WANG Fang,CUI Wei-cheng,HUANG Xiao-ping

(1 China Ship Scientific Research Center,Wuxi 214082,China;2 State Key Lab of Ocean Engineering,Shanghai Jiao Tong University,Shanghai 200030,China)

Evaluation of Surface Crack Shape Evolution Using the Improved Fatigue Crack Growth Rate Model

WANG Fang1,CUI Wei-cheng1,HUANG Xiao-ping2

(1 China Ship Scientific Research Center,Wuxi 214082,China;2 State Key Lab of Ocean Engineering,Shanghai Jiao Tong University,Shanghai 200030,China)

Many investigators have carried out the study on crack shape evolution law but some deficiencies are observed from the predicted results obtained using the existed methods.In this paper,an improved fatigue crack growth model was proposed.The nonlinear effect of the material,the closure behavior of the crack front and its distribution along the crack front are sufficiently considered in this model,which will be used to evaluate the surface crack shape evolution.Moreover,by introducing the concept of equivalent thickness at each point and the assumption of crack propagation at normal direction,the effect of different stress state at each point on crack shape evolution law is reasonably considered and the surface retardation due to boundary effect is successfully predicted.The evaluation precision of crack shape evolution law from this model is highly improved compared with the test data and the predicted curve from Paris law.

the improved fatigue crack growth rate model;surface crack;crack shape evolution

Biography:WANG Fang(1979-),female,Ph.D.senior engineer of CSSRC.

1 Introduction

Different types of defects always existed in welded components as stiffened panels and tubular sections in ship and offshore structures.The fatigue life of the components will be affected by the majority of initial defects largely.It has been widely accepted that those initial defects can be treated as two-or three-dimensional surface cracks when investigating the fatigue life of the structures.The propagation lives of surface cracks are significant and should not be neglected during investigating the whole life of the structure.That indicates the deficiencies of the traditional fatigue life assessment method based on stress-life curves(S-N curves)which ignores the whole failure process of the welded components[1].Then the approach based on fracture mechanics has been highlighted in fatigue life of welded structures.And evaluating surface crack shape evolution law is the basis of precise fatigue life prediction[2-3].

Initial surface cracks are usually treaded as semi-elliptical crack in shape and assumed to keep the simi-elliptical shape but with different shape ratio during propagation.The changes in defect aspect ratio depend principally upon initial configuration,relative crack depth and loading condition,but also depend weakly upon stress ratio,loading frequency,growth rate exponent,mean stress and the crack tip stress conditions which alter with changes in crack shape[4].Many investigators have carried out the study on predicting crack shape evolution law as the basis of fatigue life assessment[5-8].

Paris-law is always applied to estimate crack shape evolution under cyclic loading by evaluating the growth rate of the surface and deepest points[2-3].And it has been observed that the coefficients in the Paris law for the two critical points should not be equal.Some researchers deal with the problem by considering different coefficients for each point along the crack front.For example,Ref.[8]derived the coefficients for initial semi-circular cracks.To further simplify the problem,some other researchers pre-defined the shape ratio equation during propagation and use it in calculating the stress intensity factors.

However,by summarizing precious research works on the problem,several deficiencies could be noticed:(1)Almost all the research works are based on Paris law.Though many results have demonstrated the reasonability by applying Paris law but it is confined to small shape ratio a/t.Typically,when the shape ratio a/t is larger than 0.6,the large predicted error comparing to test will be found.It is because that linear elastic equation for stress intensity factor range is used in Paris law.However,when the deepest point of the crack approaches to the back of the plate,there will be an apparent strip yielding in the vicinity of the deepest point.Then the ignorance of nonlinear effect results in the deficiency.(2)The‘two-point approach’is always adopted.As introduced above,to simplify the problem,the crack profile is supposed to keep semi-elliptical.The deepest point and surface point will determine the shape of the crack while the effects of other points along the crack front are ignored.(3)The stress state effect is neglected.Qualitatively,the crack profile should be calculated by considering the stress state function along the crack front from a state of high traxiality of stress at the deepest point to a biaxial stress state at the surface point.Some researchers simplified the problem by taking for a plan strain state and a plan stress state respectively at the deepest point and the surface point.But it is not the reality and Paris law is difficult to describe the stress state effect.(4)The boundary effect at the surface crack will not be reflected by the above three assumptions.Crack growth retardation phenomenon is observed during test in the vicinity of surface point along the crack front,which is the boundary effect.

