path dependence of truss-like mixed mode cohesive laws

Engineering Fracture Mechanics 91 (2012) 117–132

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Engineering Fracture Mechanics

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Path dependence of truss-like mixed mode cohesive laws

Stergios Goutianos, Bent F. Sørensen ⇑Department of Wind Energy, Section of Composites and Materials Mechanics, Technical University of Denmark, Risø Campus, Frederiksborgvej 399,DK-4000 Roskilde, Denmark

a r t i c l e i n f o

Article history:Received 8 July 2011Received in revised form 6 December 2011Accepted 16 February 2012

Keywords:Cohesive lawCohesive elementPath dependence

0013-7944/$ - see front matter � 2012 Elsevier Ltdhttp://dx.doi.org/10.1016/j.engfracmech.2012.02.011

⇑ Corresponding author.E-mail address: [email protected] (B.F. Sørensen).

a b s t r a c t

A general theoretical analysis is presented to prove that, under mixed mode fracture, truss-like mixed mode cohesive laws (cohesive laws that are coupled in a special manner suchthat the traction vector follows the separation/opening vector) are path independent onlyin the limited case where the fracture resistance (and effective traction) is independent ofthe phase angle of openings. To verify the theoretical analysis, a specific class of truss-likecohesive laws, coupled with a failure criterion for damage initiation and an effective open-ing displacement is used. It is shown analytically and numerically that these cohesive lawsare path dependent.

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1. Introduction

Mixed mode fracture is often observed along weak planes in layered structures e.g. along interfaces between individuallayers or along adhesive joints. Under mixed mode cracking, the fracture process zone is subjected to both normal and tan-gential separations. The concept of modelling fracture by the use of cohesive laws, i.e. describing the fracture process ofmaterials through traction–separation relationships, was introduced by Dugdale [1] and Barenblatt [2]. Needleman [3] intro-duced a mode I cohesive law model in a continuum mechanics finite element model. Since then, cohesive laws have beenwidely used in advanced numerical models of materials and structures, i.e. Tvergaard and Hutchinson [4], Yang et al. [5],Mohammed and Liechti [6], Jacobsen and Sørensen [7]. Needleman [8], and Xu and Needleman [9,10] generalised cohesivelaws to include shear tractions (mixed mode). Several types of traction–separation laws have been introduced for mixedmode fracture. They can be categorised in four classes:

(a) Coupled mixed mode cohesive laws based on a potential function [3,9–15]. The fracture resistance can be independent[11] or a function of the phase angle of the openings [14]. Potential function-based cohesive laws are path-indepen-dent. That implies that the work of the local cohesive tractions depend only on the local normal and tangential sep-arations and not on the opening path history.

(b) Mixed mode cohesive laws that are path-dependent [16,17]. Thus, they cannot be derived from a potential function.Path-dependent mixed mode cohesive laws have been proposed for the modelling of fracture processes that includehistory-dependent mechanisms such as plastic deformation and frictional sliding [17].

(c) Uncoupled mixed mode cohesive laws for which the mode I and mode II cohesive laws are completely independent ofeach other. This implies that the normal traction depends only on the normal separation and the shear traction only onthe tangential separation (sliding). A sub-class of uncoupled cohesive laws is weakly coupled cohesive laws for which

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Nomenclature

B local mode mixity ratioD damage variableE Young’s modulusG energy release rateGc critical energy release rate (fracture energy)JR fracture resistance (work of cohesive tractions)J/c

work of separationJcI work of separation for pure mode I

JcII work of separation for pure mode II

JI/c

mode I part of the work of cohesive tractions

JII/c

mode II part of the work of cohesive tractionsK elastic stiffness of bi-linear cohesive lawKI mode I stress intensity factorKII mode II stress intensity factorWpath work of cohesive tractionsWcn!t

work of cohesive tractions (normal opening followed by tangential sliding)Wct!n

work of cohesive tractions (tangential sliding followed by normal opening)dn normal openingd�n end-openingdmax

n maximum normal opening

doIn normal opening for damage initiation (pure mode I)

df In critical opening for complete failure (pure mode I)

don normal opening for damage initiation (mixed mode)

dfn critical opening for complete failure (mixed mode)

dt tangential slidingd�t end-slidingdoII

t sliding for damage initiation (pure mode II)

df IIt critical sliding for complete failure (pure mode II)

dot sliding for damage initiation (mixed mode)

dft critical sliding for complete failure (mixed mode)

dm magnitude of effective openingd�m magnitude of effective end-openingdmax

m maximum effective openingdo

m effective opening for damage initiation

dfm effective opening for complete failure

/ phase angle of opening/⁄ phase angle of end-openingw mode mixityw⁄ phase angle of tractions at the end of cohesive zone~w phase angle of the stress intensity factorsg mixed mode interaction parameter (Benzeggagh and Kenane criterion)U potential functionrn normal tractionr�n normal traction at the end of the cohesive zoner̂n peak normal traction value (pure mode I)rt shear tractionr�t shear traction at the end of the cohesive zoner̂t peak shear traction value (pure mode II)re magnitude of effective tractionrc

e magnitude of effective traction for damage initiation

118 S. Goutianos, B.F. Sørensen / Engineering Fracture Mechanics 91 (2012) 117–132

fracture (complete failure, i.e. zero tractions) under mixed mode occurs when a failure criterion is satisfied [18,19].Such a fracture criterion provides a weak coupling between the two independent cohesive laws. Uncoupled mixedmode cohesive laws are actually derivable from a potential function.

S. Goutianos, B.F. Sørensen / Engineering Fracture Mechanics 91 (2012) 117–132 119

(d) Other mixed mode cohesive laws [20–23]. Some mixed mode cohesive laws can be constructed knowing the mode Iand mode II cohesive laws and using two interaction criteria: one for the onset of cohesive failure (the peak traction)and one for complete failure [20]. These interaction criteria provide a coupling between the mode I and mode II cohe-sive laws under mixed mode fracture. A special case of mixed mode cohesive laws, which we will call truss-like cohe-sive laws, are cohesive laws for which the traction vector follows the separation vector; a behaviour that resembles atruss behaviour. The present study investigates the behaviour of truss-like mixed mode cohesive laws.

