Stress–Strain Behavior

The neo-Hookean stress–strain law was used for the sake of simplicity in specifying material behavior, and in light of the fact that operating strains in the tire are generally modest. The neo-Hookean strain energy potential W is defined in terms of the material parameter C₁₀ and the principal stretches λ₁, λ₂, and λ₃,

Crack Growth Rate Law

The beneficial effects of a fatigue threshold [34–36] are neglected in this work. Instead, a conservative and simple approach to modeling crack development is taken, following the pioneering work of Thomas [37]. The fatigue crack growth rate r in rubber is characterized in terms of its dependence on the energy release rate T. A power law is employed to describe this relationship, as follows

Where many authors write this law with two material parameters, one possessing unusual units involving the exponent F, we choose to write the power law in terms of three material parameters: the fracture mechanical strength T_c, the powerlaw slope F, and the value of the crack growth rate r_c, at which the powerlaw intersects a vertical asymptote placed at T_c. This choice gives friendlier units and leaves clear the physical meaning of all of the parameters. The physical meaning of these parameters is illustrated in Fig. 2.

View Figure

• Download as Powerpoint
• Download Figure

Crack Precursor Size

Microstructural features that serve as precursors to fatigue cracks are small enough and frequent enough that one may assume a precursor exists at every point of the material. Precursors occur in a wide range of sizes, but only the largest of these end up developing into cracks. It is therefore sufficient to know the material's largest typical precursor and safest to assume that this particular precursor size is representative of any point in the material. Crack precursors have elsewhere been identified as having sizes c₀ in the range 10 × 10⁻³ < c₀ < 100 × 10⁻³ mm [27–31].

The fatigue life is herein defined as the number of cycles N required to grow a crack precursor from its initial size c₀ to its end-of-life size c_f. We have used here c_f = 1 mm. In practice, so long as c₀ ≪ c_f, the computed life N is insensitive to the choice of c_f. The crack growth rate r depends on the energy release rate T, which in turn depends on the crack size c.

Strain Crystallization

In a dynamic cycle, if the minimum load never unloads to zero (i.e., a nonrelaxing load), natural rubber and its blends may exhibit a phenomenon known as strain crystallization. Fatigue performance when strain crystallization is present in an elastomer is vastly improved relative to the case where crystallization is not present [38–42].

A simple and accurate approach to quantify this effect is to regard the powerlaw slope F of the crack growth rate law as a function F(R) of the ratio R = T_min/T_max. In the Endurica CL fatigue solver, this crystallization function may be specified using a simple table of values x(R), which is related to the slope F(R) through the transform:

x(R) was derived from results reported originally by Lindley [39,40] for natural rubber. These are replotted in Fig. 3. The equivalent fully relaxing energy release rate T_eq to be used for calculating crack growth rate from Eq. (2) is

View Figure

• Download as Powerpoint
• Download Figure

For the case of a non–strain-crystallizing elastomer, the equivalent energy release rate T_eq may be computed without any material parameters as

Spatial Modeling and Discretization

A cross section of the finite element mesh is shown in Fig. 4, along with a side-view of the mesh. The cross-section mesh consisted of 2980 nodes and 2098 elements of types CAX4H and CAX3H. The three-dimensional (3D) tire was obtained by revolving the cross section around the axle, using 31 elements around the circumference for each cross-section element. Cords in the tire were modeled via rebar elements [43]. In order to better capture the plane stress conditions on the free surfaces of the model, the external surfaces of the model were skinned with membrane elements. These were specified with negligible thickness and properties equal to the underlying solid elements to which they are attached. This practice ensures that results recovered from element centroids are associated accurately to the free surface and that they satisfy exactly the plane stress condition. Local, element-corotational coordinate systems were specified for all elements in the model, and all strains recovered for fatigue analysis were recovered in local coordinates.

View Figure

• Download as Powerpoint
• Download Figure

Operating Conditions

Tire operation was simulated at seven distinct conditions starting at 66% of rated load and ranging up to 170% of rated load. The tire inflation pressure was 837 kPa. The strain history in the rolling condition was estimated—for the sake of simplicity—by assuming the circumferential variation of the strain tensor history in static loading was equal to the time variation of the strain tensor during one tire revolution, an assumption commonly invoked for rolling resistance calculations [17,44,45]. Strain history for every element in the tire cross section was captured from the finite element analysis and passed on for fatigue analysis with Endurica CL. After executing the fatigue analysis, strain histories for several of the most critical locations in the tire were recovered and plotted. The strain histories are shown in Fig. 5. The critical locations correspond to those identified and discussed in the Results and Discussion section. They illustrate the varied and multiaxial nature of the strain histories that must be analyzed when computing tire durability.

