A computational model of ceramic microstructures subjected to multi-axial dynamic loading
Introduction
Being naturally brittle, ceramics undergo fragmentation when subjected to multiaxial loading. The shear resistance of confined ceramics plays a major role in the prevention of ballistic penetration, wear of bearings, fracture of prosthetic devices, fracture and wear of coatings, etc. Therefore, understanding the shear resistance of ceramics is important for estimating and improving their strength. It is known that investigating experimentally the behavior of ceramics is a difficult task. Moreover, experiments do not always provide direct information on crack densities and their evolution. The implementation of an iterative computational/experimental procedure requires reliable material models, physically motivated, incorporating microfailure and macrofracture at various size scales.
Attempts have been made to model the inelastic constitutive behavior of ceramics in the presence of cracks, and to validate the models through simulation of plate and rod impact experiments. Available models for the failure of ceramics are continuum damage theories (Addessio and Johnson, 1989, Curran et al., 1990, Espinosa, 1995, Johnson and Holmquist, 1992), which are based on homogenizing the cracked solid and finding its response by degrading the elasticity of the material. The fundamental assumption in these models is that the inelastic strains are caused by microcracks whose evaluation during loading degrades the strength of the material. This degradation is defined in terms of reduced moduli whose evaluation under compressive, as well as tensile loading, is formulated using the generalized Griffith criterion. In addition, some of these models account for the initiation of cracks, coalescence, friction between fragments in the comminuted zone, etc. However, some of these phenomenological models cannot describe damage induced anisotropy, their parameters are difficult to measure experimentally, and do not explicitly consider the discrete nature of fracture through crack growth and coalescence.
In spite of the above developments, continuum models have been criticized because they require assumptions on the size and distribution of microcracks to start with, and because they cannot describe the growth of dominant cracks leading to failure, which are not suitable to homogenization. Models based on a discrete approach (Camacho and Ortiz, 1996, Espinosa et al., 1998b, Miller et al., 1999, Xu and Needleman, 1995; etc.) nucleate cracks, and follow their propagation and coalescence during the deformation process. This is a phenomenological framework where the fracture characteristics of the material are embedded in a cohesive surface traction-displacement relation.
During the last few years, the mechanical behavior of polycrystalline ceramics has been studied quite extensively on a microstructural base. The influence of microscopic heterogeneities on the overall behavior, depends on morphological characteristics such as size, shape, lattice orientation and spatial distribution of different material properties. In our view, the calculation of stress and strain distributions in real and idealized microstructures can increase the understanding of the different mechanisms that control macroscopic response. Furthermore, these micromechanical simulations can be useful for quantification and determination of failure mechanisms, as well as the derivation of evolution equations to be used in continuum models, (Curran et al., 1990, Espinosa, 1995, Espinosa et al., 1998b). In this way, bridging between length scales can be accomplished.
In the case of metal–matrix composites, Ghosh and Yunshan (1995) and Ghosh et al. (1997) developed a material based Voronoi Cell Finite Element Model (VCFEM) in an attempt to overcome difficulties in modeling arbitrary microstructures by conventional finite element methods. Voronoi cells are utilized to obtain stereologic information for the different morphologies. The microscopic analysis is conducted with the Voronoi cell finite element model while a conventional displacement based FEM code executes the macroscopic analysis. This method has been tested in several heterogeneous microstructures. Discrete microcracking was not explicitly included in this model.
Onck and Van der Giessen (1999) proposed a microstructurally-based modeling technique to study the intergranular creep failure of polycrystals by means of grain elements. A crack tip process zone was used in which grains and their grain boundaries were represented discretely, while the surrounding undamaged material was described as a continuum. The constitutive description of tile grain boundaries accounted for the relevant physical mechanisms, i.e. viscous grain boundary sliding, the nucleation and growth of grain boundary cavities, and microcracking by the coalescence of cavities. Discrete propagation of the main crack occurred by linking up of neighboring facet microcracks.
