Bernhard A. Schrefler

THE CARBONATION OF CONCRETE AND THE MECHANISM OF MOISTURE, HEAT AND CARBON DIOXIDE FLOW THROUGH POROUS MATERIALS

Abstract The governing equations of moisture, heat and carbon dioxide flows through concrete within the framework of a distributed parameter model are described. The coupling terms and the non-linearity of the problem are taken into account and a numerical procedure based on the finite element method is developed to solve the set of equations. The influence of relative humidity and temperature is investigated and typical results are presented. Comparisons with experimental tests are also carried out and one example is presented in detail in order to show the reliability and the effectiveness of the proposed numerical model.

Thermo‐hydro‐mechanical analysis of partially saturated porous materials

Presents a fully coupled numerical model to simulate the slow transient phenomena involving heat and mass transfer in deforming partially saturated porous materials. Makes use of the modified effective stress concept together with the capillary pressure relationship. Examines phase changes (evaporation‐condensation(, heat transfer through conduction and convection, as well as latent heat transfer. The governing equations in terms of gas pressure, capillary pressure, temperature and displacements are coupled non‐linear differential equations and are discretized by the finite element method in space and by finite differences in the time domain. The model is further validated with respect to a documented experiment on partially saturated soil behaviour, and the effects of two‐phase flow, as compared to the one‐phase flow solution, are analysed. Two other examples involving drying of a concrete wall and thermoelastic consolidation of partially saturated clay demonstrate the importance of proper physical modelling and of appropriate choice of the boundary conditions.

Thermodynamic approach to effective stress in partially saturated porous media

Abstract The form of the equilibrium effective stress acting on the solid phase of a porous medium containing two immiscible fluid phases is derived. The derivation makes use of the postulation of the thermodynamics of the system at the macroscale, a scale on the order of tens of pore diameters. The postulation at this scale facilitates the identification of the fraction of the solid surface in contact with each fluid phase as being the appropriate coefficient weighting each of the fluid phase pressures analogous to the Bishop parameter. In addition, the curvature of the surface of the solid phases is shown to impact the pressure exerted on the solid phase by the fluid. For the special case of low saturations when the wetting phase may be considered to be present only as a film on the solid phase, the macroscale disjoining pressure is found to modify the equilibrium form of the effective stress. In addition to the equilibrium effective stress, which is related to the forces acting on the interface between the solid phase and the fluids, the appropriate relation between the fluid pressures at the fluid–fluid interface is obtained. This analysis leads to the expression for the capillary pressure as a function of the phase pressures and the disjoining pressure.

A fully coupled model for water flow and airflow in deformable porous media

A fully coupled model is developed to simulate the slow transient phenomena (consolidation) involving flow of water and air in deforming porous media. The model is of the Biot type and incorporates the capillary pressure relationship. The finite element method is used for the discrete approximation of the partial differential equations governing the problem. The temporal discretization error, iteration error and stability error are evaluated. The model is validated with respect to a documented experiment on semisaturated soil behavior. Other examples involving an air storage problem in an aquifer and a flexible footing resting on a semisaturated soil are also presented.

A fully coupled dynamic model for two-phase fluid flow in deformable porous media

Abstract A fully coupled dynamic model is presented for the analysis of water and air flow in deforming porous media, in fully or partially saturated conditions. The solid displacements and the pressures of fluids are taken as primary unknowns of the model. The finite element method is used for the discrete approximation of the partial differential equations governing the problem. The mathematical framework and the numerical implementation of the model are given in detail and the adopted approximations are put into evidence. First the model is validated with respect to documented experiments on partially saturated soil behaviour in quasi-static condition. Then the results of a full dynamic analysis are shown and discussed. In this paper, merits and drawbacks of the proposed model are highlighted.

2 — D model for carbonation and moisture/heat flow in porous materials

The previously-published one-dimensional finite element model for the analysis of the carbonation mechanism is extended to two-dimensional problems. The governing equations for the propagation of aggressive agents through concrete are rewritten for two-dimensional domains. A comparison is made with the one-dimensional model and some examples are developed to test the method. Finally, the study of a reinforcing bar placed at the comer of a concrete structure is presented in detail to show that the proposed numerical model is able to demonstrate the effects of the multidimensional moisture, heat and carbon dioxide transport through concrete.

