Dynamics of void collapse in shocked energetic materials: physics of void–void interactions

This work presents the response of a porous energetic material subjected to severe transient loading conditions. The porosities, represented by voids, entirely change the response of an otherwise homogeneous material. The variations in terms of energy distribution and maximum temperature reached in the material in the presence of heterogeneities (voids) but in the absence of chemical reactions are studied. This study also accounts for void–void interactions to enhance the understanding of the localization of energy in the material. It is observed that relative position of voids can have important consequence on energy distribution as well as rise in temperature of the energetic material. The relative position of voids further influences the interaction of secondary shock waves generated during the collapse of one void with the downstream voids. This interaction can either enhance or diminish the strength of the shock depending on the location of downstream voids. This work also reveals that the findings from mutual void–void interactions can be used to study systems with multiple voids. This is shown by analyzing systems with 10–25 % void volume fraction. The effect of void–void interactions are connected to the overall response of a chemically inert porous material to imposed transient loads.     

Fluid capacity distributions of random porous media

As a quantitative measure of the microstructure in a statistically homogeneous porous material, we introduce the notion of thefluid capacity at a specified length scale . In two dimensions, fluid capacity is the void space per unit area for a square of side and in three dimensions it is the void space per unit volume for a cube of side . The most random distribution of fluid capacity, for a prescribed mean fluid capacity, corresponds to an exponential distribution. The distribution of fluid capacity is important during unstable fluid displacements in porous media where viscous fingering occurs. For a material with an exponential fluid capacity distribution, an unstable displacement process can be simulated by simple stochastic algorithms related to diffusion-limited aggregation. We measure the two-dimensional fluid capacity distributions of published cross-section photomicrographs of sandstone, salt, and packed beds of glass beads, for various length scales A. The form of the distribution depends upon the magnitude of the length scale . For the sandstone and salt packs, appropriate length scales are found on which the fluid capacity has, to a good approximation, an exponential distribution. An exponential distribution appears to be inappropriate for the packed bed of glass beads on all length scales.     

Statistical Synthesis of Imaging and Porosimetry Data for the Characterization of Microstructure and Transport Properties of Sandstones

The microstructure of a suite of sandstone samples is quantitatively analyzed using a method which combines information from thin section micrographs of the pore space with mercury injection porosimetry in a statistical framework. This method enables the determination of a continuous distribution of pore sizes ranging from few nanometre to several hundred micrometre. The data obtained unify fractal and Euclidean aspects of the void space geometry, yield estimates of the pore-to-throat aspect ratio and challenge the ability of commonly used network models to describe fluid percolation in multiscale porous media. Application of critical path analysis to the prediction of flow permeability and electrical conductivity of sandstone core samples using the new information produces results comparable to those obtained by the classical approach—a fact attributed to the presence of macroscopic heterogeneity at the scale of several millimetres.     

A constitutive model for plastically anisotropic solids with non-spherical voids

Plastic constitutive relations are derived for a class of anisotropic porous materials consisting of coaxial spheroidal voids, arbitrarily oriented relative to the embedding orthotropic matrix. The derivations are based on nonlinear homogenization, limit analysis and micromechanics. A variational principle is formulated for the yield criterion of the effective medium and specialized to a spheroidal representative volume element containing a confocal spheroidal void and subjected to uniform boundary deformation. To obtain closed form equations for the effective yield locus, approximations are introduced in the limit-analysis based on a restricted set of admissible microscopic velocity fields. Evolution laws are also derived for the microstructure, defined in terms of void volume fraction, aspect ratio and orientation, using material incompressibility and Eshelby-like concentration tensors. The new yield criterion is an extension of the well known isotropic Gurson model. It also extends previous analyses of uncoupled effects of void shape and material anisotropy on the effective plastic behavior of solids containing voids. Preliminary comparisons with finite element calculations of voided cells show that the model captures non-trivial effects of anisotropy heretofore not picked up by void growth models.     

Analytical Determination of Flow Patterns in a Double Porosity Medium Containing Ellipsoidal Occluded Macro-Voids

This article presents the analytical study of fluid flow in a porous medium presenting pores of two different length scales: at the smallest or microscopic scale, the presence of connected voids confers a porous medium structure to the material investigated, while at the upper or mesoscopic scale, occluded macro-pores are present. This microstructure is employed to represent the progressive opening of inter-aggregate pore spaces observed in natural compacted montmorillonites polluted by heavy metal ions. Three-dimensional analytical expressions are rigorously derived for the pore fluid velocity and excess pore fluid pressure within the porous matrix, around an occluded ellipsoidal inter-aggregate void. The eccentricity ratio is employed to characterize the geometrical shape of the ellipsoidal void, while its size is characterized by the macro-porosity. Confrontations are made with numerical solutions in order to investigate the applicability of the analytical pressure and velocity solutions to microstructures of finite size.     

