Biaxial/Multiaxial Fatigue and Fracture: 31 (European Structural Integrity Society)

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The Chairman and the Scientific Council of the Polish Group of Fracture were elected during the conference, who will serve a two-year term ending in Andrzej Neimitz from Kielce University of Technology was again elected chairman. Members of the Scientific Council became: Ewald Macha from the Opole University of Technology 4. Dariusz Rozumek from the Opole University of Technology 7. On behalf of the Organizers we would like to invite you to take part in it. Ewald Macha Opole University of Technology. The main objectives of the Project are: Development of new procedures for accurate determination of fatigue characteristics of materials with controlled strain energy density parameter at the strength test stand.

Development of a spectral method for fatigue life assessment of materials under multiaxial random loading in frequency domain. Improvement of procedures of fatigue damage map determination for machine components and structures subjected to multiaxial service loading by calculations with FEM or BEM and spectral methods. Among the most important results, publication of habilitation dissertations: Organization of two International Conferences: Energy-based approach to multiaxial fatigue using the critical plane.

As in the previous period , the main bjective of the project was development of the existing and formulation of new algorithms for estimation of fatigue life of machine components and structures subjected to multiaxial service loading, based on mechanisms of crack initiation and propagation, identified during tests.

The energy-based parameter for description of fatigue properties of materials under random loading, used for generalization of multiaxial fatigue failure criteria, was analyzed and verified. The test results obtained under cyclic, constant and variable amplitudes, random, proportional and non-proportional loading were used for validation of the generalized criterion of maximum shear and normal strain energy density in the critical plane.

Much progress was made in fatigue life evaluation of some chosen welded joints in simple and complex states of stress and in development of the spectral method for fatigue life assessment in materials under multiaxial random loading in frequency domain. Spectral Method in Multiaxial Random Fatigue. Energy based approach to multiaxial fatigue using the critical plane. Stage 4 The main objectives of the project are the same as in the stages 1, 2 and 3, but we are going to concentrate on the following tasks: Development of new procedures for precise determination of the fatigue characteristics of materials with a controlled strain energy density parameter at the strength test stand.

Up to now, the energy fatigue characteristics of materials, expressed by the amplitude of strain energy density Wa-Nf has been indirectly determined, i. Thus, a more precise and direct determination of the energy characteristics becomes an important task for an energy-based approach to multiaxial fatigue. No stress relaxation was observed as shown in Fig. As welded and stress relieved specimens were tested and the results are shown in Fig. The value of To has been calculated without taking into account the residual stresses.

On the other hand, calculations have been performed on the same as welded elementary structure D, considering the initial value of the residual hydrostatic pressure of 66 MPa at the hot spots, as measured experimentally. Figure 9 b shows the new calculation and test results: Therefore, taking into account the residual stresses gives a much more accurate fatigue life prediction.

Application of the proposed methods to Sonsino's results The previous computational method was applied to analyse experimental results obtained by CM. The studied specimen is a tube of 1cm thick welded on a plate of 2. Its geometry is presented in Fig. Several loading conditions were applied to this specimen: The numerical model is built following the recommended meshing rules shown in Fig.

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For in phase loadings pure bending, pure torsion, in phase bending-torsion , as the principal stress directions do not vary, a direct calculation is sufficient. One calculates successively the values of the principal geometrical stress, and then derives To. For out of phase loadings, as the principal stress directions and amplitudes vary at any time of the cycle, it is necessary to apply the general Dang Van procedure.

One has to maximise the parameter d of Eq. The maximum occurs at a definite instant t of the loading cycle. Residual stresses distribution in the vicinity of the hot spots in the D elementary structure along Li and L2 lines X-ray measures. Tube on plate Dang Van's loading path for out of phase combined torsion and bending: Fayard and previously presented in Fig. The upper part of this curve is nevertheless limited by the yield stress of the material.

The curve fits very well the points resulting from Sonsino's study. These points are situated slightly above this curve. This is due to the choice of the failure criterion: Results of Sonsino's through crack fatigue tests compared with the design curve TQ-N Another way to illustrate the results is using the Dang Van's diagram Fig. The remarkable result is that the proposed method is predictive and very robust since various thicknesses, various geometries of welded structures as well as various loading paths are experienced with success.

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Mean values of Sonsino's through crack fatigue test results at 10 cycles presented in the Dang Van's diagram and compared to the fatigue criterion at 1 million cycles. Researchers of Institut de Soudure of France I. It is not possible for instance to correlate quantitatively the local geometry and the residual stress distribution to the fatigue behaviour. The use of a local approach allows better physical interpretation [15]. In order to be able to take into account the effect of residual stresses and the weld geometry, a systematic research program was undertaken by the I.

Fillet welds obtained by one pass or by three passes were simulated for a S steel with a bainitic transformation at cooling. Two sides one-pass fillet welds were first considered and tested. Experimental X-ray measurements were carried out to verify the validity near the surface only of the prediction. One-side fillet welds carried out in three passes were also examined.

The welded pieces were 20mm thick and were free of any clamping. Figure 14 shows all the experimental and calculation fatigue results in the Dang Van's diagram. Each point corresponds to the fatigue limit loading defined as 2. PSA's fatigue criterion at 1 million cycles is also plotted. One can observe that the experiments of the I. Thus, the optimisation of the welding process can be investigated efficiently.

These methods are in reality only applicable when the geometry or the loading is simple. Mechanical structures usually have complex shapes and they have to undergo complex multiaxial loading. This is particularly true for automotive welded structures where fatigue assessment of such specimens is easier using multiaxial approaches. The reason which motivated the present research is to propose another type of fatigue computational method.

The proposed structural approach presented in this paper is based on the use of a mutiaxial parameter derived from Dang van's fatigue criterion. The use of this design parameter allows a very good fatigue life prediction of complex automotive thin sheet welded structures. Moreover, the methodology has been successfully applied to interpret the experimental results of Sonsino on stress relieved thick welded structures submitted to different in phase and out of phase multiaxial loadings. Furthermore, experiments performed by the Institut de Soudure of France in order to quantify the quality of the weld and the influence of the residual stresses which have a great importance on the fatigue resistance, have been predicted with quite a good accuracy.

The application of the method for predicting the fatigue resistance of a great number of welded specimens and structures, which present large differences in geometry and loading, is very encouraging for the development of a general methodology for the fatigue assessment of welded structures. Fatigue 18, 3, 2. Proceedings of the international conference, Steel in Marine Structures, , Paris.

Courses and Lectures N , Ed. Partie I, approche de I'endommagement. Revue frangaise de Mecanique, 4, 3. Fatigue, 17, 1, Fatigue Design 95, Ed. Courses and Lectures, , Ed. The component is loaded by nonproportional random sequences of bending and torsion as measured during operation. The stresses in the welded structure are calculated using finite element analysis. The structure has been meshed following the n w guideline for application of the hot spot stress approach. The fatigue lifetime of the welded structure is evaluated using the hot spot stresses in conjunction with the critical plane approach to account for multiaxial fatigue.

