Elsevier

Mechanics of Materials

Volume 35, Issues 3–6, March–June 2003, Pages 333-364
Mechanics of Materials

A grain level model for the study of failure initiation and evolution in polycrystalline brittle materials. Part I: Theory and numerical implementation

https://doi.org/10.1016/S0167-6636(02)00285-5Get rights and content

Abstract

A model is presented to analyze material microstructures subjected to quasi-static and dynamic loading. A representative volume element (RVE) composed of a set of grains is analyzed with special consideration to the size distribution, morphology, chemical phases, and presence and location of initial defects. Stochastic effects are considered in relation to grain boundary strength and toughness. Thermo-mechanical coupling is included in the model so that the evolution of stress induced microcracking, from the material fabrication stage, can be captured. Intergranular cracking is modeled by means of interface cohesive laws motivated by the physics of breaking of atomic bonds or grain boundary sliding by atomic diffusion. Several cohesive laws are presented and their advantages in numerical simulations are discussed. In particular, cohesive laws simulating grain boundary cracking and sliding, or shearing, are proposed. The equations governing the problem, as well as their computer implementation, are presented with special emphasis on selection of cohesive law parameters and time step used in the integration procedure. This feature is very important to avoid spurious effects, such as the addition of artificial flexibility in the computational cell. We illustrate this feature through simulations of alumina microstructures reported in part II of this work. A technique for quantifying microcrack density, which can be used in the formulation of continuum micromechanical models, is addressed in this analysis. The density is assessed spatially and temporally to account for damage anisotropy and evolution. Although this feature has not been fully exploited yet, with the continuous development of cheaper and more powerful parallel computers, the model is expected to be particularly relevant to those interested in developing new heterogeneous materials and their constitutive modeling. Stochastic effects and other material design variables, although difficult and expensive to obtain experimentally, will be easily assessed numerically by Monte Carlo grain level simulations. In particular, extension to three-dimensional simulations of RVEs will become feasible.

Introduction

Accurate modeling of inelasticity and failure of brittle materials is key to the design of microelectronic devices, machining of ceramics and ceramic composites, design of microelectromechanical systems and armor systems. Many theories have been developed at various size scales from homogenized solids to grain level and atomistic modeling. In this article we present the details and features of the grain level model developed by Espinosa and co-workers in their study of dynamic failure of brittle materials.

A variety of continuum damage models have been developed over the last decade (Bazant and Oh, 1985; Addessio and Johnson, 1999; Curran et al., 1990; Espinosa, 1995; Johnson and Holmquist, 1992). These models 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 inelastic strains are caused by microcracks, whose evolution during loading degrades the strength of the material. This degradation is defined in terms of reduced moduli whose evolution, under compressive, tensile, and mixed loading, is formulated using the generalized Griffith criteria. In addition, some of these models account for the initiation of cracks, coalescence, friction between fragments in the comminuted zone, etc. With the exception of the multiple-plane model, these phenomenological models cannot describe damage induced anisotropy and their parameters are difficult to identify experimentally. Furthermore, these continuum models require assumptions on the initial size and distribution of microcracks and they cannot fully describe the growth of dominant cracks leading to macroscopic failure, which are not suitable to homogenization.

To overcome these limitations, models based on a discrete approach were developed, (Camacho and Ortiz, 1996; Espinosa et al., 1998a; Miller et al., 1999; Xu and Needleman, 1995). In these models, nucleation, propagation and coalescence of cracks during the deformation process is an outcome of the simulation. Discrete models are based on a phenomenological framework where the fracture characteristics of the material are embedded in a cohesive surface traction–displacement relation. 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 because their energy-based models dealt with the onset of the fragmentation event, but they did 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 flaw strength distribution.

Within the framework of cohesive interface elements the two most noteworthy cohesive failure models available in the literature are the intrinsic exponential potential-based law used by Xu and Needleman (1995), and the extrinsic linear law developed by Camacho and Ortiz (1996). The distinction between these two approaches is associated with the way the damage initiation process is modeled. In the extrinsic case, the stress-based failure criterion is external to the cohesive element. When the tractions acting along the interface between two volumetric elements have reached a critical value, the interface is allowed to open in accordance with a prescribed traction–separation relation by introducing additional nodes, along the failed interface, coupled by a cohesive law. In the intrinsic approach, the failure criterion is incorporated within the constitutive model of the cohesive elements. Failure is integrated into the cohesive law by increasing the cohesive tractions from zero to a failure point at which the tractions reach a maximum before gradually decreasing back to zero values. Implementation of the intrinsic method in a finite element analysis requires that the cohesive elements be present between the volumetric elements from the beginning of the analysis, unlike the extrinsic approach, where a cohesive element is introduced in the mesh only after the corresponding interface is predicted to start failing. This adds some artificial flexibility to the solid, which could alter wave speeds and induce spreading of the wave. We will get back to this feature in relation to the bilinear cohesive law used in our model.

