Analytical solution of the spherical indentation problem for a half-space with gradients with the depth elastic properties

The contact problem for the impression of spherical indenter into a non-homogeneous (both layered and functionally graded) elastic half-space is considered. Analytical methods for solving this problem have been developed. It is assumed that the Lame coefficients vary arbitrarily with the half-space depth. The problem is reduced to dual integral equations. The developed methods make it possible to find the analytical asymptotically exact problem solution, suitable for a PC. The influence of the Lame coefficients variation upon the contact stresses and size of the contact zone with different radius of indenter as well as values of the impressing forces are studied. The effect of the non-homogeneity is examined. The developed method allows to construct analytical solutions with presupposed accuracy and gives the opportunity to do multiparametric and qualitative investigations of the problem with minimal computation time expenditure.     

Mechanics of indentation for piezoelectric thin films on elastic substrate

Frictionless normal indentation problem of rigid flat-ended cylindrical, conical and spherical indenters on piezoelectric film, which is either in frictionless contact with or perfectly bonded to an elastic half-space (substrate), is investigated. Both conducting and insulating indenters are considered. With Hankel transform, the general solutions of the homogeneous governing equations for the piezoelectric layer and the elastic half-space are presented. Using the boundary conditions for a vertical point force or a point electric charge, and the boundary conditions on the film/substrate interface, the Green’s functions can be obtained by solving sets of simultaneous linear algebraic equations. The solution of the indentation problem is obtained by integrating these Green’s functions over the contact area with unknown surface tractions or electric charge distribution, which will be determined from the boundary conditions on the contact surface between the indenter and the film. The solution is expressed in terms of dual integral equations that are converted to a Fredholm integral equation of the second kind and solved numerically. Numerical examples are also presented. The comparison between two film/substrate bonding conditions is made. It shows that the indentation rigidity of the film/substrate system is lower when the film is in frictionless contact with the substrate. The effects of the Young’s modulus and Poisson’s ratio of the elastic substrate, indenter electrical condition and indenter prescribed electric potential on the indentation responses are presented.     

A new formulation for solving 3-D time dependent rolling contact problems of a rigid body on a viscoelastic half-space

The paper deals with a new formulation for solving the rolling contact problem without friction of a rigid body on a viscoelastic half-space in three dimensions. Assuming that the material behavior is independent of time for a sufficiently short time duration, the viscoelastic contact problem is transformed into elastic like problems. Then the contact problem is solved using a direct numerical method at each time step. The convergence of the method in time and space is good for a spherical indenter. The dissymmetry of the contact patch due to hysteresis was found in three dimensions for the spherical indenter and two cylinders of different width. Finally the method was tested for a sinusoidal varying speed and shows a good efficiency.     

The problem of a slender die

The paper deals with the contact behaviour of a slender die indenting an elastic half-space. It is shown that the problem of determining the pressure on the elastic half-space may be reduced (with an error exponentially small relative to the elongation) to a single-variable integral equation, whose solution is commonly represented by an asymptotic series in a small parameter. It was shown for a die of oval form that, depending on the type of contact region, either an increase or decrease in the force acting from the elastic half-space on the die upon approaching the end-points of the die are possible.     

Wedge indentation into elastic-plastic single crystals. 2: Simulations for face-centered cubic crystals

The asymptotic stress and deformation fields associated with the contact point singularity of a nearly-flat wedge indenter impinging on a specially-oriented single face-centered cubic crystal are derived analytically in a companion paper. In order to investigate the extent of the asymptotic fields, the indentation process is simulated numerically using single crystal plasticity. The quasistatically translating asymptotic fields consist of four angular elastic sectors separated by plastically deforming sector boundaries, as predicted in the companion paper. The asymptotic stress distributions are in accord with the analytical predictions. In addition, simulations are performed for a wedge indenter with a 90° included angle in order to investigate the consequences of finite deformation and finite lattice rotation. Several salient features of the deformation field for the nearly-flat indenter persist in the deformation field for the 90° wedge indenter. The existence of the salient features is validated by comparison to experimental measurements of the lower bound on geometrically necessary dislocation (GND) densities.     

Axisymmetric contact problem for an elastic half-space and a circular cover plate

The contact interaction problem for a thin circular rigid cover plate and an elastic half-space loaded at infinity by a tensile force directed in parallel to the boundary of the half-space is considered. It is assumed that the cover plate is not resistant to bending deformations. The problem can be reduced to an integral equation of the first kind whose kernel has a logarithmic singularity. The equation is solved approximately by the Multhopp-Kalandia method. The resulting approximate solution is compared with the previously obtained asymptotic solution.     

