Rubber Block Research Articles

Compression of Bonded Annular Rubber Blocks

The load-deflection relation and stress distribution of bonded annular rubber blocks are first derived by an approximate-linear approach. The solution approaches the exact relation asymptotically as the block thickness reduces. A general formula for the nonlinear load-deflection relation is then presented, based on a variational method, in which no specific form of the strain-energy function is required. An explicit approximate analytical solution is obtained by applying the general formula to a neo-Hookean material. The approximate solutions are compared with the finite-element analysis. The relationships among the stiffness, the shape factor s , and the diameter ratio β are explored in detail and some interesting phenomena are observed. For example, a centrally placed tiny hole in a very thin circular disk can cause a sharp drop in its stiffness.

Neural network modelling of a compliant device to enhance and simplify compliant force control

A method is offered that uses an artificial neural network (ANN) to simplify the problem of mapping a robot tool position in space when using compliant force control. For this case, the compliant device (CD) is a moulded rubber block situated between the manipulator end flange and the tool. A force sensor is mounted between the tool and the CD. It is necessary to be able to relate the recorded contact forces and torques (hereafter referred to as “forces”) to the changes in translations and orientations (hereafter referred to as “positions”) of the tool relative to the manipulator. Usually, a complex Newtonian model would be required to achieve this. For this paper, to overcome the difficulties of accurately modelling the non-linear and highly coupled characteristics of the CD, back propagating ANNs have been trained to relate the forces to positions for real data sets. The finished ANN is to be used as part of a computer model of the overall manipulator system. This paper assesses the ability of the ANN to model the CD. In particular, three issues are addressed: first, the comparison of the ANN with a conventional spring type model; second, the verification of the ANN with true data sets; third, the need to filter the ANN output to avoid problems from outliers.

Compression of Rubber Blocks

Abstract Many features of the compression of rubber blocks can be deduced by means of a simple analytical treatment which, although approximate, appears to give reasonably satisfactory predictions for stiffness and interfacial stresses. The theory has been redeveloped and refined many times; some principal contributions are noted in References 1–14. While the analysis has been frequently repeated, it is interesting to note that what is probably the first, and in some ways the most comprehensive, theoretical discussion, did not refer specifically to rubber blocks; it dealt with the elastic properties of the periodontal membrane in connection with the tightness of teeth. Apart from an extensive review in connection with the use of rubber compression springs for bridge bearings, general results for small compressions of rubber sheets and blocks have not been collected together previously in one easily-available place, as far as I am aware. A review is therefore given here, with some remarks on its limitations. Attention is drawn to the surprisingly large effect of relatively small departures from absolute incompressibity of rubber, and of the use of frictional restraints instead of perfect bonding at loaded surfaces.

Adhesion between a rubber and a grafter surface

We discuss the role of «connectors» between two rubber blocks or between a rubber and a solid. Early descriptions of these systems were restricted to very dilute connectors, working independently. In the present work, we investigate theoretically the role and the limits of the surface concentration of connectors.

Pull-out of short rods and fibres

An experimental study has been carried out of the mechanics of pull-out of short nylon rods embedded in a rubber block, considered as a model of fibres embedded in an elastic resin. Two stages of pull-out were observed. First, rubber became detached from the base of the rod creating a large internal cavity. This process was apparently initiated by internal rupture of rubber under the rod, and occurred when the local triaxial tension reached a value similar to the tensile (Young's) modulus of the rubber. After cavitation, a cylindrical debond propagated up the rod, starting at the embedded end and ending in complete pull-out. Pull-out forces were found to be consistent with a simple fracture mechanics treatment based on the elastic compliance of partially debonded specimens, both measured and calculated by finite element analysis. The fracture energy for debonding is deduced to be about 300 J m−2. The effects of varying rod diameter and depth of embedment are shown to be in satisfactory agreement with theory.