Other developments have also contributed to advances in the simulation of both quasi-static and dynamic fracture events. Theses include meshless methods (Belytschko et al., 1996, Belytschko et al., 2000; Belytschko and Tabbara, 1996), extended finite element methods, (Belytschko et al., 2001), atomistic modeling of fracture (Abraham et al., 1994, Abraham et al., 1998; Gao, 1996; Gumbsch et al., 1997), and the availability of massively parallel computational environments necessary for complex dynamic failure problems. In the case of atomistic simulations, it should be noted that the development of algorithms for bridging length scales is needed and remains under intense investigation (Kohlhoff et al., 1991; Miller et al., 1998).

During the last few years, the mechanical behavior of polycrystalline ceramics has been studied quite extensively on a microstructural bases. This inevitably requires spanning multiple length scales. Instead of proceeding phenomenologically, the principles and tools of mechanics are brought to bear on phenomena occurring at the microscale. The observer then steps back and the microscopic features blur into macroscopic fields governed by a different set of “effective” laws. The determination of these effective properties from first principles is one of the principal objectives of micromechanics. A related endeavor is the use of the knowledge base thus acquired for the design of microstructures resulting in improved material properties.

The main characteristic of material microstructure models is the capability to include, in an explicit form, the heterogeneities of the material, such as grain shape, size and orientation, second phases, voids, flaws, etc. Some models include “ad hoc” finite elements to represent heterogeneities. For instance, Ghosh and Yunshan (1995) and Ghosh et al. (1997) developed a material based Voronoi cell finite element model (VCFEM) to study metal–matrix composites. Onck and Van der Giessen (1999) introduced the “grain element” where the grain boundaries account for viscous grain boundary sliding, and nucleation and growth of voids. Discrete propagation of the main crack occurred by linking up of neighboring facet microcracks.

Among the material microstructure models based on the standard finite element method, including cohesive interfaces, one can mention the work by Zhou and Zhai (1999) and Zhai and Zhou (2000), who analyzed the dynamic crack propagation in ceramic composites using the cohesive finite element model proposed by Needleman (1988) and Xu and Needleman (1995), and the work by Helms et al. (1999), where the cohesive interfaces were embedded along grain boundaries. Zavattieri and Espinosa (2001) simultaneously performed similar analysis applied to the modeling of microcracking of ceramics. Stochastic effects were included in these analyses and comparisons with experiments were performed.

Other methods based on a statistical approach, such as the model by Ostoja-Starzewki (1998) and Ostoja-Starzewki and Wang (1999) achieved bridging between micromechanical and continuum models where the microstructural material randomness is considered below the level of a single finite element. Likewise, Mullen et al. (1997) developed a finite element-based Monte Carlo that can be used to predict scatter in the nominal elastic constants and fracture of thin films. 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.

Other models which include microcracking at the grain level can be found in the literature. Grah et al. (1996) conducted computer simulations of polycrystalline materials using a spring-network model for arbitrary in-plane crystal anisotropy. A detailed study of the interrelated physical mechanisms that result in failure modes in crystalline materials with high angle grain boundaries has been conducted by Zikry and Kao (1996). Kim et al. (1996) studied crack propagation in alumina ceramics. In their work, the competition between intergranular and transgranular propagation was utilized to determine the crack path.

Despite all these advances in the area of micromechanics, bridging between micro- and macroscales still remains one of the most challenging goals. Although in some instances comparison with experimental findings and microscopy studies have been done, the majority of the contributors omit comparison and correlation with experimental data. The accuracy of micromechanical models in capturing experimental data was assessed by Zavattieri and Espinosa (2002) and Zavattieri and Espinosa (2001). In order to provide a powerful tool in understanding the mechanisms that lead to macroscopic failure and to refine theories of damage utilized in continuum, or continuum/discrete models, a grain level micromechanical model is presented in this paper to assess intergranular microcrack initiation and evolution. A representative volume element (RVE) of an actual microstructure, subjected to multi-axial dynamic loading, is considered for the different analyses. An elastic-anisotropic 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 variation of this micromechanical model for quasi-static calculations is also discussed in this paper for the simulation of cooling, and resulting thermal residual stresses, during the material fabrication process. Residual stresses due to mismatch between thermal expansion coefficients of adjacent grains and phases can result in spontaneous microcracking (Tvergaard and Hutchinson, 1988). An implicit incremental algorithm for modeling thermal effects is included together with the formulation of non-linear cohesive interfaces.

