Populations Of Voids Research Articles

The role of secondary voids in the mechanism of ductile fracture at a crack tip

The mechanics and mechanisms of ductile fracture, ahead of a blunting crack tip, have been studied extensively. Several computational studies analyzing the effect of initial void volume fraction, void shape, void spatial distribution and the mode of loading (KI,KII etc.) on crack-void interaction and its consequence on the mechanism of fracture have been reported. The influence of small size secondary voids on the failure of ligament between the large primary voids and, hence, on fracture toughness has been analyzed using the Gurson-type homogenized models of ductile fracture. In the present work, the two populations of voids ahead of a crack tip are modeled discretely. A plane strain, central line cracked boundary layer model under small-scale yielding is considered. The role of initial shape and spatial distribution of secondary voids, matrix strain hardening and mode of imposed loading in the mechanism of ductile crack growth initiation and advance is analyzed in detail. For completeness sake, numerical calculations are also performed using a homogenized representation of the secondary voids. The results so obtained are then compared with the predictions based on discrete modeling of the secondary voids. Our numerical studies revealed that plastic flow localization resulting from a small initial volume fraction of favorably distributed secondary voids may alter the path of crack growth initiation and advance, thus, influencing the ductile fracture toughness.

Analysis of the mechanical behavior of porous materials containing two populations of voids under dynamic spherical loading

A computational homogenization analysis is performed on three-dimensional representative volume elements (RVE) that contain two distinct populations of voids, to demonstrate the influence of the interaction between cavities. RVEs are constructed as cubic elastic perfectly-plastic matrices embedding two families of spherical voids, and subjected to dynamic loading under homogeneous kinematic boundary conditions. Multiple microstructure models are considered by varying the number, position, and size of voids to evaluate the micro-inertia contribution to the overall macroscopic stress. Velocity fields within numerical RVEs are investigated to reveal the interaction of voids and their key role in the dynamic macroscopic response of the porous material. Based on numerical simulation results, a homogenization analytical model considering void interaction is proposed to describe the mechanical behavior under dynamic loadings. This model relies on adjusting the strain rate level at the boundary of unit cells composing the porous material. Comparison between our numerical and analytical results with those obtained using the classical Taylor homogenization scheme highlights the limitations of the Taylor model in the case of porous materials under dynamic loading.

Effect of Shape and Distribution of Secondary Voids on Ductile Crack Path

Ductile fracture in structural metals and alloys comprises of three distinct stages: namely nucleation, growth, and coalescence of voids. In this failure mode voids, either pre-existing in the material or nucleated during deformation grow until they coalesce to form a continuous fracture path. The larger voids nucleating early in the deformation history are often referred to as the primary voids whereas the smaller ones (typically around 0.1 to 0.01 times the size of the primary voids) nucleate much later and are described as the secondary voids. As the imposed deformation increases, voids grow, and the ductile crack path is influenced by the interaction and coalescence of the two-scale voids. The present work numerically models the interaction between the two populations of pre-existing voids lying ahead of the crack tip. The influence of secondary voids attributes, in particular, the shape and distribution of secondary voids on ductile crack propagation is analysed. A plane-strain modified boundary layer (MBL) model, under small-scale yielding condition, subjected to remote mode-I loading is analysed. Both primary and secondary voids lying near the crack tip are modelled explicitly. In the absence of secondary voids, as expected, the crack path is controlled by the distribution of primary voids. The presence of small secondary voids, however, influences the crack growth and, hence, the crack driving force. Various initial configurations of secondary voids are modelled and some details of crack propagation and the numerically computed driving force versus crack extension (J-Δa) curves are presented.

On the role of shape and distribution of secondary voids in the mechanism of coalescence

Ductile fracture of metals at room temperature has been studied extensively. Since localization of plastic flow is often viewed as a precursor to ductile fracture, several computational studies analyzing the influence of stress triaxiality and the Lode parameter on the critical strain for onset of localization have been reported. The implications of strain localization in materials containing randomly distributed voids for the mode of failure, that is, internal necking or shear banding have also been discussed. Apart from the influence of stress state, it is also of interest to model the effect of fine-scale microstructural attributes on the mode of failure and, hence, on material’s ductility. Therefore, in the present work, the mechanism of ductile fracture of an elastic–plastic solid containing two populations of pre-existing voids is modeled numerically. In the undeformed state, all the voids belonging to each family are assumed to have identical shape and size. Two different periodic spatial arrangements of primary voids are considered such that, in the absence of secondary voids, one of the configurations leads to internal necking whereas the other one exhibits a tendency for shear localization as the mode of coalescence for the larger voids. Small size secondary voids are then introduced explicitly and the role of shape and distribution of these sub-microscopic voids in the mechanism of coalescence and final failure is illustrated under plane strain as well as axisymmetric tension with a superposed hydrostatic stress. Depending on the microstructural attributes of the secondary voids, the mode of primary void coalescence is observed to shift from internal necking to void-sheeting and vice-versa. The implications of this shift in the mode of void coalescence on mesoscopic ductility is examined and a comparison of numerical predictions is made with the available experimental results.

