CALTECH ASCI TECHNICAL REPORT 094 caltechASCI/2000.094
Three Dimensional cohesive modeling of dynamic mixed-mode fracture G. Ruiz, A. Pandolfi, M. Ortiz
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2001; 52:97–120 (DOI: 10.1002/nme.273)
Three-dimensional cohesive modeling of dynamic mixed-mode fracture Gonzalo Ruiz1 , Anna Pandol62 and Michael Ortiz 3; ∗; † 1 E.
T. S. de Ingenieros de Caminos; Canales y Puertos; Universidad de Castilla-La Mancha; 13071 Ciudad Real; Spain 2 Dipartimento di Ingegneria Strutturale; Politecnico di Milano; 20133 Milano; Italy 3 Graduate Aeronautical Laboratories; California Institute of Technology; Pasadena; CA 91125; U.S.A.
SUMMARY A cohesive formulation of fracture is taken as a basis for the simulation of processes of combined tension-shear damage and mixed-mode fracture in specimens subjected to dynamic loading. Our threedimensional 6nite-element calculations account explicitly for crack nucleation, microcracking, the development of macroscopic cracks and inertia. In particular, a tension-shear damage coupling arises as a direct consequence of slanted microcrack formation in the process zone. We validate the model against the three-point-bend concrete beam experiments of Guo et al. (International Journal of Solids and Structures 1995; 32(17=18):2951–2607), John (PhD Thesis, Northwestern University, 1988), and John and Shah (Journal of Structural Engineering 1990; 116(3):585 – 602) in which a pre-crack is shifted from the central cross-section, leading to asymmetric loading conditions and the development of a mixed-mode process zone. The model accurately captures the experimentally observed fracture patterns and displacement 6elds, as well as crack paths and crack-tip velocities, as a function of precrack geometry and loading conditions. In particular, it correctly accounts for the competition between crack-growth and nucleation mechanisms. Copyright ? 2001 John Wiley & Sons, Ltd. KEY WORDS:
tension-shear damage coupling; mixed mode dynamic fracture; three-point-bend concrete beams; cohesive models; 3D 6nite element
1. INTRODUCTION Brittle materials, such as concrete, mortar, rocks and ceramics, often develop complex fracture patterns when they are subjected to high rates of loading. The elastic heterogeneities and weak interfaces present in these materials on the microscale often result in the nucleation of microcracks, which may in turn coalesce into large structural cracks. Such cracks follow ∗ Correspondence
to: Michael Ortiz; Graduate Aeronautical Laboratories; California Institute of Technology; Firestone Flight Sciences; Mail Stop 105-50; Pasadena; CA 91125; U.S.A. † E-mail:
[email protected] Contract=grant sponsor: Caltech’s ASCI=ASAP Center for the Simulation of the Dynamic Properties of Solids Contract=grant sponsor: DirecciGon General de EnseHnanza Superior, Ministerio de EducaciGon y Cultura
Copyright ? 2001 John Wiley & Sons, Ltd.
98
G. RUIZ, A. PANDOLFI AND M. ORTIZ
meandering paths and undergo frequent branching depending upon the loading and boundary conditions. Cracks may link up with free surfaces or with each other to form fragments. In concrete, the microstructural length scale is commensurate with the aggregate size and, thus, fracture processes often occur on a scale comparable to the geometrical dimensions of the structure. In addition to the energy required to fracture a material, other sources of energy dissipation may operate simultaneously, including plastic work, viscosity, and heat conduction. Finally, microinertia often plays a signi6cant role in shaping the eJective macroscopic behaviour of solids at high rates of deformation (e.g. References [1–5]). Fracture mechanics speci6cally addresses the issue of whether a body under load will remain intact or whether a new free surface will form. However, classical fracture mechanics assumes a pre-existing dominant crack and thus foregoes the issue of nucleation. Additionally, the conditions for crack growth are typically expressed in terms of parameters characterizing the amplitude of autonomous near-tip 6elds, which restricts the scope of the theory. e.g. by requiring that the plastic or process zone be small relative to geometrical dimensions such as the crack or ligament sizes. For concrete, these conditions are rarely realized in practice. In classical dynamic fracture mechanics, the fracture criterion and crack-tip equation of motion must also account for the microinertia which accompanies the motion of the crack tip. An alternative approach to fracture, which overcomes some of the aforementioned diMculties, is based on the use of cohesive models [6–35]. In these theories, fracture is regarded as a gradual process in which the separation of the incipient crack Nanks is resisted by cohesive tractions. The relation between the cohesive tractions and the opening displacements is governed by a cohesive law. Cohesive models furnish a complete theory of fracture which is not limited by any consideration of material behaviour, 6nite kinematics, non-proportional loading, dynamics, or the geometry of the specimen. In addition, cohesive theories 6t naturally within the conventional framework of 6niteelement analysis, and have proved eJective in the simulation of complex fracture processes including: the fragmentation of brittle materials [24; 36]; dynamic fracture and fragmentation of ductile materials [31; 37]; and the dynamic Brazilian test for concrete [32]. However, Camacho and Ortiz [24] have noted that the accurate description of fracture processes by means of cohesive elements requires the resolution of the characteristic cohesive length of the material. In some materials, such as ceramics and glass, this length may be exceedingly small. Thus calculations based on cohesive elements inevitably have a multi-scale character, in as much as the numerical model must resolve two disparate length scales commensurate with the macroscopic dimensions of the solid and the cohesive length of the material. In this paper, we use cohesive theories of fracture to simulate processes of tension-shear damage and mixed-mode fracture in concrete specimens subjected to dynamic loading. We adopt a simple cohesive law proposed by Camacho and Ortiz [24] which accounts for tensionshear coupling through the introduction of an eJective scalar opening displacement. The form of the eJective opening displacement allows for diJerent weights to be applied to the normal and tangential components of the opening displacement vector. This simple device also permits according the material critical normal and shear tractions bearing arbitrary ratios. This Nexibility is particularly important in the case of concrete, where the aggregate interlocking mechanism may result in critical shear tractions greatly in excess of the tensional strength of the material. The cohesive behaviour of the material is assumed to be rigid, or perfectly coherent, up to the attainment of an eJective traction, at which point the cohesive surface Copyright ? 2001 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2001; 52:97–120
THREE-DIMENSIONAL COHESIVE MODELING
99
