Fracture toughness evaluation of natural rubber

A method for estimating the fracture toughness of rubber-like materials is presented. Experimental data of a notched natural rubber sheet is analysed by application of the path-independent J-integral. A finite element code for large elastic deformations is used to evaluate the deformed shape of the rubber at crack growth initiation.Discussed is a stability criterion based on the existence of a critical value of J in relation with the experimental results.     

Two illustrations of rapid crack growth in a pre-stressed compressible neo-Hookean material

An unbounded isotropic compressible neo-Hookean solid is initially in equilibrium under uniform tensile (possibly large) pre-stress. In one case, plane strain conditions generate slit crack growth at a constant sub-critical rate; in the other, axial symmetry produces penny-shaped crack growth. The procedure of superposing infinitesimal deformations upon those that are large is carried out in terms of tractable exact full-field solutions.These solutions are examined apart from a specific fracture mechanics model, nevertheless, they show that pre-stress induces, in addition to the expected anisotropy, a critical value above which a negative Poisson effect occurs. It is also found that dilatational, rotational and Rayleigh wave speeds decrease, and that the decrease is greater for the plane strain state associated with slit crack growth than for the axially symmetric state of the penny-shaped crack.Dynamic stress intensity factors are also extracted, and found to fall below those for a linear isotropic solid at the same pre-stress and crack growth rate. Moreover, the range of growth rates for sub-critical crack propagation is also decreased.     

Elastic instabilities in rubber

Materials that undergo large elastic deformations can exhibit novel instabilities. Several examples are considered here: development of an aneurysm on inflating a cylindrical rubber tube; non-uniform stretching on inflating a spherical balloon; expansion of small cavities in rubber blocks when they are subjected to a critical amount of triaxial tension or when they are supersaturated with a dissolved gas; wrinkling of the surface of a block at a critical amount of compression; and the sudden formation of “knots” on twisting stretched cylindrical rods. These various deformations are analyzed in terms of simple strain energy functions using Rivlin's theory of large elastic deformations. The theoretical results are then compared with experimental measurements of the onset of unstable states. Such comparisons provide new tests of Rivlin's theory and, at least in principle, critical tests of proposed strain energy functions for rubber. Moreover, the onset of highly non-uniform deformations has serious implications for the fatigue life and fracture resistance of rubber components.     

Failure analysis of stainless steel at elevated temperatures

In this paper, particular emphasis has been put on gathering information on the phenomena that take place at the crack tip of a crack propagating at 1100°F. Since the experimental program was directed toward studying crack propagation in tubing, the tests were conducted on rings.From the experimentally obtained data and from the correlation with the theoretically predicted values, the following picture emerges for the fracture behavior with full plasticity present. There is a region surrounding the crack tip where very large plastic deformations take place. This region is surrounded by a much larger region where the loading is nearly proportional and the behavior can be predicted well by the results of the deformation theory of plasticity and the theory of singularity fields. As the crack propagation initiates, there is a drastic change in the crack-tip configuration. The crack tip does not blunt and a fairly sharp crack-tip region is observed. The crack tip carries a large deformation field of a far more localized nature than that observed at the initiation of the crack growth.Paper was presented at 1978 SESA Spring Meeting held in Wichita, KS on May 14–19.     

The stability of growth of a through-wall circumferential crack in a pipe subject to combined bending and tensile loadings

The stability of growth of a through-wall circumferential crack in a pipe is analysed for the case where the material has a high crack growth resistance, the analysis being based on the tearing modulus procedure. Rotations and lateral displacements are applied at the ends of the pipe, and this allows the combined effects of bending and tensile loadings on the stability of crack growth to be assessed. The general conclusion is that tensile loadings can have an adverse effect on crack stability, in accord with the conclusion reached in the author's earlier studies of plane strain crack growth in a beam. The stability results are compared with those obtained by Tada, Paris and Gamble, who allowed the tensile loadings to affect the position of the neutral axis, but did not consider instability in terms of the deformations produced by these loadings.     

