fractional calculus derivation of a rate-dependent ppr

Int J Fract (2019) 216:1–29https://doi.org/10.1007/s10704-018-00334-w

ORIGINAL PAPER

Fractional calculus derivation of a rate-dependentPPR-based cohesive fracture model: theory, implementation,and numerical results

Oliver Giraldo-Londoño · Glaucio H. Paulino · William G. Buttlar

Received: 30 December 2017 / Accepted: 27 November 2018 / Published online: 28 March 2019© Springer Media B.V., onderdeel van Springer Nature 2019

Abstract Rate-dependent fracture has been exten-sively studied using cohesive zone models (CZMs).Some of them use classical viscoelastic material mod-els based on springs and dashpots. However, such vis-coelastic models, characterized by relaxation functionswith exponential decay, are inadequate to simulate frac-ture for a wide range of loading rates. To improvethe accuracy of existing models, this work presents amixed-mode rate-dependent CZM that combines thefeatures of the Park–Paulino–Roesler (PPR) cohesivemodel and a fractional viscoelastic model. This typeof viscoelastic model uses differential operators ofnon-integer order, leading to power-law-type relaxationfunctions with algebraic decay. We derive the model inthe context of damage mechanics, such that undam-aged viscoelastic tractions obtained from a fractionalviscoelasticmodel are scaled using two damage param-eters. We obtain these parameters from the PPR cohe-sive model and enforce them to increase monotoni-cally during the entire loading history, which avoidsartificial self-healing. We present three examples, two

O. Giraldo-Londoño · G. H. Paulino (B)School of Civil and Environmental Engineering, GeorgiaInstitute of Technology, Atlanta, GA, USAe-mail: [email protected]

W. G. ButtlarSchool of Civil and Environmental Engineering, University ofMissouri, Columbia, MO, USA

used for validation purposes and one to elucidate thephysical meaning of the fractional differential opera-tors. We show that the model is able to predict rate-dependent fracture process of rubber-like materials forawide range of loading rates and that it can capture rate-dependent mixed-mode fracture processes accurately.Results from the last example indicate that the order ofthe fractional differential operators acts as a memory-like parameter that allows for the fracture modeling oflong- and short-term memory processes. The ability offractional viscoelastic models to model this type of pro-cess suggests that relaxation functions with algebraicdecay lead to accurate fracture modeling of materialsfor a wide range of loading rates.

Keywords PPR cohesive zone model · Fractionalviscoelasticity · Fractional calculus · Fractionaldifferential equations · Rate-dependent fracture ·Damage mechanics · Fracture mechanics

1 Introduction

The fracture process in many materials is inherentlyrate-dependent. This rate dependence is due to eitherbulk-material viscoelasticity, the bond-breakage pro-cess in front of the crack tip, or a combination (Bažantand Li 1997). Understanding the underlying mech-anisms that produce rate dependence in the fractureprocess is of importance in several industries, includ-ing automotive, aeronautical, and civil, among others.

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2 O. Giraldo-Londoño et al.

For example, the fracturing process of structural adhe-sives in the automotive industry exhibits rate depen-dency (Marzi et al. 2009a, b; May et al. 2015), and thussuch phenomenon must be accounted for in the designof those adhesives. Likewise, fracture in bituminousmaterials, such as asphalt concrete, is rate dependent(Kim et al. 2009). Despite extensive previous researchin this area, the processes governing rate-dependentfracture are still not fully understood, and thus we rec-ognize the need to perform more studies pertaining tothis subject.

Studies conducted in the 1960s and 1970s exploredrate-dependence of fracture (Williams 1963, 1965;Knauss 1970; Mueller and Knauss 1971; Maugisand Barquins 1978). Many of those studies, althoughimportant for explaining some of the mechanisms thatgovern rate-dependence of fracture, mainly focused onsimple geometries and considered a small fracture pro-cess zone (FPZ). Those approaches are not suitable tosolve problems involving more complex geometries orloading conditions or problems in which the size ofthe FPZ is significant. To solve more complex prob-lems, alternative approaches such as cohesive zonemodels (CZMs) are desirable because they can sim-ulate nonlinear material separation and a large fractureprocess zone (Park and Paulino 2011). Several rate-independent CZMs have been proposed that assumepotential functions based on specific-degree polyno-mials (Needleman 1987; Freed and Banks-Sills 2008)or exponential-based functions (Needleman 1990a, b;Xu and Needleman 1993). In these models, the shapeof the softening portion of the cohesive relationshipcannot be easily modified. Several studies, however,have shown that the shape of the softening portion ofthe CZM significantly affects the simulated fracturebehavior (Volokh 2004; Alfano 2006), which indicatesthat fracture simulations require a model capable ofcontrolling the softening shape of the CZM.

To address the limitations of existing CZMs, thisstudy uses the Park–Paulino–Roesler (PPR) potential-based CZM (Park et al. 2009; Park and Paulino 2012).The PPR model is a potential-based model of generalpolynomial order able to control the shape of the soft-ening portion of the traction–separation relationships inboth the normal and tangential directions. The ability tocontrol the softening shape of the traction–separationrelations allows thePPRmodel to simulate awide rangeof failure responses (e.g., quasi-brittle). In addition tohave the ability to control the softening shape of the

traction–separation curves, the PPR model can consis-tently handle different fracture energies and cohesivestrengths in modes I and II, which is essential to modelmixed-mode fracture (Park et al. 2009).

The original PPR cohesive zone model, as wellas many other CZMs, is not able to simulate rate-dependent fracture in its primary form. However,with some additional considerations, rate-independentCZMs can be used to simulate rate-dependent frac-ture phenomena. For example, a study by Coriglianoet al. (2003) investigated the rate-dependent debondingprocess in composite materials using an exponential-type cohesive model. They enhanced the exponen-tial CZM with a dimensionless interface parameterthat was a function of the crack opening velocity.Alternatively, Zhou et al. (2005) used a phenomeno-logical CZM to study dynamic crack propagation ofpre-strained PMMA plates. Using a bilinear traction–separation relationship, they constructed a cohesivemodel in which the final crack opening width (i.e., thecrack opening width in which tractions vanish) was afunction of the crack tip velocity. They chose this func-tion such that the fracture energy increased exponen-tially as the crack tip velocity increasedwhile assumingthat the cohesive strength remained constant. In con-trast,Marzi et al. (2009a) incorporated rate-dependenceinto their CZM through phenomenological expressionsfor peak traction and critical energy release rate. Theyconstructed these expressions based upon experimen-tal observations from tapered double cantilever beamtests subjected to various applied loading rates.

Other phenomenological models have been reportedin the literature and are of interest in the present inves-tigation. For instance, to address the rate-dependentmixed-mode crack propagation of epoxy adhesives,Makhecha et al. (2009) presented two cohesive modelsbased onperturbations of an exponential function. Theyconstructed the models based on experimental obser-vations of mode I fracture tests performed on compacttension specimens. To simulate the high-speed frac-ture behavior of a steel alloy, Valoroso et al. (2014)used a cohesive element in which rate dependence wascaptured through an amplification function of the crit-ical energy release rate. May et al. (2015) developed amixed-mode rate-dependent cohesive zone model thatincorporated their experimental results on both tapereddouble cantilever beam (TDCB) and tapered end-notched beam bonded with a crash-optimized adhe-sive. The experiments showed that, as the crack open-

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Fractional calculus derivation of a rate-dependent PPR-based cohesive fracture model 3

ing rate increased, fracture toughness and cohesivestrength in both the normal and tangential directionsalso increased. Their model represented both the frac-ture energy and cohesive strengths with phenomeno-logical expressions in terms of the crack opening rate.

The rate-dependent CZMs discussed above are typ-ically derived from a rate-independent CZM by defin-ing their fracture energy, cohesive strength, and/orfinal crack opening as a function of the crack open-ing rate. Therefore, those models may neglect somephysical processes related to rate-dependent fracture.Other models account for physical processes, such thatthe FPZ is modeled as a viscoelastic media. For exam-ple, some models assume that the FPZ consists of vis-coelastic fibrils that becomedamagedover time (Rahul-Kumar et al. 1999; Allen and Searcy 2001) and othersassume that FPZ is represented by nonlinear softeningsprings and dashpots (Xu et al. 2003a, b). They con-structed amode I rate-dependentCZMby replacing oneof the springs of a standard linear solid (SLS) modelwith a nonlinear softening spring, whose traction–separation behavior followed an exponential function.Although theirmodel was able to fit the obtained exper-imental results, it predicts infinite fracture energy asthe opening rate approaches infinity. Using a similarapproach, Musto and Alfano (2013) studied the rate-dependent debonding process in a double cantileverbeam (DCB) bonded by an interface of SBR/NR-blendrubber. They developed their model using conceptsfrom both linear viscoelasticity and damage mechan-ics by first obtaining normal tractions from an SLSmodel, and then scaling themusing a damage-type vari-able consistent with a rate-independent bilinear CZMin mode I. However, their numerical results were not ingood agreement with experimental data obtained fromDCB tests performed at various loading rates (Mustoand Alfano 2013).

The studies discussed above use classical viscoelas-ticmodels that consist of springs and dashpots, and thusthey have limited ability to predict the fracture behav-ior for a wide range of loading rates. Recently, Mustoand Alfano (2015) proposed a rate-dependent CZMusing fractional viscoelastic theory (Mainardi 2010).Similarly to their previous work (Musto and Alfano2013), they used an SLS model, but instead of usinga dashpot in the Maxwell arm, they used a more gen-eral element whose constitutive relationship is definedin terms of differential operators of non-integer order(Mainardi 2010). This generalized element, typically

called springpot, can interpolate between a spring anda dashpot. In contrast to their earlier work (Musto andAlfano 2013), results using the fractional viscoelasticmodel (Musto and Alfano 2015) closely matched theresults of experiments performed on DCB test speci-mens.

It is not surprising that the new cohesive zone modelpresented by Musto and Alfano (2015) is able to cap-ture the fracture response of an elastomeric material fora wide range of loading rates. That is because rheolog-ical models of fractional order, compared to classicalviscoelastic models, are able to describe more com-plex viscoelastic behavior. In particular, fractional vis-coelasticmodels are able tomodel both short- and long-term hereditary phenomena, which is possible becausethese models are characterized by relaxation func-tions of algebraic decay instead of exponential decay1

(Mainardi 2010), which is consistent with observationsmade for several materials (e.g., see Kapnistos et al.2008; Olard and Di Benedetto 2003; Davis et al. 2006,to name a few). As discussed in the subsequent section,viscoelastic models of fractional order have success-fully been used to represent the viscoelastic behavior ofseveral materials such as asphalt, polymers, biologicaltissue, among others. These attributes have motivatedus to use a fractional viscoelasticmodel to represent therate-dependent response of our mixed-mode cohesiveelement.