Additionally, a model has been created to calculate fatigue lifetime based on local elastic stresses. The accuracy of the calculations is discussed using corresponding experimental fatigue life results. Summarising overview and detailed description of the several procedures are given in [1, 2, 3, 4]. In the automotive sector there is a need for computer-aided methods to shorten development time of products. In the last years, significant developments on theoretical assessment of welded automotive components have been achieved in connection with the hot 24 G. Further experience gathered on thin-walled plane structures of commercial vehicle components under proportional constant amplitude normal stresses [7] revealed that fatigue analysis based on hot spot stresses is capable of handling such components.

The local stress approach to fatigue of welded joints represents an alternative to the hot spot stress approach. Its main advantage can be seen in the existence of an experimentally verified universal constant amplitude local stress-life curve for steel welds. However, the determination of local stresses causes a clearly increasing numerical effort. This paper deals with the critical plane approach in connection with numerically determined hot spot stresses; particularly, its application to nonproportional variable amplitude loading is shown and discussed.

In the latter part of the paper the application of the local stress approach is shown. A comparison of fatigue lives gained from hot spot stresses, local stresses and experimental investigation, respectively, will close the paper. This component serves to stabilise the driver's cab of trucks. The mechanical behaviour and the fatigue life of the component shown in Fig.

Component under investigation Figure 2 shows the test rig with two actuators at the top left-hand side of the component capable for introducing lateral forces and forces resulting from suspension. The test track driving program includes quasi-static manoeuvres like cornering, braking during cornering and braking at straight-driving as well as straight-driving over rough road segments like potholes, washboards, belgian block, country roads and rough highways. The running time for the experimental simulation of one block on the test rig amounts to approximately 28 minutes. Load configuration on the test rig Fig.

Figure 4 [8] shows various ways to model cover plate endings. Solid element e solid element modelling Fig. Various modelling approaches for cover plate endings [8] To take advantage of the experience from a previous investigation [7] on the applicability of the hot spot stress approach, simple 4-node shell elements have been used to model the tube, 8-node volume elements have been used for the forged arms.

Both meshes are tied together with constraint equations. The weld itself is modelled by increasing the shell element's thickness. In accordance with Niemi's suggestions, the element length has been set to 2. Figure 5 shows the finite element mesh used together with the load configuration. At six different locations along the length of the tube measured by means of strain gauges and calculated strains have been compared, showing an overall good agreement.

Additionally, numerical studies with various element lengths reported in Ref. This undercut has a distance of 12 mm to the surface of the forged component and corresponds to the fifth element ring starting from the end of the tube. For the nonproportional loading situation investigated here, fatigue life calculations are carried out for all shell elements of the fifth ring, separately for the outside and inside surface of the shell.

In general, calculations of this kind can be made in accordance with conventional approaches based on integral criteria such as energies, equivalent values of stresses or strains or approaches which assess the stress and strain state acting in certain directions critical plane approach. In the case of the critical plane approach, it is assumed that a microcrack that forms at the surface propagates on a preferred plane on which the normal stress-strain or shear strain response or a combination of both reaches its maximum. Using wide-ranging experimental information, Sonsino [11, 12] reports that conventional hypotheses often fail in the case of nonproportional loading.

For this reason the critical plane approach is used in this examination. Figure 6 shows the weld detail with the distribution of stress Jy at bending, where the y-direction corresponds to the tube axis. The subscript p denotes the angle between the zy- and actual plane under consideration. Stress distribution Gy at bending. To determine the failure-critical element, damage calculations have been performed for each element. The calculated maximum of damage identifies the failure-critical element and the fatigue life of the component.

Since various multiaxial fatigue criteria are currently being proposed within the context of the critical plane approach, in practice the user has to rely on gathered experience with the criteria available. In this investigation, two different criteria offered for multiaxial random loading in the context of the applied software [13] are employed.

In the first case the normal stress acting perpendicularly on the critical plane pure mode I crack configuration is regarded as the fatigue failure criterion. In the second case it is the shear stress mode n and HI crack configuration. Criteria for which combinations of both stresses are proposed to be used for nonproportional loading cases, e.

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Apart from this, failure criteria of this kind require further material characteristics which are also not available here, and their determination would increase the experimental effort significantly. This life curve is obtained by regression analysis of experimentally determined fatigue life results from various proportionally stressed welded thin plates obtained in a previous investigation [7].

Of course, hot spot stresses had been determined using the same element type and mesh refinement as in the present study in order to allow for transferring allowable hot spot stresses. This life curve provides a slope of k However, further insights into the mechanics and the theoretical-physical background which account for the different slopes in the various cases of combined normal and shear stresses, are not given in [16].

Hot spot stress-life curve If the shear stress is used as the failure criterion, a difficulty appears that no hot spot shear stress life curve determined experimentally is available at present.

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In order to be able to calculate a fatigue life, a nominal shear stress-life curve determined by Sonsino [17] by testing torsionally loaded tubes welded to plates has been taken as base here to calculate the fatigue life. Sonsino's curve has been re-calculated into a corresponding hot spot shear stress-life curve by means of finite element analysis of the welded tube-plate joints, whereby the same rules of modelling have been used as the ones for the welded detail investigated here.

The re-calculated shear stress-life curve is plotted as a dashed line in Fig. Hot spot stress approach results Figure 8 shows results of the calculated damage of the shell elements representing the weld ring evaluated using the normal stress failure criterion. Numerical analysis identifies the critical weld element, particularly the tube's inside surface, where the weld root is situated, to be the failure-critical location.

Outside surface Inside surface Failure-critical element Fig. Detail of a failed component The calculated and experimentally determined fatigue lives are established at two different load levels. Load levels are noted within this paper as normalised load factors. Calculated and experimentally determined lives are plotted in Fig. Good agreement is achieved using the normal stress failure criterion mode I in conjunction with the life curve of Ref. With decreasing load level, deviations between experimental and calculated results can be observed.

They amount to life-factors of 2. It should be taken into account that most of the damaging cycles of the load sequences corresponding to load levels 1. Other assumptions for the calculational treatment of the small cycles, e. However, further theoretical investigations and more experimental results at higher and lower load levels are required to lighten the question of the appropriate slope, but they have not been performed here due to the enormous effort.

The fatigue lives calculated using the shear stress failure criterion modes n and IE are significantly higher than the experimental ones. Experimental and calculated fatigue lives with the hot spot stress approach Load level 1. Comparison between calculated and experimentally determined fatigue lives Discussion of the calculated fatigue lives using the hot spot stress approach This investigation yields initial experience as to the applicability and accuracy of the critical plane approach supported by numerically determined hot spot stresses for nonproportionally loaded bending and torsion welded components of commercial vehicles.

