A finite deformation framework for mechanism-based constitutive models of the dynamic behavior of brittle materials

Abstract

A finite deformation mechanism-based thermodynamically consistent constitutive framework is presented for describing the dynamic behaviors of brittle materials under impact loading. The framework is developed based upon a multiplicative decomposition of the deformation gradient in terms of multiple mechanisms, including recoverable elasticity, crack-induced damage, and other inelastic mechanisms such as subgrain and granular plasticity. The finite deformation kinematics that captures the multiple mechanisms is structured within a thermodynamically consistent framework, and the consequent coupling of the various mechanisms is articulated. Specific constitutive equations are formulated for a Mie–Grüneisen equation of state, micromechanics-based dynamic-fracture-induced damage growth, subgrain or lattice plasticity for slip and other deformation modes, and granular plasticity for granular flow and pore collapse post-fragmentation. Using hot-pressed silicon carbide (SiC-N) as the model material, this integrative model is calibrated using available experiments that interrogate specific mechanisms. The effects of loading rate, the influence of confinement, and the path-dependent constitutive behaviors of the material predicted by the model are demonstrated. The model performance at the application scale is then evaluated by simulating previously performed sphere-on-cylinder impact experiments.

Introduction

Computational models for the responses of brittle materials (e.g.,ceramics and rocks) under impact loading are important for a variety of applications in geology, planetary science, and structural engineering. Traditionally, phenomenological models have been popular due to their computational efficiency and robustness when used for large scale simulations. Examples include the JH-1, JH-2, and JHB models byHolmquist and Johnson, 2002, Holmquist and Johnson, 2005 and Johnson and Holmquist (1994) for ceramics, glass, and other brittle materials, and the Kayenta model byBrannon et al. (2009) for rock and rock-like engineered materials. Although widely used, such phenomenological models (a) suffer from a lack of predictive ability when used outside of the loading scenarios against which they are calibrated, (b) suffer from a lack of experimental data that can be used for calibration, and (c) cannot be used to design new materials, and must be recalibrated every time that there are changes in the material microstructure. We focus here on developing a physics-based or mechanism-based model that is built upon specific mechanisms and that permits high-fidelity computational simulations of complex impact problems once the material itself has been characterized. Such physics-based models have the disadvantage of being relatively expensive from a computational viewpoint, but have the advantages of being able to address a wide range of loading conditions with a single parameterization of the material microstructure and of providing guidance to the development of improved materials (i.e.,materials design). For example, constitutive models based on the micromechanics of crack nucleation and growth(Basista and Gross, 1998, Chen and Ravichandran, 2000, Huang and Subhash, 2003, Deshpande and Evans, 2008, Hu et al., 2015, Bhat et al., 2012, Tonge and Ramesh, 2016) have been shown to be useful for providing insights into experimental observations and guidelines for material design.

The typical physical processes (“mechanisms”) involved in the dynamic deformation and failure of brittle materials are summarized in Fig.1, with the baseline behavior represented by elasticity and the equation of state. Broadly, these mechanisms include fracture, plasticity (including so-called quasiplasticity mechanisms like amorphization), granular flow, and porosity evolution (both pore generation and pore compaction).

The most striking feature of brittle materials is the presence of flaws or defects, which work as stress concentrators and usually lead to failure initiation when stressed. Several micromechanics models for failure under compressive loading are in common use for both slit-like defects(Ashby et al., 1986, Horii and Nemat-Nasser, 1986) and pore-like defects (Sammis and Ashby, 1986, Katcoff and Graham-Brady, 2014). Fracture processes are often considered collectively in the sense of damage, and damage evolution is typically described in terms of the evolution of the number and size of cracks and pores. Approaches based on the effective medium method such asBudiansky and O’Connell, 1976, Kachanov, 1980 and Grechka and Kachanov (2006) or energy arguments such as(Basista and Gross, 1998, Deshpande and Evans, 2008, Bhat et al., 2012) have been proposed to describe the effect of such damage on the elasticity. These fracture processes, and as a consequence damage processes, can be directly linked to the defect populations within the material, e.g.Ramesh et al. (2015).