An USAXS-SAXS study of nano-TATB under uniaxial die pressures

The microstructure of a compressed explosive solid is closely related to its shock sensitivity and mechanical properties. In this study, ultra-small-angle and small-angle X-ray scattering (USAXS and SAXS) techniques were combined to explore the hierarchical microstructure of die-pressed 2,4,6-trinitro-1,3,5-benzenetriamine (TATB) discs obtained from nanostructured TATB (nano-TATB) powder as the precursor. Using the Guinier-Porod model, the pseudo-invariants, and Porod's law, this study analyzed the microstructures of the materials on a nanometer scale to track the changes in void size, porosity, and interfacial area, which reflected the response of TATB under applied pressures of 1, 2, 5, 10, 15 ​kN and 30 ​kN. Results show that there existed three populations of voids in the measured q range. The intergranular voids with sizes of tens of nanometers were sensitive to low pressures (<15 ​kN) and presented a smooth interface with the TATB matrix. The intragranular voids with sizes of 7–8 ​nm exhibited a fairly small volume-filling ratio under high pressures (>15 ​kN), as indicated by the decrease in the volume fractal dimension. Porosities of the voids with sizes of 1–900 ​nm, which were determined by the pseudo-invariants obtained from the scattering data, decreased from 6% to 1% as the pressures increased from 1 ​kN to 30 ​kN. The response of these structural parameters to external pressures implies that the main densification mechanisms under die compression include the flow, fracturing, and plastic deformation of the TATB granules. This study provides a direct insight into the structural evolution of TATB during compression.

Densification of Two Forms of Nanostructured TATB under Uniaxial Die Pressures: A USAXS-SAXS Study.

Sequential ultra-small-angle and small-angle and X-ray scattering (USAXS and SAXS) measurements of hierarchical microstructure of a common energetic material, the high explosive 2,4,6-Triamino-1,3,5-trinitrobenzene (TATB), were performed to follow the microstructure evolution upon applied pressure. The pellets were prepared by two different routes-die pressed from a nanoparticle form and a nano-network form of TATB powder. The derived structural parameters, such as void size, porosity, and the interface area, reflected the response of TATB under compaction. Three populations of voids were observed in the probed q range from 0.007 to 7 nm-1. The inter-granular voids with size larger than 50 nm were sensitive to low pressures and had a smooth interface with the TATB matrix. The inter-granular voids with size of ~10 nm exhibited a less volume-filling ratio at high pressures (>15 kN) as indicated by a decrease of the volume fractal exponent. The response of these structural parameters to external pressures implied that the main densification mechanisms under die compaction were the flow, fracture, and plastic deformation of the TATB granules. Compared to the nanoparticle TATB, the applied pressure strongly influenced the nano-network TATB due to its more uniform structure. The findings and research methods of this work provide insights into the structural evolution of TATB during densification.

Numerical simulations and modeling of the effective plastic flow surface of a biporous material with pressurizedintergranular voids

This study is devoted to the effective plastic flow surface of a biporous polycrystalline material, with an intragranular porosity due to spherical voids, and an intergranular porosity due to larger elongated voids along the grain boundaries. These two populations of voids (or bubbles) with well separated scales and shapes are saturated by a fluid and therefore are subjected to internal pressures. The effect of the intragranular voids is modeled through a GTN (Gurson–Tvergaard–Needleman) criterion in the matrix. Numerical simulations are performed with a FFT-based (Fast Fourier Transforms) method. Particular attention is paid to the effect of the distribution of the intergranular bubbles on the effective plastic flow surface. Different microstructures with different volume fractions and sizes for the intergranular bubbles are tested under three loading conditions (the mean size of the grains being fixed). Two main results are exhibited. First, it is shown that the effect of the relative size of the intergranular bubbles on the effective plastic flow surface depends on the loading direction. Secondly, a comparison is made with the analytical model of (Vincent et al., 2014) and a correction of the porosity relative to the intergranular bubbles is introduced in this analytical model in order to take into account the specific distribution of the intergranular bubbles along the grain boundaries. This correction is expressed as a sum of two power law functions, each of them being significant either for low or for large values of the porosity of intergranular bubbles.

