Showing papers on "Representative elementary volume published in 2011"

Analytical derivation of tortuosity and permeability of monosized spheres: A volume averaging approach

Abstract: Macroscopic properties of granular materials are important in modeling a variety of flow and transport phenomena in many fields of science. Determination of these parameters has always been an issue among both researchers and engineers, mainly in view of tortuosity and permeability. This paper presents analytical functions for the tortuosity and permeability of monosized sphere arrays based on a volume averaging approach and eliminates some ambiguities by modification of the original representative elementary volume model. Veracity of the proposed formulations has been illustrated through comparisons with the latest available results on the subject. Good agreement is found.

Overall dynamic constitutive relations of layered elastic composites

Abstract: A method for the homogenization of a layered elastic composite is presented. It allows direct, consistent, and accurate evaluation of the averaged overall frequency-dependent dynamic material constitutive relations without the need for a point-wise solution of the field equations. When the spatial variation of the field variables is restricted by Bloch-form (Floquet-form) periodicity, then these relations together with the overall conservation and kinematical equations accurately yield the displacement or stress mode-shapes and, necessarily, the dispersion relations. The method can also give the point-wise solution of the elastodynamic field equations (to any desired degree of accuracy), which, however, is not required for the calculation of the average overall properties. The resulting overall dynamic constitutive relations are general and need not be restricted by the Bloch-form periodicity. The formulation is based on micromechanical modeling of a representative unit cell of the composite. For waves in periodic layered composites, the overall effective mass-density and compliance (stiffness) are always real-valued whether or not the corresponding unit cell (representative volume element used as a unit cell) is geometrically and/or materially symmetric. The average strain and linear momentum are coupled and the coupling constitutive parameters are always each others' complex conjugates. We separate the overall constitutive relations, which depend only on the composition and structure of the unit cell, from the overall field equations which hold for any elastic composite; i.e., we use only the local field equations and material properties to deduce the overall constitutive relations. Finally, we present solved numerical examples to further clarify the structure of the averaged constitutive relations and to bring out the correspondence of the current method with recently published results.

Approximation of random microstructures by periodic statistically similar representative volume elements based on lineal-path functions

Abstract: For the direct incorporation of micromechanical information into macroscopic boundary value problems, the FE2-method provides a suitable numerical framework. Here, an additional microscopic boundary value problem, based on evaluations of representative volume elements (RVEs), is attached to each Gauss point of the discretized macrostructure. However, for real random heterogeneous microstructures the choice of a “large” RVE with a huge number of inclusions is much too time-consuming for the simulation of complex macroscopic boundary value problems, especially when history-dependent constitutive laws are adapted for the description of individual phases of the mircostructure. Therefore, we propose a method for the construction of statistically similar RVEs (SSRVEs), which have much less complexity but reflect the essential morphological attributes of the microscale. If this procedure is prosperous, we arrive at the conclusion that the SSRVEs can be discretized with significantly less degrees of freedom than the original microstructure. The basic idea for the design of such SSRVEs is to minimize a least-square functional taking into account suitable statistical measures, which characterize the inclusion morphology. It turns out that the combination of the volume fraction and the spectral density seems not to be sufficient. Therefore, a hybrid reconstruction method, which takes into account the lineal-path function additionally, is proposed that yields promising realizations of the SSRVEs. In order to demonstrate the performance of the proposed procedure, we analyze several representative numerical examples.

Multiscale modeling of the nonlinear response of nano-reinforced polymers

Abstract: The present study uses a nonlinear representative volume element (RVE) to investigate the effective mechanical properties of a nano-reinforced polymer system. Here, the RVE represents the reinforcing carbon nanotube (CNT), the surrounding polymer matrix, and the CNT–polymer interface. Due to the inherent nanoscale involved in simulating CNT structures, an atomistic description is incorporated via the atomistic-based continuum multiscale modeling technique. In this way, the continuum constitutive relations are derived solely from atomistic formulations. The nonlinear response of armchair and zigzag nanotubes and their nano-reinforced polymer equivalents are considered and presented. The results reveal that reinforcing polymeric matrices with 1 to 10 vol% CNTs can result in upward of approximately 23- and 8-fold increases in the tensile and shear stiffness, respectively. These results have a direct bearing on the design and development of nano-reinforced composites.

Effective Elastic Properties of Cracked Rocks — An Overview

Abstract: Upper crustal rocks contain cracks of diverse sizes, shapes and orientations. Predicting their influence on the effective elastic properties of a rock poses a challenging problem of considerable interest, in particular to seismologists. In geophysics, theoretical work on this problem is usually complicated by the presence of pore fluids or when the defect-free matrix of a rock is elastically anisotropic, as in the case of shales for example. The first challenge arising in this context is the identification of microstructure-sentive parameters in terms of which the effective elastic constants are to be expressed. Simple parameters such as volume fraction or crack density (as usually defined) may not suffice for capturing the way in which the complex microstructure of a rock determines its effective elasticity. Defect interactions present another challenge and a number of approximate theoretical schemes have been designed to deal with this problem, but their applicability is not always clear. This chapter offers a critical assessment of these questions, putting the emphasis on microcracks as microstructural elements that often have a dominant effect on the overall elasticity. The issues of matrix anisotropy and frequency effects are examined. Because elastic wave velocities carry microstructural information, it is possible and of great interest to extract from them information on pore and crack content, as well as on the presence and properties of pore fluids. Problems related to multiple defects have been discussed in a broader context of mechanics of materials for over half a century. One of the goals of the present work is to bridge a gap in this area between geophysics and the general mechanics of materials. The former does not always utilize the results of the latter; the latter often neglects specific complexities of the former.