Dynamic crack growth along a polymer composite-Homalite interface

Dynamic crack growth along the interface of a fiber-reinforced polymer composite-Homalite bimaterial subjected to impact shear loading is investigated experimentally and numerically. In the experiments, the polymer composite-Homalite specimens are impacted with a projectile causing shear dominated interfacial cracks to initiate and subsequently grow along the interface at speeds faster than the shear wave speed of Homalite. Crack growth is observed using dynamic photoelasticity in conjunction with high-speed photography. The calculations are carried out for a plane stress model of the experimental configuration and are based on a cohesive surface formulation that allows crack growth, when it occurs, to emerge as a natural outcome of the deformation history. The effect of impact velocity and loading rate is explored numerically. The experiments and calculations are consistent in identifying discrete crack speed regimes within which crack growth at sustained crack speeds is possible. We present the first conclusive experimental evidence of interfacial crack speeds faster than any characteristic elastic wave speed of the more compliant material. The occurrence of this crack speed was predicted numerically and the calculations were used to design the experiments. In addition, the first experimental observation of a mother-daughter crack mechanism allowing a subsonic crack to evolve into an intersonic crack is documented. The calculations exhibit all the crack growth regimes seen in the experiments and, in addition, predict a regime with a pulse-like traction distribution along the bond line.     

多喷口高效能厚翼的研究

提出了以下高效能翼型的思想：用多喷口小速度切向吹气控制厚翼上的流动分离,使流动接近于理想流状况,以产生大升力,小阻力;因多喷口小速度吹气耗能小,故翼型的有效升阻比可以很大．基于雷诺平均N－S方程进行了数值模拟实验．主要结果表明：对于厚度为0.4的儒氏翼型,在升力系数高达3．5时,有效升阻比可达约50（单喷口吹气约为23）;对于厚度为0．4的"升力体"翼型,在升力系数达2．2时,有效升阻比可达40（喷口吹气约为10）．     

Crack in a material-bond lattice

A square-cell lattice is considered consisting of point masses at its knots connected by linearly elastic bonds of nonzero density. Steady-state crack propagation is studied. A general relation between the knot mass and the bond mass is assumed; however, a detailed analytical examination is made for the material-bond lattice with no concentrated masses. It is assumed that the crack divides the bond in half, and the broken bonds remain in the lattice structure. In this model, the fracture energy of the bond is ignored, and hence the local fracture energy of the lattice is zero. The classical formulation in terms of critical stresses is accepted. The macrolevel energy release does exist. The macrolevel energy release rate as a function of the crack speed is found and compared with that for the massless-bond lattice of the same averaged density. While in the main, the dependencies for these two models are similar, there are some essential differences. For the lattice with no concentrated masses this function appears discontinuous. There exists a region where the crack speed is insensitive to the variation of the macrolevel energy release rate. The admissible regions of the crack speeds for the considered two lattice models differ greatly. For the massless-bond lattice this region is rather wide, while for the other it is very narrow. Mathematically, it is of interest that some details of the factorization depend on whether the ratio of the crack speed to the wave speed is rational and, if so, whether it can be represented as a ratio of two odd numbers.     

Buckling pressure of “pumpkin” balloons

This paper presents a computational study of the critical buckling pressure of pumpkin balloons, which consist of a thin, compliant membrane constrained by stiff meridional tendons. The n-fold symmetric shape of a pumpkin balloon with n identical lobes is exploited by adopting a symmetry-adapted coordinate system, which leads to the tangent stiffness matrix in an efficient block-diagonal form; the smallest eigenvalue of a particular block leads to the buckling pressure for the balloon. Two different types of balloon design are considered. Extensive results are obtained for the buckling pressures of a set of 10 m diameter experimental balloons and also for an 80 m diameter flight balloon. The key findings are as follows: the same type of buckling mode, forming four circumferential waves is critical for most of the balloons that have been analysed; balloons with flatter lobes are more stable, and the buckling pressure varies with an inverse power-law of the number of lobes; increasing the Young’s modulus, the Poisson’s ratio of the membrane, or the diameter of the end fitting has the effect of increasing the buckling pressure; but increasing the axial stiffness of the tendons has the effect of decreasing the buckling pressure.     