Verification of the transferability of micromechanical parameters by cell model calculations with visco-plastic materials

Finite element (FE) calculations of a cylindrical cell containing a spherical hole have been performed under large strain conditions for varying triaxiality with three different constitutive models for the matrix material, i.e. rate independent plastic material with isotropic hardening, visco-plastic material under both isothermal and adiabatic conditions, and porous plastic material with a second population of voids nucleating strain controlled. The “mesoscopic” stress-strain and void growth responses of the cell are compared with predictions of the modified Gurson model in order to study the effects of varying triaxiality and strain rate on the critical void volume fraction. The interaction of two different sizes of voids was modelled by changing the strain level for nucleation and the stress triaxiality. The study confirms that the void volume fraction at void coalescence does not depend significantly on the triaxiality if the initial volume fraction of the primary voids is small and if there are no secondary voids. The strain rate does not affect f_c either. The results also indicate that a single internal variable, f, is not sufficient to characterize the fracture processes in materials containing two different size-scales of void nucleating particles.     

Improved Interpretation of Mercury Intrusion and Soil Water Retention Percolation Characteristics by Inverse Modelling and Void Cluster Analysis

This work addresses two continuing fallacies in the interpretation of percolation characteristics of porous solids. The first is that the first derivative (slope) of the intrusion characteristic of the non-wetting fluid or drainage characteristic of the wetting fluid corresponds to the void size distribution, and the second is that the sizes of all voids can be measured. The fallacies are illustrated with the aid of the PoreXpert^® inverse modelling package. A new void analysis method is then described, which is an add-on to the inverse modelling package and addresses the second fallacy. It is applied to three widely contrasting and challenging porous media. The first comprises two fine-grain graphites for use in the next-generation nuclear reactors. Their larger void sizes were measured by mercury intrusion, and the smallest by using a grand canonical Monte Carlo interpretation of surface area measurement down to nanometre scale. The second application is to the mercury intrusion of a series of mixtures of ground calcium carbonate with powdered microporous calcium carbonate known as functionalised calcium carbonate (FCC). The third is the water retention/drainage characteristic of a soil sample which undergoes naturally occurring hydrophilic/hydrophobic transitions. The first-derivative approximation is shown to be reasonable in the interpretation of the mercury intrusion porosimetry of the two graphites, which differ only at low mercury intrusion pressures, but false for FCC and the transiently hydrophobic soil. The findings are supported by other experimental characterisations, in particular electron and atomic force microscopy.     

Effect of a nonuniform distribution of voids on the plastic response of voided materials: a computational and statistical analysis

This study investigates the overall and local response of porous media composed of a perfectly plastic matrix weakened by stress-free voids. Attention is focused on the specific role played by porosity fluctuations inside a representative volume element. To this end, numerical simulations using the Fast Fourier Transform (FFT) are performed on different classes of microstructure corresponding to different spatial distributions of voids. Three types of microstructures are investigated: random microstructures with no void clustering, microstructures with a connected cluster of voids and microstructures with disconnected void clusters. These numerical simulations show that the porosity fluctuations can have a strong effect on the overall yield surface of porous materials. Random microstructures without clusters and microstructures with a connected cluster are the hardest and the softest configurations, respectively, whereas microstructures with disconnected clusters lead to intermediate responses. At a more local scale, the salient feature of the fields is the tendency for the strain fields to concentrate in specific bands. Finally, an image analysis tool is proposed for the statistical characterization of the porosity distribution. It relies on the distribution of the ‘distance function’, the width of which increases when clusters are present. An additional connectedness analysis allows us to discriminate between clustered microstructures.     

Advective fluxes in multiphase porous media under nonisothermal conditions

We consider the case in which more than one fluid phase occupies the void space of a porous medium. The advective flux law is formulated for a fluid phase, under nonisothermal conditions and with the presence of solutes in the fluid phases. The derivation of the flux laws is based on an approximated version of the averaged balance equation for linear momentum. Taking into account momentum transfer through the interface between the fluid phases, leads to coupling between the flow in adjacent phases. Fluxes are also shown to depend on the surface tension at the interface between the adjacent fluid phases. Since the latter depends on temperature and solute concentration in the two phases, the advective flux is shown to depend on both temperature and solute concentration gradients in the two phases. A preliminary order of magnitude analysis gives conditions under which the coupling phenomena are not negligible. The approach is applied to the unsaturated zone, as a typical example of a multiphase porous medium.The main conclusion is that the well known Darcy law for single phase flow, may have to be modified for a multi fluid phase system, especially when temperature and solute concentration are not uniform.     

