Low-Cycle Fatigue Crack-Propagation Behavior for Ship Cracked Plate Considering Accumulative Plastic Damage

DENG Jun-lin,YANG Ping,CHEN Yuan

(1a.Key Laboratory of High Performance Ship Technology(Wuhan University of Technology),Ministry of Education,

Wuhan 430063,China;1b.School of Transportation,Wuhan University of Technology,Wuhan 430063,China;2.Department of Naval Architecture&Ocean Engineering,Qinzhou University,Guangxi 535000,China)

Low-Cycle Fatigue Crack-Propagation Behavior for Ship Cracked Plate Considering Accumulative Plastic Damage

DENG Jun-lin1b,2,YANG Ping1a,b,CHEN Yuan1b

(1a.Key Laboratory of High Performance Ship Technology(Wuhan University of Technology),Ministry of Education,

Wuhan 430063,China;1b.School of Transportation,Wuhan University of Technology,Wuhan 430063,China;2.Department of Naval Architecture&Ocean Engineering,Qinzhou University,Guangxi 535000,China)

The fracture failure of ship structure is often the coupling result of low-cycle fatigue damage and accumulative incremental plastic damage.Low-cycle fatigue crack propagation process is gradual separation results in front of crack tip material with the decreases of the stiffness and the continuous loss of the ductility of the crack tip.A prediction model of low-cycle fatigue crack growth rate based on accumulative incremental plastic damage at crack tip is presented in this study.In order to validate the model and to calibrate the model parameters,the low-cycle fatigue crack propagation experiment was carried out for notch cracked plate specimen made of Q235 steel.The effects of stress ratio and crack closure on low-cycle fatigue crack propagation were investigated by elastic-plastic finite element stress-strain analysis of a cracked component.A good comparison was found between predictions and experimental results,which show that it is important for considering the accumulative incremental plastic damage at crack tip to predict the low-cycle fatigue crack propagation life of hull cracked plate.

hull cracked plate;constant amplitude load;accumulative plastic damage;low-cycle fatigue crack propagation life;crack closure

0 Introduction

Fatigue crack growth rate plays a decisive role in the structural life design,which is a key way to evaluate the fatigue and fracture properties of structural materials.Paris[1]formula laid the foundation about the prediction of fatigue crack propagation rate of various structural materials,however,limited to linear elastic theory,it is difficult to generalize for the low-cycle fatigue crack growth driving force of the crack tip in plastic deformation.The low-cycle fatigue crack growth is mainly in the plastic zone,the stress of the crack tip is complex,and the traditional fatigue crack growth rate formula is difficult to apply.To solve these problems,domes-tic and overseas researchers recently did a lot of research,different characterization parameters is proposed to describe the low-cycle fatigue crack extension,such as CTOD[2],plastic strain range Δε[3],cyclic J integral amplitude ΔJ[4].Proposing these parameters played a great role in promoting research on low-cycle fatigue crack propagation.However,Because of the limitations of these parameters,many researchers try to establish the corresponding low-cycle fatigue crack growth rate prediction model[5-6].

Studies[7]have pointed out that the influence of accumulative incremental plastic damage should be considered in the study of the low-cycle fatigue life of ship structures.The accumulative plastic deformation which will produce additional damage and reduce the low-cycle fatigue life of the hull structure is produced at the crack tip.Therefore,the low-cycle fatigue crack growth life analysis method of considering the accumulative damage of the crack tip is more realistic to evaluate the overall fracture load bearing capacity of the ship hull plate members.

Accordingly,the accumulative plastic deformation near crack tip decreases structure material stiffness on the crack tip plastic zone,then exacerbates the ductility loss of structure material in the crack tip plastic zone and causes the fatigue damage of crack tip by large scale yielding under low-cycle fatigue loading.This study explores the accumulative incremental plastic damage mechanics at crack tip,and presents the prediction model of low-cycle fatigue crack propagation life on the crack tip monotonic plastic zone considering the effects of accumulative plastic damage at the Gauss integral point of the crack tip of notch cracked plate under the asymmetrical stress cyclic loading.

Noroozi et al[8]proposed that engineering structure materials can be regarded as a series of medium geometric elements,the geometric element size design for ρ and fatigue crack can be analyzed as a top notch geometry element of the same size.As shown in Fig.1,the fatigue crack growth is the result of the separation of the fatigue unit ahead of the notch.In the above view,in this study we will discretize the structural material near crack tip into a continuous adjacent fatigue element,so the lowcycle fatigue crack growth can be regarded as a discontinuous process of fatigue fracture failure of these discrete fatigue units.Combined with the Miner linear accumulative damage criterion,when the accumulative fatigue damage of the fatigue element reaches a critical value,the fatigue crack is broken,and the crack tip moves forward.

1 Theoretical analysis

1.1 Low-cycle fatigue damage at crack tip

Fig.1 The fatigue crack equivalent diagram

Under the action of low-cycle fatigue,the fatigue crack tip will lead to high stress,which will lead to a large range of plastic yielding in the local area of the crack tip.According to the distance from the crack tip,the local area of the crack tip can be divided into elastic and plastic regions,as shown in Fig.2.

