A fracture evolution procedure for cohesive materials

A fracture evolution procedure for cohesive materials
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   International Journal of Fracture  110:  241–261, 2001.© 2001  Kluwer Academic Publishers. Printed in the Netherlands. A fracture evolution procedure for cohesive materials S. MARFIA and E. SACCO  Dipartimento di Meccanica, Strutture, A. & T., Università di Cassino, Cassino Via G. Di Biasio 43,03043 Cassino, Italy (Author for correspondence; e-mail: Received 14 February 2000; accepted in revised form 31 January 2001 Abstract.  The present paper deals with the problem of the evaluation of the softening mechanical response of cohesive materials under tensile loading. A nonlinear fracture mechanics approach is adopted. A new numericalprocedure isdeveloped to study theevolution of thecrack processes for 2D solids. Theproposed algorithm isbasedon the derivation and use of the fracture resistance curve, i.e., the  R -curve, and it takes into account the presenceof the process zone at the crack tip. In fact, assuming a nonlinear constitutive law for the cohesive interface,the procedure is able to determine the  R -curve, the process zone length and hence the mechanical response of any material and structure. Numerical applications are developed for studying the damage behavior of a infinitesolid with a periodic crack distribution. Size effects are investigated and the ductile-brittle transition behavior formaterials characterized by the same crack density is studied. The results obtained adopting the proposed procedureareingood accordance withtheresultsrecovered through nonlinear step bystep finiteelement analyses. Moreover,the developed computations demonstrate that the procedure is simple and efficient. Key words:  Cohesive materials, fracture, numerical procedure,  R -curve. 1. Introduction Cementitious materials, such as concrete and masonry, are characterized by cohesive or qua-sibrittle behavior, with development of cracks also for low values of positive deformations;hence, such materials present limited tensile strength, with softening response.The mechanical behavior of these materials can be studied using the concepts of the frac-ture mechanics. The classical linear elastic fracture mechanics (LEFM) is suitable only forbrittle materials; hence, it is not always able to reproduce the process of crack propagationfor cementitious materials. In fact, these materials present a process zone at the crack tipneighborhood characterized by a not negligible size. Thus, it is necessary to consider thedamage effects occurring at the process zone, according to the nonlinear fracture mechanics(NLFM).The effects of the presence of the process zone can be taken into account extending somebasic principles of LEFM and developing suitable cohesive fracture models. In particular,several NLFM approaches have been proposed in the literature, introducing the concept of the effective or equivalent elastic crack. Swartz et al. (1978) used the concept of the effectiveelastic crack to compute the fracture toughness in concrete through LEFM equations, oncethe effective crack extension at the peak load was experimentally measured. Nallathambi andKarihaloo (1986) proposed a model similar to the Swartz one, measuring the equivalent crack extension in a different way. Jenq and Shah (1985a, b) developed a two-parameter modelto predict the peak load of a concrete structure. The two parameters considered in the lattermodel are the stress intensity factor and the crack opening displacement at the critical state,i.e., at the peak load.  242  Sonia Marfia and Elio Sacco It can be noted that, in LEFM as well as in NLFM, the fracture evolution can be inves-tigated using a local stress approach or a global energy criterion. In particular, in LEFM thelocal approach is based on the Irwin stress criterion, using the concept of the stress intensityfactor, while the global approach considers an energetic balance according to the Griffithenergy release rate criterion. In NLFM, cohesive models are introduced to develop local stressanalyses, based on strength criteria, or to derive global energy approaches, which lead to the  R -curve method.The energy Griffith criterion can be preferred for studying the behavior of brittle materials,since the stress intensity factors depend on the geometry of the problem and they are notalways known a priori.The local stress analysis is generally performed to investigate on the fracture evolutionfor cohesive materials. In this framework several models and numerical procedures havebeen proposed in the literature. Among the others, Hillerborg (1984) proposed a numericalmethod to predict the softening and the fracture of concrete structures, based on the fictitiouscrack model within the finite element method. Ingraffea and Saouma (1984) and Ingraffeaand Gerstle (1985) proposed a crack propagation model, which consists in a discrete fractureapproach with a remeshing technique in the finite element framework, able to follow anycrack path. Carpinteri and co-workers developed a cohesive crack model and appropriatealgorithms for studying the softening and snap-back instability in cohesive solids (Carpin-teri, 