0

Microsoft word vu van thai luan van thac si

76 4 0
  • Microsoft word   vu van thai   luan van thac si

Tài liệu hạn chế xem trước, để xem đầy đủ mời bạn chọn Tải xuống

Tài liệu liên quan

Thông tin tài liệu

Ngày đăng: 12/05/2022, 10:56

VIET NAM NATIONAL UNIVERSITY HO CHI MINH CITY HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY VU VAN THAI THE EXTENDED MESHFREE METHOD FOR CRACKED HYPERELASTIC MATERIALS PH NG PHÁP KHƠNG L IM R NG CHO BÀI TỐN N T TRONG V T LI U SIÊU ÀN H I MAJOR: ENGINEERING MECHANICS MAJOR CODE: 8520101 MASTER THESIS HO CHI MINH CITY, January 2022 CƠNG TRÌNH TR Cán b h NG C HOÀN THÀNH T I I H C BÁCH KHOA – HQG – HCM ng d n khoa h c: TS Nguy n Thanh Nhã Cán b ch m nh n xét 1: PGS.TS Nguy n Hoài S n Cán b ch m nh n xét 2: TS Nguy n Ng c Minh Lu n v n th c s đ c b o v t i Tr ngày 15 tháng 01 n m 2022 ng i h c Bách Khoa, HQG Tp HCM Thành ph n H i đ ng đánh giá lu n v n th c s g m: Ch T ch H i Th Ký H i ng: PGS TS Tr ng Tích Thi n ng: TS Ph m B o Toàn Ph n Bi n 1: PGS TS Nguy n Hoài S n Ph n Bi n 2: TS Nguy n Ng c Minh y Viên: TS Nguy n Thanh Nhã Xác nh n c a Ch t ch H i đ ng đánh giá LV Tr ngành sau lu n v n đ c s a ch a (n u có) CH T CH H I PGS TS Tr NG ng Tích Thi n ng Khoa qu n lý chuyên TR NG KHOA KHOA H C NG D NG PGS TS Tr ng Tích Thi n I H C QU C GIA TP.HCM NG I H C BÁCH KHOA TR C NG HÒA XÃ H I CH NGH A VI T NAM c l p - T - H nh phúc NHI M V LU N V N TH C S H tên h c viên: V V N THÁI MSHV: 1970187 Ngày, tháng, n m sinh: 28/11/1991 N i sinh: Kiên Giang Chuyên ngành: C K THU T Mã s : 8520101 I TÊN (Ph TÀI : The extended meshfree method for cracked hyperelastic materials ng pháp không l i m r ng cho toán n t v t li u siêu đàn h i) II NHI M V VÀ N I DUNG: Xây d ng ph ng pháp không l i cho toán bi n d ng l n c a v t li u siêu đàn h i, toán n t v t li u siêu đàn h i Tính tốn tr ng chuy n v , ng su t, tích phân J, h s k so sánh v i l i gi i tham kh o ánh giá k t qu thu đ c t ph ng pháp đ c đ xu t III NGÀY GIAO NHI M V : 06/09/2021 IV NGÀY HOÀN THÀNH NHI M V : 22/05/2022 V CÁN B H NG D N: TS Nguy n Thanh Nhã Tp HCM, ngày 09 tháng 03 n m 2022 CÁN B H NG D N TS Nguy n Thanh Nhã TR CH NHI M B MÔN ÀO T O PGS TS V Cơng Hịa NG KHOA KHOA H C NG D NG Acknowledgement The completion of this thesis could not has been possible without guidance of my thesis supervisor Dr Nha Thanh Nguyen I would like to express my sincere gratitude to him for his continuous support, patience, enthusiasm during the process of my Master study Besides my thesis supervisor, I am very grateful to the lecturers of Department of Engineering Mechanics for their lectures, advice while I am studying Master program I am also thankful to my friends Master Vay Siu Lo, Master student Dung Minh Do, Master student Binh Hai Hoang for their listening and comments, which help me have more ideas to write my thesis Finally, I sincerely and genuinely thank my dear parents, my siblings, my beautiful wife, and my lovely daughter for their love, care, and giving me motivation throughout my life This thesis is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2019.237 i Abstract The simulation of finite strain fracture is still an open problem and appeal to many researchers in computational engineering field due to its complication of modeling and finding solution In this thesis, the non-linear fracture analysis of rubber-like materials is studied The extended radial point interpolation method (XRPIM), which combines both the Heaviside function and the branch function is employed to capture the discontinuous deformation field, as well as stress singularity around the crack tip in a hyperelastic material with incompressible state The support domains are generated to approximate displacement field and its derivatives using shape function of radial point interpolation method (RPIM) For the analysis