begins to open. The cohesive law is rendered irreversible by the assumption of linear unloading to the origin. In calculations, we allow for decohesion to occur along element boundaries only. Initially, all element boundaries are perfectly coherent and the elements are conforming in the usual sense of the displacement 6nite-element method. When the critical cohesive traction is attained at the interface between two volume elements, we proceed to insert a cohesive element at that location. Crack nucleation is thus accounted for and falls within the scope of the analysis. The cohesive element subsequently governs the opening of the cohesive surface. It is important to note that the cohesive size of concrete scales with the aggregate size. Since, as noted earlier, this cohesive length must be resolved numerically, the element size must be of the order of the aggregate size over the cohesive zone. Thus, the insertion of a new cohesive element may be regarded as the nucleation of one additional microcrack. These microcracks may subsequently coalesce among themselves, resulting in the nucleation and growth a structural crack. Since the fracture surface is con6ned to inter-element boundaries, the structural cracks predicted by the analysis are necessarily rough. Cracks in concrete, as well as in materials which crack by intergranular fracture, exhibit considerable roughness on the scale of the aggregate or grain size. The surface roughness adds to the fracture area and, consequently, increases the speci6c fracture energy. Severe roughness may also toughen the material by the aggregate interlocking and crack bridging mechanisms. By the choice of an element size comparable to the aggregate size, the numerical model furnishes a simple description of the actual crack-surface roughness which is observed experimentally and the attendant eJective toughening of the material. It is also important to note that, owing to the presence of profuse microcracking in the vicinity of the crack tip, the e.ective fracture behaviour of the material may diJer signi6cantly from that which obtained by exercising the cohesive law directly. In particular, when a precrack is subjected to pure mode-II loading a number of slanted microcracks are inevitably formed ahead of the crack tip, roughly at 45◦ to the plane of the pre-crack. The subsequent behaviour of the crack is then sensitively dependent on the extent of normal compression, which tends to close the trailing microcracks and constrains them to undergo frictional sliding. Evidently, this type of behaviour is not ‘hardwired’ into the class of cohesive laws adopted here, but it is predicted by them. Likewise, the eJective dynamic behaviour of the material is predicted as an outcome of the calculations, instead of being built into the cohesive law as a rate-sensitivity eJect [1; 38–47]. In particular, we assume the cohesive law to be rate independent. Cohesive theories, in addition to building a characteristic length into the material description, endow the material with an intrinsic time scale as well [24]. This intrinsic time scale permits the material to discriminate between slow and fast loading rates and, in materials such as concrete, ultimately allows for the accurate prediction of dynamic fracture properties [32]. We validate the predictions of the model against the experimental data of Guo et al. [1], and John and Shah [2; 3]. These tests were concerned with three-point-bend beam (TPB) specimens containing a pre-crack shifted from the central cross-section, leading to asymmetric loading conditions and resulting in mixed-mode fracture. Specimens were tested dynamically by means of a drop-weight device [1], or a modi6ed Charpy tester [2; 3]. By increasing its oJset, the pre-crack is subjected to increasingly mode-II fracture conditions, and the trajectory of the crack varies accordingly. In addition, the experimental data is indicative of a competition between two fracture mechanisms: the growth of the pre-crack; and the nucleation and Copyright ? 2001 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2001; 52:97–120
100
G. RUIZ, A. PANDOLFI AND M. ORTIZ
subsequent growth of a crack within the central cross-section of the specimen. The fracture behaviour of the specimen is also observed to be sensitively dependent on the rate of loading. These and other features of the experimental set-up and the observed behaviour of the specimens furnish an exacting test of the validity of the model. Our calculations show that the model is indeed capable of closely capturing the observed fracture patterns, crack trajectories and crack-tip displacement 6elds corresponding to diJerent experimental con6gurations. In particular, it correctly accounts for the relative stability regimes of the competing pre-crack growth vs crack nucleation mechanisms. The simulations also return accurate histories of crack extension and velocity, and the computed loads agree closely with the reported experimental load histories. The organization of the paper is as follows. A brief outline of the cohesive model and its 6nite-element implementation employed in calculations is given in Section 2 for completeness and subsequent reference. In Section 3 we review the set-up of the experiments of Guo et al. [1], and John and Shah [2; 3] which we aim to simulate. In Section 4 we collect or estimate the material properties used in subsequent calculations, with particular attention given to the tension-shear coupling constant . We devote Section 5 to an assessment of the accuracy aJorded by the discretization used in calculations. Selected results of the calculations and comparisons with experimental data are presented in Section 6. Finally, a summary and some concluding remarks are collected in Section 7. 2. COHESIVE MODEL OF FRACTURE For completeness and subsequent reference, in this section we summarize the main features of the cohesive law used in calculations. An extensive account of the theory and its 6nite-element implementation may be found elsewhere [24; 29]. A simple class of mixed-mode cohesive laws accounting for tension-shear coupling (see Camacho and Ortiz [24] and others [28; 29]) is obtained by the introduction of an eJective opening displacement , which assigns diJerent weights to the normal n and sliding S opening displacements (1) = 2 2S + n2 ; n = T · n; S = |TS | = |T − n n| Here n and S are the normal and tangential opening displacements across the cohesive surface, respectively. Assuming that the cohesive free-energy density depends on the opening displacements only through the eJective opening displacement , the cohesive law reduces to [24; 29] t t = (2 TS + n n) where t=
−2 |tS |2 + tn2
(2)
(3)
is an eJective cohesive traction, to be related to by a reduced cohesive law, and tS and tn are the shear and the normal tractions, respectively. From this relation, we observe that the weighting coeMcient de6nes the ratio between the shear and the normal critical tractions. Copyright ? 2001 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2001; 52:97–120
THREE-DIMENSIONAL COHESIVE MODELING
101
Figure 1. Loading–unloading rule from a linearly decreasing loading envelope expressed in terms of an eJective opening displacement and traction t.
