Phase-field modelling of fracture in soft solids

In this thesis, we investigate the fracture of soft solids following the phase‑field approach. We present a unified modelling framework based on an incremental variational principle and employ it to study two classes of materials. First, we examine materials exhibiting rate‑dependent behaviour, including both viscous effects in the bulk and the rate‑dependence of fracture resistance. We study the general coupling and show that a contribution to the fracture driving force arising from viscous dissipation can lead to unrealistic model predictions. Moreover, based on an experimental study, we model the rate‑ and temperature‑dependent ductile‑to‑brittle fracture transition of toffee. We demonstrate that this phenomenon is essentially caused by the complex interplay of highly dissipative viscous deformation processes in the bulk and the rate‑dependent fracture resistance. Second, we investigate solids that are incompressible and hyperelastic in the absence of damage. To this end, we propose a novel invariant‑based isochoric physics‑augmented neural network (PANN), which serves as an efficient high‑fidelity model of the bulk and enables the decoupled description of the volumetric and isochoric responses. By combining the PANN with the phase‑field approach, we establish a hybrid fracture model. We demonstrate that a controlled weakening of incompressibility is essential for preventing numerical instabilities in fracture simulations, requiring a volumetric–isochoric decomposition. As a foundation, we provide a representation of the set of admissible isochoric deformation invariants and study the role of I1 and I2 in models of incompressible hyperelasticity.:Nomenclature vii 1 Introduction 1 1.1 Mechanical behaviour of soft solids 1 1.2 Approaches to model deformation and fracture of soft matter 3 1.3 Objective and structure of this work 7 2 Foundations of continuum mechanics 10 2.1 Kinematics and diffuse crack representation 10 2.1.1 Configurations and motion 10 2.1.2 Phase-field representation of fracture 15 2.1.3 Deformation and strain 16 2.1.4 Volumetric-isochoric decomposition 19 2.2 Stress, mass and energy 20 2.2.1 Stress 20 2.2.2 Mass and energy 22 2.3 Balance laws and dissipation inequality 23 2.3.1 Balance laws of continuum mechanics 23 2.3.2 Energy balance and Clausius-Duhem inequality 25 2.4 Fundamentals of material modelling 27 2.4.1 Classification of material behaviour 27 2.4.2 Concepts of material modelling 28 3 A unified modelling framework for deformation and fracture of soft materials 31 3.1 Pseudo-potential and variational principle 31 3.1.1 General structure of the unified model 32 3.1.2 Free energy 34 3.1.3 Viscous dissipation potential 39 3.1.4 Fracture dissipation potential 41 3.2 Governing equations and physical consistency 43 3.2.1 Stationarity condition and variational weak forms 43 3.2.2 Modifications of the phase-field evolution 45 3.2.3 Governing differential equations 48 3.2.4 Thermodynamic consistency 49 3.3 Aspects of finite element implementation 51 3.3.1 General approach 51 3.3.2 Compressible materials with rate-dependent behaviour 52 3.3.3 Incompressible materials with rate-independent behaviour 54 4 Rate-dependent fracture phenomena in soft materials 56 4.1 Model specification for materials with rate-dependent behaviour 56 4.1.1 Free energy 56 4.1.2 Viscous dissipation 57 4.2 General analysis of the coupling between viscoelasticity and fracture 59 4.2.1 Parameter identification from experimental data 59 4.2.2 Model validation and analysis of the viscous driving force 64 4.2.3 Investigation of the rate-dependent fracture toughness 71 4.3 Ductile-to-brittle fracture transition of toffee 73 4.3.1 Material composition and specimen preparation 73 4.3.2 Deformation behaviour at room temperature 75 4.3.3 Rate-dependent fracture phenomena at room temperature 77 4.3.4 Temperature influence on the mechanical behaviour 83 5 Novel neural network-based isochoric hyperelastic potential 87 5.1 Integration into the unified modelling framework 87 5.2 Kinematic considerations 88 5.2.1 Set of admissible invariants 89 5.2.2 Invariants for specific load cases 92 5.3 Novel isochoric physics-augmented neural network material model 93 5.3.1 Requirements on hyperelastic potentials 93 5.3.2 Architecture of the physics-augmented neural network 95 5.3.3 Key properties of the proposed isochoric potential 96 5.4 Model performance and role of the invariants 97 5.4.1 Training procedure 98 5.4.2 Treloar’s data on natural rubber 100 5.4.3 Human cortex benchmark data of Budday et al. 104 5.4.4 Alexander’s data on neoprene 104 6 Hybrid modelling of fracture in incompressible rubber 108 6.1 Model specification for fracture of incompressible matter 108 6.2 Simulation of rubber fracture based on Treloar’s data 109 6.2.1 Parameterisation of the model of the bulk 109 6.2.2 Hybrid simulation of fracture 109 6.3 Validation of the model against Hocine’s fracture experiments 116 6.3.1 Parameterisation of the model of the bulk 116 6.3.2 Validation of the hybrid fracture model 116 7 Concluding remarks 119 7.1 Key contributions 119 7.2 Directions for future research 121 Bibliography 123 A Fréchet, Gâteaux and tensor derivatives 138 A.1 Fréchet and Gâteaux derivatives 138 A.1.1 General definition of the derivatives 138 A.1.2 Component-wise differentiation in product spaces 139 A.2 Differentiation of tensor functions 140 A.2.1 (Almost) no constraints between the coordinates of the arguments 140 A.2.2 Symmetric arguments 142 B Algorithmic aspects of finite viscoelasticity 145 B.1 Integration of the evolution of viscous deformation 145 B.2 Stress tensor and material tangent 148 C Additional details on the modelling of incompressible hyperelasticity 152 C.1 Deformation-pressure coupling for models without Flory split 152 C.2 Explicit expression for fout(I1) 153 C.3 Stress tensor for simple shear deformations 153 C.4 Hocine’s SENT test: comparison to a Neo-Hookean model of the bulk 154