3D Contact Problems Research Articles

Peridynamic contact models for fracture analysis based on the micro-beam bond

Contact problems are ubiquitous in engineering systems. Establishing contact models to predict the crack propagation and fracture is significant. Therefore, the present study proposes two peridynamic contact models for two-dimensional (2D) and three-dimensional (3D) contact problems based on the micro-beam bond and the Hertz theory, using the relative position of particles and the contact micromoduli to calculate the contact forces. Firstly, the strain is generated on the contact surface when two bodies come into contact. Simultaneously, the contact stiffness is obtained by the Hertz theory, and the contact forces are calculated according to the relationship between the strain of contact surface and the contact stiffness. Next, based on the peridynamic theory, the contact horizon and contact micromoduli are defined for the calculation of contact forces. Finally, the contact micromoduli for 2D and 3D contact problems are derived by equating the contact forces obtained from both theories. Additionally, this study defines the correction coefficients to reduce the calculation error of the proposed models and investigates the influence of different contact horizons on the calculation of contact forces. To validate the effectiveness of the proposed contact models, three numerical examples are compared with the results from the finite element method or experiments. It is demonstrated that the proposed contact models accurately calculate the contact forces and predict the crack propagation.

Semi-Analytical Model for 3D Multilayered Rolling Contact Problems With Different Creepage Combinations

Abstract In this work, a semi-analytical model replicating 3D rolling contact problems with elastic coating layers has been proposed. The dimension reduced boundary element-based model rests on the conventional half-space assumptions, with influence coefficients describing the relationship between the stress and the displacement. In this model, the influence coefficient describes the stress–strain relationship for multilayered materials, which is derived from the corresponding frequency–response functions. The governing equations for 3D rolling contact problems are solved efficiently using the conjugate gradient method with the classical fast Fourier transformation (FFT)-based technique to accelerate the calculation of the integral equation relating the stresses to the strains, and all the possible creepages, i.e., the longitudinal, lateral, and spin creepages, are considered. The results obtained with the present numerical model are compared with existing results, and good agreement is found. Some selected results are presented to show the effect of the material elasticity and the thickness of the coating, and the layering, on the traction distribution and the traction coefficient under different creepage combinations. In this way, this work provides important information related to the composition of layered coatings in rolling contact applications.

Capturing Animation-Ready Isotropic Materials Using Systematic Poking

Capturing material properties of real-world elastic solids is both challenging and highly relevant to many applications in computer graphics, robotics and related fields. We give a non-intrusive, in-situ and inexpensive approach to measure the nonlinear elastic energy density function of man-made materials and biological tissues. We poke the elastic object with 3d-printed rigid cylinders of known radii, and use a precision force meter to record the contact force as a function of the indentation depth, which we measure using a force meter stand, or a novel unconstrained laser setup. We model the 3D elastic solid using the Finite Element Method (FEM), and elastic energy using a compressible Valanis-Landel material that generalizes Neo-Hookean materials by permitting arbitrary tensile behavior under large deformations. We then use optimization to fit the nonlinear isotropic elastic energy so that the FEM contact forces and indentations match their measured real-world counterparts. Because we use carefully designed cubic splines, our materials are accurate in a large range of stretches and robust to inversions, and are therefore "animation-ready" for computer graphics applications. We demonstrate how to exploit radial symmetry to convert the 3D elastostatic contact problem to the mathematically equivalent 2D problem, which vastly accelerates optimization. We also greatly improve the theory and robustness of stretch-based elastic materials, by giving a simple and elegant formula to compute the tangent stiffness matrix, with rigorous proofs and singularity handling. We also contribute the observation that volume compressibility can be estimated by poking with rigid cylinders of different radii, which avoids optical cameras and greatly simplifies experiments. We validate our method by performing full 3D simulations using the optimized materials and confirming that they match real-world forces, indentations and real deformed 3D shapes. We also validate it using a "Shore 00" durometer, a standard device for measuring material hardness.

