Some phenomenological models, such as a shell model , a ‘string’ model , or a multi-component composite model have provided a good insight on the way tyres carry load and respond to vibratory inputs. Other thermal models, often
using the Finite Difference Method have provided ways to predict the operating
temperature of a tyre or curing thermodynamics.
The current method of choice for modelling tyres is FEA. It has been many years since
the first publication of a thermoviscoelastic tyre model using FEA .
It is now well recognised that mechanical and thermal models of tyres developed using
this method are excellent predictors of the quasi-static internal stress state of a tyre .
However, computational resources have only recently become available to allow
this technique to be used routinely in tyre design.
Use of FEA has illustrated both the non-sinusoidal nature of the stress-strain cycle as
well as the three-dimensional nature of the stress and strain tensors throughout the tyre
structure. Thermal models for tyres, based on FEA models, can predict temperatures
during curing. With an appropriate hysteretic model of the materials, the analyses can
also predict tyre operating temperatures
Modelling a complex structure
One of the major obstacles in structural analysis of tyres was shared by many other
engineering structures: the shape and material distribution did not permit a solution
from a closed form analytical technique. The FEA methodology overcame this obstacle
by permitting the structure to be subdivided into small elements for which the analytical
techniques would apply. The concatenation of the solutions for the small elements was mathematically shown to converge to the theoretically correct answer as the number of
elements approaches infinity.
Initial use of the FEA technique was done without the digital computer. However, the
analytical difficulties were exchanged for tedious accounting for the individual elements.
In order to take advantage of computational resources specialised geometric approaches,
using ‘Jacobian’ matrices to transform among coordinate systems were developed.
There are a number of different coordinate systems used within a given FEA computer
programme. There is a global coordinate system associated with the structure, and various local coordinate systems associated with each element, with the materials, and with the loading axes. Figure 11.12 illustrates a tyre with a typical global coordinate system. One representative element from that tyre is shown with its own elemental coordinate system The RCT (Radial – Circumferential – Thickness) system is useful for the performing tire stress analysis. The RCT system provides a locally orthogonal, curvilinear, coordinate system that remains oriented with the materials in each element so that related elements may be identified around the tyre circumference.
Material response model
A Material Response Model is needed to predict internal stresses from the strains. The
most common such material response model is linear elasticity represented by a material modulus. Viscoelastic, hyperelastic, and other nonlinear material models may also be
available for use with the computation of tyre stress-strain behaviour. The model chosen
for the material response determines the specifics of the required inputs for the available
FEA computations, as well as creating limitations on how the outputs from the model
may be interpreted.
A Material Response Model is also required to obtain tyre thermal response from
temperature inputs, and for the computation of mechanical hysteresis or rolling resistance.
The key to the accurate predictions from a computational model of the tyre is the
mathematical representation of the material in a form that can be used by the
computational model. Current research in this area has provided various material models
that are functions of strain, stress, temperature, frequency, and history.
Ideally, laboratory testing of the tyre materials would provide values for parameters that
can be used with the three-dimensional stress-strain cycle computed by a model. Many
published works focus on representing tyre materials for a specific set of loading cases.
While such work provides helpful insight for specific cases, it does not provide the material parameters that are needed for a general model of the tyre behaviour.
Mechanical response computation
When an appropriate material response model is used, the computation of the mechanical
response of the tyre structure can be very accurate. Many FEA practitioners have studied
the relationship between stress analysis models and tyre behaviours. A detailed dynamic
analysis of the tyre, using complex material response models, such as non-linear viscoelastic theory, often results in an unreasonable time for computing the tyre response. Therefore, sometimes a mechanical response is approximated from experimental parameters combined with FEA, such as the estimation of tyre rolling resistance using a ‘loss parameter’ for each material
Thermodynamic considerations
For tyre curing analysis, the computation of the tyre temperature as a function of time is
a result of the heat energy input, ambient temperatures, and the thermal boundary
conditions. This type of analysis contains many significant mathematical considerations
within the various computer programs. While the external appearance of the results and
their application may be different, usually the thermal results do not contain significant
technical differences.
For tyre heat buildup analysis, the energy input is from the mechanical hysteresis (tyre
rolling loss) but there are some additional considerations. First, the appropriate heat
transfer coefficients must be used in the modelling process based on the tyre velocity. A
second effect is that of the temperature on the material stiffness and hysteresis.
The heat transfer coefficient includes the velocity and geometry dependence of heat transfer to the surroundings. A power law relationship is commonly observed between fluid velocity and the heat transfer coefficient. Several authors have reported that the power law relationship is valid for tyres . However, the large variation among published coefficients for tyre heat transfer illustrates the difficulty of obtaining precise heat transfer coefficients for a specific tyre. The computation of the steady state heat generation rate of a tyre uses only the loss generated by the materials and does not
require the use of heat transfer coefficients for the tyre. A tyre’s operating temperature
should not be used to compute the material hysteresis because it is uncommon to have
published values of heat transfer coefficients available for the tyre size under consideration.
Where published values of heat transfer coefficients are available they may be used to
estimate a tyre’s temperature from computed energy losses for comparison with
experimental results
Hybrid modelling of tyres
There are several potential approaches to combining computational models of tyres.
There is some research being reported on ‘total physics’ models that combine multiple
sets of physical equations at a low level within a single computer programme.
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