An Arbitrary Lagrangian Eulerian formulation for tire production simulation

https://doi.org/10.1016/j.finel.2022.103742Get rights and content

Highlights

  • A new algorithm for the projection of history variables in an Arbitrary Lagrangian Eulerian framework.

  • A finite strain thermo-mechanically consistent material model for rubber curing.

  • Applied ALE framework to the tire production process.

Abstract

Tire production is a complex process due to large deformations, highly non-linear uncured rubber material and large temperature. It can be observed, that the production conditions have a strong influence on the cured tire behaviour and should be studied to minimize possible defects in the final product. In this contribution, an Arbitrary Lagrangian Eulerian (ALE) formulation is presented to overcome convergence issues stemming from large distortion of elements in a pure Lagrangian description. In an ALE formulation, the computational mesh is not fixed in space and can move relatively to the material motion, which allows to control the distortion of the mesh by a smoothing algorithm. Coupling the material motion with the newly obtained mesh is done by an advection algorithm to project the internal variables of a thermo-mechanically consistent material model. Based on the assumption that the internal variables are projected directly from the integration points, an additional mesh is generated with the integration points of the old mesh as grid. To show the capabilities of the presented algorithm, several numerical examples are shown ranging from a simple forging example to a complex tire production process.

Introduction

Rubber above its glass transition temperature is widely used in industry and is by far the most important application as material for tires. Rolling tires and vulcanized rubber are investigated for a long time and a large variety of material models is present to describe its visco-elastic and incompressible behaviour. In recent years, researchers and tire industry are interested as well in research on the production process of the tire to obtain information on the material distribution due to in-moulding and on the pre-stress condition and its influence on the final tire. In its uncured phase, rubber shows a very soft and sticky behaviour. In [1], several experiments are carried out to investigate the rate-dependent behaviour. Due to the sticky properties, the preparation of the samples is carried out at very low temperature. The first approach to model the behaviour is based on a generalized Maxwell model, capturing nicely the rate-dependent behaviour neglecting the plastic deformation observed in the experiments. In a more recent work [2], the visco-elastic and the elasto-plastic behaviour of the uncured rubber is included. The assumption of decoupling time-dependent and time-independent behaviour is introduced and the material formulation is based on the micro-sphere model [3]. The proposed models are able to represent the behaviour at large strains and for complex deformation states. However, the curing kinematics and the phase change of the rubber are neglected.

During the tire production process, a green uncured tire is pushed into a heating press under high pressure and temperature in its final form. After being exposed to high temperature, the vulcanization process starts and changes the properties from soft elasto-visco-plastic to stiffer visco-elastic material. Experimentally, this transition can be observed by a moving die rheometer (MDR) test at prescribed temperatures. The time- and temperature-dependent description of this process can be found in [4] for the isothermal case. Many formulations have been presented, for example [5], [6], [7], all use an incremental formulation, which is applicable for non-isothermal cases. The mechanical description of the rubber curing process is challenging and only few models are available in the literature. During vulcanization, the molecular chains of the rubber build cross-links with each other resulting in a much stiffer behaviour. Due to the cross-links, the deformation state of the uncured rubber is fixed and will be the new equilibrium configuration for the cured rubber. In [8], a continuously evolving model is presented for rubber curing by adding additional springs in the model as the state of cure increases. The included springs increase the stiffness of the rubber and affect the stress response only from the deformation after been introduced. This approach is motivated by the increase of cross-links during vulcanization. However, the thermo-mechanical consistency is not addressed due to the additional energy from the added springs. In [9], a simple thermo–mechanically consistent model is presented. It consists of a hyperelastic spring connected to a modified dashpot. In the uncured phase, the dashpot represents the plastic behaviour and the lower stiffness is taken into account. While the state of cure is increasing, the dashpot gets fixed, representing the new equilibrium state. Another model based on the conservation of energy is developed in [10]. Two separate models for the uncured and cured rubber phase are introduced and combined according to the current state of cure. While the state of cure is increasing, a modified curing strain evolves based on the assumption that the free energy remains constant during vulcanization. Therefore, thermo-mechanical consistency is fulfilled and a new equilibrium state is achieved. Due to the consistent derivation of the algorithmic tangents, the model can represent large deformations and the tire production simulation. However, during tire production simulation, the Lagrangian mesh distorts heavily and cannot be used for any further simulations. In this contribution, an Arbitrary Lagrange Eulerian (ALE) formulation is introduced to overcome these challenges.

