Numerical simulation of the dynamic
compression of a 6061-T6 aluminum metallic
foam
Biswajit Banerjee and Anup Bhawalkar
Center for the Simulation of Accidental Fires and Explosions
University of Utah
7th World Conference on Computational Mechanics, 2006
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
1 / 23
Outline
1
Motivation
Previous Work
2
Approach
3
The Plasticity Model
4
Creation of Foam Microstructures
5
Crushing of Foam Microstructures
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
2 / 23
Motivation
Why Aluminum Foams?
Low weight to volume ratio.
High weight to specific
mechanical stiffness.
Used for:
High-capacity impact
absorption.
Acoustic and thermal
control.
(Physics Today (2002), 55, 37-42)
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
3 / 23
Motivation
Why Aluminum Foams?
Low weight to volume ratio.
High weight to specific
mechanical stiffness.
Used for:
High-capacity impact
absorption.
Acoustic and thermal
control.
(Physics Today (2002), 55, 37-42)
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
3 / 23
Motivation
Why Aluminum Foams?
Low weight to volume ratio.
High weight to specific
mechanical stiffness.
Used for:
High-capacity impact
absorption.
Acoustic and thermal
control.
(Physics Today (2002), 55, 37-42)
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
3 / 23
Motivation
Why Aluminum Foams?
Low weight to volume ratio.
High weight to specific
mechanical stiffness.
Used for:
High-capacity impact
absorption.
Acoustic and thermal
control.
(Physics Today (2002), 55, 37-42)
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
3 / 23
Motivation
Why Aluminum Foams?
Low weight to volume ratio.
High weight to specific
mechanical stiffness.
Used for:
High-capacity impact
absorption.
Acoustic and thermal
control.
(Physics Today (2002), 55, 37-42)
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
3 / 23
Motivation
Challenges
Determination of dynamic structure/property relations.
Homogenization of nonlinear dynamic processes is nontrivial.
We would like to determine whether gases inside closed-cell
foams have a significant effect.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
4 / 23
Motivation
Challenges
Determination of dynamic structure/property relations.
Homogenization of nonlinear dynamic processes is nontrivial.
We would like to determine whether gases inside closed-cell
foams have a significant effect.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
4 / 23
Motivation
Challenges
Determination of dynamic structure/property relations.
Homogenization of nonlinear dynamic processes is nontrivial.
We would like to determine whether gases inside closed-cell
foams have a significant effect.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
4 / 23
Motivation
Previous Work
Previous Work: Models
Gibson and Ashby (1997):
Detailed exposition of structure-property relations in cellular
solids.
Simplified microstructures.
Ashby et al. (2000):
A design guide for metal foams.
A small-strain plasticity model was presented.
Deshpande and Fleck (2000):
Detailed study of yield surfaces.
Improved plasticity model for small strains.
Schmidt (2004):
Model of elastic-plastic anisotropy in metal foams.
Large strain effects included.
Benke and Weichert (2005):
Model incorporates effect of fluid pressure.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
5 / 23
Motivation
Previous Work
Previous Work: Models
Gibson and Ashby (1997):
Detailed exposition of structure-property relations in cellular
solids.
Simplified microstructures.
Ashby et al. (2000):
A design guide for metal foams.
A small-strain plasticity model was presented.
Deshpande and Fleck (2000):
Detailed study of yield surfaces.
Improved plasticity model for small strains.
Schmidt (2004):
Model of elastic-plastic anisotropy in metal foams.
Large strain effects included.
Benke and Weichert (2005):
Model incorporates effect of fluid pressure.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
5 / 23
Motivation
Previous Work
Previous Work: Models
Gibson and Ashby (1997):
Detailed exposition of structure-property relations in cellular
solids.
Simplified microstructures.
Ashby et al. (2000):
A design guide for metal foams.
A small-strain plasticity model was presented.
Deshpande and Fleck (2000):
Detailed study of yield surfaces.
