Contact Boundary Conditions Research Articles

Modeling of Current Density Boundary Conditions for a Heterodimensional Contact of 3D-Metal to 2D-Semiconductor

We have investigated semi-classical carrier transport at the heterodimensional contact between a three-dimensional (3D)-metal and two-dimensional (2D)-semiconductor system. Formulated for easy inclusion in numerical device simulators, current density boundary conditions for the heterodimensional contact are theoretically derived for both electrons and holes. Unlike conventional metal-semiconductor-metal (MSM) photodetectors, in which planar surface electrodes are fabricated to establish contact with the semiconductor, a novel MSM device is considered by utilizing recessed electrodes that directly contact the 2D-quantum well. The newly derived current density boundary conditions have been incorporated in a commercial numerical device simulator to investigate the steady-state and transient behavior of a novel quantum-well MSM photodetector. Results for devices with 1-μm electrode spacing at 2 V DC bias indicate a nominal 3-dB bandwidth of 25 GHz and low dark currents in the quantum well of the order of a few femto-amps.

Non-linear vibrations of a beam with cantilever-Hertzian contact boundary conditions

The non-linear vibrations of a linear beam with cantilever-Hertizian contact boundary conditions are investigated. The method of multiple scales is used to analyze this problem in which it is assumed that the beam remains in contact with the moving surface at all times. One primary result from this analysis is the amplitude–frequency relation for the various flexural modes. The amplitude–frequency curves exhibit softening behavior as expected. The amount of softening is shown to depend on the linear contact stiffness as well as the specific mode. In addition, the associated non-linear normal modes of this system are derived. The modes include a non-linear modification to the linear, harmonic component as well as a static offset term and second and third harmonic components.

On the Low Velocity Impact Response of Laminated Composite Plates using the P-Version Ritz Method

Low energy impact in composite laminates is often a crucial and destructive loading condition since it leads to significant internal damage, undetectable by visual inspection. Low velocity impact upon a laminated plate imposes a complex stress state mainly due to the structural heterogeneity resulting by the ply orientation of the constituent laminae and the contact boundary conditions, which lead to a loading path that varies with the impact energy and the properties of the impactor-impacted plate system. The present work deals with the development of a numerical scheme for the calculation of the dynamic response of any type of laminated composite plates under low-velocity impact. The governing non-linear, second order differential equations are derived using p-Ritz admissible polynomial functions and the elastoplastic version of the Hertzian contact law. The dynamic response of fully clamped, cross ply and angle ply composite plates are investigated.

Contact between a smooth microsphere and an anisotropic rough surface

This article discusses the effects of asperities on elastic and adhesive contact between a smooth sphere and a rough surface. Two numerical methods are introduced: an asperity-superposition method and a direct-simulation method. In the first method, geometric parameters such as asperity heights, orientations, and radii of curvature are identified by a least-squares regression of neighboring surface heights measured using an atomic force microscope. The rough surface is reconstructed by the superposition of these asperities. The modeling of adhesive and elastic contacts begins with the modeling of a single parabolic-shaped asperity contact. A generalized JKR model for an arbitrary parabola is developed to suit this purpose. The contact between the rough surface (represented by the supposition of parabolic-shaped asperities) and the sphere consequently is modeled bythe mapping and integration of individual asperity contacts. In the second method, pure-elastic contact is modeled by half-space elastic theory. A contact-search algorithm is used to find solutions on the displacement and the contact-pressure distribution that satisfy both the load-displacement equation and the contactboundary conditions. Results from both methods are compared to reveal the effects of asperities on adhesion and elastic-contact pressure.

Finite element analysis of tube flaring process with a conical tool

The extended r min technique is incorporated into an incrementally updated Lagrangian formulation (ULF) of an elasto-plastic finite element computer code in order to handle contact boundary conditions to analyse the axisymmetric tube flaring process with a conical tool. A modified Coulomb friction law was adopted to calculate the influence of the friction coefficient on the tube flaring process. The effects of size and mechanical properties of tubes, as well as lubricants and tool semi-angle on flaring load were studied. It was found that good lubrication is effective in reducing flaring load. An optimum tool semi-angle (tool load is lower) in flaring is determined by frictional and bending activity at the tool inlet. In addition, the forming behaviour of the tube end is investigated to understand whether outward curling takes place and how much influence the tool semi-angle and tube size (initial thickness/initial mean diameter of tube) have on outward curling mode.

