Nonlinear compressive deformation of viscoelastic porous materials

Mechanics of Materials 2 (1983) 239-247 North-Holland

239

NONLINEAR C O M P R E S S I V E DEFORMATION OF VISCOELASTIC P O R O U S MATERIALS *

R.M. CHRISTENSEN and W.W. FENG Composites & Polymers Technology,. Lawrence Lwermore National Laboratoo: Livermore. CA .04550. U.S.A. Received 5 July 1983

Two nonlinear viscoelastic solutions are presented for a porous, foam type material. The first is for the hydrostatic compression of the foam, while the second is for the one-dimensional compression of an infinite slab of the material. Relative to the particular viscoelastic constitutive equation employed, the first solution is exacL whereas the second, obtained through the use of a physical hypothesis, is approximate. Both solutions predict the macroscopic, average stress in the foam. required bs' the corresponding deformation processes. The theoretical results are compared with experimental measurements on a foam material at a porosity of 48.5%. The effect of material hysteresis is revealed and discussed.

I. Introduction

The present work is concerned with the prediction of the compressive behavior of polymeric, foam type (porous) materials. The specific objective is to model the hysteretic behavior of such materials, whereby, in a cycle of strain application and removal, there is a net loss of mechanical energy to dissipation. Thus the resulting analysis is necessarily one of viscoelastic type. Furthermore, it is herein requ/red to examine the large deformation of such materials, with the amount of compressive volume change approaching the porosity of the material. It follows that the resulting analysis must be nonlinear, since non-infinitesimal deformations are involved. The past work on the deformation of potuu~ ll~atc,la/~ piin-~ily ha~ been of an elastic type. Several exact, nonlinear solutions are collected and displayed by Truesdell and Noll (1965). Nonlinear viscoelastic solutions for foam materials appear to be quite unusual. The complications are twofold. First, there is no agreement upon which of many di-'ferent types of nonlinear viscoelastic models may be applicable. Second, there is the considerable complications of dealing with the joint effects of time dependency in the material model and nonlinear kinematics. ~,levertheless, both complications will be addressed here, and an exact solution will be obtained. Recent work in the problem of i aterest has been done by Feng and Christensen (1982). Specifically, they solved the problem of the one-d', nensional compression of a slab of elastic porous material for which deformation in the two orthogonal directions is prevented. This difficult problem does not admit an exact solution. An approximate procedure, involving certain assumptions on the kinematics was employed by the present authors (Feng and Christensen, 1982). The final step in that derivation involved the use of a work-energy principle to obtain the macroscopic stress-deformation character of the foam. It is logical to consider the extension of the methodology of Feng and Christensen (1982) to accommodate the viscoelastic effects of interest in the present work. Unfortunately, an extension of the elastic method to the viscoelastic case is not possible, because the work-energy principle does not admit generalization to viscoelasticity. Thus, another method must be found. * This research has been performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under contract No. W-7405-ENG-48. 0167-6636/83/$3.00 © 1983, Elsevier Science Publishers B.V. (North-Holland)

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R.M. Christensen, H~.W. Feng / Viscoelastic porous materials

The approach taken here involves the development of the exact solution for a particular viscoelastic problem. This solution will be of interest in its own right, but, after obtaining it, it will be modified through approximate means to also obtain the solution for the one-dimensional compression of the slab. The problem for which the exact solution will be obtained is that of the spherically symmetric compression of a spherical annulus of incompressible, viscoelastic material. The viscoelastic stress-strain constitutive relation to be employed is that developed by Christensen (1980) for application to elastomeric materials. Another significant restriction concerns the degree of porosity of the foam material. Specifically an intermediate range of porosities will be considered, such that the complications of cell geometry and cell deformation inherent in extremely porous materials are not pertinent.

2. Spl~..rical symmetry problem We assume the porous material to have a geometry type consistent with the composite spheres model of Hashin (1962). With this idealization, the macroscopic stress-strain behavior of the foam can be obtained from the analysis of a single spherical annulus cell, as in Fig. 1. The problem of interest here is the spherical compression of the foam material, and thereby the spherical compression of the annulus in Fig. 1. The material will be taken to be incompressible, and the nonlinear constitutive relation developed by Christensen'(1980) will be employed, i.e.,

O,j(t)f --PS,j+ X,,K(t)Xj,L(t)[goSKa + fo'g,(t--') ~EKL(') aT d Tl,

(1)

where cartesian tensor notation is employed with XK denoting the initial, undeformed configuration, while relates to the deformed configuration. Symbol p is the hydrostatic pressure, go is an elastic constant while gl(t) is the viscoelastic relaxation function with

x,(t)

lim gl (t) - 0.

