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Damage and crack modeling in single-edge and double-edge notched concrete beams
Engineering Fracture Mechanics 65 (2000) 247±261
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Damage and crack modeling in single-edge and doubleedge notched concrete beams M.G.D. Geers a,b,*, R. de Borst c, R.H.J. Peerlings a a
Faculty of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands b Faculty of Civil Engineering, Royal Military Academy, Avenue Renaissance 30, 1000 Brussels, Belgium c Faculty of Aerospace Engineering, Delft University of Technology, P.O. Box 5058, 2600 GB Delft, The Netherlands
Abstract The numerical modeling of damage and crack propagation in concrete and concrete structures has evolved considerably in the past years. In this contribution, a higher order continuum model is used to model the failure behavior of single-edge notched (SEN) and double-edge notched (DEN) concrete beams loaded in four-point-shear. Dierent types of boundary conditions, i.e. with freely rotating, ®xed or constrained loading supports, are investigated and the experimentally observed curved crack paths are compared with the numerical simulations. The in¯uence of the ratio of the compressive strength and the tensile strength is scrutinized and its relation with the failure mechanism is investigated. It is shown that an isotropic gradient-enhanced damage model permits to obtain a good agreement between experimental results and numerical simulations. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Damage mechanics; Failure assessment; Crack growth; Mixed mode fracture; Higher order continuum
1. Introduction The numerical modeling of concrete failure has experienced a signi®cant progress in the past decades. Dierent approaches have been adopted, leading to dierent models, such as fracture energy models, smeared crack models, Cosserat models, plasticity models, rate-dependent models, nonlocal damage models, gradient-enhanced models and others. In many of these models, concrete is considered as a quasi-brittle material in which strain softening plays a dominant role. The necessity of higher order continuum models for the description of the failure process within a continuum mechanics framework, has been recognized by * Corresponding author. Tel.: +31-40-2475076; fax: +31-40-2447355. E-mail address: [email protected] (M.G.D. Geers). 0013-7944/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 9 9 ) 0 0 1 1 8 - 6
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Nomenclature SEN DEN CM CM s e D 4 C E n eeq e eq c k ki k a b
single-edge notched double-edge notched crack mouth opening displacement crack mouth sliding displacement Cauchy stress tensor in®nitesimal strain tensor damage variable elasticity tensor Young's modulus Poisson's ratio local equivalent strain nonlocal equivalent strain gradient material parameter history variable damage initiation history parameter tension±compression sensitivity ®rst damage evolution parameter second damage evolution parameter
many authors [1±4]. These higher order descriptions typically include a nonlocal eect into the constitutive description, e.g. nonlocal damage models, gradient damage models, nonlocal or gradient plasticity models. From the numerical point of view, it has been shown that these models solve the ill-posedness of the boundary value problem. A microstructural parameter is, therefore, incorporated into the constitutive model, which is commonly denoted as an intrinsic length scale. These enriched models are, therefore, related to the micromechanics of the dominant failure processes. In the present analysis, a gradient-enhanced damage formulation will be used to model failure of notched concrete beams. Concrete beams under mixed-mode loading are typical benchmark problems that have been investigated frequently in literature [5±10]. Many of these analyses lead to contradictory conclusions, where dierent failure modes were observed depending on the experimental setup that has been used. In this paper, a comparison will be made between computational simulations and available experimental results [11]. Schlangen [11] used a speci®c experimental setup in which the loading frame could be adapted in order to obtain freely rotating and ®xed loading supports. The experimentally observed failure mode strongly depends on the type of boundary conditions that have been applied. This explains why contradictory results have been obtained in literature. In this paper, attention is focused on the damage development in single-edge and double-edge notched concrete beams, where dierent type of boundary conditions have been used. 2. Gradient-enhanced damage for concrete fracture The gradient-enhanced damage model is commonly formulated as an isotropic strain-based damage theory, in which the nonlocal eect has been incorporated by the addition of gradient terms in the constitutive description [12]. The formulation is based on a total stress±strain relation according to: s 1 ÿ D 4 C:e
1
M.G.D. Geers et al. / Engineering Fracture Mechanics 65 (2000) 247±261
249
with s the Cauchy stress tensor, e the in®nitesimal strain tensor, 4 C the fourth-order elastic stiness tensor and D the scalar-valued damage variable. In order to be compliant with the second principle of thermodynamics, the rate of the damage variable for an isotropic material with no permanent strains must satisfy _ Dr0
2
Furthermore, the damage variable D as de®ned by Eq. (1) must satisfy 0RDR1
3
The damage parameter D is coupled with the deformation history of the material through a monotonically increasing deformation history parameter k: This scalar history parameter characterizes the ultimate nonlocal equivalent strain e eq that the material has experienced in its loading history. The evolution of k upon damage evolution is usually expressed by the Kuhn±Tucker relations ÿ _ 4 kr0, e eq ÿ kR0, k_ e eq ÿ k 0 The nonlocal characteristic e eq is computed from the ®eld of local equivalent strains eeq in the vicinity of the material point via a partial dierential equation of the Helmholtz type [12]: e eq ÿ cr2 e eq eeq
5
The solution of Eq. (5) requires the incorporation of appropriate boundary conditions [12]. This equation introduces a new material parameter c, the gradient parameter, which is related to the square of the intrinsic length scale of the nonlocal continuum. Suggestions for the measurement of the characteristic length of a nonlocal continuum were made by BazÏant and Pijaundier-Cabot [13], Carmeliet [14] and Geers et al. [15]. Speci®c experimental procedures and methods have to be setup in agreement with the physical phenomena which govern the localization and fracture process. This can be done by direct measurements in the process zone or indirectly, e.g. via the size eect. Usually, complex tests are necessary which involve an iterative procedure to determine the individual parameters on a statistical basis. An important aspect for concrete fracture resides in the precise de®nition of eeq , which has to be computed from the strain tensor e in each material point. In the de®nition proposed by Peerlings et al. [4], the relative contributions of the tensile and compressive strain components dier, which can have a major impact on the failure mode that is obtained. Similar propositions can be found in [16,17]. The equivalent strain de®nition used for concrete reads s 2 kÿ1 1 kÿ1 6k J1 J1 J2 6 eeq 2k 1 ÿ 2n 2k 1 ÿ 2n 1 n 2 where the following strain tensor invariants have been used J1 tr e e1 e2 e3 i 1 1h J2 tr e e ÿ tr2 e e1 ÿ e2 2 e2 ÿ e3 2 e1 ÿ e3 2 3 3
7
The parameter k k1fcc =fct has been introduced to control the dierent sensitivity to tensile and compressive strains. The dependency of the equivalent strain on the strain components is illustrated in Fig. 1, which shows iso-eeq curves for dierent values of k:
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Fig. 1. Iso-equivalent strain curves in the principal strain space.