Two key concepts in the present study are the phase angle of the openings, /, and the phase angle of the cohesive trac-tions, w. The phase angle of the openings, /, is defined as:

/ ¼ tan�1 dt

dn

� �; ð1Þ

where dn and dt are the normal (opening) and tangential (sliding) separations (opening displacements), respectively, at a gi-ven position within the cohesive zone. The phase angle of the cohesive tractions, w, is defined as:

w ¼ tan�1 rt

rn

� �; ð2Þ

where rn and rt are the normal traction and shear traction, respectively. In generally, both / and w are functions of positionwithin the cohesive zone. Furthermore, in general / differs from w. However, as described above, for truss-like cohesive laws,/ = w, so that they vary in the same fashion within a cohesive zone.

Experimental studies have shown that under mixed mode, the fracture energy (the work of separation) depends on thephase angle of openings, /. Usually the fracture energy increases with increasing / [24–27]. Van den Bosch et al. [17] haveshown that the cohesive zone model of Xu and Needleman [9,10] predicts that energy is still required to break the cohesivezone in normal direction even after complete shear separation when the fracture energy varies with /; this is considered asbeing an unrealistic behaviour. The widely used mixed mode potential function-based cohesive law of Tvergaard and Hutch-inson [11] is formulated such that the fracture resistance (the work of the cohesive tractions for given openings) is the samefor all values of / due to the form of the potential function used.

One of the advantages of truss-like cohesive law formulations is that the fracture energy variation as a function of / canbe easily specified [20]. However, it appears that little work has been done in examining their performance in details undermixed mode conditions; this is the aim of the present work.

It should be emphasised that for truss-like cohesive laws there is a coupling through the failure criterion for the onset offailure and through the effective opening for complete failure, df

m, which can depend on /. Using truss-like cohesive laws inmixed mode fracture simulations involving large scale bridging, Sørensen et al. [28] found that a change of the peak tractionvalues of modes I and II (changing the onset of failure) and the corresponding critical openings for complete failure in suchway that the fracture energy remained unchanged, resulted in a change of the predicted component strength. Motivated bythese results, Turon et al. [29] have shown recently that truss-like cohesive zone models [20,21] predicted different overallmaximum load for different maximum values of the cohesive tractions under mixed mode fracture even in the case a smallscale fracture process zone, i.e. under the conditions of Linear Elastic Fracture Mechanics (LEFM). They proposed a method-ology, based on modification of the critical tractions (the peak traction at the onset of damage initiation), in order to obtainthe correct energy dissipation. Although, this method yields accurate numerical results, the approach lacks physical founda-tion since the peak traction values of the mode I and mode II cohesive laws are independent material properties that can bemeasured experimentally, see e.g. Sørensen and Jacobsen [30]. Therefore, it is desirable that they are not treated as adjust-able parameters especially for large scale fracture process zones.

In this work, we show that these problems of the truss-like cohesive laws stems from their path dependence. First, it isshown, in a general framework, that truss-like cohesive zones are path independent only in the case where the mixed modefracture resistance is exactly the same for all phase angles of openings. Next, a specific truss-like cohesive law [20,21], asimplemented in the commercial finite element code Abaqus [31], is examined to verify that this truss-like cohesive law ispath dependent when the fracture resistance varies with the phase angle of openings / and to investigate if the effect ofpath-dependence is large or insignificant.

The path independence of cohesive laws may be a useful characteristic when modelling mixed mode fracture. WithinLEFM, the criterion for crack growth under mixed mode can be written as (plane stress):

Gð~wÞ ¼ K2I þ K2

II

E¼ Gcð~wÞ; ð3Þ

where G denotes the energy release rate and Gc is the critical energy release rate (fracture energy), which is a function of thephase angle of the stress intensity factors, ~w, (sometimes called the mode mixity) [32,33]. Under the conditions of LEFM, thephase angle of the stress intensity factors equals the phase angle of the crack tip stresses, Eq. (2), ~w ¼ w.

In Eq. (3), KI and KII are the mode I and mode II stress intensity factors and E the Young’s modulus. The underlying premiseof LEFM is that the fracture process zone (and associated crack tip plastic zone) is so small in size that it is embedded withinthe so-called K-dominant region, the region around the crack tip in which the stress field closely follows the singular stress

Fig. 1. Three loading histories for an LEFM problem, all reaching Gc at the same mode mixity: application first of KI followed by KII (normal opening followedby tangential sliding, denoted as n ? t), application first of KII followed by KI (denoted as t ? n), and simultaneous application of KI and KII (proportional,denoted as p).


field associated with the mode I and mode II stress intensity factors, KI and KII. Then, the external load and geometry com-municate with the fracture process zone through the K-dominant region.

The criterion (Eq. (3)) applies independently of the loading history, implying that the criterion predicts the onset of crackgrowth at the same value of Gc irrespective of whether KI and KII are increased simultaneously (proportional), KI is appliedfirst and KII next, or visa versa as shown schematically in Fig. 1. This is not only a consequence of linearity (the final stressstate is obtained by superposition) but also the chosen formulation, Eq. (3). In other words, the LEFM fracture criterion isindependent of the load and crack opening history. Had the LEFM crack growth criterion been formulated such that it de-pended on the loading/opening history, the fracture energy of a given mode mixity would be different for different openingpaths; the criterion (Eq. (3)) ensures that the fracture energy depends only on the mode mixity, not the opening path. Whenmodelling fracture using a cohesive zone model (which is more general than LEFM: there are no restrictions to small scalefracture process zone and small scale yielding), the counterpart to the history-independent fracture criterion is that thecohesive laws are derived from a potential function.

