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Axisymmetric indentations of an elastic half-space by (a) a rigid sphere, (b) a rigid cone, and (c) a flat-ended rigid cylinder
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The conventional contact mechanics does not account for surface tension, however which gets important for micro-or nano-sized contacts. In the present paper, the influences of surface tension on the indentations of an elastic half space by a rigid sphere, cone and flat-ended cylinder are investigated, and the corresponding singular integral equatio...
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... on the surface elasticity theory, we consider the indenta- tions of an isotropic elastic half-space by a rigid sphere with a radius R, a rigid cone with a semiangle a, and a flat-ended rigid cylinder with a base radius a, respectively, as shown in Fig. 1. Denote the shear modulus and Poisson's ratio of the elastic half- space by G and , respectively. For each indentation, a Cartesian coordinate system (O-rz) is established with the origin O located at the initial contact point, the r and z axes along and perpendicu- lar to the initial surface of the half-space, respectively. A ...
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... AFM probing experiments on cells demonstrated that the classical Hertzian theory is insufficient to interpret the indentation load-depth data and unveiled the critical effect of membrane tension [19] . Besides, theoretical contact models were developed for the indentation of elastic continuum with constant surface tension [20][21][22] . It was found that the contact responses depart evidently from the classical when the indentation size is comparable with or smaller than the elastocapillary length defined by the ratio of surface tension to elastic modulus. ...
... Based on the solution of unit normal point force, the indentation problem can be mathematically formulated by [20] ...
... Such linearity still holds when the membrane tension is included, whereas the slope is modified through a non-dimensional function g 3 ( ω) (see in the Appendix), i.e. P = 2 aE * δg 3 ( ω) [20] . Dividing the indentation load by depth, the contact stiffness of flat-ended cylindrical indentation is simply given by ...
The inherent membrane tension of biological materials could vitally affect their responses to contact loading but is generally ignored in existing indentation analysis. In this paper, the authors theoretically investigate the contact stiffness of axisymmetric indentations of elastic solids covered with thin tensed membranes. When the indentation size decreases to the same order as the ratio of membrane tension to elastic modulus, the contact stiffness accounting for the effect of membrane tension becomes much higher than the prediction of conventional contact theory. An explicit expression is derived for the contact stiffness, which is universal for axisymmetric indentations using indenters of arbitrary convex profiles. On this basis, a simple method of analysis is proposed to estimate the membrane tension and elastic modulus of biological materials from the indentation load-depth data, which is successfully applied to analyze the indentation experiments of cells and lungs. This study might be helpful for the comprehensive assessment of the mechanical properties of soft biological systems. STATEMENT OF SIGNIFICANCE: This paper highlights the crucial effect of the inherent membrane tension on the indentation response of soft biomaterials, which has been generally ignored in existing analysis of experiments. For typical indentation tests on cells and organs, the contact stiffness can be twice or higher than the prediction of conventional contact model. A universal expression of the contact stiffness accounting for the membrane tension effect is derived. On this basis, a simple method of analysis is proposed to abstract the membrane tension of biomaterials from the experimentally recorded indentation load-depth data. With this method, the elasticity of soft biomaterials can be characterized more comprehensively.
... A pioneering research in this respect can be traced back to Hajji [33], in which Green's function of an elastic half-plane with surface tension was solved when measuring the elastic modulus of lung lobes with thin film coatings. Similar works considering the effect of surface tension in axisymmetric indentation [34] and adhesive contact problems were also reported in recent years [35][36][37]. Most existing contact models with surface effect were established within the framework of G-M theory, which can be classified into three types. ...
The fretting contact behavior of nanomaterials is significantly influenced by surface effect. A model of fretting contact between a nano-sized rigid cylindrical indenter and an elastic half-plane is established based on Gurtin-Murdoch (G-M) surface elasticity theory, with which the surface effects on the stress and displacement distributions and the size of stick region in the contact zone are studied. It is found that the surface effect induces an additional traction besides the external force applied by punch, which leads to smoother stress and displacement distributions. The normal surface-induced traction related to the residual surface stress is opposite to the externally applied compression, which results in a material stiffening in the contact zone so that the contact radius, normal displacement and normal stress decrease compared with classical predictions. The tangential surface-induced traction is opposite to the externally applied frictional stress, leading to reductions of the shear stress and tangential displacement in the contact zone. Furthermore, the surface effect leads to three possible states in the contact zone, including complete slip, partial slip and complete stick, instead of the solely partial slip state in classical fretting contact models. Among them, the complete stick is more beneficial for inhibiting the wear of contact devices, which can be realized by reducing the indenter size. The present research does not only help ones to better understand the physical mechanism in nano-scale fretting contact problems, but should also guide the anti-wear design in nano-electro-mechanical (NEMs) systems.
