Porous Material Model Research Articles

Implicit yield function formulation for granular and rock-like materials

The constitutive modelling of granular, porous and quasi-brittle materials is based on yield (or damage) functions, which may exhibit features (for instance, lack of convexity, or branches where the values go to infinity, or ‘false elastic domains’) preventing the use of efficient return-mapping integration schemes. This problem is solved by proposing a general construction strategy to define an implicitly defined convex yield function starting from any convex yield surface. Based on this implicit definition of the yield function, a return-mapping integration scheme is implemented and tested for elastic–plastic (or -damaging) rate equations. The scheme is general and, although it introduces a numerical cost when compared to situations where the scheme is not needed, is demonstrated to perform correctly and accurately.

Ductile Shear Damage Model for Porous Materials

Ductile Shear Damage Model for Porous Materials

Modeling of laser-driven water-confined shocks into porous graphite

This paper presents a laser-driven water-confined shock experiment into a commercial grade of porous graphite. An intensity of about 3 GW/cm2 led to a pressure above 2 GPa on the front surface of the 0.46 mm sample. The rear surface velocity, recorded by a Velocity Interferometer System (VISAR), reached 325 m/s. Two classical models for porous materials are discussed. The first one uses plates of dense graphite spaced out in order to obtain the correct average density. The second one models a continuous material and includes an experimental compaction curve of our porous graphite. They were implemented into hydrocodes and both gave quite correct maximum free surface velocities and shock break-out instants. Nevertheless, the continuous representation appeared to be more efficient to reproduce the experimental free surface velocity ramp. Discussions on the laser-matter interaction modeling are also provided. Finally, a protocol for the simulation of future laser experiments is proposed.

A binary array method for calculating thermal conductivity of porous materials

The present research on thermal conductivity of porous materials is hampered by their highly random internal structure. Until now, the researches could not describe internal structure of porous materials, but macro calculation expressions were established. In order to solve the above problem, this paper presents a method of calculation that applies a binary array. This method can describe the internal structure of porous materials, reflects its randomness, and calculates the thermal conductivity of porous materials. The technique selects large enough porous material models and divides them in small enough grids. Each grid needs a binary conversion process to determine if the grid is matrix or pore. Finally, a grid array composed of matrix and pore is established. Based on this grid array, this study sets the initial and boundary conditions, and then calculates the thermal conductivity of porous materials by a numerical calculation according to internal heat-transfer mechanism. This research adopts the described method to calculate the thermal conductivity of porous aluminum. The result is consistent with that of the experimental measurement, thereby verifying the accuracy and reliability of this method.

Structural characterization of amorphous materials applied to low-k organosilicate materials

We present a methodology to create computational atomistic-level models of porous amorphous materials, in particular, an organosilicate structure for ultra-low dielectric constant (ULK) materials known as “SiCOH.” The method combines the ability to satisfy geometric and chemical constraints with subsequent molecular dynamics (MD) techniques as a way to capture the complexities of the porous and amorphous nature of these materials. The motivation for studying ULK materials arises from a desire to understand the origin of the material's weak mechanical properties. The first step towards understanding how these materials might behave under processing conditions that are intended to improve their mechanical properties is to develop a suitable computational model of the material and hence is the focus of this paper. We define the atomic-scale topology of ULK materials that have been produced by chemical vapor deposition-like experimental techniques. Specifically, we have developed a method of defining the initial atom configurations and interactions, as well as a method to rearrange these starting configurations into relaxed structures. The main advantage of our described approach is the ability of our structure generation method to maintain a random distribution of relevant structural motifs throughout the structure, without relying on large unit cells and periodic boundaries to approximate the behavior of this complex material. The minimization of the different models was accomplished using replica exchange molecular dynamics (REMD). Following the generation of the ‘equilibrium’ configurations that result from REMD for a ULK material of a pre-specified composition, we demonstrate that its structural properties, including bonding topology, porosity and pore size distribution are similar to experimentally used ULK materials.

How Rocks Break

A new computational model of porous materials like sandstone shows what happens microscopically when the material is stressed to the breaking point.

