Cosserat model for periodic masonry deduced by nonlinear homogenization

The paper deals with the problem of the determination of the in-plane behavior of periodic masonry material. The macromechanical equivalent Cosserat medium, which naturally accounts for the absolute size of the constituents, is derived by a rational homogenization procedure based on the Transformation Field Analysis. The micromechanical analysis is developed considering a Cauchy model for masonry components. In particular, a linear elastic constitutive relationship is considered for the blocks, while a nonlinear constitutive law is adopted for the mortar joints, accounting for the damage and friction phenomena occurring during the loading history. Some numerical applications are performed on a Representative Volume Element characterized by a selected commonly used texture, without performing at this stage structural analyses. A comparison between the results obtained adopting the proposed procedure and a nonlinear micromechanical Finite Element Analysis is presented. Moreover, the substantial differences in the nonlinear behavior of the homogenized Cosserat material model with respect to the classical Cauchy one, are illustrated.     

A discrete model for long time sintering

A discrete model for the sintering of polydisperse, inhomogeneous arrays of cylinders is presented with empirical contact force-laws, taking into account plastic deformations, cohesion, temperature dependence (melting), and long-time effects. Samples are prepared under constant isotropic load and are sintered for different sintering times. Increasing both external load and sintering time leads to a stronger, stiffer sample after cooling down. The material behavior is interpreted from both microscopic and macroscopic points of view.Among the interesting results is the observation that the coordination number, even though it has the tendency to increase, sometimes slightly decreases, whereas the density continuously increases during sintering—this is interpreted as an indicator of reorganization effects in the packing. Another result of this study is the finding that strongly attractive contacts occur during cool-down of the sample and leave a sintered block of material with almost equally strong attractive and repulsive contact forces.     

A brick model for asperity sintering and creep of APS TBCs

A micromechanical model is developed for the microstructural evolution of an air plasma sprayed (APS), thermal barrier coating: discrete, brick-like splats progressively sinter together at contacting asperities and also undergo Coble creep within each splat. The main microstructural features are captured: the shape, orientation and distribution of asperities between disc-shaped splats, and the presence of columnar grains within each splat. Elasticity is accounted for at the asperity contacts and within each splat, and the high contact compliance explains the fact that APS coatings have a much lower modulus (and thermal conductivity) than that of the parent, fully dense solid. The macroscopic elastic, sintering and creep responses are taken to be transversely isotropic, and remain so with microstructural evolution. Despite the large number of geometric and kinetic parameters, the main features of the behaviour are captured by a small number of characteristic material timescales: these reveal the competition between the deformation mechanisms and identify the rate controlling processes for both free and constrained sintering. The evolution of macroscopic strain, moduli and asperity size is compared for free and constrained sintering, and the level of in-plane stress within a constrained coating is predicted.     

Determination of added fluid area in the homogenization model of beam bundles

A unique scalar parameter arises in the 3-D homogenization model for the beam bundle, which has the significance of the added fluid area fraction. The parameter is determined by solving a local problem defined on a unit cell, and its relation to the porosity of the bundle is investigated in this paper. This is made possible by obtaining an analytical solution of the local problem based on Weierstrass’s doubly periodic functions.     

A phase field model of electromechanical fracture

Structural reliability analyses of piezoelectric solids need the modeling of failure under coupled electromechanical actions. However, the numerical simulation of failure due to fracture based on sharp crack discontinuities may suffer in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase field. In this work, we develop a framework of diffusive fracture in piezoelectric solids. We start our investigation with the definition of a crack surface functional of the phase field that Γ-converges for vanishing length-scale parameter to a sharp crack topology. This functional provides the basis for the definition of suitable dissipation functions which govern the evolution of the crack phase field. Based on experimental results available in the literature, we suggest a non-associative dissipative framework where the fracture phase field is driven by the mechanical part of the coupled electromechanical driving force. This accounts for a hierarchical view that considers (i) the decrease of stiffness due to mechanical rupture as the primary action that is followed by (ii) the decrease of electric permittivity due to the generated free space. The proposed definition of mechanical and electrical parts of the fracture driving force follows in a natural format from a kinematic assumption, that decomposes the total strains and the total electric field into energy-enthalpy-producing parts and fracture parts, respectively. Such an approach allows the insertion of well-known anisotropic piezoelectric storage functions without change. We end up with a three-field-problem that couples the displacement with the electric potential and the fracture phase field. The latter is governed by a micro-balance equation, which appears in a very transparent form in terms of a history field containing a maximum fracture source obtained in the time history of the electromechanical process. This representation allows the construction of a very robust algorithmic treatment based on a operator split scheme, which successively updates in a typical time step the history field, the crack phase field and finally the two piezoelectric fields. The proposed model is considered to be the canonically simple scheme for the simulation of diffusive electromechanical crack propagation in solids. We demonstrate its modeling capacity by means of representative numerical examples.     

