Polymer Mechanics, Deformations, and Failure of XLPE · Polymer Mechanics, Deformations, and...

Polymer Mechanics, Deformations, and Failure of XLPE

Jorgen [email protected]

Abstract

XLPE is an important material for many high voltage applications where its excelent electrical prop-erties and low cost make it ideal for use as the insulating material. When used in these industrialapplications the material is subjected to both electrical and mechanical loads, and although the pri-mary objective of the XLPE is to withstand high electrical loads, it is also important that the me-chanical response of the material is sufficiently good. Common problems in these applications include:low stiffness at high temperatures, deformational changes due to creep, residual stresses causing aninhomogeneous material response, growth of damage in the form of voids and crazes due to applied de-formations (such as cable bends), etc. The objective of this report is both to review different problemsthat can occur and also discuss techniques that can be used to predict the behavior of XLPE whensubjected to mechanical loads and deformations.

c©Jorgen S. Bergstrom, [email protected]

1

Contents

1 INTRODUCTION 3

2 DEFORMATIONAL BEHAVIOR 3

2.1 HYPERELASTICITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 LINEAR VISCOELASTICITY THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 NON-LINEAR VISCOELASTICITY THEORY . . . . . . . . . . . . . . . . . . . . . . . 142.4 LARGE STRAIN STATE VARIABLE BASED MODELS . . . . . . . . . . . . . . . . . 15

3 INFLUENCE OF PHYSICAL AGING ON DEFORMATIONAL BEHAVIOR 16

3.1 AMORPHOUS POLYMERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 SEMICRYSTALLINE POLYMERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 DEFECT GENERATION DUE TO MECHANICAL LOADS AND DEFORMA-

TIONS 20

4.1 CRAZING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 VOID GROWTH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5 MACROSCOPIC FAILURE DUE TO MECHANICAL LOADS AND DEFORMA-

TIONS 24

5.1 BRITTLE FAILURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.2 DUCTILE FAILURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.2.1 Tearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.3 FATIGUE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 PHYSICAL PROPERTIES 28

A Laplace Transformation 31


2

1 INTRODUCTION

XLPE (cross-linked low density polyethylene) is used as the insulating material in many cutting edgehigh-voltage applications and the success of these applications hinges on a good understanding of boththe electrical and mechanical properties of the insulator. Despite the ubiquitous use of XLPE as aninsulating material, the understanding of the material properties and how it can be modeled is a fieldthat is far from being completely developed or even well understood. The goal of this report is topresent the results from a literature study that focuses on the mechanical behavior of XLPE underdifferent loading conditions. The emphasis in the presentation has been on the operational details ofdifferent modeling techniques thereby allowing for a direct comparison between the different modelsand also to enable the selection of the most appropriate model for a given problem statement.

The results from a literature search using the INSPEC and Compendix databases for the years 1970-1999 gave the results shown in Table 1. From the table it is clear that a substantial amount of articles

Key word Number of referencesPEX 91XLPE 2995

crosslinked polyethylene 1110cross-linked polyethylene 1296high voltage polyethylene 15

mechanic? 576933

Table 1: Results from a literature search.

have referred to the material XLPE, but when looking in more detail a surprisingly small amount ofexperimental data for the deformational behavior of XLPE have been presented in the literature.

The report is structured as follows, first will general aspects of the deformational behavior bepresented in Section 2. The change in material behavior due to physical aging is the topic of Section 3,and defect generation due to mechanical loads is discussed in Section 4. Different macroscopic failuremodels are discussed in Section 5, and the report is ended in Section 6 with a list of physical propertiesof XLPE.

2 DEFORMATIONAL BEHAVIOR

The discussion of the deformational behavior will be divided into a few subsections: focusing bothon the experimentally observed mechanical behavior and the different proposed theoretical models.The aim of these sections is to present/summarize a number of constitutive theories that have beenproposed in the literature and that seem to be useful to predict the deformational behavior of XLPE.It is interesting to note that when going through the literature it appears that no specific model forXLPE exists, and in fact, XLPE has not been one of the model materials to which the proposed modelis compared. Despite this there are still a relatively large number of modeling approaches that canbe applicable and the goal here is to present/summarize some of the more interesting models. Beforegoing through the details of the different techniques it is beneficial to point out that many of the


3

models have different domains of applicability: some work best at small deformations and others atlarge deformations; for some models it is an elaborate task to map out material functionals, etc.

One complicating effect that needs to be considered when studying the mechanical behavior of apolymer is the influence of eventual residual stresses. The residual stresses in a material can be definedas a stress system satisfying internal equilibrium, with no external loads or temperature gradients.Residual stresses may be classified into two broad and somewhat overlapping groups. The macroscopicresidual stresses are of long range type, extending over a macroscopic dimension of the part. Themicroscopic residual stresses are of short range and are usually confined to parts of a material wherea homogeneous view of strain is no longer possible. In most of the following, the residual stresses areassumed to either be zero or at least being well quantified.

2.1 HYPERELASTICITY

When modeling the mechanical behavior of a crosslinked polymer (such as XLPE) above its glasstransition temperature, the first model to consider should probably be a hyperelasticity based model.These models are both very simple to use and also to find the material constants for. The characteristicfeature of a hyperelasticity based model is that the strain energy density of the material is a functiononly of the current deformation state. Quite a number of model of this type have been developedthroughout the years (Gent [1996], Arruda and Boyce [1993], Wu and van der Giessen [1993], Ogden[1972], Mooney [1940], Yeoh [1993], etc.). Most of these models are already implemented in manycommercial FEM packages such as ABAQUS, and can be readily used. Which model one decidesto use is to some extent a matter of personal preference, it should be noted, however, that if thedeformations are only moderately large many models coincide with the simple neo-Hookean, which cantherefore be considered a good choice in these applications. Since there are no complicated or difficultaspects of hyperelastic models, they will not be reviewed here.

One serious limitation of this class of models is that they are based on non-linear elasticity, to predicttime-dependence or plasticity it is necessary to consider a more complicated model; one possible choiceis linear viscoelasticity, which is the topic of the next section.

2.2 LINEAR VISCOELASTICITY THEORY

When subjected to an external loading agent many materials do not exhibit a purely elastic or purelyviscous response, but a combination of the two. The simplest way to model this behavior is throughlinear viscoelasticity. This material representation has been used and studied extensively for manyyears, but despite this there are some aspects of the theory that can be difficult to grasp at first, andalso since other more complicated models often are generalizations of this theory, a relatively detailedderivation is given below.

The foundation of linear viscoelasticity theory is the Boltzmann [1876] superposition principle whichis discussed next.

Boltzmann’s Superposition Principle.One way to state the Boltzmann’s superposition principle is through the statement:


4

Each loading step makes an independent contribution to the final state.

This idea can be used to formulate an integral representation of linear viscoelasticity. The strategy isto perform an experiment in which a step function in strain is applied, ε(t) = ε0H(t), and the stressresponse σ(t) is measured. Then a stress relaxation modulus can be defined by E(t) = σ(t)/ε0. Notethat ε0 does not have to be infinitesimal due to the assumed superposition principle.

