Fracture and Fragmentation of Thin-Shells

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

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

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

4. Cohesive Thin-Shell Kinematics Reference Configuration • Reference configuration • Deformed configuration • Thin-shell constraint: Director a3 is normal to the middle surface (Kirchhoff-Love) cohesive surface Deformed Configuration cohesive surface- cohesive surface+

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

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

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

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

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

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

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

12. Petaling of Circular Al 2024-0 Plates • Geometry Plate diameter 139.7 mm Hole diameter 5.8 mm Thickness 3.175 mm • Visco-plastic shell Mass density 2719 kg/m3 Young’s modulus 6.9·104 MPa Yield stress 90 MPa • Linear irreversible cohesive law Cohesive stress 140 MPa Fracture energy 2.75 Nm • Loading Prescribed vertical velocity vmax·(r-30.0) for r < 30.0 0.0 m/s for r > 30.0

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

14. Circular Plate - Convergence 21376 elements 5344 elements vmax = 600 m/s, time = 30 μs

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

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

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