In the previous study,an improved fatigue crack growth model is proposed based on the work of McEvily et al[9]and the capability of the model has been demonstrated[10-11].It will be used in present paper to evaluate the surface crack shape evolution.The nonlinear effect in the vicinity of crack front and crack closure effect will be sufficiently considered.By introducing the concept of equivalent thickness at each point,the different stress state effect along crack front is taken into account.Surface retardation due to boundary effect can be successfully predicted.The evaluation precision of crack shape evolution law from the current model is highly improved compared with the test data and the predicted curve from Paris law.

2 Problem description

Stress concentration magnification factor of semi-elliptical cracks at the toe of fillet welded joints can be expressed as follows:

where,Kplateis the stress intensity factor in the vicinity of crack tips in a plate and Mkis a magnification factor for welded joints.Bowness and Lee[12]recommended an empirical expression of Mkof semi-elliptical crack.And the calculation methods for stress intensity factors of surface cracks are introduced in Ref.[13].

As an example,the surface crack in a T-joint is illustrated in Fig.1.And the symbols for joint and crack geometry parameters are labeled in the figure and it is supposed that the joint is subjected to tension stress σtenand bending stress σben.

Newman-Raju[5]proposed the empirical stress-intensity factor equation for the surface crack,

3 Basic expression of the improved crack growth rate model

A good crack growth rate relation should be established for accurate prediction of fatigue life based on fatigue crack propagation theory.As one of the preferable fatigue crack growth models,the modified constitutive relation developed by McEvily and his co-workers[9]is able to explain different fatigue phenomena.Recently,we further improved McEvily’s model for the purpose of explaining more fatigue phenomena[12-13].

The improved crack growth model can be described as:

where A is a material-and environmentally-sensitive constant of dimensions(MPa)-2;m is a constant representing the slope of the corresponding fatigue crack growth rate curve;n is the index indicating the unstable fracture;KCis the plane stress fracture toughness of the material;KICis the plan stain fracture toughness of the material;KCfis the fracture toughness of the material under fatigue loading;reis an empirical material constant of the inherent flaw length of the order of 1μm;a is the modified crack length which is equal to replus the actual crack length;σmaxis the maximum applied stress,σminis the minimum applied stress;Y()a is a geometrical factor;Y re()is a geometrical factor when a is equal to re;R is the stress ratio(=σmin/σmax);ΔKeffis the effective range of the stress intensity factor;ΔKeffthis the effective range of the stress intensity factor at the threshold level;Kopis the stress intensity factor at the opening level;α′is the crack tip stress/strain constraint ratio,which is 1 for the plane stress state and 1/1-2( )ν for the plane strain state;σuis the ultimate strength of the material;σYis the yield strength of the material;n′is the hardening exponent of the power-law material;ν is the Poisson’s ratio.It can be noticed that Kmax,fopand KCfare all the functions of crack length a.The effect of n is significant only in the unstable propagation region;a constant value of 6 is recommended for a quick and simple engineering analysis.

Combining Eqs.(2)and(3),the value of Y(a) for surface cracks in a plain plate can be expressed as:

4 Equivalent thickness

Eqs.(3)-(5)are mostly used in the cases of one or two dimensional problems.In order to extend the concepts in the model into surface cracks,the concepts of ‘thickness’ at each point along crack front should be re-defined.The concept of equivalent thickness[3]will be applied into the analysis here.Fig.2 shows the schematic illustration for calculation of equivalent thickness at each point.If there is a semi-elliptical crack Ai-O-Ciin a plate,then the general equation of the semi-elliptical shape is

The equation of the tangent line can be written as,

The equation of the normal line can be written as:

then the coordinates of Pi+1,jcan be obtained,

The propagated crack shape can be depicted by calculating the coordinates of enough points along the new crack front.Typically,the shape of the new crack front will not keep semi-elliptical.Then the function of stress intensity factor coefficient Y should be calculated by finite element analysis at each step.To save calculation time,it is assumed that Y for the new front is still calculated according to that for semi-elliptical shape but the ellipse should be obtained by fitting the calculated points.The parameters of new semi-elliptical crack will be used for next step of calculation then the Eqs.(1)and(2)can be used reasonably.Accordingly,the crack shape ratio evolution can be obtained through ‘cycle-by-cycle’ calculation.

