Stress Void Research Articles

Hydrogen in Electrolessly Deposited Copper Films -Additive’s Effect on Hydrogen Concentration and Structure of Films-

Electroless deposition of copper is an important process in manufacturing various electrical connections. In this process, hydrogen is generated concurrently with the deposition of copper atoms by the oxidation of reducing agent (formaldehyde). While most of them are released as molecular hydrogen bubbles, some of them are co-deposited in copper films. Hydrogen co-deposition has been recognized as cause of residual stress generation, ductility loss, and void formation in the films. However, the detailed mechanism of such hydrogen-induced phenomenon has not been clarified. We previously reported that the room-temperature grain growth observed in electrodeposited copper films is caused by hydrogen-induced superabundant vacancy-hydrogen clusters [1,2]. In this study, the atomistic state (interstitial, vacancy-hydrogen clusters, and in nanovoids) and amount of hydrogen in the electrolessly deposited copper films have been analyzed by thermal desorption spectroscopy (TDS) [3]. The hydrogen concentration is much higher than the solubility in bulk copper, and decreased by aging at room temperature. Lattice expansion and compressive residual stress of deposited copper films are decreased with hydrogen concentration, which exists at interstitial sites, by the aging. Additives for electroless deposition reduce the hydrogen concentration and the compressive residual stress of deposited Cu films.AcknowledgementsThe present work was partly supported by JSPS KAKENHI (JP22K04763).[1] N. Fukumuro, T. Adachi, S. Yae, H. Matsuda, and Y. Fukai, Trans. Inst. Met. Finish., 89, 198 (2011). DOI: 10.1179/174591911X13082997023873[2] N. Fukumuro, H. Yoshida, T. Yamazaki, Y. Fukai, and S. Yae, J. Japan Inst. Met. Mater., 80, 736 (2016). DOI: 10.2320/jinstmet.JD201608[3] N. Fukumuro, K. Tohda, and S. Yae, J. Electrochem. Soc., 169, 122505 (2022). DOI: 10.1149/1945-7111/acaa63

A Comprehensive Microstructure-Aware Electromigration Modeling Framework; Investigation of the Impact of Trench Dimensions in Damascene Copper Interconnects.

As electronic devices continue to shrink in size and increase in complexity, the current densities in interconnects drastically increase, intensifying the effects of electromigration (EM). This renders the understanding of EM crucial, due to its significant implications for device reliability and longevity. This paper presents a comprehensive simulation framework for the investigation of EM in nano-interconnects, with a primary focus on unravelling the influential role of microstructure, by considering the impact of diffusion heterogeneity through the metal texture and interfaces. As such, the resulting atomic flux and stress distribution within nano-interconnects could be investigated. To this end, a novel approach to generate microstructures of the conductor metal is presented, whereby a predefined statistical distribution of grain sizes obtained from experimental texture analyses can be incorporated into the presented model, making the model predictive under various scales and working conditions with no need for continuous calibration. Additionally, the study advances beyond the state-of-the-art by comprehensively simulating all stages of electromigration including stress evolution, void nucleation, and void dynamics. The model was employed to study the impact of trench dimensions on the dual damascene copper texture and its impact on electromigration aging, where the model findings were corroborated by comparing them to the available experimental findings. A nearly linear increase in normalized time to nucleation was detected as the interconnect became wider with a fixed height for aspect ratios beyond 1. However, a saturation was detected with a further increase in width for lines of aspect ratios below 1, with no effective enhancement in time to nucleation. An aspect ratio of 1 seems to maximize the EM lifetime for a fixed cross-sectional area by fostering a bamboo-like structure, where about a 2-fold of increase was estimated when going from aspect ratio 2 to 1.