implementation, total Largange formulation is taken into XRPIM and the numerical integration is performed by Gaussian Quadrature The tearing energy that controls the fracture of rubber-like materials is investigated by computing J-integral which is commonly used in linear fracture mechanics k parameter that is constant for a given state of strain and the displacement field surrounding two crack edges are also studied Moreover, the behavior of a hyperelastic solid with both compressible and nearly-incompressible state are analyzed by using integrated radial basis functions (iRBF) meshfree method The efficiency and accuracy of the presented method are demonstrated by several numerical examples, in which results are compared with the reference solutions ii Tóm t t lu n v n Mô ph ng phá h y bi n d ng l n v n m t v n đ m thu hút nhi u nhà nghiên c u l nh v c c h c tính tốn s ph c t p vi c mơ hình hóa tìm l i gi i Lu n v n th c hi n nghiên c u s phá h y phi n c a v t li u nh cao su b ng vi c s d ng ph ng pháp n i suy m h ng kính m r ng (XRPIM), có s k t h p hàm “Heaviside” hàm “Branch” đ bi u di n s b t liên t c c a tr ng chuy n v s suy bi n c a tr n t v t li u siêu đàn h i ng ng su t xung quanh đ nh v t tr ng thái không nén đ c Các mi n ph tr đ c t o đ x p x tr ng chuy n v đ o hàm c a chúng thông qua vi c s d ng hàm d ng c a ph ng pháp n i suy m h tích, XRPIM đ ng kính (RPIM) c áp d ng vào công th c “Largange” t ng tích phân s đ th c hi n b ng “Gaussian Quadrature” N ng l v t li u nh cao su đ đ th c thi s phân c ng xé ki m soát s phá h y c a c kh o sát thông qua vi c tính tích phân J, đ i l ng c s d ng r ng rãi c h c phá h y n tính Lu n v n c ng th c hi n kh o sát v tr ng chuy n v lân c n mép v t n t thông s k, đ i l s đ i v i m t tr ng thái bi n d ng đ ng x c a m t v t r n siêu đàn h i nén đ c b ng ph ng pháp không l c cho Ngồi ra, lu n v n c ng trình bày v c hai tr ng thái nén đ c g n nh không i s d ng hàm c s h phân (iRBF) S hi u qu xác c a ph ví d s , k t qu đ ng h ng ng pháp đ c gi i thích thơng qua c so sánh v i l i gi i tham chi u iii ng kính tích Declaration I declare that this thesis is the result of my own research except as cited in the references which has been done after registration for the degree of Master in Engineering Mechanics at Ho Chi Minh city University of Technology, VNU – HCM, Viet Nam The thesis has not been accepted for any degree and is not concurrently submitted in candidature of any other degree Author V V n Thái iv Contents List of Figures vii List of Tables x List of Abbreviations and Nomenclatures xi INTRODUCTION 1.1 State of the art 1.2 Scope of the study 1.3 Research objectives 1.4 Author’s contributions 1.5 Thesis outline METHODOLOGY 2.1 Hyperelastic material 2.1.1 Constitutive equations of hyperelastic material 2.1.2 Fracture analysis of hyperelastic material 11 2.2 Meshfree shape functions construction 13 2.2.1 Radial Point Interpolation Method (RPIM) 13 2.2.2 integrated Radial Basis Functions Method (iRBF) 17 2.3 The XRPIM for crack problem in hyperelastic bodies 22 2.3.1 Enriched approximation of the displacement field by XRPIM 22 2.3.2 Weak form for nonlinear elastic problem and discrete equations 24 IMPLEMENTATION 29 v 3.1 Numerical implementation procedure 29 3.2 Computation procedure of K maxtrix and fint matrix 30 3.3 Computation procedure of B matrix and O matrix 31 NUMERICAL EXAMPLES 34 4.1 Non-cracked hyperelastic solid 34 4.1.1 Inhomogeneous compression problem 34 4.1.2 Curved beam problem 39 4.2 Cracked hyperelastic solid 42 4.2.1 Rectangular plate with an edge crack under prescribed extension 42 4.2.2 Square plate with an edge crack under prescribed extension 44 4.2.3 Nonlinear Griffith problem 47 4.2.4 Square plate with an inclined central crack 52 CONCLUSION AND OUTLOOK 55 5.1 Conclusions 55 5.2 Future works 56 List of Publications 