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Research Papers

Modeling of Dynamically Loaded Open-Cell Metallic Foams: Yielding, Collapse, and Strain Rate Effects

[+] Author and Article Information
Pedro A. Romero

Department of Mechanical and Aerospace Engineering, Rutgers University, Piscataway, NJ 08854promero@rci.rutgers.edu

Winston O. Soboyejo

Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08540soboyejo@princeton.edu

Alberto M. Cuitiño1

Department of Mechanical and Aerospace Engineering, Rutgers University, Piscataway, NJ 08854cuitino@jove.rutgers.edu

1

Corresponding author.

J. Appl. Mech 77(3), 031002 (Jan 26, 2010) (8 pages) doi:10.1115/1.4000386 History: Received March 04, 2008; Revised May 22, 2009; Published January 26, 2010; Online January 26, 2010

Open-cell metallic foams exhibit properties desirable in engineering applications requiring mitigation of the adverse effects resulting from impact loading; however, the history dependent dynamic response of these cellular materials has not been clearly elucidated. This article contributes an approach for modeling the response of dynamically loaded open-cell metallic foams from ligament level to unit cell level to specimen level. The effective response captures the localized chaotic collapse phenomena through ligament reorientation at cell level while maintaining the history of plastic deformation at ligament level. First, the phenomenological elastoplastic constitutive behavior of the ligaments composing the unit cell is modeled. Then, using the constitutive ligament model, the effective unit cell response is obtained from a micromechanical model that enforces the principle of minimum action on a representative 3D unit cell. Finally, the macroscopic specimen response is predicted utilizing a finite element analysis program, which obtains the response at every Gauss point in the mesh from the microscopic unit cell model. The current communication focuses on the ability of the model to capture the yielding and collapse behaviors, as well as the strain rate effects, observed during impact loading of metallic foams.

Copyright © 2010 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Open-cell metallic foam (6101 Al alloy foam, 40 ppi, 4% density): (a) cellular structure and (b) typical quasistatic mechanical response

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Figure 2

Localization of deformation during quasistatic compression of F-as fabricated, T6-strengthened, O-annealed metallic open-cell foams as reported in Ref. 9

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Figure 3

Histograms from Ref. 9 showing the heterogeneous distribution of the local vertical strain field for the foams (F-as fabricated, T6-strengthened, O-annealed) presented in Fig. 2: (a) 14.3% applied vertical compression and (b) 28.3% applied vertical compression

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Figure 4

Schematic of the macroscopic continuum view and the microscopic cellular structural view for foam material composed of small cells relative to specimen size. The affine fields such as u and F prescribed at macroscopic point X are trickled down to the microscopic ligament midpoints. The unit cell vertex moves an additional amount χ resulting in nonaffine deformation within the microstructure.

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Figure 5

Depiction of the local kinematic evolution of the length of each half ligament li and the angle between any two ligaments ψij. The axial strain for any ligament i is defined as ϵi=Δλi and the change in angle between any two ligaments i and j is defined as αij=ψij−Ψij.

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Figure 6

The axial constitutive behavior of the struts composing each unit cell of an open-cell metallic foam as predicted by Eq. 10,12. The variables in Eq. 10,11,12 were taken as initial length Lo=1.0 mm, initial strut radius ro=0.3 mm, E=50.0 GPa, σyo=50.0 MPa, exponent r=4, axial exponent m=4, ϵpo=0.005, and ϵ̇po=0.10. (a) The loading axial constitutive response for tensile loading. (b) The axial constitutive response for loading and unloading for both tensile and compressive loadings.

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Figure 7

The bending constitutive behavior of the struts composing each unit cell of an open-cell metallic foam as predicted by Eqs. 20,21,22. The variables in Eqs. 20,21,22 were taken as initial length Lo=1.0 mm, initial strut radius ro=0.3 mm, E=50.0 GPa, σyo=50.0 MPa, exponent r=4, exponent m=20, αpo=0.005, and α̇po=0.10. (a) The loading bending constitutive response for tensile loading. (b) The bending constitutive response for loading and unloading for both tensile and compressive loadings.

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Figure 8

The predicted evolution of the unit cell configuration during compressive loading. The variables in Eqs. 10,11,12,20,21,22 were set to initial strut length Lo=1.0 mm, initial strut radius ro=0.3 mm, E=50 GPa, σyo=50 MPa, axial exponent r=4, axial exponent m=4, ϵpo=0.005, ϵ̇po=0.10, bending exponent r=4, bending exponent m=20, αpo=0.005, and α̇po=0.10. (a) 0% deformation, (b) 10% deformation, (c) 30% deformation, and (d) 50% deformation.

