Far-field Shear Stress Research Articles

Effective elastic properties of 3D lattice materials with intrinsic stresses: Bottom-up spectral characterization and constitutive programming

Analytical investigations to characterize the effective mechanical properties of lattice materials allow an in-depth exploration of the parameter space efficiently following an insightful, yet elegant framework. 2D lattice materials, which have been extensively dealt with in the literature following analytical as well as numerical and experimental approaches, have limitations concerning multi-directional stiffness and Poisson's ratio tunability. The primary objective of this paper is to develop mechanics-based formulations for a more complex analysis of 3D lattices, leading to a physically insightful analytical approach capable of accounting the beam-level mechanics of pre-existing intrinsic stresses along with their interaction with 3D unit cell architecture. We have investigated the in-plane and out-of-plane effective elastic properties to portray the physics behind the deformation of 3D lattices under externally applied far-field normal and shear stresses. The considered effect of beam-level intrinsic stresses therein can be regarded as a consequence of inevitable temperature variation, pre-stress during fabrication, inelastic and non-uniform deformation, manufacturing irregularities etc. Such effects can notably impact the effective elastic properties of lattice materials, quantifying which for 3D honeycombs is the central focus of this work. Further, from the material innovation viewpoint, the intrinsic stresses can be deliberately introduced to expand the microstructural design space for effective elastic property modulation of 3D lattices. This will lead to programming of effective properties as a function of intrinsic stresses without altering the microstructural geometry and lattice density. We have proposed a generic spectral framework of analyzing 3D lattices analytically, wherein the beam-level stiffness matrix including the effect of bending, axial, shear and twisting deformations along with intrinsic stresses can be coupled with the unit cell mechanics for obtaining the effective elastic properties.

Time development of live-bed scour around an offshore-wind monopile under large current–wave ratio

Scour around a circular monopile in coastal regions has been investigated extensively over the past decades, but the time development of scour depth around an offshore-wind monopile under large current–wave ratio still lacks a predictive model. By considering the conservation of sand volume and adopting the conventional exponential law for temporal variation, a semi-empirical model, which has three parameters, i.e., an equilibrium scour depth, a shape coefficient and a scour characteristic time scale, is developed for predicting the time development of scour depth around a monopile under live-bed conditions of combined wave–current flows. A series of laboratory experiments was conducted in current-only and wave–current flows to obtain data for model calibration and validation. Experimental results indicate that adding weak waves on current accelerates scour development, which is successfully captured by the proposed model through using the far-field bottom shear stress as a key model input. The overall model inaccuracy is within 25%, and the model’s applicability is further confirmed by field measurements from an offshore wind farm in east China. This model can help to determine the timing of installing scour protection around offshore monopiles, especially for the circumstances with very strong local sediment transport (live-bed).

Analytical solution for deep circular tunnels covered by an isolation coating layer subjected to far-field shear stresses

Tunnels built in areas subject to earthquake activity must withstand seismic loadings. Covering tunnel liners with a coating layer will be a possible way to mitigate seismic damage to tunnels. In this paper, an analytical solution is developed for the seismic response of deep circular tunnels covered by an isolation layer. Since the cross-section dimension of tunnels is normally much smaller than the wavelength of ground peak velocities, the inertial forces can be neglected, and the structure can be designed using the pseudo-static approach, where the seismic-induced loads or deformations can be approximated by far-field shear stresses. The ground and the isolation layer are assumed to be elastic, homogeneous, and isotropic in plane strain conditions, and the tunnel lining is represented as an elastic shell. Both the full-slip and no-slip conditions are considered for the contact between the tunnel and the isolation layer, while the interface between the isolation layer and the ground is assumed to be continuous. The relative stiffness method, proposed by Einstein and Schwartz (1979), is employed to obtain the closed-form solutions for tunnel distortion and internal forces, including axial force, bending moment, and shear force. The proposed solution is verified by providing comparisons between its results and those from the known results in literature and the Finite Element program. Parametric analyses are presented where the seismic mitigation effects of the isolation layer with different properties such as thickness, elastic modulus, and Poisson’s ratio. Results show that the elastic modulus and thickness of the isolation layer, as well as the tunnel-isolation layer interface conditions (i.e., full-slip and no-slip), have significant influences on the seismic mitigation effect, except for the Poisson’s ratio of the isolation layer. The proposed solution can be used as an effective tool for the design optimization of tunnel structures with an isolation layer.

