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Strain crystallization functions for 6 rubbers.

Under certain conditions, Natural Rubber can endure surprisingly large loads. The effect has origins in NR’s tendency to partially crystallize under nonrelaxing loads. We used a tabular function to define the dependence of the crystallization on the R ratio (where R = Tmin / Tmax) for a series of rubbers with varying degrees of the crystallization effect. For each elastomer in the series, the stress-strain, crack growth, and crack precursor size properties have been held constant. For each, we computed the Haigh diagram, which maps out the dependence of the uniaxial fatigue life as a function of mean strain and strain amplitude. The calculation reveals that the larger the crystallization effect, the more life benefit that can be derived from operating at moderate mean strains. It also shows that the effect is highly nonlinear and limited. Eventually, increasing the mean strain beyond certain limits will be harmful.

Haigh diagram showing dependence of log(fatigue life) on mean strain and strain amplitude for a strongly crystallizing rubber. Red = short life, blue = long life.

Haigh diagram for crystallizing rubber similar to that reported in Lindley’s 1973 experiments on gum natural rubber.

Haigh diagram for very weakly crystallizing rubber.

Contours of fatigue life show the number of repeats of the displacement history that will be endured before cracks appear.

Critical plane analysis is essential when analyzing fatigue performance under complex loading. This cradle mount is subject to vertical loading from the weight of the transmission, shocks from bumps in the road, as well as drive-train vibrations. We ran a Finite Element Analysis of the mount using the displacement histories for the X, Y, and Z directions as shown below. The top mounting surface is free to rotate under load. We then used the Endurica fatigue solver to calculate fatigue life (as the number N of repeats of the given load history) for each element in the model. The results are displayed as color contours on the mount. The failure mode is initiation of a crack at the upper left corner of the mount. Once the critical location in the mount is identified, then diagnostic outputs are generated for that location. Here, we checked the driving force on a crack precursor at the critical location (ie the cracking energy density), as well as the crack open/close state. The cracking energy density history allows you to see which time in the history does the most damage to the part (the peak at 53 sec). Tracking whether the crack is open or closed ensures proper accounting for the beneficial effects of compressive loads. Only the part of the load that opens or shears the crack precursor does damage. In this analysis, the crack remains open at all times except for a brief period around 40-41 sec. This indicates an opportunity to improve part life by modifying geometry or loads to induce crack closure at critical times.

History of displacements imposed on the Fine Element Model of the cradle mount.

Local load history experienced at the location of shortest life, on the most critical plane orientation.

Crack open/close state history at the critical location, on the critical plane. Blue=open, white closed.

Two channel loading history

Rainflow counting accurately identifies loading and unloading events within a complex duty cycle. This bushing is subject to a two channel loading history of radial forces in the X-direction and axial forces in the Y-direction as shown below. We ran a Finite Element Analysis of the bushing to calculate the stresses and strains on the bushing. We then performed a critical plane analysis using the Endurica fatigue solver to obtain the local load history shown below. We then used the Endurica fatigue solver to parse the load signal into a list of discrete events using the rainflow counting algorithm.  Each event contributes to the total rate of crack growth during a complete cycle. The list of the five most damaging events are shown below with references to their start and end in the signal time increment and the crack growth rate of each event. The most damaging events are quickly identified in the loading signal allowing for further opportunities to optimize the design to improve fatigue life. The bushing is shown below at the peak instant for each of the five most damaging events.

Local load history experienced at the location of shortest life, on the most critical plane orientation. All events are labeled. The top five most damaging events are colored red.

IPoint # Peak CED (MPa) R ratio FCGR (mm/cyc) at C0 Increment Start Increment End
34 0.0833 0 1.01E-5 1 247
9 0.0591 0.013 1.51E-6 57 88
32 0.0253 0.180 8.13E-12 373 390
21 0.0468 0.323 3.50E-14 295 301
13 0.0721 0.449 1.38E-16 185 199

Table listing top 5 critical events calculated using rainflow counting.

#34: Deformation and strains on bushing at increment 247

#9: Deformation and strains on bushing at increment 57

#32: Deformation and strains on bushing at increment 390

#21: Deformation and strains on bushing at increment 301

#13: Deformation and strains on bushing at increment 185

Contours of computed fatigue life in a truck tire for straight-ahead rolling conditions.

The stresses in a rolling tire combine tension from the inflation pressure, compression in the tire footprint and between the bead and rim, and shearing and bending in the carcass and shoulder.  Endurica’s temperature-dependent elastomer material models, support for multiple materials in a single analysis, and critical plane analysis methodology are ideal for the analysis of crack development in tires.  The critical plane analysis procedure enables an accurate accounting to be made of the effects of crack closure during rolling.  In particular, for this reason, the critical plane analysis method is superior to prior art fatigue analysis approaches based on scalar invariants such as strain energy density.  The analysis provides a wealth of information about the detailed failure mechanics of each point in the tire.  Outputs include the fatigue life for each element, the orientation of the critical plane in each element, and the history of local loading experienced by crack precursors on the critical plane.  The use of a fracture mechanical material definition of fatigue behavior in the model means that both materials engineers and tire development engineers can work from a common understanding of the material, and that efficient methods are available to measure the properties required to build the simulation.

Endurica predicts the change of failure mode from belt edge cracking at lower loads to sidewall cracking at higher loads.
Local crack precursor loading on the critical plane as a function of circumferential location around the tire, at 3 different loads (66%, 100%, and 170% rated load).

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