Energy release rates in rubber during dynamic crack propagation

The theoretical understanding of the fracture mechanics of rubber is not as well developed as for other engineering materials, such as metals. The present study is intended to further the understanding of the dissipative processes that take place in rubber in the vicinity of a propagating crack tip. This dissipation contributes significantly to the total fracture toughness of the rubber and is therefore of great interest from a fracture mechanics point of view. To study this, a computational framework for analysing high-speed crack growth in a biaxially stretched rubber under plane stress is therefore formulated. The main purpose is to investigate the energy release rates required for crack propagation under different modes of biaxial stretching. The results show, that inertia comes into play when the crack speed exceeds about 50 m/s. The total work of fracture by far exceeds the surface energy consumed at the very crack tip, and the difference must be attributed to dissipative damage processes in the vicinity of the crack tip. The size of this damage/dissipation zone is expected to be a few millimetres.     

Investigation of stress-strain behavior of single walled carbon nanotube/rubber composites by a multi-scale finite element method

The excellent properties of carbon nanotubes have generated technological interests in the development of nanotube/rubber composites. This paper describes a finite element formulation that is appropriate for the numerical prediction of the mechanical behavior of rubber-like materials which are reinforced with single walled carbon nanotubes. The considered composite material consists of continuous aligned single walled carbon nanotubes which are uniformly distributed within the rubber material. It is assumed that the carbon nanotubes are imperfectly bonded with the matrix. Based on the micromechanical theory, the mechanical behavior of the composite may be predicted by utilizing a representative volume element. Within the representative volume element, the reinforcement is modeled according to its atomistic microstructure. Therefore, non-linear spring-based line elements are employed to simulate the discrete geometrical structure and behavior of the single-walled carbon nanotube. On the other hand, the matrix is modeled as a continuum medium by utilizing solid elements. In order to describe its behavior an appropriate constitutive material model is adopted. Finally, the interfacial region is simulated via the use of special joint elements of variable stiffness which interconnect the two materials in a discrete manner. Using the proposed multi-scale model, the stress-strain behavior for various values of reinforcement volume fraction and interfacial stiffness is extracted. The influence of the single walled carbon nanotube addition within the rubber is clearly illustrated and discussed.     

High strain rate response of rubber membranes

This paper investigates the propagation of axisymmetric waves of finite deformation in polyisoprene rubber membranes subjected to high speed impact. High speed photography is used to monitor the motion and to determine the evolution of stretch and particle velocity in membranes at impact speeds of up to 160 m/s, producing a maximum stretch >8. A constitutive model is developed through a semi-inverse method correlating experimental results with simulations. The potential for formation of wrinkles is also addressed.     

Experimental and theoretical study on large amplitude vibrations of clamped rubber plates

In this paper, the large amplitude forced vibrations of thin rectangular plates made of different types of rubbers are investigated both experimentally and theoretically. The excitation is provided by a concentrated transversal harmonic load. Clamped boundary conditions at the edges are considered, while rotary inertia, geometric imperfections and shear deformation are neglected since they are negligible for the studied cases. The von Kármán nonlinear strain-displacement relationships are used in the theoretical study; the viscoelastic behaviour of the material is modelled using the Kelvin-Voigt model, which introduces nonlinear damping. An equivalent viscous damping model has also been created for comparison. In-plane pre-loads applied during the assembly of the plate to the frame are taken into account. In the experimental study, two rubber plates with different material and thicknesses have been considered; a silicone plate and a neoprene plate. The plates have been fixed to a heavy rectangular metal frame with an initial stretching. The large amplitude vibrations of the plates in the spectral neighbourhood of the first resonance have been measured at various harmonic force levels. A laser Doppler vibrometer has been used to measure the plate response. Maximum vibration amplitude larger than three times the thickness of the plate has been achieved, corresponding to a hardening type nonlinear response. Experimental frequency-response curves have been very satisfactorily compared to numerical results. Results show that the identified retardation time increases when the excitation level is increased, similar to the equivalent viscous damping but to a lesser extent due to its nonlinear nature. The nonlinearity introduced by the Kelvin-Voigt viscoelasticity model is found to be not sufficient to capture the dissipation present in the rubber plates during large amplitude vibrations.     