Dynamic crack growth under Rayleigh wave

A nonuniform crack growth problem is considered for a homogeneous isotropic elastic medium subjected to the action of remote oscillatory and static loads. In the case of a plane problem, the former results in Rayleigh waves propagating toward the crack tip. For the antiplane problem the shear waves play a similar role. Under the considered conditions the crack cannot move uniformly, and if the static prestress is not sufficiently high, the crack moves interruptedly. For fracture modes I and II the established, crack speed periodic regimes are examined. For mode III a complete transient solution is derived with the periodic regime as an asymptote. Examples of the crack motion are presented. The crack speed time-period and the time-averaged crack speeds are found. The ratio of the fracture energy to the energy carried by the Rayleigh wave is derived. An issue concerning two equivalent forms of the general solution is discussed.     

Examining a modified algorithm of smoothed particle hydrodynamics for a high velocity perforation of an aluminum beam

The wide spreading of utilizing of smoothed particle hydrodynamics (SPH) for numerical studies of the complex and high rate deformations of continuums, led the current study to gain a more reliable simulation by employing a modified compressible smoothed particle hydrodynamics (MCSPH) algorithm which could be a more accurate and stable technique in high tension regions, in despite of incompressible standard SPH. The main feature of the modified compressible SPH algorithm relies on a three steps solution procedure to calculate the pressure gradient, the deviatoric stress tensor, and the body forces separately. This algorithm is free of any artificial viscosity in its formulations, as well as welcoming to compressible effects which permits the pressure shock waves in high rate plastic deformation. To examine the accuracy of the algorithm, a benchmark problem of colliding rubber cylinders was simulated first and then a high velocity perforation process of an aluminum beam struck by a rigid projectile was simulated in various projectile speeds, and the failure response of the beam in each case was accompanied by crack propagation process. The prominent capability of the utilized MCSPH can be more illustrated when it was used in simulation of thickness crack propagation a tiny crack paths and defragmentation which can be encountered as a not easy numerical case study. The adequate assurance has been more fortified when the results were compared to those reported from a Finite Element method study.     

Maximally stable lobed balloons

This paper is concerned with the optimization of the cutting pattern of n-fold symmetric super-pressure balloons made from identical lobes constrained by stiff meridional tendons. It is shown that the critical buckling pressure of such balloons is maximized if the unstressed surface area of the balloon is minimized under a stress constraint. This approach results in fully stressed balloon designs that in some cases have a smaller unstressed surface area than the corresponding axisymmetric surface that is in equilibrium with zero hoop stress. It is shown that, compared to current designs, the buckling pressures can be increased by up to 300% without increasing the maximum stress in the lobe.     

Some Remarks on the Effects of Inertia and Viscous Dissipation in the Onset of Cavitation in Rubber

Through direct comparisons with experiments, Lefèvre et al. (Int. J. Frac. 192:1–23, 2015) have recently confirmed the prevailing belief that the nonlinear elastic properties of rubber play a significant role in the so-called phenomenon of cavitation—that is, the sudden growth of inherent defects in rubber into large enclosed cavities/cracks in response to external stimuli. These comparisons have also made it plain that cavitation in rubber is first and foremost a fracture process that may possibly depend, in addition to the nonlinear elastic properties of the rubber, on inertial effects and/or on the viscous dissipation innate to rubber. This is because the growth of defects into large cavities/cracks is locally in time an extremely fast process.The purpose of this Note is to provide insight into the relevance of inertial and viscous dissipation effects on the onset of cavitation in rubber. To this end, leaving fracture properties aside, we consider the basic problem of the radially symmetric dynamic deformation of a spherical defect embedded at the center of a ball made up of an isotropic incompressible nonlinear viscoelastic solid that is subjected to external hydrostatic loading. Specifically, the defect is taken to be vacuous and the viscoelastic behavior of the solid is characterized by a fairly general class of constitutive relations given in terms of two thermodynamic potentials—namely, a free energy function describing the nonlinear elasticity of the solid and a dissipation potential describing its viscous dissipation—which has been shown to be capable to describe the mechanical response of a broad variety of rubbers over wide ranges of deformations and deformations rates. It is found that the onset of cavitation is not affected by inertial effects so long as the external loads are not applied at a high rate. On the other hand, even when the external loads are applied quasi-statically, viscous dissipation can greatly affect the critical values of the applied loads at which cavitation ensues.     

