Void Growth Rate Research Articles

Integrated model for simulating Coble creep deformation and void nucleation/growth in polycrystalline solids - Part I: Theoretical framework

This study proposes an integrated model for simulating Coble creep deformation and void nucleation/growth in a 3D polycrystalline solid. Part I of the paper provides a theoretical framework for the proposed model. A representative volume element approach is employed to predict the effects of 3D polycrystalline morphology. The model comprises two distinct but interconnected stages: deformation and void nucleation/growth. The deformation stage of the proposed model comprises two sub-models: grain boundary (GB) migration and GB diffusion. The void nucleation/growth stage is composed of three consecutive calculations: void nucleation, void growth, and postprocessing. The process simulated in the void nucleation/growth stage is driven by relative GB velocity of the respective GBFs and diffusional fluxes between adjacent GBFs, which are provided from the deformation stage. The void nucleation rate is quantified using the relative GB velocity, and the initial void size is determined based on the stability condition for void existence derived from Helmholtz free energy. The void growth rate is evaluated by the atomic diffusion on the GBF and the surface of each void.

On the role of geometrically necessary dislocations in void formation and growth in response to shock loading conditions in wrought and additively manufactured Ta

This study investigates the role of geometrically necessary dislocations (GNDs) and microstructure on void nucleation and growth in wrought and additively manufactured (AM) tantalum subjected to high-strain rate loading. Multi-modal 3D data was collected using TriBeam tomography to calculate GND densities and their spatial relationship to voids. A microstructural comparison between the wrought and AM samples identified distinct void shapes and locations, with intragranular voids and more spherical voids frequently observed in the AM dataset. Results indicate that voids preferentially form at both high-angle grain boundaries and low-angle subgrain boundaries, the latter of which are frequently observed in the AM material. Through a radial distribution analysis of all voids in the datasets, significant GND localization to near-void-surface regions was observed in both samples. 3D crystal plasticity simulations were employed to extend the experimental observations, revealing higher void growth rates in [111] oriented grains when compared to [001] grains. The simulations also suggest that GNDs can be generated as part of the void growth process, with more GND accumulation for growth in a [111] grain than a [001] grain. These findings provide valuable insights into the links between nanoscale void nucleation, mesoscale void growth, and microstructural effects in dynamically loaded tantalum.

The mechanistic origins of heterogeneous void growth during ductile failure

In 1969, Rice and Tracey proposed that the rate of void growth during ductile failure is a strong function of the hydrostatic stress state. Numerous in-situ x-ray computed tomography (XCT) studies demonstrated that the average rate of void growth is well-predicted by modified versions of the Rice-Tracey equation. However, recent in-situ XCT studies of void growth demonstrated that individual voids grow in highly heterogeneous manners in a way that is not predicted by Rice-Tracey or similar models. Model-based studies using crystal plasticity finite element (CP-FE) suggest that local effects of grain orientation play a strong role during void growth, but we lack experimental data to test this hypothesis. The present study leverages recent advances in laboratory-based diffraction contrast tomography (Lab-DCT) and in-situ XCT to systematically examine the effects of grain orientation on void growth experimentally in an Al-2219 alloy. CP-FE modeling was used to assess the local stress states associated with voids and their influence on void growth rates. These data indicate that void growth is not simply controlled by the local hydrostatic stress or stress triaxiality. Additionally, no clear relationship between void growth rates and grain orientation were observed. Instead, void growth rates were highly stochastic in ways that could not be directly linked to the local stress or strain states. The combination of experimental and modeling data suggests that our current assumptions regarding the mechanisms of void growth are flawed and that additional factors, which may include the local dislocation density, affect the rate of void growth.

