Strain Energy Concentration Research Articles

Combining experimental and modelling approaches to understand the expansion of lattice parameter of epitaxial SrTi1-xTaxO3 (x = 0–0.1) films

Cubic SrTiO3 is widely used as a model perovskite oxide due to its exotic optical, electronic, magnetic, superconducting, and ferroelectric properties, which are dominantly influenced by its lattice parameter. This paper reports on the experimental observation of Ta5+ dopant-induced expansion of the perpendicular lattice parameter (a⊥) of (001) heteroepitaxial SrTixTa1-xO3 (x = 0–0.1) films grown at substrate temperature (Tg) of 750 °C under 5 × 10-5 mbar oxygen partial pressure (pO2). The experimental results demonstrate the synthesis of very high-quality epitaxial thin films in which almost all the Ta5+ dopants are substituted. However, not all the carriers are activated due to compensating defects. All these films have smooth surfaces and are stoichiometric in compositions. The value of a⊥ increases linearly with increasing Ta or free carrier concentrations. This expansion is modelled via density functional theory (DFT). The influences of three effects are considered in the present modelling: (i) ionic radius of Ta dopant (ii) deformation potential due to hydrostatic strain energy (iii) concentration of free carrier. Further calculations of a⊥ are also performed by considering the influence of substrate and the results overestimate the experimental values by ∼ 1 %. Further experiments on other perovskite systems are required to establish the efficacy of these modelling approaches.

Influence of water on mechanical behavior of surrounding rock in hard-rock tunnels: An experimental simulation

To investigate the influence of water on the mechanical behavior of rock surrounding hard-rock tunnels, a series of uniaxial and true-triaxial compression tests were performed on red sandstone samples with two different water contents (natural water content (NWC) and saturated water content (SWC)). The samples taken were cubic samples containing a circular hole and cylindrical samples. During the true-triaxial tests, hole failure was monitored and recorded in real-time with in-house developed monitoring equipment. The effects of water on the stress, energy, and fracture characteristics of rock failure in hard-rock tunnel were determined. The results indicate that, after reaching the SWC state, the strength and elastic modulus of red sandstone were reduced, and the shear characteristics became more obvious. The failure mode of the NWC holes was primarily slab ejection, while the failure mode of the SWC holes was primarily slab flaking. Water changes the mesoscopic mechanism of spalling and exhibits a double effect on hard-rock tunnels. The mechanisms of water on rockburst prevention are to reduce residual elastic strain energy, avoid excessive concentration of strain energy, and increase rockburst resistance. The ratio of the far-field maximum principal stress to the uniaxial compressive strength can be used as an index to evaluate the stability of hard-rock tunnels. The results help to rethink the influence of water on underground hard rock engineering, such as the failure mechanism of surrounding rock and the analysis of tunnel (or caverns) stability in water-rich stratum, and the mechanism of water on the rockburst prevention.

Monitoring and modelling stress state near major geological structures in an underground coal mine for coal burst assessment

Stress states and geotechnical conditions often change significantly near major geological structures (e.g. faults, shear zones, dykes) which is the cause of most major mine instabilities and/or safety hazards including coal burst, roof falls, water inrush, gas outburst, etc. In order to understand and quantify the stress state near major geological structures, an integrated study had been conducted, which included a comprehensive field monitoring program in the vicinity of a dyke in an Australian underground coal mine; detailed analysis of monitoring data to identify the stress anomalies near the dyke; and three-dimensional numerical modelling to investigate the stress distribution and the strain energy concentrations for the purpose of coal burst risk assessment.The field monitoring program included installing microseismic geophones, stressmeters and extensometers in the roadway roofs and coal pillars, aiming to obtain seismic and stress change data during longwall mining. The monitoring results indicate that the stress state was clearly different on the inbye and outbye sides of the dyke. The inbye side had a much higher stress increase than the outbye side during longwall mining. It is apparent that the dyke had caused a stress redistribution in its vicinity, which led to a stress concentration in the “hanging wall” and stress release in the “footwall”.Three-dimensional numerical modelling was conducted with both a panel-scale large model and a roadway-scale small model to investigate the stress distribution and strain energy concentration in the vicinity of the dyke during longwall mining. The modelling results agreed well with the monitoring data, further clarifying the mechanisms of the dyke influenced stress distribution.

