Spheroidal Pores Research Articles

Analytic solutions for effective elastic moduli of isotropic solids containing oblate spheroid pores with critical porosity

AbstractAccurate characterization for effective elastic moduli of porous solids is crucial for better understanding their mechanical behaviour and wave propagation, which has found many applications in the fields of engineering, rock physics and exploration geophysics. We choose the spheroids with different aspect ratios to describe the various pore geometries in porous solids. The approximate equations for compressibility and shear compliance of spheroid pores and differential effective medium theory constrained by critical porosity are used to derive the asymptotic solutions for effective elastic moduli of the solids containing randomly oriented spheroids. The critical porosity in the new asymptotic solutions can be flexibly adjusted according to the elastic moduli – porosity relation of a real solid, thus extending the application of classic David‐Zimmerman model because it simply assumes the critical porosity is one. The asymptotic solutions are valid for the solids containing crack‐like oblate spheroids with aspect ratio < 0.3, nearly spherical pores (0.7 < < 1.3) and needle‐like prolate pores with > 3, instead of just valid in the limiting cases, for example perfectly spherical pores (= 1) and infinite thin cracks (0). The modelling results also show that the accuracies of asymptotic solutions are weakly affected by the critical porosity and grain Poisson's ratio , although the elastic moduli have appreciable dependency of and . We then use the approximate equations for pore compressibility and shear compliance as inputs into the Mori–Tanaka and Kuster–Toksoz theories and compare their calculations to our results from differential effective medium theory. By comparing the published laboratory measurements with modelled results, we validate our asymptotic solutions for effective elastic moduli.

Modeling the ionic diffusion coefficient of unsaturated hardened cement paste: A micromechanical approach

Most of the concrete structures in service are unsaturated, estimating the ionic diffusion coefficient of unsaturated cement-based material is of paramount significance to assessment of corrosion of reinforced concrete. Taking account of the contributions from different water configurations at multi-scale (water films, interlayer water, large gel pore (LGP) water and capillary pore water) and the corresponding diffusion mechanisms of each type of water, a micromechanical model for estimating the ionic diffusion coefficient of unsaturated hardened cement paste (UHCP), is developed in this study. Model results show that the interconnectivity of the spheroidal capillary pore water is closely related to the aspect ratio of spheroidal capillary pore, the slender the prolate pore, the higher the interconnectivity of the capillary pore water; when the capillary pore water is totally drained, water films and interlayer water within C-S-H governs the ionic diffusion, the higher the water to cement ratio (w/c), the less the contribution of interlayer water, and the greater the contribution of water film, to ionic diffusion within UHCP. Application on saturated hardened cement paste and UHCP shows the capability of the model to accurately reproduce the experimental results.

Effective yield strength of a saturated porous medium with a spheroidal meso-pore and spherical micro-pores

A macroscopic yield criterion has been derived in the present work for a double saturated porous medium with a spheroidal pore at the mesocale and spherical pores at the microscale. These two types of pores are well separated at two different scales. The meso spheroidal pore saturated by a pore pressure which is different from the one in the micro spherical pores. A Drucker-Prager type criterion is adopted for the solid phase at the microscopic scale to describe its asymmetric behavior between tension and compression. The methodology to formulate this criterion is based on the limit analysis approach of a spheroidal volume containing a confocal spheroidal pore subjected to a uniform strain rate boundary conditions. The matrix at the mesoscopic scale obeys to a general elliptic yield criterion. Based on a two-step homogenization step, the influence of meso-pore shape (spherical, prolate or oblate), micro-porosity, meso-porosity and the effect of pore pressures at different scales are taken into account explicitly by this macroscopic yield criterion.