In this study, we combine the features of both thePPRpotential fracturemodel and a fractional viscoelas-ticmodel to derive a rate-dependentmixed-modeCZM.We develop the cohesive fracture model in the contextof damagemechanics. First, we obtain undamaged nor-mal and tangential tractions from a mixed-mode frac-tional SLS model and scale them with two scalar dam-age parameters that we obtain from the mixed-modePPR cohesive fracture model by Park et al. (2009).Unlike previous studies that only consider mode I frac-ture (Musto and Alfano 2015), the present model isable simulate the rate-dependent bond-breakage pro-cess in front of the crack tip under mixed-mode condi-tions. Using the presentmodel, we are also able tomod-ify the shape of the softening portion of the traction–separation curves through two shape parameters andto consider different fracture energies and cohesivestrengths in modes I and II consistently.

1 Relaxation functions of algebraic decay are ubiquitous in clas-sical viscoelastic models that are based on springs and dashpots.

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Another novelty of the present formulation is amethod that we introduce to compute the fractionalderivatives, which allows for a variable time step. Incontrast to traditional methods for numerical compu-tation of fractional derivatives (e.g., see Musto andAlfano 2015; Schmidt and Gaul 2002), our abilityto use a variable time step facilitates implementationof our rate-dependent CZM into into a finite element(FE) code that uses variable time stepping algorithms.The results from our model resemble experimentallyobserved results from both mode I and mixed-modefracture tests and provide insight regarding meaningand implications of fractional differential operators inrate-dependent fracture analysis. The framework pre-sented in this study can be used to extend existingrate-independent mixed-mode CZMs to capture rate-dependent fracture processes.

The remainder of this paper is organized as fol-lows. First, Sect. 2 provides some theoretical back-ground of fractional calculus and fractional viscoelas-ticity, and shows that fractional viscoelastic modelslead to power-like relaxation functions, typical of long-term hereditary phenomena. Section 3 presents detailsof the formulation of the mixed-mode rate-dependentCZM. Section 4 presents three comprehensive exam-ples, and Sect. 5 provides some concluding remarks.Three appendices complement the manuscript.

2 Preliminary theoretical considerations

This section describes some of the main conceptsrelated to fractional calculus and their applicationto fractional viscoelasticity. First, we review theRiemann–Liouville fractional integral and use it toobtain the Caputo fractional derivative. Then, wediscuss the numerical solution of the Caputofractional derivative, followed by applications offractional calculus in viscoelasticity. Finally, we usefractional viscoelasticity to generalize the standardlinear solid model, typically used in viscoelasticity.For additional details about fractional calculus and/orfractional viscoelasticity, we refer the interested read-ers to Oldham and Spanier (1974), Miller and Ross(1993), Podlubny (1998), Mainardi (2010).

2.1 Basic concepts of fractional calculus

Consider the Cauchy formula for repeated integrationshown below:

0 Int f (t) =

∫ t

0

∫ τ1

0· · ·

∫ τn−1

0f (τn)dτn · · · dτ1

= 1

(n − 1)!∫ t

0(t − τ)n−1 f (τ )dτ. (1)

This formula represents the nth repeated integral of thefunction f(τ ), τ ∈ (0, t). For notation purposes, we usethe left subscript, 0, and right subscript, t , to indicatethat the lower and upper limits of the integral aboveare τ = 0 and τ = t , respectively, and thus this nota-tion is used in all equations hereafter. We can extendCauchy’s formula from positive integer values of n toany positive real value v by substituting (n − 1)! with�(v) in Eq. (1), inwhich�(·) denotes theGamma func-tion. This substitution leads to the so-called Riemann–Liouville fractional integral of order v > 0,

0 Ivt f (t) = 1

�(v)

∫ t

0(t − τ)v−1 f (τ )dτ,

t > 0, v > 0. (2)

By construction, when v = n ∈ N, where N isthe set of natural numbers, the Riemann–Liouvillefractional integral reduces to the Cauchy formula forrepeated integration. This definition of a fractional inte-gral allows us to obtain differential operators of frac-tional order (Mainardi 2010).

One of these differential operators is the Caputofractional derivative of order v > 0, which is obtainedby applying the Riemann–Liouville integral operatorof order (m − v) to the integer derivative of order mof a function f (t), where m − 1 < v < m, m ∈ N,leading to

0Dvt f (t) = 0 I

m−vt ◦ Dm

t f (t)

=

⎧⎪⎪⎨⎪⎪⎩

1

�(m − v)

∫ t

0

f (m)(τ )

(t − τ)v+1−mdτ if m − 1 < v < m, m ∈ N

dv f (t)

dt vif v ∈ N

.

(3)

The superscript v in 0 I vt f (t) (Eq. 2) and 0Dv

t f (t)(Eq. 3) represents the order of the fractional integraland fractional derivative, respectively. Note that whenv /∈ N, the Caputo fractional differential operatorshown in Eq. (3) (as well as other fractional differentialoperators) becomes non-local because it depends on thebehavior of f (τ ) for τ ∈ [0, t]. In the next subsection,we show different methods used to compute the Caputofractional derivative and discuss the method adopted inthis study.

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2.2 Numerical approximation of the Caputo fractionalderivative

We present several numerical methods used to evaluatefractional differential operators and fractional differen-tial equations followed by a numerical scheme that weintroduce to solve the Caputo fractional derivative. Theintroduced numerical scheme allows for variable step,which is suitable for implementation in a finite elementcode using variable time-stepping algorithms. Severalalgorithms have been proposed to evaluate fractionalderivatives and to solve fractional differential equa-tions. Diethelm et al. (2002) introduced a predictor-corrector approach that aims to solve a fractional dif-ferential equation of the form Dv

t y(t) = f (t, y(t)).Additional numerical techniques to solve fractional dif-ferential equations and to evaluate fractional deriva-tives is presented by Diethelm et al. (2005). To eval-uate the Caputo fractional derivative based on mea-sured data with noise, Murio (2006) presented a regu-larization technique that leads to a stable evaluation ofsuch derivative. Krishnasamy et al. (2017) presenteda methodology to solve fractional differential equa-tions that is based on fractional Taylor basis approx-imations.

The numerical approaches described above typi-cally use a constant time step to evaluate the fractionalderivatives. Here, we present an approach that allowsus to use a variable time step to evaluate the Caputoderivative of order v ∈ (0, 1). Consider the followingCaputo fractional derivative of order v ∈ (0, 1) of afunction f (t) evaluated at t = tN ,

0Dvt f (tN ) = 1

�(1 − v)

∫ tN

0

f (τ )

(tN − τ)vdτ, 0 < v < 1. (4)

To find an approximate expression for this fractionalderivative, we first divide the time interval (0, tN ) intoN segments, which need not be equally spaced, asshown in Fig. 1.

t0 = 0 t1 t2 t3 t4 t5 tN−1 tt

N

Fig. 1 Time discretization used to evaluate the Caputo fractionalderivative in Eq. (4). In contrast to the constant time step requiredin theGrünwald–Letnikov fractional derivative (Uchaikin 2013),our method employs a variable time step (see “Appendix A”)

Next, we approximate f (τ ) in each time interval(t j−1, t j ), j = 1, . . . , N , with its backward finitedifference approximation, and substitute it into Eq. (4),which leads to the following sumof piecewise integrals:

0Dvt f (tN ) ≈ 1

�(1 − v)

N∑j=1

f (t j ) − f (t j−1)

t j − t j−1

∫ t j

t j−1

(tN − τ)−vdτ. (5)

After some algebraic maneuvering, Eq. (5) is rewrittenas

0Dvt f (tN ) ≈ f (tN ) − S f

�(2 − v)�t vN, (6)

where

S f = f (tN−1) +N−1∑j=1

[f (t j ) − f (t j−1)

]A j ,

A j = (tN − t j )1−v − (tN − t j−1)1−v

(t j − t j−1)�t−vN

,

�tN = tN − tN−1. (7)

“Appendix A” presents a detailed assessment of thenumerical scheme discussed above, which we use toestimate the Caputo fractional derivative of a functionf (t).

2.3 Fractional calculus and viscoelasticity

Here, we show the relationship between the Caputofractional derivative and the behavior of linearly vis-coelastic materials with long-term memory, such aspolymers. The relaxation behavior of polymers is nat-urally described by power-law type relaxation func-tions over a wide range of frequencies (Rubinstein andObukhov 1993; Kapnistos et al. 2008; Knauss et al.2008; Milner and Newhall 2010; Uchaikin 2013). Inits simplest form, a relaxation function of this type canbe defined as

G(t) = a

�(1 − v)t−v , 0 < v < 1, t > 0, (8)

where a/�(1 − v) is a constant measuring the “rigid-ity” of the material, and v is a relaxation parameter.In contrast to relaxation functions based on exponen-tial kernels, a power-law relaxation function under-goes algebraic decay, which we can use to representmaterials with long-term memory. From Eq. (8), the

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stress–strain relationship for this type of material isgiven by

σ(t) =∫ t

0G(t − τ)ε(τ )dτ = a

�(1 − v)∫ t

0(t − τ)−v ε(τ )dτ, 0 < v < 1, t > 0, (9)

where ε(τ ), τ ∈ [0, t] is the strain history up to time t .Without loss of generality, we consider only the one-dimensional case (Padovan 1987; Schmidt and Gaul2002).

Comparing Eqs. (3) and (9), one can observe thatthe stress, σ(t), for a material following a power-lawtype relaxation function such as that in Eq. (8) can bewritten in terms of a Caputo fractional derivative withexponent v ∈ (0, 1) (i.e., using m = 1 in Eq. (3)) asfollows:

σ(t) = a 0Dvt ε(t), (10)

where

0Dvt ε(t) = 1

�(1 − v)

∫ t

0

ε(τ )

(t − τ)vdτ, 0 < v < 1.

(11)

Therefore, we can use a stress–strain relationshipbased on Caputo fractional differential operators tomodel viscoelastic materials with long-term memory.A constitutive equation of the form shown in Eq. (10)was introduced by Scott-Blair (1947) to describe aviscoelastic material with behavior between that ofa Hookean solid and that of a Newtonian fluid. Infact, Eq. (10) represents the behavior of a Hookeansolid when v → 0, and a Newtonian fluid whenv → 1. The material represented by Eq. (10) givesrise to the more general element known as spring-pot (Mainardi 2010). As depicted in Fig. 2, a spring-pot can be used to represent either a spring or adashpot, depending on the order of the fractionalderivative.