First pseudo-grain failure model for inelastic composites with misaligned short fibers

Abstract: This paper presents a new model and an experimental investigation of the elastic–plastic flow until failure in short-fiber reinforced thermoplastics typically produced by injection molding. The distribution of fiber orientations and lengths is reproduced statistically within a representative volume element (RVE) of the composite microstructure. Then, the RVE is decomposed fictitiously into pseudo-grains (PGs) inside of which the fiber orientation and aspect ratio are unique. An incremental Mori–Tanaka model is used to predict the phase averages of the stresses and strains inside each PG. Damage intervenes in a second homogenization step: the macroscopic stress accounts for the fact that PGs fail one after the other in function of the fiber orientation and the applied strain mode. Hence, the model is called “first pseudo-grain failure model” by analogy with the “first ply failure model” in laminated composites. An evaluation of the proposed model against experimental data is conducted for short-glass-fiber reinforced polyamide 6,6 (PA6,6). It is shown that the model yields satisfactory predictions of the response under uniaxial tension of composite samples with different fiber contents cut under various directions relative to the main injection flow direction.

A coupled two-scale computational scheme for the failure of periodic quasi-brittle thin planar shells and its application to masonry

Abstract: A coupled two-scale framework is presented for the failure of periodic masonry shell structures, in which membrane-flexural couplings appear. The failure behaviour of textured heterogeneous materials such as masonry is strongly influenced by their mesostructure. Their periodicity and the quasi-brittle nature of their constituents result in complex behaviours such as damage-induced anisotropy properties with localisation of damage, which are difficult to model by means of macroscopic closed-form constitutive laws. The multi-scale computational strategies aim at solving this issue by deducing a homogenised response at the structural scale from a representative volume element (RVE), based on constituents properties and averaging theorems. The constituents inside the RVE may be modelled using any closed-form formulation, depending on the physics to represent. Scale transitions for homogenisation towards a Kirchhoff-Love shell behaviour were recently proposed. The microstructure is represented by a unit cell on which a strain-periodic displacement field is imposed. The localisation of damage at the structural scale is represented by means of embedded strong discontinuities incorporated in the shell description. Based on an assumption of single period failure, the behaviour of these discontinuities is extracted from further damaging RVEs, denoted as localising volume elements (LVEs). An acoustic tensor-based criterion adapted to shell kinematics is used to detect the structural-scale failure and find its orientation. For the material behaviour of the coarse-scale discontinuities, an enhanced upscaling procedure based on an approximate energy consistency has been proposed recently for the in-plane case and is extended to the out-of-plane case. Such a multi-scale scheme can be implemented using parallel computation tools. The corresponding multi-scale simulation results are compared to direct fine-scale computations used as a reference for the case of masonry, showing a good agreement in terms of load bearing capacity, of failure mechanisms and of associated energy dissipation.

A determination of the minimum sizes of representative volume elements for the prediction of cortical bone elastic properties

Abstract: At its highest level of microstructural organization—the mesoscale or millimeter scale—cortical bone exhibits a heterogeneous distribution of pores (Haversian canals, resorption cavities). Multi-scale mechanical models rely on the definition of a representative volume element (RVE). Analytical homogenization techniques are usually based on an idealized RVE microstructure, while finite element homogenization using high-resolution images is based on a realistic RVE of finite size. The objective of this paper was to quantify the size and content of possible cortical bone mesoscale RVEs. RVE size was defined as the minimum size: (1) for which the apparent (homogenized) stiffness tensor becomes independent of the applied boundary conditions or (2) for which the variance of elastic properties for a set of microstructure realizations is sufficiently small. The field of elastic coefficients and microstructure in RVEs was derived from one acoustic microscopy image of a human femur cortical bone sample with an overall porosity of 8.5%. The homogenized properties of RVEs were computed with a finite element technique. It was found that the size of the RVE representative of the overall tissue is about 1.5 mm. Smaller RVEs (~0.5 mm) can also be considered to estimate local mesoscopic properties that strongly depend on the local pores volume fraction. This result provides a sound basis for the application of homogenization techniques to model the heterogeneity of cortical microstructures. An application of the findings to estimate elastic properties in the case of a porosity gradient is briefly presented.