Shallow tunnels misaligned with geostatic principal stress directions: Analytical solution and 3D face effects

Rock masses often present strong stress anisotropy, especially at shallow depths; thus, the alignment of a tunnel with one of the principal stress directions is unlikely. As a result, far-field out-of-plane shear stresses are present and, yet, their effects are often neglected in tunnel design because it is commonly assumed that the tunnel is aligned with one of the principal stress directions. The paper presents an analytical solution to determine the axial displacements and the axial shear stresses around shallow tunnels not aligned with the principal stresses in elastic ground, when subjected to a far-field shear stress. The analytical solution is developed using complex variable analysis and conformal mapping techniques to consider the presence of the ground surface. The analytical solution can be added to the Verruijt and Booker (2000) solution to determine the full 3D stress and displacement fields far-behind the face of unsupported shallow tunnels subjected to a general state of stress. In addition, 3D FEM models are conducted to investigate the effects of the far-field shear stress near the face of the tunnel. The far-field shear stress induces asymmetric ground deformations near the face, which remain far-behind the face when the tunnel is supported, and when the ground is elastoplastic.

Analytical solution for tunnels not aligned with geostatic principal stress directions

It is well-known that rock masses may present marked stress anisotropy. However, most of the tunnel analyses (numerical and analytical) assume the tunnel axis aligned with one of the principal stress directions. When this is not the case, axial shear stresses appear, which then are neglected, as it is done in all analytical solutions available for tunnel analysis. Existing solutions may consider advanced nonlinear ground behavior (i.e. elastic-brittle-plastic with e.g. Hoek and Brown failure criteria), linear-elastic ground with transversely anisotropic properties, seismic loading, groundwater and support, etc., but all consider that the axis of the tunnel aligns with one of the principal far-field stresses. This is also what is generally assumed when conducting more sophisticated, three dimensional numerical analyses. In this paper, an analytical solution to calculate the stresses and displacements induced by far-field axial shear stresses is presented. Solutions for supported and unsupported tunnels are provided. The proposed analytical solution can be combined with the classical Kirsch and Einstein-Schwartz solutions to determine the complete stress and displacement fields around the tunnel. Further, the effects of stress anisotropy are discussed.

Viscoelastic solutions for stresses and displacements around non-circular tunnels sequentially excavated at great depths

This research study presents analytical solutions for the stresses and displacements around deeply buried non-circular tunnels, taking into account the viscoelasticity of the ground, and the sequential excavation of the tunnels’ cross-sections. General initial far-field stress states are assumed, and the time-dependent pressures exerted at the internal tunnel boundaries are found to account for the support effects or water pressures of the hydraulic tunnels. Then, solutions are derived for tunnels with a time-varying sizes and/or shape, by assuming the time-dependent functions specified by the designers. The analytical solutions for the stresses and displacements around elliptical and square tunnels are specifically presented for linearly viscoelastic models using a Muskhelishvili complex variable method and Laplace transform techniques. For validation purposes, numerical analyses are performed for the excavations of elliptical and square tunnels in rock which are simulated by Poynting–Thomson or generalized Kelvin viscoelastic models. Good agreements are observed between the analytical and numerical results of this study. Then, parametric analyses are carried out in order to investigate the effects of the far-field shear stress, along with the distribution forms of the internal pressures, on the ground displacements and stresses. The proposed analytical solutions can be employed to accurately predict the stress concentrations, as well as the time-dependent displacements around deeply buried elliptical or square-shaped tunnels. Furthermore, it is confirmed that this study’s described methodology may be potentially applied to obtain analytical solutions for other arbitrary shaped tunnels sequentially excavated in viscoelastic rock.

