Small-scale crack blunting at a bimaterial interface with Coulomb friction

The deformation field near the tip of a tensional crack impinging upon a normally loaded bimaterial frictional interface is studied. By allowing the two materials to slide with respect to each other along the interface the high stresses around the crack tip are relaxed. The region of slip as well as the stress distribution inside the slip zone is obtained as a function of the far field loading parameter.     

The use of Castigliano’s theorem in Coulomb friction problems

This paper presents a systematic way of analyzing the statics of elastic beams with Coulomb friction by means of Castigliano’s theorem. For this purpose, the handling of the equilibrium conditions in Castigliano’s theorem is unified first. It is shown that all equations for the unknown reaction forces and the displacements of force application points can be derived from an augmented energy function in which the force and moment balance equations are introduced via Lagrange multipliers. As a consequence, the case differentiation between stick and slip in Coulomb’s friction laws can be described in a natural way, where the set of equations determining the unknown reaction forces of an elastic body with several frictional supports is always obtained from the same function for any stick-slip sequence. To demonstrate the benefit of this approach, multiple statically indeterminate, semi-circular beams are discussed.     

Dynamics of inelastically colliding spheres with Coulomb friction: Relaxation of translational and rotational energy

We investigate the free cooling of inelastic rough spheres in the presence of Coulomb friction. Depending on the coefficients of normal restitution ε and Coulomb friction μ, we find qualitatively different asymptotic states. For nearly complete normal restitution (ε close to 1) and large μ, friction does not change the cooling properties qualitatively compared to a constant coefficient of tangential restitution. In particular, the asymptotic state is characterized by a constant ratio of rotational and translational energies, both decaying according to Haff's law. However, for small ε and small μ, the dissipation of rotational energy is suppressed, so that the asymptotic state is characterized by constant rotational energy while the translational energy continues to decay as predicted by Haff's law. Introducing either surface roughness for grazing collisions or cohesion forces for collisions with vanishing normal load, causes the rotational energy to decay according to Haff's law again in the asymptotic long-time limit with, however, an intermediate regime of approximately constant rotational energy. Received: 19 November 1999     

Stochastic analysis of strongly non-linear elastic impact system with Coulomb friction excited by white noise

The stochastic response of frictionally damped strongly non-linear elastic impact oscillator subjected to white noise excitation and its stochastic bifurcation are considered. By the stochastic averaging method based on generalized harmonic function, one can obtain the stationary probability density function of this system. The effects of system parameters on the responses are investigated and the analytical results were verified by comparing with numerical results from Monte Carlo simulations. Stochastic bifurcations are discussed through a qualitative change of the stationary probability distribution, which indicates that the coefficient of friction, damping constant of the elastic impact force respectively, can be treated as bifurcation parameters.     

Interfacial debonds of layered anisotropic materials using a quasi-static interface damage model with Coulomb friction

The resistance to mode I failure of rubbers is studied by submitting single edge notch samples to uniaxial tension. Reproducing the seminal work of Rivlin and Thomas (J Polym Sci 10:291–318, 1953), single edge notch tension specimens, presenting notches of various lengths, are stretched until break. A styrene butadiene rubber, unfilled and filled with carbon-black, and an unfilled rubber from the latter mentioned work, were considered. When the notch is smaller than one fifth of the sample width, mode I crack opening is observed, leading to catastrophic failure that creates smooth mirror-like crack surfaces. Nonetheless, the experimental force-elongation responses show that the mode I critical energy release rate cannot be calculated by a classical Griffith elastic failure analysis. When notches are longer, the SENT samples are not submitted to pure uniaxial tension only. Structural bending leads to uncontrolled mixed mode crack propagation. The surfaces created when the long notches propagate are rough and bifurcations are witnessed for the filled rubbers.     

