Various multiscale methods are reviewed in the context of modelling mechanical and thermo-mechanical responses of composites. They are developed both at the material level and at the structural analysis level, considering sequential or integrated kinds of approaches. More specifically, such schemes like periodic homogenization or mean field approaches are compared and discussed, especially in the context of non linear behaviour. Some recent developments are considered, both in terms of numerical methods (like FE2) and for more analytical approaches based on Transformation Field Analysis, considering both the homogenization and relocalisation steps in the multiscale methodology. Several examples are shown

SUMMARYPart 1 of the paper presents a new numerical model of hygro-thermal and hydration phenomena in concrete at early ages and beyond. This is a solidification-type model where all changes of material properties are expressed as functions of hydration degree, and neither as maturity nor as equivalent hydration period as in maturity-type models. A mechanistic approach has been used to obtain the governing equations, by means of an averaging theory of Hassanizadeh and Gray, also called hybrid mixture theory. The developments start at the micro-scale and balance equations for phases and interfaces are introduced at this level and then averaged for obtaining macroscopic balance equations. Constitutive laws are directly introduced at macroscopic level. The final equations, mass (water species and dry air), energy and momentum balance equations, have been written in terms of the chosen primary variables: gas pressure, capillary pressure, temperature and displacements. An evolution equation for the internal variable, hydration degree, describes hydration rate as a function of chemical affinity, considering in addition to the existing models, an effect of the relative humidity on the process. The model takes into account full coupling between hygral, thermal and chemical phenomena, as well as changes of concrete properties caused by hydration process, i.e. porosity, density, permeability, and strength properties. Phase changes and chemical phenomena, as well as the related heat and mass sources are considered. Two examples showing possibilities of the model for analysis of autogenous self-heating and self-desiccation phenomena, as well as influence of the

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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.

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SUMMARYIn Part I of this paper (Int. J. Numer. Meth. Eng., in print) a mechanistic model of hygro-thermochemical performance of concrete at early ages has been introduced. Additionally, as compared to the existing models (e.g. J. Eng. Mech. (ASCE) 1995; 121(7):785-794; 1999; 125(9):1018-1027), an effect of relative humidity on cement hydration rate and associated hygro-thermal phenomena have been taken into account. Here we deal with mechanical performance of concrete at early ages and beyond, and in particular, evolution of its strength properties (aging) and deformations (shrinkage and creep strains), described by using the effective stress concept. This allow us for explanation and modelling of phenomena known from experiments, like drying creep (e.g.

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.

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