Closed-form solutions of hole distortion for use in deep-hole drilling measurements of residual stress in orthotropic plates

The measurement of residual stress using the deep-hole drilling method relies on the evaluation of the distortion of a hole in a plate under the action of far-field direct and shear stresses. While closed-form solutions exist for the isotropic materials, in previous work for orthotropic materials, finite element analysis has been used to find the distortion. In this technical note, Lekhnitskii’s analysis is used to find closed-form solutions for the distortion of a circular hole in an orthotropic plate. The results are compared with those of finite element analysis for a range of material properties with excellent agreement.

Thermally driven migration of ice-stream shear margins

AbstractIce-stream shear margins are the lateral boundaries of narrow, fast-flowing bands of ice within an ice sheet. We develop a theory for the migration of shear margins over time driven by viscous dissipation of heat within the ice, focusing on widening of the ice stream. The location of the margin is modelled as a transition from a cold to a temperate ice-sheet bed, and simultaneously as the transition from no slip to free slip at the same location. The temperature field in the ice is affected by intense shear heating as well as by the migration velocity of the margin (i.e. by the widening rate of the ice stream); if migration is too fast, there is little time for the ice to warm up and the margin remains cold, causing the bed to freeze. This suppresses widening. Conversely, if the migration speed is too slow, the ice in the margin warms up, causing the bed on the far side of the cold–temperate transition to reach the melting point, and migration to speed up. Using a Wiener–Hopf method, we show that for a given far-field shear stress, geothermal heat flux, and ice geometry, there is a single migration velocity that balances the two effects and permits widening at a steady rate. This velocity increases with the far-field lateral shear stress imposed by the ice stream, which controls shear heating in the margin. Our results also indicate that (i) a region of temperate ice must form in the margin, and that (ii) lateral advection of ice may play a significant role in controlling migration speeds.

Drained and undrained response of deep tunnels subjected to far-field shear loading

An analytical solution for a rectangular opening in an infinite elastic medium subjected to far-field shear stresses has been derived for drained and undrained loading conditions. A number of numerical simulations has been conducted to determine the distortion of a rectangular structure in an infinite elastic medium under far-field shear stresses also for drained and undrained conditions and when there is full slip or no slip at the ground–structure interface. The results show that the shape of the opening has a minor influence on the structure’s deformations and that full-slip conditions result in lower deformations. Undrained conditions tend to reduce distortions when the structure is more flexible than the ground, but tend to increase them for stiffer structures. A comparison between results obtained for a rectangular lined opening and for a circular lined opening are presented, and show that deformations of a rectangular structure with no-slip can be estimated from equations derived for a circular opening with an incompressible liner and also with no-slip. The effects of flexibility, slip condition at the interface, and drained or undrained loading are different for circular tunnels than for rectangular tunnels.

A lamellar inhomogeneity near a multiphase reinforcement

In composites, the stress intensity factors (SIFs) of a lamellar inhomogeneity near a multiphase reinforcement are of interest. Based on extension of Eshelby’s equivalent inclusion method, a unified approach is presented to study the effect of a multiphase inhomogeneity on the SIF at the tip points of two- and three-dimensional lamellar inhomogeneities under nonuniform far-field loadings. Alteration of the SIF due to the presence of a coating layer around the inhomogeneity is addressed. Furthermore, the effect of geometry and stiffness of each phase of a multiphase reinforcement on the mixed mode SIFs of a lamellar inhomogeneity is investigated. In contrast to cracks whose SIFs are the same for uniaxial and multiaxial far-field loadings, all axial far-field applied stresses, which are parallel or perpendicular to the anticrack plane, result in the square root stress singularity at the anticrack tip points. However, only those components of the far-field shear stress whose couple vector is perpendicular to the anticrack plane would generate nonvanishing mixed mode SIFs, whereas for the shear components with couple vectors parallel to the anticrack plane, the SIF vanishes.

