STRESS RELAXATION IN SOLIDS

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STRESS RELAXATION IN SOLIDS

J. Kubát, M. Rigdahl

To cite this version:

J. Kubát, M. Rigdahl. STRESS RELAXATION IN SOLIDS. Journal de Physique Colloques, 1987,

48 (C8), pp.C8-3-C8-13. �10.1051/jphyscol:1987801�. �jpa-00227105�

JOURNAL DE PHYSIQUE

C o l l o q u e C8, s u p p l 6 m e n t a u n 0 1 2 , Tome 48, decembre 1987

STRESS RELAXATION IN SOLIDS

J. KUB~T' and M. RIGDAHL'

. "

' ~ h a l m e r s University of Technology, Department of Polymeric Materials, S-412 96 Gothenburg, Sweden

t *

Swedish Pulp and Paper Research Institute, Paper Technology Department, Box 5604, S-114 86 Stockholm, Sweden

Abstract

A cooperative model for stress relaxation is described. It is based on a two-level system where stimulated emission of phonons may occur and was formulated in order to circumvent some of the problems encountered with the theory of stress- -dependent thermal activation. The cooperative model also provides some insight into the well-documented universal similarity in s t r e s s relaxation behaviour among different materials. The dynamic-mechanical response of the cooperative model is also analyzed in some detail. The experimental s t r e s s relaxation behaviour of composites based on high density polyethylene is described and it is suggested that the internal .stress concept can be a valuable tool when trying to quantify the effect of fillers and surface treatments of fillers on the long-term mechanical p r o ~ e r t i e s of polymeric composites.

Introduction

This paper is a brief account of a cooperative model proposed by the authors to describe the features of the stress relaxation process in solids [ I ] . In the first place, the model was formulated in order to circumvent the problems encountered when using the commonly accepted theory of stress-dependent thermal activation

(SDTA) [2-41. The basic formula of the SDTA-concept is a s follows

Here

A

denotes the time derivative of the s t r e s s , u, v the activation volume, k the Boltzmann constant, and T the absolute temperature. Equation (1) repre- sents a unique relation between

A

and o ; its inability to distinguish between the role played by o when the initial s t r e s s of the relaxation experiment is changed, or when it simply decreases with time, has been the source of some confusion

[5,6]. The main reason for this is an observed approximately Linear scaling

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987801

C8-4 JOURNAL DE PHYSIQUE

of the relaxation curves with regard to the initial s t r e s s , a t least a t low stress levels. Such behaviour was also the basis of the theory of linear viscoelasticity (LVE). On the other hand, when the ;(a)-behaviour during a single relaxation experiment is considered, eq. (1) provides a correct description of the logarithmic time law, i.e. o

-

log t , normally observed in relaxing solids.

The theory of relaxation time spectra, RTS, on which LVE is based, is a purely formal instrument which does not yield any insight into the underlying physical mechanisms.

The cooperative model

Several cooperative models have been proposed in the past [7-101. Basically, many of them lead to formulae similar to eq. (1), which implies a restricted flexi- bility in taking account of the double role played by the s t r e s s , described above (i.e. the linear effect of oo in experiments with varying oo-values, and

A -

exp (vo/kT) during each experiment).

The cooperative model proposed by the authors does not suffer from the above shortcomings. Its structure is very simple, the basic assumption being that a spontaneous transition of a flow unit to a lower energy ( s t r e s s ) level may induce any number of transitions of other flow units. The relaxation implies that the upper energy level is successively depleted with regard to the unrelaxed flow units. The model is thus of the two-level type. Physically, the induction mechan- ism can be understood, a t least qualitatively, by the stimulated emission effect characteristic of the phonons emitted during each transition.

Since the induction mechanism results in, not only single transitions, but also double, triple etc. transitions, we obtain a spectrum of relaxation times of the type

Here, T denotes the relaxation time of the single transitions, .r/2 of the double etc. Obviously, a system relaxing in such a way that every transition encompasses two flow units will be characterized by a relaxation time equal to ~ 1 2 . For transi- tions occurring in clusters of size s the relaxation time will be T / S . In expression ( 2 ) , s m denotes the maximum cluster size.

The above behaviour can be formalized by the following set of equations

relating the transition rate of single, double etc. transitions,

",

to the corre- sponding numbers of flow units in the upper level, ns, which in the course of time will fall to the lower level a s clusters of varying size. Equations (3) thus correspond to a rather simple form of RTS, expressing the simple fact that larger clusters will deplete the upper level a t a higher rate. For simplicity, we have re- placed in eqs. ( 3 ) the stress by the corresponding numbers of flow units, assuming that every unit in the upper level makes the same contribution to the stress.

In eqs.(3) the contribution of each cluster mechanism is not defined. In order to determine the shape of the spectrum, we assume that the phonons bringing about the cooperation effect a r e clustered in the same way a s the transitions they induce.