Analysis and structural optimisation of bed structure for CNC lathe based on modal strain energy sensitivity

Based on the modal analysis of the finite element model of the bed, the position of the modal strain energy (MSE) concentration in the first mode is selected, and the sensitivity analysis method is used to optimise the design size of the bed. The optimisation results show that the first-order MSE can be distributed evenly after optimisation, and the quality of the bed can be reduced. At the same time, the blindness of structural optimisation can be reduced and the optimisation efficiency can be improved. This optimisation design process and method can be extended to other machine tool parts, and provide the important approach and theory reference for machine tool dynamic characteristics improvement and structure optimisation.

A new spallation mechanism of thermal barrier coatings and a generalized mechanical model

Multilayer thermal barrier coating (TBC) systems typically consist of three layers of materials: A thermal barrier top coat (TC), a thermally-grown oxide (TGO), and a bond coat (BC) in addition to the substrate. Local strain energy concentrations, called ‘pockets of energy concentration (PECs)’ in this work, often occur around the interface between the TGO and the BC. They have various causes, including local phase changes, and non-uniform creep and plastic relaxation. It is discovered that both PECs and buckling drive the spallation of a TBC in a new spallation mechanism. A PEC-based mechanical model is developed that describes, explains and predicts how blisters nucleate in a TBC under constant biaxial compressive residual stress, steadily and then unsteadily grow, and finally spall off. Two conditions are established for the occurrence of TBC spallation, which depend on the compressive residual strain energy density in the TC and the TGO, and the interface fracture toughness. Experimental validation of the model was performed using aircraft jet engine turbine blades with electron beam physical vapor deposition (EBPVD) TBCs. The predictions from the developed PEC-based mechanical model for the radii of spallation in the TBC are in a good agreement with experiment results.

Optimization of dynamics of non-continuous rotor based on model of rotor stiffness

Abstract Applying a heavy load to the rotor system of aero-engine can drastically reduce the stiffness in rotor joints, which is called “non-continuity” of the rotor and can affect the joint dynamics. Although the dynamics of a rotor system with multiple joints may be negligibly perturbed by a decrease in joint stiffness, this phenomenon is much more striking in advanced, next-generation aero-engines because their rotor structure is lighter and their loads are heavier. We thus investigate the mechanism leading to non-continuity in rotor joints to clarify how joint stiffness deteriorates upon applying a heavy load to rotor joints. This investigation combines a numerical model of joint stiffness with a model of rotor stiffness, which we use to study the dynamic characteristics of a rotor system. The results of the numerical simulation of rotor dynamics reveal that a decrease in joint stiffness leads to a significant concentration of strain energy around the joints when the rotor undergoes bending-mode vibrations, which decreases the critical speed. To avoid loss of joint stiffness and suppress rotor vibration, we develop a method to optimize the support stiffness of the rotor system and demonstrate its effectiveness experimentally.

Slip-band distributions and microstructural fading memory beneath the firn–ice transition of polar ice sheets

The Antarctic Ice Sheet is a continental ice mass with circa 23 million gigatons of ice, which represent roughly 67 % of world’s freshwater supply. This colossal mass of ice is by no means static, as the old ice slowly creeps under its own weight towards the ocean, while new ice is continually formed through the sintering of snow deposited on the ice sheet surface. A crucial role in this metamorphism is played by firn, which is the porous material in an intermediate state between the granular snow and the solid polycrystalline ice. Understanding the snow–firn–ice metamorphism is essential not only for a precise determination of the mechanical (creep) properties of polar ice, but also for comprehending the formation and decay of climate proxies widely used in ice-core studies. This work investigates the transition from firn to ice through the spatial and directional distributions of slip bands in bubbly ice. The analysis of high-resolution micrographs of ice sections extracted from the EPICA-DML Deep Ice Core allows us to identify a clear influence of strain-induced anisotropy (viz. c-axis preferred orientations) on the evolution of slip-band inclinations in deep bubbly ice. In contrast, we discover an unanticipated behaviour of slip bands in shallow bubbly ice, which prompts the introduction of the hypothesis of microstructural fading memory and the definition of a stabilization zone that may penetrate hundreds of metres into the bubbly ice. Within this stabilization zone, highly localized concentrations of strain energy and internal stresses once generated by force chains in the ancient firn are gradually redistributed by the newly formed bubbly-ice microstructure. We show that this hypothesis is compatible with the localized dynamic recrystallization episodes observed in polar firn (even at temperatures close to −45∘C), and it may also explain the sluggish rotation of c-axes observed in the upper hundreds of metres of polar ice sheets.

