Simulation of Deformation and Fracture in Very Large Shell Structures by Brandon Louis Talamini Submitted to the Department of Aeronautics and Astronautics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2015 c Massachusetts Institute of Technology 2015. All rights reserved. ○ Author................................................................ Department of Aeronautics and Astronautics August 17, 2015 Certified by. Raúl A. Radovitzky Professor of Aeronautics and Astronautics Thesis Supervisor Committee Member. David L. Darmofal Professor of Aeronautics and Astronautics Committee Member. Tomasz Wierzbicki Professor of Mechanical Engineering Committee Member. John W. Hutchinson Professor, Harvard University, Engineering and Applied Science Accepted by . Paulo C. Lozano Associate Professor of Aeronautics and Astronautics Chair, Graduate Program Committee 2 Simulation of Deformation and Fracture in Very Large Shell Structures by Brandon Louis Talamini Submitted to the Department of Aeronautics and Astronautics on August 17, 2015, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Although advances in computing have increased the limits of three-dimensional com- putational solid mechanics, structural elements remain essential in the practical de- sign of very large thin structures such as aircraft fuselages, ship hulls, automobiles, submarines, and pressure vessels. In many applications, fracture is a critical design concern, and thus the ability to numerically predict crack propagation in shells is a highly desirable goal. There are relatively few tools devoted to computational shell fracture, and of the existing approaches, there are two main defects: First, the ex- isting methods are not scalable, in the sense of parallel computing, and consequently simulation of large structures remains out of reach. Second, while the existing ap- proaches treat in-plane tensile failure, fracture due to transverse shearing has largely been ignored. In this thesis, a new computational framework for simulating deformation and fracture in large shell structures is presented that is well-suited to parallel computa- tion. The scalability of the framework derives from the combination of a discontinuous Galerkin (DG) finite element method with an interface element-based cohesive zone representation of fracture. This representation of fracture permits arbitrary crack propagation, branching, and merging, without on-the-fly mesh topological changes. Furthermore, in parallel computing, this propagation algorithm is indifferent to pro- cessor boundaries. The adoption of a shear-flexible shell theory is identified as a necessary condition for modeling transverse shear failure, and the proposed method is formulated accord- ingly. Locking is always an issue that emerges in numerical analysis of shear-flexible shells; here, the inherent flexibility afforded by DG methods in the choice of approx- imation spaces is exploited to prevent locking naturally, without recourse to mixed methods or reduced integration. Hence, the DG discretization elegantly solves both the problems of scalability and locking simultaneously. A stress resultant-based cohesive zone theory is proposed that considers trans- verse shear, as well as bending and in-plane membrane forces. The theory is quite general, and the specification of particular constitutive relations, in the form ofre- 3 sultant traction-separation laws, is independent of the discretization scheme. Thus, the proposed framework should be extensible and useful for a variety of applications. A detailed description of the implementation strategy is provided, and numerical examples are presented which demonstrate the ability of the framework to capture all of the relevant modes of fracture in thin bodies. Finally, a numerical example of explosive decompression in a commercial airliner is shown as evidence that the pro- posed framework can successfully perform shell fracture simulations of unprecedented size. Thesis Supervisor: Raúl A. Radovitzky Title: Professor of Aeronautics and Astronautics Committee Member: David L. Darmofal Title: Professor of Aeronautics and Astronautics Committee Member: Tomasz Wierzbicki Title: Professor of Mechanical Engineering Committee Member: John W. Hutchinson Title: Professor, Harvard University, Engineering and Applied Science 4 Contents 1 Introduction 19 1.1 Review of computational approaches for shell fracture . 25 1.1.1 Continuum damage models . 26 1.1.2 Cohesive zone models . 36 1.2 Objective and thesis overview . 48 2 A theory of shear deformable shells with cohesive zone fracture 53 2.1 Basic kinematics . 54 2.1.1 Tangent spaces . 56 2.1.2 Metric tensor and elements of surface length and area . 57 2.1.3 Motions . 59 2.1.4 Changes of frame . 60 2.2 Balance laws . 61 2.2.1 Admissible variations . 61 2.2.2 Principle of virtual power . 62 2.2.3 Frame indifference of the internal power . 64 2.2.4 Reduced form of the internal power. Effective resultants. 