To develop a model capable of predicting the stress response due to an arbitrary strain history,start by decomposing the strain history into a sum of infinitesimal strain steps:

ε(t) =∑

i

∆εiH(t− τi) (1)

the stress response can therefore directly be written

σ(t) =∑

i

∆εiE(t− τi). (2)

In the limit as the number of strain increments goes to infinity, the stress response (2) becomes

σ(t) =∫ t

−∞E(t− τ)dε(t) =

∫ t

−∞E(t− τ)

dε(τ)dτ

dτ. (3)

Note that if E(t) = E0, then σ(t) = E0ε(t). If only the response in one loading mode (i.e. uniaxialloading) is of interest, then it is sufficient to determine the stress relaxation modulus for that loadingmode and then by using Equation (3) the response due to any imposed deformation can be obtained.

If Equation (3) is generalized to a three-dimensional deformation state for an isotropic material,the following relationship is obtained:

T(t) =∫ t

02µ(t− τ)edτ + I

∫ t

0K(t− τ)φdτ (4)

where µ(t) is the stress relaxation shear modulus, e the rate of change of deviatoric strains, K(t) thestress relaxational bulk modulus, and φ the rate of change of volumetric strains. That is, only tworelaxation moduli need to be determined to predict any arbitrary deformation. The correspondingrelationship for a general anisotropic material is

σij(t) = Eijkl(0)εkl(t) +

t∫0

εkl(t− s)dEijkl(s)

dsds. (5)

Note, to be able to predict the stress response at any arbitrary strain history it is sufficient to knowthe stress relaxation modulus, and to be able to follow a stress history the creep compliance needs tobe known.

Example 1. Consider once again a general one-dimensional loading case. To exemplify the model


5

consider first a case in which

E(t) =

E0 exp(−t/τ0) if t ≥ 0,

0 if t < 0(6)

and

ε(t) =

0 if t < 0,

ε0t if t ≥ 0,(7)

see Figure 1. Equation (3) then gives

Figure 1: Relaxation modulus as a function of time.

σ(t) =∫ t

0E0ε exp

[− t− τ

τ0

]dτ,

yielding

σ(t) = E0ετ0

[1− exp

(−t

τ0

)], or σ(t) = E0ε0τ0

[1− exp

(−ε

ετ0

)]. (8)

The result is plotted in Figure 2, illustrating that small τ0 corresponds to fast relaxation. From (8) it isalso clear that at a fixed time, the stress response is always predicted to be proportional to the appliedstrain rate; but at a fixed strain, the stress is not a linear function of the applied strain. Furthermore,by introducing ε ≡ ετ0 it is clear that for ε � ε the stress σ → E0ε = E0ετ0. Also, when the appliedstrain is equal to ε the stress will have reached 63% of its final value.

Finally, even if a relaxation spectrum is chosen, the Boltzmann’s superposition principle does notallow for a sigmoidal shaped stress-strain response. �


6

Figure 2: Stress response.

Another commonly used functional form of the relaxation modulus is the stretched exponential1:

E(t) = E0 exp

[−(

t

τ0

)β]

(9)

which is plotted in Figure 3. With this representation, the relaxation modulus decreases down to 36%of its original value at the time t = τ0, and β specified the shape of the relaxation function. If we again

Figure 3: Stretched exponential stress relaxation modulus as a function of time.

consider a constant strain rate loading situation, the stress response becomes

σ(t) =∫ t

0E0ε exp

[−(

t

τ0

)β]

dt

1The corresponding form of the creep compliance is J(t) = J0 exp(t/τ0)γ


7

which is not easy to express in elementary functions but can be solved numerically, see Figure 4.

Figure 4: Stress response using the stretched exponential stress relaxation modulus, τ0 = 1 s.

Example 2. As a last example consider

G(t) =

G0 exp(−α0t) if t ≥ 0,

0 if t < 0(10)

and

ε(t) =

0 if t < 0,

ε0 sin(ω0t) if t ≥ 0,(11)

giving

σ(t) =∫ t

0G0 exp(−αt)ε0ω0 cos(ω0τ)dτ

=G0ε0ω0

α20 + ω2

0

[α0 cos(ω0t) + ω0 sin(ω0t)− α0 exp(−α0t)] . (12)

�

It is also possible to formulate the whole theory for the case when the stress is the driving quantity.If in this case the creep compliance is defined as J(t) = ε(t)/σ, the resulting strain can be calculatedfrom

ε(t) =∫ t

−∞J(t− τ)

dσ(τ)dτ

dτ. (13)

Storage and Loss Moduli.As a final illustration will also be shown how the theory can be applied to a sinusoidally driving strain


8

state specified byε(t) = ε0 sin(ωt). (14)

The resulting stress response can be written:

σ(t) =∫ ∞

0G(s)ωε0 cos [ω(t− s)] ds (15)

where s ≡ t− τ . This equation can be expanded into

σ(t) = ε0 sin(ωt)[ω

∫ ∞

0G(s) sin(ωs)ds

]+ ε0 cos(ωt)

[ω

∫ ∞

0G(s) cos(ωs)ds

]. (16)

Note that the integrals only converge if

lims→∞

G(s) = 0. (17)

By defining two frequency-dependent functionals: the storage modulus G′(ω) and the loss modulusG′′(ω), the stress response can be written

σ(t) = ε0

[G′(ω) sin(ωt) + G′′(ω) cos(ωt)

]. (18)

or alternatively

σ(t) = σ0 sin(ωt + δ) (19)

= σ0 sin(ωt) cos δ + σ0 cos(ωt) sin δ (20)

hence

ε0G′(ω) = σ0 cos δ (21)

ε0G′′(ω) = σ0 sin δ (22)

givingG′′

G′ = tan δ (23)

It is often convenient to introduce a complex variable based notation

G∗ =σ∗

ε∗= G′ + iG′′. (24)

It is also possible to consider a stress driven oscillation giving a complex compliance:

J∗ =ε∗

σ∗=

1G∗ = J ′ − iJ ′′. (25)


9

The relationships between J ′, J ′′, G′ and G′′ are:

J ′ =G′

G′2 + G′′2 (26)

J ′′ =G′′

G′2 + G′′2 (27)

G′ =J ′

J ′2 + J ′′2(28)

G′′ =J ′′

J ′2 + J ′′2(29)

Spectra.Several different means of specifying viscoelastic mechanical properties have been given. Specifically,relaxation functions, creep functions and complex moduli have been discussed. Another way of char-acterizing the material response is through spectra. To introduce the concept of a relaxation spectrumconsider the relaxation modulus

G(t) =∑

n

Gn exp[−t

τn

]. (30)

In the limit as n →∞ the summation is replaced by an integral:

G(t) =∫ ∞

0f(τ) exp

[−t

τ

]dτ. (31)

The function f(τ) is called the relaxation time spectrum. In practice it is often more convenient to usea logarithmic time scale

G(t) =∫ +∞

−∞H(τ) exp

[−t

τ

]d(ln τ) + G(∞). (32)

Similarly, a retardation time spectrum can be defined by

J(t) =∫ +∞

−∞L(τ)

[1− exp

(−t

τ

)]d(ln τ) + J(∞). (33)

It is possible to solve for H(t) and L(t) by using Laplace transforms, see Christensen [1982] for details.

