and shown to be determined by the angles between directions of the crack and tension

A numerical analytic method for solving boundary-value problems of linear anisotropic viscoelasticity

The paper proposes an approach to solving boundary-value problems for linear viscoelastic orthotropic bodies based on the method of operator continued fractions and the boundary-element method. A problem-solving algorithm and a procedure to estimate the possible error are outlined. The solution for a viscoelastic orthotropic plate with a rigid circular inclusion under uniaxial tension is obtained as an example and compared with available ones

Limiting state of an orthotropic plate weakened by a periodic system of collinear cracks

On the basis of a modified δc-model of cracks, we study the limiting state of an orthotropic plate made of a material satisfying the general strength condition and weakened by a system of collinear cracks. The relations for the determination of major parameters of the model of cracks (the size of process zones, stresses in these zones, and the crack-tip opening displacements) are deduced. The mechanism of fracture of the plate containing a periodic system of collinear cracks is investigated. The influence of the degree of anisotropy and geometric parameters of the problem on the formation of the process zones and limiting state of the plate is revealed. The region of safe loading of an orthotropic viscoelastic plate with cracks is determined. The influence of the rheological parameters of the material on the region of safe loading is analyzed. Numerous modern works in the field of fracture mechanics are devoted to the development of new approaches to the investigation of fracture processes in various materials (metals, polymers, composites, rocks, etc.) under the action of external loads of various types [1–5]. The extensive application of elements made of anisotropic materials explains the necessity of investigation of their behavior under loads close to critical [6 – 8]. In [9], on the basis of a modified Leonov – Panasyuk – Dugdale model, the solutions of the problems of fracture were constructed for orthotropic plates of elastic, elastoplastic, and viscoelastic materials weakened by rectilinear cracks. In a similar statement, the solution of the problem of fracture of an orthotropic plate weakened by a periodic system of collinear cracks was obtained in [10]. In what follows, we investigate the limiting state of an orthotropic plate of viscoelastic material weakened by a periodic system of collinear cracks.

Limit equilibrium of an orthotropic plate weakened by a periodic row of collinear cracks

A modified δ c -model is used to study the limiting state of an orthotropic plate weakened by a periodic row of collinear cracks and satisfying a general failure criterion. The failure mechanism of the plate is analyzed.Astudy is made of the effects of the degree of orthotropy, the biaxiality of external loading, and the geometrical parameters on the fracture process zones at the crack tips and the limiting state of the plate. The safe loading of an orthotropic viscoelastic plate with a periodic row of collinear cracks is examined. The effect of the rheological parameters on the safe-load region is studied

Nonlinear vibrations and dynamic stability of viscoelastic orthotropic rectangular plates

This paper describes the analyses of the nonlinear vibrations and dynamic stability of viscoelastic orthotropic plates. The models are based on the Kirchhoff–Love (K.L.) hypothesis and Reissner–Mindlin (R.M.) generalized theory (with the incorporation of shear deformation and rotatory inertia) in geometrically nonlinear statements. It provides justification for the choice of the weakly singular Koltunov–Rzhanitsyn type kernel, with three rheological parameters. In addition, the implication of each relaxation kernel parameter has been studied. To solve problems of viscoelastic systems with weakly singular kernels of relaxation, a numerical method has been used, based on quadrature formulae. With a combination of the Bubnov–Galerkin and the presented method, problems of nonlinear vibrations and dynamic stability in viscoelastic orthotropic rectangular plates have been solved, according to the K.L. and R.M. hypotheses. A comparison of the results obtained via these theories is also presented. In all problems, the convergence of the Bubnov–Galerkin method has been investigated. The implications of material viscoelasticity on vibration and dynamic stability are presented graphically.

Growth of a mode II crack in an orthotropic plate made of a viscoelastic composite material

The growth of a straight mode II crack in a viscoelastic orthotropic plate is examined. The plate material is modeled by a viscoelastic anisotropic medium. The shear displacement in the fracture process zone is determined as a function of time using the corresponding elastic solution, the Volterra principle, and the method of operator continued functions. The time dependence of the crack length is constructed as integral equations of three phases of stable growth. The solution of these equations gives kinetic curves