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Atomic force microscopy (AFM) studies of living biological cells is one of main experimental tools that enable quantitative measurements of deformation of the cells and extraction of information about their structural and mechanical properties. However, proper modelling of AFM probing and related adhesive contact problems are of crucial importance for interpretation of experimental data. The Johnson-Kendall-Roberts (JKR) theory of adhesive contact has often been used as a basis for modelling of various phenomena including cell-cell interactions. However, strictly speaking the original JKR theory is valid only for contact of isotropic linearly elastic spheres, while the cell membranes are often prestressed. For the first time, effects caused by molecular adhesion for living cells are analytically studied taking into account the mechanical properties of cell membranes whose stiffness depends on the level of the tensile prestress. Another important question is how one can extract the work of adhesion between the probe and the cell. An extended version of the Borodich-Galanov method for non-direct extraction of elastic and adhesive properties of contacted materials is proposed to apply to experiments of cell probing. Evidently, the proposed models of adhesive contact for cells with prestressed membranes do not cover all types of biological cells because the structure and properties of the cells may vary considerably. However, the obtained results can be applied to many types of smooth cells and can be used to describe initial stages of contact and various other processes when effects of adhesion are of crucial importance. [ABSTRACT FROM AUTHOR]

The adhesive contact problem of a half-space indented by a rigid cylindrical punch in the theory of couple-stress elasticity is investigated. Both the size effect and adhesion are considered to analyze the two-dimensional frictionless contact problem. The method of Mindlin's stress function and the technology of Fourier transform are applied to reduce the mixed boundary value problem to the singular integral equations. Efficient computational techniques is used to obtain the influence of the size effect and adhesion parameter on the adhesive contact behavior. Results indicated that the size effect play an important role in contact with adhesion and can help understanding the mechanism of adhesion. [ABSTRACT FROM AUTHOR]

A generalized double-Hertz (D-H) model has been proposed to consider the adhesive contact between an elastic cylinder and an elastic half space under inclined forces. The normal traction is exactly the same as that in the conventional D-H model. The shear traction of finite value is distributed into a slipping zone and a non-slipping zone. In the slipping zone, the shear traction is proportional to the compressive pressure. With the model, adhesive contact behaviour between cylinders has been numerically illustrated. The shear-induced peeling has been demonstrated. The value of the ratio for shear traction to normal traction larger than friction coefficient has been found in part of the non-slipping zone. Those altogether are consistent with experiments. [ABSTRACT FROM AUTHOR]

An existing hyperelastic membrane model for graphene calibrated from ab-initio data (Kumar and Parks, 2014) is adapted to curvilinear coordinates and extended to a rotation-free shell formulation based on isogeometric finite elements. Therefore, the membrane model is extended by a hyperelastic bending model that reflects the ab-inito data of Kudin et al. (2001). The proposed formulation can be implemented straight-forwardly into an existing finite element package, since it does not require the description of molecular interactions. It thus circumvents the use of interatomic potentials that tend to be less accurate than ab-initio data. The proposed shell formulation is verified and analyzed by a set of simple test cases. The results are in agreement to analytical solutions and satisfy the FE patch test. The performance of the shell formulation for graphene structures is illustrated by several numerical examples. The considered examples are indentation and peeling of graphene and torsion, bending and axial stretch of carbon nanotubes. Adhesive substrates are modeled by the Lennard–Jones potential and a coarse grained contact model. In principle, the proposed formulation can be extended to other 2D materials. [ABSTRACT FROM AUTHOR]

Recently proposed formulation of the boundary element method for adhesive contacts has been generalized for contacts of power-law graded materials with and without adhesion. Proceeding from the fundamental solution for single force acting on the surface of an elastic half space, first the influence matrix is obtained for a rectangular grid. The inverse problem for the calculation of required stress in the contact area from a known surface displacement is solved using the conjugate-gradient technique. For the transformation between the stresses and displacements, the Fast Fourier Transformation is used. For the adhesive contact of graded material, the detachment criterion based on the energy balance is proposed. The method is validated by comparison with known exact analytical solutions as well as by proving the independence of the mesh size and the grid orientation. [ABSTRACT FROM AUTHOR]

