Dynamic Constitutive Response of Polymeric Foams Subjected to Direct Impact Loading Behrad Koohbor a, Addis Kidane a, Wei-Yang Lu b a. Department of Mechanical Engineering, University of South Carolina, Columbia, SC b. Sandia National Laboratories, Livermore, CA

American Society for Composites 30th Technical Conference Michigan State University, East Lansing, MI September 2015

Introduction and Motivation Foams are engineering material of choice for applications requiring energy absorption and/or structural stability with reduced weight. Often used in automotive and safety applications.

Many of their applications entail High Strain Rate loading conditions. Image courtesy of ERG Aerospace corporation

Introduction and Motivation There are major challenges in the study of polymeric foams under dynamic loading conditions:

1) Low Impedance nature of the material 2) Change of Density during deformation

Challenge #1 Low Impedance nature of the material Which results in the

Delayed stress/strain equilibrium Low transmitted signal

Proposed Solutions: Pulse Shaping o

To increase the rise time and increase the stress uniformity in the specimen

Polymeric Bars or hollow tubes (applicable in SHPB) o

To reduce the impedance mismatch between the specimen and the bars, in order to acquire transmitted signal

Thin Specimens o

Shorten the wave reverberation time

Challenge #1 Low Impedance nature of the material Which results in the

Delayed stress/strain equilibrium Low transmitted signal

Proposed Solutions: Pulse Shaping o

To increase the rise time and increase the stress uniformity in the specimen

Polymeric Bars or hollow tubes (applicable in SHPB) o

To reduce the impedance mismatch between the specimen and the bars, in order to acquire transmitted signal

Thin Specimens o

Shorten the wave reverberation time

Challenge #1 Low Impedance nature of the material Which results in the

Delayed stress/strain equilibrium Low transmitted signal

Proposed Solutions: Pulse Shaping

Polymeric Bars or hollow tubes (applicable in SHPB) Thin Specimens o

Shorten the wave reverberation time

Challenge #1 Low Impedance nature of the material Which results in the

Delayed stress/strain equilibrium Low transmitted signal

Proposed Solutions: Pulse Shaping

Polymeric Bars or hollow tubes (applicable in SHPB) Thin Specimens o

Shorten the wave reverberation time

Number of cells is not large enough to represent the material response at continuum scale

Challenge #1 Low Impedance nature of the material Which results in the

Delayed stress/strain equilibrium Low transmitted signal

Proposed Solutions: Pulse Shaping

Polymeric Bars or hollow tubes (applicable in SHPB) Thin Specimens Objective of this Work:

Taking advantage of full-field measurements to calculate and include the effect of inertia stress into the analysis, as suggested in the literature*, **. * Pierron F, Zhu H, Siviour C. 2014 Beyond Hopkinson’s bar. Phil. Trans. R. Soc. A 372 ** Othman R, Aloui S, Poitou A. 2010 Identification of non-homogeneous…Polym. Test. 29

Theoretical Approach General Dynamic Stress Equilibrium:

ij , j bi ai

(1)

Uniaxial Compression, Absence of

Body Force Shear Stresses

z a z z

(2)

Acceleration = 0: z cons.

Acceleration ≠ 0:

z

z2

z

(3)

a z dz z2

z1

z1

(4)

Theoretical Approach L

z L, t z 0, t

L

0

, t a z , t d

Stress at position x=L and time t Stress measured at position x=0 and time t Inertia stress, which includes: • Variation of ρ from z=0 to z=L • Variation of a from z=0 to z=L

Challenge #2 Change of density during deformation (Compressibility)

Proposed Solutions: A one-dimensional model proposed to enable the calculation of local density, as a function of initial density (ρ0), local axial strain (εz), and local plastic Poisson’s ratio (ν) Assumptions o Conservation of mass

z, t 0 exp z z, t 2 z ,t 1 Local density at position z and time t

Initial Density

Local axial strain at position z and time t

z, t

d r z, t d z z, t

Local plastic Poisson’s ratio at position z and time t

Compressibility model

z, t 0 exp z z, t 2 z ,t 1 Local density at position z and time t

Initial Density

Local axial strain at position z and time t

Direct Impact Experiments using Shock Tube Piezotronic load-cells inserted behind the specimen (z = 0) High strength aluminum projectile utilized Different number of Mylar diaphragms (Strain rate applied on the specimen (ε): 2460 s-1)

H0 = 25.4 mm D0 = 25.4 mm ρ0 = 560 kg/m3 (35 pcf)

Direct Impact Experiments using Shock Tube Stereovision high speed camera system used to capture the full-field deformation response: Camera system -------------- Photron SA-X2 Cameras Resolution --------------------- 384×264 pix2 Frame rate -------------------- 100,000 fps (10 µs temporal resolution) Stereo angle ------------------ 16.1o stereo angle (Cameras mounted vertically) Illumination system --------- High intensity LED white light System triggered using oscilloscope Load data and images acquired simultaneously using high speed DAQ