2. General description of cohesive laws

2.1. Mixed mode cohesive laws derived from a potential function

The problem taken up is a planar (two dimensional) cohesive zone problem illustrated in Fig. 2a. The approach followedhere applies both to plane stress and plane strain conditions. The entire fracture process can be described by a mixed-modecohesive law, which includes the crack tip fracture process (bypassing a singular crack tip stress field) and crack bridgingphenomenon [34,14]. By evaluating the path-independent J integral [35] locally around the cohesive zone, the J integral be-comes [34]:

Fig. 2.tangent

JR ¼Z d�t

0rtðdn; dtÞddt þ

Z d�n

0rnðdn; dtÞddn; ð4Þ

where d�n and d�t are the end-opening and end-sliding of the cohesive zone, respectively (see Fig. 2a). Note that both rn and rt

are functions of dn and dt. The J integral result (Eq. (4)) can be interpreted as the work (per unit fracture area) of the cohesivetractions at the end of the cohesive zone. This result holds for any values of d�n and d�t . JR calculated according to Eq. (4) iscalled the fracture resistance. When the cohesive zone is fully developed (d�n ¼ df

n and d�t ¼ dft , where df

n and dft are the values

of dn and dt where the cohesive tractions vanish), JR equals the work of separation, J/c, also called the fracture energy.

(a) (b)Schematic illustration of a crack experiencing large scale bridging under mixed mode crack opening displacement. (a) Definition of normal,ial, and effective end-opening displacements and (b) integration path for the J integral.


Starting from the cohesive tractions, they must fulfill the following condition to be derivable from a potential function[36]:

@rn

@dt¼ @rt

@dn: ð5Þ

However, we can also start from a potential function. If it is assumed that the tractions are derived from a potential function,U:

Uðdn; dtÞ ^ Uð0;0Þ ¼ 0; ð6Þ

Then, the normal, rn, and shear, rt tractions can be taken to be functions of both dn and dt but independent of position withinthe cohesive zone:

rnðdn; dtÞ ¼@Uðdn; dtÞ

@dn; rtðdn; dtÞ ¼

@Uðdn; dtÞ@dt

: ð7Þ

From Eqs. (4), (6) and (7), the J integral becomes:

JR ¼ U d�n; d�t

� ��Uð0;0Þ ¼ U d�n; d

�t

� �: ð8Þ

This result follows from the assumption that the tractions are derived from a potential function. Then, the work of the trac-tions is independent of the opening path history and equal to the difference in the potential from the end and beginning ofthe path [37].

Finally, from Eqs. (7) and (8), the following expressions for the cohesive tractions at the end of the cohesive zone can beobtained:

r�n ¼ rn d�n; d�t

� �¼@JR d�n; d

�t

� �@d�n

; r�t ¼ rt d�n; d�t

� �¼@JR d�n; d

�t

� �@d�t

: ð9Þ

Here, an asterix indicates the position of the end of the cohesive zone. However, as the cohesive laws are taken to be thesame at any position within the cohesive zone, the cohesive laws at the end-openings (Eq. (9)) must be identical to the cohe-sive law at any position within the cohesive zone.

2.2. Cohesive laws in polar form

The magnitude of the opening displacement, denoted the effective displacement, is defined as [38]:

dm ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2

n þ d2t

q; ð10Þ

whereas from Eq. (1) the normal opening and tangential sliding are given by:

dn ¼ dm cos /; dt ¼ dm sin /: ð11Þ

The magnitude of the traction, defined as the effective traction, re, is given by:

re ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

n þ r2t

q: ð12Þ

In the following, we wish to express the openings and cohesive tractions in polar form, i.e. in terms of the magnitude andphase angle. The phase angle is of the openings, /, is defined in Eq. (1) and the phase angle of the traction, w, is definedin Eq. (2).

The end-opening and end-sliding in Cartesian form, d�n and d�t , can be transformed to polar form:

d�m ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid�n

2 þ d�t2

q; /� ¼ tan�1 d�t

d�n

� �; ð13Þ

where d�m is end-opening magnitude and /⁄ is the phase angle of the end-openings. Finally, the phase angle of the tractionvector at the end of the cohesive zone is defined as:

w� ¼ tan�1 r�tr�n

� �: ð14Þ

Then, for d�n – 0 and d�t – 0, the partial derivatives in the Cartesian space d�n � d�t , Eq. (9), can be obtained from the polard�m � /� coordinate system using the chain rule:

r�n ¼@JR

@d�n¼ @JR

@d�m

@d�m@d�n

þ @JR

@/�@/�

@d�n¼ cos /�

@JR

@d�m� sin /�

d�m

@JR

@/�ð15Þ

r�t ¼@JR

@d�t¼ @JR

@d�m

@d�m@d�t

þ @JR

@/�@/�

@d�t¼ sin /�

@JR

@d�mþ cos /�

d�m

@JR

@/�: ð16Þ

Fig. 3. Schematic illustration of bi-linear traction–separation laws (normal and tangential directions).


3. Specific relationships between cohesive tractions: truss-like cohesive laws

As mentioned in the Introduction, for truss-like cohesive laws the phase angle of the cohesive traction vector, w, and thephase angle of the openings, /, must be identical for any point within the cohesive zone, w = /. This also holds for the end-openings. Thus, the direction of cohesive tractions at the end of the cohesive zone must follow the direction of the end-openings:

w� ¼ /� ) tan w� ¼ tan /�: ð17Þ

Then, by substituting tan/⁄, r�n and r�t from Eqs. (14)–(16), respectively, into Eq. (17) we obtain:

sin /�@JR

@d�mþ cos /�

d�m

@JR

@/�¼ cos /�

@JR

@d�m� sin /�

d�m

@JR

@/�

� �tan /� ð18Þ

which is reduced to:

ðcos2 /� þ sin2 /�Þ @JR

@/�¼ 0) @JR

@/�¼ 0; ð19Þ

This implies that when w⁄ = /⁄ (truss-like cohesive laws) the tractions can be derived from a potential function only when JR

is independent of the phase angle of the openings. Note that it is the fracture resistance (the work of the cohesive tractions),calculated from Eq. (4), not just the fracture energy (the work of separation), that must be independent of /⁄ for the tractionsto be derivable from a potential function. This completes the proof (Eq. (19)). A different proof, based on Eq. (5), is given inAppendix A.