The paper presents solutions to the periodic adhesive contact problem between an elastic anisotropic layer and a wavy rigid half-space. Different systems including rigid, soft substrate and infinite elastic layer with orientation effect are considered in the paper. The periodic Green function in the form of Fourier series is obtained to formulate an integral equation of the contact pressure. The relation between the contact pressure and the relative approach is derived analytically. The results show that the adhesion effect depends strongly on both material orientation of the elastic layer and the substrate stiffness. These results are in particular useful for the design of new tunable adhesive systems. [ABSTRACT FROM AUTHOR]

The unilateral axisymmetric frictionless adhesive contact problem for a toroidal indenter and an elastic half-space is considered in the framework of the Johnson-Kendall-Roberts theory. In the case of a semi-fixed annular contact area, when one of the contact radii is fixed, while the other varies during indentation, we obtain the asymptotic solution of the adhesive contact problem based on the solution of the corresponding unilateral non-adhesive contact problem. In particular, the adhesive contact problem for Barber's concave indenter is considered in detail. In the case when both contact radii are variable, we construct the leading-order asymptotic solution for a narrow annular contact area. It is found that for a v-shaped generalized toroidal indenter, the pull-off force is independent of the elastic properties of the indented solid. [ABSTRACT FROM AUTHOR]

A two-dimensionally adhesive contact model is established, in which a gas-filled elastic membrane adheres on a stretched substrate. The free energy of the system is achieved, minimization of which leads to the relationship between the contact width and the global substrate strain. The contact solution exhibits three distinct regimes characterized by two threshold strains: (i) the contact size is hardly affected by the external loading acted on the substrate when the global substrate strain is below the first threshold value; (ii) the size of the contact is reduced quickly as the force is between the two threshold levels; (iii) the contact size tends to vanish when the global substrate strain exceeds the second threshold level. All the results share a number of common features with the experimental observation of cell orientation on a stretched substrate. Effects of the internal pressure, the tensile stiffness of the membrane and the interface work of adhesion on the two threshold levels are further discussed. The finding in the present paper should be helpful for deep understanding of adhesion mediated deformation sensing mechanism by which cells can detect mechanical signals in extracellular matrices and the design of adhesion mediated deformation sensors. [ABSTRACT FROM AUTHOR]

In this study, the frictional contact with partial slide between two dissimilar elastic cylinders is considered. According to the Spence׳s self-similarity condition, a system of singular integral equations is constructed with respect to the normal pressure and the shear traction in the contacting area. Based on the Goodman׳s hypothesis, the preceding system is uncoupled. From this, the tangential load in the central sticking zone is possible to be obtained analytically by means of the theory on the singular integral equation. Besides, a nonlinear equation in regard to the ratio of the adhesive and slip zone sizes is derived on the basis of the continuity of the tangential load. The sticking zone size can thus be determined by solving the nonlinear equation mentioned above iteratively. The problem in question is additionally solved by utilizing numerical method to make verification and validation of the theory and the related prevision found in the present paper. Numerical examples are provided to instantiate the theroy and method proposed. [ABSTRACT FROM AUTHOR]

Abstract: Adhesive contact between a rigid nanofiber and an incompressible elastic substrate is studied. A new expression of adhesive pressure, which accounts for the exact geometry of the fiber and the deformation of the substrate, is derived from the elementary Lennard–Jones (L–J) potential. This enables that the phenomenon of adhesion saturation for small fiber radii is predicted in a natural way. Numerical computations also show the validity of the well-known Derjaguin’s approximation in a quite broad region down to fiber radii of a few nanometers. These results are expected useful for such applications as nanoindentation, nanopunch, and bio-inspired adhesion. [Copyright &y& Elsevier]

The contact model for rough surfaces with power-law axisymmetric asperities in the presence of adhesion is developed. The extended JKR adhesive contact model for power-law axisymmetric asperities, denoted as JKR-n, developed by Zheng and Yu (2007, "Using the Dugdale Approximation to Match a Specific Interaction in the Adhesive Contact of Elastic Objects," J. Colloid Interface Sci., 310, pp. 27-34} is utilized to investigate the adhesive contact of rough surfaces. The JKR-n adhesive contact model generalizes the most adopted JKR model for spherical objects of n = 2. This work compares the effect of surface roughness on the adhesion force for rough surfaces with various power-law axisymmetric asperities. It is found that shapes of the asperities influence the pull-off forces greatly during the separation of rough surfaces. A general adhesion parameter that includes the shape index of asperities is proposed, and it can be used to characterize the adhesion performance of rough surfaces. [ABSTRACT FROM AUTHOR]

