constitutive model captured all of the foam behaviors that had been observed in the experiments. .... Modification to Step Function in Yield Equation . . . . . 46.
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SAND86-2927 * UC-71 Unlimited Release Printed April 1987 c
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A Phenomenological Constitutive Model for Low Density Polyurethane Foams
Michael K. Neilsen, Harold S. Morgan, Raymond D. Krieg
Prepared by Sandia National Laboratories Albuquerque, New Mexico 87 185 and Livermore, California 94550 for the United States Department of Energy under Contract DE-AC04-76DP00789
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Issued by Sandia National Laboratories, operated for the United States Department of Energy by Sandia Corporation. NOTICE: This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that ita use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government, any agency thereof or any of their contractors or subcontractors. The views and opinions expressed herein do not necessarily state or reflect those of the United States Government, any agency thereof or any of their contractors or subcontractors.
Printed in the United States of America Available from National Technical Information Service U.S. Department of Commerce 5285 Port Royal Road Springfield, VA 22161 NTIS price codes Printed copy: A05 Microfiche copy: A01
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Distribution Category UC-71 SAND86-2927 U n l i m i t e d Release P r i n t e d A p r i l 1987 A Phenomenological C o n s t i t u t i v e Model f o r Low D e n s i t y P o l y u r e t h a n e Foams"
M i c h a e l K . N e i l s e n , H a r o l d S . Morgan, and Raymond 0. K r i e g A p p l i e d Mechanics D i v i s i o n I Sandia N a t i o n a l L a b o r a t o r i e s A1 buquerque, NM 87185 ABSTRACT R e s u l t s f r o m a s e r i e s o f h y d r o s t a t i c and t r i a x i a l compression t e s t s which were p e r f o r m e d on p o l y u r e t h a n e foams a r e p r e s e n t e d i n t h i s r e p o r t . These t e s t s i n d i c a t e t h a t t h e v o l u m e t r i c and d e v i a t o r i c p a r t s o f t h e foam b e h a v i o r a r e s t r o n g l y coupled. This c o u p l i n g b e h a v i o r c o u l d n o t be c a p t u r e d w i t h any o f s e v e r a l commonly used p l a s t i c i t y models. Thus, a new c o n s t i t u t i v e model was developed. T h i s new model was based on a d e c o m p o s i t i o n o f t h e foam response i n t o two p a r t s : (1) response o f t h e polymer s k e l e t o n , and ( 2 ) response o f t h e a i r i n s i d e t h e c e l l s . The a i r c o n t r i b u t i o n was c o m p l e t e l y v o l u m e t r i c . The new c o n s t i t u t i v e model was implemented i n two f i n i t e element codes, SANCHO and PRONTO. R e s u l t s f r o m a s e r i e s o f a n a l y s e s completed w i t h t h e s e codes i n d i c a t e d t h a t t h e new c o n s t i t u t i v e model c a p t u r e d a l l o f t h e foam b e h a v i o r s t h a t had been observed i n t h e e x p e r i m e n t s . F i n a l l y , a t y p i c a l dynamic p r o b l e m was analyzed u s i n g t h e new c o n s t i t u t i v e model and o t h e r c o n s t i t u t i v e models t o demonstrate d i f f e r e n c e s between t h e models. R e s u l t s f r o m t h i s s e r i e s o f analyses i n d i c a t e d t h a t t h e new c o n s t i t u t i v e model generated displacement and a c c e l e r a t i o n p r e d i c t i o n s t h a t were between p r e d i c t i o n s o b t a i n e d u s i n g t h e o t h e r models. T h i s r e s u l t was expected.
* T h i s work p e r f o r m e d a t Sandia N a t i o n a l L a b o r a t o r i e s s u p p o r t e d b y t h e U.S. Department of Energy under c o n t r a c t number DE-AC04-76DP00789.
ACKNOWLEDGEMENTS The a u t h o r s acknowledge t h e e a r l y c o n t r i b u t i o n s o f R i c h a r d Yoshimura, Karl S c h u l e r a l s o p a r t i c i p a t e d i n t h e e a r l y model development stages. The NMERI/CERF t e s t s were completed by Debbie Hamberg a t t h e U n i v e r s i t y o f New Mexico. D u r i n g t h e i m p l e m e n t a t i o n o f t h e c o n s t i t u t i v e model i n t h e f i n i t e element codes, v a l u a b l e a s s i s t a n c e was p r o v i d e d by Mike Stone, 1521, and Lee T a y l o r , 1523. The e f f o r t s o f t h e above r e s e a r c h e r s a r e g r a t e f u l l y acknowledged.
6323, Rod May, 7544 and K a r l S c h u l e r , 1522, who d e f i n e d t h e t e s t s .
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CONTENTS Page
.
............................ Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix F i g u r e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . T e s t D e s c r i p t i o n and R e s u l t s . . . . . . . . . . . . . . . . . . . . 3 . A p p l i c a t i o n o f E x i s t i n g C o n s t i t u t i v e Models t o Low-Density P o l y u r e t h a n e Foams . . . . . . . . . . . . . . . . . . . . . . . . . 4 . New C o n s t i t u t i v e Model f o r Low-Density P o l y u r e t h a n e Foams . . . . . 5 . S o l u t i o n o f a T y p i c a l Dynamic Problem u s i n g t h e New C o n s t i t u t i v e Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 . C o n c l u s i o n s and F u t u r e Work . . . . . . . . . . . . . . . . . . . . 7 . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements
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12 13 15
25 29 53 60
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FIGURES Page Figure 2.la. F i g u r e 2.lb. F i g u r e 2.2a. F i g u r e 2.2b. F i g u r e 2.3a. F i g u r e 2.3b. F i g u r e 2.4a. F i g u r e 2.4b. F i g u r e 2.5. F i g u r e 2.6a. F i g u r e 2.6b. F i g u r e 3.1. Figure 4.1 F i g u r e 4.2a.
. A x i a l Responses f r o m A l l T e s t s on Foam 6602 . . . . V o l u m e t r i c Responses f r o m A l l T e s t s on Foam 6703 . . . A x i a l Responses f r o m A l l T e s t s on Foam 6703 . . . . V o l u m e t r i c Responses f r o m A l l T e s t s on Foam 9703 . A x i a l Responses f r o m A l l T e s t s on Foam 9703 . . . . V o l u m e t r i c Responses f r o m A l l T e s t s on Foam 9503 . . . A x i a l Responses from A l l T e s t s on Foam 9503 . . . H y d r o s t a t i c T e s t R e s u l t s f o r Foam 6704 . . . . . . . V o l u m e t r i c Responses f r o m A l l T e s t s on Foam 9505 . . . A x i a l Responses from A l l T e s t s on Foam 9505 . . . . Y i e l d S u r f a c e i n P r i n c i p a l S t r e s s Space [7] . . . . A i r C o n t r i b u t i o n f o r 9505 Foam ($I = 0.09) . . . . V o l u m e t r i c Responses f r o m A l l T e s t s on Foam 6602
. . . . . . . . . . . . .
. . . . . . . . . . . . .
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18 18
19 19 20 20
21 21 22
23 23 27 32
S k e l e t o n V o l u m e t r i c Responses f r o m A l l T e s t s on Foam6602
34
F i g u r e 4.2b.
Skeleton
34
F i g u r e 4.3a.
S k e l e t o n V o l u m e t r i c Responses f r o m A l l T e s t s on Foam6703 . . . . . . . . . . . . . . . . . . . . .
35
F i g u r e 4.3b.
S k e l e t o n A x i a l Responses f r o m A l l T e s t s on
35
F i g u r e 4.4a.
S k e l e t o n V o l u m e t r i c Responses f r o m A l l T e s t s on Foam9703
36
F i g u r e 4.4b.
Skeleton
36
F i g u r e 4.5a.
S k e l e t o n V o l u m e t r i c Responses f r o m A l l T e s t s on Foam9503 . . . . . . . . . . . . . . . . . . . . .
37
F i g u r e 4.5b.
S k e l e t o n A x i a l Responses f r o m A l l
37
F i g u r e 4.6.
S k e l e t o n V o l u m e t r i c Responses f o r
........................ A x i a l Responses f r o m A l l T e s t s on Foam 6602 . . . ... Foam 6703 . . .
........................ A x i a l Responses f r o m A l l T e s t s on Foam 9703 . . .
6
... T e s t s on Foam 9503 . . . Foam 6704 . . . . . . .
38
FIGURES CONTINUED
Page F i g u r e 4.7a.
S k e l e t o n V o l u m e t r i c Responses f r o m A l l T e s t s on . . Foam9505.. .
..... ....... .
...... .. ..... ..... .....
39
F i g u r e 4.7b.
S k e l e t o n A x i a l Responses f r o m A l l T e s t s on Foam 9505.
39
F i g u r e 4.8a.
Foam Parameter A as a F u n c t i o n o f D e n s i t y R a t i o
42
F i g u r e 4.8b.
Foam Parameter B as a F u n c t i o n o f D e n s i t y R a t i o
F i g u r e 4 . 8 ~ . Foam Parameter C as a F u n c t i o n o f D e n s i t y R a t i o
. . ... . . .. . . . .
42 43
F i g u r e 4.8d.
Foam E l a s t i c Modulus E as a F u n c t i o n o f D e n s i t y R a t i o
43
F i g u r e 4.9.
F l o w c h a r t o f New C o n s t i t u t i v e Model
44
F i g u r e 4.10.
M o d i f i c a t i o n t o Step Function i n Y i e l d Equation
F i g u r e 4.11a.
Comparison o f A n a l y t i c a l and Experimental R e s u l t s V o l u m e t r i c Responses f r o m H y d r o s t a t i c T e s t s on Foam6602.. . . . . . . . .
......
F i g u r e 4.11b. F i g u r e 4.12a. F i g u r e 4.12b. F i g u r e 4.13a.
Comparison of A n a l y t i c a l and E x p e r i m e n t a l R e s u l t s V o l u m e t r i c Response f r o m U n i a x i a l T e s t on Foam 6602
...
49
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49
Comparison o f A n a l y t i c a l and Experimental R e s u l t s A x i a l Response f r o m U n i a x i a l T e s t on Foam 6602.
-
Comparison o f A n a l y t i c a l and Experimental R e s u l t s V o l u m e t r i c Response f r o m T r i a x i a l T e s t on Foam 6602 . . .. . . . (P=10 p s i ) .
..
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...
50
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50
Comparison o f A n a l y t i c a l and Experimental R e s u l t s A x i a l Response f r o m T r i a x i a l T e s t on Foam 6602 . . . . . . . . (P=10 p s i ) .
...
..
-
Comparison o f A n a l y t i c a l and Experimental R e s u l t s V o l u m e t r i c Response f r o m T r i a x i a l T e s t on Foam 6602 . . . . . .. . (P=15 p s i ) .
51
Comparison o f A n a l y t i c a l and Experimental R e s u l t s A x i a l Response f r o m T r i a x i a l T e s t on Foam 6602 (P=15 p s i ) . . . . . . . . .
51
Comparison o f A n a l y t i c a l and Experimental R e s u l t s V o l u m e t r i c Response f r o m T r i a x i a l T e s t on Foam 6602 . . . . . . . . . (P=20 p s i ) .
52
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F i g u r e 4.15a.
48 48
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F i g u r e 4.14b.
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F i g u r e 4.14a.
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46
Comparison o f A n a l y t i c a l and Experimental R e s u l t s A x i a l Responses f r o m H y d r o s t a t i c T e s t s on Foam 6602
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F i g u r e 4.13b.
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FIGURES CONTINUED
Page F i g u r e 4.15b.
Comparison o f A n a l y t i c a l and Experimental R e s u l t s A x i a l ResDonse from T r i a x i a l T e s t on Foam 6602 (P=20 p s i ) . . .
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F i g u r e 5.1.
Finite
F i g u r e 5.2.
D i s p l a c e d Shapes o f F i n i t e Element Model a t Maximum Crush-up
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Close-up of D i s p l a c e d Foam Layer i n F i n i t e Element Model a t Maximum Crush-up .
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F i g u r e 5.4.
Displacement o f
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F i g u r e 5.5.
Acceleration o f
F i g u r e 5.3.
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... ............... Steel Cylinder . . . . . . . . . . . . . Steel Cylinder . . . . . . . . . . . . .
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APPENDIX FIGURES Page Figure A.la.
Comparison o f A n a l y t i c a l and E x p e r i m e n t a l R e s u l t s V o l u m e t r i c Responses f r o m H y d r o s t a t i c T e s t s on Foam6703..
-
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F i g u r e A.lb. F i g u r e A.2a.
F i g u r e A.2b. F i g u r e A.3a.
Comparison o f A n a l y t i c a l and E x p e r i m e n t a l R e s u l t s A x i a l Responses f r o m H y d r o s t a t i c T e s t s on Foam 6703
64
...
64
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65
Comparison o f A n a l y t i c a l and Experimental Resu t s - A x i a l Response f r o m T r i a x i a l T e s t on Foam 6703 (P=15 p s i )
65
Comparison o f A n a l y t i c a l and Experimental R e s u l t s V o l u m e t r i c Response f r o m T r i a x i a l T e s t on Foam 6703 (P=20 p s i ) .
66
Comparison o f A n a l y t i c a l and E x p e r i m e n t a l Resu t s V o l u m e t r i c Response f r o m T r i a x i a l T e s t on Foam 6703 (P=15psi).
...
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F i g u r e A.3b. F i g u r e A.4a.
Comparison o f A n a l y t i c a l and E x p e r i m e n t a l R e s u l t s - A x i a l Response f r o m T r i a x i a l T e s t on Foam 6703 (P=20 p s i )
...
