Numerical simulation of the dynamiccompression of a 6061-T6 aluminum metallic

foam

Biswajit Banerjee and Anup Bhawalkar

Center for the Simulation of Accidental Fires and ExplosionsUniversity 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 MotivationPrevious 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?

(Physics Today (2002), 55, 37-42)

Low weight to volume ratio.High weight to specificmechanical stiffness.Used for:

High-capacity impactabsorption.Acoustic and thermalcontrol.

B. Banerjee, A. Bhawalkar (Univ. of Utah) Simulation of Al Foam WCCM VII - 2006 3 / 23

Motivation

Why Aluminum Foams?

(Physics Today (2002), 55, 37-42)

Low weight to volume ratio.High weight to specificmechanical stiffness.Used for:

High-capacity impactabsorption.Acoustic and thermalcontrol.

B. Banerjee, A. Bhawalkar (Univ. of Utah) Simulation of Al Foam WCCM VII - 2006 3 / 23

Motivation

Why Aluminum Foams?

(Physics Today (2002), 55, 37-42)

Low weight to volume ratio.High weight to specificmechanical stiffness.Used for:

High-capacity impactabsorption.Acoustic and thermalcontrol.

B. Banerjee, A. Bhawalkar (Univ. of Utah) Simulation of Al Foam WCCM VII - 2006 3 / 23

Motivation

Why Aluminum Foams?

(Physics Today (2002), 55, 37-42)

Low weight to volume ratio.High weight to specificmechanical stiffness.Used for:

High-capacity impactabsorption.Acoustic and thermalcontrol.

B. Banerjee, A. Bhawalkar (Univ. of Utah) Simulation of Al Foam WCCM VII - 2006 3 / 23

Motivation

Why Aluminum Foams?

(Physics Today (2002), 55, 37-42)

Low weight to volume ratio.High weight to specificmechanical stiffness.Used for:

High-capacity impactabsorption.Acoustic and thermalcontrol.

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-cellfoams 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-cellfoams 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-cellfoams 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 cellularsolids.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 cellularsolids.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 cellularsolids.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 cellularsolids.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 cellularsolids.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 cellularsolids.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 cellularsolids.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 cellularsolids.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 cellularsolids.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 cellularsolids.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 cellularsolids.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 cellularsolids.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 cellularsolids.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 cellularsolids.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 etal. (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 etal. (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 etal. (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 etal. (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 etal. (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 etal. (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 etal. (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 etal. (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 etal. (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 PointMethod (MPM).(Sulsky et al., 1995,1996).Fluid dynamics with the multimaterial Implicit ContinuousEulerian (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 PointMethod (MPM).(Sulsky et al., 1995,1996).Fluid dynamics with the multimaterial Implicit ContinuousEulerian (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 PointMethod (MPM).(Sulsky et al., 1995,1996).Fluid dynamics with the multimaterial Implicit ContinuousEulerian (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 PointMethod (MPM).(Sulsky et al., 1995,1996).Fluid dynamics with the multimaterial Implicit ContinuousEulerian (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 PointMethod (MPM).(Sulsky et al., 1995,1996).Fluid dynamics with the multimaterial Implicit ContinuousEulerian (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 PointMethod (MPM).(Sulsky et al., 1995,1996).Fluid dynamics with the multimaterial Implicit ContinuousEulerian (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 PointMethod (MPM).(Sulsky et al., 1995,1996).Fluid dynamics with the multimaterial Implicit ContinuousEulerian (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 ThresholdStress model for 6061-T6 aluminum.Creation of foam microstructures by pressurization ofbubbles of soft material.Simulation of foam crush at various strain-rates andtemperatures 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 ThresholdStress model for 6061-T6 aluminum.Creation of foam microstructures by pressurization ofbubbles of soft material.Simulation of foam crush at various strain-rates andtemperatures 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 ThresholdStress model for 6061-T6 aluminum.Creation of foam microstructures by pressurization ofbubbles of soft material.Simulation of foam crush at various strain-rates andtemperatures 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(εp, ε̇, T , p) = (σa + Siσi + Seσe)µ(T , p)

µ0(1)

The athermal part contains the effect of grain size.The scaling factors depend on strain-rate, temperature, andpressure.Uses an empirical strain hardening model (modified Vocemodel)

dσe

dεp= θ0(T ) [1− f (σe)] (2)

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(εp, ε̇, T , p) = (σa + Siσi + Seσe)µ(T , p)

µ0(1)

The athermal part contains the effect of grain size.The scaling factors depend on strain-rate, temperature, andpressure.Uses an empirical strain hardening model (modified Vocemodel)

dσe

dεp= θ0(T ) [1− f (σe)] (2)

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(εp, ε̇, T , p) = (σa + Siσi + Seσe)µ(T , p)

µ0(1)

The athermal part contains the effect of grain size.The scaling factors depend on strain-rate, temperature, andpressure.Uses an empirical strain hardening model (modified Vocemodel)

dσe

dεp= θ0(T ) [1− f (σe)] (2)

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(εp, ε̇, T , p) = (σa + Siσi + Seσe)µ(T , p)

µ0(1)

The athermal part contains the effect of grain size.The scaling factors depend on strain-rate, temperature, andpressure.Uses an empirical strain hardening model (modified Vocemodel)

dσe

dεp= θ0(T ) [1− f (σe)] (2)

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 shearmodulus model (Nadal and Le Poac, 2003; Guinan and Steinberg, 1974)

µ(T , p) =1

J (T/Tm)

[(µ0 +

∂µ

∂pp

η1/3

)(1−

TTm

)+

ρ

Cmkb T

](3)

