On the asymptotic behavior of solutions of anisotropic viscoelastic body

The quasistatic problem of a viscoelastic body in a three-dimensional thin domain with Tresca’s friction law is considered. The viscoelasticity coefficients and data for this system are assumed to vary with respect to the thickness ε. The asymptotic behavior of weak solution, when ε tends to zero, is proved, and the limit solution is identified in a new data system. We show that when the thin layer disappears, its traces form a new contact law between the rigid plane and the viscoelastic body. In which case, a generalized weak form equation is formulated, the uniqueness result for the limit problem is also proved.

VARIATIONAL ANALYSIS OF A FRICTIONAL CONTACT PROBLEM WITH WEAR AND DAMAGE

We study a quasistatic problem describing the contact with friction and wear between a piezoelectric body and a moving foundation. The material is modeled by an electro-viscoelastic constitutive law with long memory and damage. The wear of the contact surface due to friction is taken into account and is described by the differential Archard condition. The contact is modeled with the normal compliance condition and the associated law of dry friction. We present a variational formulation of the problem and establish, under a smallness assumption on the data, the existence and uniqueness of the weak solution. The proof is based on arguments of parabolic evolutionary inequations, elliptic variational inequalities and Banach fixed point.

Tsunami Efficiency Due to Very Slow Earthquakes

Often, tsunami “sources” have been treated as a quasistatic problem. Initial studies have demonstrated that, for earthquake rupture velocities in the span of 1.5–3 km/s, the kinematic and static part of the tsunami can be treated separately. However, very slow earthquake rupture velocities in the span of 0.1–1 km/s have not been included in tsunami analytical or numerical modeling. Here, we calculated the tsunami efficiency, extending Kajiura’s definition for different models. We demonstrated that rupture velocity cannot be neglected for very slow events, that is, rupture velocities slower than 0.5 km/s. We also examined the relation of magnitude, earthquake rupture velocity, and tsunami amplitude to the efficiency of very slow tsunamigenic earthquakes. Hypothetical megathrust earthquakes (Mw>8.5) with very slow rupture velocities amplify energy from 10 to 60 times larger than moderate to large earthquakes (7.0<Mw<8.5) in the direction of rupture propagation.

RESEARCH OF THE PROBLEM OF LOSS OF STABILITY OF CYLINDRICAL THINWALLED STRUCTURES UNDER INTENSE LOCAL TEMPERATURE EXPOSURE

Presently, three-dimensional printing technology is developing rapidly. This happens due to the need to create products of complex shape, the production of which by existing standard methods is very difficult and disadvantageous, and sometimes technically impossible. The study aims to investigate the problem of temperature stability of a thin-walled cylindrical structure under unsteady local thermal exposure, simulating the motion of a laser beam spot along one of the ends. To solve the problem, the finite element method was used. Since the geometric areas of calculation were parameterized, then for each set of parameters, its own finite element mesh was used for each type of analysis. A parametric spatial finite element model of cylindrical shell pinched at the bottom end was constructed. The local moving heat flux acts on the opposite side, simulating the movement of a laser beam during the additive formation of a thin-walled element. Numerical solutions of the nonstationary dynamic heat conduction problem were obtained, spatiotemporal temperature dependences were obtained, and solutions of the quasistatic problem of the loss of both local and complete loss of stability of the equilibrium state of the cylinder at various times due to the occurrence of local compressive stresses during intense heating were obtained. The dependences of critical heat flux power corresponding to the loss of stability on the cylinder wall thickness and its height were obtained. The above results can be used as calibration parameters of the process of creating thin-walled structures using additive technologies of Laser Melting class for products that require increased requirements for manufacturing accuracy.

Influence of spatial inhomogeneity of environment temperature on temperature oscillations and thermocyclic stresses in elastic half-space

Elastic half-space on which surface the heat transfer follows the Newton-Richmann law where enviroment temperature is a periodic function of time and spatial variable is investigated. A temperature field and a stressed deformed state of half-space is found after the solution of a quasistatic problem of thermoelasticity. The solution is received in the form of double Fourier series.

Solutions to the Quasistatic Problem from the theory of Inelastic Deformations with Linear Growth Condition

This paper refers to standard models in the theory of inelastic deformations. We assume that non-linear inelastic constitutive function is of monotone type, that the growth condition holds and that the model is quasistatic. Initial, generic problem is transformed into an evolution equation in a maximal monotone field. Then we find solutions with very low regularity requirements of the forces acting on a body

The Hamiltonian System Method for the Stress Analysis in Axisymmetric Problems of Viscoelastic Solids

With the use of the Laplace integral transformation and state space formalism, the classical axial symmetric quasistatic problem of viscoelastic solids is discussed. By employing the method of separation of variables, the governing equations under Hamiltonian system are established, and hence, general solutions including the zero eigensolutions and nonzero eigensolutions are obtained analytically. Due to the completeness property of the general solutions, their linear combinations can describe various boundary conditions. Simply by applying the adjoint relationships of the symplectic orthogonality, the eigensolution expansion method for boundary condition problems is given. In the numerical examples, stress distributions of a circular cylinder under the end and lateral boundary conditions are obtained. The results exhibit that stress concentrations appear due to the displacement constraints, and that the effects are seriously confined near the constraints, decreasing rapidly with the distance from the boundary.