Coulomb friction controlled bridging stresses and crack resistance of ceramic matrix composites

Bridging stresses in ceramic matrix composites (CMC) are calculated taking into account the effect of Poisson contraction of fibre and matrix during loading and a two parameter Weibull distribution of fibre strength. A parameter study is performed in order to assess the influence of the constituent properties. The bridging stress curve provides the link from the micromechanical analysis to the prediction of macroscopic composite behaviour. The load–deflection behaviour of pre-notched CMC specimens is derived by application of the method of weight functions. The predictions are compared with experimental results giving information about the actual properties of the material.     

Contribution to the problem of tensile and bending-test method for SiC-fibre-reinforced glass

The Young's moduli of SiC-fibre-reinforced DURAN glass composites with different fibre concentrations obtained from tensile and three-point-bending tests were shown to be in good agreement, while the values of strength and bendover stress and strain were larger in the bending test than in the tensile test. It is assumed that this difference is mainly a consequence of the larger effectively stressed sample volume in the tensile-stress test, compared to the bending test, and it is a consequence of the shift of the neutral line in the bending test towards the compressive-stress region during the fracture process. The second consequence is the reason why the samples behave in a much more ductile way in the bending test and are brittle in the tensile test. The advantages and disadvantages of the two methods are discussed and demonstrated with special regard to the composites mentioned.     

A thermal problem of friction for a half-space with a crack

We investigate the effect of local frictional heating of the surface of a half-space on the stress intensity factors at the vertices of a surface cut.     

A numerical algorithm for stress integration of a fiber-fiber kinetics model with Coulomb friction for connective tissue

Complex behaviour of connective tissue can be modeled by a fiber-fiber kinetics material model introduced in Mijailovic (1991), Mijailovic et al. (1993). The model is based on the hypothesis of sliding of elastic fibers with Coulomb and viscous friction. The main characteristics of the model were verified experimentally in Mijailovic (1991), and a numerical procedure for one-dimensional tension was developed considering sliding as a contact problem between bodies. In this paper we propose a new and general numerical procedure for calculation of the stress-strain law of the fiber-fiber kinetics model in case of Coulomb friction. Instead of using a contact algorithm (Mijailovic 1991), which is numerically inefficient and never enough reliable, here the history of sliding along the sliding length is traced numerically through a number of segments along the fiber. The algorithm is simple, efficient and reliable and provides solutions for arbitrary cyclic loading, including tension, shear, and tension and shear simultaneously, giving hysteresis loops typical for soft tissue response. The model is built in the finite element technique, providing the possibility of its application to general and real problems. Solved examples illustrate the main characteristics of the model and of the developed numerical method, as well as its applicability to practical problems. Accuracy of some results, for the simple case of uniaxial loading, is verified by comparison with analytical solutions.     

Numerical analysis for dynamic contact between high‐speed wheel and elastic beam with Coulomb friction

For numerical analysis of the dynamic contact between a high‐speed wheel and an elastic beam, the equation of motion of each body is time integrated by a simple ODE solution technique and frictional contact conditions are imposed by the augmented Lagrange multiplier method using the contact errors defined in this work. For the stability of the numerical solution, the velocity and acceleration contact conditions as well as the displacement contact condition are imposed with special consideration for the high‐velocity contact point moving on the deformed beam. Especially, it is shown that the Coriolis and centripetal accelerations of the contact point moving rapidly on the deformed beam play crucial roles for the stability of the solution. It is also shown that, for a wheel rolling on a beam with friction, the acceleration constraint in the tangential direction is important for the stability of the solution. Copyright © 2008 John Wiley & Sons, Ltd.     

A numerical stress calculation procedure for a fiber-fiber kinetics model with Coulomb and viscous friction of connective tissue