Improved plasticity model for small strains.
Schmidt (2004):
Model of elastic-plastic anisotropy in metal foams.
Large strain effects included.
Benke and Weichert (2005):
Model incorporates effect of fluid pressure.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
5 / 23
Motivation
Previous Work
Previous Work: Models
Gibson and Ashby (1997):
Detailed exposition of structure-property relations in cellular
solids.
Simplified microstructures.
Ashby et al. (2000):
A design guide for metal foams.
A small-strain plasticity model was presented.
Deshpande and Fleck (2000):
Detailed study of yield surfaces.
Improved plasticity model for small strains.
Schmidt (2004):
Model of elastic-plastic anisotropy in metal foams.
Large strain effects included.
Benke and Weichert (2005):
Model incorporates effect of fluid pressure.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
5 / 23
Motivation
Previous Work
Previous Work: Models
Gibson and Ashby (1997):
Detailed exposition of structure-property relations in cellular
solids.
Simplified microstructures.
Ashby et al. (2000):
A design guide for metal foams.
A small-strain plasticity model was presented.
Deshpande and Fleck (2000):
Detailed study of yield surfaces.
Improved plasticity model for small strains.
Schmidt (2004):
Model of elastic-plastic anisotropy in metal foams.
Large strain effects included.
Benke and Weichert (2005):
Model incorporates effect of fluid pressure.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
5 / 23
Motivation
Previous Work
Previous Work: Models
Gibson and Ashby (1997):
Detailed exposition of structure-property relations in cellular
solids.
Simplified microstructures.
Ashby et al. (2000):
A design guide for metal foams.
A small-strain plasticity model was presented.
Deshpande and Fleck (2000):
Detailed study of yield surfaces.
Improved plasticity model for small strains.
Schmidt (2004):
Model of elastic-plastic anisotropy in metal foams.
Large strain effects included.
Benke and Weichert (2005):
Model incorporates effect of fluid pressure.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
5 / 23
Motivation
Previous Work
Previous Work: Models
Gibson and Ashby (1997):
Detailed exposition of structure-property relations in cellular
solids.
Simplified microstructures.
Ashby et al. (2000):
A design guide for metal foams.
A small-strain plasticity model was presented.
Deshpande and Fleck (2000):
Detailed study of yield surfaces.
Improved plasticity model for small strains.
Schmidt (2004):
Model of elastic-plastic anisotropy in metal foams.
Large strain effects included.
Benke and Weichert (2005):
Model incorporates effect of fluid pressure.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
5 / 23
Motivation
Previous Work
Previous Work: Models
Gibson and Ashby (1997):
Detailed exposition of structure-property relations in cellular
solids.
Simplified microstructures.
Ashby et al. (2000):
A design guide for metal foams.
A small-strain plasticity model was presented.
Deshpande and Fleck (2000):
Detailed study of yield surfaces.
Improved plasticity model for small strains.
Schmidt (2004):
Model of elastic-plastic anisotropy in metal foams.
Large strain effects included.
Benke and Weichert (2005):
Model incorporates effect of fluid pressure.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
5 / 23
Motivation
Previous Work
Previous Work: Models
Gibson and Ashby (1997):
Detailed exposition of structure-property relations in cellular
solids.
Simplified microstructures.
Ashby et al. (2000):
A design guide for metal foams.
A small-strain plasticity model was presented.
Deshpande and Fleck (2000):
Detailed study of yield surfaces.
Improved plasticity model for small strains.
Schmidt (2004):
Model of elastic-plastic anisotropy in metal foams.
Large strain effects included.
Benke and Weichert (2005):
Model incorporates effect of fluid pressure.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
5 / 23
Motivation
Previous Work
Previous Work: Models
Gibson and Ashby (1997):
Detailed exposition of structure-property relations in cellular
solids.
Simplified microstructures.
Ashby et al. (2000):
A design guide for metal foams.
A small-strain plasticity model was presented.