Orientation effects, hydrostatic response, and figures of merit in 2–2-type composites based on relaxor-ferroelectric single crystals

The article reports new results on the hydrostatic performance of laminar piezo-active composites based on domain-engineered relaxor-ferroelectric single crystals (SCs) with the perovskite-type structure. Layers of the composites are either single-crystal poled along [011] (where rotations of the main crystallographic axes around two co-ordinate axes are considered) or porous polymer, with aligned spheroidal pores. These composites are described by 2–2–0 connectivity. An influence of the rotation of the main crystallographic axes and porosity is analyzed, and maxima of the hydrostatic piezoelectric coefficient figure of merit and electromechanical coupling factor of the composites are found as functions of a few parameters. New diagrams are built to show regions of the large (over 1000 pC/N) at variations of two rotation angles in the single-crystal layers. Large values of the hydrostatic figure of merit ∼ (10−10 – 10−9) Pa−1 and electromechanical coupling factor are achieved in specific volume-fraction and rotation-angle ranges. These and other large values of the hydrostatic parameters of the studied 2–2-type composites are important for effective hydroacoustic applications.

Numerical Investigation of the Multiscale Characteristics of Elastoplasticity in Microheterogeneous Continuum Media Using a Simple Hollow Sphere Model

In this paper, we investigate the fundamental characteristics of plasticity in microheterogeneous continuum materials. Plasticity in such materials is a multiscale phenomenon with contrasting characteristics at different scales. A hollow sphere model, including a matrix embedding an oblate spheroidal pore, is considered as a sample representative elementary volume (REV) of a microheterogeneous material where, at the local level (microscale), a simple Drucker–Prager plasticity model is assumed for the matrix obeying the classical continuum theory of plasticity. The plasticity model is Lode angle independent with associated plastic flow and no hardening. We perform finite element numerical simulations on the REV to investigate the characteristics of the homogenized plasticity of the REV. It is seen that the homogenized plasticity shows complex characteristics such as Lode angle dependency, presence of cap, hardening, and nonassociativity. Well-established as these aspects are in the literature, they are often presumed in the plasticity models of heterogeneous materials in an ad hoc form. In contrast, we present here a systematic and lucid way to capture these aspects as emergent phenomena of heterogeneous microstructures. This has crucial implications in guiding the theoretical constitutive developments for elastoplastic materials toward adopting a multiscale approach that properly considers the effect of microstructure and naturally captures these aspects without the need for ad hoc assumptions. A fundamental assumption in the classical theory of plasticity is the plastic flow rule postulate, which states that the direction of plastic strain increment is governed only by the current state variables, independent of the loading increment direction. We show, through a stress probing exercise in 3D principal stress space, that the flow rule, assumed for the microscale plasticity, is not valid for the macroscopic plasticity. Indeed, the homogenized plastic flow direction in heterogeneous materials is governed by the current state variables and loading increment direction. Finally, we show that the plastic response is also a function of the stress path taken prior to loading. As such, nonclassical characteristics are observed for the plasticity of microheterogeneous materials. These nonclassical characteristics for the simple REV considered in this work (the hollow sphere model) align with some of the results observed in experiments or only reported for microdiscrete materials; herein, we show they are also valid for a broader range of materials endowed with microstructure, i.e., microcontinuous materials, and, as such, the fundamental assumptions in the classical theory of plasticity require revision before being applicable to the plastic behavior of heterogeneous materials.

A micromechanical analysis of strain concentration tensor for elastoplastic medium containing aligned and misaligned pores

In this paper, making use of Eshelby tensor for the elastoplastic medium, will be presented an approach for the strain concentration tensor for materials that present spherical or spheroidal pores. The pores are embedded in a homogeneous isotropic matrix which presents an elastoplastic behavior, governed by the von Mises yield criterion and isotropic hardening. In this context, it will also be analyzed the behavior of a porous material through the influence of the extraction of the isotropic part of the anisotropic operator. The effective properties of the material in each incremental step are obtained through the micromechanical Mori-Tanaka's model. Through elastoplastic strain concentration tensor for aligned or misaligned pores, it was observed that the pore direction represents a greater influence on the elastoplastic behavior of the material studied than aspect ratio (length divided by pore diameter). This approach has the advantage of being easily incorporated into conventional mean-field homogenization (MFH) schemes for the case where inclusion stiffness is neglected and presents different geometries or directions in 3D space.

