The prediction of fatigue and failure of a disc brake system is fundamental
in assessing product performance. Disc brakes operate by pressing a set of
brake pads against a rotating disc. The friction between the pads and the disc
causes deceleration. The brake system then converts the kinetic energy of
vehicle motion into heat. Severe temperature changes as well as mechanical
loadings cause inelastic deformation and circumferential tensile stress in the
disc, which may eventually lead to the failure of the disc.
The traditional way of analyzing this kind of problem is to use a Lagrangian
approach in which the mesh used to discretized the disc rotates relative to the
brake assembly. Since many revolutions are typically required to reach the
state of interest to the analyst, this approach is prohibitively expensive and
cumbersome. The steady-state transport analysis capability in
Abaqus/Standard
(Steady-State Transport Analysis),
which uses the Eulerian method in which the finite element mesh of the disc
does not rotate relative to the brake assembly but the material “flows” through
the mesh, provides a cost-effective alternative approach. The paths that the
material points follow through the mesh are referred to as streamlines. This
kinematic description converts the moving disc brake problem into a pure
spatially dependent simulation. Thus, the mesh has to be refined only in a
fixed region where the brake pads are in contact with the disc initially.
Geometry and model
The model analyzed in this example is a solid disc. An axisymmetric model is
created to define the cross-sectional geometry of the disc, as shown in
Figure 1.
The disc has a thicker friction ring connected to a conical section that, in
turn, connects to an inner hub. The inner radius of the friction ring is 86.5
mm, the outer radius is 133.0 mm, and the ring is 13.0 mm thick. The conical
section is 27.2 mm deep. The inner radius of the conical section is 64.75 mm,
the outer radius is 86.5 mm, and the section is 6.4 mm thick. The conical
section has a thinner section out to a radius of 71.25 mm, which has a
thickness of 4.5 mm. The hub has an inner radius of 33.0 mm, an outer radius of
71.25 mm, and is 6.2 mm thick. Symmetric model generation (Symmetric Model Generation)
is used to create a three-dimensional disc, as shown in
Figure 2,
by revolving the two-dimensional cross-section about the symmetry axis and to
create the streamlines needed for the steady-state transport analyses in this
example. There are eight elements through the thickness of the friction ring,
four elements through the thickness of the hub, and four elements through the
conical section. There are 40 element sectors in the circumferential direction
of the disc, with a more refined mesh used in the region with higher thermal
and stress gradients. The model consists of 9440 first-order forced
convection/diffusion bricks (DCC3D8) in the heat transfer analysis, giving a total of about 11520
degrees of freedom; and it consists of 9440 first-order bricks (C3D8) in the subsequent steady-state transport analysis, giving a
total of about 34560 degrees of freedom.
The disc pads are not modeled in the example. Instead, the thermal and
mechanical interactions between the disc and the pads are represented by the
application of appropriate distributed heat fluxes in the heat transfer
analysis and by the application of appropriate concentrated loads in the
steady-state transport analysis, respectively.
Material properties
The disc is made of metallic material, with a Young's modulus of 93.5 GPa, a
yield stress of 153 MPa, a Poisson's ratio of 0.27, and a coefficient of
thermal expansion of 11.7 × 10–6 per °C at room temperature. In this
example the dissipation of the frictional heat-generated temperature
fluctuates, ranging from a minimum value of 40°C to a maximum value of 560°C
over the entire braking cycle. The temperature distribution when the disc is
heated to its peak value is shown in
Figure 3.
Under such operating conditions plastic deformation, as well as creep
deformation, is observed. The two-layer viscoelastic-elastoplastic model, which
is best suited for modeling the response of materials with significant
time-dependent behavior as well as plasticity at elevated temperatures, is used
to model the disc (see
Two-Layer Viscoplasticity).
This material model consists of an elastic-plastic network that is in parallel
with an elastic-viscous network. The Mises metal plasticity model with
kinematic hardening is used in the elastic-plastic network, and the power-law
creep model with strain hardening is used in the elastic-viscous network.
Because the elastic-viscoplastic response of the material varies greatly over
this temperature range, temperature-dependent material properties are
specified.
The thermal properties for the disc are temperature dependent with a
conductivity of 51 × 10–3W/mm per °C, a specific heat of 501 J/kg
per °C, and a density of 7.15 × 10–6 kg/mm at room temperature.
Problem description and loading
A simulation of braking a solid disc rotating initially at an angular
velocity of 155.7 rad/sec is performed. The braking time is approximately 5
seconds, followed by a cooling period of 600 seconds. A sequentially coupled
thermomechanical analysis is performed on the solid disc using the Eulerian
approach: a forced convection/diffusion heat transfer analysis is followed by a
steady-state transport analysis. Heat fluxes with film condition and prescribed
mass flow velocity through user subroutine
UMASFL are applied to the thermal model, which consists of three
steps. The first step, which lasts 0.2 seconds, simulates the response of the
disc under constant distributed fluxes and a constant angular velocity. The
second step involves 4.8 seconds during which the distributed fluxes and the
angular velocity are decreased linearly to small values near zero at the end of
the step. The final step, which lasts 600 seconds, simulates the continued
cooling in the model. The resulting temperatures obtained during the heat
transfer analysis are applied to the subsequent mechanical analysis, which
involves five steady-state transport analysis steps.