Several studies have shown that a springpot can berepresented as an infinite series of springs and dash-pots arranged in a hierarchical manner (Schiessel andBlumen 1993; Schiessel et al. 1994, 1995; Schiesseland Blumen 1995). The hierarchical representation ofa springpot explains why, as compared to classical vis-coelastic models, those of fractional order can simulatecomplex viscoelastic responses using a small amountof parameters. A springpot can be used to generalize

Spring DashpotSpringpot(v = 0) (0 < v < 1) (v = 1)

η ηη, v

σ = ηε σ = ηεσ = η 0Dvt ε

Fig. 2 Stress–strain relationships represented by a springpot.When v → 0, a springpot takes the form of a standard spring,and when v → 1 the springpot takes the form of a standarddashpot

classical viscoelastic models such as Maxwell, Voigt,and Zener (or SLS) models. The generalization can beachieved by substituting the dashpots (found in clas-sical models) with springpots (Mainardi 2010). Thesetypes of rheological models have been used to accu-rately simulate the behavior (in frequency domain)of asphalt (Olard and Di Benedetto 2003), acous-tical damping materials (Gourdon et al. 2015), andpolymers (Kontou and Katsourinis 2016). In studiesby Olard and Di Benedetto (2003), Gourdon et al.(2015), the springpot is referred to as a paraboliccreep element. These types of models have also beenused to accurately represent the viscoelastic behavior(in time domain) for biological tissue (Davis et al.2006; Craiem et al. 2008; Dai et al. 2015), polymers(Welch et al. 1999; Kapnistos et al. 2008), amongother materials. In this work, we use a fractional gen-eralization of the SLS model because it demonstratespromise as a tool for the realistic simulation of frac-ture in rate-dependent materials (Musto and Alfano2015).

2.4 Fractional SLS model

A schematic representation of a one-dimensional frac-tional SLS model is illustrated on the top-right cornerof Fig. 3. Parameter η is a viscosity-like quantity andv ∈ (0, 1) defines the order of the Caputo fractionalderivative. Both parameters η and v dictate the behav-ior of the springpot. In addition, parameter E definesthe stiffness of the equilibrium spring (i.e., the lowerspring in the schematic of the fractional SLS model)and parameter c controls the stiffness (cE) of the springin the fractionalMaxwell element (i.e., the upper spring

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Time (s)10-3 10-2 10-1 100 101 102 103

Nor

mal

ized

rela

xatio

n m

odul

us,

0.5

0.6

0.7

0.8

0.9

1η , v cE

E

σ

ε

v = 0 3ˆ .

v = 0 5ˆ .

v = 0 7ˆ .

v = 0 9ˆ .

G(t

)/(1

+c)

E

Fig. 3 Relaxation modulus of a fractional SLS model for vari-ous orders v of the Caputo fractional derivative. For illustrationpurposes, the relaxation functions shown here are obtained usingEq. (12), with E = 1, c = 1, and η = 1 (online version in color)

in the schematic of the fractional SLS model). Whenε → 0 (slow limit), the stiffness of the fractional SLSmodel is E , and when ε → ∞ (fast limit), its stiffnessis (1 + c)E .

The viscoelastic response of the fractional SLSmodel is characterized by its relaxationmodulus,whichis given as (Mainardi 2010)

G(t) = E[1 + cEv

(−(t/τ )v

)], (12)

where:

τ = [η/(cE)

]1/v and Ev(t) =∞∑n=0

tn

�(vn + 1).

(13)

Here, Ev(t) is the Mittag-Leffler function (Mainardi2010). The behavior of the relaxationmodulus given byEq. (12) is illustrated in Fig. 3 for several values of v.Note that as v increases, we observe more stress relax-ation for t > τ (materials with short-term memory),and as v decreases, we observe less relaxation for t > τ

(materialswith long-termmemory). Therefore, one canconclude that, with the proper choice of v, a fractionalSLS model can simulate both materials with short- andlong-termmemory.Materialswith the shortestmemorycorrespond to v → 1, in which the relaxation functionfrom Eq. (12) reduces to the relaxation function for aclassical SLS model2, i.e., G(t) = E(1 + ce−t/τ ).

2 To obtain the standard SLS expression from Eq. (12), we eval-uate the Mittag-Leffler function for v = 1, which reduces to

A study by Adolfsson et al. (2005) numericallyshowed that to obtain the same relaxation function ofa fractional SLS model, a classical standard viscoelas-tic model would need an infinite number of Maxwellelements. This finding is another indication that a rhe-ological model of fractional order, compared to a clas-sical rheological model, can represent complex vis-coelastic behavior for a wide range of frequencies (ortime) with fewer input parameters, which is one of thereasons for adopting the fractional SLS model in ourformulation.

3 PPR-based rate-dependent CZM usingfractional calculus

We construct a new mixed-mode rate-dependent CZMin the context of damage mechanics. We begin byobtaining undamaged tractions Sk from a fractionalSLS model (Musto and Alfano 2015; Mainardi 2010)and then scale them with two scalar damage-typeparameters dk extracted from the PPR cohesive zonemodel by (Park et al. 2009). Here, subscript k takesvalues k = n and k = t , which refer to quantitiesevaluated in the normal and tangential directions to thecrack plane, respectively. We use the undamaged trac-tions, Sk , and the damage parameters, dk , to obtain thecohesive tractions,

Tk = (1 − dk) Sk, (k = n, t). (14)

The model can be visualized as two fractional SLSmodels, each coupled in series with a damage element,as illustrated in Fig. 4b. In the following subsection,we present details of the derivation of the undamagedtractions, Sk , which are obtained from the normal andtangential fractional SLS models. Next, we elaborateon the derivation of the damage parameters. Finally, wepresent the final expressions for the proposed mixed-mode rate-dependent cohesive tractions, Tk , and thecorresponding material tangent matrix.

3.1 Undamaged tractions Sk

To obtain the undamaged viscoelastic tractions in bothnormal and tangential directions to the crack plane, we

the Taylor series expansion of the exponential function. That is,Ev=1(t) = ∑∞

n=0tn

�(n+1) = ∑∞n=0

tnn! = et .

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ηt, v cEt

Et

ηn, v

cEn

En

Sn

St

Δn

Δt

Sn

St

Crackfaces

ηt, v cEt

Et

dt

ηn, v

cEn

En

dn

Damage elements

Tn

Tt

Δn

Δt

Tn

Tt

Crackfaces

(b)(a)

Fig. 4 Rheological representation of the mixed-model rate-dependent CZM: a schematic illustration of two fractional SLSmodels used to obtain the normal and tangential undamaged trac-

tions, Sn and St ; and b schematic illustration of two fractionalSLSmodels, each coupled in series with a damage element, usedto obtain the normal and tangential cohesive tractions, Tn and Tt .

use two fractional SLS models as those depicted inFig. 4a. We assume that in the undamaged stage, nor-mal crack opening displacements produce only normaltractions, and tangential crack opening displacementsproduce only tangential tractions. In other words, weassume no coupling between normal and tangentialtractions in the undamaged stage. To couple normal andtangential cohesive tractions, we employ two damageparameters, as we will discuss in detail in the followingsubsection.We obtain the undamaged viscoelastic tractions, Sk ,from the solution of the fractional differential equa-tions

Sk + λk 0Dvt Sk = Ek�k + γk 0D

vt �k, (k = n, t),

(15)

where:

λk = ηk/(cEk), and γk = ηk(1 + c)/c. (16)

To find a solution to Eq. (15), we approximatethe fractional derivatives numerically. If the fractionalderivatives are approximated using traditional meth-ods, such as the Grünwald–Letnikov approximation(Uchaikin 2013; Musto and Alfano 2015) which usesa constant time step, we expect to encounter a mis-match between the time step used to compute the frac-tional derivative and that used in the finite element (FE)

model. The mismatch in time steps is more prominentwhen simulating fracture because the FE model oftenrequires adaptive time-stepping algorithms. Becauseof the mismatch between time steps, one needs tointerpolate between the FE model results at each loadstep to approximate the fractional derivatives, whichintroduces additional errors to the numerical solution.For this reason, we approximate the fractional deriva-tives using a method that allows for variable time step,according to Sect. 2.2.

To solve the fractional differential equation (15), weuse the approximate expression for the Caputo frac-tional derivative given in Eq. (6) to evaluate 0Dv

t Skand 0Dv

t �k . After some algebraic maneuvering, theexpression that we obtain for the undamaged tractions,Sk , are3:

Sk(tN ) = Ek�k + �SHk , (k = n, t), (17)

where

Ek =(1 + cλk

λk + �(2 − v)�t vN

)Ek,

3 We can use an alternative model (e.g., see Giraldo-Londoñoet al. 2018) to estimate the rate-dependent undamaged tractions,which requires no computation of fractional derivatives.

123


�SHk = λk SSk − γk S�k

λk + �(2 − v)�t vN, (18)

and

SSk = Sk (tN−1) +N−1∑j=1

[Sk(t j ) − Sk(t j−1)

]A j ,

S�k = �k (tN−1) +N−1∑j=1

[�k(t j ) − �k(t j−1)

]A j .

(19)

The terms �SHk in Eq. (17) account for the history

of tractions and crack opening displacements betweent = 0 and t = tN−1.

3.2 Normal and tangential damage parameters dk

As illustrated by Fig. 4b, we adopt two damage param-eters, dk , which we obtain from the PPR cohesive zonemodel (Park et al. 2009) as conducted in a study bySpring et al. (2016). The damage parameters, dk , arecomputed as

dk (κn, κt ) = 1 − Tk (κn, κt )

Ekκk, (k = n, t), (20)

where Tk(κn, κt ) are the normal and tangential cohesivetractions from the rate-independent PPR cohesive zonemodel (Park et al. 2009), which are given by

Tn(κn, κt ) = �n

δn

[m

(1 − κn

δn

)α (m

α+ κn

δn

)m−1

−α

(1 − κn

δn

)α−1 (m

α+ κn

δn

)m]

×[�t

(1 − κt

δt

)β (n

β+ κt

δt

)n

+ 〈φt − φn〉]

,

(21)

Tt (κn, κt ) = �t

δt

[n

(1 − κt

δt

)β (n

β+ κt

δt

)n−1

−β

(1 − κt

δt

)β−1 (n

β+ κt

δt

)n]

×[�n

(1 − κn

δn

)α (m

α+ κn

δn

)m

+ 〈φn − φt 〉]

.

(22)

As discussed by Spring et al. (2016), the damageparameters in Eq. (20) and the cohesive tractions inEqs. (21) and (22) are functions of two kinematic quan-tities, κn and κt . The first represents the maximum nor-mal crack opening in the loading history, and the secondthe maximum absolute tangential crack opening in theloading history. These quantities satisfy the followingrelations (Spring et al. 2016):

κ i+1n = max

{κ in,�

i+1n

}and

κ i+1t = max

{κ it ,

∣∣∣�i+1t

∣∣∣}

, (23)

where in the context of afinite element solution scheme,the superscripts i and (i + 1) refer to the solutions attime increments i and (i + 1), respectively.

