A solution method for dynamic contact problems

An efficient method is presented for analyzing the transient dynamic contact problems of elastic bodies in this paper. This approach exploits the Lagrange multiplier concept and a special time integration algorithm. Due to the introduced high-frequency dissipation in this time integration algorithm, this method can lead to the effective analysis of real response of elastic bodies with dynamic surface contact constraints. The results of numerical examples show that this method can avoid the weakness of the classical Lagrange multiplier method in dealing with dynamic contact problems with relatively high inertial forces. Stable results can be provided when the time integration step size is small. The properties of this method have also been discussed in this paper.     

Stress intensity factors for an interfacial crack between an orthotropic half-plane bonded to a dissimilar orthotropic layer with a punch

The plane strain problem of determining Stress Intensity Factors (SIF) for a moving interfacial Griffith crack between an elastic orthotropic half-plane and a dissimilar orthotropic layer with a moving punch situated along the boundary of the layer have been considered. The problem is reduced to the solution of three simultaneous singular integral equations with Cauchy-type singularities. Expressions for SIF for the case of a general loading are obtained. Numerical results for some particular cases are also presented graphically.     

A Dirichlet–Neumann type algorithm for contact problems with friction

Domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations. We introduce a new algorithm for the numerical solution of a nonlinear contact problem with Coulomb friction between linear elastic bodies. The discretization of the nonlinear problem is based on mortar techniques. We use a dual basis Lagrange multiplier space for the coupling of the different bodies. The boundary data transfer at the contact zone is essential for the algorithm. It is realized by a scaled mass matrix which results from the mortar discretization on non-matching triangulations. We apply a nonlinear block Gauss–Seidel method as iterative solver which can be interpreted as a Dirichlet–Neumann algorithm for the nonlinear problem. In each iteration step, we have to solve a linear Neumann problem and a nonlinear Signorini problem. The solution of the Signorini problem is realized in terms of monotone multigrid methods. Numerical results illustrate the performance of our approach in 2D and 3D. Received: 20 March 2001 / Accepted: 1 February 2002 Communicated by P. Deuflhard     

Bending of an axisymmetrically loaded thin circular plate on a layered elastic half-space

The problem of interaction between an axisymmetrically loaded thin circular plate and a supporting elastic medium is reduced to that of solving an integral equation for the unknown normal contact pressure. The supporting medium is an isotropic elastic layer of constant thickness lying, with or without friction, on a semi-infinite isotropic elastic base or on a rigid base. For the solution of the resulting integral equation an effective numerical procedure is employed and some numerical results are presented.     

Numerical analysis and simulations of contact problem with wear

This paper presents a quasistatic problem of an elastic body in frictional contact with a moving foundation. The model takes into account wear of the contact surface of the body caused by the friction. We recall existence and uniqueness results obtained in Sofonea et al. (2017). The main aim of this paper is to present a fully discrete scheme for numerical approximation together with an error estimation of a solution to this problem. Finally, computational simulations are performed to illustrate the mathematical model.     

Finite element mesh for a complete solution of a problem with a singularity

It is well known that the stress field at the tip of a crack in an elastic body exhibits a singularity. In this paper, a complete elastic solution for a center cracked plate loaded under uniform tension is obtained by a finite element plane-stress analysis using constant strain elements. No a priori assumptions about the form of the stress singularity are required. The numerical results are compared with the exact analytic solution. It is shown that a particular mesh arrangement in which the size of the elements decreases in a geometric series as the crack tip is approached yields stress and strain fields which are accurate over the entire plate, even at distances very close to the crack tip. The effects of changes in mesh arrangements on the accuracy of the solution are considered. The computations are carried out on the CRAY-1 computer and the advantages of vectorization are discussed for this problem.     

