Viscoelastic Half-plane Research Articles

Surface Green's functions for an anisotropic viscoelastic half-plane and their application to contact problems

In this paper, the elastic-like surface Green's functions for an anisotropic viscoelastic half-plane are derived using the time-stepping method. Using the elastic-like surface Green's functions as the core analytical solutions, we develop semi-analytical models (SAMs) and apply them to solve two different contact problems with anisotropic viscoelastic materials. As new modeling approaches, the SAMs developed here can provide fast and efficient approaches to solving contact problems. These methods enable us to consider contact problems with generally anisotropic viscoelastic solids, in which the contact surface is frictional and either smooth or rough, and the applied loads and boundaries can be time-variant. The correctness of the derived surface Green's functions is demonstrated by comparing the numerical results obtained by SAMs and those achieved from the analytical solutions or boundary element methods. Using the obtained numerical results, the impacts of time step size, anisotropy, frictional coefficient, roughness, and applied loads on the contact responses are further analyzed and discussed.

Exploring the dynamics of viscoelastic adhesion in rough line contacts

Modeling the adhesion of viscoelastic rough surfaces is a recent challenge in contact mechanics. Existing models have primarily focused on simple systems with smooth topography or single roughness scale due to the co-action of roughness and viscoelasticity leading to elastic instabilities and rate-dependent behavior, resulting in complex adhesion dynamics. In this study, we propose a numerical model based on a finite element methodology to investigate the adhesion between a randomly rough profile and a viscoelastic half-plane. Approach-retraction simulations are performed under controlled displacement conditions of the rough indenter. The results demonstrate that viscous effects dampen the roughness-induced instabilities in both the approach and retraction phases. Interestingly, even when viscous effects are negligible, the pull-off stress, i.e., the maximum tensile stress required to detach the surfaces, is found to depend on the stiffness modulus and maximum load reached during the approach. Furthermore, when unloading is performed from a relaxed state of the viscoelastic half-plane, both adhesion hysteresis and pull-off stress are monotonic increasing functions of the speed. Conversely, when retraction begins from an unrelaxed state of the material, the maximum pull-off stress and hysteretic loss are obtained at intermediate velocities.

On the Role of Roughness in the Indentation of Viscoelastic Solids

A numerical boundary element methodology is employed to understand how fractality intervenes when a 1D rigid rough profile indents a linear viscoelastic half-plane. The focus is, in particular, on the viscoelastic dissipation and how this is influenced by the profile statistical parameters, namely, the mean square roughness h_{rm rms} of the profile, the mean square slope m_{2} and the Hurst exponent H. Our numerical investigation, properly supported by a dimensional analysis, reveals that, in the one-dimensional case under investigation, the leading role is played by h_{rm rms} and, thus, mainly by the large scales of the rough spectrum. Clearly, on an experimental level, this implies that a simple measure of the roughness parameter h_{rm rms} is sufficient to determine the viscoelastic dissipation.

Dynamic Stress-Deformed States of a Circular Tunnel of Small Position Under Harmonic Disturbances

There are many underground tunnels of various shapes located in seismically active areas that need to be protected from seismic impacts. The paper considers the impact of harmonic waves on a cylindrical shell located in a viscoelastic half-plane. The study's main purpose is to determine the stress-strain state of a cylindrical shell when exposed to harmonic waves. The basic equation of viscoelasticity in displacements with the corresponding boundary conditions is obtained. The problem posed is solved in mixed potentials that satisfy the wave equation with complex parameters. The solution is expressed in terms of special Bessel and Hankel functions. As a result of multiple reflections, a system of algebraic equations with complex coefficients is obtained. In the future, this system is solved by the Gauss method with the selection of the main element. The analytical solution is obtained in infinite series, the convergence of which is investigated numerically. The numerical results were obtained using the MATLAB software package. The reliability of the research results is confirmed by good agreement with theoretical and experimental results and those obtained by other authors.