1989, 1990; Carpinteri and Colombo, 1989). They analyzed the size effects of fracturemechanics, investigating on the transition from ductile to brittle structural behavior related tothe size scale. Then, they extended the proposed procedures for studying crack propagation infiber-reinforced materials (Bosco and Carpinteri, 1995; Carpinteri and Massabó, 1996).From a computational point of view, the NLFM local stress analysis can be solved devel-oping appropriate techniques for the time integration of the evolutive nonlinear constitutivelaw. The solution at each time step is computed through iterative procedures, which can needsignificant computational efforts. In fact, in order to reach accurate results, many iterationshave to be performed.In the framework of nonlinear fracture mechanics, the global analysis for cohesive ma-terials requires the evaluation of the process zone length and the determination of the  R -curve, which is not available for any structure, since it depends on the particular material andgeometry of the specimen.The  R -curve models consider the crack grow resistance ( R ) as a function of the equivalentcrack length and represent an extension of the Griffth criterion. Hence, from a computationalpoint of view, it appears interesting to develop numerical procedures based on the use of thefracture resistance curve able to describe the crack propagation in cohesive materials. In fact,several scientific contributions, regarding the definition, the determination and the use of the  R -curve, can be found in the literature.The concept of the  R -curve was initially introduced by Irwin to study the crack grow inmetals; then, this approach has been adopted for cementitious composites and ceramics. Theproblem of the  R -curve determination has been treated by several authors in the literature.In particular, Cook et al. (1987) proposed a procedure for evaluating the  R -curve defined byfour material constants, based on the cohesive crack. Cook and Clarke (1988) approximatethe  R -curve as a power function with two geometry dependent constants. Bazant et al. (1986)and Bazant and Kazemi (1990) determined a geometry dependent  R -curve, based on the sizeeffects; moreover, they assumed that the  R -curve is constant after the peak load. Ouyang et al.   A fracture evolution procedure for cohesive materials  243(1990) and Ouyang and Shah (1991) computed the  R -curve with two geometry dependentparameters, solving a differential equation.In the last years, the interest to the application of the concepts related to the  R -curve isdecreased. This is mainly due to the fact that the fracture resistance curve does not representan objective property of the material, since it is related to the size and to the geometry of theparticular structure. Thus, experimental results devoted to determine the  R -curve for a mater-ial, has limited possibility of application. In fact, the resistance function derived in laboratoryis a property of the particular specimen, and it cannot be used to evaluate the crack growth fora different structure, also made of the same material.On the other hand, for cohesive material the objective material properties are the tensilestrength and the critical fracture energy characterizing the constitutive law of the process zone.The aim of the work is to develop a new numerical procedure for the evolutive analysisof cohesive crack. The proposed procedure accounts for the presence of the process zone,considering a cohesive interface at the crack tip. The crack propagation is analyzed by a globalenergy criterion within the  R -curve method. In fact, a cohesive stress-crack opening consti-tutive law is considered. The objective interface material properties, i.e., the tensile strengthand the critical fracture energy, are used to derive the fracture resistance curve associatedto a structural geometry. Thus, the developed procedure is able to evaluate the process zonelength, the  R -curve and, hence, the mechanical tensile response of solids characterized by anyparticular geometry and material. In the present paper only a two-dimensional problem with aMode Iof fracture is considered, since it represents the main fracture evolution incementitiousmaterials.The paper is organized as follows. Initially, some basic concepts of the fracture mechanicsfor brittle and quasibrittle materials are briefly reported. Then, a fracture evolution procedurefor brittle materials is presented. A new numerical technique, able to evaluate the overallresponse of cohesive materials, is proposed. Finally, some numerical results are developed toassess the effectiveness of the numerical procedure, reporting comparisons with results avail-able in the literature or obtained using a finite element code with a time integration procedure.In particular, numerical applications are developed for studying the damage behavior of ainfinite solid with a periodic crack distribution. Moreover, size effects are investigated andthe ductile-brittle transition behavior for materials characterized by the same crack density isstudied. 