0　Introduction

Increase in the demand for higher reliability of Engineering structures and components has driven attention on the quality of weld and the weld quality assessment methods[1-2]. Defects in welded connections are inevitable and are difficult to control. These defects can be found out only after fabrication by inspection and involves several constraints to repair[3]. These issues leads to the adaptation of ‘fitness for purpose’ concepts in which acceptable level of defects that do not reduce the life are estimated. In structures subjected to fatigue loading, presence of any acceptable defect can grow to a critical size leading to failure. Qasim et al[4] describes fatigue as a significant parameter to be considered in the performance of mechanical components subjected to constant and variable amplitude loading. Therefore it is evident that the effects of welding on the structures subjected to fatigue loading need to be considered for safe and economic design. The fatigue life of welded structure is often dominated by crack initiation and growth from welding defects such as lack of fusion, cracks, porosity, lack of penetration, and inclusions. In the past, S-N curve based analysis is the only engineering tool which is used to evaluate the fatigue life which is the number of cycles of loading for initiating a crack. And the crack propagation was not considered to predict the life of the component.

At present,Linear elastic fracturemechanics(LEFM) made possible on the study of crack growth behavior and hence helps in the prediction of remaining lifeof the components.Vencislav et al.[5]and Satoyukitanaka et al.[6]assessed the fracture behavior of a pressure vessel by the basic LEFM approach for the tress intensity actor (SIF). Crack growth analysis using LEFM approach uses the model of the structure under study with initial weld defects considered as a crack for evaluation.Mukhtaretal[7] adopted FEM for the estimation of SIF of cruciform joint and predicted the direction of crack propagation using maximum normal stress criterion. Behaviour of fatigue crack growth (FCG) rate in different materials wasstudied by Paris and Erdogan[8] for understanding itsfailuremechanisms. Bouchard et al.[9] defines the direction of fatigue crack growth propagation as the perpendicular direction to the maximum stress at the crack tip.

Guha et al.[10] discuss about the initiation and propagation of crack from lack of penetration(LOP) which occurs in the weld root. Hobbacher[11] discuss about the determination of initial crack size must be determined by techniques like nondestructive inspection. The imperfections obtained from NDT inspection have to be transformed into an elliptical crack which can be considered for further fracture analysis. Pugno et al.[12] discusses the determination of initial and final crack length from the threshold stress intensity and fracture toughness value of the material respectively. Derived the procedure for the evaluation of fatigue crack simulation and crack growth rates (CGR) using the stress intensity factors. Magudeesran et al.[13] predicts that nearly 70% of fatigue crack occurs in the welded joint with abrupt change in section produced by excess weld reinforcement, slag inclusion, undercut, and lack of penetration.Effect of weld defects like lack of penetration (LOP) and lack of fusion (LOF) on the fatigue behaviour was studied bySanders et al.[14]on the fatigue behavior of Al-5083-0 double-V groove butt welds. They concluded that the above defects seriously reduce the fatigue life of the welded member.

3.3.1 鸿图嶂观花植物 观花植物是指其花或花序具有一定观赏价值的植物，包括色彩鲜艳或色彩变化、花型奇特、花型或花序特大、植物姿态及花序优美、着生状态特别、芳香、花的质地特异等[13]。

The prediction offatigue crack growth(FCG) in welded joints by the fracture mechanics approach will depend on loading/boundary conditions, specimen and weld bead geometry, size of initial defect and material. For standard components with standard crack geometries, the crack growth can be predicted using the stress intensity factor using the available standard analytical equations. Whereas, for non-standard geometries, finite element analysis(FEA) softwareisused to predict the stress intensity factor and thereby the crack growth.

This paper focuses on the prediction of the influence of weld defect on fatigue life of a ring type specimen which replicates the behavior of a pressure vessel. The crack grow analysis was performed by FEA software Franc 3D which calculates the stress intensity factor and fatigue life based on M-integral approach. The life of the component being analyzed is assumed to be consumed when the crack attains its critical length or half of its thickness whichever is greater.