Comparison o f A n a l y t i c a l and Experimental R e s u l t s V o l u m e t r i c Responses f r o m H y d r o s t a t i c T e s t s on Foam9703..
-
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F i g u r e A.4b. F i g u r e A.5a. F i g u r e A.5b. F i g u r e A.6a.
67
Comparison o f A n a l y t i c a l and E x p e r i m e n t a l R e s u l t s - A x i a l Responses f r o m H y d r o s t a t i c T e s t s on Foam 9703
67
Comparison o f A n a l y t i c a l and Experimental R e s u l t s V o l u m e t r i c Response f r o m U n i a x i a l T e s t on Foam 9703
68
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Comparison o f A n a l y t i c a l and Experimental R e s u l t s Response f r o m U n i a x i a l T e s t on Foam 9703.
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,
Axial
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Comparison o f A n a l y t i c a l and E x p e r i m e n t a l R e s u l t s V o l u m e t r i c Response f r o m T r i a x i a l T e s t on Foam 9703 (P=6psi).
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F i g u r e A.6b. F i g u r e A.7a.
F i g u r e A.7b.
66
68
69
Comparison o f A n a l y t i c a l and Experimental R e s u l t s - A x i a l Response from T r i a x i a l T e s t on Foam 9703 (P=6 p s i ) .
69
Comparison o f A n a l y t i c a l and E x p e r i m e n t a l R e s u l t s V o l u m e t r i c Response f r o m T r i a x i a l T e s t on Foam 9703 (P=lO p s i ) . . . . . . . . . . . . . . . . . . . . . . .
70
... .
Comparison o f A n a l y t i c a l and Experimental R e s u l t s - A x i a l Response f r o m T r i a x i a l T e s t on Foam 9703 (P=10 p s i ) . . .
9
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APPENDIX FIGURES CONTINUED
Page F i g u r e A.8a.
Comparison o f A n a l y t i c a l and Experimental Resu t s V o l u m e t r i c Response f r o m T r i a x i a l T e s t on Foam 9703 ( P 4 5 psi).
................. .....
F i g u r e A.8b. F i g u r e A.9a.
Comparison o f A n a l y t i c a l and Experimental R e s u l t s - A x i a l Response f r o m T r i a x i a l T e s t on Foam 9703 (P=15 p s i )
71
Comparison o f A n a l y t i c a l and Experimental R e s u l t s V o l u m e t r i c Responses f r o m H y d r o s t a t i c T e s t s on Foam9503..
72
...
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F i g u r e A.9b. F i g u r e A.lOa.
Comparison o f A n a l y t i c a l and Experimental R e s u l t s Responses f r o m H y d r o s t a t i c T e s t s on Foam 9503
-
Axial
......
Comparison o f A n a l y t i c a l and Experimental R e s u l t s V o l u m e t r i c Response f r o m T r i a x i a l T e s t on Foam 9503 (P=10 p s i ) .
.......................
F i g u r e A.lOb. Figure A.lla.
F i g u r e A.12a.
Comparison o f A n a l y t i c a l and Experimental R e s u l t s V o l u m e t r i c Response f r o m T r i a x i a l T e s t on Foam 9503 (P=20 p s i ) .
74
Comparison o f A n a l y t i c a l and Experimental R e s u l t s - A x i a l Response f r o m T r i a x i a l T e s t on Foam 9503 (P=20 p s i )
74
Comparison o f A n a l y t i c a l and Experimental R e s u l t s V o l u m e t r i c Response f r o m T r i a x i a l T e s t on Foam 9503 (P=31 p s i ) .
75
...
F i g u r e A. 12b. Comparison o f A n a l y t i c a l and Experimental R e s u l t s - A x i a l Response f r o m T r i a x i a l T e s t on Foam 9503 (P=31 p s i )
...
Comparison o f A n a l y t i c a l and Experimental R e s u l t s V o l u m e t r i c Responses f r o m H y d r o s t a t i c T e s t s on Foam6704..
-
Comparison o f A n a l y t i c a l and Experimental R e s u l t s Responses f r o m H y d r o s t a t i c T e s t s on Foam 6704 . .
- Axial ....
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F i g u r e A.13b.
73 73
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F i g u r e A.13a.
72
Comparison o f A n a l y t i c a l and Experimental R e s u l t s - A x i a l Response f r o m T r i a x i a l T e s t on Foam 9503 (P=10 p s i )
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Figure A.llb.
71
F i g u r e A. 14a. Comparison o f A n a l y t i c a l and Experimental R e s u l t s V o l u m e t r i c Responses f r o m H y d r o s t a t i c T e s t s on Foam9505..
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F i g u r e A. 14b. Comparison o f A n a l y t i c a l and Experimental R e s u l t s Responses f r o m H y d r o s t a t i c T e s t s on Foam 9505
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10
76 76
77
Axial
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F i g u r e A. 15a. Comparison o f A n a l y t i c a l and Experimental R e s u l t s V o l u m e t r i c Response f r o m U n i a x i a l T e s t on Foam 9505
75
...
77 78
APPENDIX FIGURES CONTINUED Page F i g u r e A.15b. F i g u r e A.16a.
Comparison o f A n a l y t i c a l and Experimental R e s u l t s - A x i a l Response f r o m U n i a x i a l T e s t on Foam 9505.
78
Comparison o f A n a l y t i c a l and Experimental R e s u l t s V o l u m e t r i c Response f r o m T r i a x i a l T e s t on Foam 9505 (P=6psi).
79
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F i g u r e A.16b.
Comparison o f A n a l y t i c a l and Experimental R e s u l t s - A x i a l Response f r o m T r i a x i a l T e s t on Foam 9505 (P=6 p s i ) .
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F i g u r e A. 17a. Comparison o f A n a l y t i c a l and Experimental R e s u l t s V o l u m e t r i c Response f r o m T r i a x i a l T e s t on Foam 9505 (P=20 p s i ) .
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F i g u r e A.17b. F i g u r e A.18a.
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80
Comparison o f A n a l y t i c a l and Experimental R e s u l t s - A x i a l Response f r o m T r i a x i a l T e s t on Foam 9505 (P=20 p s i )
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Comparison of A n a l y t i c a l and Experimental R e s u l t s V o l u m e t r i c Response f r o m T r i a x i a l T e s t on Foam 9505 (P=30 p s i ) .
81
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F i g u r e A. 18b. Comparison o f A n a l y t i c a l and Experimental R e s u l t s - A x i a l Response f r o m T r i a x i a l T e s t on Foam 9505 (P=30 p s i )
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TABLES Page T a b l e 2.1. T a b l e 4.1. T a b l e 5.1. T a b l e 5.2.
. M a t e r i a l P r o p e r t i e s f o r NMERI/CERF Foams . . . . . M a t e r i a l P r o p e r t i e s Used i n Dynamic Analyses . . . R e s u l t s f r o m Dynamic Analyses . . . . . . . . . . . T e s t s Performed on Low D e n s i t y Polyurethane Foams
12
. . . .
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16 40
55 56
1. Rigid, closed-cell,
INTRODUCTION
p o l y u r e t h a n e foams a r e used i n a v a r i e t y of
app i c a t i o n s as house i n s u l a t i o n , f i l l e r s and r i g i d i z e r s i n a i r p l a n e w ngs, f o r m a t e r i a l packaging, and i n impact l i m i t e r s f o r s h i p p i n g c o n t a i n e r s [l]. Low d e n s i t y p o l y u r e t h a n e foam i s used i n impact l i m i t e r s i n a v a r i e t y o f n u c l e a r waste s h i p p i n g c o n t a i n e r s [2,3].
During a hypothetical nuclear
waste s h i p p i n g a c c i d e n t , t h e foam i s expected t o absorb a s i g n i f i c a n t amount o f i m p a c t energy by undergoing l a r g e i n e l a s t i c volume r e d u c t i o n s . Consequently, t h e c r u s h i n g o f p o l y u r e t h a n e foams must be w e l l c h a r a c t e r i z e d if t h e o v e r a l l response o f a s h i p p i n g c o n t a i n e r i s t o be p r o p e r l y p r e d i c t e d
f o r various accident scenarios. U n f o r t u n a t e l y , t e s t d a t a on t h e c r u s h i n g o f low d e n s i t y foams a r e q u i t e limited.
M a n u f a c t u r e r s o f p o l y u r e t h a n e foams u s u a l l y p r o v i d e mechanical
p r o p e r t i e s f o r t h e i r p r o d u c t s based on r e s u l t s f r o m unconfined, u n i a x i a l , compressive t e s t s .
The u n i a x i a l s t r e s s - s t r a i n responses a r e easy t o
measure, b u t t h e y p r o v i d e minimal i n f o r m a t i o n on t h e v o l u m e t r i c response o f foam and no i n f o r m a t i o n on t h e i n t e r a c t i o n s between t h e v o l u m e t r i c and d e v i a t o r i c ( s h e a r ) responses.
I n response t o t h i s need f o r e x p e r i m e n t a l
d a t a , Sandia N a t i o n a l L a b o r a t o r i e s d e c i d e d i n 1979 t o p e r f o r m e x t e n s i v e l a b o r a t o r y t e s t s t o c h a r a c t e r i z e t h e b e h a v i o r o f low d e n s i t y p o l y u r e t h a n e foams.
A t t h e r e q u e s t o f t h e T r a n s p o r t a t i o n System Development Department,
now 6320, members o f t h e E n g i n e e r i n g A n a l y s i s Department, now 1520, d e f i n e d a s e r i e s o f h y d r o s t a t i c and t r i a x i a l compression t e s t s t o be performed on foams s u p p l i e d by General P l a s t i c s M a n u f a c t u r i n g Company.
The t e s t s were
p e r f o r m e d by t h e New Mexico E n g i n e e r i n g Research I n s t i t u t e (NMERI) a t t h e i r C i v i l E n g i n e e r i n g Research F a c i l i t y (CERF) i n 1979 and 1980.
The t e s t
p r o c e d u r e s and r e s u l t s were never f o r m a l l y documented b u t were r e p o r t e d i n a l e t t e r f r o m NMERI/CERF t o Sandia N a t i o n a l L a b o r a t o r i e s [4].
A v a r i e t y o f c o n s t i t u t i v e models have been developed t o p r e d i c t t h e b e h a v i o r o f foams w i t h i n a v a r i e t y o f l o a d ranges.
F o r example, a l i n e a r
e l a s t i c c o n s t i t u t ve model f o r foams has r e c e n t l y been developed by K r a y n i k L
and Warren [5]
a t Sandia N a t i o n a l L a b o r a t o r i e s .
T h i s model a c c u r a t e l y
d e s c r i b e s t h e beh v i o r o f foams i n t h e l i n e a r e l a s t i c regime.
13
In this
r e p o r t , a phenomenological c o n s t i t u t i v e model t h a t a c c u r a t e l y p r e d i c t s t h e l i n e a r and p o s t - y i e l d b e h a v i o r o f l o w - d e n s i t y , p o l y u r e t h a n e foams i s presented. The purpose o f t h i s r e p o r t i s t o p r o v i d e formal documentation o f t h e r e s u l t s f r o m t h e NMERI/CERF t e s t s and t o p r e s e n t a c o n s t i t u t i v e model which c a p t u r e s t h e foam b e h a v i o r e x h i b i t e d i n t h e t e s t s .
F i r s t , t h e NMERI/CERF
t e s t s a r e d e s c r i b e d and t h e measured foam responses a r e presented.
Next,
t h r e e d i f f e r e n t c o n s t i t u t i v e models t h a t have been used i n t h e p a s t t o model foam b e h a v i o r a r e shown t o have shortcomings when a p p l i e d t o t h e foam b e h a v i o r observed i n t h e e x p e r i m e n t a l NMERI/CERF t e s t s .
T h i s i s f o l l o w e d by
t h e p r e s e n t a t i o n o f a new c o n s t i t u t i v e model which accounts f o r foam b e h a v i o r observed i n t h e experiments. model i s a l s o described.
Numerical i m p l e m e n t a t i o n o f t h e new
Then, a h y p o t h e t i c a l impact problem i s analyzed
w i t h t h e new model, and t h e r e s u l t s a r e compared t o r e s u l t s o b t a i n e d w i t h (1) a c o n v e n t i o n a l d e v i a t o r i c p l a s t i c i t y model and (2) a combined v o l u m e t r i c p l a s t i c i t y w i t h p r e s s u r e dependent d e v i a t o r i c p l a s t i c i t y model.
This report
ends w i t h a few c o n c l u s i o n s about t h e use o f t h e new model and a d i s c u s s i o n o f model l i m i t a t i o n s and f u t u r e work.
14
2.
TEST DESCRIPTION AND RESULTS
S i x d i f f e r e n t General P l a s t i c s foams w i t h i d e n t i f i c a t on numbers
(ID'S)
o f 6602, 6703, 9503, 9703, 6704, and 9505 were used i n t h e NMERI/CERF t e s t s .
.
The d e n s i t i e s of t h e s e foams, as i n d i c a t e d by t h e l a s t d i g t o f t h e i r ranged f r o m 2 l b / f t 3 t o 5 l b / f t 3 .
A l l samples f r o m each foam t y p e were
The samples were o r i e n t e d such t h a t t h e axes o f
taken from a s i n g l e block.
t h e c y l i n d e r s were i n t h e " r i s e " d i r e c t i o n o f t h e foam. b l o c k s were a v o i d e d i n o r d e r t o i n s u r e homogeneity. p e r f o r m e d on a l l s i x foams.