Moving finite element analysis for the elastic beams in contact problems

The contact problem of a linear elastic beam with a rigid barrier is considered, in which the contact surface is assumed to be frictionless. The problem contains the characteristic of moving boundaries as the contact regions expand or reduce in size when the external loads alter. Starting from the variational principles, we derived the interface equations as well as the governing equation. In addition to matching the deflections and the slopes with the rigid barrier at the marginal nodes of the contact regions, the interface equations also have to be satisfied there. In essence, the interface equations are designed to locate the unknown and moving marginal nodes. Although it is confined within the framework of small deformation and linear elastic material behavior, the problem exhibits high nonlinearity due to the moving nature of the marginal nodes. In this paper, we present a moving finite element analysis employing incremental procedures and an iterative numerical scheme to tackle the problem. A fixed number of two-node beam elements are used in the moving finite element analysis and the size of the elements varies as the loads alter. A couple of examples, whose exact solutions are obtainable, are chosen to demonstrate the accuracy and efficiency of the proposed numerical algorithm.     

On almost constant contact stress distributions by shape optimization

This paper addresses the problem of finding shapes of contacting bodies avoiding undesirable stress concentrations. It has previously been shown that designing the shape of a rigid body in contact with a fixed linear elastic body by minimizing the equilibrium potential energy under an isoparametric constraint results in a uniform contact pressure distribution. As an extension of this result, it is shown here that when the shape of an elastic body in contact with a flat rigid foundation is chosen on the same premises, the uniform pressure distribution is found only if displacement gradients can be considered small. From the point of view of applications, an important conclusion is that this smallness holds in a case when linear elasticity is physically valid.     

Vibration of a rigid body on an elastic half-plane

An analytic-numerical method for the analysis of a dynamically loaded rigid body supported on an elastic half-plane is presented. Two cases are considered: (1) vertical and rocking vibrations of a rigid body in smooth contact with the surface of the half-space, and (2) vertical, horizontal and rocking vibrations of a rigid body in rough contact. Numerical procedures are described to determine the contact stress distribution and the displacements at the surface away from the body.     

An evolutionary shape optimization for elastic contact problems subject to multiple load cases

Most structures in the real life are subject to multiple load cases. This paper aims at extending the evolutionary structural optimization (ESO) algorithm to optimal contact shape design for elastic bodies under the multiple load cases. To evaluate the reference stresses of each contact node in a finite element framework, an extreme stress criterion (the worst case design) and a weighted average criterion (Pareto design) are presented. In the extreme stress method, the highest nodal contact stress under all load cases is adopted as the reference level. In the weighted average method, the weighted sum of nodal contact stresses over all the load cases is regarded as the reference. It is found that these two criteria can produce different results. In this paper, the examples are presented to demonstrate some new features of contact shape optimization in the presence of the multiple load cases.     

Development of special fastener elements

A special finite element (FASNEL) is developed for the analysis of a neat or misfit fastener in a two-dimensional metallic/composite (orthotropic) plate subjected to biaxial loading. The misfit fasteners could be of interference or clearance type. These fasteners, which are common in engineering structures, cause stress concentrations and are potential sources of failure. Such cases of stress concentration present considerable numerical problems for analysis with conventional finite elements. In FASNEL the shape functions for displacements are derived from series stress function solutions satisfying the governing difffferential equation of the plate and some of the boundary conditions on the hole boundary. The region of the plate outside FASNEL is filled with CST or quadrilateral elements. When a plate with a fastener is gradually loaded the fastener-plate interface exhibits a state of partial contact/separation above a certain load level. In misfit fastener, the extent of contact/separation changes with applied load, leading to a nonlinear moving boundary problem and this is handled by FASNEL using an inverse formulation. The analysis is developed at present for a filled hole in a finite elastic plate providing two axes of symmetry. Numerical studies are conducted on a smooth rigid fastener in a finite elastic plate subjected to uniaxial loading to demonstrate the capability of FASNEL.     

The displacement field associated with line forces in a cracked orthotropic body

An explicit result is presented for the two-dimensional Green function for an orthotropic body containing a crack along an orthotropic plane and with its tip in an orthotropic direction. A formal solution and a numerical program for its calculation are presented for a crack on a plane rotated about an orthotropic direction. The use of such Green functions to couple a finite element mesh to a surrounding elastic continuum, thereby shortening finite element calculations for composite bodies with macroscopic orthotropic symmetry, is discussed.     

