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  Cohesive mixed mode fracture modelling and experiments Rasmus Walter, John F. Olesen * Department of Civil Engineering, Technical University of Denmark, Brovej, Building 118, DK-2800 Kgs. Lyngby, Denmark a r t i c l e i n f o  Article history: Received 13 September 2005Received in revised form 4 August 2008Accepted 11 August 2008Available online 31 August 2008 Keywords: Nonlinear fracture mechanicsSteel–concrete interfaceMixed mode fracture a b s t r a c t A nonlinear mixed mode model srcinally developed by Wernersson [Wernersson H. Frac-ture characterization of wood adhesive joints. Report TVSM-1006, Lund University,Division of Structural Mechanics; 1994], based on nonlinear fracture mechanics, is dis-cussed and applied to model interfacial cracking in a steel–concrete interface. The modelis based on the principles of Hillerborgs fictitious crack model, however, the Mode I soft-ening description is modified taking into account the influence of shear. The model couplesnormal and shear stresses for a given combination of Mode I and II fracture. An experimen-tal set-up for the assessment of mixed mode interfacial fracture properties is presented,applying a bi-material specimen, half steel and half concrete, with an inclined interfaceand under uniaxial load. Loading the inclined steel–concrete interface under differentangles produces load–crack opening curves, which may be interpreted using the nonlinearmixed mode model. The interpretation of test results is carried out in a two step inverseanalysis applying numerical optimization tools. It is demonstrated how to perform theinverse analysis, which couples the assumed individual experimental load–crack openingcurves. The individual load–crack opening curves are obtained under different combina-tions of normal and shear stresses. Reliable results are obtained in pure Mode I, whereasexperimental data for small mixed mode angles are used to extrapolate the pure Mode IIcurve.   2008 Elsevier Ltd. All rights reserved. 1. Introduction Since Hillerborg et al. [2] introduced the fictitious crack model, discrete Mode I cracking in concrete has been the subjectof intensified research, which has demonstrated the usefulness of the concepts of this cohesive crack model. The focus of thisstudy is the influence of shear on the process of discrete cracking, which in the terminology of fracture mechanics is calledmixed mode cracking. Mixed mode cracking will be treated both theoretically and experimentally. In the present study themixedmode interfacial crackingof a steel–concreteinterface is infocus, where the interface is definedas a regionof concretemortar near the boundary between the two materials. Experimental experience shows that interfacial cracking of a steel–concrete interface usually occurs at a certain distance from the physical boundary between the two materials, cf. RILEMTC-108 [3]. Physically, the interfacial transition zone between concrete and steel has a finite thickness on the micro scale,which is related to the penetration of the cement paste into the rough steel surface. In the present study interfacial crackingis defined as taking place close to or inside the interfacial transition zone.Recordings of Mode I behavior of steel–concrete interfaces have already been made, see e.g. Walter et al. [4]. Less exper-imental research has been carried out on mixedmode crackingof cement-basedinterfaces,however, several numerical mod-els to describe interfacial mixed mode crackingbased on nonlinearfracture mechanics have been proposed, see e.g. Lourenço 0013-7944/$ - see front matter    2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.engfracmech.2008.08.009 *  Corresponding author. Tel.: +45 45251700; fax: +45 45883282. E-mail address:  jfo@byg.dtu.dk (J.F. Olesen).Engineering Fracture Mechanics 75 (2008) 5163–5176 Contents lists available at ScienceDirect Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech  and Rots [5] or Cervenka et al. [6]. In the present study mixed mode modelling is based on a model srcinally presented by Wernersson [1]. To introduce the concepts and ideas of this mixed mode model, a brief summary of the concepts of couplingstress intensity factors in linear elastic fracture mechanics (LEFM) for mixed mode loading is given. These concepts are wellunderstood and the summary explains how some of these concepts can be adopted to nonlinear cohesive crack modelling.When describing mixed mode cracking in linear elastic fracture mechanics the phase angle  w k  is introduced, which is a func-tion of the Mode I and II stress intensity factors  K  I  and  K  II , w k  ¼ arctan  K  II K  I    ð 1 Þ Thus, the phase angle is directly related to the stress state in the vicinity of the crack tip. Relating the phase angle to the totalcritical energy release rate shows two