The edges o f t h e
H y d r o s t a t i c t e s t s were
A l s o , s e v e r a l t r i a x i a l compression t e s t s were
p e r f o r m e d on a l l foams e x c e p t 6704. T a b l e 2.1.
ID'S,
A m a t r i x o f t h e t e s t s i s shown i n
The numbers i n t h e t a b l e r e p r e s e n t t h e number o f t e s t s performed
f o r a given t e s t condition. I n t h e h y d r o s t a t i c t e s t s , c y l i n d r i c a l t e s t samples w i t h nominal h e i g h t s and d i a m e t e r s o f 5.60
i n . and 2.73
in.,
r e s p e c t i v e l y , were j a c k e t e d w i t h a
v e r y t h i n l a t e x j a c k e t and p l a c e d i n a 2000 p s i t r i a x i a l p r e s s u r e chamber t y p i c a l l y used f o r s o i l t e s t s . sealed.
The c e l l was t h e n f i l l e d w i t h w a t e r and
The samples were a l l o w e d t o f l o a t f r e e l y i n t h e p r e s s u r e c e l l .
4 i n . d i a m e t e r p i s t o n d i s p l a c i n g a t a r a t e o f a p p r o x i m a t e l y 0.002 t h e n used t o i n c r e a s e t h e p r e s s u r e w i t h i n t h e c e l l .
A
i n / s e c was
C e l l p r e s s u r e was
measured w i t h a 200 p s i p r e s s u r e gage, and volume changes o f t h e samples were d e t e r m i n e d f r o m p i s t o n d i s p l a c e m e n t measurements.
The volume
measurements were c o r r e c t e d f o r expansion o f t h e p r e s s u r e c e l l and c o m p r e s s i b i l i t y o f t h e water.
Volume changes f r o m t h e s e sources were shown
t o be n e g l i g i b l e compared w i t h volume changes o f t h e samples. The t r i a x i a l t e s t s were performed i n a n o t h e r p r e s s u r e vessel t y p i c a l l y used f o r t e s t i n g s o i l s .
I n t h e s e t e s t s , j a c k e t e d samples were l o a d e d
h y d r o s t a t i c a l l y t o a prescribed c o n f i n i n g pressure. d i s p l a c e m e n t was a p p l i e d a t a r a t e o f 0.9 t h e foam.
Then, a d d i t i o n a l a x i a l
in/sec i n the r i s e d i r e c t i o n o f
The a x i a l l o a d and d i s p l a c e m e n t were measured and c o n v e r t e d t o
a x i a l s t r e s s and s t r a i n .
Measurements o f t h e change i n volume o f t h e w a t e r
s u r r o u n d i n g t h e samples were used t o d e t e r m i n e changes i n sample volume. S t a n d a r d u n i a x i a l compression t e s t s were p e r f o r m e d on u n j a c k e t e d samples o f t h r e e o f t h e foams.
15
I n a l l h y d r o s t a t i c t e s t s and i n a l l t r i a x i a l t e s t s , except t h e u n i a x i a l t e s t s , t h e samples were j a c k e t e d so a i r c o u l d n o t escape.
A l l t e s t s were
performed a t room temperature and a t v e r y low s t r a i n r a t e s .
Consequently,
no d a t a on temperature o r s t r a i n r a t e e f f e c t s were obtained.
T a b l e 2.1.
T e s t s Performed on Low D e n s i t y Polyurethane Foams
I
NUMBER OF TESTS FOAM
TRIAXIAL CONFINING PRESSURE (PSI)
HYDROSTATIC
ID
O*
6
10
15
20
30
31
6602
1
0
1
2
1
0
0
6703
0
0
0
1
1
0
0
9703
1
1
1
1
0
0
0
9503
0
0
1
0
1
0
1
6704
0
0
0
0
0
0
0
9505
1
1
0
0
1
1
0
~~~
*
Uniaxial test
The c o m b i n a t i o n o f h y d r o s t a t i c and t r i a x i a l t e s t s i n T a b l e 2.1 was chosen t o p r o v i d e s u f f i c i e n t d a t a t o d e s c r i b e b o t h t h e v o l u m e t r i c and d e v i a t o r i c (shear) b e h a v i o r s o f t h e foams and t h e c o u p l i n g between t h e two responses.
The v o l u m e t r i c response i s d e f i n e d as t h e r e l a t i o n s h i p between
p r e s s u r e , p, and volume s t r a i n , 7 , where
p = --*kk
2.1
3 16
and
2.2 uij
and c i j
basis.
a r e t h e foam s t r e s s and s t r a i n components i n an o r t h o n o r m a l
C o n v e n t i o n a l summation n o t a t i o n i s used t h r o u g h o u t t h i s e n t i r e
report.
and t h e d e v i a t o r i c s t r a i n
The d e v i a t o r i c s t r e s s components, Sij,
components, eij,
a r e d e f i n e d as
sij --
uij
+
psij
2.3
-
^= sij
2.4
and
€
Result
ij
f r o m t h e v a r i o s t e s t s p e r f o r m e d on t h e General P l a s t i c s foams
a r e p r e s e n t e d i n F i g u r e s 2.1
- 2.6.
B o t h t h e v o l u m e t r i c and a x i a l s t r e s s -
a x i a l s t r a i n responses a r e shown f o r each t e s t .
Only h y d r o s t a t i c t e s t s were performed on t h i s
6704 Foam i n F i g u r e 2.5. foam.
The o n l y e x c e p t i o n i s f o r
The foam a x i a l s t r e s s - a x i a l s t r a i n responses a r e p r e s e n t e d f o r use i n
t h e development o f t h e new c o n s t i t u t i v e model p r e s e n t e d i n a l a t e r s e c t i o n . The t r i a x i a l t e s t s c o n s i s t e d o f two phases:
an i n i t i a l h y d r o s t a t i c l o a d i n g
phase f o l l o w e d by a t r i a x i a l l o a d i n g phase.
The t r i a x i a l t e s t r e s u l t s i n
F i g u r e s 2.1
- 2.6
do n o t s t a r t a t z e r o s t r e s s and s t r a i n because t h e
response f r o m t h e i n i t i a l h y d r o s t a t i c l o a d i n g phase was measured o n l y a t t h e end o f t h e h y d r o s t a t i c phase o f t h e s e t e s t s .
The v o l u m e t r i c and a x i a l
s t r a i n s p r e s e n t e d i n t h e s e f i g u r e s a r e compressive and a r e d e f i n e d by (Vo
-
V)/Vo
and ( L o
-
L)/Lo,
r e s p e c t i v e l y , where V i s t h e volume o f t h e
sample, L i s t h e sample h e i g h t , and t h e s u b s c r i p t 0 i n d i c a t e s t h e i n i t i a l value o f the quantity.
The s t r e s s e s a r e a l s o compressive.
The h y d r o s t a t i c
r e s u l t s were p l o t t e d on t h e a x i a l s t r e s s - a x i a l s t r a i n p l o t s by assuming t h a t t h e a x i a l s t r e s s was equal i n magnitude t o t h e p r e s s u r e and t h e s t r a i n was i s o t r o p i c such t h a t t h e a x i a l s t r a i n was g i v e n by t h e f o l l o w i n g e q u a t i o n
1/3
(Lo
LO
L)
=
1-[1(vOvO17
"'1
2.5
0
0.
.1
F I G U R E 2.la.
0
UNIAXIAL TEST
A
TRIAXIAL TEST. P = 18 PSI
8
TRIAXIAL TEST. P = 15 PSI
0
TRIAXIAL TEST, P = 20 PSI
.2
.6
.4
UOLUME STRAIN
.5
.3 ( (
U0-U
)AJQ 1
Volumetric Responses from All Tests on Foam 6 6 0 2
-
UNIAXIAL TEST
A
TRIAXIAL TEST. P = 10 PSI TRIAXIAL TEST. P = 15 PSI
0
fa.
HYDROSTATIC TESTS
0
0
.1 .05 QX I AL STRAIN
F i g u r e 2.lb.
HYDROSTATIC TESTS
TRIAXIQL TEST. P = 20 PSI
.ls ((
.2
Lo-L )/La
.3
.25 )
A x i a l R e s p o n s e s f r o m A l l T e s t s o n F o a m 6602
18
50
40 P R
E S 30
s
U R E
20 P S
i
10
0
TRIFIXIAL TEST.
P = 15 P S I
A
T R I A X I A L TEST.
P = 20
PSI
0
.2
0.
.6
UOLUHE STRAIN
FIGURE 2.2a.
5
.3
1
( (
UQ-U )/U0 1
V o l u m e t r i c R e s p o n s e s f r o m A l l T e s t s o n Foam 6 7 0 3
-
HYDROSTATIC TESTS
80 A X
I A L
60 S T R E S
40
P 5 I
20
0
0.
. Q5
.1
AXIFlL STRAIN
FIGURE 2 . 2 b .
.2 .15 ( (Lo-L )/Lo 1
.3
.25
A x i a l R e s p o n s e s f r o m A l l T e s t s on F o a m 6703
19
0
UNIAXIAL TEST
A
TRIAXIAL TEST. P = 6 PSI
0
TRIAXIAL TEST. P = 10 PSI
-
0
TRIAXIAL TEST. P = 15 PSI
-
.2
0.
OOLUWE
.6
.4
.5
.3
. 1
F I G U R E 2.3a.
HYDROSTATIC TESTS
STRAIN
( (
Ug-U )/UQ 1
Volumetric Responses from All Tests o n Foam 9 7 0 3
-
HYDROSTATIC TESTS
A
UNIAXIQL TEST TRIAXIAL TEST. P = 6 PSI
0
TRIAXIAL TEST. P = 10 PSI
0
TRIAxIAL TEST. P = 15 PSI
0
.1
0. .05
( (
.3 .25
.15
AXIAL STRAIN
FIGURE 2 . 3 b .
.2
-
La-L ) / L a )
Axial Responses from All Tests on Foam 9703
20
50
40
P R E
s 30 S
U R
E 20
-
P
s I
HYDROSTQTIC TESTS TRIQXIAL TEST. P = 10 PSI
0
io
TRIAXIAL TEST. P = 28 PSI TRIQXIAL TEST. P = 31 PSI
A 0
0 0.
.2
.4
.1
.3
UOLUME STRAIN
F I G U R E 2.4a.
((
HYDROSTATIC TESTS
0
TRIQXIAL TEST. P = 10 PSI
A
TRIQXIQL TEST. P = 20 PSI
0
TRIQXIQL TEST. P = 31 PSI
.a5
.15
FIXIAL STRAIN
F I G U R E 2.4b.
UQ-U )/Ug 1
V o l u m e t r i c R e s p o n s e s f r o m A l l T e s t s on F o a m 9503
-
0.
.6 .5
( (
.25
La-L )/La 1
A x i a l Responses f r o m All T e s t s o n Foam 9503
21
100
,
,
,
,
1
.
.
.
.
I
.
I
0
80
no
-
A
0 0
0
A
P
R E
s
S U R E
O
60
-
0 Do
O
A 4Q
-
A
A
A
0
A
A
~
O
0 -0
-
~
-
A
5 I
A
FIGURE 2 . 5 .
-
A
00
P
20
A A
A
A
O
O
U Q 00 OO
-
SAMPLE A
A
SAMPLE 6
0
SAMPLE C
-
H y d r o s t a t i c Test R e s u l t s f o r F o a m 6 7 0 4
22
E
1
,00
801 60
-
P S
I
20
Lh-
HYDROSTATIC TESTS
0
UNIFIXIAL TEST
A
TRIAXIAL TEST. P = 6 PSI
0
TRIAXIAL TEST. P = 20 PSI TRIAXIAL TEST. P = 30 PSI
0
'
.
-
I
s
0.
FIGURE 2 . 6 b .
.
I
I
.
I
.
I
I
,
.2 .1
FIGUR.E 2 . 6 a .
.
.
.
.
I
,
.
,
.
,
,
.3 UOLUHE STRAIN ( C U0-U )/Ua
.
,
.6
.4
.5 )
Volumetric R e s p o n s e s from All T e s t s o n Foam 9505
Axial Responses from All Tests on Foam 9505
23
The r e s u l t s i n F i g u r e s 2.1
-
2.6 i n d i c a t e o n l y a small amount of
s c a t t e r between t h e t h r e e h y d r o s t a t i c t e s t s performed on each foam.
The
volume s t r a i n s measured a t t h e ends o f t h e h y d r o s t a t i c l o a d i n g phases o f t h e t r i a x i a l t e s t s a r e a l s o c o n s i s t e n t w i t h t h e v o l u m e t r i c s t r a i n s measured i n the hydrostatic tests.
However, t h e p r e s s u r e a t which v o l u m e t r i c y i e l d i n g
o c c u r s i s h i g h l y dependent on t h e t e s t c o n d i t i o n s .
For 6602 foam, t h e
p r e s s u r e a t y i e l d under h y d r o s t a t i c l o a d i n g i s a p p r o x i m a t e l y 15 p s i whereas t h e p r e s s u r e a t y i e l d under u n i a x i a l l o a d i n g i s about 8 p s i and f o r t h e t r i a x i a l t e s t w i t h a c o n f i n i n g p r e s s u r e o f 10 p s i i s a p p r o x i m a t e l y 15 p s i . I n t h e t r i a x i a l t e s t s w i t h c o n f i n i n g p r e s s u r e s o f 15 p s i and 20 p s i , t h e foam a c t u a l l y y i e l d s t w i c e , once a t 15 p s i d u r i n g t h e h y d r o s t a t i c phase o f t h e t e s t s and t h e n a g a i n a t 19.5 p s i f o r t h e t e s t w i t h 15 p s i c o n f i n i n g p r e s s u r e and a t 25 p s i f o r t h e t e s t w i t h 20 p s i c o n f i n i n g p r e s s u r e .