The size effect on void growth in ductile materials

We have extended the Rice-Tracey model (J. Mech. Phys. Solids 17 (1969) 201) of void growth to account for the void size effect based on the Taylor dislocation model, and have found that small voids tend to grow slower than large voids. For a perfectly plastic solid, the void size effect comes into play through the ratio εl/R₀, where l is the intrinsic material length on the order of microns, ε the remote effective strain, and R₀ the void size. For micron-sized voids and small remote effective strain such that εl/R₀?0.02, the void size influences the void growth rate only at high stress triaxialities. However, for sub-micron-sized voids and relatively large effective strain such that εl/R₀>0.2, the void size has a significant effect on the void growth rate at all levels of stress triaxiality. We have also obtained the asymptotic solutions of void growth rate at high stress triaxialities accounting for the void size effect. For εl/R₀>0.2, the void growth rate scales with the square of mean stress, rather than the exponential function in the Rice-Tracey model (1969). The void size effect in a power-law hardening solid has also been studied.     

Dynamic void nucleation and growth in solids: A self-consistent statistical theory

We present a framework for a self-consistent theory of spall fracture in ductile materials, based on the dynamics of void nucleation and growth. The constitutive model for the material is divided into elastic and “plastic” parts, where the elastic part represents the volumetric response of a porous elastic material, and the “plastic” part is generated by a collection of representative volume elements (RVEs) of incompressible material. Each RVE is a thick-walled spherical shell, whose average porosity is the same as that of the surrounding porous continuum, thus simulating void interaction through the resulting lowered resistance to further void growth. All voids nucleate and grow according to the appropriate dynamics for a thick-walled sphere made of incompressible material. The macroscopic spherical stress in the material drives the response in all volume elements, which have a distribution of critical stresses for void nucleation, and the statistically weighted sum of the void volumes of all RVEs generates the global porosity. Thus, macroscopic pressure, porosity, and a distribution of growing microscopic voids are fully coupled dynamically. An example is given for a rate-independent, perfectly plastic material. The dynamics of void growth gives rise to a rate effect in the macroscopic material even though the parent material is rate independent.     

The void-size effect on plastic flow localization in the Gurson model

Recent studies have shown that the size of microvoids has a significant effect on the void growth rate. The purpose of this paper is to explore whether the void size effect can influence the plastic flow localization in ductile materials. We have used the extended Gurson‘s dilatational plasticity theory, which accounts for the void size effect, to study the plastic flow localization in porous solids with long cylindrical voids. The localization model of Rice is adopted, in which the material inside the band may display a different response from that outside the band at the incipient plastic flow localization. The present study shows that it has little effect on the shear band angle.     

孔洞缺陷对3D打印Ti-6Al-4V合金疲劳试样应力分布的影响

3D打印金属技术因其个性化及可用于加工复杂零件等显著优点,在医用骨植入体领域得到了快速发展,但3D打印金属材料的孔洞缺陷所引起的应力集中现象严重降低了其疲劳强度,限制了3D打印生物金属材料的运用。本文针对3D打印Ti-6Al-4V合金超声疲劳试样,分析了Micro CT扫描试样得到的三维图像,获得了试样内孔洞缺陷的数量与体积;选择体积分数占比最大的孔洞,采用有限元方法分析了三种不同孔洞分布形式下的局部应力集中现象。研究发现,因空间位置的不同,独立的孔洞、接近自由表面的孔洞、相邻的孔洞三种不同孔洞的分布情况的应力集中系数差异显著。研究结果在一定程度上解释了目前EBM技术打印Ti-6Al-4V合金的孔洞缺陷如何对材料受力后的局部应力情况产生影响。     

Axisymmetric deformation of power-law solids containing a dilute concentration of aligned spheroidal voids

T response of an incompressible power-law matrix containing a dispersion of aligned, spheroidal voids is investigated. Attention is restricted to dilute concentrations of voids and to axisymmetric deformation of the solid. The essential step in the analysis is the solution of a kernel problem for an isolated void, and this solution is obtained accurately and efficiently using a Ritz procedure developed for this purpose. Results for macroscopic strain-rates are presented for void shapes ranging from penny-shaped cracks to infinitely long circular cylinders and for a wide range of triaxialities and matrix hardening exponents. These results are used to assess the role of void shape on the overall response of porous solids.     