We need to satisfy the relations in Eq. (23) to ensurethat the damage parameters monotonically increasethroughout the loading history (i.e., di+1

n ≥ din anddi+1t ≥ dit ), which is consistent with the assump-tion that no self-healing takes place during fracture.We use Fig. 5 to illustrate the monotonicity conditionimposed on the damage parameters. The figure showstypical normal and tangential PPR cohesive tractionsTk(κn, κt ) (Fig. 5a, b) and their corresponding damageparameters dk(κn, κt ) (Fig. 5c, d). We observe that thetwo damage parameters start at zero (when no damageoccurs), and monotonically increase to one as the kine-matic quantities (κn, κt ) increase. When both the nor-mal and tangential damage parameters approach one,the rate-independent PPR cohesive tractions vanish,indicating complete damage.

The terms Ek in Eq. (20) represent the PPR initialstiffness in the normal and tangential directions, respec-tively. These expressions were obtained in a study bySpring et al. (2016) and are provided below:

En = −�n

δ2n

(mα

)m−1(m + α)

[�t

(n

β

)n

+ 〈φt − φn〉]

,

Et = −�t

δ2t

(n

β

)n−1

(n + β)[�n

(mα

)m + 〈φn − φt 〉].

(24)

The parameters α, β, m, n, �n , and �t , used inEqs. (21)–(21) and in Eq. (24), are the same as in thePPR cohesive model (Park et al. 2009). Parameters α

and β, both greater than or equal to one, control thesoftening shape of the traction–separation relationshipin the normal and tangential directions, respectively.The softening portion of the traction–separation rela-tionship is convex if α, β > 2 or concave if α, β < 2.

123


Fig. 5 Typicaltraction–separationrelationships from the PPRcohesive model, and theircorresponding normal andtangential damageparameters: a normalcohesive traction; btangential cohesive traction;c damage parameter in thenormal direction; and ddamage parameter in thetangential direction. For thesake of illustration, thecohesive parameters usedhere are φn = 200 N/m,φt = 400 N/m,σmax = 40 MPa,τmax = 30 MPa, α = 4,β = 1.5, λn = 0.2, andλt = 0.3

(b)(a)(M

Pa)

(MPa

)

Tn Tt

(µm) (µm)κn κt(µm) (µm)κn κt

(d)(c)

(µm) µ

dn dt

(µm) (µm)κn κt(µm) (µm)κn κt

010

20

65

40

420 0

010

10

20

155 10

30

50 0

010

0.5

65

1

420 0

010

0.5

155 10

1

50 0

Moreover, the softening behavior is nearly linear if α,β = 2. Parameters m and n are functions of the shapeparameters α and β and two initial slope indicators, λnand λt , as shown below:

m = α (α − 1) λ2n

1 − αλ2n, n = β (β − 1) λ2t

1 − βλ2t. (25)

The initial slope indicators correspond to the ratiobetween the critical crack opening widths and the finalcrack opening widths:

λn = δnc/δn, λt = δtc/δt , (26)

where the critical crack opening widths δnc and δtc cor-respond to the crack opening at which the peak trac-tions are reached in the normal and tangential direc-tions, respectively. The final crack opening widths δnand δt are obtained as

δn = φn

σmaxαλn(1 − λn)

α−1( α

m+ 1

) ( α

mλn + 1

)m−1,

δt = φt

τmaxβλt (1 − λt )

β−1(

β

n+ 1

) (β

nλt + 1

)n−1

.

(27)

Energy constants �n and �t are functions of thenormal and tangential fracture energies (i.e., φn and

φt ), respectively. According to Park et al. (2009), theenergy constants are

�n = (−φn)〈φn−φt 〉/(φn−φt )

( α

m

)m,

�t = (−φt )〈φt−φn〉/(φt−φn)

(β

n

)n

, (28)

when the normal and tangential fracture energies differ(φn = φt ), and

�n = −φn

( α

m

)m, �t =

(β

n

)n

, (29)

when the normal and tangential fracture energies are thesame (φn = φt ). The bracket 〈·〉 in Eq. (28) correspondsto theMacaulay bracket, 〈x〉 = 1

2 (x + |x |).

3.3 Rate-dependent cohesive tractions

We obtain the mixed-mode rate-dependent cohesivetractions of the present study by substituting Eqs. (17)and (20) into Eq. (14), which yields

Tk = Tk (κn, κt )

Ekκk

(Ek�k + �SH

k

), (k = n, t). (30)

123


Fig. 6 Typical behavior ofour rate-dependent cohesivemodel for several prescribedconstant crack openingrates: a normal cohesivetraction and b tangentialcohesive traction. Therepresentative parametersused here are φn = 10 N/m,φt = 20 N/m,σmax = 4 MPa,τmax = 3 MPa, α = 5,β = 1.5, λn = 0.2,λt = 0.2, c = 1, v = 0.2,and λn = λt = 1.3 sv

(online version in color) Normal opening,(b)(a)

( m)

(MPa

)

10 4110-210-410-610-10Rate (mm/s):

( m)

(MPa

)

Tn

Tt

increasesincreases

00 1 2 3 4 5 6 7 0 2 4 6 8 10

2

4

6

8

0

1

2

3

4

5

6

µµ

Δn ΔtΔnΔt

Δn Δt

(b)(a)

Mod

e I f

ract

ure

ener

gy (N

/m)

Mod

e II

fract

ure

ener

gy (N

/m)

Normal opening rate, (mm/s) (mm/s)10-20 10-10 100 1010 1020

10

12

14

16

18

20

10-20 10-10 100 1010 102020

25

30

35

40

Δn Δt

Δn Δt

Fig. 7 Fracture energy obtained for several prescribed constant crack opening rates: a mode I fracture energy; and b mode II fractureenergy. The model parameters that we used to obtain these figures are the same as those used to generate Fig. 6

As shown in Fig. 6, we illustrate our rate-dependentcohesive model by plotting the normal and tangen-tial cohesive tractions as functions of the crack open-ing displacements for several constant crack openingrates in pure modes I and II. The results show that thecohesive tractions increase as the crack opening rateincreases. Increase in cohesive tractions as a functionof crack opening rate has been observed experimen-tally in polyurea/steel interfaces (Zhu et al. 2009). In theslow limit case (i.e., when �k → 0), themaximumnor-mal and tangential cohesive tractions approach Tmax

n =σmax = 4 MPa and Tmax

t = τmax = 3 MPa, respec-tively. These peak tractions correspond to the nor-mal and shear cohesive strengths that we adopted forthe rate-independent PPR cohesive model (Park et al.2009), which dictates the behavior of the damage ele-ment discussed in the previous subsection. For the fast

limit case (i.e., when �k → ∞) the normal and tan-gential cohesive tractions approach Tmax

n = 8 MPaand Tmax

t = 6 MPa, respectively, which correspond toTmaxn = (1+ c)σmax and Tmax

t = (1+ c)τmax, respec-tively.

We use the results in Fig. 6 to obtain mode I and IIfracture energies for several constant prescribed nor-mal and tangential crack opening rates, respectively.Figure 7 shows the computed fracture energies versusthe crack opening rates. As one can observe in the fig-ure, both mode I and II fracture energies grow in a sig-moidal shape as the crack opening rates increase. Forthe slow-limit case, the mode I and II fracture energiesapproach φn = 10 N/m and φt = 20 N/m, respectively.These are the fracture energies that we used to definethe damage element of the previous subsection. Forthe fast-limit case, the mode I and II fracture energies

123


approach (1 + c)φn and (1 + c)φt , respectively. Notethat, in contrast to previous studies that assume that thefracture energies increase to infinity as the crack open-ing rates increase, e.g., seeZhouet al. (2005), ourmodelpredicts finite fracture energies as the crack openingrates approach infinity. This feature of our model isintegral to the representation of real materials. Besidesthe aforementioned limit behaviors, we observe thatboth mode I and II fracture energies increase exponen-tially for a certain range of crack opening rates (linearportions of Fig. 7a, b), which agrees with experimen-tal observations in several studies (e.g., see Zhou et al.2005; Rahul-Kumar et al. 1999).

The sigmoidal behavior of the mode I and II frac-ture energies as a function of the loading rate is suit-able to model elastomeric materials, but it is inade-quate to model the behavior of glassy polymers, whichtypically become more brittle as the crack openingrate increases (e.g., see Williams 1972; Johnson andRadon 1972; Maugis 1985; Webb and Aifantis 1995;de Gennes 1996). The decrease in fracture energy withincreasing crack opening rate may be due to increas-ing temperatures in the fracture process zone (Williams1972; Johnson and Radon 1972). An alternative expla-nation for this phenomenon was recently presentedby Alfano and Musto (2017). They indicated that ifthe total elastic energy is considered as the thermo-dynamic driving force for damage growth, one mayobserve that the behavior of the fracture energy can benon-monotonic, which is observed in glassy polymers.Although such consideration could be incorporated, itis out of the scope of the present study, and thus thereader is directed to the aforementioned references.

We test the consistency of our model using the non-proportional loading/unloading crack-opening pathshown in Fig. 8. In that path, theCZM is loaded inmodeI, then it is unloaded and finally is loaded in mode IIuntil complete failure occurs. The amount of loadingin mode I is chosen such that damage in the normaldirection is dn =96% (i.e., almost complete failure).The cohesive tractions that we computed based on thegiven crack-opening path are shown in Fig. 9. Fromthese results, we observe that, as we load and unloadin mode I (i.e., region A–B–C from Fig. 8), we obtainan expected traction–separation curve. That is, the nor-mal cohesive traction, Tn , increases until it reaches apeak traction and then decreases (region A–B), whichis followed by a linear unloading (region B–C). Oncewe start loading in mode II (point C), the tangential

Time (s)

Δn

Δt

(μm

)

0 500

2

4

6

8

10

100 150 200 250 300 350

Crac

k op

enin

g

Normal opening,Tangential opening,

B

CA

D

Fig. 8 Non-proportional crack-opening path used to verify theconsistency of the rate-dependent cohesive zone model. Theloading path consists of mode I loading (A–B), and then bymodeI unloading (B–C), followed by mode II loading (C–D)

cohesive tractions, Tt , also behave as expected untilcomplete failure occurs.

