Solving initial–boundary-value creep problems

Abstract

The Galerkin-Bubnov method with global approximations is used to find approximate solutions to initial-boundary-value creep problems. It is shown that this approach allows obtaining solutions available in the literature. The features of how the solutions of initial-boundary-value problems for oneand three-dimensional models are found are analyzed. The approximate solutions found by the Galerkin-Bubnov method with global approximations is shown to be invariant to the form of the equations of the initial-boundary-value problem. It is established that solutions of initial-boundary-value creep problems can be classified according to the form of operators in the mathematical problem formulation Keywords: creep, initial-boundary-value problem, Galerkin-Bubnov method, creep of rods, creep of turbine blades, three-dimensional creep problemIntroduction. Applications of creep theory are closely related to many scientific and engineering problems such as life prediction for machine parts. Modeling creep in structures is reduced to solving initial-boundary-value problems to describe the time dependence of damage and irreversible creep strains in bodies [1,2,[11][12][13][17][18][19][20]. In these papers, as in the majority of others, creep problems are solved by the direct Ritz method, the finite-element method, and the time continuation method [3] for finding the stationary points of the variational functional.The modeling of creep is improved by sophisticating the equations of state and introducing new internal state variables such as damage and aging parameters describing meso-structural changes in a material due to creep. It is also important to develop efficient methods for solving nonlinear initial-boundary-value creep problems. The objective of the present paper is to find, using the Bubnov-Galerkin method, numerical analytic solutions of initial-boundary-value creep problems, which was not discussed in the literature.The Bubnov-Galerkin method is widely used to find numerical analytic solutions of the differential equations of initial-boundary-value problems in mechanics such as dynamics of fluid and elastic bodies [16]. This method does not require formulating a variational problem. And it does not exist in the case of creep for that matter because the work of stress during creep is dissipation energy rather than a potential. This is one of the main advantages of the Bubnov-Galerkin method formulated as partial differential equations. The available methods, which are mainly numerical, tabulate solutions in discrete space at discrete instants of time. To obtain them, use is made of various variational formulations of the creep problem. In fact, an elastic potential with additional vectors of fictitious forces (creep strains) is used, by analogy with the functionals of plasticity theory. The McComb-Schlechte-Sanders functional set on rates of stresses, strains, and displacements [13] may be considered the most successful. However, it is not used because of complexity and nonextremality.

Influence of the temperature state on the damageability due to the creep of claddings of cylindrical fuel elements

“…The aim of computer simulations is to obtain the quantity estimations about the temperature influence on the damageability of the zirconium cladding with cylindrical shape widely used in modern reactors. To reach this aim we will consider the computer simulations based on the mathematical model (1)- (8) and numerical solution scheme (11)- (14) for the zirconium cladding with the parameters typical for modern nuclear reactors used the pressurized light water as the heat carrier: a = 3, 855mm, b = 4, 55mm, p a = 6M P a, p b = 16M P a, T b = T a − 25K.…”

“…The simplest practical way to substantiate number n of the nodes is to make the series of computer simulations with different numbers of nodes to compare each other the results of these simulations. It is naturally to use first of all the rupture time (10) of the cladding as the integrated characteristic of the numerical solutions for the differential equations (1), (5) The analysis of the results of the computer simulations obtained for the different number n of the nodes shows that the proposed numerical scheme (11)- (14) with the relatively low numbers n of nodes allows to obtain the numerical solution of the problem (1)- (8) with errors of the approximations available for engineering applications, and number n = 52 is sufficiently for computer simulations of the cladding.…”

“…We will use Merson method [13] to solve the initial-value problem (12), because this method gives us the effective way to define the integrating step on time corresponding to increasing velocities of the creep deformations and the damage accumulations on the third stage [14]. Thus, we have the effective scheme (11)- (14) to numerical solve the equations (1), (5) necessary for the computer simulations of the cladding.…”

Application of the method of weighted residuals for determining the distribution of temperatures and radiation creep strains in plates

“…Застосу-вання цього методу в завданнях механіки твердого деформівного тіла та пов'язаних з нею напрямках [1,3,4] доводить його достатню точність та ефектив-ність.…”

Application of fractional exponential hereditary kernels in the nonlinear theory of viscoelasticity

“…Some relevant results are reported in [4,7,11] and reflect a method of determining the parameters of fractional exponential hereditary kernels and solving problems of viscoelasticity for polymers and composites under static and cyclic tension and compression.…”