2.1 Material characterisation

In the traditional modelling method of tyre cord-rubber composites, the shear stress between cord elements is calculated by establishing fine mesh in the cord region at the macro level. This modelling method makes the calculation very inefficient, and it is also impossible to analyse the influence of the structural characteristics of the cord-rubber composite components on tyre performance at the microscopic level. In previous tyre analyses, the properties of rubber cord composites need to be obtained through expensive material testing, and this method is not efficient. In this paper, a micromechanical analysis theory proposed by Chamis et al. [Reference Chamis, Abdi and Garg22] is employed to simulate the belt layer region of aircraft tyres. By analysing the in-plane performance derived from five standard American Society for Testing and Materials (ASTM) tests, the mechanical properties of the cord rubber composite can be reverse-engineered, effectively capturing the influence of failure in cord-rubber composite on the impact behaviour during aircraft tyres landing. These tests include longitudinal and transverse tension (ASTM D3039), longitudinal and transverse compression (ASTM D6641) and in-plane shear (ASTM D3518).

Following this reverse engineering process, the effective Young’s modulus, Poisson’s ratio, and the longitudinal tensile and compressive strengths of the cord/rubber composite can be assayed. The predictive equations, grounded in micromechanics theory, estimate the ply modulus, Poisson’s ratio, and strengths as outlined in the Ref. [Reference Murthy and Chamis23].

Modification for voids in tyres cord and rubber volume fractions:

(1) \begin{align}{\bar k_c} = \left( {1 - {k_v}} \right){k_c}\end{align}

(2) \begin{align}{\bar k_r} = \left( {1 - {k_v}} \right){k_r}\end{align}

where ${k_c}$ , ${k_r}$ and ${k_v}$ are cord, rubber and void volume fractions, respectively. ${\bar k_c}$ , ${\bar k_r}$ are the corrected parameters in tyres analysis.

Ply Poisson’s ratio:

(3) \begin{align}{v_{12}} = {v_r} + {\bar k_c}\left\{ {{v_{c12}} - {v_r}} \right\}\end{align}

(4) \begin{align}{v_{13}} = {v_{12}}\end{align}

(5) \begin{align}{v_{23}} = {\bar k_c}{v_{c23}} + {\bar k_r}\left( {2{v_r} - \frac{{{E_{22}}}}{{{E_{11}}}}{v_{12}}} \right)\end{align}

where ${v_r}$ , ${v_{c12}}$ and ${v_{c23}}$ are the rubber Poisson’s ratio, cord Poisson’s ratio along the in-plane direction and cord Poisson’s ratio along the transverse-normal direction, respectively. ${E_{11}}$ is the ply stiffness along the longitudinal direction, ${E_{22}}$ is the ply stiffness along the transverse direction.

Ply stiffness:

(6) \begin{align}{E_{11}} = {\bar k_c}{E_{c11}} + {\bar k_r}{E_r}\end{align}

(7) \begin{align}{E_{22}} = \frac{{{E_r}}}{{1 - \sqrt {{{\bar k}_c}} \left\{ {1 - \frac{{{E_r}}}{{{E_{c22}}}}} \right\}}}\end{align}

(8) \begin{align}{E_{33}} = {E_{22}}\end{align}

where ${E_r}$ is the rubber stiffness, ${E_{c11}}$ is the cord stiffness along the longitudinal direction and ${E_{c22}}$ is the cord stiffness along the transverse direction.

Ply shear modulus:

(9) \begin{align}{G_{12}} = \frac{{{G_r}}}{{1 - \sqrt {{{\bar k}_c}} \left\{ {1 - \frac{{{G_r}}}{{{G_{c12}}}}} \right\}}}\end{align}

(10) \begin{align}{G_{22}} = \frac{{{G_r}}}{{1 - \sqrt {{{\bar k}_c}} \left\{ {1 - \frac{{{G_r}}}{{{G_{c23}}}}} \right\}}}\end{align}

(12) \begin{align}{G_{13}} = {G_{12}}\end{align}

where ${G_r}$ is the rubber shear modulus, ${G_{c12}}$ is the cord shear modulus along the in-plane direction and ${G_{c23}}$ is the cord shear modulus along the transverse-normal direction.

Ply longitudinal tensile strength:

(13) \begin{align}{S_{11T}} = {\bar k_c}{S_{c11T}}\end{align}

where ${S_{c11T}}$ is the cord tensile strength along the longitudinal direction.

Ply longitudinal compressive strength:

(14) \begin{align}{S_{11C}} = {\bar k_c}{S_{c11C}}\end{align}

where ${S_{c11C}}$ is the cord compressive strength along the longitudinal direction.

