CALTECH ASCI TECHNICAL REPORT 093 caltechASCI/2000.093
Three Dimensional Finite-Element Simulation of the Dynamic Brazilian Tests on Concrete cylinders G. Ruiz, M. Ortiz, A. Pandolfi
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2000; 48:963–994
Three-dimensional ÿnite-element simulation of the dynamic Brazilian tests on concrete cylinders Gonzalo Ruiz 1 , Michael Ortiz2;∗;† and Anna Pandolÿ 3 2
1 Departamento de Ciencia de Materiales; Universidad Polità ecnica de Madrid; 28040 Madrid; Spain Graduate Aeronautical Laboratories; California Institute of Technology; Pasadena; CA 91125; U.S.A. 3 Dipartimento di Ingegneria Strutturale; Politecnico di Milano; 20133 Milano; Italy
SUMMARY We investigate the feasibility of using cohesive theories of fracture, in conjunction with the direct simulation of fracture and fragmentation, in order to describe processes of tensile damage and compressive crushing in concrete specimens subjected to dynamic loading. We account explicitly for microcracking, the development of macroscopic cracks and inertia, and the e ective dynamic behaviour of the material is predicted as an outcome of the calculations. The cohesive properties of the material are assumed to be rate-independent and are therefore determined by static properties such as the static tensile strength. The ability of model to predict the dynamic behaviour of concrete may be traced to the fact that cohesive theories endow the material with an intrinsic time scale. The particular conÿguration contemplated in this study is the Brazilian cylinder test performed in a Hopkinson bar. Our simulations capture closely the experimentally observed rate sensitivity of the dynamic strength of concrete in the form of a nearly linear increase in dynamic strength with strain rate. More generally, our simulations give accurate transmitted loads over a range of strain rates, which attests to the ÿdelity of the model where rate e ects are concerned. The model also predicts key features of the fracture pattern such as the primary lens-shaped cracks parallel to the load plane, as well as the secondary profuse cracking near the supports. The primary cracks are predicted to be nucleated at the centre of the circular bases of the cylinder and to subsequently propagate towards the interior, in accordance with experimental observations. The primary and secondary cracks are responsible for two peaks in the load history, also in keeping with experiment. The results of the simulations also exhibit a size e ect. These results validate the theory as it bears on mixed-mode fracture and fragmentation processes in concrete over a range of strain rates. Copyright ? 2000 John Wiley & Sons, Ltd. KEY WORDS:
∗ †
concrete; fracture; cohesive elements; dynamic strength; mixed mode fracture; size e ect; strain rate e ect
Correspondence to: Michael Ortiz, Graduate Aeronautical Laboratories, California Institute of Technology, Firestone Flight Sciences Laboratory, Pasadena, CA 91125, U.S.A. E-Mail:
[email protected]
Contract=grant sponsor: DirecciÃon General de Ense˜nanza Superior, Ministerio de EducaciÃon y Cultura, Spain Contract=grant sponsor: Army Research Oce; contract=grant number: DAAH04-96-1-0056
Copyright ? 2000 John Wiley & Sons, Ltd.
Received 29 January 1999 Revised 15 September 1999
964
G. RUIZ, M. ORTIZ AND A. PANDOLFI
1. INTRODUCTION The dynamic behaviour of brittle materials, including glasses, ceramics, rocks and concrete, subjected to high strain rates often involves the development of complex fracture and fragmentation patterns. Cracks may nucleate from pre-existing aws or defects which populate the material in large numbers, or at stress concentrations induced by heterogeneities present in the material on the microscale. Once nucleated, fracture may proceed in a distributed fashion, as microcracking, or in the form of a few dominant macrostructural cracks. The paths followed by such cracks are often tortuous and undergo frequent branching, specially under dynamic 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 fracture processes often occur on a scale comparable to the geometrical dimensions of the structure. In addition to the energy required to fracture the material, other sources of dissipation often operate simultaneously, including plastic working, viscosity and heat conduction. Finally, microinertia often plays a signi cant role in shaping the e ective macroscopic behaviour of solids at high rates of deformation (e.g. References [1; 2]). Several avenues have been traditionally followed in order to model this complex behaviour. One avenue is to attempt to bury all dissipative mechanisms into the constitutive relations. However, in practice the e ective behaviour of systems with complex microsctructures can only be characterized analytically, if at all, by recourse to sweeping simplifying assumptions. One particularly vexing diculty inherent to constitutive descriptions concerns their inability to endow brittle materials with a well-de ned fracture energy, i.e. a material-characteristic measure of energy dissipation per unit area of crack surface. Thus, the conventional thermodynamic route to the formulation of general constitutive relations regards dissipation as an extensive quantity possessing a well-de ned density per unit volume. Attempts to build fracture into such formulations by means of a softening stress–strain law leads to pathologies such as a spurious scaling of the e ective fracture energy with the discretization size. The formulation of appropriate functional forms of the energy of a solid which allow for both bulk and fracture-like behaviour is a challenging mathematical problem which is presently the subject of active research (e.g. References [3; 4]). Fracture mechanics speci cally addresses the issue of whether a body