67 2.2.5 Strong form of the equations of motion . 70 2.3 Thermodynamics: free energy imbalance . 70 2.4 Constitutive laws . 71 2.5 Variational principle for the quasi-static problem . 73 2.6 Cohesive zone model of fracture in shells . 74 2.6.1 Cohesive zone constitutive theory . 76 5 2.6.2 Specification of the traction-separation law . 77 2.6.3 Activation criterion . 81 3 Prototype discretization for plates 83 3.1 Reissner plate theory . 84 3.2 Discontinuous Galerkin interpolation . 87 3.3 Discontinuous Galerkin gradient operators . 88 3.3.1 Specification of the numerical traces . 91 3.3.2 Illustration of the DG gradient . 92 3.4 Discrete energy minimum principle . 93 3.4.1 Interpolation of the lifting operators . 95 3.5 Numerical assessment of accuracy and locking . 96 4 Shell discretization 101 4.1 Interpolation of kinematic fields . 103 4.1.1 Mid-surface position interpolation . 104 4.1.2 Interpolation of the director . 105 4.1.3 Interpolation of director variations . 106 4.2 Discontinuous Galerkin derivative operator . 109 4.2.1 Specification of the numerical traces . 112 4.2.2 Discrete representation of the lifting operator . 113 4.3 Discontinuous Galerkin method for quasi-static shell problems . 117 4.3.1 Element internal forces . 119 4.3.2 Stabilization internal forces . 122 4.3.3 External forces . 122 4.3.4 Iterative solution procedure . 122 4.3.5 Bulk element stiffness matrix . 125 4.3.6 Stability interface element stiffness matrix . 127 4.4 Kinematic update . 128 4.5 Time integration for dynamics . 128 4.6 Parallel implementation of explicit dynamics . 133 6 4.7 Implementation of fracture model . 134 4.7.1 Activation criterion . 135 4.7.2 Activation procedures . 136 4.7.3 Cohesive zone nodal forces and stiffness matrix . 139 4.7.4 Quasi-static fracture solution procedure . 140 5 Numerical results 143 5.1 Implementation . 144 5.2 Verification and benchmark tests . 146 5.2.1 Patch tests . 146 5.2.2 Clamped plate problem of Chinosi and Lovadina plate bending 148 5.2.3 Clamped cylinder . 149 5.2.4 Free cylinder . 150 5.2.5 Scordelis-Lo roof . 152 5.3 Nonlinear benchmarks . 155 5.3.1 Roll-up of a beam . 155 5.3.2 Twisting of a bar . 156 5.3.3 Collapse of an elastic-perfectly plastic plate . 157 5.3.4 Pinched open hemisphere . 159 5.3.5 Bending to membrane transition in a circular plate . 161 5.3.6 Free vibration of a cantilever beam . 164 5.3.7 Impulsively loaded elastic-perfectly plastic cylindrical panel . 168 5.3.8 Parallel quasi-static example: nonlinear conjugate gradient solver169 5.3.9 Scalability test . 173 5.3.10 Uniaxial fracture test . 174 5.4 Applications . 175 5.4.1 Mode III tearing of a plate . 175 5.4.2 Beam blast experiments of Menkes and Opat . 179 5.4.3 Wet sand blast of a plate . 183 5.4.4 Explosive decompression in a commercial airliner . 188 7 6 Conclusions 195 6.1 Summary . 195 6.2 Recommendations for future work . 196 A Constitutive Laws 199 A.1 Isotropic elasticity . 199 A.2 Resultant plasticity . 200 B Linearized theory 205 8 List of Figures 1-1 Catastrophic failures of tank cars carrying anhydrous ammonia from a derailment near Minot, North Dakota on January 18, 2002 (Image source: NTSB report [1]). 21 1-2 Fuselage damage on the Boeing 737-200 aircraft of Aloha Airlines flight 243, April 28, 1988 (Image source: NTSB report [2]). 21 1-3 The MV Rena ran aground off near New Zealand and ultimately broke into two sections (Image source: New Zealand Defence Force online image collection [3], shared under Creative Commons 2.0 License). 22 1-4 Basic fracture modes of thin bodies. 23 1-5 Kinematics of (a) thin plate theories and (b) shear-flexible plate theo- ries. The transverse displacement is u, the rotation of the cross section about the normal fiber is 휃x, and the transverse engineering shear strain is 훾x..................................... 24 1-6 Impulsively loaded aluminum beam experiments of Menkes and Opat (1973). These post mortem photographs show the transition from ten- sile failure to shear-off with increasing impulse. Image source: Menkes and Opat [4] . 26 1-7 Ductility curves of Johnson and Cook [5], expressed as equivalent plas- tic strain at fracture vs. stress triaxiality. Experimental data and modeling fit. Note exclusion of the experimental torsion data for4340 steel, which shows much lower ductility than predicted. 30 1-8 Ductility curves of Bao and Wierzbicki [6] . 31 9 1-9 Depiction of ductility as viewed by two state of the art ductile fracture models. (a) Fracture locus of Bai and Wierzbicki; image taken from [7]. Failure strain 휖¯f is a function of triaxiality 휂 and Lode parameter 휃¯. (b) Predictions of equivalent plastic strain at localization by Nahshon and Hutchinson’s extended Gurson plasticity model; image taken from [8]. L is the Lode parameter. (Note these plots are only representative of the underlying models, and are not calibrated to a particular material, nor to each other.) .