Contact Tractions Research Articles

Shape optimization in contact problems with desired contact traction distribution on the specified contact surface

An efficatious technique is developed to deal with the shape optimization in conctact problems with desired contact traction on a specified contact surface. The error function with appropriate weighting coefficients for smoothing local and global contact traction distribution is proposed. Since the mapping from the space of design variables to the space of error function is usually not differentiable, a Genetic Algorithm is introduced without using the gradient calculations. The boundary element technique together with the transformation matrix method are employed for the contact traction analysis. To demonstrate the versatility and applicability of this approach, two examples with desired traction distributions of elliptical and rectangular types on the specified contact surface are solved respectively.

Shape optimization in contact problems with desired contact traction distribution on the specified contact surface

Shape optimization in contact problems with desired contact traction distribution on the specified contact surface

Shear dominated transonic interfacial crack growth in a bimaterial I-II. Asymptotic fields and favorable velocity regimes

Motivated by experimental observations of transonic crack tip speeds (Lambros and Rosakis, 1994c, J. Mech. Phys. Solids 43(2), 169–188), the problem of intersonic interfacial crack growth in an elastic-rigid bimaterial system is analysed. Following the analytical procedure employed in Liu et al. (1993, J. Mech. Phys. Solids 41, 1887–1954), the two-dimensional in-plane asymptotic deformation field surrounding the tip of a crack propagating intersonically along an elastic-rigid bimaterial interface, is obtained. The theoretical results show that the near-tip stress field does not exhibit oscillations, while a stress singularity weaker than 0.5 still exists and is a function of the crack tip speed. In addition, due to the intersonic nature of crack growth, a singular line emanating from the moving crack tip is present in the near-tip field. Across this line, stresses and particle velocities suffer infinite jumps. The theoretical analysis also shows that the near-tip deformation field is shear dominated. It is also shown that in the velocity range c s < v < √2 c s , either crack face contact or negative normal tractions ahead of the crack tip exist. Visual evidence of such contact is reported in Part I of this study. These observations, together with additional experimental results of Part I, lead to the conclusion that crack growth is favorable in the velocity regimes 0 < v < c s and √2 c s < v < c 1.

Quadratic programming contact formulation for elastic bodies using boundary element method

A method for the analysis of contact of deformable bodies based on the boundary element method (BEM) has been presented in this paper. The contact problem is stated in the form of a convex quadratic programming (QP) problem written in terms of the contact tractions on the contact surface. A strategy for the incorporation of the BEM contact analysis into models whose domain may be discretized using the finite element method (FEM) has been investigated. A discussion concerning the merits of the proposed approach is provided and several examples are presented to illustrate the validity of the method. HE frictionless contact of deforming elastic bodies is studied in this paper using the boundary element method (BEM). Because of the underlying complexity of the contact problems, only a limited number of analytical solutions, which are restricted to simple geometries and loading conditions, exist in the literature. An overview of the existing analytical solutions may be found in Ref. 1. A numerical discretization of the governing relationships must be employed for the analysis of complex contact configurations. The numerical method used for such discretization in the past has mainly been the finite element method (FEM), which has been employed for obtaining approximations of the solutions to contact variational inequalities.2 A solution to the variational inequalities may be obtained by a constrained minimization of the governing functionals, such as the total potential energy functional, over an admissible set of functions. 2 The fact that domain-based functionals are easier to model with the FEM and the wide applicability of the method have resulted in significant research on contact problems based on the FEM formulations. The contact constraints in such formulations may be introduced by 1) methods where the kinematic contact constraints are directly imposed in an incremental loading of the structure,37 2) Lagrangian-based methods,810 and 3) penalty methods.2'11 Alternative minimum principles defined on the contact boundary also provide effective methods for contact analysis. These formulations may be efficiently discretized using both the FEM and the BEM; for example, see Refs. 12-14. The use of the BEM is advantageous because of the accuracy of the boundary stresses that it provides and the reduced dimensionality of the resulting discretized BEM models compared with those obtained for the domain-based methods.15 The solution to the contact problems may also be approached directly by satisfying the complementarity conditions on the contact boundary.16 Contact problems can be alternatively stated in the form of mathematical programming (MP) problems. Such formulations are derived either from boundary variational formulations or from the direct discretization of the contact complementarity conditions. Since the first publications1719 on the subject, considerable research has been performed, mainly in FEM framework.12'13'20 Recent studies14' 16<21-22 have illustrated the potential and the advantages of the combined BEM and MP approach for the analysis of contact

