Numerical evaluation of the temperature field of steady-state rolling tires

Rubber is the main component of pneumatic tires. The tire heating is caused by the hysteresis effects due to the deformation of the rubber during operation. Tire temperatures can depend on many factors, including tire geometry, inflation pressure, vehicle load and speed, road type and temperature and environmental conditions. The focus of this study is to develop a finite element approach to computationally evaluate the temperature field of a steady-state rolling tire. For simplicity, the tire is assumed to be composed of rubber and body-ply. The nonlinear mechanical behavior of the rubber is characterized by a Mooney–Rivlin model while the body-ply is assumed to be linear elastic material. The coupled effects of the inflation pressure and vehicle loading are investigated. The influences of body-ply stiffness are studied as well. The simulation results show that loading is the main factor to determine the temperature field. The stiffer body-ply causes less deformation of rubber and consequently decreases the temperature.     

An evolution equation based approach to topology optimization

The objective of topology optimization is to find a mechanical structure with maximum stiffness and minimal amount of used material for given boundary conditions [2]. There are different approaches. Either the structure mass is held constant and the structure stiffness is increased or the amount of used material is constantly reduced while specific conditions are fulfilled. In contrast, we focus on the growth of a optimal structure from a void model space and solve this problem by introducing a variational problem considering the spatial distribution of structure mass (or density field) as variable [3]. By minimizing the Gibbs free energy according to Hamilton's principle in dynamics for dissipative processes, we are able to find an evolution equation for the internal variable describing the density field. Hence, our approach belongs to the growth strategies used for topology optimization. We introduce a Lagrange multiplier to control the total mass within the model space [1]. Thus, the numerical solution can be provided in a single finite element environment as known from material modeling. A regularization with a discontinuous Galerkin approach for the density field enables us to suppress the well-known checkerboarding phenomena while evaluating the evolution equation within each finite element separately [4]. Therefore, the density field is no additional field unknown but a Gauß-point quantity and the calculation effort is strongly reduced. Finally, we present solutions of optimized structures for different boundary problems. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

不可压缩层合橡胶圆管径向膨胀的稳定性分析

研究由两类不可压缩的橡胶材料组成的层合圆柱形管道,在内表面受到突加的径向压力作用时的膨胀机理．建立了问题的数学模型;利用材料的不可压缩条件、边界条件以及圆管的径向位移和径向应力的连续性条件将相应的控制方程约化为一个二阶非线性常微分方程,并得到了该方程的首次积分．给出了管道拟静态膨胀和动态膨胀的定性分析,特别地,结合数值算例讨论了材料参数、结构参数以及径向压力对管道径向膨胀和非线性周期振动的影响．     

Dynamic analysis of multi-directional functionally graded annular plates

Dynamic analysis of multi-directional functionally graded annular plates is achieved in this paper using a semi-analytical numerical method entitled the state space-based differential quadrature method. Based on the three-dimensional elastic theory and assuming the material properties having an exponent-law variation along the thickness, radial direction or both directions, the frequency equations of free vibration of multi-directional functionally graded annular plates are derived under various boundary conditions. Numerical examples are presented to validate the approach and the superiority of this method is also demonstrated. Then free vibration of functionally graded annular plates is studied for different variations of material properties along the thickness, radial direction and both directions, respectively. And the influences of the material property graded variations on the dynamic behavior are also investigated. The multi-directional graded material can likely be designed according to the actual requirement and it is a potential alternative to the unidirectional functionally graded material.     

Micromechanical analysis of the stress transfer in single-fiber composite: The influence of the uniform and graded interphase with finite-thickness

A theoretical model is developed to analyze the stress transfer between fiber and matrix through the interphase with finite thickness. The Young's modulus of interphase is assumed to be homogeneous uniform or power-graded along radial direction while other material parameters are constants. The bonds between fiber and interphase as well as between interphase and matrix are perfect. The geometrical equations are strictly satisfied except that the radial displacement gradient with respect to the axial direction is neglected, as its magnitude is much smaller than that of the axial displacement gradient with respect to the radial direction. The equilibrium equations along radial direction are strictly satisfied, while the equilibrium equations along axial direction are satisfied in the integral forms. In addition, both the interfacial displacement and stress continuity conditions as well as stress boundary conditions are enforced exactly. Two coupled 2nd-order ordinary differential equations can be obtained in terms of average axial stresses in fiber and matrix. Finite element analysis (FEA) with refined mesh for single-fiber composite containing uniform interphase with finite thickness is developed to validate the present model. Series of parameter studies are performed to investigate the influence of interphase properties and thickness as well as the fiber volume content and model length on the stress distribution in composites.     

