International Journal of Fracture Volume 159 Issue 1 2009 [Doi 10.1007_s10704-009-9384-x] a. D. Drozdov; J. DeC. Christiansen -- Creep Failure of Polypropylene- Experiments and Constitutive

8/13/2019 International Journal of Fracture Volume 159 Issue 1 2009 [Doi 10.1007_s10704-009-9384-x] a. D. Drozdov; J. DeC

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Int J Fract (2009) 159:6379DOI 10.1007/s10704-009-9384-x

ORIGINAL PAPER

Creep failure of polypropylene: experiments

and constitutive modelingA. D. Drozdov J. deC. Christiansen

Received: 14 January 2009 / Accepted: 27 July 2009 / Published online: 26 August 2009 Springer Science+Business Media B.V. 2009

Abstract Observations are reported on isotacticpolypropylene in uniaxial tensile tests with variousstrain rates, relaxation tests with various strains, andcreep tests with various stresses at ambient tempera-ture. Constitutive equations are derived for the visco-elasticviscoplastic responses and damage of asemicrystallinepolymer at three-dimensional deforma-tions. Adjustable parameters in the stressstrain rela-tions are found by tting the experimental data. Themodel is applied to predict creep-failurediagrams in theentire interval of stresses. A phenomenologicalapproach is proposed to determine a knee stress, atwhich transition occurs from ductile to brittle rupture.Accuracy of this method is evaluated by numerical sim-ulation.

Keywords Polypropylene Viscoelasticity Viscoplasticity Damage Creep failure

1 Introduction

This paper deals with two issues: (1) experimentalinvestigation and modeling of the viscoelastic and

A. D. Drozdov ( B )Danish Technological Institute, Gregersensvej 1,2630 Taastrup, Denmark e-mail: [email protected]

J. deC. ChristiansenDepartment of Production, Aalborg University,Fibigerstraede 16, 9220 Aalborg, Denmark

viscoplastic responsesof a semicrystallinepolymer and(2) application of the constitutive equations to lifetimeprediction under conditions of creep failure.

Evaluation of lifetime of polymer structures is oneof the key issues in their design as some areas of engineer-ing demand lifetimes of 1050 years (Guedes 2006 ;Orici et al. 2008 ). Creep-failure experiments serveas an effective method for assessment of long-termstrength under time-independent loads. In one-dimen-sional tests, samples are subjected to tensile deforma-tion with a constant engineering stress until theirrupture. Data are presented in the form of a graph,where is plotted versus time to failure t f . The depen-denceof time to failureon tensile stress isapproximatedby theEyringlaw (developed within thekinetic conceptof strength Zhurkov 1964 )

t f = t 0f exp 0

, (1)

where 0 and t 0f are positive parameters. Formula (1)means that observations in creep-failure tests (plottedin semi-logarithmic coordinates) lie on a straight line

= 0 1 log t f , (2)

where 0 and 1 are material constants, and log =log 10 . Although Eq. 2 is simple, determination of coef-cients 0 and 1 requires a numberof long-termexper-iments to be performed as accuracy of these tests isrelatively low (due to large discrepancies between mea-surements of t f on different samples).

Two empirical methods were suggested to assess thecoefcient 1 in independent experiments (when this

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64 A. D. Drozdov, J. deC. Christiansen

quantity is known, only short-term creep-failure testswith sub-yield tensile stresses are needed to nd theother coefcient 0 ). The rst is grounded on compar-ison of observations in creep-failure tests and tensiletests with various strain rates ( Klompen et al. 2005 ;Janssen et al. 2008 ). Under uniaxial tension, the yield

stress y (the point of maximum on a stressstrain dia-gram) weakly grows with strain rate following thedependence ( Bauwens-Crowet et al. 1974 )

y = y0 + y1 log , (3)

where y0 and y1 arematerial constants (Fig. 7 below).Observations reported by Klompen et al. (2005 ) andJanssen et al. (2008 ) show that

y1 = 1 , (4)

which allows the amount of time-consuming creep-

failure experiments to be reduced noticeably by usingresults of tensile tests with various strain rates.

The other approach is based on the analysis of obser-vations in short-term creep tests with various tensilestresses . When the strain in a creep test is plottedversus time t , three stages of creep are distinguished.The strain rate monotonically decreases with time atthe rst stage (primary creep), remains independent of time and equal to its minimum value min ( ) at theother stage (secondary creep), and strongly grows atthe stage of ternary creep (Figs. 8, 9 below). According

to the concept of diffusive viscosity ( Greenwood 1976 ,1990 ) the product min t f remains practically indepen-dent of applied stress [experimental verication of this statement on a polypropylene composite was givenin Little et al. (1995 )]. Expressing t f from the equalitymin t f = C , where C is a constant, and inserting theresult into Eq. 2, we arrive at

= 0 + 1 log min , 1 = 1 . (5)

Equation 5 can be employed for experimental determi-nationof 1 inEq. 2 by usingobservations in short-term

creep tests only.Although Eqs. 4 and 5 were proposed as empiri-cal relations, they can be derived within the conceptof inelastic deformations ( Bodner 1992 ). Splitting thetotal strain into the sum of elastic and inelastic strains

= e + i (6)

and assuming elastic strain e to be connected with ten-sile stress by the Hooke law

e = E

, (7)

where E stands for the Youngs modulus, while the rateof inelastic strain to obey the Eyring equation

d idt

= 0i sinh (c ), (8)

where c and 0i are material constants, we arrive at the

stressstrain relationd dt

+ E 0i sinh (c ) = E ddt

. (9)

Disregarding the durations of primary and ternarycreeps compared with that of the secondary creep,assuming i to grow within the second stage of creepow from its initial value i0 to its nal value i1 , where

i0 and i1 are independent of stress (both hypothesesare fairly well conrmed by observations, see Figs. 8,9), keeping in mind that ddt =

d idt in a creep test, and

integrating Eq. 8, we obtain

i = 0i sinh (c ) t f . (10)

with i = i1 i0 . Keeping in mind that sinh (c ) 12 exp (c ) at c 1, we transform Eq. 10 into Eq. 2with

0 = 1c

ln2 i

0i, 1 =

ln10c

. (11)

Under tension with a constant strain rate , the yieldpoint is determined from the condition d dt = 0. Thisequality together with Eq. 9 results in sinh (c y ) =

/ 0i , which implies Eq. 3 with

y0 = 1c

ln20i

, y1 = ln10

c. (12)