On the role of secondary voids and their distribution in the mechanism of void growth and coalescence in porous plastic solids

Detailed three-dimensional finite element based cell model studies are carried out to understand the role of secondary voids and their distribution in the mechanism of void growth and coalescence in a porous plastic solid. In contrast to most of the previous studies, the two populations of voids (primary and secondary) are modeled explicitly, with more realistic arrangements of the later. Numerical studies revealed that the spacing of the secondary voids sets the length scale at which localization of plastic flow occur. For a given volume fraction of the secondary voids, the mesoscopic ductility depends on the void distribution. In particular, the void arrangements leading to closely spaced secondary voids result in an early onset of plastic flow localization. The mesoscopic ductility, for a fixed arrangement of secondary voids, is observed to depend on the initial volume fraction and also on the location of the secondary voids in the ligament between the primary voids. For the stress triaxiality regime (1 ≤ T ≤ 3), and for the various distributions of the secondary voids analyzed in the present study, some variation in the total void volume fraction corresponding to the onset of flow localization is observed. The variation, however, is not so large as to forbid associating the critical void volume fraction with the void coalescence criterion. Thus, as a practical approach, an average value of this critical porosity may serve as the criterion for simulating accelerated void growth accompanying coalescence in a porous plastic solid with two populations of voids, using some phenomenological constitutive framework.

Measuring baryon acoustic oscillations from the clustering of voids

We investigate the necessary methodology to optimally measure the baryon acoustic oscillation (BAO) signal, from voids based on galaxy redshift catalogues. To this end, we study the dependency of the BAO signal on the population of voids classified by their sizes. We find for the first time the characteristic features of the correlation function of voids including the first robust detection of BAOs in mock galaxy catalogues. These show an anti-correlation around the scale corresponding to the smallest size of voids in the sample (the void exclusion effect), and dips at both sides of the BAO peak, which can be used to determine the significance of the BAO signal without any priori model. Furthermore, our analysis demonstrates that there is a scale dependent bias for different populations of voids depending on the radius, with the peculiar property that the void population with the largest BAO significance corresponds to tracers with approximately zero bias on the largest scales. We further investigate the methodology on an additional set of 1,000 realistic mock galaxy catalogues reproducing the SDSS-III/BOSS CMASS DR11 data, to control the impact of sky mask and radial selection function. Our solution is based on generating voids from randoms including the same survey geometry and completeness, and a post-processing cleaning procedure in the holes and at the boundaries of the survey. The methodology and optimal selection of void populations validated in this work have been used to perform the first BAO detection from voids in observations, presented in a companion paper.

An elastic–plastic model for porous rocks with two populations of voids

In the present work, a new elastic–plastic model is proposed for a class of porous rock-like materials with two populations of pores at different scales. This model is based on the closed-form plastic criterion which was established from a nonlinear homogenization procedure in our previous work (Shen et al., 2014) . This criterion explicitly takes into account the effects of two populations of voids, respectively distributed at the microscopic and mesoscopic scales. In order to consider the plastic compressibility and pressure dependency, the solid phase at the microscopic scale is assumed to obey to a Drucker–Prager criterion. The constitutive model is completed by a non-associated plastic flow rule and an isotropic hardening law, which are defined in a phenomenological way. The proposed model is applied to describe the macroscopic mechanical behavior of the Lixhe chalk with different confining pressures. Comparisons between numerical results and experimental data are presented for the verification of the proposed model.

Void coalescence in ductile solids containing two populations of voids

Many metallic alloys contain a primary and a secondary population of voids, the difference in size reaching up to orders of magnitude. The present study extends Thomason’s and Benzerga−Leblond’s criteria for the onset of coalescence of primary voids to incorporate the effects of a secondary population. Plastic limit load analysis is performed within a finite element (FE) framework by using three dimensional (3D) cubic unit cells. The macroscopic stress state of the unit cells corresponds to axisymmetric tension with Σ22>Σ11=Σ33. The extended versions of both criteria are able to successfully predict the critical stress for the onset of coalescence.

