Fracture and Fragmentation of Thin-Shells Fehmi Cirak Michael Ortiz, Anna Pandolfi California...
-
date post
21-Dec-2015 -
Category
Documents
-
view
214 -
download
0
Transcript of Fracture and Fragmentation of Thin-Shells Fehmi Cirak Michael Ortiz, Anna Pandolfi California...
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
GeometryPlate 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 - Convergence
21376 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