Research paperA micromechanics-based fractional frictional damage model for quasi-brittle rocks
Introduction
Quasi-brittle rocks, such as granite, marble and tight sandstone, are extensively encountered in engineering practice. Proper modeling and simulation of their mechanical behaviors are of particular significance in the design and maintenance of engineering structures. It is known that quasi-brittle rocks present a strongly nonlinear mechanical responses, including volumetric dilatancy, strain-hardening/softening, etc. The development of constitutive models for adequately describing the entire stress–strain curve of quasi-brittle rocks remains a challenge, especially for the nonlinear dilatancy behavior (Zhao and Cai, 2010).
Based on the principle of maximum plastic work, dilatancy describes an increase in volume accompanied by plastic deformation in the process of shear deformations (Hill, 1998, Kruyt and Rothenburg, 2016). Generally, the dilatancy ratio, namely the ratio of the plastic volume strain increment to the plastic shear strain increment, is employed to delineate the dilatancy (Wan and Guo, 1998, Wang et al., 2019). Then, the dilatancy ratio is physically specified by the direction of plastic flow, whose determination depends heavily on the selection of the plastic flow law (Collins, 2002, Mo and Yu, 2017). With regard to conventional constitutive models with the associated plastic flow, it can be postulated that the direction of plastic flow coincides with the normal of the loading surface, which, however, is often unsuitable for characterizing the volume change under the condition of triaxial loading (Kim and Lade, 1988). To address this shortcoming, the non-associated flow rule is cast into the constitutive formulation, where a suitable plastic potential function needs to be established. Yet, the formulation of such a function on the basis of test observations is laborious and leads to add some physically meaningless parameters (Desai, 1980, Krenk, 2000). Alternatively, the approach based on a fractional operation was developed to determine the plastic flow direction without additional plastic potential. Specifically, the framework of fractional plasticity, firstly introduced by Sumelka, 2014b, Sumelka, 2014a with fixed lower and upper integral limits for viscoplasticity materials, and then extended by Sun and Xiao, 2017, Sun et al., 2018, Sun et al., 2019, Qu et al., 2019, Song et al., 2019, Liang et al., 2020 and Sun et al. (2020) for diverse materials, becomes a powerful tool in place of plastic potential function.
In micromechanical models, the primarily admitted mechanisms concern frictional sliding along closed cracks and micro-crack growth at microscopic scale (Walsh, 1980). In this context, several micromechanical models have been proposed. For example, linear fracture mechanics is introduced to establish the relations between microscopic physical processes and macroscopic inelastic responses (Prat and Bažant, 1997, Pensée et al., 2002); the homogenization technique is applied to capture the macroscopic mechanical behaviors of rock-like materials (Zhu et al., 2008, Markenscoff and Dascalu, 2012, Cao et al., 2020). However, the mechanical response enjoys a forceful nonlinearity in the coupling process of two physical mechanisms mentioned previously. At the current stage, an associated flow rule is adopted for describing friction-induced plasticity, because rational derivation of plastic potential is still quite difficult in a purely micromechanical context. In this sense, we are now interested to incorporate the fractional operation into some micromechanical formulations.
On numerical aspects, diverse algorithms have been developed for correct integration of elastoplastic constitutive equations, which can be mainly factored into three categories, that are explicit, implicit and semi-implicit schemes. With regard to the explicit scheme, the calculation error arises from the former step and then accumulates in the whole process, thus bringing about a tough issue that the yield stress is away from the loading surface (Zienkiewicz et al., 1969, Sloan, 1987). As regards the fully implicit algorithm, the hardening rule is assessed in the case of unknown stress, which leads to the requirement of solving the system of non-linear equations iteratively (Schofield and Wroth, 1968, Simo and Ortiz, 1985). Lastly, the semi-implicit scheme is developed for integrating common non-smooth elastic–plastic models (Tu et al., 2009) and applied to deal with plastic models which include two yield functions in Ghaei et al. (2010). However, by now, further efforts are still mandatory for developing a simple but valid novel numerical approach for strongly coupled plasticity-damage models.
This paper is devoted to formulating a micromechanics-based elastoplastic damage model for quasi-brittle rocks by combining the fractional plasticity theory and the linear homogenization method. Meanwhile, a straightforward and novel explicit return mapping (ERM) method is proposed for numerical purposes. The paper is structured as follows. In Section 2, we present the formulation of the constitutive model. Section 3 provides the implementation details of the ERM numerical algorithm. Section 4 outlines the identification of the model’s parameters. Comparisons between numerical simulations and test observations are reported in Section 5. Finally, some concluding remarks are made in Section 6.
Section snippets
Problem statement
Quasi-brittle rocks contain various heterogeneities (voids and cracks) and present strongly nonlinear mechanical behaviors, making it difficult and also unnecessary to take into account the local behavior of each heterogeneity. An effective medium is thus usually used instead of the real one. Quasi-brittle rocks under external loads can be approximated as a composite containing a matrix phase and a number of microcracks (Horii and Nemat-Nasser, 1983, Zhu et al., 2011). For simplicity, assume
Numerical implementation
In non-linear finite element analysis, integrating the constitutive relation is the kernel for obtaining the unidentified stress increment. Generally speaking, we accept the classification of integrating approaches, namely explicit, implicit and semi-implicit algorithms. Inspired by previous works (Halilovič et al., 2009, Zhu et al., 2016), an explicit return mapping (ERM) integration algorithm combined with the plasticity-damage coupling procedure is proposed for achieving a more
Identification of the model’s parameters
The proposed model contains only eight material/model’s parameters and each parameter has its clear physical meaning corresponds to the macroscopic mechanical response. The calibration of these parameters can be carried out by applying traditional triaxial compression tests under different confining pressures.
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The Young’s modulus and Poisson’s ratio , can be obtained from the linear part of stress–strain curves in the pre-peak phase from traditional triaxial compression experiments. As far
Model validation
In this section, numerical simulations on Beishan granite and Shandong sandstone are conducted to evaluate the predictive capability of the proposed model and the effectiveness of the developed algorithm. It is worth noting that only cracking-related mechanisms are taken into account, while other mechanisms such as pore collapse and plastic slip of crystals are ignored.
Concluding remarks
In this work, a novel constitutive model was formulated for mechanical behaviors of quasi-brittle rocks by combining the Mori–Tanaka homogenization method and the fractional plasticity theory. The proposed approach is efficient for micro-mechanical modeling of mechanical behaviors of quasi-brittle rocks by taking into account the coupling between frictional sliding and crack growth. With the help of the introduced fractional plastic flow, the volume compression/dilation transition property can
CRediT authorship contribution statement
Peng-Fei Qu: Data curation, Formal analysis, Investigation, Methodology, Software, Validation, Writing – original draft. Qi-Zhi Zhu: Conceptualization, Funding acquisition, Methodology, Supervision, Visualization, Writing – review & editing, Resources. Lun-Yang Zhao: Supervision, Writing – review & editing. Ya-Jun Cao: Supervision, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work has been jointly supported by the Fundamental Research Funds for the Central Universities, China (Grant No. B210203014), the National Key Research and Development Program of China (Grant No. 2017YFC1501102), Guangdong Basic and Applied Basic Research Foundation, China (Grant No. 2019A1515110626) and China Post-doctoral Science Foundation funded project (Grant Nos. 2020M682707, 2021T140219).
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