Microstructural characterization and density change of 304 stainless steel reflector blocks after long-term irradiation in EBR-II

While thin reactor structural components such as cladding and ducts do not experience significant gradients in dpa rate, gamma heating rate, temperature or stress, thick components can develop strong local variations in void swelling and irradiation creep in response to gradients in these variables. In this study we conducted microstructural investigations by transmission electron microscopy of two 52 mm thick 304-type stainless steel hex-blocks irradiated for 12 years in the EBR-II reactor with accumulated doses ranging from ∼0.4 to 33 dpa. Spatial variations in the populations of voids, precipitates, Frank loops and dislocation lines have been determined for 304 stainless steel sections exposed to different temperatures, different dpa levels and at different dpa rates, demonstrating the existence of spatial gradients in the resulting void swelling. The microstructural measurements compare very well with complementary density change measurements regarding void swelling gradients in the 304 stainless steel hex-block components.The TEM studies revealed that the original cold-worked-state microstructure of the unirradiated blocks was completely erased by irradiation, replaced by high densities of interstitial Frank loops, voids and carbide precipitates at both the lowest and highest doses. At large dose levels the amount of volumetric void swelling correlated directly with the gamma heating gradient-related temperature increase (e.g. for 28 dpa, ∼2% swelling at 418 °C and ∼2.9% swelling at 448 °C). Under approximately iso-thermal local conditions, volumetric void swelling was found to increase with dose level (e.g. ∼0.2% swelling at 0.4 dpa, ∼0.5% swelling at 4 dpa and ∼2% swelling at 28 dpa). Carbide precipitate formation levels were found to be relatively independent of both dpa level and temperature and induced a measurable densification. Void swelling was dominant at the higher dose levels and caused measurable decreases in density. Void swelling at the lowest doses was larger than might be expected based on the dpa level, an observation in agreement with earlier studies showing that the onset of void swelling is accelerated by decreasing dpa rates.

Nanovoid morphology and distribution in deformed HDPE studied by magnified synchrotron radiation holotomography

The individual morphology and size distribution of nano-cavities in a necked tensile specimen made of semicrystalline high-density polyethylene (HDPE) have been observed and quantified in three dimensions using magnified synchrotron radiation holotomography. Here, using a voxel size of 59.7 nm, permanent voids in the unloaded state after yielding are observed that can only be assessed in averaged manner at finer length scales using small angle X-ray scattering (SAXS) techniques. Observation of the same sample using typical synchrotron radiation tomography with micron resolution is also given highlighting the novel opportunities arising with magnified synchrotron radiation holotomography in comparison. Two populations of voids elongated in loading direction are observed: A population of large cavities that has nucleated around inclusions and a population of small cavities. All cavities are equiaxed in radial directions and elongated in loading direction which is consistent with SAXS findings of other studies. Void volume fraction, void size distributions and characteristic dimensions of the voids are assessed using 3D image analysis.

Effective flow surface of porous materials with two populations of voids under internal pressure: I. A GTN model

This study is devoted to the effective plastic flow surface of a bi-porous material saturated by a fluid. The material under consideration exhibits two populations of voids. The smaller voids are spherical voids whereas the larger ones are spheroidal and randomly oriented inside the material. These two populations of voids are subjected to internal pressures due to the presence of gases. Approximate models for the effective plastic flow surface of such a bi-porous saturated material have previously been proposed in Vincent et al. (2009), where a three-scale homogenization procedure has been performed: first, smearing out all the small spherical bubbles using a Gurson-like matrix, and second, smearing out the intergranular ellipsoidal bubbles. Our objective here is to derive a simple analytical expression of the effective flow surface, starting from one of these previous models, obtained by generalizing the approach of Gologanu et al. (1994) to compressible materials. The main contributions of the present paper are: (1) an expression for the average dilatation-rate in the matrix, (2) an approximation of the effective flow surface in the form of a Gurson–Tvergaard–Needleman criterion. The accuracy of this new model is assessed in a companion paper by comparison with full field numerical simulations.

Effective flow surface of porous materials with two populations of voids under internal pressure: II. Full-field simulations

This study is devoted to the effective plastic flow surface of a bi-porous material saturated by a fluid. Highly irradiated uranium dioxide is a typical example of such a material. In part I of this study, a GTN-type approximation of the effective plastic flow surface has been derived. In this second part, the predictions of this new model are compared with full-field numerical simulations performed with a numerical method based on Fast Fourier Transforms. This method is successfully applied to voided materials with a Gurson matrix where the voids are subjected to internal pressure. Different microstructures containing a large number of spherical or ellipsoidal voids are investigated. The deviation from isotropy of their mechanical response is measured by a new criterion.

