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.     

On the contact mechanics of a rolling cylinder on a graded coating. Part 1: Analytical formulation

In this paper, the fully coupled rolling contact problem of a graded coating/substrate system under the action of a rigid cylinder is investigated. Using the singular integral equation approach, the governing equations of the rolling contact problem are constructed for all possible stick/slip regimes. Applying the Gauss–Chebyshev numerical integration method, the governing equations are converted to systems of algebraic equations. A new numerical algorithm is proposed to solve these systems of equations. Both the coupled and the uncoupled solutions to the problem are found through an implemented iterative procedure. In Part I of this paper, the analytical formulation of the rolling contact problem and the discretization of the governing equations are introduced for all assumed stick/slip regimes. A detailed discussion of the proposed numerical algorithm, the iteration procedure and the numerical results, obtained using the analytical formulation, are given in Part II.     

巴西圆盘中心裂纹摩擦接触问题的逐点Lagrange乘子法

采用逐点Lagrange乘子法求解巴西圆盘中心裂纹在压剪荷载作用下裂纹面可能发生的摩擦接触问题。为了避免传统的Lagrange乘子法中总刚度阵求逆的困难,将Lagrange乘子逐点转到局部坐标系下,采用Gauss-Seidel迭代法求解法向和切向乘子,同时注意在求解的过程中对切向乘子约束修正,待所有点乘子求解完成后再变换到整体坐标系下迭代求解位移。与传统接触算法相比,该算法无需对总刚度阵求逆,降低了求解规模,提高了计算效率。通过该方法计算了巴西圆盘中心裂纹两种典型情况下的应力强度因子,计算结果与文献比较,吻合良好。考虑不同荷载角和裂纹长度对位移,应力强度因子和接触区的影响,并对不同摩擦系数下应力强度因子的影响进行了分析。结果表明:忽略裂纹接触摩擦作用,应力强度因子可能被高估。     

A general boundary element method approach to the solution of three-dimensional frictionless contact problems

A direct boundary element method is presented for three-dimensional stress analysis of frictionless contact problems. The isoparametric formulation of the boundary element method is implemented for the general case of contact in the absence of friction, which is limited to linear elastic homogeneous and isotropic materials. An iterative procedure is employed to determine the correct size of the contact zone by finding a boundary solution compatible with the contact condition. The applicability of the procedure is tested by application to three problems of advancing and conforming contact. The computed results are compared with numerical and analytical solutions where possible.     

On the influence of clearance in orthotropic disc-pin contacts

Solutions to contact problems are important in mechanical as well as in civil engineering, and even for the most simple problems there is still a need for research results. In the present paper we use an alternative super element procedure to solve directly, without iteration and incrementation, an orthotropic disc-pin contact problem.

The most simple solutions are named Hertz solutions (from 1882), and we use one of these solutions for comparison with finite element results. As a function of the total contact force we find (inversely) the size of the contact area, the distribution of the contact pressure, and the contact compliance. In models of finite size the compliance depends on the flexibility of the total model, including the boundary condition of the model, and therefore disagreement with the locally based analytical models is expected.

The examples of an earlier paper were restricted to axisymmetric problems with isotropic, elastic materials and excluding friction. In the present paper we restrict to the two-dimensional problems of conforming cylindrical contact with focus on the orthotropic disc-pin contact where the hole is in an orthotropic disc, i.e., in a non-isotropic material. Especially the indentation (penetration, normal approach) is badly estimated by the analytical methods and therefore deserves special attention. Results from a number of parameter studies of the influence of clearance is presented, and from this follows that some almost linear relations are found. It is concluded that a simple analytical Hertz formula is useful, but it cannot give detailed information.     

