J Fail. Anal. and Preven. (2012)12: 267-272 DOI10.1007/sl1668-0129555-3 TECHNICAL ARTICLE-PEER-REVIEWED mpact Simulation on Ductile Metal Pipe with Polymer Coating by a coupled finite element and meshfree Method Yi Gong Zhen-Guo Yang. Yu-Fei Wang Submitted: 12 September 2011/in revised form: 4 January 2012/Published online: 13 March 2012 C ASM International 2012 Abstract It is common knowledge that conventional exchange in petrochemical and power generation industries finite element method(FEM) has intrinsic limitations in [1, 2]. Its effect, particularly on the pipes in the familiar nalyzing large deformation problems like high-velocity form of metal substrate and with polymer coating for, impact, explosion, etc. because of mesh distortion and respectively, imparting structural strength and corrosion tangling: while these problems can be easily avoided by the resistance, deserves to be studied in-depth for prevention of meshfree method (MM), the latter involves greater com- failures like erosion damage [3], and even separation putation time. Therefore, in this article, in order to between two such parts. In general, research studies on simultaneously utilize the respective advantages of the two impact behavior on materials involve two types of methods nethods, a coupled simulation method between both FEM experiments and numerical simulations. In terms of the and MM was employed to analyze the high-velocity impact former one, a wealth of analytic models and relevant pre- on ductile metal pipe with polymer coating. The impacted diction equations has been obtained and reported, but most area with large deformation was discretized by SPH of them have limited applications since they are not able to (smoothed particle hydrodynamics) particles, a classic cover all kinds of target materials, e. g, ductile or brittle meshfree model and the re g section was modeled by [4, 5], with or without coatings [6]. As for the latter one, FEM meshes. By this method, the interfacial shear stresses besides the common superiorities like cost-, effort-, and between the coating and the substrate and the residual time-saving properties, finer meshes and less computation stresses beneath the contact points were studied, which times can be usually achieved by means of the prevailing results were compared with sole FEM and MM too loss of the targets and the final morphologies of their sur faces[9-12]. Nevertheless, the simplified 2D FEM model Keywords Finite nt· Meshfree has to obey certain assumptions as plane strain, plane stress, Smoothed particle amIcs(SPH)·Im axisymmetric, etc, and has the difficulty to solve the problem of multi-particle impact: the 3D FEM model also has its own limitations when under high-velocity impact, Introduction such as distortion of Lagrange meshes during large defor mation, and decrease of simulation accuracy due to cess that hits targets with oarseness of the 3D grids force and causes degradation on their surfaces, is a common In recent decades. the methods(△MMs)[13 action on pipes with applications in fluid delivery and heat gridless models of which relized by a set of scat- tered particles rather than of continuous meshes have been rapidly developed for specific applications iong. Z.-G. Yang(<) Department of Materials Science, Fudan University including crack propagation [14, 15], large deformation, Shanghai 200433, People's Republic of China explosion, fluids [16], impact [17, 18], and so on, which are e-mail:zgyang@fudan.edu.cn always encountered with mesh distortion and tanglin Spring
TECHNICAL ARTICLE—PEER-REVIEWED Impact Simulation on Ductile Metal Pipe with Polymer Coating by a Coupled Finite Element and Meshfree Method Yi Gong • Zhen-Guo Yang • Yu-Fei Wang Submitted: 12 September 2011 / in revised form: 4 January 2012 / Published online: 13 March 2012 ASM International 2012 Abstract It is common knowledge that conventional finite element method (FEM) has intrinsic limitations in analyzing large deformation problems like high-velocity impact, explosion, etc. because of mesh distortion and tangling; while these problems can be easily avoided by the meshfree method (MM), the latter involves greater computation time. Therefore, in this article, in order to simultaneously utilize the respective advantages of the two methods, a coupled simulation method between both FEM and MM was employed to analyze the high-velocity impact on ductile metal pipe with polymer coating. The impacted area with