Fracture and Fatigue Emanating from Stress Concentrators
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The ratio of the local fracture strain and critical gross strain gives the value of the elastoplatic stress concentration kH. This value has been plotted versus the stress triaxiality as shown in figure 6. The elastoplastic strain concentration factor increases with stress triaxiality and has a value between 2 and 3.
A mutual influence of the stress triaxiality and the critical strain has been seen by Shockey et al [6. The variation of the ratio Rc R0. Rc with the stress triaxiality for static and dynamic loading R0. H f notched axisymmetric specimen, XC 18 steel. This is one of the basic reasons for knowing the size of plastic zone at the notch root. More information can be found in the literature for the size of crack tip plastic zone rather than the notch root plastic zone.
The plastic zone diameter Ry is given for cracks by the general formulae: Ry E '. E takes the particular value of 0. This load corresponds to a net stress MPa, according to relationship 6.
These two methods are compared. G x40 Figure 6. From Figure 6. Good agreement can be found between experimental and computed evaluation of the plastic zone size. Plastic zone size mm 2. This evolution seems insensitive to notch. The plastic zone size in the y direction can be expressed through the notch stress intensity factor by the relationship: K 2. The variation of the parameter A with the elastoplastic stress concentration factor is presented in figure 6.
Plastic failure occurs for a ligament size less than a critical value which can lead to a new definition of the critical defect size. The strain distribution at the notch tip can be presented by a maximum and a notch strain intensity factor. The critical notch strain intensity factor can be used as a fracture toughness value. The notch ductility factor can be considered as another measure of fracture toughness. K and al. Journal of Mech.
Steel Research, pp. Journal of Mechanics and Physics of Solid,19,p R and Engineering Fracture Mechani 6. N and Merkle. R and Tracey. A and al. Journal of Mechanical Physics of Solids,vol. Previous works in the field of mechanical testing have foundthe lack of correlation between the static and dynamic test results. Earlier at the French Commission of Testing Methods in , Mr Le Chatelier proposed the use of notched specimens for themeasurement of the resistance to fracture.
This type of specimen has been used from by Mr Auscher, a ship building engineer at Indret France. The geometry of the specimens was a 20 mm square section with a one millimetre deep triangular notch on each of the four faces. Specimens were clamped, and drop weight tests done with a 1 kilogram hammer falling successively from different heights at the free end of the specimen.
This end was located mm from the clamped section. This method has the disadvantage that it requires several tests in order to obtain a consistent result. The idea of measuring the residual force after fracture and consequently the work done for fracture, was introduced by Russel in the USA and by Fremont in France.
Russell presented a paper in at the American Society of Civil Engineers in which he described the use of a pendulum. The work done for fracture was established from the difference between the initial and the final height. The pendulum built by Charpy is presented in figure 7. The weight of the pendulum was 50 kilograms and the distance between the axes to the knife was 4 metres.
The pendulum was seated on a block of masonry of five cubic metres. The specimen was seated on two supports made in a plate of 1, Kilograms. After impact the height was measure directly because the velocity of the pendulum was slow and the difference of height between arrival and departure multiplied by the weight of the pendulum gives a measure of the work done for fracture. Fragments were thrown to a distance between only 2 or 3 metres. Figure 7. The unit is expressed as kilogram-metres per square centimetre.
It seems that at that time the concept of brittle fracture was not very clear. Charpy pointed out that the test on notched specimens is not a test for brittle fracture but rather a test which allowed the classification of metals either as having a high resilience or a low resilience. However, despite the introduction during the s of mechanical fracture testing to measure the resistance to crack growth, the Charpy impact test remains, as it gives a simple inexpensive method of classifying a materials resistance to brittle fracture.
Nowdays, the tendency is also to use this test as a measure of fracture toughness. Comparison of the two methods requires the consideration of two major differences: i Charpy tests use a notched specimen, fracture mechanics tests use a pre-cracked specimen but some Charpy specimens are now pre-cracked , ii fracture mechanics testing is generally performed using quasi-static loading conditions, Charpy tests are dynamic tests.