4. Bi-linear truss-like mixed mode cohesive laws

In this section, a type of widely used truss-like mixed mode bi-linear cohesive laws [20,21], as implemented in the com-mercial finite element code Abaqus [31], is investigated with respect to the analysis presented in the previous section. Thebi-linear cohesive laws are shown in Fig. 3 for pure mode I (normal opening) and mode II (sliding). The pure mode I traction–separation law has an initially rising part with an elastic stiffness of K. At a certain peak traction value, r̂n, damage initiates.The corresponding opening is doI

n ¼ r̂n=K. After damage has initiated, the traction decays linearly with increasing normalopening dn until the opening becomes equal to the critical opening, df I

n at which rn is zero (It should be noted that in mostcases the first elastic part is introduced for purely numerical reasons. Then doI

n is usually very small, so that the traction–sep-aration law is practically a linear softening law). Similarly, for pure mode II, damage initiates at r̂t and the correspondingsliding is doII

t ¼ r̂t=K . The critical sliding for complete failure is df IIt .

It is shown in Appendix B that these mixed mode cohesive laws are truss-like cohesive laws, i.e. / = w, whereas in Appen-dix C it is proved analytically that they are path independent only when:

@dom

@/¼ 0 and

@dfm

@/¼ 0; ð20Þ

where dom is the effective opening for damage initiation and df

m is the effective opening for complete failure. Thus, the truss-like bi-linear cohesive laws described here can be obtained from a potential function (path independent cohesive laws) onlywhen both the cohesive law parameters do

m and dfm are independent of the phase angle of openings /. This result is consistent

with the general proof in Section 3.

a

+

b

+

c

Fig. 4. Different loading paths (displacement controlled): (a) normal opening, followed by tangential sliding cn?t, (b) tangential sliding followed by normalopening ct?n, and (c) proportional loading cp.


5. Numerical verification

The results presented above are verified numerically by the commercial finite element code Abaqus. In order to check thepath dependence of the truss-like bi-linear cohesive laws described in Section 4, three opening paths are chosen to the samenormal and tangential separations as shown in Fig. 4. If the mixed mode cohesive law is path independent, the work of sep-aration should be the same for the three different opening paths.

5.1. Method of calculating the work of separation

The two dimensional finite element model consists of a single zero thickness cohesive element (COH2D4) connecting twosquare plane stress reduced integration continuum elements (CPS4R). Appropriate displacement boundary conditions areapplied at the solid elements to generate the opening paths shown in Fig. 4. A large value for the Young’s modulus of the

continuum elements was selected in order to constrain the deformation only in the cohesive element E ¼ 103Kdfn

� �.

Since all nodes of the cohesive element undergo the same normal and tangential separations, the work of separation (perunit area) of the cohesive tractions of the finite element model is calculated as the sum of the work of normal and shear trac-tions, respectively:

Wpath ¼Z df

n

0rn ddn þ

Z dft

0rt ddt ; ð21Þ

where dfn and df

t are the normal and sliding openings at which the normal and shear tractions reduce to zero, respectively.Thus values of rn, rt, dn and dt are obtained from the finite element simulations and integrated numerically for the threeopening paths.

5.2. Results for linear variation of fracture energy

First, the case where the peak values of tractions are equal for pure mode I and mode II is examined ðr̂n ¼ r̂tÞ. Then, dom

(but not dfm) is independent of /. In the finite element model, the fracture energy is specified as a function of the phase angle

of openings and it is an input parameter. In the following, a linear variation of the fracture energy from mode I to mode II isassumed:

J/c¼ Jc

I 1þ JcII

JcI� 1

� �/

90�

; 0� 6 / 6 90�; ð22Þ

where JcI is the fracture energy for pure mode I and Jc

II the fracture energy for pure mode II.

Fig. 5. Cartesian dn–dt plane for mixed mode fracture for four different JcII=Jc

I ratios ðr̂t ¼ r̂nÞ. The mixed mode fracture energy, J/c, varies linearly from Jc

I toJc

II .

Fig. 6. Normalised fracture energy, Wpath, along paths cp,cn?t and ct?n, with the fracture energy, J/c, at mode mixities / = 15� and 45� as a function of JIIc

=JIc

ðr̂t ¼ r̂nÞ. The mixed mode fracture energy, J/c, varies linearly from Jc

I to JcII .


Fig. 5 shows the normalised Cartesian dn–dt plane for four different JcII=Jc

I ratios. Each solid line represents the critical nor-

mal dfnð/Þ

� �and tangential df

t ð/Þ� �

openings for a certain JcII=Jc

I ratio. Fig. 5 was constructed using Eqs. (B10), (B13) and (D6).

The path-dependence of the truss-like mixed mode cohesive laws, described in Section 4, can be seen graphically by firstfocusing at the results for Jc

II=JcI ¼ 4. Then, from Eq. (22) the work of separation along the proportional path between O

and A (/ = 45�) is J/c¼ Jc

I 1þ ð4� 1Þ 12

� �¼ 2 1

2 JcI . For the path cn?t complete fracture occurs during normal opening and thus

the work of separation is JcI . This value is obviously different from the work of separation of J/c

¼ 2 12 Jc

I obtained for the pro-portional path between O and A.

