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William V. Mars,
1
Yintao Wei,
2
Wang Hao,
3
and Mark A. Bauman
4
Computing Tire Component Durability via Critical
Plane Analysis
REFERENCE: Mars, W. V., Wei, Y., Hao, W., and Bauman, M. A., ‘‘Computing Tire
Component Durability via Critical Plane Analysis,’’ Tire Science and Technology, TSTCA,
Vol. 47, No. 1, January–March 2019, pp. 31–54.
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.
KEY WORDS: fatigue, multiaxial, critical plane analysis, durability
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
1
Corresponding author. Endurica LLC, 1219 West Main Cross, Suite 201, Findlay, Ohio
45840, USA. Email: wvmars@endurica.com
2
Department of Automotive Engineering, Tsinghua University, State Key Laboratory of
Automotive Safety and Energy, Beijing 100084, China
3
e-Rubber, Room B609, No. 1, Qinghua East Road, Haidian District, Beijing 100083, China
4
Endurica LLC, 1219 West Main Cross, Suite 201, Findlay, Ohio 45840, USA
31
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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.
FIG. 1 — Tire warranties, 1960–present.
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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 Wis defined
in terms of the material parameter C
10
and the principal stretches k
1
,k
2
, and k
3
,
W¼C10ðk2
1þk2
2þk2
33Þð1Þ
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 rin 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
TABLE 1 — Material Properties Assumed for Finite Element and Fatigue Analyses.
Tire
component(s)
Compound
C
10
(MPa)
T
c
(kJ/m
2
)
r
c
(mm/cyc) F
0
Strain
crystallizing
c
0
(mm)Polymer
a
Carbon
black
b
Abrasion gum,
belt NR N330 6.6 285 0.01 2.00 Yes 0.100
Apex NR N330 11.8 285 0.01 2.00 Yes 0.030
Carcass NR N330 8.2 285 0.01 2.00 Yes 0.010
Reinforcement NR N330 8.2 285 0.01 2.00 Yes 0.100
Sidewall NR/BR
blend
N330 4.5 16 0.01 2.93 Yes 0.100
Tread SBR N330 5.7 15 0.01 4.95 No 0.100
Innerliner IIR N990 8.2 45.9 0.01 2.98 No 0.100
Shoulder wedge NR N990 6.6 164 0.01 2.00 Yes 0.008
a
NR, Natural Rubber; SBR, Styrene Butadiene Rubber; IIR, Butyl Rubber.
b
All compounds were filled at 50 parts per hundred of rubber by mass.
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r¼rc
T
Tc
F
ð2Þ
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.
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
FIG. 2 — Parameters of the Thomas fatigue crack growth rate law.
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have elsewhere been identified as having sizes c
0
in the range 10 310
3
,c
0
,
100 310
3
mm [27–31].
The fatigue life is herein defined as the number of cycles Nrequired to
grow a crack precursor from its initial size c
0
to its end-of-life size c
f
. We have
used here c
f
¼1 mm. In practice, so long as c
0
c
f
, the computed life Nis
insensitive to the choice of c
f
. The crack growth rate rdepends on the energy
release rate T, which in turn depends on the crack size c.
N¼Zcf
c0
dc
rðTÞð3Þ
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 Fof the crack growth rate law as a function F(R) of the ratio
R¼T
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
FIG. 3 — Strain-crystallization function x(R) for Lindley’s 1973 measurements on natural rubber
[39, 40].
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slope F(R) through the transform:
xðRÞ¼1F0
FðRÞð4Þ
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
Teq ¼10
logðTÞxðRÞlogðTcÞ
1xðRÞð5Þ
For the case of a non–strain-crystallizing elastomer, the equivalent energy
release rate T
eq
may be computed without any material parameters as
Teq ¼ð1RÞTmax ð6Þ
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
FIG. 4 — Isometric and cross-section views of 1200R20 TBR tire model finite element mesh.
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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
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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 /and h.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.
FIG. 5 — Strain history tensor components at 100% rated load at critical locations in various tire
components. The footprint of the tire is centered at 0 degrees circumferential position.
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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]
dWc¼Sdeð7Þ
where Sis the traction vector associated with the specified material plane, and
deis the strain increment vector on the specified material plane. The energy
release rate Tof the specified crack orientation, at crack size c, was estimated
through the following relationship:
T¼2pWccð8Þ
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 Sassociated with the plane of interest may be resolved into one
component acting normal to the plane, and two components acting in shear on
FIG. 6 — Typical results of critical plane analysis. The fatigue life is computed for each possible
crack orientation. Each point on the sphere represents a unit normal vector for a given orientation,
and is colored according to life. The orientation predicted to give the shortest life is identified as the
critical plane.
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the plane. When computing W
c
, only the tensile and shear parts of Sare used.
Any part of the strain work associated with the compressive component of Sis
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,
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FIG. 7 — Effect of operating vertical load on computed fatigue life of shortest lived tire components.
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FIG. 8 — Evolution of tire failure mode with increases in operating vertical load.
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FIG. 9 — Strain energy density amplitude change with increases in vertical load.
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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 n¼hþ0.493, þ0.082, þ0.866iplane 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 Rratio that a crack will
experience during its cycle, taking due account of the Rratio 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
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FIG. 10 — Belt package durability analysis results.
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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
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FIG. 11 — Sidewall durability analysis results.
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FIG. 12 — Innerliner durability analysis results.
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the cycle, except for a brief moment upon exiting the footprint (i.e., þ22.5 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
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FIG. 13 — Tread durability analysis results.
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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.
Future development of the model will address leveraging its critical plane
analysis framework in combination with a microsphere-based model of energy
dissipation [60–62], and toward incorporating aging effects into the material
model [63–65].
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[53]A
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54 TIRE SCIENCE AND TECHNOLOGY
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