Axisymmetric frictionless indentation of a rigid stamp into a semi-space with a surface energetic boundary

An axisymmetric problem for a frictionless contact of a rigid stamp with a semi-space in the presence of surface energy in the Steigmann–Ogden form is studied. The method of Boussinesq potentials is used to obtain integral representations of the stresses and the displacements. Using the Hankel transform, the problem is reduced to a single integral equation of the first kind on a contact interval with an additional condition. The integral equation is studied for solvability. It is shown that for the classic problem in the absence of surface effects and for the problem with the Gurtin–Murdoch surface energy without surface tension, the obtained equation represents a Cauchy singular integral equation. At the same time, for the Gurtin–Murdoch model with a non-zero surface tension and for the general Steigmann–Ogden model, the problem results in the integral equation of the first kind with a weakly singular or a continuous kernel, correspondingly. Hence, the contact problem is ill-posed in these cases. The integral equation of the first kind with an additional condition is solved approximately by using Gauss–Chebyshev quadrature for evaluation of the integrals. Numerical results for various values of the parameters are reported.

Investigation of the shapes of cuts in a plate in contact with a rigid stamp

A mathematical model of a contact interaction between a plate and rigid stamp is derived taking into account physical and design details. The plate is considered to have a crack, that changes its form. The problem of the contact is evaluated based on the theory of variational inequalities. The shape of the stamp is assumed to be perpendicular to the plate surface and the Poisson’s ratio is between 0 and 0.5. Analytical formulation of the study consists of transformation equation, boundary conditions and integral equation. The result is used in maximization and minimization problems for choosing extremal shape of the vertical break in the plate.

Method for solving plane unsteady contact problems for rigid stamp and elastic half-space with a cavity of arbitrary geometry and location

In the work, the process of unsteady contact interaction of rigid stamp and elastic half-space having a recessed cavity of arbitrary geometry and location with a smooth boundary was investigated. Three variants of contact conditions are considered: free slip, rigid coupling, and bonded contact. The method for solving the problem is constructed using boundary integral equations. To obtain boundary integral equations, the dynamic reciprocal work theorem is used. The kernels of integral operators are bulk Green functions for the elastic plane. Because of straight-line approximations of the domain boundaries with respect to the spatial variable and straight-line approximations of the boundary values of the desired functions with respect to time, the problem is reduced to solving a system of algebraic equations with respect to the pivotal values of the desired displacements and stresses at each time interval. One of the axes is directed along the regular boundary of half-space, the second - deep into half-space.

Frictional contact problem of an anisotropic laterally graded layer loaded by a sliding rigid stamp

This study proposes analytical and computational methods for the solution of the sliding frictional contact problem of an anisotropic laterally graded layer loaded by an arbitrarily shaped rigid stamp. The plane-strain orthotropy prevails in the layer which is bonded to a rigid foundation. Each of four orthotropic stiffness coefficients is exponentially varied through the lateral direction of the elastic layer. The Fourier transformations of the field variables are employed in the formulation. The gradient of a displacement component on the surface is then converted to a singular integral equation of the second kind. The singular integral equation is solved by means of the Gauss–Jacobi quadrature integration techniques, a collocation method, and a recursive integration method for the Cauchy integral considering the flat and triangular stamp profiles. The finite element method solutions of the same contact problems are performed using the augmented Lagrange method which is implemented in virtue of ANSYS design parametric language. An iterative algorithm is additionally utilized for the (incomplete) triangular stamp problem to conveniently reach the solutions for predetermined contact lengths. The convergence and comparative analyses are carried out to elucidate the trustworthiness of the analytical and computational methods proposed. Moreover, the parametric analyses infer that the contact-induced damage risks can be effectively alleviated upon tuning the degree of orthotropy and the lateral heterogeneity of the elastic layer.