Wu and Niu, 1995a, Wu and Niu, 1995b presented a micromechanical model of the fracture of polycrystalline ice. Their model is based on a statistical description of the ice microstructure, which contains crystals of random sizes and orientations and a random distribution of grain boundary crack precursors. The analysis takes into account microstructural stresses originating from the elastic anisotropy of the constituent crystals. Friction in precursors and crack–crack interactions are also considered. The model is applied to several microstructures generated from a graph model. It was shown that the critical crack density, and critical damage, were not appropriate descriptors of failure, that the compressive strength was strongly dependent, on the microstructural variations and that crack–crack interactions were very important in compressive fracture.
Kim et al. (1996) studied crack propagation in alumina ceramics. The competition between intergranular and transgranular propagation was utilized to determine the crack path.
Miller et al. (1999) considered models based on energy balance and compared their predictions of fragment size to the results of numerical simulations. They found differences due to the fact that the energy-based models deal with the onset of the fragmentation event, but they do not include the time dependence of the process. Therefore, they proposed a model that included the time history of the fragmentation process and parameters such as the speed of crack propagation, and the strength and flaw distribution.
In this paper, a micro-mechanical finite element, modeling of ceramic microstructures under dynamic loading is presented to assess intergranular microcrack initiation and evolution. A representative volume element of an actual microstructure, subjected to compression–shear dynamic loading, is considered for the analysis. A large deformation elastic–anisotropic viscoplasticity model for the grains, incorporating grain anisotropy by randomly generating principal material directions, is included. Cohesive interface elements are embedded along grain boundaries to simulate microcrack initiation and evolution. Their interaction and coalescence are a natural outcome of the calculated material response.
A systematic and parametric study of the effect of different factors is carried out. The effects of interface element parameters, grain anisotropy, grain size and a stochastic distribution of interface properties are studied in terms of microcrack initiation and evolution and crack density. The pulverization of the material upon unloading is examined with the microstructural model. The qualitative and quantitative results presented in this article are intended to provide valuable insight for developing more refined continuum theories on fracture properties of ceramics.
Section snippets
Computational model
The finite element analysis of the initial boundary value problem is performed using a total Lagrangian continuum approach with a large deformation elastic–anisotropic model.
A displacement based finite element formulation is obtained from the weak form of the momentum balance or dynamic principle of virtual work. The weak form at time t in total Lagrangian co-ordinates, (i.e., referred to the reference configuration), is given by
Case study: pressure–shear experiment
Plate impact experiments offer unique capabilities for the characterization of advanced materials under dynamic loading conditions. These experiments allow high stresses, high pressures, high strain rates and finite deformations to be generated under well characterized conditions. They all rely on the generation of one-dimensional waves in the central region of the specimen in order to allow a clear interpretation of the experimental results and the mathematical modeling of the material
Results and discussion
We shall focus on the study of the variation of geometrical and physical parameters that characterize the ceramic microstructure and their effect on the microstructure response.
In principle, the response of the piece of ceramic considered may depend on several factors such as grain anisotropy, interfacial strength, representative computational cell size, shape and size of the grains, etc.
In order to validate our model, microstructure response should not depend on numerical parameters such as
Concluding remarks
A model was presented for the dynamic finite element analysis of ceramic microstructures subjected to multi-axial dynamic loading. The model solves an initial-boundary value problem using a multi-body contact model integrated with interface elements. It simulates microcracking at grain boundaries and subsequent large sliding opening and closing of the microcracks. Numerical results are shown in terms of microcrack patterns and evolution of crack density. Simulations with different values of
Acknowledgements
H.D. Espinosa would like to thank J. Hutchinson for providing insight on the role of the cohesive law shape and parameters in the prediction of fragmentation. He also acknowledges the support from the Division of Engineering and Applied Sciences at Harvard University during his sabbatical visit. The authors acknowledge the discussion and collaboration provided by S. Dwivedi in the implementation of the computational model. They would also like to thank the ARO and DoD HPCMP for providing
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