Pergamon Int. J. Impact Engn 9 Vol. 21, No. 5, pp. 369 386. 1998

c. 1998 Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain

PII: S0734-743X(97)00087-0 0734-743X/98 $19.00 + 0.00

CONSTITUTIVE MODELING OF POLYMERIC FOAM MATERIAL SUBJECTED TO DYNAMIC CRASH LOADING

JUN ZHANG*'y, NOBORU KIKUCHIt, VICTOR LI+ +, ALBERT YEE§ and GUY NUSHOLTZ¶I

tDepartment of Mechanical Engineering and Applied Mechanics, +Department of Civil and Environmental Engineering, §Department of Material Science and Engineering, The University of Michigan, Ann Arbor, MI, 48109, U.S.A.; and ¶Chrysler Motor Corporation, Auburn Hills,

MI 48326, U.S.A.

(Received 19 July 1996; in revised form 5 December 1997)

Summary--This paper describes detail work on constitutive law modeling of low-density polymeric foam materials. Selected experimental results on low-density polyurethane (PU), polypropylene (PP), and polystyrene (PS) foams are presented. A rate-dependent hydrodynamic constitutive equation is presented for rigid polymeric foams. Focus has been placed on modeling of strain rate dependency and temperature effect on polymeric foams subjected to high rate impact loading. Numerical implementation procedure for the constitutive model is described. The constitutive model has been implemented into finite-element program as a user-defined material subroutine. Numerical examples are provided to validate the model under simple and complex loading conditions. © 1998 Published by Elsevier Science Ltd. All rights reserved

a,b a (~vv), b (e~.p) C1, C2 Dijlm F Fo g(6, ¢) G , K L(T) n(e)

P Pc0 q, w

P, Pn+ 1

t r ia l P n + 1

Sn+ 1

At Tg Tr

O~

~o

8 d p

^

NOTATION

strain rate dependency coefficients volumetric strain-hardening parameters WLF function coefficients elasto-plastic stiffness tensor yield function at an arbitrary strain rate yield function at quasi-static strain rate plastic flow potential function elastic shear and bulk moduli for isotropic foams WLF shifting factor power coefficient for power law rate dependency hydrostatic pressure initial hydrodynamic compressive strength stress correction factors hydrostatic pressure from previous time step updated hydrostatic pressure trial hydrostatic pressure deviatoric stress tensor deviatoric stress tensor from previous time step updated deviatoric stress trial deviatoric stress time step increment polymer glass transition temperature reference temperature yield ellipsoid center volumetric plastic flow coefficient longitudinal strain rate quasi-static longitudinal strain rate deviatoric strain rate plastic deviatoric strain rate volumetric strain rate plastic volumetric strain rate plastic flow rate tensor

*Corresponding author.

369

370 J. Zhang et al.

~zzp ~xxp ~yyp

~d.n+ 1/2

~v,n+ 1/2

1!

Vp a(~.) o-0(e) (~n+ 1 ^trial 0"n+ 1

Gy0

o-vm

~o

d~

longitudinal strain under uniaxial compression transverse strain under uniaxial compression effective plastic strain rate deviatoric strain rate tensor at half time step volumetric strain rate at half time step plastic flow consistency parameter elastic Poisson's ratio plastic Poisson's ratio nominal stress response at an arbitrary strain rate nominal stress response at a quasi-static strain rate updated stress trial stress initial uniaxial compressive strength von Mises effective stress Jaumann objective stress rate internal state variable initial shear strength coefficients for rate-dependent plasticity spin tensor

I N T R O D U C T I O N

Low-density polymeric foams have found use in a wide range of engineering applications due to their excellent impact energy-absorbing capability. In the automobile industry, for example, the new provisions in the Federal Motor Vehicle Safety Standards (FMVSS) require the use of polymeric foam materials inside motor vehicles to protect the passengers during traffic accidents. In this application, foams are subjected to high-speed large deformations. On the other hand, vehicle design requires repetitive numerical simulation using a finite element program such as LS-DYNA3D [1] for cost-effective crashworthiness evaluation. The designing of the polymeric foam cushion system requires the knowledge of the constitutive behavior of the material under various deformation rates and temperatures. Eventually, the material constitutive must be coded into finite-element programs such as LS-DYNA3D for crashworthiness analysis.

