Void growth and coalescence in single crystals

Void growth and coalescence in single crystals are investigated using crystal plasticity based 3D finite element calculations. A unit cell involving a single spherical void and fully periodic boundary conditions is deformed under constant macroscopic stress triaxiality. Simulations are performed for different values of the stress triaxiality, for different crystal orientations, and for low and high work-hardening capacity. Under low stress triaxiality, the void shape evolution, void growth, and strain at the onset of coalescence are strongly dependent on the crystal orientation, while under high stress triaxiality, only the void growth rate is affected by the crystal orientation. These effects lead to significant variations in the ductility defined as the strain at the onset of coalescence. An attempt is made to predict the onset of coalescence using two different versions of the Thomason void coalescence criterion, initially developed in the framework of isotropic perfect plasticity. The first version is based on a mean effective yield stress of the matrix and involves a fitting parameter to properly take into account material strain hardening. The second version of the Thomason criterion is based on a local value of the effective yield stress in the ligament between the voids, with no fitting parameter. The first version is accurate to within 20% relative error for most cases, and often more accurate. The second version provides the same level of accuracy except for one crystal orientation. Such a predictive coalescence criterion constitutes an important ingredient towards the development of a full constitutive model for porous single crystals.     

Effects of void band orientation and crystallographic anisotropy on void growth and coalescence

The effects of void band orientation and crystallographic anisotropy on void growth and linkage have been investigated. 2D model materials were fabricated by laser drilling a band of holes into the gage section of sheet tensile samples using various orientation angles with respect to the tensile axis normal. Both copper and magnesium sheets have been studied in order to examine the role of crystallographic anisotropy on the void growth and linkage processes. The samples were pulled in uniaxial tension inside the chamber of an SEM, enabling a quantitative assessment of the growth and linkage processes. The void band orientation angle has a significant impact on the growth and linkage of the holes in copper. As the void band orientation angle is increased from 0° to 45°, the processes of coalescence and linkage are delayed to higher strain values. Furthermore, the mechanism of linkage changes from internal necking to one dominated by shear localization. In contrast, the void band orientation does not have a significant impact on the void growth and linkage processes in magnesium. Void growth in these materials occurs non-uniformly due to interactions between the holes and the microstructure. The heterogeneous nature of deformation in magnesium makes it difficult to apply a coalescence criterion based on the void dimensions. Furthermore, the strain at failure does not show a relationship with the void band orientation angle. Failure associated with twin and grain boundaries interrupts the plastic growth of the holes and causes rapid fracture. Therefore, the impact of the local microstructure outweighs the effects of the void band orientation angle in this material.     

Modeling of ductile fracture: Significance of void coalescence

In this paper void coalescence is regarded as the result of localization of plastic flow between enlarged voids. We obtain the failure criterion for a representative material volume (RMV) in terms of the macroscopic equivalent strain (E_c) as a function of the stress triaxiality parameter (T) and the Lode angle (θ) by conducting systematic finite element analyses of the void-containing RMV subjected to different macroscopic stress states. A series of parameter studies are conducted to examine the effects of the initial shape and volume fraction of the primary void and nucleation, growth, and coalescence of secondary voids on the predicted failure surface E_c(T, θ). As an application, a numerical approach is proposed to predict ductile crack growth in thin panels of a 2024-T3 aluminum alloy, where a porous plasticity model is used to describe the void growth process and the expression for E_c is calibrated using experimental data. The calibrated computational model is applied to predict crack extension in fracture specimens having various initial crack configurations and the numerical predictions agree very well with experimental measurements.     

Temperature dependent bulge test for elastomers

The bulge test is a particularly convenient testing method for characterizing elastomers under biaxial loading. In addition, it is convenient to utilize this test for validating material models in simulation due to the heterogeneous strain field induced during inflation. During the bulge test the strain field for elastomers covers uniaxial tension at the border to pure shear and equibiaxial tension at the pole. Elastomeric materials exhibit a hyperelastic material behavior, with a dependency on temperature and loading rate. The temperature effect on the mechanical behavior during biaxial loading is considered in the present study. A bulge test setup combined with a temperature chamber is developed in order to characterize this effect, and an exemplary temperature dependent characterization of a poly(norbornene) elastomer is performed with this setup. The equibiaxial stress–strain curves measured at 60 °C, 20 °C and −20 °C are presented.     