57 REFERENCES 58 vi List of Figures 2.1: Undeformed and deformed geometries of a body 2.2: Contour used for J-intergal 13 2.3: Local support domains and field node for RPIM 14 2.4: Local support domains and field node for iRBF 18 2.5: Field node surrounding the crack line 23 2.6: Distance r and angle  of xk in local coordinate system 24 2.7: 2D hyperelastic solid with a crack and boundary conditions 24 3.1: The algorithm of Numerical implementation procedure 32 3.2: The algorithm for computing B matrix and O matrix 33 4.1: Inhomogeneous compression problem 35 4.2: Percent of compression at point M for various values of distributed force in the compressible inhomogeneous compression problem 36 4.3: Percent of compression at point M for various values of distributed force in the nearly-incompressible inhomogeneous compression problem 36 4.4: Deformed configuration of the plate in the compressible state with f = 200 N/mm2 (magenta grid indicates the undeformed configuration of the plate) 37 4.5: Deformed configuration of the plate in the nearly-incompressible state with f = 250 N/mm2 (magenta grid indicates the undeformed configuration of the plate) 37 vii Table 4.1: The effect of domain size chosen to compute J-integral on its results Crack length Size of J-integral domains a 2.5 × dc 3.5 × dc 4.5 × dc 5.0 × dc 6.0 × dc 0.5 4.262 4.191 4.169 4.185 4.171 0.6 4.808 4.655 4.627 4.633 4.650 0.7 4.990 4.920 4.946 4.978 5.020 0.8 5.415 5.216 5.212 5.248 5.297 0.9 5.325 5.292 5.347 5.384 5.450 1.0 5.707 5.488 5.501 5.545 5.608 4.2.3 Nonlinear Griffith problem In this section, a Griffith cracked membrane with dimensions W = H = mm subjected to two fundamental loadings as shown in Figs 4.18 (a) and (b) is considered to investigate k parameter which relates the energy loss due to the crack and strain energy of the sample without crack The membrane has a central crack of length a = 0.5 mm According to [3], k is defined as k J 2Wc (4.1) where W and c stand, respectively, for the uniform strain energy density of the sample in the undeformed state and the initial half length of the crack For these examples, a MPa tensile loading is prescribed and the symmetry of the model is ignored to check the difference of J-integral for both tips The domain size chosen to compute J-integral is 6dc The model uses 40 × 40 scatter nodes as Fig 4.19 for simulation and the shear modulus of incompressible Neo-Hookean model is chosen as 0.4225 MPa 47 Figure 18: Nonlinear Griffith problem: (a) uniaxial extension; (b) equibiaxial extension Figure 19: Nodal distribution of nonlinear Griffith problem For uniaxial extension, the evolution of J-integral for both tips versus the stretch level is depicted in Fig 4.20 and compared with XFEM approach [3] Then k parameter of the right crack tip as a function of stretch level is computed based on Eq (4.1) and plotted in Fig 4.21 The result of k parameter of the right crack tip is compared with that obtained by Lake [23] and Yeoh [24] In 1970, Lake derived k for a central crack help in simple extension based on the energy loss due to the opening of a crack to the energy that required to closed it again k   (4.2) where  is the principal extension ratios k parameter is also approximated by Yeoh in 2002 using crack surface displacement technique with assuming that the crack 48 opens to from an elliptical opening of length 2a* and height 2b* as Fig 4.22 He obtained k  y b* (4.3) 4Wc where W and c are defined in Eq (4.1)  y is the applied stress in the normal direction to the crack and given as    W     (4.4)    WI1   y2 WI  (4.5)  y   y   y2    I1  W  y I for simple tension,  y   y   y5  for equibiaxial tension In Eqs (4.4) and (4.5), I1 and I2 are defined in Chapter Figure 20: Variations of J-integral with respect to the elongation in the case of uniaxial extension