Growth rate of spherical voids with non-comoving dark matter and baryons

We present numerical solutions to Einstein’s equations describing large spherical cosmic voids constituted by two components: dark matter and baryons, with a non-vanishing initial relative velocity, in an asymptotically homogeneous background compatible with the ΛCDM concordance model. We compute numerically the evolution of such configurations in the dark matter frame, with a hypothetical homogeneous distribution of baryons, but respecting the values dictated by the concordance model for the average baryon-to-dark matter density ratio. We reproduce the well-known formation of overdensities at the edge of the void and recover the Lemaître–Tolman–Bondi solutions in the comoving limit of our simulations. We compute the average growth factor of matter fluctuations and find that it departs significantly from the linear perturbative prescription even in the comoving case, where the non-linearity of inhomogeneities has an impact.

A Noncontact Method for Estimating Thin Metal Film Adhesion Strength Through Current-Induced Void Growth

Abstract Studies have reported that the electromigration-induced void growth velocity in metal thin films is inversely related to the adhesion strength of the metal thin film with the base and passivation layers. It was also observed that the contribution of interface adhesion strength to electromigration resistance decreases with an increase in temperature. In this study, an expression is derived for the diffusive void growth velocity induced by electromigration from a generalized thermodynamically consistent continuum-based theory for reaction–diffusion driven solid-state interface evolution. This relation captures the effect of adhesion with the base and passivation layers on electromigration resistance of thin metal films. Electromigration experiments were carried out at elevated temperatures and high current density to induce voiding in thin Cu metal film deposited on a base layer of TiN and passivated with TiN or SiNx. The degradation of interface adhesion strength with temperature is modeled using an Andrade-type of relationship. The void growth rates characterized in these experiments are combined with the expression for void growth rate to estimate the interface adhesion strength for the Cu–TiN and Cu–SiNx interfaces. The methodology for estimating the adhesion strength of the metal-passivation layer interface is validated through comparison with interface adhesion strengths from mechanical de-adhesion tests reported in the literature.

Predicting porosity distribution effects on the orientation induced plastic anisotropy of ductile solids: A crystal plasticity investigation

This paper investigates the role of crystallographic orientation and heterogeneous void distribution on void growth using three-dimensional crystal plasticity simulations. Face-centered cubic crystal structure is chosen as the model system. The role of crystallographic orientation on void growth is mapped by modeling single crystals as cubic unit cells with an embedded spherical void. Reminiscent of real aggregates, porosity is then re-distributed in single crystals to assess the role played by heterogeneous distribution of voids. Simulations are coupled with statistical analysis tools to decode the complex interplay involved in orientation dependent anisotropic plasticity and stochastic parameters governing heterogeneous void distribution like void size and placement. The applicability of the obtained results for single crystals is appraised for polycrystals as well by embedding the porous crystal in a randomly textured polycrystalline aggregate. Statistical analysis is repeated for all the considered orientations of the central grain to highlight the intricacies associated with plastic anisotropy-randomness in the heterogeneous microstructure. The results reveal interesting variations in mechanical strength, ductility and void growth rate in the three-dimensional orientation space. Analysis of configuration-property linkage reveals critical variables regulating the material strength and ductility for single crystals as well as polycrystals. Ramifications of the obtained results in designing strong and tough ductile components for practical applications are discussed.

Void nucleation at dislocation boundaries aided by the synergy of multiple dislocation pile-ups

Void nucleation is of great significance in understanding ductile fracture. Recent experiments have shown that voids are nucleated via vacancy condensation and dislocation boundaries are the main nucleation sites. However, it is unclear what role is played exactly by dislocation boundaries in promoting void nucleation. Here we propose a new mechanism for dislocation boundary-induced void nucleation and develop a corresponding model based on the classical nucleation theory and vacancy diffusion theory. The model suggests that void nucleation is mainly influenced by hydrostatic stress, temperature, and relative vacancy concentration, whose contributions are systematically studied. It is also suggested that the vacancy formation energy and the interaction energy of hydrostatic stress and vacancy, which are absent in the previous models and introduced in ours, exhibit a clear tendency to lower the activation free energy barrier. Analysis of the nucleation kinetic suggests that the growth rate of void depends on the vacancy diffusion coefficient and vacancy concentration; the higher the values of these parameters, the faster the growth rate of the void. The kinetic feasibility of the newly proposed mechanism is examined using three-dimensional discrete dislocation dynamics simulations. The results predict that the size of incipient voids nucleated at the dislocation boundary is ∼35 nm, which is consistent with the experimental characterization value of ∼50 nm. Finally, when the relaxation of the dislocation boundary is considered, the synergistic effect is weakened.