Friction drive of an SAW motor. Part III: modeling

A 2-layer modeling method of friction drive of a surface acoustic wave motor is proposed. The surface layer accounts for the previously proposed point-contact friction drive model, which was generalized to correspond spatially to the underlying layer that is comprised of a 3-D elasticity field. A method to determine stiffness through the use of analytical solutions of 3-D contact problems bridges the 2 layers. Because the determined stiffness expresses the accuracy of the results regarding either layer, the validity of the results concerning the stiffness and the resulting stress field was evaluated by comparison with the results of finite element analysis. Furthermore, we executed numerical simulations by using the friction drive model, which were compared with the measured displacements of the frictional surface of the slider. The simulation accurately represented the normal displacement of the frictional surface; the modeling procedure in the normal direction was found to be reliable. However, because the friction coefficient drastically changes the tangential displacement, we could not discuss the reliability of the modeling procedure in the tangential direction. A thorough discussion of the friction drive would thus require further investigation of the friction phenomena.     

Relative Slip on Contact Surface under Partial Slip Fretting Fatigue Condition

Abstract:  A combined experimental–numerical approach was utilized to characterize the relative slip along the contact surface and its features under the partial slip fretting fatigue condition. Relative displacements at two locations on the substrate (specimen) and fretting pad were measured in fretting fatigue tests. These measurements were then utilized to validate finite element analysis. Effects of the coefficient of friction on the relative slip and contact condition were investigated. The stress state along the contact surface was also investigated. Two contact geometries were analysed: cylinder-on-flat and flat-on-flat. There was no change in relative displacement between locations away from the contact surface because of the change in the coefficient of friction, while relative slip on the contact surface was affected by coefficient of friction. In addition, stick/slip sizes were affected by the change in coefficient of friction. Comparison between present and previous finite element models showed that stress state, as well as a critical plane-based crack-initiation parameter, was not much different between these approaches, while relative slip on the contact surface changed considerably.     

An alternative formulation for quasi-static frictional and cohesive contact problems

It is known by Engineering practitioners that quasi-static contact problems with friction and cohesive laws often present convergence difficulties in Newton iteration. These are commonly attributed to the non-smoothness of the equilibrium system. However, non-uniqueness of solutions is often an obstacle for convergence. We discuss these conditions in detail and present a general algorithm for 3D which is shown to have quadratic convergence in the Newton–Raphson iteration even for parts of the domain where multiple solutions exist. Chen–Mangasarian replacement functions remove the non-smoothness corresponding to both the stick-slip and normal complementarity conditions. Contrasting with Augmented Lagrangian methods, second-order updating is performed for all degrees-of-freedom. Stick condition is automatically selected by the algorithm for regions with multiple solutions. The resulting Jacobian determinant is independent of the friction coefficient, at the expense of an increased number of nodal degrees-of-freedom. Aspects such as a dedicated pivoting for constrained problems are also of crucial importance for a successful solution finding. The resulting 3D mixed formulation, with 7 degrees-of-freedom in each node (displacement components, friction multiplier, friction force components and normal force) is tested with representative numerical examples (both contact with friction and cohesive force), which show remarkable robustness and generality.     

Axisymmetric contact of an elastic layer underlain by rigid base

Axisymmetric contact problems of an elastic layer are studied in this paper. The lower surface of the layer is either bonded to or in smooth contact with a rigid base. The upper surface is in tensionless smooth contact with a rigid flat cylinder, a rigid sphere, an elastic sphere or a circular plate, respectively. A general method is developed so that all these related problems are treated in a straightforward and like manner. In comparison to the other methods used in previous papers, the simplicity and high numerical accuracy are the major advantages of this method. The convergence is proved by comparing the numerical values to some available analytical solutions. Extensive numerical results for contact radii, displacements and contact pressures are provided.     

