Parabolic Indenter Research Articles

Nonlinear indentation of second-order hyperelastic materials

The classical problem of indentation on an elastic substrate has found new applications in the field of the Atomic Force Microscopy. However, linearly elastic indentation models are not sufficiently accurate to predict the force–displacement relationship at large indentation depths. For hyperelastic materials, such as soft polymers and biomaterials, a nonlinear indentation model is needed. In this paper, we use second-order elasticity theory to capture larger amplitude deformations and material nonlinearity. We provide a general solution for the contact problem for deformations that are second-order in indentation amplitude with arbitrary indenter profiles. Moreover, we derive analytical solutions by using either parabolic or quartic surfaces to mimic a spherical indenter. The analytical prediction for a quartic surface agrees well with finite element simulations using a spherical indenter for indentation depths on the order of the indenter radius. In particular, the relative error between the two approaches is less than 1% for an indentation depth equal to the indenter radius, an order of magnitude less than that observed with models which are either first-order in indentation amplitude or those which are second-order in indentation amplitude but with a parabolic indenter profile.

Contact Problem of Sliding a Parabolic Indenter on a Heterogeneous Base

Contact Problem of Sliding a Parabolic Indenter on a Heterogeneous Base

Contact interaction of a predeformed plate which lies without friction on rigid base with a parabolic indenter

Within the framework of linearized formulation of a problem of the elasticity theory, the stress-strain state of a predeformed plate, which is modeled by a prestressed layer, is analyzed in the case of its smooth contact interaction with a rigid axisymmetric parabolic indenter. The dual integral equations of the problem are solved by representing the quested-for functions in the form of a partial series sum by the Bessel functions with unknown coefficients. Finite systems of linear algebraic equations are obtained for determination of these coefficients. The influence of the initial strains on the magnitude and features of the contact stresses and vertical displacements on the surface of the plate is analyzed for the case of compressible and incompressible solids. In order to illustrate the results, the cases of the Bartenev – Khazanovich and the harmonic-type potentials are addressed.

Контактные свойства градиентных материалов с высоким показателем градиентности

An adhesive contact between an incompressible parabolic indenter and a half-space is investigated, the modulus of elasticity of which is a power-law function of depth. A diagram showing the areas corresponding to a stable solution of the contact problem for high gradientness indexes is shown. It is shown that the adhesive contact between materials with a high gradientness indexes is not destroyed when they are moved at any distance from each other.

Frictional Energy Dissipation in a Contact of Elastic Bodies Subjected to Superimposed Normal and Tangential Oscillations

Frictional damping in elastic contact of a parabolic indenter subjected to a combination of oscillations in normal and tangential directions is numerically simulated. The dissipated energy first increases linearly with coefficient of friction, then decreases linearly, and finally reaches a constant value. These three regions correspond to the states of complete slip, partial slip and complete stick. All three asymptotical dependencies can be described analytically. The dissipated energy in a dimensionless form is function of the ratio of normal oscillation amplitude and mean indentation depth, the ratio of change in contact area and sticking area, and phase shift between normal and tangential oscillation. Master curves are suggested.

Boundary Element Calculations for Normal Contact of Soft Materials With Tensed Surface Membrane

This work considers the non-adhesive frictionless contact problem of soft materials with surface being tensed by equi-biaxial tension. The boundary element method based on Fast Fourier Transform and conjugate gradient algorithm is extended to deal with this problem. By comparing with existing analytical solutions for the axisymmetric contact between a rigid parabolic indenter and an elastic half space with constant membrane/surface tension, our numerical simulations are validated having great accuracy and efficiency. Then, with this numerical tool, we further calculate the contact responses of a soft substrate indented by a smooth indenter with general quadric profile and a rough indenter with self-affine fractal surface, respectively. Some nontrivial contact behavior resulted from the presence of membrane tension are demonstrated and discussed.

Adhesion between a Rigid Indenter and an Elastic Half-Space for Incompressible Gradient Media with a High Gradientness Index

The contact problem of a cylindrical indenter and gradient medium is considered with allowance for adhesion forces between surfaces of contacting bodies. An analytical solution of the problem is found for the case of a gradient medium with a power dependence of the elastic modulus on the normal coordinate. The case of incompressible media with a Poisson ratio of 1/2 is described in detail. It is shown that the behavior of the system in this case can qualitatively differ from the case of homogeneous media. The influence of the surface roughness, which is modeled in the form of axisymmetric waviness is considered. Special attention is paid to the contact between a parabolic indenter and a gradient medium; the differences from the case of a cylinder are shown.

Адгезия между жестким индентором и упругим полупространством для несжимаемых градиентных сред с высоким показателем градиентности

Contact problem of cylindrical indenter and graded media with account of adhesion forces between two contacted bodies is considered. Analytical solution of the problem for the case of graded media with power-law dependence of elastic modulus on normal coordinate have been found. The case of uncompressible media with Poisson ratio 1/2 was considered in details. It is shown, that in this case the behavior of system can be qualitatively different from the case of homogeneous media. Influence of roughness of the surface is considered. Surface roughness is presented as axially symmetric “waviness”. The contact of parabolic indenter and a gradient media was considered. Differences from the case of a cylinder is shown.

