1. Fracture and Fragmentation of Thin-Shells Fehmi Cirak Michael Ortiz, Anna Pandolfi California Institute of Technology

2. Detonation Driven Fracture n Advanced computational models are required for computing the interaction of a detonating fluid and its products with a fracturing thin-shell tube n Required features of a a thin-shell fragmentation finite- element code n Large visco-elasto-plastic deformations n Crack initiation, propagation, turning, and branching n Contact mechanics n Eulerian-Lagrangian fluid-shell coupling n Mesh adaptation near the crack tips Tearing of an aluminum tube by gaseous detonation Courtesy of J. Shepherd and T. Chao

3. Building Blocks n Subdivision Thin-Shell Finite Elements n The Kirchhoff-Love type thin-shell equations, the appropriate mechanical model for thin-shells, are discretized n Reference and deformed thin-shell surface is approximated with smooth subdivision surfaces n The resulting finite elements are efficient as well as robust (no locking!) n Cohesive Model of Fracture n Fracture is modeled as a gradual process with cohesive tractions at the crack flanks n Parameters such as peak stress and fracture energy can be incorporated n Computationally more tractable than fracture mechanics n No assumptions about the shell constitutive model n No ambiguities for dynamics and plasticity

4. Cohesive Thin-Shell Kinematics n Reference configuration n Deformed configuration n Thin-shell constraint: Director a 3 is normal to the middle surface (Kirchhoff- Love) Reference Configuration Deformed Configuration cohesive surface + cohesive surface - cohesive surface

5. n Deformation gradient n Displacement jumps at the crack flanks n Elastic potential energy of the shell with embedded cohesive surface n Minimum potential energy leads to a discrete set of equations n Away from the crack flanks, conforming FE approximation requires smooth shape functions n At the crack flanks, proper transfer of the cohesive tractions and coupled forces necessary Cohesive Thin-Shell Mechanics

6. Smooth Subdivision Shape Functions n Subdivision schemes provide smooth shape functions in the topologically irregular setting n On regular patches, smooth quartic box-splines are used n On irregular patches, Loop's subdivision scheme leads to regular patches n References: n F. Cirak, M. Ortiz, Int. J. Numer. Meth. Engrg. 51 ( 2001 ) n F. Cirak, M. Ortiz, P. Schröder, Int. J. Numer. Meth. Engrg. 47 ( 2000) Regular patch Irregular patch after one level of subdivision

7. Subdivision Thin-Shell FE n Initial and deformed shell surface is approximated with a subdivision surface n Vertex positions of the control mesh are the only degrees of freedom n Same degrees of freedom like finite elements for solids / fluids n Element integrals are evaluated with an efficient one point quadrature rule n Exact kinematics for large deformations and strains n Arbitrary 3-d constitutive models n Hyper-elasticity: St. Venant, Neo-Hookean, Mooney-Rivlin n Visco-plasticity

8. Fracture in 1-D Non-local subdivision shape functions Cohesive tractions and coupled forces Cohesive tractions couple the displacements and rotations of the left and right crack flank Fractured beam with ghost elements

9. n Each element is considered separately and cohesive elements are introduced on all edges Cohesive Subdivision Thin-Shell FE Displacement jumps activate cohesive tractions Reference configuration Director (Normal) jumps activate cohesive coupled forces

10. Linear Cohesive Law n The relation between cohesive traction and opening displacement at the edge is governed by the cohesive law n Decomposition of the opening displacement after activation n Effective opening displacement n Effective opening traction n Cohesive traction

11. Computational Challenges n The proposed fragmentation strategy increases the initial number of vertices by approximately six n Parallelism and high-end computational tools are crucial n Current parallelization strategy n Partition the control mesh and identify one-layer of elements at the processor boundaries n Distribute the partitioned mesh n Add to the boundaries ghost elements for enforcing boundary conditions n Fragment each element patch n Introduce at the element boundaries cohesive elements

12. Petaling of Circular Al 2024-0 Plates n Geometry Plate diameter139.7 mm Hole diameter 5.8 mm Thickness 3.175 mm n Visco-plastic shell Mass density2719 kg/m 3 Young’s modulus 6.9·10 4 MPa Yield stress 90 MPa n Linear irreversible cohesive law Cohesive stress 140 MPa Fracture energy 2.75 Nm n Loading Prescribed vertical velocity v max ·(r-30.0)for r < 30.0 0.0 m/sfor r > 30.0

13. Circular Plate - Snapshots Time = 26.85 μsTime = 35.85 μs Time = 8.93 μs Time = 17.93 μs v max = 600 m/s

14. Circular Plate - Convergence 21376 elements5344 elements v max = 600 m/s, time = 30 μs

15. Circular Plate – Impact Velocities Time = 60 μs, 5344 elements v max = 150 m/sv max = 300 m/s

16. n Validation with archival plate petaling data n Fluid-shell coupled simulation for modeling the bulging and venting at the crack flaps during detonation driven fracture n The current coupling algorithm needs to be extended to deal with fluid on both sides of the thin-shell n Scalable parallelization n Adaptive mesh refinement and coarsening close to the crack front Outlook

17. Towards Coupled Fragmentation n Coupled simulation of airbag deployment Time = 4.25 msTime = 8.16 ms Time = 12.13 msTime = 18.02 ms Joint work with Raul Radovitzky