The damage D progressively reduces the intrinsic stiness of the material [18]. The damage parameter D is computed directly from a damage evolution law, which describes the mechanical in¯uence of the material degradation. The parameters in the damage evolution law must provide the necessary ¯exibility to model the mechanical response of the material. The used damage evolution law for concrete is ki 1 ÿ a a eÿb kÿki 8 D1ÿ k where coecients a and b are the material parameters. The threshold value for damage initiation ki is the initial value for k at time t 0: It is the ultimate equivalent strain that characterizes the elastic material behavior prior to occurrence of damage. More computational details can be found in [12].
3. Beams loaded with freely rotating supports 3.1. Single-edge notched beam The single-edge notched (SEN) beam subjected to an antisymmetric four-point-shear loading, has been analyzed experimentally and numerically by Schlangen [11]. The geometry and the loading conditions of the specimen are given in Fig. 2. The thickness of the SEN beam equals 50 mm. The antisymmetric loading of the beam results in a curved crack path, which starts from the right corner of the notch and which ends at the lower right loading platen. The beam has been modeled with 5 5 mm2 eight-noded quadrilateral elements for the coarse mesh and 2.5 2.5 mm2 elements for the ®ne mesh. The material parameters used in this model [4] are: Young's modulus E = 35,000 MPa, Poisson's ratio v 0:2, gradient parameter c = 1 mm2, compression±tension sensitivity k 15, damage initiation history parameter ki 6 10ÿ5 , damage evolution law parameters a 0:96 and b 100: The two central loading platens have been modeled as rigid bodies through the introduction of appropriate tyings between the nodal displacements. One nodal force is then centrally applied on the rigid platen. The loading platens at the lower left and upper right extremity of the beam were modeled with a nodal force and a nodal support, respectively. A plane stress situation has been assumed. Damage initiates at the right corner of the notch and at the bottom of the beam, opposite to the upper central loading platen. The damage growth at the bottom is arrested, while it keeps growing at the notch. The experimentally observed curved crack path, as observed by Schlangen [11], is shown in Fig. 3. The evolution of the damage growth in the beam is depicted in successive stages in Fig. 4, with
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251
Fig. 2. Single-edge notched beam con®guration: beam thickness = 50 mm; width loading platens = 20 mm.
the corresponding levels computed with the coarse mesh (see also Fig. 6). The obtained damage pro®le is in good agreement with the experimentally observed crack path of Fig. 3. The secondary damage zones at the opposite edge of the central loading platens have been con®rmed experimentally. Fig. 5 shows the load P versus the crack mouth opening displacement (CMOD) for the simulation and experiments. The quality of the ®t is apparent. The mesh dependence has been investigated by comparing the results for two dierent mesh sizes for which the same damage distribution was found. The load±CMSD curves for both meshes are shown in Fig. 6. The good quality of the simulation is mainly attributable to the proper de®nition of the equivalent strain (6) which has been utilized in the numerical analysis. Other de®nitions fail to predict the correct crack path [4].
Fig. 3. Single curved crack in a rotating SEN beam [11].
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Fig. 4. Damage evolution in an SEN beam with rotating supports P 40 kN (top ®gure) 4 37 kN 4 7 kN (bottom ®gure)).
3.2. Double-edge notched beam The double-edge notched (DEN) beam, which has been proposed and tested by BazÏant and Pfeier [5,19] to study the shear fracture of concrete, has been analyzed several times in literature [6±8,11]. The specimen was tested by Schlangen [11] in a symmetric test machine. The DEN specimen, loaded in fourpoint-shear, is shown in Fig. 7. The loading platens are positioned adjacent to the notch, which results in a smaller shear zone than that observed in the SEN test. Consequently, the load is applied close to the notch (10 mm), and failure by shearing towards the notch is a possible failure mode in the absence of friction in the supports. In the experiment, the support platens are subjected to rotational and translational frictions. The inclusion of some frictional aspects in the FEM model for the loading platens seems, therefore, appropriate. This frictional eect turns out to be particularly important for the DEN beam with freely rotating supports. Small rotational springs can be added to the support platens or a small eccentricity of the loading force on the platen can be easily introduced, while the positioning of the platens remains unaected. The latter option has been adopted in this analysis. The introduction
Fig. 5. P±CMOD curve.
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253
Fig. 6. Mesh independence.
of a small eccentricity in the model also permits to take into account the deviations with respect to the assumed antisymmetry. Indeed, the heterogeneous character of the concrete material and the presence of aggregates with variable diameters in the vicinity of the loading platens will always provoke a more or less pronounced eccentric transfer of the load through the specimen. If the inherent eccentricity is totally neglected, an antisymmetric solution is found that consists of two curved cracks which originate from each notch. If the compression±tension strength ratio k is smaller than 15, the beam fails by shearing and compression failure at the notches. However, by adding a small eccentricity to the loading force, the failure mode becomes asymmetric and stays asymmetric upon further loading. The damage evolution in the beam is depicted in Fig. 8 for an eccentricity of 2.5 mm. A reduced eccentricity equal to 1 mm is shown in Fig. 9. A crack originates from each notch, but the lower crack is arrested, while the other crack continues to grow. The solution thus found corresponds well with the experimental solutions where a long crack and a short crack was found, see Fig. 10. This ®gures corresponds to three experiments that were taken under the same test conditions, but for which the heterogeneous character
Fig. 7. Con®guration of the DEN shear test: beam thickness = 37.5 mm; width loading platens = 20 mm.
