Axisymmetric Indentation Research Articles

Axisymmetric Indentation of Circular Rigid Plate on Layered Elastic Halfspace with Transverse Isotropy

This paper investigates the contact problem of a layered elastic halfspace with transverse isotropy under the axisymmetric indentation of a circular rigid plate. Fourier integral transforms and a backward transfer matrix method are used to obtain the analytical solution of the contact problem. The interaction between the rigid plate and the layered halfspace can be expressed with the standard Fredholm integral equations of the second kind. The induced elastic field in the layered halfspace can be expressed as the semi-infinite integrals of four known kernel functions. The convergence and singularity of the semi-infinite integrals near or at the surface of the layered halfspace are resolved using an isolating technique. The efficient numerical algorithms are used and developed for accurately calculating the Fredholm integral equations and the semi-infinite integrals. Numerical results show the correctness of the proposed method and the effect of layering non-homogeneity on the elastic fields in layered transversely isotropic halfspace induced by the axisymmetric indentation of a circular rigid plate.

Squeeze film characteristics of a rigid lubricated axisymmetric indenter punch onto unbonded and bonded soft elastic layers

This paper investigates the squeeze film behavior of a rigid indenter punch with parabolodial surface profile which interacts with a thin, soft, linearly elastic layer through a thin lubricant film. The layer has infinite radial extent and is either bonded or unbonded to a rigid foundation. Slow and fast squeeze film regimes of lubrication are defined and delineated based on the dry contact radius of a spherical indenter and the radial extent of the lubricant film. Favorable agreement with relevant experimental studies supports the formulation method and regime classification.

Stress tensor in the linear viscoelatsic incompressible half-space beneath axisymmetric bodies in normal contact

We present a simple method to determine the stress state in a linear viscoelastic half-space in frictionless unilateral normal contact with an axisymmetric rigid indenter for arbitrary loading histories. The procedure consists in the superposition of elastic solutions for flat punch or parabolic contact along the line load within the framework of the method of dimensionality reduction. We provide simple expressions for all stress components and the hydrostatic pressure gradient, which is of particular interest in the biomechanical field. The method brings advantages especially in its numerical application, since no convolution in time over the entire contact history is necessary.

Axisymmetric Indentations of an Elastic Half-Space With Tensed Surface/Membrane in the Johnson–Kendall–Roberts Adhesive Approximation

Abstract Owing to the significant effects of adhesive force and surface/membrane tension, the classical contact models often fail to describe the indentation responses of soft materials and biological systems. This work addresses the axisymmetric indentation of an elastic substrate with constant surface/membrane tension by a spherical, conical, or cylindrical flat indenter in the Johnson–Kendall–Roberts adhesive approximation. On the basis of non-adhesive contact solutions accounting for the surface/membrane tension effect, explicit expressions for the external load and depth with respect to the contact radius are derived for the adhesive contact cases, which act as the theoretical fundamental for the accurate analysis of indentation tests. Despite using different correction functions, the results for spherical indentation are consistent with the solution of previous studies. It is found that the role of surface/membrane tension in the adhesive contact behavior is controlled by a dimensionless parameter. As the parameter gets larger, the pull-off force and the contact size at zero-external load for spherical and conical indentations are smaller, whereas the pull-off force for cylindrical flat indentation is higher.

Identifiability of soft tissue constitutive parameters from in-vivo macro-indentation