Numerical modeling of solid state sintering using heterogeneous packing and diffusion processes

Physical modeling and computer simulation for the description of solid state low stresses sintering process have been performed. The first step consists in fitting heterogeneous particles together in a three-dimensional virtual box, with shape and size distributions given as parameters. These particles are then submitted to gravity, contact boundary conditions, friction forces, and Van Der Waals interaction. Thus, one dispose of a realistic dense packing. Starting from size distribution and fitting parameters, it is then possible to obtain the same densities and the same configurations as those observed in the real case. The densification process is then simulated. The constitutive model is based on volume, surface, grain boundary and gaseous diffusion, using laws derived from local energetic considerations and avoiding strong hypothesis. This approach allows to validate work on sintering theory, based on particular cases or with unchecked assumptions. Important post-processing investigations let us to identify constants and pertinent options for the homogenization problem which will led us to a macroscopic description of the sintering process. The paper describes laws and algorithms used in the proposed modeling. Results are shown to illustrate solid state sintering of powder aggregates.

A Mathematical Programming Incremental Formulation for Unilateral Frictional Contact Problems of Linear Elasticity

In this article a detailed analytical formulation of the unilateral contact boundary conditions with Coulomb's law of dry friction is first attempted and the quasi-static contact problem between 3-D elastic bodies is studied thereafter. Discretizing the bodies by the Finite Element Method, introducing fictitious contact bonds and using the concept of the equivalent structural system, an incremental Nonlinear Complementarity Problem is finally formulated. Then, using additional simplifying assumptions, this problem can be transformed into an incremental Linear Complementarity Problem.

Algebraic elimination of slide surface constraints in implicit structural analysis

AbstractSlide surface and contact boundary conditions can be implemented via Lagrange multipliers in the algebraic equations in implicit structural analysis. This indefinite set of equations is difficult to solve by iterative methods and is often too large to be solved by direct methods. When there are m constraints and there exists a set of m variables where each variable is only involved in a single constraint, we advocate a direct elimination technique which leaves a sparse, positive definite system to solve by iterative methods. We prove that the amount of ‘fill‐in’ created by this process is independent of the size of the slide surfaces. In addition, the eigenvalues of the reduced matrix do not differ significantly from the eigenvalues of the unconstrained matrix. This method can be extended to the case where constrained surfaces intersect and leads to a graph theoretic approach for determining which variables can be eliminated efficiently for constraints with more general structure. Published in 2003 by John Wiley & Sons, Ltd.

축대칭 변형체의 마찰 접촉문제에 관한 유한요소 해석

This paper is concerned with the numerical analysis of frictional contact problems in axisymmetric bodies using the rigid-plastic finite element method. A contact finite element method, based on a penalty function, are derived from variational formulations. The contact boundary condition between two deformable bodies is prescribed by the proposed algorithm. The program which can handle frictional contact problem is developed by using pre-existing rigid-plastic finite element code. Some examples used in this paper illustrate the effectiveness of the proposed formulations and algorithms. Efforts focus on the deformation patterns, contact force, and velocity gradient through the various simulations.

Existence of Global Weak Solutions for Coupled Thermoelasticity with Barber's Heat Exchange Condition

The existence of global weak solutions for coupled thermoelasticity with the nonlinear contact boundary condition and Barber’s heat exchange condition is proved via the Faedo-Galerkin, monotonicity and compactness methods. Some a priori bounds obtained with Gronwalls inequality in connection with the embedding and trace theorems lead to accomplishing a generalization of our previous study [5]. The heat-exchange coefficient associated with Barber’s heat exchange condition is dependent only on the normal displacement. This dependence is described by a bounded Lipschitz function. Moreover, this study is some extension of works due to Andrews et al. [3] and Elliot et al. [12]. 2000 Mathematics Subject Classification. 35B45, 35K05, 35L55, 35Q72, 73C35, 73B30, 73C35.