Relationships Between Different Viscoelastic Functions.The formal relationship between the relaxation modulus G(t) and the creep compliance J(t) can bederived by taking the Laplace transformation of Equations (3) and (13)

σ(s) = sG(s)ε(s) (34)

ε(s) = sJ(s)σ(s) (35)

which can be rewritten asG(s) =

1s2J(s)

. (36)


10

Based on the Laplace limit theorems it can be shown [Christensen, 1982] that:

limt→0

J(t) = limt→0

1G(t)

(37)

limt→∞

J(t) = limt→∞

1G(t)

(38)

Also since L−1(1/s2) = t we directly get:∫ t

0G(t− τ)J(τ)dτ =

∫ t

0J(t− τ)G(τ)dτ = t. (39)

Differential Form of the Integral Representation.In this section will be shown that the integral form

σ(t) =∫ t

−∞G(t− τ)

dε(τ)dτ

dτ. (3-rep)

can also be written in differential form as

P (D)σ(t) = Q(D)ε(t) (40)

where P (D) and Q(D) are polynomials of D ≡ d/dt.To show this start by taking the Laplace transform of Equation (40):

P (s)σ(s)− 1s

N∑k=1

pk

k∑r=1

srσ(k−r)(0) = Q(s)ε(s)−N∑

k=1

qk

N∑r=1

srεk−r(0). (41)

The Laplace transform of (3) isσ(s) = sG(s)ε(s), (34-rep)

demonstrating that the two forms are equal if

sG(s) =Q(s)P (s)

(42)

andN∑

r=k

prσr−kij (0) =

N∑r=k

qrεr−k(0), k = 1, 2, . . . , N (43)

which is a restriction on the initial conditions.

Rheological Models—Maxwell Model.The Maxwell rheological model constitutes a linear spring (σ = Eε1) and a linear dashpot (ε2 = σ/η)in series. The rate of change in strain of the system is given by

ε =σ

E+

σ

η(44)


11

which is the differential representation. Now consider a stress relaxation experiment ε(t) = ε0H(t)giving

dσ

dt+

E

ησ = 0, (for t > 0)

having the solution

σ(t) = σ0 exp[−t

η/E

]. (45)

Hence the stress relaxation function for the Maxwell model is

E(t) = E0 exp[−t

η/E

].

In this section the key aspects of linear viscoelasticity theory has been presented. It has been shownthat to characterize the material only one2 functional form needs to be determined. The materialfunctional can be determined through one creep, stress relaxation or oscillatory experiment. Oncethe material dependent functional has been determined the response due to any arbitrary imposeddeformation or loading can be directly calculated.

The Use of Shift Functions to Generalize Linear Viscoelasticity Theory

For many materials the domain in which linear viscoelasticity theory give good predictions is unfortu-nately relatively small. Based on experimental observations it turns out that the influence of variationsin external parameters (such as temperature and aging time), which contribute to the limited applica-bility of the theory, can be accounted for by using a shift function approach. The reason for the successof this simple idea is that when the material functional is plotted using appropriate log-scales, changesin these external parameters do not change the shape of the plotted curve to any significant degree,only shift it.

Time-Temperature Equivalence.So far the discussion has been for a general viscoelastic material and can therefore be applied also toa polymer. But when considering polymers a number of complications become apparent. One of themore important issues that need to be recognized is the strong temperature dependence of the material.It has been shown experimentally that in many cases the temperature dependence can be modeled bya scaling of time using what has been termed the time-temperature equivalence. The basis for thisprinciple is shown in Figure 5 illustrating that if the experimentally observed stress relaxation modulusis plotted as a function of logarithmic time, the shape of the resulting curves is the same for a wideinterval of temperatures. In fact, the only significant difference between the curves is a horizontal shift.This observation suggests that if the relaxation modulus is known at one temperature (i.e. the ‘mastercurve’ is known) then the relaxation modulus at any other temperature is obtained if the horizontalshift factor aT = aT (T ) is known. Any material that has this property is called a rheologically simplematerial.

2Only one material function is required for an incompressible isotropic material subjected to a one-dimensional loadingsituation. For an isotropic material subjected to a general loading situation two material functionals need to be determined,and for an anisotropic material the number of functional are dependent on the the material symmetry.


12

Figure 5: Dependence of stress relaxation modulus on time and temperature.

The time shifts can be written

log tT0 − log tT = log aT (46)

where tT is the time at temperature T and tT0 the time at temperature T0. Equation (46) givesaT = tT0/tT or tT = tT0/aT , hence the behavior at a temperature of T becomes exactly the same asthe behavior at the reference temperature T0 if the time is accelerated by the factor aT . In general thetemperature is a function of time T = T (tT0), so instead of tT = tT0/aT it is necessary to write

dtT =dtT0

aT (T (tT0))(47)

giving

tT =∫ tT0

0

dt′T0

aT (T (t′T0))

. (48)

The effective time experienced by the material–the material time–is a function of temperature andwall clock time. For a rheologically simple material, the scaling of time with temperature occurs in allviscoelastic quantities such as G, J , J ′, J ′′, tan δ, etc. And the scaling constant must be the same forall quantities for the material to be rheologically simple.

Example 3. To exemplify this idea consider a simple Maxwell element: G(t) = G0 exp(−t). If the


13

material is rheologically simple the relaxation modulus becomes

G(t, T ) = G0 exp[−aT (T )t].

The influence of aT (T ) on the stress-strain behavior was examined in Example 1. �

One commonly quoted representation of aT is the WLF-equation [Williams et al., 1955]:

log aT (T ) =C1(T − T0)C2 + T − T0

(49)

where C1 = 17.4 and C2 = 51.6 K. This relationship is often used for amorphous polymers in thetemperature range T ∈ [Tg − 50 K, Tg + 50 K].

Vertical Shifts.As will be discussed in more detail below, the stress relaxation modulus (and also the creep compliance)curves when plotted as a function of logarithmic time often turn out to have the same shape not onlyfor different temperatures but also for variations in other parameters (such as aging time). But tocreate a master curve in these cases it is often necessary to also use vertical shifts (on a log-scale), seeFigure 6,

Figure 6: Vertical shifts of the relaxation modulus.

log Gα − log Gα0 = log b (50)

giving

b =Gα

Gα0. (51)

In summary, the integral formulation

σ(t) =∫ t

−∞G(t− τ)

dε(τ)dτ

dτ. (3-rep)

becomes

σ(t) =∫ t

0b(t)G(t(t)− t(τ))

dε(τ)dτ

dτ (52)

when both vertical shift b(θ1(t)) and horizontal shift t(t) = t(θ2(t)), where θ1 and θ2 are externalparameters such as temperature, are considered.

2.3 NON-LINEAR VISCOELASTICITY THEORY

It has been known for a long time that in many practical applications of polymers, linear viscoelasticitytheory does not give a very accurate representation of the material response. And for this reason anumber of different extensions to the Boltzmann’s superposition principle have been developed (Smith[1962], Leaderman [1962], Bernstein et al. [1963], Ward and Wolfe [1966], etc.). In the following willone additional model developed by Schapery [1969] be discussed.