Plastic deformation and damage modeling of AA7075 synthetic 3D microstructure created using generative AI

3D microstructures provide valuable insight into material behavior which is essential in elucidating microstructural phenomena, such as particle morphology and void damage, and consequent macroscopic material response. However, creating 3D microstructures is extremely laborious and expensive, requiring complex microstructural characterization and imaging techniques such as Focussed-Ion Beam based Scanning Electron Microscopy (FIB-SEM) or X-ray Computed Tomography (XCT). To this end, synthetic 3D microstructures were rapidly generated from orthogonal 2D images using SliceGAN, which proved a practical and cost-effective method. In this study, multiple synthetic microstructures of AA7075-O, a complex microstructure of various strengthening precipitates within a softer aluminum matrix, were post-processed, meshed, and modeled for different damage behavior in FEA using advanced constitutive material models. Subsequently, the synthetic and real microstructures were qualitatively and quantitatively analyzed for their elastoplastic deformation and ductile void damage responses.This study illustrates the viability of an integrated AI-FE methodology in studying microstructural micromechanics, demonstrating that synthetic microstructures exhibited a very similar stress-strain response, especially when using a free boundary condition, and comparable stress distribution and void damage, albeit with some discrepancies. Also, it emphasizes the influence of particle morphology on strength and damage, where highly irregular particles play a dual role in increasing strain hardening by restricting matrix flow at the cost of increased ductile damage induced by decohered particles. Lastly, the more advanced FE models, with multiple voiding mechanisms, reduced the discrepancy between real and synthetic microstructures compared to simpler models

Deep learning-based analysis of true triaxial DEM simulations: Role of fabric and particle aspect ratio

This study investigates the influence of micro-scale entities such as inherent and induced fabric anisotropy on the stress–strain behaviour of granular assemblies. In tandem with this exploration, our objective is to formulate a novel correlation that quantifies the evolution of fabric tensor across diverse loading paths. This correlation can be introduced to enhance the micro-mechanical insights in conventional constitutive models. Employing the Discrete Element Method (DEM), we simulate the drained and undrained responses of 680 transversely isotropic particulate assemblies with diverse initial fabrics and particle Aspect Ratio (AR) under true triaxial loading conditions. We consider a second-order fabric tensor based on inter-particle contact orientations to trace fabric evolution during loading. To account for fluid–solid interaction under undrained conditions, we adopted the DEM-Coupled Fluid Method (CFM). The simulation results highlight the significant influence of the Lode angle, particle shape and initial fabric on the stress–strain behaviour of granular materials, with fabric evolution primarily affected by the stress state, void ratio and particle AR. Lastly, we propose a new correlation for the quantification of the fabric tensor using a multi-layer feed-forward neural network. The satisfactory performance of the suggested correlation is demonstrated through a comparison between DEM data and predicted fabric tensor values.

Static and dynamic strength properties of highly structured clay-rich diatomite in Shengzhou, China

Diatomite is a widespread geological substance found in coastal and lake areas worldwide. Although diatomites are increasingly used in geotechnical engineering construction, a significant knowledge gap exists regarding their mechanical responses to static and dynamic loadings, particularly in high-speed railway projects. This study investigates the static and dynamic behaviors of diatomites through extensive experiments, considering both natural and remolded states. It focuses on representative diatomite specimens obtained from the Shengzhou region in China, an area intersected by the Hang-Shao-Tai high-speed railway. The results of uniaxial compression and undrained triaxial shear tests show that the structural degradation of diatomite significantly impacts its stress-strain responses, failure patterns, and mechanical parameters under static and dynamic loading conditions. The structural effects of natural diatomite tend to degrade as the confining pressure level increases, resulting in corresponding changes in its mechanical parameters. Undrained triaxial shear tests conducted on natural diatomite reveal that the inclinations of the shear band closely correspond to those predicted by the Mohr-Coulomb solution. Additionally, we present a modified hyperbolic-type model to evaluate the damping ratio of diatomites. Furthermore, a power-law model based on Hardin's formula was used to predict the maximum shear modulus of diatomites under varying stress levels and void ratios. Microstructural and chemical analyses were conducted using scanning electron microscopy and X-ray diffraction to gain insights into the observed structural effects at the microscale.