An improved atomistic simulation based mixed-mode cohesive zone law considering non-planar crack growth

A novel and improved atomistic simulation based cohesive zone law characterizing interfacial debonding is developed which explicitly accounts for the non-planarity of the crack propagation. Group of atoms in the simulation constituting cohesive zones which are used to obtain local stress and crack opening displacement data are determined dynamically during the non-planar crack growth as they cannot be determined apriori. The methodology is used to study the debonding of Σ5 (2 1 0)/[0 0 1] symmetric tilt grain boundary interface in a Cu bicrystal under several mixed mode loading conditions. Simulations show that such bicrystalline specimen exhibits three types of energy dissipative mechanisms – shear coupled GB migration (SCM) away from the crack-tips, change in spacial orientation of GB structural units rendering highly disordered grain boundary near the crack tips and brittle intergranular fracture. Which combination of these three deformation mechanism will be active influencing the degree of non-planarity of the crack propagation at various stages of loading depends on the loading mode-mixity. As the ratio of shear component of the loading parallel to the GB plane and normal to the tilt axis with respect to the normal loading increases (thereby increasing the mode-mixity), overall strain-to-failure also increases and SCM tends to become the dominant deformation mechanism. Through this framework, analytical functional forms and parameters describing cohesive laws for both normal and shear traction as a function of the mode-mixity of the loading and crack opening displacement are predicted.     

On the accuracy of self-consistent elasticity formulations for directionally solidified polycrystal aggregates

In this work, the elastic properties of directionally solidified (DS) polycrystal aggregates are investigated through a combination of analytical and numerical approaches. The effects of crystallographic misorientations and grain aspect ratios of aggregates with ellipsoidal shaped grains are first examined following a self-consistent approach. Finite element techniques are then used to examine the effects of grain size on the elastic properties of the aggregate and to assess the accuracy of the self-consistent predictions. To that purpose, a finite element procedure is presented to generate numerically realistic 3D DS microstructures from electron back-scatter diffraction (EBSD) lattice orientation measurements on an arbitrary cross-section of a DS material. The elastic stiffnesses predicted numerically and analytically are then compared with experimental data on a Ni-base DS alloy tested uniaxially along arbitrary orientations. The general trend predicted analytically was found to be consistent with the numerical and experimental results. Furthermore, an increase in the misorientation between the [0 0 1] axis of each DS grain with respect to the grain growth direction was found to decrease the elastic anisotropy of the DS material.     

考虑三轴约束时孔洞的聚合机理及有效能量准则

通过体胞分析方法,对不同状孔洞在从光滑试样到裂纹试样的三轴应力场中的聚合机理进行了较精解的有限元分析,计算结果表明：（１）孔洞的相互靠近和横向扩展是导致相邻孔洞发生内颈缩聚合的两种基本机制,在应力三维度Ｒσ等于１．２５附近,这两种机制发生较明显的变化。（２）单纯以孔洞体积分数ｆＣ概念为基础的材料破坏参数一般敏感于应力三维度,不能很好地预报不同三轴应力场中材料的破坏,在此基础上,提出了描述孔洞聚合的     

Void fraction and pressure drop during external upward two-phase crossflow in tube bundles – part I: Experimental investigation

The present paper is the part I of a broad study concerning void fraction and pressure drop for air-water upward external flow across tube bundles. Experimental results were obtained for liquid and gas superficial velocities ranging from 0.02 to 1.50 m/s and 0.20 to 10.00 m/s, respectively. Void fraction measurements were performed for bubbly flow using a capacitive probe. The test section consisted of a triangular tube bundle counting with 19 mm OD tube and transverse pitch of 24 mm. Initially, the paper describes the test facility and the data regression and experimental procedures. Then, the pressure drop and void fraction measurements are validated based on tests for single-phase flow and quiescent liquid conditions, respectively. Finally, the experimental data are presented and analyzed. In the second part of this study (Part II), a literature review on predictive methods for void fraction and pressure drop is presented. Additionally, these methods are compared with the database presented in Part I and new predictive methods for void fraction and frictional pressure drop are proposed.     