A practical iterative procedure to estimate seismic-induced deformations of shallow rectangular structures

An iterative procedure is proposed to estimate seismic-induced distortions of cut-and-cover rectangular structures. The procedure is based on an existing analytical solution for deep rectangular structures subjected to far-field shear stress which assumes elastic behavior of the soil and structure, tied contact at the soil–structure interface, and static loading. The new proposed procedure builds on the analytical solution and approximates dynamic response with a pseudo-static analysis and incorporates soil-stiffness degradation through an iterative scheme where the soil shear modulus is changed in each iteration based on the shear strain of the soil obtained in the previous iteration. The presence of the ground surface and slip at the soil–structure interface are neglected in the method proposed, but their effects are shown to be small and have compensating results when soil nonlinearity is introduced. Predictions obtained from the analytical solution have been verified by a series of numerical tests, which include the response of the Daikai subway station during the 1995 Kobe earthquake in Japan and the Los Angeles Civic Center subway station subjected to the 1994 Northridge earthquake in California. The relative errors in terms of deformation between analytical and numerical results are smaller than 15%. The procedure results in stresses on the structure that compare well with those obtained with the numerical method when there is no slip between soil and structure. If slip is allowed, the analytical solution overpredicts tensile normal stresses and underpredicts compressive normal stresses.

Yield point of metallic glass

Shear bands form in most bulk metallic glasses (BMGs) within a narrow range of uniaxial strain ε y ≅ 2%. We propose this critical condition corresponds to embryonic shear band (ESB) propagation, not its nucleation. To propagate an ESB, the far-field shear stress τ ∞ ≈ Eε y/2 must exceed the quasi-steady-state glue traction τ glue of shear-alienated glass until the glass transition temperature is approached internally due to frictional heating, at which point ESB matures as a runaway shear crack. The incubation length scale l inc necessary for this maturation is estimated to be ∼10 2 nm for Zr-based BMGs, below which sample size-scale shear localization does not happen. In shear-alienated glass, the last resistance against localized shearing comes from extremely fast downhill dissipative dynamics of timescale comparable to atomic vibrations, allowing molecular dynamics (MD) simulations to capture this recovery process which governs τ glue. We model four metallic glasses: a binary Lennard-Jones system, two binary embedded atom potential systems and a quinternary embedded atom system. Despite vast differences in the structure and interatomic interactions, the four MD calculations give ε y predictions of 2.4%, 2.1%, 2.6% and 2.9%, respectively.

Analytical solution for deep rectangular structures subjected to far-field shear stresses

Underground structures located in seismic areas have to support the static loads transferred from the surrounding ground under normal working conditions, as well as the loads imposed by any seismic event. Typically underground structures have cross section dimensions much smaller than the wave length of ground peak velocities, in which case inertial forces can be neglected and the structure can be designed using a pseudo-static analysis, where the seismic-induced loads or deformations can be approximated by a far-field shear stress or strain. Current close-form solutions for deep rectangular structures subjected to a far-field shear stress are approximations that do not consider all the relevant variables. An analytical solution is presented in this paper for deep rectangular structures with a far-field shear stress. Complex variable theory and conformal mapping have been used to develop the solution, which is applicable to deep rectangular structures in a homogeneous, isotropic, elastic medium. The solution shows that the deformations of the structure depend on the relative stiffness between the structure and the surrounding ground, and on the shape of the structure. The analytical solution has been verified by comparing its predictions with results from a finite element method and from previously published data.

Effect of internal stresses on thermo-mechanical stability of interconnect structures in microelectronic devices

Interconnect structures in microelectronic devices can deform via unusual, scale-sensitive phenomena due to thermo-mechanical loads sustained during processing, or during service as part of a microelectronic package. Examples include creep/plasticity of Cu interconnect lines embedded in a dielectric layer at the back-end of Si chip, and diffusionally accommodated sliding at Cu–dielectric interfaces. These effects may result in in-plane (IP) changes in Cu line dimensions, cause strain incompatibilities between Cu and LKD in the out-of-plane (OOP) direction, and cause Cu lines to migrate or crawl under far-field shear stresses imposed by the package. In this paper, a shear-lag based model is utilized to simulate IP and OOP deformation in a Cu–LKD interconnect structure on a Si substrate under thermal cycling conditions associated with processing. A separate model, which simulates IP deformation of Cu interconnects embedded in LKD under thermo-mechanical cycling conditions imposed when the chip is attached to a package, is also presented. The models, which incorporate a constitutive interfacial sliding law developed previously, help rationalize experimental atomic force microscopy (AFM) observations of inelastic strain accrual in Cu lines, and the development of dimensional incompatibility between adjoining components in devices, during thermal cycling.