The purpose of the first step in the mechanical analysis is to obtain a
steady-state solution for a disc under constant concentrated loads due to the
application of the brake pads to the disc. There is only one increment in this
step. A constant temperature of 40°C is used, and a constant angular transport
velocity of 155.7 rad/sec is specified.
The second step obtains a series of quasi-steady-state transport solutions
under different temperature loading passes through the disc. The pass-by-pass
steady-state transport analysis technique is used for this purpose. This step
lasts 0.2 seconds with a constant angular velocity of 155.7 rad/sec throughout
the entire step. Several increments are involved, with each increment
corresponding to a complete temperature loading pass through the disc. The
temperature values obtained during the first step of the heat transfer analysis
are read into this step.
The third step also obtains a series of quasi-steady-state transport
solutions under different temperature loading passes through the disc. However,
this step involves 4.8 seconds over which the angular velocity is decreased
linearly from 155.7 rad/sec at the beginning of the step to a small value close
to zero at the end of this step. There are several increments in this step,
with each increment corresponding to a complete temperature loading pass
through the disc. The temperature values obtained during the second step of the
heat transfer analysis are read into this step.
The fourth step obtains a steady-state solution for the disc due to the
removal of the concentrated loads. There is only one increment in this step.
The last step obtains a series of quasi-steady-state transport solutions
when the disc cools down. There are several increments over a step period of
600 seconds. The temperatures obtained during the last step of the heat
transfer analysis are read into this step. Since the angular velocity is very
small, this step essentially simulates a long-term elastic-plastic response for
the disc.
Solution controls
Since the modified Newton's method is used in a steady-state transport
analysis, more numerical iterations are necessarily required to obtain a
converged solution. To decrease the computational time required for the
analysis due to the unnecessary cutback of the increment size, the time
incrementation control parameters are used to override the default values (see
Time Integration Accuracy in Transient Problems).
Results and discussion
One of the considerations in the design of a disc brake system is the stress distribution and
deformation in the region where the brake pads are applied. Circumferential tensile stress,
which may cause the fracture of the disc, will develop, making this region critical in the
design. The results shown in Figure 6 through Figure 11 are measured in this region (element 7817, integration point 5; see point A in Figure 2). The temperature in this region (node 7820) is shown in Figure 4 as a function of the time over the entire braking process. Figure 5 shows the von Mises stress distribution just before the distributed loads are removed and
the cooling period starts (Step 3, increment 60).
Figure 6,
Figure 7,
and
Figure 8
show the evolution of the circumferential stress, circumferential plastic
strain, and circumferential viscous strain, respectively, as a function of the
time throughout a complete braking cycle. A tensile stress of 54 MPa is
developed after the disc is cooled down completely. Both plastic strain and
viscous strain reach their saturation levels during the cooling period. The
time evolution of the circumferential stress versus the circumferential plastic
strain, shown in
Figure 9,
is obtained by combining
Figure 6
with
Figure 7.
Similarly, the time evolution of the circumferential stress versus the
circumferential viscous strain, shown in
Figure 10,
is obtained by combining
Figure 6
with
Figure 8.
The shapes of the stress-strain curves represent the plastic and viscous
energies dissipated, respectively, over an entire braking cycle. These
dissipated energies, which could be used to predict the fatigue life for the
disc, are shown in
Figure 11
as a function of the time.
Acknowledgements
SIMULIA
gratefully acknowledges PSA Peugeot Citroën
and the Laboratory of Solid Mechanics of the Ecole Polytechnique (France) for
their cooperation in developing the Eulerian algorithm for steady-state
transport analysis and for supplying the geometry and material properties used
in this example.
Three-dimensional model for the mechanical analysis.
References
Maitournam, M.
H., “Formulation
et Résolution Numérique des Problèmes Thermoviscoplastiques en Régime
Permanent,” Thèse de l’Ecole des Ponts et
Chaussées, 1989.
Nguyen-Tajan, T.
M.L., “Modélisation
Thermomécanique des Disques de Frein par une Approche
Eulérienne,” Thèse de l’Ecole
Polytechnique, 2002.
Figures
Figure 1. Mesh for the axisymmetric model. Figure 2. Mesh for the three-dimensional model. Figure 3. Temperature distribution when the disc is heated to its peak
value. Figure 4. Temperature at node 7820 as a function of time during the entire
braking period. Figure 5. Von Mises stress distribution (Step 3, increment 60) just before the cooling period
starts. Figure 6. Evolution of the circumferential stress as a function of time. Figure 7. Evolution of the circumferential plastic strain as a function of
time. Figure 8. Evolution of the circumferential viscous strain as a function of
time. Figure 9. Evolution of the circumferential stress versus the circumferential
plastic strain. Figure 10. Evolution of the circumferential stress versus the circumferential
viscous strain. Figure 11. Evolution of the plastic dissipated energy and the viscous dissipated
energy as a function of time.