Impact problems often involve both high confining pressures and high strain rate deformations. Under such conditions, various inelastic deformation mechanisms have been observed(LaSalvia and McCauley, 2010). The development of advanced material characterization techniques makes it possible to observe various inelastic deformation mechanisms in ceramics, including amorphization(Chen et al., 2003), slip(Hirsch et al., 2018), twinning(Chen et al., 2006), and phase transformations(Ueno et al., 1992). For instance,Heard and Cline (1980) observed a brittle-to-ductile transition in BeO and AlN ceramics under highly confined quasi-static compression and identified slip by dislocation generation and motion as the activated mechanism at high pressure. Similarly,Vajdova et al. (2012) observed that twinning is operative in hydrostatically and triaxially compressed limestone. Traces of dislocation slip and stacking faults have been also found in various ceramics such as AlON(Paliwal et al., 2008) and SiC(Shih et al., 2000) under dynamic compression. Such inelastic deformation mechanisms have been also found in brittle solids under shock compression. Insensitivity of the shear strength to pressure was observed in AlN and Al${}_2$O${}_3$ subjected to shock loading, suggesting similar plastic deformation mechanisms(Grady, 1998, Rosenberg et al., 1991, Kipp and Grady, 1994, Chen et al., 2006). Similarly, the splitting of the shock-wave front into an elastic precursor and a plastic compressive wave has been observed in granite(Yuan and Prakash, 2013).

Such observations demonstrate that it is important to incorporate these inelastic mechanisms in physics-based constitutive formulations to allow modeling of the inelastic responses of brittle solids in the high strain rate and high pressure regime. Depending on the applications, melting, vaporization and other phase transformation phenomena may also need to be modeled(Jutzi et al., 2015).

In addition, microcracking and other damaging processes lead eventually to fragmentation and comminution of the brittle solid into a granular material(Hogan et al., 2017). The mechanics of granular materials has been widely studied in the civil engineering community and has been also considered within impact problems(Curran et al., 1993, Deshpande et al., 2011, Tonge and Ramesh, 2016, Jutzi et al., 2015, Bažant and Caner, 2013). Frictional sliding between fragments and particle rotations(Curran et al., 1993, Zhai et al., 2019) accompanied by competition between dilatancy and pore compaction results in granular flow, and this provides a mechanism for large scale deformations and the significant energy dissipation during an impact event. The associated dilatancy and pore-collapse during compression-shear loading also changes the pressure on the surrounding material(Curran et al., 1993).

Lattice plasticity, non-statistical damage mechanics and granular flow were incorporated in a small-strain formulation byDeshpande et al. (2011), which is an excellent example of a mechanism-based formulation. However, the statistical distributions of defects in ceramics dominate their behavior, and finite deformation kinematics and the corresponding couplings must also be incorporated in order to address the response during a high speed and high pressure impact event(Meyers, 1994).

The mechanisms activated during dynamic impact are also often intertwined (for instance, microcracking can lead to the elevation of local stresses(Horii and Nemat-Nasser, 1986) and reduce the effective area for plastic deformation(Park and Voyiadjis, 1998)). The evolution of the inelastic mechanisms may also induce additional damage resulting from slip band-nucleated cleavage(Lankford et al., 1998) or sliding along amorphization bands(Zeng et al., 2019a). In prior efforts(Zeng et al., 2019a), we have incorporated amorphization, damage and granular flow into a mechanism-based formulation, but the important mechanism of dislocation-based plasticity and its coupling with the other mechanisms during finite deformations has not been integrated into such models.

In this work, we develop a thermodynamically consistent finite deformation framework to incorporate these mechanisms and the effect of mechanism coupling into constitutive formulations for brittle solids under impact loading. The paper is structured as follows. The modeling framework that incorporates finite deformation kinematics and thermodynamic consistency is first presented. The framework description is followed by a detailed development of the material model including the constitutive formulations for elasticity and equation of state, microcrack-informed damage, subgrain plasticity, and granular plasticity. Note that we limit our discussion to the most commonly observed mechanisms for brittle materials under dynamic impact, but the framework and process is general. In addition, given the trade-off between computational efficiency and fidelity, we ensure that key physical mechanisms are captured but allow phenomenological descriptions of specific mechanisms when necessary. We then demonstrate the modeling framework by considering a specific advanced ceramic known as SiC-N, a hot-pressed silicon carbide that is used in impact applications. The model parameters are identified and calibrated using existing primary laboratory measurements on the material. The calibrated mechanism-based model is then used to demonstrate the ability of the model to capture various responses including shock response, and rate-, pressure-, and path-dependence. Finally, the model performance at the application scale is evaluated by simulating a complex impact experiment.