在航太、造船和海洋工程等領域，結構損傷（如裂紋）為這類大型焊接結構不可避免的議題，其中船板結構於複雜的海洋負載下，裂紋擴張的速率會嚴重影響結構物的剩餘壽命，因此分析具損傷之結構物的疲勞破壞為本領域極為重要的部分。傳統數值方法如有限元素法於分析位移不連續即結構損傷部位，會出現應力奇異性等破壞力學挑戰，因此基於積分方程形式的近場動力學方法被引入以有效解決上述數值問題。但近場動力學考慮所有物質點與其特定範圍內的其他物質點間的交互作用導致其計算時間較長，因此本研究提出一種混合維度近場動力學與有限元素法耦合模型，結合兩種方法的優點來計算破壞力學問題同時提高計算效率。藉由短時間取得的破壞參數預測結果來實現對於結構損傷的疲勞破壞分析，此研究可評估結構於損傷狀態下的裂紋擴張情形，對本領域發展為一重要進展。

Traditional numerical methods, such as the Finite Element Method (FEM), face challenges in analyzing displacement discontinuities, such as those occurring at damaged regions of a structure, leading to issues like stress singularities in fracture mechanics. To address these numerical challenges, the Peridynamics method, based on integral equations, is introduced as an effective solution. However, Peridynamics considers the interaction between all material points and other material points within a specified horizon, resulting in longer computational times. To improve computational efficiency, this study proposes a Mixed-Dimensional Peridynamics-Finite Element Method (MD PD-FEM), which combines the advantages of both methods for solving fracture mechanics problems. By rapidly obtaining fracture parameter predictions, fatigue failure analysis of structural damage is performed. This study enables the assessment of crack propagation in damaged structures and represents a significant advancement in the field. The paper verifies the static response of structures using 2D and 3D PD, as well as 2D PD-FEM and MD PD-FEM, for cases including edge-through cracks and surface cracks. The results, based on quick and accurate crack opening displacement (COD) predictions, show good agreement with experimental data for the number of loading cycles.

[1] ISSC committee III.2 report: Fatigue and fracture. Proceedings of the 20th International Ship and Offshore Structures Congress (2018)
[2] C.Y. Ji, H.Z. Xue, X.H. Shi, O. Gaidai. Experimental and numerical study on collapse of aged jacket platforms caused by corrosion or fatigue cracking. Engineering Structures, 112:14-22 (2016)
[3] Y.M. Zhang, M. Fan, Z.M. Xiao, W.G. Zhang. Fatigue analysis on offshore pipelines with embedded cracks. Ocean Engineering, 117:45-56 (2016)
[4] W. Huang, N. Sridhar. Fatigue failure risk assessment for a maintained stiffener-frame welded structure with multiple site cracks. International Journal of Applied Mechanics, 8:1650024 (2016)
[5] D.T. Ngoula, H.T. Beier, M. Vormwald. Fatigue crack growth in cruciform welded joints: Influence of residual stresses and of the weld toe geometry. International Journal of Fatigue, 101:253-262 (2017)
[6] W. Mao, Z. Li, V. Ogeman, J.W. Ringsberg. A regression and beam theory based approach for fatigue assessment of containership structures including bending and torsion contributions. Marine Structures, 41:244-266 (2015)
[7] Z. Ding, Z. Gao, X. Wang, Y. Jiang. Modeling of fatigue crack growth in a pressure vessel steel Q345R. Engineering Fracture Mechanics, 135:245-258 (2015)
[8] I. Lotsberg, G. Sigurdsson, A. Fjeldstad, T. Moan. Probabilistic methods for planning of inspection for fatigue cracks in offshore structures. Marine Structures, 46:167-192 (2016)
[9] P.W. Harper, S.R. Hallett. Advanced numerical modelling techniques for the structural design of composite tidal turbine blades. Ocean Engineering, 96:272-283 (2015)
[10] G. Liu, B. Zhong, X. Tian, P. Chen, W. Mu. Numerical analysis on the HSS and SIF of multi-planar DX-joint welds for offshore platforms. Ocean Engineering, 127:258-268 (2016)