under load will remain intact or new free surface will form. However, the classical theory of fracture mechanics is predicated around the assumption of a pre-existing dominant crack. Under these and other restrictive assumptions, fracture mechanics successfully predicts when a crack will grow, in what direction and how fast, and how the results of laboratory experiments can be scaled up to structural dimensions. However, the reliance on a pre-existing dominant crack entirely forgoes the issue of nucleation, which is for the most part foreign to classical fracture mechanics. Situations, such as this arise in fragmentation or crushing, involving many intersecting cracks also fall outside the purview of classical fracture mechanics. Additionally, the conditions for crack growth are typically expressed in terms of parameters characterizing the amplitude of autonomous near-tip elds. The applicability of these criteria is therefore contingent upon the existence and establishment of such near-tip elds. This in turn severely restricts the scope of the theory, e.g. by requiring that the plastic or process zone be small relative to any limiting geometrical dimension of the solid such as the ligament size, conditions which are rarely realized in materials such as concrete. In addition, the structure of the autonomous near-tip elds depends sensitively on the constitutive behaviour of the material, which inextricably ties the fracture criteria to the constitution of the material. In dynamic fracture, the fracture criterion is additionally responsible for accounting for the microinertia which accompanies the Copyright ? 2000 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2000; 48:963–994
THREE-DIMENSIONAL FINITE-ELEMENT SIMULATION
965
motion of the crack tip. Another complicating factor is non-proportional and cyclic loading, such as arises in high-cycle fatigue, which renders important near-tip parameters, such as the J -integral, inapplicable. An alternative approach to fracture is based on the use of cohesive models to describe processes of separation leading to the formation of new free surface [5–30]. Cohesive theories of fracture address many of the same issues contemplated by classical fracture mechanics while e ectively overcoming most of the diculties alluded to above. Indeed, cohesive models furnish a complete theory of fracture, and permit the incorporation into the material description of bona de fracture parameters such as a fracture energy and a spall strength. By focusing speci cally on the separation process, a sharp distinction is drawn in cohesive theories between fracture, which is described by recourse to cohesive laws, and bulk material behaviour, which is described through an independent set of constitutive relations. The use of cohesive models is therefore not limited by any consideration of material behaviour, nite kinematics, non-proportional loading, dynamics, or the geometry of the specimen. Another appealing aspect of cohesive laws as models of fracture is that they t naturally within the conventional framework of nite element analysis. However, one important numerical requirement pertaining to the use of cohesive theories is that the cohesive zones must be adequately resolved by the discretization in order to obtain mesh-size-independent results [23]. In materials for which the characteristic cohesive length is small, this may necessitate substantial mesh re nement near crack tips. In materials such as concrete, by contrast, the characteristic cohesive length is comparatively large, typically of the order of 10 cm, and the mesh-size requirements are often less severe. The feasibility of using cohesive theories of fracture for the direct simulation of fragmentation processes in brittle materials subjected to impact was established by Camacho and Ortiz [23]. In this approach, individual cracks are tracked as they nucleate, propagate, branch and possibly link up to form fragments. Clearly, it is incumbent upon the mesh to provide a rich enough set of possible fracture paths, an issue which may be addressed within the framework of adaptive meshing. The ensuing granular ow of the comminuted material is also simulated explicitly. In this manner, the daunting task of developing constitutive relations which account for crushing and fragmentation is avoided altogether. The delity of cohesive elements in applications involving dynamic fracture of ductile materials has recently been investigated by Pandol et al. [30], who have simulated the fragmentation of expanding aluminium rings. The numerical simulations have been found to be highly predictive of a number of observed features, including: the number of dominant and arrested necks; the fragmentation patterns; the dependence of the number of fragments and the fracture strain on the expansion speed; and the distribution of fragment sizes at xed expansion speed. Pandol et al. [31] have additionally shown that simulations of dynamic crack propagation based on cohesive models are highly predictive of a number of observed features, including: the crack growth initiation time; the trajectory of the propagating crack tip; and the formation of shear lips near the lateral surfaces. Repetto et al. [32] have applied cohesive models to the simulation of failure waves in glass rods subjected to impact. Their calculations correctly capture the development and propagation of a sharp failure wave, its propagation speed and the bursting of the comminuted material following the passage of the failure wave. The simulations of dynamic fracture of concrete available in the literature commonly model the tensile strength as an increasing function of strain rate [33–43]. In addition, the fracture energy is often presumed to be constant and independent of strain rate [33; 44]. These rate-dependency Copyright ? 2000 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2000; 48:963–994