Fracture and healing of elastomers: A phase-transition theory and numerical implementation

A macroscopic theory is proposed to describe, explain, and predict the nucleation and propagation of fracture and healing in elastomers undergoing arbitrarily large quasistatic deformations. The theory, which can be viewed as a natural generalization of the phase-field approximation of the variational theory of brittle fracture of Francfort and Marigo (1998) to account for physical attributes innate to elastomers that have been recently unveiled by experiments at high spatio-temporal resolution, rests on two central ideas. The first one is to view elastomers as solids capable to undergo finite elastic deformations and capable also to phase transition to another solid of vanishingly small stiffness: the forward phase transition serves to model the nucleation and propagation of fracture while the reverse phase transition models the possible healing. The second central idea is to take the phase transition to be driven by the competition between a combination of strain energy and hydrostatic stress concentration in the bulk and surface energy on the created/healed new surfaces in the elastomer.From an applications point of view, the proposed theory amounts to solving a system of two coupled and nonlinear PDEs for the deformation field and an order parameter, or phase field. A numerical scheme is presented to generate solutions for these PDEs in N=2 and 3 space dimensions. This is based on an efficient non-conforming finite-element discretization, which remains stable for large deformations and elastomers of any compressibility, together with an implicit gradient flow solver, which is able to deal with the large changes in the deformation field that can ensue locally in space and time from the nucleation of fracture. The last part of this paper is devoted to presenting sample simulations of the so-called Gent–Park experiment. Those are confronted with recent experimental results for various types of silicone elastomers.

Geometry effect on mechanical performance and fracture behavior of micro-scale ball grid array structure Cu/Sn–3.0Ag–0.5Cu/Cu solder joints

Solder joint integrity has long been recognized as a key issue affecting the reliability of integrated circuit packages. In this study, both experimental and finite element simulation methods were used to characterize the mechanical performance and fracture behavior of micro-scale ball grid array (BGA) structure Cu/Sn–3.0Ag–0.5Cu/Cu solder joints with different standoff heights (h, varying from 500 to 100μm) and constant pad diameter (d, d=480μm) and contact angle under shear loading. With decreasing h (or the ratio of h/d), results show that the stiffness of BGA solder joints clearly increases with decreasing coefficient of stress state and torque. The stress triaxiality reflects the mechanical constraint effect on the mechanical strength of the solder joints and it is dependent on the loading mode and increases dramatically with decreasing h under tensile loading, while the change of h has very limited influence on the stress triaxiality under shear loading. Moreover, when h is decreased, the concentration of stress and plastic strain energy along the interface of solder and pad decreases, and the fracture location of BGA solder joints changes from near the interface to the middle of the solder. Both geometry and microstructure greatly affect the shear behavior of joints, the average shear strength shows a parabolic trend with decreasing standoff height. Furthermore, the brittle fracture of BGA solder joints after long-time isothermal aging was investigated. Results obtained show that, under the same shear force, the stress intensity factors, KI and KII, and the strain energy release rate, GI, at the Sn–3.0Ag–0.5Cu/Cu6Sn5 interface and in the Cu6Sn5 layer obviously decrease with decreasing h, hence brittle fracture is more prone to occur in the joint with a large standoff height.

Influence of Geostress Orientation on Fracture Response of Deep Underground Cavity Subjected to Dynamic Loading

Deep underground cavity is often damaged under the combined actions of high excavating-induced local stresses and dynamic loading. The fracturing zone and failure type are much related to the initial geostress state. To investigate the influence of geostress orientation on fracture behaviours of underground cavity due to dynamic loading, implicit to explicit sequential solution method was performed in the numerical code to realize the calculation of geostress initialization and dynamic loading on deep underground cavity. The results indicate that when the geostress orientation is heterotropic to the roadway’s floor face (e.g., 30° or 60°), high stress and strain energy concentration are presented in the corner and the spandrel of the roadway, where V-shaped rock failure occurs with the release of massive energy in a very short time. When the geostress orientation is orthogonal to the roadway (e.g., 0° or 90°), the tangential stress and strain energy distribute symmetrically around the cavity. In this regard, the stored strain energy is released slowly under the dynamic loading, resulting in mainly parallel fracture along the roadway’s profile. Therefore, to minimize the damage extent of the surrounding rock, it is of great concern to design the best excavation location and direction of new-opened roadway based on the measuring data ofin situgeostresses.