Simulation of the frictional stick-slip instability

A lattice solid model capable of simulating rock friction, fracture and the associated seismic wave radiation is developed in order to study the origin of the stick-slip instability that is responsible for earthquakes. The model consists of a lattice of interacting particles. In order to study the effect of surface roughness on the frictional behavior of elastic blocks being rubbed past one another, the simplest possible particle interactions were specified corresponding to radially dependent elastic-brittle bonds. The model material can therefore be considered as round elastic grains with negligible friction between their surfaces. Although breaking of the bonds can occur, fracturing energy is not considered. Stick-slip behavior is observed in a numerical experiment involving 2D blocks with rough surfaces being rubbed past one another at a constant rate. Slip is initiated when two interlocking asperities push past one another exciting a slip pulse. The pulse fronts propagate with speeds ranging from the Rayleigh wave speed up to a value between the shear and compressional wave speeds in agreement with field observations and theoretical analyses of mode-II rupture. Slip rates are comparable to seismic rates in the initial part of one slip pulse whose front propagates at the Rayleigh wave speed. However, the slip rate is an order of magnitude higher in the main part of pulses, possibly because of the simplified model description that neglected intrinsic friction and the high rates at which the blocks were driven, or alternatively, uncertainty in slip rates obtained through the inversion of seismograms. Particle trajectories during slip have motions normal to the fault, indicating that the fault surfaces jump apart during the passage of the slip pulse. Normal motion is expected as the asperities on the two surfaces ride over one another. The form of the particle trajectories is similar to those observed in stick-slip experiments involving foam rubber blocks (Bruneet al., 1993). Additional work is required to determine whether the slip pulses relate to the interface waves proposed by Brune and co-workers to explain the heat-flow paradox and whether they are capable of inducing a significant local reduction in the normal stress. It is hoped that the progressive development of the lattice solid model will lead to realistic simulations of earthquake dynamics and ultimately, provide clues as to whether or not earthquakes are predictable.

Frictional heat generation and seismic radiation in a foam rubber model of earthquakes

Results from a study of stick-slip particle motion at the interface between two stressed foam rubber blocks indicate that normal vibrations and interface separation are an important part of the stick-slip process in foam rubber. The dimension of the dynamic slip pulse is small compared to the dimension of the model (approximately 10 cm vs. 200 cm) consistent with the abrupt-locking slip pulse model ofBrune (1970, 1976), andHeaton (1990). A comparison of frictional heat generation between stable-sliding and stick-slip foam rubber models indicates a linear relation between the temperature increase on the fault surface (for a given distance of slip) and the driving shear force for the stable-sliding model, while for the stick-slip model there is essentially no variation in frictional heat generation with an increase in shear stress. We performed experiments to investigate the ratio of normal motion to shear motion at different levels of normal stress in the stick-slip foam rubber model. Preliminary result indicate that the normal component of the particle motion increases more rapidly with increasing normal stress than the shear component. The phenomenon of interface separation and normal vibrations may thus explain some of the most frustrating problems in earthquake mechanics, e.g., the heat flow paradox, the long-term weakness of major active faults, and anomalousP-wave radiation.

Strain energy release rates for internal cracks in rubber blocks

Strain energy release rates have been calculated by Finite Element Analysis (FEA) for growth of a small crack in a cylindrical rubber block under tension. The crack was assumed to lie in the center of the block in a plane perpendicular to its axis. The rubber was taken to be virtually incompressible in bulk and non-linearly elastic. Only small changes were found in the stress-concentrating effect of the crack for imposed tensile strains of up to 150 per cent. Secondly, the rubber cylinder was assumed to be bonded between two parallel flat rigid plates. The stress-concentrating effect of the crack was found to increase markedly as the thickness of the bonded rubber block was reduced, but again it was only slightly dependent on the applied tensile strain.

Expansion of a cavity in a rubber block under unequal stresses

Elastic expansion of a small spherical void in the interior of a rubber block has been investigated by finite element analysis (FEA). The block was subjected to a far-field tensile or compressive stress while an internal pressure was applied to the cavity. The rubber was assumed to be virtually incompressible in volume and neo-Hookean in elastic behavior. Critical stress states were determined at which the void would become indefinitely large-a form of elastic instability. An applied tensile stress was found to lower the critical inflation pressure for instability, in agreement with experimental observations [1] whereas a compressive stress increased it. The critical mean stress was lowest for an isotropic stress system (when the applied tension was zero). It was higher when either compressive or tensile stresses were applied, i.e. under non-isotropic stresses, in agreement with the analysis of Hou and Abeyaratne [2]. However, the present results show a considerably smaller effect of tensile stress than predicted by Hou and Abeyaratne.