Modeling of interfacial sliding and film crawling in back-end structures of microelectronic devices

The fine-scale of interconnect structures in the back-end of modern microelectronic devices makes them susceptible to unusual, scale-sensitive deformation phenomena during processing or service because of internal stresses induced by thermal expansion mismatch between adjoining materials. During thermo-mechanical cycling associated with processing or service, dimensional changes may occur in Cu interconnect lines embedded in a low-K dielectric (LKD) due to plasticity/creep, strain incompatibilities may arise between Cu and LKD due to diffusionally accommodated interfacial sliding, and Cu lines may crawl or migrate via plastic deformation and interfacial sliding under far-field shear stresses imposed by the package. Although small, these effects can have a pronounced effect on component reliability. This paper presents shear-lag based modeling approaches to simulate out-of-plane (OOP) strain incompatibilities which arise within a single-layer Cu-LKD back-end structure (BES) during back-end processing, and in-plane (IP) deformation and migration of Cu interconnects within the BES after the chip is attached to a flip-chip package. Both models incorporate a previously developed constitutive interfacial sliding law, and help rationalize experimentally observed interfacial strain incompatibilities within Cu-LKD BES.

Mechanical behavior of Σ tilt grain boundaries in nanoscale Cu and Al: A quasicontinuum study

Molecular simulations using the quasicontinuum method are performed to understand the mechanical response at the nanoscale of grain boundaries (GBs) under simple shear. The energetics and mechanical strength of 18 Σ 〈1 1 0〉 symmetric tilt GBs and two Σ 〈1 1 0〉 asymmetric tilt GBs are investigated in Cu and Al. Special emphasis is placed on the evolution of far-field shear stresses under applied strain and related deformation mechanisms at zero temperature. The deformation of the boundaries is found to operate by three modes depending on the GB equilibrium configuration: GB sliding by uncorrelated atomic shuffling, nucleation of partial dislocations from the interface to the grains, and GB migration. This investigation shows that (1) the GB energy alone cannot be used as a relevant parameter to predict the sliding of nanoscale high-angle boundaries when no thermally activated mechanisms are involved; (2) the E structural unit present in the period of Σ tilt GBs is found to be responsible for the onset of sliding by atomic shuffling; (3) GB sliding strength in the athermal limit shows slight variations between the different interface configurations, but has no apparent correlation with the GB structure; (4) the metal potential plays a determinant role in the relaxation of stress after sliding, but does not influence the GB sliding strength; here it is suggested that the metal potential has a stronger impact on crystal slip than on the intrinsic interface behavior. These findings provide additional insights on the role of GB structure in the deformation processes of nanocrystalline metals.

Diffusionally accommodated interfacial sliding in metal-silicon systems

The kinetics and mechanism of diffusionally accommodated interfacial sliding (interfacial creep) under far-field shear and normal stresses was studied, based on diffusion-bonded Al-Si-Al sandwich specimens. A previously developed interfacial creep law [Funn and Dutta, Acta Mater 1999; 47: 149], which proposed that interfaces may slide via interface-diffusion controlled diffusional creep, was experimentally validated by carrying out a systematic parametric study. In agreement with the model, the Si-Al interfaces slid via diffusional creep ( n = 1) under the influence of an effective shear stress, which depends on the far-field shear and normal stresses, as well as the interfacial topography. Compressive stresses acting normal to the interface lowered the effective shear stress, resulting in a threshold effect, thus reducing the sliding rate. The rate of sliding was controlled by diffusional mass transport through a thin amorphous, O-rich interfacial layer, under the influence of local interfacial stress gradients, which arose due to the topological features of the interface. Instances of interfacial sliding in the absence of interfacial de-cohesion, which have been noted in composites, thin-film systems, etc., may be explained by the present mechanism, which also offers an alternative rationalization of threshold behavior during diffusional flow (besides interface-reaction control).