FFT-Based Numerical Method for Nonlinear Elastic Contact

In theoretical research pertaining to sealing, a contact model must be used to obtain the leakage channel. However, for elastoplastic contact, current numerical methods require a long calculation time. Hyperelastic contact is typically simplified to a linear elastic contact problem, which must be improved in terms of calculation accuracy. Based on the fast Fourier transform, a numerical method suitable for elastoplastic and hyperelastic frictionless contact that can be used for solving two-dimensional and three-dimensional (3D) contact problems is proposed herein. The nonlinear elastic contact problem is converted into a linear elastic contact problem considering residual deformation (or the equivalent residual deformation). Results from numerical simulations for elastic, elastoplastic, and hyperelastic contact between a hemisphere and a rigid plane are compared with those obtained using the finite element method to verify the accuracy of the numerical method. Compared with the existing elastoplastic contact numerical methods, the proposed method achieves a higher calculation efficiency while ensuring a certain calculation accuracy (i.e., the pressure error does not exceed 15%, whereas the calculation time does not exceed 10 min in a 64 × 64 grid). For hyperelastic contact, the proposed method reduces the dependence of the approximation result on the load, as in a linear elastic approximation. Finally, using the sealing application as an example, the contact and leakage rates between complicated 3D rough surfaces are calculated. Despite a certain error, the simplified numerical method yields a better approximation result than the linear elastic contact approximation. Additionally, the result can be used as fast solutions in engineering applications.

Duality Method for Solving 3D Contact Problems with Friction

The article studies a 3D contact problem with Coulomb friction for an elastic body resting on a rigid support. The solution of the quasi-variational formulation of the problem is defined as a fixed point of some mapping that associates the given force of the normal reaction of the support with the value of the normal stress in the contact zone. The fixed point is sought by the method of successive approximations, the convergence of which is proved using modified Lagrange functionals. The results of the numerical solution using finite element modeling and the proximal gradient method are presented.

Duality Method for Solving 3D Contact Problems with Friction

Duality Method for Solving 3D Contact Problems with Friction

Response of elastic/viscoelastic layers on an elastic half-space in rolling contacts: Towards a new modeling approach for elastohydrodynamic lubrication

The 3D quasi-static contact problem of a rigid sphere rolling on an elastic half-space covered by a thin viscoelastic/elastic layer is studied as representative of soft layered contacts in engineering, physics, and biomedical applications, as well as for its potential merit to model Elasto-Hydrodynamically Lubricated (EHL) contacts in specific EHL film behavior suggested in earlier research. The viscoelastic layer behavior is modeled with a standard linear solid (SLS) model with a single relaxation time. Two approaches are used, the foundation (Winkler) reduced model assuming unidirectional stress–strain behavior normal to the surface only and the Papkovich–Neuber potential model which, based on the complete Navier–Cauchy equations, accounts for non-local support and bending effects. The models are validated against literature and compared. Whereas coated/layered contact problem studies mostly consider relatively thick coatings, and results focus on the pressure distribution and the contact area size, in this paper we consider layers with a thickness much smaller than the contact radius, i.e. of the order of a thickness of a conventional EHL film. The details of the layer deformation, pressure profiles and subsurface stresses are presented and interpreted in terms of the underlying physics, in particular the thin layer limit common also in thin fluid layers. It is shown that two dimensionless parameters, the ratio of the elastic modulus between the layer and the substrate and the ratio of the layer thickness to the corresponding Hertz contact width, dominate the systematic response for elastic layered contact problems. And four dimensionless parameters with two extras, the Deborah number based on the Hertz contact width and the ratio of the two elastic limits of the SLS viscoelastic material, are needed and analyzed for viscoelastic layered contacts. The results presented provide a good framework for the understanding and interpretation of the effects of viscoelastic layers on the deformation and pressure distribution of contact problems. Finally, the capability of actually modeling EHL film behavior via a viscoelastic layer suggested by van Emden et al. (2017) is discussed.