The damage mechanism of the metastable β-titanium alloy Ti5321 with specific microtexture using synchrotron X-ray microtomography

In order to investigate damage mechanism of metastable β-titanium alloy Ti5321 with bimodal microstructure and specific microtexture, in situ tensile test was performed facilitating with synchrotron X-ray microtomography. The damage evolution of metastable β-titanium alloy Ti5321 can be predicted in three dimensions by proper micromechanics-based modelling methods, the Rice and Tracey model and Huang model. Through comparisons of the Rice and Tracey model and Huang model validated using the values of stress triaxiality calculated from experimental data and Finite Element (FE) simulations, the prediction accuracy of growth rates of voids can be enhanced. The results also show that the void growth is strongly related to microtexture of α plates. It is found that the growth path of largest voids is along α plates which have their c-axes with similar inclination angle of ∼45° from rolling direction (RD). Electron backscatter diffraction (EBSD) was also used to reveal the damage mechanism. This result is important enlightenment for study of damage mechanism of metastable β-titanium alloys.

Modeling of the Atomic Structure in the Vicinity of Spherical Voids in Aluminum and the Calculation of Anisotropy of the Void Growth Rate

Modeling of the Atomic Structure in the Vicinity of Spherical Voids in Aluminum and the Calculation of Anisotropy of the Void Growth Rate

Simulation of the atomic structure near spherical void in aluminum and the calculation of the anisotropy of the void growth rate

We simulate structure in the vicinity of different size nanovoids using a new variant of the Molecular Statics, wherein atomic structure in the vicinity of nanovoids and the parameters that define the displacements of atoms placed in elastic continuum around main computation cell are determined in a self-consistent manner. Then, the previously obtained kinetic equations are applied to calculate the shifting rate of the elements of the void surface in certain crystallographic directions. These equations take into account the dependence of the vacancy flux on the deformation fields. The displacement rates in different crystallographic directions for bcc and fcc metals are significantly different. The results show that the effects studied by computer simulation can lead to a change in the shape of initially spherical nanopores and cause their transformation into cuboidal pores in metals with a cubic structure under irradiation.

Void growth in ductile materials with realistic porous microstructures

In this paper, we have investigated void growth in von Mises materials which contain realistic porous microstructures. For that purpose, we have performed finite element calculations of cubic unit-cells which are subjected to periodic boundary conditions and include porosity distributions representative of three additively manufactured metals. The initial void volume fraction in the calculations varies between 0.00564% and 1.75%, the number of actual voids between 14 and 5715, and the pores size from 2.3μm to 110μm. Several tests with different void sizes and positions have been generated for each of the three porous microstructures considered, and for each test we have performed several realizations with different spatial arrangement of the voids. The simulations have been carried out with random spatial distributions of pores and with clusters of the same size and different void densities. The macroscopic stress state in the unit-cell is controlled by prescribing constant triaxiality and Lode parameter throughout the loading. Calculations performed exchanging the loading directions for a given distribution of void sizes and positions have shown that the porous microstructure makes the macroscopic strain softening of the unit-cell (slightly) anisotropic. Moreover, the results obtained with the realistic porous microstructures have been compared with unit-cell calculations having an equivalent single central pore, and with calculations in which the material behavior is modeled with Gurson plasticity. It has been shown that both initial void volume fraction and spatial and size distribution of voids affect the macroscopic response of the porous aggregate and the void volume fraction evolution. Moreover, the calculations with random spatial distribution of voids have brought out that different tests of the same microstructure carry significant variations to the effective behavior of the porous aggregate, and that the interaction between neighboring pores dictates the volume evolution of individual voids, especially at higher macroscopic triaxiality. The calculations with clusters have shown that pores clustering promotes localization/coalescence due to increased interaction between the voids, which results in an increased growth rate of voids in clusters with large number of pores. Moreover, the results for the evolution of the distribution of plastic strains in the unit-cell have provided quantitative indications of the role of porous microstructure on the development of heterogeneous plastic strain fields which promote macroscopic strain softening. Namely, the accelerated growth rate of the plastic strains near the voids which indicates the onset of localization/coalescence has been shown to occur earlier as the number of voids in the microstructure increases.