... By employing the method of integral transformation, the surface Green functions of elastic half-plane/space with surface tension were formulated [8,[25][26][27]. On this basis, twodimensional and axisymmetric indentation problems were addressed in the absence of adhesive forces [28][29][30][31][32]. The solutions were later used to elucidate the size-dependent contact deformation observed in experiments and simulations [33][34][35]. ...
... Membrane. The non-adhesive spherical, conical, and cylindrical flat indentations of an elastic substrate with tensed surface/membrane have been carefully investigated in the literature, see e.g., Fig. 2 Virtual loading-unloading process to give the final contact state in the JKR adhesive approximation in Ref. [28]. Based on the surface Green function of Hajji [8], the axisymmetric contact problems can be described by the following singular integral equation ...
... (20)) can be efficiently solved by applying a numerical method based on the Gauss-Chebyshev quadrature. Details of the numerical algorithm can be found in the previous work [28]. It has been recognized that the effects of surface/membrane tension on the contact response can be characterized by the ratio of intrinsic length s to contact radius a (denote ω = s/a) [28][29][30][31][32]. Based on the numerical results, we can derive explicit expressions for the depth-contact radius relation and the loadcontact radius relation as follows: ...
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.
... Stress field around cylindrical nanopore by various models of surface elasticity where s C is the Cauchy surface stress tensor in the current configuration. Based on a similar equation with the Piola-Kirchhoff surface stress tensor of the first kind s instead of s C , some boundary value problems have been solved, e.g., a nanosized hole of an arbitrary shape in an elastic half-plane [7], an elliptical hole in an anisotropic half-plane [9], a nanosized spheroidal cavity in an isotropic elastic medium [49,75], an arbitrarily shaped nanosized hole in elastic plane [72], two nanoscale holes of irregular shapes in an infinite elastic matrix [73], pure bending of Al nanowires [74], the indentation of an isotropic elastic half-space by a rigid axisymmetric body [39,77]. ...
Surface elasticity models including the Gurtin–Murdoch theory within the framework of continuum mechanics are analyzed and applied to the 2-D boundary value problem of a circular cylindrical nanopore being in an elastic body under remote loading. A brief overview of these models and their various applications is provided in the paper. Assuming that the surface of the nanopore is free from an external load and incorporating surface stresses, the general boundary equation is formulated in terms of the unknown complex displacement. It is shown that the boundary equation of each model is a particular case of the general one. The solution of the problem in the general case including all models leads to the singular integro-differential equation which is explicitly evaluated. The final solution is presented for the stress field by means of elementary functions. The effect of each model on the stress field arising around the nanopore due to uniaxial remote loading is numerically investigated. It is disclosed that some models display both the great deviations and qualitative differences from the Gurtin–Murdoch model.
... It would be more difficult to implement this method for poroelastic [36,60,61] and magneto-electro-elastic [62] materials. A much more difficult open problem concerns the indentation stiffness of an elastic solid with surface effects, which was analytically studied so far only in the axisymmetric cases [63][64][65] (see also Yuan and Wang [66]), where a boundary element method has been suggested to cope with non-axisymmetric contacts. It is to note that the crack problems with surface elasticity effects with the analysis of the stress singularity near the crack tip were recently considered in the literature [67,68]. ...
The incremental normal contact compliance of an elastic body beneath a frictionless indenter of arbitrary smooth shape is considered under the assumption of a simply connected domain of contact. An approximate expression for the contact stress-intensity factor in conjunction with the Griffith–Irwin formula for the variation of the elastic energy under a small perturbation of the contact contour is used to estimate the contact compliance. It is shown that the Sevostianov–Kachanov variational approach allows to derive upper and lower bounds, whose arithmetic mean is found to possess high accuracy for contacts of regular polygonal form. A number of possible generalizations are briefly discussed.
... Pinyochotiwong et al. [60] considered both in-plane surface stress and out-of-plane residual tension in the study of frictionless indentation and reported more prominent size-dependent responses than existing studies. Long and Wang [61] and Long et al. [62] used the solution given by Hajji [63] for a vertical point load on a stretched surface to numerically solve the Hertzian contact and indentation problem with the contribution of surface tension, respectively. Long et al. [62] developed a closed-form expression between indentation load and surface tension for the case that the substrate deformation is negligible. ...