In the plastic zone,the crack tip stress field can be divided into the monotonic plastic zone related to the maximum stress intensity factor Kmaxand the cyclic plastic zone related to the stress intensity factor ΔK.The monotonic plastic zone rmand the cyclic plastic zone rcrespectively are as follows[9]:

Fig.2 The plastic zone at the front of the crack

where σycσyc′is the cyclic yield strength;rcis the cyclic plastic zone radius of the crack tip,and the rmis the monotonic plastic zone radius;n′is the cyclic hardening exponent;for the plane stress component,κ=1.

The crack tip stress field can be divided into elastic and plastic form.According to the different regions of the crack tip,the stress and strain field distribution of the crack tip is proposed.Fig.3 shows the crack tip respectively in cyclic loading in tension and pressure load force diagram,according to the equation of Creager-Paris[9],we can draw a crack sharp linear elastic stress field.

On the symmetry axis of the fatigue crack tip,θ=0,the simplified elastic stress and strain field at crack tip could be obtained.The elastic stress field in the plane stress state at crack tip,which is represented by the radius of the plastic zone at the tip of the fatigue crack,is shown in the plane stress state:where ΔK is the stress intensity factor range;E is the modulus of elasticity;μ is Poisson’s ratio;r is the expansion of the materials to the flaw tip distance;r0is crack tip blunting radius.

Fig.3 The schematic diagram of stress at crack tip under tension/compression loading

According to the Neuber’s local stress and strain rule,the elastic strain energy of the tip of the low-cycle fatigue crack can be equal to the plastic strain in the same loading condition and the same fatigue crack area.That is:

There is a significant passivation phenomenon near crack tip.The plastic stress field near the crack tip can be obtained by interpolation method in plane stress state.According to Eq.(1),the real cyclic stress-strain field near crack tip represented by the crack tip plastic zone radius is shown as:

where σyc′is the cyclic yield strength;rcis the size of the crack tip cyclic plastic zone;rmis the crack tip plastic zone size;n′is the cyclic hardening exponent;k′is the cyclic hardening coefficient.

According to Rice’s hypothesis,the passivation radius r0is equal to half of the crack tip opening displacement CTOD,which can be expressed as follows[10]:

The crack tip plastic zone under cyclic loading is divided into the cyclic plastic zone and the monotonic plastic zone.When the stress ratio R≥0,rm≥rc,based on the finite element analysis results,this study makes the following assumptions:

(1)Outside the monotonic plastic zone,σ,ε,Δσ and Δε are elastic,which are determined by the Eq.(3);

(2)On the cyclic plastic zone,σpand εpare plastic,which are determined by the Eq.(5);

(3)Between the cyclic plastic zone and monotone region,σ,ε,Δσ and Δε are elastic plastic coexistence region,based on the results of finite element analysis,Δσ and Δε are obtained by the nonlinear interpolation function.

The Manson-Coffin model can be used to describe the low-cycle fatigue behavior of ship structure,we can get its life relationship as follows:

where σf′is the fatigue strength coefficient; εf′is the fatigue ductility coefficient;b is the fatigue strength index;c is the fatigue ductility index;Nfis the fatigue life.

For the problem of low-cycle fatigue of ship structures,the elastic component of the fatigue crack tip is negligible,so the plastic component of the Eq.(9)is only considered.According to the above ideas,substituting the acquired plastic strain component into Eq.(9),we can obtain the low-cycle fatigue life(Nf)of the corresponding hull crack plate as follows:

1.2 Accumulative incremental plastic damage at crack tip

Based on the chaboche nonlinear hardening model[11],in the previous study[12],the author pointed out under non symmetric low-cycle fatigue loading,notched plates at the notch root will produce accumulative incremental plastic deformation and eventually lead to a gap in the ductile failure;combined with the Newton Raphson iterative method,respectively,the change of the stiffness of the notch root zone and the accumulative incremental plastic deformation after n+1 cycles are obtained.Combined with the geometrical shape of the crack,the low-cycle fatigue crack can be considered as a special notch with a short half axis of the notched plate,namely:b=0,using the same Newton-Raphson iteration procedure and method,similarly,we can obtain stiffness changes in local region of the low-cycle fatigue crack tip and corresponding accumulative incremental plastic deformation expression after n+1 cycles,respectively as follows:

where:

According to Eq.(11),the theoretical solution of accumulative incremental plastic deformation near the crack tip after n+1 cycle of low-cycle fatigue load can be obtained by using the Newton-Raphson iterative method.The corresponding accumulative incremental plastic deformation expression near crack tip after n+1 cycle is as below:

After the n+1 cycle,the corresponding plastic strain increment can be obtained;by updating the corresponding parameters in turn,the plastic strain increments under each corresponding cycle can be obtained.

The accumulation of the crack tip plastic deformation will produce hull structure of the material ductility loss which will cause accumulative incremental plastic damage of ship structure,in the stress control cycle,for the asymmetrical stress cycle in which mean stress is not zero,in order to consider the accumulative plastic strain we can use the model proposed by Xia[13]that can calculate the failure cycles controlled by the accumulative plastic strain.

so:

where ε˙r=dεr/dN is accumulative incremental plastic deformation of the crack tip under cyclic loading determined by Eq.(12); ε˙ris accumulative strain rate of accumulative plastic strain and steady development;κr,β is related material constants obtained by fitting test.