Characterization of bubble mobility in channel flow with fibrous porous media walls

In composites processing, resin is introduced into a fibrous domain to cover all the empty spaces between the fibers. It is important to extract air bubbles from the domain before the resin solidifies. Failure to do so will entrap these voids in the final part, which is detrimental to its performance. Hence, there is a need to understand bubble motion in a fibrous porous domain in which the bubbles move with the resin in channels surrounded by fibrous walls. A rising bubble model is presented that consists of a single spherical void in a cylindrical axisymmetric two-phase domain of resin and air surrounded by porous media boundaries. The motion of a bubble in a channel flow with porous boundaries is modeled by replacing the walls with a slip velocity. Focus is on how the porous media permeability influences the bubble motion. A parameter called bubble mobility is defined as the ratio of bubble rise velocity to the resin free surface velocity. Results suggest that fabric permeability and fluid properties can be optimized to increase bubble mobility and ultimately lead to reduction in void content during composites processing.     

Compaction of a mixture of copper and molybdenum nanopowders modeled by the molecular dynamics method

A problem of compacting a mixture of copper and molybdenum nanopowders under the action of external loading generated by a spherical piston is solved by the molecular dynamics method. Interatomic interaction is calculated with the use of a multiparticle potential obtained by the embedded atom method. It is shown that compaction leads to significant deformations in copper, resulting in the loss of the crystalline structure; copper nanoparticles melt and fill the entire porous space. Molybdenum particles are deformed to a much smaller extent; they are not destroyed and preserve their crystalline structure. Under high loading, there appear voids in copper at the stage of compact extension; these voids rapidly grow in size and coagulate into one large void located in the nanocell center. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 11–23, September–October, 2008.     

Image force on a straight dislocation emitted from a cylindrical void

The image force exerted by the free surface of a cylindrical circular void on a nearby straight dislocation depends on whether the dislocation has arrived at its location by the emission from the surface of the void, or by the glide from infinity. In the context of elasticity theory, in the first case, the dislocation has been created by imposing the displacement discontinuity along the cut from the free surface of the void to the center of the dislocation, and, in the second case, from the center of the dislocation to infinity. The explicit expressions for the two corresponding image forces are derived and compared. It is shown that the attraction from the free surface of the void is stronger in the first case, particularly for smaller voids. Furthermore, in the case of dislocation emitted from the surface of the void, the interaction energy depends on the cut used to impose the displacement discontinuity, but not in the case of a dislocation approaching the void from infinity. The relevance of the obtained results for the materials science problems is discussed.     

Plastic potential and yield function of porous materials with aligned and randomly oriented spheroidal voids

Based on an energy approach, the plastic potential and yield function of a porous material containing either aligned or randomly oriented spheroidal voids are developed at a given porosity and pore shape. The theory is applicable to both elastically compressible and incompressible matrix and, it is proved that, in the incompressible case, the theory with spherical and aligned spheroidal voids also coincides with Ponte Castaneda's bounds of the Hashin-Shtrikman and Willis types, respectively. Comparison is also made between the present theory and those of Gurson and Tvergaard, with a result giving strong overall support of this new development. For the influence of pore shape, the yield function and therefore the stress-strain curve of the isotropic porous material are found to be stiffest when the voids are spherical, and those associated with other pore shapes all fall below these values, the weakest one being caused by the disc-shaped voids. The transversely isotropic nature of the yield function and stress-strain curves of a porous material containing aligned pores are also demonstrated as a function of porosity and pore shape, and it is further substantiated with a comparison with an exact, local analysis when the void shape becomes cylindrical.     

一维流体饱和粘弹性多孔介质层的动力响应

本文研究了不可压流体饱和粘弹性多孔介质层的一维动力响应问题。基于粘弹性理论和多孔介质理论,在流相和固相微观不可压、固相骨架服从粘弹性积分型本构关系和小变形的假定下,建立了不可压流体饱和粘弹性多孔介质层一维动力响应的数学模型,利用Laplace变换,求得了原初边值问题在变换空间中的解析解,并利用Laplace逆变换的Crump数值反演方法,得到原动力响应问题的数值解。数值研究了饱和标准线性粘弹性多孔介质层的动力响应,分析了固相位移、渗流速度、孔隙压力及固相有效应力等的响应特征。结果表明,与不可压流体饱和弹性多孔介质相同,不可压流体饱和粘弹性多孔介质中亦只存在一个纵波,并且固相骨架的粘性对动力行为有显著的影响。