Three-dimensional contact stress analysis of a composite laminate with bolted joint

Based on an incremental restricted variational principle formulated and the transformation matrix derived from three-dimensional contact kinematic conditions, a general but effective finite element technique satisfying various contact conditions is established for the three-dimensional contact stress analysis of a composite laminate with bolted joint. The double-lap bolted joint in the thin graphite/epoxy laminate of [ 45 0 /– 45 90 ] s lay-up or in the thick glass-reinforced polyester laminate of various stacking sequences subjected to a bolt load and/or a clamping torque is considered in the analysis. The effects of friction, clearance, bolt elasticity, stacking sequence and clamp-up on the contact tractions around the bolted joint are studied systematically. The distribution of contact tractions through the thickness of the bolt-laminate interface is also shown. The delamination onset of the composite laminate can thus be evaluated accordingly. Several examples are carried out to demonstrate the applicability of the technique developed. Good agreement between the present computed results and the referenced experimental results is noted. The present work would be of help for the design of a composite laminate with bolted joint.

Numerical and Experimental Failure Analysis of Composite Laminates with Bolted Joints under Bending Loads

A three-dimensional contact mechanics analysis has been successfully made on the composite laminates with bolted joints under bending loads. The AS4/3501-6 laminates of [08/908] s and [908/08] s lay-up with two types of countersunk bolted joints, MS-24UNF-3A (ψ = 80°) and NAS-24UNJF-3A (ψ = 100°), subjected to bending loads are analyzed, respectively. The effects of friction, stacking sequence and countersunk bolt head on the contact tractions around the bolted joint are studied systematically. The progressive failure analysis based on the maximum stress theory, Ye delamination criterion and the complete ply failure approach is performed for predicting the maximum bending load that the laminate can sustain before it catastrophically fails. Toward this end, experimental tests were also conducted to measure the maximum bending load for the laminates with the countersunk bolted joints under bending loads. From the results, it is shown that the joint configuration of [908/08] s laminate and NAS-24UNJF-3A (ψ = 100°) countersunk bolt has more higher bending strength while the joint configuration of [08/908] s laminate and NAS-24UNJF-3A (ψ = 100°) countersunk bolt has more higher bending stiffness. Good agreement between the computed and experimental results shows the applicability of the technique developed. The present work should be of help for the design of composite laminates with the countersunk bolted joints.

Interface yield map of a hard coating under sliding contact

The von Mises stress distributions in a hard coating and in elastic sliding contact are analysed by the finite element method. An elliptical distribution of normal and traction contact pressure is assumed for the analysis of von Mises stress for various combinations of coating thickness, friction coefficient and elastic modulus of the coating and substrate. The position of yield is found with the calculated result and the local yield map is introduced in relation to the yield strength ratio of the coating to the substrate and the ratio of the coating thickness to the half-contact width. The local yield maps show that yield at the coating-substrate interface in the substrate is the most general case under a wide range of contact conditions. An interface yield map for identification of the yield regime and the no-yield regime at the coating-substrate interface in the substrate is introduced in relation to the normalized critical maximum contact pressure and the ratio of the coating thickness to the half-contact width, and a sample equation is introduced for the calculation of the critical maximum contact pressure for the yield at the coating-substrate interface in the substrate based on the map.