large deformation was discretized by SPH (smoothed particle hydrodynamics) particles, a classic meshfree model, and the remaining section was modeled by FEM meshes. By this method, the interfacial shear stresses between the coating and the substrate and the residual stresses beneath the contact points were studied, which would have referenced values in analyzing failure modes of components with similar composite structure. Then, the results were compared with sole FEM and MM too. Keywords Finite element Meshfree Smoothed particle hydrodynamics (SPH) Impact Introduction Impact, the dynamic process that hits targets with great force and causes degradation on their surfaces, is a common action on pipes with applications in fluid delivery and heat exchange in petrochemical and power generation industries [1, 2]. Its effect, particularly on the pipes in the familiar form of metal substrate and with polymer coating for, respectively, imparting structural strength and corrosion resistance, deserves to be studied in-depth for prevention of failures like erosion damage [3], and even separation between two such parts. In general, research studies on impact behavior on materials involve two types of methods: experiments and numerical simulations. In terms of the former one, a wealth of analytic models and relevant prediction equations has been obtained and reported, but most of them have limited applications since they are not able to cover all kinds of target materials, e.g., ductile or brittle [4, 5], with or without coatings [6]. As for the latter one, besides the common superiorities like cost-, effort-, and time-saving properties, finer meshes and less computation times can be usually achieved by means of the prevailing two-dimensional (2D) finite element method (FEM) [7, 8], while the 3D FEM model could even give the real weight loss of the targets and the final morphologies of their surfaces [9–12]. Nevertheless, the simplified 2D FEM model has to obey certain assumptions as plane strain, plane stress, axisymmetric, etc., and has the difficulty to solve the problem of multi-particle impact; the 3D FEM model also has its own limitations when under high-velocity impact, such as distortion of Lagrange meshes during large deformation, and decrease of simulation accuracy due to coarseness of the 3D grids. In recent decades, the meshfree methods (MMs) [13], gridless models of which are discretized by a set of scattered particles rather than a series of continuous meshes, have been rapidly developed for specific applications including crack propagation [14, 15], large deformation, explosion, fluids [16], impact [17, 18], and so on, which are always encountered with mesh distortion and tangling Y. Gong Z.-G. Yang (&) Y.-F. Wang Department of Materials Science, Fudan University, Shanghai 200433, People’s Republic of China e-mail: zgyang@fudan.edu.cn 123 J Fail. Anal. and Preven. (2012) 12:267–272 DOI 10.1007/s11668-012-9555-3
268 J Fail. Anal and Preven.(2012)12: 267-272 problems in FEM. SPH (smoothed particle hydrodynamics) Table 1 Material constants and Johnson-Cook parameters in the [19, 20], DEM(diffuse element method) [21], EFG model (element-free Galerkin)[22], RKPM(reproducing kernel Materials E GPa D A MPa B MPa n particle method)[23]. MLSRKM(moving least-square reproducing kernel method)(24, 25), etc are all the rep- 2024-T3Al 710.34 369 6840730. I7 resentative models in MMs. However, such MMs usually PU coatin 2.30.151461500.4980.097 ost more computation times than FEM. Consequently, E and D denote the elastic modulus and Poisson,s ratio several algorithms coupled with both FEM and MM have been put forward to utilize the respective advantages of each of them [26-31. In this article, a coupled simulation method between FEM and the meshfree model (MM) SPH was employed via the impact effect on ductile metal pipe with polymer coating [32]. Specifically, the impacted area(involving both the coating and the substrate) with large deformation was dis- cretized by SPH particles while the other section with les deformation was still modeled by FEM meshes. Two dif- ferent impact angles of 90 and 45(representing the normal and the oblique impacts, respectively) were imposed, and their functions on energy evolution, plastic strain, and stresses distribution of the targets during the impact process ere systematically analyzed. Finally, results of the normal impact with this coupled method were also compared with