1　Linear elastic fracture mechanics

Combining Eq.(1) and Eq.(2), rearranging the terms and integrating between the limits of initial crack length ‘ai’ and the critical crack length ‘ac’, the expression for fatigue life can be obtained as given in Eq.(3).

(1)

Where ai is critical crack length (mm), Kthis threshold stress intensity factor (MPa·mm1/2). σ0is endurance limit of the material (MPa).

Pressure vessel is a critical component fabricated by rolling and welding the sheet metal and is subjected to higher stresses as well as fatigue loading in many cases. Possibility of defective weldments causes severe impact on the structural integrity of the pressure vessel as the weld defect leads to the origin of crack. In order to simply the computational and experimental effort of the fatigue analysis of the pressure vessel, it has been proposed to design an equivalent ring type specimen which exactly simulates the global stress pattern of the vessel. Based on Roark’s formula[19]and plane stress condition the dimensions and load to induce a stress of 100 MPa on a ring type specimen which was arrived. Based on this, a flat plate of cross section 12 mm × 6 mm was rolled and welded (Fig.3) to get the ring specimen with mean radius 200 mm.

(2)

Where da is change in crack length, dN is change in life cycle, C and m are material constants,stress intensity factor (K) varies with respect to shape of geometry and type of crack and some common cases are given in Table 1.

Table 1　Stress intensity factor for different crack and geometry

TypeofcrackStressintensityfactorKEdgecrackoflength‘a’inasemi⁃infiniteplate1．12σ　πaCentralpenny⁃shapedcrackofradius‘a’ininfinitebody2σ　aπCentrecrackoflength‘2a’inaplateofwidth‘W’σ　WtanπaWTwosymmetricaledgecracks，eachlength‘a’，inplateoftotalwidth‘W’σ　WtanπaWæèçöø÷+0．1sin2πaWæèçöø÷æèçöø÷

Fracture mechanics approach is generally used to predict theremaining life of a structure with a pre-existing crack. LEFM approach predicts the fatigue crack growth propagation based on stress intensity factor (SIF), K which again depends on the geometry and load on the structure. In case of cyclic loading, range of SIF, K is used for the study. Developments in fracture mechanics happened after the studies by Irwin[15] who have introduced the SIF ‘K’and stated that when K reaches a critical value called as fracture toughness of the material, fracture occurs. SIF defined by Irwin is given in Eq.(6).

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(3)

Where ai is initial crack length in mm, ac is critical crack length in mm, NT is number of cycles to reach critical crack length, Ni is number of cycles to initiate a crack (from S-N curve).

Subjected to the constraints; 0<a/c<2 and c/b<0.5 provided that a/t< 1.

(4)

Where ΔK is range of stress intensity factor (MPa·mm1/2), ΔK=Kmax-Kmin,σ is applied stress range(N·mm-2),σ=σmax-σmin, a is crack length (mm), Y is dimensionless function which depends on the geometry ofthe cracked body considered.

The fracture toughness value of the materials[2] is used to predict the critical crack length of a material as given in Eq.(5).

(5)

Where ac is critical crack length (mm), Kc is fracture toughness (MPa·mm1/2), σ is maximum stress or yield strength of materials (MPa).

The crack initiation stage in welded joints due to welding defects such as LOF, porosity and LOP is smaller compared to the crack propagation stage.As discussed, prediction of FCG is easier for standard structures and in case of complex structures, FEA software is used. Franc 2D/3D software is a tool used to simulate crack growth in 2D/3D engineering structures subjected to complex loading conditions.The initial defect/crack in the structure which can be predicted from NDT should be modeled as an equivalent crack in the finite element model. The SIF which is the depended factor for crack FCG is evaluated by M-integral method in the software[17]. Al-Mukhtar et al.[7] used the FRANC2D software to predict the stress intensity factor during the crack propagation phase in standard weld geometries.This paper focuses on the demonstration of the usage of FEM software to study the behaviour of FCG due to weld defects and its validation using numerical analysis. Accordingly a standard FCG problem defined by Newman et al.[18] and given in Eq.(7) is considered and the fatigue life is predicted analytically using LEFM discussed in this section and also using Frank3D software.