1.3 Low-cycle fatigue crack propagation rate model

In traditional critical damage model[14],the crack growth size is equal to the cyclic plastic zone radius in front of crack tip in which the plastic accumulative damage is ignored between the cyclic plastic zone and the monotonic plastic zone.It is pointed out that the plastic accumulative damage between the cyclic plastic zone and the monotonic plastic zone at the crack tip is not negligible and should be considered by Castro[15].

According to the Castro’s[15]assumption,the low-cycle fatigue crack growth size is equal to the monotonic plastic zone radius in this study,the monotonic plastic yield zone at the crack tip is composed of a series of micro fatigue units δρ,as shown in Fig.4.

Fig.4 The material unit model at front of crack-tip

The crack tip extends forward to moving a fatigue unit δρ in each constant amplitude loading cycle,and it is considered that the plastic accumulative damage outside the monotonic plastic zone is negligible.Therefore,a total of fatigue units rc/δρ in the monotonic plastic zone need to be considered in a cyclic fatigue loading.It is assumed that the total damage in the low-cycle fatigue crack growth process is composed of the low-cycle fatigue damage Dfand the accumulative incremental plastic damage Dr:where Dfis the low-cycle fatigue damage without consideration of accumulative plastic damage;Dris the additional damage caused by accumulative plastic damage.

The additional damage caused by the influence of the accumulative incremental plastic damage near the crack tip is obtained by using the linear accumulative damage rule.According to Eq.(10)and Eq.(14),the accumulative damage caused by the low-cycle fatigue load on the first i fatigue unit is:

where Nfis a failure cycle controlled by low-cycle fatigue,and Nris the failure cycle of accumulative plastic strain control.

After the N times load cycles,the total accumulative damage of all fatigue units rc/δρ in the monotonic plastic zone of the original fatigue crack tip is as follows:

When the accumulative damage of fatigue unit in the monotonic plastic zone reaches a critical value D0,the crack tip moves forward to extend a size of monotonic plastic zone,and the corresponding cyclic number is the low-cycle fatigue crack propagation life N that propagate a distance of the monotonic plastic zone.

Through a lot of experiments,Jiang[16]found the low-cycle fatigue crack growth can be seen as a process of the low-cycle fatigue crack repeated initiation.According to the point proposed by Hurley et al[17],the nominal fatigue crack growth size is equal to the crack tip plastic zone radius in the fatigue crack propagation direction.The structure materials near the crack tip are discretized into a continuous adjacent fatigue unit,and the low-cycle fatigue crack propagation is regarded as a discontinuous process of fatigue fracture failure.

When the accumulative coupled damage of fatigue unit in the monotonic plastic zone of the crack tip reaches a critical value,the fatigue unit ruptured causes the crack tip move forward.The average accumulative damage unit in the whole low-cycle fatigue propagation region is defined as:and the low-cycle fatigue propagation rate is as follows:

The Eq.(18)is the prediction model for the low-cycle fatigue crack growth rate of the hull crack plate considered the accumulative incremental plastic damage at crack tip.In this model,the plastic accumulative damage caused between the cyclic plastic zone and the monotonic plastic zone at crack tip is considered,so it is more comprehensive to evaluate the low-cycle fatigue crack propagation rate of cracked plate.

2 Experimental investigations

2.1 Experimental setup

The fatigue crack growth experiments were made on 7 mm thick hull notch cracked plate specimens with other dimensions shown in Fig.5.The specimen and the test setup are shown in Fig.6.The steel Q235,is a low carbon steel which is widely used in hull structures,was employed to the tests,where the uniaxial tensile stress-strain curve of Q235 steel are obtained by tensile test as shown in Fig.7 and the basic material mechanics properties of Q235 steel are obtained by tensile test as shown in Tab.1.Chemical composition(in%wt)of this material is:C 0.18,Si 0.43,Mn 1.4,P 0.02,S 0.014 and V 0.13.The crack planes in the specimens were orientated perpendicular to the rolling direction of the plates.The fatigue crack growth experiments were performed in air and at room temperature using a computer controlled servohydraulic test machine,MTS322 250 kN.The crack length was simultaneously measured using strain extensometer with a 10 mm gauge length,±1 mm range,and 0.01%extensometer strain control accuracy whose operating temperature ranges from_80_C to 200_C,as shown in Fig.6.A uniaxial controlled load with maximum applied load 7.2 kN was used to control the total stress range and a tensile-compression loading with a triangular waveform was used to ensure that the strain rate remained constant in a loop beginning with the tensile load.

Fig.5 A notch cracked plate specimen(notch diameter is 4.8 mm,thickness is 7 mm)

Fig.6 Test setup diagram of specimen

Fig.7 Tensile stress strain curves of Q235 steel

Tab.1 The mechanics properties of Q235

A 0.25 Hz frequency was chosen based on other low-cycle fatigue test results available in the literature.The specimens were tested in cyclic loading,considering the influence of compressive stress,the tests were conducted under constant amplitude loading with different stress ratios and mean stress to study the low-cycle fatigue crack growth.Each test was performed to failure and a mean of three measurements under each condition was taken.