Equation 4 follows from Eqs. 11 and 12. To developEq. 5, it sufces to note that d idt = min during thesecondary stage of creep and to apply Eq. 10 with

i = C .Experimental data on polymers and polymer com-

posites (Barton and Cherry 1979 ; Lu and Brown 1990 ,1991 ; Teoh et al. 1992 ; Ben Hadj Hamouda et al. 2001 ,2007 ; Krishnaswamy 2005 , 2007 ) demonstrate thatEq. 2 predicts theeffectof stress ontimetofailure t f atrelatively large (sub-yield) stresses only. When obser-vations at all stresses < y are taken into account, thecurve ( log t f ) consists of two segments (correspond-ing to large and small stresses) described by straightlines(with lowandhighslopes, respectively) connectedby a short interval along which transition from the rstto the second regime of failure occurs (Figs. 15, 16and 17 below). The presence of a knee point on the

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Creep failure of polypropylene 65

( log t f ) diagrams is associated with (1) accumula-tion of damage under deformation and (2) transitionfrom the ductile (high stresses) to brittle (low stresses)regime of rupture. These two phenomena are distin-guished by which constitutive relations are modiedwhen an explicit dependence on history of deformation

is introduced. When Eq. 8 is revised by replacing thestress with an effective stress = /( 1 ) , where is some damage variable that reects nucleation,growth and coalescence of micro-voids and micro-cracks ( Kachanov 1999 ), changes in slope of a creep-failure diagram are attributed to material damage(Leckieand Hayherst1974 ; Wang andHabraken1996 ).When the inelastic strain at fracture i1 is assumed todecrease with time under deformation, one can speak about material embrittlement (Golub et al. 2000 ). Bothapproaches are limited in the sense that they adequatelyapproximate experimental data at large times (exceed-ing the knee point), but cannot predict these observa-tions based on measurements in short-term tests. Fora description of physically-based approaches to treat-ment of material damage, thereader is referredto Mish-naevsky and Brondsted (2008 ).

From the standpoint of applications, the most inter-esting parameter is the knee stress k that character-izes limits of applicability of Eq. 2. Assessment of time to failure at small stresses (far below the kneepoint) appears to be of secondary importance as it oftenexceeds design lifetimes of polymer structures.

To the best of our knowledge, no empirical tech-niqueshave been developed to predict theknee point ona creep-failure diagram by using observations in short-term tests only. Conventional approached are groundedon the time-temperature superposition principle (thatpermits the duration of creep-failure tests to be reducednoticeably by performing experiments at elevated tem-peratures and re-calculating their results to ambienttemperature) or its modications (the so-called accel-erated creep-failure tests), see Zornberg et al. (2004 ),Bueno et al. (2005 ), Alwis and Burgoyne (2006 , 2008 ),Kongkitkul and Tatsuoka (2007 ), Cai et al . (2008 ).Advantages andshortcomingsof these methods are dis-cussed in Kongkitkul and Tatsuoka (2007 ).

The following phenomenological approach is pro-posed to calculate the knee stress within the Eyringmodel. It is based on the observation (Fig. 10) that Eq. 8correctly ts the experimental dependence i( ) onlywhen different parameters 0i and c are used at lowand high stresses . Matching the data at low and high

stresses separately results in two theoretical curves thatintersect at some critical stress cr . Identifying this crit-ical stress with k , one can predict the knee point on acreep-failure diagram.

Anadvantageof this method is that it allows the kneestress to be found by using observations in short-term

creep tests without need to perform time-consumingexperiments. Itsshortcomingsarethat (1) this approachis model-oriented in the sense that it is grounded on theEyring equation 8 that (a) poorly describe observationsin tensile tests with constant strain rates in the post-yield region and (b) totally fail to predict experimentaldata in relaxation tests, and (2) no rational explanationis provided for the equality

cr = k . (13)

The objective of this study is three-fold: (1) to reportobservations on isotacticpolypropylene in uniaxial ten-sile tests with various strain rates, relaxation tests atvarious strains, and creep tests at various stresses, (2)to derive a constitutive model for the viscoelastic andviscoplastic responses and damage of semicrystallinepolymers that adequately describes the experimentaldata and to nd its adjustable parameters by tting theobservations, and (3) to demonstrate that Eq. 13 is ful-lled when the knee point on a creep-failure diagramis determined within the model.

The exposition is organized as follows. Experimen-

tal data are reported in Sect. 2. Constitutive equationsare developed in Sect. 3. Adjustable parameters in thestressstrain relations are found in Sect. 4. Results of numerical simulation for creep-failure tests are pre-sented in Sect. 5. Some concluding remarks are for-mulated in Sect. 6.

2 Experimental results

Isotactic polypropylene Moplen HP400R (density0.905 g / cm 3 , melt ow rate 25g/10 min (230 C,2.16 kg), tensile modulus 1.35 GPa) was purchasedfrom Basell Polyolens (Basell, Switzerland). Dumb-bell specimens (ASTM standard D638) with the cross-sectional area 9 .84 mm 3.95 mm were molded byusingthe injection-molding machine Ferromatic K110/ S60-2K.

Mechanical tests were conducted at ambient tem-perature with the help of a universal testing machineInstron-5568 equipped with electro-mechanical sen-sors for the control of longitudinal strains in the active

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Fig. 1 Stress versus tensile strain . Circles : experimental dataina tensile test witha cross-head speed of 10 mm/min. Solid line :results of numerical simulation

Fig. 2 Stress versus tensile strain . Symbols : experimentaldata ( tensile test with the cross-head speed 100 mm/min, , , , loading paths of relaxation curves). Solid line : pre-

diction of the model

zone of samples. The tensile force was measured by astandard load cell.

The experimental program involved three series of tensile tests. Each test was repeated on three differentsamples. Average (over three sets of data) observationsare presented in Figs. 1, 2, 3 and 4.

In the rst series of experiments, samples werestretched with constant cross-head speeds of 10 and

Fig. 3 Stress versus relaxation time t . Symbols : experimentaldata in tensile relaxation tests at various tensile strains . Solid lines : results of numerical simulation

Fig.4 Strain versus creep time t . Symbols : experimental data

in tensile creep tests with various engineering stresses MPa.Solid lines : predictions of the model

100 mm/min up to their necking. These cross-headspeeds corresponded to the strain rates = 2.1 10 3

and = 2.1 10 2 s 1 , respectively. The experimentaldata arereportedin Figs. 1 and 2, wherethe engineeringstress (the ratio of axial force to cross-sectional areaof specimens in the stress-free state) is plotted versus

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tensile strain . The following conclusions are drawnfrom these gures:

1. The stressstrain dependencies are strongly non-linear: the engineering stress (1) increases with

at small strains, (2) reaches its maximum y at

the yield strain y 0.1, and (3) weakly decreaseswith strain in the post-yield region.2. Necking of specimens occurs at a strain n 0.2

that slightly decreases with strain rate . I t isobserved on a stressstrain diagram as a pro-nounced reduction in engineering stress . Aftera neck is formed, the stress becomes practicallyindependent of .

