Shell-Fluid Coupled Simulation of Detonation-

Driven Fracture and Fragmentation

Fehmi Cirak and Ralf Deiterding

California Institute of Technology

Third M.I.T. Conference on Computational Fluid and Solid Mechanics, June 14-17, 2005

F. Cirak

Modeling and simulation challengesDuctile mixed mode fracture with large deformationsSuccessive change of the mesh topologyFluid-shell interaction under changing mesh topology

Al 6061-T6 Tube Fracture (J. Shepherd)

Fractured tubes

Experiments courtesy of J. Shepherd, Caltech

Ductile fracture

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Fractured Thin-Shell Kinematics

Kirchhoff-Love assumption: Director a3 is normal to the deformedmiddle surface

Reference configuration Deformed configuration

Cohesiveinterface

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Fractured Thin-Shell Equilibrium

Shell and cohesive interface contribute to the internalvirtual work

Shell internal virtual work consists of a membrane and bending term

Cohesive internal virtual work consists of a tearing, shearing, and hinge term

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Subdivision FE-Discretization

Away from crack flanks, conforming FE discretizationrequires smooth shape functions

On regular patches, quartic box-splines are used

On irregular patches, subdivision schemes are used (here Loop's scheme)

Regular patch for element J Irregular patch for element K

J K

F. Cirak, M. Ortiz, P. Schröder, Int. J. Numer. Meth. Engrg. 47 (2000)

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Discontinuous Shape Functions

Pre-fractured patches operate independently forinterpolation purposes

Edge opening displacements and rotations activatecohesive tractions

Membrane mode

JK

Element J Element K

Bending modeShear mode

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Cohesive Interface Model

Membrane, shear, and bending tractions are computedby numerical integration over shell thickness

At each quadrature point a conventional irreversible cohesive model is used

Conformity prior to crack initiation can be enforced

Displacement jump

Tra

ctio

n

Linearly decreasing envelope

Displacement jump

Tra

ctio

n

Smith-Ferrante envelope

F. Cirak, M. Ortiz, A. Pandolfi, CMAME (2005)

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0.75

0.85

0.95

1.05

1.15

1.25

100 1000 10000 100000

Penalty parameter [x 69000]

Norm

aliz

ed m

ax.

defle

ctio

nSimply Supported Plate

Geometry:

Length 1.0

Thickness 0.1

Linear elastic material:

Young’s modulus 69000

Poisson’s ratio 0.3

displacements fixeddisp

lacements

fixed

pressure loading fractured

non-fractured

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Fluid-Shell Coupling: Overview

High speed flows interacting with thin-shells effectivelyrequire a coupled Eulerian-Lagrangian approach

In Eulerian formulations, mesh points are fixedIn Lagrangian formulations, mesh points follow the trajectories of materialpoints

Eulerian-Lagrangian couplingArbitrary Lagrangian Eulerian method

High accuracy, but algorithmically challenging for shells with large deformations

Interface tracking and Interface capturing schemesAlgorithmically very robustWell established in Cartesian mesh based Eulerian fluid codesRecently applied to fluid-solid coupling

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Gas Dynamics

Compressible inviscid fluid flow (Euler equations)

Specific total energy

Equation of state for perfect gas

Mass conservation

Momentum conservation

Energy conservation

ratio of specific heats

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Gas Dynamics - Discretization

Euler equations in conservation law form

Finite volume discretization on a Cartesian grid

Reduced to one dimensional problems along eachcoordinate axis using dimensional splitting

Fluxes are computed by solving local Riemann problems

For additional features see amroc.sourceforge.net

mesh size

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Thin-shell and fluid equations are integrated with an explicit timeintegration scheme

Coupling is achieved by enforcing:Continuity of normal velocityContinuity of traction normal componentUnconstrained tangential slip

Explicit Fluid-Shell Coupling -1-

Eulerian

“ghost” fluid domain

Eulerian

“real” fluid domain

Lagrangian

thin-shell domain

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Explicit Fluid-Shell Coupling –2-

Enforcing the interface conditions on the fluid gridthrough ghost-cells

Ghost cell values are extrapolated from the values at the shell-fluid interfaceNormal velocity modifications in the ghost cells

Corresponds to reflecting the normal fluid velocity component in a moving localcoordinate frame attached to the shell

Enforcing the interface conditions on the shellInterpolated pressures from the fluid mesh are applied as external tractionboundary conditions to the shell

extrapolated fluid velocity

extrapolated shell velocity

normal and tangent to the interface

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Airbag – Geometry and Discretization

Shell Mesh: 10176 elements

0.74 m

0.86 m0.4

9 m

0.86 m

0.123 m

0.0

25 m

Fluid Mesh: 48x48x62 cells

F. Cirak, R. Radovitzky, C&S (2005)

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Airbag – Limit Surface

total simulation

time

approx. 23 ms

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Airbag – Kinetic Energy

total simulation

time

approx. 23 ms

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Fluid Induced Fracture

Cohesive interface model:

Linearly decreasing envelope

with loading and unloading

2365 m/sPmax =

2.1, 5, 6 MPa

4.2 cm

61.0 cm

Initial crack (0.35 cm)

thickness = 0.89 mm

Material model for Al 6061-T6:

J2 - plasticity with viscosity

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Coupled Simulation in Elastic Regime

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Circumferential strain at x=28.2cm

The fluid grid initialized with normal shock conditions

In contrast to the computation, the experiment performedwith a detonation wave

Coupled Simulation in Elastic Regime

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Coupled Simulation with Fracture

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Coupled Simulation - Threshold

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Coupled Simulation - Close-up

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Coupled Simulation - Pmax = 6.0MPa

20736 shell elements, 80x80x640 fluid cells

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Computations performed on an Intel Xeon Cluster41 processors for the fluid29 processors for the shellTotal computing time ~8h

Partitioning of the fluid domain

Partitioning of the shell domain alc@livermore

mcnano@caltech

Computational Specifics

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Conclusions

The shell and the fluid, as well as their coupledinteraction, are considered in full detail

The proposed coupling approach is very robust andefficient

No remeshingAlgorithmic coupling with minor modifications of shell and fluid solvers

Although first steps towards validation are encouraging,detonation tube experiments are far too challenging tosimulate. Currently, a new set of shock tube experimentsare done by J. Shepherd at Caltech.