The real-time information such as time,load,crack tip opening displacement and extensometer strain was recorded.After the fatigue failure,fractographs and microscopic changes were examined using LEO Electron Microscopy 1530 VP field emission scanning electron microscope(FESEM)to analyze the failure mechanisms.An example of the fracture process and fracture morphologies of low-cycle fatigue crack propagation is shown in Fig.8.

First by sticking to the crack tip strain,we measured to get accumulative plastic deformation at the crack tip and the corresponding relationship between the deformation rate and the number of cycles under different stress ratio and mean stress and provide validation for the verification of the model;then by the scale line prefabricated on the initial crack tip and com-bined with distribution in the sample on the high times magnifying glass,we can observe and record a crack moving cycles of a tick spacing and obtain low-cycle fatigue crack growth rate curve.

By connecting the projector with the magnifying glass,the position of the current crack tip can be obtained clearly and timely.When the fatigue load is stuck in the cyclic vale,the horizontal coordinate of the crack tip is recorded,and the crack length corresponding to the cycle can be obtained by subtracting the horizontal coordinate from the prior determination of the fatigue crack initiation and crack.The crack growth rate can be obtained by the increment of the crack length divided by the number of load cycles experienced by two adjacent crack measurements:

where ai+1and aiare measured values of two adjacent crack length.Crack propagation velocity calculated by Eq.(19)is mean velocity of extension from aito ai+1.ΔKeffalso uses the mean crack length aˉ=(ai+1+ai)/2 of two measurements;specific calculation expressions can refer to the study of Ref.

[18].

The effective stress intensity factor range of the crack tip can be obtained by the following equation[18]:

2.2 Fracture morphology analysis

Fig.8 shows examples of the low-cycle fatigue crack propagation process of specimen from Q235 steel inspected during a fatigue test with R=-1.Observation of the surfaces in the fatigue test,did not detect subcracks other major cracks.The irregularity on the fracture surface was more significant in the fatigue test with different stress ratios.Fig.8 shows the fracture morphology of specimens.The fracture morphology in the fatigue test showed transgranular fracture.It can be found that the accumulation of plastic deformation in the crack tip region was obvious,in the case of higher stress ratio,larger damage was produced in the process of cyclic loading,which led to the decrease of the ability to resist deformation for the material.With the increasing of stress ratio,the larger accumulative plastic deformation will lead to earlier failure.When the accumulative plastic deformation in front of crack tip is up to the critical value,the crack tip is gradually moved downward from a vertical scale to the next one with performance for the crack propagation.

Fig.8 Fracture process and fracture morphology of low-cycle fatigue crack growth

Therefore,it is necessary to consider the effect of accumulative incremental plastic damage at the Gauss integral point of the low-cycle fatigue crack tip in the study of the low-cycle fatigue crack growth rate of the cracked plate under asymmetrical stress cyclic loading.

2.3 Experimental result and discussion

According to the conventional mechanical properties of Q235 steel obtained by the test in Tab.1,Figs.9 and 10 summarize low-cycle fatigue crack growth rate of notch cracked plate specimen under different stress ratio and mean stress.Since no pre-cracks were used in the fatigue crack growth experiments,the significant influence of the notch geometry was obviously reflected by the early stage of crack growth.The stabilized fatigue crack growth rate was achieved when the crack propagates out of the notch affected zone.

Fig.9 Comparison curve of the low-cycle fatigue crack growth rate of specimens under different stress ratios

Fig.10 Comparison curve of the low-cycle fatigue crack growth rate of specimens under different mean stress

The curves of Figs.9 and 10 show a decrease in the fatigue crack growth rate with the decrease of stress ratio R,as expected.The R-effect can be attributed to the influence of mean tensile stress on fatigue life.A higher mean tensile stress results in a shorter fatigue life,which corresponds to a faster fatigue crack growth rate.Specimens that have been deformed by the fatigue experiments show large plastic deformation in the case of ship cracked plate specimen.The results indicate that the accumulative incremental plastic damage is an appropriate parameter for characterizing the fatigue crack tip state,and show that the accumulative plastic damage near the crack tip accelerates the low-cycle fatigue crack growth rate of hull cracked plate.The accumulative plastic deformation is larger,the low-cycle fatigue crack growth rate of hull cracked plate increased more significantly.It is showed that the accumulative plastic damage should be considered in the low-cycle fatigue crack growth rate analysis of hull cracked plate.

From Figs.9 and 10,it can be seen that the test results are basically coincide with the prediction curve,and the scatter is small,and shows that the prediction model in this study considering the effect of accumulative incremental plastic damage at crack tip can accurately reflect the low-cycle fatigue crack propagation behavior of notch cracked plate.

