Contribution to fatigue damage prediction of thin shell finite element models

The paper deals with the implementation of the chosen fatigue damage criterions into FE analysis. Our study considers a shell finite element structural analysis in conjunction with the multiaxial rainflow counting and the fatigue damage prediction. The computational analysis considers stochastic response under random excitation in time domain and damage calculation based on so called critical plane approach (CPA).


Introduction
Fatigue damage of structural features is a complex physical process which is governed by a great number of parameters related to, for example, local geometry and material properties of the structural region surrounding the crack growth path.There are different approaches and methods which can be used in fatigue life predictions [1].
It is commonly recognized that it is impossible for a physical model to account for all fatigue influencing parameters, thus a lot of approximate models have been conceived for practical fatigue assessments.In every stadium of fatigue damage cumulation dominates a definite mechanism controlled by more or less known and verified rules.There exists the stage of microplastic process in total volume of material with following stage of fatigue crack nucleation and stage of their growing with more or less detailed zoning.Despite of this research no results have been achieved, which could be considered as successful ones.This applies mainly to the cases of random and combined stress, where today's procedures used in one axis stress analysis fails [2].
There are plenty of hypotheses used for evaluating a degree of damage caused by variable load [3][4][5].Life prediction methods which presume homogeneous material (free from cracks, inclusions or defects) at the outset of the investigation can be divided into strain-based (low-cycle fatigue) and stress-based (high-cycle fatigue) methods.This study assumes stress-based approaches use the elastic stress range (or amplitude) as the governing load parameter.At a sufficient load level, which may result in a fatigue life of approximately 10 7 cycles, a threshold referred to as the fatigue or endurance limit can be seen for many materials.
Research of mechanism and processes of fatigue failure of materials achieves great advance, however there still doesn't exist general failure model, which should be applicable for different conditions of activity.There is needed an integration for such a procedures into a modern systems of computing aided design (CAD) [6][7][8] in relationship to methods of strength computing transferred by finite element method (FEM) [9][10][11].

Thin shell stress calculation
We will consider well-known shell finite elements (Kirchhoff's or Mindlin's formulation) [1,12].The stiffness parameters depend on material constants and element geometry, mainly on its thickness.At first we have to prepare the stress calculation process.This process is based on the expression of the j-th element membrane forces and bending moments (without shear forces) [13,14], i.e.

 
( The auxiliary matrices I m and I b can be calculated only using the numerical approach.Further details about E m, E b , D, B m , B b , u el and t are presented in [10,15] Let's build new material and auxiliary matrices where the matrix I 3 is the classical unit matrix.Then (3) can be rewritten as follows 3 Chosen approaches for multiaxial fatigue damage calculation Let's now focus on the cumulative damage counting by using multiaxial rainflow decomposition of the stress response.It should be noted that the fatigue damage calculation of the machine parts is generally problematic because the results are considerable changed in the principal stresses [2,4,5].Using FE analysis we can get six components of the stress-time function (multiaxial stress) but it is very difficult to obtain an equivalentuniaxial load spectrum by reason of comparison with applied computational fatigue curve.
In our case the rainflow analysis for random stresses known in classic uniaxial form as von Mises or Tresca hypotheses is impossible.It means that the important goal of this part will be to propose some approaches to estimate the high-cycle fatigue damage for multiaxial stresses caused by random vibration analysed structure [10,16,17].Generally we can apply two fundamental approaches for multiaxial rainflow counting:  Critical Plane Approach (CPA) [13], and  Integral Approach (IA) [10].It is well-known that the Wöhler curve (Fig. 1, sometimes called S-2N curve) is basic source of getting information of the material fatigue life.Generally the S-2N curve is statistically evaluating by experimental fatigue curve.This is a graph of the magnitude of a cyclical nominal stress  A against the logarithmic scale of cycles to failure 2N f .It is advantage to show it in logarithmical or semi logarithmical coordinates.

Fig. 1. S-2N curve
The  A -2N f relation can be written as follows where  f is the fatigue stress coefficient, 2N f is number of cycles to failure, b is fatigue strength exponent and  A is stress amplitude to failure.Some researches the relationship (7) rewrite into following form where m = -(1/b) and Considering mean stress modified version of the stress amplitude (using Goodman, Soderberg, Geber), Eq. ( 8) can be rewritten as follows If k = 1 and R F = R E (yield stress) the Soderberg's model is used, if k = 1 and R F = R M (strength limit) the Goodman's model is used and if k = 2 and R F = R M the Geber's model is used.Using the linear Palmgren-Miner law we can calculate fatigue damage for stress amplitude  Ai as follows

Findley hypothesis
Findley has assumed the critical plane as a plane with maximum shear stress, i.e. the fatigue equivalent shear stress can be written as follows [10,18] where k is Findley's factor which value for tough metal can be about 0.3.Using von Mises relationship between normal and shear stresses and classic plane stress analysis for top or bottom element surface it is possible rewrite (9) into following form The damage calculation can be realised using Eq. ( 10).Presented relationships have been applicable for FE analyses.Numerical and experimental tests have confirmed that the factor k = 0.3 was overstated [4,13] by author.

Dang Van hypothesis
Dang Van again has assumed the critical plane with shear stress but with difference in factor k, which can be calculated from normal and shear fatigue limit, i.e.
where  C is shear (torsional) fatigue limit,  C is normal (axial) fatigue limit,  1 ,  2 ,  3 are principal stresses.Relationship ( 11) is possible to use like that Using von Mises hypothesis we can get [4] 232 0 Using by application of the shell stress theory and Eqs.6a,b we can obtain Relations ( 12) and ( 17) present equivalent stresses applicable for rainflow decomposition for both proportional and non-proportional loading.The cumulative damage calculation can be realised using Eq. ( 10) again.

HMH modified hypothesis
Applying von Mises equivalent stress for CPA we can obtain following relationship or in detail where In this case it should be noted that computational approach depends on a searching process of a critical plane normal vector n CPA .By reason rainflow analysis it is very important to know the sign of the calculated equivalent stress therefore the sign of this stress will be defined by sign of normal component.For searching process was used optimizing tools in Matlab [7] and optimizing problem for cumulative damage function is usually formulated as follows for unknown vector n CPA and stresses on bottom surface.The same computational process we can have to realise on top element surface.

Damage calculation using IA
The fundamental idea is to count rainflow cycles on all linear combinations of the stress random vector components [10], i.e.Hence the next goal will be to find extreme value of the estimated damage for vector c and i-th element, i.e.

Cumulative Damage Calculations
In Table 1 is shown estimate of cumulative fatigue damage for ribbed beam.Graphical representations of cumulative fatigue damage of analyzed shell structure for three chosen hypothesis are shown in Fig. 6, Fig. 7 and Fig. 8.

Conclusion
The goal of this study has been to realized an overview and propose of possibilities of computational fatigue damage (life) prediction of modeled shell structure with help of finite element method.There was used a software facility interface between MATLAB and ADINA software.
that the parameters c i belong to a hypersphere  if the stress state is biaxial (e.g.thin shell finite element) the stress components can be written under the form of three dimension vector  = [ x ,  y ,  xy ] T and the equivalent stress will be calculated as follows