A cohesive plastic and damage zone model for dynamic crack growth in rate-dependent materials

Mode I steady-state dynamic crack growth in rate-dependent viscoplastic solids containing damage, under small scale yielding conditions, is analyzed based on a modified cohesive zone model. A multi-scale approach is used to describe the entire non-linear zone consisting of a plastic region and a damage region, each of which has its own constitutive law. Traction in the damage region is characterized by a softening power-law, in terms of the ultimate strength, a softening index and a rate sensitivity factor. In the plastic region, the cohesive law is assumed to be both strain hardening and rate dependent. The critical crack opening displacement at the physical crack-tip controls crack growth. The governing integral equations are derived and solved by a collocation method combined with associated boundary conditions. Numerical results are presented for the traction and opening profiles along the cohesive zone, the fracture energy and lengths of the damage and non-linear zones at different crack speeds and for different material parameters. The importance of factors, such as material softening, plastic deformation, crack speed and viscosity, is identified by parametric studies. In addition, the competition of plastic flow and material damage, and its effect on crack growth, are discussed.     

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.     

Three-dimensional modeling of intersonic shear-crack growth in asymmetrically loaded unidirectional composite plates

An anisotropic cohesive model of fracture is applied to the numerical simulation of Coker and Rosakis experiments (2001). In these experiments, a unidirectional graphite–epoxy composites plate was impacted with a projectile, resulting in an intersonic shear-dominated crack growth. The simulations account for explicit crack nucleation––through a self-adaptive remeshing procedure––crack closure and frictional sliding. The parameters used in the cohesive model are obtained from quasi-static fracture experiments, and successfully predict the dynamic fracture behavior. In keeping with the experiments, the calculations indicate that there is a preferred intersonic speed for locally steady-state growth of dynamic shear cracks, provided that sufficient energy is supplied to the crack tip. The calculations also show that the crack tip can attain speeds in the vicinity of the longitudinal wave speed in the direction of the fibers, if impacted at higher speeds. In addition, a double-shock which emanates from a finite size contact region behind the crack tip is observed in the simulations. The predicted double-shock structure of the near-tip fields is in close agreement with the experimental observations. The calculations additionally predict the presence of a string of surface hot spots which arise following the passage of the crack tip. The observed and computed hot spot structures agree both in geometry as well as in the magnitude of the temperature elevation. The analysis thus suggests intermittent friction as the origin of the experimentally observed hot spots.     

Assessment of Digital Image Correlation as a method of obtaining deformations of a structure under fluid load

Digital Image Correlation (DIC) is employed for the measurement of full-field deformation during fluid–structure interaction experiments in a wind tunnel. The methodology developed for the wind tunnel environment is quantitatively assessed. The static deformation error of the system is shown to be less than 0.8% when applied to a curved aerofoil specimen moved through known displacements using a micrometre. Enclosed camera fairings were shown to be required to minimise error due to wind induced camera vibration under aerodynamic loading. The methodology was demonstrated using a high performance curved foil, from a NACRA F20 sailing catamaran, tested within the University of Southampton RJ Mitchell, 3.5 mx2.4 m, wind tunnel. The aerodynamic forces induced in the wind tunnel are relatively small, compared with typical hydrodynamic loading, resulting in small deformations. The coupled deflection and blade twist is evaluated over the tip region (80–100% Span, measured from the root) for a range of wind speeds and angles of attack. Steady deformations at low angles of attack were shown to be well captured however unsteady deformations at higher angles of attack were observed as an increase in variability due to hardware limitations in the current DIC system. It is concluded that higher DIC sample rates are required to assess unsteady deformations in the future. The full field deformation data reveals limited blade twist for low angles of attack, below the stall angle. For larger angles, however, there is a tendency to reduce the effective angle of attack at the tip of the structure, combined with an unsteady structural response. This capability highlights the benefits of the presented methodology over fixed-point measurements as the three dimensional foil deflections can be assessed over a large tip region. In addition, the methodology demonstrates that very small deformations and twist angles can be resolved.     

