Classical Hertz Contact Research Articles

A comprehensive isogeometric analysis of frictional Hertz contact problem

The development of computational methods has advanced the solving of complex contact problems. Despite this, reproducing the analytical solution of a classical Hertz contact problem using computational methods remains challenging. Finite element method (FEM) has gained popularity, with many studies solving frictionless Hertz contact problems. However, the more general Hertz contact problem with finite friction lacks an analytical solution and is more complex. FEM was first used for this in the 1970s. Recently, isogeometric analysis (IGA) has shown enhanced efficiency than FEM. However, comprehensive studies on the frictional Hertz contact problem using IGA are lacking. The present study aims to fill this gap by examining a benchmark example and exploring contact simulation parameters using the developed in-house MATLAB® code.

Nonlinear frictional contact behavior of porous polymer hydrogels for soft robot

Focusing on the oblique frictional contact problem of hydrogel for soft robot, a numerical contact model is developed to analyze the nonlinear behaviors including local contact large deformation and friction effect of hydrogel soft material during oblique contact. Based on the constitutive relation of hyperelastic material, the updated free energy function of hydrogel is derived. The contact algorithm is given to compute the contact responses of both collinear contact example and oblique contact example. The applicability of classical Hertz contact theory is discussed. The influence of coefficient of friction on the stress distribution of contact zone and the contact states of contact surface are also investigated. The numerical results show that the material nonlinearity and the geometric nonlinearity (i.e. large deformation) of hydrogel lead to the invalidation of classical Hertz contact theory. For the oblique contact, the internal stress grads of hydrogel will be redistributed as the coefficient of friction increases. Meanwhile, the maximum stress position moves from underside to the contact surface. Two high stress zones occur inside and on the contact surface at the same time, respectively. Besides, when the coefficient of friction is smaller (i.e. <i>μ</i> < 0.05), for the collinear contact all contact points on the contact surface are in critical state from static friction to dynamic friction. But for the oblique contact, some contact points are always in stable static friction state.

The Hertz contact problem in couple-stress elasticity

The classical three-dimensional Hertz contact problem is re-examined in the present work within the context of couple-stress elasticity. This theory introduces characteristic material length scales that emerge from the underlying microstructure and has proved to be very effective yet rather simple for modeling complex microstructured materials. An exact solution of the axisymmetric contact problem is obtained through Hankel integral transforms and singular integral equations. The goal is to examine to what extend this gradient theory captures the experimentally observed indentation response and the related size effects in materials with microstructure. Furthermore, the present work extends the classical Hertz contact solution in the framework of a generalized continuum theory providing thus a useful theoretical background for the interpretation of spherical indentation tests of microstructured materials. The results obtained in the present work can be used in order to model nano/micro indentation experiments of several materials, such as polymers, ceramics, composites, cellular materials, foams, masonry, and bone tissues, of which their macroscopic response is influenced by the microstructural characteristic lengths.

A numerical model for calculation of the restitution coefficient of elastic-perfectly plastic and adhesive bodies with rough surfaces

A contact mechanics model based on Boundary Element Method (BEM) is used to simulate the contact of rough spheres having elastic-perfectly plastic adhesive behaviour. The model has been verified for smooth surfaces with the classical Hertz contact behaviour in elastic regime, JKR theory in the adhesive regime and Thornton and Ning's analytical solution in the elasto-plastic adhesive regime. In this paper, the effect of yield stress, interface energy and surface roughness on the Coefficient of Restitution (COR) is investigated and reported. The results show that surface roughness dramatically affects the COR and higher roughness values lead to lower CORs in general. In addition, it is found that in the presence of surface roughness, the COR varies based on different maximum indentation depths in 3 different stages; the asperity dominant, bulk elastic dominant and bulk plastic dominant. Interestingly, it was shown that there is a critical indentation depth beyond which the effect of surface roughness will disappear and rough surfaces act like smooth ones. This critical indentation depth is proportional to the Root Mean Square roughness (Rq) value of the surface. Numerical results suggest that the yield stress influences the COR and higher yields stress results in higher COR for both rough and smooth surfaces. Furthermore, the effect of interface energy on the COR for rough surfaces is minimal and only significant at low indentations.