Solution of Contact Problem for Supporting Node of Beam Hinged Plate

The paper considers a solution of contact problem for hinged supporting node of beam floor slab (coating). The main goal is to determine a stress state of the area where a plate rests on the wall. In this case, a number of problems are solved: construction of reactive pressure diagrams in the area of direct plate and wall contact, clarification of the calculated plate span, influence of contact zone size on a value of maximum bending moment in the middle of the plate, determination of contact area at various indices of flexibility and comparison of the obtained results with the known solution of rigid stamp and elastic quarter-plane interaction. The calculation has been carried out by the Zhemochkin method, its implementation for the given task corresponds to a mixed method of structural mechanics. As an illustration, the calculation has been performed on a concentrated load applied in the middle of the plate span. In the course of the study, it has been established that when a reinforced concrete slab rests on concrete and brick walls, the contact zone reduces itself to two Zhemochkin sections. When a flexibility index is decreased that corresponds to slab support on a less rigid base, the contact area is increased, and that, in its turn, has an influence on an increase of the calculated slab span and the bending moment in the middle of the slab. In the case of an absolutely rigid plate support (flexibility index is equal to zero), the contact stresses reach their maximum value at the point of plate edge contact and elastic quarter-plane. The nature of the diagram is confirmed by an analytical dependence of contact stress distribution obtained by Aleksandrov V. M. when solving a problem of pressing a rigid stamp into an edge of an elastic wedge.

Frictionless contact of a rigid stamp with a semi-plane in the presence of surface elasticity in the Steigmann–Ogden form

In this paper, the surface elasticity in the form proposed by Steigmann and Ogden is applied to study a plane problem of frictionless contact of a rigid stamp with an elastic upper semi-plane. The results of this work generalize the results for contact problems with Gurtin–Murdoch elasticity by including additional dependency on the curvature of the surface. The mechanical problem is reduced to a system of singular integro-differential equations, which is further regularized using the Fourier transform. The size dependency of the solutions of the problem is highlighted. It is observed that the curvature dependence of the surface energy is increasingly important at small scales. The numerical procedure of the solution of the system of singular integro-differential equations is presented, and numerical results are obtained for different values of the mechanical parameters.

Dynamics of the layered earthen cloth timber-carrying roads.

Представлена математическая модель для динамического прочностного расчета конструкций лесовозных дорог на слоистом земляном полотне с целью повышения надежности и долговечности. Динамическая реакция земляного полотна определяется при действии на жесткий штамп, расположенный на поверхности полотна, движущейся нагрузки, имитирующей перемещение транспорта. Расчеты выполняются методом конечных элементов. Обосновывается применение метода конечных элементов для расчета переходных динамических процессов в слоистом земляном полотне. Методика расчета построена на конкретном примере слоистого земляного полотна. Строится ряд решений динамической задачи при изменяющейся скорости движения нагрузки по штампу. Отношения вертикальных динамических и статической составляющих перемещений штампа дают коэффициенты динамичности земляного полотна в зависимости от скорости движения нагрузки. The mathematical model for dynamic strength calculation of designs timber-carrying roads to a layered earthen cloth is submitted with the purpose of increase of reliability and durability. Dynamic reaction of an earthen cloth is defined at action on the rigid stamp located on a surface of a cloth, the moving loading simulating moving of transport. Calculations are carried out by a finite elements method. Application of a finite elements method for calculation of transitive dynamic processes in a layered earthen cloth is proved. The design procedure is constructed on a concrete example of a layered earthen cloth. A line of decisions of a dynamic problems is under construction at changing speed of movement of loading on a stamp. Relations of the vertical dynamic and static displacements, making movings of a stamp, give factors of dynamism of an earthen cloth depending on speed of movement of loading.

The Frictional Contact Problem of a Rigid Stamp Sliding over a Graded Medium

In this study, the contact problem for a graded elastic half-plane in frictional contact with a rigid stamp is considered. The plane contact problem is assumed to be linear elastic and the Poisson's ratio is assumed to be constant. Analytical formulation of the study includes Fourier transforms of the governing equations and boundary conditions. The resulting integral equation is solved numerically. Contact pressure, in-plane stress and the stress intensity factor at the sharp edges of the contact are evaluated and demonstrated for various stamp profiles. The results are compared with a closed form solution for homogeneous isotropic half-plane indented by rigid stamps. The effects of the nonhomogeneity parameter, coefficient of friction and stamp profiles on the contact and in-plane stresses are analyzed in detail.