Polymeric foam is composed of a large amount of microscopic polymer cellular struc- tures. Figures 1 and 2 show scanning electron microscopes (SEM) of a closed-cell polysty- rene foam (density 1.6 kg m/m 3) and a open-cell polyurethane foam (density 6.9 kg m/m3). Polymers are very temperature-sensitive materials. Based on the post loading behavior at room temperature (20°C), polymeric foams are catalogued as rigid (elasto-plastic) and flexible (elastic) foams.

Mechanism-based constitutive models for polymeric foams have been well documented by Gibson and Ashby [2]. The failure envelope of low-density rigid foam under multi-axial loading conditions has been studied by Gibson e t al. [3] and Triantafillou et al. [4]. Puso and Govindjee [5] presented an orthotropic rate-independent plasticity and a visco- plasticity model by extending the work by Gibson e t al. [3] for rigid polymeric foams. They further implemented the constitutive model in the computer code DYNA3D. Other re- searchers have taken the phenomenological modeling approach which only describes the inelastic response of foams under uniaxial compression conditions [-6 8]. This approach is not generally applicable for multi-axial loading conditions. Constitutive equations based on continuum constitutive theories have also been investigated. Roscoe's critical state theory [9] has found successful applications for porous materials. Krieg [10] developed a plasticity theory for soils and crushable foams. This plasticity theory has a yield surface which is a parabola of revolution about the hydrostatic axis with a planar cap on the normally open end. Neilsen et al. [-11] developed a plasticity theory for a semi-rigid polyurethane foams which has six intersecting planar yield surfaces forming a cubic shape in the principal stress space.

The remainder of this paper is organized as follows: in Section 2, the experimental program and selected testing results are presented; in Section 3, a rate-dependent

Constitutive modeling of polymeric foam material 371

Fig. 1. SEM of closed-cell polystyrene foam (density 1.6 kgm/m3).

Fig. 2. SEM of open-cell polyurethane foam (density 6.9 kgm/m3}.

hydrodynamic rigid foam constitutive model is presented; the numerical implementation procedure for the constitutive model is outlined in Section 4; model verification and numerical examples are shown in Section 5; conclusions from this study are summarized in Section 6.

372 J. Z h a n g et al.

2. EXPERIMENTAL INVESTIGATION

2.1. Testin 9 equipment

In order to cover a wide range of loading speeds, material tests were conducted on two types of testing machines: (1) an electro-hydraulic Instron 1331 machine equipped with a temperature chamber for low rate tests and (2) a pneumatically driven ICI impact machine for high-rate impact tests. The Instron 1331 machine can obtain a loading speed from quasi- static (0.08 mm/s, ASTM Standard D1621) to intermediate rate at 250 mm/s with a closed- loop servo-controlled system. Higher loading rate from 3 to 10 m/s can be obtained by ICI impact machine. For tests on ICI machine, stress-strain history is recorded from an accelerometer and a quartz load cell (22,240 N force capacity) through a transient data recorder. Although the open-loop ICI machine does not provide a constant load rate in the entire impact process, the recorded velocity history showed a broad plateau which was taken to compute the average strain rate.

2.2. Experimental program

Table 1 summarizes the entire experimental program. Foam material used for auto- mobile padding applications is mainly subjected to compressive loads. Under impact, the stress state of the foam is dominated by compression and shear. Tensile properties becomes less relevant. Focus is placed on material stress strain behaviors under (1) uniaxial com- pression, (2) shear, and (3) hydrostatic compression. According to ASTM Standard D1621, foam specimens were cut in dimensions of 50 x 50 x 50 mm 3 for compression and tension tests and 50 x 50 x 100mm 3 for simple shear tests. Four different cross-head rates of 8.0 x 10 -5, 4.0 x 10 -3, 0.229, and 4.45 m/s (corresponding to strain rates 1.6 x 10 -3, 0.08, 4.6, and 88 l/s) were used in uniaxial compression tests. All specimens were loaded with up to 80% volumetric strain in compression tests and loaded until fracture, in tension and simple shear tests. For simple shear tests, foam specimens were glued in-between two L-shaped loading fixtures made of steel. For hydrostatic compression tests, foam specimens were wrapped by latex rubber and immersed into a specially designed hydrostatic

Table 1. S u m m a r y of exper iment p r o g r a m

Test M o d e F o a m Type P P foam PS foam P U foam

Dens i ty (kg /m 3) 3.0 4.9 1.6 6.4 6.9 9.6

Rigid Rigid Rigid Rigid Flexible Flexible

St ra in rate (s -1)

