薄板件自然疲勞裂紋擴展SBFEM仿真方法

李翰超，杜平安，楊 丹，于亞婷，劉建濤

(1.電子科技大學機械與電氣工程學院 成都 611731；2.西南交通大學機械工程學院 成都 610031)

Thin-walled plates and shells have been extensively used in civil,automotive,aerospace and many other disciplines because of their outstanding designability,high stiffness-to-weight ratio and excellent strength-to-weight ratio[1-3].However,in practical applications,the actual usage lifetime of a number of these components is much shortened due to their sensitivity to the unavoidable imperfections,microcracks,accumulated damage in service and so forth.It is of particular importance to understand the physical essence of fatigue crack initiation and growth process inside such structures and to accurately predict their fatigue life so as to realize safe applications and prevent catastrophic accidents.

Up to now,various experimental,theoretical,and numerical works[4-15]have been conducted in this aspect,while quantifying the crack growth behavior together with the service lifetime of actual thin-walled structures is still a challenging problem for mechanics theories and engineering applications.In this study,our attention is mainly focused on the simulation of NFCG process of thin-walled plates and predicting their fatigue life under various cyclic loading conditions in the scaled boundary finite element method(SBFEM) framework.

Different from the finite element method(FEM),extended finite element method and meshfree method[12,14-17],SBFEM is a semi-analytical approach combining the advantages of specificly FEM and the boundary element method.The finite element discretization is employed along the circumferential direction parallel to the external boundary of a certain domain.Then,the governing partial differential equations of boundary value problem(BVP) are transformed into a set of ordinary differential equations with the radial coordinate as the independent variable.Hence,the resulting physical fields along the radial direction are analytical,while those in the circumferential direction are numerical.Further,the mesh generation efficiency of complex structures is substanntially improved with greatly reduced unknown numbers,as only the external boundary necessitate being discretized.Therefore,these merits make the SBFEM a promising strategy in modelling singularity ascribed to cracks[18],analyzing damage in huge structures[19],reliability analysis[20]and so forth.

SBFEM was first developed by Wolf and Song[21]to deal with the unbounded BVPs,and its theoretical foundation was further enriched.[10,11,18,22].Afterwards,this method was extended to evaluating the stress intensity factor(SIF),cohesive effect,interactions and T-stress around the crack tip inside homogeneous materials or between two dissimilar materials[20,23-26],to solve couple-field problems[27-30],and construct polygon elements[13,31-33].However,quite a few works[13,34]have been conducted so far on predicting the fatigue lifetime of thin-walled plates taking account of the integrated effects of cyclic loading conditions and specimen characteristics in the framework of SBFEM.Hence,this work aims to fill in this gap,and elaborate a computational scheme to simulate the NFCG phenomenon with SBFEM and predict the fatigue lifetime of specimen with different thicknesses under various cyclic loading conditions.

The structure of this paper is organized as follows.A NFCG model is first introduced to account for the combined effect of the loading conditions and the specimen characteristics.Hereafter,the governing formulations of SBFEM is briefly invoked and a computational strategy is developed to simulate the NFCG process and to predict the fatigue lifetime of thin-walled plates.The simulation scheme is summarized as a flow chart to clearly illustrate the whole process.

1 Natural Fatigue Crack Propagation Considering the Loadings and Specimen Characteristics

Here,we consider a 2D continum structureΩbounded by the external boundaryΓshown in Fig.1 as an illustration,and the boundary conditions are given as

Experimental observations[4,5,16,34-36]indicate that the fatigue crack growth in practical components is affected by various factors,including the loading sequence and ratios,microdefects,specimen geometry,material properties and so forth.Many NFCG models[6-8,37-40]have been proposed thus far.In this study,the following NFCG model is adopted to characterize the fatigue crack growth rates of the whole process[8,39].

wherec1andm1～m3are the model coefficients to be determined based on the experimental data.Physically,m2andm3determine the manner of da/dNapproaching the cracking threshold ΔKthand the material fracture roughnessKICin Fig.2,respectively.ΔKdenotes the variation of the stress intensity factor(SIF),andRis the cyclic stress ratio.