[11] E. Madenci, E. Oterkus. Peridynamic theory and its applications. Springer (2014)
[12] S.A. Silling, O. Weckner, E. Askari, F. Bobaru. Crack nucleation in a peridynamic solid. International Journal of Fracture, 162:219-227 (2010)
[13] A. Agwai, I. Guven, E. Madenci. Predicting crack propagation with peridynamics: a comparative study. International Journal of Fracture, 171:65-78 (2011)
[14] Y.D. Ha, F. Bobaru. Studies of dynamic crack propagation and crack branching with peridynamics. International Journal of Fracture, 162:229-244 (2010)
[15] E. Oterkus, I. Guven, E. Madenci. Impact damage assessment by using peridynamic theory. Central European Journal of Engineering, 2:523-531 (2012)
[16] E. Oterkus, I. Guven, E. Madenci. Fatigue failure model with peridynamic theory. IEEE Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems, 1-6 (2010)
[17] Macek, R.W., Silling, S.A.: Peridynamics via finite element analysis. Finite Elem. Anal. Des. 43(15), 1169–1178 (2007)
[18] Kilic, B., Madenci, E.: Coupling of peridynamic theory and the finite element method. J. Mech. Mater. Struct. 5(5), 707–733 (2010)
[19] Lubineau, G., Azdoud, Y., Han, F., Rey, C., Askari, A.: A morphing strategy to couple non-local to local continuum mechanics. J. Mech. Phys. Solids 60(6), 1088–1102 (2012)
[20] Han, F., Lubineau, G., Azdoud, Y., Askari, A.: A morphing approach to couple state-based peridynamics with classical continuum mechanics. Comput. Methods Appl. Mech. Eng. 301, 336–358 (2016)
[21] Liu, W., Hong, J.W.: A coupling approach of discretized peridynamics with finite element method. Comput. Methods Appl. Mech. Eng. 245, 163–175 (2012)
[22] Seleson, P., Beneddine, S., Prudhomme, S.: A force-based coupling scheme for peridynamics and classical elasticity. Comput. Mater. Sci. 66, 34–49 (2013)
[23] Galvanetto, U., Mudric, T., Shojaei, A., Zaccariotto, M.: An effective way to couple fem meshes and peridynamics grids for the solution of static equilibrium problems. Mech. Res. Commun. 76, 41–47 (2016)
[24] Zaccariotto, M., Mudric, T., Tomasi, D., Shojaei, A., Galvanetto, U.: Coupling of fem meshes with peridynamic grids. Comput. Methods Appl. Mech. Eng. 330, 471–497 (2018)
[25] Madenci, E., Barut, A., Dorduncu, M., Phan, N.D.: Coupling of peridynam ics with finite elements without an overlap zone. Paper presented at 2018 AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con ference, Kissimmee, Florida, 8–12 January 2018 (2018)
[26] Anicode, S.V.K., Madenci, E.: Seamless coupling of bond- and state-based peridynamic and finite element analyses. Mech. Mater. 173, 104433 (2022)
[27] D'Elia, M., Li, X., Seleson, P., Tian, X., Yu, Y.: A review of local-to-nonlocal coupling methods in nonlocal diffusion and nonlocal mechanics. J. Peridynam. Nonlocal Model. 4, 1–50 (2021)
[28] Wang, J., Lou, Z., Min, X., Zou, J.: A DOF expanding method for connecting solid and shell element. Comm. Numer. Methods Engrg. 12(6), 321–330 (1996)
[29] McCune, R.W., Armstrong, C.G., Robinson, D.J.: Mixed-dimensional coupling in finite element models. Int. J. Numer. Methods Eng. 49(6), 725–750 (2000)
[30] Shim, K.W., Monaghan, D.J., Armstrong, C.G.: Mixed dimensional coupling in f inite element stress analysis. Eng. Comput. 18, 241–252 (2002)
[31] Yu, Y., Chan, T.H.T., Sun, Z.H., Li, Z.X.: Mixed-dimensional consistent coupling by multi-point constraint equations for efficient multi-scale modeling. Adv. Struct. Eng. 15(5), 837–853 (2012)