The evolution of damage parameters, dn and dt ,that we obtain from the crack-opening path depictedin Fig. 8 is illustrated in Fig. 10. The figure shows thatboth normal and tangential damageparameters increasein region A–B (for pure mode I loading), which isexpected because both normal and tangential damageparameters are coupled through the kinematic quanti-ties κn and κt—see Eq. (20). That is, loading in the nor-mal direction affects the behavior of theCZMin the tan-gential behavior and vice-versa. In addition, both dam-age parameters remain constant in region B–C (unload-ing), which is consistent with our assumption that noself-healing is considered in the model. Once mode IIloading begins (point C), both normal and tangentialdamage parameters increase monotonically until com-plete failure occurs. Because at point C the tangentialdamage parameter, dt , is close to one, the tangentialcohesive tractions predicted for the model are signifi-cantly smaller than the rate independent shear cohesivestrength, τmax = 80 MPa, used to define the damageelement. This is an expected response in a physical sys-tem in which both normal and tangential damage arecoupled.

3.4 Material tangent matrix and related remarks

To implement our model in a finite element softwaresuch asABAQUS (2011), one needs to obtain themate-rial tangent matrix. To obtain such matrix, we differen-

123


Fig. 9 Cohesive tractionsin the a normal and btangential directionsobtained for thenon-proportionalcrack-opening path ofFig. 8. The CZM parametersused here areφn = 200 N/m,φt = 900 N/m,σmax = 40 MPa,τmax = 80 MPa, α = 3,β = 1.5, λn = 0.2,λt = 0.3, c = 1, v = 0.2,and λn = λt = 1.5 sv

Tangential opening, Δt (μm)Normal opening,(b)(a)

Nor

mal

trac

tion,

Δn (μm)0

00 5 10 15

0

2

4

6

8

10

12

20

40

60

2 4 6 8Tn

(MPa)

Tang

entia

l tra

ctio

n,Tt

(MPa)

B

CA C

D

Time (s)

Dam

age

para

met

er

0 100 200 300 400 5000

0.5

1

dnNormal,dtTangential,

B C

A

D

Fig. 10 Evolution of normal and tangential damage parameters,dn and dt , obtained for the non-proportional crack-opening pathof Fig. 8. The damage parameters are obtained from Eq. (20)using the CZM parameters from Fig. 9

tiate the cohesive tractions, Tk , in Eq. (30), with respectto the crack opening displacements,�k , which leads to

D(�n,�t ) =[Dtt Dtn

Dnt Dnn

], (31)

where

Dtt = ∂Tt∂�t

= StEtκt

[Dnn(κn, κt ) − Tt (κn, κt )

κt

]

∂κt

∂�t+ Et Tt (κn, κt )

Etκt, (32)

Dtn = ∂Tt∂�n

= StEtκt

Dtn(κn, κt )∂κn

∂�n, (33)

Dnt = ∂Tn∂�t

= SnEnκn

Dnt (κn, κt )∂κt

∂�t, (34)

Dnn = ∂Tn∂�n

= SnEnκn

[Dtt (κn, κt ) − Tn(κn, κt )

κn

]

∂κn

∂�n+ En Tn(κn, κt )

Enκn. (35)

Parameters Dtt , Dtn , Dnt , and Dnn are the componentsof the material tangent matrix for the rate-independentPPR cohesive zonemodel Park and Paulino (2012), andare shown in “Appendix B”.

We use the material tangent matrix D(�n,�t ) forFE implementation purposes, and so we refer inter-ested readers to Park and Paulino (2012) for a detailedprocedure regarding the FE implementation of a two-dimensional linear cohesive element. Note that weincluded all loading–unloading–reloading scenarios inEqs. (32)–(35) through the values of ∂κn/∂�n and∂κt/∂�t , which we obtain from Eq. (23) as follows:

∂κn

∂�n=

{1 if �n = κn

0 if �n < κn,

∂κt

∂�t=

{�t/ |�t | if |�t | = κt

0 if |�t | < κt. (36)

Equations (36) show that the kinematic quantities κnand κt only vary under pure loading conditions, but theyremain constant if unloading/reloading occurs. The val-ues of ∂κn/∂�n and ∂κt/∂�t (which dictate the evo-lution of κn and κt ) are illustrated in Fig. 11.

4 Examples and discussion

Here, we present and discuss three examples to test ourmixed-mode rate-dependent CZM. In the first example,we validate our model by comparing numerical resultswith those from rate-dependent DCB tests conductedbyMusto andAlfano (2015). In the second example,weconduct a rate-dependent mixed-mode test on a com-pact tension/shear (CTS) specimen and compare ournumerical results with experimental results obtained

123


Δn

Δt

κn

κt

∂κn

∂Δn

= 1

∂κt

∂Δt

= 0

∂κn

∂Δn

= 1

∂κt

∂Δt

= 1

∂κn

∂Δn

= 0

∂κt

∂Δt

= 0

∂κn

∂Δn

= 0

∂κt

∂Δt

= 1

Δt

Δn

Δt

Δn

Δt

Δn

Δt

Δn

Fig. 11 Given the maximum normal and tangential crack open-ing during the loading history (reference point), four possibleloading scenarios may occur. For pure loading (white region),both ∂κn/∂�n = 1 and ∂κt/∂�t = 1; for partial unloading(light gray regions), ∂κn/∂�n = 1 and ∂κt/∂�t = 0 when�n > κn and �t < κt , and ∂κn/∂�n = 0 and ∂κt/∂�t = 1

when �n < κn and �t > κt ; and for pure unloading (dark grayregion), both ∂κn/∂�n = 0 and ∂κt/∂�t = 0. This illustrationshows only positive values of �t ; therefore, to account for bothpositive and negative values of �t , the values of ∂κt/∂�t shownhere need to be multiplied by sign(�t )—adapted from Springet al. (2016)

by Liechti and Wu (2001). In the third example, weperform a parametric analysis to investigate the effectof the model parameters in the behavior of a fiber rein-forced composite. We use the third example to eluci-date the meaning of the order of the Caputo fractionalderivative and its relevance in rate-dependent fracturesimulations4.

4.1 Double cantilever beam test

In this example, we compare the numerical resultsfrom our model to those from DCB tests performed atvarious loading rates. We obtain the experimentalresults from a study by Musto and Alfano (2015), whoperformed several DCB tests with geometry and loadapplication points, as depicted in Fig. 12a. The geome-try of the DCB tests consists of two steel arms partiallybonded with a styrene butadiene rubber interface. Eachsteel arm is 8mm thick, 25mmwide, and 200mm long,and the rubber interface is 1 mm thick, 120 mm long.Musto and Alfano (2015) performed DCB debondingtests at several prescribed constant cross-head displace-

4 To obtain the results shown in all the examples, we imple-mented our rate-dependent CZM as a user-defined element(UEL) subroutine in the commercial FE software ABAQUS(2011).

ment rates ranging from 0.01, 0.1, 1, 10, 100, to 500mm/min.

To compare our results to those of the experimentaldata fromMusto and Alfano (2015), we created six FEmodels, each of which corresponds to one of the pre-scribed cross-head displacement rates. The mesh andboundary conditions for each FE model are illustratedin Fig. 12b. We impose a constant velocity boundarycondition at the load application points. Themagnitudeof the velocity imposed at these points corresponds toeach of the prescribed cross-head displacement rates inthe DCB tests.

In addition to mesh and boundary conditions, wedefine material properties in our FE models. We modelthe steel arms using a linear elastic material withYoung’s modulus of 200 GPa and Poisson’s ratio of0.3. In addition,wemodel the rubber interface usingourmixed-mode rate-dependent CZM and select an appro-priate set of model parameters to match the experimen-tal results by Musto and Alfano (2015). Table 1 showsthe set of input parameters for the rate-dependent ele-ments used in our FE models. Because the DCB testsare dominated bymode I fracture, we arbitrarily choosethe same model properties in the normal and tangentialdirections. Ideally, those arbitrary parameters shouldbe obtained from proper fracture testing (Reeder andCrews 1990).

123


Fig. 12 Description of thedouble cantilever beam(DCB) model: a geometryand load application points(all dimensions are in mm)and b finite element meshand boundary conditions.We discretize the steel armsusing 800 four-nodequadrilateral plane strainelements and the rubberinterface using 60zero-thicknessrate-dependent cohesiveelements

Rubber layerSteel arms

1

8

120200

Load

Load

V

V

800 bulk elements 60 cohesive elements

(a)

(b)

Table 1 Fracture parameters for the cohesive elements used tomodel the rubber interface in the DCB test

Mode I fracture energy, φn (N/mm) 2.8

Mode II fracture energy, φt (N/mm) 2.8

Normal cohesive strength, σmax (MPa) 1.2

Tangential cohesive strength, τmax (MPa) 1.2

Non-dimensional shape parameters, α = β 3.5

Initial slope indicators, λn = λt 0.45

Order of the fractional derivative, v 0.3

Stiffness parameter, c 8.5

Normal and tangential springpot viscosityparameter, λ = ηk/(cEk)

3.5

Figure 13 shows both numerical and experimen-tal results for the DCB tests at various loading rates.The results show that the peak load increases as theapplied cross-head displacement rate increases. How-ever, for the chosen set of fracture parameters, ourmodel slightly underestimates the experimental peakload for lower rates (e.g., less than 10 mm/min) andslightly overestimates the experimental peak load forhigher rates (e.g., more than 10mm/min). Although wecould improve our model predictions by further cali-brating the set of fracture parameters (Shen and Paulino2011a, b), this is beyond the scope of the present study.

Figure 14 presents a typical result fromone of the FEsimulations,whichwe use to illustrate the delaminationprocess in one of the DCB tests. This figure shows theevolution of the von Mises stress and the deformationhistory of one of the DCB tests corresponding to anapplied rate of 1 mm/min. Here, we observe that thevon Mises stress increases between points A and C,

decays smoothly between points C and E, and dropsdrastically after point E. The drop in stresses is con-sistent with the drop in the applied load, which resultsfrom the loss of the ligament length bonding the twosteel arms.

4.2 Mixed-mode rate-dependent rubber/metaldebonding

In this example, we simulate the quasi-static mixed-mode debonding process between rubber and metal.The numerical results obtained from our mixed-moderate-dependent CZM are compared against the exper-imental results obtained by Liechti and Wu (2001).Based on their study, we use the compact tension/shear(CTS) specimen to simulate the rate-dependent frac-ture test. The CTS specimen geometry (Wu 1999) andthe finite element discretization are shown in Fig. 15.