The finite element results concerning the failure-critical elements along the weld ring and the crack initiation starting from the weld root correspond well to the experimental observations. Comparison of experimental and calculation fatigue lives shows that the critical plane approach taking the hot spot normal stress mode I as failure criterion indicates a slight trend to conservative calculations at lower load levels.

At load level 1. With regard to future applications it must be pointed out that the experience with the existing failure approaches and criteria has still to be extended significantly. The results presented are encouraging because it seems possible to describe the mechanical behaviour and fatigue life if results from similar components with approximately the same type of stress state, weld geometry, and welding procedure are available and if it is possible to take recourse to this experience. This includes both the type of finite element modelling and the constant amplitude stress-life curve used for the prediction.

For the component under consideration here, a modification of the local weld geometry had been proposed in order to avoid fatigue failure from the weld root. Moreover, the fabrication process for the design shown in Fig. The new design of the weld detail is shown in Figs. The main limitation of the hot spot stress approach lies in its inability to distinguish between the two design details.

In such a case, the local stress approach offers an alternative. Stresses calculated based on the Theory of Elasticity for the notch at the weld undercut and weld root, respectively, are supposed to characterise the fatigue behaviour. Within this concept the notch root radius is set Evaluation of Fatigue of Fillet Welded Joints in Vehicle Components Under Multiaxial Service Loads 31 to 1mm in accordance with Radaj's suggestions [2] based on the "worst case" concept.

The stress-life curves corresponding with the acting local stresses and describing the failure behaviour of steel welds are reported by Olivier [19] for normal stresses and Olivier and Amstutz [20] for shear stresses. Finite element modelling A very fine mesh of elements is necessary to determine the local notch root stress of this weld. To restrict the numerical expense, it is recommended to apply a submodelling technique for its determination. The submodel is a very detailed finite element model of the weld geometry and its neighbourhood.

In practical engineering applications, it is not possible today to model a complete component or even vehicle down to the last detail. As mentioned above, all weld notch roots have to be modelled with a radius of 1mm. To achieve sufficient accuracy, at least 8 elements with quadratic form functions have to be used to model a notch root quarter circle. The boundary conditions of the submodel are provided by the coarse model - here the model applied for hot spot stress calculation - as the displacements at the submodel's boundary. Only the first 20mm of the tube needed to be modelled on forged arm.

This is sufficient to get undisturbed stresses in the weld notch on the tube. Only a cylindrical detail is modelled from the forged arm. The centerline of the cylindrical detail is the centerline of the tube, too. Starting from an area meshed with plane elements, the submodel has been generated by rotating this area around the tube axis. The complete submodel of the new design of the fillet weld is shown in Fig. A transverse section is shown in Fig.

Complete submodel of the weld Fig. Transverse section Local stress and fatigue life calculation The elastic notch stresses at the welding undercut and the weld root are used to determine the fatigue behaviour of the component. Figure 13 shows the distribution of the nodal solution stress Jy at bending. The position of failure cannot be predicted from a single stress state because of the nonproportional load situation in case of interaction of both load cases. Distribution of the normal stress Oy at bending, complete submodel and transverse section I Fig.

Distribution of the shear stress 2. This T-N curve only depends on the state of stress ratio R. It is described by the slope of A: According to Olivier [19], these values do not depend on residual stresses for plate thicknesses between 8mm and 15mm, similar to the ones in the present investigation.

The critical element and corresponding fatigue lives have been calculated applying three different multiaxial fatigue criteria. The first criterion -maximum principal stress - is very suitable to limit the failure-critical region because of the short computing time. Nevertheless, it is considered that life prediction should be based on a critical plane criterion.

In this investigation the same two critical plane criteria already used for the hot spot stress approach - normal stress mode I and shear stress mode n and HI -are applied. Both the principal stress and the normal stress critical plane criterion require a constant amplitude normal stress-life curve. This curve is plotted in Fig. The shear stress-life curve plotted as dashed line is used in connection with the shear stress critical plane criterion.

In all cases Miner's rule has been applied for the damage accumulation. Local stress-life curves Local stress approach results Taking the load-time sequences shown in Fig. It is sufficient to calculate these damage sums for notch root elements only. Calculated lifetime results for all three mulitaxial criteria are listed in Table 2. The table contains the results for the two load levels 1. Generally, the calculated critical position and that is to say the critical element depends on the fatigue criterion used. However, within this investigation, identical elements could be found as fatigue-critical for the criteria applied.

This is related to the fact that one of the two load cases, here bending, is dominating. The element suffering the highest stress amplitudes at bending is predicted as failure-critical by any criterion, additional torsion merely adds minor shear stress amplitudes which are nearly constant around the weld undercut ring. We face a typical situation for multiaxial fatigue in practice: Unfortunately, it is very difficult to oversee the situation without prior in-depth analysis. The forged arm is also plotted for a better clarification of the position of the failure-critical element.

This element is situated on the welding undercut to the tube. Another region with nearly identical calculated lifetime is at the welding undercut to the forged arm. This is the position of the failure-critical element under torsion. Because of irregularities of the manufactured welds it is significant to regard both regions with high damage as failure-critical. The elements in the weld root region are not critical. Good agreement is observed between the calculated fatigue lives for the different multiaxial criteria.

The reason is that the uniaxial local stress situation prevailing here results in comparable predicted lives when evaluated with the shear stress-life curve. For comparison purposes, the old design as shown in Fig. Figure 17 shows the plane model before rotating. In accordance with Radaj's suggestions [2, 3] all notches are modelled with radius 1 mm. The weld root notch is modelled as a circle [20]. Figure 18 shows the complete submodel for the old design. Differences between the two variants are limited to the detail shown in Fig.

Plane submodel of the older geometry Fig. Submodel of the older geometry after rotating 36 a SAVAIDIS ETAL Local stress and fatigue life calculation for the submodel of the old design The displacements taken from the coarser model used for hot spot stress calculations have been applied to the submodel's boundaries. The stress calculation has been performed for both load cases separately. Possible contact between free surfaces of the tube and the forged arm has not been taken into consideration.

It is worth noting that consideration of such geometric nonlinearities would result in the fact that the superposition is no longer valid. Accordingly, increasing numerical expense would be vast. Neglecting this effect allows the commercial software [13] to manage the handling of the nonproportional superposition of the load cases in order to calculate local stress-time histories for all stress components. Figure 19 shows a cut through the model with plotted normal stresses Cy at bending.

It is shown that the root of the fillet weld is very highly stressed, too. Because of the notch geometry and the weakening of the cross section the shear stress at the weld root under pure torsion is higher than in the new design. Calculated failure-critical locations, criterion maximum principal stress Table 3. The experimental results are plotted as dots. Comparison between calculated and experimental fatigue lives.

The hot spot stress approach in conjunction with the criterion critical plane - normal stress shows a slight trend to conservative calculations at lower load levels. Lifetimes predicted with the local stress approach are generally very conservative here. The predicted fatigue lives for the new geometry are slightly shorter than the ones calculated for the old geometry. However, the advantage of the new geometry compared to the old geometry is that the weld root is no longer failure-critical. Good agreement exists between the two criteria critical plane normal stress and maximum principal stress.

