Networks and Heterogeneous Media
December 2014 , Volume 9 , Issue 4
Special issue on the mathematics of concrete
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Although the concrete is a simple man-made material with initially-controlled composition (for instance, all ingredients are known beforehand, the involved chemical mechanisms are well studied, the mechanical strength of test samples is measured accurately), forecasting its behaviour for large times under variable external (boundary) conditions is not properly understood. The main reason is that the simplicity of the material is only apparent. The combination of the heterogeneity of the material together with the occurrence of a number of multiscale phase transitions either driven by aggressive chemicals (typically ions, like in corrosion situations), or by extreme heating, or by freezing/thawing of the ice lenses within the microstructure, and the inherent non-locality of the mechanical damage leads to mathematically challenging nonlinear coupled systems of partial differential equations (PDEs).
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Failure of quasi-brittle materials such as concrete needs a proper description of strain softening due to progressive micro-cracking and the introduction of an internal length in the constitutive model in order to achieve non zero energy dissipation. This paper reviews the main results obtained with the non local damage model, which has been among the precursors of such models. In most cases up to now, the internal length has been considered as a constant. There is today a consensus that it should not be the case as models possess severe shortcomings such as incorrect averaging near the boundaries of the solid considered and non local transmission across non convex boundaries. An interaction-based model in which the weight function is constructed from the analysis of interaction has been proposed. It avoids empirical descriptions of the evolution of the internal length. This model is also recalled and further documented. Additional results dealing with spalling failure are discussed. Finally, it is pointed out that this model provides an asymptotic description of complete failure, which is consistent with fracture mechanics.
We study a two-scale homogenization problem describing the linearized poro-elastic behavior of a periodic two-component porous material exhibited to a slightly compressible viscous fluid flow and a first-order chemical reaction. One material component consists of disconnected parts embedded in the other component which is supposed to be connected. It is shown that a memory effect known from the purely mechanic problem gets inherited by the reaction component of the model.
We study spring-block systems which are equivalent to the P1-finite element methods for the linear elliptic partial differential equation of second order and for the equations of linear elasticity. Each derived spring-block system is consistent with the original partial differential equation, since it is discretized by P1-FEM. Symmetry and positive definiteness of the scalar and tensor-valued spring constants are studied in two dimensions. Under the acuteness condition of the triangular mesh, positive definiteness of the scalar spring constant is obtained. In case of homogeneous linear elasticity, we show the symmetry of the tensor-valued spring constant in the two dimensional case. For isotropic elastic materials, we give a necessary and sufficient condition for the positive definiteness of the tensor-valued spring constant. Consequently, if Poisson's ratio of the elastic material is small enough, like concrete, we can construct a consistent spring-block system with positive definite tensor-valued spring constant.
The subject of the present paper is the derivation and asymptotic analysis of a mathematical model for the formation of a mushy region during sulphation of calcium carbonate. The model is derived by averaging, with the use of the multiple scales method, applied on microscopic moving - boundary problems. The latter problems describe the transformation of calcium carbonate into gypsum on the microscopic scale. The derived macroscopic model is solved numerically with the use of a finite element method. The results of some simulations and a relevant discussion are also presented.
In this paper we deal with a one-dimensional free boundary problem, which is a mathematical model for an adsorption phenomena appearing in concrete carbonation process. This model was proposed in line of previous studies of three dimensional concrete carbonation process. The main result in this paper is concerned with the existence and uniqueness of a time-local solution to the free boundary problem. This result will be obtained by means of the abstract theory of nonlinear evolution equations and Banach's fixed point theorem, and especially, the maximum principle applied to our problem will play a very important role to obtain the uniform estimate to approximate solutions.
In this paper we consider a three-component reaction-diffusion system with a fast precipitation and dissolution reaction term. We investigate its singular limit as the reaction rate tends to infinity. The limit problem is described by a combination of a Stefan problem and a linear heat equation. The rate of convergence with respect to the reaction rate is established in a specific case.
When dealing with concrete materials it is always a big issue how to deal with the moisture transport. Here, we consider a mathematical model for moisture transport, which is given as a system consisting of the diffusion equation for moisture and of the ordinary differential equation which describes a hysteresis operator. In  we already proved the existence of a time global solution of an initial boundary value problem of this system, however, the uniqueness is obtained only for one dimensional domains. The main purpose of this paper is to establish the uniqueness of a solution of our problem in three dimensional domains under the assumption of the smooth boundary and initial data.
We study the homogenization of a reaction-diffusion-convection system posed in an $\varepsilon$-periodic $\delta$-thin layer made of a two-component (solid-air) composite material. The microscopic system includes heat flow, diffusion and convection coupled with a nonlinear surface chemical reaction. We treat two distinct asymptotic scenarios: (1) For a fixed width $\delta>0$ of the thin layer, we homogenize the presence of the microstructures (the classical periodic homogenization limit $\varepsilon\to 0$); (2) In the homogenized problem, we pass to $\delta\to 0$ (the vanishing limit of the layer's width). In this way, we are preparing the stage for the simultaneous homogenization ($\varepsilon\to 0$) and dimension reduction limit ($\delta\to 0$) with $\delta=\delta(\epsilon)$. We recover the reduced macroscopic equations from  with precise formulas for the effective transport and reaction coefficients. We complement the analytical results with a few simulations of a case study in smoldering combustion. The chosen multiscale scenario is relevant for a large variety of practical applications ranging from the forecast of the response to fire of refractory concrete, the microstructure design of resistance-to-heat ceramic-based materials for engines, to the smoldering combustion of thin porous samples under microgravity conditions.
We study the solvability and homogenization of a thermal-diffusion reaction problem posed in a periodically perforated domain. The system describes the motion of populations of hot colloidal particles interacting together via Smoluchowski production terms. The upscaled system, obtained via two-scale convergence techniques, allows the investigation of deposition effects in porous materials in the presence of thermal gradients.
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