(2)

t " * OO

E~cL is the strain 2EKt. = X,.jcXi.L-- 8KL.

Deformation measure

given by (3)

Relation (1) was developed for application to elastomers; in fact, for sufficiently slow processes, the integral term in (1) becomes negligibly small and the remaining terms constitute the theory of rubber

Fig. 1. Spherical annulus.

R.M. Christensen, W.W. Feng / Viscoelastw porous materiaL~

241

elasticity (Treloar, 1975). In spherical coordinates, with point symmetry, the Cauchy stress (1) takes the form

o,,=-p+3,2r[go+½gt*(~2,-1)],

oss=o,~ = - p + ~ 2 [ g o + ½ g l * ( X 2 - 1 ) ] ,

(4)

where the stretch ratios are

X, = d r / d R ,

X s = ~,, -- r / R ,

(5)

and the convolution operation is given by 1) = fotg,(t-- ~)

g,,()2_

8(~2(¢) aT

1) dr.

(6)

In this spherically symmetric problem, the material incompressibility condition is satisfied by

R ~ - r3(t) = R~ - r3(l) = R32 - r23(l),

(7)

and the quasi-static equilibrium equation by do,,

2

dr + r ( ° ' r - ° e e ) = °

or

dar,

(S)

The boundary conditions are specified by stresses upon the inner and outer boundaries, as at

r = rI:

or, = 01,

a t r = r2 :

Orr =

°2"

(9)

The substitution of stresses (4) into the equilibrium equation (8) leads to the form --_

R 3

R 4

1

+ 2 R} 23 [ g' o [ ( R ) 4 - ( .- r ~ ) 2 ] + ½ ( R )r4 g , , ( R ) 4 - - ~ ( R ) 2 g l . ( R ) 2

R

(10)

Upon using (7) in (10) we write this simply as

dp/dr = F(R),

(11)

where F ( R ) also has a dependence upon either r, or r 2. Convert the derivative in (11) into one with respect to R, then integrate and evaluate the constant of integration and the deformed state radius r I or r 2 from the boundary conditions (9). The final form is expressed as

°2°.

(R141j ,

,,R2'l(j where the interrelation between rl and r 2 comes from (7) and eliminating r 1 leaves (12) with one unknown variable, r2. If rz = rz(t) is specified, then (12) gives the boundary stress difference directly. Alternatively, if the boundary stresses are specified, then (12) may be inverted to give r2(t), the general state of deformation follows from (7). Our interest will be in specifying the deformation, and then evaluating the boundary stresses from (12).

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R.M. Christensen. W. W. Feng / Viscoekzsticporous materials

As an example we consider a single term relaxation function g,(t)

=

Ce - ' / ' ,

(13)

where r is the relaxation time. it is not necessary to recalculate the entire hereditary integrals in the convolution operations in (12) at each new value of time, t. The following recursion formula can be proved directly, and it greatly facilitates the numerical evaluation of (12). The formula, appropriate to (13), is given by

A~L(t,,+,)=e -.X,/,~,~KL,It,)+Ce-_~,/Z,[Eh.L(t,,

+, ) - EKt.(t,,)]

,

(]4)

where

t,,+l=t,+At and AKt.(t)= gl(t--

f0

~ dr. *) OE"L(r)

(15)

Consider the case of a ramp cycle of deformation shown in Fig. 2. Take the porosity level as 50%, thus ( R t / / R 2 ) 3 = 0.5. Taking the pressure on the inside as zero, then the resulting pressure on the outside of the annulus is as shown in Fig. 2. This stress is thus the macroscopic, average stress in the foam material. The increasing values of C / g o in Fig. 2 correspond to an increasing viscoelastic effect in relation to purely rubber elasticity type behavior. Cross plotting stress vs. deformation gives the results shown in Fig. 3. At a fixed value of the material constant, as C / g o = 1.0, the effect of porosity is as shown in Fig. 4. As can be seen from Fig. 3, the size of the hysteresis loop has an obvious relation to the viscoelastic constant, C. The

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Fig. 2. Ramp half cycle and stress response, porosity = 50%. Fig. 3. Stress vs. deformation for properties variation, porosity = 50%.

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R.M. Christensen. W. IV. Feng / Viscoelastic porous materials I

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Fig. 4. Stress vs. deformation for porosity variation, C/g o = 1.0.

Fig. 5. Infinite slab and spherical cell.

time duration of the ramp in Fig. 2 is selected to give dissipative effects by "tuning' to the relaxation time of the material. This point will be discussed further in the next section. Of course, ramp cycles could be taken to be very slow, for a fixed relaxation time, in which case the response would be entirely elastic. The porosity variation, as shown in Fig. 4, has a profound effect upon the pressure required to produce the deformation.