Constitutive equations for coupled flows in clay materials

Abstract: [1] We first upscale the local transport (Stokes and Nernst-Planck) equations to the scale of a single capillary saturated by a binary 1:1 electrolyte. These equations are then upscaled to the scale of a network of tortuous capillaries embedded in a homogeneous and continuous mineral matrix, including the influence of the distribution of pore sizes but excluding the effect of connectivity between the pores. One of the features of our theory is to account for transport along the mineral surface in the so-called Stern layer because of recent evidence that this mechanism is effective in describing frequency-dependent electrical conductivity. Real clay materials are, however, not described by a set of capillaries, so we have to modify the model to include the effect of transversal dispersivity, for example. We found no evidence for transport in the Stern layer because of the discontinuity of the solid phase at the scale of a representative elementary volume in clay materials. The effect of the diffuse layer is accounted for through the use of a Donnan equilibrium approach to determine the effective concentrations of the ions in the pore space, which are different from the ionic concentrations of an ionic reservoir in local equilibrium with the porous material. We found that the diffuse layer controls various transport properties, including, for example, the DC electrical conductivity, the osmotic efficiency coefficient, the streaming potential coupling coefficient, and the macroscopic Hittorf numbers. Comparison to a large data set of experimental data, mainly on clay materials, confirms the validity of the derived relationships used to describe the material properties entering into the constitutive equations.

Micromechanical multiscale model for alkali activation of fly ash and metakaolin

Abstract: The process of alkali activation of fly ash and metakaolin is examined in the view of micromechanics. Elasticity is predicted via semi-analytical homogenization methods, using a combination of intrinsic elastic properties obtained from nanoindentation, evolving volume fractions and percolation theory. A new quantitative model for volume fraction is formulated, distinguishing the evolution of unreacted aluminosilicate material, solid gel particles of N-A-S-H gel, and open porosity, which is partially filled with the activator. The stiffening of N-A-S-H gel is modeled by increasing the fraction of solid gel particles. Their packing density and intrinsic elasticity differ in N-A-S-H gels synthesized from both activated materials. Percolation theory helps to address the quasi-solid transition at early ages and explains a long setting time and the beneficial effect of thermal curing. The low ability of N-A-S-H gel to bind water chemically explains the high porosity of Ca-deficient activated materials. Micromechanical analysis matches well the elastic experimental data during the activation and elucidates important stages in the formation of the microstructure.

The poroelastic role of water in cell walls of the hierarchical composite “softwood”

Abstract: Wood is an anisotropic, hierarchically organized material, and the question how the hierarchical organization governs the anisotropy of its mechanical properties (such as stiffness and strength) has kept researchers busy for decades. While the honeycomb structure of softwood or the chemical composition of the cell wall has been fairly well established, the mechanical role of the cell wall water is less understood. The question arises how its capability to carry compressive loads (but not tensile loads) and its pressurization state affect mechanical deformations of the hierarchical composite “wood”. By extending the framework of poro-micromechanics to more than two material phases, we here provide corresponding answers from a novel hierarchical set of matrix-inclusion problems with eigenstresses: (i) Biot tensors, expressing how much of the cell wall water-induced pore pressure is transferred to the boundary of an overall deformation-free representative volume element (RVE), and (ii) Biot moduli, expressing the porosity changes invoked by a pore pressure within such an RVE, are reported as functions of the material’s composition, in particular of its water content and its lumen space. At the level of softwood, where we transform a periodic homogenization scheme into an equivalent matrix-inclusion problem, all Biot tensor components are found to increase with decreasing lumen volume fraction. A further research finding concerns the strong anisotropy of the Biot tensor with respect to the water content: Transverse components increase with increasing water content, while the relationship “longitudinal Biot tensor component versus volume fraction of water within the wood cell wall” exhibits a maximum, representing a trade-off between pore pressure increase (increasing the longitudinal Biot tensor component, dominantly at low water content) and softening of the cell wall (reducing this component, dominantly at high water contents). Soft cell wall matrices reinforced with very stiff cellulose fibers may even result in negative longitudinal Biot tensor components. The aforementioned maximum effect is also noted for the Biot modulus.

The effects of CNT waviness on interfacial stress transfer characteristics of CNT/polymer composites

Abstract: Based on the new simplified 3D Representative Volume Element (RVE) for a wavy carbon nanotube (CNT), an analytical model has been developed to study the stress transfer in single-walled carbon nanotube (SWCNT) reinforced polymer composites. The model is capable of predicting axial as well as interfacial shear stresses along a wavy CNT embedded in a matrix. Based on the pullout modeling technique, the effects of waviness, aspect ratio, CNT diameter, volume fraction, Poisson’s ratio and matrix modulus on axial and interfacial shear stresses have also been analyzed in details. The results of the analytical model are in good agreements when compared with the corresponding results for straight CNTs.

Prediction of the cohesive strength for numerically simulating composite delamination via CZM-based FEM

Abstract: The cohesive strength is an important parameter in numerically modeling composite delamination via CZM (cohesive zone model) based FEM. A micromechanical model is proposed to predict the cohesive strength based on the periodic RVE technique. A periodic displacement boundary condition has been presented on the assumption that the RVE is orthotropic in the sense of overall response. The cohesive strengths of T700/QY8911 and AS4/PEEK laminates at various fibers cross-angles are gained by FEM. With the predicted cohesive strengths the FEM simulations on Mixed-Mode bending (MMB) and seven-point bending test are presented, and the results are in fair agreement with experimental observation.