that of the sole FEm and the sole SPh method to discuss their Fig. 1 Schematic diagram of the impact model individual advantages and disadvantages Fig. 1. Specifically, the impacted area was modeled as SPH particles including 1, 350 for polymer coating and 3, 600 for metal substrate, and their uniform masses were, respectively, Modeling 3.733 10 and 1.385 x 10 kg per particle. The other section of the target was still modeled by FEM meshes with The impact process was simulated by means of 3D explicit element of SOLID 164. The impact velocity was 80 m/s, and dynamic analysis in ANSYS/LS-DYNA I0.0. The Johnson- two impact angles of 90 and 45 were exerted, respectively, Cook(-C)(33, 34 ]viscoplastic material model was applied representing the normal and the oblique impacts. The solu- for the flow stress behavior of the target materials and the tion time was set 1.5 x t. where t was the time that the von Mises flow stress was accordingly expressed in Eq 1: erodent particle needed to contact the target surface Gr=A+B(py 1+cIn(p As for the constrained conditions, the tied-nodes-to- surface contact was established between the sph scheme (Eg 1) and the FEM meshes to couple the two sections. Besides, the where A and B are the yield stress constant and the strain dent particle and the SPH section. In order to simplify the hardening constant; n,C, and m are constants; EP is the simulation, only a half-model was evaluated, and hence, the equivalent plastic strain; aP is the plastic strain rate, and Eo is constraints and SPH symmetry planes were set for the FEM the reference strain rate. T and Tm are the temperature and and SPH sections at the boundaries to achieve the symmetry the melting point of the target material, while T is the room conditions. Also, all of the bottom and outside nodes of the temperature. Aluminum alloy 2024-T3 Al and polyurethane target materials were defined to non-reflecting boundaries (PU) were chosen as the metal substrate and the polymer coating, respectively, material constants as well as J-C parameters of which are listed in Table 1 [9, 35]. The ero- Results and Discussion dent was 2 mm(diameter)aluminum spherical particle with density of 2, 770 kg/mand elastic modulus of 71 GPa. oupled Method In terms of the model geometry, thicknesses of the Al substrate and PU coating were 1. 4 and 0. I mm, respectively. During the simulation process, the internal and the kinetic The size of the whole target was 1.5 66 mm, as shownin energies evolutions of the normal (90%)and the oblique
problems in FEM. SPH (smoothed particle hydrodynamics) [19, 20], DEM (diffuse element method) [21], EFG (element-free Galerkin) [22], RKPM (reproducing kernel particle method) [23], MLSRKM (moving least-square reproducing kernel method) [24, 25], etc. are all the representative models in MMs. However, such MMs usually cost more computation times than FEM. Consequently, several algorithms coupled with both FEM and MM have been put forward to utilize the respective advantages of each of them [26–31]. In this article, a coupled simulation method between FEM and the meshfree model (MM) SPH was employed via the commercial software ANSYS/LS-DYNA to study the impact effect on ductile metal pipe with polymer coating [32]. Specifically, the impacted area (involving both the coating and the substrate) with large deformation was discretized by SPH particles while the other section with less deformation was still modeled by FEM meshes. Two different impact angles of 90 and 45(representing the normal and the oblique impacts, respectively) were imposed, and their functions on energy evolution, plastic strain, and stresses distribution of the targets during the impact process were systematically analyzed. Finally, results of the normal impact with this coupled method were also compared with that of the sole FEM and the sole SPH method to discuss their individual advantages and disadvantages. Modeling The impact process was simulated by means of 3D explicit dynamic analysis in ANSYS/LS-DYNA 10.0. The Johnson– Cook (J–C) [33, 34] viscoplastic material model was applied for the flow stress behavior of the target materials, and the von Mises flow stress was accordingly expressed in Eq 1: rf ¼ A þ Bðe p Þ n ½ 1 þ c ln e_ p e_0 1 T Tr Tm Tr m ðEq 1Þ where A and B are the yield stress constant and the strain hardening constant; n, c, and m are constants; ep is the equivalent plastic strain; e_ p is the plastic strain rate, and e_0 