FATIGUE PROPAGATION Ð SOME NOTES

Transcription

1 FATIGUE PROPAGATION Ð SOME NOTES Anders Ekberg Dep. of Solid Mechanics, Chalmers Univesity of Technology INTRODUCTION AND PURPOSE This report is part of the graduate course FATIGUE AND FRACTURE MECHANICS. This is a literature course during summer of -97. The contents of the course should be equivalent to 6 weeks of full time work. The literature in the course is chosen in order to put the emphasis on multiaxial fatigue and fracture mechanics. In this fatigue propagation part, several papers and books on fatigue propagation, mainly with a fracture mechanics approach, are used. Note that the comments reflect my thoughts on the paper. I may well have misunderstood some of the contents etc. Also, I have included my own associations and comments in the notes below (not always explicitly stated). So, read the following with a suspicious mind. This paper is produced using FrameMaker 5.. on a PowerMac. The file has then been saved as a postscript -file and moved to a HP-workstation using Fetch Finally, it has been converted to a pdf-file using Adobe Distiller 3.0. The document uses the fonts New Century Schoolbook, Times and Symbol.. BASIC FATIGUE CRACK PROPAGATION The following notes are mainly from Ch.0 in [].

2 2 (0) BASIC ANALYSIS OF CRACK GROWTH RATE The area of fracture mechanics analysis of cracks is fairly well established as long as restrictions are made to small scale plasticity, constant amplitude, fairly large crack size and uniaxial loading (my comment). The fracture mechanics approach is based on the assumption that the crack tip conditions are uniquely defined by a single loading parameter, e.g. the stress intensity factor. In the case of propagation of cracks, the range of the stress cycle is used. Also, the mid value of stress will have an influence (for instance du to the effect of hydrostatic stress acting on the crack), and the stress history may have an interest. Under these assumptions, the crack propagation can be characterized by the relationship = f ( D K, R, H) dn where is the crack growth per cycle, D K º ( K, and H is dn max Ð K min ) R K min º K max a history term, which may have influence if K max or K min varies during the load history. Note that () violates the assumption that the conditions at the crack tip is uniquely defined by single loading parameter (if H=0). I.e. two configurations with identical K max and K min values may experience different crack growth rates. This will complicate the analysis considerably, which is the reason why the assumption H=0 often is made. However, this will lead to approximate expressions in the case of variable amplitude loading. One equation of the type shown in, is Paris law, (), which can be written as = C D K m dn where C and m are material parameters. m varies from m»2 to 7. Note that the physical unit of C depends on the magnitude of m. Note that this equation does not include any influence of previous loading history (i.e. H=0). Also, there is no influence of the R-ratio (which for example has the consequence that the influence of a superposed hydrostatic stress is neglected). If the crack growth is studied experimentally, a plot of ( ) ( dn) vs. DK will have a shape as shown in FIG.. From this plot, it can be concluded that Paris law should mainly be used to model crack growth in region II. There are several other models that are aimed at model all (or some parts of) the ( ) ( dn) - DK relationship. The fatigue life can be directly estimated by integrating Paris law. However, this procedure presumes that region II includes the dominating part () (2)

3 3 (0) of the fatigue life. Two interesting features of the curve in FIG., are the existence of a crack growth treshold K th and the existence of a critical value K c. If the stress intensity range do not exceed K th, there will be no propagation of existing cracks. At the other extreme, K c, K max will approach the fracture toughness and the material will fail. Paris law presumes elastic conditions. In order to take plastic deformation into account, the J-integral can be applied, i.e. the crack growth rate can be approximated by the expression = C D J m dn (3) log dn I II III logdk DK th K c Fig. Fatigue crack growth behavior in metals as described by the crack growth rate (/dn) vs. the width of the stress intensity factor during one loading cycle (DK). CRACK GROWTH TRESHOLD Elber, in 970, discovered that crack closure exist in cyclic loading, even for loads that are greater than zero. This crack closure will decrease the fatigue life by reducing the effective stress intensity range. In FIG. 2, the stress intensity during some load cycles is plotted. If the crack does not close, the stress intensity range will be DK º K max Ð K min, but since the crack closes at K=K op, the effective stress intensity range will be proposed a modified crack growth law m = C D K dn eff DK eff º K max Ð K op. Elber thus (4) There are several mechanisms that can give rise to crack closure. One is