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Figure 9

The effective response for foams with elastic cell walls. The values for the variables in Eqs. 10,11,12,20,21,22 were set to initial length Lo=1.0 mm, initial strut radius ro=0.3 mm, E=10 GPa, σyo=0.5 GPa, axial exponent r=1, axial exponent m=1, ϵpo=0.005, ϵ̇po=0.10, bending exponent r=1, bending exponent m=1, αpo=0.005, and α̇po=0.10. (a) The effective cell and plateau response for foams composed from struts with a high initial yield stress such that the response does not involve any plastic deformation. (b) The load and unload responses when the cell walls (struts) do not undergo any plastic deformation.

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Figure 14

Simulation of impact at a constant downward velocity of v=10.0 m/s on a homogeneous elastoplastic 2D open-cell foam specimen with a high initial yield stress resulting in a small amount of plastic deformation. The snapshots were taken at different times corresponding to different average vertical strains during the dynamic compression. The colors represent the local deformation in the vertical direction and the black lines describe the FEM mesh. The snapshots clearly demonstrate the transition from (nearly) homogeneous deformation to heterogeneous deformation (mixture of collapsed and uncollapsed regions) and back to nearly homogeneous (completely collapsed) deformation.

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Figure 15

Simulation of impact at a constant downward velocity of v=10.0 m/s on a homogeneous elastoplastic 2D open-cell foam specimen with a low initial yield stress resulting in a large amount of plastic deformation. The snapshot were taken at different times corresponding to different average vertical strains during the dynamic compression. The colors represent the local deformation in the vertical direction and the black lines describe the FEM mesh. The snapshots clearly demonstrate the transition from (nearly) homogeneous deformation to heterogeneous deformation (mixture of collapsed and uncollapsed regions) and back to nearly homogeneous (completely collapsed) deformation.

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Figure 13

The effect of the initial rate of plastic deformation (ϵ̇opandα̇op) on the response of metallic open-cell foams. The foam density was taken as 100 kg/m3, the applied strain was maintained at 1000 s−1, and the values for the variables in Eqs. 10,11,12,20,21,22 were set to initial strut length Lo=1.0 mm, initial strut radius ro=0.3 mm, E=50 GPa, σyo=50 MPa, axial exponent r=4, axial exponent m=4, ϵpo=0.005, bending exponent r=4, bending exponent m=20, and αpo=0.005. (a) The effective cell response (solid line) and the effective plateau (dashed line) for foams loaded at different initial rates of plastic deformation. (b) The effect of initial rate of plastic deformation on the predicted effective plateau stress for metallic open-cell foams.

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Figure 12

The effect of applied strain rate on the response of metallic open-cell foams. The foam density was taken as 100 kg/m3 and the values for the variables in Eqs. 10,11,12,20,21,22 were set to initial strut length Lo=1.0 mm, initial strut radius ro=0.3 mm, E=50 GPa, σyo=50 MPa, axial exponent r=4, axial exponent m=4, ϵpo=0.005, ϵ̇po=0.10, bending exponent r=4, bending exponent m=20, αpo=0.005, and α̇po=0.10. (a) The effective cell response (solid line) and the effective plateau (dashed line) for foams loaded at different strain rates ϵ̇. (b) The effect of the applied strain rate ϵ̇ on the predicted effective plateau stress for metallic open-cell foams.

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Figure 11

The effect of initial yield stress on the response of metallic open-cell foams. The foam density was taken as 100 kg/m3 and every run was carried out at a strain rate of 1 s−1. The values for the variables in Eqs. 10,11,12,20,21,22 were set to initial strut length Lo=1.0 mm, initial strut radius ro=0.3 mm, E=50 GPa, axial exponent r=4, axial exponent m=4, ϵpo=0.005, ϵ̇po=0.10, bending exponent r=4, bending exponent m=20, αpo=0.005, and α̇po=0.10. (a) The effective cell response (solid line) and the effective plateau (dashed line) for foams composed of struts with different initial yield stress σyo. (b) The effect of the initial yield stress σyo on the predicted effective plateau stress for metallic open-cell foams.

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Figure 10

The effective response for foams with elastoplastic cell walls. The values for the variables in Eqs. 10,11,12,20,21,22 were set to initial strut length Lo=1.0 mm, initial strut radius ro=0.3 mm, E=50 GPa, σyo=50 MPa, axial exponent r=4, axial exponent m=4, ϵpo=0.005, ϵ̇po=0.10, bending exponent r=4, bending exponent m=20, αpo=0.005, and α̇po=0.10. (a) The effective cell and plateau response for foams composed from struts with low initial yield stress such that the response involves a lot of plastic deformation. (b) The load and unload responses when the cell walls (struts) undergo plastic deformation.

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