In brittle materials, this ratio may be estimated by imposing lateral con6nement on specimens subjected to high-strain-rate axial compression [48; 49]. It also roughly de6nes the ratio of KIIc to KIc of the material. We assume the existence of a loading envelope de6ning a relation between t and under the conditions of monotonic loading. We additionally assume that unloading is irreversible. A simple and convenient type of irreversible cohesive law is furnished by the linearly decreasing envelope shown in Figure 1, where c is the tensile strength and c the critical opening displacement. Following Camacho and Ortiz [24] we assume linear unloading to the origin, Figure 1. Upon closure, the cohesive surfaces are subjected to the contact unilateral constraint, including friction. We regard contact and friction as independent phenomena to be modeled outside the cohesive law. Friction may signi6cantly increase the sliding resistance in closed cohesive surfaces. In particular, the presence of friction may result in a steady—or even increasing—frictional resistance while the normal cohesive strength simultaneously weakens. Cohesive theories introduce a well-de6ned length scale into the material description and, in consequence, are sensitive to the size of the specimen (see, e.g. Reference [8]). The characteristic length of the material may be expressed as ‘c =
EGc fts2
(4)
where Gc is the fracture energy and fts the static tensile strength. Camacho and Ortiz [24] have noted that, when inertia is accounted for, cohesive models introduce a characteristic time as well. This characteristic or intrinsic time is tc =
cc 2fts
(5)
where is the mass density and c the longitudinal wave speed. Owing to this intrinsic time scale, the material behaves diJerently when subjected to fast and slow loading rates. This sensitivity to the rate of loading confers cohesive models the ability to predict subtle features of the dynamic behaviour of brittle solids, such as crack-growth initiation times and Copyright ? 2001 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2001; 52:97–120
102
G. RUIZ, A. PANDOLFI AND M. ORTIZ
propagation speeds [37], and the dependence of the pattern of fracture and fragmentation on strain rate [31]. In addition, they account for the dynamic strength of brittle solids, i.e. the dependence of the dynamic strength on strain rate [32]. In calculations, we allow for decohesion to occur along element boundaries only. Initially, all element boundaries are perfectly coherent and the elements are conforming in the usual sense of the displacement 6nite element method. When the critical cohesive traction is attained at the interface between two volume elements, we proceed to insert a cohesive element at that location. The cohesive element subsequently governs the opening of the cohesive surface. Details of the 6nite-element implementation may be found elsewhere [24; 29]. 3. EXPERIMENTAL SET-UP We speci6cally aim to simulate the experiments reported by Guo et al. [1], and John and Shah [2; 3]. Both sets of tests were carried out on prismatic plain concrete beams loaded in a three-point bend con6guration. The specimens were notched at an oJset with respect to their central cross-section to induce mixed-mode crack growth. Guo et al. [1] used a drop-weight tester with a 10-kg impactor. In order to provide an averaged displacement history curve at the load point, the concrete beam was instrumented at the contact point with a dynamic load cell, plus two non-contact displacement gauges at the load point on both sides. The crack extension and velocity throughout the tests were measured by two diJerent methods. On one side of the beam, 6ve strain gauges were glued along the expected crack path; on the other side a moirGe diJraction grating of 600 lines= mm spanning the area was transferred to the previously polished surface of the beam along the expected crack path. Upon contact with a push rod, the drop weight triggered several Nashes and cameras which recorded the horizontal and vertical moirGe patterns, from which the 6eld of displacements around the crack—and the crack pattern itself—could be obtained. Details on this experimental procedure can be found in several papers by Kobayashi and coworkers [50–52], or in References [53; 54]. Gopalaratnam et al. [55] instrumented a Tinius Olsen 64 Charpy tester and adapted it to a three-point bending mixed-mode test on concrete specimens. Here we simulate John and Shah’s experiments [2; 3] performed with this apparatus. John and Shah measured the history of the load transmitted by the 27.3-kg tup of the pendulum, as well as the load exerted by the anvil supporting the specimen. Several strain gauges glued to the specimen allowed measurement of the strain rates. The crack advance and velocity were not recorded in these experiments. Nevertheless, the trajectories of the cracks through the beams could be recovered by the post-mortem examination of the specimens. The geometry and dimensions of the prismatic beams used in both experimental programmes are depicted in Figure 2. Guo et al. extended the initial 19-mm machined notch by approximately 13 mm by stable symmetric loading. It should be noted that the relative oJset, = 2 oJset=L, of the notch with respect to the central cross-section was systematically varied in John and Shah’s beams, with values: = 0; 0:5; 0:7; 0:72; 0:77 and 0:875. In order to reduce the inertial eJects on the measured load and to minimize the inNuence of the contact surface inhomogeneities on the test results, both Guo et al. [1] and John and Shah [2; 3] inserted a rubber pad between the impactor and the specimens. Guo et al. [1] recorded and reported the entire displacement history for the specimen point under the Copyright ? 2001 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2001; 52:97–120
THREE-DIMENSIONAL COHESIVE MODELING
103
Figure 2. Geometry and dimensions of the beams tested by; (a) Guo et al. [1]; and (b) John and Shah [2; 3].
Figure 3. Prescribed displacements under the load point, based on data by by Guo et al. [1].
impactor (Figure 3). The load history, also reported in Reference [1], provides a convenient basis for the validation of numerical results. John and Shah’s tests are characterized by the strain rate, measured by a gauge located at the central cross-section of an unnotched beam, on the bottom side [2; 3; 55]. The strain rate is in turn related to the height from which the tup is dropped. After a transient eJect due to the rubber pad, the strain rate reaches an ostensibly constant value. In calculations, we prescribe a constant velocity at the point of impact chosen so as to match the reported ˙ Taking into account the strain rate, ˙N , imposing an uniform velocity at the impact point, . geometry of the beam, the two values are related by the formula L2 ˙ ˙ = 6D
(6)
where L and D are the beam span and depth. For the reported ˙N = 0:3= s, (6) gives ˙ = 0:03 m= s. Copyright ? 2001 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2001; 52:97–120
104
G. RUIZ, A. PANDOLFI AND M. ORTIZ
4. MATERIAL CHARACTERIZATION Table I summarizes the main properties of the concrete used in the experimental programmes of Guo et al. [1], and John and Shah [2; 3]. The only property measured by Guo et al. by means of an independent test was the compressive strength. The remaining material properties were identi6ed by Guo et al. using inverse numerical analysis techniques. The values of the parameters were calibrated on the basis of two-dimensional dynamic 6nite-element analyses carried out using a cohesive model of fracture in conjunction with a prescribed crack path [1]. Speci6cally, Guo et al. used the load-line displacement and the horizontal and vertical crack opening displacement records, corresponding to the dynamic drop-weight tests to calibrate a linear cohesive law. By way of contrast, the material properties supplied by John and Shah were measured experimentally, with the sole exception of the tensile strength. We have estimated the tensile strength following the recommendations set forth in the CEB-FIP Model Code [56]. The shear critical traction is not documented in the original works of Guo et al. [1] or John and Shah [2; 3]. The material is assumed to obey the linear irreversible cohesive law shown in Figure 1, with the tensile strength taken as the cohesive strength of the material. In addition, in the calculations of interest here, and for the materials under consideration, the bulk behaviour of concrete can be idealized as elastic to a good approximation, (cf., e.g. Reference [57; Section7:1:6]). Indeed, the specimen is dominated by tensile cracking and the compressive strength of the material is not reached. As a slight and probably inconsequential improvement, in our calculations we assume that the material obeys J2 -plasticity following the attainment of its compressive strength. Details of the speci6c implementation of J2 -plasticity employed in calculations may be found elsewhere [58]. It should be carefully noted that, under the conditions envisioned in this study, and more generally when dealing with the fracture of concrete, conventional concepts from linear– elastic fracture mechanics must be applied with great caution or not at all. Thus, computing the characteristic length from (4) using the properties listed in Table I, we obtain 400 and 100 mm for the materials of Guo et al. and John and Shah, respectively. This characteristic length is larger than the depth of the beams, 95.25 and 76:2 mm, respectively. The condition of the specimens simulated here corresponds therefore to a ‘fully yielded’ situation and linear– elastic fracture mechanics, which fundamentally rests on the small-scale yielding condition, does not apply. By way of contrast, cohesive theories of fracture are not so constrained, and can be applied to situations such as the ones envisioned here, in which the size of the process zone is comparable to the ligament length or any other limiting geometrical dimension. Further discussions of this and related issues may be found in References [21; 57; 59]. Table I. Concrete parameters. Guo et al. [1] Maximum aggregate size (mm) Elastic modulus (GPa) Compressive strength (MPa) Tensile strength (MPa) Fracture energy (N=m) Copyright ? 2001 John Wiley & Sons, Ltd.
6.4 32.3 33.8 3.1 120
John and Shah [2; 3] 9.5 29.0 39.7 3.0 31.1
Int. J. Numer. Meth. Engng 2001; 52:97–120
THREE-DIMENSIONAL COHESIVE MODELING
105
Figure 4. (a) Geometry of the double-edge notched fracture specimens used by Nooru-Mohamed and van Mier [63; 65]; and (b) schematic of the loading and boundary conditions.