Deshpande and Fleck (2000):
Detailed study of yield surfaces.
Improved plasticity model for small strains.
Schmidt (2004):
Model of elastic-plastic anisotropy in metal foams.
Large strain effects included.
Benke and Weichert (2005):
Model incorporates effect of fluid pressure.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
5 / 23
Motivation
Previous Work
Previous Work: Models
Gibson and Ashby (1997):
Detailed exposition of structure-property relations in cellular
solids.
Simplified microstructures.
Ashby et al. (2000):
A design guide for metal foams.
A small-strain plasticity model was presented.
Deshpande and Fleck (2000):
Detailed study of yield surfaces.
Improved plasticity model for small strains.
Schmidt (2004):
Model of elastic-plastic anisotropy in metal foams.
Large strain effects included.
Benke and Weichert (2005):
Model incorporates effect of fluid pressure.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
5 / 23
Motivation
Previous Work
Previous Work: Models
Gibson and Ashby (1997):
Detailed exposition of structure-property relations in cellular
solids.
Simplified microstructures.
Ashby et al. (2000):
A design guide for metal foams.
A small-strain plasticity model was presented.
Deshpande and Fleck (2000):
Detailed study of yield surfaces.
Improved plasticity model for small strains.
Schmidt (2004):
Model of elastic-plastic anisotropy in metal foams.
Large strain effects included.
Benke and Weichert (2005):
Model incorporates effect of fluid pressure.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
5 / 23
Motivation
Previous Work
Previous Work: Models
Gibson and Ashby (1997):
Detailed exposition of structure-property relations in cellular
solids.
Simplified microstructures.
Ashby et al. (2000):
A design guide for metal foams.
A small-strain plasticity model was presented.
Deshpande and Fleck (2000):
Detailed study of yield surfaces.
Improved plasticity model for small strains.
Schmidt (2004):
Model of elastic-plastic anisotropy in metal foams.
Large strain effects included.
Benke and Weichert (2005):
Model incorporates effect of fluid pressure.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
5 / 23
Motivation
Previous Work
Previous Work: Models
Gibson and Ashby (1997):
Detailed exposition of structure-property relations in cellular
solids.
Simplified microstructures.
Ashby et al. (2000):
A design guide for metal foams.
A small-strain plasticity model was presented.
Deshpande and Fleck (2000):
Detailed study of yield surfaces.
Improved plasticity model for small strains.
Schmidt (2004):
Model of elastic-plastic anisotropy in metal foams.
Large strain effects included.
Benke and Weichert (2005):
Model incorporates effect of fluid pressure.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
5 / 23
Motivation
Previous Work
Previous Work: Experiments and Simulations
Deshpande and Fleck (2000), Paul and Ramamurty (2000),
Danneman and Lankford (2000), Tan et al.(2005), Mukai et
al. (2005):
High strain-rate experiments on Al foams.
Contradictory statements on strain-rate sensitivity.
General consensus is that strain-rate dependence is small.
Deqing et al. (2005):
Effect of cell size of Al foam properties (experiments).
Decrease in energy absorption with increasing cell size!
Issen et al. (2005), Schmidt (2004):
Observed localized compaction bands in closed cell foams.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
6 / 23
Motivation
Previous Work
Previous Work: Experiments and Simulations
Deshpande and Fleck (2000), Paul and Ramamurty (2000),
Danneman and Lankford (2000), Tan et al.(2005), Mukai et
al. (2005):
High strain-rate experiments on Al foams.
Contradictory statements on strain-rate sensitivity.
General consensus is that strain-rate dependence is small.
Deqing et al. (2005):
Effect of cell size of Al foam properties (experiments).
Decrease in energy absorption with increasing cell size!
Issen et al. (2005), Schmidt (2004):
Observed localized compaction bands in closed cell foams.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
6 / 23
Motivation
Previous Work
Previous Work: Experiments and Simulations
Deshpande and Fleck (2000), Paul and Ramamurty (2000),
Danneman and Lankford (2000), Tan et al.(2005), Mukai et
al. (2005):
High strain-rate experiments on Al foams.