An incremental relaxation finite element analysis of viscoelastic problems with contact and friction

Based on the principle of virtual work, a general but effective finite element technique associated with an incremental relaxation procedure is established for the analysis of viscoelastic contact problems with friction. A generalized Maxwell model is used to model the viscoelastic constitutive equations of which the relaxation function is represented by the sum of a series of decaying exponential functions of time. The contact behaviors are accurately performed through a transformation matrix method. By the incremental relaxation procedure developed, the stress of each element of the generalized Maxwell model at instant t ( N+1) includes two parts: (1) the part decaying from the stress at instant t ( N) during the time interval [ t ( N) , t ( N+1) ], and (2) the incremental stress induced from the incremental strain applied at instant t ( N) . The total stress at instant t ( N+1) can thus be computed by the sum of the stress of the elements of the generalized Maxwell model and is only dependent on the previous stress level and the incremental strain applied at instant t ( N) . Hence, both the total strain and displacement at each instant are not used in the total stress calculation. In addition, the numerical errors caused by selecting larger time increments can also be avoided. Finally, the influence of friction on the contact traction distributed on the contact surface at various time instants is presented.

Effect of shot peening on fatigue behavior in aluminum alloys: Wagner, L. and Mueller, C. Mater. Manuf. Proc. 1992 7, (3), 423–440

Suitability of different multi-axial parameters in predicting fretting fatigue life of Ti–6Al–4V specimens has been investigated. Ameliorating effect of surface treatments on fretting fatigue has been studied. In simple uni-axial/multi-axial fatigue tests, nucleation as well as propagation of cracks occur under the influence of identical stresses. Hence nucleation accounts for most of the total life. Fretting fatigue crack nucleation occurs due to very large contact stresses, effect of which is felt only close to the surface (due to steep gradients). Propagation mostly occurs due to lower stresses in the bulk of the material (negligible influence of contact tractions) and forms a significant portion of total life. Total life has to be taken as sum of initiation life calculated from different multi-axial fatigue parameters and propagation life from conventional fracture mechanics approach. Steep stress gradients necessitate the adoption of a statistics based approach to predict the crack initiation life, based on an assumed distribution of flaws. The quality of comparison between predicted and experimentally observed failure lives provides confidence in the notion that conventional fatigue life prediction tools can be used to assess fretting fatigue failure. Effect of surface treatments like shot-peening with or without additional surface coatings on total life of the specimen and on friction coefficient has been studied.

Boundary element analysis for contact problems with friction

A boundary element method associated with the transformation matrix derived from the geometric kinematic conditions of contact components is developed for the study of the contact behavior of elastic contact problems with friction. Owing to the fact that contact problems are inherently nonlinear and irreversible, an incremental approach is considered. Using the method developed, the degrees of freedom can be largely reduced, especially for the problem with complicated geometry. The mismatching of contact nodes is allowable and the contact traction is determined directly without tedious iterative procedures. Several examples are analyzed to demonstrate the high efficiency and applicability of the present technique. The computed results are found to agree well with other analytical, numerical or experimental solutions. The influence of friction on contact behavior is also presented.

Treatment of contact traction at the die-workpiece interface for the elastic analysis of die deformation

A method to calculate the contact traction on the die-workpiece interface is proposed. A systematic method for performing the elastic finite element analysis of the die deformation by using the information obtained from the finite element analysis of the workpiece deformation is proposed. The proposed method can be applied to both the two-dimensional and three-dimensional analyses of die deformation independently of the friction modelling and the constitutive equations. A rigid-plastic finite element analysis of workpiece deformation is carried out for the upsetting of a cylindrical workpiece. In order to check the validity of the present method for both flat and curved contact surfaces, the elastic finite element analysis of die deformation is then performed for flat square block forging as well as for a bevel gear forging (Yoon and Yang, Int. J. Mech. Sci. 32, 277–291, 1990) [8]. From the computation it is shown that the error due to the transfer of data from the workpiece mesh to the die mesh does not influence appreciably the elastic deformation of the die. The proposed method can be applied to arbitrarily curved die geometries in three-dimensional metal forming problems.

