On the Influence of Interlayer Nonuniformity on Contact Characteristics during the Interaction of a Pipe and a Rigid Cylindrical Insert

2021 ◽  
Vol 887 ◽  
pp. 706-710
Author(s):  
Kirill E. Kazakov
Keyword(s):  
Integral Equation ◽  
Analytical Solution ◽  
Contact Problem ◽  
Analytical Methods ◽  
Integral Operators ◽  
Contact Stresses ◽  
Coating Properties ◽  
Discontinuous Functions ◽  
Different Types ◽  
Separate Factor

Contact problem for viscoelastic aging pipe with a longitudinally nonuniform thin elastic internal coating and a rigid cylindrical insert is considered in the paper. The basic integral equation with integral operators of different types (mixed integral equation) is given. It's analytical solution for contact stresses in insert area is presented. The solution is constructed in such a way that the function describing the inner coating nonuniformity is distinguished by a separate factor. This fact allows one to perform accurate calculations even in cases where the coating properties are described by rapidly changing and even discontinuous functions. Other known analytical methods do not allow one achieving such a results.

scholarly journals Wear of elastic tube with nonuniform coating by rigid bush

2020 ◽  
Vol 162 ◽  
pp. 02002 ◽  
Author(s):  
Kirill E. Kazakov
Keyword(s):  
Analytical Solution ◽  
Special Form ◽  
Special Functions ◽  
Contact Stresses ◽  
Elastic Tube ◽  
Analytical Representation ◽  
Standard Methods ◽  
In Series ◽  
Separate Factor ◽  
Over Time

This article is devoted to the statement and construction of analytical solution of the wearcontact problem for a rigid bush and elastic pipe with a coating in the case when the coating is nonuniform. The presence of nonuniformity leads us to the necessity of constructing a solution in a special form over special functions, since standard methods does not allow us to effectively take into account the complex properties of the coating. Analytical representation for contact stresses under the bush is presented in series with separate factor, which connect with complex properties of coating. This allows provide effective calculation even if these properties are described by rapidly changing or discontinuous functions. It is also shown that contact stresses will be negligible over time.

scholarly journals Nonsteady frictional heating on a sliding contact

2018 ◽  
Vol 226 ◽  
pp. 03030
Author(s):  
Vladimir B. Zelentsov ◽  
Boris I. Mitrin
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Contact Problem ◽  
Half Plane ◽  
Frictional Heating ◽  
Contact Stresses ◽  
Sliding Contact ◽  
Singular Equation ◽  
Static Contact ◽  
Class Of Functions

We consider quasi-static contact problem on frictional heating on a sliding contact of a rotating rigid cylinder and a half-plane. The cylinder is pressed towards the half-plane material. The problem is reduced to solution of a singular integral equation with respect to contact stresses. Solution of the singular equation is looked for in a class of functions limited on the edge, with two additional conditions to determine timedependent boundaries of the contact area. Temperature at the contact and inside the half-plane is determined in terms of contact stresses.

scholarly journals Contact problem for foundations with multilayer nonuniform coatings of variable thickness

2020 ◽  
Vol 162 ◽  
pp. 02004
Author(s):  
Kirill E. Kazakov ◽  
Svetlana P Kurdina
Keyword(s):  
Mathematical Model ◽  
Integral Equation ◽  
Analytical Solution ◽  
Contact Problem ◽  
Variable Thickness ◽  
Complex Shape ◽  
Integral Conditions ◽  
High Quality ◽  
Main Structure ◽  
Coating Layers

The article is devoted to study of the contact problem for a punch with a complex shape and a base with a coating consisting of nonuniform layers of variable thickness. Such a foundations are often found in practice. Coating layers in them can play a role, for example, heat or electric insulators. Such layers can be used as protection against mechanical stress on the main structure. Mathematical model of the problem is constructed. It is a mixed integral equation containing functions that describe the properties and thicknesses of the layers, as well as the shapes of the contacting bodies, and additional integral conditions. Analytical solution of the problem is presented for one of the formulation options. In the resulting solution, the functions associated with the properties and forms of bodies are distinguished by separate terms. This allows one to perform high-quality calculations and analysis of the behavior of the punch on the layer, even if these functions are rapidly changing.