Contradictory statements on strain-rate sensitivity.
General consensus is that strain-rate dependence is small.
Deqing et al. (2005):
Effect of cell size of Al foam properties (experiments).
Decrease in energy absorption with increasing cell size!
Issen et al. (2005), Schmidt (2004):
Observed localized compaction bands in closed cell foams.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
6 / 23
Motivation
Previous Work
Previous Work: Experiments and Simulations
Deshpande and Fleck (2000), Paul and Ramamurty (2000),
Danneman and Lankford (2000), Tan et al.(2005), Mukai et
al. (2005):
High strain-rate experiments on Al foams.
Contradictory statements on strain-rate sensitivity.
General consensus is that strain-rate dependence is small.
Deqing et al. (2005):
Effect of cell size of Al foam properties (experiments).
Decrease in energy absorption with increasing cell size!
Issen et al. (2005), Schmidt (2004):
Observed localized compaction bands in closed cell foams.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
6 / 23
Motivation
Previous Work
Previous Work: Experiments and Simulations
Deshpande and Fleck (2000), Paul and Ramamurty (2000),
Danneman and Lankford (2000), Tan et al.(2005), Mukai et
al. (2005):
High strain-rate experiments on Al foams.
Contradictory statements on strain-rate sensitivity.
General consensus is that strain-rate dependence is small.
Deqing et al. (2005):
Effect of cell size of Al foam properties (experiments).
Decrease in energy absorption with increasing cell size!
Issen et al. (2005), Schmidt (2004):
Observed localized compaction bands in closed cell foams.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
6 / 23
Motivation
Previous Work
Previous Work: Experiments and Simulations
Deshpande and Fleck (2000), Paul and Ramamurty (2000),
Danneman and Lankford (2000), Tan et al.(2005), Mukai et
al. (2005):
High strain-rate experiments on Al foams.
Contradictory statements on strain-rate sensitivity.
General consensus is that strain-rate dependence is small.
Deqing et al. (2005):
Effect of cell size of Al foam properties (experiments).
Decrease in energy absorption with increasing cell size!
Issen et al. (2005), Schmidt (2004):
Observed localized compaction bands in closed cell foams.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
6 / 23
Motivation
Previous Work
Previous Work: Experiments and Simulations
Deshpande and Fleck (2000), Paul and Ramamurty (2000),
Danneman and Lankford (2000), Tan et al.(2005), Mukai et
al. (2005):
High strain-rate experiments on Al foams.
Contradictory statements on strain-rate sensitivity.
General consensus is that strain-rate dependence is small.
Deqing et al. (2005):
Effect of cell size of Al foam properties (experiments).
Decrease in energy absorption with increasing cell size!
Issen et al. (2005), Schmidt (2004):
Observed localized compaction bands in closed cell foams.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
6 / 23
Motivation
Previous Work
Previous Work: Experiments and Simulations
Deshpande and Fleck (2000), Paul and Ramamurty (2000),
Danneman and Lankford (2000), Tan et al.(2005), Mukai et
al. (2005):
High strain-rate experiments on Al foams.
Contradictory statements on strain-rate sensitivity.
General consensus is that strain-rate dependence is small.
Deqing et al. (2005):
Effect of cell size of Al foam properties (experiments).
Decrease in energy absorption with increasing cell size!
Issen et al. (2005), Schmidt (2004):
Observed localized compaction bands in closed cell foams.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
6 / 23
Motivation
Previous Work
Previous Work: Experiments and Simulations
Deshpande and Fleck (2000), Paul and Ramamurty (2000),
Danneman and Lankford (2000), Tan et al.(2005), Mukai et
al. (2005):
High strain-rate experiments on Al foams.
Contradictory statements on strain-rate sensitivity.
General consensus is that strain-rate dependence is small.