is the reference strain rate. T and Tm are the temperature and the melting point of the target material, while Tr is the room temperature. Aluminum alloy 2024-T3 Al and polyurethane (PU) were chosen as the metal substrate and the polymer coating, respectively, material constants as well as J–C parameters of which are listed in Table 1 [9, 35]. The erodent was 2 mm (diameter) aluminum spherical particle with density of 2,770 kg/m3 and elastic modulus of 71 GPa. In terms of the model geometry, thicknesses of the Al substrate and PU coating were 1.4 and 0.1 mm, respectively. The size of the whole target was 1.5 9 696 mm, as shown in Fig. 1. Specifically, the impacted area was modeled as SPH particles including 1,350 for polymer coating and 3,600 for metal substrate, and their uniform masses were, respectively, 3.733 9 107 and 1.385 9 106 kg per particle. The other section of the target was still modeled by FEM meshes with element of SOLID 164. The impact velocity was 80 m/s, and two impact angles of 90 and 45 were exerted, respectively, representing the normal and the oblique impacts. The solution time was set 1.5 9 t, where t was the time that the erodent particle needed to contact the target surface. As for the constrained conditions, the tied-nodes-tosurface contact was established between the SPH scheme and the FEM meshes to couple the two sections. Besides, the eroding-nodes-to-surface contact was defined between erodent particle and the SPH section. In order to simplify the simulation, only a half-model was evaluated, and hence, the constraints and SPH symmetry planes were set for the FEM and SPH sections at the boundaries to achieve the symmetry conditions. Also, all of the bottom and outside nodes of the target materials were defined to non-reflecting boundaries. Results and Discussion Coupled Method During the simulation process, the internal and the kinetic energies evolutions of the normal (90) and the oblique Table 1 Material constants and Johnson-Cook parameters in the model Materials E, GPa t A, MPa B, MPa n cm 2024-T3 Al 71 0.34 369 684 0.73 0.0083 1.7 PU coating 2.3 0.15 146 150 0.498 0.097 … E and t denote the elastic modulus and Poisson’s ratio Fig. 1 Schematic diagram of the impact model 268 J Fail. Anal. and Preven. (2012) 12:267–272 123
J Fail. Anal. and Preven. (2012)12: 267-272 Kinetic Energy of 45" Impac (45%)impact samples are presented in Fig. 2. As shown in 0.04 Internal Energy of 45'"Impact the curves, the initial kinetic energies of the erodent pa Kinetic Energy of 90 Impact ticles converted into the internal energies of the target Internal Energy of 90 Impact he rebound kinetic energies of the particles. As for ductile target, the internal energy commonly appears in the form of plastic deformation. Thus, based on the fact as shown in 002 Fig. 2 that the internal energy absorbed by target under 90 was higher than that under 45 it can be concluded that greater plastic deformation was induced from normal impact than from oblique impact. This phenomenon has been verified through Fig. 3 as well: larger deformed area 0.00 of the coatings after normal impact than after oblique Impact Figure 4a and b show the interfacial shear stresses dis- Fig 2 Intemal and kinetic energies evolutions under normal and tributions between coating and substrate during impact blique impact The shear stresses of the two samples both reached their 1460 7300 5840 81 t1. Ous Pa)0000 1.0ps 022 = 8760 g760 2921 29 1460 1168 1168 300 5840 t=3.0 Pa)0000 MPa)0000 168 22 4381 t7Ous t=7.Ous P)0000 Fig 3 Stress distribution of the coating:(a) normal impact;(b)oblique impact Spring
(45) impact samples are presented in Fig. 2. As shown in the curves, the initial kinetic energies of the erodent particles converted into the internal energies of the targets and the rebound kinetic energies of the particles. As for ductile target, the internal energy commonly appears in the form of plastic deformation. Thus, based on the fact as shown in Fig. 2 that the internal energy absorbed by target under 90 was higher than that under 45, it can be concluded that greater plastic deformation was induced from normal impact than from oblique impact. This phenomenon has been verified through Fig. 3 as well: larger deformed area of the coatings after normal impact than after oblique impact. Figure 4a and b show the interfacial shear stresses distributions between coating and substrate during impact. The shear stresses of the two samples both reached their Fig. 3 Stress distribution of the coating: (a) normal impact; (b) oblique impact Fig. 2 Internal and kinetic energies evolutions under normal and oblique impact J Fail. Anal. and Preven. (2012) 12:267–272 269 123