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Fig. 8. Damage in the DEN beam with rotating supports, 2.5 mm eccentric load P 25 kN (top ®gure) 4 7 kN (bottom ®gure)).
of each specimen leads to a slightly dierent response. These experiments show that the length of the secondary crack varies (smallest for specimen 2). This eect is well retrieved with the applied eccentricity in the model, which is well understood if the material is heterogeneous. The larger the eccentricity, the large the heterogeneity in the vicinity of a loading platen. The length of the short crack clearly depends on the applied eccentricity which was given to the loading point. A smaller eccentricity induces a longer secondary crack, as noticed from Figs. 8 and 9. This observation is con®rmed if the load±CMOD/ CMSD curves of the simulations and experiments are compared. It can be noticed from the numerical
Fig. 9. Damage in the DEN beam with rotating supports, 1 mm eccentric load (P = 18 kN).
Fig. 10. Experimental crack patterns [11].
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Fig. 11. Numerical results.
solution in Fig. 11 that larger eccentricities cause smaller maximum loads. This observation is con®rmed by the experimental results in Fig. 12, where the numbers next to the curves correspond to the numbers of the experiments indicated in Fig. 10. The eccentricity in the experiments is caused by friction in the supports and the heterogeneity of the concrete material. From this analysis, it may be concluded that small disturbances of the boundary conditions or inherent material heterogeneities have an important in¯uence on the ®nal failure mode. This explains why dierent crack lengths have been reported experimentally. 4. Beams loaded with ®xed supports In order to study the in¯uence of the boundary conditions, experimental analyses have been carried out on SEN and DEN beams with a ®xed support loading system instead of freely rotating supports [11]. The ®xation of the supports was carried out by mounting diagonal bars to the loading frame. The use of this ®xed loading frame invariably introduces friction in the support platens. Furthermore, additional constraints were imposed on the rotations of the supports. The ®xation of the supports
Fig. 12. Experiments [11].
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Fig. 13. Damage in the SEN beam with ®xed supports P 21 kN).
introduces a constraint with respect to the horizontal displacement and the rotations of the supports. It is found that the exact boundary conditions which are used to model these constraints are important and may lead to dierent failure modes. In the FEM model, the boundary conditions have to be modi®ed accordingly. To model the ®xation of the supports linear constraints, nodal restraints or spring elements were introduced in the FEM model. However, it is not possible to give a complete quantitative analysis of the load±CMOD/CMSD results without taking into account the frictional eects in the supports, which have an important in¯uence on measured forces and displacements. The study is, therefore, limited to a qualitative interpretation of the computed failure modes and damage distributions. 4.1. Single-edge notched beam The ®xed loading frame of the SEN beam was modeled with sti linear springs between the loaded nodes of interconnected loading platens combined with linear constraints in order to inhibit the free rotation of the platens. This ®xed loading system leads to the same curved crack as observed in the freely rotating specimen and the appearance of a second ¯exural crack which starts at the unnotched side of the specimen. The damage pro®le in the beam is shown in Fig. 13. The experimental crack pattern corresponding to these boundary conditions is depicted in Fig. 14, which con®rms the predicted damage zone. However, it should be remarked that the obtained solution depends on the applied boundary conditions and on the stiness of the constraining springs. If these springs are too sti, a splitting crack is found at the center of the beam, as depicted in Fig. 15. 4.2. Double-edge notched beam The DEN beam under ®xed loading conditions is a particularly interesting case, since several failure modes were then found for it. Experimentally, two failure patterns were observed which are shown in
Fig. 14. Cracks in the SEN beam with ®xed supports [11].
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Fig. 15. Splitting crack in the SEN beam with ®xed supports P 8 kN).
Fig. 16. DEN double curved crack mode [11].
Figs. 16 and 17. The ®rst failure mode consists of two antisymmetric curved cracks which originate from each notch, see Fig. 16. The second mode is characterized by the presence of a splitting crack in the center of the beam, as shown in Fig. 17. In the experiment, cracks under the loading platens were only observed sporadically, while they are more prone to appear in a numerical simulation in which contact between concrete and loading platen cannot be described exactly. It may, therefore, be necessary to inhibit the numerical damage growth in
Fig. 17. DEN splitting crack mode [11].
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Fig. 18. Double curved damage zones in the DEN beam with ®xed supports P 18 kN).
the vicinity of the loading platens which is due to the horizontal constraints. The ®xed support loading frames were modeled with constraints between the two nodes of each loading frame. However, the relative motion of the loading frames with respect to one another, which typically occurs with some slip in the supports, is not constrained. With these boundary conditions, signi®cant damage occurs under the loading platens and double curved crack mode comes out naturally (Fig. 18). Note that the same result can be obtained if sti ¯exible springs are used to model the ®xed loading frames. The cracks associated
Fig. 19. Splitting crack in the DEN beam with ®xed supports (P = 31 kN pre-peak (top ®gure) 4 37 kN post-peak 4 31 kN 4 15 kN (bottom ®gure)).
