Solution Of Dynamic Contact Problems Research Articles

ON CONTACT PROBLEMS FOR TWO STAMPS AND A NEW TYPE OF CRACK MODEL

In this work, for the first time, an exact solution of the contact problem of interaction with a multilayer base of two semi-infinite stamps, the ends of which are parallel to each other, is constructed. Stamps are assumed to be absolutely rigid, and the distance between them can have any finite value. The task is an important stage in the algorithm for constructing models of a new type of crack in materials of different rheologies. The mechanism of destruction of the medium by cracks of a new type is radically different from the mechanism of destruction of the medium by Griffiths cracks, and has so far been poorly studied. Griffiths formed his cracks with a smooth border as a result of compression from the sides of an elliptical cavity in the plate. Cracks of a new type have a piecewise smooth border, resulting from the replacement of an ellipse with a rectangle compressed from the sides. The problem considered in the article can be considered as the result of the formation of a new type of crack with absolutely rigid banks and deformable lower boundary. Thanks to it, after the solution, it becomes possible to switch to deformable stamps and a crack of a new type in the rheological medium. The solution of this problem turned out to be possible due to the construction of exact solutions of the Wiener–Hopf integral equations on a finite segment. The paper shows how the solution of one of the previously unsolved problems allows us to investigate and solve exactly other problems, to identify previously unknown properties and resonances. As a result of constructing an exact solution to the problem, the fact that the solution of dynamic contact problems for stamp systems is not unique was confirmed and a dispersion equation for finding resonant frequencies was constructed.

Scalable TFETI with optional preconditioning by conjugate projector for transient frictionless contact problems of elasticity

The FETI based domain decomposition method is adapted to implement the time step of the Newmark scheme for the solution of dynamic contact problems without friction. If the ratio of the decomposition and discretization parameters is kept uniformly bounded, then the cost of the time step is proved to be proportional to the number of nodal variables. The algorithm uses our in a sense optimal MPRGP algorithm for the solution of strictly convex bound constrained quadratic programming problems with optional preconditioning by the conjugate projector to the subspace defined by the trace of the rigid body motions on the artificial subdomain interfaces. The proof of optimality combines the convergence theory of our MPRGP algorithm, the classical bounds on the spectrum of the mass and stiffness matrices, and our theory of the preconditioning by a conjugate projector for nonlinear problems. The results are confirmed by numerical solution of 2D and 3D dynamic contact problems.

Решение динамической контактной задачи о взаимодействии полубесконечных штампов с упругой полуплоскостью с учётом фактора износа.

Решение динамической контактной задачи о взаимодействии полубесконечных штампов с упругой полуплоскостью с учётом фактора износа.

Finite elements in space and time for dynamic contact problems

AbstractDynamic obstacle and Signorini problems are discretized by continuous and discontinuous finite elements in space and time. The resulting discrete problems are attributed to a standard implicit time‐stepping scheme through relaxation of impact phenomena and suitable numerical integration. The method can cope with dynamic contact problems, which is shown by an analysis of a model problem. Moreover, numerical examples demonstrate that it is actually able to approximate the solution of dynamic contact problems, which are not fully covered by the theory. Copyright © 2008 John Wiley & Sons, Ltd.

Study of Iterative Algorithms for Solution of Dynamic Contact Problems for Elastic Cracked Bodies

A plane problem is solved for the contact interaction between the faces of a rectilinear crack under the action of a normally incident harmonic tension–compression wave. Iterative algorithms are presented to solve the problem for both given initial distribution of contact forces and given initial discontinuity in the displacements of the crack faces. The convergence rates of the algorithms, the maximum contact forces, and displacement discontinuities are compared.

A numerical solution for dynamic contact problems satisfying the velocity and acceleration compatibilities on the contact surface

For the numerical solution of dynamic contact problems, correct contact points and displacements are determined by iteratively reducing the displacement error vector monotonically toward zero. And spurious oscillations are prevented from the solution by enforcing the velocity and acceleration compatibilities of the contact points with the corresponding error vectors. Economic computation is possible because the accelerated iterative schemes are used and because decomposition of large matrix is not required in the iterations for the contact analysis of elastic bodies. Numerical simulations are conducted to demonstrate the accuracy of the solution and the necessity of the velocity and acceleration compatibilities on the contact surface.

Periodic convolution equations on segments which occur in elasticity theory and mathematical physics: PMM, vol. 35, n≗ 5, 1971, pp. 861–868

Integral convolution equations given on a system of segments are considered; the kernel of the integral equation is a periodic function with period T The unique solvability of the equations is established for kernels of a general kind, an approximate factorization is given a foundation, and a method for construction of the approximate solution is also indicated. Integral equations of the kind mentioned occur in mixed problems of elasticity theory and mathematical physics, posed for finite bodies [1, 2], as well as for infinite bodies with periodic boundary conditions [3]. The result herein, just as in [4], is necessary for the correct formulation and solution of dynamic contact problems.