Simultaneous Measurements of Elastic Wave Velocity and Porosity of Epidosites Collected From the Oman Ophiolite: Implication for Low VP/VS Anomaly in the Oceanic Crust

AbstractGeophysical surveys of the oceanic crust indicate that hydrothermal circulation universally occurs in seismic layer 2, which results in a low seismic velocity and high VP/VS due to the occurrence of cracks. However, the anomalously low VP/VS observed at the layer 2/3 transition cannot be explained by the crack model, because the effective medium theory predicts an increase in VP/VS due to crack development. In this study, we present the first evidence that shows the low VP/VS in the oceanic crust is caused by epidotization due to upward fluid flow in hydrothermal systems. Simultaneous measurements of elastic wave velocity and porosity of epidosites collected by the Oman Drilling Project show that quartz precipitation and spheroidal pores results in low VP/VS, in contrast to diabases that contain thin cracks. The presence of spheroidal pores in epidosites is supported by CT imaging, and is consistent with predictions from the effective medium model.

Multi-scale description of hydro-mechanical coupling in swelling clays. Part I: Nonlinear poroelasticity

This paper presents a micromechanical description of the swelling behavior of partially saturated clays following a two-stage homogenization procedure on a three-dimensional Representative Elementary Volume (REV) of clay. At the nano-scale, the REV of clay particles includes idealized parallel clay platelets and oblate spheroidal pores in between them that are saturated with an electrolyte solution. Swelling forces at all spacing ranges, including the crystalline and osmotic regimes, are considered. At the micro-scale, the REV of clay is comprised of dispersed clay aggregates and partially saturated (variably-sized) spherical pores in between them. At issue, is reconstructing the couplings of mechanical, hydraulic and electrochemical forces at the macroscopic level and their effect on the overall behavior of clay. In Part I, we focus on reversible deformation mechanisms in clay. A generalized nonlinear form of the classical Biot’s poroelasticity relation is obtained with additional terms accounting for capillary and swelling stresses. Capillary stress emerges from interaction of fluid phases at different scales, and takes into account the surface tension effects. The swelling part on the other hand, correlates with the net disjoining hydration and electrochemical forces at the nano-scale. As a result, the stress description of clays is embedded with microstructural information. In addition, a robust localization procedure is utilized to track the microstructural changes associated to micro-porosity and clay platelet spacing. The nonlinearity of the stress–strain description is shown to originate from the dependency of the particles’ elasticity on spacing with electrochemical origins. Finally, base-line features of the model are investigated through material point simulations of a clay swelling test. The discrepancies between the model predictions and experimental measurements are resolved in Part II of this paper by considering also the plastic deformations within the framework developed in Part I.

Structural Design Method for Constructions: Simulation, Manufacturing and Experiment

The development of additive manufacturing technology leads to new concepts for design implants and prostheses. The necessity of such approaches is fueled by patient-oriented medicine. Such a concept involves a new way of understanding material and includes complex structural geometry, lattice constructions, and metamaterials. This leads to new design concepts. In the article, the structural design method is presented. The general approach is based on the separation of the micro- and macro-mechanical parameters. For this purpose, the investigated region as a complex of the basic cells was considered. Each basic cell can be described by a parameters vector. An initializing vector was introduced to control the changes in the parameters vector. Changing the parameters vector according to the stress-strain state and the initializing vector leads to changes in the basic cells and consequently to changes in the microarchitecture. A medium with a spheroidal pore was considered as a basic cell. Porosity and ellipticity were used for the parameters vector. The initializing vector was initialized and depended on maximum von Mises stress. A sample was designed according to the proposed method. Then, solid and structurally designed samples were produced by additive manufacturing technology. The samples were scanned by computer tomography and then tested by structural loads. The results and analyses were presented.

The effect of pore shape on the Poisson ratio of porous materials

A brief review is given of the effect of porosity on the Poisson ratio of a porous material. In contrast to elastic moduli such as K, G, or E, which always decrease with the addition of pores into a matrix, the Poisson ratio may increase, decrease, or remain the same, depending on the shape of the pores, and on the Poisson ratio of the matrix phase, . In general, for a given pore shape, there is a unique critical Poisson ratio, , such that the addition of pores into the matrix will cause the Poisson ratio to increase if , decrease if , and remain unchanged if . The critical Poisson ratio for spherical pores is 0.2, for prolate spheroidal pores is close to 0.2, and tends toward zero for thin cracks. For two-dimensional materials, for circular pores, 0.306 for squares, 0.227 for equilateral triangles, and again approaches 0 for thin cracks. The presence of a “trapped” fluid in the pore space tends to cause to increase, and for the range of parameters that may occur in rocks or concrete, this increase is more pronounced for thin crack-like pores than for equi-dimensional pores. Measurements of the Poisson ratio therefore may allow insight into pore geometry and pore fluid. If the matrix phase is strongly auxetic, small amounts of porosity will generally not cause the Poisson ratio to become positive.