Un iax ia l 1.60 × 10 3 • • • • • •

compress ion 8.00 × 10-1 • • • • • •

4.60 • • • • • • 8.80 × 101 • • • • •

Hydros t a t i c 4.00 x 10- 3 • • • •

compress ion 2.00 x 10 1 • • • • 1.15 x 101 • • • •

Un iax ia l 1.60 x 10- 3 • •

tens ion 8.00 × 10- 1 • •

4.60 • • 8.80 x 101

Simple 1.60 x 10 3 • • • • shear 8.00 x 10- t • • • •

4.60 • • • • 8.80 x 101

Constitutive modeling of polymeric foam material 373

Fig. 3. Hydrostatic compression device.

compression chamber filled with water. Air is allowed to escape from the specimens through an air vent on the lid of the chamber. Figure 3 shows the cylindrical device designed for the hydrostatic compression test.

2.3. Experimental result

Experimental work reveals that, at room temperature (20°C), the polypropylene foams are semi-rigid foam with only 20% permanent strain after being unloaded from 80% applied strain. The polystyrene foams are rigid (crushable) foams with no resilience while polyurethane foams are flexible with full but retardative resilience. Every test was repeated twice for data repeatability. Under compression, foam specimens showed macroscopically uniform deformation without noticeable bulging and distortion. A typical foam stress- strain response under uniaxial compression exhibits three regimes: (1) a rough- ly linear elastic regime at low stress due to cellular wall elastic bending, (2) a plateau regime corresponding to cell wall buckling (for flexible PU foams) or plastic yielding (for rigid PP and PS foams), and (3) a steeply rising hardening regime due to consolidation of the foam. Figure 4 shows the PP foam (4.9 kg m/m 3) specimen sequentially deformed with negligible Poisson's effect under uniaxial compression. Figures 5 7 show the stress-strain responses of PU foam (6.9 kgm/m3), PP foam (4.9 kgm/m3), and PS foam (1.6 kgm/m 3) under compression with varying loading rate. The stress-strain curve of foam under hydrostatic compression is similar to that under uniaxial compression. Figures 8 and 9 depict the hydrostatic compressive stress-strain response of the PU foam (6.9 kg m/m 3) and the PP foam (4.9 kgm/m3). As we can see, foam mechanical properties are extremely rate sensitive. The broad stress plateau is believed to be caused by cellular wall buckling. The stress rises sharply as a result of material consolidation (Figs 10 and !1). The failure mechanism under tension and shear is different from that under compression. Under tension and shear, the cellular wall will align towards the maximum principal stress direction. The foam will eventually tear apart as a result of cell fracture for rigid foams or plastic necking for flexible foams. Figure 12 depicts the tensile stress strain response of the PU foam (6.9 kgm/m3). Figure 13 shows shear stress-strain response of the PS foam (1.6 kg m/m3).

374 J. Zhang et al.

"iF- - l i!

Fig. 4. Deformation uniformity of polypropylene foam (density 4.9 kgm/m 3) with near zero plastic Poisson's ratio under uniaxial compression.

0.12

0.10

0.07

0.05 oo

i I

+ 2 . 2 9 10 -1 m/s

• 4.00 10 ̀3 m/s

---o-- 8.00 104 m/s

o ....................... i 0.00

0.0 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 Strain

Fig. 5. Stress strain response of polyurethane foam (density 6.9 kg m/m 3) under uniaxial compression.

r.~

2.00

1.50

1.00

0.50

0.00 0.0 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9

Strain

Fig. 6. Stress-strain response of polypropylene foam (density 4.9 kgm/m 3) under uniaxial compression.

0.80

0.70

0.60

0.50

0.40

0.30

Consti tutive modeling of polymeric foam material 375

0.20

0.10

0.00 0.0 0.1 0.2 0.3 0.5

Strain

0.6 0.7 0.8 0.9

Fig. 7. Stress strain response of polystyrene foam (density compression.

1.6 k g m / m 3) under uniaxial

1.00 , , T - - - 7

" 1 o _ " i ----°--z" v = 1.15 101 sec -I i ~" 0.80 .t Eov=2.0010-1sec-11 .......................... i ~ ...................... 1

+ 'v--40010 sec- 1 ! 0.60 . . . . . . .

0.40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.20 .........

0.00 0.0 0.2 0.4 0.6 0.8 1.0

Volumetric Strain -- dV / V 0

Fig. 8. Stress strain response of polyurethane foam (density 6.9 k g m / m 3) under hydrostatic compression.