3. One-dimensional compression problem Next we consider the case of the compressive deformation of an infinite slab of the foam material, as shown in Fig. 5. Macroscopically, the deformation is of a uniform, one-dimensional nature. However, on the scale of the single spherical cell, as in Fig. 1, the resulting deformation would be extremely complex, possessing only axial, rather than spherical symmetry. This complex problem, in the elastic case, was solved by a rather involved approximate means by Feng and Christensen (1982). As discussed in S~,ction 1, that elastic method does not admit generalization to viscoelasticity. Accordingly we must find a new method to analyze this problem in the viscoelastic case. We begin with a hypothesis, as follows. We assume that at equal volume change in the spherical deformation problem and in the one-dimensional deformation problem, the work done is the same for both problems. First we will examine the c(msequence of the work equivalency hypothesis insofar as macroscopic stress is concerned, thereafter we will test the hypothesis in the case of infinitesimal deformation. First relate volume change in the two problems through

( H - h ) / H = ( B 3 - b 3 ) / B 3,

(16)

where now B = R 2 and b = r 2 in the previous notation (see Fig. 5). The work requirement is expressed as (17)

WIsLAB--__ WIsPHERE"

The increment of work per unit initial volume, for the two problems is given by dWIsLA n ----O d ( H - h ) / H

and

dWIsPnERE ----(3b2/B3)p d ( B - b).

(18)

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R.M. Christensen, W. W. Feng / Viscoelastic porous materials

Equating the two forms in (18) and using (16) gives the simple result o =p.

(19)

Thus, at equal volume change, and a hypothesis of equivalent work done, the macroscopic, average stresses in the two problems are equal. We will use this hypothesis to obtain the solution to the one-dimensional deformation problem, from the solution to the spherical problem of Section 2. Before proceeding with the general nonlinear problem we examine the problem in the infinitesimal case. The work determining stress component in one-dimensional deformation is specified by o = (x +

(20)

where Av is the volume change and X and ~ are the Lam6 constants. Correspondingly, the stress due t," spherical compression is given by

p = kAc

(21)

where again Av is the volume change and k the bulk modulus. Now, from the elastic relations it follows that + 2ja = k + ~t.

(22)

Thus we see that the additional factor of ~/~ in (22) means that the stress in the spherical compression problem is a lower bound to the stress in the one-dimensional problem at equal volume change. The following question remains: Is the lower bound close to or far from the actual result? We can gain information on that problem by appeal to some linear theory composite material results. In the case of a porous medium governed by the composite spheres model, and s, = J2, the results of Hashin (1962) and Hashin and Shtrikman (1963) give the rigorous forms

k/l~m = 4(1 - c)/(3c),

~t/~tm ~ 3(1 - c ) / ( 3 + 2c),

(23)

where k and p are the effective properties of the medium, and/t m is the shear modulus of the constituent material, with c being the volume fraction of the pores. Use relations (23) and the standard elasticity relation h = k - -~ to form

(h+ 2~t)/k ~ (3 + 5c)/(3 + 2c),

(24)

For c ffi 0 the ratio in (24) is 1, thus the stress in spherical and the one-dimensional problem are identical. At the practical case of 50% porosity, (24) gives

(~ + 2tz)/kl~.l/2 ~ ~. Thus, the stress in the spherical problem would underestimate the one-dimensional problem stress by somewhat less than 27%. As the porosity decreases, the error becomes more tightly bounded, i.e., smaller. These results only apply to the infinitesimal case; however, we can give some corresponding results in the nonlinear case. Continuing the restriction to elasticity, it is simple to establish that, at equal volume change, the average stress in the spherical problem is a lower bound to the stress in the one-dimensional problem. Furthermore, as the compression continues, the infinitesimal results suggests that the difference between the two solutions diminishes. This is physically reasonable since the resistance to pore collapse is inherently a volumetric effect. With this guidance and bounds reasoning in the elastic case, we proceed to the viscoelastic case and use the equivalent volume change criterion to interrelate the stresses in the two problems. Using (16), in the notation of the previous section, we have

h/H

=

(r21R 2)3.

(25)

R.M. Christensen, W. W. Feng / Viscoelastic porous materials

245

Specifying the amount of compression of the slab through h/H, then solving (25) for r2/R 2, and substituting into (12), while using (7), gives the average pressure solution p = - 0 2 , with o I = 0. The complete solution for the average, macroscopic stress in the one-dimensional problem is now obtained. One last matter remains before considering some experimental results. Further work with the solution will be restricted to the case of the relaxation function characterized by a single exponential. Of necessity then, the relaxation time should be selected to be in the proper range relative to the excitation. We again consider the ramp deformation program shown in Fig. 2. Obviously the characteristic time of the program will excite only a portion of the relaxation spectrum of the material. We seek to determine the excited part of the relaxation spectrum. Take t o to be the duration of the half cycle program of Fig. 2. We use the present viscoelasticity theory to evaluate the relaxation time ~- that 'tunes' most closely to the excitation. The uniaxial form of (1), with stress-free lateral surfaces, is given by

ot,=go(X2-~)-~fo'g,(t-T)A(~ldT+½X2fo'gi(t-T)A(A2)

dT.