Free surface on a simple fluid between rotating eccentric cylinders. Part II: Experiments

This paper summarizes the results from an experimental investigation of the effects of eccentricity and rotational speed on the free surface shape on a viscoelastic liquid between eccentric cylinders. In the experimental geometry, the inner cylinder rotates and the outer cylinder is stationary. The experiments show that there is a circumferential pressure gradient (the lubrication effect) which has a dominant influence on the free surface shape at all eccentricities and rotational speeds. For a liquid with small normal stress effects, the normal-stress induced component of the deformation tends to be overwhelmed by the lubrication effect, whereas a liguid with large normal stress effects exhibits characteristics normal-stress induced deformations at small eccentricities and rotational speeds. There is good agreement between experiment and second order predictions for the large normal stress liquid under these conditions. The ranges of eccentricities and rotational speeds for which second order theory describes the low normal stress liquid appear to be much more limited and are difficult to reproduce experimentally.     

Prestretch effect on snap-through instability of short-length tubular elastomeric balloons under inflation

The inflated elastomeric balloon structures are widely used in engineering fields such as elastomeric actuators and artificial muscles. This study, involving both experiment and modeling, is focused on the prestretch effect on non-linear behavior of inflated short-length tubular elastomeric balloons. In the experiment, the prestretched tubular elastomeric balloon is subjected to air pressure while the two ends are fixed with rigid tubes. The shape evolutions of the tubular elastomeric balloons are illustrated. The non-axisymmetric bulging is observed in the inflated tubular balloon with small prestretch. An analytical model based on continuum mechanics is developed to investigate the inflation behavior of the tubular balloons, and the analytical results agree well with the experimental observation. Analysis shows that snap-through instabilities may happen during the inflation of the tubular balloon. Prestretch along the axis of the tubular balloon can suppress instability during inflation and regulate the reaction force along the axial direction. This work can guide the future application of tubular balloons in elastomeric actuators and artificial muscles.     

Experimental Investigation and Finite Element Analysis of Fatigue Crack Growth in Pipes Containing a Circumferential Semi-elliptical Crack Subjected to Bending

In this paper, a circumferential external surface flaw in a metallic round pipe under cyclic bending loading is considered. Because of very rapid changes in the geometrical parameters around the crack front region, the mesh generation of this region must be done with great care. This may lead to an increase in the run time which makes it difficult to reach valid results and conclusions. Because of the advantages of the sub-modeling technique in problems which need very high mesh density, this method is used. Stress intensity factors in mode I condition are determined using three-dimensional finite element modeling with 20 node iso-parametric brick elements in the ANSYS 9.0 standard code and the singular form of these finite elements at the crack front. In order to estimate the analysis error, the structural parameter error in energy norm criterion was used. Because of the advantages of non-dimensional analysis, this method is employed, and the stress intensity factors are normalized. For the analysis of the fatigue crack growth, the Paris law is used. The propagation path of the surface flaw is obtained from the diagram of aspect ratio versus relative crack depth. The fatigue crack growth analysis (the relative crack depth against loading cycles diagram) of different initial crack aspect ratio under cyclic loading is also considered. Fatigue shape development of initially semi-elliptical external surface defects is illustrated. The effect of the Paris exponent (material constant) on fatigue crack propagation is shown as well. Moreover, the fatigue crack growth of several specimens is assessed experimentally using a manually-constructed experimental set up. Finally, the experimental results obtained by cyclic bending loading tests are compared with the numerical results. The experimental results show good conformity with the finite element results.