Contact between 3D beams with rectangular cross‐sections

In this paper frictionless contact between 3D beams is analysed. The beam model is used in which large displacements but small strains are allowed. The element is derived on the basis of updated Lagrangian formulation using physical shape functions with shear effect included. An effective contact‐search algorithm, which is necessary to determine an active set for the contact contribution treatment, is elaborated. The contact element uses the same set of physical shape functions as the beam element. A consistent linearization of contact contribution is derived and expressed in suitable matrix form, easy to use in FEM approximation. Several numerical examples depict the efficiency of the presented approach. Copyright © 2002 John Wiley & Sons, Ltd.     

Frictional contact formulation using quadratic programming

A new solution procedure for contact problems in elasticity with prescribed normal tractions on contact surface has been proposed in this paper. The procedure is based on the boundary element method and quadratic programming. It is next used in a two step solution algorithm for the analysis of contact problems with friction. Several numerical examples are presented and compared with results obtained using alternative solution methods.This research has been supported in part of the National Science Foundation Presidental Young Investigator award MSS-9057055 with D. Oscar Dillon as the program manager, and by the U.S. Department of Energy, under contact DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc., by an appointment to the Oak Ridge National Laboratory Postdoctoral Research Associates Program administered jointly by the Oak Ridge Institute for Science and Education and Oak Ridge National Laboratory.     

Frictional contact analysis of composite materials

In this paper, the highly non-linear frictional contact problems of composite materials are analysed. A proportional loading, the potential contact zone method and finite element analysis are used to solve the problems. A tree-like searching method is used to obtain the solution of the parametric linear complementary problem, which may overcome the anisotropic properties of contact equations caused by composite materials. In the frictional contact analysis of composite materials, the distributions of normal contact pressures, tangential contact stresses and relative tangential displacements are presented for different contact material systems and different coefficients of friction. The results show that the solutions in the paper have good agreement with Hertzian solutions. The influence of different contact material systems and different coefficients of friction on the contact stresses and displacements is large. As a numerical example, ball-indentation tests of composite materials are modelled by the three-dimensional finite element method.     

Contact with friction between beams in 3‐D space

In this paper a formulation to deal with friction between straight beams undergoing large displacements in 3‐D space is proposed. The detection of the contact point and the computation of the amount of sliding are carried out using a completely symmetric treatment between the two contacting beams. Starting from the virtual work equation the consistent linearization of the frictional contact contribution is computed and the complete equation set is arranged in matrix form suitable for FE implementation. Some numerical examples are added to show the effectiveness of the method. Copyright © 2000 John Wiley & Sons, Ltd.     

Frictional contact of flexible and rigid bodies

 This paper is about planar frictional contact problems of both flexible and rigid bodies. For the flexible case a nonlinear finite element formulation is presented, which is based on a modified Coulomb friction law. Stick-slip motion is incorporated into the formulation through a radial return mapping scheme. Linearly interpolating four node elements and three node contact elements are utilized for the finite element discretization. The corresponding tangent stiffness matrices and residual vectors of the equations of motion are presented. In the rigid body case the contact problem is divided into impact and continual contact, which are mathematically described by linear complementarity problems. The impact in normal direction is modeled by a modified Poisson hypothesis, which is adapted to allow multiple impacts. The formulation of the tangential impact is grounded on Coulombs law of friction. The normal contact forces of the continual contact are such that colliding bodies are prevented from penetration and the corresponding tangential forces are expressed by Coulombs law of friction. Examples and comparisions between the different methods are presented. Received: 10 January 2001     

Frictional mortar contact for finite deformation problems with synthetic contact kinematics

In this paper we present a mortar based method, for frictional two dimensional contact problems. It is based on the work by Tur et al. (Comput Methods Appl Mech Eng 198(37–40):2860–2873, 2009) and uses the same concentrated integration scheme as well as a non regularized tangential contact formulation based on Lagrange multipliers only. We abstract the contact kinematics to a rather synthetic formulation. Therefore we are able to use two different methods of defining the normal field on the discretized surface normal: The popular method of averaged non-mortar side normal and the rather simple non continuous mortar side normal field. The problem is solved with a fixed point Newton–Raphson procedure and for both normal fields the full linearizations are derived. With numerical examples we show the performance of the more concise formulation of the non averaged non continuous mortar side normal field.     