Heat partition in dry orthogonal cutting of unidirectional CFRP composite laminates

Carbon fiber reinforced polymer (CFRP) components can be generally prepared near-net-shape, however, they still need machining after manufacturing to meet the geometrical accuracy with excellent surface quality required for assembly. Due to the cutting temperature is prone to exceed the glass-transition temperature of the resin, the irreversible chemical and mechanical degradation are therefore difficult to avoid. It becomes a crucial challenge to eliminate the particular thermal effect on the composite machining process. In comparison with the temperature measurement, the analytical model not only can reveal the physical essence of thermal effect, but also can predict the temperature field distribution to provide the reasonable cutting parameters. In particular, a key parameter for calculating the cutting temperatures is the heat partition ratio. However, the research work on this issue was rarely found. In this paper, a fiber orientation-based analytical model was developed to predict the heat partition ratio based on the classical Hertz contact theory. The finite element model was also built with the validation of the experimental measurement from the thermal imaging tests. The results suggest the heat partition ratio is mainly determined by the cutting parameters. Moreover, the fiber orientations have a remarkable impact on this ratio. Due to the heat partition ratio is considerably larger in a CFRP workpiece than the cutting tool, more heat energy was transferred to the CFRP during machining. Therefore, a small depth of cut leads to a reduced tendency for thermal effect on the CFRP composites.

Contact Behaviour of Composite Laminate under Quasi-Static Indentation Load

A viable solution to the ever-demanding weight-saving target in aerospace industry is the replacement of conventional engineering alloys with composite materials in primary structures. A major concern to the effective use of composite laminates is the substantial reduction in the compressive strength when the contact force has exceeded the delamination threshold load (DTL). This paper focuses on the study of the contact behavior of composite laminates under quasi-static indentation (QSI) forces. The effect of damage initiation and growth on contact behavior has been investigated via detailed assessment of the relation between the indentation force and the dent depth. Different phases corresponding to undamaged, local damage, global damage, and final failure of the laminate have been identified. A modification to the classical Hertz contact law has been proposed to account for the matrix material resistance to plastic deformation.

Application of Contact Theory to Evaluation of Elastic Properties of Low Consolidated Porous Media

A new approach to the calculation of the elastic bulk modulus of low consolidated porous media is developed based on a physical consolidation model of rocks and the classical Hertz contact theory. The derived analytical relationships for the elastic bulk modulus, which take into account some micro-structural characteristics of packing, are compared with theoretical predictions from various micromechanics theories, Hashin-Shtrikman strict bounds as well as with experimental results available for low consolidated granular materials. The latter comparison demonstrates a good agreement.

Nanoindentation of polydimethylsiloxane elastomers: Effect of crosslinking, work of adhesion, and fluid environment on elastic modulus

With the potential to map mechanical properties of heterogeneous materials on a micrometer scale, there is growing interest in nanoindentation as a materials characterization technique. However, nanoindentation has been developed primarily for characterization of hard, elasto-plastic materials, and the technique has not been validated for very soft materials with moduli less than 5 MPa. The current study attempted to use nanoindentation to characterize the elastic moduli of soft, elastomeric polydimethylsiloxane (PDMS) samples (with different degrees of crosslinking) and determine the effects of adhesion on these measurements using adhesion contact mechanics models. Results indicate that nanoindentation was able to differentiate between elastic moduli on the order of hundreds of kilo-Pascals. Moreover, calculations using the classical Hertz contact model for dry and aqueous environment gave higher elastic modulus values when compared to those obtained from unconfined compression testing. These data seem to suggest that consideration of the adhesion energy at the tip-sample interface is a significantly important parameter and needs to be taken into account for consistent elastic modulus determination of soft materials by nanoindentation.

Finite element analysis of Hertz contact problem with friction

A new finite element method of solution for the design and analysis of frictional contact problems is presented. The nonlinear problem is solved by imposing geometric constraints on the pseudo equilibrium configuration, defined as a configuration at which the compatibility conditions are violated. The compatibility conditions are satisfied through a load step applied to the pseudo equilibrium configuration. The contact conditions of sticking, slipping, rolling or tension release is determined from the relative magnitudes of the normal and tangential global nodal forces. The classical Hertz contact problem is analyzed with and without friction between the contacting bodies and good agreement is found between the analytical and the numerical solutions.

A Mathematical Model of Produce Damage Mechanisms

ABSTRACT A mathematical model, enabling prediction of produce bruise size and depth of fruit deformation is developed. The maximal pressure at the center of the contact circle is also computed. The model assumes spherical fruit shape, and considers configurations of two fruits in contact, a fruit in contact with a rigid plane and a fruit in a spherical cup, such as in tray packs. The material properties of fruits are expressed in terms of a non-linear relaxation modulus while comparing results to the classical Hertz contact problem of perfectly elastic and linear viscoelastic spheres. Equations are developed for constant load forces, e.g., dead weight, continuous sinusoidally varying force, a transient force pulse and combinations thereof. These enable comparison of bruise size, caused by static and dynamic loads of the same magnitude.

Some useful results in the classical Hertz contact problem

A resumé is given of the results available in the literature for the stresses resulting from frictionless loading by Hertz pressures. The problem is posed formally in tensor notation and new convenient forms are given for the complete stress field, in the general elliptical-patch case.