Effective elastic properties of particulate-reinforced compossite materials

A numerical approach for determination of the effective properties of particulate composite materials has been developed. A representative volume element (RVE) of the composite material is analyzed with help of the finite-element method. Uniform boundary displacements or tractions are applied on the boundaries of the RVE for introducing the known average strain in the RVE. Local stress and strain distributions in the RVE are calculated using the finite-element method. Different effective elastic constants can be calculated by averaging the local fields corresponding to different sets of boundary conditions. The present approach allows us to determine the effective properties of particle-reinforced composites with acceptable accuracy. The calculated effective properties of the composite are between the upper and lower Hashin—Shtrikman bounds. The results based on the present approach lead to higher stiffness of composites in comparison with analytical approaches.Institute fur Werkstoffwissenschaften, Fachberech Werkseoffwissenschaften, Martin-Luther-Universität Halle-Wittenberg, D-06099 Halle, Germany. Published in Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 450–459, July–August, 1997.     

A new Chebyshev spectral approach for vibration of in-plane functionally graded Mindlin plates with variable thickness

This paper presents a new and simple approach for vibration analysis of in-plane functionally graded (IPFG) plates with variable thickness based on the Chebyshev spectral method. Both the material properties and the thickness which vary in the plane of the plate are approximated by high-order Chebyshev expansions. Gauss-Lobatto sampling is adopted for spatial discretization. A consistent governing equation in discrete form is derived by utilizing Lagrange’s equation for all kinds of IPFG plates whose material property functions and thickness function are square-integrable and infinitely differentiable in the domain. Its mass matrix is diagonal and stiffness matrix is symmetric. Classical and point-supported boundary conditions are incorporated through projection matrices. This approach is independent of the type of material gradation, meshfree, and flexible to adjust the computation cost and precision according to needs. A series of numerical examples involving different kinds of material gradations, thickness variations, and boundary conditions are carried out to demonstrate the validity of the proposed method. The results obtained from the present method show a good convergence and agree with those in literature very well.     

辛体系下THz波在碳纳米管阵列中的传播

对于一个被周期性平行有限长碳纳米管阵列填充的平面波导,本文基于平行碳纳米管阵列的等效介质模型,忽略其空间色散,考虑了电磁波的损耗,从而得到填充介质的介电特性,并将电磁波在波导中的传播导入到Hamilton体系,同时考虑两侧边界条件均为理想导电边界条件,从而在辛理论框架下求解本征值方程,得到了电磁波传播色散关系.分析可知,存在一个窄的频段,电磁波基模无法传播,然而在频段外,电磁波基模传播具有极其低的损耗,这使得碳纳米管阵列具有宽频带传播的特性,这些特性使得碳纳米管阵列相比于传统材料具有更优的传播特性.     

On the viscoelastic properties of prestrained rubber

The material behaviour of rubber is investigated for a special type of deformations namely small amplitude vibrations superposed on finite static prestrain. By restricting the magnitude of the prestrain it is possible to derive a simplified constitutive equation which can be used to solve simple boundary value problems. On this theoretical background the stiffness of two types of sample geometries were computed and measured in a vibration experiment. The resulting expression for the dynamic sample stiffness contains besides the classical relaxation modulus three other complex, frequency dependent moduli and the static prestrain.     

Development of size-dependent consistent couple stress theory of Timoshenko beams

In this paper, a linear size-dependent Timoshenko beam model based on the consistent couple stress theory is developed to capture the size effects. The extended Hamilton's principle is utilized to obtain the governing differential equations and boundary conditions. The general form of boundary conditions and the concentrated loading are employed to determine the exact static/dynamic solution of the beam. Utilizing this solution for the beam's deformation and rotation, the exact shape functions of the consistent couple stress theory (C-CST) is extracted, which leads to the stiffness and mass matrices of a two-node C-CST finite element beam. Due to the complexity and high computational cost of using the exact solution's shape functions, in addition to the Ritz approximate solution, a two primary variable finite element model of C-CST is proposed, and the corresponding general deformation and rotation fields, shape functions, mass and stiffness matrices are calculated. The C-CST is validated by comparing the prediction of different beam models for a benchmark problem. For the fully and partially clamped cantilever, and free-free beams, the size dependency of the formulations is investigated. The static solutions of the classical and consistent couple stress Timoshenko beam models are compared, and a criterion for selecting the proper model is proposed. For a wide range of material properties, the relation between the beam length and length scale parameter is derived. It is shown that the validity domain of the consistent couple stress Timoshenko model barely depends on the beam's constituent material.     

Experimental Investigations on and Modelling of the Transient Phenomena of the Payne-Effect.

The wide majority of industrially–used rubber is filled with a considerable amount of active fillers like carbon black or silica. Due to this, the material is strengthend and mechanical key features like stiffness and strength are significantly increased. In contrast to unfilled rubber, filled elastomers show a pronounced amplitude dependence, which is widely known as Fletcher–Gent or Payne effect. Besides that, some recently published works show a significant history dependence of this effect with distinctive relaxation phenomena. In the present work, some experiments on typical tyre rubber compounds with focus on these amplitude dependent phenomena are presented. On this basis, an appropriate thermodynamic consistent phenomenological material model of finite viscoelasticity is introduced. In order to incorporate the history dependent phenomena of the amplitude dependence, this model is generalized with intrinsic time scales on the basis of inner structure variables, which are a measure of the materials microstructure. The performance of the model is critically demonstrated by a few simulation results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An iterative elastic SBFE approach for collapse load analysis of inelastic structures