Modelling of frequency- and amplitude-dependent material properties of filler-reinforced rubber

Filler-reinforced elastomers are extremely complicated materials with pronounced deformation and temperature history-dependent material properties. In the current paper, the dynamic material behaviour is investigated and modelled. To this end, a carbon black-filled rubber compound is loaded with harmonic deformations under different frequencies and amplitudes and the stationary stress response is evaluated in terms of the storage and the loss modulus or, equivalently, in terms of the dynamic modulus and the loss angle. In this essay, detailed experimental investigations of the dynamic material properties of carbon black-filled elastomers are provided and a new three-dimensional constitutive approach of finite nonlinear viscoelasticity to represent the observed material behaviour is developed.     

Stress response of a rubber block with frictional contact under axial loading

Rubber blocks (or springs) are structural components widely used in many applications. Design characteristics of a rubber block under axial loading are an apparent modulus (or compression modulus) and normal and shear stresses on contact surfaces. These are affected by a contact condition of the rubber block in contact with two rigid plates and the shape of the block. The problem of a rubber block bonded to two rigid plates has been solved using various approaches. In contrast, for a rubber block whose one surface is bonded to a rigid plate and the other surface in contact with a frictional surface, there is little work in spite of practical applications. For this contact condition, approximate solutions for rectangular blocks in plane strain and for axi-symmetric discs are derived under the assumption of Coulomb frictional contact. The problem is treated as an extension of the problem of an incompressible rubber block bonded to two rigid plates with one of the plates having a frictional interaction with the rubber block. In the linear range of deformation, finite element analysis and experimental results for rubber blocks with shape factors ranging from 1 to 6 are compared for the validation of analytic results. It is found that friction coefficients play important roles in the design characteristics of the rubber block.     

The deformation and fracture of balloons

Inflation of balloons provides a straightforward way of achieving large biaxial deformations. Previous studies have shown that when a balloon bursts, crack propagation occurs at very high speed – much higher than would be expected from the low strain modulus and elastic wave velocity of the rubber. The present paper is concerned with studies of the deformation and fracture of cylindrical balloons. On inflation, the deformations of such a balloon pass through an unstable region but subsequently increase monotonically with pressure. In this relatively high pressure region, the ratio of the longitudinal and circumferential extension ratios is broadly in accord with expectations from high-strain elasticity theory when the ratio of the corresponding stresses is taken into account. On bursting, crack speeds up to around 300 m/s in this region. It is shown that these speeds are in accord with large increase in incremental moduli for the highly-strained rubber. Marked changes in crack tip profile observed at very high crack speeds are consistent with control of the rate of growth by inertia rather than by the viscoelastic properties of the rubber (as is believed to be the case at lower speeds). Consistent with this, various elastomers having different glass transition temperatures show similar crack growth behaviour in the very high speed region.     

Asymptotic mechanical fields at the tip of a mode I crack in rubber-like solids

Asymptotic analyses of the mechanical fields in front of stationary and propagating cracks facilitate the understanding of the mechanical and physical state in front of crack tips, and they enable prediction of crack growth and failure. Furthermore, efficient modelling of arbitrary crack growth by use of XFEM (extended finite element method) requires accurate knowledge of the asymptotic crack tip fields. In the present work, we perform an asymptotic analysis of the mechanical fields in the vicinity of a propagating mode I crack in rubber. Plane deformation is assumed, and the material model is based on the Langevin function, which accounts for the finite extensibility of polymer chains. The Langevin function is approximated by a polynomial, and only the term of the highest order contributes to the asymptotic solution. The crack is predicted to adopt a wedge-like shape, i.e. the crack faces will be straight lines. The angle of the wedge and the order of the stress singularity depend on the hardening of the strain energy function. The present analysis shows that in materials with a significant hardening, the inertia term in the equations of motion becomes negligible in the asymptotic analysis. Hence, there is no upper theoretical limit to the crack speed.     

Deflections of a rubber membrane

This paper deals with the problem of the transverse deflection of a natural rubber membrane that is fixed along a circular boundary. Uniaxial experiments were performed in order to characterize the constitutive behaviour of the rubber material in terms of several constitutive models available in the literature. These constitutive models were used to develop computational estimates for the quasi-static load-displacement response of a rigid spherical indentor that deflects the rubber membrane in a controlled fashion and to determine the deflected shape of the membrane at specified load levels. Both axisymmetric and asymmetric deflections of the rubber membrane were investigated. The paper provides a comparison of the experimental results for the membrane deflections with results derived from computational simulations.     