2　Examples for the prediction of FCG

Consider a semi-elliptical surface crack with dimension ‘2a’ and ‘2c’in a finite plate of dimension 2b× 2h×t (Fig.1) for which the stress intensity factor[19]is given in Eq.(6).The initial dimensions of semi elliptical crack problem are a=c=1 mm in a finite geometry of 2b=2h=100 mm and t=10 mm as shown in Fig.1.

然而，PBL教学模式在具体运行过程中也暴露出一些问题：（1）一些学生由于对传统的教育模式的依赖，自学能力差，难以适应PBL 教学模式。（2）由于PBL 教学强调的是针对问题进行学习，可能使得学生将注意力集中在解决问题上，而降低了对理论知识学习的要求，最终导致知其理论基础不扎实，失去了进一步发展的根基。（3）PBL 教学目前还没有统一的标准，教材、教师绝大多数都是各个院校自己编写和选拔的，其水平参差不齐，有些甚至制约了PBL 教学的效果[12-14]。

(6)

The initial crack size can be determined by nondestructiveevaluations like radiography testing and ultrasonic testing. In normal structures, the initial crack length is also be predicted empirically from Eq.(4).

where Q is the shape factor of an elliptical crack given in Eq.(7), FS is the boundary correction factor given in Eq.(8).

(7)

FS=[M1+M2+M3(a/t)2]g×fφ×fw

(8)

Fig.1　Semi elliptical surface crack in a finite body[18]

where

where Uy1and Uy2are composed by the first and the last rows of Uy,respectively.

The deviation in the results of fatigue life predicted by analytical and numerical methods is 1.85% and this validation is helps to predict the SIF and the fatigue life of nonstandard geometry.

Final crack size or the critical crack length is considered as 60% of total thickness of the plate that cause an unstable fracture. Stress of 100 MPa is applied at the boundary parallel to semi elliptical crack and the life for the propagation of crack at the incre-ment of 0.5 mm upto 6 mm is determined by Paris law, given in Eq.(6). Also the problem is modeled in Franc 3D software with crack and the calculated value of number of cycles for each incremental crack length is estimated (Table 2 & Fig.2).

日子一天天过去，我自以为与叶霭玲的关系是虚的，与白丽筠的关系才是实的，有着实质性内容。但是，就像性生活不能公之于众那样，我与白丽筠的关系也是放不上台面的。相反，我与叶霭玲的关系却有两家的父母亲属做见证，有着不容怀疑的社会属性。因而事实上，我与叶霭玲的关系才是实的，与白丽筠反倒是虚的。

Fig.2　Crack length vs fatigue life

Table 2　Comparison of analytical solution with FEA result

Cracklengtha/mmParametersFSgfφfwQStressintensityfactor/(MPa·mm1/2)AnalyticalKforeachcrackDKforcrackincrementNumericalDKforcrackincrementFatiguelife/cyclesAnalyticalNumericalErrorforfatiguelife(%)1．01．041．01112．46118．81118．81117．350001．51．051．01112．46146．06132．44143．504131644411246．772．01．051．01112．46169．55157．81169．646573626940535．582．51．061．01112．46190．83180．19195．798213928543254．013．01．071．01112．46210．72200．77220．659399729600462．143．51．071．01112．46229．69220．20241．79102985410388430．874．01．081．01112．46248．07238．88262．11110025910994150．084．51．101．02112．46266．06257．06282．43115675411474450．815．01．111．02112．46283．82274．94298．38120293111856341．445．51．121．02112．46301．43292．62307．72124123212192941．766．01．131．02112．46318．96310．19315．57127338612519781．68

3　Casestudy on the prediction FCG in non-standard geometry

Paris[16] relates fatigue crack growth rate, da/dN with the range of SIF, K which is called as Paris equation as given in Eq.(2).

