## ABSTRACT

Tire developers are responsible for designing against the possibility of crack development in each of the various components of a tire. The task requires knowledge of the fatigue behavior of each compound in the tire, as well as adequate accounting for the multiaxial stresses carried by tire materials. The analysis is illustrated here using the Endurica CL fatigue solver for the case of a 1200R20 TBR tire operating at 837 kPa under loads ranging from 66 to 170% of rated load. The fatigue behavior of the tire's materials is described from a fracture mechanical viewpoint, with care taken to specify each of the several phenomena (crack growth rate, crack precursor size, strain crystallization, fatigue threshold) that govern. The analysis of crack development is made by considering how many cycles are required to grow cracks of various potential orientations at each element of the model. The most critical plane is then identified as the plane with the shortest fatigue life. We consider each component of the tire and show that where cracks develop from precursors intrinsic to the rubber compound (sidewall, tread grooves, innerliner) the critical plane analysis provides a comprehensive view of the failure mechanics. For cases where a crack develops near a stress singularity (i.e., belt-edge separation), the critical plane analysis remains advantageous for design guidance, particularly relative to analysis approaches based upon scalar invariant theories (i.e., strain energy density) that neglect to account for crack closure effects.

## Introduction

Providing for adequate durability inevitably ranks high on tire development agendas. Owing to the highly competitive nature of the tire market and continual innovation on the part of manufacturers, developers face constant demand either for greater durability or for lower tire cost without sacrificing durability. This is evident, as shown for example in the evolution of tire warranties over the last few decades; see Fig. 1. Developers are responsible to design against the eventuality of crack development in each of the various components of a tire [*1 *,*2 *]. The task requires knowledge of the stress–strain and fatigue behaviors of each compound in the tire, analysis of the loads carried by each tire component, and calculation of the damaging effects of operation under load.

The concepts and procedures for characterizing tire material behavior [*3 *–*13 *] and for calculating tire durability [*14 *–*24 *] have largely been established. Even when new, tire materials contain microscopic features that have the potential to develop as cracks [*25 *]. Whether or not these crack precursors actually develop and the rate at which they do develop depends upon the mechanical properties of the rubber and upon the driving forces experienced by the crack precursor. The characterization methods required for tire durability analysis include measurement of cyclic stress–strain behavior, of the fatigue crack growth rate curve, and of the typical size of crack precursors.

The forces driving crack development at any location in the tire can be computed via finite element analysis. If it is desired to include the crack as a feature of the finite element mesh, then methods based upon the fracture mechanical energy release rate may be applied [*15 *,*24 *,*26 *]. Alternatively, when crack precursors are small relative to other tire dimensions and strain gradients [*27 *–*31 *], the loads on a virtual small crack at the center of each finite element centroid may be analyzed without including the crack as an explicit feature of the finite element mesh [*19 *]. A novel aspect of the present analysis is the application of critical plane analysis [*32 *], which considers explicitly which crack orientation at each point will receive the most damage, given the stress history.

The purpose of this work is to illustrate how these procedures may be deployed and used for routine tire development programs.

## Material Modeling

The basic mechanical behaviors that govern tire durability include the stress–strain behavior, the fatigue crack growth rate law, and the size of crack precursors [*33 *]. In addition, some tire compounds exhibit strain crystallization, whose effect on the crack growth rate law must be considered in any analysis of durability. In the following sections, we define the models selected for specifying the mechanical behavior and we give in Table 1 a summary of the parameter values chosen for each tire compound.

### 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 *C*_{10} and the principal stretches *λ*_{1}, *λ*_{2}, and *λ*_{3},

### 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*, at which the powerlaw intersects a vertical asymptote placed at

_{c}*T*. 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.

_{c}### 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 *c*_{0} in the range 10 × 10^{−3} < *c*_{0} < 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 *c*_{0} to its end-of-life size *c _{f}*. We have used here

*c*= 1 mm. In practice, so long as

_{f}*c*

_{0}≪

*c*, the computed life

_{f}*N*is insensitive to the choice of

*c*. The crack growth rate

_{f}*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

For the case of a non–strain-crystallizing elastomer, the equivalent energy release rate *T*_{eq} may be computed without any material parameters as

## Finite Element Analysis

The calculations reported are for a 1200R20 TBR truck tire.

### 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.

### 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.

## Fatigue Analysis

A fatigue analysis was made for the strain history recovered at the centroid of each finite element of the tire model using the Endurica CL fatigue solver. The analysis considers the effects of multiaxial, variable amplitude straining on durability. For the sake of giving a clear and concise account of the fatigue analysis method, this study is focused exclusively on mechanical effects, and it neglects thermal effects. The material properties used in this analysis are specified for room temperature.

### 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.

### 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 *d***ɛ** 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.

## Results and Discussion

Critical plane analysis has been applied for all of the components in the subject tire. Because it does not require the inclusion of cracks as meshed features of the finite element model, the method is quite simple to apply as a postprocessing step once the tire model has been built and executed via standard procedures. This avoids the computational costs and convergence risks associated with including a crack in the finite element mesh. It may also conveniently be applied in combination with Futamura's deformation index concept [*54 *–*56 *] to quickly understand how compound stiffness variations might impact durability [*57 *]. The life dependence on vertical load of the most critical tire locations is plotted in Fig. 7. Despite the fact that thermal effects have been neglected, the result that the outermost belts have the shortest lifetime at typical operating conditions seems consistent with typical experience, as does the result that extreme loads eventually cause sidewall failures. The critical locations in the tire can be easily identified by plotting contours of fatigue life, as shown in Fig. 8. Because strain energy density is somewhat popular as an indicator of potential fatigue life, Fig. 9 gives contour plots of the amplitude of the strain energy density for comparison.

For each of the components analyzed, the following results are presented:

A contour plot showing the distribution of fatigue life and location of first crack initiation, for the case of 100% rated load.

A view of the unit damage sphere showing the distribution of fatigue lives across all possible critical plane orientations at the centroid of the finite element with the shortest life. The unit normal vector of the critical plane is also noted. Colors on the unit sphere indicate the base 10 logarithm of fatigue life, with red indicating the shortest lives and blue the longest.

A plot showing the history of the cracking energy density experienced on the critical plane during one tire revolution. This plot also shows the crack open/close state on the critical plane at the element with shortest fatigue life.

A plot showing the effect of rated vertical tire load on the calculated fatigue life of the tire component.

### 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.

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.

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.

### 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.

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.

## Conclusion

Critical plane analysis offers analysts the capability to accurately consider the effects of multiaxial straining on the development of crack precursors during tire operation. It fully accounts for the fact that crack precursors may initially appear in any orientation, but in the end prefer an orientation that maximizes damage. The analysis provides a detailed account of how multiaxial loads are experienced by a given precursor. For each strain history analyzed, the critical plane is identified, and the effects of crack closure are considered. Also, the effects of strain crystallization/nonrelaxing energy release rate history are considered. The analysis uses a material definition and a damage calculation that are rooted in well-accepted principles of fracture mechanics, providing a widely practiced path for material characterization, as well as the high efficiency and accuracy that are possible from correctly implemented fracture mechanics experiments.

The procedures applied here predicted commonly observed trends in the durability and failure mode of the tire considered across all tire components. The method is quite general in scope, and has the inherent advantage of not requiring the insertion of a crack into the meshed geometry. Using the method, a modest analysis effort yields a wealth of information about the durability and failure mechanics of all of the tire components.

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