[32] Garusi, E., Tralli, A.: A hybrid stress-assumed transition element for solid-to beam and plate-to-beam connections. Comput. Struct. 80(2), 105–115 (2002)
[33] Chavan, K.S., Wriggers, P.: Consistent coupling of beam and shell models for thermo-elastic analysis. Int. J. Numer. Methods Eng. 59(14), 1861–1878 (2004)
[34] Tsai, P., Chen, W.H.: The combination of different type elements using a lagrangian multiplier technique for static and dynamic structural analysis. Comput. Struct. 21(3), 493–500 (1985)
[35] Nguyen, V.P., Kerfriden, P., Claus, S., Bordas, S.P.A.: Nitsche's method method for mixed dimensional analysis: conforming and non-conforming continuum beamandcontinuum-plate coupling. Preprint at https://arxiv.org/abs/1308.2910 (2013)
[36] Yamamoto, T., Yamada, T., Matsui, K.: Numerical procedure to couple shell to solid elements by using nitsche's method. Comput. Mech. 63, 69–98 (2019)
[37] Osawa, N., Hashimoto, K., Sawamura, J., Nakai, T., Suzuki, S.: Study on shell–solid coupling fe analysis for fatigue assessment of ship structure. Mar. Struct. 20(3), 143–163 (2007)
[38] Pagani, A., Carrera, E.: Coupling three-dimensional peridynamics and high-order one-dimensional finite elements based on local elasticity for the linear static anal ysis of solid beams and thin-walled reinforced structures. Int. J. Numer. Methods Eng. 121(22), 5066–5081 (2020)
[39] S.A. Silling, A. Askari. Peridynamic model for fatigue cracking. Sandia Report, (2014)
[40] G. Zhang, Q. Le, A. Loghin, A. Subramaniyan, F. Bobaru. Validation of a peridynamic model for fatigue cracking. Engineering Fracture Mechanics, 162:76-94 (2016)
[41] J. Jung, J. Seok. Mixed-mode fatigue crack growth analysis using peridynamic approach. International Journal of Fatigue, 103:591-603 (2017)
[42] C.T. Nguyen, S. Oterkus, E. Oterkus. An energy-based peridynamic model for fatigue cracking. Engineering Fracture Mechanics, 241:107373 (2021)
[43] B. Liu, R. Bao, F. Sui. A fatigue damage-cumulative model in peridynamics. Chinese Journal of Aeronautics, 34:329-342 (2021)
[44] F. Wang, Y. Ma, Y. Guo, W. Huang. Studies on quasi-static and fatigue crack propagation behaviours in friction stir welded joints using peridynamic theory. Advances in Materials Science and Engineering, 5105612 (2019)
[45] K. Hong, S. Oterkus, E. Oterkus. Peridynamic analysis of fatigue crack growth in fillet welded joints. Ocean Engineering, 235:109348 (2021)
[46] X. Ma, L. Wang, J. Xu, Q. Feng, L. Liu, H. Chen. A two-dimensional ordinary state-based peridynamic model for surface fatigue crack propagation in railheads. Engineering Fracture Mechanics, 265:108362 (2022)
[47] J. Han, G. Wang, X. Zhao, R. Chen, W. Chen. Modeling of multiple fatigue cracks for aircraft wing corner box based on non-ordinary sate peridynamics. Metals, 12:1286 (2022)
[48] Y. Liu, L. Deng, W. Zhong, J. Xu, W. Xiong. A new fatigue reliability analysis method for steel bridges based on peridynamic theory. Engineering Fracture Mechanics, 236:107214 (2020)
[49] Nguyen, C.T., Oterkus, S., Oterkus, E.: An energy-based peridynamic model for fatigue cracking. Eng. Fract. Mech. 241, 107373 (2021)
[50] Sajith S, Murthy K, Robi P. Experimental and numerical investigation of mixed mode fatigue crack growth models in aluminum 6061–T6. Int J Fatigue, 130:105285 (2020)
[51] X. Ma, W. Yin, Y. Wang, L. Liu, X. Wang, Y. Qian. Fatigue failure analysis of U75V rail material under I+II mixed-mode loading: Characterization using peridynamics and experimental verification. Int J Fatigue, 185:108371 (2024)