The estimation of the tension-shear coupling parameter , Equation (1), which de6nes the ratio between the shear and the normal critical traction for the cohesive model is of particular concern for concrete and requires some care. We have already remarked that roughly reNects the ratio of mode-II to mode-I fracture toughness. There is experimental evidence [60–69] that suggests that the intrinsic fracture toughness of concrete, i.e. the critical stress intensity factor required to advance a semi-in6nite crack within its plane in the absence of kinking, is signi6cantly larger in pure mode-II than in pure mode-I owing to the interlocking of aggregate particles, which suggests a large value of . Indeed, Swartz et al. [60] calculated the ratio KIIc =KIc by analytical and numerical simulations of several mixed-mode tests. Their estimated values range between 3 and 6, depending on the type of specimen and on the underlying evaluation method. We provide next a quantitative estimation of the from the mixed-mode experiments performed by Nooru-Mohamed and van Mier [63; 65]. Nooru-Mohamed and van Mier tested double-edge notched specimens subject to mixed-mode load (Figure 4(a)). Three diJerent types of concrete were used, with maximum aggregate size equal to 2; 12 and 16 mm, respectively. The experimental set-up was designed to transmit independently the axial and the shear loads, following predetermined mixed-mode load paths, as sketched in Figure 4(b). We focus our attention on a particular load path, in which an increasing axial load was applied up to the point when a prescribed average crack width was obtained, followed by the complete unloading of the specimen (Figure 5(a)). An increasing shear load with displacement control was subsequently applied, keeping the axial load to zero and allowing for dilatancy of the crack, up to complete failure of the specimen (Figure 5(b)). The 6rst phase of the axial loading process generates a straight crack joining the two notches of the specimen. The material at the crack surfaces is therefore decohered and softened by tension, but its strength is not completely eliminated. During the application of the shear stress, the crack opens in pure mode-II, until the slip opening is too large and some dilatancy occurs (Figure 5(c)). Figure 6 shows the crack patterns obtained by this loading process for three diJerent values of the mode-I maximum crack opening ( = 50; 100 and 150 m) on 200 × 200 × 50 mm specimens of the 12-mm maximum aggregate size concrete. It should be noted that throughout the test the crack is subjected to pure-mode loading, 6rst in mode-I and later in mode-II. This load path has a straightforward interpretation within the context of the irreversible cohesive law shown in Figure 5(d). Thus, after the applied traction reaches the tensile strength Copyright ? 2001 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2001; 52:97–120
106
G. RUIZ, A. PANDOLFI AND M. ORTIZ
Figure 5. Load path followed to determine : (a) axial loading and unloading; followed by; (b) loading in shear up to failure; and (c) history of the displacements and s ; while (d) shows the load path deduced from the cohesive law.
Figure 6. Crack patterns corresponding to experiments by Nooru-Mohamed and van Mier [63; 65].
Copyright ? 2001 John Wiley & Sons, Ltd.
Figure 7. Upward view of some of the meshes used to simulate: (a) experiments of Guo et al.; and (b) John and Shah tests ( = 0:5). Int. J. Numer. Meth. Engng 2001; 52:97–120
THREE-DIMENSIONAL COHESIVE MODELING
107
of the specimen, a cohesive crack opens up to the prescribed opening displacement, thereby reducing the normal bearing capacity of the specimen to Pn . At the onset of unloading, the average normal traction is tn = Pn =A, A being the area of the unbroken ligaments. The specimen is then reloaded in shear up to a load Ps , corresponding to a shear traction ts = Ps =A, and brought to failure along the descending cohesive envelop. The axial load Pn at the maximum displacement and the shear peak load Ps correspond to the same point in the ‘eJective’ cohesive envelop, Figure 5(d). Thus, according to (3), the value of is =
Ps Pn
(7)
Applying (7) to the experimental results of Nooru-Mohamed and van Mier [63; 65], we obtain = 4; 6:6 and 6 for the 2; 12 and 16-mm maximum aggregate size concretes, respectively. These results provide an indication of the range of the tension-shear coupling parameter for concrete. These values are consistent with the aforementioned results obtained by Swartz et al. [60] for the ratio KIIc =KIc . Based on these considerations, in our simulations we set = 5. 5. COMPUTATIONAL MESH AND CONVERGENCE TESTS Figure 7(a) shows the mesh used in our simulations of Guo et al. experiments, whereas Figure 7(b) shows one of the meshes used in the simulations of John and Shah’s tests, corresponding to a relative oJset = 0:5. The boundary surfaces and the interior of the specimens are meshed automatically by an advancing front method [70] using 10-node quadratic tetrahedra. In order to cut down on the size of the calculations, the 6nite-element meshes are designed so as to be 6ne and nearly uniform on and in the vicinity of the expected crack paths. The element size equals half the maximum aggregate size in the 6ne mesh region and gradually coarsens away from that region up to the maximum aggregate size. It should be carefully noted that coarse mesh has a somewhat lower numerical toughness than the 6ne mesh [24] and, consequently, does not introduce a barrier for the propagation of the path. Thus, the presence of a 6ne mesh region and the mesh gradation used in the calculations does not bias the path of the crack. It bears emphasis that the maximum mesh size is 1=25 of the characteristic cohesive length (4) of the material and may, therefore, be expected to yield objective and mesh-size insensitive results [24]. In order to verify this point, we have compared solutions obtained from two diJerently re6ned meshes. The tests correspond to John and Shah’s specimens for = 0 and ˙N = 1= s. The results of the two tests are displayed in Figure 8. The reaction histories at the supports diJer negligibly up to the third peak. The waves in the curves are due to the oscillation of the beam and are also in keeping with the experiments. The diJerence in cohesive energy consumption in the two meshes is also very small, about 1 per cent. Figure 8 compares the front view of the crack advance for the two meshes at two diJerent stages of the analysis, 250 and 600 s after the impact of the tup. Also shown in the 6gure are level contours of damage, de6ned as the fraction of expended fracture energy to total fracture energy per unit surface, or critical energy release rate. The crack advance and the damage distribution comparison provide a further indication of the good resolution provided by the coarse mesh. Copyright ? 2001 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2001; 52:97–120
108
G. RUIZ, A. PANDOLFI AND M. ORTIZ
Figure 8. Crack pro6les and load history for meshes with diJerent element sizes (notched specimen).