Computational homogenization of plastic porous media with two populations of voids

The macroscopic yield response of random porous media containing two populations of spherical voids is investigated via large volume computations. The computed yield surface is compared to analytical criteria recently developed for the above-mentioned porous media. To overcome the observed discrepancies, the analytical models are modified by introducing additional parameters which are numerically derived. These parameters depend neither on the void volume fraction nor on the void size. Large volume computations are also used to build the yield surface for porous media containing a single population of randomly distributed spherical voids. The results show that, for an identical fraction of porosities, there is no significant difference between a double and a single population of voids. This result implies that the yield response of porous media containing two populations of voids may be also described by analytical criteria developed in the case of a single population.

Size effects in ductile failure of porous materials containing two populations of voids

Abstract The ductile failure behavior of porous materials containing two populations of voids of different size is investigated numerically by means of 3D cell model calculations. In contrast to previous studies a non-local Gurson model is used to describe the secondary void population in the matrix material. Due to the internal length scale incorporated in the non-local model, it is possible to describe the size of the secondary voids in the matrix material. The results are obtained for loading states with different stress triaxialities and Lode parameters. The influence of the size of the secondary voids is analyzed and it is shown that larger secondary voids lead to a higher stress carrying capacity. This size effect is studied for different primary void arrangements. Furthermore, the strain and primary void volume fraction at the onset of coalescence are presented and cross references to experimental findings are drawn.

Observations on Mode I ductile tearing in sheet metals

Abstract Cracked ductile sheet metals, subject to Mode I tearing, have been observed to display a variety of fracture surface morphologies depending on the material properties, and a range of studies on the fracture surface appearance have been published in the literature. Whereas classical fractures such as cup-cone, cup–cup and slanting are widely observed, the phenomenon of a slanted crack which systematically “flips” back and forth in a roughly 45° orientation has only recently been reported. The present study aims to add details and understanding to this crack growth phenomenon – through experimental testing and comparison with published results. A series of crack propagation tests have been carried out, where cracks are driven many plate thicknesses under global Mode I loading. The current study employs both the edge crack specimen (ECS) loaded in combined in-plane bending and extension, and the double edge notched tension (DENT) specimen loaded in extension. Clear evidence of the alternating slant failure is observed for a normal strength steel and details on the phenomenon are brought out by grinding and polishing cross-sections in successive steps along the crack growth direction. Moreover, these results are compared to a soft aluminum (1050A) which predominantly shows cup–cup or slant failure. In general, both materials fail by the mechanism of void growth to coalescence (fractographs are included), and evidence points toward cup–cup being favored by materials that experience extensive amounts of void growth (thus severe thinning in the process zone), whereas slanting is typical in high strength materials that rapidly nucleate large populations of smaller voids.

Modeling ductile fracture in metals involving two populations of voids – influence of continuous nucleation of secondary voids upon growth and coalescence of primary voids

This paper is devoted to the study of the influence of the nucleation of small, secondary voids upon the growth and coalescence of large, primary cavities in porous ductile metals. For this purpose, a simple model is considered: a hollow sphere subjected to hydrostatic loading whereby the central hole simulates a large, primary void, and the presence of secondary voids in the surrounding matrix is accounted for by the Gurson–Tvergaard–Needleman homogenized model for plastic porous materials. Continuous nucleation of small voids in the matrix is described by the straightforward phenomenological formula proposed by Pineau and Joly. Initially, the problem is solved analytically, then numerically when a non-uniform distribution of secondary voids develops in the matrix. These elements are used to define a simplified model in which only a small number of internal parameters describe the growth of primary voids coupled with the nucleation, growth and coalescence of secondary voids. Comparisons between the results derived from this simplified model and those obtained through the numerical computations are used to demonstrate the accuracy of the proposed model.

Cavitation during deformation of polymers on the example of polypropylene

AbstractCavitation during uniaxial deformation of isotactic polypropylene (PP) was studied in samples having different amounts of α and β crystals. It has been shown that deformation is always accompanied by neck formation. Because the very beginning the plastic deformation is accompanied by cavitation. The scale of cavitation, determined with the use of X‐ray scattering and volume measurements increases with an increase of the content of β phase in the examined samples. Cavities change their shape with deformation—at first they are elongated perpendicularly to the direction of stretching, next, at the strain of 1.5, they reorient in the direction of deformation, which is connected with changes in the surrounding lamellar structure. The calculations of Guinier's radius showed that two populations of voids are present in deformed PP samples, characterized by the gyration radii of 5 and 13 nm for small deformations. Estimations based on the volume increase and sizes of cavities indicate that typically more than one cavity are present in amorphous layers adhering to lamella surfaces. © 2012 Wiley Periodicals, Inc. J Appl Polym Sci, 2012