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Fig. 20. Splitting crack in the ®xed DEN beam with k equal to 10 P 18 kN (top ®gure) 4 17 kN 4 15 kN (bottom ®gure)).
with the computed damage in the neighbourhood of the supports can also be noticed in the experimental crack distribution shown in Fig. 16. Splitting cracks can be obtained by using a sti loading frame and by inhibiting damage in the immediate vicinity of the platens. This is done by restraining all the horizontal displacements of the loaded nodes. In this case, the double curved crack is arrested due to the new con®guration of the horizontal forces in the specimen. The splitting crack arises at the center of the beam, i.e. between the two other cracks. The development of the splitting crack is shown in Fig. 19 for a compression±tension sensitivity ratio k equal to 15. After the double curved crack has been arrested, damage starts to grow near the notches and propagates towards the center of the beam, which results in the characteristic Sshape of the splitting crack. As soon as the damage (which is still far below unity) has reached the center of the beam, the damage will further increase at the center and extend back towards the notches. The ®nal damage distribution corresponds with the experimental S-shape, which is given in Fig. 17. The development of this S-shaped crack depends on the value of the compression±tension sensitivity ratio k: If the concrete is characterized by a smaller value of k, e.g. 10, damage initiates from the center of the beam instead from the notches, as shown in Fig. 20. Obviously, the value of k has an important in¯uence on the ultimate load, the propagation of the crack and the characteristic S-shape, which is less pronounced for a smaller compression±tension sensitivity ratio.
5. Conclusions The damage initiation and propagation in SEN and DEN concrete beams has been investigated numerically, where a comparison has been made with the available experimental results. The freely
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rotating and a ®xed loading frame have been modeled in order to study the in¯uence of the boundary conditions on the computed damage bands. A higher order continuum damage model was used successfully to describe most of the experimentally observed phenomena: single curved cracks (SEN rotating), straight cracks (SEN ®xed), asymmetric double curved cracks (DEN rotating), antisymmetric double curved cracks (DEN ®xed) and splitting cracks with a characteristic S-shape (DEN ®xed). The signi®cant in¯uence of the boundary conditions in the loading supports on the ®nal failure mode has been emphasized, and the eect of the sensitivity to compressive strains versus tensile strains has been illustrated. In spite of the assumption of isotropy, a gradient-enhanced damage model permits to obtain reliable results in view of the prediction of the failure behavior of concrete. The incorporation of the correct compression±tension sensitivity plays a dominant role in the propagation of damage and the ®nal curved shape of the macroscopically observed crack.
References [1] Pijaudier-Cabot G, Dube JF, La Borderie C, Bode L. Damage models for concrete in transient dynamics. In: BazÏant ZP, Bittnar Z, JiraÂsek M, Mazars J, editors. Fracture and Damage in Quasibrittle Structures Ð Experiment, Modeling and Computer Analysis. Proceedings of the US±Europe Workshop. London: E & FN Spon, 1994. p. 201±15. [2] de Borst R, Pamin J, Peerlings RHJ, Sluys LJ. On gradient-enhanced damage and plasticity models for failure in quasi-brittle and frictional materials. Comp Mech 1995;17:130±41. [3] Geers MGD, de Borst R, Brekelmans WAM, Peerlings RHJ. Strain-based transient-gradient damage model for failure analyses. Computer Methods in Applied Mechanics and Engineering 1998;160(1/2):133±54. [4] Peerlings RHJ, de Borst R, Brekelmans WAM, Geers MGD. Gradient-enhanced damage modelling of concrete fracture. Mechanics of Cohesive-Frictional Materials 1998;3:323±42. [5] BazÏant ZP, Pfeier PA. Test of shear fracture and strain-softening in concrete. In: Proceedings of the 2nd Symposium of the Interaction of Non-Nuclear Munitions with Structures Panama City Beach, FL. 1985. [6] Ingraea AR, Panthaki MJ. Analysis of shear fracture tests of concrete beams. In: Finite element analysis of reinforced concrete structures. Tokyo: ASCE, 1985. p. 151±73. [7] Rots JG, Kusters GMA, Blauwendraad J. Strain-softening simulations of mixed-mode concrete failure. In: Shah SP, Swartz SE, editors. Proceedings SEM/RILEM International Conference on Fracture of Concrete and Rock. New York: Springer± Verlag, 1987. p. 175±89. [8] Swartz SE, Taha NM. Preleminary investigation of the suitability of the Iosipescu test specimen for determining mixed mode fracture properties of of concerete. Technical Report 191, Kansas State University, Manhattan, KS, 1987. [9] Carpinteri A, Ferrara G, Melchiorri G. Single edge notched specimen subjected to four point shear: an experimental investigation. In: Shah SP, Swartz SE, Barr B, editors. Fracture of concrete and rock ± recent developments. London/New York: Elsevier Applied Science Publishers, 1989. [10] Ferrara G. On the accuracy of the mixed mode tests performed at the Enel-Cris laboratories. Technical Report 4275, EnelCris, Milan, Italy, 1991. [11] Schlangen E. Experimental and numerical analysis of fracture processes in concrete. Ph.D. Thesis, Delft University of Technology, 1993. [12] Peerlings RHJ, de Borst R, Brekelmans WAM, de Vree JHP. Gradient-enhanced damage for quasi-brittle materials. Int J Num Meth Eng 1996;39:3391±403. [13] Bazant ZP, Pijaudier-Cabot G. Measurement of characteristic length of nonlocal continuum. Journal of Engineering Mechanics 1989;115(4):755±67. [14] Carmeliet J. Inverse identi®cation of gradient damage parameters from localization phenomena in quasi-brittle materials. In: Owen DRJ, Onate E, Hinton E, editors. Computational Plasticity, Fundamentals and Applications, Proceedings of COMPLAS V. Barcelona: CIMNE, 1997. p. 798±803. [15] Geers MGD, de Borst R, Brekelmans WAM, Peerlings RHJ. Validation and internal length scale determination for a gradient damage model: application to short glass®bre-reinforced plypropylene. International Journal of Solids and Structures 1999;36:2557±83. [16] Mazars J. A description of micro- and macro-scale damage of concrete structures. Engineering Fracture Mechanics 1986;25(5/ 6):729±37.