Abstract: In the last decade, instrumented macro- and nano-indentation tests have become useful tools for probing mechanical properties of materials. In this context, little in-depth work has been done for soft bio-materials like biofilms, tissues, gels, cells, etc. Such materials show large elastic strains when subjected to relatively fast loading (excluding viscoelastic phenomena). The present work focuses on the adhesive contact of such materials. Flat punches of circular imprint are used and are subjected to pullout normal forces or bending moments. The materials are modeled as hyperelastic (fast-testing conditions), using the Mooney–Rivlin (M–R) strain energy density function. We examine both incompressibility and compressibility issues. The contact problem of half-space materials interacting with rigid indenters is solved explicitly for moderate deformations and is compared with finite element results. Experiments are conducted with an artificial material (gel, reinforced with talc powder) that is modeled as a hyperelastic material. The present work is expected to extend indentation testing to important technologies like medical applications (health monitoring of tissues) and food industries (quality control of various production stages). It is concluded that the adhesive contact can be used to estimate the initial elastic modulus of soft substrates, but not all the material constants required by the hyperelastic models. [Copyright &y& Elsevier]

Abstract: In this paper, adhesive contact of a rigid cylinder on an elastic power-law graded half-space is studied analytically with the theory of weakly singular integral equation and orthogonal polynomial method. Emphasis is placed on the coupling effect between tangential and normal directions which was often neglected in previous works. Our analysis shows that the coupling effect tends to reduce the contact area in the compressive regime. The effect of bending moment on the adhesion behavior is also examined. Like a pull-off force, there also exists a critical bending moment at which the cylinder can be bended apart from the substrate. However, unlike pull-off force, the critical bending moment is insensitive to the gradient exponent of the graded material. [Copyright &y& Elsevier]

Abstract: This paper presents a study on the adhesive contact of a rigid sphere with an elastic–perfectly plastic half-space. Analytical solutions for contact force and contact radius relation, contact force and contact displacement relation during both loading and unloading are derived. From the present solutions the plastic pull-off force can be calculated analytically. The present study shows that the plastic pull-off force calculated with considering the variation of curvature within the contacted surface is higher than that calculated based on a constant curvature taken at the contact centre, and specially if the contact radius is calculated based on the Hertzian solution for elastic contacts then the obtained plastic pull-off force will be less than a half of the present plastic pull-off force. [Copyright &y& Elsevier]

The Hertz type contact problems and the analytical treatment of depth-sensing nanoindentation are under consideration. Fundamental relations of nanoindentation tests are derived for various boundary conditions within the contact region. First, a frictionless contact problem for a convex punch of revolution is studied and the connection between Galin and Sneddon solutions is shown. The Bulychev–Alekhin–Shorshorov (BASh) relation that is commonly used for evaluation of elastic modulus of materials by nanoindentation, is discussed. An analogous relation is derived that is valid for adhesive (no-slip) contact. Similarly to the Pharr–Oliver–Brotzen frictionless analysis, the obtained relation is independent of the geometry of the punch. Further, we study solutions to adhesive contact for punches whose shapes are described by monomial functions and obtain exact solutions for punches of arbitrary degrees of the monoms. These formulae are similar to the formulae of the frictionless Galin solutions and coincide with them when the material is incompressible. Finally, indenters having some deviation from their nominal shapes are considered. It is argued that for shallow indentation where the tip bluntness is on the same order as the indentation depth, the indenter shapes can be well approximated by non-axisymmetric monomial functions of radius. In this case problems obey the self-similar laws. Using one of the authors’ similarity approach to three-dimensional contact problems and the corresponding formulae, other fundamental relations are derived for depth of indentation, size of the contact region, load, hardness, and contact area, which are valid for both elastic and non-elastic, isotropic and anisotropic materials for various boundary conditions. In particular, it is shown that independently of the boundary conditions, the current area of the contact region is a power-law function of the current depth of indentation whose exponent is equal to a half of the degree of the monomial function of the shape. [Copyright &y& Elsevier]

Exemplified by adhesive contact of a rigid sphere with equiaxially stretched elastomeric substrates, the effect of finite pre-strain on adhesion behavior is studied based on the JKR model. The analysis indicates that the pre-strain reduces the contact area between the sphere and the substrate, but does not alter the adhesion strength. [ABSTRACT FROM AUTHOR]