Elasticity Does Not Necessarily Break Down in Nanoscale Contacts

Atomistic structures can have (sharp) features that are not accounted for in standard continuum theories. A prominent example is a Hertzian contact in which, however, the indenting tip is cut out of a crystal, whereby the tip acquires a discretized height profile. The microscopic stresses observed for such quantized indenters show sharp stress peaks at the edges of the height steps so that the stress profiles differ from those produced by smooth, parabolic indenters. Such deviations are frequently misinterpreted as the breakdown of continuum theory at the nanoscale. In this Letter, the stress peaks are confirmed to also occur in a continuum treatment containing steps. In addition, it is shown that analytical solutions for smooth tips can compare extremely well to those with steps if both stress fields are passed through the same (Gaussian) filter smearing out the features in real space with a resolution close to the broadest terrace of the quantized tip. Related statements are shown to also hold for the stress distribution function of randomly rough indenters with quantized height profiles.

NUMERICAL IMPLEMENTATION OF FRETTING WEAR IN THE FRAMEWORK OF THE MDR

Two numerical methods are proposed to improve accuracy of the numerical calculation of fretting wear in the framework of the Method of Dimensionality Reduction (MDR). Due to the singularity of the transformation equations, instabilities appear at the border between the stick and slip regions after many transformations from the one-dimensional to the three-dimensional contact and back. In these two methods, the transformation equations are reformulated to weaken the singularity of the integrals and a stable simulation of fretting wear is realized even with the wear models which go beyond the classical Archard law. With an example of dual-oscillation, we show the change in the worn profile of a parabolic indenter as well as the stress distribution on the contacting surface during the oscillating cycles under the Archard’s law of wear and Coulomb’s law of friction.

A STUDY ON GROSS SLIP AND FRETTING WEAR OF CONTACTS INVOLVING A POWER-LAW GRADED ELASTIC HALF-SPACE

For the steady wear state of two contact problems involving power-law graded materials, closed-form solutions are derived in terms of pressure distribution and limiting shapes of profile. Both gross slip of an initially flat-ended cylindrical punch on a power-law graded half-space and the load-controlled fretting wear under partial slip of an initially parabolic indenter are studied. In the case of gross slip at fixed penetration depth there exists a certain exponent of elastic inhomogeneity, for which the effective volume change takes its maximum value. To minimize wear due to fretting under partial slip, an amplitude dependent design of the material gradient is necessary. For large amplitudes of the tangential force a gradient ranged from a soft surface to a hard ground is beneficial, small amplitudes require a reverse gradient characterized by a hard surface and a soft ground. However, the choice of the material gradient also has a decisive influence on the strength of stress singularities at the contact edge and thus the initiation of fretting fatigue cracks, which is why it is discussed in more detail.

Adhesive Strength of Contacts of Rough Spheres

The adhesive contact between a parabolic indenter with superimposed roughness and an elastic half space is studied in the JKR-limit (infinitely small range of action of adhesive forces) using the boundary element method with mesh-dependent detachment criterion suggested in 2015. Three types of superimposed roughness are considered: one- and two-dimensional waviness and randomly rough roughness. It is shown that in the case of regular waviness, the character of adhesion is governed by the Johnson adhesion parameter. For our randomly rough surfaces a new adhesion parameter has been identified numerically, which uniquely determines the adhesive strength of the contact.

A new approach for closed-form analytical solution of two-dimensional symmetric double contacts and the comparison with finite element method

In this paper, a new procedure is developed for the solution of a general two-dimensional uncoupled symmetric double contact problem with smooth contact zones in which the indenter geometry is described by a piecewise biquadratic function. This procedure gives an approximate closed-form solution for any smooth indenter profile. In order to evaluate the accuracy of this approach, it is applied to the symmetric indentation of a flat surface by two rigidly interconnected parabolic indenters and results are compared with the exact unclosed-form solution. Moreover, this procedure is applied to the symmetric indentation of a flat surface by two rigidly interconnected cylinders to compare the results with the finite element solution obtained by the finite element method software, ABAQUS. The results showed that in comparison with the finite element method, this procedure is a fast and highly accurate method with low complexity that makes feasible the possibility of determining approximate closed-form solution for a wide range of indenter geometries with a concavity between two symmetric contact zones; hence it can be useful in practical issues.

Cylindrical lateral depth-sensing indentation of anisotropic elastic tissues: Effects of adhesion and incompressibility

ABSTRACTA two-dimensional frictionless adhesive contact problem for a parabolic indenter pressed against an orthotropic elastic layer resting on a smooth rigid substrate is studied in the framework of the Johnson–Kendall–Roberts (JKR) theory. In the case of a relatively small contact zone with respect to the layer thickness, the fourth-order asymptotic solution (up to terms of order ) is obtained, and the pull-off force is expanded in terms of the non-dimensional measure of the work of adhesion. In particular, a pinch/compression method for soft tissue is considered, and the testing methodology is suggested based on a least-squares best fit of the first-order asymptotic model to the depth-sensing indentation data for recovering two of the three independent elastic moduli which characterize an incompressible transversely isotropic material. The case of a weakly compressible material, which is important for biological tissues, is also discussed. The developed asymptotic model can be effectively used for small values of a certain dimensionless parameter, which is proportional to the work of adhesion and the indenter radius squared, on the one side, and inversely proportional to the effective elastic modulus and the elastic layer thickness cubed, on the other.