In order to verify the accuracy of the coarse mesh in the presence of crack nucleation, we have carried out additional tests without a pre-notch. Figure 9 shows the numerical results for both the 6ne and coarse meshes. As in the pre-notch case, we again 6nd that the load curves match up to the peak load. However, the cohesive energy consumption is signi6cantly larger for the 6ne mesh. This discrepancy is also apparent in the level contours in damage, Copyright ? 2001 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2001; 52:97–120
THREE-DIMENSIONAL COHESIVE MODELING
109
Figure 9. Crack pro6les and load history for meshes with diJerent element sizes (unnotched specimen).
which reveal a broader microcracked zone for the 6ne mesh; and in the crack pro6le, which is sharp in the case of the coarse mesh and comparatively rougher and unsteady for the 6ne mesh. The larger extent of microcracking which accompanies nucleation in the 6ne mesh in turn accounts for the higher level of energy dissipation. Copyright ? 2001 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2001; 52:97–120
110
G. RUIZ, A. PANDOLFI AND M. ORTIZ
These observations provide an illustration of our incomplete understanding of issues of numerical convergence of cohesive 6nite-element models, specially in the presence of nucleation, branching and fragmentation. Thus, while it appears quite clear that 6nite-element solutions begin to exhibit strong convergence once the cohesive length is resolved (e.g. Reference [24]), the situation is far more uncertain for more complex fracture processes. It is not entirely clear at this time whether the increase of microcracking as the mesh is re6ned in the tests just discussed is a numerical artefact or reNects the actual physics of the crack nucleation process in concrete. Or, in this latter case, how and in what sense the 6nite-element solution converges in the limit of in6nite re6nement. For instance, numerical experiments [32] seem to indicate that, while the local details of the microcracking pattern are vastly non-unique, certain microscopic measures, such as energy dissipation, are quite reproducible and appear to converge properly under mesh re6nement. These issues clearly suggest themselves as worthwhile subjects for future research. 6. SIMULATION RESULTS Selected results of the calculations and comparisons with experimental data are presented in this section. 6.1. Crack pattern Figure 10 shows four snapshots of the simulation of one of John and Shah’s Charpy tests corresponding to a relative oJset = 0:5 [2; 3]. The snapshots record the evolution of the crack pattern. Speci6cally, the 6gure shows the intersection of the cohesive elements with the surfaces of the specimen. As may be seen from the 6gure, the dominant fracture mode for this geometry consists of the extension of the pre-notch towards the point of application of the load. This structural crack is accompanied by a certain amount of distributed microcracking and profuse branching, as expected in concrete. However, it should be carefully noted that some of the microcracks and crack branches are partially failed and should not be construed as completely formed free surfaces. Simultaneously with the growth of the main crack, a diJuse microcracking zone forms around the tensile 6ber of the central cross-section. There is also evidence of distributed damage at the point of contact with the impactor. These distributed damage zones act as additional energy sinks. The tensile microcrack zone remains diJuse throughout the simulation and fails to develop into a structural crack. Thus, Figure 10 suggests that two failure modes are in competition: the extension of the pre-notch and the nucleation of a new crack within the loading plane. For the particular geometry shown in Figure 10, the pre-notch extension mode clearly wins out at the expense of the nucleation mode. Figure 11 compares the experimentally and predicted crack paths in two cases: the 6rst for the geometry of Guo et al., Figure 11(a), and the second for John and Shah, Figure 11(b). The results of the calculations are shown as level contours of cohesive damage. Superimposed on the 6gures are the experimental observed bounds of the crack trajectories (thick solid line). In Figure 11(b), the straight line emanating from the notch tip is a linear–elastic fracture mechanics prediction for the crack-growth direction reported by John and Shah [2; 3]. As is evident from these 6gures, the predicted crack paths are in excellent agreement with the experimentally observed crack trajectories. Copyright ? 2001 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2001; 52:97–120
THREE-DIMENSIONAL COHESIVE MODELING
111
Figure 10. Snapshots of the simulation of John and Shah’s Charpy tests on concrete specimens [2; 3], showing the evolution of the crack pattern.
Figure 11. Computed damage distribution (shaded mesh) compared to the experimental crack pattern (thick solid lines) by: (a) Guo et al. Hawkins [1]; and (b) John and Shah [2; 3]. Copyright ? 2001 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2001; 52:97–120
112
G. RUIZ, A. PANDOLFI AND M. ORTIZ
6.2. In1uence of the notch o.set on the crack pattern and failure mode As already mentioned, the TPB specimens tested by John and Shah had a notch oJset from the center of the beam [2; 3]. The notch oJset was primarily intended as a means of varying the mode-II=mode-I ratio. Thus, for instance, an increase in the notch oJset was observed to result in a corresponding increase in the kinking angle for crack-growth initiation, which is indicative of a higher mode mixity. Interestingly, as already mentioned at high oJsets John and Shah observed a change in failure mode from pre-notch extension to crack nucleation within the plane of the load. This exchange of stability was observed to occur at a critical oJset of roughly 0.7. Figure 12 displays computed crack patterns for relative oJsets = 0; 0:5; 0:6 and 1, respectively, and thus illustrates the dependence of the predicted failure mode on notch oJset. The calculations are in good qualitative agreement with the experimental observations of John and Shah [2; 3], although the critical oJset for the exchange of stability is somewhat underpredicted at about 0.6. However, it should be noted that the critical oJset depends sensitively on the tensionshear coupling parameter . Thus, a high (low) value of describes a material which is stronger in shear (compression) than in compression (shear). Correspondingly, a high value of inhibits the development of mode-II cracks and thus promotes the nucleation of a mode-I crack within the central cross section. Contrariwise, a low value of favours the extension of the notch. This dependence is shown in Figure 13, which collects the crack patterns obtained in simulations of John and Shah’s test [2; 3] with a 6xed relative oJset = 0:5 and for values of = 0:1; 1; 5 and 10. In particular, a transition in failure mode from pre-notch extension to crack nucleation is observed between values of = 5 and 10. It follows from this analysis that the critical oJset value = 0:7 observed by John and Shah may be matched exactly by suitably increasing the value of above the value = 5 used in calculations, but this enhancement will not be pursued here.
Figure 12. Numerical simulations of the three-point-bend experiments of John and Shah [2; 3]. EJect of the pre-notch oJset on crack pattern and failure mode. Copyright ? 2001 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2001; 52:97–120
THREE-DIMENSIONAL COHESIVE MODELING
113
Figure 13. Numerical simulations of the three-point-bend experiments of John and Shah [2; 3]. EJect of the tension-shear coupling constant on crack pattern failure mode.