4 4 (0) plasticity induced crack closure, where compressive residual stresses, stemming from plastic deformation due to high tensile load in previous load cycles, closes the crack. K max DK DK eff K op K min Closed crack Fig. 2 Stress intensity range in cyclic loading and the influence of crack closure on the effective stress intensity range. The crack closure concept can be used to explain crack propagation thresholds in a qualitative way as follows. Assume a modified Paris law, i.e. m = C D K dn eff = Cæ DK eff D K è DK ø ö m (5) Cææ K max Ð K op ö DK ö m K C max K = = ææ Ð op ö DK èè DK ø ø èè Ð DKø ø K max K min K C op ç ç æ Ð ö ç ç æ DK ö m K = = Cææ Ð op ö DK ç ç K Ð min DK èè Ð R DKø ø èè ø ø K max Where, the definitions R=K min /K max has been used.treshold is obtained when DK eff =0, which is equivalent to setting the inner parenthesis in (5) equal to zero, i.e. ö m ö m K op Ð æ ö = 0 DK Ð R è ø th = K op ( Ð R) DK th (6) At this stress intensity level, the crack will not open at all during a load cycle

5 and the crack will thus not propagate (note that K op has been assumed constant in (6)). The other extreme occurs when the inner parenthesis equals unity, i.e. 5 (0) K op Ð æ ö = DK Ð R èdkø K op = = K æ op --- Ð ö èr ø Ð Ð R (7) In this case no crack closure will occur in the load cycle and DK eff =DK. The relation (5) is plotted in FIG. 3. Note that the treshold limit is dependent on the R-ratio in such a manner that a superposed static tension (i.e. a positive R) gives a lower crack growth treshold and a superposed static compression (i.e. a negative R) gives a higher treshold. In other words, a superposed static compression reduces the risk of fatigue crack growth. Fig. 3 Simulation of crack growth treshold using the crack closure concept. In reality, K op is not a material parameter (constant), but depends also on the type of loading, as well as on history effects. The actual mechanisms of crack growth threshold is not entirely due to crack closure, but also depends on the microstructure. Comment: On top of this mechanical treshold, there is also a micro structural treshold, where the small crack (from less than one to some grain diameters in size) can be arrested due to interaction with grain bounries etc. (see McDowell). There exists a number of limitations to the concept of DK eff. Some of which are The equations for determining K op are mostly empirical and only reliable to a particular loading regime. Fatigue ta can not give an unanimous estimation of K eff (since the estimation ck eff, where c is a constant, will also produce a correla-

6 6 (0) tion of the fatigue ta. History effects are almost never taken into consideration when K eff is applied. In reality, K op depends on the previous history. VARIABLE AMPLITUDE LOADING As described above, there is a history effect involved in the propagation of cracks. The mechanism behind this is similar to the mechanism of plasticity induced crack closure, as described on page 3. Consider an overload applied to a crack. The overload will cause plastic flow in an area ahead of the crack tip. Because of redistribution of stresses in unloading, there will be a compressed zone just ahead of the crack tip. If the overload is high enough, there can even be a compressive yield zone ahead of the crack, see FIG. 4. This will lead to a retartion in the crack growth rate, since the compressive zone will both reduce the effective stress intensity factor (due to crack closure) and also reduce the tensile stress ahead of the crack tip in the following load cycles. Loading Unloading s Y crack Plastic zone (tension) crack s Y Plastic zone (compression) Fig. 4 The influence of a compressive overload. One model for analyzing this behavior is the Wheeler model, which compares a current plastic zone (which is the plastic zone ahead of the crack tip due to the current load cycle) to the tensile plastic zone due to the overload. Retartion effects are assumed to take place only when the current plastic zone is within the overloading plastic zone. The retartion factor is defined by Wheeler through the relations æ ö = F èdnø R F æ D a+ rd c, where (8) R dn R ö g = è rd o ø g is a fitting parameter that depends on material properties and stress

7 7 (0) spectrum. The diameters of the plastic zones are given by rd o æ K o ö 2 = d bpè ø and c æ K max ö 2 = bpè ø (9) s Y s Y K o is the stress intensity factor at overload, b is 2 for plane stress and 6 for plane strain. Da is defined in FIG. 5. d o crack Da d c Fig. 5 Plastic zone size due to overload, and plastic zone due to the current loading. History effects can be rather pronounced in variable amplitude loading if the rate of change of the stress intensity (dk/) is low or the variable amplitude loading is not confined to single overloads. In the latter case, there are of course marked history effects as long as the crack is propagating within the plastic zone stemming from the overload. Luckily, simple fatigue laws, that do not take retartion effects into account, are usually conservative. However, the most accurate means of quantifying fatigue life under variable amplitude loading is toy a cycle.-by-cycle integration of a crack propagation law that takes retartion effects into account. SHORT AND LONG CRACKS Short cracks can be divided into two categories, microstructurally short and mechanically short cracks. The microstructurally short cracks have a size up to some grain diameters. They are interacting closely with the microstructure. This leads to severe problems in determining the crack growth behavior since, at this scale, the material can not be considered homogenous. This leads to a oscillatory growth rate behavior as the crack propagates through grains and bounries. More information is given in [2]. Cracks shorter than some mm:s are often considered mechanically short. They are long enough for continuum theory to be applicable (i.e. the surrounding material to be considered homogenous), but the cracks do not behave as long cracks. They typically grow faster than long cracks which experience similar