3 Numerical analysis and discussion

In the present study,the commercial finite element software ABAQUS is employed,in material model von Mises yield function,associated flow rule and Chaboche kinematic hardening model[11]is used.Chaboche kinematic hardening model is an advanced material model which is capable to capture basic cyclic plastic response of materials like Bauschinger effect,plastic shakedown,ratcheting resulting from asymmetric cycles of stress,and mean stress relaxation resulting from an asymmetric strain cycles.The constitutive equations contain 11 material parameters,namely,E,v,k,b,Q,C1,α1,C2,α2,C3and α3.The kinematic hardening behavior is described by C1;α1;C2;α2;C3and α3,where α1,α2and α3are the saturated values of the kinematic hardening variables,and C1,C2and C3indicate the speed with which the saturation is reached.The isotropic hardening is depicted by Q and b,the initial size of the yield surface is represented as k,E is the Young’s modulus,v is the Poisson’s ratio.The parameter values,optimized from the uniaxial test data of Q235 at room temperature,are listed in Tab.2.The above material model is adopted in the finite element software ABAQUS.Comparisons of the experimental result and the model simulations are given in Fig.11 for the stress-controlled cyclic test.

Tab.2 Values of the material parameters of Q235 steel for the viscoplastic constitutive material model

3.1 The numerical model

A standard notch cracked plate specimen is considered and the finite element meshes for the specimen,are shown in Fig.12 by using the commercial finite element software ABAQUS.Because of the thin thickness and loading conditions,finite element modeling of the notch cracked plate specimen is treated as plane stress case.Four-node plane-stress elements with full integration(CPS4)are used in FE mesh model.The refined mesh elements are used for the region near the crack tip to simulate the high stress and strain gradients.The element size near the crack tip is chosen to be 55 lm to obtain relatively high accurate stress-strain response.

Fig.11 Comparision of experiment and simulation of stress-strain response for a specimen

Fig.12 Finite element model for a specimen and the refinement mesh for crack growth region

In the current FE study,crack growth is simulated by debonding in the mesh model.The applied load is simulated for two cycles,the function of the first cycle is to allow the crack tip to fully extend and complete the redistribution of the stress-strain field around the crack tip,the second cycle is assumed to represent the stress and strain field ahead of the crack tip until the crack grows again by one element size.The crack growth procedure is simulated by releasing a sequence of nodes along the path of crack growth at a given rate (2.75×10_6 m/cycle).The nodes to be released are initially bonded to a rigid surface set on along the symmetry axis.There is no tendency of relative slip between the contact surfaces and therefore,the friction coefficient is set to be zero.A criterion of crack length versus time is used to control the crack growth rate,i.e.,the debonding of nodes from the rigid surface.The nodes are released just after the accumulative coupling damage reaching a critical value.As the crack grows from one node’s position to the next,the force carried by the node is gradually released to zero over a half cycle.Contact elements are used between the crack surfaces and a rigid surface is constructed along the symmetry axis to prevent the overlapping of crack surfaces during unloading.

The size of the low-cycle fatigue crack propagation is the monotonic plastic zone of crack tip,and it can believe that the accumulative coupling damage of accumulative incremental plastic damage and low-cycle fatigue damage cause fatigue failure of structural materials,when the accumulative coupling damage of fatigue unit in the monotonic plastic zone of the low-cycle fatigue crack tip reaches a critical value,the fatigue unit ruptured causes the crack tip move forward extensions.

Using the method proposed by Zheng[18]to periodically release the fatigue crack tip node by artificially controlling to control the low-cycle fatigue crack growth rate in this study,we can obtain the accumulative coupling damage at the Gauss integral point of crack tip under corresponding cyclic loading,and the number of cycles of releasing the nodes unit by writing the Post processing program.Then we could calculate the corresponding low-cycle fatigue crack transient growth rate.

3.2 Stress and strain field near the crack tip

The author’s previous study[12]had shown that there is a significant accumulative incremental plastic deformation phenomenon at static crack tip under low-cycle fatigue load,that low-cycle fatigue crack propagation is the result of the separation of the crack tip material.In this study,a series of numerical simulations are conducted to study the stress and strain fields at the crack tip under low-cycle fatigue load.

Through the observation of a special unit of crack tip expansion path,we can study and analyze stress and strain field of the expansion of the crack tip,in order to ensure to observe stable crack propagation and there are at least eight units from the special unit to the crack tip.Crack propagation will be simulated by node expansion technique of sequential release of crack surface,when accumulative fatigue damage of the crack tip node reaches a critical value,we will release the nodes,we will definite contact element of the crack face to prevent the overlap of the upper and lower crack elements during the unloading process[19].3.2.1 Stress strain field near the crack tip

In order to study stress strain behavior of extended crack tip,under the action of non symmetrical stress cycle under low-cycle fatigue load which ratio R=0.1 of 0.25 Hz,we select three position points of different distances(AO=14.8 μm,BO=9.3 μm,CO=3.8 μm),which is near crack tip Gauss integral point in the expansion process path,stress strain hysteresis loop is shown in Fig.13;the Gauss points near the crack tip located in the original crack tip in polar coordinates(r=14.8 μm,θ=45°).

In Fig.13,we can see that the accumulative incremental plastic deformation in front of crack tip is similar to the static crack in low-cycle fatigue crack propagation process.In addition,the stress and strain at the crack tip gradually increase with the decrease of the distance between the Gauss point O and the crack tip,and when the crack tip is closer to the Gauss integral points O,the stress and strain field of crack tip is larger,the accumulative incremental plastic deformation of crack tip is more obvious.

