A thermodynamic modeling of creep in rock salt

doi:10.1016/j.ijrmms.2022.105298

International Journal of Rock Mechanics and Mining Sciences

Volume 162, February 2023, 105298

https://doi.org/10.1016/j.ijrmms.2022.105298 Get rights and content

Abstract

A number of constitutive models have been proposed along time in order to describe the mechanical behavior of rock salt. The basic formulation of these models relies, in general, on the description of deformation mechanisms in the macro or the micro scale. The present note aims at providing a framework for the development of models to describe the creep behavior of rock salts based upon continuum thermodynamics principles. This note provides the theoretical basis and, based on the proposed framework, presents the development of a two-mechanism model that describes transient and steady-state creep of rock salt. Results obtained with this model are presented as well as a discussion on the main features of the proposed formulation.

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:

•
It is a solid theoretical basis able to predict a surprising variety of phenomena.
•
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.
•
It only requires the specification of two scalar functions, rather than a set of tensorial relationships.
•
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 ${\dot{ε}}_{i j}$ is decomposed into three parts: ${\dot{ε}}_{i j} = {\dot{ε}}_{i j}^{e} + {\dot{ε}}_{i j}^{t} + {\dot{ε}}_{i j}^{s}$ where ${\dot{ε}}_{i j}^{e}$ is the elastic strain rate, ${\dot{ε}}_{i j}^{t}$ and ${\dot{ε}}_{i j}^{s}$ 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: $ψ = ψ_{e} (ε_{i j}^{e}) + ψ_{t} (ε_{i j}^{t})$

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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A thermodynamic modeling of creep in rock salt

Abstract

Introduction

Section snippets

Theoretical framework

Results and discussion

Conclusions

Declaration of competing interest

Acknowledgments

Int J Rock Mech Min Sci

Int J Rock Mech Min Sci

Int J Rock Mech Min Sci Geomech Abstracts

Eng Geol

Comput Geotech

Comput Struct

Mech Res Commun

Int J Rock Mech Min Sci

Int J Solid Struct

J Mech Phys Solid

Int J Rock Mech Min Sci

Int J Rock Mech Min Sci Geomech Abstracts

Tectonophysics

Steady-state creep of rock salt: improved approaches for lab determination and modelling

Rock Mech Rock Eng

Comparison of advance constitutive models for the mechanical behavior of rock salt – results from a joint research project. I. Modeling of deformation processes and benchmark calculations

Joint projects on the comparison of constitutive models for the mechanical behavior of rock salt. Overview of the models and results of 3-D benchmark calculations