On the Incorporation of Frictional Effects in the Tire/Ground Contact Area

Abstract This paper introduces a three-dimensional finite element model of a tire with a detailed tread block structure to more accurately describe the stress distribution in the tire/ground contact area. Because of the importance of properly modeling the contact, several preliminary studies are detailed. First, the frictional contact between a rib and the ground is presented using a plane strain model. In particular, the edge effects are studied. Second, a combined experimental and numerical method for determining the coefficient of friction between a rubber block and a glass plate is described. A friction law is developed which includes the well-known dependence of the coefficient of friction on the normal stress. Finally, the three-dimensional tire model is discussed and results are given.

The Effect of Compressibility on the Stress Distributions in Thin Elastomeric Blocks and Annular Bushings

The effect of the bulk compressibility of elastomers on the response of rubber blocks and bushings bonded to platens is in vestigated. Closed-form solutions for the stresses and deformations within the elastomer are presented for the case of rigid adherends. It is shown that even with relatively small shape factors, the compressibility can significantly affect the apparent stiffness. A finite element analysis shows that the closed-form solution accurately predicts the stress distribution for rigid adherends, but also reveals that platen deformations in realistic systems may significantly alter the distributions.

Analysis of the Contact Deformation of Tread Blocks

Abstract The deformation behavior of a tire in contact with the roadway is complicated, in particular, under the traction and braking conditions. A tread rubber block in contact with the road undergoes compression and shearing forces. These forces may cause the loss of contact at the edges of the block. Theoretical analysis based on the energy method is presented on the contact deformation of a tread rubber block subjected to compressive and shearing forces. Experimental work and numerical calculation by means of the finite element method are conducted to verify the predicted results. Good agreement is obtained among these analytical, numerical, and experimental results.

The combined compression and shear of a rectangular rubber block

For long rectangular rubber blocks with metal plates bonded to their upper and lower surfaces, approximate force-deflection relations are obtained for the problem of combined compression (or tension) and shear. The results obtained generalise that of Klingbeil and Shield for the case of pure compression alone, as well as incorporating the well known simple shear result. However, unlike the pure compression solution, horizontal and vertical force resultants on the free surfaces cannot both be set to zero, which is a well known characteristic of simple shearing. Here the strategy is adopted of equating to zero only the horizontal force resultants on the free surfaces so that the pure compression approximation of Klingbeil and Shield emerges in the event of zero shear. The force-deflection relations so obtained are illustrated graphically.

Numerical Prediction of the Aramid-Cord Load in the Goodrich Block Fatigue Test

Abstract In the Goodrich Block Fatigue test, tire cords are tested for their (compression) fatigue behavior. The test specimen, a rubber block with a cord in its center, is cyclically extended and compressed. After a certain running time the retained cord strength is measured. The local cord load cannot be measured directly. Therefore, a calculation procedure is provided to predict the local stress and strain along the cord due to clamping, elongation and compression of the specimen. To this end, a few—for aramid only four—basic linear finite-element computations are needed. Calculation results show that clamping causes a complex load distribution along the cord. This load is not uniform but varies along the axis, and the center of the cord is already in compression. By subsequently elongating the specimen, this compressive region decreases while compression of the specimen leads to an increase. From the standard aramid test conditions, it is apparent that the cord remains compressed in the center. The cord part beginning at the end always remains under tension. Furthermore, there is a small cord section at some distance from the center which is cyclically stretched and compressed during the dynamic load. Experiments show that a fatigued aramid cord mostly breaks in this area, where the cord is subjected to both tension and compression. By combining calculation results with experimentally determined breaking positions of cords, accurate information can be obtained about the loading conditions which are most severe. These findings lead to a better understanding of the failure mechanism of tire cords in the Goodrich Block Fatigue test.