3. Additional tests with higher strain rates (observa-tions arenot presented) show a transition from duc-tile rupture (necking) to brittle fracture of samples

at the cross-head speeds above 200 mm/min.The other series of tests involves 4 relaxation tests

with the strains = 0.031, 0.057, 0.114, and 0.143. Ineach test, a sample was stretched with the cross-headspeed of 100 mm/min up to the required strain. After-wards, the strain was xed, and a decay in engineeringstress was measured as a function of time t . Withreference to the protocol ASTM E328, the durationt rel = 20 min of short-term relaxation tests was cho-sen. Experimental data in relaxation tests are depictedin Figs. 2 and 3. In Fig. 2, stressstrain diagrams alongthe loading paths of the relaxation curves are reported.In Fig. 3, the engineeringstress is plotted versus loga-rithm of relaxation time t = t t 0 , where t 0 denotes aninstant at which the strain reached its required value.

The following conclusions are drawn from Fig. 3:

1. At strains in the sub-yield region ( < y ), therelaxation curves are practically parallel to eachother.

2. With thegrowthof strain in thepost-yield region,the slope of the curve ( log t ) monotonically

increases.The last series of experiments involves three creep

tests with the engineering stresses = 12 .2, 20.7, and27.0MPa. In each test, a specimen was stretched withthe cross-head speed of 100 mm/min until the stressreaches its required value. Afterwards, the engineeringstress was xed, and an increase the strain wasmeasured as a function of time. Following the protocolASTM D2990, the duration of creep tests t cr = 20min was chosen. Experimental data in creep tests are

reported in Fig. 4, where the strain is plotted versuslogarithm of creep time t = t t 0 (t 0 denotes an instantwhen the stress reached its maximum value).

The following conclusions are drawn from Fig. 4:

1. At small stresses , an increase in strain with

time is rather modest.2. With an increase in stress, the strain rate growspronouncedly, especially in the vicinity of a rup-ture point.

3. At the stress = 27 MPa, necking of samplesoccurred before the creep test nished.

Figure 2, where experimental stressstrain diagramsare presented on different samples, demonstrates highrepeatability of measurements. Deviations betweenobservations in tensile tests on different samples donot exceed 3% (except for strains in the close vicin-ity of the necking point n , where these discrepanciesgrowup to 10%). The same conclusion it true for obser-vations in relaxation tests (Fig. 3) and in a creep testwith = 12 .2 MPa (Fig. 4): appropriate relaxation andcreep curves on different specimens practically coin-cide (the difference is less than 3%). Observations ondifferent samples in creep tests with = 20.7 and27.0 MPa show some discrepancies, in particular, at rel-atively large times t . Deviations of experimental datafrom the average curves (depicted in Fig. 4) do notexceed 10%.

3 Constitutive equations

Our aim now is to develop constitutive equations for asemicrystalline polymer at arbitrary three-dimensionaldeformations with small strains. Connement to smallstrains appears to be natural as the purpose of the modelis to describe the mechanical response below the neck-ing point with n 0.2. To reduce the number of adjustable parameters in the stressstrain relations, weapply a homogenization concept according to whicha semicrystalline polymer with a complicated micro-structure is modeled as an equivalent one-phase con-tinuum ( Bergstrom et al. 2002 ). An incompressibleheterogeneous non-afne transient network of exiblechains subjected to damage is chosen as the equivalentmedium.

The incompressibility hypothesis is introduced tosimplify derivations only. It reects the fact thatpolypropylene is a weakly compressible polymer: its

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Poissons ratio ranges from 0.40 to 0.44 and slightlyincreasesunder stretching ( KolarikandPegoretti 2006 ).

Two types of chains are distinguished in an equiv-alent network: permanent and temporary ( Tanaka andEdwards 1992 ). Permanent chains do not detach fromtheir junctions, whereas temporary chains are rear-

ranged. When an end of an active temporary chain sep-arates from its junction at some instant 1 , the chainis transformed into the dangling state. When the freeend of a dangling chain merges with the network atinstant 2 > 1 , the chain returns into the active state.Attachment and detachment of temporary chains occurat random times being driven by thermal uctuations.

An inhomogeneous equivalent network consists of meso-regions with various activation energies for rear-rangement of chains. In the stress-free state, the rateof separation of active chains from their junctions in ameso-domain with activation energy u is governed bythe Eyring equation = exp [ u /( k BT )], where stands for a temperature-independent attempt rate, T isthe absolute temperature, and k B denotes Boltzmannsconstant. Conning ourselves to isothermal processesat a xed temperature T and introducing the dimen-sionless energy v = u /( k BT ) , we re-write this relationin the form

= exp ( v). (14)

Deformation of the network induces transformation of

meso-regions which is reected by changes in theiractivation energy. Activation energy of a meso-domain(with energy v in the reference state) at instant t 0reads v = A( t )v , where A( t ) is a scalar function thatobeys the initial condition A(0) = 1. It follows fromthis equality and Eq. 14 that

( t , v) = exp[ A( t )v ]. (15)

Equation 15 reects oneaspectof damageof an equiva-lent network: acceleration of its time-dependentresponse due to a decrease in activation energy forrearrangement of chains. To describe the other aspectof damage, we denote by N 1 and N 2 the numbers of permanent and temporary chains per unit volume of the network in its reference state. For an equivalentnetwork with no damage, these quantities are indepen-dent of time. For a network with damage, N 1 and N 2slowly decrease with time under deformation. Evolu-tion of these parameters is described by the differentialequationsd N 1dt

= ( t ) N 1 ,d N 2dt

= ( t ) N 2 (16)

with the initial conditions N 1 (0) = N 01 and N 2 (0) = N 02 , where N

01 and N

02 are constants, and the function

( t ) will be determined in what follows.Non-afnity of the equivalent network means that

junctions between chains slide with respect to their ref-erence positions under loading. Sliding (plastic ow)

of junctions is described by the strain tensor p . It isassumed that the rate of sliding of junctions is propor-tional to the rate of macro-deformationd pdt

= ( t )d dt

, (17)

where denotes the strain tensor for macro-deforma-tion, and the function ( t ) (1) equals zero in thereference state (junctions do not slide at very smalldeformations), (2) monotonically increases with strain(acceleration of plastic ow under loading), and (3)tends to its ultimate value = 1 at large deforma-tions (the rate of developed plastic ow coincides withthat of macro-deformation).