J Fail. Anal and Preven.(2012)12: 267-272 maximums at 3 us after beginning of the impact, but their residual stress. This mechanism has been verified from locations were, respectively, about 0.3 mm(normal)and Fig. 5 showing that higher(compressive)residual stresses 0.2 mm(oblique)away from the contact points. This result accumulated in the upper part of the metal substrate( the revealed that the crater area under oblique impact was range from 0. 1 to 0.5 mm in x-axis), and then transformed relatively smaller, testifying the phenomena in Fig 3. into the lower(tensile) residual stresses when the depth However, values of these shear stresses, which were grew to the lower part of the substrate(the range from 0.5 responsible for the buckling of the coating, were greater to 1.5 mm in x-axis). This fact revealed that the ductile under oblique impact than those under normal impact. In metal substrate provided adequate structural strength under other words, under oblique impact, higher possibilities impact. Meanwhile, comparatively speaking, the residual would be introduced for separation between coating and stress from normal impact(90%)was a little bit larger than substrate, and also for removal of coating material after that from oblique impact. This phenomenon may be attributed to the fact described as follows as was dis- Actually, for both normal and oblique impacts, great cussed above, the greater the plastic deformation that compressive stresses were engendered upon coatings and occurred in the target under normal impact, the larger the substrates because of erodent particles striking. Through residual stress that was left the shot peening experiment on ductile material ANSI 4340 steel, Torres and Voorwald [36] pointed out that such Comparison with FEM compressive stress only located at limited areas on the target surface, the compressive residual stress increased In this section, results of the normal impact from the below the surface until reaching a maximum depth, and coupled method will be compared to those from the sole then tended to decrease. transforming into the tensile fem and the sole sph models table 2 tabulates the dif ferences between the three models. Indeed. the computation time of sole FEM model was much less than those with SPH particles, and the resulting curve was even smoother. as seen in Fig. 5. However. obvious distortion occurred on the FEM meshes within the impacted area Fig. 6a), while the crater of the SPh particles was free of this problem(Fig. 6b). This could be explained by the fact that the FEM elements were connected by nodes, and thus its result was relatively smooth but inaccurate because of element distortion under such high-velocity impact. Com- prehensively speaking, the coupled method yielded t=9gs」 reasonable results. and also involved cost-effective com- Distance(mm) SPH90° △SPH45° 一FEM90° Distance(mm) Distance beneath the contact point(mm) Fig 4 Interfacial shear stresses between coating and substrate: (a) Fig. 5 Residual stresses beneath the contact points(along the path of
maximums at 3 ls after beginning of the impact, but their locations were, respectively, about 0.3 mm (normal) and 0.2 mm (oblique) away from the contact points. This result revealed that the crater area under oblique impact was relatively smaller, testifying the phenomena in Fig. 3. However, values of these shear stresses, which were responsible for the buckling of the coating, were greater under oblique impact than those under normal impact. In other words, under oblique impact, higher possibilities would be introduced for separation between coating and substrate, and also for removal of coating material after impact. Actually, for both normal and oblique impacts, great compressive stresses were engendered upon coatings and substrates because of erodent particles striking. Through the shot peening experiment on ductile material ANSI 4340 steel, Torres and Voorwald [36] pointed out that such compressive stress only located at limited areas on the target surface, the compressive residual stress increased below the surface until reaching a maximum depth, and then tended to decrease, transforming into the tensile residual stress. This mechanism has been verified