The clustering of phonons can be treated in terms of the theory of unrestricted partitions, due to its equivalence with B-E-statistics. As shown in [ I ] , the expec- tance of a cluster of size s in an unrestricted partition of the number N is

1

with B' = ( 6 ~ / n ' ) ' , a result which also follows from the Hardy-Ramanujan formula [ I l l . Obviously, the quantity to be partitioned in the present case is

-

ati, where a is a constant converting the total flow rate a t a given time to a pure number; a thus has the dimension of time at the same time a s it defines the conerence period during which the partitioning process takes place. To calculate the relaxation kinetics we thus rewrite e q . ( 4 ) to apply to

-

^ai, giving

with p = (-6aA/n2)$. The quantity - a ';ls is the number of clusters of size s falling down to the lower level within the coherence period a. The cluster numbers are converted to the corresponding flow rate by

J O U R N A L D E P H Y S I Q U E

yielding for the contribution of the s-clusters to the flow rate

An interesting property of this expressions is that for s < 6

which implies that the contribution of different cluster sizes to the total flow rate is approximately constant, a t least when s does not exceed 8.

It is then a straightforward matter to calculate the n(t)-dependence, i.e. the shape of the actual relaxation curve. For t = 0 one obtains

with y denoting Eulers constant, 0.5772.

. . .

For the time variation of n (or s t r e s s ) the above formulae give

a k n z @ { $ [ @ ( I + k t ) ]

-

$(1

+

@ kt)}. (10)

Here JI is the psi(digamma) function; its close similarity to In t shows that the cooperative model reproduces the o (log t ) normally observed in experiments.

It can further be shown [12] that the flow rate corresponding to the cooperative model can also be written a s

which is mother demonstration of the similarity to B- E-statistics

.

Similarity of the stress relaxation process in varlous solids

It has been shown in numerous publications earlier, cf. e . g . [5, 131 that the maximum slope F of the o(ln t ) curves obeys the relation

with o

*

denoting the initial effective stress. This latter quantity, being equal

0

to the difference between the applied and the internal s t r e s s , is crucial in the present context. Normally, little attention is paid to i t , since its determination

implies measuring the relaxation process until a state of equilibrium is reached (internal s t r e s s equals the equilibrium s t r e s s attained after a sufficiently long period of time). Equation (12) has been found valid both for polycrystalline and single crystal metal solids, a s well a s amorphous and semicrystalline polymers. To these can be added other types of solids, such a s ionic crystals, alloys etc.

There is. today no theoretical approach to the stress relaxation process explaining the similarity between various solids as expressed by eq.(12). However, the co- operative model appears to show some promise in this respect. A s shown in ref.

[14], the value of n / F which according to eq.(12) should be 10, was reproduced accurately on the basis of the cooperative model. The method used was to compare second derivatives of n with regard to t obtained from -the distribution ( 7 ) and from the approximation ( 8 ) , the maximum cluster size being identified with p.

At present, further work is in progress to investigate the properties of the co- operative model with special regard to the similarity expressed by eq.(12).

Recalling e q . ( l ) , the basic equation of the SDTA-concept, it may be remarked that the application of the similarity (12) to this relation produces the improbable re- sult

according to which the activation volume would be independent of the structural details of the relaxing solid [5].

Dynamic response of the cooperative model

The dynamic response of the cooperative model can be evaluated in a rather straightforward manner. The discrete distribution of relaxation times H ( r ) corre- sponding to eq.(7) is given b y , cf. [12, 157

with .rs given by e q . ( 2 ) with s varying from 1 to s m . Assuming that the concepts of linear viscoelasticity a r e applicable the storage and loss moduli E1(w) and E"(w), can be evaluated from

JOURNAL DE _PHYSIQUE

and

where w is the angular frequency. This evaluation can be performed numerically but useful analytical approximations of E 9 ( w ) and Elf( U ) are derived in ref. [15].

It should be understood that the values of E7(w) and EI7(w) calculated here are not the measurable moduli but quantities that should vary with the frequency in the same way a s the measured values.

Figures 1 and 2 show the calculated values of E7(w) and E"(w) as functions of the frequency w

.

The parameters have here been chosen to comply with eq. ( 1 2 ) .

C V)

.-

_C

3

.- !?

.-

([I

-

g 10

.

-4 -2 0 2

Log frequency s-I

Figure 1. E f ( w ) as a function of the frequency w according to e q s . ( l 4 ) and (151.

Log frequency s-I

Figure 2 . E f t ( W ) as a function of the frequency w according to eqs. (14) and ( 1 6 ) .

The maximum slope of the E'(1n w)-curve is here given by a similar numerical relation to eq. ( 1 2 ) . Thus according to the cooperative model there also exists a counterpart to eq. (12) in the frequency dependence of the dynamic-mechanical properties.

