Computational Physics Department
Peridynamic Modeling of Structural Damage and Failure Stewart Silling Sandia National Laboratories
[email protected] Simon Kahan Cray Inc.
[email protected] April 21, 2004 Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. /home/sasilli/emu/salishan1/salishan1.frm
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Department of Energy Joint DOE/DoD Munitions Technology Program The Boeing Company US Nuclear Regulatory Commission
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Contributors Abe Askari, Boeing Phantom Works Tony Crispino, Century Dynamics, Inc. Paul Demmie, Sandia Bob Cole, Sandia Prof. Florin Bobaru, University of Nebraska Sponsors
Why another method? Theory Examples Parallelization and performance EMU on the Cray MTA-2
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Computational Physics Department
Outline
u ˜
-
u ˜
+
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Develop a model in which exactly the same equations hold everywhere, regardless of any discontinuities. ♦ To do this, get rid of spatial derivatives.
Goal
• Special techniques of fracture mechanics are cumbersome.
discontinuities.
• Classical continuum mechanics uses partial differential equations. • But the partial derivatives do not exist along cracks and other
Computational Physics Department
Need for a new theory: Why is fracture a hard problem?
Basic idea of the peridynamic theory Computational Physics Department
• Equation of motion:
ρu˙˙ = L u + b ˜ ˜ ˜ where L u is a functional. ˜ • A useful special case: L u ( x, t ) = ∫ f ( u ( x', t ) – u ( x, t ), x' – x ) dV x'. ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ R˜ where x is any point in the reference configuration, and ˜ f is a vector-valued function. ˜ More concisely: L = ∫ f ( u' – u, x' – x ) dV '. ˜ ˜ ˜ ˜ ˜ R˜
• f is the pairwise force
˜ function. It contains all constitutive information. • It is convenient to assume that f vanishes outside some ˜ δ. horizon
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R
δ
f ˜
x' x ˜ ˜
Microelastic materials Computational Physics Department
• Simplest class of constitutive models: microelastic: ♦ There exists a scalar-valued function w , called the micropotential, such that
∂w ( η, ξ ) f ( η, ξ ) = ∂ η ˜ ˜ ˜ ˜ ˜ ˜
♦ Interaction between particles is equivalent to an elastic spring. ♦ The spring properties can depend on the reference separation vector.
ξ+η ˜ ˜
u' ˜ x' ˜
u ˜
ξ ˜
x ˜ ♦ Work done by external forces is stored in a recoverable form
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Some useful material models Computational Physics Department
• Microelastic ♦ Each pair of particles is connected by a spring. • Linear microelastic ♦ The springs are all linear. • Microviscoelastic need to store bond data! ♦ Springs + dashpots • Microelastic-plastic ♦ Springs have a yield point • Ideal brittle microelastic: springs break at some critical stretch ε.
Spring force
Yielded Unload Failed
ε
Elastic
Yielded
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Spring stretch
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*J. Fineberg & M. Marder, Physics Reports 313 (1999) 1-108
Experiment*
Calculated damage contours
Crack growth direction
• Plate is stretched vertically. • Code predicts stable-unstable transition.
Computational Physics Department
Dynamic fracture in a PMMA plate
• Code predicts correct crack angles*. • Crack velocity ~ 900 m/s.
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*J. F. Kalthoff & S. Winkler, in Impact Loading and Dynamic Behavior of Materials, C. Y. Chiem, ed. (1988)
Maraging steel plate
Notches
Computational Physics Department
Dynamic fracture in a tough steel: mode transition
Peridynamic fracture model is “autonomous” Computational Physics Department
• Cracks grow when and where it is energetically
favorable for them to do so. • Path, growth rate, arrest, branching, mutual interaction are predicted by the constitutive model and equation of motion (alone). • No need for any externally supplied relation controlling these things. • Any number of cracks can occur and interact.
•
Interfaces between materials have their own bond properties.
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1.70 ms
1.36 ms
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1. Not all of the target is shown.
0.68 ms
0.38 ms
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2.05 ms
1.02 ms
• Peak force occurs at about 0.4ms (end of drilling phase):1
Computational Physics Department
Example: Perforation of thin ductile targets
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Weak fiber
Weak matrix
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Weak interface
Initial condition
DCB test on a composite
• Crack path, growth, and stability depend only on material properties. • No need for separate laws governing crack growth.
Computational Physics Department
Example: Composite material fracture
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Computational Physics Department
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Example: Dynamic fracture in a balloon
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Damage on surface
Pullout damage
Computational Physics Department
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Exit crater
Exit debris
Example: Penetration into reinforced concrete
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• Cracks interact with each other and eventually join up.
“Experiment”
upward.
• Elastic sheet is held fixed on 3 sides. Part of the 4th side is pulled
Computational Physics Department
Example: Membrane fracture
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Stretching of a network of fibers (courtesy of Prof. F. Bobaru, University of Nebraska)
Self-shaping of a fiber due to interactions between different parts
• Fibers can interact through long-range (e.g. van der Waals) or contact. f
Computational Physics Department
Examples: Mechanics of fibers
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*simulation of full-scale experimental data in open literature (Sugano et al., Nuclear Engineering and Design 140 373-385 (1993).
• F4 into a 3.6m thick concrete block*
Computational Physics Department
Aircraft impact onto structures
Numerical solution method for dynamic problems Computational Physics Department
• Theory lends itself to mesh-free numerical methods. • No elements. • Changing connectivity. • Brute-force integration in space. Δx j δ i
i
ρu˙˙ = ˜
j
∑
i
x –x