Ply transverse tensile strength:

(15) \begin{align}{S_{22T}} & = \left\{ {1 - \left( {\sqrt {{V_c}} - {V_c}} \right)\left( {1 - {E_r}/{E_{c22}}} \right)} \right\}\nonumber\\& * {S_{rT}}\left( {\left[ {1 - \sqrt {4{V_v}/\pi \left( {1 - {V_c}} \right)} } \right]} \right)\end{align}

where ${V_c}$ is the cord volume fraction, ${S_{rT}}$ is the rubber tensile strength and ${V_v}$ is the void volume fraction.

Ply transverse compressive strength:

(16) \begin{align}{S_{22C}} & = \left\{ {1 - \left( {\sqrt {{V_c}} - {V_c}} \right)\left( {1 - {E_r}/{E_{c22}}} \right)} \right\}\nonumber\\& * {S_{rS}}\left( {\left[ {1 - \sqrt {4{V_v}/\pi \left( {1 - {V_c}} \right)} } \right]} \right)\end{align}

where ${S_{rS}}$ is the rubber compressive strength.

Ply in-plane shear strength:

(17) \begin{align}{S_{12S}} & = \left\{ {1 - \left( {\sqrt {{V_f}} - {V_f}} \right)\left( {1 - {G_r}/{G_{c22}}} \right)} \right\}\nonumber\\& * {S_{rS}}\left( {\left[ {1 - \sqrt {4{V_v}/\pi \left( {1 - {V_f}} \right)} } \right]} \right)\end{align}

where ${S_{rS}}$ is the rubber shear strength.

As demonstrated, the ply strength of tyres can be calculated by analysing the tensile, compressive and shear strengths of both the cord and rubber components. Specifically, the longitudinal tensile and compressive strengths of the layers are predominantly determined by the characteristics of the cord, whereas the transverse tensile, compressive and in-plane shear strengths of the layers are primarily influenced by the rubber’s mechanical properties.

2.2 Multi-scale progressive analysis

Cord-rubber composites have multi-scale characteristics, including the macro-scale of the laminate structure, the mesoscale of the component materials and microscopic scales such as pore spaces and microcracks. Due to its multi-scale characteristics, the damage, failure and stiffness reduction of cord-rubber composite structures in aircraft tyres during impact loading also shows strong multi-scale characteristics. The composite material micromechanics method considers the damage and failure behaviour of the component materials at the microscopic scale under external loads. The existing computing power and calculation efficiency are limited if the analytical model is established entirely from the microscopic scale. Therefore, it is necessary to develop a multi-scale analysis method to establish the connection between macroscopic structures and microscopic physical phenomena to ensure the high efficiency of macroscopic analysis and the accuracy of microscopic models. This work uses a multi-scale progressive analysis method to consider the effects of cord-rubber composite microscopic void volume fraction, cord volume fraction and material scale factor of composites on tyre deformation history and overall impact force; the basic calculation process is shown in Fig. 1.

Figure 1.

The aircraft tyre multi-scale progressive analysis process.

The multi-scale progressive analysis of cord-rubber composite structure in aircraft tyres is mainly divided into three steps:

1. (1) Analyse the macrostructure of aircraft tyres, taking into account the macrostructure design, material properties, plying methods and overall influence of the loading mode of the structure on the stress and strain distribution inside the structure.

2. (2) Analyse the key tyre belt region using the multi-scale analysis method. The loading conditions of the macroscopic structural elements or integration points are transferred to the microscopic unit cell model. The maximum stress — or strain-based failure criterion is used to judge the failure of the rubber and cord, respectively.

3. (3) The material stiffness of the damaged corresponding element is reduced, and after homogenisation processing, the macroscopic properties of the damaged material are assigned to the corresponding element in the macrostructure. The macroscopic structure is reanalysed and iterated until convergence, at which point it enters the next iteration step.

This paper uses the commercial finite element analysis software ABAQUS/Standard to complete the multi-scale analysis and realise the coupling among different scale models. The information transfer between the macroscopic structural finite element model and the microscopic finite element model is established through a user-defined subroutine UMAT (User subroutine to define a Material’s mechanical behaviour) in ABAQUS. The finite element calculation framework for the multi-scale analysis of aircraft tyres is shown in Fig. 2.

Figure 2.

The finite element calculation framework for multi-scale analysis of aircraft tyre.

The following functions are mainly implemented in UMAT [Reference Behroozinia, Taheri and Mirzaeifar24]:

1. (1) SDV (state-dependent variables) is used in UMAT to record the damage status of macroscopic and corresponding microscopic model elements. In instances where damage is induced by tension, the stiffness of the macro model element is diminished to 1% of its initial elastic modulus. If the damage results from shear, the stiffness is reduced to 10% of the initial elastic modulus, and if it stems from compression, the reduction is to 20% of the initial elastic modulus.

2. (2) Transfer the failed microscopic model element information recorded in SDV to the microscopic model, reduce the material properties of the corresponding microscopic element, and apply the load conditions.