Numerical study on coalescence of two pre-existing coplanar flaws in rock

Crack propagation and coalescence processes are the fundamental mechanisms leading to progressive failure processes in rock masses, in which parallel non-persistent rock joints are commonly involved. The coalescence behavior of the latter, which are represented as pre-existing coplanar flaws (cracks), is numerically investigated in the present study. By using AUTODYN as the numerical tool, the present study systematically simulates the coalescence of two pre-existing coplanar flaws in rock under compression. The cumulative damage failure criterion is adopted in the numerical models to simulate the cumulative damage process in the crack initiation and propagation. The crack types (shear or tensile) are identified by analyzing the mechanics information associated with the crack initiation and propagation processes. The simulation results, which are generally in a good accordance with physical experimental results, indicate that the ligament length and the flaw inclination angle have a great influence on the coalescence pattern. The coalescence pattern is relatively simple for the flaw arrangements with a short ligament length, which becomes more complicated for those with a long ligament length. The coalescence trajectory is composed of shear cracks only when the flaw inclination angle is small (such as β ? 30°). When the pre-existing flaws are steep (such as β ? 75°), the coalescence trajectory is composed of tensile cracks as well as shear cracks. When the inclination angle is close to the failure angle of the corresponding intact rock material, and the ligament length is not long (such as L ? 2a), the direct shear coalescence is the more favorable coalescence pattern. In the special case that the two pre-existing flaws are vertical, the model will have a direct tensile coalescence pattern when the ligament length is short (L ? a), while the coalescence between the two inner flaw tips is not easy to achieve if the ligament length is long (L ? 2a).     

Irreversible thermodynamics of triple junctions during the intergranular void motion under the electromigration forces

A rigorous reformulation of internal entropy production and the rate of entropy flow is developed for multi-component systems consisting of heterophases, interfaces and/or surfaces. The result is a well-posed moving boundary value problem describing the dynamics of curved interfaces and surfaces associated with voids and/or cracks that are intersected by grain boundaries. Extensive computer simulations are performed for void configuration evolution during intergranular motion. In particular we simulate evolution resulting from the action of capillary and electromigration forces in thin film metallic interconnects having a “bamboo” structure, characterized by grain boundaries aligned perpendicular to the free surface of the metallic film interconnects. Analysis of experimental data utilizing previously derived mean time to failure formulas gives consistent values for interface diffusion coefficients and enthalpies of voids. 3.0 × 10⁻⁶ exp(−0.62 eV/kT) m² s⁻¹ is the value obtained for voids that form in the interior of the aluminum interconnects without surface contamination. 6.5 × 10⁻⁶ exp(−0.84 eV/kT) m² s⁻¹ is obtained for those voids that nucleate either at triple junctions or at the grain boundary-technical surface intersections, where the chemical impurities may act as trap centers for hopping vacancies.     

Bubble–wall interaction and two-phase flow parameters on a full-scale boat boundary layer

Full scale bubbly flow experiments were performed on a 6 m flat bottom survey boat, measuring the void fraction, bubble velocity and size distributions as the bubbles naturally entrained at the bow of the boat interact with the boat’s boundary layer. Double-tip sapphire optical probes capable of measuring bubbles down to 50 μm in diameter were specifically designed and built for this experiment. The probes were positioned under the hull at the bow near the bubble entrainment region and at the stern at the exit of the bottom flat plate. Motorized positioners were used to vary the probe distance to the wall from 0 to 50 mm. The experiments were performed in fresh water (Coralville Lake, IA) and salt water (Panama City Beach, FL), at varying velocities with most data analysis performed at 10, 14 and 18 knots. The results indicate that the bubbles interact significantly with the boundary layer. At low velocity in fresh water, bubble accumulation under the hull and coalescence are evident by the presence of large bubbles at the stern. At high speeds bubble breakup dominates and very small bubbles are produced near the wall. It is also observed that salt water inhibits coalescence, even at low boat speeds. The void fraction increases with speed beyond 10 knots and peaks near the wall. Bubble velocities show slip with the wall at all speeds and exhibit large RMS fluctuations, increasing near the wall.     

Developing structure of two-phase flow in a large diameter pipe at low liquid flow rate

In order to develop the interfacial area transport equation for the interfacial transfer terms in the two-fluid model, accurate data sets on axial development of local parameters such as void fraction, interfacial area concentration, interfacial gas velocity and Sauter mean diameter are indispensable to verify the modeled source and sink terms in the interfacial area transport equation. From this point of view, local measurements of both group 1 spherical/distorted bubbles and group 2 cap/slug bubbles in vertical upward air–water two-phase flow in a large diameter pipe with 200 mm in inner diameter and 26 m in height were performed at three axial locations of z/D = 41.5, 82.8 and 113 as well as 11 radial locations from r/R = 0–0.95 by using four-sensor probe method. Here, z, r, D and R are the axial distance from the inlet, radial distance from the pipe center, pipe diameter and pipe radius, respectively. The liquid flow rate and the void fraction ranged from 0.0505 m/s to 0.312 m/s and from 1.98% to 32.6%, respectively in the present experiment. The flow condition covered extensive region of bubbly flow, cap turbulent flow as well as their transition. The extensive analysis on the radial profiles of local flow parameters and their axial developments demonstrate the development of interfacial structures along the flow direction due to the bubble coalescence and breakup and the gas expansion. The significant decrease in void faction and interfacial area concentration and the increase in Sauter mean diameter and interfacial velocity were observed when the gradual flow regime transition occurred. Finally, the net change in the interfacial area concentration due to the bubble coalescence and breakup was quantitatively investigated in the present paper to reflect the true transfer mechanisms in observed two-phase flows.     