 A material model based on fiber–fiber Coulomb and viscous friction for connective tissue was introduced in references Mijailovic (1991) and Mijailovic et al. (1993). A numerical procedure was developed in reference Mijailovic (1991) by considering sliding of fibers as a contact problem between bodies, and results were verified by experiments. Another numerical approach, much more stable and efficient, for stress calculation in case of Coulomb friction only, was proposed in reference Kojic et al. (1998). This paper represents an extension of the numerical algorithm of reference Kojic et al. (1998) to include viscous friction between fibers. The complex history of sliding is traced by dividing the current sliding length between fibers on a number of segments. The computational procedure is simple, numerically reliable and provides stress calculation for arbitrary cyclic loading of connective tissue material in tension, shear, and tension and shear simultaneously. Typical hysteresis loops can be reproduced by using the proposed numerical algorithm. The model is implemented into a finite element program PAK (Kojic et al., 1996), with a possibility of solving real problems of connective tissue. Solved examples illustrate the main features of the fiber–fiber kinetics model and of the developed numerical procedure. Received: 1 February 2002 / Accepted: 14 October 2002     

Meshless shape design sensitivity analysis and optimization for contact problem with friction

 In this paper, a continuum-based shape design sensitivity formulation for a frictional contact problem with a rigid body is proposed using a meshless method. The contact condition is imposed using the penalty method that regularizes the solution of variational inequality. The shape dependency of the contact variational form with respect to the design velocity field is obtained. The dependency of the response with respect to the shape of the rigid body is also considered. It is shown that the sensitivity equation needs to be solved at the final converged load step for the frictionless contact problem, whereas for the frictional contact case the sensitivity solution is needed at the converged configuration of each load step because the sensitivity of the current load step depends on that of the previous load step. The continuum-based contact formulation and consistent linearization is critical for accurate shape design sensitivity results. The accuracy of the proposed method is compared with the finite difference result and excellent agreement is obtained for a door seal contact example. A design optimization problem is formulated and solved to reduce the contact gap opening successfully in a demonstration of the proposed method.     

Asymptotic analysis of solutions of a thin film lubrication problem with Coulomb fluid-solid interface law

The asymptotic behavior of a Stokes flow with Coulomb free boundary friction condition when one dimension of the fluid domain tends to zero is studied. The strong convergence of the velocity is proved, a specific Reynolds equation is obtained, and the uniqueness of the limit velocity and pressure distributions is established.     

Contribution of ultrasonic to microstructure and mechanical properties of tilt probe penetrating friction stir welded joint

Tilt probe penetrating friction stir welding(PFSW)was an innovative technology proposed in recent years to avoid the formation of kissing bond in the root of joint.However,with the heat input decreasing,"S"line or zigzag line was easily introduced in the PFSW joint.In this study,ultrasonic enhanced tilt probe penetrating friction stir welding(U-PFSW)was developed to solve this problem and achieve improved joint mechanical properties.The experimental results confirmed that U-PFSW was a potent technology to completely clear the original butt surface,providing a crucial prerequisite for the achievement of high-strength joint.The application of ultrasonic improves the joint tensile strength and fracture elongation from 336 MPa and 4.3％to 359 MPa and 6.8％,respectively.Furthermore,the strength of stir zone was also increased from 391 MPa in PFSW to 420 MPa in U-PFSW.Analyses of texture and precipitate indi-cated that the dynamic recrystallization(DRX)and precipitation strengthening were both enhanced by the ultrasonic.Ultrasonic-enhanced DRX enabled a complete elimination of the"S"line;the enhanced precipitation strengthening by vacancy-induced mechanism in U-PFSW was the intrinsic reason for the significantly improved mechanical properties.     

The contact problem of a rigid stamp with friction on a functionally graded magneto-electro-elastic half-plane

We consider in this study the frictional sliding contact problem between a functionally graded magneto-electro-elastic material and a perfectly conducting rigid punch subjected to magneto-electro-mechanical loads. The problem is formulated under plane strain conditions. Using Fourier transform, the resulting plane magneto-electro-elasticity equations are converted analytically into three coupled singular integral equations in which the unknowns are the normal contact stress, the electric displacement and the magnetic induction. These integral equations are then solved numerically to obtain the distributions of the normal contact stress, electric displacement and magnetic induction at the surface of the graded medium. The main objective of this paper is to study the effect of the non-homogeneity parameter, the friction coefficient and the elastic, electric and magnetic coefficients on the surface contact pressure, electric displacement and magnetic induction distributions for the case of flat and circular punch profiles.