Semi-Analytical Modeling Of Three-Dimensional Contact Problem Within The Framework Of Couple Stress Elasticity

The contact problem is one of the typical mechanical problems in engineering practice.The investigations on various contact problems are fundamental scientific researches and are of great significance to practical engineering.With the development of micro-/nano-technology,micro-/nano-structures are widely used in various advanced fields.However,the strong size effects introduced by the reduction of the structure size has become an important mechanical problem that cannot be ignored.Experimental studies have pointed out that the size effects in the contact problem widely exist in various materials,so-called indentation size effects.The indentation size effect generally means a stiffer response as the indentation depth decreases.However,the classical elastic theory,due to the shortage of the scale-related parameters,cannot capture the effects caused by scale changes.On the other hand,the general threedimensional contact problem,because of its complex form,is hard to obtain analytical solutions.In addition,the numerical method,such as the finite element method,is timeconsuming in dealing with the three-dimensional contact problem.Therefore,the investigation,especially the theoretical investigation,of the contact involving size effects currently is still very limited.Motivated by the abovementioned reasons,our purpose is to establish a set of three-dimensional semi-analytical contact models involving the size effects and other common impact factors.The analytical part of the present work is mainly based on the framework of the classical couple stress theory.And the conjugate gradient method,the Fourier transform,and the Laplace transform are employed in the numerical process.With these contact models,the contact behaviors affected by the size effects,the motion states,the different fields,and the material constants will be deeply investigated in detail.The present work mainly includes:(1)Based on the couple stress theory,the frequency response functions of the elastic half-space under vertical load are obtained.Then,with the aid of the conjugate gradient method and the fast Fourier transform,the semi-analytical contact model of the rigid indenter and the elastic half-space,which can accurately and efficiently predict the effects of the characteristic material length on the three-dimensional contact behaviors,is established.Numerical results show that the semi-analytical model can well describe the size effects existing in the small-scale contact.And a stiffer response is obtained with the decrease of the indentation depth.(2)Based on the couple stress theory,the frequency response functions of the elastic film acted by the vertical load and bonded on a rigid substrate are obtained,and the corresponding semi-analytical contact model is subsequently established.In addition to the size effects of the film,the elasticity of the indenter and the thickness of the film are considered in this contact model.Parametric analyses are performed to study the dependence of the contact behaviors on the different parameters.And the sensitivity of all independent parameters is presented to find the dominating impact factors.Finally,the combined effects of the couple stress and the film thickness are summarized in one behavior map to illustrate the region that the couple stress effects or the film thickness must be taken into account.(3)Based on the couple stress theory,the frequency response functions of the elastic half-space under both the vertical and horizontal load are given,and the corresponding semi-analytical sliding contact model is subsequently established.It is assumed that the indenter moves quasi-statically on the surface of the half-space and the relationship of the pressure and the friction in the contact region follows the Coulomb friction law.Besides the characteristic material length,a new scale-related parameter,named as a transverse coefficient,is introduced,which describes the effects of the transverse curvature on principle couple stress.The dependences of the pressure and the stress field on the characteristic material length,the surface friction coefficient,as well as the transverse coefficient are presented.The obtained results show that the transverse coefficient has no effects on the pressure distributions but significantly affects the stress field.(4)Based on the couple stress elastodynamic theory,the frequency response functions of the half-space with a steady moving vertical load are obtained with the help of the Galilean transform and the Helmholtz resolution theorem.And the corresponding semi-analytical contact model of an indenter steady sliding on the surface of the half-space is established.The combined effects of the speed-induced“softening” and the indentation depth-induced “stiffening” are investigated in detail.The sensitivity of the indenter speed and the characteristic material length on the contact response is presented in the cases of different speed ratios and dimensionless characteristic material lengths.Parametric analyses show that the characteristic material length significantly affects the contact responses under the low-speed case.In contrast,the indenter speed becomes the dominant factor when speed is high.(5)Based on the couple stress thermoelastic theory,the frequency response functions of the half-space under the vertical load and heat flux are obtained.And the corresponding semi-analytical contact model of a hot flat-end cylinder pressed on a half-space is established.The state parameter that analytical got in the two-dimensional case is successfully employed here to describe the three-dimensional contact state.And based on the numerical model,the effects of the characteristic material length,the Poisson's ratio,and the state parameter on the surface pressure are investigated in detail.(6)Based on the couple stress theory and linear viscoelastic theory,the frequency response functions of the half-space with moving load are obtained by employing the elastic-viscoelastic correspondence principle.The total loading time is divided into many time steps.And in each time step,the viscoelastic response can be determined using the conjugate gradient method and fast Fourier transform.By using the numerical integration method to deal with the time integration,the semi-analytical contact model of a ball rolling on the surface of a viscoelastic half-space is established,which the transient response in each time step can be obtained.The dependences of the pressure distributions and the viscoelastic dissipation on the characteristic material length,the rolling speed,and the loading time are investigated.