Bevel edge preparation was carried out on the end surface of the ring specimen as shown in Fig.3 and TIG welding. Material properties and chemical composition of parent material and weldment are given in Table 3. The Paris constants for the material is taken as m=3 and c=5.21×10-13.

Fig.3　Schematic of ring specimen

Table 3　Chemical compositions and mechanical properties

DescriptionChemicalcompositions(wt%)CSiMnFeOthersMechanicalpropertiesYieldstrength/MPaTensilestrength/MPaElongation(%)Parentmaterial(AISI1023)0．2370．1250．426990．21236042515%Weldmaterial(E7018)0．0710．4601．210980．25051257728．49

Fatigue life of the ring specimen with weld defect is carried out by numerical analysis. The finite element model of the ring specimen with weldment was modeled in CREO parametric software. The loading calculated using Roark’s formula (194N) is given as a fluctuating load at the top of the ring and the bottom portion of the ring type specimen is constrained in all degrees of freedom. Static analysis is carried out on the ring specimen using FEA software ANSYS and linked with Franc3D for fatigue analysis.

Weld defects like lack of penetration, lack of fusion, porosity are incorporated as elliptical crack at weld zone for further crack growth analysis. Weldments are tested for any weld defects using radiographic testing (Fig.4). Cluster porosity was model as a crack of 6 mm length and 2 mm depth, LOP was model as a crack of 8 mm length and 2 mm depth, LOF was model as a crack of 6 mm length and 2 mm depth (Fig.5).

对蔬菜产业发展的认识还有差距，重视程度不够，一些乡（镇）和职能部门对蔬菜产业发展的思路不够清晰，没有制定出指导当前和今后一个时期蔬菜产业发展的具体规划和实施方案；部分乡村干部的行政管理责任没有落实到位，说的多、管的少，没有建立严格的责任考评机制，一些基层干部不善于学习设施蔬菜种植新技术，对设施蔬菜的种植技术一知半解，缺乏对蔬菜种植的有力指导。

Fracture analysis of ring type specimens was carried out till the critical crack length of 4 mm beyond which the crack propagation will be unstable and the results are shown in Table 4 and Fig.6. From the results, it could be noted that Lack of Fusion is the most influencing defect on the fatigue life compared to other two defects.

Fig.4　Radiographic image of welded ring type specimen

Fig.5　FEM of ring specimen with crack

Table 4　Fatigue life of ring type specimen

TypeofwelddefectMeasuredsizeofinitialcrack/mmFatiguelife/cyclesLackoffusion(LOF)c-6，a-21644701Lackofpenetration(LOP)c-8，a-21890843Porosityc-6，a-22254706

Fig.6　Fatigue life of ring type specimen

4　Conclusions

This research aims at the evaluation of fatigue life of complex and nonstandard geometries with various weld defects like lack of fusion, lack of penetration and porosity using numerical analysis. Experimental evaluation on fatigue life of nonstandard components is time consuming and not cost effective. Analytical modeling of the weld joint can be carried out with stress intensity approach (LEFM) for determining the fatigue life of standard components. An attempt has been made to estimate the fatigue life based on LEFM and compared the life estimated using numerical analysis package Franc3D. The methodology has been extended for a pressure vessel simplified as a ring type specimen with same stress pattern. The ring specimen has been incorporated with weld defects as equivalent crack and the fatigue life has been estimated for the different weld defects. It was observed that the fatigue life of specimen with lack of fusion defect is less compared with other type of defects.

References

[1]　Dijkstra O D, Snijder H H, Rongen H J M. Assessment of the remaining fatigue life of defective welded joints. IABSE Reports, 1990, 59: 177-188.

[2]　Sumesh A, Thekkuden D, Thomas B B,et al. Acoustic signature based weld quality monitoring for SMAW process using data mining algorithms.Applied Mechanics and Materials, 2015, 813-814:1104-1113.