3.2.2 Accumulative plastic deformation near the crack tip

(1)The effect of stress ratio:According to the finite element calculation model given in Fig.12,Fig.14 shows the curve of the accumulative incremental plastic strain and the accumulative incremental plastic strain rate at the Gauss integral point of the extended fatigue crack tip under different stress ratios.

Fig.14 The curve of accumulative plastic strain at crack tip under different stress ratios

Fig.15 The curve of accumulative plastic strain at crack tip under different mean stress

From Fig.14,the accumulative plastic strain near the crack tip increased with the increase of the circulation and stress ratio under the asymmetric low-cycle fatigue loading with different stress ratio and the same maximum stress σmax,and the stress ratio is larger,the accumulative plastic strain also is relatively larger,but with the gradual increase of circulation cycle,the accumulative incremental plastic strain rate at the tip of the fatigue crack changes from quickly decreasing in the beginning stage to stabling.The results show that the stress ratio has more obvious influence on accumulative incremental plastic strain of the hull plate.

(2)The effect of mean stress:According to the finite element calculation model given in Fig.12,the low-cycle fatigue crack growth calculation is carried out.Fig.15 shows the curve of the accumulative incremental plastic strain and the accumulative incremental plastic strain rate at the Gauss integral point of the extended fatigue crack tip under different mean stresses.

From Fig.15,the accumulative plastic strain near the Gaussian integral point at crack tip increases with increasing of the number of cycles under cyclic loading of constant stress amplitude σawith different mean σm,and the mean stress is larger,the accumulative plastic strain is relatively higher,but with the gradual increase of circulation cycle,the accumulative incremental plastic strain rate at the tip of the fatigue crack changes from quickly decreasing in the beginning stage to stabling.The results show that the mean stress has comparatively obvious influence on accumulative incremental plastic strain of local position near the crack tip.

In summary,the accumulative plastic deformation obviously exists near the crack tip under asymmetric low-cycle load,and which leads to the final separation of the material on the crack tip propagation path and the crack tip moves forward to expansion of mobile.A similar conclusion is drawn by Zheng et al[18,20]by the analysis of the low-cycle fatigue crack propagation based on the accumulative plastic dissipation energy density of the crack tip.Compared to the external load stress ratio,the effect of the local stress ratio of the crack tip is more significant than that of the accumulative plastic deformation of the crack tip,which should be considered in the analysis of fatigue crack growth.At the same time,the difference of mean stress also influences the accumulation of plastic deformation at the tip of the fatigue crack.

3.2.3 Crack tip plastic zone

The local plastic zone radius at crack tip is a reasonable base for assessing the low-cycle fatigue crack propagation life.The plastic zone radius at the crack tip in a tension-compression loading cyclic includes the forward plastic zone induced by the tensile loading and the reverse plastic zone induced by unloading.In the case of the corresponding load,the plastic zone of the crack tip is calculated by the finite element method and the corresponding plastic zone is shown in Figs.16 and 17.

Fig.16 The crack tip plastic zone radius contour under tension-compression cyclic loading

Fig.17 The plastic zone radius in a tensioncompression loading cyclic

Fig.18 Residual stress distribution at crack tip along the crack propagation direction under different stress ratios

The crack tip plastic zone can be divided into positive plastic zone and reverse plastic zone under cyclic loading.The reverse plastic zone appeared at the crack tip after unloading from the tension load.From Fig.17,we can see under the tension and compression cycle load,the reverse plastic zone radius is roughly 1/4 posi-tive plastic zone radius,Chen et al[21]also draw the approximate conclusion through finite element analysis of the center-through cracked plate.

3.2.4 Normal stress distribution near the crack tip

The distribution of the normal stress along the crack path at the minimum load for the final step of the low-cycle fatigue crack growth is presented in Fig.18 for different stress ratios.

Fig.18 shows that obvious compressive stress field appeared behind the crack tip under the minimum load in cyclic loading,and the fatigue causes the corresponding crack closure and the compressive stress at the crack tip position reaches the maximum,the large pressure stress will cause the crack tip blunting effect.Pressure stress gradually transition to the normal stress at the crack tip with the increase of distance from the crack tip,the local compressive stress in front of crack tip is mainly residual compressive stress induced by accumulative plastic deformation caused by maximum tensile load.

3.2.5 Crack closure behavior

As the crack propagates,it creates a plastic wake zone behind the crack tip,which promotes premature crack closure during a cycle.The crack closure behavior will decrease the low-cycle fatigue crack growth rate,and which is strongly dependent on stress ratios and plays a significant role on the stress-strain field ahead of the crack tip.In the present study,plasticity induced crack closure has been taken into account during the simulation.Fig.19 presents the crack opening displacement(COD)under different stress ratios.

Fig.19 The crack opening displacement for different stress ratios

Fig.20 The curves of crack opening stress with the change of Δa/rpunder different a/W

Fig.19 shows that the crack is still open when the stress ratio is R=0 and the load is unloaded to zero,and the opening displacement of each point at the end of the crack decreases with the increase of the pressure load.For open crack,stress concentration exists at the crack tip when the load is loaded,and causes the reverse yield near the crack tip.When the cyclic load is unloaded from the maximum tensile load to zero,the elastic deformation disappears,and the COD of each point on the surface of the crack is only produced by the residual plastic deformation.In the R＜0,the COD decreased gradually with increasing of the compressive stress ratio,when R=-1 reached the maximum compressive stress,the part away from the crack tip is in the closed state,crack closure first started from crack tip back a distance position,and gradually moves towards the crack tip;at the same time,due to the local residual strain appears at the crack tip,which causes crack open displacement is greater than zero and is in an open state in the process of unloading.Zhang by studying on Aluminum Alloy also obtained the conclusion that crack will remain open in R＜0 when the loading unloading to zero[22].

