Advanced material modeling for virtual tire development

Introduction The tire industry is undergoing a significant transformation, shifting towards virtual prototyping to reduce development time, costs, and environmental impact associated with physical testing [2]. This transition to digital twin technology necessitates highly accurate constitutive models capable of capturing the complex behavior of carbon black-reinforced rubber compounds used in tire manufacturing. However, current models often fall short in accurately predicting the nonlinear, viscoelastic, and rate-dependent behavior of filled elastomers across the wide range of loading conditions and frequencies encountered in tire applications. This research addresses this critical gap by developing advanced material models for virtual tire development. The main research objectives are: • To develop a unified theoretical framework for nonlinear viscoelasticity that encompasses existing models. • To create robust computational tools for identifying these advanced material models from the experiments. • To explore innovative data-driven approaches to nonlinear viscoelasticity that complement traditional phenomenological modeling to overcome their limitations in describing the complex nonlinear behaviour of filled rubber. Theoretical Framework This thesis builds upon and extends several key theories in the field of viscoelasticity and rubber mechanics. The foundation is laid with the Parallel Rheological Framework (PRF) [3], a computational framework that extends the assemblage of rheological elements used in the linear theory to nonlinear viscoelasticity. Here the PRF is systematically developed from 1D linear viscoelasticity to 3D non-linear finite viscoelasticity [4] by consistently invoking the principle of virtual power (PVP) and the dissipation inequality in the spirit of seminal work of Gurtin [5]. Central to this development is the multiplicative decomposition of the deformation gradient (Bilby-Kröner-Lee decomposition) [6–11], which extends to finite deformation the additive split between elastic and viscous deformation. The PRF incorporates objective rates of elastic strain measures [12], such as the Oldroyd derivative, to ensure frame invariance of the constitutive equations [5]. The thesis extends these theories by developing a generalized framework [13] that encompasses various existing models as special cases, including the Reese and Govindjee [14], Bergström and Boyce [15], Kumar and Lopez-Pamies [16], and Yoshida and Sugiyama [17] models. Particular attention is given to the formulation of non-linear viscosity functions that capture phenomena like shear-thinning and strain-induced softening, crucial for accurately modeling filled rubber under dynamic loading conditions [18]. This unified approach allows for a systematic comparison of different models and provides a flexible structure for developing new constitutive relationships. Furthermore, this framework can be easily used as theoretical foundations of data-driven modeling, that combines the multiplicative structure of the PRF with neural networks to model complex nonlinear viscoelastic behaviour. An innovation in this work is the derivation of closed-form solutions for simple shear and torsion of thin-walled cylinders under large deformations. These solutions are obtained through a semi-inverse approach, where the form of the solution is postulated based on physical insights, and then verified to satisfy the governing equations. Unlike in hyperelastic model where simple shear is determined by a single parameter, this work demonstrates that for inelastic materials, the solution requires three coupled evolution equations. The derivation is crucial for the numerical routine developed to calibrate the constitutive models. Experimental Characterization of Filled Rubber The experimental foundation of this research is built upon a comprehensive characterization of filled rubber compounds, with a focus on capturing their complex viscoelastic behavior across a wide range of loading conditions and different carbon black content. Dynamic Mechanical Analysis (DMA) forms the cornerstone of this characterization, providing crucial insights into the material’s response under various strain amplitudes and frequencies. Central to this experimental approach are simple shear sweep tests, conducted on filled elastomers. These tests subject the material to sinusoidal strain inputs, described by γ(t) = γs + γd sin(ωt), where the strain amplitude γd is systematically varied while maintaining a constant frequency. The experimental data is analyzed to extract the storage modulus G′ and loss modulus G′′, which represent the real and imaginary components of the complex modulus G(ω), respectively. This methodology allows for a detailed investigation of amplitude-dependent phenomena, particularly the Payne effect, which is characterized by a significant reduction in storage modulus with increasing strain amplitude and represents one of the main challange for currently available models. In addition to dynamic mechanical analysis, constant shear rate tests have been explored. These tests apply continuous shear rates and capture both the initial elastic response and viscous flow, revealing the material’s non-linear properties. Lower shear rates result in a smoother, more gradual stress-strain curve, whereas higher rates cause increased stiffening. This behavior is particularly prominent in filled rubber samples. In this type of tests, the modulus characterizing the short time response as well the long-term one can be readily measured [18]. The equilibrium spring in the Maxwell rheological model is directly associated with the terminal modulus, representing the