[3]　Maddox S J. Fatiguelife prediction methods in welded joints: case studies. Advances in Fatigue Science and Technology, 1989, 159: 569-583.

[4]　Qasim B, Emad K. Effect of V notch shape on fatigue life in steel beam made of high carbon steel alloy AISI 1078. International Journal of Modern Engineering Research, 2014, 4(6): 39-46.

[5]　Vencislav G, Stojan S, Aleksandar S, et al. Structural integrity assessment of pressure vessels with defect in welded joints. Scientific Technical Review, 2007, 57(3-4): 32-42.

[6]　Satoyuki T,Takahiro K,Hiroshi O.Study on crack propagation simulation of surface crack in welded joint structure, Marine structure, 2014, 39: 315-334.

[7]　Al-Mukhtar A M, Henkel S, Biermann H, et al. A finite element calculation of stress intensity factors of cruciform and butt welded joints for some geometrical parameters. Jordan Journal of Mechanical and Industrial Engineering, 2009, 3(4): 236-245.

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

[9]　Bouchard P O, Bay F, Chastel Y, et al. Crack propagation modelling using an advanced remeshing technique.Computer Methods in Applied Mechanics and Engineering, 2000, 189(3):723-742.

[10]　Balasubramanian V, Guha B.A new model to predict the fatigue life of flux cored arc welded cruciform joints containing LOP defects. Metals and materials, 1998, 4(4): 662-666.

[11]　Hobbarher A. Fracture and fatigue of welded joints and structure. Philadelphia: Woodhead Publishing Limited, 2011.

[12]　Pugno N, Ciavarella M, Cornetti P, et al. A generalized Pari’s law for fatigue crack growth. Journal of the Mechanics and Physics of Solids, 2006, 54(7): 1333-1349.

[13]　Magudeeswaran G, Balasubramanian V, Madhusudhan R G. Effect of welding processes and consumables on fatigue crack growth behavior of armour grade quenched and tempered steel joints.Defiance Technology, 2014, 10(1): 47-59.

[14]　Sanders W W, Lawrence F V. Fatigue behavior of aluminum alloy weldments. Fatigue Testing of Weldments, 1978, 648: 22-34.

[15]　Irwin G R. Analysis of stresses and strains near the end of a crack traversing a plate. Journal of Applied Mechanics, 1957, 24: 361-457.

[16]　Paris P C. The growth of crack due to variations in loads.Bethlehem: Lehigh University, 1960.

[17]　Warzynek P A,Carter B J,Banks-sills L.The M-integral for computing stress intensity factors in generally anisotropic materials.Washington: NASA Consultant Reports,2005.

[18]　Newman J C, Raju I S.Stress-intensity factor equations forcracks in three-dimensional finite bodiessubjected to tension and bending loads.Computational methods in the mechanics of fracture, 1986, 2: 311-334.

[19]　Budynas R G, Young W C. Roark's formulas for stress and strain.New York: McGraw-Hill Company, 2011.

6.轻度霉变玉米的处理。对轻度霉变的玉米可先粉碎后用清水浸泡,料水比为1:3(反复换水,直到浸泡水无色为止),之后烘干再与预混料混合使用。同时增加饲料中复合维生素,以加强肝的解毒功能,在配合全价日粮时减少玉米用量,添加5%的植物脂肪或固体磷脂以补充不足部分的能量,减少中毒机会。

色彩本身并没有先天的特定意义，它是在自然和人类社会演进和运用过程中形成了它的形态、功能和寓意性的内涵。因此，色彩在雕塑艺术中的运用和发挥是与创作者希望表达或揭示的意涵相互关联的，也是与作品应对的特定时代、场域及特定语境相对应的。犹如我们在古埃及的墓穴彩塑，庞培的装饰壁画或中国的古代的漆器、陶瓷等艺术那样，不同的色彩及相互关系均与其人文理念和美学内涵有着相应的关系。雕塑的材质、表面的色彩等形式元素均与其某种表征性或符号性的涵义相关。