14

The Schapery Representation.Schapery used the same approach as Leaderman in dividing the compliance into elastic, Ju, and delayedresponse, ∆J(t) = J(t)− Ju. The creep strain is then taken to be given by

ε(t) = g0(σ)Juσ(t) + g1(σ)∫ t

−∞∆J(τ − τ ′)

dg2(σ(t′))dt′

dt′ (53)

where τ is the reduced time given by

τ = τ(t) =∫ t

0

dt′

aσ(σ(t)). (54)

Note that g0(σ), g1(σ), g2(σ), and aσ(σ) are all functions of σ. Similarly, the stress relaxation basedrepresentation can be written

σ(t) = he(ε)Grε(t) + h1(ε)∫ t

−∞∆G(τ − τ ′)

dh2(ε(t′))dt′

dt′. (55)

This representation incorporates time-temperature equivalence through horizontal shifts aσ, and ver-tical shifts through h1(ε).

2.4 LARGE STRAIN STATE VARIABLE BASED MODELS

A different approach to model the time- and temperature-dependent behavior of polymeric materials isto use a state variable based framework of the type that has been used to study the large strain plasticdeformation of metals. This class of models has been shown to successfully capture the dependence ofdeformation state, time, and temperature of many polymers both below the glass transition temperature[Boyce et al., 1992, Dooling and Buckley, 1998, Buckley and Jones, 1995] and above the glass transitiontemperature [Llana and Boyce, 1999, Bergstrom and Boyce, 1998, Sweeney and Ward, 1996]. Thestructure of all these models is similar in that they all use a temperature, strain state, and pressuredependent flow rule that captures the non-linear viscoelastic behavior. To illustrate a typical modelframework will the large strain kinematics of one model [Boyce et al., 1992] be presented.

In this model the deformation gradient acting on the material point of interest is decomposed intoelastic and plastic parts F = FeFp (the nomenclature used here is the same as Gurtin [1981]). Thevelocity gradient can then be written

L = FF−1 = Le + FeLpFe−1 (56)

where Lp = Dp +Wp and the plastic spin can be taken to be 0 [Boyce et al., 1989]. The Cauchy stressis given by

T =1JLe [lnUe] , (57)

and the rate kinematics by:DP = γpN, (58)


15

where N is the tensorial direction and is given by

N =1√2τ

T∗′. (59)

The convected stress T∗′is determined by

T∗′= T− 1

JFeBFeT . (60)

The plastic shear strain rate, γ, is given by a scalar equation Argon [1973]

γp = γ0 exp[−As

kθ

(1−

(τ

s

)5/6)]

. (61)

Strain softening is modeled by taking the athermal shear stress s to evolve to a preferred state sss

s = h

(1− s

sss

)γp. (62)

The effective equivalent shear stress is found from the tensorial difference between the total stress T

and the convected back stress on the nonlinear spring element B.

τ =[12T∗′ ·T∗′

]1/2

. (63)

The deviatoric back stress tensor B is taken by the hyperelastic 8-chain representation, here representedin its functional form

B = f(Fe). (64)

This type of state variable based approach can be effectively incorporated into a finite element package,and the material constants are simple to find through a few experimental tests [Boyce et al., 1992].

3 INFLUENCE OF PHYSICAL AGING ON DEFORMATIONAL

BEHAVIOR

The deformational behavior of polymeric materials is often influenced by physical aging. This depen-dence is manifested through material stiffening and embrittlement. The following subsections presentdifferent approaches to model this behavior.

3.1 AMORPHOUS POLYMERS

Shift-factor approach.Through the introduction of shift-factors in Section 2 all operational details of linear viscoelasticitytheory has been presented. The goal of the current section is to discuss in more detail the materialscience aspects of physical aging. It is well known that above Tg, temperature affects creep because the


16

equilibrium free volume is temperature-dependent. Below Tg, the free volume becomes time-dependent,thus causing aging. Experimentally observed creep compliance curves often have the shape shown inFigure 7. The influence of aging time on the shift is often characterized through a shift rate:

Figure 7: Influence of physical aging on the creep compliance.

µ ≡ −d(log ate)d(log te)

, (65)

which can also be writtenate =

(te0te

)µ

(66)

where µ is a constant between 0 and 1 (should be close to one), te0 is the aging time at which themaster curve was obtained, and te is the aging time of interest. Above Tg where aging has no timedependence, µ is zero. Just below Tg it rapidly increases to about unity. At low temperatures the agingbegins to cease and µ decreases. Based on experimental observations it is known that a variation inte induces an almost horizontal shift, whereas an additional change in temperature also produces anadditional vertical shift

J(T, te; t) = B(T )J(Tr, ter; a(T, te)t). (67)

Note that this approach is only valid for momentary creep curves (t � te). Also note that the shiftdecreases with increasing stress, independent of polymer, deformation state and temperature.

Some general guidelines: the temperature range in which aging occurs is generally between Tβ andTg; the creep compliance can often be written in the following mathematical form

J(t) = J0 exp[(

t

t0

)m],

where m ≈ 0.3. Unlike amorphous polymers, filled elastomers and semicrystalline polymers age attemperatures above their Tg, see Section 3.2.

Finally, the long term behavior cannot be predicted using this approach, since in this case agingwill change the material properties with time. Hence, linear viscoelasticity is inappropriate underthese conditions. A generalization of the presented ideas applicable also for long term behavior will bepresented next.

Long Term Behavior.As was mentioned in the previous subsection, the Boltzmann’s superposition principle is not applicablein long term tests due to material aging. The aging manifests itself through an actual change of theshape of the creep curves, making a shift-based approach less accurate. The influence of long termaging, however, can be modeled from short-term aging by using the time scaling principle discussedbelow.

From experimental results it is known that the vertical shift factor hardly changes with te, and that


17

the shift rate µ stays more or less constant with te. Consider a creep test started at an aging time ofte, then at creep time t, the total aging time is (t + te) and Equation (66) gives

a(t) =(

tete + t

)µ

. (68)

The effective time interval at time t is dτ = a(t)dt giving

τ =∫ τ

0a(t′)dt′

which can be written

τ =

te ln[1 + t/te], if µ = 1

te1−µ

[(1 + t

te

)1−µ− 1]

, if µ < 1.(69)

Procedure to determine long-term creep from tests of short duration.

• The first step is to determine the shift rate due to aging, µ. Start by heating the sample to thedesired temperature T , and then measuring the creep or stress relaxation at various aging timeste. Let te increase by a factor of 2− 3 per test, and stop the tests at about 10% to 20% of te. Alltests are preferably done with one sample.

• The second step is to measure the momentary creep compliance, Jte(t). Correct the obtaineddata for aging by using (69). Determine the master curve at an aging time of te by horizontaland vertical shifts. Note, if enough temperatures are tested, the time-temperature shift factorrelationship can also be obtained.

• The last step is to determine the long-term compliance from Equation (69).

A simplified procedure is to measure the creep compliance and plot J as a function of log t. Ter-minate the test when the creep curve has become nearly straight. Predict the long term behavior bylinear extrapolation.

Yet another method has been developed by Bradshaw and Brinson (1997,1999). Their approachis to run cyclic creep tests for an extended amount of time, using the same procedure as presentedabove, but then they use all the experimental data to determine the appropriate shift functions; henceallowing for a more accurate determination of the material parameters.