1.20

1.00

0.80

0.60

0.40

0.20

0.00 0.0

+ e " = 1.15 101 sec -1 v

~'v = 2.00 10 -1 sec -l

+ e " v = 4.00 10 -3 sec -1

0.1 0.8 0.9 0.2 0.3 0.5 0.6 0.7

Volumetric Strain -- dV / V 0

Fig. 9. Stress-strain response of polypropylene foam (density 4.9 k g m / m 3) under hydrostatic compression.

376 J. Zhang et al.

Fig. 10. SEM of consolidated polypropylene foam (4.9 kgm/m3).

Fig. 11. SEM of consolidated polyurethane foam (6.9 kgm/m3).

2.4. Strain rate sensit ivity characterizat ion

The experimental results have shown that stress-strain response of polymeric foam materials are dependent on the strain rate. Motivated by the work of Nagy et al. [12] the rate sensitivity is assumed by the power law:

\~o)

Consti tutive modeling of polymeric foam material 377

0.30 , , , , , , , , , ,- : T - - -

-+-229101 m s] : / 0.25 • 4.oo lo-3m,s I ........ J

0 2 0

0.15

0.10

0.05

0.00 , ~ , , , , - ~

0.00 0.10 0.20 0.30 0.40

Strain

Fig. 12. Stress strain response of polyurethane foam (density 6.9 k g m / m 3) under uniaxial tensions.

0.12

0.08

0.06 L

0.04

0.02

0.00 0.00

+ 2 . 2 9 10 q rrds [

A 4.00 10 .3 m]s [

---o-- 8.00 10 -5 m/s [

0.05 0.10 0.15 0.20

Shear Strain

Fig. 13. Stress-strain response of polystyrene foam (density 1.6 k g m / m 3) under simple shear.

where a0(e) represents the nominal stress response at a constant quasi-static strain rate; ~,0, and n(~) = a + be is the power coefficient for rate dependency. In this case, ~o is assumed to be the lowest strain rate 1.60 x 10- 3 1/s. a and b are material constants. To obtain material constants, stress cr is plotted against the strain rate ~ on the log-log scale for different strain levels. For selected foams, it is noticed that the data form a family of straight lines with tangent n that is approximately a linear function of strain ~,. Figure 14 illustrated this phenomenon by polypropylene foam (density 4.9 kgm/m3). Using the linear-regression program, n(e.) is fitted as a linear form of e. Table 2 shows the calculated n(~) for selected polymeric foam materials.

2.5. Temperature effect characterization

Experimental results also show that temperature has major effects on the constitutive behavior of the polymeric foam material. Figure 15 shows the stress-strain response of polypropylene foam (4.9 kgm/m 3) at various temperatures under uniaxial compressive strain rate 4.6 1/s. As we can see, polymeric foam becomes softer as temperature rises. This implies that the energy-absorbing capability is reduced as temperature rises. According to WLF equation [13], the following phenomenological constitutive equation is taken to describe the t ime-temperature effects. This allows the modulus in very short-time or extremely long-time tests at one temperature to be obtained at a more reasonable time at

378 J. Zhang et al.

r~

101

J e 5% Strain ~- 50% Strain I i

- - ~ - - 10% Strain - ~ .- 60% Strain |

20% Strain - . v . - 7 0 % Strain | i

~ - - 30% Strain - - v - 80% Strain | - . u . - 40% Strain [ !

10 ° ,,- . . . . . . . . . . --,, . . . . . . . . . . . • ! • - v

f i } _ . . . . . ~- . . . . . . _-121 . . . . . . G

e - - - - - _ - - - - - - . ©

~ . ~ _ . . ~ - - o .

1 0 -1 , , , , , , , , J , , , . . . . . I , , , , , , , , I , , , , , , .

10 -3 10 .2 10 l 100 101

Strain Rate (1/sec)

Fig. 14. Power law strain rate sensitivity of polypropylene foam (density 4.9 kgm/m3).

Table 2. Power law rate sensitivity power coefficient for selected foams

Type of foam n(~)

Polypropylene foam (4.9 kg m/m 3) 0.037 + 0.0136e Polyurethane foam (9.6 kgm/m 3) 0.307 + 0.105% Polystyrene foam (1.6 kgm/m 3) 0.016 + 0.0833e

3.00 ' ' ' r ' ' ' I . . . . . . b '

• -200 C I lftli 2 . 5 0 - - -o - - O ° C ' I . . . . . . . . . . . . . . . . .... l i f t

25 C 2 . 0 0 - - e - - 5 0 o c ................ ............. :

- - - " - - 8 0 ° ~" 1 . 5 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.00 L ............................. ........................................... i . . . . . . . . . . . . . . . . ......

