Keywords: optimization, low cycle fatigue, notch correction, plasticity models.

Geometric discontinuities in machine component such as holes, fillets, grooves, etc., shortly notches, are the source of stress or strain concentration. The machine components generally work under cyclic stresses. The fatigue phenomenon is created by these stresses and is observed particularly in the presence of notches. The fatigue damages are unavoidable and the unwanted damages have a destructive effect on structural integrity and service conditions. The fatigue life of a notched component is generally shorter than that of an unnotched element. In practice, the total fatigue life, with the number of cycles , of machine or structural elements is given as a sum of two portions. The first one corresponds to the initiation stage (fatigue crack initiation), the second part corresponds to the subsequent fatigue crack propagation . For non sharply notched parts, 50%, (some sources report, that 85-90%) of the whole lifetime is connected with the initiation phase. Hence, the problem of predicting the critical number of loading cycles corresponding to crack initiation in a machine element is of fundamental importance for rational design with specified service life. It is expedient to distinguish between high-cycle (classic) and low-cycle fatigue. When the elastic local stress and strain exceed the elastic limit, an elasto-plastic stress evolution occurs. The crack initiation is then dependent on the plastic dissipated energy and the stress at the notch root. A closely related problem is that of rational design of notch shape in order to maximize the critical number of cycles corresponding to crack initiation.

Generally the value of the fatigue life depends on several factors: (1) geometry of the structure , (2) material properties , (3) history of the response of the structure to external loading and (4) boundary conditions , i.e. . Today, it is very well known, that by proper modification of the shape of notched parts (shape optimization) we can significantly reduce the peak stress and increase significantly the lifetime of machine parts. In this paper the aim of the optimal design is to provide proper shape of notches or component boundaries to increase the number of cycles corresponding to crack initiation i.e. for a given boundary conditions , external loading and material properties find a such shape of notched part for which for which , (initiation stage) with constraints , where is a boundary shape to be modified, and is a given variation domain of . The stress-strain behaviour in the notch tip of elastic-plastic bodies is approximated by the generalized Neuber's or Glinka's rule [1,2]. The incremental Neuber's or Glinka's rule is associated with multisurface plasticity model of Mróz [1,2,3]. In the Mróz model the uniaxial stress-strain material curve can be represented in the 3D space by a set of work-hardening surfaces (in 2D ellipses). The process terminates with a critical plane damage criterion to assess fatigue lives. The present analysis is composed of three steps: i) specification of elastoplastic steady cyclic states at the notch root using the multisurface hardening model and also a simplified method for which the elastic solution is transformed into the corresponding elastoplastic state by applying the proper mapping rule onto the hardening surfaces (nested surfaces); ii) specification of the crack initiation condition using the accumulated plastic dissipation or critical plane concept with respective stress and strain components acting on the physical plane; iii) formulation of the optimization problem with constraint set on the number of cycles corresponding to crack initiation. These three steps are mathematically formulated and the numerical iterative procedure is proposed. Because the objective function is not differentiable, using "bound formulation" the max min problem is transformed to the simple max problem with extra constraints on at some critical points on the boundary (the BEM or the FEM nodes around notch tip). The modified shape of the notch contour is defined by Bezier's curves. The sequential linear programming (SLP) method is adopted as the optimization procedure. The objective function is linear and only constraints should be linearized. The stress field (the Neuber's or Glinka's rule links the fictitious linear elastic stress-strain response in the notch tip with the actual elasto-plastic stress strain) is evaluated using the BEM method.

The above notch correction and plasticity models have been applied to shape optimization of notches in uniaxially or multiaxially loaded machine parts. There is observed a significant increase in the number of cycles corresponding to damage initiation in machine elements..

1

G. Glinka, A. Buczynski, A. Ruggeri, "Elastic-plastic stress-strain analysis of notches under non-proportional loading paths", Archive of Mechanics, 52, 589-607, 2000.

2

T.E. Langlais, J.H. Vogel, D.F. Socie, T.S. Cordes, "A multiaxial fatigue life prediction program", Fatigue Design of Components, ESIS Publication 22, Elsevier, Amsterdam, 85-95, 1997.

3

Z. Mróz, "Multisurface hardening model for monotonic and cyclic response of metals", Handbook of Materials Behaviour Models, Academic Press, 223-231, 2001.

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