Deqing et al. (2005):
Effect of cell size of Al foam properties (experiments).
Decrease in energy absorption with increasing cell size!
Issen et al. (2005), Schmidt (2004):
Observed localized compaction bands in closed cell foams.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
6 / 23
Approach
Computational Tools
Uintah Computational Framework.
Parallel multiphysics framework.
Large deformation solid mechanics with the Material Point
Method (MPM).
(Sulsky et al., 1995,1996).
Fluid dynamics with the multimaterial Implicit Continuous
Eulerian (ICE) algorithm.
(Kashiwa et al., 2000).
Fluid-structure interaction on a common grid. (Kashiwa et al., 2000;
Guilkey et al., 2006).
Rate-dependent elastic-plastic stress computation. (Maudlin and
Schiferl, 1996).
SCIRun visualization tools.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
7 / 23
Approach
Computational Tools
Uintah Computational Framework.
Parallel multiphysics framework.
Large deformation solid mechanics with the Material Point
Method (MPM).
(Sulsky et al., 1995,1996).
Fluid dynamics with the multimaterial Implicit Continuous
Eulerian (ICE) algorithm.
(Kashiwa et al., 2000).
Fluid-structure interaction on a common grid. (Kashiwa et al., 2000;
Guilkey et al., 2006).
Rate-dependent elastic-plastic stress computation. (Maudlin and
Schiferl, 1996).
SCIRun visualization tools.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
7 / 23
Approach
Computational Tools
Uintah Computational Framework.
Parallel multiphysics framework.
Large deformation solid mechanics with the Material Point
Method (MPM).
(Sulsky et al., 1995,1996).
Fluid dynamics with the multimaterial Implicit Continuous
Eulerian (ICE) algorithm.
(Kashiwa et al., 2000).
Fluid-structure interaction on a common grid. (Kashiwa et al., 2000;
Guilkey et al., 2006).
Rate-dependent elastic-plastic stress computation. (Maudlin and
Schiferl, 1996).
SCIRun visualization tools.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
7 / 23
Approach
Computational Tools
Uintah Computational Framework.
Parallel multiphysics framework.
Large deformation solid mechanics with the Material Point
Method (MPM).
(Sulsky et al., 1995,1996).
Fluid dynamics with the multimaterial Implicit Continuous
Eulerian (ICE) algorithm.
(Kashiwa et al., 2000).
Fluid-structure interaction on a common grid. (Kashiwa et al., 2000;
Guilkey et al., 2006).
Rate-dependent elastic-plastic stress computation. (Maudlin and
Schiferl, 1996).
SCIRun visualization tools.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
7 / 23
Approach
Computational Tools
Uintah Computational Framework.
Parallel multiphysics framework.
Large deformation solid mechanics with the Material Point
Method (MPM).
(Sulsky et al., 1995,1996).
Fluid dynamics with the multimaterial Implicit Continuous
Eulerian (ICE) algorithm.
(Kashiwa et al., 2000).
Fluid-structure interaction on a common grid. (Kashiwa et al., 2000;
Guilkey et al., 2006).
Rate-dependent elastic-plastic stress computation. (Maudlin and
Schiferl, 1996).
SCIRun visualization tools.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
7 / 23
Approach
Computational Tools
Uintah Computational Framework.
Parallel multiphysics framework.
Large deformation solid mechanics with the Material Point
Method (MPM).
(Sulsky et al., 1995,1996).
Fluid dynamics with the multimaterial Implicit Continuous
Eulerian (ICE) algorithm.
(Kashiwa et al., 2000).
Fluid-structure interaction on a common grid. (Kashiwa et al., 2000;
Guilkey et al., 2006).
Rate-dependent elastic-plastic stress computation. (Maudlin and
Schiferl, 1996).
SCIRun visualization tools.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
7 / 23
Approach
Computational Tools
Uintah Computational Framework.
Parallel multiphysics framework.
Large deformation solid mechanics with the Material Point
Method (MPM).