8 8 (0) DK-levels. This is due to the plastic zone size which, for small cracks, is significant compared to the crack size (compare with the Irwin plastic zone correction). Also, the crack closure load is higher for small cracks (especially for low DK magnitudes), which leads to a higher DK eff -value than for corresponding long cracks. Comment: The latter can be motivated by the fact that a short crack has a larger portion of uncracked materia in its surrounding that unloads the crack faces. For small magnitudes of DK (region I in FIG. ), the crack propagation is, as mentioned above, difficult to predict. It is very dependent on microstructure and flow properties of the material and the growth may experience an arrest. The crack growth rate is sensitive to the size of the grains in this region. However, it is rather difficult to predict in which manner, since finer grains will lead to a closer spacing of grain bounries, which the crack has to break through. Also, the yield stress will probably increase. On the other hand, the roughness of the crack will decrease with decreased grain size an this will lead to less roughness induced crack closure and higher DK eff magnitudes. For larger magnitudes of DK, the crack growth rate will be governed by a power law (such as Paris law). In this region (region II in FIG. ), the crack growth rate is fairly insensitive to the microstructure (however, the constants m and C are different for different materials). If the stress intensity ratio is increased even further (region III in FIG. ), the crack growth rate will accelerate. Finally fracture will occur. The behavior of this fracture is rather sensitive to the microstructure and flow properties of the material. DESIGNING AGAINST FATIGUE Ð DAMAGE TOLERANCE METHODOLOGY The steps in this design process is as follows. For a certain component, subjected to a certain loading, a critical flaw size, a c, is estimated. If a flaw of this size exists in the component, it will lead to fracture. A safety factor is applied to the critical flaw size in order to define a tolerable flaw size a t. Preferably, this size should be chosen such that it would take a reasonable amount of time to propagate the crack from a t to a c. Now, the component is subjected to inspections and an initial flaw size a 0 is set as the largest flaw size this inspection could have missed (note that this value should be much larger than the detectability limit, which is the smallest crack that might be detected ). The time to grow the crack from a 0 to a t is computed. The time in-between the inspections should be lesser than this time. Preferably, one inspection should be able to miss the crack without a failure of the

9 9 (0) component (i.e. the time should be at least twice the growth time). If a crack is detected, there are two possibilities. The first is to repair the component (or take it out of service). The second possibility is to make a new inspection at a time which is smaller than the computed time for the crack to grow from the detected length to a t. This approach can be combined with an analysis to ensure that the failure of a component will not lead to catastrophic failure of the structure. REFERENCES. Andersson T.L., FRACTURE MECHANICS, CRC-Press, 2. McDowell, D. L., BASIC ISSUES IN THE MECHANICS OF HIGH CYCLE METAL FATIGUE, Int. J of Fract., vol.80, pp.03-45, 996 APPENDIX MATLAB code used to plot diagrams in the text above. CRACKCLOSE.M % MATLAB-program to study the influence of crack % closure on crack propagation treshold. % The equation plotted is (0.7) in Andersson - % ÒFracture MechanicsÓ % I.e. /dn/c=( (/(-R) - Kop/dK )*dk )^m % -> /dn/(c*kop^m)=/(-r)*(dk^m/kop^m) -dk^(m-)/kop^(m-) % Assignment of constants R0=0; R2=.2; R4=.4; R6=.6; R_2=-.2; R_4=-.4; R_6=-.6; m=3; % Use dk/kop as parameter dkkop=0:0.05:0;

10 0 (0) % Calculate results for different R-ratios % Use the expression above to calculate the nondimensional % crack growth rate dn0=./(-r0).*dkkop.^m-dkkop.^(m-); dn2=./(-r2).*dkkop.^m-dkkop.^(m-); dn4=./(-r4).*dkkop.^m-dkkop.^(m-); dn6=./(-r6).*dkkop.^m-dkkop.^(m-); dn_2=./(-r_2).*dkkop.^m-dkkop.^(m-); dn_4=./(-r_4).*dkkop.^m-dkkop.^(m-); dn_6=./(-r_6).*dkkop.^m-dkkop.^(m-); % Plot the results in a log-log-diagram loglog(dkkop,dn_6,õ--õ,dkkop,dn_4,õ:õ,dkkop,dn_2,õ*õ,... dkkop,dn0,õoõ,dkkop,dn2,õxõ,dkkop,dn4,õ.õ,dkkop,dn6) axis ([ ]) legend(ôr=-0.6õ,õr=-0.4õ,õr=-0.2õ,õr=0õ,õr=0.2õ,õr=0.4õ,õr=0.6õ) xlabel(ô\deltak\divkopõ) ylabel(ô\divdn \times \div(c\timeskop^m)õ)