The prediction of the mechanical loss factor from the cooperative model may now easily be obtained from the definition tan6 = E"(w)/E1(w). In ref. [15] it is shown that agreement between the experimental values of tan6 and those calculated from the cooperative model is rather good in the case of high density polyethylene ( H D P E ) a t room temperature.

Stress relaxation in polymeric composites

'Che internal stress level oi, i.e. the s t r e s s level approached in a stress relaxa- tion experiment after very long times, is an important parameter since it has a strong influence on the shape of the relaxation curves. It also corresponds to the load-bearing capability of the polymer after very long times, and in a way the 0.-concept can be used to characterize the long-term properties of polymers and

1

other materials. Today polymeric composites play an increasingly important role a s engineering materials and it is thus important to investigate how the incorpora-

JOURNAL DE PHYSIQUE

tion of fillers and reinforcing elements affects the viscoelastic properties. The ui- -concept can be a useful tool in such an analysis.

Figure 3 shows how the stress relaxation behaviour of HDPE is affected when 15 and 3 0 % ( b y volume) of clay is added to the polymer. The stress level approaches a higher value a t longer times, i.e. the internal stress increases, a s the filler con- tent increases. The general relation e q . ( 1 2 ) appears however not to be affected by the filler addition [ 1 6 ] .

..

_*.. ^{1 I I I I}

0 HDPE

A H D P E * 1 5 % c l a y

V HDPE + 3 0 % c l a y

. .

-

^'e

-

-

^{I I I} I ^I I

log

t

s

Figure 3 . Stress relaxation curves for HDPE filled with 15 and 30 % ( b y volume) of clay. The initial strain was approximately 0 . 6 %.

In the absence of residual stresses, the internal stresses discussed here are de- formation-induced, i.e. they a r e related to the initial deformation ^E~ of the speci- men when performing the relaxation experiment [13]. Figure 4 shows the rela- tion between oi and G~ for HDPE and HDPE filled with 15 % glass fibres. Evidently the glass fibre reinforcement produces a higher oi-value for a given elongation.

Surface treatment of the glass fibres with silanes yielded a further increase in internal stresses in this case. From the linear part of these curves a modulus may be defined a s

1 I

0 HDPE

o HDPE + 1 5 % G F

-

HDPE + 1 5 % G F , s i l a n e t r .

- -

-

_m

-

- -

eon

-

0

- -

Figure 4 . The internal s t r e s s ai vs. the initial strain for HDPE and HDPE filled with 15 % untreated and silane surface-treated glass fibres.

If the increase in the modulus E. due to the filler incorporation is the same as

1

the corresponding increase in the Young's modulus, the filler can be regarded a s inert. This is however not observed in many cases [16, 171. It is not unusual that the modulus Ei increases faster with the filler content than E . This result may be due to the formation of an interphase region in the matrix close to the filler surface with properties different from those of the bulk matrix, cf. refs.

[18-201. A reduction of the mobility of the molecules close to the filler surface

may result from this giving a higher o -value and also a corresponding reduction i

in creep r a t e [16, 171.

The effect of the filler on the long-term mechanical properties (stress relaxation) may be characterized by the ratio [ 1 7 ]

where the subscript o refers to the corresponding moduli for unfilled specimens.

It should be mentioned that the oi-values used here refer to infinite relaxation time and were obtained by an extrapolation procedure suggested b y Li [ 2 1 ] . When

JOURNAL DE PHYSIQUE

r

> 1 , this is interpreted as being the result of interactions occurring a t the filler/

/polymer interface, e.g. the formation of an interphase region. It is also to be expected that surface treatments of the filler can affect the r-value. Table 1 gives r-values for HDPE filled with a number of different fillers [17].

Table 1. The ratio

r

and the apparent thickness AR of an interphase region uni- formly covering the filler particles for a series of HDPE-composites [17].

Incorporated fillerladditive

r

AR, nm

Unfilled HDPE CaC03 15 % HDPE with wax

CaC03 15 % (HDPE with wax) Clay 15 %

Clay 15 % (silane treated) Clay 30 %

Clay 30 % (silane treated) Glass spheres 2 0 %

Glass spheres 20 % (silane treated)

A s can be seen, the r-value is in many cases greater than one. The silane treatment of the glass spheres was very effective in increasing the r-value (last two rows in Table 1 ) . In this case a n azide functional alkoxysilane, which may be able to bond with the polyethylene chain [ 2 2 ] was used. This silane was not used with the clay filler. Table 1 also gives values of the estimated (from s t r e s s relaxation data) thickness of an interphase region uniformly covering the filler particles.

In conclusion it may be said that the r-vaue ( o r the oi-concept) may prove to be a valuable tool when trying to assess the effect of fillers and different s u r - face treatments on the long-term mechanical properties of composites. More work is however required to substantiate this important point further.

Acknowledgements

The authors thank Drs. Ch. Hogfors and C. -G. Ek for many valuable discussions.

Thanks a r e also due to the Swedish Board for Technical Development for financial support.

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107,

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