Analytical and numerical study of the effects of an elastically-linked body on sloshing

In order to investigate the effects of an elastically-linked moving body on liquid sloshing inside a tank, an analytical formulation and a numerical approach were proposed to assess hydrodynamic loads in a partially filled rectangular tank with a body connected to the tank by springs. The analytical approach was developed based on the potential theory to calculate fluid velocity field, and the dynamics of the liquid sloshing coupled to the moving body are described as a mechanical system with two degrees of freedom. The coupling between the fluid and the moving body is given by a damping force calculated based on the body geometry and the fluid velocity field. The proposed numerical approach is based on the Moving Particle Semi-implicit (MPS) method, which is a Lagrangian particle-based method and very effective to model nonlinear hydrodynamics due to fluid–structure interaction. In the numerical approach, the rigid body is modeled as a cluster of particles and the motions are calculated considering its mass, moment of inertia, hydrodynamic loads and springs restoring forces. The elastic link between the body and tank is modeled by applying Hooke’s law. Simple cases of floating body motion were used to validate the numerical method. Finally, analytical and numerical results were compared. Despite its simplicity, the analytical approach proposed in the present work is an efficient approach to provide qualitative understanding and a first estimate of the moving body effects on the sloshing inside the tank. On the other hand, the numerical approach can provide more detailed information about the coupling phenomena, and it is an effective mean for the assessment of the reduction of the sloshing loads due to the moving body with elastic link. Finally, the effectiveness of the concept as a sloshing suppressing device is also investigated.     

A fixed-grid model for simulation of a moving body in free surface flows

A two-dimensional computer model is developed to simulate free surface flow interaction with a moving body. The model is based on the cut-cell technique in a fixed-grid system. In this model, a body is approximated by the partial cell treatment (PCT), in which an irregular body is represented by the volumetric fraction of solid in Cartesian cells. The body motion is tracked by Lagrangian method whereas the fluid motion around the body is solved by Eulerian method. The concept of “locally relative stationary (LRS)” is introduced in this study. In the LRS method, a source term is added locally to the conventional continuity equation on body surfaces to take account of body motions, which subsequently affects the computational results of fluid pressure and flow velocity around the body. The LRS method is incorporated into an earlier Reynolds averaged Navier-Stokes (RANS) equations model developed by Lin and Liu [A numerical study of breaking waves in the surf zone. J Fluid Mech 1998;359:239-64]. The new model is capable of simulating generic turbulent free surface flows and their interaction with a moving body or multiple moving bodies. A series of numerical experiments have been conducted to verify the accuracy of the model for simulation of moving body interaction with a free surface flow. These tests include the generation of a solitary wave with the prescribed wave paddle movements, water exit and water impact and entry of a horizontal circular cylinder, fluid sloshing in a horizontally excited tank, and the acceleration/deceleration of an elliptical cylinder near a water surface. Excellent agreements are obtained when numerical results are compared to available analytical, experimental, and other numerical results. The model is a simple-to-implement computational tool for simulating a moving body in turbulent free surface flows.     

Fatigue life prediction under cyclic thermal loads using the boundary elements method for two-dimensional problems

This study describes a sub-domain boundary element procedure for predicting the fatigue life of elastic media under cyclic transient thermal loads. The procedure assumes an initial crack in a two-dimensional medium, and evaluates the crack extension along a path defined by the maximum circumferential stress criterion. Partial closure may occur on the growing crack faces due to thermal loading. Appropriate thermal and mechanical boundary conditions are imposed on the numerical model to account for the contact state. The analysis requires solutions to the equations of quasi-static thermo-elasticity, and uses an iterative procedure to detect the contact region. We investigate cases of pure opening or mixed mode fracture, demonstrating the effects of crack closure on the predicted fatigue life. We compare the obtained results with those of other publications.     

An Algorithm for Compliant Contact Between Complexly Shaped Bodies

A new contact algorithm designed for multibody dynamics is presented. It is based on representation of the body surfaces by polygon meshes and contact force determination by the elastic foundation model. Areal discretisations of the contact patches are constructed using methods closely related to computer graphics, e.g. collision detection based on bounding volume hierarchies and generation of subdivision surfaces by means of boundary representation data structures. Two examples prove the robustness of the method for complexly shaped bodies causing multiple and multiply bordered contact patches and conforming contacts.     

