Anthony R. Ingraffea Symposium September 27, 2014 State- Based Peridynamic Lattice Modeling of Reinforced Concrete Structures Walter Gerstle Department of Civil Engineering University of New Mexico, U.S.A. 1
Scope Claude- Louis Navier and Augustin- Louis Cauchy, 1821-1827 State- Based Peridynamic Lattice Model Elasticity Damage Plasticity Examples Conclusions Navier Cauchy 2
Navier, 1821 3
Navier s assumption of spatial continuity 4
Does this look continuous to you? http://reinforced- concrete.blogspot.com/2010_10_01_archive.html 5
Cauchy s Concept of Stress R 2 Traction: Stress: R 1 6
Continuous function 7
Discontinuous Displacement across Crack u(x) is continuous above crack u(x) is discontinuous across crack 8
There is no physical reason why solids should deform continuously (in space) So why do we teach continuum mechanics? Because continuum mechanics yields analytical solutions (for a few simple problems) Computers are now ubiquitous But computers use discrete arithmetic So let us use discrete (particle) mechanics But we want some regular structure for our particles Let us use a particle lattice 9
Discrete models for solid mechanics 1941, Hrennikoff, A., Solution of problems of elasticity by the framework method 1977, Burt, N. J. & Dougill, J. W., Progressive Failure in a Model Heterogeneous Medium 1989, Herrmann, H. J.; Hansen, A. & Roux, S., Fracture of disordered, elastic lattices in two dimensions 1992, Schlangen, E. and J. G. M. Van Mier modeled the concrete as discrete lattice of beams. 1993, Itasca consulting group, Universal Discrete Element Code (UDEC). 1995, Jirásek, M. and Z. P. Bažant used a particle model with random geometry. 1998, Bolander, J. E. and S. Saito modeled the concrete as spring networks with random geometry. 2011, Cusatis, G. and coworkers proposed a meso-scale model for concrete in which they modeled the aggregates and mortar matrix. 2012, Potyondy, D. O. The Bonded-Particle Model (many others) 10
Peridynamics peri = near dynamic = force In 2000, S. A. Silling of Sandia National Laboratories introduced continuum peridynamic theory. In 2007, Silling introduced state- based continuum peridynamic theory. Peridynamics reformulates continuum mechanics theory to overcome deficiencies in modeling of deformation discontinuities. 11
Peridynamic physical description Terminology for peridynamic model. Peridynamic mathematical description (Equation of motion for particle i) 12
Peridynamic states The original peridynamic model included only a bond- based model for materials. In 2007, S. A. Silling introduced the concept of peridynamic states to generalize the original bond- based peridynamic model. In peridynamic state- based model, the interaction between two particles i and j, also, depends upon the state of other particles, k, in the neighborhood. State- based peridynamics is capable of modeling broader material behaviors such as plasticity. 13
Lattice Model 1 2 3 Y 1 Face-Centered Cubic (FCC) Z X 2 3 Hexagonal Close- Packed (HCP) Newton s laws are applied to each lattice particle 14
Quasi- brittle materials deform discontinuously in most regimes of interest. Original peridynamics has not discarded the continuum paradigm entirely. Computational implementation of peridynamics requires ad hoc discretization decisions. The lattice model discards the spatial continuum completely. The lattice model is computationally straightforward. 15
Solid models, like computer graphics, pixilated edge vertex volume face 16
Implemented so far: Constitutive models for concrete Linear elasticity Damage Plasticity Damping 17
Face- Centered Cubic Particle Lattice 18
Stretch State, Y 19
Force State, T 20
Relationship between stretch state, Y, and strain, ε For a homogeneous (small) strain field, (For a spatially non- homogeneous strain field, stretch state and strain are not directly comparable.) 21
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Linear Elastic Lattice Model 23
Decomposition of Stretch and Force States 24
Tensile Damage 1 1 S 1 S 2 S 3 Damage Potential versus Elastic Volumetric Stretch S 1 S 2 S 3 Bond Force versus Elastic Volumetric Stretch 25
Plasticity 26
Damping Internal damping force External damping force 27
Modeling of reinforcing bars and bond Steel particles interact with concrete particles within material horizon. Steel is one- dimensional elasto- plastic peridynamic lattice. Stiffness of peridynamic steel- concrete bonds similar to that of concrete- concrete bonds. Bond between steel and concrete is elastic in compression, but sustains no tension. Bond of reinforcement (red) to concrete (black) using peridynamic interactions (tan). 28
Reinforced Concrete Beam 29
Show Movies of Reinforced Concrete Beam in Bending and Torsion 30
(SPLM = State- based Peridynamic Lattice Model) SPLM can simulate major features of concrete: (Dynamics, stability, large deformations, damage, fracture, plasticity, post- peak mechanisms) Simplicity of SPLM makes it potentially applicable for practicing engineers: (Meshless method, no convergence studies needed, few material parameters) SPLM requires a lot of computational power, which is becoming cheap. SPLM competes with traditional approaches: (continuum mechanics, finite elements, fracture mechanics) SPLM in the classroom? Conclusions 31
Acknowledgements Dr. Susan Atlas, Department of Physics and Astronomy, UNM Dr. Stewart Silling, Sandia National Laboratories Center for Advanced Research Computing, UNM Graduate Students: Nicolas Sau, Eduardo Aguilera, Navid Sakhavand, Vijay Janardhanam, Kiran Tuniki, Asifur Rahman, Hossein Honarvar, Raybeau Richardson, Aziz Asadollahi 32