But this was to be expected in a predominantly locally uniaxial situation. Evaluation of Fatigue of Fillet Welded Joints in Vehicle Components Under Multiaxial Service Loads 39 Validation of calculated fatigue results Calculated fatigue lives or damage sums are influenced by several factors which are worth to be discussed in the following: As mentioned previously, the local stress approach is applied in practice in connection with the submodelling technique.

Within the latter, the usual procedure is to take displacements and rotations from a coarse mesh as boundary conditions for the finely meshed submodel. Attention should be paid to the requirement that the stiffnesses of the models do not differ too much. For example, a very stiff submodel will yield much higher stresses under applied displacements than a compliant submodel would. The higher stresses occurred for the stiffer new design variant.

This is the reason for the lower predicted lives for this design. Some inherent uncertainties are obviously linked with the submodelling technique itself. Calculated results also depend on the multiaxial fatigue criterion used. In general nonproportional loading cases, criteria based on conventional equivalent stresses Tresca, von Mises are inappropriate.

In special cases with locally proportional stress situation, they might work to a limited extent as well. The hypothesis of the effective equivalent stress introduced by Sonsino and Klippers [21] showed good predictions for welded flange-tube joints from fine-grained steel FeE under bending and torsion with constant and variable amplitudes. As mentioned above, these cases - dominating uniaxial stresses for example - have some practical relevance.

In case of doubts on local stress states, critical plane criteria should be preferred: The first one should normally be used when normal stresses are dominant. The second criterion is appropriate in the case of dominating shear stresses. Scatter of fatigue lives is an unavoidable matter of fact. It should be taken into account when comparing calculated and experimentally determined lives. The number of tested components here is quite low; thus, even mean values of lives are subject to uncertainties. On the other hand, the baseline stress-life curves for prediction, Figs.

Thus, a factor of 2 to 3 in lives can easily arise from this fact and can be qualified as minor inaccuracies. At last, the real fatigue life mainly depends on the geometrical form and quality of a single weld. It is possible to model the real welded form or geometry from a drawing. In this report the geometry of already existing welds is modelled.

However, the plane surface of the weld and the notch radius 1mm are idealisations. But geometry and stress distribution depend on each other. Therefore, the element with maximum damage sum is not the only one to be looked at. Other elements with similar damage sums should also be regarded as failure-critical as well. These failure-critical locations can be determined quite accurately using the local stress approach.

Using the hot spot stress approach, only one critical location on the tube has been detected. This location is verified and one more location has been detected with the local stress approach. The corresponding finite element results for the older geometry verified the weld root as failure-critical.

This is in good agreement to the experimental results. After modification of the tube to the new geometry the weld root is no longer failure-critical. Figure 23 shows a tested component with new geometry. A further effect is shown: The crack initiation does not start from the weld at all. The notch at the tapering of the tube far away from the weld undercut is failure-critical. Finally, this result prevented the presentation of experimental fatigue lives for the new design for the weld undercut.

On the other hand the result reveals some aspects of the problems when dealing with life prediction in an industrial environment: As the tapered tube is known from experience to survive quite a number of truck lives further investigations on the component have minor priority. Nevertheless, gaining experience with life prediction techniques is the prerequisite to apply them in design process. Tapering of tube Forged arm Failure-critical location calculated with hot spot approach and local approach Fig.

The stresses in the welded structure have been calculated using finite element analysis. The fatigue life has been determined theoretically, by means of the hot spot and local stresses in conjunction with the critical plane approach, and experimentally. Though the experimental data base is quite narrow, basic trends can be derived from the present investigation concerning the possible field of application and capability of the theoretical approaches calculating fatigue life of vehicle components.

A coarse finite element model of the component has been created in accordance with the n w guideline for application of the hot spot stress approach. Comparison of numerically and experimentally determined end-results fatigue lifetimes shows that the coarse model is suitable to determine lifetimes with the hot spot stress approach, if a reliable hot spot stress-life curve at constant amplitude loading is existing for the detail investigated. Because of the coarse mesh, the geometry of the weld is not modelled in detail. For instance, in this investigation only one failure-critical location has been detected.

If experimental results are missing, appropriate S-N curves from publications, e. In cases of multiaxial loading causing normal and shear stresses, attention must be paid to the slope of the S-N curve used, since various suggestions are reported in the literature. The major advantage of the hot spot stress approach is a relatively low expense to model and calculate. The coarse finite element model is unsuitable for usage in conjunction with the local stress approach. Therefore, a submodel of the failure-critical detail has been created here.

Due to the finer mesh, the number of elements is increasing. The results obtained here show that weld geometry optimisation is only possible with the local stress approach. In general, higher numerical expense does only make sense, if failure-critical locations should be investigated more exactly. In the majority of engineering applications, a finer mesh or more detailed model is often an unrealisable option due to technical and commercial reasons. To calculate fatigue lives in accordance with the local stress approach, no experimental input data are required, when using the universal local a-N curve.

Woodhead Publishing Limited, Cambrigde. Bo vet-Griff on, M. Fatigue assessment of welded automotive aluminium components using the hot spot approach. Fatigue design criterion for welded structures. For example, Table 1 reflects threshold conditions derived from the experimental results of Zhang 6 and Eqs. The maximum non-propagating fatigue crack represents the fatigue limit strain range.

It can be seen that the non-propagating crack length af exceeds the average ferrite grain size da 37 lam in medium carbon steel in the transverse direction, with a standard deviation of 18 gm. However, there is a significant difference between microstructural barriers d and the threshold short crack length of Stage II for strain ranges close to the fatigue limit. What is the mechanism of microstructural short crack propagation beyond the first grain to reach the Stage II threshold? To answer this question it is necessary to consider basic microstructural aspects of fatigue crack growth and fatigue resistance of metals and develop the representation of microstructural barriers.

The transcrystalline crack that becomes the short fatigue failure crack is the one located in the largest ferrite grain, because it grows fastest. It is clear that Modelling Threshold Conditions Thus D, the strong barrier gives an absolute upper bound to weak barrier values dm.

The transition of short crack propagation from Stage I to Stage II is determined by an equality of crack rates from Eqs. The number of microstructurally short crack steps depends on a material's microstructure, and also on the applied strain range because Stage I is limited by the transition of crack propagation to Stage II after the mechanical threshold has been exceeded. The experimental results lie between two curves described by the Tresca and Rankine criteria. The Rankine and Tresca curves were derived from Zhang's push-pull crack growth results, using Eq.

The location of the experimental curve is dependent on the applied strain range and can be determined by the threshold conditions. Rankine's criterion correlates with experimental results for the low strain range regime. The use of Tresca's criterion is more justified in the high strain range regime of loading. It is clear that it is necessary to modify these criteria and the parameters of Eq. The following procedure is proposed to establish the modified criterion and parameters of Eq. It is considered that two points will create a modified threshold criterion. The first point is determined by Rankine's criterion at the fatigue limit strain range Ayf.