The Connection Between Electromigration Resistance and Thin-Film Adhesion and Their Degradation With Temperature

Growth of voids driven by electromigration in current carrying metal thin films is a common mode of failure in microelectronic devices. Studies have shown that electromigration driven void growth rate in such devices is inversely related to the interface adhesion strength between the current carrying metal thin films and the passivation layers on top of the metal films. In this paper, we characterize the degradation in electromigration resistance with temperature in metallic thin films with passivation and argue on a general mathematical form for connecting the electromigration resistance to adhesion strength. Test structures with thin film Cu metal lines are designed and fabricated for performing electromigration experiments and characterizing the void growth rate. Tests are carried out on devices with Cu thin film having TiN and SiN <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</sub> as passivation layers, over a range of current densities and temperatures. The growth rates of the voids in the Cu lines undergoing electromigration are analyzed using classical models from literature. The experimental observations on void growth rate with increase in current density and temperature are validated by comparing the data to those from prior literature. The void growth is also simulated and the void interface is tracked using an explicit boundary tracking computational technique. It is observed that the contribution of interface adhesion strength to electromigration resistance decreases with increase in temperature. That is, the experiments suggest that the passivation layers cease to provide resistance to electromigration with increasing temperature. Based on the test results, we develop a general form of a mathematical model to describe the degradation in electromigration resistance with metal film adhesion to the passivation layer.

In Situ/Operando Studies for Reduced Eletromigration in Ag Nanowires with Stacking Faults

AbstractIn this study, the effect of stacking faults (SFs) on electromigration in silver nanowires (AgNWs) in particular, with respect to their effects on necking and void growth, is investigated. The galvanic replacement reaction is used to synthesize the AgNWs in bulk at low cost. By varying the concentration of silver nitrate, AgNWs are obtained with and without SFs. In situ TEM analysis provides strong evidence that the SFs can effectively suppress the migration of surface atoms. Furthermore, an investigation of the void growth process reveals that SF facets parallel to the {111} plane contribute to the anisotropic change in morphology and slow down the rate of void growth by 135 times. Thus, planar defects can be beneficial to extending the lifetimes of devices by causing intrinsic changes to the material properties.

Use Of Micro-Void Content Growth Rates To Validate And Add Value To Electrical Insulation Tan-Delta Aging Testing

Our research as well as others has shown that micro-void content in electric insulation polymers grows in a way that can be correlated to the degree of aging. Specific results of our experiments combined with research by others have led us to conclude that a promising technique for predicting remaining life in electric cable insulation, based on micro-void content and proximity to void limiting parameters, can be developed. This approach involves use of acoustic or optical microscopy to establish an estimate of void content in polymers by determining micro-void sizes and density. Separate research is used to establish limiting values for percent void content correlating to material failure. The mode of failure varies depending on the applied voltage regime. For example, at medium and high voltage levels, partial discharge detection can be considered indicative of pending end of life. Whereas for low voltage regimes, brittleness to the point of cracking susceptibility would allow the potential for moisture ingress and shorting and can be considered end of life. It has been separately shown that void growth rate is a function of temperature and radiation dose rate both during normal and nuclear accident conditions and is predictable based on the known polymer chemical degradation equations, which produce gaseous products in the form of oxygen, water vapor, carbon dioxide, and carbon monoxide. Thus if end of life void content is known, the degree of void content growth occurring during a design basis accident is properly accounted for, and void content growth rates during normal temperature and radiation conditions are considered, then remaining life in electrical insulation can be accurately predicted. In recent years, several techniques have been proposed to assess electrical insulation aging. One of the more promising approaches for use with medium voltage cable is the tan-delta technique. This paper will demonstrate how the mirco-void content approach can be used to validate other techniques such as tan-delta and add additional meaning and value not otherwise available from tan-delta alone.