Finite‐difference modeling of SH‐wave propagation in nonwelded contact media

Many geological structures of interest are known to exhibit fracturing. Fracturing directly affects seismic wave propagation because, depending on its scale, fracturing may give rise to scattering and/or anisotropy. A fracture may be described mathematically as an interface in nonwelded contact (i.e., as a displacement discontinuity). This poses a difficulty for finite‐difference modeling of seismic wave propagation in fractured media, because the standard heterogeneous approach assumes welded contact. In the past, this difficulty has been circumvented by incorporating nonwelded contact into the medium parameters using equivalent medium theory. We present an alternate method based on the homogeneous approach to finite differencing, whereby nonwelded contact boundary conditions are imposed explicitly. For simplicity, we develop the method in the SH‐wave case.In the homogeneous approach, nonwelded contact boundary conditions are discretized by introducing auxiliary, so‐called fictitious, grid points. Wavefield values at fictitious grid points are then used in the discrete equation of motion, so that the time‐evolved wavefield satisfies the correct boundary conditions. Although not as general as the heterogeneous approach, the homogeneous approach has the advantage of being relatively simple and manifestly satisfying nonwelded contact boundary conditions. For fractures aligned with the numerical grid, the homogeneous and heterogeneous approaches yield identical results. In particular, in both approaches nonwelded contact results in a larger maximum stable time step size than in the welded contact case.

An intermediate crack model for flaws in piezoelectric solids

The aim of this paper is to study a crack model for piezoelectric bodies. In recent years it became evident that the electrically impermeable or the perfect electric contact boundary conditions on the crack faces are inadequate for many physical situations, over- or underestimating the electric field influence on the propagation process. The crack model here investigated is intermediate between these two limit cases. The generalized plane problem for an infinite piezoelectric body with a central crack is converted into a system of integro-differential equations, then reduced to an integro-differential equation similar to Prandtl's equation of aerodynamics. For poled ceramics with transversely isotropic symmetry, the integral equation is numerically solved using quadrature formulae for both plane and antiplane states of deformation. The energy release rate calculated with the discrete solution is then compared with that given by the exact solution for an elliptic hole embedded in the infinite piezoelectric body. A range of values of the cavity thickness is found, for which the considered crack model is a good approximation of the exact two-body problem, while the impermeable and the perfect contact models are not appropriate.

Generic unfolding of the thermoelastic contact instability

The classic model of a one-dimensional thermoelastic rod suspended between a hot and cold wall is revisited. In this model, the rod is held in place at the cold end, while at the hot end it is allowed to separate from or make contact with the wall. When the model includes the contact and gap dependent thermal boundary condition known as the Barber condition it serves to illustrate the well-known thermoelastic contact instability. All previous studies of this instability have focused upon the symmetric case where, as a control parameter is varied, the system undergoes a pitchfork bifurcation. That is, a new pair of linearly stable steady-state solutions bifurcate symmetrically from a previously unique solution which has changed from stable to unstable. Here, it is shown that this behavior is not generic. Rather, for typical contact resistance functions, a fold bifurcation is encountered. This represents a generic unfolding of the classic pitchfork bifurcation and contains the pitchfork as a special case.

Elasto-plastic Finite-Element Analysis of the Axisymmetric Tube-Flaring Process with Conical Punch

The incremental updated Lagrangian formulation of an elasto-plastic finite-element computer code has been incorporated into the extend r min technique in order to handle the contact boundary condition as applied to the analysis of the axisymmetric tube-flaring process with a conical punch. A modified Coulomb's friction law was also adopted to calculate the influence of friction coefficients on the tube-flaring process. Effects of the size and the mechanical properties of tubes, lubricants and the punch semi-angle on the flaring load were studied. It was found that good lubrication and tubular materials with a high strain hardening coefficient are both effective in reducing the flaring load. An optimum punch semi-angle (punch load is lower) in flaring is determined by work on frictional and bending at the punch inlet. In addition, the effect of springback on the tube diameter or angle of final deformation, under different friction coefficients after unloading, was also explored.