Rigid-elastic modeling of meshing gear wheels in multibody systems

In many applications in mechanical engineering, gear wheels are used to transmit power between rotating shafts and, therefore, the ability to incorporate them into multibody systems and to simulate contact between them has become an essential topic in multibody dynamics.However, in some applications gear wheels may not be considered as being perfectly rigid. Due to the effect of contact forces there occur relevant deformations in meshing teeth and it is required for a high quality of the analysis to introduce some elasticities in the model of meshing gear wheels. Therefore, in this work elastic elements between the teeth and the body of each gear wheel are considered. This approach is especially well suited for multibody systems since it is a compromise between a totally rigid model and a fully elastic model allowing the simulation of large motions with many revolutions while still important elasticities are considered. The teeth and the body of each gear wheel are still modelled as being rigid but they are connected to each other by elastic elements. In doing so, an efficient and physically motivated algorithm is described and implemented in order to find the effects of multi-tooth contact as well as backlash and left and right hand side contact of the meshing teeth. Some examples compare the simulation results of rigid, partially elastic and fully elastic models.     

基于倍频双晶复合传感器的构件疲劳微裂纹非线性超声检测

通过对固体中非线性超声传播模型的研究,分析了裂纹静态压力与超声波作用力对裂纹超声非线性响应的影响,在此基础上,建立了反映裂纹区应力-应变非线性关系的弹性接触机制超声非线性响应模型;以及反映裂纹闭合状态转换的碰撞接触机制超声非线性响应模型。基于上述理论,通过实验研究发现裂纹尖端区的二次谐波激发效率与裂纹的开口区和闭合区及裂纹最终扩展的极限长度有关。因此,可使用二次谐波激发效率作为定量表征金属试件疲劳微裂纹缺陷的特征参数,实验中使用了自主研制倍频双晶复合换能器,这种倍频双晶复合换能器在工程实际应用中作在线检测更为方便、实用。     

Some theorems about spectrum and finite element approach for eigenvalue problems for elastic bodies with voids

The paper concerns eigenvalue problems for elastic bodies with voids in contact with massive rigid plane punches. The linear theory of elastic materials with voids according to the Cowin–Nunziato model is used. A variational principle is constructed which has the properties of minimality, similar to the well-known variational principle for problems with pure elastic media. The discreteness of the spectrum and completeness of the eigenfunctions are proved. As a consequence of variational principles, the properties of an increase or a decrease in the natural frequencies, when the mechanical and “porous” boundary conditions and the modulus of elastic solid with voids change, are established. A finite element method is proposed for numerical solution of eigenvalue problems for elastic media with voids. Some effective block algorithms for finite element eigenvalue problems with partial coupling are described. Numerical experiments are presented for determining the first eigenfrequencies of an axisymmetric elastic body with voids.     

An automatic incrementation technique for contact problems with friction

A numerical method for analysis of elastostatic contact problems with friction has been developed. This class of problems are load history dependent because of the irreversible nature of frictional forces. An automatic incrementation technique of the applied load has been developed and implemented in the algorithm. The method is a direct method based on an iterative procedure applied to a set of linear equations established with the finite element method. The size of the applied load increments, automatically chosen by the algorithm, is in general influenced both by the nature of the problem and of the discretization of the bodies involved. The frictional forces occurring in the slip zone of the contact area are treated as known tangential forces calculated from the normal forces in the previous iteration. This piecewise linear treatment of the frictional contact problem requires an innermost iteration loop over the applied tangential force.The tangential force must coincide with the coefficient of friction times the normal force obtained in the last iteration, otherwise a new tangential force has to be calculated and the system of equations must be solved for a new right hand side vector. The automatic incrementation technique is based on the fact that each iteration is a linear problem. A tentative load increment is used in the solution of a certain iteration. A linear scaling of this solution is performed afterwards. A load scale factor is calculated in each contact node pair where a change of contact condition will occur. The change in contact status corresponding to the node pair with the smallest load scale factor is the only change which is accomplished in a certain iteration. The uniqueness of this kind of contact problem with friction has not been mathematically proven for a general case.The method has been applied to a case of loading and unloading of an elastic halfspace by a rigid cylindrical stamp and compared to solutions by Spence and Turner.