Effect of grain size on damage and failure in two-phase materials: Homogenized CuPb

It is well known that spall failure strongly depends on the microstructure of a material. There have been numerous studies to study the effect of grain size on the overall spall strength and the total amount of damage in single element metals like copper. However, such systematic studies remain rare in two-phase materials and alloys. In this work, two incipient spall experiments were performed on a Cu–1%Pb alloy to understand the effect of grain size on the damage and failure in a two-phase material. Overall, these results showed that even though the spall strength did not change as a function of grain size, there were significant differences in the total amount of damage as a function of grain size. A clear increase in the total damage present in the material was seen as the grain size was increased from 32 to either 70 or 75 μm in either of the experiments. This difference was attributed to variations in the void growth rate as the grain size was increased.

Phase-field simulation of void evolution in UO&lt;sub&gt;2&lt;/sub&gt; under applied stress

Owing to the migration and aggregation of point defects produced by cascade collision, a large number of cavities form easily during irradiation of the uranium dioxide (UO&lt;sub&gt;2&lt;/sub&gt;) that is an important nuclear fuel. In addition, cavities are also inevitably introduced into the ceramic fuel during sintering. Moreover, the creep strain and thermal strain, caused by the extreme environment of high temperature and strong irradiation, significantly increase the applied stress of nuclear fuel. Therefore, it is crucial to investigate the microstructure evolution of the cavities in UO&lt;sub&gt;2&lt;/sub&gt; fuel under applied stress. In this work, a phase-field model of void evolution in UO&lt;sub&gt;2&lt;/sub&gt; under applied stress is established. Firstly, the elastic equilibrium equation is solved by the perturbation-iterative method, and the stress distribution around a single void under applied stress is calculated. The results show that the stress concentration is observed at the edge of the void, and the simulated stress distribution is consistent with the theoretically analytical results. Then, the evolution processes of a single void under different applied stresses are simulated by the phase-field model. The results show that the growth rate of void increases with the augment of applied stress. Finally, the effect of applied stress on grain growth and void evolution in polycrystalline are also studied. The results show that the applied stress will accelerate the void growth. With the increase of the applied stress, the effect of the applied stress on accelerating the void evolution increases.

Influence of various material parameters on void growth in amorphous glassy polymers

Glassy polymers were developed in the 1950s and in the present era, they are used in various applications. Over the past decades, their usage has increased considerably and thus there is a need to capture their behaviour in various conditions to extend their applications in diverse new domains. However, the implementation of lack of accurate material models in commercial FEA software has limited the research work in this area. Like ductile metals, glassy polymers also follow the three stages of failure: void nucleation, void growth, and coalescence. The current study aims to solve this problem with the help of a user subroutine in ABAQUS. VUMAT subroutine in ABAQUS explicit software is written based on the existing theories, which are validated from experimental data collected from earlier literature. Successful implementation of the subroutine is followed by studying the influence of different material parameters and loading rate on the material response. Furthermore, void growth study is done with cell model geometry, where one percent void is considered. Two materials chosen for the study are polycarbonate (PC) and polymethyl methacrylate (PMMA). Void growth rate is studied with various strain ratios. Softening and hardening parameters in material models are also varied to do a comparative void growth rate study. It has been observed that the void growth is also affected by the loading rate. The initial void is also varied to see the effects of its size over the void growth phenomena.