Microstructure and mechanical and thermal shock properties of hierarchically porous ceramics

Hierarchically porous Al2O3 ceramics with a porosity of 30–69% containing small intergranular pores and large pores created by a pore-forming agent were fabricated by gel-casting followed by partial sintering with starch addition. The influence of various pore types on the mechanical and thermal shock properties of the ceramics was subsequently investigated. A thermal shock parameter K was introduced to evaluate the dependence of strength degradation rate on the thermal shock temperature difference, and a new method for calculating the intergranular porosity and extrinsic porosity of the porous ceramics was developed. A low thermal shock strength degradation rate was obtained at a high ratio of fracture toughness (KIC) and elasticity modulus (E) of the material, which was achieved by decreasing the relative fraction of spheroidal extrinsic pores. However, the presence of these pores increased both the KIC and E values.

Analytical Method for Evaluating Calcium Diffusion Coefficient of Partially Leached Cement Paste

This paper aims at presenting an analytical method for evaluating the calcium diffusion coefficient of partially leached cement paste. The analytical method is developed based on a two-scale model of partially leached cement paste. On the first scale, the solid phase in partially leached cement paste is modeled as a two-phase composite material, composed of a matrix of calcium silicate hydrate (C─S─H) gel and inclusions of unhydrated cement and undissolved calcium hydroxide (CH) crystal. The relative calcium diffusion coefficient is derived analytically by the differential effective medium scheme. On the second scale, partially leached cement paste is treated as a system consisting of the solid phase and spheroidal capillary pores; the percolation behavior of capillary pores near the critical volume fraction is considered. In combination with percolation theory, the effective medium approach is modified to evaluate the calcium diffusion coefficient of partially leached cement paste. The depolarization factor is expressed in terms of the critical volume fraction of capillary pores, and the percolation exponents are calibrated from computer simulation data. Finally, the validity of the analytical method is verified with computer simulation data and experimental results collected from the literature. It is concluded that the proposed analytical method can predict the calcium diffusion coefficient of partially leached cement paste with reasonable accuracy.

The influence of the pore shape on the bulk modulus and the Biot coefficient of fluid-saturated porous rocks

Fluid-saturated rocks are multi-phasic materials and the mechanics of partitioning the externally applied stresses between the porous skeleton of the rock and the interstitial fluids has to take into consideration the mechanical behaviour of the phases. In these studies the porosity of the multi-phasic material is important for estimating the multi-phasic properties and most studies treat the porosity as a scalar measure without addressing the influence of pore shape and pore geometry. This paper shows that both the overall bulk modulus of a porous medium and the Biot coefficient depend on the shape of the pores. Pores with shapes resembling either thin oblate spheroids or spheres are considered. The Mori–Tanaka and the self-consistent methods are used to estimate the overall properties and the results are compared with experimental data. The pore density and the aspect ratio of the spheroidal pores influence the porosity of the geomaterials. For partially saturated rocks, the equivalent bulk modulus of the fluid–gas mixture occupying the pore space can also be obtained. The paper also examines the influence of the pore shape in estimating the Biot coefficient that controls the stress partitioning in fluid-saturated poroelastic materials.

Comparison of effective parameters of lead-free 1–3-type composites based on ferroelectric single crystals

The article reports results of the comparative study on the performance of lead-free 1–3-type composites based on domain-engineered lead-free ferroelectric single crystals. Of interest are the 1–3–0 ferroelectric single crystal/porous polymer and 1–0–3 ferroelectric single crystal/piezoelectric single crystal/polymer composites with heterogeneous matrices. In the 1–3–0 composite this matrix is a polymer medium with the system of spheroidal air pores. In the 1–0–3 composite the polymer matrix contains a system of spheroidal piezoelectric single crystal inclusions, and the arrangement of these inclusions is similar to that of the pores in the matrix of the related 1–3–0 composite. Volume-fraction and aspect-ratio dependences of the piezoelectric properties, squared figure of merit and electromechanical coupling factors of the 1–3-type composites are analysed, and comparison of the effective parameters of the composites is made. The role of the composite microgeometry in forming the piezoelectric anisotropy and squared figures of merit is discussed. Due to large electromechanical coupling factors, squared figures of merit and anisotropy factors, the studied lead-free 1–3-type composites are of interest as advanced materials suitable for piezoelectric transducer, sensor and energy-harvesting applications.