0.50 ..........

0.00 0.0 0.2 0.4 0.6 0.8

Strain (mm/mm)

Fig. 15. Temperature effect on polypropylene foam (density 4.9 kg m/m 3) stress-strain response at 4.6 1/s.

o the r t empera tu re s . Th e t e m p e r a t u r e - d e p e n d e n t f unc t i on L(T) is def ined by the W L F shi f t ing factor

L ( T ) = e x p [ £ 1 ( T - - - ~ - - T r ) . ]

C2+ T - T J ' (2)

where C1 a n d C2 are m a t e r i a l c o n s t a n t s to be d e t e r m i n e d f rom t e m p e r a t u r e effect experi- ments . T , is the reference t e m p e r a t u r e (often selected as p o l y m e r glass t r a n s i t i o n t emper - a tu re Tg). W h e n T equa l s to Tr, L(T) becomes uni ty . T h e n the c o m b i n e d rate a n d t e m p e r a t u r e p h e n o m e n o l o g i c a l cons t i t u t i ve e q u a t i o n becomes

\~o1

Constitutive modeling of polymeric foam material 379

3 . 0 0 -20c ~ "20 C' test I . L KI - - o c o OC, test I @ /~

2 . 5 0 H 25 c ~ 25 c , test | [~iiit / ~

i i i [ ] ~l ....... 50C O 50C test I ~ / 4

2.o0 L I - 0 ,tost I i ..... /

O.O0 ~o, ,: . . . . . . . . . . . . . . i . . . .

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Nominal Strain

Fig. 16. Comparison of model prediction and experimental result of polypropylene foam (density 4.9 kgm/m 3) at strain rate 4.6 l/s.

To examine the validity of Eqn (3), the PP foam (4.9 kg m/m 3) was used as an example. The material parameters C1 and Cz were determined from the quasi-static (1.60 x 10 -3 l/s) curves at - 20°C and 80°C. The calculated values are: C~ = 6.52°C and C2 = 468.T'C. Figure 16 shows the comparison between model predictions and experimental results at a strain rate of 4.6 1/s.

3. CONSTITUTIVE M O D E L I N G

Experimental results show that all polymeric foam materials in this study are all isotropic. In this section, a rate-dependent isotropic foam constitutive model is presented.

3.1. Elastic response

The initial deformation stage of a rigid foam is roughly linear elastic. Using the objective stress rate in a co-rotational reference system, the constitute law is expressed in terms of an objective stress rate 6 v,

6 v =- D i j l m ~ l m , (4a)

where Dij~,, is the elasto-plastic stiffness tensor and ~m is the strain rate tensor. This relation is decomposed into deviatoric and volumetric components:

b~j) = 2G(~a -/,Up) - K(i:,. - ~ ' :vp) I , (4b)

where ~d, ~dp, ~'v, and ~,.p, are the rates of deviatoric strain, plastic deviatoric strain, volumetric strain and plastic volumetric strain tensors, respectively. In this study, for frame-indifference reason, ~v is the Jaumann stress rate tensor defined by (d},

b v = arij + alkCOkj -- COik%, (5)

where d) is the spin tensor, aij is the Cauchy stress rate and a~j is the Cauchy stress.

3.2. Yield locus

For porous material, polymeric foam yield function should include a hydrostatic stress term due to compressibility. Experimental work shows that the initial yield envelope is approximately a quadratic surface elongated along the hydrostatic axis truncated on compression and tension directions in the stress space. This assumption is confirmed by

380 J. Zhang et al.

many other experimental studies [10, 2, 4]. In this study, the yield locus is defined by a single ellipse spanned on the plane of effective stress O-vm and hydrostatic pressure p:

[p x0(e.~p)]2 2 - - O'vm F - F o - 4 1 = 0 , (6)

a(c~.p) b(e,,p)

where Xo(~p), a(e~p) and b(e,.p) are three material parameters that define the center and the lengths of the major and minor axes of the yield ellipse. These parameters are variables which are functions of total plastic volumetric strain t~,,p that describes the so-called consolidation phenomenon. Therefore, the yield ellipse is extendible in the a,m - p stress space as a porous material becomes consolidated under compression. The three consolida- tion variables can be uniquely defined by a combination of any three testing modes. In our study, these tests include (1) uniaxial compression, (2) hydrostatic compression, and (3) simple shear. The initial yield ellipse and evolution of the yield locus of the PP foam (4.9 kgm/m 3, 5% relative density) are plotted in Fig. 17. The variable material constants allow us to incorporate volumetric strain-hardening property as the foam is being compressed.