Effect of shape on void growth: A coupled Extended Finite Element Method (XFEM) and Discrete Dislocation Plasticity (DDP) study

Voids are one of the many material defects present at the microscopic length scale. They are primarily responsible for the formation of cracks and hence contribute to ductile fracture. Circular voids tend to deform into elliptical voids just before their coalescence to form cracks. The principle aim of this study is to investigate the effect of void shape on the micro-mechanism of void growth by using Discrete Dislocation Plasticity simulations. For voided crystals, conventional DDP produces a continuous slip step throughout the material even if a dislocation escapes from a non-convex domain. To overcome this issue, the Extended Finite Element Method (XFEM) is used here to incorporate the displacement discontinuity. Different aspect ratios of elliptical voids are considered under uniaxial and biaxial deformation boundary conditions. The results suggest that voids having the largest surface area tend to have maximum growth rate as compared to void with lower surface area, i.e. “larger is faster”. Under biaxial loading, a higher magnitude of strain hardening, and void growth rate are observed as compared to uniaxial loading. The results also suggest that the orientation of slip planes as well as voids, affect the overall plastic behavior of the voided-ductile material. Furthermore, circular void tends to induce minimum growth rate but have the maximum strain hardening effect as compared to other void shapes under both loading conditions. The results of this study provide a deeper understanding of ductile fracture with applications in manufacturing industry, aerospace industry and in the design of nano/micro-electromechanical devices i.e. NEMS/MEMS.

Role of crystallographic orientation on intragranular void growth in polycrystalline FCC materials

In this work, we study the effect of crystallographic orientation and applied triaxiality on the growth of intragranular voids. Two 3D full-field micromechanics methods are used, the dilatational visco-plastic fast-Fourier transform (DVP-FFT) and the crystal plasticity Finite Elements (CP-FE), both of which incorporate a combination of crystalline plasticity and dilatational plasticity. We demonstrate with several select cases that predictions of void growth from both formulations agree qualitatively. With the more computationally efficient DVP-FFT, additional effects of polycrystalline microstructure and the influence of nearest neighborhood are investigated. Crystals bearing a single intracrystalline void are studied in three types of 3D microstructural environments: isolated single crystals, individual equal-sized grains within a regular polycrystal, and individual variable sized grains within a polycrystal with grains and voids randomly located. We show that loading type plays a significant role. In strain-rate controlled conditions, voids in the hardest [111]-crystals grow the fastest in time, whereas in stress-controlled conditions, voids in the softest [100]-crystal grow the fastest in time. The analysis reveals that on average void growth is slower for the same starting orientation in the polycrystal than in the single crystal. We find that at the highest triaxiality tested that the correlation between crystal orientation and void growth rate in the polycrystal strengthens, drawing closer to that seen in the isolated single crystals. These results and model can help guide the microstructural design of polycrystalline materials with high strength and damage-tolerance in high-rate deformation.

Investigating the formation mechanism of void lattice in tungsten under neutron irradiation: from collision cascades to ordered nanovoids

The “void lattice” formed by periodic arrangements of voids, usually replicating the symmetry and crystallographic orientation of the host matrix, is an interesting phenomenon in materials under neutron irradiation. In this work, taking tungsten (W) as an example, we explore the formation mechanism of the void lattice using an object kinetic Monte Carlo (OKMC) model together with the collision cascades simulated by the molecular dynamics method. The entire formation processes from the chaotic neutron irradiation defects to the observable void lattice are reproduced via OKMC simulation, which could be divided into three stages: nucleation, incubation and growth. It is found that both the one-dimensional (1D) migration of SIAs and the fraction of clustered vacancies in cascades play a critical role in the formation of the void lattice. On the one hand, the 1D migration of SIAs leads to the mutual protection of voids aligned in <111> directions. The self-shielded voids may therefore grow faster than the unaligned ones. On the other hand, a moderate fraction of clustered vacancies in cascades guarantees the stable nucleation and growth rates of voids. Once the density of the <111> aligned voids reaches a critical value, the shrinkage rate of the unaligned voids will overwhelm their growth rate, leading to the formation of void-free channels and thus the void lattice. Our results reveal the synergistic effects of the fraction of clustered vacancies in cascades and 1D migration of SIAs for the formation of the void lattice in W under neutron irradiation, which improves the fundamental understanding of the self-assembled microstructures in irradiated materials.