In Eq.(2),Urefers to the effective stress intensity factor range ratio defined by

whereγis the crack opening ratio.To properly describe the combined effect ascribed to the loading conditions and the specimen characteristics,γis defined in this study as[8,41]

The coefficientsA0～A3are evaluated by

withvthe material Poisson’s ratio andtrethe relative thickness of thin-walled plates expressed as

wherecis a material-property-related constant,tis the specimen thickness andkmatis defined as

whereσfis the material flow stress,andσsis the material yielding stress.

Then,the foregoing fatigue crack propagation length Δai+1can be explicitly expressed as

where ΔKirepresents the variation of the mixed-mode SIF[42]of the(i+1)-th cycle,and is further written as

In this study,the maximum circumferential stress criterion[43]is adopted to characterize the fatigue crack growth direction

Here,we assume that the fatigue cracks under consideration propagate along a straight line within one loading cycle.Combining(3)～(10),we can quantitatively determine the directional FCG velocities,the cracking direction and the location of the newly-formed crack tipPi+1.

It is noted that the foregoing NFCG model takes explicitly account of the influence of the loading conditions by(3),the material property and specimen geometry effects by(2),(6) and(7),and the cracking physics near ΔKthandKIC.Hence,it is of great convenience in applying this model for specimen design,material selection,and NFCG rate and fatigue lifetime estimation in practical applications.In the following sections,the NFCG model in Eqs.(1)～(10)will be numerically implemented in the SBFEM context.

2 Simulation of the Fatigue Crack Growth

As stated previously,SBFEM shows various advantages over other strategies in simulating natural fatigue crack growth and in predicting fatigue lives due to its accurate description of SIFs at the crack tip,fewer DOFs are needed for simulation and analytical physical fields along the radial direction.In this section,a computational scheme is developed using SBFEM to simulate the NFCG process of 2D thinwalled structures.

2.1 Governing Formulations of 2D Continum Containing Natural Fatigue Cracks

For brevity,the 2D continum in Fig.1 is taken again as an example.Here,the quasi-static elastic problem is addressed,and the body force ofΩis assumed to be negligible.Thus,the equilibrium equation of a pointxinΩis written as

withσ=[σxx,σyy,σxy]Tbeing the stress vector and

The stress vectorσat the pointxis related to the strain vectorε=[εxx,εyy,εxy]Tby

whereDis the second-order elastic matrix of the material formingΩ.

The strain vectorεcan be rewritten in terms of the displacement vectoru=[ux,uy]Tas

The displacementuand tractionfsatisfy the relations presented in Eq.(1) on the external boundaryΓand fulfill the formula below on the cracking surfaceΓi:

Equations(11)～(15) together with(1) constitute the strong governing formulation of the boundary value problem in Fig.1.Then,the weak governing equation of the BVP in Fig.1 is derived by using the virtual work theorem[22]or the weighted residual formulation[10,11]as

wherexsrefer to the points on the boundaryГ.

Then,the whole domainΩis discretized into polygon elements of any number of sides,e.g.Fig.3.Simultaneously,a scaling center(e.g.O(i)) is defined for each element and specifically,the whole boundary of the element is visible from its scaling center.A local polar coordinate system is defined such that the radial axisξdirects from the scaling centerO(i)towards the element boundary,and its value is 0 at the scaling centre and is 1 at the element boundary; the circumferential axis is denoted asηand physically represents the distance between a particular point and the origin point selected on the element boundary.

Accordingly,the displacement vector of any point(ξ,η) in thei-th subdomainDican be explicitly approximated by

wheremidenotes the total number of nodes on external boundary ofDi;Nk(η) is the shape function associated with thek-th node and related to the circumferential coordinate merely;uk(ξ) is the radial displacement function associated with thek-th boundary node and defined by[10,11,22]

whereck_jis the boundary condition dependent constant and physically represent the contribution of each mode to the final solution;λjandφk_jare the positive eigenvalues and the associated eigenvectors of a standard eigenproblem[43]and satisfy the internal equilibrium along the ξ direction[42].