3.3. Plastic flow law

Low-density foam materials usually exhibit negligible lateral bulging or near zero Poisson's ratio under simple compression. This indicates that foam materials, unlike metals or other J2-type plastic materials, possess both volumetric plastic flow and shear flow when plasticity is initiated. In the present study, the flow rule is defined by a non-associative flow potential in terms of hydrostatic and effective stresses

2 g(~., (~) = j ~ p 2 + O.vm ' (7)

where ~ is a material parameter that controls volumetric plastic flow. ~b is a general internal state variable, which is defined as the plastic volumetric strain in this case. The plastic strain rate tensor obeys the consistency condition

% =/" 0~ ' (8)

1.4 , ! . . . . . . . ! .... ]

1.2- / : Uniaxial Cgmpression

$ i ~ 1 : . . . . . . . -

~ 0'6 ~ T ~ i o n ~ 0.4 e Lrni en~ )

o -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 H y d r o s t a t i c P r e s s u r e P ( M P a )

Fig. 17. Yield locus evolution as polypropylene foam (density 4.9 kg m/m 3) consolidated.

Constitutive modeling of polymeric foam material 381

where ~p is the plastic flow rate tensor and )~ is the plasticity consistency parameter. The plasticity theory requires the stress state to be either inside or on the yield surface and )~ can grow only when the stress state is on the yield surface. This allows us to solve ). by invoking the consistency condition Eqn (8) and the yield criterion Eqn (6).

For uniaxial compression in the z-direction, the plastic Poisson's ratio Vp is defined as

~xxp = i~')'yp = - - Vp~zzp (9)

and the plastic volumetric strain rate is written as

~.p = (1 - 2Vp)~_,zv. (10)

Substituting Eqn (7) into Eqn (8), we have

~,p = A L [- (~O'vm ~P] 29 L2a, ,m~-a + 2 ~ p ~ . (11)

This can be rewritten as

~p=)t 3 I 2~ T- ] 29 s - - 9 P J" (12)

For the uniaxial deformation case, the longitudinal and volumetric plastic strains are found by decomposition of Eqn (12):

~zzp = ). Szz --~-9 p , (13)

~p i:~.p = - z - - . (14)

g

The value of the longitudinal deviator s~z is - 2p. Substituting Eqns (13) and (14) into Eqn (10), we have

9(1 - 2vv) - (15)

2(1 + Vp)

If ~. = 9, there will be no lateral plastic deformation resulting from uniaxial compression. In this case, the measured engineering stress is identical to true stress. To be physically admissible, the values of hydrostatic pressure multiplier should be limited within the range of 0 ~< ~ ~< 9. The upper limit corresponds to zero Poisson's ratio. The lower limit corre- sponds to incompressible J2 plasticity. For crushable foams with zero plastic Poisson's ratio, the flow potential can be written as [14]

2 g(6, c~) = x/gp 2 + a,.m. (16)

It should be noted that plastic flow potential g(&, ~) is also the function of total volumetric plastic strain or relative density of the foam.

3.4. Volumetric hardenin 9 and strain rate dependence

Polymeric foams become consolidated under hydrostatic compression. The volumetric hardening properties can be defined by varying material parameters Xo (gyp), a(evp) and b(~:vp ) to expand the yield envelope with increasing volumetric plastic strain. These three variables are calculated from stress-strain curves from three different testing modes.

382 J. Zhang et al.

The phenomenological model with combined strain-rate hardening and temperature softening effects is given by Eqn (3). In the numerical implementation, by ignoring the temperature factor, the power-law Eqn (3) can be rewritten as

~p = Foo - 1) , (17)

where ~k and q are material parameters for strain rate sensitivity, F and F0 are yield functions under an arbitrary plastic strain rate and a reference quasi-static plastic strain

rate, respectively, and ~p is the effective plastic strain rate defined by ~,p = N~ 3 c'ij c ' i j"

4. NUMERICAL IMPLEMENTATION

The constitutive equation derived in the previous section has been implemented into explicit finite-element program LS-DYNA3D. This program allows the user to add a user material subroutine to define new material constitutive models.

4.1. Elas t i c trial s t ress

First, the decomposed objective elastic trial stresses are calculated as

^trial Sn+l = S,, + 2G~d, ,+I /2At ,

trial P,+ 1 = P, - K ~ , , + 1~2At.