966
G. RUIZ, M. ORTIZ AND A. PANDOLFI
laws are necessarily empirical and endeavor to model the e ective or macroscopic behaviour of concrete under dynamic loading. In this paper we investigate the feasibility of using cohesive theories of fracture, in conjunction with the direct simulation of fracture and fragmentation, in order to describe processes of tensile damage and compressive crushing in concrete specimens subjected to dynamic loading. We account explicitly for microcracking, the development of macroscopic cracks and inertia, and the e ective dynamic behaviour of the material is predicted as an outcome of the calculations. Indeed, the cohesive properties of the material are assumed to be rate-independent, and are therefore determined by static properties such as the static tensile strength. The ability of model to predict key aspects of the dynamic behaviour of concrete, such as the strain-rate sensitivity of strength may be traced to the fact that cohesive theories, in addition to building a characteristic length into the material description, they endow the material with an intrinsic time scale as well [23]. This intrinsic time scale permits the material to discriminate between slow and fast loading rates and ultimately allows for the accurate prediction of the dynamic strength of the material as a function of strain rate and other rate e ects. The particular con guration contemplated in this study is the Brazilian cylinder test performed in a Hopkinson bar. Dynamic Brazilian tests performed using a split-Hopkinson pressure bar (SHPB) have been proposed as a convenient means of determining the tensile strength of cohesive materials [41; 37; 36; 45–47]. These tests have yielded a wealth of information on the dynamic behaviour of concrete. For instance, it is well-established that the e ective tensile strength of the material increases with strain rate [41; 37; 36; 33]. By contrast, the fracture energy is ostensible insensitive to strain rate [33; 44]. Detailed measurements of the near-tip stress elds attendant to cracks propagating in mode I and mixed mode [35] have shown that the length of the cohesive or crack-bridging zone at a crack tip decreases with increasing strain rate. Dynamic Brazilian tests therefore furnish a convenient yet exacting validation test for cohesive theories of fracture applied to concrete. Our simulations give accurate transmitted loads over a range of strain rates, which attests to the delity of the model where rate e ects are concerned. The model also predicts key features of the fracture pattern such as the primary lens-shaped cracks parallel to the load plane, as well as the secondary profuse cracking near the support. The primary cracks are predicted to be nucleated at the centre of the circular bases of the cylinder and to subsequently propagate towards the interior, in accordance with experimental observations. The primary and secondary cracks are responsible for two peaks in the load history, also in keeping with experiment. The results of the simulations also exhibit a size e ect, i.e. a dependence of the e ective behaviour on the size of the specimen [48]. These results validate the theory as it bears on mixed-mode fracture and fragmentation processes in concrete over a range of strain rates. Our simulations also provide useful insights into the interpretability of the dynamic Brazilian test, and the sensitivity of the measurement to details of the experimental design, issues which have been addressed extensively in the experimental literature [49–52]. The organization of the paper is as follows. A brief outline of the particular form of the cohesive models and their nite-element implementation employed in calculations is given in Section 2. In Section 3 we begin by compiling the main parameters which de ne the simulations, including the specimen geometry, material parameters, load and boundary conditions, and the details of the mesh design. The section closes with a detailed comparison of the results of the calculations with corresponding experimental observations, including load and energy histories, crack patterns and estimated crack velocities. In Section 4 we proceed to explore the e ect of variations in selected parameters of the model, including strain rate, specimen size, the width of the bearing Copyright ? 2000 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2000; 48:963–994
THREE-DIMENSIONAL FINITE-ELEMENT SIMULATION
Figure 1. Cohesive surface traversing a 3D body.
967
Figure 2. Loading–unloading rule from linearly decreasing loading envelop expressed in terms of an e ective opening displacement and traction t.
strips, specimen size and the ratio of modes II to I toughness. Finally, a summary and some concluding remarks are collected in Section 5.
2. FINITE-ELEMENT MODEL By way of a general framework, we start by considering a deformable body occupying an initial con guration B0 ⊂ R3 . The boundary @B0 of the body is partitioned into a displacement boundary @B0 ;1 and a traction boundary @B0 ; 2 . The body undergoes a motion described by a deformation mapping e : B0 × [0; T ] → R3 , where [0; T ] is the duration of the motion, under the action of body forces 0 b and prescribed boundary tractions t applied over @B0 ; 2 . The attendant deformation gradients are denoted F and the rst Piola–Kirchho stress tensor P (cf. e.g. Reference [53]). In addition, the solid contains a collection of cohesive cracks. The locus of these cracks on the undeformed con guration is denoted S0 (Figure 1). Under these conditions, the weak form of linear momentum balance, or virtual work expression, takes the following form: Z
Z · W − P · ∇0 W] dV0 − [0 (b − e) B0
Z t ·