Large Strain State Variable Based Models.When searching the literature, only one model [Hasan and Boyce, 1993] was found that attempts tocapture the influence of physical aging on the deformational behavior by using a state variable basedkinematics framework of the type discussed in Section 2.4. In this model the flow rule is augmentedwith a more explicit consideration of the number of shear transformation sites D:

γ = γ0νGD exp(−∆Gf

kθ

). (70)


18

The number density of potential shear transformation sites is modeled as a scalar-valued internal statevariable which evolve with physical aging (time and temperature) and inelastic straining:

D = D(θ, t) + D(γ, D). (71)

Physical aging will result in a temperature-dependent evolution (generally a decrease) in D with time.More generally, a contribution to aging due to the presence of stress (in particular pressure) may alsobe incorporated. As a first attempt [Hasan and Boyce, 1993] used the first order kinetics:

dD

dγp= −D −D∞

τp, (72)

where D∞ is the steady-state number density of regions capable of undergoing shear transformations(probably weakly temperature dependent) and τp is the characteristic ‘time’ for evolution of D duringplastic deformation.

3.2 SEMICRYSTALLINE POLYMERS

From experimental data it is known that semi-crystalline polymers age both below and above Tg. Oneway to explain this is the argument used by [Struik, 1978] that the crystals disturb the amorphous phaseand reduce the segmental mobility. This reduction will be at its maximum in the immediate vicinity ofthe crystals, and only at large distances from the crystals will the properties of the amorphous phasebecome equal to those of the bulk material. The main consequence of this in-mobilization is that theglass transition will be extended towards the high temperature side. Above Tg of the bulk amorphousmaterial, some parts of the amorphous phase are rubbery while other parts are still glassy.

In modeling this behavior, [Struik, 1978] assumes that the crystalline phase behaves as an inertfiller. For simplicity, the distribution in Tg’s is modeled by assuming that there are only two clearlydistinct Tg’s: TL

g and THg . Furthermore, the creep compliance J is simply taken as the sum of the two

parts: J(t) = JL(t) + JH(t). Using this approach four different temperature domains are introduced:(1) T < TL

g ; (2) T ≈ TLg ; (3) TL

g < T < THg ; (4) T ≥ TH

g . The behavior of these regions will bediscussed separately.

1. (T < TLg ): At temperatures well below TL

g , the creep and aging behavior of the semicrystallinepolymer will be similar to that of amorphous polymers. In the upper part of region 1, the shapeof J(t) is dominated by JL(t). The effect of aging on the total compliance J(t) can be representedas a horizontal shift due to JL(t) combined with a vertical shift due to JH(t). The long termcreep can be modeled using the same methods as for amorphous polymers.

2. (T ≈ TLg ): Same behavior as in region 1, but with a different shape of the creep curves. Experi-

mental data have shown that the long term creep curves are more or less straight for t � te, thissuggests that the linear extrapolation method developed for amorphous polymers can be applied.

3. (TLg < T < TH

g ): All aging effects arise from JH(t). The total compliance will shift upward tothe right. Time-temperature superposition is not possible. The long term creep behavior is the


19

same as in region 2.

4. (T ≥ THg ): The entire amorphous phase becomes rubbery. The aging effect disappear.

For both amorphous and semi-crystalline polymers, high stresses can erase previous aging, i.e. the shiftrate µ = d(log a)/d(log te) decreases with increasing σ. Application of a high stress is to some extentcomparable to a brief heating to above Tg, followed by a quench. The large deformation processesgenerate free volume, and the aging is partially erased. After some time, the free volume productioncan no longer compete with the normal volume-relaxation process.

4 DEFECT GENERATION DUE TO MECHANICAL LOADS AND

DEFORMATIONS

Up till now only homogeneous aspects of the deformational behavior have been discussed. Under manydifferent loading and deformation conditions, however, defects can be generated. And the the topic ofthis section is to discuss conditions for this to occur.

A defect can be defined as an unwanted irregularity in a material on the microscale (or larger).This irregularity causes the material to become inhomogeneous3. Typical examples of defects include:variations in cross-linking density, residual stresses, inclusions, voids, asperities, fissues, contamination,precipitate particles, dislocations, etc. Some of these defects are generated during the manufacturingprocess and therefore not explicitly considered here (e.g. scorching, interface problems such as asper-ities, fissues, contamination, inclusions). Their implicit influence on the growth of other defects suchas voids, however, is important. The focus of this section is on the generation and growth of defectsdue to mechanical loads and deformations, i.e. on microvoids and crazes and how they become voidsand cracks.

4.1 CRAZING

Consideration of the plastic deformation of polymers has traditionally been concerned only with shearyielding. This occurs at constant volume and can take place uniformly throughout the sample. Certainpolymers, particular thermoplastics in the glassy state, are capable of undergoing a localized formof plastic deformation known as crazing. This is found to take place only when there is an overallhydrostatic tensile stress and the formation of crazes causes the material to undergo a significantincrease in volume. The crazes appear as small crack-like entities which are usually initiated onthe specimen surface and are oriented perpendicular to the largest principal tensile stress. Closerexamination shows that they are regions of cavitated polymer and not true cracks, although the crackswhich lead to eventual failure of the specimen usually nucleate from pre-existing crazes.

In practice, the use of many glassy polymers is often limited by their tendency to undergo crazingat relatively low stresses in the presence of crazing agents. As with other mechanical properties, theformation of crazes in polymers depends upon testing temperature and the rate or time-period of

3Note that with this definition, multi phase materials (such as composites) do not necessarily contain defects.


20

loading. The craze initiation stress drops with increasing testing temperature and with decreasingstrain-rate.

As crazing is a yield phenomenon there have been attempts to establish a craze criterion in thesame way that Tresca and von Mises criteria have been used for shear yielding. It is known thatcrazing occurs typically about one-half of the yield stress of the polymer. This is in the elastic regionof material response, but the occurrence of crazing does not usually cause any detectable change inslope of the stress-strain curve as the volume fraction of crazed material is initially very low.

It has been found that the craze profile ahead of a crack is similar to the plastic zone model proposedby Dugdale [1960] for metals. In this model the stress singularity at the crack tip is canceled by thesuperposition of a second stress field with compressive stresses along the length of the crack. Thiscompressive stress, which is taken to be constant, is identified with the craze stress. The length of thecraze zone ahead of an active crack has been shown by Rice [1968] to be given by

R =π

8·K2

IC

σ2c

, (73)

and the crack opening displacement (COD) by

δt =K2

IC

σcE∗ . (74)

Experiments have shown that for many polymers the COD has very little dependence on temperatureand strain rate. Using this approach the existence of a brittle-ductile transition can be directly under-stood: both yielding and crazing, which are competing processes, are energy activated with differenttemperature and strain rate dependence. Experiments have indicated that when occurring in a bulkspecimen, crazes contain about 50% polymer and 50% void.