For
t h e s e two t r i a x i a l t e s t s , t h e d a t a i n d i c a t e t h a t when t h e a d d i t i o n a l a x i a l l o a d s a r e f i n a l l y a p p l i e d , t h e foam has h i g h e r r e s i s t a n c e t o t h e a x i a l l o a d s than t o continued h y d r o s t a t i c loading.
This type o f behavior i s not
commonly observed f o r most m a t e r i a l s and i s an i n d i c a t i o n o f t h e unusual c o u p l i n g between t h e v o l u m e t r i c and shear responses o f t h e foam.
This
c o u p l i n g can a l s o be seen i n t h e a x i a l s t r e s s - a x i a l s t r a i n curves i n Figures 2.lb
-
2.6b.
24
APPLICATION OF EXISTING CONSTITUTIVE MODELS
3.
TO LOW-DENSITY POLYURETHANE FOAMS The l o g i c a l f i r s t s t e p i n t h e development of a c o n s t i t u t i v e model f o r l o w d e n s i t y p o l y u r e t h a n e foams i s t o t r y t o f i t e x i s t i n g c o n s t i t u t i v e models
I n t h i s s e c t i o n , t h r e e t y p e s o f models w h i c h have been used i n t h e p a s t t o model low d e n s i t y foam b e h a v i o r [SI a r e e v a l u a t e d w i t h
t o t h e t e s t data.
r e s p e c t t o t h e NMERI/CERF d a t a .
The t h r e e t y p e s o f models c o n s i d e r e d i n
t h i s s e c t i o n i n c l u d e : (1) a u n i a x i a l c r u s h model, ( 2 ) a c o n v e n t i o n a l d e v i a t o r i c p l a s t i c i t y model w h i c h i s commonly used t o d e s c r i b e t h e b e h a v i o r o f m e t a l s , and ( 3 ) a " s o i l s " model w h i c h combines v o l u m e t r i c p l a s t i c i t y w i t h p r e s s u r e dependent d e v i a t o r i c p l a s t i c i t y .
3.1.
U n i a x i a l Crush Model The u s u a l method o f t e s t i n g foam i s i n a c o n v e n t i o n a l u n i a x i a l
compression t e s t .
T h i s i s a c o n v e n i e n t method because t h e most common
a p p l i c a t i o n f o r foam c r u s h i n g i n v o l v e s a u n i a x i a l crush.
The u n i a x i a l
s t r e s s - s t r a i n c u r v e can be d i r e c t l y used t o compute d i s p l a c e m e n t s f r o m l o a d s o r v i c e versa.
T h i s s i m p l e u n i a x i a l model has a severe l i m i t a t i o n , however,
whenever t h e foam i s used i n a m u l t i a x i a l c r u s h mode.
The u n i a x i a l b e h a v i o r
c a n n o t b e a p p l i e d by s i m p l y i g n o r i n g s t r e s s e s i n t h e o t h e r d i r e c t i o n s . i s seen i n F i g u r e 2.lb.
This
where t h e a x i a l s t r e s s - a x i a l s t r a i n c u r v e i s shown
t o be n o t u n i q u e b u t i n s t e a d v e r y s e n s i t i v e t o t h e a p p l i e d p r e s s u r e .
It i s
o b v i o u s t h a t a m u l t i a x i a l model i s necessary i f m u l t i a x i a l l o a d i n g i s in v o l ved.
3.2.
C o n v e n t i o n a l D e v i a t o r i c P l a s t i c i t y Model A second method o f m o d e l i n g foam i s t o use a c o n v e n t i o n a l p l a s t i c i t y
model, w h i c h i s t h e s i m p l e s t m u l t i a x i a l model.
U n i a x i a l y i e l d s t r e n g t h s can
be measured and g e n e r a l i z e d t o m u l t i a x i a l c o n d i t i o n s u s i n g c o n v e n t i o n a l d e v i a t o r i c p l a s t i c i t y assumptions.
One o f t h e assumptions w h i c h must be
e v a l u a t e d , however, i s t h a t a model o f t h i s t y p e a l l o w s o n l y e l a s t i c volume
25
strains.
The h y d r o s t a t i c d a t a i n F i g u r e s 2 . l a t o 2.6a
indicate that the
foams undergo l a r g e p l a s t i c volume s t r a i n s when s u b j e c t e d t o s u f f i c i e n t load.
Thus, c o n v e n t i o n a l d e v i a t o r i c p l a s t i c i t y models a r e unable t o c a p t u r e
t h e p l a s t i c v o l u m e t r i c b e h a v i o r o f p o l y u r e t h a n e foams.
Another assumption
made i n c o n v e n t i o n a l p l a s t i c i t y models i s t h a t t h e v o l u m e t r i c and d e v i a t o r i c responses a r e n o t coupled.
That i s , d e v i a t o r i c l o a d i n g i s assumed t o have
no e f f e c t on v o l u m e t r i c b e h a v i o r , and d e v i a t o r i c y i e l d i s n o t a f f e c t e d by t h e pressure.
I f a d e v i a t o r i c v o l u m e t r i c d e c o m p o s i t i o n were v a l i d , a l l o f
t h e v o l u m e t r i c responses i n F i g u r e s 2 . l a t o 2.6a would be t h e same regardless o f the load history.
The curves i n F i g u r e s 2 . l a t o 2.6a
indicate
t h a t t h e v o l u m e t r i c response i s c l e a r l y dependent on l o a d h i s t o r y and t h e occurrence o f d e v i a t o r i c loading.
Thus, c o n v e n t i o n a l d e v i a t o r i c p l a s t i c i t y
models f a i l t o c a p t u r e two i m p o r t a n t f e a t u r e s o f p o l y u r e t h a n e foam b e h a v i o r , v o l u m e t r i c p l a s t i c i t y and v o l u m e t r i c - d e v i a t o r i c c o u p l i n g .
3.3.
Combined V o l u m e t r i c P l a s t i c i t y w i t h P r e s s u r e Dependent D e v i a t o r i c P l a s t i c i t y Model A n o t h e r c l a s s o f rnultiaxia1,models
which combine v o l u m e t r i c p l a s t i c i t y
w i t h p r e s s u r e dependent d e v i a t o r i c p l a s t i c i t y were examined n e x t .
These
models were a t t r a c t i v e because t h e y have f e a t u r e s w h i c h d e v i a t o r i c p l a s t i c i t y models do n o t ; namely, c a p a b i l i t i e s f o r v o l u m e t r i c p l a s t i c i t y and c o u p l i n g between t h e v o l u m e t r i c and d e v i a t o r i c responses. model o f t h i s t y p e , developed by K r i e g [ 7 ]
A particular
f o r s o i l and c o n c r e t e , was
examined i n d e t a i l f o r i t s a p p l i c a b i l i t y t o foam.
I n t h i s model, t h e y i e l d
f u n c t i o n i s assumed t o be s e p a r a b l e i n t o t h e p r o d u c t o f d e v i a t o r i c and volumetric parts.
The v o l u m e t r i c y i e l d f u n c t i o n i s independent o f t h e
d e v i a t o r i c stresses, but the deviatoric p o r t i o n o f the y i e l d function i s dependent on t h e p r e s s u r e .
The shape o f t h e d e v i a t o r i c y i e l d s u r f a c e i s a
p a r a b o l o i d o f r e v o l u t i o n about t h e p r e s s u r e a x i s as shown i n F i g u r e 3.1. The v o l u m e t r i c ,
eV,
and d e v i a t o r i c ,
eS, y i e l d
f u n c t i o n s a r e g i v e n by t h e
f o l l o w i n g equations
ev
= P
-
3.1
f(d
26
HVDROSTAT
F I G U R E 3.1.
Y i e l d S u r f a c e i n P r i n c i p a l Stress Space
27
[7]
Os
=
J~
-
( a o + alp
+ a2p 2 1
where p i s t h e p r e s s u r e ( d e f i n e d i n E q u a t i o n 2 . 1 ) ,
3.2 f i s a function o f the
v o l u m e t r i c s t r a i n and d e f i n e s t h e m a t e r i a l s v o l u m e t r i c s t r e s s - s t r a i n behavior,
J2 i s t h e second i n v a r i a n t of t h e d e v i a t o r i c s t r e s s e s , and ao, al
and a2 a r e m a t e r i a l c o n s t a n t s .
S i n c e f i n E q u a t i o n 3.1 i s d e f i n e d f r o m
v o l u m e t r i c s t r e s s - s t r a i n d a t a , t h i s model c a p t u r e s t h e v o l u m e t r i c p l a s t i c i t y o f p o l y u r e t h a n e foam.
However, i n t h i s model, t h e v o l u m e t r i c response i s
c o n s i d e r e d t o be independent o f t h e d e v i a t o r i c response.
T h i s assumption i s
o b v i o u s l y n o t v a l i d f o r t h e d a t a p r e s e n t e d i n F i g u r e s 2 . l a t o 2.6a. N e i t h e r u n i a x i a l models, c o n v e n t i o n a l d e v i a t o r i c p l a s t i c i t y models, n o r models w h i c h combine v o l u m e t r i c p l a s t i c i t y w i t h p r e s s u r e dependent d e v i a t o r i c p l a s t i c i t y a r e a p p r o p r i a t e f o r l o w d e n s i t y p o l y u r e t h a n e foams. T h e r e f o r e , a new c o n s t i t u t i v e model was developed and i s p r e s e n t e d i n t h e next section.
28
4.
NEW CONSTITUTIVE MODEL FOR LOW DENSITY POLYURETHANE FOAMS
The f i r s t s t e p i n t h e development o f a new c o n s t i t u t i v e model f o r l o w d e n s i t y p o l y u r e t h a n e foams was t o examine t h e i n d i v i d u a l components o f t h e foam s t r u c t u r e .
Each o f t h e foams used i n t h e NMERI/CERF t e s t s was a c l o s e d
c e l l foam w i t h a i r i n s i d e t h e c e l l s . s t r u c t u r a l components: a i r i n s i d e t h e foam.
T h e r e f o r e , each foam c o n s i s t e d o f two
(1) t h e polymer s t r u c t u r e o r s k e l e t o n and (2) t h e I n a p p l i c a t i o n s where t h e a i r c o u l d n o t escape f r o m
the skeleton during loading, the a i r could carry a substantial p a r t o f the load.
I n a l l o f t h e NMERI/CERF t e s t s , e x c e p t t h e u n i a x i a l t e s t s , t h e
samples were j a c k e t e d and a i r c o u l d n o t escape.
Thus, a model w h i c h
c o n s i d e r e d t h e c o n t r i b u t i o n o f t h e a i r t o t h e o v e r a l l s t r u c t u r a l response o f t h e foams was a p p r o p r i a t e f o r t h e foam b e h a v i o r e x h i b i t e d i n t h e NMERI/CERF tests. T o t a l foam response can be decomposed i n t o t h e response o f t h e s k e l e t o n and t h e response o f t h e a i r i n t h e f o l l o w i n g manner.
Since, t h e a i r does
n o t r e s i s t any shear d e f o r m a t i o n , t h e a i r c o n t r i b u t i o n i s c o m p l e t e l y volumetric.
F o r convenience, t h e s k e l e t o n i s assumed t o occupy t h e same
space as t h e foam.
T h i s i m p l i e s t h a t t h e s k e l e t o n s t r a i n components a r e
e q u a l t o t h e foam s t r a i n components.
A l s o , t h e foam s t r e s s components, uij,
a r e g i v e n by t h e f o l l o w i n g e q u a t i o n
(7
i j = u;;
+
air
u
'i j
4.1
where u;; a r e t h e s k e l e t o n s t r e s s components and u a i r6 r e p r e s e n t s t h e a i r ij c o n t r i b u t i o n t o t h e normal s t r e s s components. To b e t t e r u n d e r s t a n d t h i s e q u a t i o n , c o n s i d e r a h y d r o s t a t i c compression t e s t i n w h i c h t h e foam sample i s j a c k e t e d and t h e a i r i s n o t a l l o w e d t o escape.
I f t h e s k e l e t o n was
s t r u c t u r e d so t h a t i t c o u l d n o t c a r r y any l o a d , t h e n t h e e x t e r n a l p r e s s u r e aPP i e d t o t h e foam w o u l d equal t h e i n t e r n a l a i r p r e s s u r e . I n o t h e r words, t h e foam s t r e s s components would equal t h e a i r c o n t r i b u t i o n . T h i s foam wou d n o t be a b l e t o r e s i s t any d e v i a t o r i c l o a d i n g .
I n most foams, however,
t h e s k e l e t o n i s s t r u c t u r e d so t h a t i t can c a r r y l o a d and t h e c o n t r i b u t i o n o f t h e s k e l e t o n must be added t o t h e a i r c o n t r i b u t i o n t o d e t e r m i n e how much
29
l o a d t h e foam can c a r r y .
I n t h e n e x t s e c t i o n , an e x p r e s s i o n f o r t h e a i r
c o n t r i b u t i o n as a f u n c t i o n o f foam s t r a i n components i s d e r i v e d . The i d e a l gas law was used t o d e r i v e an e x p r e s s i o n f o r t h e a i r air contribution, u I f t h e foam i s compressed f r o m i n i t i a l volume Vo t o a
.
f i n a l volume o f VI,
t h e volume s t r a i n , y , can be expressed as 4.2
The volume o f t h e foam i s equal t o t h e sum o f volume o f t h e polymer f r o m and t h e volume o f t h e a i r t r a p p e d i n s i d e , Vair.
w h i c h i t i s made, Vp,
The
volume o f t h e polymer i s f i x e d when t h e foam i s manufactured and m e r e l y changes i t s p o s i t i o n as t h e foam deforms.