Frictional Hertzian contact between a laterally graded elastic medium and a rigid circular stamp

This study investigates the problem of sliding frictional contact between a laterally graded elastic medium and a rigid circular stamp. Analytical and computational methods are developed to evaluate the contact stresses. In the analytical formulation, spatial variation in the shear modulus of the graded medium is represented by an exponential function, and Poisson’s ratio is taken as a constant. Coulomb’s dry friction law is assumed to hold within the contact area. The two-dimensional plane elasticity problem is formulated utilizing Fourier transforms, and the resulting Cauchy-type singular integral equation of the second type is solved by applying an expansion–collocation technique. The finite element method is used in the computational analysis of the contact problem. In the finite element model, continuous variation of the shear modulus is taken into account by specifying this property at the centroid of each finite element. The finite element-based solution procedure is verified by making comparisons to the results obtained through the analytical method. Numerical results generated for the laterally graded medium with an exponential variation in the shear modulus illustrate the influences of lateral gradation and coefficient of friction upon the contact stress distributions. The capability of the proposed finite element method is further demonstrated by providing numerical results for a laterally graded medium whose shear modulus is represented by a power function.     

Frictional contact induced crack initiation in incompressible substrate

Surface crack initiation in an incompressible substrate induced through frictional fretting contact is analyzed using an energy-based fracture mechanics model. A closed-form energy release rate for surface crack initiation at the contact boundary has been derived with the crack growth angle determined by the mixed mode singular stress field at the contact edge. The driving forces in the form of J_i-integral, the critical energy release rate and the critical load for crack initiation from the crack free surface have been formulated. The relations between the friction coefficient and crack initiation angle, critical load have been specified.     

Frictional contact finite elements based on mixed variational principles

This paper presents an original approach to the numerical modelling of unilateral contact by the finite element method. The main point is the development of mixed contact finite elements in which the displacement field and the contact stress field (pressure and friction shear) are discretized independently. The theory, based on variational principles, is first presented in the framework of infinitesimal deformations and, subsequently, is extended to large inelastic strains.     

Non-smooth Nonlinear Equations Methods for Solving 3D Elastoplastic Frictional Contact Problems

Based on the non-smooth nonlinear equations method for modeling three-dimensional elastic frictional contact problems (hereafter called NNEM), the extension to elastoplastic case in which the material nonlinearity is also involved is presented in this paper. Two approaches which combine two methods for solving elastoplastic problem with NNEM are proposed. A Numerical example is given to demonstrate the validation of the approaches.     

Analysis of Bernoulli beams with 3D stochastic heterogeneity

The behavior of stochastically heterogeneous beams, composed of isotropic sub-elements of randomly distributed stiffness is studied. Cross sectional as well as longitudinal heterogeneity are included. Average displacements, reaction forces and their statistical variance are found analytically by a functional perturbation method. Ratio of sub-element to beam characteristic size is not negligible and the use of an equivalent homogeneous structure with the classical effective material properties is not sufficient. The major aim is to study the relation between various microstructure properties (grain size, shape, modulus, statistical correlation lengths etc.) and the overall behavior of linear elastic Bernoulli beams. For the statically determinate case, only cross sectional 2D microstructure statistics is found to affect the elastic response, so that an equal average displacement can be achieved by an equivalent, non-isotropic homogeneous beam. For the indeterminate case, the average values of macro properties are affected by the 3D morphological features. Therefore, the proper equivalent homogeneous beam has to include non-local elastic properties. A simple reciprocal relation, connecting two separate loading systems is found, relating their external forces and displacement statistical variances. Morphological parameters, like two point probability moments, used in the final results are derived analytically, and their physical interpretations are discussed.     