Relationship between paddy soil adhesion to steel and to rubber

Forty-three pairs of samples concerning adhesion of paddy soil to steel and to rubber are used to identify the relationship between these two groups of variables. Results show that there is no significant difference between them, implying that they are roughly equal to each other in engineering practice.     

Problems in determining the elastic strain energy function for rubber

Lightly crosslinked natural rubber can be stretched by 600% or more, and recovers almost completely. It is often regarded as a model highly elastic material and characterized by a strain energy function to describe its stress-strain behavior under various types of deformation. A number of such functions have been proposed; some of them appear in current finite element programs. They are usually validated by comparison with measured stress-strain relations by Treloar [7] [L.R.G. Treloar, Stress-strain data for vulcanized rubber under various types of deformation, Trans. Faraday Soc. 40 (1944) 59-70] and Jones and Treloar [15] [D.F. Jones, L.R.G. Treloar, The properties of rubber in pure homogeneous strain, J. Phys. D Appl. Phys. 8 (1975) 1285-1304]. But Treloar pointed out that the relations at high strains became markedly irreversible, and he did not assign a strain energy function for strains greater than about 300%. Rivlin's universal relation between torsional stiffness and tensile stress [14] [R.S. Rivlin, Large elastic deformations of isotropic materials. Part V1: further results in the theory of torsion, shear and flexure, Philos. Trans. R. Soc. A 243 (1949) 251-288] is applied here to show that a typical elastic solid cannot be described by any strain energy function at strains greater than about 300%. Elastic strain energy functions for higher strains, or for other rubbery materials, are thus of doubtful value unless evidence for reversibility of stress-strain relations is adduced or the applicability of a strain energy function is demonstrated.     

Non-linear stresses in a rubber cylinder sheared by pressure at one end

Normal stresses are set up by shearing a rubber block or tube. They depend strongly on the end conditions, even for relatively long specimens [A.N. Gent, J.B. Suh, S.G. Kelly III, Mechanics of rubber shear springs, Int. J. Non-Linear Mech. 42 (2007) 241-249; J.B. Suh, A.N. Gent, S.G. Kelly III, Shear of rubber tube springs, Int. J. Non-Linear Mech. 42 (2007) 1116-1126]. We have now examined a solid rubber cylinder bonded within a rigid cylindrical tube and subjected to pressure at one end. In this case, the correct end conditions for a simple shear deformation are met, at least approximately. Theoretical analysis and finite element calculations show that inwardly directed second-order stresses are set up at the wall, in contrast to the outwardly directed stresses generated by shearing a block or tube. However, for the particular geometry considered, the stresses were rather small in comparison with the applied pressure. Conditions are described under which they would be significantly larger. Stresses in a non-linearly viscous fluid under steady shear flows are expected to be similar, depending strongly on the geometry, end shapes and stress conditions.     

An elementary molecular-statistical basis for the Mooney and Rivlin-Saunders theories of rubber elasticity

By relaxing the assumption that the end-to-end vectors of molecules transform as macroscopic material line elements, we arrive at a generalization of the molecular-statistical theory of rubber elasticity. This generalization includes as special cases continuum-mechanical theories proposed by Mooney and by Rivlin and Saunders as improvements upon the classical neo-Hookean theory.     