A coupled FETI-BDNM for solving 3D elastic frictional contact problem

Based on the B-differentiable Newton method (BDNM) and finite element tearing and interconnecting (FETI) algorithm, a coupled approach named FETI-BDNM is developed to solve the 3D elastic contact problem with friction. In this method, the contact bodies are decomposed into a number of subdomains. The interfaces between the two adjacent substructures will be skillfully considered as the special contact surfaces that are always in the stick state. By resorting to the generalized inverse of stiffness matrices of the subdomains, the relationships between the relative displacements and forces on contact/interface, i.e., contact flexibility matrices will be computed efficiently. The contact equations, displacement continuity equations and equilibrium equations for each subdomain are expressed in the form of B-differentiable equations (BDEs) uniformly, which can be solved by the damped Newton-Raphson method with 1D Armijo type linear search. The proposed method makes it possible to solve large-scale contact problems without compromising the solution accuracy compared to the original BDNM, and the good convergence of BDEs and parallel scalability of the FETI algorithm will be preserved. Numerical examples validate the accuracy, efficiency, robustness and effects of the number of subdomains on the computational time of the proposed method.

A novel SAM/X-FEM coupling approach for the simulation of 3D fatigue crack growth under rolling contact loading

The present work proposes a novel efficient numerical approach for the simulation of three-dimensional propagation of non-planar frictional crack under Rolling Contact Fatigue (RCF). The developed model relies on a global–local strategy involving the Semi-Analytical Method (SAM), dedicated to the resolution of 3D contact problems, and the eXtended-Finite Element Method (X-FEM), dedicated to the solving of 3D crack problems. Similar to a submodeling technique, it consists in performing a local X-FEM analysis of the 3D RCF crack problem by means of boundary conditions extracted from a prior global SAM analysis which solves the contact problem between two semi-infinite bodies without considering the crack. A powerful procedure for the transfer of relevant mechanical quantities between the SAM and X-FEM models is developed. It allows the use of non-matching and non-conforming discretization scheme for the global SAM model and the local X-FEM model, particularly worthwhile for the pre-processing stage. An initial arbitrary stress profile can be considered to investigate the influence of residual stresses on crack behavior. The non-intrusive SAM/X-FEM coupling is integrated into a fully automatic crack propagation algorithm. This provides a powerful and user-friendly tool suitable for industrial applications, which easily handles 3D long non-planar frictional crack growth in the region of interest. After introducing some details concerning the implementation of the model, a validation of the innovative SAM/X-FEM coupling is performed by using results from the literature. Furthermore, a practical example demonstrates the great potential of this novel numerical technique to simulate in a fast, robust and accurate way the 3D complex behavior of fatigue crack under moving contact. • Simulation of the 3D frictional crack propagation under rolling contact conditions. • Non-intrusive global–local approach based on the coupling of SAM and X-FEM. • Field transfer procedure allowing independent meshes for the SAM and X-FEM models. • Fast, efficient and robust resolution coupled with a pre-processing time saving. • Influence of residual stresses on crack behavior highlighted.

A virtual element method for 3D contact problems with non-conforming meshes

The Virtual Element Method is ideal for problems with non-conforming meshes since nodes can be inserted at the virtual element boundary to obtain a conforming mesh structure, without changing the ansatz of the element. The contact conditions can be employed on different enforcing strategies. This feature can be explored also for three-dimensional contact problems. In this work, we introduce a new framework, including a remeshing strategy, which leads to conforming meshes. Various numerical examples in 3D will be shown, including standard cases like the contact patch test and Hertzian contact.