Measurement of the liquid phase mass in gas-liquid sprays by X-ray attenuation

An optical method for measuring the mass of liquid phase in a spray has been developed into an X-ray method. An X-ray of about 6 keV is used to measure the mass distribution of liquid phase in the spray formed by a concentric injector with water and gaseous nitrogen at one atmosphere of pressure. The possibility of measuring the mass of gaseous oxygen and hydrogen is discussed.     

Distribution-enhanced homogenization framework and model for heterogeneous elasto-plastic problems

Multi-scale computational models offer tractable means to simulate sufficiently large spatial domains comprised of heterogeneous materials by resolving material behavior at different scales and communicating across these scales. Within the framework of computational multi-scale analyses, hierarchical models enable unidirectional transfer of information from lower to higher scales, usually in the form of effective material properties. Determining explicit forms for the macroscale constitutive relations for complex microstructures and nonlinear processes generally requires numerical homogenization of the microscopic response. Conventional low-order homogenization uses results of simulations of representative microstructural domains to construct appropriate expressions for effective macroscale constitutive parameters written as a function of the microstructural characterization. This paper proposes an alternative novel approach, introduced as the distribution-enhanced homogenization framework or DEHF, in which the macroscale constitutive relations are formulated in a series expansion based on the microscale constitutive relations and moments of arbitrary order of the microscale field variables. The framework does not make any a priori assumption on the macroscale constitutive behavior being represented by a homogeneous effective medium theory. Instead, the evolution of macroscale variables is governed by the moments of microscale distributions of evolving field variables. This approach demonstrates excellent accuracy in representing the microscale fields through their distributions. An approximate characterization of the microscale heterogeneity is accounted for explicitly in the macroscale constitutive behavior. Increasing the order of this approximation results in increased fidelity of the macroscale approximation of the microscale constitutive behavior. By including higher-order moments of the microscale fields in the macroscale problem, micromechanical analyses do not require boundary conditions to ensure satisfaction of the original form of Hill's lemma. A few examples are presented in this paper, in which the macroscale DEHF model is shown to capture the microscale response of the material without re-parametrization of the microscale constitutive relations.     

A model for brittle fracture based on the hybrid phase field model

We propose a phase field model for crack propagation based on the hybrid model and justify the model by constructing a family of asymptotic solutions.     

A tensor estimator for the homogenization of absolute permeability

An estimator for an effective permeability tensor based on one-phase incompressible flow is presented. Effective large-scale permeability tensors are well approximated by rough approximations to the fine-scale pressure. The estimator works for all kinds of heterogeneous reservoirs and is fairly independent of boundary conditions.     

A one-dimensional model for incoherent phase changes

We present here a simplified version of the model of incoherent solid-solid transitions with mass diffusion developed by Gurtin and Cermelli in [3]. An incoherent phase change is always associated with some kind of defect production at the interface: we consider here a one-dimensional continuum, so that the resulting equations allow study to be made of the influence of volume (vacancy) production on the evolution of the system.

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Sommario In questo lavoro viene presentato un adattamento del modello di transizione di fase incoerente sviluppato da Gurtin e Cermelli in [3]. Una transizione incoerente è sempre associata alla produzione di un qualche tipo di difetto all'interfaccia: consideriamo qui un modello semplificato di continuo unidimensionale, in modo da poter studiare l'effetto dei difetti di volume (lacune) sull'evoluzione del sistema.