The most well known are presented in figure 7. The stress distribution for the stress normal to notch plane Vyy at the notch tip for the three types of specimen Charpy U,V and Schnadt are presented in Figure 7. It can be seen that the maximum stress is higher for Schnadt specimens highest notch acuity and the lowest value is obtained from the U specimen greatest notch radius. The position of this maximum stress Xm moves far away from the notch tip when the notch acuity decreases.
In Table 7. It should be noted that the elastoplastic stress concentration is higher when the notch radius is small and is practically independent of the load level exception of Charpy U. Table 7. From this table we note that the position of maximum stress moves away from the notch tip when the notch radius increases. The elastic and elastic-plastic normal stress distribution at the notch root exhibits a decreasing dependence with distance from the notch tip that is relatively complicated.
As can be seen in this figure, the stress distribution is of classical form as previously described in Chapter 4. We can divide this stress distribution into 4 regions. A graphical procedure for the determination of Xef described previously is applied. The effective distance is related to the minimum value of the relative stress gradient F. Charpy V notch specimens made of a CrMoV steel a with fine carbide FC microstructure yield stress MPa were tested statically in bending at a selected temperature in the lower shelf region.
The tensile stress distribution at the notch was calculated using FEM. A 2D model, was used under plane strain conditions for the elastic plastic analysis. The flow behaviour was computed using incremental plasticity adopting a Von Mises criterion. The correlation between experimental and calculated results was good. Several tests were carried out at room temperature indicating a fracture load in the range [ We note that this distance is greater than the microstructural unit grain size for example.
The effective stress is defined as the average stress inside the fracture process zone. For this material the mean value of the effective stress is MPa which can be compared to the average maximum local stress at fracture Vmax of MPa. The fracture toughness is given by : K cU 3. The transition temperature is not intrinsic to the material as we can see on figure 7. From figure 7. Tests have been carried out using a steel St [7. It is well known that ductile fracture is sensitive to stress triaxiality as shown by Rice and Tracey [7.
Notch Effects in Fatigue and Fracture
The stress triaxiality is defined as the ratio of the hydrostatic stress to the equivalent Von Mises stress E Vm. For this reason, ductile fracture initiates at the point of maximum triaxiality which changes with temperature. When the temperature increases, this point moves toward the notch tip and the brittle part of the fracture surface decreases. This ductile initiation is following by brittle propagation.
At high temperatures the second transition appears and corresponds to a propagation transition. When the conditions of constraint at the tip of the running fibrous crack are sufficient, brittle fracture initiates. When these conditions cease to be satisfied because of the increasing temperature, the fracture surface becomes totally fibrous. The role of triaxiality on fracture initiation in Charpy test has been discussed independently by several authors [7. The use of pre-cracked Charpy specimens has allowed the connection with Fracture Mechanics concepts.
The classical impact pendulum measures the energy used for fracture. By the addition of instrumentation to this equipment it is possible to record the load and the displacement versus time. The force measurement is carried out by strain gauges bonded to both sides of the hammer. These strain gauges are connected to a voltage supply and amplifiers. The calibration of the electrical signal derived from the strain gauges is made in two ways: -i static calibration by loading the instrumented tup by compression on a mechanical testing device; -ii dynamic loading based on the area under the registered load displacement curve until critical displacement dc being equal to the work done for fracture; dc U 7.
These pulses are produced when two pins on the hammer go through an optical trigger device. The displacement is measured via an angle measurement device and converted into linear displacement knowing the velocity of the hammer. The time to fracture initiation is generally measured by a magnetic emission probe [7. Two examples of recorded data are presented in figure 7. After a linear increase and several load oscillations owed to reflecting shock waves, the load increases to a maximum. When fracture occurs in the ductile-brittle transition regime, initiation appears after the maximum load and is characterised by a sudden drop.