Analysis of 2D contact problems under cyclic loads using IGABEM with Bézier decomposition

Non-uniform rational B-splines (NURBS) are a convenient way to integrate CAD software and analysis codes, saving time from the operator and allowing efficient solution schemes that can be employed in the analysis of complex mechanical problems. In this paper, the Isogeometric Boundary Element Method coupled with Bézier extraction of NURBS and conventional BEM are used for analysis of 2D contact problems under cyclic loads. A node-pair approach is used for the analysis of the slip/stick state. Furthermore, the extent of the contact area is continuously updated to account for the nonlinear geometrical behaviour of the problem. The Newton–Raphson’s method is used for solving the non-linear system. A comparison to analytical results is presented to assess the performance and efficiency of the proposed formulation. Both BEM and IGABEM show good agreement with the exact solution when it is available. On most examples, they are equivalent with some advantage for IGABEM, though the former is slightly more accurate in some situations. This is probably due to the smoothness of NURBS not being able to describe sharp edges on tractions. As expected, IGABEM incurs in higher computational cost due to the basis being more complex than conventional Lagrangian polynomials.

PROGRAM CALCULATION OF EFFECTIVE CONTACT AND BENDING CRITERIONAL STRESSES OF TEETH OF NOVIKOV GEARING

The article presents the full version of the NOVGOST computer program for determining the effective contact and bending phase criterional stresses of Novikov gearing teeth with basic profile according to GOST 15023–76, developed in the system MAPLE 17. The program reproduces the process of real multipoint engagement of Novikov gearing, which takes into account the inevitable errors in the manufacture and assembly of a gear pair, dependent on the assigned degree of accuracy, as well as the rigidity of the teeth themselves and the associated parts of the gearbox – shafts and bearings. The program is based on the results of solving a 3D contact problem to determine the phase dependences of the effective contact and bending stresses, as well as rigidity of teeth from the ends of the teeth, when the contact area is positioned anywhere along the length of the tooth, including the end of gear rim, taking into account the influence of the latter on the listed parameters through the coefficients of influence. With the help of the presented program partial forces in real multipoint engagement are determined, acting on each contact area and, In this way, phase criterional stress are calculated, determining the load-bearing capacity and service life of Novikov՚s gearing. The program laid the opportunity to set the parameter of the longitudinal modification of the teeth, optimal selection of which allows significantly reduce the criterional stress and thereby significantly improve working capacity projected gearing. The program can be very useful for designing and manufacturing in production competitive in the global market gearboxes with high technical economic indicators, equipped with Novikov gearing.

Signorini-Type Problems over Non-Convex Sets for Composite Bodies Contacting by Sharp Edges of Rigid Inclusions

A new type of non-classical 2D contact problem formulated over non-convex admissible sets is proposed. Specifically, we suppose that a composite body in its undeformed state touches a wedge-shaped rigid obstacle at a single contact point. Composite bodies under investigation consist of an elastic matrix and a rigid inclusion. In this case, the displacements on the set, corresponding to a rigid inclusion, have a predetermined structure that describes possible parallel shifts and rotations of the inclusion. The rigid inclusion is located on the external boundary and has the form of a wedge. The presence of the rigid inclusion imposes a new type of non-penetration condition for certain geometrical configurations of the obstacle and the body near the contact point. The sharp-shaped edges of the obstacle effect such sets of admissible displacements that may be non-convex. For the case of a thin rigid inclusion, which is described by a curve and a volume (bulk) rigid inclusion specified in a subdomain, the energy minimization problems are formulated. The solvability of the corresponding boundary value problems is proved, based on analysis of auxiliary minimization problems formulated over convex sets. Qualitative properties of the auxiliary variational problems are revealed; in particular, we have found their equivalent differential formulations. As the most important result of this study, we provide justification for a new type of mathematical model for 2D contact problems for reinforced composite bodies.

Stable node-based smoothed finite element method for 3D contact problems

Stable node-based smoothed finite element method for 3D contact problems

Effect of surface relief on sliding friction of viscoelastic bodies

A model for calculating the sliding friction force between a surface with regular relief and a viscoelastic foundation in dry conditions is suggested. The Winkler type viscoelastic foundation with both normal and tangential compliances is considered. The local Coulomb friction law is assumed between the surfaces. The total friction is calculated including the contribution of both local adhesive friction and hysteretic losses in the viscoelastic material when elements of the surface relief cyclically deform the material during sliding. The contact problem is solved by the strip method in which the area of interaction is divided into strips parallel to the direction of sliding, and a 2D contact problem is solved in each strip in the closed form. Two types of periodic profile of the moving surface are considered—a profile with trapezoidal asperities and a doubly periodic sinusoidal profile. The model is applied to analyze the influence of the parameters of surface relief on the coefficient of sliding friction for various values of external load and sliding velocity.