Thus, t h e volume s t r a i n becomes
4.3 or
Y
=(-
( , -) l
e)
+
4.4
The denominator o f E q u a t i o n 4.4 can be expressed as
(I+$)=
where
1 1- 9
4.5
i s t h e volume f r a c t i o n o f s o l i d m a t e r i a l .
I f t h e i d e a l gas law i s
expressed as
P
airVair =
-,30
4.6
where pair
i s t h e a i r pressure,
"air
i s t h e a i r volume, n i s t h e mole
f r a c t i o n o f a i r , R i s t h e u n i v e r s a l gas c o n s t a n t , and T i s t h e a b s o l u t e t e m p e r a t u r e , E q u a t i o n 4.4 can be r e w r i t t e n as
4.7
and r e a r r a n g e d t o g i v e
4 .a
The p r e s s u r e pr:i t h e foam.
i s t h e i n t e r n a l a i r p r e s s u r e when no l o a d i s a p p l i e d t o
Thus, pyir
can be expressed as
air P1 =
air Po
-
air
4.9
(7
The n e g a t i v e s i g n i s needed i n f r o n t o f uair because po a i r and p;ir p o s i t i v e i n compression whereas uair i s p o s i t i v e i n t e n s i o n .
are
Substitution
o f E q u a t i o n 4.9 i n t o E q u a t i o n 4.8 l e a d s t o an e x p r e s s i o n f o r t h e a i r air contribution, u
.
4.10 F o r a p r e s c r i b e d foam volume s t r a i n , 7 , E q u a t i o n 4.10 d e s c r i b e s t h e s t r e s s c a r r i e d by t h e a i r .
Note t h a t f o r i s o t h e r m a l c o n d i t i o n s when t h e foam
volume s t r a i n i s equal t o z e r o t h e a i r c o n t r i b u t i o n i s a l s o equal t o zero. A l s o , t h e a i r c o n t r i b u t i o n approaches i n f i n i t y as t h e foam volume s t r a i n approaches 6 volume.
-
1 o r i n o t h e r words as t h e foam volume approaches t h e polymer
A p l o t o f t h e a i r c o n t r i b u t i o n as a f u n c t i o n o f volume s t r a i n i s
shown i n F i g u r e 4.1.
F o r a p p l i c a t i o n s i n w h i c h t h e a i r can escape f r o m t h e
foam, t h e s t r e s s c a r r i e d by t h e a i r can be n e g l e c t e d by s e t t i n g pri; t o zero.
equal
5Q0
400 P
R
E
s
300
S
U
R E 200 P 5 I
10Q
0 .2
0.
.1
F I G U R E 4.1
.7
.5 ( (
UQ-U )/U0
.9
)
A i r C o n t r i b u t i o n f o r 9 5 0 5 Foam
32
1.
.8
.6
.4
.3 UOLUME STRRIN
(4
= 0.09)
The response o f t h e s k e l e t o n can now be d e t e r m i n e d f r o m t h e NMERI/CERF tests.
S i n c e t h e foam and t h e s k e l e t o n occupy t h e same volume, t h e foam and
s k e l e t o n s t r a i n s a r e t h e same.
Also, t h e s k e l e t o n s t r e s s components can be
d e r i v e d from Equation 4.1 w r i t t e n i n t h e f o l l o w i n g form sk 'i j
=
(7
ij
- u
a i r6 ij
4.11
The s k e l e t o n s t r e s s components a r e d e t e r m i n e d by s u b t r a c t i n g t h e e x p r e s s i o n g i v e n by E q u a t i o n 4.10 f o r t h e s t r e s s c a r r i e d by t h e a i r f r o m t h e foam normal s t r e s s components.
The s k e l e t o n responses have been f o u n d i n t h i s
way u s i n g t h e d a t a p l o t t e d i n F i g u r e s 2.1 4.2
-
4.7.
-
2.6 and a r e shown i n F i g u r e s
E q u a t i o n 4.10 s h o u l d r a t h e r a c c u r a t e l y d e s c r i b e t h e a i r b e h a v i o r
-
and F i g u r e s 4.2
4.7 d e s c r i b e t h e s k e l e t o n b e h a v i o r .
The v o l u m e t r i c responses f o r h y d r o s t a t i c l o a d i n g i n d i c a t e t h a t t h e s k e l e t o n may s o f t e n o r harden s l i g h t l y a f t e r y i e l d .
I n addition, the
v o l u m e t r i c s k e l e t o n responses a r e a f f e c t e d by t h e d e v i a t o r i c l o a d i n g conditions. tests.
Furthermore, t h e l a t e r a l s t r a i n s a r e z e r o f o r t h e u n i a x i a l
T h i s i m p l i e s t h a t P o i s s o n ' s r a t i o f o r t h e s k e l e t o n i s equal t o z e r o .
T h a t i s , t h e s k e l e t o n response i n a p r i n c i p a l s t r e s s d i r e c t i o n i s n o t a f f e c t e d by t h e o t h e r p r i n c i p a l s k e l e t o n s t r e s s e s .
The s k e l e t o n responses
f o r 6602 Foam i n F i g u r e 4 . 2 i n d i c a t e t h a t f o r h y d r o s t a t i c l o a d i n g t h e y i e l d s t r e s s can be expressed as a f u n c t i o n o f t h e volume s t r a i n , 7.
If the
l o a d i n g i s d e v i a t o r i c , t h e a x i a l y i e l d s t r e s s appears t o be equal t o t h e a x i a l y i e l d s t r e s s f o r h y d r o s t a t i c l o a d i n g p l u s a constant.
Thus, t h e y i e l d
s t r e s s i n each p r i n c i p a l s t r e s s d i r e c t i o n can be expressed as g
= A < II'>
+
B (1
+C
y)
4.12
where 11' i s t h e second i n v a r i a n t o f t h e d e v i a t o r i c s t r a i n s , < > i s t h e h e a v y s i d e s t e p f u n c t i o n , 7 i s t h e volume s t r a i n o r f i r s t i n v a r i a n t o f t h e foam s t r a i n s , and A,
B, and C a r e c o n s t a n t s .
Constant B i s t h e y i e l d s t r e s s
o f t h e s k e l e t o n f o r p u r e l y h y d r o s t a t i c l o a d i n g , and t h e p r o d u c t o f B and C i s t h e s l o p e o f t h e s k e l e t o n v o l u m e t r i c response a f t e r y i e l d i n g f o r p u r e l y h y d r o s t a t i c lobding.
C o n s t a n t A i s equal t o t h e d i f f e r e n c e between t h e
a x i a l y i e l d s t r e s s f o r h y d r o s t a t i c l o a d i n g and t h e a x i a l y i e l d s t r e s s f o r
33
30
P 25 R
-
-
-
0
E
0
HYDROSTATIC TESTS UNIAXIAL TEST
A
T R I A X I A L TEST. P = 10 PSI
-
8
TRIAXIAL TEST, P = 15 PSI
0
TRIAXIAL TEST. P = 20 PSI
-
S
5 I
0.
.2
4
. I
.3
UOLUME STRA I N
FIGURE 4.2a.
.ti 5
( (
UQ-U )/UQ
1
Skeleton Volumetric Responses from A l l Tests on F o a m 6 6 0 2
A
0
UNIAXIAL TEST
A
TRIAXIRL TEST. P = 1Q PSI
0
X
I
0
A L S T R
HYDROSTATIC TESTS
TRIAXIAL TEST. P = 15 PSI
-
TRIAXIAL TEST. P = 20 PSI
.
0 0
-
0
E
S S
0
P
s
AXIAL STRQIN c ( L ~ - L) / L ~)
FIGURE 4.2b.
Skeleton A x i a l Responses from All Tests on F o a m 6 6 0 2
34
50
40
HYDROSTATIC TESTS
0
TRIQXIQL TEST. P
A
T R I A X I A L TEST. P = za PSI
15 PSI
P R
E S 30
s
U
R E 20
P S I
10
0
STRA I N
UOLUHE
FIGURE 4.3a.
,
,
,
,
I
.5
.3
. 1
100
.6
.4
.2
0.
UQ-U )/U0
( (
>
Skeleton Volumetric Responses from A l l T e s t s on F o a m 6 7 0 3 .
'
.
,
I
.
.
.
.
,
.
.
.
.
I
.
.
,
.
l
.
,
.
.
L
8Q A
HYDROSTATIC TESTS
0
TRIAXIAL TEST. P = 15 PSI
A
TRIAXIAL TEST. P = 20 PSI
X
I FI L
s
60
-
40
-
20
-
I
I:
T R
E
S
P S I
0.
.
.l
AXIAL STRAIN
FIGURE 4 . 3 b .
.2 .15
.05
( (
.3
.25
La-L )/Lo 1
Skeleton Axial Responses f r o m A l l T e s t s on F o a m 6703 35
-
HYDROSTATIC TESTS
0
UNIAXIAL TEST
b
TRIAXIQL TEST. P = 6 PSI
o T R I A X I A L TEST. P
TRIAXIAL TEST. P = 15 PSI
0
0.
.2
. 1
= 10 PSI
.6
.4
UOLUHE STRA IN
.3 ((
UQ-U )/U0 1
.5
Skeleton Volumetric Responses from
FIGURE 4 . 4 a .
A l l Tests o n F o a m 9 7 0 3
l
.
'
.
I
I
.
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.
X
I A L
i
.
.
.
-
-
A30
l
a--
-
E
S
s
-
.
.
.
.
HYDROSTATIC TESTS UNIAXIAL TEST TRIAXIAL TEST. P = 6 PSI
0
TRIAXIAL TEST. P = 10 PSI TRIAXIQL TEST, P = 15 PSI
0
A
10
l
0 A
E T 30 S - 00 T
R 20
.
-1
-
acP003000a30ooaooO
0
J
0 0
-
I
FIGURE 4 . 4 b .
Skeleton A x i a l Responses from A l l Tests on Foam 9 7 0 3 36
s0
,
.
l
. , . . . . l , . . . l . . . . l . . . . , . . . . ,
-
HYDROSTATIC TESTS
-
0
TRIAXIAL TEST, P = 10 PSI
A
TRIAXIAL TEST. P
0
TRIAXIAL TEST. P = 31 PSI
P R E S 30 S U
20
PSI
i
R E
20 P 5 I
10
0 0.
.2
.5
.3 UOLUME
F I G U R E 4.5a.
.6
.4
.1 STRAIN
( (
Ug-U )/UQ
)
S k e l e t o n Volumetric R e s p o n s e s from
A l l T e s t s o n Foam 9 5 0 3
0
-
A 60
m
D
HYDROSTATIC TESTS TRIAXIAL TEST. P = 10 PSI'
A
TRIAXIAL TEST. P = 20 PSI
0
TRIAXIAL TEST. P = 31 PSI
X
A*
I FI L S T
R 40 E
S
S
0
P s 20 I
0 0.
1 .05
FlXIAL STRAIN
FIGURE 4 . 5 b .
.2 .15 ( (
3
.25 Lo-L
)/La
)
S k e l e t o n A x i a l Responses from A l l T e s t s o n Foam 9 5 0 3
37
FIGURE 4 . 6 .
Skeleton Volumetric Responses for Foam 6 7 0 4
38
1 20
100
80
60
P
40
S
-
I
20
HYDROSTATIC TESTS
0
UNIAXIAL TEST
A
TRIAXIAL TEST. P = 6 PSI
0
TRIAXIAL TEST. P = 20 PSI
0
TRIAXIAL TEST. P = 30 PSI
-
0 0.
.2
FIGURE 4 . 7 a .
1
.5
.3
STRAIN
UOLUHE
140
c
.4
. 1
( (
UQ-U )/U0 1
Skeleton Volumetric Responses from All Tests on Foam 9505
'
*
"
1
~
"
'
1
"
~
'
1
"
'
~
1
"
"
-
120
A X
I
100
A L S
T
80
-
60
-
R
E
P
-
40
5
HYDROSTATIC TESTS UNIAXIAL TEST
A
TRIAXIAL TEST. P = 6 PSI
0
TRIAXIAL TEST. P = 20 PSI
0
TRIAXIQL TEST. P = 30 PSI
I
20
0
1
0.
,
,
,
-
0
.
1
.
.1
.
.
.
1
1
.2
39
1
1
,
1
*
1
.
1
. 3
d e v i a t o r i c loading.
The f i r s t t e r m i n E q u a t i o n 4.12 i s a c t i v e o n l y i f t h e
loading i s deviatoric.
The p r i n c i p a l s k e l e t o n s t r e s s e s must be l e s s t h a n o r
I f the p r i n c i p a l skeleton stresses are less
e q u a l t o t h e y i e l d f u n c t i o n g. than g, the behavior i s e l a s t i c .
I f the p r i n c i p a l skeleton stresses are
e q u a l t o g, t h e b e h a v i o r may be p l a s t i c .
Constants B and C a r e determined
f r o m a h y d r o s t a t i c t e s t , and c o n s t a n t A i s d e t e r m i n e d f r o m a t r i a x i a l o r uniaxial test,
Thus, a t l e a s t one h y d r o s t a t i c t e s t and one t r i a x i a l o r
u n i a x i a l t e s t a r e needed t o c h a r a c t e r i z e t h e foam b e h a v i o r .
Material
p r o p e r t i e s f o r t h e v a r i o u s foams ( T a b l e 4.1) were determined f r o m t h e h y d r o s t a t i c and u n i a x i a l s k e l e t o n d a t a .