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[17] Saouridis C, Mazars J. Prediction of the failure and size eect in concrete via bi-scale damage approach. Engineering Computations 1992;9:329±44. [18] Kachanov LM. Time of the rupture process under creep conditions. Tzv Akad Nauk SSR Otd Tekh 1958;8:26±31. [19] BazÏant ZP, Pfeier PA. Shear fracture test of concrete. Materials and Structures (RILEM) 1986;110(19):111±21.
www.elsevier.com/locate/engfracmech
Damage and crack modeling in single-edge and doubleedge notched concrete beams M.G.D. Geers a,b,*, R. de Borst c, R.H.J. Peerlings a a
Faculty of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands b Faculty of Civil Engineering, Royal Military Academy, Avenue Renaissance 30, 1000 Brussels, Belgium c Faculty of Aerospace Engineering, Delft University of Technology, P.O. Box 5058, 2600 GB Delft, The Netherlands
Abstract The numerical modeling of damage and crack propagation in concrete and concrete structures has evolved considerably in the past years. In this contribution, a higher order continuum model is used to model the failure behavior of single-edge notched (SEN) and double-edge notched (DEN) concrete beams loaded in four-point-shear. Dierent types of boundary conditions, i.e. with freely rotating, ®xed or constrained loading supports, are investigated and the experimentally observed curved crack paths are compared with the numerical simulations. The in¯uence of the ratio of the compressive strength and the tensile strength is scrutinized and its relation with the failure mechanism is investigated. It is shown that an isotropic gradient-enhanced damage model permits to obtain a good agreement between experimental results and numerical simulations. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Damage mechanics; Failure assessment; Crack growth; Mixed mode fracture; Higher order continuum
1. Introduction The numerical modeling of concrete failure has experienced a signi®cant progress in the past decades. Dierent approaches have been adopted, leading to dierent models, such as fracture energy models, smeared crack models, Cosserat models, plasticity models, rate-dependent models, nonlocal damage models, gradient-enhanced models and others. In many of these models, concrete is considered as a quasi-brittle material in which strain softening plays a dominant role. The necessity of higher order continuum models for the description of the failure process within a continuum mechanics framework, has been recognized by * Corresponding author. Tel.: +31-40-2475076; fax: +31-40-2447355. E-mail address: [email protected] (M.G.D. Geers). 0013-7944/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 9 9 ) 0 0 1 1 8 - 6
248
M.G.D. Geers et al. / Engineering Fracture Mechanics 65 (2000) 247±261
Nomenclature SEN DEN CM CM s e D 4 C E n eeq e eq c k ki k a b
single-edge notched double-edge notched crack mouth opening displacement crack mouth sliding displacement Cauchy stress tensor in®nitesimal strain tensor damage variable elasticity tensor Young's modulus Poisson's ratio local equivalent strain nonlocal equivalent strain gradient material parameter history variable damage initiation history parameter tension±compression sensitivity ®rst damage evolution parameter second damage evolution parameter
many authors [1±4]. These higher order descriptions typically include a nonlocal eect into the constitutive description, e.g. nonlocal damage models, gradient damage models, nonlocal or gradient plasticity models. From the numerical point of view, it has been shown that these models solve the ill-posedness of the boundary value problem. A microstructural parameter is, therefore, incorporated into the constitutive model, which is commonly denoted as an intrinsic length scale. These enriched models are, therefore, related to the micromechanics of the dominant failure processes. In the present analysis, a gradient-enhanced damage formulation will be used to model failure of notched concrete beams. Concrete beams under mixed-mode loading are typical benchmark problems that have been investigated frequently in literature [5±10]. Many of these analyses lead to contradictory conclusions, where dierent failure modes were observed depending on the experimental setup that has been used. In this paper, a comparison will be made between computational simulations and available experimental results [11]. Schlangen [11] used a speci®c experimental setup in which the loading frame could be adapted in order to obtain freely rotating and ®xed loading supports. The experimentally observed failure mode strongly depends on the type of boundary conditions that have been applied. This explains why contradictory results have been obtained in literature. In this paper, attention is focused on the damage development in single-edge and double-edge notched concrete beams, where dierent type of boundary conditions have been used. 2. Gradient-enhanced damage for concrete fracture The gradient-enhanced damage model is commonly formulated as an isotropic strain-based damage theory, in which the nonlocal eect has been incorporated by the addition of gradient terms in the constitutive description [12]. The formulation is based on a total stress±strain relation according to: s 1 ÿ D 4 C:e
1
M.G.D. Geers et al. / Engineering Fracture Mechanics 65 (2000) 247±261
249
with s the Cauchy stress tensor, e the in®nitesimal strain tensor, 4 C the fourth-order elastic stiness tensor and D the scalar-valued damage variable. In order to be compliant with the second principle of thermodynamics, the rate of the damage variable for an isotropic material with no permanent strains must satisfy _ Dr0
2
Furthermore, the damage variable D as de®ned by Eq. (1) must satisfy 0RDR1
3
The damage parameter D is coupled with the deformation history of the material through a monotonically increasing deformation history parameter k: This scalar history parameter characterizes the ultimate nonlocal equivalent strain e eq that the material has experienced in its loading history. The evolution of k upon damage evolution is usually expressed by the Kuhn±Tucker relations ÿ _ 4 kr0, e eq ÿ kR0, k_ e eq ÿ k 0 The nonlocal characteristic e eq is computed from the ®eld of local equivalent strains eeq in the vicinity of the material point via a partial dierential equation of the Helmholtz type [12]: e eq ÿ cr2 e eq eeq
5
The solution of Eq. (5) requires the incorporation of appropriate boundary conditions [12]. This equation introduces a new material parameter c, the gradient parameter, which is related to the square of the intrinsic length scale of the nonlocal continuum. Suggestions for the measurement of the characteristic length of a nonlocal continuum were made by BazÏant and Pijaundier-Cabot [13], Carmeliet [14] and Geers et al. [15]. Speci®c experimental procedures and methods have to be setup in agreement with the physical phenomena which govern the localization and fracture process. This can be done by direct measurements in the process zone or indirectly, e.g. via the size eect. Usually, complex tests are necessary which involve an iterative procedure to determine the individual parameters on a statistical basis. An important aspect for concrete fracture resides in the precise de®nition of eeq , which has to be computed from the strain tensor e in each material point. In the de®nition proposed by Peerlings et al. [4], the relative contributions of the tensile and compressive strain components dier, which can have a major impact on the failure mode that is obtained. Similar propositions can be found in [16,17]. The equivalent strain de®nition used for concrete reads s 2 kÿ1 1 kÿ1 6k J1 J1 J2 6 eeq 2k 1 ÿ 2n 2k 1 ÿ 2n 1 n 2 where the following strain tensor invariants have been used J1 tr e e1 e2 e3 i 1 1h J2 tr e e ÿ tr2 e e1 ÿ e2 2 e2 ÿ e3 2 e1 ÿ e3 2 3 3
7
The parameter k k1fcc =fct has been introduced to control the dierent sensitivity to tensile and compressive strains. The dependency of the equivalent strain on the strain components is illustrated in Fig. 1, which shows iso-eeq curves for dierent values of k:
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Fig. 1. Iso-equivalent strain curves in the principal strain space.