3D Contact Problem with Adhesion for Two-Layered Elastic Half-Space

3D Contact Problem with Adhesion for Two-Layered Elastic Half-Space

Analysis of mechanical deformation effect on the voltage generation of a vertical contact mode triboelectric generator

One of the associated factors that controls the performance of a triboelectric generator (TEG) is the mechanical deformation of the dielectric layer. Therefore, a good contact model can be a prominent tool to find a more realistic and efficient way of determining the relationships between the contact and electrical output of the generator. In this study, experiments are conducted on a vertical contact mode triboelectric generator under an MTS machine. The open-circuit voltages are measured at different loads imposed by the MTS by controlling the cyclic displacement of the top tribo layer of the generator. A finite-element-based theoretical model is developed to explain the behavior of the generator during the experiments. The 2D-contact problem of the micro-structured tribo layers is simulated and then the contact results are integrated into 3D to find the actual contact area between the two surfaces. These numerical contact results improve the existing theoretical model by evaluating the correct surface charge density and contact area as a function of the input parameters. The excellent agreement between our experimental and theoretical results illustrates that theoretical modeling could be used as a robust approach to predict the mechanical and electrical performance of TEGs. In addition, some parametric studies of the harvester are presented here for different geometrical parameters of the microstructures.

Analytical approach in determination of the surface layer parameters of machine parts hardened by a moving elastic indenter

The reducing of machine time during surface hardening by smoothening is possible by increasing in the contact length between the hardening element and part surface. We study the case: the indenter-working surface is a fragment of a cylindrical surface, which the symmetry axis is parallel to the axis of symmetry of the part hardened surface. We suggest that such a process scheme is corresponds to a 2D contact problem in which a cylindrical stamp moves in an inertial coordinate system along an elastic half-plane (circle). We tasking the boundary conditions in the contact zone, using the complex potentials developed in the works of L.N. Galin.

The effect of conduction and convective conditions at interstitial regions on 3D thermoelastic contact problems

Contact variables can be greatly affected by small temperature variations occurring in the contact zone and the surrounding areas. So it is essential to take into account the temperatures and heat fluxes on these regions to obtain the deformations and tractions that appear in the bodies. This work studies how the 3D thermomechanical contact variables are affected by the thermal conductive and convection boundary conditions at the interstitial regions (i.e., possible heat interchange between the bodies due to a material or a fluid contained in the separation zones between them). For this purpose, considering the previous works of Giannopoulos and Anifantis, this work extends their 2D incremental contact methodology based on the Boundary Element Method to solve steady-state 3D thermoelastic contact problems where interstitial conductive and convective boundary conditions are considered. So the equilibrium configuration of the system is determined incrementally by solving the uncoupled thermal and thermoelastic problems for each incremental thermal and mechanical load. The proposed scheme is validated solving several benchmark problems presented in the literature and later applied to solve different 3D thermoelastic contact problems where the response is clearly influenced by the boundary conditions at interstitial regions.

A coupled damage model and a semi-analytical contact solver to simulate butterfly wing formation around nonmetallic inclusions

This paper presents a damage modeling approach to simulate the formation of butterfly wings in rolling contact fatigue coupled with a fast method for solving 3D contact problems with the presence of a spherical heterogeneity. The damage model is implemented in a semi-analytical model using the Eshelby’ s equivalent inclusion method in the contact solver. The proposed technique can easily include the effects of heterogeneous inclusions to the homogeneous solution in the contact algorithm. Contact pressure and subsurface stress field computation time is kept small due to the use of 3D and 2D Fast Fourier Transforms. Cuboidal inclusions with the same size as the discretization of the half-space are superimposed to represent the microstructural alterations during butterfly wings formation.