Discrete element study of the effect of particle shape on critical state characteristics and constitutive parameters of sand

Based on the discrete element particle flow software PFC3D, four irregular particles with the same aspect ratio, convexity, and sphericity as the binary pictures are constructed. To examine the influence mechanisms of particle shape on mechanical properties, microstructure, and critical characteristics, an index F characterizing the regularity of the particle shape is introduced. By examining the link between the regularity index and the constitutive parameter, a statistical model is developed to predict the peak friction angle while considering the impacts of relative density and particle shape. The findings demonstrate that decreasing the value of the regularity index always causes an increase in the stress ratio, shear dilatancy, and void ratio. In p'-q, e-logp’, and MCN-logp’ spaces, the distribution pattern of critical state lines is affected by the particle shape. At the same strain instant, when the particle becomes more irregular, the distribution direction of the maximum contact force of irregular particles tends to be more horizontal to resist the greater external loads. In comparison to spherical particles, the Δq/p' and Δφp of irregular particles increase linearly with the increase of ΔF. The measured and predicted values of the peak friction angle for the statistical model are essentially consistent. The critical parameters ecs、ηcs、φcs、MCNcs decrease linearly with the increase of the regularity index. The above conclusions serve as a guide for revising the empirical formula.

Analytical analysis of spherical cavity expansion considering particle breakage effect of sand and its application for CPT tests

Particle breakage in sand significantly influences the stress and deformation predictions of the spherical cavity expansion theories. However, existing theories often overlook or lack a quantitative description of particle breakage, leading to significant errors in the actual results. Based on the Simple Critical State Sand (SIMSAND) model, considering the particle breakage effect, the traditional Euler’s problem of the drainage spherical cavity expansion is converted into a set of first-order ordinary differential equations described by Lagrangian. Fontainebleau sand is taken as an example, and the influence of the initial stress of the sand on the stress state and void ratio is studied. An analysis of the relative breakage surrounding the cavity reveals that the primary impact zone extends to a distance thrice the expansion radius. An axisymmetric model of the cone penetration test is established, the analytical and numerical solutions of expansion stress and cone tip resistance are analysed and the correctness of the spherical cavity expansion theory is verified. Furthermore, the cone tip resistance results, derived from the theory of spherical cavity expansion, are consistent with the results of numerical calculations. Notably, when the particle breakage effect is not considered, the calculated results of cone tip resistance significantly increase.

Equivalent Compression Curve for Clay–Sand Mixtures Using Equivalent Void-Ratio Concept

Clay–sand mixtures are deposited worldwide, such as naturally sedimentary soils and dredged clays in human activities. These mixtures usually have a fine fraction above the transitional fine content, and the compressibility is dominated by clay matrix and affected by sand fraction. In this paper, an explicit equivalent compression curve is proposed for clay–sand mixtures. This is done by directly incorporating the equivalent void ratio into the intrinsic compression model of the clay matrix. The roles of initial and evolving soil structure developed during the compression process are described by a structural variable, which can be directly computed from the experimental data. Experimental data compiled from various literature are used for evaluating the accuracy of the suggested model. They indicate that the proposed relationship of stress versus equivalent void ratio is simple yet effective to evaluate the compressibility of clay–sand mixtures with a wide spectrum of sand fraction. The model is also practically useful for estimating the compressibility of other gap-graded soils, such as clay–gravel mixtures. The proposed stress versus equivalent void-ratio equation is an elementary function and can be readily implemented into general elastoplastic models.

Assessment of micro-indentation method in the characterization of DP600 behavior using a proposed martensite distribution parameter in FEM simulation approach

This work proposes a simulation approach to characterize the behavior of DP600 steel subjected to micro indentation tests. Axisymmetric and three-dimensional artificial RVEs were generated with statistical data from metallographic images. FEM simulations of the micro indentation tests used the Gurson model and Rodriguez-Gutierrez equation. There is proposed a new parameter called spherical radius to quantify the closeness of the hard particles to the indenter tip. The simulated 3D curves were compared with respect to a reference curve of DP600, obtaining relative errors below 20 percent. Results show that the hard particles near the indenter affect the stress, strain, and micro void fields by changing the indentation response behavior. The spherical radius allows capturing the effect of the hard particles closeness because when its value is low, or the hard particles are closer to the indenter tip, the relative error of the average curve decreases due to the hard particles are constraining the deformation caused by the indenter in the material.