Integral-based constitutive equation for rubber at high strain rates

High-speed experiments were conducted to characterize the deformation and failure of Styrene Butadiene Rubber at impact rates. Dynamic tensile stress–strain curves of uniaxial strip specimens and force–extension curves of thin sheets were obtained from a Charpy tensile impact apparatus. Results from the uniaxial tension tests indicated that although the rubber became stiffer with increasing strain rates, the stress–strain curves remained virtually the same above 280 s⁻¹. Above this critical strain rate, strength, fracture strain and toughness decreased with increasing strain rates. When strain rates were below 180 s⁻¹, the initial modulus, tensile strength and breaking extension increased as the strain rate increased. Between strain rates of 180 and 280 s⁻¹, the initial modulus and tensile strength increased with increasing strain rates but the extension at break decreased with increasing strain rates. A hyper-viscoelastic constitutive relation of integral form was used to describe the rate-dependent material behavior of the rubber. Two characteristic relaxation times, 5 ms and 0.25 ms, were needed to fit the proposed constitutive equation to the data. The proposed constitutive equation was implemented in ABAQUS Explicit via a user-defined subroutine and used to predict the dynamic response of the rubber sheets in the experiments. Numerical predictions for the transient deformation and failure of the rubber sheet were within 10% of experimental results.     

Pressure-sensitive ductile layers – I. Modeling the growth of extensive damage

Polymeric adhesives sandwiched between two elastic substrates are commonly found in multi-layers and IC packages. The non-elastic deformation and flow stress of such adhesive joints are highly pressure-sensitive. In this work, we study the effects of pressure-sensitivity, α, and plastic dilatancy, β, on void growth and coalescence ahead of a crack in ductile adhesive joints. To this end, a single layer of discrete voids is placed ahead of the crack in a pressure-sensitive dilatant adhesive sandwiched between two elastic substrates. The adhesive joint is subjected to small-scale yielding conditions. Using an associated flow rule (α = β), we show that pressure-sensitivity not only intensifies damage levels but also increases its spatial extent several fold. The damage level as well as its spatial extent is found to be even greater when a non-associated flow rule (β < α) is deployed. A reduction in the damage process zone’s thickness further increases the voiding activity in the adhesive, thereby resulting in brittle-like failure. This work also examines the fracture toughness trends using a material failure criterion for crack growth.     

Modelling of the plastic flow of trip-aided multiphase steel based on an incremental mean-field approach

An incremental mean-field model is developed for the prediction of transformation induced plasticity (TRIP) in multiphase steel. The partitioning of strain between softer and harder constituents is computed based on an elastic-plastic Mori–Tanaka approach that accounts for the progressive transformation of austenite into martensite. The latter transformation is predicted using an energy-balance criterion that is formulated at the level of individual austenite grains. The model has been tested against experimental data. Macroscopic stress-strain curves and rate of martensite formation have been measured on sheet samples subjected to various loading modes: uniaxial tension, simple shear, and (in-plane) uniaxial compression. These experiments were performed at 20 °C and the uniaxial tensile test was repeated at ?30 °C. The mean-field model produces fair predictions of the macroscopic hardening resulting from TRIP on the condition that a sufficient proportion of the load is carried by the very hard martensite inclusions. Such prediction implies that one accounts for the stress heterogeneity across the ferrite-based matrix. At the same time, the model reproduces the elastic lattice strains and the plastic elongation which are measured within the phases by neutron diffraction and by image correlation in a scanning electron microscope, respectively. The model can be used in finite element simulations of forming processes which is illustrated in a study of necking of a cylindrical bar under uniaxial tension.     