A quasistatic viscoelastic contact problem with friction

We consider a mathematical model which describes the bilateral frictional contact of a viscoelastic body with an obstacle. The viscoelastic constitutive law is assumed to be nonlinear and the friction is described by a nonlocal version of Coulomb's law. A weak formulation of the model is presented and an existence and uniqueness result is established when the coefficient of friction is small. The proof is based on classical results for elliptic variational inequalities and fixed point arguments. We also consider a model for the wear of the contacting surface due to friction. In the case of sliding contact we obtain a new nonstandard contact boundary condition. We prove the existence of the unique weak solution to the problem with the same restriction on the friction coefficient.

The diffraction of elastic waves by an imperfect fusion bond

We consider diffraction by a semi-infinite crack located along a fusion interface between two differing elastic media. Two types of crack, namely open and partially closed cracks, are investigated. An open crack is modelled by a stress-free contact boundary condition and a partially closed crack is modelled by a spring contact boundary condition. For the latter, the jump in the stress across the crack is assumed to be proportional to the jump in the displacement across the crack. This situation arises in, for example, a K-weld where the fine grain of the parent material (for example, ferritic or forged austenitic steel) is in stark contrast with the coarse-grained weld metal (for example, austenitic weld metal). In the metal weld the direction of the grain axis varies through the metal. However, diffraction is a local phenomenon and so the austenitic steel is assumed to have a zonal axis so that it may be modelled by a transversely isotropic composite. The ferritic or forged austenitic steel will be modelled as an isotropic material.

A contact algorithm for smoothed particle hydrodynamics

This paper describes the development and testing of a contact algorithm for Smoothed Particle Hydrodynamics (SPH). The treatment of contact boundary conditions in SPH has not been adequately addressed, and the development of the normalised smoothing function approach has highlighted the need for correct treatment of boundary conditions. A particle to particle contact algorithm was developed for 2D. The penalty formulation was used to enforce the contact condition, and several equations for the penalty force calculation were considered. The contact algorithm was tested for 1- and 2D problems for the velocity range between 0.2 and 4.0 km/s to determine the best penalty force equation and the best approach for applying the contact force. The tests showed that the zero-energy mode problem in the SPH method had to be addressed, as contact excited a zero-energy mode that caused non-physical motion of particles. The test results were compared to the DYNA3D results for the same problems.

A penalty approach for contact in smoothed particle hydrodynamics

Summary This paper describes the development and testing of a contact algorithm for SPH. The treatment of contact boundary conditions in SPH has not been adequately addressed, and the development of the normalised smoothing function approach has highlighted the need for correct treatment of boundary conditions. A particle to particle contact algorithm was developed for 2D. The penalty formulation was used to enforce the contact condition, and several equations for the penalty force calculation were considered. The contact algorithm was tested for one and two dimensional problems for the velocity range between 0.2 and 4.0 km/s to determine the best penalty force equation and the best approach for applying the contact force. The tests showed that the zero-energy mode problem in the SPH method had to be addressed, as contact excited a zero-energy mode that caused non-physical motion of particles. The test results were compared to the DYNA3D results for the same problems. Background to boundary condition treatment in SPH is given in Section 1. In Section 2 the development of a contact algorithm for SPH in 1 and 2D is described. Testing of the contact algorithm in 1 and 2D is described in Section 3. Conclusions are presented in Section 4.

Solution of a contact problem for a plate with a deformable insert

Contact problems with friction are solved for a rectangular plate with a circular hole into which a ring plate (insert) is placed with a small clearance. Two versions of contact boundary conditions are formulated. According to the proposed approximate formulation of the problem, the boundary conditions in both versions are satisfied not at the actual contact points but at specified pairs of points. Therefore, it is sufficient to determine attachment, slip, contact, and contact-free regions on just one of the contacting contours. The finite-element method and the Boussinesq principle are used to solve the problem. One of the versions of boundary conditions, compared to the other, gives smaller values for the strain energies of the plate and insert, the stress-concentration coefficient, and the lengths of attachment and contact regions.

Incidence of flexural gravitational waves on the line of contact of two floating ice plates of different thickness

We analyze the influence of a rift in an ice field on the propagation of flexural gravitational waves in a basin of finite constant depth. The ice cover is simulated by two floating semiinfinite elastic plates of different thickness. We studied the dependence of the amplitude coefficients of reflection and transmission of waves incident on the rift on the frequency of running waves, the thickness of ice on both sides of the rift, and the type of contact boundary conditions at the rift.