3.1 Rearrangement of a transient network

Rearrangement of a transient network is described bythe function n(t , , v ) that equals the number (per unitvolume) of temporary chains at time t 0 that havereturned into the active state before instant t andbelong to a meso-domain with activation energy v (in

the reference state). In particular, the numberof tempo-rary chains in meso-domains with activation energy vat time t reads n (t , t , v) , and the number of temporarychains that were active in the reference state and havenot separated from their junctions until time t is givenby n ( t , 0, v) . Thenumber of temporary chains that wereactive at the initial instant and detach from their junc-tionswithin the interval [t , t + dt ] reads n / t ( t , 0, v)dt , the number of dangling chains that return into theactive state within the interval [, + d ] is given by R(,v) d with

R(,v) = n

(t , , v ) | t = , (18)

and the number of chains that merged (for the last time)with the network within the interval [, + d ] anddetachfrom their junctionswithin theinterval [t , t + dt ]equals 2n / t ( t , , v ) dt d . The number n (t , v)ofactive temporarychains (per unit volume) that belongto meso-domains with activation energy v at time t isgiven by

n( t , v) = N 2( t ) p(v), (19)

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where p(v) stands for the (time-independent) distribu-tion function of temporary chains.

Detachment of active chains from their junctions isdescribed by the equations

n

t

(t , 0, v) = [ ( t , v) + ( t )] n( t , 0,v),

2n t

( t , , v ) = [ ( t , v) + ( t )]n

( t , ,v) .

(20)

Therst term in thesquare brackets reects reductioninthenumberof active chains due to their separation from junctions, while the other characterizes disappearanceof junctions driven by damage. Equations 20 state thatthe rate of separation of active chains from their junc-tions is proportional to thenumberof active chains in anappropriate meso-region. Integrating Eq. 20 with initialconditions 18 and 19, keeping in mind that n(t , t , v) =n (t , v) , and using Eq. 16, we nd that

n( t , 0, v) = p(v) N 2 (0) exp

t

0

( ( s , v) + ( s )) ds ,

n

( t , , v ) = p(v) N 2 () (, v) exp

t

( ( s , v) + ( s )) ds . (21)

3.2 Stressstrain relations

At small strains, the strain tensor for elastic deforma-tion e reads

e( t ) = (t ) p( t ). (22)

It follows from Eqs. 17 and 22 that

d edt

( t ) = (1 ( t ))d dt

( t ). (23)

The strain energy of a chain is given by w = 12 e :e , where stands for rigidity, and the colon denotesconvolution. Assuming the energy of inter-chain inter-action to be accounted for by the incompressibilitycondition only, we calculate the strain energy densityper unit volume of the network as the sum of strainenergies of active chains

W (t ) = 12

N 1( t ) +

0

n (t , 0, v) dv e( t ) : e( t )

+

0

dv

t

0

n

(t , , v ) e( t ) e ( )

: e( t ) e( ) d . (24)

The rst term in Eq. 24 equals the strain energy of permanent chains and temporary chains that have notbeen rearranged within the interval [0, t ], while the lastterm expresses the strain energy of chains that have lastmerged with the network at various instants [0, t ].It is presumed that the stress totally relaxes in a dan-gling chain before it merges with the network, whichmeans that the strain energy (at time t ) of a chain trans-formedinto theactive stateat time < t depends on therelative elastic strain tensor e (t , ) = e ( t ) e( ) .

For isothermal deformation of an incompressiblemedium, the ClausiusDuhem inequality reads

Q = dW dt

+ : d dt

0,

where Q stands for internal dissipation per unit volumeand unit time, and denotes the deviatoric componentof the stress tensor . Substituting Eq. 24 into this rela-tion, we conclude that the ClausiusDuhem inequalityis satised for an arbitrary deformation process, pro-vided that

( t ) = P ( t ) I + (1 ( t ))

( t ) e (t )

0

p(v) dv

t

0

() (, v)

exp

t

( ( s , v) + ( s )) ds e ( ) d ,

(25)

where P ( t ) stands for an unknown pressure, I denotestheunit tensor, = N 2 /( N 1 + N 2) is a constant, and thefunction ( t ) = [ N 1 (t ) + N 2 (t )] is governed by thedifferential equation

ddt

= ( t ), ( 0) = 0 (26)

with 0 = N 01 + N 02 .

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3.3 Material functions and adjustable parameters

Constitutive equation 25 together with kinematicEqs. 22 and 23, evolution equation 26 for damage andEq. 15 involve 4 material functions p(v), A( t ) ,( t ) ,and ( t ) .

With reference to the random energy model ( Derrida1980 ), weadopt thequasi-Gaussian formula for thedis-tribution function p,

p(v) = p0 exp v2

2 2 (v 0),

p(v) = 0 (v < 0).(27)

An advantage of Eq. 27 is that it contains theonly adjustable parameter > 0. The pre-factor p0is determined from the normalization condition

0 p(v) dv = 1.

The rate of damage is described by

= B qeq , (28)

where B and q arepositive constants, and eq = 32 :

12 stands for the equivalent stress.

The function that describes sliding of junctionsobeys the differential equation

ddt

= a eq (1 b e eq ) 2 , (29)

where a and b are positive constants, e eq = 23 e :

e12 , and eq = 23

d dt :

ddt

12. Although Eq. 29 is

treated as a phenomenological relation, some physicalmeaning may be ascribed to this kinetic equation of thesecond order. To justify this order, one can speculatethat sliding of junctions in the amorphous phase accel-erates slippage of crystalline lamellae, which, in turn,results in the growth of rate of plastic ow in amor-phous regions. The presence of b in Eq. 29 indicatesthat does not reach its ultimate value = 1 dueto breakage of crystallites whose remains slow down

sliding in the amorphous matrix.Mechanically-induced decay in activation energies

of meso-regions is described by

A = A0 A1 exp K w 2p , (30)

where A0 , A1 , and K are positive coefcients, and

wp( t ) = t

0 () : dp

dt ( ) d denotes the specic work of plastic deformations. Equation 30 is fullled untilsome instant atwhich itsright-hand side vanishes, while A = 0 afterwards.

The constitutive equations involve 11 material con-stants 0 , , , , a , b , A0 , A1 , K , B, q with the fol-lowing physical meaning:

1. The modulus 0 characterizes elastic propertiesof a virgin sample.

2. The parameters , , and describe its viscoelas-tic response; stands for a characteristic rate of relaxation, characterizes distribution of relaxa-tion times, and reects strength of the relaxationprocess.

3. The constants a and b determine rate of sliding of junctions in an equivalent network.

4. The quantities A0 , A1 , and K characterize theeffect of plastic deformation on the viscoelasticresponse.

5. The parameters B and q describe damage of a

semicrystalline polymer.The number of adjustable parameters is quite compa-rable with that in other constitutive models in visco-elastoplasticity of polymers that account for materialdamage ( Simo 1987 ; Wang and Habraken 1996 ,Schapery1999 ; Kaliskeetal.2001 ; Linand Schomburg2003 ; Marklund et al. 2008 ; Levesque et al. 2008 ).