Reliable identification of soft tissue material parameters is frequently required in a variety of applications, particularly for biomechanical simulations using finite element analysis (FEA). However, determining representative constitutive laws and material parameters is challenging and often comprises a bottleneck that hinders the successful implementation of FEA. Soft tissues exhibit a nonlinear response and are commonly modeled using hyperelastic constitutive laws. In-vivo material parameter identification, for which standard mechanical tests (e.g., uniaxial tension and compression) are inapplicable, is commonly achieved using finite macro-indentation test. Due to the lack of analytical solutions, the parameters are commonly identified using inverse FEA (iFEA), in which simulated results and experimental data are iteratively compared. However, determining what data must be collected to accurately identify a unique parameter set remains unclear. This work investigates the sensitivities of two types of measurements: indentation force-depth data (e.g., measured using an instrumented indenter) and full-field surface displacements (e.g., using digital image correlation). To eliminate model fidelity and measurement-related errors, we employed an axisymmetric indentation FE model to produce synthetic data for four 2-parameter hyperelastic constitutive laws: compressible Neo-Hookean, and nearly incompressible Mooney-Rivlin, Ogden, and Ogden-Moerman models. For each constitutive law, we computed the objective functions representing the discrepancies in the reaction force, the surface displacement, and their combination, and visualized them for hundreds of parameter sets, spanning a representative range as found in the literature for the bulk soft tissue complex in human lower limbs. Moreover, we quantified three identifiability metrics, which provided insights into the uniqueness (or lack thereof) and the sensitivities. This approach provides a clear and systematic evaluation of the parameter identifiability, which is independent of the selection of the optimization algorithm and initial guesses required in iFEA. Our analysis indicated that the indenter’s force-depth data, despite being commonly used for parameter identification, was insufficient for reliably and accurately identifying both parameters for all the investigated material models and that the surface displacement data improved the parameter identifiability in all cases, although the Mooney-Rivlin parameters remained poorly identifiable. Informed by the results, we then discuss several identification strategies for each constitutive model. Finally, we openly provide the codes used in this study, to allow others to further investigate the indentation problem according to their specifications (e.g., by modifying the geometries, dimensions, mesh, material models, boundary conditions, contact parameters, or objective functions).

Indentation of a pressurized silicon stomach model – A non-invasive study of gastric wall stiffness and pressurized gastric content stiffness

Background and aimsEvaluation of gastric wall stiffness and intragastric pressure is essential for detailed assessments of gastric accommodation. However, non-invasive assessments are needed for large scale clinical studies and none of the existing methods takes abdominal wall effect into the calculation. This study aimed to assess gastric wall stiffness and gastric content stiffness as a proxy for intragastric pressure using novel mechanical modeling and non-invasive indentation tests on a silicon stomach model. MethodsA silicon stomach model (scaling 1:1 with the human stomach) was indented using a pressure algometer at intragastric pressures from 0 to 0.8 kPa. Wall thicknesses and luminal cross-sectional areas along the stomach were measured with ultrasound images. The gastric wall stiffness was compared between measurements from tensile tests on strips cut from the silicon stomach and estimations from a shell indentation model. The pressurized gastric content stiffness was predicted from a bonded two-layer axisymmetric half-space indentation model. ResultsThe gastric wall stiffness estimated from the shell indentation model showed no difference to measurements from the mechanical tests on the cutting strips (p = 0.14). The predicted gastric content stiffness was strongly associated with the intragastric pressure (r > 0.83, p < 0.001). ConclusionsThe mechanical model developed in this study can simultaneously predict the gastric wall stiffness and the pressurized gastric content stiffness. In future studies, the method can be applied to reveal intragastric pressure conditions non-invasively via the pressurized gastric content stiffness during gastric accommodation and emptying such as with magnetic resonance imaging.

A numerical method based on Polonsky-Keer's conjugate gradient iterative scheme for solving axisymmetric contact of layered half-space

Axisymmetric contact analysis of layered half-space is of considerable interest in many tribological applications including protective hard coating, metal-matrix composites, etc. Polonsky-Keer's iterative scheme based on conjugate gradient method (CGM) is powerful for a general three-dimensional contact problem using rectangular meshes, but would inevitably lead to excessive computational burden and inaccurate boundary discretization for axisymmetric contact. In this paper, Polonsky-Keer's CG-based iterative scheme is adapted for axisymmetric indentation of a layered elastic half-space by taking advantage of annular ring discretization, where the elementary solutions for the surface deflection and subsurface stresses can be determined with assistance of the Hankel transforms. The present method can not only effectively circumvent the discretization error, but is also versatile and flexible to solve the contact between the layered materials and an arbitrary axisymmetric indenter. Parametric studies are provided to validate the present solutions, and the comparisons of the average relative errors for different types of elements are conducted to demonstrate the excellent convergence and accuracy of the proposed numerical scheme.