material’s stiffness in the long-term, steady-state response when only the equilibrium spring supports the load. In contrast, the instantaneous modulus reflects the combined stiffness of both the equilibrium spring and the non-equilibrium spring, capturing the material’s initial elastic response. In this early stage, the dashpot is effectively closed, allowing both springs to contribute to the overall stiffness. Over time, as the dashpot opens, the non-equilibrium spring relaxes, leaving only the equilibrium spring to sustain the load, which corresponds to the terminal modulus. Model Validation and Calibration The validation and calibration of the selected constitutive models form a critical component of this research. Firstly, the closed-form solutions derived for simple shear and torsion of thin-walled cylinders are rigorously compared with full 3D finite element simulations. These simulations, conducted using Comsol Multiphysics, cover a range of specimen geometries (with aspect ratios H/L varying from 1 to 50) and deformation rates (characterized by different ratios of material to deformation times, τm/τd) to validate the accuracy of the analytical solutions and delineate their range of applicability. Notably, for specimens with aspect ratios larger than 10, the closed-form solutions show excellent agreement with the computationally intensive 3D simulations, with relative errors below 5% for shear stress and 10% for normal stress differences. For such a reason the calibration of the model paramaters can be carried out by using the derived solution for simple shear, that is significantly faster than the 3D numerical simulations. The calibration of model parameters employs a Particle-Swarm-Optimization (PSO) approach, implemented using the Pymoo Python library [19]. This optimization process aims to minimize the discrepancies between the experimental data and model predictions. Several constitutive models are evaluated within this framework, including the Reese and Govindjee model [14], the Bergström and Boyce model [15], the Kumar and Lopez-Pamies model [16], and the Strain Hardening Power Law model [3]. This comparative analysis reveals that models incorporating strain dependent viscosity functions, such as the Bergström and Boyce and Kumar and Lopez-Pamies models, demonstrate superior performance in capturing the complex nonlinear behavior of filled rubber compounds, particularly in representing the Payne effect. Data-driven Modeling One of the results of the thorugh analysis and comparison of several phenomenological models carried out in this thesis is that they lack the ability of capturing the intrecate nonlinear response of filled rubber in particular under time-varying loading conditions. For this reason a novel Deep Rheological Element (DRE) to enhance the modeling of viscoelastic materials under finite strains is introduced. In this approach, the conventional dashpot in the Maxwell model is replaced with a neural network-based viscosity function. To ensure thermodynamic consistency, the network’s final layer employs a custom activation function that guarantees positive viscosity outputs. The DRE is implemented within the framework of finite strain viscoelasticity, maintaining compatibility with the multiplicative decomposition of the deformation gradient. A two-step training strategy is employed: first, the network is pre-trained on viscosity data generated from a calibrated phenomenological model; subsequently, it undergoes refinement using stress-strain histories derived from dynamic mechanical analysis tests. This approach allows for effective training with limited experimental data. Key Findings • The unified theoretical framework developed in this thesis successfully captures a wide range of nonlinear viscoelastic behaviors observed in filled rubber compounds. A comparative analysis in simple shear revealed that the Kumar and Lopez-Pamies and Bergström and Boyce models demonstrated superior performance in matching experimental data, particularly in capturing the transition of storage modulus from low to high dynamic amplitudes, known as Payne effect. • Advanced experimental protocols contributed to a better understanding of the interaction between rate and time effects in the material response. • The proposed Deep Rheological Framework, combining the structure of the Parallel Rheological Framework with neural networks (NN), showed promising results in capturing complex nonlinear viscosity functions. This hybrid approach demonstrated improved accuracy over traditional phenomenological models and addresses the challenge of maintaining physical consistency with miniminal computational effort. Future Work • Implementation as UMAT: Development of the Deep Rheological Element (DRE) as a User Material Subroutine (UMAT) for integration into finite element software, enabling more comprehensive structural simulations. • Extended Model Validation: Further validation of the developed models against full-scale tire testing data, especially for various rubber compounds, to ensure robustness and generalizability. • Advanced Rheological Frameworks: Investigation of more complex rheological structures beyond the single Maxwell element, as the current model shows limitations in capturing the full complexity of filled rubber behavior under diverse loading conditions. • Model Interpretability: Enhancing the interpretability of the Deep Rheological Element through techniques such as symbolic regression. This approach aims to discover analytical expressions for the learned viscosity functions, providing deeper insights into the physical mechanisms underlying the material behavior and facilitating the development of more transparent, physics-informed models.