The final part of the diagram corresponds to final ductile tearing. The following parameters are registered on figure 7. In the lower shelf region, crack initiation occurs without macroscopic plastic deformation. Time to initiation is relatively short and can be more than the time necessary for load stabilisation after several load oscillations due to shock wave reflections on the specimen surfaces. According to Kaltoff [7. In the transition region crack initiation usually occurs after significant plastic deformation. The dynamic critical value of the energy parameter JId is divided into an elastic and a plastic components 7.
The elastic component is derived from the dynamic critical stress intensity factor Kcd: K 2Id. The onset of stable crack extension is then measured via additional technique such as strain gauges, stretch zone measurement, COD or magnetic emission. The chemical composition is given in table 7. This material is generally fully ductile and the following parameters have been measured ,load at general yielding Fgy ,load at maximum Fm, total work done for fracture Uc, and energy dissipated Um until maximum force is reached.
The variation of energy up to maximum force is presented in Figure 7. The variation is linear and increases up to a notch radius of about 1.
Analysis of toughness on notched specimens by volumetric method
However, it appears than below notch radii less than of 1 mm a constant value of energy is obtained. Fracture toughness has been obtained from relationship: Uc 7. According to [7. For low notch radii values of JIc are considered constant, figure 7. The transition temperature determined from this test can be a tool for the design against brittle fracture.
However, this test cannot give directly the acceptable or the critical defect size. For this it is necessary to use a fracture toughness approach and determining the fracture toughness KIc. It is however to gain advantage of the use of the inexpensive and rapid Charpy test and obtain a fracture toughness via a correlation using the Charpy energy KV in Joules. Values of the notch sensitivity index. This kind of correlation was first proposed by Barsom and Rolfe [7. The correlation was different for the lower shelf; 2 K Ic 0. For this reason it is necessary, in addition, to convert Charpy energy and fracture toughness to shift the transition curve for a given value of temperature.
This method has been introduced in correlation proposed by Sanz et Al [7. The Wallim correlation [7. The concept of design presented in the last version of Eurocode 3 part 2 January [7. K I,eq is the equivalent stress intensity factor of a structural component taking into account the design stress Vd and the design value of the defect ad K I , eq M k.
At temperatures above the transition, the Eurocode Kmat curve overestimates the fracture toughness of the material. The values obtained have been plotted versus the temperature T-T on the same diagram figure 7. In this case, the correlation is relatively poor. Comparison with fracture toughness derived from Charpy energy.
Atthe same time, the introduction of linear elastic fracture mechanics led to the use of pre-cracked Charpy specimens. However the introduction of this kind of specimen has never supplanted the use of the traditional Charpy V notch. The initial U notch recommended by Charpy has been progressively discarded. This approach requires that the critical load is suitably evaluated and the free oscillations experienced by the specimen during impact are understood. The recent introduction of Notch Fracture Mechanics has provided a promising tool for this problem.
In this approach, there is no fundamental difference between fracture emanating from a notches or a crack.
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However, the effective fracture stress is sensitive to stress field conditions and particularly to relative stress gradient. It is interesting to notice that Charpy implicitly suggested that the specific work for fracture is a measure of fracture toughness. He termed this Resilience. This parameter is now realised as an energy fracture criterion in the form of JIC.
Journal of Applied Mechanics. G and Montariol. G J, Krassiko. The fatigue strength reduction factor varies with the number of cycles to failure as can be seen in figure 8. For low cycle fatigue its value is near unity; for high cycle fatigue, it increases asymptotically to the elastic stress concentration factor value. In the literature, numerous empirical formulae are available, allowing the fatigue strength reduction factor at endurance limit only to be derived from the elastic stress concentration factor. Several comments can be made which undermine this assumption: i the notch effect depends on the loading mode and specimen geometry for the same maximum stress or stress range values; ii the position of maximum elasto-plastic stress is not at the notch tip and not connected with the point of fracture or fatigue initiation ; iii the use of the elastic stress concentration coefficient kt results in over-conservative predictions vis a vis experimental results especially for low notch tip radii.