3.4 Uniaxial tension

For uniaxial tension of an incompressible medium withtensile strain (t ) , the elastic strain e( t ) obeys theequation

d edt

= (1 )ddt

. (31)

The function is determined by

ddt

= a (1 b e ) 2ddt

, ( 0) = 0. (32)

The engineering stress ( t ) reads

( t ) = (1 ( t ))

E ( t ) e( t )

0

p(v) dvt

0

E () (, v) exp

t

( ( s , v) + ( s )) ds e( ) d ,

(33)

where the Youngs modulus E ( t ) = 32 ( t ) is governedby the equation

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d E dt

= E , E (0) = E 0 (34)

with

= B q . (35)

The function is determined by Eqs. 15 and 30, where

dwpdt

= ddt

, w p (0) = 0. (36)

For short-term tests, when a decrease in the elasticmodulus is disregarded, Eq. 33 reads

( t ) = E 0(1 ( t )) e( t )

0

Z ( t , v) p(v) dv ,

(37)

where the function Z ( t , v) = t

0 (, v) exp (

t

( s , v)ds ) e ( ) d obeys the differential equation Z t

( t , v) = ( t , v) [ e (t ) Z ( t , v) ] , Z (0, v) = 0.

(38)

Under tension with relatively large strain rates, whenthe viscoelastic effects are of secondary importance,Eq. 37 is simplied

( t ) = E 0[1 ( t )] e( t ). (39)

For a relaxation test with a xed strain , the quantitiese , , and A are independent of time, and the stress

is determined by

( t ) = s0 + s1

0

p(v) exp ( (v) t ) dv, (40)

where (v) is given by Eq. 15, and

s0 = (1 ) 0 , s1 = 0 , 0 = E 0 (1 ) e .

(41)

For a creep test with a xed stress , Eq. 37 impliesthat

e( t ) =

E 0 (1 ( t ))+

0

p(v) Z (t , v) dv. (42)

The only modication of Eq. 42 for a long-term creeptest (when material damage is taken into account) con-sists in replacement of the initial Youngs modulus E 0

with the current modulus E ( t ) ,

e( t ) =

E ( t )( 1 ( t ))+

0

p(v) Z ( t , v) dv. (43)

4 Determination of adjustable parameters

The objective of this section is two-fold: (1) to ndmaterial constants in the stressstrain relations bymatching the observations depicted in Figs. 14, and(2) to assess applicability of Eqs. 4 and 5 by compar-ison of results of numerical analysis for tensile testswith various strain rates, creep tests, and creep-failuretests.

4.1 Relaxation tests

We begin with the analysis of experimental data in atensile relaxation test with = 0.057 (Fig. 3). As thisstrain is rather small, no substantial changes in activa-tion energies of meso-regions occur along the loadingpath of an appropriate relaxation curve, which impliesthat A = 1 in Eq. 15. To nd adjustable parame-ters , , and , we x some intervals [0, max ] and[0, max ], where and are located, and divide theseintervals into J = 10 subintervals by the points (i ) =i , ( j ) = j (i, j = 1, . . . , J 1) with =

max / J and = max / J . For each pair { (i ) , ( j) },

the integral in Eq. 40 is evaluated numerically by theSimpson method with v = m v, m = 0, 1, . . . , M 1,where M = 200 and v = 0.4. The coefcients s0 ands1 are determined by the least-squares method from thecondition of minimum of the function

F =k

exp ( t k ) num ( t k )2 ,

where summation is performed over all instants t k atwhich the experimental data are reported, exp is theengineering stress measured in the test, and num isgiven by Eq. 40 . The coefcient is calculated fromEq. 41. After nding the best-t values (i ) and ( j ) ,this procedure is repeated for thenew intervals [ (i 1) ,

(i + 1) ] and [ ( j 1) , ( j+ 1) ], to ensure good accuracyof tting. The best-t parameters , , and are listedin Table 1. Afterwards, the experimental data in relax-ation tests with other strains are matched by usingthese values of , , and . Each set of observations isapproximated separately by means of the above algo-rithm with theonly adjustable parameter A. Thebest-tvalues of this parameter are reported in Fig. 5.

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Table 1 Adjustable parameters in the constitutive equations

Parameter Unit Value

E 0 GPa 1.61 0.75

9.40 s 1 0.75a 16.0b 0.43 A0 1.09 A1 8.38 10 2

K MPa 2 1.88 10 1

q 0.75 B MPa

12 s 1 5.0 10 7

Fig. 5 Parameter A versus plastic work w p . Circles : treatmentof experimental data in tensile relaxation tests at various strains

. Solid line : their approximation by Eq. 30

4.2 Tensile tests with constant strain rates

We begin with approximation of the experimentalstressstrain curve at tension with the cross-head speed100 mm/min (Fig. 2). Assuming this cross-head speedto be rather large, we disregard the viscoelastic phe-nomena and employ Eqs. 31, 32 , and 39 in the ttingprocedure. The quantities E 0 , a , and b are determinedby using the following algorithm. We x some intervals[0, amax ] and [0, bmax ], where a and b are assumed tobe located, and divide these intervals into J = 10 sub-intervals by the points a (i ) = i a , b ( j ) = j b (i , j =1, . . . , J 1) with a = amax / J and b = bmax / J .For each pair {a (i ) , b ( j) }, Eqs. 31 and 32 are integrated

numerically by the RungeKutta method with the stept = 1.0 10 4 s. The modulus E 0 is found by the

least-squares technique fromthecondition of minimumof the function

F =

k

exp ( k ) num ( k )2 , (44)

where summation is performed over all strains k atwhich the data are reported, exp is the engineeringstress measured in the test, and num is given by Eq. 39.When the best-t values a (i ) and b ( j ) are determined,theabove calculationsarerepeatedforthe new intervals[a (i 1) , a (i + 1) ] and b ( j 1) , b ( j+ 1) . After nding thematerial constants E 0 , a , and b, we integrate Eqs. 31,32, 36, and 39 with these parameters along the loadingpath of each relaxation curve presented in Fig. 3 andcalculate the plastic work wp . The coefcient A is plot-

ted versus wp in Fig. 5. The experimental dependenceis approximated by Eq. 30, where the exponent K isdetermined by the method of nonlinear regression, andthe coefcients A0 and A1 are calculated by the least-squares technique. The best-t values of A0 , A1 , andK are collected in Table 1.

We proceed with tting the stressstrain curve ina tensile test with the cross-head speed 10 mm/min(Fig. 1). Adjustable parameters E 0 , a , and b are deter-mined by tting the experimental diagram depicted inFig. 1 with the help of Eqs. 31, 32, and 3638, where

and A are given by Eqs. 15 and 30, respectively.The best-t values of E 0 , a , and b are determined bythe above procedure from the condition of minimum of function 44. These quantities are listed in Table 1.