A semianalytical and finite-element solution to the unbonded contact between a frictionless layer and an FGM-coated half-plane

2017 ◽  
Vol 24 (2) ◽  
pp. 448-464 ◽  
Author(s):  
Jie Yan ◽  
Changwen Mi ◽  
Zhixin Liu
Keyword(s):  
Integral Equation ◽  
Finite Element ◽  
Singular Integral Equation ◽  
Contact Problem ◽  
Singular Integral ◽  
Material Properties ◽  
Elastic Layer ◽  
Coated Substrate ◽  
Receding Contact ◽  
Contact Size

In this work, we examine the receding contact between a homogeneous elastic layer and a half-plane substrate reinforced by a functionally graded coating. The material properties of the coating are allowed to vary exponentially along its thickness. A distributed traction load applied over a finite segment of the layer surface presses the layer and the coated substrate against each other. It is further assumed that the receding contact between the layer and the coated substrate is frictionless. In the absence of body forces, Fourier integral transforms are used to convert the governing equations and boundary conditions of the plane receding contact problem into a singular integral equation with the contact pressure and contact size as unknowns. Gauss–Chebyshev quadrature is subsequently employed to discretize both the singular integral equation and the force equilibrium condition at the contact interface. An iterative algorithm based on the method of steepest descent has been proposed to numerically solve the system of algebraic equations, which is linear for the contact pressure but nonlinear for the contact size. Extensive case studies are performed with respect to the coating inhomogeneity parameter, geometric parameters, material properties, and the extent of the indentation load. As a result of the indentation, the elastic layer remains in contact with the coated substrate over only a finite interval. Exterior to this region, the layer and the coated substrate lose contact. Nonetheless, the receding contact size is always larger than that of the indentation traction. To validate the theoretical solution, we have also developed a finite-element model to solve the same receding contact problem. Numerical results of finite-element modeling and theoretical development are compared in detail for a number of parametric studies and are found to agree very well with each other.

scholarly journals RELATIVE PORE VOLUME UNDER INDENTATION OF POROUS MATERIALS

2021 ◽  
Vol 83 (4) ◽  
pp. 462-470
Author(s):  
V.B. Zelentsov ◽  
A.D. Zagrebneva ◽  
P.A. Lapina ◽  
S.M. Aizikovich ◽  
Wang Yun-Che
Keyword(s):  
Porous Material ◽  
Contact Problem ◽  
Porous Layer ◽  
Pore Volume ◽  
Lower Boundary ◽  
Contact Stresses ◽  
Plane Contact ◽  
Plane Contact Problem ◽  
Static Contact ◽  
Rigidity Modulus

Investigation of the function of the relative volume of pores under the load action is carried out on the base of the solution of the static contact problem of the indentation of a layer made of a material with voids or unfilled pores. A rigid strip indenter with a flat base is pressed into a porous layer that is adhered to a non-deformable base along the lower boundary. The formulated 3D problem of the indentation of a porous layer is reduced to solving the plane contact problem of the indentation of a porous strip. The plane contact problem is reduced to solving an integral equation for unknown contact stresses, the solution of which is constructed by the method of successive approximations in the form of an asymptotic expansion in the dimensionless parameter of the problem. The obtained contact stresses and the force acting on the indenter made it possible to study the influence of the nonclassical moduli of the layer porous material (the connectivity modulus and pore rigidity modulus) on the main contact characteristics and on the distribution of the function of the relative pore volume. The connectivity modulus increase leads to an increase in the compliance of the layer porous material, the pore rigidity modulus increase leads to an increase in the rigidity of the layer porous material. The maximum value of the distribution function of the relative pore volume in the porous material of the layer is achieved under the indenter base centre, regardless of the change in the porous material non-classical moduli.

scholarly journals On sound waves of finite amplitude

1953 ◽  
Vol 216 (1127) ◽  
pp. 539-547 ◽  
Keyword(s):  
Integral Equation ◽  
Analytical Solution ◽  
Arbitrary Function ◽  
Initial Conditions ◽  
Finite Amplitude ◽  
Sound Waves ◽  
One Dimensional ◽  
The One ◽  
Complex Integral ◽  
Gas Cloud

An analytical solution of Riemann’s equations for the one-dimensional propagation of sound waves of finite amplitude in a gas obeying the adiabatic law p = k ρ γ is obtained for any value of the parameter γ. The solution is in the form of a complex integral involving an arbitrary function which is found from the initial conditions by solving a generalization of Abel’s integral equation. The results are applied to the problem of the expansion of a gas cloud into a vacuum.

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