typical behaviors, which can explain the difference between ductile and brittle mate-rials. Materials have been classified by He et al. [7] using the phase angle. There it is stated, that for brittle materials the en-ergy release rate increases significantly when increasing the phase angle  w k , thus the material is weaker in Mode I than inMode II. The critical energy release rate ratio between Mode II and I,  G IIc = G Ic , for concrete is larger than 1 and in some cases ithas been measured in the range of 5, see e.g. Carpenteri and Swartz [8]. So in this respect concrete may be considered as abrittle material, and consequently also the steel–concrete interface, since interfacial fracture is related to the behavior of theconcrete.A stress intensity based failure criterion, has been suggested by Wu [9] in the form, K  I K  Ic   m þ  K  II K  IIc   n 6 1 : 0  ð 2 Þ Nomenclature a  interface angle on mixed mode specimen d  mixed mode displacement d n  normal crack opening d t  tangential crack opening d measured  measured deformation d maxnk  kink point on the pure Mode II uniaxial stress–crack normal opening curve d maxtk  kink point on the pure Mode II uniaxial stress–crack tangential opening curve ^ Q  i  vector containing experimental data w  mixed mode angle w k  phase angle of stress intensity factors r  normal stress r f   peak normal stress on pure Mode I uniaxial stress–crack normal opening curve r maxk  kink point on the pure Mode I uniaxial stress–crack normal opening curve s  shear stress s f   peak shear stress on pure Mode II uniaxial stress–crack tangential opening curve s maxk  kink point on the pure Mode II uniaxial stress–crack tangential opening curve a n1  slope of pure Mode I uniaxial stress–crack normal opening curve a n2  slope of pure Mode I uniaxial stress–crack normal opening curve  A s  area of fracture zone a t1  slope of pure Mode II uniaxial stress–crack tangential opening curve a t2  slope of pure Mode II uniaxial stress–crack tangential opening curve b n2  parameter of pure Mode I uniaxial stress–crack tangential opening curve b t2  parameter of pure Mode II uniaxial stress–crack tangential opening curve D ij  stiffness components for interface description G f   fracture energy G IIf   pure Mode II fracture energy G If   pure Mode I fracture energy I   moment of inertia of fracture zone K  II  Mode II stress intensity factor K  I  Mode I stress intensity factor K  i  initial flexibility of specimen K  s  stiffness of set-up outside the fracture process zone m  exponent to couple Mode I and II n  exponent to couple Mode I and II Q  i  vector containing estimated data w  Mode I crack opening 5164  R. Walter, J.F. Olesen/Engineering Fracture Mechanics 75 (2008) 5163–5176   where  m ,  n  are material exponents. Using this criterion it is possible to define a failure criterion based on the state of stress,i.e. the combination of normal and shear stress, see for instance Carpenteri and Swartz [8]. However, LEFM is not applicablein the case of concrete fracture due to its large fracture process zone, cf. Peterson [10]. Therefore, the present study adoptssome of the concepts by applying the same form of interaction as given in Eq. (2). The present analysis is carried out usingcohesive crack modelling and is nonlinear in the sense that softening after crack initiation is included. As the interface hasreached its peak load, e.g. in uniaxial tension, the interface degradation is not necessarily abrupt as it may involve an amountof energy dissipation before complete separation.In the general theory of elasticity, materials can be classified in two main groups: hyper- and Cauchy-elastic materials.The strain energy function of a hyper-elastic material,  w s  ¼ R   r ij d  ij , is equal to a potential. Hence the strain energy functionis not dependent on the strain path. Contrary to this every state of stress for a Cauchy-material is unique and defined by thestrain path. Cracking in both hyper- and Cauchy-materials have been modelled by several authors using nonlinear fracturemechanics. Needleman [11] formulated a cohesive crack model to study interfacial debonding with hyper-elastic materialproperties. He derived normal and shear stresses from an elastic potential which only depended on the normal and tangen-tial displacements.A nonlinear interfacial mixed mode model with path dependency has been defined by Wernersson [1], based on cohesivecrack modelling. In this case, contrary to the case of a potential, the total fracture energy,  G f  , is defined by G f   ¼ Z  C ð r d d n þ s d d t Þ ð 3 Þ where  C  is the deformation path that results in complete failure of the considered crack. As reported by several authors, seee.g. Cervenka et al. [6], a cementitious interface can be described as a path dependent media, and different amounts of frac-ture energy are consumed in the cases of pure Mode I and II failures. These observations support a model taking into accountpath dependency and cohesive crack modelling with softening. 