6.3. Displacements 2eld around the crack Guo et al. [1] used a MoirGe diJraction grating in order to measure the specimen surface displacement 6eld around the crack tip. These measurements provide a wealth of data regarding the strain and stress 6elds over the crack-tip area. Figure 14 shows a typical sequence of MoirGe patterns associated with horizontal and vertical displacements in the vicinity of the main crack [1]. Regrettably, in the original paper the time elapsed between two consecutive snapshots and the displacement increment between contours is not reported. This limitation notwithstanding, the measured MoirGe interferometry patterns permit a useful qualitative comparison with the numerical predictions. Figure 15 shows synthetic MoirGe interferometry patterns constructed from the results of the calculations. We choose a time delay between consecutive images of 100 s. Solid (dashed) lines denote positive (negative) displacements. A careful comparison between Figure 14 and Figures 15(a) and 15(b) reveals an excellent overall agreement between simulation and observation. This general agreement notwithstanding, it is interesting to note that the measured patterns are smoother than the computed ones in the immediate vicinity of the crack tip and show no signs of microcracking. Following Post [53], a plausible explanation is that the MoirGe grating is strong enough to remain intact away from the main crack and thus covers the underlying microcracks or inhibits their formation altogether. It is also interesting to note in Figure 15(c), which shows the level contours of normal out-of-plane displacement on the surface of the specimen, that the heavily microcracked process zone at the tip of the crack bulges out of the specimen. This illustrates the strong dilatancy which the model predicts to occur simultaneously with intense distributed damage. 6.4. Crack extension and velocity The sequence of MoirGe patterns obtained by Guo et al. in their experiments allowed them to determine the main crack-tip trajectory, and by numerical diJerentiation, the crack-tip velocity Copyright ? 2001 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2001; 52:97–120
114
G. RUIZ, A. PANDOLFI AND M. ORTIZ
Figure 14. Sequence of MoirGe interferometry patterns of: (a) horizontal; and (b) vertical displacements [1].
history, Figure 16. The TPB specimens were also instrumented with 6ve strain gauges located over the expected path of the crack on the side of the specimen not covered by the MoirGe grating. The agreement between both sets of data attests to the accuracy of the measurements. Figure 16 also superposes the predicted crack-length and crack-tip velocity histories on the experimental data. The onset of crack growth coincides with the arrival of tensile waves to the pre-notch tip roughly 320 s after impact. The crack quickly settles down to a nearly constant velocity of approximately 170 m= s during a time interval of 200 s. At the end of this interval, the crack slows down and arrests owing to the arrival of relief waves and the interaction between the crack tip and the damaged zone under the impactor. The predicted crack-growth initiation time is consistent with observation. The agreement between the calculated and measured crack trajectories is excellent. The computed crack-tip velocity history exhibits two crack-speed peaks separated by a somewhat slower regime. This crack slow-down occurs simultaneously with a burst in microcrack activity around the crack tip. 6.5. Load history curves Figure 17 compares the experimental and computed load histories corresponding to tests of Guo et al. [1]. It should be carefully noted that the numerical model assumes a mathematically sharp and completely formed pre-notch, whereas in the experiments the pre-notch was introduced by stable symmetric loading. This procedure is likely to result in a pre-crack which is bridged by intact ligaments and, consequently, has some residual strength. Owing to this Copyright ? 2001 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2001; 52:97–120
THREE-DIMENSIONAL COHESIVE MODELING
115
Figure 15. Sequence of computed contours of (a) horizontal displacement; (b) vertical displacement; and (c) out-of-plane displacement.
diJerence in the initial con6guration of the specimen, the numerical and measured load histories are not comparable during the early stages of the test, and are shown as dashed lines in Figure 17. Beyond this point, the experimental and numerical load histories exhibit two peak loads, Figure 17. The time of occurrence and the intensity of the 6rst peak is well predicted by the model. The subsequent agreement between the experimental and calculated load histories is poor. In this regard it should be noted that the 6rst peak occurs when the amount of Copyright ? 2001 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2001; 52:97–120
116
G. RUIZ, A. PANDOLFI AND M. ORTIZ
Figure 16. (a) Crack length history measured by Guo et al. using both strain gauges and MoirGe interferometry, compared to numerical predictions; (b) measured and computed crack tip velocities.
Figure 17. Comparison between measured and computed load history. Data by Guo et al. [1].
crack extension is small, and the structure of the process zone is largely determined by the cohesive strength of the material. By contrast, the second peak occurs after substantial crack growth has taken place, and the entire cohesive law has come into play. It is thus likely that the simple linear cohesive envelop assumed in the calculations is not suMcient to match the experimental observations accurately, and that a better agreement requires the use of more elaborate cohesive laws. 7. SUMMARY AND CONCLUSIONS We have taken a cohesive formulation of fracture as a basis for the simulation of processes of combined tension-shear damage and mixed-mode fracture in solids subjected to dynamic Copyright ? 2001 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2001; 52:97–120
THREE-DIMENSIONAL COHESIVE MODELING
117
loading. The cohesive model accounts for key fracture properties such as the cohesive strength and the fracture energy, and, in conjunction with inertia, introduces characteristic length and time scales into the material description. The particular model proposed by Camacho and Ortiz [24], and subsequent extensions thereof [29–32], also accounts for tension-shear coupling and permits according the shear and tensile strengths independent values bearing arbitrary ratios. The cohesive law is assumed to be rate-insensitive and, therefore, all rate eJects predicted by the theory are a consequence of the interplay between inertia and fracture [24; 31; 32]. This is in contrast to most past simulations of the dynamic fracture of concrete, which directly model the tensile strength as an increasing function of strain rate [1; 38–47]. In calculations, the fracture surface is con6ned to inter-element boundaries and, consequently, the structural cracks predicted by the analysis are necessarily rough. However, in simulations of concrete this numerical roughness can be made to correspond to the physical roughness by the simple device of choosing the element size to be comparable to the aggregate size. This choice of element size also serves to resolve the cohesive zone size, which in concrete is a small multiple of the aggregate size. The resulting simulations thus have a multiscale character, in as much as the discretization is charged with resolving both a micromechanical scale—the cohesive lengthscale—and the structural dimension. We have validated the predictions of the model against the experimental data of Guo et al. [1], and John