The loading is applied as indicated by Liechti andWu (2001), such that we first apply a normal displace-ment,uy , in a stepwisemanner, followedby a tangentialdisplacement, ux , which is applied at a constant rate.We consider both positive and negative shear displace-ments in this study, and in both cases, we first applya normal displacement (uy = 1.5 mm) linearly overa time period of 0.05 s followed by a shear displace-ment with a rate of ux = 0.01 mm/s for the positiveshear test or ux = − 0.02 mm/s for the negative sheartest. The rates of shear displacement used here are con-sistent with the shear rates of γ = 5 × 10−3/s andγ = 1 × 10−2/s reported by Liechti and Wu (2001)for the positive and negative shear experiments, respec-tively. We wait 100 s between application of the nor-

123


Fig. 13 Load versuscross-head displacementcurves for the doublecantilever beam test atseveral applied openingrates. Experimental resultsare obtained from Mustoand Alfano (2015)

Opening, (mm)10 15 20 25 30 35 40 45

Load

, P (k

N)

0

0.5

1

1.5

2

2.5

3

0.01Numerical results for V =Experimental results for V = 0.01

0.10.1

11

1010

100100

500 (mm/min)500 (mm/min)

50

P

P

x

y

Z

100 5 15 20 25 30 35 40 450

0.5

1

1.5

2

A

B

C

D

E

F

G Opening, (mm)

Load

, P (k

N)

A

B CD

E

F G

(Avg: 75%)S, Mises

0.0 49.2 98.3147.5196.7245.8295.0344.2393.3442.5491.7540.8590.0

Fig. 14 Evolution of von Mises stresses (kPa) on a typical DCB test (left) for various points during the loading history (right) (onlineversion in color)

123


ux

uy

# bulk elements: 1,400# cohesive elements: 100

(b)

(a)

Cohesive elements Rubber layer

Rubber layer

Steel

2

22.350.8

Initial notch

ux

uy

ux

uy

Fig. 15 Geometry of the CTS specimen and finite elementdiscretization used to study the mixed-mode rate-dependentdebonding process between rubber and steel: a a 2D representa-tion of the CTS geometry (dimensions is mm), and b the finite

element mesh and boundary conditions used in the simulation.The rubber layer is discretized using 1400 four-node plane stresselements and the debonding layer using 100 zero-thickness rate-dependent (or rate-independent) cohesive elements

mal and the shear displacements, which helps to dissi-pate some of the viscoelastic stresses in the bulk mate-rial induced by the fast application of the normal dis-placement. This strategy avoids large jumps in boththe normal and shear loads that occur in the numeri-cal model when the normal displacement is applied toofast.

The behavior of the rubber (bulk material) is sim-ulated using a visco-hyperelastic material model thatuses a three-term incompressible Ogdenmodel (Ogden1972) with stored-energy density function of the form5

W =3∑

i=1

2μi

α2i

(λαi1 + λ

αi2 + λ

αi2 − 3), (37)

where λi , i = 1, 2, 3 are the principal stretches, andμi

and αi are material parameters. We use the parametersshown in Table 2 to define the hyperelastic behaviorof rubber6. The viscoelastic response of rubber is rep-resented by a 14-term Prony series, whose terms aregiven in Liechti and Wu (2001).

We simulate the behavior of the rubber/steel inter-face using the mixed-mode rate-dependent cohesiveelement derived in Sect. 3. The material parameters

5 The form of the stored energy function (37) is that used inABAQUS (2011) and differs from the form the one used byLiechti and Wu (2001).6 Note that the numerical values in Table 2 differ from thosereported byLiechti andWu (2001) because the formof theOgdenstored energy function used in their work differs from that givenin Eq. (37).

Table 2 Parameters used todescribe the hyperelasticbehavior of rubber

Term μi (MPa) αi

1 0.103 0.5

2 2.45×10−4 8.91

3 0.447 −2

Table 3 Fracture parameters used to simulate the mixed-moderate-dependent rubber/metal debonding problem

Mode I fracture energy, φn (N/mm) 0.87

Mode II fracture energy, φt (N/mm) 3.05

Normal cohesive strength, σmax (MPa) 2.0

Tangential cohesive strength, τmax (MPa) 2.4

Non-dimensional normal shape parameter, α 3.0

Non-dimensional tangential shape parameter, β 3.5

Initial slope indicators, λn = λt 0.05

Order of the fractional derivative, v 0.3

Stiffness parameter, c 1.49

Normal and tangential springpot viscosityparameter, λ = ηk/(cEk)

1.8

that we use to represent the rate-dependent debond-ing process are given in Table 3. These parameters areobtained by trial and error, such that we are able toapproximate the experimental results. We also simu-late the debonding behavior of the rubber layer usingthe rate-independent PPRmodel (Park et al. 2009; Parkand Paulino 2012). To obtain the results from the rate-

123


independent CZM, we scale the mode I and II frac-ture energies and cohesive strengths from Table 3 toaccount for the expected increase of fracture energy andcohesive strength due to the applied loading rates. Theparameters used for the rate-independent PPR modelare φn = 2.02 N/mm, φt = 7.08 N/mm, σmax = 4.64MPa, τmax = 5.57 MPa, α = 3.0, β = 3.5, andλn = λt = 0.05.

Figure 16 shows both the numerical and experimen-tal crack opening displacement (COD) profiles for thepositive and negative shear experiments, at the onset ofcrack initiation and for an instant during crack growth.Due to the finite rotations allowed in the simulations,the numerical results are able to capture the “Z”-shaped crack opening profile obtained experimentallyfor the positive shear case. The numerical CODprofilesobtained with the rate-dependent CZM agree well withthose obtained experimentally both for crack initiationand during crack growth and for both the positive andthe negative shear experiments. However, the results

from the rate-independent PPR model only agree wellwith the experimental results for the negative shearexperiment. Because the PPRmodel has no viscoelasticbehavior built in, it cannot simultaneously predict thebehavior from both positive and negative shear exper-iments using a single set of fracture parameters.

As depicted in Fig. 17, we also obtain crack separa-tion profiles in both normal and shear directions, duringcrack initiation and crack growth. For the positive shearexperiments (Fig. 17a, b),�x < �y during crack initia-tion and�x > �y during crack growth. That is becauseduring crack growth, the crack profile acquires a “Z”shape, which allows for larger displacements in thehorizontal direction than in the vertical direction. Theresults from the negative shear experiment (Fig. 17c,d) show that |�x | > �y both during crack initiationand during crack growth. That is because the negativeshear experiment requires more energy than the posi-tive shear experiment to debond the rubber layer fromthe steel substrate, and thus, at the onset of debonding,

Fig. 16 Numerical andexperimental crack-openingdisplacements for thepositive shear (top) andnegative shear (bottom)tests during: a and c crackinitiation, and b and d crackgrowth. The results for therate-independent CZM arefound using the PPR CZM(Park et al. 2009). Thelocation x = 0 mmcorresponds to the notch tip -0.5

0

0.5

1

1.5

2

COD

(mm

)

-0.5

0

0.5

1

1.5

2CO

D (m

m)

-1-2 0 1 2Location, x (mm)


(b)(a)

-0.5

0

0.5

1

1.5

2

COD

(mm

)

-0.5

0

0.5

1

1.5

2

COD

(mm

)



(d)(c)

Rate-dependent CZM ExperimentalRate-independent CZM

123


Fig. 17 Crack separationprofiles for modes I and IIobtained from the simulatedmixed-mode fracture testswith positive shear (top) andnegative shear (bottom)during: a and c crackinitiation, and b and d crackgrowth. The numericalresults for therate-independent CZM areobtained using the PPRCZM (Park et al. 2009). Inthe results displayed here,the location x = 0 mmcorresponds to the notch tip Location, x (mm)

(a)

0

1

2

3

-2 -1 0 1 2

(mm

)or

Location, x (mm)(b)

0

1

2

3

-2 -1 0 1 2

Location, x (mm)(c)

-6

-4

-2

0

2

-2 -1 0 1 2-6

-4

-2

0

2

Location, x (mm)(d)

-2 -1 0 1 2

(rate dependent)(rate dependent)

(rate independent)(rate independent)

Δx

Δx

Δy

Δx

Δy

Δy

(mm

)or

Δx

Δy

(mm

)or

Δx

Δy

(mm

)or

Δx

Δy

the rubber layer is already considerably stretched in thehorizontal direction for the negative shear experiment.As expected from the results shown in Fig. 16, the nor-mal and tangential crack opening displacements fromthe rate-independent CZM are larger in magnitude thanthose from the rate-dependent CZM.

In addition to COD and crack-opening profiles, wealso compute the time histories of applied loads in bothshear and normal directions, as shown in Fig. 18. Thepredicted shear load behaves almost linearly for boththe positive and the negative shear experiments (e.g.,see Fig. 18a and c), which agrees with observations byLiechti andWu (2001). The linear response is attributedto the behavior of the bulk material obtained from theOgden model. As indicated in Fig. 18b, the change innormal load, which is obtained by subtracting the nor-mal preload due to uy from the current normal load,agreeswellwith the experimentalmeasurements for thepositive shear experiment, but over-predicts the exper-

imental measurements for the negative shear exper-iment. Over-prediction in the negative shear experi-ment, in which large shear strains are required to causedebonding, is expected based on the findings of LiechtiandWu (2001). In Fig. 7e of their paper, they show thatthe numerical model for the bulk over-predicts normalstresses when large shear strains are applied.

4.3 Rate-dependent matrix/fiber debonding

The following example investigates the rate-dependentdebonding behavior of a matrix/fiber interface.We per-form an analysis on a unit cell in the form of a regu-lar hexahedron with a cylindrical inclusion, shown inFig. 19a. We analyze a unit cell under equi-biaxial ten-sion and assume plane strain conditions. Because ofsymmetry, wemodel a quarter of the domain (Fig. 19b).The FE mesh and boundary conditions that we use for

123


00

20

40

60

80

100

50 100 150 200 250Time (s)

(a)

Shea

r loa

d (N

)

0-80

-60

-40

-20

0

20

50 100 150 200 250Time (s)

(b)

Chan

ge in

nor

mal

load

(N)

00

20

40

60

80

100

20 40 60 80 100 120 0 20 40 60 80 100 120Time (s)

(c)

Shea

r loa

d (N

)

-80

-60

-40

-20

0

20

Time (s)(d)

Chan

ge in

nor

mal

load

(N)

Rate-dependent CZM ExperimentalRate-independent CZM

Fig. 18 Numerical and experimental load histories for themixed-mode fracture tests with positive shear (top) and nega-tive shear (bottom): a and c shear load, and b and d change

in normal load. The numerical results for the rate-independentCZM are obtained using the PPR CZM (Park et al. 2009)

# nodes: 7911# bulk elements: 7623# cohesive elements: 200

(c)(b)(a)

Fig. 19 Description of the model used to study the rate-dependentmatrix/fiber debondingprocess:aA3Drepresentationof the unit cell, b a slice containing one quarter of the cross sec-

tion used to model the geometry, and c the finite element meshand boundary conditions used in the simulations

123


this problem are shown in Fig. 19c. Ngo et al. (2010)and Park and Paulino (2012) studied this problem, butthey neglected the effects of the loading rate. The studyby Ngo et al. (2010) analyzed multiple fiber radii andthat by Park and Paulino (2012) focused on a fiberradius of ap = 2 cm. In this example, we use the samefiber radius as that used by Park and Paulino (2012) andassume that the volume fraction of the fiber equals 0.6.