... Long and Wang [61] and Long et al. [62] used the solution given by Hajji [63] for a vertical point load on a stretched surface to numerically solve the Hertzian contact and indentation problem with the contribution of surface tension, respectively. Long et al. [62] developed a closed-form expression between indentation load and surface tension for the case that the substrate deformation is negligible. This relation is similar to that for indentation of an elastic membrane. ...
... Long et al. [56,57,62] studied the effect of surface tension on indentation response by different indenters. They used a non-dimensional parameter, s, related to the surface tension, i.e., s = γ * 0 (1 − ν)/μ, in correlating indentation depth with indentation load through s/a. ...
Nanoindentation technique, which is based on Hertzian contact theory and Sneddon's solutions for the contact between a rigid indenter and an elastic half-space, has been widely investigated due to its practical importance in localized mechanical test of submicron structures. However, both the Hertzian contact theory and Sneddon's solutions do not take into account the contribution of surface stress to contact deformation. In this work, we study the effect of surface stress without the out-of-plane term on the indentation deformation of an elastic half-space by rigid, axisymmetric indenters, including flat-ended cylindrical, conical and spherical indenters. In contrast to classical theories, which are based on physically admissible condition of finite normal stress at contact edge, an alternative condition of geometrical continuity at contact edge is used to determine the contact radius. The numerical results reveal the combinational effects of the surface stress and Poisson's ratio on the load-displacement relationship for the indentation of the elastic half-space. The surface stress causes significant change in shear stress and modest variation in normal stress in the direction normal to the surface of the elastic half-space. The numerical method used in this work offers a feasible approach to study the effect of surface stress on the contact deformation of elastic substrates.
... To yield the above results, made have been used of the relation α = (1 − 2ν)/2(1 − ν). It should be noted that this result is exactly identical to Eqs. (11) of Hajji [18], which enjoys popularity in modeling nano-contact problems with surface tension [8,24,44,52,65]. ...
Surface effect have significant influence on mechanical behavior of material at nano scale. Classical solutions have limitations in predicting the micromechanics of materials due to the neglect of surface effect. This paper examines a generalized Mindlin's problem for an elastic half space with surface effect and subjected to arbitrarily orientated body forces. It employs the Gurtin and Murdoch's theory of surface mechanics to describe the surface effect. It fully considers both the residual stress and surface elastic constants. It utilizes the GKS based method to solve the boundary value problem. Analytical solutions of the full elastic field due to uniform circular loading and point loading are derived and expressed in terms of Hankel transform. These new solutions are general and can be reduced to several existing solutions for the surface loading problems and those derived based on simplified version of surface elasticity. Additionally, exact closed-form solutions for the half space with limiting boundary conditions, including traction free, rigidly fixed, inextensible and rolling surfaces can also be obtained from the present solutions. The new solutions have revealed some unique physical phenomena for incompressible solid, which are more general than existing findings. Numerical studies are performed using the solutions to investigate the elastic behavior of the half space with the presence of the surface effect. The numerical results show the substantial differences between the elastic fields predicted by present solutions and those by classical solutions without the consideration of surface effect. The new solutions for point loading can serve as Green's function to address related dislocation problems and other mixed boundary value problems in nano-scale materials, where the surface effect can play significant role on the mechanical behavior of material.
... By using the method of integral transformation, the surface displacements induced by a given pressure or a concentrated force on the surface of elastic half-space/plane with surface stress were analytically derived (Hajji, 1978;He and Lim, 2006;Wang and Feng, 2007;Koguchi, 2008;Zhou and Gao, 2013;Gao et al., 2013). Using the surface Green's functions considering surface stress, Long et al. (2017Long et al. ( , 2018 addressed the plane strain and axisymmetric indentation problems by solving the corresponding singular integral equations. Hayashi et al. (2013) analyzed the contact problems of anisotropic elastic materials with surface stress and surface elasticity. ...
... The spherical indentation problem of an elastic half-space with surface stress under pure normal load has been well addressed (Long et al., 2017;Kim and Gouldstone, 2008). The influence of surface stress can be reflected by a dimensionless characteristic scale parameter β defined by the ratio of the material intrinsic length to the contact radius. ...