from Fig. 5 showing that higher (compressive) residual stresses accumulated in the upper part of the metal substrate (the range from 0.1 to 0.5 mm in x-axis), and then transformed into the lower (tensile) residual stresses when the depth grew to the lower part of the substrate (the range from 0.5 to 1.5 mm in x-axis). This fact revealed that the ductile metal substrate provided adequate structural strength under impact. Meanwhile, comparatively speaking, the residual stress from normal impact (90) was a little bit larger than that from oblique impact. This phenomenon may be attributed to the fact described as follows. As was discussed above, the greater the plastic deformation that occurred in the target under normal impact, the larger the residual stress that was left. Comparison with FEM In this section, results of the normal impact from the coupled method will be compared to those from the sole FEM and the sole SPH models. Table 2 tabulates the differences between the three models. Indeed, the computation time of sole FEM model was much less than those with SPH particles, and the resulting curve was even smoother, as seen in Fig. 5. However, obvious distortion occurred on the FEM meshes within the impacted area (Fig. 6a), while the crater of the SPH particles was free of this problem (Fig. 6b). This could be explained by the fact that the FEM elements were connected by nodes, and thus its result was relatively smooth but inaccurate because of element distortion under such high-velocity impact. Comprehensively speaking, the coupled method yielded reasonable results, and also involved cost-effective computation time. Fig. 4 Interfacial shear stresses between coating and substrate: (a) normal impact; (b) oblique impact Fig. 5 Residual stresses beneath the contact points (along the path of z-axis) 270 J Fail. Anal. and Preven. (2012) 12:267–272 123
J Fail. Anal. and Preven. (2012)12: 267-272 Table 2 Comparison between coupled method and sole FEM, and sole SPH methods Methods Number of elements Number of SPh Results CPU FEM model Sole sph model Coupled model 19,242 4.950 352 836 The results in MPa are the equivalent stresses of the node at the impact point under normal impact Meanwhile, Yi Gong also appreciate the help fron Shaofan Li's research group in Department of Civil Engineering at University of California, Berkele e Short- term International Exchange programme fund fo Students of Fudan universit Refere I. Gong, Y, Yang, Z.G., Yuan, J Z: Failure analysis of leakage on titanium tubes within heat exchangers in a nuclear power plant. Part II. Mechanical degradation. Mater. Corros. 63, 18-28(2012) 2. Gong, Y Zhong, J, Yang, Z.G.: Failure analysis of bursting on Mater.Des.31,4258-4268(2010) 3. Wood, R.J. K: The sand erosion performance of coatings. Mater. Des.20,179-191(1999 4. Hutchings, I M. Particle erosion of ductile metals: a mechanism of material removal. Wear 27, 121-128(1974) 5. Hutchings, I.M.: Ductile-brittle transitions and wear maps for the rosion and abrasion of brittle materials. J. Phys. D Appl. Phys. 25,A212-A22l(199 6. Barkoula N.M. Kocsis. J : Review the solid le erosion of polymers an heir composites. J Mater. Sci. 37, 3807-3820(2002) 7. Shimizu, K. et al.: FEM analysis of the dependency on impact angle during erosive wear. Wear 233-235, 157-159(1999 8. Chen, Q, Li, D.Y.: Computer simulation of solid particle erosion. ear254.203-210(2003 9. Zouar, B, Touratier, M. Simulation of organic coating removal wear253,488-497(200 10. Har, J, Fulton, R.E.: A parallel finite element procedure for Fig 6 Plastic strain of the area:(a)from FEM; (b)from contact-impact problems. Eng Comput. 19, 67-84(2003) coupled method(sole SPH meth I1. Griffin, D, Daadbin, A. Datta, S: The dew f a three. dimensional finite element model for solid pa oslon on an alumina scale/MA956 substrate. Wear 256 (2004) Conclusions 12. Aquaro, D: Erosion due to the impact of solid particles of materials resistant at high temperature. Meccanica 41, 539-551 (1)A coupled model with both FEM and MM was utilized to analyze the high-velocity impact on metal 13. Nguyen, V.P., Rabczuk, T, Bordas, S, Duflot, M: Meshless substrate pipe with polymer coating, effectively a review and computer implementation aspects. Math Simulat.79,763-813(2008) avoiding the mesh distortion and tangling problems 14. Rabczuk, T, Belytschko, T: Cracking particles: a simplified in sole fem simulation meshfree method for arbitrary evolving cracks. Int J. Num methods eng.61,2316-2343(2004) (2) Two impact angles of 90 and 45 were, respec- 15. Rabczuk, T, Belytschko, T: A three-dimensional large defor- tively, applied via this coupled model to compare their influences on energy evolutions, surface mor- ation meshfree method for arbitrary evolving cracks. Comput. Method Appl.M196,2777-279902007) stresses distributions 17. Zeng. X W. Li. S.F.: A multiscale cohesive zone model and Acknowledgments This study was supported by the Shanghai mulations of fracture. Comput. Methods Appl. Mech. Eng. 199, Leading Academic Discipline Project (Project Number: B113 7-556(2010) Spring