The paper proposes a novel numerical approach that incorporates the use of a modified elastic compensation method, within a polygon scaled boundary finite element framework, to determine the maximum load capacity of structures at plastic collapse. The distinctive feature of the proposed scheme is its effective computational ability in performing a series of successive elastic analyses by systematically adjusting elastic material properties of structures up to failure. The quadtree structural discretization within a polygon scaled boundary finite element platform enables model construction of sophisticated geometries at modest computing effort and thus the effective analysis of large-scale structures. The approach overcomes the challenges associated with stress singularity and locking phenomena under incompressibility conditions, even in the presence of high-order nonlinear yield loci. The robustness and accuracy of the proposed scheme are validated through a number of benchmarks and available practical engineering applications in 2D and 3D spaces. These illustrate the influences of some key algorithmic parameters, and the satisfaction of a lower-bound limit given by the present analysis method for a sufficiently fine discretization.     

Analytical solution for the fully coupled static response of variable stiffness composite beams

An analytical solution is presented for the 3D static response of variable stiffness non-uniform composite beams. Based on Euler-Bernoulli theory, a set of governing differential equations are obtained, in which four degrees of freedom are fully coupled. For the variable stiffness beam, the governing field equations have variable coefficients reflecting the stiffness variation along the beam. Using the direct integration technique, the general analytical solution is derived in the integral form and the closed-form expressions of the obtained solutions are presented employing a series expansion approximation. The series expansion representation enables the proposed approach to be applicable for variable stiffness composite beams with arbitrary span-wise variation of properties. As an alternative solution, the Chebyshev collocation method is applied to the proposed formulation to verify the results obtained from the analytical solution. A number of variable stiffness composite beams made by fibre steering with various boundary conditions and stacking sequences are considered as the test cases. The static response are presented based on the analytical solution and Chebyshev collocation method and excellent agreement is observed for all test cases. The proposed model presents a reliable and efficient approach for capturing the complicated behaviour of variable stiffness non-uniform composite beams.     

Analysis of extension propagation process of interface crack between belts of a radial tire using a finite element method

A radial tire is a very complex structure made from rubber elastomers and fiber–rubber composite materials. During its use, extension propagation of interface crack between belts can occur, which obviously affects its durability and life. In the present paper, a new mathematical model of extension propagation of interface crack in complex composite structures is presented. The model can reveal the extension propagation dependence of interface crack on the relative size of energy release rates at the left and right crack tips and on the interfacial material properties. The extension propagation model of interface crack, Irwin’s virtual crack close technique and the finite element analysis method are used together in simulating numerically the extension propagation process of a interface crack between belts of a radial tire. The present study numerical results show that the extension propagation model of interface crack proposed in this paper can more realistically characterize the complexity of the extension propagation process of interface crack in complex composite structures.     

Application of Calderón’s inverse problem in civil engineering

In specific fields of research such as preservation of historical buildings, medical imaging, geophysics and others, it is of particular interest to perform only a non-intrusive boundary measurements. The idea is to obtain comprehensive information about the material properties inside the considered domain while keeping the test sample intact. This paper is focused on such problems, i.e. synthesizing a physical model of interest with a boundary inverse value technique. The forward model is represented here by time dependent heat equation with transport parameters that are subsequently identified using a modified Calderón problem which is numerically solved by a regularized Gauss-Newton method. The proposed model setup is computationally verified for various domains, loading conditions and material distributions.     

A novel treatment of crack boundary conditions in phase field models of fracture

In the phase field approach for fracture an additional scalar field is introduced in order to describe the state of the material between intact and fully broken. So far, for the loading dependent degradation of stiffness (damage) either the volumetric-deviatoric split of strain [1, 2] or the spectral decomposition [3, 4] is used. In contrast to such an isotropic degradation of stiffness, the fully broken state represents a crack with a particular orientation. Both aforementioned approaches do not take the crack orientation into account. This may lead to the violation of the crack boundary conditions. In order to satisfy these conditions the phase field approach is modified here by taking the orientation of the crack into account. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Frictional contact of elastomer materials on rough rigid surfaces

The frictional behavior of elastomer materials is still unexplored, but very important for many industrial applications. Special attention is turned to rubber friction on rough road tracks. Due to the non‐rigid material characteristics of the rubber it is not sufficient to use a constant friction coefficient like Coulombs law. The frictional qualities depend on many different influences like sliding velocity, applied normal stress, surface roughness, material properties and the temperature in the contact zone. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two approaches for the numerical determination of electromagnetic fields near material interfaces using radial basis functions in a meshless method

This article discusses the use of radial basis functions in the solution of partial differential equations. In particular, an investigation of how to implement boundary conditions at material interfaces is presented. A comparison is made of solutions obtained using only radial basis functions with those obtained using either jump functions or elliptical functions in conjunction with standard radial basis functions. The purpose of adding either jump functions or elliptical functions is to alleviate the inaccurate handling of the boundary conditions at material interfaces. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010