A number of different criteria for craze initiation have been proposed. Since crazing involves volu-metric expansion and cavitation a reasonable starting assumption is to assume that crazing occurs ata critical strain εc. The influence of pressure can be considerated by taking

εc = A +B

σ1 + σ2 + σ3(75)

where A and B are dependent on time and temperature, and σ1 > σ2 > σ3 are the principal stresses.The critical strain can be written

εc =1E

(σ1 − νσ2 − νσ3) , (76)

where ν is the Poissons ratio. The final craze initiation criterion therefore becomes

σ1 − νσ2 − νσ3 = C1 +C2

σ1 + σ2 + σ3. (77)

A theory for craze growth has been developed by Argon et al. [1977] who proposed that the crazefront advances by a meniscus instability mechanism in which craze tufts are produced by the repeated


21

break-up of the concave air/polymer interface at the crack tip. A theoretical treatment of this modelpredicts that the steady-state craze velocity would relate to the five-sixths power of the maximumprincipal tensile stress, a result that has been experimentally verified.

The influence of gases at sufficiently low temperatures has been shown by Brown et al. [1978] to makealmost all linear polymers craze. Parameters such as the density of the crazes and the craze velocityincrease with the pressure of the gas and decreases with increasing temperature. It was concluded thatthe surface concentration of the absorbed gas was a key factor in determining its effectiveness as acrazing agent.

4.2 VOID GROWTH

A number of different void growth criteria have been developed, as a first example is here discussedthe growth of a cavity in a neo-Hookean material.

Example 4. Consider a spherical body with radius B containing a hole with radius A. On the outsidesurface of the body is a radial Piola-Kirchhoff traction S applied. Given S, A, B, and the shearmodulus µ, the goal is to determine the radius of the cavity.

To solve this problem first introduce the following nomenclature: a = radius of the hole in deformedconfiguration, and b = radius of the body in deformed configuration. Note that uppercase variablesrefer to the undeformed configuration and lowercase variables to the deformed configuration. Theneo-Hookean material model is

T = µB− pI. (78)

The kinematics of the problem is governed by the deformation in spherical coordinates: r = f(R),θ = Θ, φ = Φ, giving the deformation gradient:

[F] =

λR 0 00 λΘ 00 0 λΦ,

(79)

where

λR =dr

dR= f ′(R), (80)

λΘ = λΦ =r

R=

f(R)R

. (81)

The incompressibility condition can be written

λRλΘλΦ = f ′f2

R2= 1

which has the solutionf(R) = r =

(R3 −A3 + a3

)1/3.


22

The Cauchy stresses are given by

TRR = µλ2R − p, (82)

TΘΘ = TΦΦ = µλ2Θ − p. (83)

In this case there is only one non-trivial equilibrium equation

dTrr

dr+

2r

(Trr − Tθθ) = 0

which can be solved for the radial stress

Trr

2µ=∫

dr

r

(λ2

θ − λ2r

).

Numerical integration of this equation together with the boundary condition Trr = 0 at r = a andthe relationship between Piola-Kirchhoff traction and Cauchy traction (S = Tb2/B2) gives the resultsshown in Figure 8. The figure shows that as the normalized cavity size goes to zero the nucleation

Figure 8: Growth of a cavity under hydrostatic tension.

stress goes to a finite value which can be shown to be

S =52µ.

�

Steenbrink et al. [1997] have developed a model for the macroscopic behavior of porous glassypolymers. Their study was motivated by the plastic deformation in voided polymer-rubber blendscaused by cavitation of the rubber particles. The role of strain localization into shear bands and theirsubsequent propagation into controlling void growth was highlighted. An approximate constitutivemodel was presented for the description of the macroscopic overall behavior of porous glassy polymers.The model includes a modification of existing porous plasticity models to account for elasticity effectson the initiation of overall plasticity which are important in polymers because of their relatively highyield strain.

Gurson [1977] derivation of an approximate yield function for a porous rigid-plastic metal has formedthe basis for many recent investigations into ductile fracture. In the model developed by Steenbrink etal., the Gurson potential is modified to

Φ =12

σ′ · σ′

τ2+ 2q1f cosh

(e ln

[1 +

√3σm

2eτ

])− [1 + (q1f)] = 0. (84)

The correction for elasticity is controlled by the parameter e, defined by

e = ln( τ

E

). (85)


23

The change of the void volume fraction f due to growth of the voids during a deformation process isgoverned by the evolution relation

f = (1− f) trDp + fe. (86)

expressing the conservation of mass, and where

fe = f03σm

4G. (87)

5 MACROSCOPIC FAILURE DUE TO MECHANICAL LOADS

AND DEFORMATIONS

The ultimate failure of a polymer can be of two types: brittle or ductile. Brittle failure is designatedwhen the specimen fails at its maximum load (and the effective strain is less than 10%). The distinctionbetween brittle and ductile failure is also manifested through energy dissipated in fracture (which canbe measured for example with Charpy and Izod tests), and the nature of the fracture surface.

The existence of a brittle-ductile transition was discussed in Section 4.1. A further argumentbecomes clear by realizing that brittle fracture and plastic flow are independent processes. Fromexperiments it is known that the brittle strength is not affected much by strain rate and temperature,but that the yield stress is a strong function of strain rate and temperature. It is also known thatcrosslinking increases the yield strength but generally does not increase the brittle strength much.Consequently there will be a temperature at which there is a transition from ductile to brittle behavior.

It is well known that the presence of a sharp notch can change the fracture mode from ductile tobrittle. A simple explanation for this has been developed by Orowan who realized that for an ideallydeep and sharp notch in an infinite solid, the plastic constrain raises the yield stress to a value ofapproximately 3σy, therefore: (1) if σB < σy, brittle behavior; (2) if σy < σB < 3σy, ductile but brittleif notched; (3) if σB > 3σy, ductile behavior.

This type of behavior is presented in Figure 9. In this figure, the dashed line represents the divisionbetween brittle and brittle when notched behavior; and the dashed-dotted line represents the divisionbetween ductile and brittle when notched behavior. In the figure is also plotted experimental data forLDPE, illustrating that LDPE is under normal conditions is a ductile material.

Figure 9: Plot of brittle strength at −180◦C against the yield stress (open circle at −20◦, and opensquare at +20◦).

5.1 BRITTLE FAILURE

Even though XLPE under normal conditions is a ductile material, brittle failure can still be an impor-tant failure mode. In fact, the Achilles heel of many polymers is their tendency to fail at relatively lowstress levels through the action of certain hostile environments. For example, brittle-type slow crackfracture is the dominant mode of failure in polyethylene pipe used for natural gas distribution. Failures


24

of this type usually occur over a relatively long period of time, and are characterized by slow crackgrowth.

To predict this type of failure, linear elastic fracture mechanics (LEFM) can be a useful tool, seeSection 5.2.1 for a general energy-based failure criterion. Attempts to quantify this phenomenon usingthe J integral approach have been hampered by the large specimen thickness required to effect planestrain conditions and because monotonic loading does not readily initiate brittle cracks in MDPE. Aprocedure to determine fracture toughness (J1c) using fatigue crack propagation has been developedStrebe and Moet [1992]. By incorporating both time and energy, a measure of the power to fracture hasbeen obtained which can differentiate between these pipe resins on the basis of their brittle crackingresistance.