The critical value of A corresponding to the begin-ning of necking

Acr = 0.43 (45)

is found from the condition of a noticeable drop of stress in the diagram depicted in Fig. 1. This value isused as a criterion of failure in all subsequent calcula-tions. Although Eq. 45 slightly differs from standardfailure criteria, combination of Eqs. 30 and 45 impliesthat an energy-based condition of rupture is used in thenumerical analysis (Brueller 1981 ).

To demonstrate the ability of the constitutive modelto predict the response in tensile tests, numerical simu-lation is conductedof thestressstrain curve for tensionof a specimen with the cross-head speed 100 mm/min.Figure 2 reveals good agreement between the experi-mental data and the results of numerical analysis.

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Fig. 6 Stress versus tensile strain . Solid lines : results of numerical simulation for uniaxial tension with the strain rates = 10 6 , 10 5 , 10 4 , 10 3 , 10 2 , and 10 1 s 1 , from bottomto top, respectively

To showthat theconstitutiveequationsprovide phys-icallyplausiblepredictions forotherstrainrates as well,we present results of numerical simulation for tensionwith thestrain rates = 10 6 , 10 5 , 10 4 , 10 3 , 10 2 ,and 10 1 s 1 in Fig. 6. Thisguredemonstrates ductilefracture of samples (with pronounced necking) under

deformation with low strain rates (below 0 .01 s 1 ) andbrittle fracture (with no necking) under stretching withrelatively high strain rates (above 0 .01 s 1 ).

For discussion of applicability of Eq. 4 to predictcreep-failure diagrams, numerical simulation is per-formed of stressstrain curves under tension with var-ious strain rates . For each strain rate, the maximumstress y on a stressstrain diagram is determined, andthis parameter is plotted versus in Fig. 7. Thisgure demonstrates good agreement between the dataand their approximation by Eq. 3, where the coef-

cients y0 and y1 are calculated by the least-squaresmethod.

4.3 Creep tests

To validate the model, numerical analysis is conductedof Eqs. 31, 32, 36, 38, and 42 for the short-term creeptests whose results aredepictedin Fig. 4. Thegoverningequations are, rst, integrated along the loading paths

Fig. 7 Yield stress

y versus strain rate

. Circles : results of numerical simulation. Solid line : their approximation by Eq. 3

of appropriate creep curves (stretching with the cross-head speed 100 min/min from zero up to an engineer-ing stress ), and afterwards, for tensile deformationsunder the constant stress. Figure 4 demonstrates goodagreement between the observations and predictionsof the model with the material parameters collected inTable 1.

Short-term creep curves are reported in Fig. 4 inthe semi-logarithmic form (log t ) . Linear plots of thesame curves (for a wider range of stresses ) arereported in Fig. 8, where three stages of creep oware clearly distinguished at relatively large (above24 MPa). Transitions from the primary to secondaryand from secondary to ternary creep ows can beobserved for all stresses . However, the smaller a ten-sile stress, the larger interval of time is necessary tostudy these transitions, see Fig. 9, where appropriatedata are presented for in the range between 14.5 and16.0 MPa.

For each creep curve depicted in Figs. 8 and 9, theinterval of secondary creep (along which the straingrows linearly with time) is tted with a straight line.The dependence of creep rate (that coincides with therate of inelastic strain i) on stress is reported inFig. 10. The data in regions of low ( 17 MPa) andhigh ( 20 MPa) stresses are approximated by Eq. 8separately. The adjustable parameters c and 0i in Eq. 8are found by the method of nonlinear regression andlisted in Table 2. Intersection of the theoretical curves

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Fig. 8 Strain versus time t . Circles : results of numerical simu-

lation forcreep testswith variousstresses MPa. Thinsolid lines :approximation of the creep curves along the intervals of second-ary creep by straight lines. Thick solid line : the critical straincharacterizing transition from the secondary to ternary creep

Fig. 9 Strain versus time t . Circles : results of numerical simu-lation forcreep testswith variousstresses MPa. Thinsolid lines :approximation of the creep curves along the intervals of second-ary creep by straight lines. Thick solid line : the critical straincharacterizing transition from the secondary to ternary creep

in Fig. 10 results in the critical stress

cr = 16 .8 MPa . (46)

Two conclusions are drawn from the data presented inFigs. 8 and 9 for the region of high stresses:

Fig. 10 Rate of inelastic strain i versus tensile stress .Circles : treatment of results of numerical simulation for tensilecreep tests. Solid lines : their approximation by Eq. 8

Table 2 Adjustable parameters c and 0i

Parameter Unit Value Remark

c MPa 1 1.00 Small stressesc MPa 1 0.57 Large stressesln 0i s

1 29 .87 Small stressesln 0i s

1 22 .59 Large stresses

1. At all stresses , (i) the secondary creep startswhen the strain reaches its initial value 0 0.046, and (ii) transition from the secondary to ter-nary creep occurs at the same strain cr 0.105.

2. At all stresses 15 MPa, duration of the sec-ondary stage of creep ow strongly exceeds thoseof the primary and ternary stages.

The latter conclusion is conrmed by the results pre-sented in Fig. 11 , where the strain rate in creep tests isplotted versus time t . This gure (1) demonstrates thatthe kinetics of decrease in strain rate is independentof tensile stress at the stage of primary creep, and(2) shows similarities between evolution of the strainrate at transition from the secondary to ternary stagesin creep tests with various stresses.

For each curve in Fig. 11 , the minimum strain ratemin is found (this strain rate coincides with the rate of secondary creep), and the dependence of stress onminimum strain rate is depicted in Fig. 12. The data

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Fig. 11 Strain rate versus time t . Solid lines : results of numer-ical simulation for creep tests with the engineering stresses =25, 26, 27, 28, 29, 30, 31, and 32 MPa, from bottom to top,respectively

Fig. 12 Stress versus minimum strain rate min . Circles :

results of numerical simulation for creep tests with various engi-neering stresses . Solid line : their approximation by Eq. 5

are approximated by Eq. 5, where the coefcients 0and 1 are calculated by the least-squares technique.

We proceed with numerical simulation of long-termcreep curves at various stresses. For each stress ,the dependence (t ) is calculated until an instant t f ,which is found from the condition that the coefcient A reaches its critical value Acr . Stress is plotted

versus t f in Fig. 13. The data in the region of largestresses ( 25 MPa) are approximated by Eq. 2,where the coefcients 0 and 1 are calculated by theleast-squares method. Figure 13 demonstrates that theresults of numerical analysis at relatively low stresses( < 22 MPa) slightly deviate from the straight line.

This means that assessment of time to failure at lowstresses without account for material damage leads tooverly optimistic estimates of lifetime, in accord withthe conclusion of Holmstrom and Auerkari (2008 ).