In the standard approach for the finite element method (FEM), the material points are connected directly to the nodes of the computational mesh. In this pure Lagrangian formulation, several advantages are present, for example the direct tracking of the boundary of the body and the history variables are assigned to a unique integration point, which does not change during the analysis. Furthermore, the solution algorithm itself is easy and the deformation can be directly computed from the reference configuration. However, for a finite strain analysis, where large deformations are present, for example in an in-moulding simulation of a green tire, the computational mesh can distort severely. In this case, the deformations cannot be approximated correctly by the elements and the solution algorithm cannot find a converged state for the principle balance laws, causing the simulation to abort. The opposite formulation is an Eulerian framework, where the spatial mesh remains fixed while the material can freely move [11]. Due to the fixed spatial mesh, no distortions can occur and the position of the material is evaluated at the nodes. The major disadvantage of this approach is the complex tracking of the boundary, due to the fixed spatial mesh. In an Arbitrary Lagrange Eulerian (ALE) formulation, the advantages of both approaches are combined. The computational mesh is able to deform and the boundary can be tracked while the material is also able to flow through the mesh. This procedure is already used, among others, in the formulation of steady state rolling tires [12], [13] and in metal forming processes [14].

For the ALE approach, two different solution algorithms are documented in literature, a fully coupled monolithic formulation [15], [16] and a staggered computation [17], [18]. Due to the rather complex formulation of the fully coupled approach, a staggered algorithm is more common and used in this contribution. In a staggered algorithm, the ALE equations are computed independent to the Lagrangian step. In other words, the smoothing of the mesh, the advection of the material history and the Lagrangian step, where an actual time step of the solution is carried out, is done one after the other. Due to this split, already available commercial or non-commercial finite element method (FEM) programs can be employed for the Lagrangian step. Instead of an adaptive mesh-refinement, where the node to element connectivity changes and a completely new mesh is used for the next time step, the ALE approach uses the same node to element connection and adjusts only the position of each node to obtain a higher quality mesh. In this contribution, the reference element technique [19] is used for the smoothing of the mesh. In this algorithm, every distorted element is compared to a perfectly smooth element and forces are computed to adjust the nodes. In other ALE approaches, a minimization problem is solved to minimize a defined potential function, for example an adapted Neo-Hookean strain energy density function [20], to overcome the distortion of the mesh.

The most important step of the ALE approach is the advection of the material history variables from one mesh to the other. Due to the different representation of the body by another mesh, inaccuracies in the projection lead to different results compared to a pure Lagrangian simulation. For hyperelastic and hyperelastoplastic materials, which are used in metal forming simulation, explicit schemes for the advection step are employed. The two main approaches are the Lax–Wendroff scheme [21] and the Godunov-like update formulation [22]. Both approaches use an approximation of the tangent field of every history variable to compute the consistent projection independently, which is disadvantageous for a material formulation with a large number of history variables. Other ALE approaches use the projection of the history variables from the integration points to the nodes of the old mesh. In this case, the position of the new integration point with respect to the old mesh needs to be found. Using the isoparametric concept and knowing the isoparametric coordinates allows to project all history variables at once to the new integration point. This algorithm is stable and can be implemented easily, however, due to the double projection, inaccuracies are introduced in the solution. Therefore, a new algorithm for projecting the history variables and the deformation history from one mesh to another is presented in this contribution. Instead of projecting the history values to the nodes, they are projected directly from the integration points of the old mesh to the new integration point. Introducing a mesh, generated by the integration points, and, computing the isoparametric coordinates with respect to this mesh, only one interpolation is needed to advect all history variable in a numerically efficient way.

In this contribution, a consistent material model describing vulcanization as well as the uncured and cured rubber phase is briefly explained. Subsequently, the used fractional-step ALE formulation is presented with a new advection algorithm for projecting the history variables. The formulation is then applied to several numerical examples, showing its benefits compared to a pure Lagrangian formulation.

Section snippets

Thermo–mechanically consistent material formulation

The phase change of the rubber is described by a material formulation based on the micro-sphere model [23]. The main idea of the material model is to combine already available approaches for uncured and cured rubber in a thermo–mechanically consistent way and to describe the permanent strain, which occurs during curing. Therefore, an internal variable c, the state of cure (SOC), is introduced. The formulation is presented briefly in this contribution. For a detailed description the reader is

Arbitrary Lagrangian Eulerian formulation

In a Lagrangian framework, the motion of the computational mesh is connected to the motion of the body. The deformation of the material is computed from an initially defined and fixed reference configuration B. In an ALE framework, the material motion is partially decoupled from the mesh. This is done by splitting the motion into a material and spatial motion, see Fig. 2. The position of any particle m can be described in the material configuration Xt=χm,tand, analogously, in the spatial

Numerical examples

The herein presented Arbitrary Lagrangian Eulerian formulation is applied to four independent numerical examples to show its advantages compared to a Lagrangian formulation and to other projection algorithms.

Conclusion

An Arbitrary Lagrangian Eulerian framework for uncured and cured rubber is introduced by an improved approach for the advection of the history variables. A thermo–mechanically consistent material model for rubber curing is presented. For the smoothing of the mesh, an algorithm by [19] is used, which provides a high quality mesh in few iterations. Due to the staggered and uncoupled ALE framework, the spatial mesh quality is much better in comparison to coupled frameworks [20], where the material

CRediT authorship contribution statement

Thomas Berger: Methodology, Software, Writing – original draft. Michael Kaliske: Reviewing and editing, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

The financial support of this research by Hankook Tire Ltd., Daejeon, South Korea , is gratefully acknowledged.

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