Research on Low Cycle Fatigue Behavior of BGA Structure Lead-free Solder Joints

A plastic strain energy density methodology is proposed to evaluate the initiation and propagation of fatigue crack in lead-free solder joints. The relationships among the plastic strain, plastic strain energy, continuum damage mechanics(CDM) and fatigue life are clarified. Crack growth correlation constants for micro-scale ball grid array(BGA) structure solder joints(with standoff height h in the range 100 to 500 μm and a pad diameter 480 μm) are determined by a combination of experimental estimation and numerical calculation. The results show that the cycle numbers of crack initiation and propagation have power function relationship with the plastic strain energy density generated in each fatigue cycle. Crack propagation rate is affected by stress triaxiality, which is dependent on loading modes, i.e., stress triaxiality increases dramatically with decreasing h under tensile load because of the mechanical constraint effect arising from interfaces and package structure, while under shear load the standoff height has very limited effect on stress triaxiality. Furthermore, crack growth correlation constants identified in solder joints with h=100 μm can be well used to predict fatigue life of solder joints with different geometries, indicating that the fatigue life prediction method proposed in this study can effectively prevent the influence of plastic strain energy concentration caused by structural and volume factors on the prediction of fatigue life of BGA structure solder joints.

Problèmes capacitaires en viscoplasticité avec effets de torsion

We study the homogenization of elasticity problems like (1) when f, g are strictly convex functions satisfying a growth condition of order p∈(1,+∞), g is positively homogeneous of degree p, kε→+∞, and Trε consists of an ε-periodic distribution of parallel fibers of cross sections of size rε≪ε. The problem (1) corresponds to a simplified model of small deformation nonlinear elasticity describing, for instance, the small deformations of an Ogdenʼs material (Ogden, 1972 [8]). When 1<p<2, it may also characterize the viscoplastic creep experienced, at high temperatures, by a metallic composite governed by the Norton–Hoff model (Friaâ, 1979 [7]). In this case, uε represents the velocity vector field. We show that if p⩽2, a concentration of strain energy appears in a small region of space surrounding the fibers. This extra contribution is characterized by a local density of the sections of the fibers with respect to some appropriate capacity depending, if p<2, on the angles of rotation of the fibers with respect to their principal axis. This rotating behavior generates, in parallel, the emergence of torsional strain energy within the fibers.

A Notion of Capacity Related to Elasticity: Applications to Homogenization

We study a notion of capacity related to elasticity which proves convenient for analyzing the concentrations of strain energy caused by rigid displacements of some infinitesimal parts of an elastic body in two or three dimensions. By way of application, we investigate the behavior of solutions to initial boundary value problems describing vibrations of periodic elastic composites with rapidly varying elastic properties. More specifically, we analyze a two-phase medium whereby a set of heavy stiff tiny particles is embedded in a softer matrix. This task is set in the context of linearized elasticity.

The Effects of Residual Stress Distributions on Indentation‐induced Microcracking in B4C–TiB2 Eutectic Ceramic Composites

The boron carbide (B4C) titanium diboride (TiB2) ceramic eutectic is being investigated for armor and tribological applications. Electron diffraction shows [110] TiB2//[211] B4C//growth direction, (0001) TiB2//(20) B4C is parallel to the interface plane, and transmission electron microscopy (TEM) imaging reveals no interface phase. Thermal residual stress distributions are calculated via finite element modeling of an experimental eutectic microstructure. The B4C matrix is found to be about 400 MPa in compression, and the TiB2 lamellae approximately 1.3 GPa in tension. Stress and strain energy concentrations are found at the tips of TiB2 lamellae. TEM of deformed materials correlates well with the finite element calculations, showing preferential fracture in areas of stress concentration. Interfacial delamination and crack deflection are also observed in deformed materials.