The foam e l a s t i c modulus, E, was
t a k e n as t h e s l o p e o f a b e s t f i t c u r v e t h r o u g h t h e d a t a i n t h e e l a s t i c regime.
Y i e l d f u n c t i o n parameter C was determined f r o m t h e s l o p e o f a b e s t
f i t c u r v e t h r o u g h t h e d a t a i n t h e p l a s t i c regime o f t h e h y d r o s t a t i c t e s t s .
T a b l e 4.1.
M a t e r i a l P r o p e r t i e s f o r NMERI/CERF Foams
B
C
4
15.5
0.738
0.035
37.0
21.3
0.218
0.050
344
19.5
11.8
1.590
0.050
9503
650
36.0
24.5
0.511
0.060
6704
2050
49.8
48.8
-0.613
0.080
9505
3010
49.2
60.8
-0.517
0.090
FOAM
E
A
6602
462
9.5
6703
644
9703
E - YOUNG'S MODULUS, p s i A
-
YIELD FUNCTION PARAMETER, p s i
B - YIELD FUNCTION PARAMETER, p s i C - YIELD FUNCTION PARAMETER
4 - VOLUME FRACTION OF SOLID M A T E R I A L 40
The n e x t s t e p i n t h e model b u i l d i n g process was t o express t h e c h a r a c t e r i z i n g parameters i n terms o f volume f r a c t i o n .
A consideration o f
t h e p h y s i c a l meaning o f t h e c o n s t a n t s showed t h a t parameters A, B and E s h o u l d approach z e r o as volume f r a c t i o n goes t o z e r o . parameters were f i t t e d w i t h a power f u n c t i o n . l i n e a r f u n c t i o n o f volume f r a c t i o n .
F o r t h i s reason t h e s e
Parameter C was f i t t e d w i t h a
T h i s l e a s t squares f i t t i n g process
produced t h e v a l u e s g i v e n i n E q u a t i o n s 4.13
-
4.16.
The f i t t e d curves and
s u p p o r t i n g d a t a a r e shown i n F i g u r e 4.8.
A =
3440
4
1.676
4.13
B =
2780
4 1.645
4.14
C
2.21
E =
454000
-
31.1 4
4
2.20
E q u a t i o n 4.16 i n d i c a t e s a t$‘“ Warren [ 5 ]
4.15 4.16
dependence f o r Young’s Modulus.
K r a y n i k and
have r e p o r t e d a t$2’o dependence f o r Young’s Modulus f o r low
d e n s i t y foams.
E q u a t i o n s 4.13 t o 4.16 c o u l d be used t o e s t i m a t e i n p u t
parameters f o r t h e new c o n s t i t u t i v e model i f t h e o n l y d a t a a v a i l a b l e f o r t h e foam was i t s volume f r a c t i o n .
However, f u r t h e r e x p e r i m e n t a l t e s t i n g s h o u l d
be completed t o improve o u r c o n f i d e n c e i n Equations 4.13 t o 4.16. The c o n s t i t u t i v e model i s summarized i n t h e f l o w c h a r t shown i n F i g u r e 4.9.
F o r a g i v e n i n i t i a l s t r e s s and s t r a i n s t a t e , t h e foam s t r e s s
components can be d e t e r m i n e d f r o m t h e e x p r e s s i o n s f o r t h e s t r e s s c a r r i e d by t h e a i r and by t h e s k e l e t o n .
I n i t i a l foam s t r e s s components and a i r
p r e s s u r e a r e used t o compute t h e i n i t i a l s k e l e t o n s t r e s s components. s t r a i n s a r e updated u s i n g t h e s t r a i n r a t e and t i m e s t e p s i z e .
Foam
Trial
s k e l e t o n s t r e s s components a r e computed by assuming t h a t t h e s k e l e t o n b e h a v i o r remains e l a s t i c o v e r t h e t i m e s t e p .
The t r i a l s k e l e t o n s t r e s s
components a r e t h e n r o t a t e d t o p r i n c i p a l s t r e s s d i r e c t i o n s , and each p r i n c i p a l t r i a l s t r e s s i s compared w i t h t h e y i e l d s t r e s s .
41
I f a principal
104 I
3
FI
P 5
i
100
I
10-2
10-1
100
UOLUME FRFICTION
F I G U R E 4.8a.
Foam Parameter A as a Function o f Densitv Ratio
104
103
6 102
P S
I
101
100 10-2
10-1
100
UOLUME FRQCTION
F I G U R E 4.8b.
Foam Parameter B as a Function of Density Ratio
42
2.
1.5
1.
c
e . 5
0.
-. 5 -1.
0.
.04
.08
.02
.06
.l
UOLUHE FRFICTION
FIGURE 4 . 8 ~ . Foam P a r a m e t e r C a s a F u n c t i o n o f D e n s i t y R a t i o
10-2
10-1
100
UOLUHE FRFICTION
FIGURE 4 . 8 d .
Foam E l a s t i c M o d u l u s E a s a F u n c t i o n o f Density Ratio
i--
7 ---- -
I
COMF'UTE: Initial Skeleton Stress
I
NOTE: Since Poisson's Ratio for the Skeletorl = 0.
I
UPDATE: Strain Components
c
I
1
COMPUTE: Principal Skeleton Stress ow, and directions
COMPUTE: Yield Stress = A(I[>+ B (1 + Cy)
COMPUTE: New acirarid Foarri Sires5 oii = & $1 + 6.. 'I (p
FIGURE 4 . 9 .
Flowchart
of
N e w Constitutive Model
44
t r i a l s k e l e t o n s t r e s s i s s m a l l e r i n magnitude t h a n t h e y i e l d s t r e s s , i t becomes t h e a c t u a l p r i n c i p a l s k e l e t o n s t r e s s a t t h e end o f t h e t i m e s t e p . P r i n c i p a l s k e l e t o n s t r e s s e s w i t h magnitudes t h a t a r e equal t o o r g r e a t e r Once
t h a n t h e y i e l d s t r e s s a r e s e t equal i n magnitude t o t h e y i e l d s t r e s s .
t h e p r i n c i p a l s k e l e t o n s t r e s s e s a r e determined, t h e y a r e r o t a t e d back t o t h e g l o b a l c o o r d i n a t e system.
The s t r e s s c a r r i e d by t h e a i r i s t h e n computed
u s i n g t h e updated volume s t r a i n .
F i n a l l y , foam s t r e s s components f o r t h i s
t i m e s t e p a r e computed by a d d i n g t h e s t r e s s c a r r i e d by t h e a i r t o t h e normal s k e l e t o n s t r e s s components. The n e x t s t e p i n t h e development o f t h e new c o n s t i t u t i v e model was i t s i m p l e m e n t a t i o n i n f i n i t e element computer codes.
The model was i n c o r p o r a t e d
i n SANCHO [8], a q u a s i s t a t i c dynamic r e l a x a t i o n code, and i n PRONTO [ 9 ] , t r a n s i e n t dynamics code.
a
The i m p l e m e n t a t i o n i n b o t h codes was r e l a t i v e l y
s t r a i g h t f o r w a r d and f o l l o w e d t h e f l o w c h a r t i n F i g u r e 4.9.
However, t h e r e
were two m i n o r m o d i f i c a t i o n s made t o t h e c o n s t i t u t i v e model d u r i n g t h e i m p l e m e n t a t i o n phase.
The f i r s t m o d i f i c a t i o n i n v o l v e d t h e s t e p f u n c t i o n
used i n t h e y i e l d c r i t e r i o n f o r t h e s k e l e t o n ( E q u a t i o n 4.12). f u n c t i o n a c t s as an " o n - o f f "
The s t e p
s w i t c h w i t h v a l u e s o f e i t h e r 0 o r 1.
The
d i s c o n t i n u o u s jumps between t h e v a l u e s o f 0 and 1 caused some convergence d i f f i c u l t i e s , b u t t h e s e d i f f i c u l t i e s were e a s i l y s o l v e d by r e p l a c i n g t h e s t e p f u n c t i o n w i t h a s t e e p s i n e f u n c t i o n w h i c h a l l o w e d a continuous v a r i a t i o n between 0 and 1 ( F i g u r e 4.10).
The second m o d i f i c a t i o n a f f e c t e d
t h e a i r p r e s s u r e c o n t r i b u t i o n and y i e l d s t r e s s e q u a t i o n s .
These e q u a t i o n s
were w r i t t e n as f u n c t i o n s o f e n g i n e e r i n g volume s t r a i n b u t t h e computer codes i n w h i c h t h i s model was implemented used l o g a r i t h m i c s t r a i n i n t h e i r c o n s t i t u t i v e model r o u t i n e s .
The e n g i n e e r i n g volume s t r a i n , 7 , can be
expressed as a f u n c t i o n o f t h e c u r r e n t l o g a r i t h m i c s t r a i n components as f o l 1 ows
y
=
e
'kk
- 1
4.17
where lija r e t h e l o g a r i t h m i c s t r a i n components, and e i s t h e base o f t h e n a t u r a l l o g a r i t h m system.
E q u a t i o n 4.17 was used i n t h e implementation o f
t h e model i n t h e computer codes t o express t h e a i r p r e s s u r e c o n t r i b u t i o n and t h e y i e l d s t r e s s as f u n c t i o n s o f t h e l o g a r i t h m i c s t r a i n components.
45
2.
-
I
1.s
-
1.
-
.
'
.
.
I
.
.
.
,
I , .
,
,
,
1
'
1
.
1
-
I
(11'
I
>
I
I
-
.5
I
-
I
I
I
- STEP
I
I
0.
1
FUNCTION SINE FUNCTION
-
-
-.s
'
----
I
.
.
-
"
~
.
1
.
.
.
.
1
.
,
.
.
1
.
.
.
.
1
I
,
,
,
I
.
,
,
,
,
-
-1.
F I G U R E 4.10.
Modification to Step Function in Y i e l d Equation
46
The l a s t s t e p i n t h e development o f t h i s new c o n s t i t u t i v e model was t o v e r i f y t h a t t h e model a c c u r a t e l y r e p r e s e n t e d t h e p o l y u r e t h a n e foam b e h a v i o r .
To v e r i f y t h e model, a s e r i e s o f a n a l y s e s was completed u s i n g t h e new c o n s t i t u t i v e model i n SANCHO and PRONTO.
T h i s s e r i e s o f analyses was
completed u s i n g an a x i s y r r r n e t r i c , one element model o f a NMERI/CERF t e s t sample.
Boundary c o n d i t i o n s on t h e model were v a r i e d t o r e p r e s e n t t h e
v a r i o u s NMERI/CERF t e s t s .
E x p e r i m e n t a l foam b e h a v i o r and c o n s t i t u t i v e model
b e h a v i o r f r o m a h y d r o s t a t i c t e s t f o r 6602 Foam a r e shown i n F i g u r e s 4.11a and 4.11b. test.
The new c o n s t i t u t i v e model a c c u r a t e l y modeled t h i s h y d r o s t a t i c
T h i s r e s u l t was expected because parameters f o r t h e c o n s t i t u t i v e
model were s e l e c t e d based on r e s u l t s f r o m t h e e x p e r i m e n t a l h y d r o s t a t i c and uniaxial tests.
Experimental foam b e h a v i o r and c o n s t i t u t i v e model b e h a v i o r
f r o m a u n i a x i a l t e s t f o r 6602 Foam a r e shown i n F i g u r e s 4.12a and 4.12b. Again, t h e c o n s t i t u t i v e model a c c u r a t e l y r e p r e s e n t e d t h e experimental foam behavior.
Experimental foam b e h a v i o r and c o n s t i t u t i v e model b e h a v i o r f r o m
t r i a x i a l t e s t s f o r 6602 Foam a r e shown i n F i g u r e s 4.13
- 4.15.
The new
c o n s t i t u t i v e model a c c u r a t e l y r e p r e s e n t e d t h e foam b e h a v i o r f o r a l l o f t h e t r i a x i a l tests.
A l s o , t h e r e was no s i g n i f i c a n t d i f f e r e n c e between r a s u l t s
o b t a i n e d u s i n g SANCHO and r e s u l t s o b t a i n e d u s i n g PRONTO.
Comparisons
between e x p e r i m e n t a l r e s u l t s and model b e h a v i o r f o r o t h e r foams a r e shown i n Appendix A.
P R E S S
U
2 20 P S
I
10
OOA
-
N M E R I K E R F TESTS
FINITE
ELEMENT ANaLYsIs
0 0.
.2
.4
.1
.3 UOLUHE STRAIN
F I G U R E 4.11a.
( (
.6
.5 UQ-U )/Ug
)
Comparison of Analytical and Experimental Results Volumetric Responses from Hydrostatic Tests on F o a m 6602
/ A 30 X
-
I R L
s T
R 20
E
-
S S
NMERI X E R F TESTS
0 0A
-
F I N I T E ELEMENT FINQLYSIS
3 AXIQL STRAIN
FIGURE 4 . 1 1 b .
( (
Lo-L )/La
)
C o m p a r i s o n of A n a l y t i c a l a n d E x p e r i m e n t a l R e s u l t s Axial Responses from Hydrostatic Tests on Foam 6602
48
20
A
-
15
NflERI/CERF
TESTS
F I N I T E ELEflENT QNALYSIS
P
R E
S S U
;10 P S I
5
0
0.
.2 .1
UOLUHE STRAIN
F I G U R E 4.12a.
A 30 X
.4
.5
.3 ( (
UQ-U )/UQ
)
C o m p a r i s o n of Analytical and E x p e r i m e n t a l Results Volumetric Response from Uniaxial Test on Foam 6602
-
!
I A
I
AXIAL S T R A I N
F I G U R E 4.12b.
