Unified Variational Theory For Fracture Phase-Field

Fracture is one of the most important mechanisms of failure in engineering materials.The establishment of a theoretically complete and numerically feasible unified method for describing and predicting the material fracture process is a core topic common to engineering mechanics and solid mechanics.The brittle fracture phase-field theory is one of the most important theories for material failure that emerged at the beginning of the 21st century.The variational description of Griffith’s ideal brittle fracture lays the mechanical foundation for the phase-field theory of brittle fracture,and the research of Boudin et al.provides a numerically feasible regularization method for the variational description.The brittle fracture phase-field theory can describe and predict the complex fracture phenomenon,including crack nucleation,extension,coalescence,and even branching.The introduction of the intrinsic characteristic length scale makes it immune to the mesh-sensitivity problem.These properties have made the phase-field theory of fracture one of the most advanced theories in the study of material failure.Brittle fracture is a special case of general cohesive fracture.Despite numerous attempts by researchers worldwide to establish a general phase-field theory for cohesive fracture,a unified,mathematically rigorous,and physically explicit theory still does not exist.The reason for this is that the existing research is generally phenomenological,and the most central physical quantity—energy density function in the phase-field fracture theory has to be postulated by rational guessing,which seriously affects the objectivity of this theory as rigorous science.In other words,the phase-field fracture theory lacks a rigorous physical foundation!At the same time,the existing fracture phase-field theory is constrained in a conservative system,and it is difficult to reflect some features in mode-Ⅱ fracture and mixed fracture.Both of these points indicate that the phase field theory of fracture is still in the process of development.For this reason,a systematic study is carried out in this thesis.In this thesis,it is found that there is a strict pairwise relationship between the core physical quantity:energy density of phase-field fracture theory and the constitutive relation of fracture—cohesive law.Mathematically,the relationship is a pair of integral transformations called Feng-Li integral transform.In general,the energy density of the phase-field fracture theory is unknown while the material’s cohesive law is known as an experimentally measurable physical relationship.Through the Feng-Li integral transform,a rigorous phase-field fracture theory can be established from the experimentally measurable physical relation,which breaks the confusion of the existing theories that rely on experience and intuition to construct models,and provides a solid physical foundation for the phase-field fracture theory.Further,this thesis proposes a variationally consistent directional energy decomposition method and gives a complete unified phase-field fracture theory by combining the Feng-Li integral transform.The new directional energy decomposition method can delicately select part of the elastic energy to couple with the phase-field(damage)according to the crack direction so that the crack driving force correctly takes into account the influence of the crack direction and can more realistically portray the energy transformation process in the fracture process.In the framework of the variational method,the expression of stress is also affected by the energy decomposition method.Under the effect of energy direction decomposition,the stress-strain relationship of the proposed theory truly reflects the transformation of the material from isotropic to orthotropic anisotropy during the fracture process.The symmetry axes of the orthogonal anisotropy are exactly the normal and tangential vectors of the crack surface,which perfectly portrays the effect of the fracture process on the stress-strain relationship.In order to extend the application of phase-field fracture theory to general dissipative systems in which the fracture energy is related to the fracture mode,a thermodynamic framework for general dissipative systems is proposed in this thesis.Based on this framework,a class of phase-field fracture methods applicable to general dissipative systems is obtained by introducing the concept of dissipative force and combining it with the Feng-Li integral transform,which can consider general mixed fracture with different fracture energy GcⅠ in tension and GcⅡ in shear.In this thesis,we find that in a general system with dissipative forces,for a physical state to remain stable,the virtual work in the vicinity of that state must satisfy an inequality,which we call the variational inequality of virtual works.Under any small perturbation,the variational inequality of virtual works is a necessary condition for the physical process to be observable in reality.In this thesis,it is demonstrated that the complete governing equations of a general dissipative system can be derived merely by the combination of the aforementioned variational inequality and the law of energy conservation,which provides a solid variational theory for the phase-field fracture method in general dissipative systems.In this thesis,a large number of numerical examples are studied,involving tension,shear,compression,and complex mixed loading conditions.The numerical results verify the effectiveness of the proposed models.Finally,possible future works are discussed.