(Sulsky et al., 1995,1996).
Fluid dynamics with the multimaterial Implicit Continuous
Eulerian (ICE) algorithm.
(Kashiwa et al., 2000).
Fluid-structure interaction on a common grid. (Kashiwa et al., 2000;
Guilkey et al., 2006).
Rate-dependent elastic-plastic stress computation. (Maudlin and
Schiferl, 1996).
SCIRun visualization tools.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
7 / 23
Approach
The Process
Determination of parameters for the Mechanical Threshold
Stress model for 6061-T6 aluminum.
Creation of foam microstructures by pressurization of
bubbles of soft material.
Simulation of foam crush at various strain-rates and
temperatures with MTS model for 6061-T6 Al.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
8 / 23
Approach
The Process
Determination of parameters for the Mechanical Threshold
Stress model for 6061-T6 aluminum.
Creation of foam microstructures by pressurization of
bubbles of soft material.
Simulation of foam crush at various strain-rates and
temperatures with MTS model for 6061-T6 Al.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
8 / 23
Approach
The Process
Determination of parameters for the Mechanical Threshold
Stress model for 6061-T6 aluminum.
Creation of foam microstructures by pressurization of
bubbles of soft material.
Simulation of foam crush at various strain-rates and
temperatures with MTS model for 6061-T6 Al.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
8 / 23
The Plasticity Model
The Mechanical Threshold Stress Model
The flow stress is given by (Follansbee and Kocks, 1988; Goto et al., 2000)
σy (p , ,
˙ T , p) = (σa + Si σi + Se σe )
µ(T , p)
µ0
(1)
The athermal part contains the effect of grain size.
The scaling factors depend on strain-rate, temperature, and
pressure.
Uses an empirical strain hardening model (modified Voce
model)
dσe
= θ0 (T ) [1 − f (σe )]
(2)
dp
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
9 / 23
The Plasticity Model
The Mechanical Threshold Stress Model
The flow stress is given by (Follansbee and Kocks, 1988; Goto et al., 2000)
σy (p , ,
˙ T , p) = (σa + Si σi + Se σe )
µ(T , p)
µ0
(1)
The athermal part contains the effect of grain size.
The scaling factors depend on strain-rate, temperature, and
pressure.
Uses an empirical strain hardening model (modified Voce
model)
dσe
= θ0 (T ) [1 − f (σe )]
(2)
dp
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
9 / 23
The Plasticity Model
The Mechanical Threshold Stress Model
The flow stress is given by (Follansbee and Kocks, 1988; Goto et al., 2000)
σy (p , ,
˙ T , p) = (σa + Si σi + Se σe )
µ(T , p)
µ0
(1)
The athermal part contains the effect of grain size.
The scaling factors depend on strain-rate, temperature, and
pressure.
Uses an empirical strain hardening model (modified Voce
model)
dσe
= θ0 (T ) [1 − f (σe )]
(2)
dp
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
9 / 23
The Plasticity Model
The Mechanical Threshold Stress Model
The flow stress is given by (Follansbee and Kocks, 1988; Goto et al., 2000)
σy (p , ,
˙ T , p) = (σa + Si σi + Se σe )
µ(T , p)
µ0
(1)
The athermal part contains the effect of grain size.
The scaling factors depend on strain-rate, temperature, and
pressure.
Uses an empirical strain hardening model (modified Voce
model)
dσe
= θ0 (T ) [1 − f (σe )]
(2)
dp
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
9 / 23
The Plasticity Model
Shear Modulus Model
We use a temperature and pressure dependent shear
modulus model (Nadal and Le Poac, 2003; Guinan and Steinberg, 1974)
!
"
!
