Numerical investigation on the dynamic behavior of advanced ceramics

Abstract

This paper studies the dynamic behavior of advanced ceramics by a finite element model. It implements the direct simulation of fracture and fragmentation together with a mixed-mode cohesive law to describe the fracture process. In particular, we simulate dynamic Brazilian tests performed with a Hopkinson bar, at strain rates ranging from 65 to 89 s⁻¹, on six different materials: three kinds of alumina with different average grain sizes and degrees of purity, a blend of alumina and zirconia, silicon carbide and boron carbide. The rate dependence of the results emerges explicitly from the calculations, thanks both to the inertia attendant to the fracture process, and to the time effect provided by the cohesive law. Indeed, the simulations give accurate values for the dynamic strength of the six ceramics under study. The simulations also predict the main features of the crack pattern.

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.