A Direct analysis of elastic contact using super elements

Solutions to contact problems are important in mechanical as well as in civil engineering, and even for the most simple problems there is still a need for research results. In the present paper we suggest an alternative finite element procedure and by examples show the need for more knowledge related to the compliance of contact surfaces. The most simple solutions are named Hertz solutions from 1882, and we use some of these solutions for comparison with our finite element results. As a function of the total contact force we find the size of the contact area, the distribution of the contact pressure, and the contact compliance. In models of finite size the compliance depends on the flexibility of the total model, including the boundary condition of the model, and therefore disagreement with the locally based analytical models is expected and found. With computational contact mechanics we can solve more advanced contact problems and treat models that are closer to physical reality. The finite element method is widely used and solutions are obtained by incrementation and/or iteration for these non-linear problems with unknown boundary conditions. Still with these advanced tools the solution is difficult because of extreme sensitivity. Here we present a direct analysis of elastic contact without incrementation and iteration, and the procedure is based on a finite element super element technique. This means that the contacting bodies can be analyzed independently, and are only coupled through a direct analysis with low order super element stiffness matrices. The examples of the present paper are restricted to axisymmetric problems with isotropic, elastic materials and excluding friction. Direct extensions to cases of non-isotropy, including laminates, and to plane and general 3D models are possible.     

Effect of crack face contact and friction on Brazilian disk specimens—A finite difference solution

A full curvilinear transformation is employed to study the effect of contact and friction on Brazilian disk specimens containing a crack and subjected to concentrated loads at angles 0° <  < 90°. Homogeneous and bimaterial disks made of glass and epoxy are considered. The effect of loading angle and friction coefficient on the stress intensity factors, as well as the contact length is studied. The results are compared to available semi-analytical and finite elements results. It is found that when the crack faces are in contact without stick zones, an increase in friction causes a decrease of the normal gap, tangential shift and stress intensity factors. When stick conditions appear in the contact zone, an increase in the coefficient of friction also results in increasing the stick zone within the contact zone.     

Shear loaded interface crack under the influence of friction: a finite difference solution

Formulation of the elastic two‐dimensional problem of contact with friction is presented. Two‐dimensional equilibrium equations and boundary conditions in an orthogonal curvilinear co‐ordinate system are written explicitly. The above formulation is solved with the aid of the finite difference technique. An iterative algorithm which does not require load increments is employed for solving interface fracture problems with contact and friction subjected to a monotonically increasing load. The J‐integral is extended for problems in which there is friction along the crack faces. Stress intensity factors are calculated by means of the J‐integral, as well as an asymptotic expansion of the tangential shift. Two problems are analysed: (1) a crack in homogeneous material in the presence of friction involving stationary contact; and (2) an interface crack in the presence of friction involving receding contact. Results are compared to those found by analytical and semi‐analytical methods which are presented in the literature, as well as to those obtained by means of the finite element method. The accuracy of the results establishes the reliability of the finite difference analysis, as well as the post‐processors. In addition, a problem involving stick conditions is considered. It is observed that with increasing friction, the normal gaps and tangential shifts decrease. The size of the contact zone increases and values of the stress intensity factor decrease. Copyright © 2004 John Wiley & Sons, Ltd.     

Conserving algorithms for frictionless and full stick friction dynamic contact problems using the direct elimination method

In this paper, conserving time‐stepping algorithms for frictionless and full stick friction dynamic contact problems are presented. Time integration algorithms for frictionless and full stick friction dynamic contact problems have been designed to preserve the conservation of key discrete properties satisfied at the continuum level. Energy and energy‐momentum–preserving algorithms for frictionless and full stick friction dynamic contact problems, respectively, have been designed and implemented within the framework of the direct elimination method, avoiding the drawbacks linked to the use of penalty‐based or Lagrange multipliers methods. An assessment of the performance of the resulting formulation is shown in a number of selected and representative numerical examples, under full stick friction and slip frictionless contact conditions. Conservation of key discrete properties exhibited by the time‐stepping algorithm is shown.     