The comparison between the simulations of two different crack lengths by the FE model in Fig.12 with the calculation results by Newman formula[23]for crack opening stress is presented in Fig.20.Fig.20 indicates that the crack opening stress first increases and then remains stable with the growth of low-cycle fatigue crack,and which is basically consistent with the Newman formula[23]by FE model in Fig.12 under low-cycle fatigue load.

3.2.6 Low-cycle fatigue crack propagation life

Combined with the prediction model of the low-cycle fatigue crack growth rate under constant amplitude low-cycle fatigue load,the low-cycle fatigue propagation life of hull cracked plate is presented as follows:

where a0is the critical length of the low-cycle fatigue crack initiation for the hull cracked plate,a0=0.5 mm;ɑfis the effective crack growth length at the end of the test obtained by experimental measurement.As the low-cycle fatigue crack propagates after the crack tip moves out of the effective crack growth length,the specimen is quickly rupture.Due to the short time,this part will not be considered.

Fig.21 Comparison of the predicted and experimental values of the low-cycle fatigue crack propagation life under different stress ratios(Left)and mean stress(Right)

From Fig.21,according to the prediction model by Eq.18 of the low-cycle fatigue crack growth rate,the predicted values and measured results are in good agreement,which shows that the proposed model based on the accumulative plastic damage near the crack tip can describe the crack evolution of low-cycle fatigue of hull notch cracked plate under constant low-cyclic fatigue load.

4 Conclusions

In the present work,a prediction model is presented to correlate the low-cycle fatigue crack growth rate based on the accumulative damage at the Gauss integral point of the crack tip of notch cracked plate under the asymmetrical stress cyclic loading,and the following concluding remarks can be drawn:

(1)The proposed model based on the accumulative plastic damage at crack tip can describe the crack evolution of low-cycle fatigue of hull notch cracked plate under constant low-cyclic fatigue load.When the accumulative plastic damage at the Gauss integral point of the crack tip reaches a critical value,the crack tip material is gradual separation,then the low-cycle fatigue crack moves forward.

In addition,the prediction model considers the effect of the accumulative plastic damage between the cyclic plastic zone and the monotonic plastic zone in front of crack tip,which results in a more comprehensive reflection of the low-cycle fatigue crack growth mechanism.

(2)The plastic deformation at each Gauss point accumulates progressively as the crack tip approaches it.Progressive ratcheting behavior has been observed in the cyclic plastic zone and that accumulative plastic strain increases with the increase of R-ratio.

The results show that the low-cycle fatigue crack growth rate of the cracked plate has been accelerated by the accumulative incremental plastic damage at crack tip.The accumulative incremental plastic deformation is larger,and the low-cycle fatigue crack propagation rate of the hull cracked plate increased more significantly.

(3)The prediction model of low-cycle fatigue crack growth rate agrees well with the lowcycle fatigue crack propagation tests.It shows that the accumulative incremental plastic damage at the Gauss integral point of the crack tip should be considered for predicting the lowcycle fatigue crack growth rate of hull cracked plate.

[1]Paris P C,Erdogan F.A critical analysis of crack propagation laws[J].Journal of Basic Engineering,1963,85(4):528-533.

[2]Hatanaka K,Fujimitsu T,Watanabe H.Growth of low-cycle fatigue crack in steels for machine structural use[J].Trans Jpn Soc.Mech.Eng.A,1986,52:579-86.

[3]Tomkins B.Fatigue crack propagation-an analysis[J].Phil.Mag.,1968,18(155):1041-66.

[4]Dowling N E.Crack growth during low-cycle fatigue of smooth axial specimen cyclic stress-strain and plastic deformation aspects of fatigue crack growth[J].ASTM STP 1977,637:97-121.

[5]Maurel V,Rémy L,Dahmen F,Haddar N.An engineering model for low-cycle fatigue life based on a partition of energy and micro-crack growth[J].International Journal of Fatigue,2009,31:952-961.

[6]Kamaya M.Low-cycle fatigue crack growth prediction by strain intensity factor[J].International Journal of Fatigue,2015,72:80-89.

[7]Kang G Z,Gao Q,Cai L X,et al.Experimental study on the uniaxial and nonproportionally multiaxial of SS304 stainless steel at room and high temperatures[J].Unclear Engineering and Design,2002,216:13-26.

[8]Noroozi A H,Glinka G,Lambert S.Prediction of fatigue crack growth under constant amplitude loading and a single overload based on elasto-plastic crack tip stresses and strains[J].Engineering Fracture Mechanics,2008,75(2):188-206.

[9]Creager M,Paris P C.Elastic field equations for blunt cracks with reference to stress corrosion cracking[J].Int.J Fract.Mech.,1967,3:247-52.

