Polymeric Foams Subjected to Direct Impact Loading Behrad Koohbor a - - PowerPoint PPT Presentation

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

Michigan State University, East Lansing, MI September 2015

American Society for Composites 30th Technical Conference

Introduction and Motivation

Image courtesy of ERG Aerospace corporation

 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.

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
• To increase the rise time and increase the stress uniformity in the specimen
• Polymeric Bars or hollow tubes (applicable in SHPB)
• To reduce the impedance mismatch between the specimen and the bars, in order to

acquire transmitted signal

• Thin Specimens
• 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
• To increase the rise time and increase the stress uniformity in the specimen
• Polymeric Bars or hollow tubes (applicable in SHPB)
• To reduce the impedance mismatch between the specimen and the bars, in order to

acquire transmitted signal

• Thin Specimens
• 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
• 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
• 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

 General Dynamic Stress Equilibrium:

i i j ij

a b     

,

 Uniaxial Compression,

Absence of Body Force Shear Stresses

z z

a z     

 Acceleration = 0:  Acceleration ≠ 0:

. cons

z 





 

2 1 1 2

z z z z z z z

dz a   

(4) (3) (2) (1)

Theoretical Approach

Stress at position x=L and time t

       



 

 

L z z z

d t a t t t L

 

      , , , ,

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

Theoretical Approach

L

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)

• Conservation of mass

Challenge #2

     

t z d t z d t z

z r

, , ,     

       

  1

, 2

, exp ,





t z z

t z t z



  

Local density at position z and time t Initial Density Local axial strain at position z and time t Local plastic Poisson’s ratio at position z and time t

Assumptions

Compressibility model

       

  1

, 2

, exp ,





t z z

t z t z



  

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)

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 Experiments using Shock Tube

Direct Impact Results

ρ0 = 640 kg/m3 (40 pcf) ρ0 = 160 kg/m3 (10 pcf)

Full-Field Density – Compressibility Model

Local density of the material was calculated using the proposed model:

       

  1

, 2

, exp ,





t z z

t z t z



  

Local density at position z and time t Initial Density Local axial strain at position z and time t

     

t z d t z d t z

z r

, , ,       

t z

r

, 

 

t z, 

 

t z

z

, 

Up to ~6% increase in the density was calculated

Local density of the material was calculated using the proposed model:

       

  1

, 2

, exp ,





t z z

t z t z



  

Local density at position z and time t Initial Density Local axial strain at position z and time t

Full-Field Density – Compressibility Model

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:

         

t t z u t z u t t z u t t z a

z z z z

        , , 2 , 1 ,

2

Evaluating the Inertia Term

       



 

 

z z z z

d t a t t t z

 

      , , , ,

   

     

 

  



n i i i i z z

s a d t a t

1

, ,     

 

Number of slices used in this work (n): 25

Thickness of each slice ≈ 1 mm

Numerical evaluation of the integral (inertia term):

Full-Field Stress and Strain

       



 

 

z z z z

d t a t t t z

 

      , , , ,

Full-Field Stress and Strain

       



 

 

z z z z

d t a t t t z

 

      , , , ,

Constitutive Response

Local stress-strain curves obtained at different locations:

Constitutive Response

Comparison with conventional measurement:

New Results – Higher Strain Rates

H = 28 mm D = 26 mm Projectile Velocity = 162 m/s H = 18 mm D = 26 mm Projectile Velocity = 123 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 = 28 mm D = 26 mm Projectile Velocity = 162 m/s H = 18 mm D = 26 mm Projectile Velocity = 123 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