5.2 DUCTILE FAILURE

The failure of PE typically occurs as a result of either a shear rupture or a sudden onset of instabilityin a previously slowly growing crack. In both cases the time to failure depends greatly on temperature.One approach to predict the lifetime of a component is the theory of crack growth in viscoelasticmedia. In this case the defects can originate from a specific part design (e.g. sharp corners), or canbe inherently present in the material (e.g. dust particles), or can be introduced during part use (e.g.scratches, cuts). The largest defect in a given stress state plays the role of a trigger for the process ofslow crack growth, leading to brittle failure.

Note, however, that ultimate properties are far less reproducible than the relationship between thestress and strain up to the breaking point, since the mechanical failure depends on quantities whichare subject to statistical fluctuations.

By studying the shear rupture, it is known that Verdu [1994]: (1) only ultimate elongation ε is apertinent variable in kinetic studies of aging; (2) ∆ε is higher for initially ductile materials than forinitially brittle; (3) the rupture envelope σ = f(ε) is often very close to the initial tensile curve.

The relation between tensile creep behavior and slow crack growth have been studied by O’Connelet al. [1995]. To determine the deformation surface they studied drawn sample of PE. By considering amechanism for craze growth the initiation and slow growth of cracks was determined. Using the modelthey showed that it is possible to find the contribution the creep of the drawn material makes to crazegrowth and ultimately to the failure of the craze.

The degradation chemical events contributing to the final failure can be of two types: homogeneouslydistributed or spatially heterogeneously distributed. And the heterogeneities can either be on themorphological or macroscopic scale.

Morphological: It is well known that molecular reactants such as oxygen or water cannot penetratethe crystalline phase, so the degradation of semicrystalline polymers is generally restricted to theamorphous phase. The consequences of the embrittlement are well understood at the qualitativelevel, but quantitative relationships are generally lacking. The problem is often complicated bythe existence of two morphological structural scales: individual lamellae and spherulites withthe corresponding brittle failure mechanisms and probably distinct chemical reactives due todifferences in molecular mobility between the intra- and interspherulitic amorphous phase.


25

Macroscopic: Diffusion-controlled degradation kinetics lead to a core-shell structure.

5.2.1 Tearing

The material behavior under these conditions was first studied by Rivlin and Thomas [1955]. Bydefining the tearing energy T as expended energy per unit thickness per unit increase in crack length,the tearing energy can be determined from a standard trousers test. The work done in the test is

∆W = 2F ·∆c

where F is the force and ∆c is the increase in crack length. The tearing energy can therefore directlybe obtained from

T =2F

t

where t is the thickness.Most molecular theories of the strength of cross-liked polymers treat rupture as a critical stress

phenomenon, and it is known that the tensile strength can be much increased by the inclusion ofreinforcing fillers.

Bueche [1961] proposed that these fillers increase the tensile strength by allowing the applied loadto be shared amongst a group of chains, thus decreasing the chance of a break to propagate. Effectsof different strain rates and temperatures can often be modeled through shift factors of WLF formsuggesting that the fracture process is dominated by viscoelastic effects.

Example 5. As an example will energetic aspects of the tearing of crosslinked polymers above Tg bestudied. Energy conservation of any macroscopic body can be written:

δUin − δUdissipated = δUstored + δUkinetic, (88)

whereδUin = energy input to the body by external loads,δUdissipated = energy dissipated in the body,δUstored = elastic energy stored in the body,δUkinetic = increase in kinetic energy of the body.

Now consider a fracture event in which the area of cracked surface increases by δA. Also in this casethe general energy conservation equation relationship (88) is valid. To study the crack growth take thepartial derivative of the energy conservation equation with respect to crack area:

∂Uin

∂A−

∂Udissipated

∂A=

∂Ustored

∂A+

∂Ukinetic

∂A(89)

which can be written∂Uin

∂A− ∂Ustored

∂A=

∂Udissipated

∂A+

∂Ukinetic

∂A. (90)


26

Assume that the change in kinetic energy with crack length can be neglected4, and further introducethe following definitions:

G ≡ ∂Uin

∂A− ∂Ustored

∂A, (91)

R ≡∂Udissipated

∂A. (92)

The parameter G is called energy release rate, toughness, or crack driving force; and R is called thefracture resistance. The fracture initiation criterion can now be written

G ≥ R. (93)

The stability of the fracture event can be assessed by studying a virtual change in crack area δA,the process is unstable if

G +∂G

∂AδA > R +

∂R

∂AδA, (94)

but since G = R at crack initiation, the criterion for unstable crack growth becomes

∂G

∂A>

∂R

∂A. (95)

If the fracture resistance is independent of the crack length, then the energy dissipation processes arelocal to the crack tip and R can also be written as Gc, the critical strain energy release rate. Notethat for a brittle material the fracture resistance is equal to the surface energy of the material. For alinearly elastic material the global energy condition (93) can also be written in terms of the strengthof the crack tip field KI giving the failure condition

KI = KIc. (96)

For a crosslinked polymer in its rubbery state the derivation presented above is still valid, but thecritical energy release rate has been termed the tearing energy.

Rivlin and Thomas [1955] showed that the crack driving force is still primarily determined by thestate of deformation in the neighborhood of the crack tip and that the change in total energy withcrack length is independent of the test piece geometry. �

5.3 FATIGUE

Polymeric parts have been used increasingly as structural elements in instruments and devices thatare subjected to long-term stress fluctuations. Such use has prompted an increased interest in theirmechanical performance when subjected to low-level load variations commonly known as fatigue. Thefatigue failure of thermoplastics generally develops in two phases. First, the material accumulatedfatigue damage in the form of nanometer-sized voids (i.e. in the initiation phase), which ultimatelyleads to the formation of visible crazes. The initiation time can be a major fraction of the total

4For moving cracks the change in kinetic energy sometimes becomes significant.


27

lifetime, especially at low stress amplitudes. The crazes further grow, form cracks and propagate (inthe propagation phase) until final failure occurs. The crack propagation of most plastics, beyond thethreshold region, can be described by Paris law. For cross-linked polymers above Tg the tearing energyis often used:

dc

dN= ATn.

It has been established that the fatigue lifetime of most polymers is very sensitive to external fields,and as a rule is much shorter than the corresponding lifetime under constant stress.

External parameters such as test frequency, ambient temperature, and stress amplitude are known tobe very important due to thermal heating. Increasing the molecular weight often causes a significantlyincreased fatigue life. It is also known that fatigue loading generally embrittles the polymer, but theexperimental information is very sparse and it is still not clear what kind of structural changes cause thedeterioration. Since physical aging also embrittles polymeric materials, the role of mechanical loadingon the aging kinetics in amorphous polymers has been extensively discussed.

It is also important to realize that many laboratory fatigue results may or may not be applicablein engineering design of a particular polymeric component. This is due to the fact that there aredifferences in test versus service cyclic frequency, specimen area-to-volume ratio, cooling conditions,waveform, and nature of loading.

By studying the changes in the mechanical properties of amorphous PC, Li et al. [1995] determinedthat during the fatigue failure initiation stage an overall embrittlement of the polymer occurs. Theirresults indicate that the fracture toughness of the material can be reduced by ∼ 35% without visibledamage. The ‘static’ mechanical characteristics, such as yield stress and strain absorbed energy, wereshown to be influenced very modestly by the cyclic fatigue. The observations are explained by thepresence of a small number of nanometer-sized voids or ‘protocrazes’, detected by transmission andscanning electron microscopy measurements. It is shown that physical aging and fatigue loading affectthe structural state of the polymer differently despite similar effects on the macroscopic mechanicalproperties.