Direct Impact Results

ρ0 = 160 kg/m3 (10 pcf)

ρ0 = 640 kg/m3 (40 pcf)

Full-Field Density – Compressibility Model Local density of the material was calculated using the proposed model:

z, t 0 exp z z, t 2 z ,t 1 Local density at position z and time t

r z, t

z z, t

z, t

Initial Density

Local axial strain at position z and time t

z, t

d r z, t d z z, t

Full-Field Density – Compressibility Model Local density of the material was calculated using the proposed model:

z, t 0 exp z z, t 2 z ,t 1 Local density at position z and time t

Initial Density

Local axial strain at position z and time t

Up to ~6% increase in the density was calculated

Full-Field Acceleration Full-field acceleration az(z,t) was calculated from the full-field displacement uz(z,t) based on a finite difference scheme:

az z, t

1 u z z, t t 2u z z, t u z z, t t t 2

Evaluating the Inertia Term Numerical evaluation of the integral (inertia term): z

z z, t z 0, t , t a z , t d 0

n

z

, t a , t d a s 0

i

z

i 1

Number of slices used in this work (n): 25 Thickness of each slice ≈ 1 mm

i

i

Full-Field Stress and Strain z

z z, t z 0, t , t az , t d 0

Full-Field Stress and Strain z

z z, t z 0, t , t az , t d 0

Constitutive Response Local stress-strain curves obtained at different locations:

Constitutive Response Comparison with conventional measurement:

New Results – Higher Strain Rates

H = 18 mm D = 26 mm Projectile Velocity = 123 m/s

H = 28 mm D = 26 mm Projectile Velocity = 162 m/s

Summary A non-parametric analysis was performed to include the concurrent influences of inertia stress and material compressibility into the dynamic deformation analysis of low impedance polymeric foams. Full-field stress-strain response of the specimen was obtained using the non-parametric analysis.

The main limitation here was the time resolution of the system, which can be overcome using ultra high speed cameras currently available. The method can be considered as a useful means to characterize low impedance and soft materials deformed at high strain rate conditions.

Acknowledgements

New Results – Higher Strain Rates

H = 18 mm D = 26 mm Projectile Velocity = 123 m/s

H = 28 mm D = 26 mm Projectile Velocity = 162 m/s

Summary A non-parametric analysis was performed to include the concurrent influences of inertia stress and material compressibility into the dynamic deformation analysis of low impedance polymeric foams. Full-field stress-strain response of the specimen was obtained using the non-parametric analysis.

The main limitation here was the time resolution of the system, which can be overcome using ultra high speed cameras currently available. The method can be considered as a useful means to characterize low impedance and soft materials deformed at high strain rate conditions.

Acknowledgements

American Society for Composites 30th Technical Conference Michigan State University, East Lansing, MI September 2015

Introduction and Motivation Foams are engineering material of choice for applications requiring energy absorption and/or structural stability with reduced weight. Often used in automotive and safety applications.

Many of their applications entail High Strain Rate loading conditions. Image courtesy of ERG Aerospace corporation

Introduction and Motivation There are major challenges in the study of polymeric foams under dynamic loading conditions:

1) Low Impedance nature of the material 2) Change of Density during deformation

Challenge #1 Low Impedance nature of the material Which results in the

Delayed stress/strain equilibrium Low transmitted signal

Proposed Solutions: Pulse Shaping o

To increase the rise time and increase the stress uniformity in the specimen

Polymeric Bars or hollow tubes (applicable in SHPB) o

To reduce the impedance mismatch between the specimen and the bars, in order to acquire transmitted signal

Thin Specimens o

Shorten the wave reverberation time

Challenge #1 Low Impedance nature of the material Which results in the

Delayed stress/strain equilibrium Low transmitted signal

Proposed Solutions: Pulse Shaping o

To increase the rise time and increase the stress uniformity in the specimen

Polymeric Bars or hollow tubes (applicable in SHPB) o

To reduce the impedance mismatch between the specimen and the bars, in order to acquire transmitted signal

Thin Specimens o

Shorten the wave reverberation time

Challenge #1 Low Impedance nature of the material Which results in the

Delayed stress/strain equilibrium Low transmitted signal

Proposed Solutions: Pulse Shaping

Polymeric Bars or hollow tubes (applicable in SHPB) Thin Specimens o

Shorten the wave reverberation time

Challenge #1 Low Impedance nature of the material Which results in the

Delayed stress/strain equilibrium Low transmitted signal

Proposed Solutions: Pulse Shaping

Polymeric Bars or hollow tubes (applicable in SHPB) Thin Specimens o

Shorten the wave reverberation time

Number of cells is not large enough to represent the material response at continuum scale

Challenge #1 Low Impedance nature of the material Which results in the

Delayed stress/strain equilibrium Low transmitted signal

Proposed Solutions: Pulse Shaping

Polymeric Bars or hollow tubes (applicable in SHPB) Thin Specimens Objective of this Work:

Taking advantage of full-field measurements to calculate and include the effect of inertia stress into the analysis, as suggested in the literature*, **. * Pierron F, Zhu H, Siviour C. 2014 Beyond Hopkinson’s bar. Phil. Trans. R. Soc. A 372 ** Othman R, Aloui S, Poitou A. 2010 Identification of non-homogeneous…Polym. Test. 29

Theoretical Approach General Dynamic Stress Equilibrium:

ij , j bi ai

(1)

Uniaxial Compression, Absence of

Body Force Shear Stresses

z a z z

(2)

Acceleration = 0: z cons.