An advanced pore-scale model for simulating water retention characteristics in granular soils

Accurately estimating the soil water characteristics curve (SWCC) has become indispensable in successfully implementing unsaturated soil mechanics into engineering practice. The objective of this paper is to present a pore-scale model for the prediction of the hysteretic SWCC of granular soils using basic properties like grain size distribution (GSD) and porosity. The Discrete Element Method (DEM) is used to model the soil structure by generating a stable assembly of spherical particles to attain a predefined grain size distribution and porosity. A network of pores and throats is extracted from this assembly using a medial axis-based network extraction technique, and adequate pore-scale mechanisms are availed to simulate two-phase flow along drying and wetting paths. The current work is further expanded to model SWCC under different void ratios. In addition, SWCCs exhibit stress-dependent behavior, and a numerical framework for modeling SWCCs under various stress conditions is proposed. The effect of particle shapes used for generating DEM samples on SWCC is also investigated. The modeled SWCC compares well with measured curves obtained for a range of granular soils from the literature. The model effectively estimates the hysteretic drying-wetting response and can capture the variation in SWCC with change in net stress and initial void ratio.

Cumulative deformations and particle breakage in calcareous sand subjected to drained high-cyclic loading: Experimental investigation

Transportation infrastructures filled with calcareous sand are subjected to a high-cyclic loading. Existing studies indicated that a key challenge in predicting the cumulative deformation of calcareous sand was a lack of thorough knowledge about the effects of various important field factors. This paper presents an experimental study on the drained monotonic and high-cyclic triaxial behavior of calcareous sand. The dependencies of cumulative deformation and particle breakage behavior on practical variables, such as mean effective stress (p'), cyclic stress ratio (ζ), void ratio (e), and stress-induced anisotropy, etc., were summarized for calcareous sand. It was found that a larger e and consolidation anisotropy led to a significant increase in the accumulated axial strain, transforming the stable shakedown behavior to transitional or unstable shakedown behaviors. The increase in anisotropy can cause a remarkable increase in dilatancy, as well as a severe failure. A phase transition state line (PTSL) was observed to exist in e-(p/pa)0.7 plane under high-cyclic loading, where the PTSL was found dependent on initial void ratios. As the p', ζ, e, and anisotropy increase, a larger particle breakage index (Br) of calcareous sand was observed, where the total input energy (Win) played a substantial role. The relationship between Br and Win was found to be unique under high-cyclic loading, regardless of the stress conditions and initial void ratios. Moreover, the relationship between Br and Win under high-cyclic loading was discovered to differ dramatically from that under monotonic loading. An appropriate equation was thus developed to describe the correlation between Br and Win for high-cyclic loading.

Effect of process parameters on porosity defects in Al-Si alloy hot rolling

Hot-rolled Al-Si alloy sheets are used to be formed Al alloy welding additions and cladding materials that are widely used in the aviation, automotive, and air conditioning industries. However, if parameters of hot-rolling process are not properly selected, porosity defects during casting will lead to cracks or fractures in Al-Si alloy sheets. This study based on meso-damage theory and established a damage equation describing porosity defects in the hot rolling process. The parameters of GTN (Gurson-Tvergaard-Needleman) void defect evolution, including the volume fraction of voids in different periods, the plastic equivalent variation of void nucleation, the standard variance of equivalent variation of void nucleation, and the correction coefficient of the material, are determined in this study. A finite element model of the porosity defects is developed based on the determined parameters. The finite element method is used to analyze the effect of different process parameters on the porosity defects and cracks in Al-Si alloy sheets during hot rolling, then we performed the hot rolling test to validate the results. The results show that the void stress and diameter are reduced and the quality of porosity defects is improved by decreasing rolling reductions and increasing the temperature. The optimized process parameters were a 20% depression and a temperature of 550°C.

A consistent calibration process for the Matsuoka–Nakai friction angle under direct simple shear conditions for clay hypoplasticity

In geotechnical engineering, direct simple shear (DSS) tests are used to determine strength and stiffness parameters of a material. DSS predictions of a constitutive model influence failure mechanisms in plane-strain, finite-element applications. In this article, we introduce a closed-form solution to determine the Matsuoka–Nakai equivalent critical friction angle from direct simple shear stress and normal stress data. In order to apply it to clay hypoplasticity (Mašín, 2013), we carry out DSS simulations to investigate rotations of principal stresses and the principal stress state at critical state for plane-strain conditions.Finally, we interpret DSS predictions of clay hypoplasticity for different overconsolidation ratios and demonstrate that the location of the CSL in the vertical stress–void ratio plane coincides with the location of the CSL in mean effective stress–void ratio plane.