3. (3) Process the microscopic model result file(.odb) in PDE, extract the stress of the microscopic model elements to conduct failure judgements on the microscopic model elements that did not fail in the previous step, and then store the failed element information in an external file. Finally, the information on the new failed element is returned to SDV.

4. (4) Return all the microscopic failure element information in the SDV to the unit cell model and apply the load conditions to solve the macroscopic elastic modulus.

3.1 The fluid cavity element simulates the effects of inflation pressure

In the analysis of automobile tyres, the effect of internal pressure is minimal due to the relatively small loads and deformations encountered during simulations. Consequently, researchers often utilise a distributed pressure approach to simulate the inflation pressure of these tyres. In contrast, aircraft tyres undergo significant deformation during operation, and this volume change substantially impacts the internal inflation pressure, which cannot be overlooked. The ABAQUS simulation platform offers a surface-based fluid cavity element capable of simulating a closed structure filled with air or liquid. This study employs the fluid cavity element to model the air pressure within aircraft tyres, capturing the variations in internal air pressure resulting from tyre deformation during landing, as shown in Fig. 3.

Figure 3.

Finite element model based on surface fluid cavity element.

Assuming that the air within the aircraft tyre behaves as an ideal gas, the Equation of state for an ideal gas is expressed as follows:

(18) \begin{align}\tilde p = \rho R\left( {\theta - {\theta ^z}} \right)\end{align}

(19) \begin{align}\tilde p = p + {p_A}\end{align}

Where $\tilde p$ is the absolute pressure, ${p_A}$ is the pressure of the surrounding environment, $p$ is the pressure inside the cavity, $R$ is the gas constant, $\theta $ is the ambient temperature, ${\theta ^z}$ is absolute zero, $R$ can also be determined by the universal gas constant $\tilde R$ and the molecular mass of the gas $MW$ :

(20) \begin{align}R = \frac{{\tilde R}}{{MW}}\end{align}

The current fluid volume can be expressed as:

(21) \begin{align}\bar V = \sum {{{\bar V}^e}} \left( {p,\theta } \right) = \sum {m^e/\rho } = m/\rho\end{align}

Then the corresponding volume pressure can be satisfied as:

(22) \begin{align}\frac{{d\bar V}}{{dp}} = - \frac{m}{{{\rho ^2}}}\frac{{d\rho }}{{dp}} = - \frac{{mR\left( {\theta - {\theta ^z}} \right)}}{{\left( {p + {p_A}} \right)}}\end{align}

The compressibility of a fluid can be expressed by its bulk modulus:

(23) \begin{align}p = - K\left( {\frac{{V - {V_0}}}{{{V_0}}}} \right) = - {K_{\rho R}}\left( {{\rho ^{ - 1}} - \rho _0^{ - 1}} \right){V_0}\end{align}

Where $K$ is the bulk modulus of the fluid, $V$ is the current volume of the fluid, $\rho $ is the density at the current pressure, ${\rho _0}$ is the density at the current temperature when the air pressure is zero, ${V_0}$ is the volume at the initial temperature when the pressure is zero.

3.2 Numerical model of the aircraft tyre

This study divides a 680 × 220 aircraft tyre model into nine sections, as illustrated in Fig. 4. The radial tyre features cords that extend to the bead at an angle of approximately 90° relative to the centreline of the tread. A circumferential belt serves to stabilise the carcass. In developing the finite element model, simplifications can be made to ensure accurate simulation results are obtained under convergence conditions, provided the actual requirements permit. The simplifications include the following:

1. (1) Consideration of only the longitudinal tread grooves.

2. (2) The assumption is that the road surface behaves as a rigid body.

Figure 4.

Finite element model of aircraft tyre section.

The Mooney-Rivlin constitutive Equation characterises the nonlinear properties of rubber. The analysis utilises a total of 22,800 C3D8R elements, which are 8-node linear hexahedra with reduced integration. The deformation potential energy within the Mooney-Rivlin model is quantified using a specific Equation:

(24) \begin{align}U = \sum\limits_{i = 1}^N {{C_{i0}}} {\left( {{{\bar I}_1} - 3} \right)^i} + \sum\limits_{i = 1}^N {\frac{1}{{{D_i}}}{{\left( {{{\bar J}_{e1}} - 1} \right)}^{2i}}} \end{align}

The strain energy potential ( $U$ ), elastic volume ratio ( ${\bar J_{e1}}$ ), first invariant of the deviatoric strain ( ${\bar I_1}$ ), shear behaviour parameter ( ${C_{i0}}$ ) and compressibility parameter ( ${D_i}$ ) are key parameters used to characterise rubber materials.