( (
La-L )/La )
Comparison of Analytical and Experimental Results Axial Response from Uniaxial Test o n Foam 6 6 0 2
49
-
40
A
NHERI/CERF
-
30 P R
FINITE
TESTS
ELEMENT a N a L w I s
E
s s
U
g
20
P S I
10
0 0.
.1
.z
.4
UOLUtlE STRAIN
F I G U R E 4.13a.
A30 X
.6
.5
.3 ((
U0-U )/U0 1
Comparison of Analytical and Experimental Results Volumetric Response from Triaxial Test o n Foam 6602 (P=10 psi)
-
I
A L
S T RZ0
E
-
S S
J A
P s 10 I
-
-
N M E R I 4 E R F TESTS
.2
.1
0.
.I5
.05 AXIAL S T R A I N
FIGURE 4 . 1 3 b .
ELEMENT a N a L w I s
FINITE
( (
1
.3
.25
La-L )/Lg 1
Comparison of Analytical and Experimental Results Axial Response from T r i a x i a l Test on Foam 6 6 0 2 (P=10 p s i ) 50
0
A
-
N M E R I K E R F TESTS F I N I T E ELEMENT QNFILYSIS
P S
I
10
0 0.
.z
.I
.4
UOLUHE STRAIN
F I G U R E 4.14a.
Gl 30 X
F1 I
.6
.3 ( (
.5 UQ-U )/Ug
)
Comparison of Analytical and Experimental Results V o l u m e t r i c R e s p o n s e f r o m T r i a x i a l T e s t o n F o a m 6602 ( P = 1 5 psi)
-
i n
A
A
A- a
-
0
A
N M E R I K E R F TESTS
-
"
F I G U R E 4.14b.
~
'
~
F I N I T E ELEMENT QNFILYSIS
~
"
'
"
~
~
'
~
-
"
'
"
"
'
'
Comparison of Analytical and Experimental Results Axial R e s p o n s e from T r i a x i a l Test o n Foam 6602 (P=15 psi)
51
-
P R
E
S S U
E 20 P S
r
i
10
A
NMERIICERF
-
TESTS
F I N I T E ELEnENT CINALYSIS
0 0.
.2
. 4
.1 UOLUME STRAIN
F I G U R E 4.15a.
fi 30
( (
3
.5
.3 U0-U )NUB
)
Comparison of Analytical and Experimental Results Volumetric Response from Triaxial Test o n Foam 6602 (P-20 p s i )
A
I A L
A
S T
2S 2 0
-
-
S
P 5 I
10
-
A
-
.z .15
.05 A X I A L STRAIN
-
TESTS
F I N I T E ELEMENT ANALYSIS
. 1
0.
F I G U R E 4.15b.
NMERIICERF
( (
. .? .25
La-L ) / L a
)
Comparison of Analytical and Experimental Results Axial Response from Triaxial Test o n Foam 6602 ( P = 2 0 psi) 52
5.
SOLUTION OF A TYPICAL DYNAMIC PROBLEM USING THE NEW CONSTITUTIVE MODEL
A t y p i c a l dynamic p r o b l e m was a n a l y z e d u s i n g t h e new foam c o n s t i t u t i v e model and PRONTO.
T h i s p r o b l e m was chosen t o demonstrate t h e c a p a b i l i t i e s
o f t h e model f o r h a n d l i n g complex s t r e s s s t a t e s .
Results from t h i s a n a l y s i s
were compared w i t h r e s u l t s o b t a i n e d u s i n g a c o n v e n t i o n a l d e v i a t o r i c p l a s t i c i t y model and a combined v o l u m e t r i c p l a s t i c i t y w i t h p r e s s u r e dependent d e v i a t o r i c p l a s t i c i t y model t o demonstrate t h e e f f e c t s o f u s i n g t h e v a r i o u s models f o r t h e foam m a t e r i a l . T h i s p r o b l e m c o n s i s t e d o f an i n f i n i t e l y l o n g s t e e l c y l i n d e r surrounded by a foam l a y e r t h a t was covered w i t h a t h i n aluminum s h e l l .
The two-
d i m e n s i o n a l , p l a n e s t r a i n f i n i t e element model shown i n F i g u r e 5.1 was used i n t h e s e analyses.
The c y l i n d e r was dropped o n t o a r i g i d s u r f a c e a t an
i n i t i a l v e l o c i t y o f 528 i n c h e s p e r second, and t h e r e s u l t i n g d e f o r m a t i o n s and a c c e l e r a t i o n s were computed. o f a n a l y s e s a r e g i v e n i n T a b l e 5.1.
M a t e r i a l p r o p e r t i e s used f o r t h i s s e r i e s The foam l a y e r was assumed t o be
9505 Foam and was modeled w i t h t h e t h r e e d i f f e r e n t c o n s t i t u t i v e models d i s c u s s e d above.
M a t e r i a l p r o p e r t i e s f o r t h e d e v i a t o r i c p l a s t i c i t y model
were based on r e s u l t s f r o m t h e u n i a x i a l NMERI/CERF t e s t on 9505 Foam.
For
t h i s model t h e m a t e r i a l was assumed t o be e l a s t i c p e r f e c t l y p l a s t i c . R e s u l t s f r o m b o t h t h e u n i a x i a l and t h e h y d r o s t a t i c NMERI/CERF t e s t s on 9505 Foam were used t o d e t e r m i n e m a t e r i a l p r o p e r t i e s f o r t h e combined v o l u m e t r i c and p r e s s u r e dependent d e v i a t o r i c p l a s t i c i t y model.
The v o l u m e t r i c response
f o r t h i s model was based on r e s u l t s f r o m t h e h y d r o s t a t i c t e s t s and t h e d e v i a t o r i c response f o r t h i s model was based on r e s u l t s f r o m t h e u n i a x i a l test.
M a t e r i a l p r o p e r t i e s f o r t h e new c o n s t i t u t i v e model were a l s o based on
r e s u l t s f r o m b o t h t h e u n i a x i a l and t h e h y d r o s t a t i c NMERI/CERF t e s t s on 9505 Foam. R e s u l t s f r o m t h i s s e r i e s o f a n a l y s e s a r e summarized i n T a b l e 5.2. R e s u l t s o b t a i n e d u s i n g t h e new c o n s t i t u t i v e model a r e between r e s u l t s o b t a i n e d u s i n g t h e c o n v e n t i o n a l d e v i a t o r i c p l a s t i c i t y model and t h e combined v o l u m e t r i c p l a s t i c i t y w i t h p r e s s u r e dependent d e v i a t o r i c p l a s t i c i t y model. The c o n v e n t i o n a l d e v i a t o r i c p l a s t i c i t y model does n o t a l l o w f o r any v o l u m e t r i c p l a s t i c i t y and i s s t i f f e r t h a n t h e o t h e r two models.
53
The
FIGURE 5.1.
F i n i t e E l e m e n t Yodel
54
T a b l e 5.1.
M a t e r i a l P r o p e r t i e s Used i n Dynamic A n a l y s e s
Aluminum ( E l a s t i c ) YOUNG'S MODULUS = lO.E+06 POISSON'S R A T I O = 0.30 D E N S I T Y = 2.5E-04
psi
Steel (Elastic)
psi
YOUNG'S MODULUS = 29.E+06 P O I S S O N ' S R A T I O = 0.30 DENSITY = 7.OE-04
2 4 lb s /in
2 . 4 lb s /in
Foam ( C o n v e n t i o n a l D e v i a t o r i c P l a s t i c i t y M o d e l ) YOUNG'S MODULUS POISSON'S RATIO DENSITY YIELD STRENGTH HARDENING MODULUS BETA
= 3010.
psi
= 0.00 = 7.5E-06 = 110. = 0. = 0.
2 . 4 lb s /in psi psi
Foam (Combined V o l u m e t r i c and D e v i a t o r i c P l a s t i c i t y M o d e l ) SHEAR MODULUS = 1505. BULK MODULUS = 1003. D E N S I T Y = 7.5E-06 YIELD FUNCTION CONSTANT - a. = 110. YIELD FUNCTION CONSTANT - al = 0. YIELD FUNCTION CONSTANT - a2 = 0.
psi 4 Psi 2 lb s /in
Foam (New C o n s t i t u t i v e M o d e l ) YOUNG'S MODULUS DENSITY OLUME FRACTION OF SOLID MATERIAL - 9 I N I T I A L A I R PRESSURE - p YIELD FUNCTION CONSTANT YIELD FUNCTION CONSTANT - B YIELD FUNCTION CONSTANT - C
= 3010. = 7.5E-06
= 0.090 = 14.7 = 49.2
= 60.8 = -0.517
55
Psi 2 4 lb s /in psi psi psi
L
T a b l e 5.2.
R e s u l t s f r o m Dynamic Analyses -
CONSTITUTIVE MODEL USED
MAX IMUM CRUSH-U P (in.)
MAXIMUM STEEL BODY ACCELERATION (9)
CONVENTIONAL D E V I A T O R I C PLASTICITY MODEL
1.93
399
NEW FOAM MODEL
2.57
275
COMBINED VOLUMETRIC AND DEVIATORIC PLASTICITY MODEL
2.84
208
combined v o l u m e t r i c p l a s t i c i t y w i t h p r e s s u r e dependent d e v i a t o r i c p l a s t i c i t y model does n o t a l l o w f o r any change i n v o l u m e t r i c response due t o t h e occurrence o f d e v i a t o r i c loading.
Experimental r e s u l t s i n d i c a t e d t h a t t h e
o c c u r r e n c e o f d e v i a t o r i c l o a d i n g would s t i f f e n t h e v o l u m e t r i c response.
The
combined v o l u m e t r i c p l a s t i c i t y w i t h p r e s s u r e dependent d e v i a t o r i c p l a s t i c i t y model i s s o f t e r t h a n t h e o t h e r two models.
The new c o n s t i t u t i v e model
c a p t u r e s b o t h v o l u m e t r i c p l a s t i c i t y and changes i n t h e v o l u m e t r i c response due t o t h e occurrence o f d e v i a t o r i c l o a d i n g .
D i s p l a c e d shapes o f t h e f i n i t e
element model a t maximum crush-up a r e shown i n F i g u r e s 5.2 and 5.3.
Plots
of d i s p l a c e m e n t and a c c e l e r a t i o n o f t h e s t e e l c y l i n d e r as a f u n c t i o n o f t i m e a r e shown i n F i g u r e s 5.4 and 5.5,
respectively.
f i l t e r e d w i t h a lowpass f i l t e r w i t h a c u t - o f f
56
The a c c e l e r a t i o n p l o t s were
frequency o f 1000 Hz.
1
conuentional deuiutoric
p l u s t i c ~ t ymodel
new f o a m mode I
combined u o l u m e t r i c
FIGURE 5.2.
and d e u i a t o r i c
Displaced Shapes of a t Maximum C r u s h - u p 57
p l u s t i c i t y model
F i n i t e Element Model
- 'c o n u e n t i o n u l d e u i u t o r i c p l u s t i c i t y m o d e l
n e w f o a m mode I
combined u o l u m e t r i c und d e u i ' u t o r i c p l u s t i c i t y model
FIGURE
5.3.
C l o s e - u p o f D i s p l a c e d Foam L a y e r i n F i n i t e E l e m e n t M o d e l a t Plaximum C r u s h - u p
58
.
D
I
s
-1.
P L
& -A
CONUENTIONAL DEUIATORIC PLASTICITY MODEL
W
NEU FOAM MODEL
Q-€I
COMGINED UOLUMETRIC AND DEUIATORIC PLASTICITY MODEL
-
A
C E -1.5 M
/
/
-
/ /
E
/
,
N T I
n
-2.5
....
-
----=I
- _- - _ _e - -
0.
.004
. OQZ
.008
TIME
FIGURE 5 . 4 .
A
E
300 400
E
L E R
A T I
.01Z
.006
.O1
seconds
Displacement of
Steel Cylinder
i
200
0
N 100
9 0
t
-1SEiI 0.
Q-€I '
1
'
I
COMGINED UOLUMETRIC AND DEUIATORIC PLASTICITY MODEL "
'
I
.004
"
.002
I
"
'
I
.008
.006
TIME
FIGURE 5 . 5 .
'
"
'
I
"
" ,012
.O1
seconds
A c c e l e r a t i o n of
59
Steel Cylinder
6.
CONCLUSIONS AND FUTURE WORK
The b e h a v i o r o f r i g i d , c l o s e d - c e l l , p o l y u r e t h a n e foam was experimentally investigated.
I t was found t h a t t h e s e foams undergo l a r g e
p l a s t i c v o l u m e t r i c s t r a i n s when s u b j e c t e d t o s u f f i c i e n t l o a d and t h a t t h e d e v i a t o r i c and v o l u m e t r i c b e h a v i o r s f o r these foams a r e coupled. A c o n v e n t i o n a l d e v i a t o r i c p l a s t i c i t y model and a combined v o l u m e t r i c p l a s t i c i t y w i t h p r e s s u r e dependent d e v i a t o r i c p l a s t i c i t y model d i d n o t c a p t u r e t h e c o u p l i n g t h a t o c c u r r e d between t h e d e v i a t o r i c and v o l u m e t r i c b e h a v i o r s i n t h e NMERI/CERF t e s t s .
T h e r e f o r e , a new c o n s t i t u t i v e model f o r
l o w d e n s i t y p o l y u r e t h a n e foams was developed.
T h i s new c o n s t i t u t i v e model
c a p t u r e d a l l foam b e h a v i o r s t h a t were observed i n t h e NMERI/CERF t e s t s . T h i s model was implemented i n two f i n i t e element codes, SANCHO and PRONTO. A t y p i c a l problem was a n a l y z e d u s i n g t h i s new c o n s t i t u t i v e model and two o t h e r c o n s t i t u t i v e models t o demonstrate d i f f e r e n c e s between t h e v a r i o u s models.