Contact with stick zone between an indenter and a thin incompressible layer

The dominant asymptotic term for the indentation of a thin elastic incompressible layer by an axisymmetric rigid indenter is considered. Complete adhesion is supposed everywhere in the contact area or else in a given inner region surrounded by an annular frictionless zone. Both the problems are formulated in the form of systems of coupled dual integral equations. Using operators transforming kernels of the Hankel transform into kernels of the Weber–Orr transform, the dual integral equations are reduced to systems of Fredholm integral equations of the second kind whose structures permit deriving asymptotic solutions. Simple expressions for the contact stresses, the penetration depth, and the contact radius in the case of an unknown contact area are obtained. Explicit formulae, derived for the flat and power law indenter profiles, allow us to analyze how stick and frictionless zones affect mechanical characteristics. Results manifest that the punch penetration exhibits strong sensitivity to contact conditions inspite of the fact that the radial traction is small. A conical indenter is less sensitive than flat-ended and spherical indenters.     

Pressure of a Narrow Band-Like Punch Upon an Elastic Half-Space

An asymptotic solution is obtained to the contact problem of a band-like punch acting upon an elastic half-space. The method of joined asymptotic expansions is used. The results of numerical calculations are presented. The efficiency of the approach is tested by comparing it with another method     

Wedge indentation into elastic-plastic single crystals, 1: Asymptotic fields for nearly-flat wedge

Asymptotic stress and deformation fields under the contact point singularities of a nearly-flat wedge indenter and of a flat punch are derived for elastic ideally-plastic single crystals with three effective in-plane slip systems that admit a plane strain deformation state. Face-centered cubic (FCC), body-centered cubic (BCC), and hexagonal-close packed (HCP) crystals are considered. The asymptotic fields for the flat punch are analogous to those at the tip of a stationary crack, so a potential solution is that the deformation field consists entirely of angular constant stress plastic sectors separated by rays of plastic deformation across which stresses change discontinuously. The asymptotic fields for a nearly-flat wedge indenter are analogous to those of a quasistatically growing crack tip fields in that stress discontinuities can not exist across sector boundaries. Hence, the asymptotic fields under the contact point singularities of a nearly-flat wedge indenter are significantly different than those under a flat punch. A family of solutions is derived that consists entirely of elastically deforming angular sectors separated by rays of plastic deformation across which the stress state is continuous. Such a solution can be found for FCC and BCC crystals, but it is shown that the asymptotic fields for HCP crystals must include at least one angular constant stress plastic sector. The structure of such fields is important because they play a significant role in the establishment of the overall fields under a wedge indenter in a single crystal. Numerical simulations—discussed in detail in a companion paper—of the stress and deformation fields under the contact point singularity of a wedge indenter for a FCC crystal possess the salient features of the analytical solution.     

功能梯度材料涂层半空间的轴对称光滑接触问题

求解了功能梯度材料涂层半空间的轴对称光滑接触问题,其中梯度层剪切模量按照线性变化,利用Hankel积分变换方法求解微分方程,将问题化为具有Cauchy型奇异核的积分方程.数值方法求解表明:功能梯度材料涂层半空间在刚性柱形压头和球形压头作用下,接触表面分布应力,接触半径以及最大压痕受材料梯度效应的影响较大.     

Dynamics of a system of remote punches on an elastic half-space

The dynamic interaction (contact) between an elastic half-space and several smooth punches is studied. It is assumed that the dimensions of the contact regions Ω_i are much smaller than the distances between them and the scale of time of the process considered is comparable with the time required for an elastic wave to travel from one region to another. An asymptotic approach to the solution of the problem is proposed and the first two terms of the asymptotic representation of the displacement in the contact region and its neighborhood are constructed. Institute of Problems of Machine Science, Russian Academy of Sciences, St. Petersburg 199178. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 6, pp. 204–210, November–December, 1999.     

Stress disturbances caused by the inhomogeneity in an elastic half-space subjected to contact loading

Inhomogeneities can increase localized stress and cause microstructural alterations to initiate fatigue failures in rolling elements under cyclic contact loading. To study the stress disturbances created by the inhomogeneity, a two-dimensional contact stress analysis is presented for a cylindrical indenter sliding on an elastic half-space containing an inhomogeneity of arbitrary shape. The boundary element method is used to analyze the contact problem, where actual contact boundary, contact pressure as well as tractions and displacements at inhomogeneity–substrate interface are determined by solving a set of integral equations numerically. Numerical results are presented to investigate effects and the stress disturbances caused by the inhomogeneity with various locations, sizes and material properties of inhomogeneity. The results also show that hard inclusions are more detrimental than soft deformable particles in rolling contact elements.     

Analysis of the impact of a sharp indenter

The present work investigates the impact of a sharp indenter at low impact velocities. A one-dimensional model is developed by assuming that the variation of indentation load as a function of depth under dynamic conditions has the same parabolic form (Kick's Law) as under static conditions. The motion of the indenter as it indents and rebounds from the target is described. Predictions are made of the peak indentation depth, residual indentation depth, contact time, and rebound velocity as functions of the impact velocity, indenter mass and target properties. Finite element simulations were carried out to assess the validity of the model for elastoplastic materials. For rate-independent materials agreement with the model was good provided the impact velocity did not exceed certain critical values. For rate-dependent materials the relationship between load and depth in the impact problem is no longer parabolic and the model predictions cannot be applied to this case. The rate-dependent case can be solved by incorporating the relationship between the motion of the indenter and the dynamic flow properties of the material into the equation of motion for the indenter.     