For each JcII=Jc

I ratio, the three opening paths shown in Fig. 4 were simulated and the results are shown in Fig. 6 for / = 15�and 45� (see Fig. 5). The work of cohesive tractions along each of the three paths is normalised with the expected fractureenergy J/c

calculated using Eq. (22). From Fig. 6 it is seen that when the loading follows the proportional path, cp, the numer-ically computed work of cohesive tractions (see Eq. (21)) equals the fracture energy J/c

(see Eq. (22)) for all JcII=Jc

I ratios. Therelative difference was below 0.1%. From Fig. 6 it is seen that when a non-proportional path is followed, the work of sepa-ration is independent of the path only when Jc

II ¼ JcI , i.e. when rc

e, dom and df

m are all independent of /. This result is in accor-dance with the analysis presented in Section 3. Fig. 6 shows that the work of cohesive tractions of different opening paths canbe significantly different.

Next, a case is presented in which the peak tractions in mode I and mode II are different. Experiments have shown [28]that the mode II peak traction value, r̂t , can be significantly higher than the mode I peak traction value, r̂n. Therefore, cal-culations as above were performed for r̂t ¼ 2r̂n, with K being independent of /. Then, do

m increases with /. Fig. 7 shows theCartesian df

n � dft plane for four different Jc

II=JcI ratios. Fig. 8 shows the fracture energy normalised with the expected fracture

energy for the three different loading paths (Fig. 4) as a function of the JcII=Jc

I ratio. The most interesting observation is thateven when Jc

II ¼ JcI , Wcn!t

– Wct!n– J/c

. This can also be seen in Fig. 7 for the t ? n path, where the cohesive element fails

Fig. 7. Cartesian dn–dt space for mixed mode fracture for four different JcII=Jc

I ratios ðr̂t ¼ 2r̂nÞ. Mixed mode fracture energy, J/c, varies linearly from Jc

I to JcII .

Fig. 8. Normalised fracture energy, Wpath, along paths cp,cn?t and ct?n, with the fracture energy, J/c, at mode mixity / as a function of JIIc

=JIcðr̂t ¼ 2r̂nÞ. The

mixed mode fracture energy varies, J/c, varies linearly from Jc

I to JcII .

Fig. 9. Normalised fracture energy, Wpath, along paths cp,cn?t and ct?n, with the fracture energy, J/c, at mode mixity / as a function of JIIc

=JIcðr̂t ¼ r̂nÞ. The

mixed mode fracture energy is given from Eq. (23). The solid circles indicate that the cohesive element has not completely failed.



completely during the tangential sliding. The results show that it is not just the fracture energy, but the entire fracture resis-tance (Eq. (4)), i.e., all cohesive parameters (rc

e, dom and df

m) that must be independent of / if the simulated fracture energyshould be opening path independent. This confirms the proof (Eq. (19)).

5.3. Results for mixed fracture energy described by the Benzeggagh and Kenane criterion

The numerical model from Section 5 is used to examine the path dependence of truss-like mixed mode cohesive law (withlinear softening) where the mixed mode fracture energy is described by the Benzeggagh and Kenane criterion [39]:

Fig. 10.values,derived

J/c¼ Jc

I þ JcII � Jc

I

� �Bg; ð23Þ

where g is a mixed mode interaction parameter (in the simulations g is equal to 2), and B is a local mixed mode ratio givenby:

B ¼ JIIð/ÞJIIð/Þ þ JIð/Þ

: ð24Þ

Similarly to the results of Figs. 6 and 8, 9 shows that this truss-like mixed mode cohesive law is also path independent onlywhen Jc

II ¼ JcI and that the effect of the fracture energy variation with /, Eqs. (22) and (23), has a relative small effect on the

difference between the simulated and specified fracture energy.

6. Discussion

6.1. Consequences of path-independence

As explained in Section 2, for a cohesive law that is derived from a potential function, the fracture resistance of a cohesivezone, Eq. (4), is independent of the opening path history. Fig. 10 shows the dn–dt plane with different openings paths to twoend opening/sliding values, d�n and d�t , illustrated by points A and B. Irrespective of the exact openings and sliding history ofthe d�n and d�t (and the opening and sliding history of all points within the cohesive zone) the same fracture resistance is at-tained as shown mathematically by Eq. (8). Likewise, the fracture resistance corresponding to the values of d�n and d�t of pointB is independent of the opening path; it just depends on the actual values of d�n and d�t at point B. This follows from Eq. (8). Thefact that the fracture resistance of a cohesive zone can be calculated from Eq. (8), requiring only the knowledge of the valuesof the end-opening and end-sliding, d�n and d�t , is convenient for both analytical and numerical modelling.

What is special for truss-like cohesive laws derived from a potential function is as follows. Still, the fracture resistanceattained at point A (Fig. 10b) is independent of the opening history. However, any combination of d�n and d�t that gives thesame value of the effective end-opening, d�m, given by Eq. (10) (shown as the circle with dashed line in Fig. 10b), now mustpossess the same fracture resistance value as point A. Likewise, all combinations of d�n and d�t that have the same value of d�mas point B has the same fracture resistance as point B (shown as the dotted line in Fig. 10b). More experimental investigationis needed to clarify whether this is a realistic behaviour.

We also note that by fulfilling Eq. (17) (and thus fulfilling Eq. (19)), Eqs. (15) and (16) reduce to:

r�n ¼ cos w�@JR

@d�mr�t ¼ sin w�

@JR

@d�mð25Þ

or, by the use of Eq. (A3):

(a) (b)

Cohesive laws derived from a potential function: Illustration of a schematic dn–dt plane with different openings paths to two end opening/slidingd�n and d�t , points A and B. (a) The attained fracture resistance of points A and B are independent of the opening paths. (b) For a truss-like cohesive lawfrom a potential function the fracture resistance is the same for same magnitude of opening (indicated by the circular arcs).