and Shah [2; 3]. These tests concerned three-point-bend beam (TPB) specimens containing a pre-crack shifted from the central cross section and subject to dynamic loading. The presence of a pre-notch in the specimen leads to asymmetric loading conditions and results in mixed-mode fracture. Our calculations show that the cohesive model is indeed capable of closely capturing the observed fracture patterns, crack trajectories and crack-tip displacement 6elds corresponding to diJerent experimental con6gurations. The simulations also return accurate histories of crack extension and velocity. The relative ease with which cohesive theories of fracture are capable of describing complex crack patterns, often involving profuse microcracking, and crack trajectories is particularly noteworthy. This ease is in contrast with the complexities inherent to the tracking mathematically sharp three-dimensional cracks by conventional linear–elastic fracture mechanics criteria [71–74]. The classical fracture mechanics paradigm becomes intractable in situations involving complex patterns of interacting cracks, e.g. in zones of distributed microcracking or in the face of fragmentation processes; and, for all practical purposes, it plainly ceases to apply when such complex fracture patterns occur in combination with inelasticity, 6nite deformations, dynamics, free surfaces and interfaces, and other similar complicating circumstances. Cohesive theories also facilitate the treatment of crack nucleation and coalescence. Our calculations demonstrate the ability of the theory to predict the formation of diJuse microcracking zones and the nucleation of structural cracks, e.g. by microcrack coalescence. These fracture mechanisms are clearly observed in the experiments of John and Shah [2; 3]. The competition between crack growth and crack nucleation, and the exchange of stability between the two failure modes at a critical pre-notch oJset, are well predicted by the theory. Finally, we conclude by suggesting several subjects for further research. Thus, while in our calculations the computed load histories agree closely with experiment up to the 6rst peak, the subsequent agreement is amenable to improvement. This most likely requires the formulation of more elaborate cohesive laws, possibly containing a larger material parameter set. In addition, while our numerical tests show strong convergence of the 6nite element Copyright ? 2001 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2001; 52:97–120
118
G. RUIZ, A. PANDOLFI AND M. ORTIZ
solutions in the presence of a sharp pre-notch, the convergence properties of the solutions when crack nucleation is involved remain to be fully ascertained. ACKNOWLEDGEMENTS
The support of Caltech’s ASCI=ASAP Center for the Simulation of the Dynamic Properties of Solids is gratefully acknowledged. Gonzalo Ruiz gratefully acknowledges the 6nancial support for his stay at the California Institute of Technology provided by the Direcci4on General de Ense5nanza Superior, Ministerio de Educaci4on y Cultura, Spain. We are indebted to Santiago Lombeyda, Research Scientist at the Center for Advanced Computing Research, Caltech, for rendering the simulation results. REFERENCES 1. Guo ZK, Kobayashi AS, Hawkins NM. Dynamic mixed mode fracture of concrete. International Journal of Solids and Structures 1995; 32(17=18):2591–2607. 2. John R. Mixed mode fracture of concrete subjected to impact loading. PhD Thesis, Northwestern University, 1988. 3. John R, Shah SP. Mixed-mode fracture of concrete subjected to impact loading. Journal of Structural Engineering 1990; 116(3):585 – 602. 4. Ravichandar K, Knauss WG. An experimental investigation into dynamic fracture, Part 1. Crack initiation and arrest. International Journal of Fracture 1984; 25(4):247–262. 5. Liu C, Knauss WG, Rosakis AJ. Loading rates and the dynamic initiation toughness in brittle solids. International Journal of Fracture 1998; 90(1–2):103–118. 6. Dugdale DS. Yielding of steel sheets containing slits. Journal of the Mechanics and Physics of Solids 1960; 8:100 –104. 7. Barrenblatt GI. The mathematical theory of equilibrium of cracks in brittle fracture. Advances in Applied Mechanics 1962; 7:55 –129. 8. Rice JR. Mathematical analysis in the mechanics of fracture. In Fracture, Liebowitz H (ed.). Academic Press: New York, 1968; 191–311. 9. Hillerborg A, Modeer M, Petersson PE. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and 6nite elements. Cement Concrete Research 1976; 6:773 – 782. 10. Rose JH, Ferrante J, Smith JR. Universal binding energy curves for metals and bimetallic interfaces. Physical Review Letters 1981; 47(9):675 – 678. 11. Carpinteri A. Mechanical Damage and Crack Growth in Concrete. Martinus NijhoJ: Dordrecht, The Netherlands, 1986. 12. Needleman A. A continuum model for void nucleation by inclusion debonding. Journal of Applied Mechanics 1987; 54:525 –531. 13. Ortiz M. Microcrack coalescence and macroscopic crack growth initiation in brittle solids. International Journal of Solids and Structures 1988; 24:231–250. 14. Willam K. Simulation issues of distributed and localized failure computations. In Cracking and Damage, Mazars J, Bazant ZP (eds). Elsevier Science: New York, 1989; 363–378. 15. Needleman A. An analysis of decohesion along an imperfect interface. International Journal of Fracture 1990; 42:21– 40. 16. Needleman A. Micromechanical modeling of interfacial decohesion. Ultramicroscopy 1992; 40:203 – 214. 17. Xu XP, Needleman A. Void nucleation by inclusion debonding in a crystal matrix. Modelling and Simulation in Materials Science and Engineering 1993; 1:111–132. 18. Xu X.P, Needleman A. Numerical simulations of fast crack growth in brittle solids. Journal of the Mechanics and Physics of Solids 1994; 42:1397–1434. 19. Tvergaard V, Hutchinson JW. The inNuence of plasticity on mixed-mode interface toughness. Journal of the Mechanics and Physics of Solids 1993; 41:1119 –1135. 20. Ortiz M, Suresh S. Statistical properties of residual stresses and intergranular fracture in ceramic materials. Journal of Applied Mechanics 1993; 60:77 – 84. 21. Planas J, Elices M, Guinea GV. Cohesive cracks as a solution of a class of nonlocal problems. In Fracture and Damage in Quasibrittle Structures. Experiment, Modelling and Computer Analysis, Bazant ZP (ed.). E & FN SPON, London 1994. 22. Xu XP, Needleman A. Numerical simulations of dynamic interfacial crack growth allowing for crack growth away from the bond line. International Journal of Fracture 1995; 74:253 – 275. 23. Ortiz M. Computational micromechanics. Computational Mechanics 1996; 18:321–338. Copyright ? 2001 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2001; 52:97–120
THREE-DIMENSIONAL COHESIVE MODELING
119