A Design Scheme for Multidisk Beier Traction Variators

The unique design feature of the Beier traction variator is that it can provide a higher power capacity per unit volume by using multiple thin disk pairs in a planetary system. Operating with a single effective traction contact and with the absence of idlers and dummy elements, promises for high efficiencies are also encouraging. In this paper, a design scheme for such mechanical drives is developed. Kinematics and kinetics of the rolling sliding disks are presented yielding actual performance ratings. Geometrical slippage in the Hertzian contact areas is considered for evaluating power losses at different load levels and transmission ratios. A torque sensitive pressure device which automatically creates contact forces is incorporated in the analysis. Constant traction coefficient pertinent to the boundary lubrication mode nearing thin fluid film friction is adopted for actual performance estimates. Among the different geometrical parameters, the bead radius of the input disk flange was found to play a major role in limiting contact stresses and torque capacities. Efficiencies of transmission, excluding shaft support bearing losses, are above ninety percent for a power of one kilowatt per contact.

Contact tractions between orthotropic materials by using the boundary element method

The boundary element formulation and the computer implementation of the 2-D contact problem with small displacements and strains between elastic orthotropic materials are presented in this paper. The computer program includes isoparametric linear, quadratic and quarter-point-traction-singular elements. The contact interface conditions are explicitly imposed. It is possible to include several contact zones with different friction coefficients between the solids. The algorithm minimizes the number of equations to be solved for each load step. Several examples have been included, specially the computation of contact tractions in composite material plates with bolted joints or the influence on the stress intensity factor of the crack closure effects.

The rigid punch problem with friction

The rigid punch problem with Coulomb's friction and a power law surface normal compliance is considered. A quasistatic variational inequality is derived. When it is discretized in time we obtain, as a step in a sequence, an incremental variational inequality. The existence of a solution of this problem is shown, and some results concerning uniqueness and regularity are considered. The duality theory of Mosco, Capuzzo-Dolcetta and Matzeu is used to derive a dual variational inequality where the dual variables may be interpreted as contact tractions. Frictionless problems are briefly discussed.

Response of an axially loaded elastic pile in a Gibson soil

An elastic analysis is presented to study the working load response of an axially loaded single elastic pile embedded in a linearly non-homogeneous incompressible elastic (Gibson) soil. The solution scheme is based on a variational method. A boundary integral representation based on exact displacement and traction Green's functions is used to model the response of the surrounding nonhomogeneous soil medium. The Green's functions are evaluated by using numerical integration techniques. The analysis ensures compatibility of displacement in the vertical and the radial directions along the true contact surface between the pile and the surrounding soil medium. Comparisons are presented to verify the accuracy of existing solutions based on approximate analysis methods. Selected numerical results are presented to portray the influence of the degree of non-homogeneity of soil on the axial stiffness and load-transfer and contact traction curves of a single elastic pile. L'article présente une analyse élastique pour étudier le comportement suis charge de service d'un pieu individuel élastique chargé axialement et fiché dans un sol élastique incompressible (Gibson) qui est non-homogéne dans le sells linéaire. La solution eat basée sur une méthode de variations. Une représentation integrale limite basée sur le déplacement précis et les fonctions de traction de Green eat employée pour modèliser la réponse du sol nonhomogéne environnant. Les fonctions de Green sont calculies au moyen de techniques d'intégration numérique. L'analyse assure compatibilité du deplacement dans les directions verticales et radiales le long de la surface vielle de contact entre le pieu et le sol environnant. Des comparaisons sont données afin de vérifier la precision des solutions existantes basées sur les méthodes d'analyse approximative. Des réultats numériques choisis sont présentés pour démontrer l'influence exercée par la non-homogénéité du sol sur la raideur axiale et sur les courbes de transfert de chargement et de traction de contact d'un pieu individuel elastique.

Study of half toroidal continuously variable transmission. 1st Report. Analysis of the contact force by means of loading cam.

In traction drives, the contact force is one of the most important factors used to obtain traction force. A loading cam and preload springs are used in the Half Toroidal CVT (Continuously Variable Transmission) for automobiles. Their use is very practical and it is considered that enough contact force is obtained on the traction contact point. Actually, on the cam surface, there is a friction loss which makes the contact force lower. In this report, the analysis of the cam force is examined and measured actual cam forces under various conditions are shown. The friction loss on the cam surface causes skew of the cam rollers. The friction coefficient is about 0.016∼0.019. The cam force is decreased by almost 20 %.

