Instytut Podstawowych Problemów Techniki

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Optimal shapes in the class of polynomial functions for rotating annular disks with respect to the mixed creep rupture time are found. Two effects leading to damage: diminishing of transversal dimensions and growth of micro-cracks are simultaneously taken into account. The first of them requires the finite strain analysis, the latter is described by Kachanov's evolution equation. Behaviour of the material is described by nonlinear Norton's law, generalized for true stresses and logarithmic strains, and the shape change law in form of similarity of true stresses and logarithmic strains deviators. For optimal shapes of the disk, changes of geometry and a continuity function are presented. The theoretical considerations based on the perception of the structural components as some highlighted objects with defined properties is presented

Słowa kluczowe:

annular disk, mixed creep rupture, optimal design

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Streszczenie:

A simulation model is presented for the creep process of the rotating disks under the radial pressure in the presence of body forces. The finite strain theory is applied. The material is described by the Norton-Bailey law generalized for true stresses and logarithmic strains. A mathematical model is formulated in the form of a set of four partial differential equations with respect to the radial coordinate and time. Necessary initial and boundary conditions are also given. To make the model complete, a numerical procedure is proposed. The given example shows the effectiveness of this procedure. The results show that the classical finite element method cannot be used here because both the geometry and the loading (body forces) change with the time in the creep process, and the finite elements need to be redefined at each time step.

Słowa kluczowe:

creep process, rotating disk, finite strain theory, simulation model

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Streszczenie:

The paper presents a simulation model for the creep process of rotating disks under radial tensional pressure subjected to of body force. The finite strain theory is applied. The material is described by the Norton - Bailey law generalized for true stresses and logarithmic strains. The mathematical model is formulated in form of set of four partial differential equations with respect to radial coordinate and time. Necessary initial and boundary conditions are also given. To make the model complete, the numerical procedure for solving this set is proposed. What is worth noticing the classical F EM is not applicable, because not only geometry, but also loading (body forces) change in time during the creep process. It would demand redefinition of finite elements at each time step. In uniaxial problem similar model was presented in [ 4 ], but now it is developed for complex stress state. Possible different formulations of initial and boundary conditions may be found in [ 5 ]. The procedure may be useful in problems of optimal design of full disks in [ 6 ]

Słowa kluczowe:

creep process, rotating disks, finite strain theory, simulation model

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Streszczenie:

The mixed creep rupture theory to the optimization problem for the complex stress state is used. The problem of optimal shape for the rotating full disk with respect to mixed rupture time is investigated. The mathematical model of mixed creep rupture is described by the system of five partial differential equations. Difficulty of the problem results from two types of nonlinearities: geometrical connected with the use of the finite strain theory and physical – the material is described by the Norton’s creep law, here generalized for true stresses and logarithmic strains. Additional time factor leads to subsequent complications. The parametric optimization describing the initial shape of the disk is applied. The obtained results are compared with the optimal disks with respect to ductile creep rupture time

Słowa kluczowe:

Mixed creep rupture, Structural optimization, Full disk

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Streszczenie:

This article deals with the influence of boundary conditions on the optimal shape of a rotating, axisymmetric annular disk of given volume that maximizes the ductile creep rupture time. The finite strain theory and physical law in form of Norton's law generalized for true stresses and logarithmic strains are applied. The optimal shape is found using parametric optimization. The initial shape of the disk is defined by class of polynomial function.

Słowa kluczowe:

Boundary conditions, Optimal design, Ductile creep rupture time, Annular disk

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The first attempt of finding of optimal shape for bars in presence of body forces with respect to mixed creep rupture is made. For given volume of the bar, distribution of initial cross-section, ensuring the longest life-time to mixed rupture is sought. The finite strain theory and physical law in form of Norton's law generalized for true stresses and logarithmic strains are applied. Using the method of parametric optimization, the best of linear and quadratic functions describing the initial shape of the bar are found. The shape of initial strength is corrected in a way leading to longer life-time. Results of both approaches are compared

Słowa kluczowe:

Mixed rupture, Structural optimization, Nonhomogeneity

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Streszczenie:

This paper presents the problem of optimal design with respect to ductile creep rupture time for rotating disk. The material is described by the Norton–Bailey nonlinear creep law, here generalized for true stresses and logarithmic strains. For complex stress states, the law of similarity of deviators, combined with the Huber–Mises–Hencky hypothesis is applied. The set of four partial differential equations describes the creep conditions of annular disk. The optimal shape of the disk is found using parametric optimisation with one free parameter. The results are compared with disks of uniform thickness.

Słowa kluczowe:

optimal design, annular disk, ductile creep rupture time

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Streszczenie:

Structural elements working under creep conditions belong to the relatively new branches of structural optimization. Among many new possibilities of optimization criteria, which offers optimization of structures in creep conditions, the most important seems to be time to rupture. Most papers on optimal structural design are based on the brittle creep rupture theory proposed by Kachanov (small strain theory). Application of the ductile rupture theory proposed by Hoff in optimization problems are rather scare as it requires finite strain theory. For the first time it was used by Szuwalski [ 1], [2] and Szuwalski, Ustrzycka [4]. The first attempt of application the mixed theory to shape optimization was made by Ustrzycka and Szuwalski [3] for bars under nonuniform tension. Here, for the first time, the optimization problem with respect mixed rupture time, is solved for the complex stress state. Application of mixed rupture theory proposed by Kachanov takes into account: geometrical changes - diminishing of transversal dimensions resulting from large strains (as in ductile rupture) and growth of microcracks (as in brittle rupture). In present paper the problem of optimal shape with respect to mixed creep rupture time for the rotating full disk is investigated. Difficulty of the problem results from two types of nonlinearities: geometrical connected with the use of the finite strain theory and physical - the material is described by the Norton’s creep law, here generalized for true stresses and logarithmic strains. The mathematical model of mixed creep rupture is described by the system of five partial differential equations in dimensionless form. The system must be integrated with respect to material coordinate (Runge – Kutta fourth order method) and to time with variable step (Euler’s method). The parametric optimization with one, or two free parameters describing the initial shape of the disk is applied. The obtained results are compared with the disks with respect to ductile creep rupture time [2]

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Streszczenie:

The problem of optimal design with respect to mixed creep rupture time is a new one. The first attempt of solution for rotation bar was made in 2010 by Szuwalski and Ustrzycka [5]. Difficulties of the problem result from physical an d geometrical nonlinearities, and were observed earlier in solution for ductile creep rupture [4], [6]. Some problems of optimal design for annular rotating disks were discussed by Farshi and Bidabadi [2]. Analytical solution for the elastic-plastic stress distribution in rotating annular disks were obtained by Ç ğ allio ğ lu et al. [1] and Gun [3]. Because of those difficulties the parametric optimization was adopted. In certain class of function describing the initial shape of the disk we are looking for optimal parameters leading to the longest time to mixed rupture under assumption of constant volume. The axially symmetric annular disk rotating with constant angular velocity is loaded by centrifugal forces resulting from the own mass of the disk and additional mass uniformly distributed at the outer edge (Figure 1). The mathematical model of mixed creep rupture is described by the system of five partial differential equations in dimensionless form. n the last equation the continuity function Ψ describing damage of material was introduced. The criterion of rupture is fulfilled when Ψ takes values 0. The optimal shape among linear functions (uniparametric optimization) and quadratic functions (biparametric optimization) was found by checking some disks in earlier predicted domain of admissible solutions. The better results were obtained using biparametric optimization. The obtained results are compared with the disks with respect to ductile creep rupture time [6].

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