Fracture and Fragmentation of Thin-Shells

Fehmi Cirak Michael Ortiz, Anna Pandolfi

California Institute of Technology

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

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

Cohesive Thin-Shell Kinematics Reference configuration

Deformed configuration

Thin-shell constraint: Director a3 is normal to the

middle surface (Kirchhoff-Love)

Reference Configuration

Deformed Configuration

cohesive surface+ cohesive surface-

cohesive surface

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

Cohesive Thin-Shell Mechanics

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)

Regular patch Irregular patch after one level of subdivision

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

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

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

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

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

Petaling of Circular Al 2024-0 Plates Geometry

Plate diameter 139.7 mmHole diameter 5.8 mmThickness 3.175 mm

Visco-plastic shellMass density 2719 kg/m3

Young’s modulus 6.9·104 MPaYield stress 90 MPa

Linear irreversible cohesive lawCohesive stress 140 MPaFracture energy 2.75 Nm

LoadingPrescribed vertical velocity vmax·(r-30.0) for r < 30.0

0.0 m/s for r > 30.0

Circular Plate - Snapshots

Time = 26.85 μs Time = 35.85 μs

Time = 8.93 μs Time = 17.93 μs

vmax = 600 m/s

Circular Plate - Convergence21376 elements 5344 elements

vmax = 600 m/s, time = 30 μs

Circular Plate – Impact Velocities

Time = 60 μs, 5344 elements

vmax = 150 m/svmax = 300 m/s

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

Outlook

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