Comparison of XFEM solution [3] with XRPIM method results 49 Figure 21: Variations of k with respect to the elongation in the case of uniaxial extension Comparison of XRPIM method results with Lake [23] and Yeoh [24] According to Fig 4.20, the maximum values of J-integral at left and right crack tip are 0.878932 and 0.878932, respectively So the evolution of J-integral for both tips versus the stretch level is closely identical Furthermore, the numerical results (both J-integral and k value) obtained from the developed method of uniaxial extension example are in good agreement with the reference solutions Figure 22: Yeoh’s assumsion of the crack’s deformation For equibiaxial extension example, to verify the assumption of Yeoh [24] about forming an elliptical opening of the crack, the deformed configuration surrounding 50 two crack edges of the plate is plotted in Fig 4.23 It is clearly to see that both results are similar to each other Figure 23: Deformed configuration surrounding two crack edges in the case of equabiaxial extension Comparison of XRPIM method results with Yeoh [24] Fig 4.24 shows k parameter calculated at the right crack tip for equibiaxial extension example The result is compared with those obtained by Legrain et al [3] and Yeoh [24] It can be seen on the chart, the result given by XRPIM is best fit with XFEM solution [3] and have a small difference with Yeoh’s solution The reason is that Eq (4.3) relies on a number of assumptions that needed to be justified One of them is the assumption of a linear relation between Fy-the force applied on dX, an element of the crack in undeformed state, to close the crack and uy-the crack surface displacement (opening) in the y-direction, for non-linear elastic materials, is admitted 51 Figure 4.24: Variations of k with respect to the elongation in the case of equibiaxial extension Comparison of XRPIM method results with Legrain [3] and Yeoh [24] 4.2.4 Square plate with an inclined central crack As the last example, this thesis considers a square plate with W = H = mm which contains an inclined central crack The problem is shown in Fig 4.25 (a), is subjected to uniaxial tension stress   0.7 MPa at both top and bottom edges The plate has an inclined central crack of length 2a equal to 0.5 mm In order to perform the analyses, the skew angle  is assumed from 00 to 900 The material is identical to one considered in previous example A system of 40 × 40 scatter nodes formed adaptively to the crack line as Fig 4.25 (b) is employed for simulation The evolution of J-integral of the right crack tip for four cases of skew angle   00 ;270 ; 450 and 630  versus applied load level is plotted in Fig 4.26 52 Figure 25: Square plate with an inclined central crack (a), Nodal distribution (b) Figure 26: Variations of J-integral of the right crack tip with respect to applied force The normalized J-integral for various cases of skew angle is shown in Fig 4.27 They are calculated by dividing the J-integral value by one of   00 The result is compared with that obtained by Legrain et al [3], who used 12734 dofs fined mesh for simulation The numerical result is in good agreement with the reference solution [3] 53 Figure 4.27: Variations of normalized J-integral value of the right crack tip with respect to skew angles 54 Chapter CONCLUSION AND OUTLOOK 5.1 Conclusions This thesis has successfully utilized meshfree method for analyzing 2D noncracked and cracked hyperelastic solid using iRBF and XRPIM, respectively There are some conclusions based on numerical results above and it can be outlined as  The proposed method has shown the reliability and accuracy for analyzing 2D non-cracked and cracked hyperelastic solid  For non-cracked problem of hyperelastic material, the convergence rate of compressible state is faster than nearly-incompressible