     

Viscosity of the liquid phase in a dispersion

The motion of a dispersion (continuous medium and particles) may be described [1] via the equations ot conservation of matter and momentum for the two phases separately. Here it is necessary to know how the viscosity, pressure in the solid, and other quantities vary with the parameters of the motion. This difficulty occurs even for the very simple model where the internal stresses in the dispersed phase are taken as zero, as there is then an uncertainty as to the viscosity of the medium, which is not a material constant and is dependent on the concentration. There is also uncertainty as to the forces of interaction between the phases. There are numerous empirical relationships for these forces, and also a theoretical one [2]. Here an analogous method is applied to derive an expression for the viscosity of the liquid. This viscosity applies to a liquid filtering through a porous medium in the particular case where the concentration is such as to produce close packing of the solid particles. The result corresponds to standard formulas in the case of low concentrations.     

A thermodynamic approach to isotropic-nematic phase transitions in liquid crystals

We propose a dynamical model for (non-isothermal) phase transitions in liquid crystals. Macroscopic motions of the liquid crystal (LC) are neglected, while the coupling with the electromagnetic field is considered. The LC is described in terms of the classical order tensor Q, which is split as Q=s N, where N is a normalized tensor; two independent evolution laws are given for s and N. The model includes an evolutive equation for the temperature field obtained from an appropriate form of the energy balance, in which the internal powers associated to the equations for s and N are accounted for. The thermodynamic restrictions in the constitutive relations which ensure the Clausius–Duhem inequality have been pointed out.     

A variational formulation for the incremental homogenization of elasto-plastic composites

This work addresses the micro–macro modeling of composites having elasto-plastic constituents. A new model is proposed to compute the effective stress–strain relation along arbitrary loading paths. The proposed model is based on an incremental variational principle (Ortiz, M., Stainier, L., 1999. The variational formulation of viscoplastic constitutive updates. Comput. Methods Appl. Mech. Eng. 171, 419–444) according to which the local stress–strain relation derives from a single incremental potential at each time step. The effective incremental potential of the composite is then estimated based on a linear comparison composite (LCC) with an effective behavior computed using available schemes in linear elasticity. Algorithmic elegance of the time-integration of J₂ elasto-plasticity is exploited in order to define the LCC. In particular, the elastic predictor strain is used explicitly. The method yields a homogenized yield criterion and radial return equation for each phase, as well as a homogenized plastic flow rule. The predictive capabilities of the proposed method are assessed against reference full-field finite element results for several particle-reinforced composites.     

A nonlinear homogenization procedure for periodic masonry

The present paper deals with the problem of the determination of the in-plane behavior of masonry material. The masonry is considered as a composite material composed by a regular distribution of blocks connected by horizontal and vertical mortar joints. The overall constitutive relationships of the regular masonry are derived by a rational micromechanical and homogenization procedure. Linear elastic constitutive relationship is considered for the blocks, while a new special nonlinear constitutive law is proposed for the mortar joints. In particular, a mortar constitutive law, which accounts for the coupling of the damage and friction phenomena occurring during the loading history, is proposed; the developed model is based on an original micromechanical analysis of the damage process of the mortar joint. Then, an effective nonlinear homogenization procedure, representing the main novelty of the paper, is proposed; it is based on the transformation field analysis, using the technique of the superposition of the effects and the finite element method. The presented methodology is implemented in a numerical code. Finally, numerical applications are performed in order to assess the performances of the proposed procedure in reproducing the mechanical behavior of masonry material.     

Modelling of functionally graded materials by numerical homogenization

Summary  In this contribution, the mechanical behaviour of different ZrO₂/NiCr 80 20 compositions is analysed and compared with experimental findings. The microwave-sintered material is found to possess a slightly dominant ceramic matrix for intermediate volume fractions. Its thermal expansion coefficient deviates from the rule of mixture. The modulus and the stress strain behaviour can be simulated by a numerical homogenization procedure, and the influence of residual stresses is found to be negligible. A newly introduced parameter (matricity) describes the mutual circumvention of the phases and is found to strongly control the stress level of the composite, globally as well as locally. Finally, a graded component and a metal/ceramic bi-material are compared for thermal as well as mechanical loading. Received 23 November 1999; accepted for publication 26 May 2000