2. Basic concepts of fracture mechanics Some of the fundamental concepts of the fracture mechanics are herein reported to set theadopted notations. The discussion is limited only to Mode I for 2D homogeneous problems,characterized by unit thickness. Classical results, which are at the basis of the evolution proce-dure developed in the next section, are recalled. Books on fracture mechanics, as for instance(Karihaloo 1998; Bazant and Planas, 1998), can be referred for more detailed information.As emphasized in the previous section, the crack evolution for brittle and cohesive materi-als can be studied adopting two different approaches: −  local stress analyses at the crack tip neighborhood, −  global energetic criteria.  244  Sonia Marfia and Elio Sacco 2.1. L INEAR ELASTIC FRACTURE MECHANICS  Local approach The local approach in LEFM is based on the use of the stress intensity factor  K I  , which is afunction of the geometry and boundary conditions for the considered problem. For cracks ininfinite media or in infinite strips,  K I   can be evaluated by the formula: K I   = F  I   σ  √  π a,  (1)where  σ   is the mean stress,  a  is the crack length and  F  I   is a geometrical coefficient thatdepends on  a  and on a characteristic size of the problem. For a single crack in an infinite solid F  I   = 1.According to the stress criterion introduced by Irwin, the crack propagation occurs whenthe stress intensity factor  K I   reaches a critical value  K IC , called fracture toughness, whichdepends on the material.The main difficulty in the application of this approach is the determination of   K I   that isknown only for some cases, as reported in proper handbooks (Murakami, 1987). Global approach The analysis of crack propagation in Mode I for brittle materials can be developed using theglobal approach based on the Griffith criterion (Bazant and Planas, 1998).For a given structure with an initial notch, a generalized force  P  , representing the loadingcondition, and the corresponding displacement  η , can be introduced. The force  P   and dis-placement  η  are related by the equation  P   =  K(a)η , where  K(a)  is the overall stiffness of the structure. Hence, the internal stored energy can be computed as  U(η,a) =  12 K(a)η 2 .In a general loading process, where both the displacement  η  and the crack length  a  canvary, the energy release rate  G  can be evaluated as: G  =−  ∂U(η,a)∂a  η = const . =− 12d K(a) d aη 2 .  (2)According to the Griffith fracture criterion, the crack propagates when the energy releaserate  G  reaches the critical values  G c . The critical energy release rate  G c , also called fractureenergy, is a material property which does not depend on the cracking history.  Local-global equivalence The stress intensity factor  K I   and the energy release rate  G  are related by the followingformula deduced by Irwin: K I   =√  E ′ G,  (3)where  E ′  =  E  for plane stress while  E ′  =  E/( 1 − ν 2 )  for plane strain, with  E  and  ν  theYoung modulus and the Poisson ratio, respectively.Taking into account Equations (2) and (3), the geometrical factor  F  I   can be determined forcracks in infinite media or in infinite strips from the relation (1) as: F  I   =  K I  σ  √  π a =√  G E ′ σ  √  π a =  BK(a)   −  12 π a d K(a) d aE ′ ,  (4)where it is assumed  σ   = P/B , with  B  the characteristic width of the structure.   A fracture evolution procedure for cohesive materials  245 Figure 1.  Damage zone at the crack tip, the bold line represents the process zone. 2.2. N ONLINEAR ELASTIC FRACTURE MECHANICS  Local approach To investigate on the crack propagation in quasibrittle materials, cohesive models have beendeveloped in the literature, for instance, by Dugdale (1960), Barenblatt (1962) and Hilleborget al. (1976). The cohesive models consider the presence of a process zone at the crack tip,where a proper constitutive law for the tensile normal stress  σ   and the relative displacement w , between the two sides of the crack, is assumed. The process zone starts to develop whenthe stress reaches the tensile strength  f  t   of the material. The relation  σ   − w  is characterizedby a softening law, so that the tensile normal stress  σ   is a decreasing function of the relativedisplacement  w , and the area under the stress-displacement function is the fracture energy  G c .Hence, an interface is introduced to model the behavior of the process zone. The interfaceis characterized by the following constitutive law: σ   = ( 1 − D) K int w,  (5)where  K int is the initial stiffness of the interface and  D  is the damage variable (0  ≤  D  ≤  1).Note that the damage parameter assumes the values: D  = 0 when  w  = w 0  and  σ   = f  t  ,D  = 1 when  w  = w c  and  σ   = 0 , 0  < D <  1 when  w 0  < w < w c  and  f  t   > σ >  0 , (6)where  w 0  and  w c  are the relative displacements corresponding to the damage beginning andthe complete final damage, respectively.An abscissa  ξ  , coaxial with the crack and with the srcin at the crack tip is introduced,as shown in Figure 1. The value of   ξ   individuates the position of the process zone. Twopossibilities can occur: −  the process zone is not completely developed, i.e. at  ξ   =  0 it is  w < w c  and hence f  t   > σ >  0 and 0  < D <  1; in this case the process zone starts at the crack tip and
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