state and this trend is similar to VDQ method Furthermore, the solution of iRBF method in a regular domain is better than an irregular domain  The displacement filed along the crack and the singular stress filed at the crack tip are described by using two sorts of enrichment functions including the Heaviside function and an asymptotic function in XRPIM It should be noticed that in linear fracture mechanics, asymptotic functions contain four terms while nonlinear fracture mechanics possesses only one  The displacement field surrounding two crack edges including vertical displacement for an edge crack and both components of displacement field for a central crack are in good agreement with the reference solutions  The evolution of J-integral is best fit with the solution of XFEM approach for almost examples In addition, the domain size chosen to calculate Jintegral has a small influence to J-integral value 55  k parameter of nonlinear Griffith problem (uniaxial and equibiaxial extension examples) is match well with XFEM solution For comparison of Yeoh and XRPIM, k at small strain agrees well with Yeoh’s theory but they have a definite difference at finite strain 5.2 Future works From the current works and obtained results, the author proposes some futures works as follows  Improve the solution of iRBF method for an irregular domain  Perform the simulation of cracked hyperelastic solid with other material models such as Mooney-Rivlin or Ogden  Develop XRPIM for more complicated applications like dynamic analysis for finite deformation or damage analysis for solid under large strain condition 56 List of Publications V V Thai, N T Nha, N N Minh, T T Thien and B Q Tinh, “A Meshfree Method Based on Integrated Radial Basis Functions for 2D Hyperelastic Bodies,” in Proc ICOMMA, 2021, pp 990-1003 N T Nha, V V Thai, L S Vay and N T Hien, “A meshless radial point interpolation method for finite deformation analysis of hyperelasticity,” Science and Technology Development Journal vol 24, pp SI18-SI24, Feb 2022 N T Nha, N N Minh, V V Thai, T T Thien and B Q Tinh, “A meshfree model enhanced by NURBS-based Cartesian transformation method for cracks at finite deformation in hyperelastic solids,” Engineering Fracture Mechanic vol 261, pp 108176, Feb 2022 57 REFERENCES [1] N Moës et al., "A finite element method for crack growth without remeshing," International Journal for Numerical Methods in Engineering vol 46, pp 131-150, Jul 1999 [2] J Dolbow et al., "Discontinuous enrichment in finite elements with a partition of unity method," Finite Elements in Analysis and Design vol 36, pp 235-260, Nov 2000 [3] G Legrain et al., "Stress analysis around crack tips in finite strain problems using the eXtended finite element method," International Journal for Numerical Methods in Engineering vol 63, pp 290-314, Mar 2005 [4] A Karoui et al., "The eXtended finite element method for cracked hyperelastic materials: A convergence study," International Journal for Numerical Methods in Engineering vol 100, pp 222-242, Jul 2014 [5] D H Hai et al., "An extended polygonal finite element method for large deformation fracture analysis," Engineering Fracture Mechanics vol 209, pp 344-368, Mar 2019 [6] M Fleming et al., "Enriched element-free Galerkin methods for crack tip fields," International Journal for Numerical Methods in Engineering vol 40, pp 1483-1504, Dec 1997 [7] Y T Gu et al., "An enriched radial point interpolation method (e-RPIM) for analysis of crack tip fields," Engineering Fracture Mechanics vol 78, pp 175-190, Jan 2011 [8] Y T Gu, "An Enriched Radial Point Interpolation Method Based on Weak-Form and Strong-Form," Mechanics of Advanced Materials and Structures vol 18, pp 578-584, Nov 2011 58 [9] G Ventura et al., "A vector level set method and new discontinuity approximations for crack growth by