(26)

Let h = 1 + ¢(t) and retain terms up to order ¢2 in (26) to obtain p,,

=

3go,(t)[1

a

-

t

¢(t)]+2fog,(t-

defT)

dr,

(27)

where P n is the stress per unit initial area. For the ramp program let the strain c be given by , ( t ) = [~t

for O~ t <~~t o,

t ~(t o -

t)

(28)

for ~t o ~< t ~< t o

which when combined with (27) gives Pl~ = 3gott(1 - i t ) + ~tCr(1 - e - ' / ' )

for0~
(29)

and PH = 3 g o ¢ ( t o - t)[1 - - ~ ( t o -- t)] - -~Cri[l + e - t / ' -

2e -"-','/2'/'1

for ½to~< t ~< t o .

Next calculate the dissipation over the cycle through the work expression fO

°

W = fo Pile(t)dt.

(30)

Substituting (28) and (29) into (30) gives

W=

~C¢i2 [ t o - ~ ' ( 3 - 4e -'°A2"' + e-'"/')].

(31)

This result is similar to that given by Yang and Chen (1982). N o w take the derivative of (31), to maximize the dissipation, as

a w / a ~ = O.

(32)

Operation (32) gives (to - 6¢) + 2(t 0 + 4~')e - ' ° / u ' ~ - (to + 2¢)e -'°/" = 0.

(33)

The solution of this equation is very close to 'r ~ -~to

(34)

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R.M. Christensen, W. IT'. Feng / Viscoelastic porous materials

since, at this value, the last two exponential terms in (33) are very small. Thus, in the comparison with experimental data to follow, the relaxation time will be taken to be ~ me duration of the half cycle ramp. Relaxation times other than this value will contribute much less to the overall dissipation, and hysteretic behavior.

4. Experimental results The program of deformation remains the half cycle ramp shown in Fig. 2. The relaxation time is taken as ~"= ~to where t o is the duration of the straining program, The material constants go and C are evaluated from the compressive ramp deformation of a small sample of solid silicone rubber. The values of go and C give a fit of the form shown in Fig. 6. The theoretical result is from (26), with the relaxation function (13). The experimental data in Fig. 6 show a considerable hysteresis effect. The duration of the ramp deformation is 1 sec. Next the same elastomer, in foam form was tested. The material has a porosity of 48.5%. The experimental data are obtained from the compression of a very thin porous sheet. The theoretical prediction is obtained by the method of the previous section, explicitly through (25), (12) and (7). The comparison between the data and the theoretical prediction is shown in Fig. 7. It may be noted that the theoretical results predict a larger hysteresis effect than is actually observed. While this difference is difficult to explain, it does demonstrate that the entire energy dissipation effect reflects an inherent material characteristic, rather than coming from extraneous sources such as friction in the apparatus.

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0.02

0.04

0.06

0.08

0.10

Displacement ~ = H - h (cm) Fig. 7. O n e - d i m e n s i o n a l c o m p r e s s i o n o f f o a m m a t e r i a l , - t h e o r e t i c a l , H = 0 . 0 0 3 m.

R.M. Christensen. W.W. Feng / V£~coelusticporous materials

References Christensen, R.M. (1980). "A nonlinear theory of viscoelasticity for application to elastomers", J. Appl. Mech. 47, 762. Feng, W.W. and R.M. Christensen (1982), "Nonlinear deformation of elastomeric foams", lnternat. J. Nonlinear Mech. i 7, 355. Hashin, Z. (1962), "'The elastic moduli of heterogeneous materials", J. Appl. Mech. 29, 143. Hashin, Z. and S. Shtrikman (1963), "A variational approach

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to the theory of the e]astic behavior of multiphase materials". J. Mech. Phys. $o/ids 11. 127. Treloar. L.R.G. (1975), The Physics of Rubber Elasticity. 3rd ed.. Clarendon Press. Oxford. Truesdell, C. and W. Noll (1965). "Non-linear field theories of mechanics", in: S. Flugge. ed., Handbuch der Physik. Springer, Berlin. Yang, T.Q. and Y. Chen (1982}. "'Stress response and energy di,~,;ipation in a linear viscoelastic material under periodic triangular strain loading", J. Pol.vmer Sci. 20. 1437.