6 PHYSICAL PROPERTIES

In this section are a few select properties of PE summarized. For more data see for example theEncyclopedia of Polymer Science.

• Low-density polyethylene is a partially (50-60%) crystalline solid melting at about 115◦C, withdensity in the range 0.91 – 0.94. It is soluble in many solvents at temperatures above 100◦C,but no room-temperature solvents exist. The relatively low crystalline melting point limits thetemperature range of good mechanical properties.

• The electrical properties of polyethylene are outstandingly good, probably ranking next to thoseof polytetrafluorethylene for high-frequency uses. In thick sections polyethylene is translucentbecause of its crystallinity, but high transparency is obtained in thin films.


28

• Polyethylene ages on exposure to light and oxygen, with loss of strength, elongation, and tearresistance. The probably point of attack is the tertiary hydrogens on the chain at branch points.Stabilizers retard the deterioration, but few are compatible enough with the polymer to do muchgood.

References

A. S. Argon. A theory for the low temperature plastic deformation of glassy polymers. Philos. Mag.,28:839–865, 1973. 16

A. S. Argon, J. C. Hannoosh, and M. M. Salama. Fracture 1977, volume 1, page 445. Waterloo,Canada, 1977. 21

E. M. Arruda and M. C. Boyce. A three-dimensional constitutive model for the large stretch behaviorof rubber elastic materials. J. Mech. Phys. Solids., 41(2):389–412, 1993. 4

J. S. Bergstrom and M. C. Boyce. Constitutive modeling of the large strain time-dependent behaviorof elastomers. J. Mech. Phys. Solids, 46:931–954, 1998. 15

B. Bernstein, E. A. Kearsley, and L. J. Zapas. Trans. Soc. Rheol., 8:391, 1963. 14

L. Boltzmann. Pogg. Ann. Phys. U. Chem., 7:624, 1876. 4

M. C. Boyce, E. L. Montagut, and A. S. Argon. The effects of thermomechanical coupling on the colddrawing process of glassy polymers. Polymer Engineering and Science, 32(16):1073–1085, 1992. 15,16

M. C. Boyce, G. G. Weber, and D. M. Parks. On the kinematics of finite strain plasticity. J. Mech.Phys. Solids, 37(5):647–665, 1989. 15

N. Brown, B. D. Metzger, and Y. Imai. J. Polymer. Sci. Polymer. Phys. Ed., 16:1085, 1978. 22

C. P. Buckley and D. J. Jones. Glass-rubber constitutive model for amorphous polymers near the glasstransition. Polymer, 36(17):3301–3312, 1995. 15

F. Bueche. Mullins effect and rubber-filler interaction. J. Appl Polymer Sci., 5:271–281, 1961. 26

R. M. Christensen. Theory of Viscoelasticity; an introduction. Academic Press, 1982. 10, 11

P. J. Dooling and C. P. Buckley. The onset of nonlinear viscoelasticity in multiaxial creep of glassypolymers: A constitutive model and its application to pmma. Polymer Engineering and Science, 38(6):892–904, 1998. 15

D. S. Dugdale. Yielding of steel sheets containing slits. J. Mech. Phys. Solids, 8:100–108, 1960. 21

A. N. Gent. Rubber Chem. Technol., 69:59, 1996. 4


29

A. L. Gurson. Continuum theory of ductile rupture by void nucleation and growth: Part i. yield criteriaand flow rules for porous ductile media. J. Engng Mater. Technol., 99:2–15, 1977. 23

M. E. Gurtin. An introduction to continuum mechanics. Academic Press, Inc., 1981. 15

O. A. Hasan and M. C. Boyce. Energy storage during inelastic deformation of glassy polymers. Polymer,34(24):5085–5092, 1993. 18, 19

H. Leaderman. Trans. Soc. Rheol., 6:361, 1962. 14

X. Li, H. A. Hristov, A. F. Yee, and D. W. Gidley. Influence of cyclic fatigue on the mechanicalproperties of amorphous polycarbonate. Polymer, 36(4):759–765, 1995. 28

P. G. Llana and M. C. Boyce. Finite strain behavior of poly(ethylene terephthalate) above the glasstransition temperature. Polymer, 40:6729–6751, 1999. 15

M. J. Mooney. J. Appl Phys., 11:582, 1940. 4

P. A. O’Connel, M. J. Bonner, R. A. Duckett, and I. M. Ward. The relationship between slow clackpropagation and tensile creep behavior in polyethylene. Polymer, 36:2355–2362, 1995. 25

R. W. Ogden. Proc. R. Soc. London, Ser. A, 326:565, 1972. 4

J. R. Rice. Fracture – An Advanced Treatise, chapter 3. Academic Press, 1968. 21

R. S. Rivlin and A. G. Thomas. Rupture of rubber. i. characteristic energy for tearing. J. Polym. Sci.,X:291–318, 1955. 26, 27

R. A. Schapery. On the Characterization of Nonlinear Viscoelastic Materials, volume 9. 1969. 14

T. L. Smith. Trans. Soc. Rheol., 6:61, 1962. 14

A. C. Steenbrink, E. Van der Giessen, and P. D. Wu. Void growth in glassy polymers. J. Mech. Phys.Solids, 45(3):405–437, 1997. 23

J. J. Strebe and A. Moet. A time dependent fracture toughnes measure for polyethylene. AMSE 1992,155:61, 1992. 25

L. C. E. Struik. Physical Aging in Amorphous Polymers and Other Materials. Elsevier ScientificPublishing Co., Amsterdam, 1978. 19

J. Sweeney and I. M. Ward. A constitutive law for large deformations of polymers at high temperature.J. Mech. Phys. Solids, 44(7):1033–1049, 1996. 15

J. Verdu. Effect of aging on the mechanical properties of polymeric materials. J.M.S.–Pure Appl.Chem., A31(10):1383–1398, 1994. 25

I. M. Ward and J. M. Wolfe. J. Mech. Phys. Solids, 14:131, 1966. 14


30

M. L. Williams, R. F. Landel, and J. D. Ferry. J. Amer. Chem. Soc., 77:3701, 1955. 14

P. D. Wu and E. van der Giessen. On improved network models for rubber elasticity. J. Mech. Phys.Solids, 41(3):427–456, 1993. 4

O. H. Yeoh. Some forms of the strain energy function for rubber. Rubber Chem. Technol., 66(5):754–771, 1993. 4

A Laplace Transformation

The Laplace transformation is defined by

L [f(t)] = f(s) =∫ ∞

0f(t)e−stdt. (97)

The Laplace transformation of a derivative is

L[df(t)dt

]= sf(s)− f(0). (98)

The Laplace transformation of a convolution integral is

L[∫ t

0f(t− u)g(u)du

]= f(s)g(s). (99)


31

Polymer Mechanics, Deformations, and Failure of XLPE · Polymer Mechanics, Deformations, and...

Documents

Transcript of Polymer Mechanics, Deformations, and Failure of XLPE · Polymer Mechanics, Deformations, and...