Vibrations of a deformed half-space with inclusion under the influence of surface waves

There is a large number of underground tunnels of various shapes located in seismic zones that need to be protected from seismic impacts. The paper considers the effect of harmonic surface waves on a cylindrical inclusion of various shapes located in a viscoelastic half-plane. The main purpose of the study is to determine the stress-strain state of the obstacle when exposed to harmonic waves. The problem is solved by the finite element method. It was found that the maximum stress concentration is allowed at long waves, and the stress concentration with increasing depth and wavelength approaches the static value of stress. The reliability of the obtained research results is confirmed by good agreement with theoretical and experimental results obtained by other authors.

Numerical Solution of Integro-Differential Equations Modelling the Dynamic Behavior of a Nano-Cracked Viscoelastic Half-Plane

Abstract The scattering of time-harmonic waves by a finite, blunt nano-crack in a graded, viscoelastic bulk material with a free surface is considered in this work. Non-classical boundary conditions and a localized constitutive equation at the interface between crack and matrix, following the Gurtin-Murdoch surface elasticity theory are introduced. An efficient numerical technique is developed using integro-differential equations along the nano-crack line that is based on an analytically derived Green‘s function for the quadratically inhomogeneous half-plane. The dependence of the diffracted and scattered waves and of the local stress concentration fields on key problem parameters such as viscosity, inhomogeneity, surface elasticity, and interaction between the nano-crack and the free surface are all examined through an extensive parametric study.

Dynamic analysis of an elastic plate on a cross-anisotropic and continuously nonhomogeneous viscoelastic half-plane under a moving load

The dynamic response of an elastic thin plate to a load moving on its surface with constant velocity is determined analytically. The plate rests on a cross-anisotropic and continuously nonhomogeneous viscoelastic half-plane soil medium. Soil nonhomogeneity is associated with elastic moduli increasing with depth. Viscoelastic effects are introduced via the hysteretic damping model. The solution of the problem is obtained with the aid of the complex Fourier series method involving the horizontal coordinate x and the time t. Thus, the governing partial differential equations of motion for the plate–soil system are reduced to an algebraic equation for the plate and a system of two ordinary differential equations with variable coefficients for the soil. This system is solved analytically by the method of Frobenius. Compatibility and dynamic equilibrium at the plate–soil interface as well as boundary conditions at the plate surface and at infinite soil depth enable one to finally obtain the response for both the plate and the soil analytically. Verification of the solution is done by means of comparisons with the known existing analytical solutions for the special cases of isotropy and homogeneity with or without the plate. Parametric studies are finally performed, and the effects of soil anisotropy, nonhomogeneity and damping on the plate and the supporting soil are assessed.

Contact problems of the sliding of a punch with a periodic relief on a viscoelastic half-plane

A method is proposed for the analytical solution of problems of the sliding of a punch with a periodic relief on a viscoelastic half-plane under conditions when they are in complete contact as well as problems on the loading of a viscoelastic half-plane with a piecewise-constant pressure that moves at a constant velocity on its surface. The problems are considered in a quasistatic formulation. The method is based on the expansion of the sought functions in trigonometric Fourier series and finding the relation between the displacements of the boundary and the contact pressures. Examples of different shapes of punches are considered and the dependencies of the contact characteristics and the mechaical component of the friction force on the sliding velocity, the viscosity parameters of the base and the geometric characteristics of the surface are studied.

Motion of a rigid punch at a low velocity along the boundary of a viscoelastic half-plane

The contact problem of the interaction of a rigid punch with a viscoelastic half-plane is considered. The dependence of the displacement of the boundary of half-plane on the normal load applied to it is determined, and the integral equation for determining the contact pressure is derived and solved by the method of “small λ”. Distributions of contact pressures under the punch are graphically represented.