Axisymmetric Problem for Indentation of Multi-Layer Poroelastic Body

We considered the axisymmetric indentation of poroelastic multi-coating bonded to poroelastic substrate indented by a rigid flat-ended cylindrical indenter. The approach that expressed poroelastic solutions in term of two displacement functions and the transfer matrix method which can formulate the effect of multi-layer composite were applied in this study. Furthermore, we adopted the analytic method that using Laplace-Hankel transform and expressing the normal contact stress at the surface layer as an appropriate series with a Chebyshev orthogonal polynomial to reduce dual integral equations to an infinite system of simultaneous equations in Laplace transform domain. The numerical results of time-dependent axial displacement of rigid cylindrical indenter were presented by performing numerical inverse Laplace transform. The results of axial displacement for one-layer bonded to rigid impervious substrate given by this study were good agreement with previously reported results.

Revisiting the theory behind AFM indentation procedures. Exploring the physical significance of fundamental equations

Fundamental contact mechanics models concerning the interaction of an axisymmetric indenter and an elastic half-space are usually employed in atomic force microscopy (AFM) indentation methods. In this paper, a simplified ‘equivalent’ physical system is used to correlate basic magnitudes such as the applied force on an elastic half space, the Young’s modulus, the contact radius and the indentation depth. More specifically, the equations correlating the above magnitudes are derived using fundamental physics instead of the typical rigorous mathematical process with a small error. In addition, the relation between a force-indentation curve and the indenter’s shape is also presented in detail in order to help students and non-specialists in contact mechanics to obtain a strong background to the AFM indentation theory.

Indentation of a circular hyperelastic membrane by a rigid cylinder

In the paper we investigate the effect of friction on the stress–strain state of a hyperelastic membrane in contact with a massive solid body. We consider the axisymmetric indentation of a thin circular rubber membrane by a cylindrical indenter for different contact conditions: dry contact, liquid lubrication, and gel lubrication. We present experimental results and a mathematical model of the indentation. We use the non-linear theory of elastic membranes and the Coulomb friction model in the simulation. We describe different contact conditions with different friction coefficients. We compare the experimental and theoretical dependences between indentation force and indenter displacement.

The surface tension effect revealed via the indentation scaling index

The axisymmetric frictionless indentation problem for an isotropic elastic half-space with the uniform surface tension is considered. The first-order regular and singular asymptotic models are constructed for the limit situations of relatively small (in the membrane limit) or large (in the Hertzian limit) values of the contact radius. By using the method of collocation, a two-parameter analytical approximate solution is developed with coefficients evaluated in triple quadratures. The accuracy of the obtained analytical models is investigated in the case of a paraboloidal indenter by comparison with numerical solutions available in the literature. Based on the asymptotic behavior of the solution in the limit situations, the so-called indentation scaling index is introduced for the analysis of the depth-sensing indentation data. By making use of the novel contact parameter it is possible to extract the surface tension effect independently of determining the reduced elastic modulus of a tested semi-infinite sample. It is shown that by employing an appropriate normalization, the contact problem can be reduced to that parameterized by only one parameter (relative contact radius), which implies that irrespectively of the combination of the material parameters, the force–displacement curve drawn in the normalized variables will be universal.

Explicit transformation between non-adhesive and adhesive contact problems by means of the classical Johnson-Kendall-Roberts formalism.

The classic Johnson-Kendall-Roberts (JKR) contact theory was developed for frictionless adhesive contact between two isotropic elastic spheres. The advantage of the classical JKR formalism is the use of the principle of superposition of solutions to non-adhesive axisymmetric contact problems. In the recent years, the JKR formalism has been extended to other cases, including problems of contact between an arbitrary-shaped blunt axisymmetric indenter and a linear elastic half-space obeying rotational symmetry of its elastic properties. Here the most general form of the JKR formalism using the minimal number of a priori conditions is studied. The corresponding condition of energy balance is developed. For the axisymmetric case and a convex indenter, the condition is reduced to a set of expressions allowing explicit transformation of force-displacement curves from non-adhesive to corresponding adhesive cases. The implementation of the developed theory is demonstrated by presentation of a two-term asymptotic adhesive solution of the contact between a thin elastic layer and a rigid punch of arbitrary axisymmetric shape. Some aspects of numerical implementation of the theory by means of Finite-Element Method are also discussed. This article is part of a discussion meeting issue 'A cracking approach to inventing new tough materials: fracture stranger than friction'.