Fig. 11.lower t


r�e ¼@JR

@d�m¼ dJR

dd�m: ð26Þ

The symbol for partial differentiation, @, can be replaced by the symbol for the differential d since the truss-like cohesive lawcan now be considered being a function of just a single parameter, the effective end-opening, d�m. The physical interpretationis that all energy associated with the fracture resistance can be associated with the effective traction, since the direction ofthe effective traction follows that of the openings.

6.2. Consequences of path-dependence

Truss-like mixed mode cohesive laws that do not fulfill Eq. (17) are path-dependent. Then, the work of the cohesive trac-tions depends on the opening path. This has a number of consequences that should be noted.

First (and probably most important), there is no guarantee that the fracture energy specified for a given phase angle ofopenings will be obtained in a model that uses mode-dependent truss-like cohesive laws. This is significant since, as men-tioned in the Introduction, truss-like cohesive laws are sometimes used to model fracture of interfaces for which the fractureenergy is specified as a function of /. However, as shown in Section 5, the resulting work of separation at a point within acohesive zone can significantly higher or lower than the specified fracture energy (see also Fig. 11). As a result, model pre-dictions of the strength of a structure using a mode-dependent truss-like cohesive law are associated with significantly lar-ger uncertainties than predictions based on path independent cohesive laws. This appears to be an unintended feature oftruss-like cohesive law that has not been noticed before. As mentioned in the Introduction, Turon et al. [29] found that mixedmode truss-like cohesive laws did not predict the work of separation and the structural strength accurately even under LEFMconditions. Fig. 10 in the study of Turon et al. [29] shows that the mode mixity of a point within an active cohesive zone in aMMB (mixed mode bending) specimen under LEFM conditions changes from dominantly mode II to dominantly mode I withincreasing opening, suggesting that the opening history of a point in the active cohesive law is non-proportional although theapplied loads are applied proportionally. Thus, the differences in the predicted work of separation of Turon et al. [29] areprobably more due to a local mode mixity history effect than due to applied loading history (in principle, for LEFM problems,the loads applied to the MMB specimen are a linear superposition of mode I and mode II loadings and they could, in principle,be applied in sequence instead of simultaneously). The local mode mixity history appears to be affected by the cohesive zoneparameters; Fig. 11 in Turon et al. [29] shows that the ratio of the peak tractions has a significant effect on the work of sep-aration. Although the cohesive law formulation of Turon et al. [29] may differ from the formulation in Abaqus that we haveused, it is tempting to try to explain the results of Turon et al. [29] by our results. From our results (e.g. Fig. 6) we observe thata non-proportional opening history that starts with tangential sliding followed by a normal opening (like Fig. 10 in Turonet al. [29]) would tend to result in a higher work of separation than specified for Jc

II > JcI . Furthermore, if the loads of the

MMB specimen had been applied non-proportionally in the study of Turon et al. [29], then there might have been a morepronounced effect.

Second, one cannot a priori calculate the work of separation of a cohesive zone for a given phase angle of opening, sincethe work of separation now depends on the history of the opening and sliding experienced by each point within the cohesivezone. For a numerical model (e.g. a finite element model), the work of the cohesive tractions then has to be calculated numer-ically as Eq. (21) for each point within the cohesive zone. Although this is feasible, it is more involving than the calculation ofthe fracture resistance of a cohesive law derived from a potential function as described in Section 6.1. However, for path-dependent cohesive laws, analytical tools, such as beam-theory based models and the J integral cannot be used. If the workof separation of the cohesive zone cannot be calculated using analytical models (even for small-scale fracture process zone,i.e. under LEFM conditions), then it becomes more difficult to assess the effects of changes in cohesive law parameters onoverall strength of structures.

Third, the use of path-dependent cohesive law in simulations of cyclic crack growth requires special care. If a given pointwithin a cohesive zone may experience different opening paths during loading (opening) and unloading (closing), the work

Sketch showing the fracture energy as a function of mode mixity. The predicted work of separation of truss-like cohesive laws can be higher orhan the specified fracture energy due to difference in the opening path history.


of the cohesive traction during loading and unloading will differ. This corresponds to generation or dissipation of energy. Thismay unfortunate for numerical models such as the finite element method that are based on energy principles. If such fea-tures are incorporated in cohesive laws, they should represent relevant physical phenomena (it may not be unphysical orunrealistic that fatigue mechanisms may release or consume energy). Obviously, modelling of fatigue crack growth cycleby cycle requires that the unloading and reloading behaviour of the cohesive law differs. Otherwise, the next cycle wouldbe exactly as the previous cycle. Then, no cyclic crack growth would occur. However, the way that the unloading and reload-ing behaviour of the cohesive law differs from each other should be formulated such that it represents the physical degra-dation of the actual fracture process zone of the material or interface in question. The unloading and reloading responseshould thus be formulated on the basis of experimental measurements.

Finally, we would like to remark that of course there are fracture mechanisms that involve history-dependent phenomenasuch as plasticity and friction. Such fracture mechanisms can be modelled by path-dependent cohesive laws. However, thepath dependency should be formulated on the basis of experimental data or micromechanical models. Path dependentbehaviour should not appear as an un-intentional complication. In order to avoid such undesirable features, it is preferredto use cohesive laws derived from a potential function unless there is clear experimental evidence that shows how the cohe-sive laws depend on the opening history. As far as the authors know, no experimental data has been published that studieswhether or not real cohesive laws are opening path dependent or not.

7. Concluding remarks

A general theoretical proof was given to show that truss-like mixed mode cohesive laws (cohesive laws for which thephase angle of tractions equals the phase angle of the openings) are inherently path dependent except the limiting casewhere the fracture resistance (and thus the mixed mode traction–separation laws) is independent of the phase angle ofthe openings. A specific bi-linear truss-like cohesive law, coupled through a failure criterion and an effective displacement,was selected to verify the theoretical analysis. It was shown analytically and numerically that the bi-linear truss-like mixedmode cohesive law is path dependent in accordance with the proof. The implication is that truss-like cohesive laws, beingsimple to implement, possess an un-intented complication that can cause differences in the overall load predictions evenunder LEFM conditions.