The viscoelastic material property response is modeled as follows [Reference Wei and Olatunbosun17]:

(25) \begin{align}{g_R}\left( t \right) = 1 - \sum\limits_{i = 1}^N {\bar g_i^p\left( {1 - {e^{ - t/\tau _i^G}}} \right)} \end{align}

where $\bar g_i^p$ and $\tau _i^G$ are the material constants determined by modeling the physical test in ABAQUS [Reference Wei and Olatunbosun17]. Table 1 provides a summary of the rubber material parameter values utilised in this study.

Table 1.

Rubber material parameter values

The reinforced material properties are assumed to be homogeneous and orthotropic. The model defines the belt layer using SFM3D4R surface elements, with the reinforced layers BELT-1 and BELT-2 comprising 2,600 and 2,400 finite elements, respectively. For the tyre bead section, 1,200 C3D8R finite elements are employed for discretisation. The steel cord is represented as an elastic material. Table 2 summarises the reinforced material parameter values utilised in this study.

Table 2.

The reinforced material parameter values

Figure 5.

Two different approaches to model the cord-rubber composite: (a)rebar element approach, and (b)micromechanical-base element approach.

To analyse the mechanical properties of cord-rubber composites, rebar elements are commonly utilised in tyre structures. In this approach, the rebar element represents the cord, while the solid element represents the rubber. However, this method cannot capture the shear response in tyre structures and evaluate the structural damage evolution of aircraft tyres during service. To address this limitation, a micromechanics-based element is introduced in this study to model the cord-rubber composite, as illustrated in Fig. 5. Subsequent tyre analyses will compare two different modelling approaches. The impact load scenario simulation was performed on a 16-core AMD Ryzen 9 3950X supercomputer (Dell), equipped with 2 CPUs operating at 1.6 GHz per core. The entire simulation process required one hour to complete.

4.1 Static load scenario validation

In this case, the tyre modelling involves two sequential stages: the generation of a two-dimensional axisymmetric inflation model and the development of a three-dimensional symmetric static load model. The two-dimensional axisymmetric inflation model is applied to the corresponding rim. The tyre is inflated to the recommended 630 kPa value specified by the manufacturer. The static loaded radius of an aircraft tyre is defined as the vertical distance from the axle centreline to the tyre surface when the tyre is initially inflated to its rated pressure and subsequently subjected to its rated load, as illustrated in Fig. 7. According to FAA regulations, both tyre pressure and load must maintain the static loaded radius within specified limits [Reference Yao, Yue and Geng20]. The study compared the influence of the fluid cavity element and uniform pressure on the sidewall in simulating tyre inflation pressure under static load conditions.

Figure 7.

The static loaded radius of an aircraft tyre.

Figure 8 displays the vertical load-deformation curves of the tyre under various vertical loads at a rated inflation pressure of 630 kPa. The static deformations of aircraft tyres are 20.0 mm, 18.1 mm, 16.2 mm, 12.1 mm and 10.0 mm at test loads of 5,000 N, 4,500 N, 4,000 N, 3,500 N, 3,000 N and 2,500 N, respectively. The change curve of the vertical load and deformation of the aircraft tyre indicates that simulating inflation pressure in the way of the fluid cavity element closely aligns with experimental results. Therefore, the volume changes caused by the deformation of aircraft tyres during operation significantly impact the internal inflation pressure, which cannot be ignored. Furthermore, the numerical simulation results exhibit strong consistency with the experimental results and previous research findings [Reference Guo, Bastien and Blundell19].

Figure 8.

Deformation versus vertical load curves in static load scenario.

4.2 Impact load scenario validation

The aircraft tyre impact process was simulated using dynamic numerical calculations in ABAQUS, as depicted in Fig. 9. In this study, a rigid model was employed to simulate non-deformable ground. The tyre-ground interaction utilised the penalty method and a friction coefficient was set to 0. A concentrated mass was placed at the tyre’s centre to simulate the aircraft’s weight, and gravitational forces were applied to the entire tyre model. Additionally, the vertical landing speed was introduced by setting a predefined initial velocity field for the model. The initial boundary conditions in the finite element analysis (FEA) aligned with those established in tyre impact experiments. The entire tyre impact process is conducted with an impact mass of 300 kg, an inflation pressure of 630 kPa, and an impact vertical velocity of 2 m/s.

Figure 9.

The aircraft tyre impact process in ABAQUS.

The tyre impact test platform measures the aircraft’s deformation and impact load using displacement and force sensors. Figure 10 displays the load curve in the impact test and the tyre load curve simulated by two different model methods. The tyre model simulation in the micromechanical-base element approach is more closely related to the experimental results than the model simulated in the rebar element approach. The time of the peak value in the numerical simulation varies slightly under two different model methods, with a certain lag observed in the model that simulates the rebar element approach. Furthermore, it is apparent that the tyre load steadily increases before reaching its peak at 0.36 s, at which point the tyre is compressed to its maximum deformation.

Figure 10.

The aircraft tyre experiment and simulation in impact load.