#
T
1
∂µ p
ρ
µ(T , p) =
1−
µ0 +
+
k T
J (T /Tm )
∂p η 1/3
Tm
Cm b
(3)
We use a pressure dependent melting temperature model
(Burakovsky et al., 2000)
"
0
1
1 µ
Tm (p) = Tm (0)
+ 4/3 0 p
ζ ζ
µ0
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
#
(4)
WCCM VII - 2006
10 / 23
The Plasticity Model
Shear Modulus Model
We use a temperature and pressure dependent shear
modulus model (Nadal and Le Poac, 2003; Guinan and Steinberg, 1974)
!
"
!
#
T
1
∂µ p
ρ
µ(T , p) =
1−
µ0 +
+
k T
J (T /Tm )
∂p η 1/3
Tm
Cm b
(3)
We use a pressure dependent melting temperature model
(Burakovsky et al., 2000)
"
0
1
1 µ
Tm (p) = Tm (0)
+ 4/3 0 p
ζ ζ
µ0
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
#
(4)
WCCM VII - 2006
10 / 23
The Plasticity Model
Model Validation: Shear and Melt Models
50
Tallon (1979)
ASMH (1996)
NP (η = 0.9)
NP (η = 1.0)
NP (η = 1.1)
45
Boehler and Ross (1997)
Burakovsky et al. (2000)
BPS Melt Model
35
4000
30
Tm (K)
Shear Modulus (GPa)
40
6000
25
20
2000
15
10
5
0
0
0.2
0.4
0.6
0.8
1
0
−50
Shear Modulus.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
0
50
100
150
Pressure (GPa)
T/Tm
Melt Temperature.
Simulation of Al Foam
WCCM VII - 2006
11 / 23
The Plasticity Model
Model Validation: Flow Stress Model
500
600
Expt.
Expt.
Expt.
Expt.
0.001/s
0.001/s
0.001/s
0.001/s
(Davidson, 1973)
(Davidson, 1973)
(Davidson, 1973)
(Green, 1966)
400
300
200
Expt. 297K 4.8e−5/s (Davidson, 1973)
Expt. 373K 5.7e−4/s (Davidson, 1973)
Expt. 473K 5.7e−4/s (Davidson, 1973)
450
400
True Stress (MPa)
True Stress (MPa)
500
300K
367K
478K
589K
350
300
250
200
150
100
100
50
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0
0
True Plastic Strain
0.1
0.15
0.2
0.25
0.3
True Plastic Strain
Strain rate = 0.0001/s.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
0.05
Strain rate = 0.0057/s.
Simulation of Al Foam
WCCM VII - 2006
12 / 23
The Plasticity Model
Model Validation: Flow Stress Model
600
Expt. 298K 1000/s (Rosenberg et al., 1986)
Expt. 463K 1000/s (Rosenberg et al., 1986)
Expt. 618K 1000/s (Rosenberg et al., 1986)
True Stress (MPa)
500
400
300
500
200
100
0
0.02
0.04
0.06
0.08
0.1
True Plastic Strain
Strain rate = 1000/s.
700
Expt. 77 K 1500/s (Lee et al., 2000)
Expt. 298 K 2000/s (Lee et al., 2000)
Expt. 373 K 1500/s (Ogawa, 2001)
True Stress (MPa)
600
True Stress (MPa)
400
0
300
200
100
500
0
400
Expt. 298 K 3000/s (Lee et al., 2000)
Expt. 298 K 8000/s (Lesuer et al., 2001)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
True Plastic Strain
300
Strain rate = 3000-8000/s.
200
100
0
0
0.05
0.1
0.15
0.2
0.25
True Plastic Strain
Strain rate = 1500-2000/s.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
13 / 23
Creation of Foam Microstructures
Bubble Creation
60
50
40
40
35
20
mm
30
25
30
mm
mm
15
10
20
20
15
10
10
5
5
0
0
5
10
mm
15
20
0
0
10
20
mm
30
40
0
0
10
20
30
mm
40
50
60
Use a Poisson process to create bubble distribution.
Periodic RVE.