The damage D progressively reduces the intrinsic stiness of the material [18]. The damage parameter D is computed directly from a damage evolution law, which describes the mechanical in¯uence of the material degradation. The parameters in the damage evolution law must provide the necessary ¯exibility to model the mechanical response of the material. The used damage evolution law for concrete is ki 1 ÿ a a eÿb kÿki 8 D1ÿ k where coecients a and b are the material parameters. The threshold value for damage initiation ki is the initial value for k at time t 0: It is the ultimate equivalent strain that characterizes the elastic material behavior prior to occurrence of damage. More computational details can be found in [12].
3. Beams loaded with freely rotating supports 3.1. Single-edge notched beam The single-edge notched (SEN) beam subjected to an antisymmetric four-point-shear loading, has been analyzed experimentally and numerically by Schlangen [11]. The geometry and the loading conditions of the specimen are given in Fig. 2. The thickness of the SEN beam equals 50 mm. The antisymmetric loading of the beam results in a curved crack path, which starts from the right corner of the notch and which ends at the lower right loading platen. The beam has been modeled with 5 5 mm2 eight-noded quadrilateral elements for the coarse mesh and 2.5 2.5 mm2 elements for the ®ne mesh. The material parameters used in this model [4] are: Young's modulus E = 35,000 MPa, Poisson's ratio v 0:2, gradient parameter c = 1 mm2, compression±tension sensitivity k 15, damage initiation history parameter ki 6 10ÿ5 , damage evolution law parameters a 0:96 and b 100: The two central loading platens have been modeled as rigid bodies through the introduction of appropriate tyings between the nodal displacements. One nodal force is then centrally applied on the rigid platen. The loading platens at the lower left and upper right extremity of the beam were modeled with a nodal force and a nodal support, respectively. A plane stress situation has been assumed. Damage initiates at the right corner of the notch and at the bottom of the beam, opposite to the upper central loading platen. The damage growth at the bottom is arrested, while it keeps growing at the notch. The experimentally observed curved crack path, as observed by Schlangen [11], is shown in Fig. 3. The evolution of the damage growth in the beam is depicted in successive stages in Fig. 4, with
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Fig. 2. Single-edge notched beam con®guration: beam thickness = 50 mm; width loading platens = 20 mm.
the corresponding levels computed with the coarse mesh (see also Fig. 6). The obtained damage pro®le is in good agreement with the experimentally observed crack path of Fig. 3. The secondary damage zones at the opposite edge of the central loading platens have been con®rmed experimentally. Fig. 5 shows the load P versus the crack mouth opening displacement (CMOD) for the simulation and experiments. The quality of the ®t is apparent. The mesh dependence has been investigated by comparing the results for two dierent mesh sizes for which the same damage distribution was found. The load±CMSD curves for both meshes are shown in Fig. 6. The good quality of the simulation is mainly attributable to the proper de®nition of the equivalent strain (6) which has been utilized in the numerical analysis. Other de®nitions fail to predict the correct crack path [4].
Fig. 3. Single curved crack in a rotating SEN beam [11].
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Fig. 4. Damage evolution in an SEN beam with rotating supports P 40 kN (top ®gure) 4 37 kN 4 7 kN (bottom ®gure)).
3.2. Double-edge notched beam The double-edge notched (DEN) beam, which has been proposed and tested by BazÏant and Pfeier [5,19] to study the shear fracture of concrete, has been analyzed several times in literature [6±8,11]. The specimen was tested by Schlangen [11] in a symmetric test machine. The DEN specimen, loaded in fourpoint-shear, is shown in Fig. 7. The loading platens are positioned adjacent to the notch, which results in a smaller shear zone than that observed in the SEN test. Consequently, the load is applied close to the notch (10 mm), and failure by shearing towards the notch is a possible failure mode in the absence of friction in the supports. In the experiment, the support platens are subjected to rotational and translational frictions. The inclusion of some frictional aspects in the FEM model for the loading platens seems, therefore, appropriate. This frictional eect turns out to be particularly important for the DEN beam with freely rotating supports. Small rotational springs can be added to the support platens or a small eccentricity of the loading force on the platen can be easily introduced, while the positioning of the platens remains unaected. The latter option has been adopted in this analysis. The introduction
Fig. 5. P±CMOD curve.
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Fig. 6. Mesh independence.
of a small eccentricity in the model also permits to take into account the deviations with respect to the assumed antisymmetry. Indeed, the heterogeneous character of the concrete material and the presence of aggregates with variable diameters in the vicinity of the loading platens will always provoke a more or less pronounced eccentric transfer of the load through the specimen. If the inherent eccentricity is totally neglected, an antisymmetric solution is found that consists of two curved cracks which originate from each notch. If the compression±tension strength ratio k is smaller than 15, the beam fails by shearing and compression failure at the notches. However, by adding a small eccentricity to the loading force, the failure mode becomes asymmetric and stays asymmetric upon further loading. The damage evolution in the beam is depicted in Fig. 8 for an eccentricity of 2.5 mm. A reduced eccentricity equal to 1 mm is shown in Fig. 9. A crack originates from each notch, but the lower crack is arrested, while the other crack continues to grow. The solution thus found corresponds well with the experimental solutions where a long crack and a short crack was found, see Fig. 10. This ®gures corresponds to three experiments that were taken under the same test conditions, but for which the heterogeneous character
Fig. 7. Con®guration of the DEN shear test: beam thickness = 37.5 mm; width loading platens = 20 mm.