24. Camacho GT, Ortiz M. Computational modelling of impact damage in brittle materials. International Journal of Solids and Structures 1996; 33(20 – 22):2899–2938. 25. Tvergaard V, Hutchinson JW. EJect of strain dependent cohesive zone model on predictions of interface crack growth. Journal de Physique IV 1996; 6:165 – 172. 26. Tvergaard V, Hutchinson JW. EJect of strain dependent cohesive zone model on predictions of crack growth resistance. International Journal of Solids and Structures 1996; 33:3297–3308. 27. Xu XP, Needleman A. Numerical simulations of dynamic crack growth along an interface. International Journal of Fracture 1996; 74:289 – 324. 28. De-AndrGes A, PGerez JL, Ortiz M. Elastoplastic 6nite element analysis of three-dimensional fatigue crack growth in aluminum shafts subjected to axial loading. International Journal of Solids and Structures 1999; 36(15):2231–2258. 29. Ortiz M, Pandol6 A. A class of cohesive elements for the simulation of three-dimensional crack propagation. International Journal for Numerical Methods in Engineering 1999; 44:1267–1282. 30. Pandol6 A, Ortiz M. Solid modeling aspects of three-dimensional fragmentation. Engineering with Computers 1998; 14(4):287–308. 31. Pandol6 A, Krysl P, Ortiz M. Finite element simulation of ring expansion and fragmentation. International Journal of Fracture 1999; 95:279 – 297. 32. Ruiz G, Pandol6 A, Ortiz M. Three-dimensional 6nite-element simulation of the dynamic brazilian tests on concrete cylinders. International Journal for Numerical Methods in Engineering 2000; 48(7):963 – 994. 33. GGalvez JC, CendGon D, Planas J, Guinea GV, Elices M. Fracture of concrete under mixed loading. Experimental results and numerical prediction. Fracture Mechanics of Concrete Structures, Proceedings FRAMCOS-3. AEDIFICATIO Publishers: Freiburg, Germany, 1998. 34. GGalvez JC, Elices M, Guinea GV, Planas J. Mixed mode fracture of concrete under proportional and nonproportional loading. International Journal of Fracture 1998; 94:267–284. 35. CendGon DA, GGalvez JC, Elices M, Planas J. Modelling the fracture of concrete under mixed loading. International Journal of Fracture 2000; 103(3):1–18. 36. Repetto EA, Radovitzky R, Ortiz M. Finite element simulation of dynamic fracture and fragmentation of glass rods. Computer Methods in Applied Mechanics and Engineering 2000; 183(1–2):3 –14. 37. Pandol6 A, Guduru PR, Ortiz M, Rosakis AJ. Finite element analysis of experiments of dynamic fracture in c300 steel. International Journal of Solids and Structures 2000; 37:3733 – 3760. 38. Yon JH, Hawkins NM, Kobayashi AS. Strain-rate sensitivity of concrete mechanical properties. ACI Materials Journal 1992; 89(2):146 – 153. 39. Yu C-T, Kobayashi AS, Hawkins NM. Energy-dissipation mechanisms associated with rapid fracture of concrete. Experimental Mechanics 1993; 33(3):205 – 211. 40. Du J, Yon JH, Hawkins NM, Arakawa K, Kobayashi AS. Fracture process zone for concrete for dynamic loading. ACI Materials Journal 1992; 89(3):252 – 258. 41. Ross A, Tedesco JW, Kuennen ST. EJects of strain rate on concrete strength. ACI Materials Journal 1995; 92:37– 47. 42. Hughes ML, Tedesco JW, Ross A. Numerical analysis of high strain rate splitting-tensile tests. Computers and Structures 1993; 47:653– 671. 43. Reinhardt HW, Weerheijm J. Tensile fracture of concrete at high loading rates taking account of inertia and crack velocity eJects. International Journal of Fracture 1991; 51:31– 42. 44. Tedesco JW, Ross CA, McGill PB, O’Neil BP. Numerical analysis of high strain rate concrete direct tension tests. Computers and Structures 1991; 40(2):313 – 327. 45. Tedesco JW, Ross CA, Kuennen ST. Experimental and numerical-analysis of high-strain rate splitting tensile tests. ACI Materials Journal 1993; 90(2):162–169. 46. Yon JH, Hawkins NM, Kobayashi AS. Numerical simulation of mode-I dynamic fracture concrete. Journal of Engineering Mechanics, ASCE 1991; 117(7):1595–1610. 47. Yon JH, Hawkins NM, Kobayashi AS. Fracture process zone in dynamically loaded crack-line wedge-loaded, double-cantilever beam concrete specimens. ACI Materials Journal 1991; 88(5):470 – 479. 48. Chen WN, Ravichandran G. Dynamic compressive behavior of ceramics under lateral con6nement. Journal de Physique IV 1994; 4:177–182. 49. Chen WN, Ravichandran G. Static and dynamic compressive behavior of aluminum nitride under moderate con6nement. Journal of the American Ceramic Society 1996; 79:579 – 584. 50. Dadkhah MS, Wang FX, Kobayashi AS. Simultaneous on-line measurement of orthogonal displacement 6elds by MoirGe interferometry. Experimental Technique 1998; 12:28 – 30. 51. Wang FX, Kobayashi AS. High-density MoirGe interferometry. Optical Engineering 1990; 29(1):38 – 41. 52. Arakawa K, Drinnon RH, Kosai M, Kobayashi AS. Dynamic fracture analysis by MoirGe interferometry. Experimental Mechanics 1991; 31(4):306 – 309. 53. Post D. Developments in MoirGe interferometry. Optical Engineering 1982; 21(3):458 – 467. 54. Epstein JS (guest editor). Special issue on MoirGe interferometry. Optics and Lasers in Engineering 1990; 12(2–3). Copyright ? 2001 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2001; 52:97–120
120
G. RUIZ, A. PANDOLFI AND M. ORTIZ
55. Gopalaratnam VS, Shah SP, John R. A modi6ed instrumented Charpy-test for cement-based composites. Experimental Mechanics 1984; 24(2):102–111. 56. CEB-FIP Model Code 1990. Final Draft, Bulletin D’Information N.203, 204 and 205, EFP Lausanne. 57. BaWzant ZP, Planas J. Fracture and Size E.ect in Concrete and Other Quasibrittle Materials. CRC Press: Boca Raton, FL, 1998. 58. Cuiti˜no A.M, Ortiz M. A material-independent method for extending stress update algorithms from small-strain plasticity to 6nite plasticity with multiplicative kinematics. Engineering Computations 1992; 9:437– 451. 59. Planas J, Elices M. Nonlinear fracture of cohesive materials. International Journal of Fracture 1991; 3:139–157. 60. Swartz SE, Lu LW, Tang LD, Refai TM. Mode II fracture parameter estimates for concrete from beam specimens. Experimental Mechanics 1988; 28(2):146 –153. 61. Swartz SE, Taha NM. Crack-propagation and fracture of plain concrete beams subjected to shear and compression. ACI Structural Journal 1991; 88(2):169 – 177. 62. van Mier JGM, Nooru-Mohamed MB. Geometrical and structural aspects of concrete fracture. Engineering Fracture Mechanics 1990; 35(4 – 5):617– 628. 63. van Mier JGM, Nooru-Mohamed MB, Timmers G. An experimental study of shear fracture and aggregate interlock in cement based composites. Heron, 1991; 36(4). 64. Hassanzadeh M. Behaviour of fracture process zones in concrete inNuenced by simultaneously applied normal and shear displacements. PhD Thesis, Lund University of Technology, 1992. 65. Nooru-Mohamed MB. Mixed-mode fracture of concrete: an experimental approach. PhD Thesis, Delft University of Technology, 1992. 66. Nooru-Mohamed NB, Schlangen E, van Mier JGM. Experimental and numerical study on the behavior of concrete subjected to biaxial tension and shear. Advanced Cement Based Materials 1993; 1(1):22–37. 67. Jia Z, Castro-Montero A, Shah SP. Observation of mixed mode fracture with center notched disk specimens. Concrete and Cement Research 1996; 26(1):125 – 137. 68. Subrumaniam KV, Popovics JS, Shah SP. Testing concrete in torsion: Instability analysis and experiments. Journal of Engineering Mechanics, ASCE 1998; 124(11):1258 – 1268. 69. Cedolin L, Bisi G, Nardello P. Mode II fracture resistance of concrete. Concrete Science and Engineering 1999; 1:37 – 44. 70. Radovitzky R, Ortiz M. Tetrahedral mesh generation based on node insertion in crystal lattice arrangements and advancing-front-Delaunay triangulation. Computer Methods in Applied Mechanics and Engineering 2000; 187(3 – 4):543–569. 71. Xu G, Bower AF, Ortiz M. The inNuence of crack trapping on the toughness of 6ber reinforced composites. Journal of the Mechanics and Physics of Solids 1998; 46(10):1815 –1833. 72. Xu G, Bower A, Ortiz M. An analysis of non-planar crack growth under mixed mode loading. International Journal of Solids and Structures 1994; 31:2167–2193. 73. Xu G, Ortiz M. A variational boundary integral method for the analysis of 3-D cracks of arbitrary geometry modeled as continuous distributions of dislocation loops. International Journal for Numerical Methods in Engineering 1993; 36:3675 – 3701. 74. Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P. Meshless methods: an overview and recent developments. Computer Methods in Applied Mechanics and Engineering 1996; 139:3 – 47.
Copyright ? 2001 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2001; 52:97–120