Modeling of Sliding of a Smooth Indenter over a Viscoelastic Layer Coupled with a Rigid Base

The article deals with constant-speed sliding of a smooth indenter along the boundary of a viscoelastic layer coupled with a rigid half-space. The problem is investigated in a quasistatic statement by constructing a solution for the case of a load sliding, distributed inside of a rectangular element, which allows using the boundary element method and an iterative procedure. The effect of sliding velocity and layer thickness on the contact pressure distribution and the deformation component of the frictional force is studied.     

Contact Problem for a Rectangular Punch on the Porous Elastic Half-Space

The paper is concerned with a contact problem about rigid rectangular punch forced into a half-space made of a linear elastic isotropic material with voids. We use a Cowin–Nunziato model for the half-space, and reduce the problem to a double Fredholm integral equation of the first kind. Then we apply two different approaches, to solve this equation. The first one is based on a direct collocation numerical technique. The second method is asymptotic, and we use a small parameter that is the relative width of the punch. Finally, compliance of the punch is determined, and results of the two different methods are compared with each other, as well as with a Sivashinsky–Panek–Kalker solution. Mathematics Subject Classifications (2000) 74M15.     

Seismic efficiency of a contact explosion and a high-velocity impact

The seismic energy transferred to an elastic half-space as a result of a contact explosion and a meteorite impact on a planet’s surface is estimated. The seismic efficiency of the explosion and impact are evaluated as the ratio of the energy of the generated seismic waves to the energy of explosion or the kinetic energy of the meteorite. In the case of contact explosions, this ratio is in the range of 10⁻⁴–10⁻³. In the case of wide-scale impact effects, where the crater in the planet’s crust is produced in the gravitational regime, a formula is derived that relates the seismic efficiency of an impact to its determining parameters. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 3–12, March–April, 2007.     

Nonstationary axisymmetric problem of the impact of a spherical shell on an elastic half-space (initial stage of interaction)

The supersonic stage of interaction (where the rate of expansion of the contact region is no less than the speed of compression waves) between a Timoshenko-type spherical shell (indenter) and an elastic half-space (foundation) is studied. The expansion of the desired functions in series in Legendre polynomials and their derivatives are used to construct a system of resolving equations. An analytical-numerical algorithm for solving this system is developed. A similar problem was considered in [1], where the original problem was replaced by a problem with a periodic system of indenters.     

Modeling of softsphere normal collisions with characteristic of coefficient of restitution dependent on impact velocity

This letter presents a theoretical model of the normal (head-on) collisions between two soft spheres for predicting the experimental characteristic of the coefficient of restitution dependent on impact velocity. After the contact force law between the contacted spheres during a collision is phenomenologically formulated in terms of the compression or overlap displacement under consideration of an elastic—plastic loading and a plastic unloading subprocesses, the coefficient of restitution is gained by the dynamic equation of the contact process once an initial impact velocity is input. It is found that the theoretical predictions of the coefficient of restitution varying with the impact velocity are well in agreement with the existing experimental characteristics which are fitted by the explicit formula.     

Nonaxisymmetric elastoplastic impact of a parabolic body on a spherical shell

A method is proposed to calculate a spherical shell under nonaxisymmetric impact of a massive body. The motion of the shell is described by momentless equations, which are solved using the Laplace transformation and an asymptotic expansion of the required quantities in a small parameter. The contact interaction force P(t) was determined for the elastoplastic model of local bearing deformation for a parabolic impactor. Plots of the solution are given. The validity of the results is confirmed by good agreement between the solution and the limiting cases — an axisymmetric impact and an impact on a half-space.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 181–186, January–February, 2005.     

Contact laws between solid particles

This paper presents a comprehensive study for the contact laws between solid particles taking into account the effects of plasticity, strain hardening and very large deformation. The study takes advantage of the development of a so-called material point method (MPM) which requires neither remeshing for large deformation problems, nor iterative schemes to satisfy the contact boundary conditions. The numerical results show that the contact law is sensitive to impact velocity and material properties. The contact laws currently used in the discrete element simulations often ignore these factors and are therefore over-simplistic. For spherical particles made of elastic perfectly plastic material, the study shows that the contact law can be fully determined by knowing the relative impact velocity and the ratio between the effective elastic modulus and yield stress. For particles with strain hardening, the study shows that it is difficult to develop an analytical contact law. The same difficulty exists when dealing with particles of irregular shapes or made of heterogeneous materials. The problem can be overcome by using numerical contact laws which can be easily obtained using the material point method.