A multiphase medium model for localisation and postlocalisation simulation in geomaterials

SUMMARY It is recalled that negative water pressures are of importance in localisation phenomena of fully saturated undrained samples of dilatant geomaterials. A model to simulate cavitation phenomena connected with such pore water tractions is developed and implemented in a simplified form in a dynamics code for partially saturated porous media. A case of localisation is studied from the onset of the instability up to the full developed shear band. The weak mesh dependence of the maximum effective plastic strain, due to the employed physical model, is also shown.

A method for 3-D hydraulic fracturing simulation

We present a method for the simulation of 3-D hydraulic fracturing in fully saturated porous media. The discrete fracture(s) is driven by the fluid pressure. A cohesive fracture model is adopted where the fracture follows the face of the elements around the fracture tip which is closest to the normal direction of the maximum principal stress at the fracture tip. No predetermined fracture path is needed. This requires continuous updating of the mesh around the crack tip to take into account the evolving geometry. The updating of the mesh is obtained by means of an efficient mesh generator based on Delaunay tessellation. The governing equations are written in the framework of porous media mechanics theory and are solved numerically in a fully coupled manner. An examples dealing with a concrete dam is shown.

THE EFFECTIVE STRESS PRINCIPLE: INCREMENTAL OR FINITE FORM?

SUMMARY Different expressions of the effective stress principle can be found in the literature, in particular some are written in finite form and others in incremental form. For the purpose of the paper we take for granted that stress-strain relationships exist or can be obtained for the effective stress coming from both formulations. We investigate the consequences of the choice of particular finite or differential forms when they are introduced in a weak form of the linear momentum balance equation of two- of three-phase porous media for its numerical solution. For partially saturated geomaterials the importance of the capillary pressure-saturation relationship is pointed out. KEY WORD: effective stress principle; stressstrain relationships; porous media

A multiphase model for three-dimensional tumor growth

Several mathematical formulations have analyzed the time-dependent behaviour of a tumor mass. However, most of these propose simplifications that compromise the physical soundness of the model. Here, multiphase porous media mechanics is extended to model tumor evolution, using governing equations obtained via the Thermodynamically Constrained Averaging Theory (TCAT). A tumor mass is treated as a multiphase medium composed of an extracellular matrix (ECM); tumor cells (TC), which may become necrotic depending on the nutrient concentration and tumor phase pressure; healthy cells (HC); and an interstitial fluid (IF) for the transport of nutrients. The equations are solved by a Finite Element method to predict the growth rate of the tumor mass as a function of the initial tumor-to-healthy cell density ratio, nutrient concentration, mechanical strain, cell adhesion and geometry. Results are shown for three cases of practical biological interest such as multicellular tumor spheroids (MTS) and tumor cords. First, the model is validated by experimental data for time-dependent growth of an MTS in a culture medium. The tumor growth pattern follows a biphasic behaviour: initially, the rapidly growing tumor cells tend to saturate the volume available without any significant increase in overall tumor size; then, a classical Gompertzian pattern is observed for the MTS radius variation with time. A core with necrotic cells appears for tumor sizes larger than 150 μm, surrounded by a shell of viable tumor cells whose thickness stays almost constant with time. A formula to estimate the size of the necrotic core is proposed. In the second case, the MTS is confined within a healthy tissue. The growth rate is reduced, as compared to the first case - mostly due to the relative adhesion of the tumor and healthy cells to the ECM, and the less favourable transport of nutrients. In particular, for tumor cells adhering less avidly to the ECM, the healthy tissue is progressively displaced as the malignant mass grows, whereas tumor cell infiltration is predicted for the opposite condition. Interestingly, the infiltration potential of the tumor mass is mostly driven by the relative cell adhesion to the ECM. In the third case, a tumor cord model is analyzed where the malignant cells grow around microvessels in a 3D geometry. It is shown that tumor cells tend to migrate among adjacent vessels seeking new oxygen and nutrient. This model can predict and optimize the efficacy of anticancer therapeutic strategies. It can be further developed to answer questions on tumor biophysics, related to the effects of ECM stiffness and cell adhesion on tumor cell proliferation.