(18)

(19)

4.2. Plas t i c correc t in 9 s t ress

The trial elastic stresses and strains keep increasing until initial yielding occurs. By using the one-step Euler backward return mapping algorithm [15], the plastic consistency parameter is calculated. After each time step, the new yield surface is updated according to the total plastic volumetric strain. Due to the adoption of the non-associative flow law, the return mapping path is not radial. The one-step Euler backward return mapping is of first-order accuracy. For infinitesimal strain increment, the numerical error can be very small [16, 17]. The allowable post-yielding stresses are determined by

O.n+ 1 ^ = an+l~trial -- 2G~dv, n+ l / zA t + Ksvp, n+ l /2At I •

By splitting Eqn (20) into deviatoric and volumetric components, it becomes

~+~ =(1-2G)~--~At)gt f2~

(1 x^trial - - W ) S n + I ,

r l ~ ~ A _ ~ trial Pn+l = 1 -- l ~ / ~ L x ~ ) p n + 1

(1 • 3 K~ \ trial - 2cx AtTd)p.+l

X

f l K~ ~ trial

(20)

(21)

(22)

where the coefficients before the trial stresses are the plastic stress correcting factors. These equations represent a stress projection from the elastic trial stress back onto the yield surface. The only unknown is the plastic flow multiplier ),.

C o n s t i t u t i v e m o d e l i n g of p o l y m e r i c f o a m m a t e r i a l 383

The plastic flow multiplier can be calculated by invoking the consistency condition [Eqn (8)] and the yield criterion [-Eqn (6)]. In this study, the elastic Poisson's ratio is nearly zero according to experimental measurement. For a crushable foam with zero elastic Poisson's ratio, the stress correcting factors for the pressure and deviator are identical when ~ _ _ 92 since

3G 9(1 - 2v) - K - 2(1 + v~' (23)

By inserting Eqns (21) and (22) into the yield criterion (6), we obtain a quadratic equation in terms of plastic stress correcting factor q

where

c l q 2 + c 2 q + C 3 = O, (24)

trial 2 p t r i a l 2 O'vm

cl - + - - , (25) b a

2Xo pt r ia l c2 - - - , (26)

C 3 - - 1. (27) a

Consequently, there are two real roots

- - C 2 -'}- x / C 2 - - 4 C l C 3

q l , 2 = 2cl (28)

The stress correcting factor should be a positive one. In a general case when the material elastic Poisson's ratio is not zero, from Eqns (29), (30)

and (6), two roots wl.2 for the resulted quadratic equation can be obtained. It should be noticed that the introduction of a non-associative flow law may cause the non-uniqueness of the solution. Since the return path is no longer radial, the return path may not intersect the yield ellipse. Therefore, a non-associative flow law does not guarantee a physically admiss- ible solution.

5. MODEL VALIDATION

The constitutive equations derived have been implemented into LS-DYNA3D using a user-defined material subroutine. Numerical simulations are conducted to validate the constitutive model under different loading conditions. The material parameters are calib- rated from the experimental program. The initial yield locus at quasi-static rate can be obtained from (1) uniaxial compressive strength aro, (2) hydrostatic compressive strength Pco and (3) simple shear strength to. For the polypropylene foam (density 4.9 kg m/m3), these parameters are ayo = 0.15 MPa, Pco = 0.074 MPa, and ro = 0.058 MPa. The post-yielding volumetric-hardening behavior is defined by three load curves, Xo(g,,p), a(gvp) and b(g~,p) in LS-DYNA3D. The strain rate dependence is defined by Eqn (17).

Numerical simulations were performed under various loading conditions. Figure 18 shows the comparison of stress-strain responses of the polypropylene foam (4.9 kg m/m 3) under uniaxial compression at three different rates. A deformed configuration without lateral expansion is also shown in the inset of Fig. 18. Figure 19 shows shear stress responses of polystyrene foam (1.6 kgm/m 3) under different rates. The after-yielding behavior is perfectly plastic because there is no volumetric strain hardening during the deformation.

High-speed hemispheric tests have also been performed to validate the foam model under more general loading conditions. In the test, a rigid hemispheric (q5127 mm, 22.2 kg m, steel) impactor is dropped onto a polypropylene foam (density 4 .9kgm/m 3) block

384 J. Zhang et al.

2.00

1 . 5 0

1.oo L r~

• ~ i 0 z 0.50 ~ . . . . . . . .

i

i ! O Numerical Result (1.6E-3 l/see)

Experimental Result (1.6E-3 l/sec) o Numerical Result (8.0E-2 l/sec)

Experimental Result (8.0E-2 l/sec) zx Numerical Result (4.6 l/sec)

Experimental Result (4.6 1/sec)

f i ,

0.00 ~ . . . . 1 0.0 0.2 0.4 0.6 0.8

Nominal Strain

Fig. 18. Model uniaxial compressive responses for polypropylene foam (4.9 kgm/m3).

