Chloé Arson

Anisotropic Damage Models for Geomaterials: Theoretical and Numerical Challenges

A new anisotropic damage model for rock is formulated and discussed. Flow rules are derived with the energy release rate conjugate to damage, which is thermodynamically consistent. Drucker–Prager yield function is adapted to make the damage threshold depend on damage energy release rate and to distinguish between tension and compression strength. Positivity of dissipation is ensured by using a nonassociate flow rule for damage, while nonelastic deformation due to damage is computed by an associate flow rule. Simulations show that the model meets thermodynamic requirements, follows a rigorous formulation, and predicts expected trends for damage, deformation and stiffness.

A Model of Damage and Healing Coupling Halite Thermo-mechanical Behavior to Microstructure Evolution

Abstract Creep processes in halite (salt rock) include glide, cross-slip, diffusion and dynamic recrystallization. Diffusive mass transfer (DMT) can result in crack rebonding, and mechanical stiffness recovery. Crack rebonding driven by DMT occurs within a few days at room temperature and low pressure. DMT is enhanced at higher temperatures, which could be beneficial for the sustainabilty of geological storage facilities in salt mines. On the one hand, visco-plastic laws relating creep microscopic processes to microstructure changes are empirical. On the other hand, theoretical models of damage and healing disconnect thermodynamic variables from their physical meaning. The proposed model enriches the framework of continuum damage mechanics with fabric descriptors. In order to infer the form of fabric tensors from microstructure observation, creep tests were carried out on granular salt under constant stress and humidity conditions. The evolution of net damage is governed by a diffusion equation, in which the characteristic time scales with the typical size of halite crystals, and the diffusion coefficient is a function of temperature. A stress path comprising a tensile loading, a compressive unloading, a creep–healing stage and a reloading was simulated. Macroscopic and microscopic model predictions highlight the increased efficiency of healing with time and temperature. The model presented in this paper is expected to improve the fundamental understanding of damage and healing in rocks at both macroscopic and microscopic levels, and the long-term assessment of geological storage facilities.

Analysis of Friction Induced Thermo-Mechanical Stresses on a Heat Exchanger Pile in Isothermal Soil

Abstract In most analytical and numerical models of heat exchanger piles, strain incompatibilities between the soil and the pile are neglected, and axial stresses imposed by temperature changes within the pile are attributed to the thermal elongation and shortening of the pile. These models incorporate thermo-hydro-mechanical couplings in the soil and within the pile foundation, but usually neglect thermo-mechanical couplings between the two media. Previous studies assume that the stress changes imposed by temperature variations in a heat exchanger pile are mainly due to the constrained thermal elongation and shortening of the pile. Also, several recent approaches utilize spring models that focus only on the soil-pile interface in modeling temperature-induced stresses in a heat exchanger pile and implicitly ignore the effect of the full displacement field on soil-pile interaction. By contrast, in this paper, interface elements are introduced in a numerical model of a heat exchanger pile, analyzed in axisymmetric and stationary conditions. The pile is subjected to a uniform temperature increase, with free top and fixed top conditions in elastic and elasto-plastic soil profiles. Simulation results show that the constrained vertical elongation is the most detrimental factor for pile foundation performance. However it is worth noticing that while mechanical constraints (e.g., fixed top and/or fixed bottom) impose maximum stress increases at the ends of the pile, interface effects result in maximum stresses around the mid-length of the pile. This preliminary study indicates that soil-pile friction does not increase pile internal stresses to the point where it would be necessary to over-dimension the foundation pile for heat exchanger use. Furthermore, one cannot expect a significant gain in foundation performance due to the improvement of soil-pile frictional resistance as a result of increased lateral stresses at soil-pile contact. Additional numerical analyses are ongoing, in order to investigate the role of the degree of fixity induced by the building on the heat exchanger pile, and to extend these preliminary analyses to transient operational modes and cyclic thermo-mechanical loading of the heat exchanger pile.