Mechanics of rubber shear springs

FEA calculations have been carried out for a model rubber shear spring, consisting of a block of a highly elastic material, bonded between two rigid parallel plates and sheared by displacing one of the plates parallel to the other in its own plane. The block was prevented from deforming in the perpendicular direction, and thus was deformed in plane strain. Stress distributions along the bond-line and the center-line are reported and compared with those expected from the theory of large elastic deformations. Unexpected tensile stresses were found to develop in the interior of the sheared block. They are attributed to the absence on the end surfaces of the stresses needed to maintain a simple shear, causing a pronounced change in the reference pressure—a consequence that is usually overlooked. Because the internal stresses are governed by the boundary conditions, they were strongly affected by the shape of the end surfaces. In addition, they were reduced markedly by assigning values to Poisson's ratio slightly lower than 0.5, thus allowing some volume expansion of the rubber. Strain energy release rates were also evaluated for growth of a crack along the bond-line, starting at the edges, and compared with those reported previously by Lindley and Teo [Energy for crack growth at the bonds of rubber springs, Plast. Rubber Mat. Appl. 4 (1979) 29-37], Muhr et al. [A fracture mechanics study of natural rubber-to-metal bond failure, J. Adhes. Sci. Technol. 10 (1996) 593-616], Gregory and Muhr [Stiffness and fracture analysis of bonded rubber blocks in simple shear, in: D. Boast, V.A. Coveny (Eds.), Finite Element Analysis of Elastomers, Professional Engineering Publications, Bury St. Edmunds, UK, 1999, pp. 265-274] and Gough and Muhr [Initiation of failure of rubber close to bondlines, in: Proceedings of the International Rubber Conference, Maastricht, Netherlands, June 2005, IOM Communications Ltd., London, 2005, pp. 165-174]. They confirm that a long crack at the compression edge will grow faster than one at the tension edge, but the results for short cracks were inconclusive.     

Analysis of extrudate swell behaviour of rubber compounds using laser scan detector

Summary Classical extrudate swell measurements such as the gravimetric or the shrinkage techniques, are difficult, complex and time consuming methods. Using laser scan detector, new instruments have been recently developed which allow the extrudate swell behaviour of rubber compounds to be analysed in a quick and reliable manner.This paper discusses the requirements for suitable analysis of the extrudate swell of rubber compounds. Due to the specific elastic response of rubbers, these requirements differ from those for thermoplastics. Recent results obtained using typical industrial formulations, such as natural rubber/polybutadiene blends, are presented and their practical significance is discussed.Paper presented at the VIIIth International Congress on Rheology, Naples, Sept. 1–5, 1980.With 3 figures     

A multiaxial criterion for crack nucleation in rubber

The present paper is a first step towards the definition of a new multiaxial fracture criterion for rubber-like materials. Assuming that elastomers are subjected to a uniform distribution of intrinsic flaws, the framework of Eshelbian mechanics is considered. More precisely, the properties of the energy-momentum tensor are thoroughly studied to derive the criterion. A basic numerical example is presented and the qualitative discrepancy between the results obtained with this criterion and those relative to more classical approaches is highlighted.     

A method for predicting the internal motion resistance of rubber-tracked undercarriages,Pt. 3. A research on bending resistance of rubber tracks

Optimizing the efficiency of rubber-tracked undercarriages requires models for calculating external and internal motion resistance, including the resistance resulting from bending of rubber tracks. The experiments on the bending resistance of rubber tracks and a new model of this phenomenon are discussed in this article. An empirical model of friction in bearings typically implemented in driving and idler wheels of rubber-tracked undercarriages is also presented. According to the sample computations carried out on the basis of these models, the efficiency of rubber-tracked undercarriages might be improved by minimizing the number and maximizing the diameter of idler wheels. Furthermore, it has been shown that increase in the initial tension and driving force transmitted by rubber tracks does not significantly affect bending resistance of these tracks; however, it results in increased friction in the driving and idler wheels’ bearings. Nevertheless, the higher the driving force transmitted by the rubber tracks, the higher the efficiency of rubber-tracked undercarriages. Consequently, since track systems of vehicles operating at relatively small drawbar pull will manifest exceptionally low efficiency, there is a serious need for optimizing them in terms of energy consumption.     

Definition of a new predictor for multiaxial fatigue crack nucleation in rubber

From an engineering point of view, prediction of fatigue crack nucleation in automotive rubber parts is an essential prerequisite for the design of new components. We have derived a new predictor for fatigue crack nucleation in rubber. It is motivated by microscopic mechanisms induced by fatigue and developed in the framework of Configurational Mechanics. As the occurrence of macroscopic fatigue cracks is the consequence of the growth of pre-existing microscopic defects, the energy release rate of these flaws need to be quantified. It is shown that this microstructural evolution is governed by the smallest eigenvalue of the configurational (Eshelby) stress tensor. Indeed, this quantity appears to be a relevant multiaxial fatigue predictor under proportional loading conditions. Then, its generalization to non-proportional multiaxial fatigue problems is derived. Results show that the present predictor, which is related to the previously published predictors, is capable to unify multiaxial fatigue data.