Axisymmetric frictionless indentation of a rigid stamp into a semi-space with a surface energetic boundary

An axisymmetric problem for a frictionless contact of a rigid stamp with a semi-space in the presence of surface energy in the Steigmann–Ogden form is studied. The method of Boussinesq potentials is used to obtain integral representations of the stresses and the displacements. Using the Hankel transform, the problem is reduced to a single integral equation of the first kind on a contact interval with an additional condition. The integral equation is studied for solvability. It is shown that for the classic problem in the absence of surface effects and for the problem with the Gurtin–Murdoch surface energy without surface tension, the obtained equation represents a Cauchy singular integral equation. At the same time, for the Gurtin–Murdoch model with a non-zero surface tension and for the general Steigmann–Ogden model, the problem results in the integral equation of the first kind with a weakly singular or a continuous kernel, correspondingly. Hence, the contact problem is ill-posed in these cases. The integral equation of the first kind with an additional condition is solved approximately by using Gauss–Chebyshev quadrature for evaluation of the integrals. Numerical results for various values of the parameters are reported.

Numerical moving mesh solution for the JKR adhesive contact between an incompressible layer and an axisymmetric rigid indenter

Recent research extended the non-adhesive contact problems between an incompressible layer and a rigid indenter to adhesive cases in the limit of the Johnson–Kendall–Roberts (JKR) model, where it simply changes the boundary condition. The governing equation of this problem is in the form of Poisson’s equation, and there are two boundary conditions, one of which serves to determine the extent of the contact area. This makes it possible to develop a numerical solution of an adhesive thin incompressible layer indentation problem. For a numerical implementation, we have devised a finite element formulation with a moving mesh technique satisfying the slope boundary condition, which determines the actual extent of the contact area. We shall apply the proposed numerical method to an adhesive contact problem by a spherical rigid indenter to demonstrate the validity of the method. Furthermore, we will compare the characteristics of the JKR indentation solutions between a half-space and a thin incompressible layer.

A Discussion Regarding the Application of the Hertz Contact Theory on Biological Samples in AFM Nanoindentation Experiments

Background: Atomic Force Microscopy (AFM) Nanoindentation procedure regarding biological samples poses significant challenges with respect to the accuracy of the provided results. These challenges are related to the inhomogeneity of biological samples, various uncertainties in experimental methods and certain approximations regarding the theoretical analysis. The most commonly used theoretical model for data processing at the linear elastic regime regarding biological samples is the Hertz model. Objective: This paper focuses on the investigation of the resulting errors of the basic equation of the Hertz theory that depend on the ratio, indentation depth/indenter’s radius regarding the Young’s modulus calculation. Methods: An extended new equation is derived which takes into account the influence of the indentation depth/indenter’s radius ratio on the calculation of the Young’s modulus and can be easily used for calculations. The derived equation is further combined with equations which take into account the shape of the sample. Results: Several examples in the literature that do not take into account the value of the ratio indentation depth/indenter’s radius are reported and the related errors are calculated and discussed. Moreover, a rational explanation, regarding the extended differences of the Young’s modulus calculations using the same experimental results when these are processed using the Hertz model and the Oliver &amp; Pharr analysis (which is the general model that applies for any axisymmetric indenter) is provided. Conclusion: A complete and reliable theoretical tool was developed (that takes into account the indentation depth/indenter’s radius ratio and the shape of the sample) which can be generally applied in order to reduce the errors produced by the current methodology (Hertz model).