View Figure

• Download as Powerpoint
• Download Figure

Critical Plane Analysis

When fatigue cracks develop from microscopic precursors in rubber, they tend to do so on specific planes that are associated with the history of applied loads. The orientation and loading experiences of such planes under the action of a given mechanical duty cycle directly govern the rate at which cracks develop, and thus ultimately the fatigue life [19,46–49]. Critical plane analysis consists of evaluating the number of cycles required for crack precursor development on each possible failure plane, and then in identifying the plane for which the shortest life is expected [32,50,51]. Each possible plane can be specified in terms of its unit normal in the undeformed configuration. The domain of the search is the unit sphere (see Fig. 6) representing all possible orientations of the crack plane, and the unit normal may equivalently be specified via the spherical coordinates φ and θ. The Endurica CL fatigue life solver has been used to perform the critical plane analysis. The unit sphere can be plotted with color contours representing fatigue life, as shown in Fig. 6.

View Figure

• Download as Powerpoint
• Download Figure

Loading Experience of the Critical Plane

In order to evaluate the life of the crack precursor plane, the loading history local to the plane must be estimated. This is accomplished by computing the cracking energy density W_c, via numerical integration of the definition [50]

where S is the traction vector associated with the specified material plane, and dɛ is the strain increment vector on the specified material plane. The energy release rate T of the specified crack orientation, at crack size c, was estimated through the following relationship:

The energy required to drive the growth of a crack precursor comes from strain energy stored in material surrounding the precursor. In general, only a part of the strain energy density is available, depending on the loading state and orientation of the crack. Equation (8) reflects that the energy release rate of a small crack surrounded by homogenously strained material scales linearly with the size of the crack. The validity of this rule for multiaxial loading cases has been established both from experience [52] and from mathematical arguments that consider the balance of configurational stresses [53].

Crack Closure

Under compression, crack closure can cause a significant portion of strain energy to remain unavailable for driving growth of crack precursors. The traction vector S associated with the plane of interest may be resolved into one component acting normal to the plane, and two components acting in shear on the plane. When computing W_c, only the tensile and shear parts of S are used. Any part of the strain work associated with the compressive component of S is excluded.

Belt Package

At low and moderate loads, the calculation identifies the steel belt endings as the most critical location for cracks in the tire. The location of first crack initiation within the belt package depends on load. At the lowest loads modeled, the crack initiates on the inboard end of the topmost belt cover strip. As the load is increased, the location switches to the outboard end of belt No. 3. With further load increases, the sidewall fatigue life eventually becomes more critical than the belt. The full results of the analysis for the belt package are shown in Fig. 10.

View Figure

• Download as Powerpoint
• Download Figure

The distribution of computed fatigue life per element is shown in the top-left panel, for the case of 100% rated load. Only elements of the belt package are shown. The unit sphere damage results of the critical plane analysis are plotted in the top right panel for the case of 100% rated load. The results are presented as a sphere on which colors represent the fatigue life of the potential crack plane. The most critical plane is noted. The analysis detects a “one-sided” shearing scenario (consistent with the fact that the 1–3 (circumference-thickness) shear component dominates at this location, see Fig. 5), in which a strong preference for the n = 〈+0.493, +0.082, +0.866〉 plane is revealed. The complementary crack at approximately 90 degrees from the critical plane, colored blue, is the least favorable for crack growth. In one-sided shearing, the plane experiencing maximum tension tends to receive significant damage, while its twin plane experiences maximum compression and therefore crack closure. Accurate accounting for crack closure effects is one of the strong benefits of critical plane analysis.

The second panel from the top shows the history of cracking energy density W_c during one revolution of the tire, on the most critical plane. The center of the footprint is at zero degrees circumferential position. The history is plotted for the case of 100% of rated load. The plot also shows the crack open/close state on the critical plane. The crack is seen to remain at least slightly open during most of its cycle. There is a brief period upon passing through the footprint when a closure event occurs, when the circumferential-thickness shearing component is at its full reversal. Critical plane analysis enables a proper accounting to be made of the effects of strain crystallization. In this case, it is recognized that the loading history is fully relaxing. Accounting for strain crystallization requires accurate identification of the R ratio that a crack will experience during its cycle, taking due account of the R ratio dependence on crack plane orientation [39].

The bottommost panel of Fig. 10 shows how the fatigue life for each component of the belt package depends on rated load. The sensitivities of each belt are quite distinct, a fact that gives rise to changes in predicted failure mode of the tire. At loads below about 70% rated load, the belt 2 ending is projected to be critical. Above 70% rated load, the belt 3 ending is most critical.