2. Mixed mode model The mixed mode model by Wernersson [1] is briefly presented here. For a full review on the mixed mode model the read-er is referred to the srcinal work by Wernersson [1]. The reason and motivation for using the present mixed mode model isthe possibility of including the fracture behavior which is expected when modelling a steel–concrete interface. The main fea-tures governing the mixed mode behavior of a steel–concrete interface are as listed:   Discrete cracking (cracking along an interface).   Stress softening (include tension and shear softening).   Mixed mode cracking (coupling of normal and tangential crack opening).   Path dependency (the amount of fracture energy consumed depends on the fracture mode and path, i.e. how Mode I and IIare combined during cracking).A limitation to the present study is that only monotonic loading is considered, thus no effects from cyclic loading are ta-ken into account. In the case of unloading, the model will follow the same path as followed during loading, however, this is aviolation of the expected material behavior, but accepted as a limitation in the present study since only monotonic crackgrowth is considered. Finally, it is assumed that no compression failure occurs, the mixed mode model only applies to tensileload under the influence of shear.  2.1. General interface description The final goal for the mixed mode modelling is the implementation of a constitutive model for discrete cracks in a finiteelement code. For instance the mixed mode model may be applied in an interface element. Usually, to model a continuousgeometry, a FE interface is modelled with a thickness of zero, and every node is associated with a normal and a tangentialdisplacement and associated normal and shear stresses, corresponding to the crackopening. This is illustrated in Fig. 1 show-ing a three node interface element. Fig. 1.  A three node interface element with node stresses and displacements indicated. R. Walter, J.F. Olesen/Engineering Fracture Mechanics 75 (2008) 5163–5176   5165  The relationship between stresses and crack opening in plane strain is given by the following expression: rs   ¼  D 11  D 12 D 21  D 22    d n d t    ð 4 Þ where  r  and  s  are the normal and shear stress, respectively. The displacements  d n  and  d t  are the normal and tangential dis-placement, respectively. The  D ij  components relate the stresses to the normal and tangential displacement,  d n  and  d t  , respec-tively. In pure elastic mode no coupling is assumed between normal and shear mode and as a consequence, the off diagonalterms are set to zero,  D 12  ¼  D 21  ¼  0. After peak stress it is important to couple the two crack modes, Mode I and II. A situationwith the off diagonal terms set to zero is equal to having two independent springs. The model derived here allows for theimplementation of a constitutive relationship to represent a general interface behavior.  2.2. Coupling of Mode I and II  Consider two stress–crack opening relationships, one for the normal opening and another for the tangential displacementof the crack:  r ð d n Þ  and  s ð d t Þ , visualized in Fig. 2. The curves are described in two parts, an elastic and a nonlinear part. Theelastic part is described as the initial ascending part from zero stress to peak stress and is characterized by a very large stiff-ness, D n  and D t  , to model initial continuous geometry of the interfacial zone. This part has no physical meaning but is purelymodelled having a slope in order for the numerical solver applied to be able to converge. In this, nonphysical elastic stage,the two modes are uncoupled. The post peak behavior is described by a descending, softening part, which relates the normaland shear stresses acting across the crack to the normal ( d n ) or tangential ( d t ) openings, respectively.Each linear segment of the complete stress–crack deformation relationship is treated individually. The indices ( k ) and( k  + 1) refer to two successive kink points, which define a linear segment on the uniaxial softening curve. The subscript’max’is applied as notation when referring to the pure Mode I and II uniaxial curves. Maximum is used, since it is assumed that the Fig. 2.  Uniaxial stress–crack opening relationships in (a) pure Mode I and (b) pure Mode II. The curves are described by a stiff, linear ascending part untilpeak stress and a multi linear post peak softening part. Fig. 3.  For a given mixed mode state the crack openings  d n  and  d t  can be found as a function of the mixed mode angle  w .5166  R. Walter, J.F. Olesen/Engineering Fracture Mechanics 75 (2008) 5163–5176 
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