In this paper, we investigate the effects of surface stress on the spherical contact problem under combined normal and lateral loads. The boundary element method based on fast Fourier transform and conjugate gradient algorithm is employed to perform the contact analysis with partial slip. It is shown that as the size of contact reduces to the characteristic scale that surface stress dominates, the contact behavior will significantly deviate from the classical model. Under the same loading condition, the tangential traction within contact region tends to distribute more evenly and the width of slip zone would be relatively smaller due to the presence of surface stress. In addition, it is found that the variations of the total tangential force and the lateral contact stiffness with respect to the lateral displacement depend strongly on the contact scale. This work provides a theoretical basis for the study of contact and friction of some soft materials and biological systems.
... for the flat coating flakes idealized as circular disk [26], and ...
Paclitaxel coated balloon catheters (PCB) were developed as a polymer-free non-implantable alternative to drug eluting stents, delivering similar drug payloads in a matter of minutes. While PCB have shown efficacy in treating peripheral arterial disease in certain patient groups, restenosis rates remain high and there is no class effect. To help further optimize these devices, we developed a scanning electron microscopy (SEM) imaging technique and computational modeling approach that provide insights into the coating micromorphology dependence of in vivo drug transfer and retention. PCBs coated with amorphous/flaky or microneedle coatings were inflated for 60sec in porcine femoral arteries. Animals were euthanized at 0.5, 24 and 72 h and treated arteries processed for SEM to image endoluminal coating distribution followed by paclitaxel quantification by mass spectrometry (MS). Endoluminal surfaces exhibited sparse coating patches at 0.5 h, predominantly protruding (13.71 vs 0.59%, P < 0.001), with similar micro-morphologies to nominal PCB surfaces. Microneedle coating covered a 1.5-fold endoluminal area (16.1 vs 10.7%, P = 0.0035) owing to higher proximal and distal delivery, and achieved 1.5-fold tissue concentrations by MS (1933 vs 1298 μg/g, P = 0.1745) compared to amorphous/flaky coating. Acute longitudinal coating distribution tracked computationally predicted microindentation pressure gradients (r = 0.9, P < 0.001), with superior transfer of the microneedle coatings attributed to their amplification of angioplasty contact pressures. By 24 h, paclitaxel concentration and coated tissue areas both declined by >93% even as nonprotruding coating levels were stable between 0.5 and 72 h, and 2.7-fold higher for microneedle vs flaky coating (0.64 vs 0.24%, P = 0.0195). Tissue retained paclitaxel concentrations at 24–72 h trended 1.7-fold higher post treatment with microneedle coating compared to the amorphous/flaky coating (69.9 vs 39.9 μg/g P = 0.066). Thus, balloon based drug delivery is critically dependent on coating micromorphologies, with superior performance exhibited by micromorphologies that amplify angioplasty pressures.
... The constant membrane tension in Hajji (1978) and Kim and Gouldstone (2008) is corresponding to the residual surface tension, and the superficial layer modeled in Argatov and Sabina (2012) and Argatov and Mishuris (2016) can be considered as a solid surface with surface elasticity. Recently, problems of surface loadings and contacts have been widely studied based on the surface elasticity theory (He and Lim, 2006;Wang and Feng, 2007;Zhou and Gao, 2013;Long et al., 2017;. It is found that the deformation behavior would be distinctly different from classical models when the size of loading is comparable or even smaller than a critical length, usually at nano/microscale. ...
... When the energy difference w = γ 1 +γ 2 -γ 12 (called work of adhesion) is appreciable and the effect of surface energy outside the contact can be neglected, adhesive contact model with specific work of adhesion would be appropriate (Johnson et al., 1971). On the other hand, if the energy difference equals to zero (w = 0) but the surface energy of one contact body is much larger than the other (i.e., γ 1 >> γ 2 and γ 1 ≈ γ 12 ), the contact problem can be addressed in the framework of surface elasticity theory (He and Lim, 2006;Wang and Feng, 2007;Zhou and Gao, 2013;Long et al., 2017;. In this work, we consider the latter circumstance with constant surface tension or, equivalently, the non-adhesive contact of solids with tensed surface membrane. ...
... The variations of indentation load and indentation depth with respect to the ratio s/a are plotted in Figures 5A,B, respectively. It is seen that our numerical results obtained from BEM calculations coincide well with existing semi-analytical solutions (Kim and Gouldstone, 2008;Long et al., 2017). The size dependent contact behavior is reproduced. ...
This work considers the non-adhesive frictionless contact problem of soft materials with surface being tensed by equi-biaxial tension. The boundary element method (BEM) 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, our numerical simulations are validated having great accuracy. Moreover, the developed BEM algorithm is applied on the calculations of elastic 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 essential contact behaviors resulted from the presence of membrane tension are presented and discussed.