Relating inhomogeneous deformation to local texture in zirconium through grain-scale digital image correlation strain mapping experiments

The effect of local texture on inhomogeneous plastic deformation is studied in zirconium subjected to uniaxial compression. Cross-rolled commercially pure Zr 702 plate that had a strong basal (0 0 0 1) texture through the plate thickness, and a non-basal texture in cross-section, was obtained. At a compressive strain rate of 1 s^?1, samples loaded either in the through-thickness or in-plane directions exhibited significant differences in yield strength, hardening response and failure mechanisms. These macroscopic differences are related to microstructural features by combining information from electron backscattered diffraction with real time in situ imaging and subsequent full-field strain measurements obtained using digital image correlation. Experimental results indicate that the through-thickness loaded zirconium samples, which show a strong basal-texture in the loading direction, do not deform homogeneously – implying the lack of a representative volume element. The detailed surface deformation fields provided by digital image correlation allow for a qualitative and quantitative analysis of the relationship between grain orientation and patterns of deformation bands that form as the precursors to development of an adiabatic shear band in the through-thickness loaded sample. For the in-plane loaded samples, inhomogeneities still exist at the microscale, but the collective behavior of several grains leads to a homogeneous response at the macroscale. It is observed that local texture for hcp polycrystals, which are significantly slip restricted, can directly affect both local and global response, even at low to moderate plastic strains.     

The degenerate scale problem for the Laplace equation and plane elasticity in a multiply connected region with an outer circular boundary

This paper investigates the degenerate scale problem for the Laplace equation and plane elasticity in a multiply connected region with an outer circular boundary. Inside the boundary, there are many voids with arbitrary configurations. The problem is analyzed with a relevant homogenous BIE (boundary integral equation). It is assumed that all the inner void boundary tractions are equal to zero, and tractions on the outer circular boundary are constant. Therefore, all the integrations in BIE are performed on the outer circular boundary only. By using the relation z * conjg(z) = a * a, or conjg(z) = a * a/z on the circular boundary with radius a, all integrals can be reduced to an integral for complex variable and they can be integrated in closed form. The degenerate scale a = 1 is found in the Laplace equation and in plane elasticity regardless of the void configuration.     

Effect of calcining temperature and time on the characteristics of Sb-doped SnO2 nanoparticles synthesized by the sol–gel method

Spherical Sb-doped SnO₂ (ATO) nanoparticles were synthesized by the sol–gel route, employing SnCl₄·5H₂O and SbCl₃ as precursors in an ethanol solution. The influences of the calcining temperature and calcining time on the crystallite size, crystallinity, lattice parameters, lattice distortion ratio and the resistivity of the ATO nanoparticles were synthetically investigated. The results suggested that the ATO nanoparticles were crystallized in a tetragonal cassiterite structure of SnO₂ with a highly (1 1 0)-plane-preferred orientation. The calcining temperature had a dominating effect on the crystallite size, crystallinity, lattice distortion ratios and resistivity of the ATO. As the calcining temperature increased, the average crystallite size increased, the crystallinity was promoted accompanied by a decrease in the lattice distortion ratio and a corresponding decrease in the resistivity of the ATO. X-ray diffraction (XRD) and Fourier transform infrared spectrophotometer (FTIR) analysis revealed that Sb ions could not entirely supplant the Sn ions in the SnO₂ lattice for a calcining time of less than 0.5 h, even at a calcining temperature of 1000 °C. The ATO nanoparticles calcined at 1000 °C for 3.0 h possessed the lowest resistivity of 10.18 Ω cm.     

Effect of elastic anisotropy on surface pattern evolution of epitaxial thin films

Stress-induced surface instability and evolution of epitaxial thin films leads to formation of a variety of self-assembled surface patterns with feature sizes at micro- and nanoscales. The anisotropy in both the surface and bulk properties of the film and substrate has profound effects on the nonlinear dynamics of surface evolution and pattern formation. Experimentally it has been demonstrated that the effect of anisotropy strongly depends on the crystal orientation of the substrate surface on which the film grows epitaxially. In this paper we develop a nonlinear model for surface evolution of epitaxial thin films on generally anisotropic crystal substrates. Specifically, the effect of bulk elastic anisotropy of the substrate on epitaxial surface pattern evolution is investigated for cubic germanium (Ge) and SiGe films on silicon (Si) substrates with four different surface orientations: Si(0 0 1), Si(1 1 1), Si(1 1 0), and Si(1 1 3). Both linear analysis and nonlinear numerical simulations suggest that, with surface anisotropy neglected, ordered surface patterns form under the influence of the elastic anisotropy, and these surface patterns clearly reflect the symmetry of the underlying crystal structures of the substrate. It is concluded that consideration of anisotropic elasticity reveals a much richer dynamics of surface pattern evolution as compared to isotropic models.