However the use of the fatigue strength concentration factor is widely used and numerous solutions for various notch geometry and loading mode are available. In general, they refer to particular value of kf defined at endurance limit: kf VD , V D, n 8. The relationship between the elastic stress concentration factor and the fatigue strength reduction factor kt may be classified into three categories depending on the assumptions used: a models using empirical relationships and based on the concept of an average stress over a given distance, b models based on the value of stress giving rise to a non-propagating short crack initiated from a notch, c models based on the localisation of fatigue damage in an effective volume.
In more recent models the influence of loading mode and specimen geometry are introduced by way of the relative stress gradient F. These different models are presented in table 8. They are numerous and give different results, especially for low notch radii and need one or more empirical constants. They incorporate one or two empirical constants depending on the mechanical characteristics of the material. Therefore it is advisable not to use the constant values determined for these steels. The effective stress range at endurance limit may be expressed by the following equation : X ef 1 ' V ef , D 8.
The ratio between these two magnitudes introduces the material plastic damage and kt the geometry effect. Introducing into the above formula the equations of Basquin and Manson: c b ' H pl H ' f. Substituting in the equation 8. Author Kuhn et Hardraht [8. U0 and Xc are small distances; A et B constants ; D slope. Table 8. The accumulation of fatigue damage depends on the stress field intensity in the damaged zone. Referring to this concept, Sheppard [8. Yao et al. An additional simplification may be provided if we consider that the fatigue process volume is cylindrical with a diameter equal to Xef.
The assumption made in this approach is that the fatigue failure needs a physical volume to occur. Its extent from the notch tip is called the effective distance. This approach was first introduced for an elastic stress distribution [ 8.
Since then, it has been subject to further development powing the increasing knowledge concerning scale and loading mode effects on fatigue. In [8. The effective stress according to the volumetric approach was first determined as the stress value corresponding to the stress distribution for the effective distance. It is now defined as the average of the weighted stresses in this volume. This weighted stress depends on the relative stress gradient in order to account for the loading mode and scale effects in fatigue.
The fatigue life duration determined by traditional methods requires the value of the fatigue strength reduction factor, which is a function of the stress concentration factor kt Figure 8. Based on the traditional approach, these geometries should have the same fatigue life duration. In fact, the fatigue life duration of each specimen type is different from the other. It should be noted that the stress distribution near the notch tip is different and U2 consequently the first derivative of the stress distribution function is also different.
This fact can explain the difference in fatigue strengths of the two specimens. The relative stress gradient is defined as the ratio between the first derivative of the stress distribution function and the value of the stress at the point. This assumption is based on an analogy with the notch plastic zone which is practically cylindrical. The fatigue process volume is the high stressed region ahead of the crack or notch tip. If we considered a typical elastoplastic stress distribution at a notch tip figure 8.
Two particular points are visible on the distribution: i the point where the stress is at maximum; ii the limit between region II and region III. The choice of the value of the effective distance was made by trial and error method. By definition, the effective stress leads to the same lifetime as that given when testing smooth specimens of the same material.
Verifications on different materials and specimen geometries [8. The weight function is assumed to be at the notch tip and at the point of maximum stress equal to unity. For this reason, the choice of the weight function has the following form: 8. The effective stress Vef is defined as the average value of the weighted stress in the fatigue process volume: 1.
In tension or bending, this stress is conventionally denoted Vyy. We can write in bi-dimensional case: 1. Two types of key-seat have been tested: Sled Runner and semi-circular end.
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The geometry of these two types of key-seat is presented in figure 8. The experimental results of these tests is presented in figure. From these results we can see that the introduction of a key-seat causes the reduction in the fatigue lifetime. However for this sled runner key-seat, the radius R has a very important role in fatigue life duration. The effective stress may then been calculated according the procedure describe in 8. The volumetric method is able to predict the fatigue life duration for any notched structure.
