Numerical investigation on the dynamic behavior of advanced ceramics
Introduction
Advanced ceramics are widely used as ballistic armors due to their excellent stiffness to weight and resistance to weight ratios. Another appealing property of advanced ceramics is their capacity for dissipating energy by fracture and friction when they are comminuted by the impact of a projectile. Thus, the dynamic mechanical response of these materials is especially important to grant the required level of protection to an armor. This is why some of the scientific research in this field has focused on the determination of their dynamic properties and the development of accurate constitutive models and failure criteria at high strain rates.
As with other quasi-brittle materials, advanced ceramics perform very well under compression, but their tensile strength is low, approximately one order of magnitude below the compressive strength. Hence most times failure takes place when the tensile strength is reached, rendering it an essential property in the modelization of these materials. Several different tests have been proposed to measure the tensile strength of ceramics and it has been found that it grows with the strain rate under every experimental configuration. In particular, the diametral compression test or Brazilian test performed by means of a Hopkinson bar is seen by some experimentalists as a convenient set up giving reliable measurements.
Gálvez, Rodrı́guez, Sánchez-Gálvez and Navarro are a case in hand. They recently carried out a series of dynamic Brazilian tests on several advanced ceramics to show the feasibility of this experimental method to get the tensile strength at high strain rates [1], [2], [3], [4]. They studied six different materials: three aluminas (Al2O3) with different grain size and degree of purity, alumina blended with zirconia (Al2O3 + ZrO2), silicon carbide (SiC) and boron carbide (B4C); the specimens were cylinders whose diameter and thickness were 8 and 4 mm respectively. The increase in the tensile strength for the dynamic tests (performed at strain rates ranging from 65 to 89 s−1) with respect to the static tests (7 to 10 × 10−7 s−1) varied from 20% for the SiC to almost 90% for the blend of alumina and zirconia. They also studied the rupture mechanisms by means of high speed photography and performed a fractographic study of the crack surfaces using a scanning electron microscope. The latter revealed that there was no hint of plastic deformation and that the fracture was predominantly transgranular, with no change in the fracture mode with the strain rate [3], [4].
Here we use their results to validate a model that was originally applied to concrete undergoing the same test conditions [5] and that also gave good results studying the propagation of dynamic cracks under mixed-mode loading [6]. It consists of a finite element model which allows fragmentation, i.e. the opening of a crack where and when tension reaches the tensile strength, and that uses a mixed-mode cohesive model to control the fracture process [7].
Cohesive models are suitable for simulating cracking processes in advanced ceramics, since these materials, in spite of their relative small grain size, develop long fracture process zones––of the order of millimeters, depending on the microstructure and geometry of the specimen––due to the bridging and interlocking of the grains in the wake of the crack [8], [9], [10], [11], [12]. These process zones constitute the main energy dissipation mechanism for this kind of materials [12], [13]. Indeed, cohesive theories applied to advanced ceramics have successfully explained the dependency of some of their properties and methods of characterization––like the R-curve––on the shape and size of the specimen [14], [15]. Furthermore, cohesive models are feasible for handling the dynamic effects appearing in advanced ceramics, for they automatically discriminate between slow and fast rates of loading [16] and thus there is no need to include the rate dependency within the description of the constitutive equations.
Another interesting feature of our finite element model is the explicit treatment of fracture and fragmentation [17]. It tracks individual cracks as they nucleate, propagate, branch and possibly link up to form fragments, the ensuing granular flow of the comminuted materialist also simulated explicitly. It is incumbent upon the mesh to provide a rich enough set of possible fracture paths since the model allows decohesion to occur along element boundaries only. However, no mesh dependency is expected as long as the cohesive elements adequately resolve the fracture process zone of the material [5], [16]. It is also interesting to note that the microinertia attendant to the material in the dynamic fragmentation process contributes to the correct simulation of the rate effects [5], [6], [16].
The simulations in this paper give a good prediction of the tensile strength for each material and come out with crack patterns very similar to the actual ones observed in the experiments. The model predicts the formation of a principal crack that nucleates in the center of the specimen and grows towards the bearing areas, as well as some secondary cracking parallel to the main crack and near to the loading areas.