R e s u l t s f r o m t h i s s e r i e s o f analyses i n d i c a t e d t h a t t h e new
c o n s t i t u t i v e model g e n e r a t e d d i s p l a c e m e n t and a c c e l e r a t i o n p r e d i c t i o n s t h a t were between p r e d i c t i o n s o b t a i n e d u s i n g t h e o t h e r two models.
This r e s u l t
was expected. Because t h e e x p e r i m e n t a l NMERIKERF t e s t s were a l l s t a t i c t e s t s t h e r e was no way t o d e t e r m i n e i f r a t e e f f e c t s were i m p o r t a n t ; t h e r e f o r e , no r a t e e f f e c t s were i n c l u d e d i n t h e new c o n s t i t u t i v e model.
I n t h e f u t u r e , dynamic
t e s t s s h o u l d be completed t o d e t e r m i n e t h e e f f e c t s of s t r a i n r a t e s .
The
e f f e c t s o f t e m p e r a t u r e changes were a l s o n o t i n v e s t i g a t e d as p a r t o f t h e NMERI/CERF t e s t s .
I n t h e new c o n s t i t u t i v e model i t was assumed t h a t t h e a i r
behaves as an i d e a l gas and t h a t temperature changes have no e f f e c t on t h e polymer s k e l e t o n .
P o l y u r e t h a n e i s expected t o have a s t r o n g t e m p e r a t u r e
dependence above t h e g l a s s y t r a n s i t i o n temperature, b u t t h e r e s u l t i n g e f f e c t upon c e l l w a l l c o l l a p s e i s unknown. laboratory tests.
T h i s s h o u l d be i n v e s t i g a t e d i n
Once t h e new c o n s t i t u t i v e model has been m o d i f i e d t o
i n c l u d e any i m p o r t a n t r a t e o r temperature e f f e c t s , i t c o u l d t h e n be used w i t h c o n f i d e n c e t o a n a l y z e dynamic events.
F u t u r e comparisons between
e x p e r i m e n t a l r e s u l t s and a n a l y s e s w i t h t h i s c o n s t i t u t i v e model would f u r t h e r i n c r e a s e c o n f i d e n c e i n i t s accuracy.
60
F i n a l l y , most s h i p p i n g c o n t a i n e r s t h a t use p o l y u r e t h a n e foam impact l i m i t e r s have a t h i n l a y e r o f some metal around t h e impact l i m i t e r . A c c u r a t e analyses o f such s h i p p i n g c o n t a i n e r s w i l l r e q u i r e t h e i m p l e m e n t a t i o n of elements t h a t a c c u r a t e l y and e f f i c i e n t l y model t h i n Sayers i n t h e f i n i t e element codes i n which t h e foam model i s used.
61
7.
REFERENCES
1.
Doyle, E. N., "The Development and Use o f Polyurethane Products," McGraw-Hill Book Co., 1971, pp. 256-263.
2.
" S a f e t y A n a l y s i s Report f o r t h e NUPAC 125-B Fuel Shipping Cask," Document Number 71-9200, Nuclear Packaging, Inc., Federal Way, Washington, May 1985.
3.
" T r a n s u r a n i c Package T r a n s p o r t e r TRUPACT S a f e t y A n a l y s i s Report f o r Packaging," SAND 83-7077, Sandia N a t i o n a l L a b o r a t o r i e s , Albuquerque, New Mexico, May 1986.
4.
Hamberg, D., NMERI/CERF, l e t t e r t o R. May, K. Schuler, and R. Yoshimura, Sandia N a t i o n a l L a b o r a t o r i e s , Albuquerque, New Mexico, February 29, 1980.
5.
K r a y n i k , A. W. and Warren, W. E., "The L i n e a r E l a s t i c P r o p e r t i e s o f Foams," Presented a t W i n t e r Meeting, S o c i e t y o f Rheology, Santa Monica, CA, January 1987.
6.
" T r a n s p o r t a t i o n System Impact L i m i t e r Design Using R i g i d Wellman, G. W., P o l y u r e t h a n e Foam," SAND84-2271, Sandia N a t i o n a l L a b o r a t o r i e s , Albuquerque, New Mexico, 1985.
7.
K r i e g , R. D., "A Simple C o n s t i t u t i v e D e s c r i p t i o n f o r S o i l s and Crushable Foams," SC-DR-72-0883, Sandia N a t i o n a l L a b o r a t o r i e s , Albuquerque, New Mexico, 1972.
8.
Stone, C. M., K r i e g , R. D., and B e i s i n g e r , Z . E., "SANCHO - A F i n i t e Element Computer Program f o r t h e Q u a s i s t a t i c , Large Deformation, I n e l a s t i c Response o f Two-Dimensional S o l i d s , " SAND84-2618, Sandia N a t i o n a l L a b o r a t o r i e s , Albuquerque, New Mexico, 1985.
9.
T a y l o r , L. M. and Flanagan, D. P., "A Two-Dimensional T r a n s i e n t S o l i d Dynamics Program," SAND86-0594, Sandia N a t i o n a l L a b o r a t o r i e s , Albuquerque, New Mexico, 1986.
62
APPENDIX A
To v e r i f y t h a t t h e new c o n s t i t u t i v e model c a p t u r e d t h e foam b e h a v i o r observed d u r i n g t h e NMERI/CERF t e s t s , a s e r i e s o f analyses was completed u s i n g t h e new model i n SANCHO and PRONTO.
T h i s s e r i e s o f analyses was
completed u s i n g an axisymmetric, one element model o f a NMERI/CERF t e s t sample.
Boundary c o n d i t i o n s on t h e model were v a r i e d t o r e p r e s e n t t h e
v a r i o u s NMERI/CERF t e s t s .
R e s u l t s f r o m t h i s s e r i e s o f analyses a r e compared
w i t h r e s u l t s f r o m t h e NMERI/CERF t e s t s i n t h i s appendix.
These r e s u l t s
i n d i c a t e t h a t t h e new c o n s t i t u t i v e model does c a p t u r e t h e foam b e h a v i o r observed d u r i n g t h e NMERI/CERF t e s t s .
.
60
,
50
-
P R 40 E
-
.
,
,
s
S U R30
E
-
,” 20
:
I
c NflER I K E R F TESTS
D 0A
‘
~
.
.
l
0.
l
.
.
.
l
,
.
.
.
I
,
.
,
.
.2 .1
F I G U R E A.la.
F I N I T E ELEflENT QNALYSIS
.4
.3
I
,
I
c
,
,
.
,
.6
Comparison of Analvtical and Experimental Results Volumetric Responses from Hvdrostatic Tests on F o a m 6703
60
5Q FI X
I A 40 L
S T
30
s s 20 P 5 I
10
0.
.1 .05 F l X I A L STRAIN
F I G U R E A.lb.
.2 . l s ( (
Lo-L
)/Lo
.3
.25 )
Comparison o f Analytical and Experimental Results Axial Responses from Hydrostatic Tests o n Foam 6703
64
60
(
r
"
'
l
"
"
I
"
"
Sb3
.
A
N H E R I K E R F TESTS
-
F I N I T E ELEMENT FINFILYSIS
P R 40
E S S U
E
P 5
30
z0
I
10
0
0.
.2
.1
UOLUME STRA I N
F I G U R E A.2a.
.4
.3 ( (
Ug-U )/U0
.6
.s
)
Comparison o f Analytical and Experimental Results V o l u m e t r i c R e s p o n s e from T r i a x i a l Test on F o a m 6703 (P=15 psi)
ylOO 1zE3
-
.
,
.
,
A
-
L FI I
,
,
l
.
l
NMERI/CERF
l
l
,
l
l
l
l
l
.
.
~
'
TESTS
F I N I T E ELEMENT FINALYSIS
80
c
AXIAL STRAIN
F I G U R E A.2b.
( (
La-L ) / L a
)
Comparison o f Analytical and Experimental Results Axial R e s p o n s e from T r i a x i a l Test o n F o a m 6703 ( P = 1 5 psi) 65
60
a
-
50
N M E R I K E R F TESTS F I N I T E ELEMENT CINQLYSIS
P
R 40 E 5
s
U
E 30 E
20
I
10
0
0.
.2
.4
. 1
FIGURE A.3a.
( (
.6
.5
.3 UOLUME S T R A I N
U0-U ) / \ l o
)
Comparison of A n a l y t i c a l and E x p e r i m e n t a l R e s u l t s V o l u m e t r i c R e s p o n s e f r o m T r i a x i a l T e s t o n Foam 6 7 0 3 (P=20 p s i )
120
a
-
100
N M E R I X E R F TESTS
F I N I T E ELEMENT CINQLYSIS
F1 X
I A L
80
5 T
E
60
5 S 40
P S I
20
Q
0.
. 1 .05 QXIAL STRAIN
FIGURE A . 3 b .
.3
.2 .15 ( ( Lo-L
)/Lo
.25 )
Comparison o f A n a l y t i c a l and Experimental R e s u l t s A x i a l R e s p o n s e f r o m T r i a x i a l T e s t o n Foam 6 7 0 3 (P=20 psi) 66
30
OOA
-
25
N M E R I X E R F TESTS F I N I T E ELEMENT QNALYSIS
P
R 20 E S S U R 15
E
P
s l0 t
5
0 0.
.2
.3 l J O L U M E STRAIN
25
,
-
( (
.5 Ug-lJ )/U0
)
Comparison o f Analytical and Experimental Results Volumetric Responses from Hydrostatic Tests o n F o a m 9703
FIGURE A.4a.
30
.6
.4
.1
,
,
,
(
.
OOA
-
,
.
,
1
"
.
.
,
' . " ' " ' " ' - / '
.
N M E R I A E R F TESTS
-
F I N I T E ELEMENT QNALYSIS
0
A X
I
-
-
-
-
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-
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Comparison of Analytical and Experimental Results A x i a l R e s p o n s e s f r o m H y d r o s t a t i c T e s t s o n F o a m 9703
67
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Comparison o f Analytical and Experimental Results Volumetric Response from Triaxial Test o n Foam 9703 (P=10 psi)
60
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Comparison of Analytical and Experimental Results Axial Response from Triaxial Test on Foam 9703 (€'=lo p s i ) 70
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Comparison of Analytical and Experimental Results Volumetric Response from Triaxial Test o n Foam 9703 ( P = 1 5 psi)
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Comparison of Analytical and Experimental Results Axial Response from Triaxial Test o n Foam 9 7 0 3 ( P = 1 5 psi) 71
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72
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Comparison of Analytical and Experimental Results Axial Response from Triaxial Test o n Foam 9503 (P=10 psi) 73
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Comparison of A n a l y t i c a l and Experimental Results Axial Responses from Hydrostatic Tests o n Foam 6 7 0 4
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77
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F I G U R E A.15a.
C o m p a r i s o n o f Analytical and Experimental Results Volumetric Response from Uniaxial Test o n Foam 9505
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Comparison o f Analytical and Experimental Results A x i a l R e s p o n s e f r o m U n i a x i a l T e s t o n F o a m 9505
78
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F I G U R E A.16a.
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Comparison of Analytical and Experimental Results Volumetric Response from Triaxial Test on Foam 9505 (P=20 p s i )
F I G U R E A.17a.
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Comparison o f Analytical and Experimental Results Axial Response from Triaxial Test o n Foam 9505 (P=20 psi) 80
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F I G U R E A.18a.
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Comparison o f Analytical and Experimental Results Axial Response from Triaxial Test o n Foam 9505 (P=30 p s i )
Distribution: U. S . Department o f Energy O f f i c e o f S c i e n t i f i c & T e c h n i c a l I n f o r m a t i o n (226) Oak Ridge, TN 37830 A t t n : DOE/OSTI-4500-R74 UC-71
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Joint Integration Office 4308 C a r l i s l e , NE Albuquerque, NM 87107 Attn: J . R o l l , Westinghouse K . M c K i n l e y , Rockwell R . M. J e f f e r s o n SANDIA INTERNAL: 1510 J. W. N u n z i a t o 1511 D. K . G a r t l i n g A. M. Kraynik 1511 W. Herrrnann ( a t i 1520 R. D. K r i e q (10) 1521 M. K . N e i l s e n ( 1 5 ) 1521 H. S . Morgan ( 5 ) 1521 1521 C . M . Stone G . W. We1 lman 1521 1522 R . C . Reuter E . D . Reedy 1522 K . W. Schu l e r 1522 1523 J. H . B i f f l e L . M. Taylor 1523 A. K . M i 1l e r 1524 K . W. Gwinn 1524 L . w. Dav ison 1530 w. c . L u t h 1540 A t t n : W . R . Wawersik 1550 R . C . Maydew R . G . Kep l e r 1810 J . G . Curro 1813 1813 W . E . Warren ( 5 ) S . A. Landenberger ( 5 ) 3141 w. L . Garner ( 3 ) 3151 3154-1 C . H. D a l i n , F o i DOE/OSTI ( 2 8 ) 3310 W. D. B u r n e t t 6000 D. L. Hartley 6300 R . W . Lynch 6320 J. E. Stiegler A t t n : T T C Master F i l e 6320 TTC L i b r a r y (25) 6321 R . E . Luna J . M . Freedman 6322 6323 G. C . Allen, J r . 6323 R . H . Yoshimura ( 5 ) 7 544 R . A . May P . W. Dean 8024 8240 C . W. Robinson 8242 L . I . Weingarten 8243 M . 1. C a l l a b r e s i