Conclusions (1) A coupled model with both FEM and MM was utilized to analyze the high-velocity impact on metal substrate pipe with polymer coating, effectively avoiding the mesh distortion and tangling problems in sole FEM simulation. (2) Two impact angles of 90 and 45 were, respectively, applied via this coupled model to compare their influences on energy evolutions, surface morphologic transformations, and shear and residual stresses distributions. Acknowledgments This study was supported by the Shanghai Leading Academic Discipline Project (Project Number: B113). Meanwhile, Yi Gong also appreciate the help from Professor Shaofan Li’s research group in Department of Civil and Environmental Engineering at University of California, Berkeley under the Shortterm International Exchange Programme Fund for Doctoral Students of Fudan University. References 1. Gong, Y., Yang, Z.G., Yuan, J.Z.: Failure analysis of leakage on titanium tubes within heat exchangers in a nuclear power plant. Part II. Mechanical degradation. Mater. Corros. 63, 18–28 (2012) 2. Gong, Y., Zhong, J., Yang, Z.G.: Failure analysis of bursting on the inner pipe of a jacketed pipe in a tubular heat exchanger. Mater. Des. 31, 4258–4268 (2010) 3. Wood, R.J.K.: The sand erosion performance of coatings. Mater. Des. 20, 179–191 (1999) 4. Hutchings, I.M.: Particle erosion of ductile metals: a mechanism of material removal. Wear 27, 121–128 (1974) 5. Hutchings, I.M.: Ductile-brittle transitions and wear maps for the erosion and abrasion of brittle materials. J. Phys. D Appl. Phys. 25, A212–A221 (1992) 6. Barkoula, N.M., Karger-Kocsis, J.: Review processes and influencing parameters of the solid particle erosion of polymers and their composites. J. Mater. Sci. 37, 3807–3820 (2002) 7. Shimizu, K., et al.: FEM analysis of the dependency on impact angle during erosive wear. Wear 233–235, 157–159 (1999) 8. Chen, Q., Li, D.Y.: Computer simulation of solid particle erosion. Wear 254, 203–210 (2003) 9. Zouari, B., Touratier, M.: Simulation of organic coating removal by particle impact. Wear 253, 488–497 (2002) 10. Har, J., Fulton, R.E.: A parallel finite element procedure for contact-impact problems. Eng. Comput. 19, 67–84 (2003) 11. Griffin, D., Daadbin, A., Datta, S.: The development of a threedimensional finite element model for solid particle erosion on an alumina scale/MA956 substrate. Wear 256, 900–906 (2004) 12. Aquaro, D.: Erosion due to the impact of solid particles of materials resistant at high temperature. Meccanica 41, 539–551 (2006) 13. Nguyen, V.P., Rabczuk, T., Bordas, S., Duflot, M.: Meshless methods: a review and computer implementation aspects. Math. Comput. Simulat. 79, 763–813 (2008) 14. Rabczuk, T., Belytschko, T.: Cracking particles: a simplified meshfree method for arbitrary evolving cracks. Int. J. Numer. Methods Eng. 61, 2316–2343 (2004) 15. Rabczuk, T., Belytschko, T.: A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Comput. Method Appl. M 196, 2777–2799 (2007) 16. Rabczuk, T., Gracie, R., Song, J.H., Belytschko, T.: Immersed particle method for fluid–structure interaction. Int. J. Numer. Methods Eng. 81, 48–71 (2010) 17. Zeng, X.W., Li, S.F.: A multiscale cohesive zone model and simulations of fracture. Comput. Methods Appl. Mech. Eng. 199, 547–556 (2010) Table 2 Comparison between coupled method and sole FEM, and sole SPH methods Methods Number of elements Number of SPH Resultsa , MPa CPU time, s FEM model 24,192 … 340 242 Sole SPH model … 24,192 354 2,334 Coupled model 19,242 4,950 352 836 a The results in MPa are the equivalent stresses of the node at the impact point under normal impact Fig. 6 Plastic strain of the impacted area: (a) from FEM; (b) from coupled method (sole SPH method was same) J Fail. Anal. and Preven. (2012) 12:267–272 271 123
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