Uses input size distribution and volume fraction.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
14 / 23
Creation of Foam Microstructures
Bubble Creation
60
50
40
40
35
20
mm
30
25
30
mm
mm
15
10
20
20
15
10
10
5
5
0
0
5
10
mm
15
20
0
0
10
20
mm
30
40
0
0
10
20
30
mm
40
50
60
Use a Poisson process to create bubble distribution.
Periodic RVE.
Uses input size distribution and volume fraction.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
14 / 23
Creation of Foam Microstructures
Bubble Creation
60
50
40
40
35
20
mm
30
25
30
mm
mm
15
10
20
20
15
10
10
5
5
0
0
5
10
mm
15
20
0
0
10
20
mm
30
40
0
0
10
20
30
mm
40
50
60
Use a Poisson process to create bubble distribution.
Periodic RVE.
Uses input size distribution and volume fraction.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
14 / 23
Creation of Foam Microstructures
Bubble Pressurization
Use fluid-structure interaction to expand bubbles.
Compressible Neo-Hookean solid with K = 0.6 MPa and µ =
0.3 MPa.
Gas inside bubbles is pressurized by adding heat.
Gas between bubbles has low pressure and temperature.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
15 / 23
Creation of Foam Microstructures
Bubble Pressurization
Use fluid-structure interaction to expand bubbles.
Compressible Neo-Hookean solid with K = 0.6 MPa and µ =
0.3 MPa.
Gas inside bubbles is pressurized by adding heat.
Gas between bubbles has low pressure and temperature.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
15 / 23
Creation of Foam Microstructures
Bubble Pressurization
Use fluid-structure interaction to expand bubbles.
Compressible Neo-Hookean solid with K = 0.6 MPa and µ =
0.3 MPa.
Gas inside bubbles is pressurized by adding heat.
Gas between bubbles has low pressure and temperature.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
15 / 23
Creation of Foam Microstructures
Bubble Pressurization
Use fluid-structure interaction to expand bubbles.
Compressible Neo-Hookean solid with K = 0.6 MPa and µ =
0.3 MPa.
Gas inside bubbles is pressurized by adding heat.
Gas between bubbles has low pressure and temperature.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
15 / 23
Crushing of Foam Microstructures
Crushing Without Interior Gas
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
16 / 23
Crushing of Foam Microstructures
Crushing With Interior Gas
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
17 / 23
Crushing of Foam Microstructures
Stress-Strain Curves
Impact Velocity = 200 m/s (~ 5000/s)
300
True Stress (MPa)
250
No Gas
With Gas
200
150
100
50
0
−50
0
B. Banerjee, A. Bhawalkar (Univ. of Utah)
0.5
1
True strain
Simulation of Al Foam
1.5
2
WCCM VII - 2006
18 / 23
Crushing of Foam Microstructures
Effect of Strain-Rate and Temperature
Impact Velocity = 20 m/s (~ 500/s)
10
8
10
v = 20 m/s
v = 200 m/s
6
True Stress (MPa)
True Stress (MPa)
8
4
2
6
4
2
0
−2
0
T = 300 K
T = 450 K
0.2
0.4
0.6
True strain
0.8
1
0
0
Effect of Strain Rate
B. Banerjee, A. Bhawalkar (Univ. of Utah)
0.2
0.4
0.6
True strain
0.8
1
Effect of Temperature
Simulation of Al Foam
WCCM VII - 2006
19 / 23
Summary
Summary
MTS plasticity model of 6061-T6 Al.
A novel way of creating a foam microstructure.
There is a clear effect of including gas in the stress-strain
curves.
Crushing of foam shows no shear banding.
Further work is needed to create a macroscopic model of
the dynamic response of Al foam.
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
20 / 23
Summary
Creation of Foam Microstructure
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
21 / 23
Summary
Crushing of Empty Foam
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
22 / 23
Summary
Crushing of Gas-Filled Foam
B. Banerjee, A. Bhawalkar (Univ. of Utah)
Simulation of Al Foam
WCCM VII - 2006
23 / 23