Meshing an idealized crack between the belts in order to calculate the energy release rate has sometimes been recommended [15] as the most accurate approach for estimating the effect of mechanical variables on the durability of the belt-edge region of the tire. This approach benefits from the fact that the energy release rate is minimally sensitive to the meshing scheme chosen by the analyst, although it may increase cost and risk in the analysis by (1) requiring the analyst to include the crack as a model feature, and (2) introducing new risks of numerical nonconvergence (particularly where compression and shearing of a crack make the solution more difficult).

An alternative practiced by some analysts [14] has been to use the distribution of strain energy density as an indicator of relative durability. This practice does not involve any special meshing or computational costs above what is normally required to solve a tire model, but it requires particularly careful control over mesh density. Refinements of the mesh invariably lead to increases of the strain energy density, since the belt edge is the location of a stress singularity. This practice also suffers from the fact that strain energy density is known to give suboptimal correlation with fatigue life, when a wide range of multiaxial states is considered in the correlation [50,58]. In particular, strain energy density fails to account properly for the beneficial effects of compression on fatigue. In compression, quite a large amount of strain energy may be stored without a damaging effect due to crack closure. Indeed, as indicated in Fig. 9, the amplitude of the strain energy density is greatest between the rim and the bead, irrespective of the tire load. In our models, the strain energy density amplitude between the rim and the bead is roughly twice the value at the belt endings. This result is not consistent with the common experience that the belt edges of the tire are observed to show the first signs of crack growth in typical durability evaluations.

Sidewall

The analysis results for the sidewall are shown in Fig. 11. Cracks were predicted to first occur on the external surface of the sidewall, near the location where maximum curvature is obtained due to bending. In practice, cracks often initiate on free surfaces [13]. For a given size of crack precursor, the energy release rate on a free surface is nearly twice the energy release rate of the same precursor in the interior. Also, the elastostatic equations naturally tend to produce solutions with stress maxima on the free surface that has maximum curvature. In order to best capture the plane stress conditions occurring on the free surface of the sidewall, the 3D sidewall elements were “skinned” with membrane elements.

View Figure

• Download as Powerpoint
• Download Figure

The unit damage sphere in the upper right panel indicates a symmetric pattern of damage, with two equally favorable, perpendicular planes on which cracks are likely to develop. The double crack plane system is typical of strain histories that involve “fully reversed” shearing [59]. The 13 shear component can be seen in Fig. 5 as fully reversed, and the critical plane is seen to have its normal at nearly ±45 degrees in this plane.

It is seen in the center panel that the crack loading history is fully relaxing, with peak load occurring at the center of the footprint. The crack open/close state on the critical plane is indicated as closed in compression during most of the cycle, except for a brief moment upon exiting the footprint (i.e., +22.5 deg). A similar plot could be made on the alternative “twin” critical plane. It shows the same trend, but with the opening event happening upon entry to the footprint (i.e., −22.5 deg).

Innerliner

Results of the analysis for the innerliner are shown in Fig. 12. Cracking near the toe of the tire profile is predicted, where the toe makes contact with the rim. The critical plane is oriented predominantly so that its normal is roughly in the 1 (circumferential) direction. The crack operates always in the open state, indicating nonrelaxing tension with peak crack load occurring at the center of the tire footprint. In this case, the 11, 22, and 12 components of the strain tensor are responsible for the damage.

View Figure

• Download as Powerpoint
• Download Figure

Tread

The results of the analysis for the tread are shown in Fig. 13. Cracks are predicted to first appear at the base of the outboard grooves. The damage sphere indicates two simultaneous critical planes. The crack load is maximum at the center of the footprint. The crack is open during most of the tire revolution, but is closed while passing through the footprint.

View Figure

• Download as Powerpoint
• Download Figure

The sensitivity of the tread fatigue life to load is seen to be remarkably low until quite high loads are reached. Perhaps this indicates that once the tire is flattened in the footprint due to contact with the road, further load serves only to change the footprint length through which the crack is closed, not the amount of bending in going from the curved to flat configuration.

Cracks on the free surface of the tread are shown to grow much more rapidly than cracks on the interior, no doubt due to the compressive states attained in the footprint, which would tend to produce crack closure on the interior of the tread elements. This is further illustrated by noting that the maximum Cracking Energy Density (CED) is attained during the tread groove's pass through the footprint, at a time when the 11 strain reaches its most compressive value and the crack on the critical plane is closed.