Thermal Conductivity Measurement of Low-k Dielectric Films: Effect of Porosity and Density

The thermal conductivity of low-dielectric-constant (low-k) SiOC:H and SiC:H thin films has been measured as a function of porosity using a heat transfer model based on a microfin geometry and infrared thermometry. Microscale specimens were patterned from blanket films, released from the substrate, and subsequently integrated with the experimental setup. Results show that the thermal conductivity of a dense specimen, 0.7 W/mK, can be reduced to as low as 0.1 W/mK by introducing 30% porosity into it. The measured thermal conductivity shows a nonlinear decrease with increasing porosity that approximately follows the porosity-weighted simple medium model for porous materials. Neither the differential effective medium nor the coherent potential model could predict the density dependence of the thermal conductivity. These results suggest that more careful consideration is required for application of generic porous materials modeling to low-k dielectrics.

Infiltration and dimensional scaling of inkjet droplets on thick isotropic porous materials

We study the imbibition of picoliter (pL)-sized inkjet droplets on controlled pore glass membranes (CPG), as a suitable model for isotropic three-dimensional porous materials. We do so using a variety of liquids, i.e., water, formamide and diiodomethane, and measure the evolution of the imbibition process using high-speed digital imaging. Here, experiments were conducted on 2–280 nm CPG membranes with drops with initial volumes ranging from 100 to 600 pL. We derive scaling laws for imbibition through dimensional analysis and advance the argument that the rate of absorption is related to two-dimensionless groups $$V = v(t) \epsilon /v_{\text{tot}} ;\,T = tkp_{\text{c}} /\mu \,\left( { \epsilon /v_{\text{tot}} } \right)^{2/3} ,$$ where v(t) is the imbibed volume, as determined from experiments, t is the time, v tot the total liquid volume, $$ \epsilon $$ the porosity, μ the liquid viscosity, k the permeability, and p c the Laplace capillary pressure. We show this scaling to well describe the system at intermediate T values and report that $$V \propto T^{0.8} .$$

Adsorption in complex porous networks with geometrical and chemical heterogeneity

We report here a simple algorithm to create 2D lattice-based models of porous deposits of preformed nanometric particles, by mimicking to some extent the physics of the actual deposition/aggregation mechanism. The heterogeneous porous networks obtained exhibit anisotropic properties unlike lattice-based models of porous materials in the existing literature, such as those of porous Vycor glass. We have then used calculations based on the mean field kinetic theory, in order to study the thermodynamics and dynamics of fluid adsorption and desorption in these lattice-based porous models. We showcase the influence of pore heterogeneity on the phase equilibrium of the confined fluid, studying both heterogeneity in pore size distribution and chemical heterogeneity of the internal surface.

Anisotropic finite-strain models for porous viscoplastic materials with microstructure evolution

In this work, we propose a constitutive model for the finite-strain, macroscopic response of porous viscoplastic solids, accounting for deformation-induced changes in the size, shape and distribution of the voids. The model makes use of consistent homogenization estimates obtained by the “iterated variational linear comparison” procedure of Agoras and Ponte Castañeda (2013) to characterize both the instantaneous effective response of the porous material and the evolution of the underlying microstructure. The proposed model applies for general, three-dimensional loading conditions and can be implemented numerically for use in standard FEM codes. We also investigate the interplay between the evolution of the microstructure and the macroscopic stress–strain response, in the context of displacement-controlled, plane strain loading (bi-axial straining) of initially isotropic, porous, rigid-plastic materials with power-law hardening. We focus on the effect of strain triaxiality, and consider both extensional and contractile loading conditions leading to porosity growth and collapse, respectively. For both types of loadings, it is found that the macroscopic behavior of the material exhibits an initial hardening regime followed by a softening regime at sufficiently large strains. Consistent with earlier models and experimental results, the softening regime for extensional loadings is a consequence of porosity growth. On the other hand, the softening behavior predicted for contractile loadings is found to be a consequence of void collapse, i.e., of rapid changes in the average shape of the pores leading to crack-like shapes. For both types of loadings, the transition from hardening to softening in the macroscopic response can be identified with the onset of macroscopic strain localization. The associated critical conditions at the onset of localization are determined as a function of the strain triaxiality. The type of localization band ranges from dilatational to compaction bands as the bi-axial straining varies from uniaxial extension to uniaxial contraction.