Lagrange multiplier approach to unilateral indentation problems: Well-posedness and application to linearized viscoelasticity with non-invertible constitutive response

The Boussinesq problem describing indentation of a rigid punch of arbitrary shape into a deformable solid body is studied within the context of a linear viscoelastic model. Due to the presence of a non-local integral constraint prescribing the total contact force, the unilateral indentation problem is formulated in the general form as a quasi-variational inequality with unknown indentation depth, and the Lagrange multiplier approach is applied to establish its well-posedness. The linear viscoelastic model that is considered assumes that the linearized strain is expressed by a material response function of the stress involving a Volterra convolution operator, thus the constitutive relation is not invertible. Since viscoelastic indentation problems may not be solvable in general, under the assumption of monotonically non-increasing contact area, the solution for linear viscoelasticity is constructed using the convolution for an increment of solutions from linearized elasticity. For the axisymmetric indentation of the viscoelastic half-space by a cone, based on the Papkovich–Neuber representation and Fourier–Bessel transform, a closed form analytical solution is constructed, which describes indentation testing within the holding-unloading phase.

Contact-induced change of the bandgap of semiconductors of the wurtzite structure

Progress in the synthesis of semiconductors of nanoscale structures has revived an interest in using semiconductors of different structures in the sensing of mechanical deformation through the change of photoluminescent characteristics under external stress/strain. In the heart of the deformation-induced change of photoluminescent characteristics is the deformation-induced change of the bandgap of semiconductors. In this work, we investigate the bandgap changes of bulk semiconductors of the wurtzite structure due to elastic indentation by a rigid, axisymmetric indenter. Using deformation potentials and the analytical solutions for the contact problems, we obtain closed-form solutions of the contact-induced change of the bandgaps for the elastic indentation by three different indenters of flat-ended, spherical, and conical shapes. For the elastic indentation by a flat-ended indenter, the bandgap changes are proportional to the indenter displacement/indentation load. For the elastic indentation by a spherical indenter, the bandgap changes are proportional to the cubic root of the indention load. For the elastic indentation by a conical indenter, the bandgap changes exhibit the characteristic of shape-similarity, independent of the indentation depth/load.

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.

Exact axisymmetric adhesive contact analysis for a pre-deformed soft electroactive half-space

In this paper, three-dimensional exact solutions of adhesive contact between a pre-deformed compressible soft electroactive half-space and an axisymmetric rigid indenter are presented. The change of surface adhesion energy during the contact is examined by using the modified JKR model, which accounts for the real contact area instead of the projected area. With the help of new results in the potential theory method, all physical (field) variables are derived in terms of elementary functions for three common types of axisymmetric indenters (flat-ended, conical, and spherical). The analytical contact relations for different indenter geometries and material properties are provided and summarized in Tables 2 and 3 to serve a solid base for revealing the underlying electromechanical mechanism of soft electroactive materials. For numerical illustration, neo-Hookean isotropic electroactive material is considered. The simulation results clearly demonstrate that both the mechanical and electric biasing fields significantly affect the indentation measurement of the electroactive material. Moreover, at either micro- or nano-scale, adhesion plays a prominent role in the indentation responses. It is of interest that, even without adhesion, the normal stress somewhere in the contact region may become tensile under a prescribed pre-stretch when the biasing electric displacement exceeds a certain value. This abnormal phenomenon actually corresponds to surface instability of the half-space under the biasing field. In the case of adhesive contact, other than the surface instability, the value of surface adhesion energy between the probe and the sample will impose a constraint on the validity of indentation analysis within the linear elastic regime.

Adhesive Interaction of Elastic Bodies with Regular Surface Relief

This paper focuses on adhesive interaction between an axisymmetric indenter and an elastic half-space taking into account the surface microrelief in the form of regularly located identical asperities covering the indenter. The microlevel contact between the asperities system and the elastic half-space is considered discrete. The solution to the problem at the macrolevel is constructed using the dependence of the effective specific adhesion force on the value of the nominal gap obtained by solving the problem at the microlevel. The constructed solution makes it possible to model the influence of microgeometry and adhesion parameters on the contact interaction of elastic bodies at the macrolevel.