New synthesis route of highly porous InxCo4Sb12 with strongly reduced thermal conductivity

Highly porous, In-filled CoSb3 skutterudite materials with an attractive thermoelectric figure of merit (ZT ~ 1) and corresponding dense samples were fabricated through the cost-effective method of reduction in oxides in dry hydrogen and the pulsed electric current sintering (PECS) method, respectively. The reduction process was described in detail using in situ thermogravimetric analysis of Co2O3, Sb2O3 and In(NO3)3·5H2O separately and in a mixture. Two methods to synthesise the same material were examined: (a) free sintering of an initially reduced powder and (b) PECS. The free-sintered materials with higher porosities (up to ~ 40%) exhibited lower values of electrical conductivity than the dense PECS samples (porosity up to ~ 5%), but the benefit of an even sixfold reduction in thermal conductivity resulted in higher ZT values. The theoretical values of thermal conductivity for various effective media models considering randomly oriented spheroid pores are in good agreement with the experimental thermal conductivity data. The assumed distribution and shape of the pores correlated well with the scanning electron microscope analysis of the microstructure. The lowest value of thermal conductivity, equal to 0.5 W/m K, was measured at 523 K for In0.1Co4Sb12 with 41% porosity. The highest value of ZTmax = 1.0 at 673 K was found for the In0.2Co4Sb12 sample in which the porosity was 36%.Graphic abstract

Effective elastic properties and thermal conductivity of isotropic rocks containing concave pores. Application to oolitic limestones

This paper focuses on effective elastic properties, and thermal conductivity of materials constituted by an isotropic homogeneous solid matrix containing both ellipsoidal and non-ellipsoidal pores. In the frame of Effective Media Theory (EMT), property contribution tensor approach extended to non-ellipsoidal inhomogeneities are used to estimate effective elastic properties and thermal conductivity. On the basis of recent results obtained by the authors, a reference 3D shape is considered: supersphere characterised by a concavity parameter and being convex or concave. Related compliance contribution tensors verifying cubic symmetry have been characterised thanks to 3 D finite element modelling and analytical approximation formula have been derived and are used in this paper. Analysis is focussed on the effect of the parameter characterising concavity on both effective elastic coefficients and thermal conductivity. Application to isotropic porous rock with two separated classes of pores, such as oolitic limestones, is considered. Numerical results are presented for a reference oolitic rock, Lavoux limestone. Maxwell, Self-Consistent and Differential schemes are compared. It is shown that concavity parameter is of major importance as may be aspect ratio of spheroidal pores, and that the three reference schemes may conduce to comparable results for the particular investigated microstructure, characterised by a high volume fraction of porous grains (oolites)

Plateau-Rayleigh instabilities in pore networks of additively manufactured polymers: A modeling perspective

Fused Filament Fabrication (FFF) additive manufacturing (AM) is a popular and widespread polymer AM method that provides rapid part production with relatively low cost prototyping. Polymeric structures made through FFF AM typically have internal pore structures which result from the manufacturing process. A Monte Carlo algorithm is implemented to simulate pore morphology evolution from build-path networks to spheroidal pores in AMed post-annealed structures. The model demonstrated that the evolution of these pore networks phenomenologically follows stages associated with the Plateau-Rayleigh instability and that thermal gradients result in migration of the spherical pores, as was observed experimentally.