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Fig. 8. Damage in the DEN beam with rotating supports, 2.5 mm eccentric load P 25 kN (top ®gure) 4 7 kN (bottom ®gure)).
of each specimen leads to a slightly dierent response. These experiments show that the length of the secondary crack varies (smallest for specimen 2). This eect is well retrieved with the applied eccentricity in the model, which is well understood if the material is heterogeneous. The larger the eccentricity, the large the heterogeneity in the vicinity of a loading platen. The length of the short crack clearly depends on the applied eccentricity which was given to the loading point. A smaller eccentricity induces a longer secondary crack, as noticed from Figs. 8 and 9. This observation is con®rmed if the load±CMOD/ CMSD curves of the simulations and experiments are compared. It can be noticed from the numerical
Fig. 9. Damage in the DEN beam with rotating supports, 1 mm eccentric load (P = 18 kN).
Fig. 10. Experimental crack patterns [11].
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Fig. 11. Numerical results.
solution in Fig. 11 that larger eccentricities cause smaller maximum loads. This observation is con®rmed by the experimental results in Fig. 12, where the numbers next to the curves correspond to the numbers of the experiments indicated in Fig. 10. The eccentricity in the experiments is caused by friction in the supports and the heterogeneity of the concrete material. From this analysis, it may be concluded that small disturbances of the boundary conditions or inherent material heterogeneities have an important in¯uence on the ®nal failure mode. This explains why dierent crack lengths have been reported experimentally. 4. Beams loaded with ®xed supports In order to study the in¯uence of the boundary conditions, experimental analyses have been carried out on SEN and DEN beams with a ®xed support loading system instead of freely rotating supports [11]. The ®xation of the supports was carried out by mounting diagonal bars to the loading frame. The use of this ®xed loading frame invariably introduces friction in the support platens. Furthermore, additional constraints were imposed on the rotations of the supports. The ®xation of the supports
Fig. 12. Experiments [11].
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Fig. 13. Damage in the SEN beam with ®xed supports P 21 kN).
introduces a constraint with respect to the horizontal displacement and the rotations of the supports. It is found that the exact boundary conditions which are used to model these constraints are important and may lead to dierent failure modes. In the FEM model, the boundary conditions have to be modi®ed accordingly. To model the ®xation of the supports linear constraints, nodal restraints or spring elements were introduced in the FEM model. However, it is not possible to give a complete quantitative analysis of the load±CMOD/CMSD results without taking into account the frictional eects in the supports, which have an important in¯uence on measured forces and displacements. The study is, therefore, limited to a qualitative interpretation of the computed failure modes and damage distributions. 4.1. Single-edge notched beam The ®xed loading frame of the SEN beam was modeled with sti linear springs between the loaded nodes of interconnected loading platens combined with linear constraints in order to inhibit the free rotation of the platens. This ®xed loading system leads to the same curved crack as observed in the freely rotating specimen and the appearance of a second ¯exural crack which starts at the unnotched side of the specimen. The damage pro®le in the beam is shown in Fig. 13. The experimental crack pattern corresponding to these boundary conditions is depicted in Fig. 14, which con®rms the predicted damage zone. However, it should be remarked that the obtained solution depends on the applied boundary conditions and on the stiness of the constraining springs. If these springs are too sti, a splitting crack is found at the center of the beam, as depicted in Fig. 15. 4.2. Double-edge notched beam The DEN beam under ®xed loading conditions is a particularly interesting case, since several failure modes were then found for it. Experimentally, two failure patterns were observed which are shown in
Fig. 14. Cracks in the SEN beam with ®xed supports [11].
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Fig. 15. Splitting crack in the SEN beam with ®xed supports P 8 kN).
Fig. 16. DEN double curved crack mode [11].
Figs. 16 and 17. The ®rst failure mode consists of two antisymmetric curved cracks which originate from each notch, see Fig. 16. The second mode is characterized by the presence of a splitting crack in the center of the beam, as shown in Fig. 17. In the experiment, cracks under the loading platens were only observed sporadically, while they are more prone to appear in a numerical simulation in which contact between concrete and loading platen cannot be described exactly. It may, therefore, be necessary to inhibit the numerical damage growth in
Fig. 17. DEN splitting crack mode [11].
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Fig. 18. Double curved damage zones in the DEN beam with ®xed supports P 18 kN).
the vicinity of the loading platens which is due to the horizontal constraints. The ®xed support loading frames were modeled with constraints between the two nodes of each loading frame. However, the relative motion of the loading frames with respect to one another, which typically occurs with some slip in the supports, is not constrained. With these boundary conditions, signi®cant damage occurs under the loading platens and double curved crack mode comes out naturally (Fig. 18). Note that the same result can be obtained if sti ¯exible springs are used to model the ®xed loading frames. The cracks associated
Fig. 19. Splitting crack in the DEN beam with ®xed supports (P = 31 kN pre-peak (top ®gure) 4 37 kN post-peak 4 31 kN 4 15 kN (bottom ®gure)).