The second point is determined by the intersection of the Tresca's criterion threshold line with the experimental threshold in the high strain range regime of Fig. Table 2 shows excellent agreement with the experimental results because the critical value of 7 lxm was selected for the torsion data in Fig. Table 2 Constants and exponents of equations 7 and 8 and microstructural fatigue limit parameters for medium carbon steel under torsion loading.

Obviously, the fatigue limit condition for combined loading differs from the fatigue limit under torsion and push-pull. To estimate the fatigue limit condition under combined loading, an approach based on the F-plane 8 can be used. Contours of constant endurance plotted on a graph of 89 versus Zl6n may be represented by the elliptical equation Modelling Threshold Conditions In the case considered here, constant endurance is referred to the fatigue limit.

Constants g and l can be obtained from Eq. Thus, the F-plane approach allows us to estimate fatigue limit strain range components for combined loading Table 3. The modified threshold line described by Eq. A modified threshold condition and an equation for Stage II short crack growth under combined loading have been derived from Tresca's and Rankine's equivalent strain criteria and two selected boundary threshold points, which are fitted to empirical torsion fatigue threshold results.

It is clear that the parameters and exponents of short crack growth equations are dependent on microstructure as well as on the type of loading. The dependence on microstructure is illustrated by the maximum non-propagating fatigue crack lengths af or distances between strong microstructural barriers , which have been derived from equations for Stage II with a growth rate equal to zero under fatigue limit strain ranges Table 4. Obviously, the change of size of non-propagating cracks for various loading conditions is connected with a change of crack growth directions between major microstructural barriers.

The high value for af in torsion reflects the anisotropy of the material, with greater microstructural dimensions for cracks growing along the longitudinal axis of the bar. A general fatigue lifetime Nr is determined by summation of lifetimes for each crack propagation stage. The number of crack steps for Stage I was taken into account in Eq.

The transition length atr is given by Eq. It is assumed that after transition cracks will remain in Stage II without reverting to Stage I under constant amplitude loading. The predicted lifetime for an equivalent regime of combined loading is compared with experimental life for regimes of torsion and push-pull loading of solid specimens Table5. The equivalent regime of combined loading was calculated by employing the F-plane approach with Eq. Theoretical results for endurance under combined load closely correlate with experimental lives for equivalent regimes of torsion and pushpull loading.

The following conclusions can be drawn from the present study: Perez and Brown M. Thesis, University of Sheffield, UK. Matvienko would like to thank the University of Sheffield and the Russian Academy of Sciences for funding a one-year visit to the Department of Mechanical Engineering, University of Sheffield. Two materials - a ductile aluminium alloy and brittle PMMA - were used to analyse damage processes under biaxial loading conditions, as tension and shear.

Experiments have shown that material properties strongly affect the damage process in both monotonic and variable loading. The explanation of this behaviour is based on an elastic-plastic FEM analysis for the aluminium alloy and a non-local damage accumulation criterion for the PMMA. High stress and strain gradients are associated with the presence of sharp corners and may be represented by various asymptotic relations depending on the opening angle, loading modes and boundary conditions.

Such corners frequently appear as weak points in engineering structures reducing their strength and durability. They can also serve as interesting objects for theoretical and experimental studies related to fatigue and fracture processes. In such cases elastic stress fields based on the solutions of the theory of elasticity are of great importance. They are also useful for local plasticity near the notch root, due to the fact that the differences between the elastic and elastoplastic strain fields some distance from the notch are small.

Two materials, one ductile and one brittle, were used. For each material and V-notch considered, the following experimental data have been collected and analysed: If the notch faces are free from loading, the asymptotic stress fields near the notch are given by Williams solution 1 , or may be also described, together with the displacement field, by Eqs 1 recently published in 2: Their values are obtained from the characteristic Eqs 2: The numerical solutions of the characteristic Eqs. Asymptotic values of the stress field near the corner strongly depend on the notch opening angle 2[3, the element shape, the loading mode and the displacement conditions imposed on the body.

Therefore approximate values of the generalised stress intensity factors are usually based on FEM results, where appropriate special finite elements fill the core region of the apex. Such numerical procedure has been applied to determine K values for plates with V-notches experimentally tested as described below. Brittle materials were represented by 5 mm thick PMMA with opening angles 21] equal to 40 and 80 degrees.

All notches were made on a vertical milling machine for a set of PMMA specimens fixed between external plates of aluminium alloy. This prevented PMMA from cracking during the cutting process. Final machining of notches was executed by means of a special cutting tool with the p r o f i l e corresponding to the notch angle. Microscopic measurements have shown that the notch root radii were much lower than 0. In order to impose biaxial loading conditions, a special device has been used together with a standard uniaxial fatigue machine. As is shown in Fig.

Four washers 4 and 5 with increased coefficient of friction are placed on both sides of the tested plate 6 and fix it firmly with four bolts 7 and nuts. In this way both extremes of the plate are always kept always parallel and the external load applied by means of this device generates a biaxial stress state. The resultant tensile and shearing forces in the element 6 can be calculated from Eqs.

All tests have been carried out on a hydraulic machine with MTS controlling system. A frequency of 3 Hz was chosen for all fatigue tests run with constant displacement aluminium and loading PMMA amplitudes applied to the device. After that, basic parameters of fatigue tests were established.

In the case of the aluminium alloy, the limit load of the plate depends on the shape of the far field plastic zones and their mutual interaction in the central cross section, showing the important role of shearing stresses. Maximum strength, obtained for between 30 and 45 degrees, corresponds to the maximal force necessary to develop and join together two plastic zones growing from the opposite notches.

This phenomenon has been confirmed by an elastoplastic FEM analysis. Notch angle 2[3, equal to 20 and 80 degrees, had no significant influence on the limit load value. However, sharper notches slightly reduced the strength of the plate. The critical load for the monotonic test of PMMA was quite different to that for the aluminium alloy.

The strength of the element depends on the notch angle and the orientation of the plate with respect to the device. Fatigue behaviour of specimens made of the aluminium alloy mainly depends on the plastic zone size and its orientation. Fatigue cracks appeared on the specimen surface as two small curvilinear cracks, in both sides of the plastic zone. After a certain number of cycles one crack dominated and developed across the plate. The crack plane was inclined 45 degrees to the lateral surface of the specimen, and 0 degrees in the middle plane, showing 3-dimensional fatigue behaviour of the damage process.

Some experimental results for the aluminium alloy are shown in Table 1. Nf Fmax [kN] Similar behaviour o f P M M A was also reported in 8 , w h e r e fracture process for cruciform specimens with an artificial crack under biaxial loading was investigated. S o m e experimental results related to the present study are p r e s e n t e d in Table 2 and Fig.