Some History-Dependent Problems for Dissimilar Cylinders With Finite Friction

This paper is concerned with the problem of contact between two dissimilar elastic cylinders. The cylinders are pressed together by a compressive force, P, and loaded by a tangential force, H, which is transmitted in the form of a distribution of frictional tractions at the interface. We assume that Coulomb’s law of friction holds at the interface. We develop a numerical algorithm which is sufficiently general to deal with an arbitrary history of normal and tangential loading, P(t), H(t) and which retains the full description of the coupling between the normal and tangential tractions. The proposed algorithm gives good numerical accuracy in comparison with previous analytical and numerical solutions of a variety of problems for normal and tangential loading of two contacting cylinders with friction. It is also found to be very efficient in the use of computer time. Results for some previously unsolved problems involving the full coupling between normal and tangential contact tractions indicate that such coupling can have a significant effect on the solution, particularly in regard to the extent of slip and stick zones within the contact area.

Tribological examination of unlubricated and graphite-lubricated silicon nitride under traction stress

The tribological features of silicon nitride under rolling contact traction stress were examined for unlubricated room temperature and graphite-lubricated high temperature conditions. Very high traction coefficients (1.0) cause tensile stress cracking at the surface with unlubricated clean surfaces. Wear occurs by fine-scale material removal, independent of the stress cracks, and results in transparent sheets of debris which appear to be silicon oxide. Graphite lubrication at high temperatures can be remarkably effective with additives taking an active role. A glassy-like surface is formed that contains graphite along with zinc and phosphate from the additive. The application of the graphite solids by burnishing is not a benign mechanical process but wears silicon nitride tribochemically. Very smooth surfaces can be generated when lubricated or unlubricated.

Localized asymmetric crack closure with stick-slip in the contact

An idealized problem is solved revealing several new contact and stick-slip configurations in partially closed cracks subjected to mixed mode in-plane loading. An initially open crack under remote tension is closed locally by two concentrated line loads whichpinch the crack shut. The reverse situation with localized separation due to line loads which pry the crack open is also investigated. The line loads may be placed such that isolated closure (or separation) removed from either one or the other, or both crack tips may be obtained. Examples of results for all of the possible contact configurations are presented. Remote shear stress is also considered, the various stick-slip patterns are delineated, and frictional history effects are discussed. Results for the actual contact and stick-slip status, contact tractions, mode I and II crack opening displacements and stress intensity factors are presented in parameter studies of the variation of the magnitude and position of the line loads. The mode I results are based on a new exact solution of the normal stress integral equations, while the mode II results are derived numerically from the shear stress integral equations with frictional shear tractions calculated from the mode I solution for the contact stress. Applications of this type of problem include reinforcement to arrest slow crack growth and mixed-mode fatigue crack closure due to either cyclic stationary loading or traveling loads.

弾性地盤上の隣接剛体構造物の動的相互作用

The objective of this study is to develop a theory for the response of a group of rigid structures with arbitrarily shaped bases attached to the surface of an elastic half space under the effects of the seismic excitation. Two methods are developed for the calculation of the dynamic response of a group of rigid structures. One is the application of a computationally efficient boundary element method. This procedure is based upon subdividing the contact area into a number of smaller triangular sub-regions, and the contact tractions and the Green's functions are assumed to be linearly varying functions of space coordinate within each sub-region. The obtained system of integral equations is discretized and reduced to a system of linear algebraic equations. The other is a much simpler method than the first one, and it is for obtaining a first order approximation. For this method only the response of a isolated structure and the Green's functions for the elastic half-space are needed. Numerical results for two cylindrical masses (Fig.2) are presented when they are excited by vertically incident plane waves. The amplitude ratio of the motion of the structure to that of the corresponding isolated structure is plotted in Fig.6 as functions of dimensionless frequency for different values of separation. A comparison of the four sets of curves of Fig.6 makes it evident that the effect of varying the mass of the structure, mc, on such a ratio is not so significant as the mass of the second structure, ma, and that the agreement between the results of the proposed two methods is reasonably good.