To show that our results are consistent with observa-tions, two long-term creep tests were conducted withtensile stresses = 17 and 20 MPa. In each test,stretching was performed with the cross-head speed100 mm/min until a required stress . Afterwards, thestress was xed, and evolution of strain with time t was monitored during 12 h. The experimental data arepresented in Fig. 14, where strain is plotted versustime t . As no creep rupture has been observed in thetest with = 17 MPa, time to failure t f is determinedby using the following algorithm. For each set of data,an appropriate creep curve is approximated by a strainline along the interval of secondarycreep (Fig. 14). Theparameter t f is calculated from theconditionthat tensilestrain at instant t f reaches its critical value cr . Theexperimental dependence ( t f ) is reported in Fig. 13(lled circles), which shows an acceptable agreementbetween the observations and predictions of the model.Although the experimental data depicted in Fig. 13cannot be treated as a strong validation of the consti-tutive equations (only two tests were carried out with-out attempts to assess statistics of creep rupture), theydemonstrate that the results of simulation do not differnoticeably from observations in long-term tests.

Comparison of Figs. 7, 12 , and 13 implies that thecoefcients y1 = 4.81 MPa , 1 = 4.92 MPa, and 1 = 4.87 MPa practically coincide (the differencebetween them is about 1%). This conrms applicabil-ity of Eqs. 4 and 5 for determination of parameter 1in Eq. 2.

5 Creep-failure tests

To evaluate the knee stress k , numerical simulation isconductedof long-termcreep testswithvariousstresses . First, Eqs. 31, 32, 3436, 38, and 43 are integratedalong the loading path of a creep curve (tension with thecross-head speed of 100 mm/min from the zero stress

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Fig. 13 Stress versus time to failure t f . Unlled circles : resultsof numerical simulation. Solid line : their approximation by Eq. 2.Filled circles : observations in long-term creep tests

Fig. 14 Strain versustime t . Circles : observations in long-termcreep tests with various stresses MPa. Thin solid lines : theirapproximation by straight lines along the intervals of second-ary creep. Thick solid line : the critical strain that characterizestransition from the secondary to ternary creep

until a required stress ). Afterwards, these relationsare integrated with a constant until the function A( t )reaches its critical value 45. The latter condition deter-mines time to failure t f . Calculations are performedwith the material constants listed in Table 1.

Fig. 15 Stress versus time to failure t f . Unlled circles : resultsofnumericalsimulation foran equivalent networkwith q = 0.75.Solid lines : their approximation by Eq. 2. Filled circles : obser-vations in long-term creep tests

The adjustableparameters B and q are chosenas fol-lows. The coefcient B is determined from the condi-tionthat thecreep-failurediagram ( log t f ) is describedby a straight line above the knee stress k (which meansthat there is no upward turn in the diagram similar tothat observed in Fig. 13 at large t f ). Results of simu-

lation show that B is practically independent of q andreads B = 5.0 10 7 MPa

12 s 1 .

Two conditions are employed to nd the best-tvalue of q . The rst requires that the knee stress k determined with the help of numerical analysis coin-cides with the critical stress cr found in Fig. 10. Thisimplies that q = 0.75. An appropriate stress-lifetimediagram is depicted in Fig. 15. It shows that (1) thecreep-failure diagram in the post-knee region is linear(in agreement with available experimental data), and(2) the phenomenological approach to determination of k as the point of intersection of two curves i( ) mea-sured at small and large stresses is in accord with theproposed constitutive model. Figure 15 reveals, how-ever, somedeviationsbetween the observations in long-term creep tests and the results of calculations.

The other condition for determination of the expo-nent q requires that the predicted creep-failure diagramin the sub-knee region coincides with the experimentaldata in additional tests. The best-t value of q foundfrom this requirement reads q = 0 .33. An appropriate

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Fig. 16 Stress versus time to failure t f . Unlled circles : resultsofnumericalsimulation foran equivalent networkwith q = 0.33.Solid lines : their approximation by Eq. 2. Filled circles : obser-vations in long-term creep tests

creep-failure curve is plotted in Fig. 16, which demon-strates that the dependence of stress on time to failureremains linear in the post-knee region, while the kneestress k = 14 .4 MPa is less than the critical stress crby about 17%.

To assess theeffect of q onknee stress k , the stressstrain relations are integrated with q = 0.5. The resultsof numerical analysis are reported in Fig. 17, whichshows that the knee stress equals k = 15.4 MPa,which is rather close to cr (the difference is 9%) on theone hand, while predictions of the constitutive modelat = 17 and 20 MPa are in good agreement withobservations in long-term creep tests, on the other.

Figures 1517 demonstrate that (1) the account formaterialdamagewith thehelp of Eqs. 16 and 28 ensuresthat creep-failure diagrams are linear both in the sub-knee and post-knee regions, and (2) the constitutivemodel implies the knee stress k that is close to thecritical stress cr found in Fig. 10.

6 Concluding remarks

Observations are reported on isotactic polypropylenein uniaxial tensile tests with various strain rates, relax-ation tests at various strains, and creep tests at variousstresses at room temperature.

Fig. 17 Stress versus time to failure t f . Unlled circles : resultsof numerical simulation for an equivalent network with q = 0.5.Solid lines : their approximation by Eq. 2. Filled circles : obser-vations in long-term creep tests

Constitutive equations are derived for the viscoelas-ticandviscoplastic responses of a semicrystallinepoly-merat an arbitrary three-dimensional deformation withsmall strains. A polymer is treated as an inhomoge-neous non-afne transient network of exible chainssubjected to damage. Damage of the network reects

two processes at the micro-level: (1) evolution of acti-vation energies for rearrangement of chains, and (2)reduction in concentration of junctions under loading.Adjustable parameters in the stressstrain relations arefound by tting the experimental data.

The model is applied to the analysis of creep rup-ture and prediction of theknee stress (that characterizestransition from ductile to brittle failure) by means of observations in short-term creep tests. Numerical sim-ulation demonstrates that

1. Equation 2 correctly describes the effect of stress

on time to failure t f when different coefcients m (m = 0, 1) are employed in the regions of duc-tile and brittle failure.

2. In the regime of ductile failure, Eqs. 4 and 5 ade-quately predict the coefcient 1 in Eq. 2.

3. Transition from ductile to brittle failure may beattributed to a slow decrease in concentration of active chains in an equivalent polymer network.

A phenomenological approach is proposed to pre-dict the knee stress. It is grounded on the observation

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that the Eyring equation 8 correctly approximates thedependence of inelastic strain rate on stress when dif-ferent parameters are used at small and large stresses.The knee stress k is identied with the critical stress cr at which these two diagrams intersect. Numericalanalysis shows that the knee stress calculated by means

of Eq. 13 differs from that found by taking into accountdamage of a polymer network by less than 20%.

Acknowledgments Financial support by the EuropeanCommission through project Nanotough213436 is gratefullyacknowledged. The authors are thankful to anonymous review-ers for valuable suggestions.