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Fig. 20. Splitting crack in the ®xed DEN beam with k equal to 10 P 18 kN (top ®gure) 4 17 kN 4 15 kN (bottom ®gure)).
with the computed damage in the neighbourhood of the supports can also be noticed in the experimental crack distribution shown in Fig. 16. Splitting cracks can be obtained by using a sti loading frame and by inhibiting damage in the immediate vicinity of the platens. This is done by restraining all the horizontal displacements of the loaded nodes. In this case, the double curved crack is arrested due to the new con®guration of the horizontal forces in the specimen. The splitting crack arises at the center of the beam, i.e. between the two other cracks. The development of the splitting crack is shown in Fig. 19 for a compression±tension sensitivity ratio k equal to 15. After the double curved crack has been arrested, damage starts to grow near the notches and propagates towards the center of the beam, which results in the characteristic Sshape of the splitting crack. As soon as the damage (which is still far below unity) has reached the center of the beam, the damage will further increase at the center and extend back towards the notches. The ®nal damage distribution corresponds with the experimental S-shape, which is given in Fig. 17. The development of this S-shaped crack depends on the value of the compression±tension sensitivity ratio k: If the concrete is characterized by a smaller value of k, e.g. 10, damage initiates from the center of the beam instead from the notches, as shown in Fig. 20. Obviously, the value of k has an important in¯uence on the ultimate load, the propagation of the crack and the characteristic S-shape, which is less pronounced for a smaller compression±tension sensitivity ratio.
5. Conclusions The damage initiation and propagation in SEN and DEN concrete beams has been investigated numerically, where a comparison has been made with the available experimental results. The freely
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rotating and a ®xed loading frame have been modeled in order to study the in¯uence of the boundary conditions on the computed damage bands. A higher order continuum damage model was used successfully to describe most of the experimentally observed phenomena: single curved cracks (SEN rotating), straight cracks (SEN ®xed), asymmetric double curved cracks (DEN rotating), antisymmetric double curved cracks (DEN ®xed) and splitting cracks with a characteristic S-shape (DEN ®xed). The signi®cant in¯uence of the boundary conditions in the loading supports on the ®nal failure mode has been emphasized, and the eect of the sensitivity to compressive strains versus tensile strains has been illustrated. In spite of the assumption of isotropy, a gradient-enhanced damage model permits to obtain reliable results in view of the prediction of the failure behavior of concrete. The incorporation of the correct compression±tension sensitivity plays a dominant role in the propagation of damage and the ®nal curved shape of the macroscopically observed crack.
References [1] Pijaudier-Cabot G, Dube JF, La Borderie C, Bode L. Damage models for concrete in transient dynamics. In: BazÏant ZP, Bittnar Z, JiraÂsek M, Mazars J, editors. Fracture and Damage in Quasibrittle Structures Ð Experiment, Modeling and Computer Analysis. Proceedings of the US±Europe Workshop. London: E & FN Spon, 1994. p. 201±15. [2] de Borst R, Pamin J, Peerlings RHJ, Sluys LJ. On gradient-enhanced damage and plasticity models for failure in quasi-brittle and frictional materials. Comp Mech 1995;17:130±41. [3] Geers MGD, de Borst R, Brekelmans WAM, Peerlings RHJ. Strain-based transient-gradient damage model for failure analyses. Computer Methods in Applied Mechanics and Engineering 1998;160(1/2):133±54. [4] Peerlings RHJ, de Borst R, Brekelmans WAM, Geers MGD. Gradient-enhanced damage modelling of concrete fracture. Mechanics of Cohesive-Frictional Materials 1998;3:323±42. [5] BazÏant ZP, Pfeier PA. Test of shear fracture and strain-softening in concrete. In: Proceedings of the 2nd Symposium of the Interaction of Non-Nuclear Munitions with Structures Panama City Beach, FL. 1985. [6] Ingraea AR, Panthaki MJ. Analysis of shear fracture tests of concrete beams. In: Finite element analysis of reinforced concrete structures. Tokyo: ASCE, 1985. p. 151±73. [7] Rots JG, Kusters GMA, Blauwendraad J. Strain-softening simulations of mixed-mode concrete failure. In: Shah SP, Swartz SE, editors. Proceedings SEM/RILEM International Conference on Fracture of Concrete and Rock. New York: Springer± Verlag, 1987. p. 175±89. [8] Swartz SE, Taha NM. Preleminary investigation of the suitability of the Iosipescu test specimen for determining mixed mode fracture properties of of concerete. Technical Report 191, Kansas State University, Manhattan, KS, 1987. [9] Carpinteri A, Ferrara G, Melchiorri G. Single edge notched specimen subjected to four point shear: an experimental investigation. In: Shah SP, Swartz SE, Barr B, editors. Fracture of concrete and rock ± recent developments. London/New York: Elsevier Applied Science Publishers, 1989. [10] Ferrara G. On the accuracy of the mixed mode tests performed at the Enel-Cris laboratories. Technical Report 4275, EnelCris, Milan, Italy, 1991. [11] Schlangen E. Experimental and numerical analysis of fracture processes in concrete. Ph.D. Thesis, Delft University of Technology, 1993. [12] Peerlings RHJ, de Borst R, Brekelmans WAM, de Vree JHP. Gradient-enhanced damage for quasi-brittle materials. Int J Num Meth Eng 1996;39:3391±403. [13] Bazant ZP, Pijaudier-Cabot G. Measurement of characteristic length of nonlocal continuum. Journal of Engineering Mechanics 1989;115(4):755±67. [14] Carmeliet J. Inverse identi®cation of gradient damage parameters from localization phenomena in quasi-brittle materials. In: Owen DRJ, Onate E, Hinton E, editors. Computational Plasticity, Fundamentals and Applications, Proceedings of COMPLAS V. Barcelona: CIMNE, 1997. p. 798±803. [15] Geers MGD, de Borst R, Brekelmans WAM, Peerlings RHJ. Validation and internal length scale determination for a gradient damage model: application to short glass®bre-reinforced plypropylene. International Journal of Solids and Structures 1999;36:2557±83. [16] Mazars J. A description of micro- and macro-scale damage of concrete structures. Engineering Fracture Mechanics 1986;25(5/ 6):729±37.
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[17] Saouridis C, Mazars J. Prediction of the failure and size eect in concrete via bi-scale damage approach. Engineering Computations 1992;9:329±44. [18] Kachanov LM. Time of the rupture process under creep conditions. Tzv Akad Nauk SSR Otd Tekh 1958;8:26±31. [19] BazÏant ZP, Pfeier PA. Shear fracture test of concrete. Materials and Structures (RILEM) 1986;110(19):111±21.