Similarly to Park and Paulino (2012), our studyassumes that the elastic modulus of the fiber is Ep =40 GPa and the elastic modulus of the matrix isEm = 20 GPa. We also assume that both the fiberand the matrix have a Poisson’s ratio νp = νm =0.25. The elastic properties that we selected for thematrix and the fiber are representative of a PolyvinylAlcohol-Engineered Cementitious Composite (PVA-ECC) (Tosun-Felekoglu et al. 2014;Kang andBolander2016). However, the purpose of the current example isnot to study a given particular composite, but insteadis to elucidate the effects of cohesive zone modelparameters, particularly those from the fractional SLSmodel.

Because the problem studied herein is dominated bymode I fracture, we arbitrarily assume the same frac-ture properties in modes I and II to define the prop-erties of the rate-dependent cohesive elements. In thepresent analysis, we fix the material properties thatdetermine the behavior of the damage elements andvary those that determine the behavior of the fractionalSLS model. We vary the properties of the fractionalSLS model in order to analyze their effects on therate-dependent debonding process at the matrix/fiberinterface. To determine the behavior of the damage ele-ments, we assume φn = φt = 100 N/m, α = β = 3,λn = λt = 0.1, and σmax = τmax = 10 MPa.

To examine the effects of the loading rate, we movethe free faces of the unit cell at several constant speedsin the directions indicated by the arrows in Fig. 19b.The FE model results allow us to obtain the aver-age macroscopic stress–strain curves for each appliedvelocity and considering the range of the fractionalSLS model parameters (i.e., for the values of λk , c,and v). We compute the average macroscopic strainsas ε = V t/L0, where V is the applied velocity at thefree faces of the unit cell, t is time, and L0 is the initial(undeformed) length of the unit cell. Furthermore, weobtain the macroscopic stresses by averaging the reac-tion forces at the nodes to which the velocity boundary

condition is applied and then normalizing with respectto the undeformed area of the free faces.

Figure 20 presents the FE results for the case ofv = 0.3. From these results, we observe that the initialslopes of the stress–strain curves are independent of theapplied velocity, which indicates that for lower strainlevels, the elastic behavior of both the matrix and theparticle control the macroscopic stress strain responseof the composite. This linear portion extends as theapplied velocity increases, but as the strain increases,the stress–strain curves deviate from their initially lin-ear behavior until the peak stress is reached. After thepeak stress is reached, themacroscopic stress decreases(stress softening), but for higher applied velocities (e.g.,for V = 1 mm/min or for V = 104 mm/min), thestresses suddenly decrease7. For all these cases, thepoint of sudden decrease corresponds to complete fail-ure of the cohesive elements. From the set of resultsdisplayed in Fig. 20, we also find that, as the appliedvelocity becomes smaller, the macroscopic responseobtained from the rate-dependent mixed-model CZMapproaches the response from the rate-independentPPR CZM (Park et al. 2009).

To further investigate the effects of both the rate-dependent model parameters and loading rates, we plotthe peak stress for the values of v, λk , and c, as a func-tion of the applied velocity, as shown in Fig. 21. Asshown by the results, the peak stress increases as theappliedvelocity increases, following a sigmoidal curve.One can also observe that, for lower applied velocities,the peak stresses approach the value ofσmax = 10MPa,which is the cohesive strength that we used as one of theinput parameters for the damage element. The resultsalso indicate that the peak stress value for low appliedvelocities is independent from either v, λk , or c, whichindicates that the slow-limit behavior is governed by thebehavior of the rate-independent PPR cohesive model(Park et al. 2009). Additionally, the numerical resultsindicate that, as the applied velocity increases, the peakstresses approach a limit value equal to (1+c)σmax. Thebehavior observed in either the slow-limit or fast-limitcase is, by construction, consistent with the expectedbehavior of our model, which in the slow-limit and fastlimit cases should behave as the rate-independent PPR

7 We observed some convergence issues at the kink points asso-ciated with failure at the matrix/fiber interface. To circumventthese convergence issues, we used a line search technique andallowed for more Newton–Raphson iterations.

123


λk = 1.5, c = 2.5

λk = 1.5, c = 5

λk = 1.5, c = 7.5

λk = 2, c = 2.5

λk = 2, c = 5

λk = 2, c = 7.5

λk = 2.5, c = 2.5

λk = 2.5, c = 5

λk = 2.5, c = 7.5

Macroscopic strain (%)0 0.05 0.1 0.15 0.2M

acro

scop

ic s

tres

s (M

Pa)

0

20

40

60

80


acro

scop

ic s

tres

s (M

Pa)

0

20

40

60

80


acro

scop

ic s

tres

s (M

Pa)

0

20

40

60

80


acro

scop

ic s

tres

s (M

Pa)

0

20

40

60

80


acro

scop

ic s

tres

s (M

Pa)

0

20

40

60

80


acro

scop

ic s

tres

s (M

Pa)

0

20

40

60

80


acro

scop

ic s

tres

s (M

Pa)

0

20

40

60

80


acro

scop

ic s

tres

s (M

Pa)

0

20

40

60

80


acro

scop

ic s

tres

s (M

Pa)

0

20

40

60

80

10− 8Rate = mm/min10− 4Rate = mm/min

10 4Rate = Rate independent

mm/min

1Rate = mm/min

Fig. 20 Macroscopic stress–strain relationships obtained usingfractional derivative of order v = 0.3. The plots are presentedfor various springpot viscosity parameters, λk , and fractionalSLS model stiffness parameter, c, considering the range ofapplied velocity, V , from 10−8 to 104 mm/min. As the applied

velocity decreases, the macroscopic response obtained from therate-dependent CZM approaches that obtained from the rate-independent PPR CZM by Park et al. (2009) (online version incolor)

model (Park et al. 2009) with initial stiffness Ek and(1 + c)Ek , respectively.

The order of the fractional derivative, v, also affectsthemacroscopic behavior of the composite structure forintermediate values of the applied velocity. Accordingto the numerical results in Fig. 21, peak stresses aremore sensitive to the applied velocity when the orderof the fractional derivative is smaller. In fact, we foundthat both peak stresses and macroscopic stress–straincurves were more sensitive to the applied velocity forlower values of v. The reason for this behavior is thatthe relaxation modulus for the fractional SLS model

depends on the value of v. As shown in Fig. 3, weobservemore stress relaxation when v increases (short-termmemorymaterials) and less stress relaxationwhenv decreases (long-term memory materials). The mem-ory effect explains why macroscopic stresses are moresensitive to the applied velocity as v decreases. Basedon this observation, we infer that v plays the role of amemory-like parameter, which enables us to simulatelong- and short-term memory fracture processes.

123


Rate (mm/min)10 -15 10 -10 10 -5 10 0 10 5

Peak

str

ess

(MPa

)

0

20

40

60

80

Rate (mm/min)10 -15 10 -10 10 -5 10 0 10 5

Peak

str

ess

(MPa

)

0

20

40

60

80

Rate (mm/min)10 -15 10 -10 10 -5 10 0 10 5

Peak

str

ess

(MPa

)

0

20

40

60

80

Rate (mm/min)10 -15 10 -10 10 -5 10 0 10 5

Peak

str

ess

(MPa

)

0

20

40

60

80

Rate (mm/min)10 -15 10 -10 10 -5 10 0 10 5

Peak

str

ess

(MPa

)

0

20

40

60

80

Rate (mm/min)10 -15 10 -10 10 -5 10 0 10 5

Peak

str

ess

(MPa

)

0

20

40

60

80

Rate (mm/min)10 -15 10 -10 10 -5 10 0 10 5

Peak

str

ess

(MPa

)

0

20

40

60

80

Rate (mm/min)10 -15 10 -10 10 -5 10 0 10 5

Peak

str

ess

(MPa

)

0

20

40

60

80

Rate (mm/min)10 -15 10 -10 10 -5 10 0 10 5

Peak

str

ess

(MPa

)

0

20

40

60

80

λk = 1.5, c = 2.5

λk = 1.5, c = 5

λk = 1.5, c = 7 .5

λk = 2, c = 2.5

λk = 2, c = 5

λk = 2, c = 7.5

λk = 2.5, c = 2.5

λk = 2.5, c = 5

λk = 2.5, c = 7.5

v = 0.3 v = 0.5 v = 0.7 v = 0.9

Fig. 21 Parametric space for macroscopic peak stress as a func-tion of the applied velocity. The plots are presented for vari-ous springpot parameters, λk , and fractional SLS model stiffnessparameter, c, considering the range for the order of the fractional

derivative from v = 0.3 to v = 0.9. Peak stresses behave in theform of a sigmoidal curve, and their magnitudes are highly influ-enced by the spring stiffness of theMaxwell arm in the fractionalSLS model (online version in color)

5 Summary and conclusions

In this study, we present a mixed-mode rate-dependentcohesive zonemodel (CZM) that combines the featuresof the Park–Paulino–Roesler (PPR) cohesive modelwith a fractional standard linear solid (SLS) model.In our model, we obtain undamaged viscoelastic trac-tions from a fractional SLS model and then scale themwith two damage parameters that are obtained from thePPR cohesive model (Park et al. 2009). We are able touse this model to accurately simulate rate-dependentfracture processes for a wide range of loading rates.The accuracy of the model is a direct result of the

fractional SLS model, which entirely governs the rate-dependence in this CZM. In addition, we successfullyapproximate experimental results obtained from therate-dependentmixed-mode fracture test byLiechti andWu (2001), which demonstrates the capability of themodel to capture rate-dependent mode-mixity depen-dence.

We analyze the behavior of the relaxation functionof the fractional SLS model to provide physical inter-pretation to the order, v, of the fractional differentialoperator. We conclude that v can be interpreted as amemory-like parameter, which allows for simulationof crack propagation problems governed by hereditary

123


phenomena with long- and short-term memory. Thisparameter strongly influences the debonding process offiber-reinforced structures for low loading rates. Whenv is small (e.g., v < 0.3), the macroscopic stress–strain behavior of the fiber-reinforced structures is sen-sitive to applied displacement rates ranging between1 × 10−12 mm/min and 1 × 104 mm/min. However,when v is larger, our results indicate that the macro-scopic stress–strain relationship is nearly insensitive tothe loading rate up to approximately 1×10−1 mm/min.Using intermediate values for v, we were able to con-trol the rate-dependence of the CZM for intermediateloading rates. From these observations, the contribu-tion of this work is is a model that accurately simulatesrate-dependentmixed-mode fracture processes for bothlong- and short-term memory materials.

