A thermodynamic modeling of creep in rock salt
Introduction
Currently, a variety of constitutive models are used to simulate the creep behavior of rock salt in underground facilities. Examples are the Sandia's Multimechanism Deformation Model,1 SUVIC,2 the viscoplastic model by Cristescu,3 the Composite Model,4 Gunther-Salzer,5 Lux-Wolters6 and Lubby2.7 Since 2004, these models have been documented and validated.8, 9, 10, 11, 12 Some are based on deformation mechanisms, others are rheological, viscoplastic or micromechanical models.
Creep constitutive models for rock materials have been developed in the framework of thermodynamics of irreversible processes.13, 14, 15 This theoretical framework can be applied to rock salt.
Thermodynamics applied to continua has evolved through the work of Coleman,16 Rice,17 Ziegler,18 Germain,19 Lemaitre and Chaboche,20 Maugin,21 and Houlsby and Puzrin.22 According to this framework, the first and second law of thermodynamics are enforced in the mathematical formulation. As a result, the whole constitutive response of the material can be obtained from two scalar valued convex functions that act as thermodynamical potentials. The main advantages of this approach are:
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It is a solid theoretical basis able to predict a surprising variety of phenomena.
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It consists of a systematic procedure for constructing constitutive models that stay compact and organized.
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It ensures thermodynamically consistent responses even in general circumstances.
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It only requires the specification of two scalar functions, rather than a set of tensorial relationships.
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It correctly models stored and dissipated energy in any irreversible process.
Our goal is to formulate a new creep model for rock salt within the framework of continuum thermodynamics. In doing so, we assume that the elastic behavior is linear and isotropic, and the creep behavior is non-linear and isochoric. To model the stationary creep, we adopt two operating deformation mechanisms, following the arguments of Refs. 23, 24, 25.
Moreover, our model includes a backstress and also partitions the creep strain into transient and steady parts. Although it may seem physically artificial, opting for the strain partitioning makes the mathematical definition of stationary creep very easy. Nonetheless, thanks to the coupling enforced in the dissipation function, transient creep and steady-state creep are not independent; they are governed by the same deformation mechanisms. By using a backstress, our model can reproduce directional hardening as well.
In the first part, we present the theoretical framework to construct the model and give explicit expressions for the thermodynamic potentials. In the second part, we evaluate the model in predicting the creep behavior of rock salt. In the Appendix, we report the analytical derivation of the constitutive equations.
Section snippets
Theoretical framework
For simplicity, we restrict the model to small deformations. According to the partitioned approach, the strain rate tensor is decomposed into three parts:where is the elastic strain rate, and are the creep rates, respectively, the transient (primary creep) and the steady-state (secondary creep).
We assume the existence of a decoupled Helmholtz free energy describing the reversible processes:
To ensure that the principles of
Results and discussion
This section presents a brief evaluation of the proposed model in predicting the creep behavior of rock salt. Rather than making accurate predictions, the objective here is to assess the general behavior of the model, capturing the main features of both transient and steady-state creep. The model requires eleven material constants to simulate the creep behavior of rock salt: K, G, c, A1, Q1, n1, M1, A2, Q2, n2, M2. Of these eleven, two define the elastic behavior, three define the primary creep
Conclusions
This note presented how to derive a double-mechanism deformation model in the framework of continuum thermodynamics. We postulated a free energy function and a dissipation function to establish a new set of creep constitutive equations for rock salt. In choosing the thermodynamic potentials, given in Equations (9), (10), we accounted for linear elasticity, non-linear isochoric creep and kinematic hardening. A superposition in the dissipation potential was made to model stationary creep
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 study was developed with the ANP research and developed incentive law n°9.478, 06/08/1997, and authors thank Repsol Sinopec Brasil for all support to this work. Also, this study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Financial Code 001.
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