Near-surface failure of layered viscoelastic materials

Within the framework of a piecewise homogeneous body model, with the use of exact equations of the geometrically nonlinear theory of viscoelastic bodies, the distribution of near-surface self-balanced normal stresses in a body consisting of a viscoelastic half-plane, an elastic locally curved bond layer, and a viscoelastic covering layer is investigated. A method for solving the problem considered by employing the Laplace and Fourier transformations is developed. Numerical results for the self-balanced normal stresses caused by a local curving (imperfection) of an elastic bond layer upon tension and compression of the body mentioned along the free face plane are presented and analyzed. The viscoelasticity of the materials is described by the Rabotnov fractional-exponential operators. A macroscopic failure criterion is proposed, and the validity of this criterion is examined.

Viscoelastic state of a semi-infinite medium with multiple circular elastic inhomogeneities

This paper is concerned with the problem of an isotropic, linear viscoelastic half-plane containing multiple, isotropic, circular elastic inhomogeneities. Three types of loading conditions are allowed at the boundary of the half-plane: a point force, a force uniformly distributed over a segment, and a force uniformly distributed over the whole boundary of the half-plane. The half-plane is subjected to farfield stress that acts parallel to its boundary. The inhomogeneities are perfectly bonded to the material matrix. An inhomogeneity with zero elastic properties is treated as a hole; its boundary can be either traction free or subjected to uniform pressure. The analysis is based on the use of the elastic-viscoelastic correspondence principle. The problem in the Laplace space is reduced to the complementary problems for the bulk material of the perforated half-plane and the bulk material of each circular disc. Each problem is described by the transformed complex Somigliana’s traction identity. The transformed complex boundary parameters at each circular boundary are approximated by a truncated complex Fourier series. Numerical inversion of the Laplace transform is used to obtain the time domain solutions everywhere in the half-plane and inside the inhomogeneities. The method allows one to adopt a variety of viscoelastic models. A number of numerical examples demonstrate the accuracy and efficiency of the method.

Local near-surface buckling of a system consisting of an elastic (viscoelastic) substrate, a viscoelastic (elastic) bond layer, and an elastic (viscoelastic) covering layer

Within the framework of a piecewise homogenous body model and with the use of a three-dimensional linearized theory of stability (TLTS), the local near-surface buckling of a material system consisting of a viscoelastic (elastic) half-plane, an elastic (viscoelastic) bond layer, and a viscoelastic (elastic) covering layer is investigated. A plane-strain state is considered, and it is assumed that the near-surface buckling instability is caused by the evolution of a local initial curving (imperfection) of the elastic layer with time or with an external compressive force at fixed instants of time. The equations of TLTS are obtained from the three-dimensional geometrically nonlinear equations of the theory of viscoelasticity by using the boundary-form perturbation technique. A method for solving the problems considered by employing the Laplace and Fourier transformations is developed. It is supposed that the aforementioned elastic layer has an insignificant initial local imperfection, and the stability is lost if this imperfection starts to grow infinitely. Numerical results on the critical compressive force and the critical time are presented. The influence of rheological parameters of the viscoelastic materials on the critical time is investigated. The viscoelasticity of the materials is described by the Rabotnov fractional-exponential operator.

Linear viscoelastic analysis of a semi-infinite porous medium

This paper considers the problem of a semi-infinite, isotropic, linear viscoelastic half-plane containing multiple, non-overlapping circular holes. The sizes and the locations of the holes are arbitrary. Constant or time dependent far-field stress acts parallel to the boundary of the half-plane and the boundaries of the holes are subjected to uniform pressure. Three types of loading conditions are assumed at the boundary of the half-plane: a point force, a force uniformly distributed over a segment, a force uniformly distributed over the whole boundary of the half-plane. The solution of the problem is based on the use of the correspondence principle. The direct boundary integral method is applied to obtain the governing equation in the Laplace domain. The unknown transformed displacements on the boundaries of the holes are approximated by a truncated complex Fourier series. A system of linear equations is obtained by using a Taylor series expansion. The viscoelastic stresses and displacements at any point of the half-plane are found by using the viscoelastic analogs of Kolosov–Muskhelishvili’s potentials. The solution in the time domain is obtained by the application of the inverse Laplace transform. All the operations of space integration, the Laplace transform and its inversion are performed analytically. The method described in the paper allows one to adopt a variety of viscoelastic models. For the sake of illustration only one model in which the material responds as the standard solid in shear and elastically in bulk is considered. The accuracy and efficiency of the method are demonstrated by the comparison of selected results with the solutions obtained by using finite element software ANSYS.