ADAPTIVE REFINEMENT AND MULTISCALE MODELING IN 2D PERIDYNAMICS

The original peridynamics formulation uses a constant nonlocal region, the horizon, over the entire domain. We propose here adaptive refinement algorithms for the bond-based peridynamic model for solving statics problems in two dimensions that involve a variable horizon size. Adaptive refinement is an essential ingredient in concurrent multiscale modeling, and in peridynamics changing the horizon is directly related to multiscale modeling. We do not use any special conditions for the "coupling" of the large and small horizon regions, in contrast with other multiscale coupling methods like atomistic-to-continuum coupling, which require special conditions at the interface to eliminate ghost forces in equilibrium problems. We formulate, and implement in two dimensions, the peridynamic theory with a variable horizon size and we show convergence results (to the solutions of problems solved via the classical partial differential equations theories of solid mechanics in the limit of the horizon going to zero) for a number of test cases. Our refinement is triggered by the value of the nonlocal strain energy density. We apply the boundary conditions in a manner similar to the way these conditions are enforced in, for example, the finite-element method, only on the nodes on the boundary. This, in addition to the peridynamic material being effectively "softer" near the boundary (the so-called "skin effect") leads to strain energy concentration zones on the loaded boundaries. Because of this, refinement is also triggered around the loaded boundaries, in contrast to what happens in, for example, adaptive finite-element methods.

Modello geodinamico dell'Area Umbro-Marchigiana e suo significato sismogenetico

Starting from the analisys of the geodynamic evolution of the Umbrian- Marchean fold belt, we attempt to outline a seismogenetic model of the area. This model rests 1) on the assumption that every deformation tends to reduce the stress by which it has been originated so that the magnitude of the vectors changes and the stress field results reoriented (Price 1959, Ramsay 1967) and 2) on the observation that Umbrian-Marchean deformations of both sedimentary mesozoic cover and crystalline ercinic basement, in spite of their disharmonious behaviour due to the inter- position ot the incompetent level of the triassic evaporites, show a common displacement field (Lavecchia & Pialli 1981 a). At present, as far as the Umbrian-Marchean geodynamic evolution is concerned, all the available geological and geophysical data are in agreement with the existence of a displacement field (two axis in a horizontal plane of apenninic and counterapenninic directions, one axis vertical) whose orientation has been kep trought space and time, while the magnitude of its vectors underwent considerable variations. In a given area, the sequence of events trough the time should be the following. At time T, a horizontal counterapenninic direction of maximum compression a, shortens the crystalline basement and causes a crustal thickening. In response to the consequent increase of the lithostatic load the crust subsides isostatically. Because the elastic warping of the lithosphere beneath the load extends beyond the actual limits of the load, a peripheral depression is created in which the foredeep trough, that migrates northeastward in advance of the deformation, grows (Price 1973). The continuous increase of lithostatic load produces, at time T2, a reorientation of the strain ellipsoid with a vertical intermediate principal direction. Transcurrent shear zones (left-lateral about E-W and right-lateral N-S) and anticlockwise rotation are generated in the crust, while an arcuate fold belt with convexity towards East, formed by periclinal interdigitate folds, develops in the cover. Such deformation reduces the magnitude of the maximum counterapenninic complessive vector (a), so that the strain ellipsoid undergoes a new reorientation (time 3) with a vertical minimum principal strain direction. Crustal thinning and apenninic horst and graben structures are so generated. Obviously all these deformational phases (tensional, transcurrent and compressional) and the associated stress regimes, should be present at a given time, throughout the space from SW to NE. Following Adriatic this scheme, the present seismicity of the Umbrian- Marchean area should be a consequence of its Adriatic geodynamic distinction in apenninic fold belt, deformed Pliocene foretrough (B) compressional zone (C) and should reflect differents stress regimes, with normal faults to the West (A), transcurrent shear zones in the middle (B) and reversed in the East (C). Such seismic zoning is in a good agreement with the distribution ot the focal mechanisms of this area (Cagnetti et Al. 1978, Lavecchia e Pialli 1981 a) which confirms the existence of a tensional stress regime with apenninic and counterapenninic direction in the Apennines and a transcurrent regime in the periadriatic area. The much less clear evidence of a compressional regime in the Adriatic aseismic zone (Gasparini, Praturlon 1981) gives a further confirmation of the proposed model if interpreted as due not to a real absence of a compressional stress, but due to the existence of aseismic creep on reversed apenninic shear zones. A much greater concentration of strain energy is infact required to begin frictional sliding on thrust faults than on normal or transcurrent faults pore fluid pressure, A, continuous ductile shearing can be stable on reversed faults, while sudden release of shear strain energy consequently seismic failure, can realize on normal and transcurrent faults.