Effective stress obtained by volumetric method reported on fatigue reference curve gives directly the fatigue life duration Figure 8. Safety factor can be introduced to get admissible stress which is the ratio of the effective stress by safety factor a value of 2 can be used. It is important to mention here that, it is not possible to apply other fatigue models to specimens with key-seats. This is because almost all of these models refer to the notch radius, which cannot be determined in the case of the specimens with key-seats.
The advantages of the volumetric method include the possibility of predicting fatigue life duration for any loading case using notched geometry structures, the absence of empirical and doubtful coefficients used in traditional methods and the opportunity to obtain fast and economical results using a Finite Element method. The geometry of these specimens is presented in Figure 8. This figure shows loading mode has a strong influence on smooth specimen fatigue life duration for the same stress amplitude. The results of this analysis are presented in table 8.
Stress range MPa Rorsion Tension-compression Rotating bending Von Mises stress for torsion Number of cycles for fatigue life duration Figure 8. In order to see the influence of the loading mode, variation of the fatigue strength reduction factor versus the number of cycles to failure has been plotted. The fatigue strength reduction factor is more important in bending and increases more rapidly with the number of cycles than for torsion.
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The principle of the volumetric method has been applied to tests performed in tensioncompression figure 8. We can note the good agreement between the prediction of the volumetric method and the fatigue reference curve smooth specimens curve. Several approaches are possible and the fatigue criterion can be a combination of the second stress invariant J2 and the hydrostatic pressure Table 8.
Authors Sines [8. In formulas 8. The basic mechanism for fatigue crack initiation is the maximum shearing stress which occurs on the most favourably oriented crystallographic plane. The maximum shear stress and the plane of maximum shear stress have to be determined in order to apply this criterion. The negative influence of hydrostatic pressure increases linearly. The best value for the critical layer is determined by a trial and error method and its value is of the order of the material characteristic, e.
The ratio between the shear and normal gross stress is 2. The geometry of the smooth and notched specimens used is described in figure 8. Shear stress range MPa 0. Using Finite Element Method and assuming elastoplastic material behaviour, maximum principal stress V1, maximum shear stress W, hydrostatic pressure Vh and relative stress gradient F were computed. An example of such computations is given in figure 8. From figure 8. In order to take into account the influence of the stress gradient, the effective maximum shear stress and hydrostatic pressure are computed; These quantities are defined as the average values of maximum shear stress and hydrostatic pressure over the effective distance which is defined as the distance of minimum relative stress gradient.
We can see that the size parameters of such elliptical representation is not constant but depends on fatigue life duration Figure 8. This can be seen in figure 8. A typical problem is that of low cycle fatigue crack initiation at stress concentration at a nozzle in a pressure vessel. Notch effects in low cycle fatigue have received little attention according to the literature. The actual trend is to treat this problem using an energy approach for low cycle fatigue.
The steel studied is a 35 NCD 16 steel French standard with the following mechanical properties: Experimental results are presented using the traditional Coffin [8. The values of these parameters are presented in Table 8. Using the cyclic stress-strain range relationship: 'V 1 n' 8. This computation gives an effective strain energy density range which is assumed constant in the effective volume. A a smooth - 0. This value is used to define the fatigue strain energy density concentration factor. Experimental results are presented in figure 8. The strain energy density at the notch tip has been computed using a finite element method.
Calculations have been made using the cyclic stress strain curve. The strain energy density range is plotted versus the distance from the notch tip in a bi-logarithmic graph. A typical example of such a curve is given in figure 8. The effective strain energy density range is the average value of the distribution over the distance Xef. The accuracy of this definition can be checked by comparison of the number of cycles to failure obtained experimentally and those obtained using the value of the effective strain energy density range.