References

Alwis KGNC, Burgoyne CJ (2006) Time-temperature superpo-

sition to determine the stress-rupture of aramid bres. ApplCompos Mater 13:249264

Alwis KGNC, Burgoyne CJ (2008) Accelerated creeptesting foraramid bres using the stepped isothermal method. J MaterSci 43:47894800

Bergstrom JS, Kurtz SM, Rimnac CM, Edidin AA (2002) Con-stitutivemodelingof ultra-high molecularweightpolyethyl-ene under large-deformation and cyclic loading conditions.Biomaterials 23:23292343

Bauwens-Crowet C, Ots J-M, Bauwens J-C (1974) The strain-rate and temperature dependence of yield of polycarbon-ate in tension, tensile creep and impact tests. J Mater Sci9:11971201

Barton SJ, Cherry BW (1979) Predicting the creep rupture lifeof polyethylene pipe. Polym Eng Sci 19:590595

Ben Hadj Hamouda H, Simoes-Betbeder M, Grillon F, BlouetP, Billon N, Piques R (2001) Creep damage mechanisms inpolyethylene gas pipes. Polymer 42:54255437

Ben Hadj Hamouda H, Laiarinandrasana L, Piques R(2007) Fracture mechanics global approach conceptsapplied to creep slow crack growth in a medium densitypolyethylene (MDPE). Eng Fracture Mech 74:21872204

Bodner SR (1992) Unied plasticity for engineering applica-tions. Kluwer, New York

BruellerOS (1981) Energy-related failurecriteriaof thermoplas-tics. Polym Eng Sci 21:145150

Bueno BS, Costanzi MA, Zornberg JG (2005) Conventional and

accelerated creep tests on nonwoven needle-punched geo-textiles. Geosynth Int 12:276287Cai H, Miyano Y, Nakada M, Ha SK (2008) Long-term fatigue

strength prediction of CFRP structure based on microme-chanics of failure. J Compos Mater 42:825844

Derrida B (1980) Random-energy model: limit of a family of disordered models. Phys Rev Lett 45:7992

Golub VP, Pogrebnyak AD, Chernetskaya EV (2000) The kinet-ics of creep embrittlement under long-term static loading.Int Appl Mech 36:795802

Greenwood GW (1976) Fracture under creep conditions. MaterSci Engng 25:241245

Greenwood GW (1990) Mechanistic interpretations of someempirical correlations in creep and creep fracture. ISIJ Int30:795801

Guedes RM (2006) Lifetime predictions of polymer matrix com-posites under constant or monotonic load. Composites A37:703715

Holmstrom S, Auerkari P (2008) Effect of short-term data on

predicted creep rupture lifePivoting effect and optimizedcensoring. Mater High Temp 25:103109Janssen RPM, Govaert LE, Meijer HEH (2008) An analytical

method to predict fatigue life of thermoplastics in uniax-ial loading: sensitivity to wave type, frequency, and stressamplitude. Macromolecules 41:25312540

Kachanov LM (1999) Rupture time under creep conditions. IntJ Fracture 97:1118

Kaliske M, Nasdala L, Rothert H (2001) On damage modellingfor elastic and viscoelastic materials at large strain. CompStruct 79:21332141

Klompen ETJ, Engels TAP, van Breemen LCA, SchreursPJG, Govaert LE, Meijer HEH (2005) Quantitative predic-tion of long-termfailure of polycarbonate.Macromolecules

38:70097017Kolarik J, Pegoretti A (2006) Non-linear tensile creep of poly-

propylene: Time-strain superposition and creep prediction.Polymer 47:346356

Kongkitkul W, Tatsuoka F (2007) A theoretical framework toanalyse the behaviour of polymer geosynthetic reinforce-ment in temperature-accelerated creep tests. Geosynth Int14:2338

Krishnaswamy RK (2005) Analysis of ductile and brittle fail-uresfromcreep rupture testing of high-density polyethylene(HDPE) pipes. Polymer 46:1166411672

Krishnaswamy RK (2007) Inuence of wall thickness on thecreepruptureperformance of polyethylene pipe.Polym EngSci 47:516521

Leckie FA,HayherstDR (1974) Creep rupture of structures. ProcRoy Soc London A 340:323347

Levesque M, Derrien K, Baptiste D, Gilchrist MD (2008) Onthe development and parameter identication of Schapery-typeconstitutive theories.MechTime-DependentMater 12:95127

Little RE, Mitchell WJ, Mallick PK (1995) Tensile creep andcreep rupture of continuous strand mat polypropylenecomposites. J Compos Mater 29:22152217

Lin RC, Schomburg U (2003) A nite elastic-viscoelastic-elas-toplastic material law withdamage: theoretical and numeri-cal aspects. Comp Meth Appl Mech Engng 192:15911627

Lu X, Brown N (1990) The transition from ductile to slow crack

growth failure in a copolymer of polyethylene. J Mater Sci25:411416Lu X, Brown N (1991) Unication of ductile failure and slow

crack growth in an ethylene-octene copolymer. J Mater Sci26:612620

MarklundE, Eitzenberger J, VarnaJ (2008) Nonlinearviscoelas-tic viscoplastic material model including stiffness degra-dation for hemp/lignin composites. Compos Sci Technol68:21562162

Mishnaevsky L, Brondsted P (2008) Micromechanicalmodelingof damage and fracture of unidirectional ber reinforcedcomposites: A review. Comput Mater Sci 44:13511359

1 3


17/17


Orici AC, Herszberg I, Thomson RS (2008) Review of meth-odologies for composite material modelling incorporationfailure. Comp Struct 86:194210

Schapery RA (1999) Nonlinear viscoelastic and viscoplasticconstitutive equations with growing damage. Int J Fracture97:3366

Simo JC (1987) On a fully three-dimensional nite-strain vis-

coelastic damage model: Formulation and computationalaspects. Comp Meth Appl Mech Engng 60:153173Tanaka F, Edwards SF (1992) Viscoelastic properties of physi-

callycross-linkednetworks. Transient networktheory. Mac-romolecules 25:15161523

Teoh SH, Cherry BW, Kaush HH (1992) Creep rupture model-ling of polymers. Int J Damage Mech 1:245256

Wang XC, Habraken AM (1996) An elastic-visco-plastic dam-age model: from theory to applications. J Phys IV C 6:549558

Zhurkov SN (1964) Kinetic concept of the strength of solids. IntJ Fracture Mech 1:311323

Zornberg JG, Byler BR, Knudsen JW (2004) Creep of geo-textiles using time-temperature superposition methods.J Geotech Geoenviron Eng 130:11581168

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Documents

International Journal of Fracture Volume 159 Issue 1 2009 [Doi 10.1007_s10704-009-9384-x] a. D. Drozdov; J. DeC. Christiansen -- Creep Failure of Polypropylene- Experiments and Constitutive