4 Model validation

Ref.[14]conducted tests on series of 7075-T6 plates with semi-elliptical crack and Ref.[3]gave predicted shape ratios using Paris law.They are depicted together with the current predicted results in Fig.3.When the shape ratio a/t is small,the results from Paris law and the present model are similar and agree well with the test data.But when the shape ratio becomes larger with crack propagation,the results from Paris law show discrepancy comparing to the test.On the contrary,the present model can give satisfactory evaluation results by considering nonlinear effect when calculating stress intensity factors.It should be noted that the values of a and c for each step in Fig.3 are the fitted one.Fig.4 gives the current predicted crack profile of the semi-elliptical crack with a0/t=0.2,a0/c0=1.0 in Ref.[14].Apparent growth retardation can be observed in the predicted fronts.

Similar conclusion can be obtained by comparing the current predicted results with the test data on 7075-T6(51)[15]and the predicted results[3]based on Paris law depicted in Figs.5 and 6.It can be seen that the present model can give more precise results comparing to Paris law.But different shape evolution tendency is observed comparing to Fig.3.The difference results from the values of initial crack shape ratios.Larger initial a/c will result in ascending trend with crack propagation while opposite trend is for small initial a/c.

5 Conclusions

In the present study,the improved fatigue crack growth model is used to evaluate the surface crack shape evolution and some useful conclusions can be drawn as follows:

(1)The strip-yield effect in the vicinity of crack front and crack closure effect could be sufficiently considered by using the nonlinear equation of stress intensity factor in present model;

(2)By introducing the concept of equivalent thickness at each point,the different stress state effect along crack front can be taken into account;

(3)Surface retardation due to boundary effect can be successfully predicted;

(4)The evaluation precision of crack shape evolution law from the current model is highly improved compared with the test data and the predicted curve from Paris law.

Acknowledgements

This study was supported by the Innovative Scholars Support Program of Jiangsu Province,Project No.BK2008004,2008-2010.

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基于改進(jìn)的統(tǒng)一疲勞裂紋擴(kuò)展速率模型的表面裂紋擴(kuò)展規(guī)律預(yù)報(bào)

王 芳1，崔維成1，黃小平2

（1中國船舶科學(xué)研究中心，江蘇 無錫 214082；2上海交通大學(xué) 海洋工程國家重點(diǎn)實(shí)驗(yàn)室，上海200030）

很多學(xué)者對(duì)表面裂紋形狀變化規(guī)律進(jìn)行了研究，但是理論上仍存在較大缺陷，因此現(xiàn)有方法預(yù)報(bào)結(jié)果的準(zhǔn)確性有待考察。文章作者們提出了一個(gè)改進(jìn)的統(tǒng)一疲勞裂紋擴(kuò)展速率模型，本模型合理考慮了材料的非線性效應(yīng)和裂紋前緣的三維約束效應(yīng)及三維約束大小在前緣各點(diǎn)的分布函數(shù)。通過引入等效厚度的概念及法線方向擴(kuò)展的假定較好地考慮了裂紋前緣各點(diǎn)對(duì)擴(kuò)展之后形狀比變化規(guī)律的影響，預(yù)報(bào)得到的裂紋前緣形狀能夠觀察到明顯的邊界點(diǎn)擴(kuò)展滯后現(xiàn)象，同時(shí)本模型預(yù)報(bào)結(jié)果與試驗(yàn)結(jié)果及傳統(tǒng)模型預(yù)報(bào)結(jié)果進(jìn)行了比較，證明本模型提高了表面裂紋擴(kuò)展規(guī)律預(yù)報(bào)的精度。

改進(jìn)的疲勞裂紋擴(kuò)展率模型；表面裂紋；裂紋形狀變化

U661.4

A

王 芳（1979-），女，博士，中國船舶科學(xué)研究中心高級(jí)工程師；

黃小平（1964-），男，博士，上海交通大學(xué)副教授，碩士生導(dǎo)師。

U661.4

A

1007-7294（2011）06-0660-09

date：2011-04-07

Supported by the Innovative Scholars Support Program of Jiangsu Province(Project No.BK2008004,2008-2010)

崔維成（1963-），男，博士，中國船舶科學(xué)研究中心研究員，博士生導(dǎo)師；