Such a comparison indicates that this definition gives relatively satisfactory results figure 8. Variation of the fatigue strength reduction factor defined from strain energy density is plotted versus the number of cycles for two notch radii and presented in figure 8. We can see that the fatigue strain energy density factor is practically constant with the number of cycles, i.
The Hot Spot approach requires the use of fatigue strength reduction factor which may be obtained from several empirical relationships which can include two or more material constants and the relative stress gradient. The volumetric method uses the assumption that the fatigue process requires a physical volume on which an effective stress range operates.
This effective stress range is an average value of the weighted stress distribution inside the fatigue process volume. This effective stress range is sensitive to hydrostatic pressure and consequently to loading mode. Notch effects still exist in low cycle fatigue and can be described using the fatigue strain energy density factor. Sines and J. Waissman eds. Chapman and Hall, London. M, Dissertation Techn. Hoschule, Stuttgart. W P and Klesnil. L, Chehimi. C,and Pluvinage G. Y and Wang. International Journal Fatigue, 15, p Niu, L.
Engineering Fracture Mechanics ,Vol. M and Pluvinage. VDI-Z, 97, 8. K, F and Skalli. ASTM, Vol. S, Journal of Materials, Vol. D and Tuller. The existence of a stress concentration is a deciding factor in the behaviour of a structure under fatigue loading. Nikei et al [9. The results presented in figure 9. The coefficient of the elastic stress concentration kt of a welded joint depends on geometry and loading. Some analytic formula can be found in the literature and allow the user to calculate the stress concentration factor; loading effects on the value of kt.
Different formulae for the stress concentration factor are proposed within the literature. Some simplified formulae do not take into account the entire range of parameters. These studies have shown that an increase in the notch radius or a decrease in the connecting angle reduces the stress concentration effect considerably. Fatigue strength reduction factor kf 0. Niu and Glinka [9. In figure 9. The major difference between the analytical formula and the finite-element calculations is that concerning the influence of the connecting radius. The influence of the stress concentration in an end to end joint compared with a plan joint is evident in figure 9.
We see that the fatigue strength of the planed joint is quite high compared with that of the rough joint. However the product of the gross stress and the stress concentration factor does not allow one to encounter the curve of the planed joint; the latter may actually be used as a reference supposed incorporating the effect of residual stresses.
This is especially true for low radii at the notch tip. For this reason, we use another empirical coefficient in fatigue to characterise the notch effect. It is symbolised kf and denoted fatigue strength reduction factor. It is determined during similar tests and defined by: kf VD V D, n 9. Some equations encountered in the literature allow one to determine the coefficient of stress concentration in fatigue, according to kt.
The fatigue notch factor increases with increase in the number of cycles and tends asymptotically towards the value of kt for a large number of cycles, as can be seen in figure 9. The relationship between the elastic stress concentration factor and the fatigue notch factor kt may be classified in three categories depending on the assumption used: that is, models using empirical relations and based on the concept of mean stress for a certain distance, and models based on non- propagating stress for a short crack initiated from a notch.
In practice design engineers tend to adopt simplified methods for evaluating fatigue life. These methods often have poor accuracy and there is a need for the use of computational methods which, although sophisticated, are more accurate and reliable. This derives directly used from the deterministic, general science approach ofthe19th century.
We have a dedicated site for Germany. A vast majority of failures emanate from stress concentrators such as geometrical discontinuities. The role of stress concentration was first highlighted by Inglis who gives a stress concentration factor for an elliptical defect, and later by Neuber With the progress in computing, it is now possible to compute the real stress distribution at a notch tip.
This distribution is not simple, but looks like pseudo-singularity as in principle the power dependence with distance remains. This distribution is governed by the notch stress intensity factor which is the basis of Notch Fracture Mechanics. Notch Fracture Mechanics is associated with the volumetric method which postulates that fracture requires a physical volume. Since fatigue also needs a physical process volume, Notch Fracture Mechanics can easily be extended to fatigue emanating from a stress concentration.
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