Frictional contact between 3D beams

 The paper deals with frictional contact between 3D beams with rectangular cross-sections. In the analysis large displacements and small strains are allowed. Hence the cross-section behaves like a rigid body undergoing displacement and rotations. Contact between the beams is assumed to be point-wise between their edges. For the friction the simple non-associated Coulomb law is adopted. The analogy to rigid plasticity is used where stick and slip conditions are considered and backward Euler scheme is applied whenever the yield condition is violated. Contact constraints for normal and tangential part are introduced using the active set strategy and the penalty method. A consistent linearisation of both contact contributions is derived and expressed in suitable matrix form, easy to use in FEM approximations. Several numerical examples including the comparison to the full 3D-solid analysis depict the efficiency of the presented approach. Received 5 May 2001     

Sliding and indentation by a rigid half-wedge with friction and displacement coupling effects

Dynamic rigid indentation induced by sliding contact in the presence of Coulomb friction is studied by considering a rigid half-wedge which simultaneously translates tangentially and normally at constant speeds with respect to the surface of an elastic half-plane. The tangential displacements of half-plane points in the contact zone are not a priori assumed to be negligible, and are thus coupled with the normal displacements in imposing the contact zone boundary conditions. The contact zone itself, as defined on the undeformed half-plane, extends at a constant rate. A complete solution to the problem shows that the coupling effects may be small for low indentation speeds and relatively flat half-wedges. The coupling effect decreases the singularity order at the half-wedge apex, while the friction increases it. Indeed, the two effects may cancel out in this and other aspects of the solution. It is also found that apex particle velocity singularities occur only due to the tangential motion of the half-wedge.     

Numerical analysis of a generalized plane ‘elastica’ with non-linear material behaviour

This article presents a numerical procedure for the analysis of a planar, continuous, flexible member. Large displacements and rotations, non-linear material behaviour and general cross sections are admitted. A shooting method is utilized along with Newton-type iteration for improved estimates. The resulting computer program is used to compare with numerical and analytical solutions of problems solved by others, and to present the solution to a problem which brings out very clearly the effects of both geometric and material non-linearities.     

A new singular boundary element for crack problems: Application to bolted joints

The present paper is concerned with the effective numerical implementation of the two-dimensional Dual Boundary Element Method to analyse the mixed-mode crack growth in bolted joints. All the boundaries are discretized with discontinuous quadratic boundary elements and the crack-tip is modeled by singular elements that exactly represent the strain field singularity 1/ . The Stress Identity Factors can be computed very accurately from the crack opening displacement at collocation points extremely close to the crack tip. Furthermore, the analysis of two-dimensional elastic contact problems is developed, with an iterative procedure. The contact equations are written explicitly with both transactions and displacements retained as unknowns. The computed results show that the proposed approach for Stress Intensity Factors evaluation is simple, produces very accurate solutions and has little dependence on the size of the elements near the crack tip. The algorithm is applied to several two-dimensional examples and the results obtained are in very good agreement with analytical solutions and experimental results carried out at the AEROSPATIALE Research Centre.     

Symmetric‐Galerkin BEM simulation of fracture with frictional contact

A symmetric‐Galerkin boundary element framework for fracture analysis with frictional contact (crack friction) on the crack surfaces is presented. The algorithm employs a continuous interpolation on the crack surface (utilizing quadratic boundary elements) and enables the determination of two important quantities for the problem, namely the local normal tractions and sliding displacements on the crack surfaces. An effective iterative scheme for solving this non‐linear boundary value problem is proposed. The results of test examples are compared with available analytical solutions or with those obtained from the displacement discontinuity method (DDM) using linear elements and internal collocation. The results demonstrate that the method works well for difficult kinked/junction crack problems. Copyright © 2003 John Wiley & Sons, Ltd.     

岩体结构面切向循环加载本构关系研究

回顾岩体结构面切向循环加载的力学试验和数值模型,并建立新的本构模型。在粘接状态下,结构面切向本构关系表示为由双曲线和直线段组成的滞回曲线。在接触状态由粘接变为滑移的过程中,峰值摩擦角以双曲线函数逐渐降低至残余摩擦角,体现出峰值剪切特性。考虑结构面切向与法向耦合的剪胀关系,建立相应的接触刚度矩阵,并引入初始剪胀角和残余剪胀角来建立分段抛物线形式的剪胀曲线。循环加载导致的磨损对结构面的摩擦和剪胀特性均产生影响,通过以切向塑性功指数函数表示摩擦角和剪胀角的磨损过程。模型在物理意义上反映了切向循环加载的特性,计算结果能较好地拟合试验曲线。