EFG," International Journal for Numerical Methods in Engineering vol 54, pp 923-944, Apr 2002 [10] N T Nha et al., “Crack growth modeling in elastic solids by the extended meshfree Galerkin radial point interpolation method,” Engineering Analysis with Boundary Elements vol 44, pp 87-97, Jul 2014 [11] S L N Wang, "A large-deformation Galerkin SPH method for fracture," Journal of Engineering Mathematics vol 71, pp 305-318, Feb 2011 [12] G Zi et al., "Extended meshfree methods without branch enrichment for cohesive cracks," Computational Mechanics vol 40, pp 367-382, 2007 [13] M D Nam and T C Thanh, “Approximation of functions and its derivative using radial basis function networks,” Applied Mathematical Modelling vol 27, pp 1997-220, Mar 2003 [14] H L H Phuc and L V Canh, “A stabilized iRBF mesh-free method for quasi-lower bound shakedown analysis of structures,” Computers and Structures vol 228, pp 106157, Feb 2020 [15] R Hassani et al., “Large deformation analysis of 2D hyperelastic bodies based on the compressible nonlinear elasticity: A numerical variational method,” International Journal of Non-Linear Mechanics vol 116, pp 3954, Nov 2019 [16] A A Griffith, “The phenomena of rupture and flow in solids,” Mathematical, Physical and engineering sciences vol 221, pp 163-198, Jan 1921 [17] R S Rivlin and A G Thoma, “Rupture of rubber I Characteristic energy for tearing,” Journal for Polymer Science vol X, pp 291-318, Mar 1953 59 [18] J R Rice, “A path independent integral and the approximate analysis of strain concentration by notches and cracks,” Journal of Applied Mechanics vol 35, pp 379-386, Jun 1968 [19] B Moran and C F Shih, “Crack tip and associated domain integrals from momentum and energy balance,” Engineering Fracture Mechanics vol 27, pp 615-642, 1987 [20] G R Liu, Meshfree methods: moving beyond the finite element method CRC Press, Boca Raton, 2010 [21] N H Kim, Introduction to Nonlinear Finite Element Analysis Springer US, New York, 2015 [22] R Rashetnia and S Mohammadi, “Finite strain fracture analysis using the extended finite element method with new set of enrichment functions,” International Journal for Numerical Methods in Engineering vol 102, pp 1316-1351, Mar 2015 [23] G J Lake, “Application of fracture mechanics to failure in rubber articles, with particularly reference to groove cracking in tyres,” in International conference on deformation, yield and fracture of Polymers, Cambridge, 1970 [24] O H Yeoh, “Relation between crack surface displacement and strain energy release rate in thin rubber sheets,” Mechanics of Materials vol 34, pp 459-474, Aug 2002 60 PH N LÝ L CH TRÍCH NGANG H tên: V V n Thái Ngày, tháng, n m sinh: 28/11/1991 N i sinh: a ch liên l c: B12F23 Chung c Lotus Sen H ng, ph D Kiên Giang ng An Bình, D An, Bình ng QUÁ TRÌNH ÀO T O N m 2009 – 2014 2019 – hi n t i Ngành h c B c đào t o C n t C k thu t N ih c ih c Khoa C Khí, Tr ng i h c Bách khoa, i h c Qu c gia Tp.HCM Cao h c Khoa Khoa H c ng D ng, Tr ng i h c Bách khoa, i h c Qu c gia Tp.HCM Q TRÌNH CƠNG TÁC N m Ch c v N i công tác 2014 – 2015 K s Thi t k ch t o máy Công ty ARICO 2015 – hi n t i Th nghi m viên QUATEST 61 ... thesis could not has been possible without guidance of my thesis supervisor Dr Nha Thanh Nguyen I would like to express my sincere gratitude to him for his continuous support, patience, enthusiasm... compression problem 35 4.2: Percent of compression at point M for various values of distributed force in the compressible inhomogeneous compression problem 36 4.3: Percent of compression... problems with incompressible hyperelastic solid The incompressible Neo-Hookean model is used for simulation and problems are considered in plane stress condition Some results of simulation for non-cracked
- Xem thêm -

Xem thêm: Microsoft word vu van thai luan van thac si ,

Từ khóa liên quan