In a similar way the fatigue damage initiation condition can be defined. The fatigue damage accumulation occurs at the notch tip, when the averaged value of the normal stress Crndistributed over the damage zone do exceeds the threshold value j2 Fatigue and Fracture of Plane Elements In this approach the fatigue crack initiates when the accumulation damage measure On on any of the physical planes reaches the critical value: The range of averaged stresses Act 0 along the characteristic damage zone do, is then given by Eq.

Their numerical values are shown in Table 3. Table 3 FEM results of generalised stress intensity factors for the tested specimens.

ESIS TC03 - Fatigue Of Engineering Materials And Structures

Theoretical results of fatigue initiation cycle number Nf, obtained from the above model, are compared to the experimental data for PMMA in Fig. Equivalent stress range vs. A non-linear system of equations, solved by means of a gradient method, gave the following values: Both dashed lines shown in Fig. This is probably due to the small singularity effect for in-plane sheafing, compared with the higher order terms of the stress field near the corner.

In the case of an aluminium alloy subjected to monotonic load representative for ductile materials the strength of the element depends on the shape of the far field plastic zones expanding from the opposite notches and joining together. Thus, notch angles equal to 20 and 80 degrees had no significant influence on the limit load value. This phenomenon has been confirmed qualitatively and quantitatively by an elastic-plastic FEM analysis.

The results of fatigue tests for the same material have shown the importance of the plastic zone sizes on the fracture process, which inside the material was initially different from that on the surfaces of the plate. Local plasticity and corresponding hardening of the material have made the early stage of crack propagation more complicated for analysis, due to the 3-dimentional nature of the damage process. Under pure shear, only one fatigue crack was initiated very easily and propagated from the notch apex through the centre of the plastic zone lying in the central plane of the element.

In the damage process of the brittle material, represented by PMMA, the elastic stress and strain fields appeared to be the most important factor. In both monotonic and fatigue tests, the crack paths were almost identical and depended on the notch opening angle 2[3 and the ratios between tension and shear, showing the most important role of normal stresses in the fracture process.

Thus, the critical load depended also on these parameters. The non-local stress damage accumulation criterion by Seweryn and Mr6z, which has appeared useful in predicting critical monotonic loads 3 , also remains in satisfactory agreement with experimental results of fatigue tests for PMMA and may be applied in estimating the fatigue crack initiation period as well as initial fracture angle.

Fatigue tests of brittle materials are generally difficult to carry out due to the very narrow range between the threshold and critical values of the stress intensity factor. Multiaxial Fatigue and Design eds Pineau A. Mroz, TU Opole, Poland, vol. Acknowledgements The investigation described in this paper is a part of research project No. The computation of the stress-strain state in the concentrator has been done and the applicability of the modified criterion of shear octahedral stresses to the assessment of the cyclic life of an element under test has been shown.

The modification of the criterion lies in the proposed method for taking into account the residual stresses as a non-uniaxial cycle asymmetry. In the above zones, the material is under conditions of nonuniform non-uniaxial stress state, which requires the assessment of its lifetime, based on the established ultimate state criteria. These criteria are most frequently established and verified given the uniform field of stresses and macroscopic volumes of the material in this field, and in this connection their formal application in the case of a local nonuniform complex stress state is not well substantiated, although in many practical problems it is this question that needs to be settled.

A structural stress concentrator with residual stresses induced in the material by machining is one of the most typical cases for structural elements. The loading of such elements results in a complex stress state which, being established in the concentrator, interacts with the volumetric field of residual stresses relaxing in the process of cyclic loading. The complexity of resolving this problem on the prediction of the cyclic life has led, in this case, to the appearance of a great number of works in this line of investigation 1 - 4.

However, the question has not yet been settled, which is effects the many-sided nature of the problem to be resolved. In the present work an attempt has been undertaken by the authors to develop the calculation-and-experimental procedure for assessing the cyclic life of an element, with a stress concentrator and residual stresses based on the material complex stress state, which is taken into account by means of calculations and the experimental justification of the applicability of the material ultimate state criteria. Assessment of the Cyclic Life It is used for manufacturing steam generator collectors during the operation of which both concentrators in the form of holes and residual stresses induced by pressing-in of pipelines are present therein.

The steel under study has the following mechanical characteristics under static loading: Test specimens were prepared in the form of a plate of rectangular section with dimensions of 14x2. A central circular hole 3 mm in diameter was made in the specimens. Cyclic loading at a frequency of 36 Hz was realised under fully-reversed tensioncompression. The application of various methods of machining result in inducing non-uniaxial residual stresses in an element subjected to machining.

In this case, the ratios of residual stress components along different axes can be different. Therefore, when preparing the test specimens with a stress concentrator, the residual stresses were induced in the concentrators by various methods providing different ratios of residual stress components along the axes. The methods for inducing residual stress are presented schematically in Fig. The residual stresses were induced immediately prior to fatigue order to reduce their possible decrease with time as a result of the relaxation processes.

This mode of loading was realised using a testing machine along the axis of the cyclic loading and a clamping device specially designed for this purpose perpendicular to the axis of cyclic loading, i. The software applied enables one to make an elastoplastic calculation of the material stress-strain state in the stress concentrator. The initial data for the calculation were the geometrical characteristics of the specimen, the material static stress-strain diagram and the specimen loading parameters.

Figure 2 presents the results of the calculation of residual stresses in the stress concentrator induced according to the schemes under study. Distribution of residual stress components through the thickness of the material in the concentrator: In the case when the residual stresses are generated according to schemes 3 and 4, the stress components Ooy far exceed the OoX components. In doing so, the calculated variation in the forces applied has been made so as to induce the static stress on the concentrator surface in the direction of the x-axis with the subsequent investigation of the influence of this static component directed perpendicular to the axis of the varying loading on the cyclic life of specimens with a concentrator.

From the results given it follows that by changing the method of generating the residual stresses and applying biaxial static load, one can obtain various cases of residual stress combinations or static ones under loading according to scheme 5 along the X and Y axes including the limiting cases where the stress component along one of the axes exceeds the component along the other axis to an extent that the latter can be neglected. The residual stress diagrams given in Fig. Therefore, later on, in the development of the model of calculation for assessing the cyclic life of an element under test, one should be familiar with the method of determining the level of residual stresses taking into account their relaxation under cyclic loading.

For this purpose, the literature data on the direct measurement of the residual stress kinetics resulting from the relaxation under the action of a cyclic load in steels of similar grade have been analysed 7. For this reason, in the subsequent assessment of the cyclic life for specimens with a concentrator and residual stresses, the fatigue damage to the material at the stage where the residual stresses reach the level of stabilisation was not taken into account.

As a result of the analysis of the data on the relaxation of residual stresses under high-cycle loading of steels, the relationship of a closed form has been obtained 7. It enables one to determine the level of the stabilised residual stress along each axis. The initial data for computation are the material static and cyclic stress-strain diagrams geometrical characteristics of the specimen and the calculation of the distribution of residual and cyclic stress components through the material depth based on these data.