Acceleration ≠ 0:

z

z2

z

(3)

a z dz z2

z1

z1

(4)

Theoretical Approach L

z L, t z 0, t

L

0

, t a z , t d

Stress at position x=L and time t Stress measured at position x=0 and time t Inertia stress, which includes: • Variation of ρ from z=0 to z=L • Variation of a from z=0 to z=L

Challenge #2 Change of density during deformation (Compressibility)

Proposed Solutions: A one-dimensional model proposed to enable the calculation of local density, as a function of initial density (ρ0), local axial strain (εz), and local plastic Poisson’s ratio (ν) Assumptions o Conservation of mass

z, t 0 exp z z, t 2 z ,t 1 Local density at position z and time t

Initial Density

Local axial strain at position z and time t

z, t

d r z, t d z z, t

Local plastic Poisson’s ratio at position z and time t

Compressibility model

z, t 0 exp z z, t 2 z ,t 1 Local density at position z and time t

Initial Density

Local axial strain at position z and time t

Direct Impact Experiments using Shock Tube Piezotronic load-cells inserted behind the specimen (z = 0) High strength aluminum projectile utilized Different number of Mylar diaphragms (Strain rate applied on the specimen (ε): 2460 s-1)

H0 = 25.4 mm D0 = 25.4 mm ρ0 = 560 kg/m3 (35 pcf)

Direct Impact Experiments using Shock Tube Stereovision high speed camera system used to capture the full-field deformation response: Camera system -------------- Photron SA-X2 Cameras Resolution --------------------- 384×264 pix2 Frame rate -------------------- 100,000 fps (10 µs temporal resolution) Stereo angle ------------------ 16.1o stereo angle (Cameras mounted vertically) Illumination system --------- High intensity LED white light System triggered using oscilloscope Load data and images acquired simultaneously using high speed DAQ

Direct Impact Results

ρ0 = 160 kg/m3 (10 pcf)

ρ0 = 640 kg/m3 (40 pcf)

Full-Field Density – Compressibility Model Local density of the material was calculated using the proposed model:

z, t 0 exp z z, t 2 z ,t 1 Local density at position z and time t

r z, t

z z, t

z, t

Initial Density

Local axial strain at position z and time t

z, t

d r z, t d z z, t

Full-Field Density – Compressibility Model Local density of the material was calculated using the proposed model:

z, t 0 exp z z, t 2 z ,t 1 Local density at position z and time t

Initial Density

Local axial strain at position z and time t

Up to ~6% increase in the density was calculated

Full-Field Acceleration Full-field acceleration az(z,t) was calculated from the full-field displacement uz(z,t) based on a finite difference scheme:

az z, t

1 u z z, t t 2u z z, t u z z, t t t 2

Evaluating the Inertia Term Numerical evaluation of the integral (inertia term): z

z z, t z 0, t , t a z , t d 0

n

z

, t a , t d a s 0

i

z

i 1

Number of slices used in this work (n): 25 Thickness of each slice ≈ 1 mm

i

i

Full-Field Stress and Strain z

z z, t z 0, t , t az , t d 0

Full-Field Stress and Strain z

z z, t z 0, t , t az , t d 0

Constitutive Response Local stress-strain curves obtained at different locations:

Constitutive Response Comparison with conventional measurement:

New Results – Higher Strain Rates

H = 18 mm D = 26 mm Projectile Velocity = 123 m/s

H = 28 mm D = 26 mm Projectile Velocity = 162 m/s

Summary A non-parametric analysis was performed to include the concurrent influences of inertia stress and material compressibility into the dynamic deformation analysis of low impedance polymeric foams. Full-field stress-strain response of the specimen was obtained using the non-parametric analysis.

The main limitation here was the time resolution of the system, which can be overcome using ultra high speed cameras currently available. The method can be considered as a useful means to characterize low impedance and soft materials deformed at high strain rate conditions.

Acknowledgements

New Results – Higher Strain Rates

H = 18 mm D = 26 mm Projectile Velocity = 123 m/s

H = 28 mm D = 26 mm Projectile Velocity = 162 m/s

Summary A non-parametric analysis was performed to include the concurrent influences of inertia stress and material compressibility into the dynamic deformation analysis of low impedance polymeric foams. Full-field stress-strain response of the specimen was obtained using the non-parametric analysis.

The main limitation here was the time resolution of the system, which can be overcome using ultra high speed cameras currently available. The method can be considered as a useful means to characterize low impedance and soft materials deformed at high strain rate conditions.

Acknowledgements