An improved nonlocal Gurson model for plastic porous solids, with an application to the simulation of ductile rupture tests

In recent work, the author and co-workers have developed a new methodology to delocalize the damage in Gurson model for porous ductile materials. The motivation was to rectify the difficulties connected to the excessive damage smoothing arising in the practical use of the original damage delocalization method. The new approach consists of delocalizing the logarithm of the damage instead of the damage itself. The relevance of the new method to avoid mesh size effects and satisfactorily reproduce typical ductile fracture experiments are explored in this work.

Computational Modeling of Porous Ceramics with Bioactive Layer

The paper deals with a creation of computational model of a high porous ceramic material. This type of material has a large-scale industrial utilization. The computational model was created based on micro-CT data in the ANSYS 14.0 software using Finite Element Method. A creation of a porous ceramic struts model which respect a micro architecture is quite difficult (computer demanding and micro-CT data). The micro-CT slices are converted into a 3D model using image processing (used software STL Model Creator). The local first principle stress was analyzed because, ceramic is the brittle material. Furthermore, the influence of the thick layer around the individual struts was analyzed.

A stress-based variational model for ductile porous materials

The main objective of this paper is to formulate a very new statically-based model of ductile porous materials having a von Mises matrix. In contrast to the Gurson’s well known kinematical approach applied to a hollow sphere, the proposed study proceeds by means of a statical limit analysis procedure. Its development and implementation require the choice of an appropriate trial stress field. The starting point is Hill’s variational principle for rigid plastic matrix. The use of a lagrangian multiplier allows to satisfy the plastic criterion in an average sense. The proposed trial stress field, complying with internal equilibrium, is composed of an heterogeneous part (exact solution for the stress field in the hollow sphere under pure hydrostatic loading) to which is added a uniform deviatoric stress field. Owing to this choice, the stress vector conditions on the void boundary are relaxed. By solving the resulting Saddle point problem, we derive closed form formula which depends not only on the first and second invariant of the macroscopic stress tensor but also on the sign of the third invariant of the stress deviator. The obtained results are fully discussed and compared to existing models, available numerical data and to Finite Elements results obtained from cell calculation carried out during the present study. Finally, we provide for the new model the macroscopic flow rule as well as the porosity evolution equations which also show very original features. Some of these features are illustrated.

Use of a porous material description of forests in infrasonic propagation algorithms

Infrasound can propagate very long distances and remain at measurable levels. As a result infrasound sensing is used for remote monitoring in many applications. At local ranges, on the order of 10 km, the influence of the presence or absence of forests on the propagation of infrasonic signals is considered. Because the wavelengths of interest are much larger than the scale of individual components, the forest is modeled as a porous material. This approximation is developed starting with the relaxation model of porous materials. This representation is then incorporated into a Crank-Nicholson method parabolic equation solver to determine the relative impacts of the physical parameters of a forest (trunk size and basal area), the presence of gaps/trees in otherwise continuous forest/open terrain, and the effects of meteorology coupled with the porous layer. Finally, the simulations are compared to experimental data from a 10.9 kg blast propagated 14.5 km. Comparison to the experimental data shows that appropriate inclusion of a forest layer along the propagation path provides a closer fit to the data than solely changing the ground type across the frequency range from 1 to 30 Hz.

Density Dependence of the Macroscale Superelastic Behavior of Porous Shape Memory Alloys: A Two-Dimensional Approach

Porous Shape Memory Alloys (SMAs) are of particular interest for many industrial applications, as they combine intrinsic SMA (shape memory effect and superelasticity) and foam characteristics. The computational cost of direct porous material modeling is however extremely high, and so designing porous SMA structure poses a considerable challenge. In this study, an attempt is made to simulate the superelastic behavior of porous materials via the modeling of fully dense structures with material properties modified using a porous/bulk density ratio scaling relation. Using this approach, direct modeling of the porous microstructure is avoided, and only the macroscale response of the model is considered which contributes to a drastic reduction of the computational cost. Foam structures with a gradient of porosity are also studied, and the prediction made using the fully dense material model is shown to be in agreement with the mesoscale porous material model.