Bulging of Rubber through Apertures

Abstract 1. When rubber bulges through an aperture, the bulge height varies linearly with applied pressure. It also varies linearly with radius or width of the aperture when the dimensions of the rubber block are much larger than the aperture, in good agreement with linear elasticity theory. When the block is not much larger than the radius or width of the aperture, a simple empirical factor provides a satisfactory correction. 2. FEM results were in good agreement with experiment in all cases and confirmed the empirical correction for relatively large apertures. 3. Bulge profiles were calculated to be elliptical in all cases. 4. Results for an elastic solid are wholly consistent with Roscoe's results for a viscous liquid. 5. Changing slip conditions at the interface had a surprisingly small effect. The exact theory, Equations (5) and (7), is based upon a nonslip condition. When a fully lubricated condition was employed instead, so that free sliding was allowed at the interface, the maximum bulge heights predicted by FEM were only about 5% higher than before for a circular aperture and about 12% higher for a slit. Experimental observations of bulge heights with dry and lubricated interfaces were in good agreement with these predictions. 6. Bulge heights through small apertures provide a simple means of measuring pressure at contact surfaces.

Friction in a Cone Clutch and in a Friction Drive With Wedge-Type Bodies

Mathematical relationships obtained on the basis of the friction theory in a wedge pair are widely used in technical calculations. With the help of them the limit circumferential force in cone clutches and friction drives with wedge-type bodies (wedge roller transmissions, variable-speed friction block belt drives, V-belt transmissions and etc.) are determined. This may seem to be a simple problem which could be solved without difficulties, however the classical equations used can not be derived in stringent terms and was not confirmed experimentally. The aim of this paper is to clarify this problem.

Fresh autogeneic, frozen allogeneic, and decalcified allogeneic bone grafts in dogs

In fully-grown mongrel dogs, diaphyseal ulnar defects 25 mm long were stabilised by screws and plates, and were temporarily filled with silicone rubber blocks. After eight weeks the block was replaced either by fresh autogeneic cancellous bone, allogeneic deep-frozen cancellous bone, allogeneic decalcified bone matrix, or bone matrix gelatin. After 24 weeks the implants were evaluated by radiography, histology, and measurements of new bone volume, using computer-assisted density registration on microradiographs. Only the autogeneic bone grafts led to healing in all instances. Bone regeneration in the other groups was not significantly better than in the sham group in which no graft was employed. Decalcified bone matrix proved ineffective.

Pull-out and fragmentation in model fibre composites

An experimental study has been carried out of debonding and fibre rupture in model composites. A single glass rod or fibre was embedded in the centre of a long transparent silicone rubber block. Strains in the rubber in close proximity to the rod or fibre were measured as the specimen was slowly stretched. Pull-out forces, strain distributions, and debonded lengths are compared with the predictions of a simple theory based on a fracture energy criterion for debonding, and taking into account friction at the debonded interface. Experiments were carried out with rods of different diameter, rubber blocks of varied cross-section, and with two levels of adhesion. By extrapolating the debonded length to zero, values of the debonding force in the absence of friction were obtained. They were in accord with fracture energies of about 50 J/m2 for weak bonding and about 200 J/m2 for strong bonding. Fibre fragmentation lengths were measured also. They were in reasonable agreement with the inferred fracture energies and the measured frictional properties of silicone rubber sliding on glass. In a separate study, it was found that the frictional stress between cast silicone rubber and glass was approximately constant, about 0.1 MPa, rather than proportional to pressure, for pressures exceeding about 0.02 MPa. This feature is attributed to a particularly smooth interface between the two materials.

Low-order mixed method finite elements in non-linear elasticity

A mixed method finite-element formulation for non-linear plane-strain elastic deformations is described. Incompressibility is built in explicitly. All fields have piecewise linear approximants over triangles except pressure, which is taken to be constant over matched pairs of triangles. The code is used in an investigation of the tilting mode of deformation of bonded rubber blocks. In particular the variation of tilting moment with aspect ratio is focused on for a range of tilting angles.

Morphologies of poly(dimethylsiloxane)-nylon-6 diblock copolymers and blends

The morphologies of poly(dimethylsiloxane)-nylon-6 diblock copolymers and binary/ternary blends with PDMS and nylon-6 homopolymers have been revealed by transmission electron microscopy (TEM). The spherulitic morphologies of materials cast from solution in 2,2,2 trifluoroethanol have been compared to the spherical microphase-separated structures obtained by annealing above the melting point of nylon-6. Either the PDMS rubber block length or the nature of the casting solvent can be employed to shift control of the morphology between mechanisms of microphase separation and crystallization. As a result, ‘path-dependent’ morphologies have been observed for PDMS-nylon-6 diblock copolymers and blends and these are explained in a unified manner via a conceptual phase diagram.