A 3D contact smoothing method using Gregory patches

In this work, a method is developed for smoothing three‐dimensional contact surfaces. The method can be applied to both regular and irregular meshes. The algorithm employs Gregory patches to interpolate finite element nodes and provide tangent plane continuity between adjacent patches. The resulting surface interpolation is used to calculate gaps and contact forces, in a variationally consistent way, such that contact forces due to normal and frictional contact vary smoothly as slave nodes transition from one patch to the next. This eliminates the ‘chatter’ which typically occurs in a standard contact algorithm when a slave node is situated near a master facet edge. The elimination of this chatter provides a significant improvement in convergence behaviour, which is illustrated by a number of numerical examples. Furthermore, smoothed surfaces also provide a more accurate representation of the actual surface, such that resulting stresses and forces can be more accurately computed with coarse meshes in many problems. This fact is also demonstrated by the numerical examples. Published in 2002 by John Wiley & Sons, Ltd.     

Interpolation of rotational variables in nonlinear dynamics of 3D beams

The formulation of dynamic procedures for three-dimensional (3-D) beams requires extensive use of the algebra pertaining to the non-linear character of the rotation group in space. The corresponding extraction procedure to obtain the rotations that span a time step has certain limitations, which can have a detrimental effect on the overall stability of a time-integration scheme. The paper describes two algorithms for the dynamics of 3-D beams, which differ in their manifestation of the above limitation. The first has already been described in the literature and involves the interpolation of iterative rotations, while an alternative formulation, which eliminates the above effect by design, requires interpolation of incremental rotations. Theoretical arguments are backed by numerical results. Similarities between the conventional and new formulation are pointed out and are shown to be big enough to enable easy transformation of the conventional formulation into the new one. © 1998 John Wiley & Sons, Ltd.     

Core compression in sandwich beams under contact loading

Failure of the core in sandwich structures under concentrated loading is of potential concern, and it is difficult to compute the core compressive stress by simple means. Contact loading adds additional complexity, as surface displacements are imposed and the contact zone size and pressure distribution is initially unknown. However contact loading is important as it is widely used in three or four point bend tests to determine failure properties, and is also typically involved in impact loading. The calculation of core compressive stress was addressed in the present work by utilizing an elasticity solution due to Pagano and Srinivas and Rao for transverse loading of layered orthotropic materials. Contact pressure distributions were obtained by systematically varying pressures and comparing the computed surface displacements with the indentor profile. The results show that the pressure distribution for an orthotropic half-space is applicable to sandwich beams over a wide range of variables. A beam-on-elastic-foundation model was found to be useful in correlating the analysis results for core compressive stress.     

Frictional contact problems of kinked cracks modelled by a boundary integral method

A numerical method is presented for the solution of two dimensional crack problems including the effects of crack kinks and frictional contact between crack faces. The metod is based on an integral equation for the resultant forces along a crack. Coulomb friction between contacting crack surfaces is taken into account. The numerical implementation is demonstrated by considerations of surface and sub-surface piece-wise straight line cracks in a half-plane. Numerical results are presented to illustrate the efficiency and the reliability of the presented method.     

Frictional contact analysis of functionally graded materials with Lagrange finite block method

Based on the one‐dimensional differential matrix derived from the Lagrange series expansion, the finite block method was recently developed to solve both the elasticity and transient heat conduction problems of anisotropic and functionally graded materials. In this paper, the formulation of the Lagrange finite block method with boundary type in the strong form is presented and applied to non‐conforming contact problems for the functionally graded materials subjected to either static or dynamic loads. The first order partial differential matrices are only needed both in the governing equations and in the Neumann boundary condition. By introducing the mapping technique, a block of quadratic type is transformed from the Cartesian coordinate of global system to the normalized coordinate with eight seeds. Time dependent partial differential equations are analyzed in the Laplace transformed domain and the Durbin's inversion method is applied to determine all the physical values in the time domain. Conforming and non‐conforming contacts are investigated by using the iterative algorithm with full load technique. Illustrative numerical examples are given and comparisons have been made with analytical solutions. Copyright © 2015 John Wiley & Sons, Ltd.