AN ELASTOPLASTIC CONSTITUTIVE MODEL FOR POROUS MATERIALS

A micromechanics-based elastoplastic constitutive model for porous materials is proposed. With an assumption of modified three-dimensional Ramberg–Osgood equation for the compressible matrix material, the variational principle based on a linear comparison composite is applied to study the effective mechanical properties of the porous materials. Analytical expressions of elastoplastic constitutive relations are derived by means of micromechanics principles and homogenization procedures. It is demonstrated that the derived expressions do not involve any additional material constants to be fitted with experimental data. The model can be useful in the prediction of mechanical properties of elastoplastic porous solids.

Theoretical Model Research of Metal Rubber Material

Metal rubber is a kind of pure metal structure of the porous material. The paper, starting from the internal structure of metal rubber, based on the theory of porous, based on dry friction nonlinear theory, establishing the unit model of porous metal rubber material, for the development of product design and application of the metal rubber material to provide a theoretical basis.

Dynamic cratering of graphite: Experimental results and simulations

The cratering process in brittle materials under hypervelocity impact (HVI) is of major relevance for debris shielding in spacecraft or high-power laser applications. Amongst other materials, carbon is of particular interest since it is widely used as elementary component in composite materials. In this paper we study a porous polycrystalline graphite under HVI and laser impact, both leading to strong debris ejection and cratering. First, we report new experimental data for normal impacts at 4100 and 4200 m s−1 of a 500-μm-diameter steel sphere on a thick sample of graphite. In a second step, dynamic loadings have been performed with a high-power nanosecond laser facility. High-resolution X-ray tomographies and observations with a scanning electron microscope have been performed in order to visualize the crater shape and the subsurface cracks. These two post-mortem diagnostics also provide evidence that, in the case of HVI tests, the fragmented steel sphere was buried into the graphite target below the crater surface. The current study aims to propose an interpretation of the results, including projectile trapping. In spite of their efficiency to capture overall trends in crater size and shape, semi-empirical scaling laws do not usually predict these phenomena. Hence, to offer better insight into the processes leading to this observation, the need for a computational damage model is argued. After discussing energy partitioning in order to identify the dominant physical mechanisms occurring in our experiments, we propose a simple damage model for porous and brittle materials. Compaction and fracture phenomena are included in the model. A failure criterion relying on Weibull theory is used to relate material tensile strength to deformation rate and damage. These constitutive relations have been implemented in an Eulerian hydrocode in order to compute numerical simulations and confront them with experiments. In this paper, we propose a simple fitting procedure of the unknown Weibull parameters based on HVI results. Good agreement is found with experimental observations of crater shapes and dimensions, as well as debris velocity. The projectile inclusion below the crater is also reproduced by the model and a mechanism is proposed for the trapping process. At least two sets of Weibull parameters can be used to match the results. Finally, we show that laser experiment simulations may discriminate in favor of one set of parameters.

A fast random walk algorithm for computing diffusion-weighted NMR signals in multi-scale porous media: A feasibility study for a Menger sponge

A fast random walk (FRW) algorithm is adapted to compute diffusion-weighted NMR signals in a Menger sponge which is formed by multiple channels of broadly distributed sizes and often considered as a model for soils and porous materials. The self-similar structure of a Menger sponge allows for rapid simulations that were not feasible by other numerical techniques. The role of multiple length scales on diffusion-weighted NMR signals is investigated.

Nonlinear Poro-Elastic Model for Unsaturated Porous Solids

A nonlinear poroelastic constitutive model for unsaturated porous materials is formulated based on a higher order formulation of free energy including mechanical and moisture contributions and the coupling between moisture and mechanics. This orthotropic model leads to the explicit formulation of the dependence of the compliance, moisture capacity, and coupling coefficient on stress and liquid pressure. The nonlinear poroelastic material properties can be easily determined from mechanical testing at different moisture content and free swelling/sorption tests. An academic example illustrates the capacity of the proposed model to describe nonlinear moisture dependent elasticity, stress dependent sorption, and swelling, also called mechano-sorption and moisture expel during mechanical loading. Two materials are analyzed in detail: wood and Berea sandstone. The poroelastic model shows a good agreement with measurements. Different moisture dependence of the elastic properties is found, with wood showing a more complex moisture/mechanical interaction. Berea sandstone is found to show an important nonlinear elastic behavior dependent on stress, similar in dry and wet conditions.