TANGENTIAL DISPLACEMENT INFLUENCE ON THE CRITICAL NORMAL FORCE OF ADHESIVE CONTACT BREAKAGE IN BIOLOGICAL SYSTEMS

The dependencies of the critical components of normal and tangential forces corresponding to the contact breakage between a parabolic indenter and an elastic half-space have been determined taking into account adhesive interaction. In order to describe the adhesive contact, the method of dimensionality reduction (MDR) and the modified rule of Hes taking into account tangential displacements have been used. The influence of the surface energy depending on the indenter separation angle has been studied.

Friction in an adhesive tangential contact in the Coulomb-Dugdale approximation

ABSTRACTWe study the problem of tangential frictional contact in the presence of adhesion. The model can be considered as a generalization of the theory by Cattaneo and Mindlin to the case where there are “long range adhesive interactions” between the contacting surfaces, which exert an additional pressure on the surfaces even in the absence of an external normal force. The adhesion forces are described by the Dugdale model and the tangential forces in the contact by Coulomb’s law of dry friction. These approximations allow obtaining an analytical solution for the tangential contact problem of a rigid parabolic indenter and a half-space. As in the case of nonadhesive contact, application of an arbitrarily small tangential force leads to slip in the narrow ring-shaped area near the contact boundary. Further increase in tangential load leads to a decrease in the radius of the stick region until sliding expands to the entire contact region. The obtained analytical solution shows that the main governing parameter of the problem is the parameter introduced by Maugis for the normal adhesive contact problem.

Analysing the effects of sliding, adhesive contact on the deformation and stresses induced within a multi-layered elastic solid

This paper presents a mathematical model of sliding, adhering contact between a rigid parabolic indenter and a multi-layered elastic solid, which is assumed to comprise of a homogeneous coating bonded through a functionally-graded transitional layer to a homogeneous substrate. The adhesive forces in this investigation are modelled using Lennard-Jones potential and an iterative algorithm is proposed that solves for the contact pressure, surface displacement and sub-surface stresses resultant within the layered solid. The effects of surface adhesion and different material properties such as varying coating/transition layer thickness and coating hardness on the solution of the contact problem are subsequently investigated in detail.The numerical approach presented in this paper demonstrates the significance of having a suitable mathematical representation for the traction distribution along the sliding, adhering contact. It is found that under weakly adhering conditions, the assumption of only Coulombic traction suffices to determine the displacements and subsurface stresses within the multi-layered solid. However, it is noted that stress concentrations within the material begin to propagate through all three layers of the elastic solid with increased surface adhesion, which could potentially induce plasticity and lead to material ploughing under sliding. The proposed model allows us to further investigate and improve our understanding of the combined effects of traction and boundary adhesion in sliding contacts, which can be used to inform the design of materials needed in such conditions.

Sliding of two smooth indenters on a viscoelastic foundation in the presence of friction

The axisymmetric contact problem of sliding of two solid parabolic indenters on a viscoelastic half-space with constant velocity is considered. Shear stresses modeling the adhesive component of the friction force act in the contact area. The model of the foundation material is described by an integral operator with an exponential kernel characterized by one relaxation time. The problem is solved by the boundary element method. The dependences of the contact characteristics on the sliding velocity, the normal load, and the distance between the centers of the indenters is analyzed. The results can be used to study the effect of the roughness elements modeled by two indenters on the contact characteristics and the deformation component of the friction force.

Cylindrical lateral depth-sensing indentation testing of thin anisotropic elastic films

A two-dimensional frictionless contact problem for a parabolic indenter pressed against an orthotropic elastic strip resting on a frictionless rigid substrate is studied. The sixth-order asymptotic solution is obtained in the case of a relatively small contact zone with respect to the elastic strip thickness. The so-called cylindrical lateral indentation test, which utilizes lateral contact of a cylindrical indenter, is developed for indentation testing of a thin transversely isotropic film with the symmetry plane orthogonal to the contact plane under the assumption that the film thickness is small compared to the cylinder indenter length. The presented testing methodology is based on a least squares best fit of the second-order asymptotic model to the depth-sensing indentation data.

On the analytical solution for sliding contact of piezoelectric materials subjected to a flat or parabolic indenter

This work presents the analytical solution of sliding contact problem of piezoelectric materials under the action of a flat or parabolic indenter. Inside the contact region, the indenter and the piezoelectric materials are in relative motion. Boundary conditions are well equipped. For a flat indenter and a parabolic indenter, non-classical singularity is presented and exact results are derived for surface contact stress, in-plane stress and electric displacement. Figures show that the influence of the friction coefficient on the stress components is more pronounced than that of the piezoelectric coefficient ratio, while the opposite trend is observed for the electric displacement. Under a sliding parabolic indenter, an increasing of the piezoelectric coefficient ratio makes the contact region become narrower.