Time-dependent fracture behavior of single-walled carbon nanotubes with and without Stone-Wales defects

Based on a concept of molecular-level strain energy concentration and a kinetic fracture model, the time-dependent fracture behavior of three different types of single-walled carbon nanotubes (zigzag, armchair, chiral) with or without Stone-Wales (SW) defects were studied. It is found that the SW defects have influences on the site of crack initiation but the path of crack propagation is closely related to the chirality of the nanotube. Defect-free armchair and chiral tubes exhibit higher static fatigue strength than the zigzag tube of similar size. All of the single-walled carbon nanotubes with SW defect are predicted to have a shorter static fatigue life compared with the defect-free tubes. Moreover, geometric form of the defects is found to influence the time-dependent behavior. Predictions with the present model agree reasonably well with experimental stress-life data of single-walled carbon nanotube rope in epoxy.

A Kinetic Model for Time-Dependent Fracture of Carbon Nanotubes

Based on the classical kinetic concept of solid fracture and a strain concentration concept, a model is proposed for predicting time-dependent fracture of carbon nanotubes. The time-dependent fracture behavior of a zigzag type single-walled carbon nanotube with a range of preexisting cracks under tension is studied by molecular mechanics simulations and a numerical scheme using crack front strain energy concentration. Results of the study quantitatively agree with a recent study on fatigue of aligned single-walled carbon nanotube bundles. It is found that the coefficient of strain energy concentration increases as a crack grows and the time-to-failure of the carbon nanotube is dominated by the lifetimes of a few bonds after initial bond dissociation when load is large while more bonds contribute to the overall lifetime when applied load is small, resulting in the logarithm of time-to-failure of carbon nanotubes being approximately linearly related to applied stress.

Multiscale fracturing as a key to forecasting volcanic eruptions

Among volcanoes reawakening after long repose intervals, the final approach to eruption (∼1–10 days) is usually characterised by accelerating rates of seismicity. The observed patterns are consistent with the slow extension of faults, which continue to grow until they connect a pre-existing array of subvertical fractures and so open a new pathway for magma to reach the surface. Rates of slow extension are here investigated assuming that gravitational loading and magma overpressure create a fluctuating stress field in the country rock. Fluctuations are due to the intermittent growth of small cracks that cannot be detected by monitoring instruments. The rate of detected events is then determined by the frequency with which the concentration of strain energy around the tips of macroscopic faults becomes large enough to permit fault extension. The model anticipates oscillations in seismic event rate about a mean trend that accelerates with time, and identifies the rate of increase in peak event rate as a key indicator of the approach to eruption. The results are consistent with earlier empirical analyses of seismic precursors and, applied to data before the andesitic eruptions of Mt Pinatubo, Philippines, in 1991, and of Soufriere Hills, Montserrat, in 1995, they suggest that, for such volcanoes, reliable forecasts of magmatic activity might eventually be feasible on the order of days before eruption.

Evaluation and application of controlling parameters for seismic events in hard-rock mines

A method is proposed for quantitative assessment and interpretation of the seismic hazard of planned excavation in a hard-rock mine. The quantitative assessment is derived from three-dimensional, elastic, boundary element modelling of the host rock mass. Parameters obtained from the modelling are associated with historical observations of seismicity in the mine to generate probabilistic relations between seismic event occurrence and event strength. The parameters used in the event spatial occurrence relations and the event strength estimates are the Factor of Safety against seismic failure for different types of seismic events, which are inferred from back analysis of field observations, and the Modelled Ground Work. The estimates of these parameters are related to the controlling quantities for a seismic event, which are a local state of stress sufficient to cause rock mass failure, and an unstable local concentration of strain energy, i.e. energy in excess of that which can be dissipated non-violently in rock mass failure. Application of the method is illustrated by reference to several case studies of mine seismicity.