Our micromechanical model provides explicit account for arbitrary microstructural morphologies and microscopic fracture patterns making easier to identify and design microstructural configurations that enhance fracture toughness, and therefore lead to improve fabrication of new single and multi-layer ceramic materials. Through the consideration of actual microstructures, the effects of various fracture mechanisms is delineated. The unique advantages of the micromechanical model proposed in this work include: (1) explicit account of real, arbitrary material microstructures, (2) explicit modeling of fracture in a non-constrained manner, therefore arbitrary crack paths or microcrack patterns are admitted, (3) direct analysis of the stochastic nature of fracture in heterogeneous microstructures, (4) analysis of the effect of residual stresses, (5) resolution of fracture explicitly over multiple length scales and free of ad hoc fracture criterion, therefore crack initiation, growth, and coalescence is a natural outcome of material response, applied loading, and boundary constrains, (6) the representative computational cells where the calculations take place are chosen such that direct comparison with experimental data can be made.

Section snippets

Grain level micromechanical model

In this section, the micromechanical finite element modeling of ceramic microstructures under dynamic loading is presented to assess intergranular microcrack initiation and evolution. A RVE of a ceramic microstructure, subjected to multi-axial dynamic loading, is considered for the analysis. The model is based on a plane strain analysis of a polycrystalline material described with a multi-body finite element mesh. Each grain is individually represented by a mesh with six-noded triangular finite

Cohesive model: contact/interface algorithm

A multi-body contact–interface algorithm to describe the kinematics at the grain boundaries is used to simulate crack initiation and propagation. An explicit time integration scheme is adapted to integrate the system of spatially discretized ordinary differential equations. Fig. 4 describes the contact model, which is integrated with interface elements to simulate microcracking at the grain boundaries and subsequent large sliding, opening and closing of the interface. The tensile and shear

Initial and boundary conditions in plate impact experiments

The grain level model being presented will be used to model plate impact wave propagation experiments. In this context, initial and boundary conditions need to be defined. In these subsections we address this feature of the simulations.

Due to specimen dimensions in plate impact experiments, such as the material grain size and the number of triangular elements per grain, certain considerations and assumptions can be made in order to avoid computationally expensive calculations. For the dynamic

Explicit integration scheme

An explicit central-difference integration algorithm is being used to integrate the system of spatially discretized ordinary differential equations in time. The algorithm, accounting for acceleration corrections due to contact, is summarized in Table 1. As in any initial boundary value problem, initial displacements and velocities, u0 and v0, are required. Initial accelerations a0 are calculated from initial applied forces fext0 and initial internal forces fint0.

At each time step n, the nodal

Modeling of stochastic effects

The interfaces between different material phases are important in determining many bulk properties. One of the simplest interface types is the boundary between two crystals of the same material. If two crystals of exactly the same orientation are brought together, they fit perfectly. However, if the crystals are slightly tilted, there is a disregistry at the interface, which is equivalent to insertion of a row of dislocations. The number of dislocations per unit length and the energy of the

Modeling residual thermal stresses and thermal induced microcracking

Throughout this study, the ceramic microstructure has been idealized as an ensemble of randomly oriented, elastically and thermally anisotropic grains with brittle intergranular interfaces. In the process of cooling, the solid is brought from its fabrication temperature down to some final conditions, typically room temperature. The stresses in the sample are assumed to be relaxed at the fabrication temperature due to creep. During this cooling process, the body develops microstructural residual

Microcrack evolution and stereology

An important feature of the proposed model is its capability of producing microcrack patterns and evolution of crack densities. The issue is how to describe these results with a single weighted parameter. In this section crack surface per unit volume is defined and its function illustrated. Consider a RVE under the dynamic conditions described in Section 4. Fig. 23 shows the evolution of crack pattern along the whole microstructure for a typical calculation. As the wave front advances, crack

Concluding remarks

A model was presented to analyze material microstructures subjected to quasi-static and dynamic loading. An RVE composed of a set of grains was introduced with special consideration to the size distribution, morphology, phases and presence and location of initial defects. Stochastic effects were considered in relation to grain boundary strength and toughness. An important feature of the model was highlighted, i.e., its capability to capture the evolution of stress induced microcracking from the

Acknowledgements

The authors acknowledge ARO and DoD HPCMP for providing supercomputer time on the 128 processors Origin 2000 at the Naval Research Laboratory-DC (NRL-DC). This research was supported by the National Science Foundation through Awards no. CMS 9523113 and CMS-9624364 (NSF-CAREER), the Office of Naval Research YIP through Award no. N00014-97-1-0550, the Army Research Office through ARO-MURI Award no. DAAH04-96-1-0331 and the Air Force Office of Scientific Research through Award no. F49620-98-1-0039.

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    Present address: General Motors Research and Development Center, 30500 Mound Road, Warren, MI 48090-9055, USA.

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