Coupled effects of stress state and void ratio on thermal conductivity of saturated soils

Soils surrounding energy geostructures (e.g. energy piles and borehole heat exchangers) always have different overburden pressures and void ratios. Their thermal conductivities are important for analysing the performance of energy geostructures such as the heat exchange rate between soils and geostructures. So far, however, the coupled effects of stress state and void ratio on thermal conductivity have not been fully understood. In this study, six series of tests were conducted to measure the thermal conductivity of three different soils at various stress states and void ratios, including reconstituted kaolin clay, compacted silt from Hong Kong and compacted Toyoura sand. The results indicate that when the stress increases from 0 to 1200 kPa, the thermal conductivity increases by 60% for the clay, 25% for the relatively looser silt, 20% for the relatively denser silt, 10% for the relatively looser sand and 7·5% for the relatively denser sand. The increment is attributed to not only a reduction of the void ratio, but also an enhancement in the inter-particle contact. Based on the experimental results, a new equation for thermal conductivity was proposed. It is able to well capture the variation of thermal conductivity with stress and void ratio using a single set of parameters.

A new ductile fracture model for structural metals considering effects of stress state, strain hardening and micro-void shape

In this study, a new micromechanically-motivated phenomenological uncoupled fracture model is proposed for predicting the ductile fracture of structural metals considering the combinative effects of stress state, strain hardening and micro-void shape. The Lee–Mearvoid growth theory, which is used for characterizing the growth of ellipsoidal voids in a hardening matrix, is modified to derive the influence function of stress triaxiality, void shape and strain hardening on material’s ductility. Meanwhile, a J2-J3 based Bai–Wierzbicki’s yield function is used as a failure criterion and incorporated into the power-law hardening function to derive the Lode dependence of fracture strain. Further, a detailed parametric study is performed to demonstrate the influence of the material parameters on the 3D fracture envelope and 2D fracture locus of the new model. It is shown that the unique feature of the new fracture model is that it constructs an explicit and quantitative relationship between the ductility, strength and microstructure of metals. All parameters in the new model have intuitive physical meanings which correspond to specific material property indies controlling the material strength and evolution of micro-void defects. In the meantime, these model parameters can be easily calibrated via conventional tests, which greatly reduces the difficulty of the calibration work and makes the model being more friendly for engineering applications. Finally, test data of three structural metals, i.e. 2024-T351, 6061-T6 aluminum alloys and Q460 high strength structural steel, obtained under two loading conditions, i.e. proportional and non-proportional loading conditions, are used to calibrate and validate the proposed model. The accuracy of the new fracture model is then investigated by comparing its prediction result with that obtained from the modified Mohr–Coulomb and Hosford–Coulomb model. It is shown that the proposed new fracture model can predict the ductility limit of the investigated structural metals accurately under a wide range of stress states and provides the same good prediction accuracy as the other two typical models, which confirms the promising prospect of the new model in practical applications.

Use of CFD-DEM to evaluate the effect of intermediate stress ratio on the undrained behaviour of granular materials

This study investigates the effect of intermediate stress ratio (b) on the mechanical behaviour of granular soil in true triaxial tests. A CFD-DEM solver with the ability to model compressible fluid and moving mesh has been developed and calibrated based on existing experimental test results on Nevada sand. The effect of b on the undrained true triaxial test, which has been neglected in the literature, was investigated using a reasonable number of models. The effects of the initial confining stress and initial void ratio also have been studied. The developed model was used to calculate the hydrodynamic forces on the particles and evaluate the ratio of the particle–fluid interaction force to the resultant force on the particles. It has been demonstrated that, in numerical studies, the effect of these forces cannot be neglected.