Fatigue curves were constructed in nominal stresses as a function of lifetime prior to initiation of a 0. Figure 3 illustrates the fatigue curves for smooth specimens curve 1 , specimens with a concentrator without residual stresses curve 2 , specimens with stress concentrators curves 3 to 7 and specimens with a concentrator with the biaxial static compression being superimposed curve 8. Fatigue curves for the investigated specimens: Based on the given experimental data, we consider the possible methods for going from fatigue curves for smooth specimens to those for specimens with concentrators and relaxed residual stresses.

In this case the nonuniformity of stress distribution through the material depth should be taken into account. This can be done in a variety of ways, namely, by the recalculation of stress-strain diagrams using the coefficients which take into account the stress gradient 8 ; by taking into account the depth of action of the stresses which exceed half the fatigue limit with the use of a relative gradient of the first principal stress 9 ; by establishing the critical depth of the surface layer at which the fatigue failure occurs and which is constant for a given material 10, Since the distributions of stresses through the depth of the material in the concentrator under conditions of elastoplastic deformation are nonuniform, the application of the first two methods is troublesome.

For this reason, the possible application of the third method of those listed above has been considered. Based on the calculation procedure applied for determining residual stress distributions, an elastoplastic analysis has been made of the stress-strain state in a stress concentrator under cyclic loading. In so doing, the cyclic stress-strain diagrams for the steel under study were used 8. Calculations were made for several levels of nominal stresses. Figure 4 presents the distribution of stress intensity amplitudes along the X-axis. The point on the specimen median plane where the stress in the concentrator is maximum for the elastic solution of the problem is taken as the X-axis origin.

For all the nominal stresses the maximum stresses are observed along the Y-axis. Here, with distance from the concentrator surface, the values of tray change non-monotonically and have a maximum at a distance of lure from the concentrator surface. The stress OaZ are lower by a factor of 3 or 4 than the value of 13ay for the corresponding Assessment of the Cyclic Life In view of the fact that the stress in the concentrator along the Y-axis is significantly higher than those along the two other orthogonal axes, the distribution of stress intensity amplitudes through the material depth given in Fig.

Distribution of stress intensity amplitudes through the depth of the material in the concentrator: The diagrams of stress intensity amplitudes Fig. As is evident from those results, the stress amplitude intensity both on the concentrator surface and down to a certain depth exceeds that in a smooth specimen for equal lives. We designated this depth by Xeq and note that on the concentrator surface X - 0 and at the maximum point of the stress intensity amplitude XXmax the values of t3ai characterising the stress-strain state in the stress concentrator correspond to much shorter lives on the fatigue curve for smooth specimens, as compared to point Xeq.

Considering the analysis made on the stresses in the concentrator, the following may be suggested: As is evident from the data in Fig. STEPURA 10,11 about the constancy of the critical depth of the material surface layer at which the stress equivalence for a smooth and notched specimen is observed. The analysis of similar results made for other materials allows one to say that for low characteristics of cyclic plasticity, the Xeq-value is close to a constant value, which is in agreement with Refs. Using the proposed relationship 7 , the components of stabilised residual stresses were determined for specimens with a concentrator and with residual stresses taking into account the components of the cyclic stresses, which are acting on the corresponding axes.

The residual stresses obtained are thereafter considered to be applied statically. They were determined for the following material depths in the concentrator: Other lives are not considered in this case in view of the similarity of the results and regularities established at the chosen life level. In the realisation of various schemes for inducing residual stresses, measures were taken to provide their combination with both different ratios along the two principal axes and a prevailing value along either individual axis.

In view of this, the application of the approaches wherein the first principal stress is used for assessing the cyclic lifetime seems to be impossible. Therefore, a search for the equivalent stresses for asymmetrical cycles presenting a superposition of cyclic non-uniaxial stress in the concentrator and non-uniaxial stabilised residual stresses will be made with the use of the stress intensity, as in the case of determining the Xeq.

In so doing, we take by convention that the intensity of residual compressive stresses is a negative value which is necessary for construction of the diagram of limiting amplitudes in terms of stress intensities referred to as the conventional diagram of limiting amplitudes. Such diagrams presented in Fig. To construct them, the data on both the specimens with a concentrator and with residual stresses and smooth specimens solid symbols and specimens with a concentrator without residual stresses open symbols were used.

From the given data the following conclusions can be drawn: The calculated and experimental data obtained are in the best agreement with the criterion of octahedral stresses in the form Analysing those deviations and the nature of residual stresses, the authors have noticed the following regularity: The indicated regularity can result from the fact that the criterion employed like criteria of similar kind 12 does not take into account the material sensitivity to the cycle asymmetry which manifests itself in the change in the amplitude of a varying stress depending on the value and sign of the mean stress: Therefore, the stress components in Eqs.

In view of the aforesaid, Eqs 2 and 3 will be written in the following form. For the calculation it is necessary to have the data on smooth specimens: Stabilised values of residual stresses are found for this depth Xeq and their consideration as a cycle asymmetry is made according to dependencies 7 and 8.

By introducing the refinements according to formula 11 and using the criterion c u r v e s '17oct VS Pmax, the N 1 value is found which corresponds to the life of the specimen with a stress concentrator and residual stresses under the action of nominal stress Crln. By specifying several values of cra , N 1, and repeating the described procedure, we get the fatigue curve for an element with a stress concentrator. The described algorithm for assessing cyclic life was realised for several variations of tested specimens with concentrators and residual stresses.

The results are given in Fig. As follows from the data obtained, the prediction is satisfactory, though an increase in the prediction error is observed with a decrease in cyclic life. We relate this fact to a larger error of the determination of the stabilised residual stresses for those lives. Assessment o f the Cyclic Life However, it has been shown that this criterion should be modified to take into account the components of the cycle asymmetry along different axes.

The results have been demonstrated on plane specimens with a central hole and residual stresses induced therein whose stabilised values are taken as the static components of the loading. On the basis of the modified criterion of the material ultimate state, an algorithm has been proposed for the calculation of high-cycle fatigue life for a typical structural element with a concentrator and residual stresses, and the agreement between the calculated and experimental results has been shown.

Here, account is taken of a certain layer of a damaged material in the stress concentrator wherein its ultimate state has been attained. Analysis of materials stress-strain state, Problemy Prochnosti, No. A survey of the state of the art, Journal of Testing and Evaluation, Vol. Young Scientists of the Inst. Ukraine in Russian , Kiev, pp. Tension-torsion low cycle fatigue tests were carried out using hollow cylinder tube specimens OD 12 mm, ID 9 mm, gage length 6.

Nonproportional strain written with only strain path and having a material constant correlated nonproportional fatigue lives within a factor of two scatter band.