Investigation of the Stress-Strain State of Viscoelastic Piecewise-Homogeneous Bodies by the Method of Boundary Integral Equations

The problem on the stress state of a viscoelastic half-plane containing a finite number of inclusions of arbitrary shape and subjected to the action of distributed tangential and normal loads on its boundary is considered. Integral representations for the displacement vector and stress tensor are obtained for the case of an ideal mechanical contact on the conjugation contour of the regions. Discrete analogues of the boundary-temporal integral equations are constructed with account for the singularities of the stress field near the corner points. A numerical calculation is performed and the mechanical effects for an epoxy matrix with metal inclusions are analyzed.

The viscoelastic deformation of an orthotropic body (half-plane) by a rigid punch

The method of operator continued fractions is used to solve the problem on the stress state in a viscoelastic orthotropic half-plane loaded by a punch at the instantt=0. The pressure in the half-plane is determined on the basis of the Volterra principle and by solving the corresponding elastic problem. The influence of the rheological parameters on the stress state of the half-plane is shown by an example for a composite material.

The stress state of a viscoelastic orthotropic half-plane loaded with a concentrated force

The operator continued-fraction method is used to solve the problem on the stress state of a viscoelastic orthotropic half-plane loaded at a momentt=0 with a force normal to the boundary of the half-plane. The stress field in the half-plane is determined based on the Volterra principle and the solution of the corresponding elastic problem. The influence of the rheological parameters on the stress state of the half-plane is shown by an example of a specific material.

The Problem of Several Indentors Moving on a Viscoelastic Half-Plane

The problem of several indentors moving on a viscoelastic half-plane is considered in the noninertial approximation. The solution of this mixed boundary value problem is formulated in terms of a coupled system of integral equations in space and time. These are solved numerically in the steady-state limit for the case of two indentors. The phenomena of hysteretic friction and interaction between the two indentors are explored.

Green's function of the stationary dynamic problem for a viscoelastic half-space

A stationary Green's function is constructed for a viscoelastic half-plane in the form of a Fourier integral, and asymptotic estimates obtained for the integrand are used to carry out a numerical realization efficiently. Green's function was constructed for an elastic half-plane in /1/, but was complicated and difficult to calculate. A simpler expression was given for this function in /2/ with reference to a source which was not easily accessible. A solution of the problem of the steady-state oscillations of a homogeneous, Isotropie viscoelastic half-plane, caused by the action of an arbitrarily oriented concentrated force varying harmonically with time, is given below. The solution is used as the basic for constructing the BEMDYST boundary-element program.

The transient quasi-static plane viscoelastic moving load problem

The problem of a moving load on a viscoelastic half-plane is considered, in the non-inertial approximation. From the viscoelastic generalization of the Kolosov-Muskhelishvli equations, an integral equation is derived, which applies to the general transient problem. In the case of frictionless contact, explicit transient corrections to the coefficient of hysteretic friction are given for large times—when these corrections are small. Particularly explicit results are given for the case where viscoelastic losses are small. It emerges that the transient corrections to the hysteretic friction coefficient are negative.

The Sliding With Coulomb Friction of a Rigid Indentor Over a Power-Law Inhomogeneous Linearly Viscoelastic Half-Plane

In a previous paper, the title problem was solved for a homogeneous power-law linearly viscoelastic half-plane. Such material has a constant Poisson’s ratio and a shear modulus with a power-law dependence on time. In this paper, the shear modulus is assumed also to have a power-law dependence on depth from the half-plane boundary. As in the earlier paper, only a quasi-static analysis is presented, that is, the enertial terms in the equations of motion are not retained and the indentor is assumed to slide with constant speed. The resulting boundary value problem is reduced to a generalized Abel integral equation. A simple closed-form solution is obtained from which all relevant physical parameters are easily computed.