Two Approaches to the Problem of High-cycle Fatigue of Materials and Structures

Creation of new types of materials and development of new methods aimed at extending the lifetime of structures are two interconnected approaches to solve the problem of metallic structures fatigue. The paper considers a method for estimating the efficiency of damping implemented in metallic structures by means of energy-intensive materials. The results of comparative calculation are given on the ultimate number of transverse vibration cycles in bridge girders. The calculation uses a linear hypothesis of fatigue-caused damage summation and a model of cyclic material degradation. Notable increase of the vibration decrement for foam-filled metallic structures is predicted, with the lifetime of products lengthening as a result.


Introduction
The problem of extending the lifetime of metallic structures of lifting and transporting machines is most often solved via technological techniques aimed at affecting the surface layer of metals or by means of diminishing variable dynamic loads. The latter method often uses structural damping [1][2][3][4]. The paper [5] discussed one of the approaches to increasing the durability of a thin-walled closedprofile beam by filling it with a low-modulus energy-intensive material. A method of evaluating the lifetime of structural elements which takes cyclic degradation of material properties into account was analyzed in [6,7]. This study uses that method for the purposes of comparatively estimating the lifetimes of unfilled and foam-filled bridge girders of an overhead crane.
The efficiency of structural damping applied to a metallic structure is assessed by running a costly experiment on the basis of a vibrorecord for free damped vibrations. Below a possibility is considered of theoretically predicting the lifetime of a filled bridge girder from a known logarithmic vibration decrement for unfilled beams. For example, the decrement 1  for the span of a bridge crane in [8]   for filled beams could be found using the basic theory of vibrations for a material particle with its mass reduced to the centre of the span.
2 Theoretical estimation of logarithmic decrement 2  a Corresponding author : oldim96@mail.ru Differential equation for damped vibrations of a material particle could be expressed as [9] where coefficient n and circular frequency p are linked to a circular frequency 0 p , determined for free sustained (undamped) vibrations by the correlation The decrement  , conditional period of vibrations T and circular frequency are interconnected via the following expression: (2) Denoting the vibration parameters for an unfilled girder by 1 in their lower indices and for a filled girder -by 2, and omitting insignificant effect of a light-weighted filler on the circular frequency 0 p , the next two expressions could be formulated: The vibration frequencies 1  and 2  for unfilled and filled beams could easily be determined using a typical computer software, for example, such as SolidWorks [10,11]. When the decrement 1  is pre-determined, Eq. 3 is completely defined. The solution of Eq. 1 for a known coefficient n provides the law of variation in point-mass displacements, and, for a linear system, even the law of variation in damped-vibration stresses where 0  is a dynamic stress related, for example, to the acceleration applied to or removed from the system when starting to lift the load or applying the brakes. Constraining the level of significant amplitudes to the fatigue limit R  , the period of their action * t could be found on the condition of Then, the number of damped vibrations caused by a single dynamic load which produces the effect of 0  stress in a critical point of a metallic structure would be expressed as In engineering calculations, 0  means the normal stress in a critical section of a beam [8,12] or the von Mises stress [13]. The latter is determined in terms of high-cycle fatigue by solving a boundary value problem from the theory of elasticity. The SolidWorks software package provides desired numerical solution using the finite-element method.

Model of material accounting for cyclic degradation
For a specified law of variation in alternating loads, the durability could easily be predicted on the basis of a linear hypothesis of fatigue-caused damage summation [14]   1 N n (6) But, neither the measure of damage n/N, nor the criterion of failure (equaling 1) do not have any physical explanation. Correcting the right part from experimental data does not guarantee to have the accuracy of prediction maintained when the conditions of testing change. The authors base their approach to the problem of metal fatigue [7] on the experiments aimed at degrading such parameters of stress-strain diagrams as ultimate strength and ultimate strain. With a force approach chosen to describe the fatigue process and an exponential function picked to approximate an experiment-based kinetic curve, this expression follows: where M N is the durability by the Weller diagram, m is an experimental material constant.
Generalization of the approach onto nonstationary load is defined on the condition of equivalence of two cyclic material states with a different history of loading: This equation helps to find an equivalent (in terms of damage) number of cycles э n in that kinetic curve to which the transition is made. The criterion of fatigue failure in Eq. 8 is generalized onto the nonstationary load in the form of a rule for intersection of an actual material strength and the level of maximal stress in the cycle.
How the lack of uniaxiality in the stressed state affects the lifetime has not been studied in a sufficient detail. It is known, for example, that the second main stress did not impact the lifetimes of pipe specimens if it was less than 0.8 of the main stress itself [15]. In that case, the kinetic curve from Eq. 7 could be used immediately in phenomenological calculation. Eqs where B t is thickness of a basic specimen.
The fatigue curve could be used both in its typical form and in any other form derived from the results of mechanical testing on the material fatigue. On the other hand, the non-linear function of Eq. 7 presumes that the damaging action of the cycle of stresses depends on where they are located in the whole spectrum of loading. Therefore, regrouping of cycles -which is a common practice in correlation and spectrum analyses of loading -is excluded. To illustrate the point, self-induced transverse vibrations of bridge girders of an overhead crane should be considered. Those emerge at the moments of starting and stopping the travelling mechanism and come in two variants -for unfilled and foam-filled girders. The symmetry in the cycle of loading (with a constant asymmetry coefficient R=-1) is provided by varying stresses in accordance with the law of Eq. 4.

Vibrations of bridge girders of overhead cranes
The dimensions of a bridge girder for an overhead crane of 19.5 m in span and 20 t in lifting capacity were found in typical strength and rigidity calculations. That beam is 1.1 m high and 0.6 m wide, and the required thickness of its wall is 9 mm. A finite-element model of an unfilled beam was built in SolidWorks (Fig. 1), followed by an assembly model with a filler.
where 1  n for the cycle-by-cycle variation in stress amplitudes (as in the example).
Based on this equation, an algorithm was developed for comparative calculation of the number of cycles prior to the failure of a metallic structure due to transverse vibrations. The algorithm uses the linear hypothesis of Eq. 6 and the model of cyclic material degradation described. For a number of steels, a numerical value of the constant m was close to 2, and, therefore, the calculation assumed m = 2. The numbers of cycles prior to the failure due to transverse vibrations for the linear damage summation NLH and in terms of the proposed model of material ND (Table 1) are expressed in blocks of vibrations caused by a single dynamic load. The table shows the interval boundaries of variation in the logarithmic decrements 1  and 2  (calculated using Eq. 3). The second column provides lifetimes in terms of cycle blocks calculated using the linear hypothesis for the specified boundary values of the decrements in the first column.
Those numbers indicate that the structural damping affects notably the lifetimes. The third column gives lifetimes for the same decrements calculated using the model which allows for the cyclic degradation of material. A substantial discrepancy is observed between the predictions of the ultimate