0.12 . . . .

/ 0.10 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ~ -

, v ~ o 0 , u ] i ~oo

, N " 0.08 ~ n ~ i . . . . . . . . . . . . . . . ; . . . . . . . . . . . .

/# Po ° ..... i w

g 0.06 .................................................................................

/'~c-,~ I o Test Result (0.0016 l/sec, I /v o I : Test Result (0,08 l/sec) I

0 . 0 4 ~o i ,~ Test Result (4.6 ,/s~) I / ~ I - - - Numerical Result (0,00161/see) I

0.02 ~ . . . . [ N u m e r i c a l Result (0.08 l/sec) I o I - - - Numerical Result (4.6 l/sec) I !

o . o o . . . . . . . i . . . . :, . . . . ] , ~ 2 0.00 0.05 0.10 0.15 0.20 0.25

Shear Strain

Fig. 19. Model shear responses of polystyrene foam (1.6 kg m/m3).

Trinse levels 0 . 0 0 0 E + ~ 0

7 . 4 1 7 E + 0 2 1.483E-+93 2.225E+S3 2.967E+03 B.788E+83 4.45BE+83

Fig. 20. Finite element mesh of PP foam (4.9 kgm/m 3) in an indentation test.

Constitutive modeling of polymeric foam material 385

f P i n g e l e v e l s - B . 9 7 1 E + 0 2

4.10BE+02 1 . 2 1 9 E + 0 3 ~ . 0 2 6 E + 0 3 E.SBaE+~3 3 . 6 4 2 E + 0 3 4 . 4 5 8 E + S B

Fig. 21. Deformed configuration of polypropylene foam (4.9 kgm/m 3) and velocity field under indentation at time t = 0.02 s.

16 103

14 103

12 103 z

10 103

8 103

6 103

4 103

2 103

0 10 ° 0 20 40 60 80 1 O0

Penetration (mm)

Fig. 22. Contact force deflection responses of polypropylene foam (4.9 kgm/m 3) in an indentation test.

(203 x 203 x 101 mm 3) with an initial speed of 4.45 m/s (10 mh). Figure 20 shows the finite- element mesh for the indentation test. Figure 21 depicts the deformed configuration of the foam block. The resultant force-deflection curves are plotted in Fig. 22. As we can see, numerical simulation predicts force-deflection response quite closely in the loading stage but not in the unloading phase. As we mentioned before, polypropylene foam (density 4.9 kg m/m 3) is a semi-rigid foam which has only 20% permanent deformation after impact. The excessive residual plastic deformation predicted by the model is mainly due to the assumption that polymeric foam plasticity initiates at the beginning of the stress plateau. According to the preliminary study on foam microscopic deformation mechanism, we understand that the initiation of the stress plateau is actually caused by microscopic elastic buckling of cellular struts. Future investigation should look into polymeric foam cell level and take sub-cellular level deformation mechanism into account.

386 J. Zhang et al.

6. C O N C L U S I O N

Based on previous discussions, we draw the following conclusions:

(1) Polymeric foam const i tut ive behavior is extremely strain rate and tempera ture dependent . Cellular buckl ing under compression initiates a long stress plateau. Fur ther

compress ion causes stress bo t tom up due to foam consol idat ion. (2) Trans i t ions of deformat ion and failure mode are observed as the loading condi t ion

changes. Foams behaving ductile under compression may fail as being britt le materials

under shear and tension. (3) A ra te-dependent elasto-plastic foam const i tut ive model has been developed. A

hydrodynamic const i tut ive model that features a single-surface yield criterion, a non- associated plastic flow law and a power-type s train-rate dependence showed reasonably good predict ion for the responses of rigid polymeric foams under simple and complex loading condit ions.

(4) Besides phenomenologica l approach, future work on foam const i tut ive model ing

should take sub-cellular level deformat ion mechanism into account.

Acknowledgements--This research has been supported by a grant from the American Automobile Manufacturers Association. Special thanks to Livermore Software Technology Corporation for the use of LS-DYNA3D software in numerical investigations presented in this study.

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