Acknowledgements The authors gratefully acknowledge sup-port by the Raymond Allen Jones Chair at the Georgia Instituteof Technology. The interpretation of the results of this work issolely that by the authors, and it does not necessarily reflect theviews of the sponsors or sponsoring agencies.

Appendix A Assessment of the numerical schemeused to estimate the Caputo fractional derivative

To verify the accuracy of the numerical scheme pre-sented in Sect. 2.2 to approximate the Caputo fractionalderivative of a function, f (t), we rewrite Eq. (6) morecompactly, as follows:

0Dvt f (tN ) ≈ 0 D

vt f (tN ) = �t−v

N

N∑j=1

( f j − f j−1)B j ,

B j =�t vN (t1−v

N , j−1 − t1−vN , j )

t j, j−1, (38)

where f j−1 = f (t j−1), f j = f (t j ), and tu,v =tu − tv, ∀ u, v ∈ 0, . . . , N .

For the numerical assessment, we use the polyno-mial function f (t) = 1

2 t2, for which the Caputo deriva-

tive is known in closed-form as

0Dvt f (t) = �(3)

2�(3 − v)t2−v .

The numerical results computed using Eq. (38) arecompared with the exact Caputo derivative for vari-ous orders of the fractional derivative, v. We use theunequally-spaced time grid shown in Fig. 22 to com-pute the fractional derivatives. As illustrated by Fig. 23,the results obtained fromEq. (38) approximate well theexact Caputo fractional derivative for all orders of thefractional derivative that we considered. To quantifythe quality of the approximation, we compute the rel-ative H1-norm of the error between the computed andexact fractional derivatives,

ε1 = ‖0Dvt f (t) − 0 Dv

t f (t)‖1‖0Dv

t f (t)‖1, (39)

in which 0 Dvt f (t) refers to the approximation of

0Dvt f (t), i.e., Eq. (38), and ‖ · ‖1 = ∫ t

0 ‖·‖dτ is theH1-norm of a given function. The computed errors aredisplayed in Table 4.

We also conduct a convergence study to determinethe rate of convergence of our numerical solution. Forthis purpose, we use a uniform grid, such that a sin-gle time step �t is used to obtain t j+1 = t j + �t . Asconducted before, we evaluate the relative H1-norm forv = 0.3, 0.5, 0.7, and 0.9.As indicated byFig. 24, ournumerical scheme has a convergence rate ofO(�t2−v ).

In addition to assessing the accuracy of the proposednumerical scheme, we provide some remarks relatedto its computational cost. For comparison purposes,we use the numerical approximation of theGrünwald–Letnikov fractional derivative (Grünwald 1867), whichhas been used in studies by Schmidt and Gaul (2002)and Musto and Alfano (2015), to name a few. Theapproximation to the Grünwald–Letnikov fractionalderivative is

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Time (s)

Fig. 22 Unequally spaced time discretization used to evaluatethe accuracy of Eq. (38) to estimate the Caputo fractional deriva-tive for a smooth function f (t). The grid consists of 101 points

(i.e., N = 100), such that ti = 2τ 2i (i = 0, . . . , 100), withτi+1 = τi + 1/N and τ0 = 0

123


0.3 0.5 0.7 0.9

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

Time (s)

v =Exact results for 0.3 0.5 0.7 0.9v =Numerical results for

0Dv tf(t

)

Fig. 23 Numerical and exact Caputo fractional derivative off (t) = 1

2 t2 obtained on the unequally-spaced time grid from

Fig. 22. We consider various orders of the fractional derivative,v ∈ (0, 1)

Table 4 Relative H1-normof the error between thecomputed and exactfractional derivatives forf (t) = 1

2 t2 and for various

orders of the fractionalderivative, v

v ε1

0.3 3.8 × 10−4

0.5 1.2 × 10−3

0.7 3.2 × 10−3

0.9 8.3 × 10−3

Dvt f (t) ≈

(t

N

)−v N−1∑j=0

Dj+1 f (t − j t/N ),

Dj+1 = �( j − v)

�(−v)�( j + 1)= j − 1 − v

jD j , D1 = 1.

(40)

To approximate a fractional derivative using eitherEq. (38) or (40), we need to sum N terms, which showsthat both equations scale equally as the number of gridpoints, N increase. However, due to the larger numberof floating point operations involved in Eq. (38), its costof evaluation is larger than that in Eq. (40). We verifiedthis observation numerically by computing the frac-tional derivative of f (t) = 1

2 t2 for various values of N

using both Eq. (38) and (40), as depicted in Fig. 25.

Appendix B Components of the material tangentmatrix for the rate-independent PPR cohesive zonemodel

Parameters Dtt , Dtn , Dnt , and Dnn in Eqs. (32)–(35)are the components of the material tangent matrix for

10 -4 10 -3 10 -2 10 -1 10 0

1

0.30.5

v =v =

0.7v =0.9v =

10 -8

10 -6

10 -4

10 -2

10 0

Time increment,

-nor

m o

f err

or,

H1

1.1

1.3

1.5

1.7

Fig. 24 Relative H1-norm of the error between the numericaland exact Caputo derivatives of f (t) = 1

2 t2, computed using

Eq. (39), for various values of �t . As observed in the results, thenumerical scheme used in this study exhibits a convergence rateof O(�t2−v )

Com

puta

tiona

l tim

e (s

)

N

10 110 -4

10 -2

10 0

10 2

10 2 10 3 10 4

Proposed schemeGrunwald-Letnikov approx.

Fig. 25 Computational cost for the evaluation of a fractionalderivative as a function of N . For either Eq. (38) (proposedscheme) or Eq. (40) (Grünwald–Letnikov approximation) andfor each value of N , the computational time is determined as theaverage time required to evaluate the fractional derivative in allgrid points

the rate-independent PPR cohesive zone model (Parkand Paulino 2012). Because κn and κt are non-negativequantities, wewrite the components of thematerial tan-gent matrix for the rate independent PPR model as

Dtt (κn, κt ) = �t

δ2t

[(n2 − n)

(1 − κt

δt

)β

(n

β+ κt

δt

)n−2

+ (β2 − β)

(1 − κt

δt

)β−2

123


(n

β+ κt

δt

)n

− 2βn

(1 − κt

δt

)β−1 (n

β+ κt

δt

)n−1]

[�n

(1 − κn

δn

)α (m

α+ κn

δn

)m

+ 〈φn − φt 〉]

, (41)

Dtn(κn, κt ) = Dnt (κn, κt )

= �n�t

δnδt

[m

(1 − κn

δn

)α (m

α+ κn

δn

)m−1

− α

(1 − κn

δn

)α−1 (m

α+ κn

δn

)m]

[n

(1 − κt

δt

)β (n

β+ κt

δt

)n−1

− β

(1 − κt

δt

)β−1 (n

β+ κt

δt

)n]

, (42)

Dnn(κn, κt ) = �n

δ2n

[(m2 − m)

(1 − κn

δn

)α (m

α+ κn

δn

)m−2

+ (α2 − α)

(1 − κn

δn

)α−2 (m

α+ κn

δn

)m

− 2αm

(1 − κn

δn

)α−1 (m

α+ κn

δn

)m−1]

[�t

(1 − κt

δt

)β (n

β+ κt

δt

)n

+ 〈φt − φn〉]

. (43)

Appendix C List of symbols

α Parameter controlling the shape of the cohe-sive softening curve in the normal direction

δn Conjugate normal final crack openingwidthδt Conjugate tangential final crack opening

widthβ Parameter controlling the shape of the cohe-

sive softening curve in the tangential direc-tion

δnc Normal crack opening width at the peaknormal traction in the original PPR model

�n Normal separation along the fracture sur-face

δn Normal final crack opening width�i+1

n Normal separation along the fracture sur-face at time increment i + 1

�in Normal separation along the fracture sur-

face at time increment iδtc Tangential crack opening width at the peak

normal traction in the original PPR model�t Tangential separation along the fracture sur-

face

δt Tangential final crack opening width�i+1

t Tangential separation along the fracture sur-face at time increment i + 1

�it Tangential separation along the fracture sur-

face at time increment i�n Energy constant in the PPR model�t Energy constant in the PPR modelηn Rigidity of the springpot in the normal

directionηt Rigidity of the springpot in the tangential

directionDnn, Dnt Normal components of the material tan-

gent stiffness matrix from the original PPRmodel

Dtn, Dtt Tangential components of the material tan-gent stiffness matrix from the original PPRmodel

Tn Normal cohesive traction in the originalPPR model

Tt Tangential cohesive traction in the originalPPR model

v Order of the Caputo fractional derivativeκn Kinematic quantity defining the maximum

normal crack opening width in the historyof loading

κ i+1n Maximum normal crack opening width at

time increment i + 1κ in Maximum normal crack opening width at

time increment iκt Kinematic quantity defining the maximum

absolute tangential crack opening width inthe history of loading

κ i+1t Maximum absolute tangential crack open-

ing width at time increment i + 1κ it Maximum absolute tangential crack open-

ing width at time increment iλn Parameter controlling the hardening slope

of the PPR model in the normal directionλt Parameter controlling the hardening slope

of the PPRmodel in the tangential direction(˙) Time derivative〈·〉 Macauley bracketD Material tangent stiffness matrixφn Fracture energy in the normal direction

(with zero tangential separation)φt Fracture energy in the tangential direction

(with zero tangential separation)σmax Cohesive strength in the normal direction

from the original PPR model

123


τmax Cohesive strength in the tangential directionfrom the original PPR model

0Dvt (·) Caputo fractional derivative of order v

0 I vt (·) Riemann–Liouville fractional integral of

order v

c Stiffness parameter for the fractional SLSmodel

Dnn, Dnt Normal components of the material tangentstiffnessmatrix for the rate-dependentCZM

dn Normal scalar damage parameter consistentwith the PPR model

Dtn, Dtt Tangential components of the material tan-gent stiffness matrix for the rate-dependentCZM

dt Tangential scalar damage parameter consis-tent with the PPR model

En Initial stiffness parameter for thePPRmodelin the normal direction

Et Initial stiffness parameter for thePPRmodelin the tangential direction

m Nondimensional exponent in thePPRmodeln Nondimensional exponent in thePPRmodelSn Normal undamaged traction from fractional

SLS modelSt Tangential undamaged traction from frac-

tional SLS modelt timeTn Rate-dependent cohesive traction in the

normal directionTt Rate-dependent cohesive traction in the tan-

gential direction

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