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The additional hardening of aluminum alloy under nonproportional straining was also discussed in relation with fatigue life. Nonproportional loading reduces the low cycle fatigue life due to the additional hardening depending on strain history, so the nonproportional parameter must take account of the additional hardening. A couple of nonproportional parameters which include the stress range or amplitude have been proposed , and stress terms in the parameters are able to be calculated using the inelastic constitutive equation , but it is not a simple procedure in general and requires many material constants.

There is no well established method of estimating nonproportional low cycle fatigue life based on only strain history. The authors 9 carried out nonproportional low cycle fatigue tests using a hollow cylinder specimen of Type stainless steel and proposed a nonproportional low cycle fatigue parameter written with only strain history. Type stainless steel is known as a material, which shows the large additional hardening under nonproportional loadings 5, Fatigue lives drastically reduced by additional hardening, which depends on strain history.

The maximum reduction is a factor of 10 when compared with the proportional fatigue life. However, the degree of additional hardening is material dependent, so that the reduction of nonproportional lives is also material dependent. The aim of this paper is to examine the nonproportional low cycle fatigue life of 80 17 --u.. Mises' equivalent total strain controlled nonproportional low cycle fatigue tests were carried out using hollow cylinder specimens with 9 mm inner diameter, 12 mm outer diameter and 6.

Test machine used was a tension-compression and reversed torsion electric servo hydraulic low cycle fatigue machine. Case 0 is a push-pull test and is the base data used for the nonproportional life prediction. Total axial strain range was varied from 0. Strain paths shown in the figure were determined so as to make clear the various effects in nonproportional straining 9.

In this paper, one cycle is defined as full straining for both axial and shear cycles. Thus, a complete straining along the strain paths shown in Fig. In Case 3 and 4, a complete cycling was counted as two cycles. The principal strain range, AeI, is determined by two strains, ei A and ei B , and the angle between them, where A and B are the times maximising the strain range in bracket in Eqs.

Nonproportional Low Cycle Fatigue In the figure, a factor of two scatter band is shown by lines based on the push-pull data, i. Case 0 data, and attached numbers denote the Case number. ASME strain range correlates fatigue lives unconservatively for some Cases by more than a factor of two. The lowest fatigue lives occurred in Case 13, i. The significant reduction in fatigue life also occurred in Case 10 and 12, box paths, as well as circular path.

Specimen shape and strain paths are the same as those in this study. The figure shows the same trend of the data correlation as that in Fig. Comparison of the results between Fig. Thus, nonproportional strain parameter must take account of these two factors. Fatigue lives of A1 alloy, Fig. The results in these two figures indicate that the small reduction in LCF life occurs for small additional hardening material and the large reduction for large additional hardening material.

The maximum principal stress range is a suitable parameter for the former material but is not for the latter material. Reduction in nonproportional LCF life is connected with the degree of additional hardening In nonproportional loading, the principal strain direction is changed with proceeding cycles, so the maximum shear stress plane is changed continuously 46 T. This causes an interaction between slip systems and which results in the formation of small cells 10,11 for Type steel.

Large additional hardening occurred by the interaction of slip systems for that steel. No large interaction occurred in A1 alloy since dislocations change their glide planes easily following the variation of the maximum principal strain direction Table 1 Summary of the test results. Other literature reported that aluminum alloys show little or no additional hardening 3,4,10 , while Type stainless steel usually gives a significant additional hardening 5,10, Thus, the nonproportional LCF parameter must include a parameter, which expresses the amount of additional hardening.

For Type stainless steel, it was 0. T is time for a cycle and fNP is normalised by T and ei max. The reason for making fNP integral form is that the experimental results indicate that the nonproportional LCF life is significantly influenced by the degree of principal strain direction change and strain length after the direction change. The value of fNP takes zero for proportional straining. The parameter given by Eq. The authors 13 proposed another nonproportional strain on the basis of the equivalent strain based on crack opening displacement COD strain to improve the data correlation of proportional LCF lives.

The nonproportional strain range based on COD is defined similar to Eq. Constants, [3 and m', in Eq. In these figures, a and b are the correlation of fatigue lives for A1 alloy and Type steel, respectively. Therefore, the two nonproportional strains proposed by Eqs. These equations have only one material constant, which is determined by the stress range ratio under 90 degrees out-of-phase and proportional loadings.

Fatigue lives of aluminum alloy were reduced by nonproportional loading but the reduction was not so large as that of Type stainless steel. The two nonproportional strains were applied nonproportional LCF lives of the two materials. The scatter of the data was within a factor of two for both the materials and which indicates the two strains are suitable parameter for correlating nonproportional LCF data of small and large additional hardening materials.

Seminar on Multiaxial Plasticity, France, Benallal et al. On the basis of this analysis, a general fatigue criterion is formulated for multiaxial stress. The existing multiaxial criteria of integral approach and of the critical plane approach can be derived as special cases from the general fatigue criterion.

On this basis, a new modification of shear stress intensity hypothesis SIH that provides satisfactory agreement between experimental and calculated results is proposed. As a rule, the multiaxial stress state is of a very complex nature. The individual stress components may vary in a mutually independent manner or at different frequencies, for instance, if the flexural and torsional stresses on shaft are derived from two vibrational systems with different natural frequencies.

For assessing this multiaxial stress, the classical multiaxial criteria, such as the von Mises criterion or the maximum shear stress criterion, are not directly applicable. This is illustrated in Fig. The second case involves a pulsating tensile normal stress, Crx, and a compressively pulsating normal stress Cry, Fig.

In both load cases, the principal stresses exhibit the same variation with time. In accordance with the classical multiaxial criteria, the same equivalent stresses are calculated in both cases. The endurance limits are very different, however, as shown by experiments 1. This is explained by the fact that the principal direction can vary in the case of multiaxial 55 56 J. A variable principal direction is not taken into account by the classical fatigue criteria. For calculating the endurance limit in the case of multiaxial stresses, a number of multiaxial criteria have been developed during the past decades 2, These developments are even more comprehensive, as indicated by recent studies 13, The multiaxial criteria differ considerably in formulation, in the range of applicability, and in the reliability of prediction.

Furthermore, they also involve highly different physical interpretations, if such an interpretation has been considered and indicated at all in formulating the hypothesis. As a matter of principle, the known multiaxial fatigue criteria can be subdivided into hypotheses of the critical plane approach, hypotheses of integral approach, as well as empirical criteria. In the case of integral approach, the equivalent stress is calculated as an integral of the stresses over all intersection planes of a volume element; compare with the hypothesis of the effective shear stress 2 and the shear stress intensity hypothesis SIH 3, In the case of the critical intersection plane approach, only the intersection plane with the critical stress combination is considered, for instance, with the modified shear stress hypothesis proposed by McDiarmid.

In the present publication, the weakest link theory is first analysed. Subsequently, the relationship between the weakest link theory and the classical multiaxial criteria is explained. The multiaxial fatigue criteria of the critical intersection plane approach Weakest Link Theory On the basis of this analysis, a general multiaxial criterion is formulated for arbitrary multiaxial stresses.

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