Combining Eqs.(12)～(18) and assembling the stiffness matrix and the loadings of all polygon subdomains together following the procedure presented in Refs.[22,43],we finally obtain the equilibrium equation of the aforementioned BVP:

whereKis the global stiffness matrix related to the geometry and the material of the BVP in question,dcontains all unknows at the nodes,andFis the global external force vector.

Solving Eq.(19) delivers the nodal displacements on the element boundary.Then,invoking the relations in Eqs.(17) and(18),we can determine the resulting displacement and stress of any point in the domain by

whereB1(η) andB2(η) are the strain-displacement matrices[22,43].Note that these two matrice are functions of the circumferential coordinateηonly,which indicates that the displacement and stress solutions by SBFEM are analytical along the radial directionξ.

Practically,the propagation direction of natural fatigue cracks is largely influenced by the external loadings,the material property,the geometry of the structures,and the local defects.Fig.4 visualizes the schematic diagram of a generic cracked body modelled by the SBFEM.To evaluate the SIFs of cracks of this kind,the Williams eigenfunction expansion around the crack tip[24,42]is first applied and the stresses of the point Q at the crack surface direction on the element boundary as Fig.4 in the global.Cartesian coordinate system are then transformed to the normal stressσn(mode I) and the shear stressτn(mode II) defined on the cracking sruface plane.Finally,the mixed model SIFs are determined by

whereL0is the distance between the crack tip and the pointQwhose location doesn’t necessarily coincide any existing node.The stresses ofQcan be conveniently recovered by using Eq.(20).

2.2 Numerical Implementation of Natural Fatigue Crack Propagation

From many literatures[6-8,37-41]and experimental observations,it is known that the NFCG includes three typical stages shown in Fig.2.The slow growth stage near the cracking threshold ΔKth(Stage I),the steady growth stage(Stage II) and the fast fracturing stage near the fracturing point((1?R)KIC)(Stage III).At stage I,the fatigue crack propagation rate da/dNis so small that the crack length increment Δaduring one loading cycle is much smaller compared with the side length of an subdomain.Hence,the crack of this stage is assumed to propagate within one subdomain.Two situations are considered in this study about this stage:1) as the variation Δθcofθccomputed by Eq.(10)between the current loading cycle and the former one is smaller than a critical value θc,say 1o,the fatigue crack is considered to propagate along the same direction and the node at the fatigue crack tip moves to a new position determined based on Eqs.(8)～(10),see Fig.5; 2) otherwise,a new nodal point is first added at the newly-generated crack tip and two new elements are added between this point and the original crack tip.Meanwhile,a line parallel to and another one atN6perpendicular to the new crack segement(N3andN7in Fig.6) are constructed to divide the original two domains into 4 parts and two new nodal points are added simultaneously at the intersection points of the subdomain boundaries as Fig.6.Hereafter,the new line elements are added accordingly.In this study,the fatigue cracks of all instances under consideration are modeled along the element boundary,as in Fig.3 and Fig.4,so as to guarantee the visuality of the whole boundary of the element from it scaling center when curved cracks are involved.

At stage II,the remeshing and optimization strategy[43]is appropriate for modeling the fatigue crack propagations and adopted in this study.As for the last stage,nature fatigue cracks under one cyclic loading may propagate through several subdomains,see the dashed lines in Fig.7a and Fig.8a.To properly handle this issue,two cases are considered here and the remeshing strategies in Fig.7b and Fig.8b are employed to generate the discretized models for simulations with SBFEM.

Up to now,the computational strategy using the SBFEM to numerically simulate the 2D NFCP phenomenon has been introduced completely,and an in-house package may be developed by using C++,MATLAB or others to perform the simulations.The whole process is summarized as a flow chart presented in Fig.9.

3 Conclusion

In this work,we established a computational methodology framed within SBFEM to simulate the natural fatigue crack growth and to predict the fatigue lifetime of thin-walled plates considering the influence of the loading conditions and the specimen characteristics simultaneously.To be noted,the effects of loading conditions,specimen thickness and material properties could be studied with our elaborated methodology to serve as a beneficial tool for quantitatively studying the natural fatigue crack propagations and designing practical specimens.

4 Acknowledgements

This work was funded by the Young Scientific Innovation Team of Science and Technology of Sichuan(2017TD0017).