Acknowledgments

The authors thank the anonymous reviewers for many insightful comments and suggestions. BFS was supported by theDanish Centre for Composite Structures and Materials for Wind Turbines (DCCSM), Grant No. 09-067212 from the DanishCouncil for Strategic Research.

Appendix A. An alternative proof

The differential form

dJ ¼ rn ddn þ rt ddt; ðA1Þ

is exact (meaning that a potential function exists for the differential) if [36]:

(a) the domain is simply connected (in two dimensions, it means that the domain should be without holes) and

(b) that the following relationship is full filled:

@rn

@dt¼ @rt

@dn: ðA2Þ

Using the normal and shear tractions from Eqs. (2) and (12) rewritten in terms of the magnitude and the phase angle:

rn ¼ re cos w rt ¼ re sin w; ðA3Þ

we can transform the terms in Eq. (A2) to polar form using the chain rule:

@rn

@dt¼ @rn

@dm

@dm

@dtþ @rn

@/@/@dt

¼ @re

@dmcos w sin /þ @re

@/sin wþ re

@ cos w@/

� �cos /dm

; ðA4Þ

@rt

@dn¼ @rt

@dm

@dm

@dnþ @rt

@/@/@dn

¼ @re

@dmsin w cos /� @re

@/cos wþ re

@ sin w@/

� �sin /dm

: ðA5Þ


Inserting Eqs. A4 and A5 into Eq. (A2) leads to:

@re

@dmðcos w sin /� sin w cos /Þ þ re

dmcos /

@ cos w@/

þ sin /@ sin w@/

� �¼ 1

dm

@re

@/ð� cos w cos /� sin w sin /Þ; ðA6Þ

which can be rewritten as:

@re

@/¼ dm tanðw� /Þ @re

@dm� re

cosðw� /Þ cos /@ cos w@/

þ sin /@ sin w@/

� �: ðA7Þ

Eq. (A7) holds for any cohesive law that is derivable from a potential function, since it is equivalent to Eq. (A2).For a truss-like cohesive law:

w ¼ /: ðA8Þ

Inserting Eq. (A8) into Eq. (A7) and performing the differentiation with respect to / in the parenthesis to the right leads to:

@re

@/¼ 0: ðA9Þ

Eq. (A9) shows that truss-like cohesive laws can be derived from a potential function, only when the effective traction, re, isindependent of the phase angle of the opening, /, implying that the mixed mode cohesive law (and thus the fracture resis-tance) is independent of /. In other words, for truss-like mixed mode cohesive laws that are derived from a potential func-tion the effective traction re must be a function of the magnitude of the opening, dm, only.

Appendix B. Proof of truss-like behaviour for bi-linear cohesive laws

In order to follow the derivation, it should be realised that the following parameters depend on / and dm:

dn; dt; don; d

ot ; d

fn; d

ft ;rn;rt ;re; ðB1Þ

whereas the parameters that depend only on / are:

dom; d

fm;r

ce;D;w: ðB2Þ

However, the following parameters do not depend on / and dm (parameters for pure mode I and mode II cohesive laws):

doIn ; d

oIIt ; d

f In ; d

f IIt ;K; r̂n; r̂t: ðB3Þ

For pure mode I, the linear softening is described through a damage variable, D 2 [0, 1] which is given by:

D ¼df I

n dmaxn � doI

n

� �dmax

n df In � doI

n

� � ; ðB4Þ

where dmaxn is the maximum opening attained in the loading history. For monotonic loading dmax

n ¼ dn. In the linear softeningpart, the pure mode I traction–separation law is given by:

rnðdnÞ ¼ ð1� DÞKdn: ðB5Þ

From Eqs. (B4) and (B5), the cohesive normal traction, rn, can be written as a function of the normal opening, dn (pure modeI):

rnðdnÞ ¼ r̂ndf I

n � dn

df In � doI

n

: ðB6Þ

For pure mode II, similar equations as above apply. The traction–separation relationship in the tangential direction is givenby:

rtðdtÞ ¼ ð1� DÞKdt : ðB7Þ

Eqs. (B5) and (B7) represent pure mode I and pure mode II cohesive laws, respectively.For mixed mode openings, the cohesive tractions are coupled through the damage variable D and a failure criterion (for

damage initiation) to account for mixed mode fracture effects. Similar to Eq. (B4), the damage parameter, D, for mixed modefracture is defined as:

D ¼df

m dmaxm � do

m

� �dmax

m dfm � do

m

� � ; ðB8Þ

where dmaxm the maximum effective opening attained in the loading history (under monotonic loading dmax

m ¼ dm). One of themost widely used failure criterion is [40]:


rn

r̂n

� �2

þ rt

r̂t

� �2

¼ 1: ðB9Þ

The critical effective traction for damage initiation can be derived from Eq. (B9) using Eqs. (2) and (12):

rce ¼

r̂nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos2 wþ r̂n

r̂t

� �2sin2 w

r ; ðB10Þ

which shows the dependence of rce on w and therefore (by Eq. (B2)) on /.

For mixed mode, the normal and shear tractions are also given by Eqs. (B5) and (B7) with D given by Eq. (B8). Using Eqs.(B5) and (B7), dn and dt to be expressed in terms of rn and rt. Then, the effective displacement, dm, defined in Eq. (10) can bewritten as:

dm ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirn

ð1� DÞK

� �2

þ rt

ð1� DÞK

� �2s

: ðB11Þ

Eq. (B11) with Eq. (12) gives:

dm ¼1

ð1� DÞK re: ðB12Þ

The normal opening and the tangential displacement, using Eq. (A3), are:

dn ¼re cos wð1� DÞK

dt ¼re sin wð1� DÞK : ðB13Þ

From Eq. (B13) is easily seen that tan/ = dt/dn = tanw and thus / = w. This demonstrates that for this mixed mode cohesivelaw formulation the traction vector follows the opening vector. The bi-linear cohesive law described here is thus a truss-likecohesive law. Then, according to the general proof in Section 3, the cohesive laws cannot be derived from a potential functionunless the cohesive law parameters are independent of /.