[10]Rice J R.Stresses due to a sharp notch in a work-hardening elastic-plastic material loaded by longitudinal shear[J].J App.Mech.,1967,34:287-298.

[11]Chaboche J L,Nouailhas D.Constitutive modeling of ratcheting effects1.Experimental facts and properties of the classical models[J].Journal of Engineering Materials and Technology-Transactions of the ASME,1989,111(4):384-392.

[12]Deng Junlin,Du Bo,Yang Ping.Research on the fracture toughness for shipcracked plates based on the accumulative increment plastic deformation[J].J Ship Mech.,2016,20(6):748-757.

[13]Xia Z,Kujawki D.Effect of mean stress and ratcheting strain on fatigue life of steel[J].Int.J Fatigue,1996,18:335-341.

[14]Cojocaru D,Karlsson A M.Assessing plastically dissipated energy as a condition for fatigue crack growth[J].International Journal of Fatigue,2009,31(7):1154-1162.

[15]de Castro J T P,Meggiolaro M A,de Oliveira Miranda A C.Fatigue crack growth predictions based on damage accumulation calculations ahead of the crack tip[J].Computational Materials Science,2009,46(1):115-123.

[16]Jiang Y,Ding F,Feng M.An approach for fatigue life prediction[J].Journal of Engineering Materials and Technology,2007,129(2):182-189.

[17]Hurley P J,Evans W J.A new method for predicting fatigue crack propagation rates[J].Materials Science and Engineering A,2007,466:265-273.

[18]Zheng X,Cui H,Su X.Numerical modeling of fatigue crack propagation based on the theory of critical distances[J].Engineering Fracture Mechanics,2013,114:151-165.

[19]Oliva V,Kunes I.FEM analysis of cyclic deformation around the fatigue crack tip after a single overload[J].ASTM Special Technical Publication,1993,1211:72-90.

[20]Zhao L G,Tong J.A viscoplastic study of crack-tip deformation and crack growth in a nickel-based superalloy at elevated temperature[J].Journal of the Mechanics and Physics of Solids,2008,56:3363-3378.

[21]Chen Jingjie,Huang Yi,Dong Leilei.A study on evaluation method of crack tip reverse plastic zone size for the center cracked steel plate model under tension-compression cyclic loading[J].Engineering Fracture Mechanics,2015,133:138-151.

[22]Zhang J Z,He X D,Sha Y,et al.The compressive stress effect on fatigue crack growth under tension-compression loading[J].International Journal of Fatigue,2010,32(2):361-367.

[23]Newman Jr J C.A crack opening stress equation for fatigue crack growth[J].Int.J Fract.,1984,24:R131-135.

考慮塑性損傷的船體裂紋板低周疲勞裂紋擴展行為研究

鄧軍林1b，2， 楊 平1a，b， 陳 遠1b

（1a.高性能船舶技術(shù)教育部重點實驗室（武漢理工大學(xué)），武漢430063；1b.武漢理工大學(xué) 交通學(xué)院，武漢 430063；2.欽州學(xué)院 船舶與海洋工程系，欽州535000）

船舶結(jié)構(gòu)的擴展斷裂失效往往是低周疲勞破壞和累積遞增塑性破壞耦合作用的結(jié)果，疲勞裂紋的擴展就是裂紋尖端前緣材料剛度不斷降低延展性不斷耗失而逐漸分離的結(jié)果。基于彈塑性斷裂力學(xué)理論，文章提出了考慮累積塑性損傷的低周疲勞裂紋擴展速率預(yù)測模型。通過低周疲勞裂紋擴展試驗擬合出模型相關(guān)材料參數(shù)并驗證預(yù)測模型的合理性。通過系列有限元計算對平均應(yīng)力及應(yīng)力幅值的影響因素進行了數(shù)值分析。該模型的計算結(jié)果與已有實驗結(jié)果基本吻合；對合理預(yù)估船體裂紋板的常幅低周疲勞裂紋擴展壽命有重要意義。

船體裂紋板；常幅疲勞載荷；累積塑性損傷；低周疲勞裂紋擴展壽命；裂紋閉合效應(yīng)

TG113.25 U661.42

A

鄧 軍林（1984-），男，武漢理工大學(xué)交通學(xué)院博士后；

楊 平（1955-），男，武漢理工大學(xué)交通學(xué)院教授，博士生導(dǎo)師；

陳 遠（1992-），男，武漢理工大學(xué)交通學(xué)院碩士研究生。

TG113.25 U661.42 Document code:A

10.3969/j.issn.1007-7294.2017.12.007

date：2017-06-10

Supported by The National Natural Science Foundation of China(No.51479153);the Provincial Natural Science Foundation of Guangxi(No.2016GXNSFAA380033);the basic ability promotion program for young and middle-aged teacher of University in Guangxi(No.2017KY0809)

Biography：DENG Jun-lin(1984-),male,post-doctoral,E-mail:junlin.deng@163.com;

YANG Ping(1955-),male,professor/tutor,E-mail:pyang@whut.edu.com;

CHEN Yuan(1992-),male,graduate student,E-mail:chendayuans@163.com.

1007-7294（2017）12-1507-20