Chemo-mechanical evolution of pore space in carbonate microstructures upon dissolution: linking pore geometry to bulk elasticity†

One of the challenges faced today in a variety of geophysical applications is the need to understand the changes of elastic properties due to time-variant chemomechanical processes. The objective of this work is to model carbonate rock elastic properties as functions of pore geometry changes that occur when the solid matrix is dissolved by carbon dioxide. We compared two carbonate microstructures: porous micrite (“mudstone”) and grain-supported carbonate (“packstone”). We formulated a mathematical model that distinguishes the effects of microporosity and macroporosity on stiffness changes. We used measures of mechanical and chemical porosity changes recorded during injection tests to compute elastic moduli and compare them to moduli obtained from wave velocity measurements. In mudstones, both experimental and numerical results indicate that bulk moduli change by less than 5%. The evolution of elastic moduli is controlled by macropore enlargement. In packstones, model predictions underestimate changes of elastic moduli with total porosity by 10% to 80%. The total porosity variation is 60% to 75% smaller than the chemical porosity variation, which indicates that pore expansion due to dissolution is counterbalanced by pore shrinkage due to compaction. Packstone elastic properties are controlled by grain sliding. The methodology presented in this paper can be generalized to other chemomechanical processes studied in rocks, such as dislocations, glide, diffusive mass transfer, recrystallization, and precipitation.

Micromechanics based discrete damage model with multiple non-smooth yield surfaces: theoretical formulation, numerical implementation and engineering applications

The discrete damage model presented in this paper accounts for 42 non-interacting crack microplanes directions. At the scale of the representative volume element, the free enthalpy is the sum of the elastic energy stored in the non-damaged bulk material and in the displacement jumps at crack faces. Closed cracks propagate in the pure mode II, whereas open cracks propagate in the mixed mode (I/II). The elastic domain is at the intersection of the yield surfaces of the activated crack families, and thus describes a non-smooth surface. In order to solve for the 42 crack densities, a Closest Point Projection algorithm is adopted locally. The representative volume element inelastic strain is calculated iteratively using the Newton–Raphson method. The proposed damage model was rigorously calibrated for both compressive and tensile stress paths. Finite element method simulations of triaxial compression tests showed that the transition between brittle and ductile behavior at increasing confining pressure can be captured. The cracks’ density, orientation, and location predicted in the simulations are in agreement with experimental observations made during compression and tension tests, and accurately show the difference between tensile and compressive strength. Plane stress tension tests simulated for a fiber-reinforced brittle material also demonstrated that the model can be used to interpret crack patterns, design composite structures and recommend reparation techniques for structural elements subjected to multiple damage mechanisms.

Mechanistic Analysis of Rock Damage Anisotropy and Rotation Around Circular Cavities

We used the differential stress-induced damage (DSID) model to predict anisotropic crack propagation under tensile and shear stress. The damage variable is similar to a crack density tensor. The damage function and the damage potential are expressed as functions of the energy release rate, defined as the thermodynamic force that is work-conjugate to damage. Contrary to the previous damage models, flow rules are obtained by deriving dissipation functions by the energy release rate, and thermodynamic consistency is ensured. The damage criterion is adapted from the Drucker–Prager yield function. Simulations of biaxial stress tests showed that: (1) three-dimensional states of damage can be obtained for three-dimensional states of stress; (2) no damage propagates under isotropic compression; (3) crack planes propagate in the direction parallel to major compression stress; (4) damage propagation hardens the material; (5) stiffness and deformation anisotropy result from the anisotropy of damage. There is no one-to-one relationship between stress and damage. We demonstrated the effect of the loading sequence in a two-step simulation (a shear loading phase and a compression loading phase): the current state of stress and damage can be used to track the effect of stress history on damage rotation. We finally conducted a sensitivity analysis with the finite element method, to explore the stress conditions in which damage is expected to rotate around a circular cavity subject to pressurization or depressurization. Simulation results showed that: (1) before damage initiation, the DSID model matches the analytical solution of stress distribution obtained with the theory of elasticity; (2) the DSID model can predict the extent of the tensile damage zone at the crown, and that of the compressive damage zone at the sidewalls; (3) damage generated during a vertical far-field compression followed by a depressurization of the cavity is more intense than that generated during a depressurization of the cavity followed by a vertical far-field compression.