Estimating pore volume compressibility by spheroidal pore modeling of digital rocks

The real pores in digital cores were simplified into three abstractive types, including prolate ellipsoids, oblate ellipsoids and spheroids. The three-dimensional spheroidal-pore model of digital core was established based on mesoscopic mechanical theory. The constitutive relationship of different types of pore microstructure deformation was studied with Eshelby equivalent medium theory, and the effects of pore microstructure on pore volume compressibility under elastic deformation conditions of single and multiple pores of a single type and mixed types of pores were investigated. The results showed that the pore volume compressibility coefficient of digital core is closely related with porosity, pore aspect ratio and volumetric proportions of different types of pores. (1) The compressibility coefficient of prolate ellipsoidal pore is positively correlated with the pore aspect ratio, while that of oblate ellipsoidal pore is negatively correlated with the pore aspect ratio. (2) At the same mean value of pore aspect ratio satisfying Gaussian distribution, the more concentrated the range of pore aspect ratio, the higher the compressibility coefficient of both prolate and oblate ellipsoidal pores will be, and the larger the deformation under the same stress condition. (3) The pore compressibility coefficient increases with porosity. (4) At a constant porosity value, the higher the proportion of oblate ellipsoidal and spherical pores in the rock, the more easier for the rock to deform, and the higher the compressibility coefficient of the rock is, while the higher the proportion of prolate ellipsoidal pores in the rock, the more difficult it is for rock to deform, and the lower the compressibility coefficient of the rock is. By calculating pore compressibility coefficient of ten classical digital rock samples, the presented analytical elliptical-pore model based on real pore structure of digital rocks can be applied to calculation of pore volume compressibility coefficient of digital rock sample.

The static and dynamic shear-tension mechanical response of AM Ti6Al4V containing spherical and prolate voids

The dynamic tensile response of additively manufactured Ti6Al4V shear-tension specimens containing discrete artificial pores was investigated under quasi-static and dynamic loading. Specimens containing spherical and spheroidal pores were designed with one or three pores, whose total volume fraction was kept equal to that of the single pore, albeit with variable spacing between the pores. Specimens containing prolate pores were designed with different pore orientations with respect to the shear direction. For the geometrical configurations investigated, it was found that the very presence of the pore (or pores) reduces the displacement to failure, compared to the dense specimens, in both quasi-static and dynamic regimes. The shape of the pore, the number of pores and their distribution, have a minor effect on results, in both loading-rate regimes. Similarly, there is no sensitivity to the orientation of the prolate pore, for the investigated spheroid and gauge dimensions.

Model of Forming Isotropic and Anisotropic Graphite Under High Temperatures and Fluences Neutron Irradiation

A model is proposed for describing the shape change of isotropic and anisotropic graphite under the influence of high temperatures and high neutron radiation fluences. The model is based on a new approach, which uses the following provisions: description of the near-pore neighborhood in graphite as a solid solution using the theory of phase transformations of the first kind; consideration of a new phase as a spheroidal pore of small eccentricity, flattened along the direction of greatest stress; taking into account the clustering of carbon atoms at fluences of more than . The graphite non-isotropy is characterized by different pore sizes, different diffusion coefficients, the lengths of the paths of graphite atoms along and across volume of the sample, which in turn depend on the temperature of the sample. It is proposed that the initial element on the basis of which a new phase will be formed is the spheroidal pore of small eccentricity, flattened along the direction of the highest tension. A kinetic equation that describes the diffusion of pores under the influence of high temperatures and intense neutron fluxes is obtained. Initially, the pores are in a field of predetermined stresses oriented along and across the sample. The contribution of diffusion processes is due to the term proportional to the pore distribution function in the sample, and the effect of the neutron flux is described by an additional term in the kinetic equation. The obtained kinetic equation for anisotropic graphite can be transformed for isotropic graphite. For isotropic and anisotropic graphite, model solutions have been obtained that characterize the change in its volume with time under the influence of a neutron flux and high temperature. It is shown that the magnitude of the change in the relative volume of reactor graphite for isotropic graphite exceeds a similar value for anisotropic graphite. Theoretical confirmation of the laws governing the swelling of anisotropic graphites under the influence of large neutron fluences and in the high-temperature field, previously obtained by other authors, is obtained: longitudinal compression of anisotropic graphite samples corresponds to a change in the linear dimensions of isotropic graphites; the transverse compression of anisotropic graphite samples is less than the change in the longitudinal linear dimensions of isotropic graphites.