The paper is organized as follows: a brief account of the main assumptions of the model and of its finite element implementation is given next. Section 3 describes the experimental set-up (3.1), the specimen geometry and material parameters (3.2), the load and boundary conditions (3.3), the mesh used in the simulations (3.4), and the simulation results (3.5). Finally, in Section 4, we draw several conclusions regarding the applicability of cohesive models to study the dynamic behavior of advanced ceramics.
Section snippets
Finite element model
For completeness and posterior reference, in this section we summarize the main features of the cohesive law used in the calculations. An extensive account of the theory and its finite element implementation may be found elsewhere [7], [16].
A simple class of mixed-mode cohesive laws accounting for tension–shear coupling (see Camacho and Ortiz [16] and others [7], [18]) is obtained by the introduction of an effective opening displacement δ, which assigns different weights to the normal δn and
Simulation of the dynamic behavior of advanced ceramics
The simulations in this paper refer to experiments reported by Rodrı́guez et al. [1], Gálvez et al. [2], Gálvez [3] and Gálvez et al. [4]. The experiments consist of Brazilian tests performed with a Hopkinson bar on six different advanced ceramics. As follows we briefly describe the experimental set-up (3.1), the specimen geometry and material parameters (3.2), the load and boundary conditions (3.3), the mesh used in the simulations (3.4), and the simulation results (3.5).
Summary and conclusions
We used cohesive theories of fracture, in conjunction with the direct simulation of fracture and fragmentation, to describe processes of tensile damage and compressive crushing in advanced ceramics subjected to dynamic loading. Indeed, cohesive models are suitable for simulating cracking processes in advanced ceramics, since these materials develop long fracture process zones––two or three orders of magnitude longer than the average grain size––due to the bridging and interlocking of the grains
Acknowledgements
We are indebted to Prof. Michael Ortiz for providing us with access to his numerical codes at the Graduate Aeronautical Laboratories, California Institute of Technology, as well as for his advice during the development of this work. We are also thankful to Dr. Francisco Gálvez and Prof. Jesús Rodrı́guez for providing us with some records of their tests and insights into their excellent experimental work. Rena C. Yu and Gonzalo Ruiz thank the Ministerio de Educación, Cultura y Deporte, Spain,
References (33)
- et al.
A three-dimensional analysis of crack trapping and bridging by tough particles
J. Mech. Phys. Solids
(1991) Dynamic fracture of ceramics and ceramic composites
Mater. Sci. Engng. A
(1991)- et al.
Fracture process zone modeling of monolithic Al2O3
Engng. Fract. Mech.
(1999) - et al.
Computational modelling of impact damage in brittle materials
Int. J. Solids Struct.
(1996) - et al.
Elastoplastic finite-element analysis of three-dimensional fatigue crack growth in aluminum shafts subjected to axial loading
Int. J. Solids Struct.
(1999) - et al.
Three dimensional cohesive-element analysis and experiments of dynamic fracture in C300 steel
Int. J. Solids Struct.
(2000) - et al.
Mixed-mode fracture in biaxial stress state––Application of the diametral-compression (Brazilian disk) test
Engng. Fract. Mech.
(1987) - et al.
Splitting tests: an alternative to determine the dynamic tensile strength of ceramic materials
J. Phys. IV
(1994) - et al.
Tensile measurements of ceramic materials at high rates of strain
J. Phys. IV
(1997) - Gálvez F. Caracterización Mecánica de Materiales Cerámicos Avanzados a Altas Velocidades de Deformación, PhD thesis,...
Influence of the strain rate on the tensile strength in aluminas of different purity
J. Phys. IV
Three-dimensional finite-element simulation of the dynamic Brazilian tests on concrete cylinders
Int. J. Numer. Meth. Engng.
Three-dimensional cohesive modeling of dynamic mixed-mode fracture
Int. J. Numer. Meth. Engng.
Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis
Int. J. Numer. Meth. Engng.
R-curve behavior of Al2O3 ceramics
Acta Metall. Mater.
Process zone of polycrystalline alumina
Exp. Mech.
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