Appendix C. Proof of path dependency of bi-linear truss-like mixed mode cohesive laws

We now proceed to show analytically that the cohesive law as formulated above cannot be derived from a potential func-tion. Recall that in general the cohesive law parameters rc

e, dom and df

m are functions of /, i.e. rce ¼ rc

eð/Þ, dom ¼ do

mð/Þ anddf

m ¼ dfmð/Þ.

To show that the particular truss-like mixed mode cohesive law described here is path-dependent, we need to show thatthe Eq. (5) is not satisfied.

The partial derivatives of the tractions in Eq. (5) can be obtained from Eqs. (B5) and (B7):

@rn

@dt¼ �Kdn

@D@dt

; ðC1Þ

@rt

@dn¼ �Kdt

@D@dn

; ðC2Þ

where @D/@dn and @D/@dt are obtained from Eq. (B8):

@D@dn¼ � @d

fm

@dn

dom dm � do

m

� �dm df

m � dom

� �2 �@do

m

@dn

dfm df

m � dm

� �dm df

m � dom

� �2 þdf

mdom

d3m df

m � dom

� � dn; ðC3Þ

@D@dt¼ � @d

fm

@dt

dom dm � do

m

� �dm df

m � dom

� �2 �@do

m

@dt

dfm df

m � dm

� �dm df

m � dom

� �2 þdf

mdom

d3m df

m � dom

� � dt : ðC4Þ

Substituting Eqs. (C1)–(C4) into Eq. (5) (the requirement for the existence of a potential function) gives:

@dfm

@dndm � do

m

� �do

m þ@do

m

@dndf

m dfm � dm

� � !dt ¼

@dfm

@dtdm � do

m

� �do

m þ@do

m

@dtdf

m dfm � dm

� � !dn: ðC5Þ

This Equation must hold true for any value of dn and dt or alternatively any value of dm and /. The partial derivatives @dfm=@dt

and @dfm=@dn can be written in polar form as:


@dfm

@dt¼ @d

fm

@dm

@dm

@dtþ @d

fm

@/@/@dt¼ @d

fm

@dmsin /þ @d

fm

@/cos /dm

; ðC6Þ

and@df

m

@dn¼ @d

fm

@dm

@dm

@dnþ @d

fm

@/@/@dn¼ @d

fm

@dmcos /� @d

fm

@/sin /dm

: ðC7Þ

When expressed in polar form, the openings dfm and do

m depend only on /, i.e.:

@dfm

@dm¼ 0 ^ @do

m

@dm¼ 0: ðC8Þ

Inserting Eq. (C8) into Eqs. (C6) and (C7) gives:

@dfm

@dt¼ @df

m

@/cos /dm

; ðC9Þ

@dfm

@dn¼ � @d

fm

@/sin /dm

: ðC10Þ

The partial derivatives @dom=@dn and @do

m=@dt can be transformed in a similar way.Transforming Eq. (C5) to polar coordinates using Eqs. (C9) and (C10) gives:

@dfm

@/cos /dm

dm � dom

� �do

m þ@do

m@/

cos /dm

dfm df

m � dom

� �� @df

m@/

sin /dm

dm � dom

� �do

m �@do

m@/

sin /dm

dfm df

m � dm

� � ¼ sin /cos /

; ðC11Þ

which can be written as:

@dfm

@/¼ �

dm dfm � dm

� �do

m dm � dom

� � @dom

@/: ðC12Þ

Eq. (C12) should hold for any values of dm and /. From Eq. (C8) we note that dfm depends only on /. Thus, the left hand side of

Eq. (C12) depends on / only, whereas the right hand side depends on both / and dm. Thus it follows from Eq. (C8) that Eq.(C12) holds true for any value of dm only if Eq. (20) is satisfied.

Appendix D. Stress-separation laws and work of separation for bi-linear truss-like mixed mode cohesive laws

Using Eqs. (B12) and (B13), the damage variable, D, from Eq. (B8) is (with w = /):

D ¼df

m dn � dom cos /

� �df

m � dom

� �dn

¼df

m dt � dom sin /

� �df

m � dom

� �dt

: ðD1Þ

Inserting Eq. (D1) into Eqs. (B5) and (B7) we obtain respectively:

rn ¼Kdo

m

dfm � do

m

dfm � dm

� �cos / ðD2Þ

rt ¼Kdo

m

dfm � do

m

dfm � dm

� �sin /: ðD3Þ

The cohesive tractions reduce to zero when dm ¼ dfm. At this point dn ¼ df

n ¼ dfm cos / and dt ¼ df

t ¼ dfm sin /. Then, the work of

the cohesive tractions for the modes I and II components of the traction vector is:

JI/c¼ 1

2rc

e cos /� �

dfm cos /

� �¼ 1

2rc

edfm cos2 /

JII/c¼ 1

2rc

e sin /� �

dfm sin /

� �¼ 1

2rc

edfm sin2 /: ðD4Þ

The total work of separation (the fracture energy) at the phase angle of opening /, is:

J/c¼ JI

/cþ JII

/c¼ 1

2rc

edfm: ðD5Þ

It is clear that J/cdepends on / when df

m and rce are functions of / (and vice versa). Then, from Eq. (D5), the effective opening

for complete failure can be determined:

dfm ¼

2J/c

rce: ðD6Þ

Thus, in general, dfm depends on /.


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path dependence of truss-like mixed mode cohesive laws

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