A MULTIPHASE ANALYSIS FOR ENVIRONMENTAL IMPACT ASSESSMENT WITH θ-STOCK FINITE ELEMENT PROGRAM

In the THM modeling of multiphase medium, the coupling effects of skeleton, suction, and temperature have been integrated via the concept of state surfaces of void ratio and degree of saturation. Based on proposed formulation, a fully coupled numerical model for the behavior of soil deformation, water flow, air flow, heat flow in unsaturated soil has been developed and integrated in a finite element code θ-Stock by the first author. This program is conceived with this idea that it will be able to analyze the response of a soil in different states of humidity to mechanical, thermal loading, and also damage phenomena. Damage model is dedicated to unsaturated brittle rocks. It mixes phenomenological and micromechanical concepts and is formulated based on the use of independent state variables. The expression of the liquid permeability is modified in order to represent the influence of fracturing on interstitial fluid flows. The final matrix form of established field equations of the proposed model for unsaturated case has been encoded for this particular purpose, in a finite element program which had been developed for dry and saturated soils previously.

Micro-Macro Analysis and Phenomenological Modelling of Salt Viscous Damage and Application to Salt Caverns

AbstractThis paper aims to gain fundamental understanding of the microscopic mechanisms that control the transition between secondary and tertiary creep around salt caverns in typical geological storage conditions. We use a self-consistent inclusion-matrix model to homogenize the viscoplastic deformation of halite polycrystals and predict the number of broken grains in a Representative Elementary Volume of salt. We use this micro-macro modeling framework to simulate creep tests under various axial stresses, which gives us the critical viscoplastic strain at which grain breakage (i.e., tertiary creep) is expected to occur. The comparison of simulation results for short-term and long-term creep indicates that the initiation of tertiary creep depends on the stress and the viscoplastic strain. We use the critical viscoplastic deformation as a yield criterion to control the transition between secondary and tertiary creep in a phenomenological viscoplastic model, which we implement into the Finite Element Method program POROFIS. We model a 850-m-deep salt cavern of irregular shape, in axis-symmetric conditions. Simulations of cavern depressurization indicate that a strain-dependent damage evolution law is more suitable than a stress-dependent damage evolution law, because it avoids high damage concentrations and allows capturing the formation of a damaged zone around the cavity. The modeling framework explained in this paper is expected to provide new insights to link grain breakage to phenomenological damage variables used in Continuum Damage Mechanics.

Modeling the Influence of Thermo-Mechanical Crack Opening and Closure on Rock Stiffness

A thermodynamic framework is proposed to model the effect of mechanical stress and temperature on crack opening and closure in rocks. The model is based on Continuum Damage Mechanics with damage defined as the second-order crack density tensor. The free energy of damaged rock is expressed as a function of deformation, temperature and damage. The damage criterion controls mode I crack propagation, captures temperature-induced decrease of rock toughness, and accounts for the increase of energy release rate necessary to propagate cracks induced by damage. Crack closure is modeled through unilateral effects produced on rock stiffness. Simulations show that: (1) under anisotropic mechanical boundary conditions, crack closure occurs during cooling, (2) the thermo-mechanical strain energy necessary to close cracks during cooling is larger than the strain energy needed to close the cracks by mechanical compression. Parametric study highlights the thermo-mechanical stress redistributions occurring during closure. The proposed framework is expected to bring new insights in the design and reliability assessment of geotechnical reservoirs and repositories.

Simulation of the Unsaturated Excavation Damage Zone Around a Tunnel Using A Fully Coupled Damage-Plasticity Model

During tunnel excavation, stress redistribution produces plastic deformation and damage around the opening. Moreover, the surrounding soil can be either saturated or unsaturated. Suction has a significant influence on the mechanical behaviour of geomaterials. Depending on their stress state and on their moisture content, clay-based materials can exhibit either a ductile or a brittle behaviour. Plasticity leads to permanent strains and damage causes the deterioration of the soil elastic and hydraulic properties. The damage-plasticity model proposed in this work is formulated in terms of a damaged constitutive stress, defined from the principle of Bishops hydro-mechanical stress (for unsaturated conditions), and from the principle of damaged effective stress used in Continuum Damage Mechanics. The evolution laws are obtained by using the principle of strain equivalence. This hydro-mechanical damage-plasticity model was implemented in a Finite Element code. The excavation of a tunnel is simulated at different constant suctions. The results obtained illustrate the influence of suction on the development of plastic and damaged zones.