A state-dependent subloading constitutive model with unified hardening function for gas hydrate-bearing sediments

Natural gas hydrate in ocean sediments and permafrost areas may become a significant potential energy resource. Since the hydrate dissociation may affect the stability of production wells and even lead to geological hazards, it is essential to study the mechanical properties of gas hydrate-bearing sediments (GHBS) for the efficient and safe extraction of gas hydrate. This study presents an extended subloading Modified Cam-Clay model for clayey-silty and sandy GHBS. The state-dependent unified hardening function and equivalent skeleton void ratio are newly introduced to consider the coupled effects of stress level, void ratio, dilatancy, and hydrate saturation on the mechanical behavior of GHBS. Based on the theory of hyperelasticity, an elastic constitutive relation accounting for the influences of cementation caused by hydrate formation and sediment structure change caused by hydrate dissociation is established by using a stiffness evolution function related to hydrate saturation. The bonding and debonding law of strength reflecting the hydrate cementation and its degradation are used. The model is applied to simulate different triaxial tests of GHBS, and its performance in predicting the isotropic compression, strain hardening and softening, shear contraction and dilation, and collapse induced by hydrate dissociation is investigated.

AXIAL COMPONENT OF PLASTIC MODULUS OF SAND UNDER SLOW PERIODIC LOAD

Soil deformation under slow periodic load is of particular importance to predict the functionality of soil structure in a variety of engineering practices. However, lacking quantification on plastic deformation hinders the deformation analysis of soil subjected to slow periodic load. The axial component of plastic modulus is pivotal and is determined to calculate axial plastic strain and overall soil deformation. In this study, a static triaxial compression system was designed to perform a series of static cyclic triaxial tests and cyclic confined compression tests on a silica sand. The results showed that plastic strain increments decreased gradually, as the loading cycle proceeded, and finally only the elastic strain increment existed. The empirical equation of axial component of plastic modulus was formulated considering the effects of stress states, void ratios, and loading cycles. In combination with the description of elastic modulus, the axial strain increment in each loading cycle was formulated. In addition, in the unloading process of loading cycle, the volume contraction phenomenon was explained by unloading plastic strain and microscopic particle motion based on the present results.

A Novel Approach to the Analysis of the Soil Consolidation Problem by Using Non-Classical Rheological Schemes

The paper presents classical and non-classical rheological schemes used to formulate constitutive models of the one-dimensional consolidation problem. The authors paid special attention to the secondary consolidation effects in organic soils as well as the soil over-consolidation phenomenon. The systems of partial differential equations were formulated for every model and solved numerically to obtain settlement curves. Selected numerical results were compared with standard oedometer laboratory test data carried out by the authors on organic soil samples. Additionally, plasticity phenomenon and non-classical rheological elements were included in order to take into account soil over-consolidation behaviour in the one-dimensional settlement model. A new way of formulating constitutive equations for the soil skeleton and predicting the relationship between the effective stress and strain or void ratio was presented. Rheological structures provide a flexible tool for creating complex constitutive relationships of soil.

Casting voids in nickel superalloy and the mechanical behaviour under room temperature tensile deformation

Abstract The microstructure of a second-generation nickel base superalloy is studied using X-ray computed tomography (XCT) and scanning electron microscopy (SEM). The as-cast material contains 0.15 (±0.001) vol% voids and these are distributed in the inter-dendritic region. The volume fraction of the voids increases to 0.21 (±0.001) vol% after tensile deformation. Surface observations show evidence of dislocation emissions from the void surface, a mechanism possibly facilitates the expansion of the voids and contributes to the increased void volume fraction. Phenomenological parameters such as stress triaxiality, often believed to control void growth, are investigated through crystal plasticity simulation and compared with literature reported data. The results indicate weak correlation between stress triaxiality and void growth, but this may be possibly due to the lack of data at higher level of plastic deformation, which is limited by the ductility of the material. The distribution of the stress triaxiality field within the sample is heterogeneous and the peak of the triaxiality field is a function of the ratio between notch diameter and sample width. A smaller notch diameter to sample width ratio tend to distribute the triaxiality peaks towards the centre of the sample but also lead to higher strain localisation, an effect that results in early sample failure.