We use a pressure dependent melting temperature model(Burakovsky et al., 2000)

Tm(p) = Tm(0)

[1ζ

+1

ζ4/3

µ′

0µ0

p

](4)

B. Banerjee, A. Bhawalkar (Univ. of Utah) Simulation of Al Foam WCCM VII - 2006 10 / 23

The Plasticity Model

Shear Modulus Model

We use a temperature and pressure dependent shearmodulus model (Nadal and Le Poac, 2003; Guinan and Steinberg, 1974)

µ(T , p) =1

J (T/Tm)

[(µ0 +

∂µ

∂pp

η1/3

)(1−

TTm

)+

ρ

Cmkb T

](3)

We use a pressure dependent melting temperature model(Burakovsky et al., 2000)

Tm(p) = Tm(0)

[1ζ

+1

ζ4/3

µ′

0µ0

p

](4)

B. Banerjee, A. Bhawalkar (Univ. of Utah) Simulation of Al Foam WCCM VII - 2006 10 / 23

The Plasticity Model

Model Validation: Shear and Melt Models

0 0.2 0.4 0.6 0.8 1

0

5

10

15

20

25

30

35

40

45

50

T/Tm

Sh

ear M

odu

lus (G

Pa)

Tallon (1979)ASMH (1996)

NP (η = 0.9)

NP (η = 1.0)

NP (η = 1.1)

Shear Modulus.

Tm (K

) 0

2000

4000

6000

−50 0 50 100 150

Boehler and Ross (1997)Burakovsky et al. (2000)

BPS Melt Model

Pressure (GPa)

Melt Temperature.

B. Banerjee, A. Bhawalkar (Univ. of Utah) Simulation of Al Foam WCCM VII - 2006 11 / 23

The Plasticity Model

Model Validation: Flow Stress Model

0 0.02 0.04 0.06 0.08 0.1 0.120

100

200

300

400

500

600

True Plastic Strain

Tru

e S

tress (M

Pa)

Expt. 300K 0.001/s (Davidson, 1973)Expt. 367K 0.001/s (Davidson, 1973)Expt. 478K 0.001/s (Davidson, 1973)Expt. 589K 0.001/s (Green, 1966)

Strain rate = 0.0001/s.

0 0.05 0.1 0.15 0.2 0.25 0.30

50

100

150

200

250

300

350

400

450

500

True Plastic Strain

Tru

e S

tress (M

Pa)

Expt. 297K 4.8e−5/s (Davidson, 1973)

Expt. 373K 5.7e−4/s (Davidson, 1973)

Expt. 473K 5.7e−4/s (Davidson, 1973)

Strain rate = 0.0057/s.

B. Banerjee, A. Bhawalkar (Univ. of Utah) Simulation of Al Foam WCCM VII - 2006 12 / 23

The Plasticity Model

Model Validation: Flow Stress Model

0 0.02 0.04 0.06 0.08 0.1

0

100

200

300

400

500

600

True Plastic Strain

Tru

e S

tress (M

Pa)

Expt. 298K 1000/s (Rosenberg et al., 1986)

Expt. 463K 1000/s (Rosenberg et al., 1986)

Expt. 618K 1000/s (Rosenberg et al., 1986)

Strain rate = 1000/s.

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)

0

100

200

300

400

500

600

700

0 0.05 0.1 0.15 0.2 0.25

Tru

e S

tress (M

Pa)

True Plastic Strain

Strain rate = 1500-2000/s.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

100

200

300

400

500

True Plastic Strain

Tru

e S

tress (M

Pa)

Expt. 298 K 3000/s (Lee et al., 2000)

Expt. 298 K 8000/s (Lesuer et al., 2001)

Strain rate = 3000-8000/s.

B. Banerjee, A. Bhawalkar (Univ. of Utah) Simulation of Al Foam WCCM VII - 2006 13 / 23

Creation of Foam Microstructures

Bubble Creation

0 5 10 15 200

5

10

15

20

mm

mm

0 10 20 30 400

5

10

15

20

25

30

35

40

mm

mm

0 10 20 30 40 50 600

10

20

30

40

50

60

mm

mm

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

0 5 10 15 200

5

10

15

20

mm

mm

0 10 20 30 400

5

10

15

20

25

30

35

40

mm

mm

0 10 20 30 40 50 600

10

20

30

40

50

60

mm

mm

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

0 5 10 15 200

5

10

15

20

mm

mm

0 10 20 30 400

5

10

15

20

25

30

35

40

mm

mm

0 10 20 30 40 50 600

10

20

30

40

50

60

mm

mm

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

0 0.5 1 1.5 2−50

0

50

100

150

200

250

300

True strain

Tru

e S

tres

s (M

Pa)

Impact Velocity = 200 m/s (~ 5000/s)

No GasWith Gas

B. Banerjee, A. Bhawalkar (Univ. of Utah) Simulation of Al Foam WCCM VII - 2006 18 / 23

Crushing of Foam Microstructures

Effect of Strain-Rate and Temperature

0 0.2 0.4 0.6 0.8 1−2

0

2

4

6

8

10

True strain

Tru

e S

tres

s (M

Pa)

v = 20 m/sv = 200 m/s

Effect of Strain Rate

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

True strainT

rue

Stre

ss (

MPa

)

Impact Velocity = 20 m/s (~ 500/s)

T = 300 KT = 450 K

Effect of Temperature

B. Banerjee, A. Bhawalkar (Univ. of Utah) 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-straincurves.Crushing of foam shows no shear banding.Further work is needed to create a macroscopic model ofthe 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