破壞力學的凝聚區模型考量了材料間的內聚力，並將彈性理論中裂縫尖端產生的無限大應力以有限的破壞強度取代之。然而，以凝聚區模型結合有限元素模擬裂縫開裂需預先設定開裂方向，事先於裂縫成長路線上鋪排凝聚區元素，完全未破壞材料與完全破壞的尖銳裂縫間的不連續，更是增加模擬上的困難。近年相場模型的出現，以連續性的相場函數描述部分破壞的程度，模糊化原先尖銳裂縫的不連續面，使裂縫開裂過程較不受限於網格元素的鋪設，也降低數值模擬的難度。
本文使用有限元素軟體COMSOL以特定的相場模型模擬平面破壞問題。此相場模型特別之處在於可與傳統凝聚區模型近似，設定破壞強度並於裂縫生成時維持模糊裂縫帶寬。本文分析對稱三點彎矩試驗與不對稱三點彎矩試驗等平面裂縫問題。在對稱三點彎矩試驗中，分別以施力點位移控制和裂縫開口位移控制進行裂縫成長模擬，並將極限載重結果與文獻數值解比較。不對稱三點彎矩試驗的模擬結果則是驗證了範圍參數以及破壞強度的選擇只要保持合理的比例關係，極限載重差距不大而裂縫開裂路線則不受影響。與實驗數據比較後，發現模擬結果之裂縫開裂路線稍微偏右；極限載重則因幾何上的不同，會相等於實驗解或是略低於實驗解。

In fracture mechanics, the cohesive zone model replaces the infinite stress in the process zone near crack tip in elasticity by finite cohesive strength. However, to make cracks propagate in simulation, the cohesive zone elements need to be embedded along the predicted path. Besides, the discontinuity between undamaged part and sharp crack arises the difficulties in simulation. The phase-field model developed in recent years approximates discontinuous crack faces by continuous phase field, so that the difficulty in simulation has been lowered. In this study, the phase-field model is applied in the FEM software COMSOL to simulate crack propagation in plane problems, i.e., a symmetric and an asymmetric three-point bending fracture test. In the first problem, the load-point displacement and the crack mouth opening displacement controls were both used in simulation. The choices of length parameter in the phase field model affected the width of crack propagation, but do not influence the path of crack propagation because of constant crack width. Also, the choices of length parameter slightly affected the ultimate load. Nonetheless, as long as the length parameter is reasonably selected, the simulation by the phase field model can give very good prediction for fracture behavior compared to other numerical solution or experimental data.

1. Griffith, A.A. “The Phenomena of Rupture and Flow in Solids”. Philos. Trans. R. Soc. Lond, 221, pp. 163-198. 1921.
2. Barenblatt, G.I. “The Formation of Equilibrium Cracks During Brittle Fracture. General Ideas and Hypotheses, Axially-Symmetric Cracks”. Appl. Math. Mech. 23. pp. 622-636. 1959.
3. Francfort, G., Marigo, J. “Revisting Brittle Fracture as an Energy Minimization Problem”. J. Mech. Phys. Solids, 46, pp. 1319-1342. 1998.
4. Bourdin, B., Francfort, G., Marigo, J.-J. “Numerical Experiments in Revisited Brittle Fracture”. J. Mech. Phys. Solids, 48, pp. 797-826. 2000.
5. Miehe, C., Welschinger, F., Hofacker, M. “Thermodynamically Consistent Phase-Field Models of Fracture: Variational Principles and Multi-Field FE Implementations”. Int. J. Number. Methods Engrg., 83, pp. 1273-1311. 2010.
6. Ambati, M., Gerasimov, T., de Lorenzis, L. “A Review on Phase-Field Models for Brittle Fracture and a New Fast Hybrid Formulation”. Comput. Mech., 55, pp. 383-405. 2015.
7. Kuhn, C., Müller, R. “A Phase Field Model for Fracture”. Proc Appl Math Mech., 8, pp. 10223–10224. 2008.
8. Borden, MJ., Hughes, TJR., Landis, CM., Verhoosel, CV. “A Higher-order Phase-field Model for Brittle Fracture: Formulation and Analysis Within The Isogeometric Analysis Framework”. Comput Methods Appl Mech Eng 273. pp.100–118. 2014.
9. Wu, J.Y., V.P. Nguyen. “A Length Scale Insensitive Phase-Field Damage Model for Brittle Fracture”, J. Mech. Phys. Solids, 119, pp. 20-42. 2018.
10. Wu, J.Y. “A Unified Phase-Field Theory for the Mechanics of Damage and Quasi-Brittle Failure in Solids”. J. Mech. Phys. Solids, 103, pp. 72-99. 2017.
11. Mumford, D., Shah, J. “Optimal Approximation by Piecewise Smooth Functions and Associated Variational Problems”. Comm. Pure Appl. Math., 42, pp. 577-685. 1989.
12. Ambrosio, L., Tortorelli, V.M. “Approximation of Functional Depending on Jumps by Elliptic Functional Via T-Convergence”. Comm. Pure Appl. Math., 43, pp. 999-1036. 1990.
13. Lorentz, E., Godard, V. “Gradient Damage Models: Towards Full-Scale Computations”. Comput. Methods Appl. Mech. Eng, 200, pp. 1927-1944. 2011.
14. Wu, J.Y., Chen, W.X. “Phase-Field Cohesive Zone Modeling of Multi-Physical Fracture in Solids and The Open-Source Implementation in COMSOL MULTIPHYSICS”. Theor. Appl. Fract. Mech., 111, pp.1-21. 2021.
15. Tada, H. “The Stress Analysis of Cracks Handbook”, Third Edition, Del Research Corporation, Hellertown, PA, pp. 58-60. 1973.
16. Ingraffea, A., & Grigoriu, M. “Probabilistic Fracture Mechanics: A Validation of Predictive Capability”, Tech, DTIC Document, 155, pp. 36-51. 1990.
17. COMSOL MULTIPHYSICS, 5.5 Documentation. 2019.