A numerical investigation on the effect of the inflow conditions on the self-similar region of a round jet

In this paper we consider the direct numerical simulation (DNS) of a spatially developing free round jet at low Reynolds numbers. Simulation of a spatially evolving flow such as the jet requires boundary conditions, which allow entrainment into the turbulent flow across the lateral boundaries of the computational domain. The boundary conditions which satisfy this requirement are so-called traction free boundary conditions. After showing that these boundary conditions lead to a correct behavior of the velocity near the lateral boundary of the jet, we will consider the DNS of the jet flow at a Reynolds number of 2.4×103 and compare the results with experimental data obtained by Hussein et al. [J. Fluid Mech. 258, 31 (1994)] and by Panchapakesan and Lumley [J. Fluid Mech. 246, 197 (1993)]. The results of our numerical simulations agree very well with the experimental data. Next we use the DNS to investigate the influence of the shape of the velocity profile at the jet orifice on the self-similarity scaling for the far-field velocity and shear stress profile. Evidence is presented in support of the suggestion by George [Advances in Turbulence (Springer, New York, 1989)] that the details of self-similarity depend on the initial conditions. This fact implies that there may exist no universally valid similarity scaling for the free jet.

Longitudinal shear transfer in fiber optic sensors

A coated optical fiber embedded in a host composite matrix is analysed to study the strain transfer to, and the stress and strain concentrations caused by, the optical fiber. The optical fiber and its coating are modeled as concentric, circular inclusions embedded in an isotropic homogeneous matrix. A case of a far-field longitudinal shear stress parallel to both the optical fiber and the structural fibers is considered. The effects of the modulus of the coating as well as its thickness is studied for the strain transfer and the stress and strain concentrations. It is shown that when the coating is softer than the matrix, a thinner coating will induce more shear transfer; when the coating is stiffer than the matrix, a thicker coating will induce more shear transfer. It is also shown that the maximum shear strain is induced in the optical fiber when the shear modulus of the coating is the geometric mean of the shear moduli of the fiber and the matrix.

Method for calculating the interlaminar stresses in symmetric laminates containing a circular hole

An efficient approximate solution for three-dimensional stress distributions around a circular hole in symmetric laminates under a set of far-field in-plane stresses is presented. Stress functions for each ply are assumed according to the boundary-layer equilibrium equations. In addition, all the boundary conditions for each ply and traction continuity at the ply interface are exactly satisfied. Eventually, the unknown parameters in stress functions are determined by the minimization of complementary energy of the whole laminate. Numerical examples are presented in comparison with given literature which shows that the present method is efficient. I. Introduction D UE to stiffness discontinuity between plies, it is easy to cause interlaminar stress concentrations near the freeedge region in composite laminates. Such stresses will commence delamination, matrix crackings, and failure of laminates, especially under fatigue loading. Since 1970, numerous investigations15 have used various methods to determine the interlaminar stresses at the straight free edge of composite laminates. Pipes and Pagano1 adopted the anisotropic elasticity theory in conjunction with the finite difference method to analyze a simple four-ply laminate. Wang and Crossman2 used a finite element approach to investigate this same problem. Wang and Choi3'4 derived an analytical solution based on Lekhnitskii's stress potentials and the theory of anisotropic elasticity to determine the exact order of stress singularity at the free edges of laminate. In an effort to develop an efficient method to deal with thick laminates (say 100 plies), Kassapoglou and Lagace5 used the force balance method in conjunction with the principle of minimum complementary energy to obtain an analytical solution for interlaminar stresses at straight free edges. Owing to the complicated geometry for the curved free edges as compared with the straight free edges, little work has been done for composite laminates with curved free edges. Basically, the analysis of straight free edge may be assumed as a problem with two-dimensional stress and strain variations. However, the curved free edge is a typical three-dimensional problem. This difference has rendered the analysis for the curved free edge more difficult than the straight free edge. A boundary-layer theory based on the perturbation technique for isotropic elastic plates with a circular hole developed by Reiss6 has been extended to composite laminates by Tang.7-8 This approach is based on identifying the boundary-layer problem as two equivalent problems, namely, a modified torsion problem and a modified plane strain problem. However, Tang's solution can only satisfy part of boundary conditions in an average sense, which may result in unreliable stress results near the free edges.9 In addition, Tang's solution will be very tedious for laminates with numerous plies. Zhang and Ueng10 proposed a simplified method to investigate the effect of the ratio of hole radius to laminate thickness on the interlaminar stress distributions around a hole in a (0/90 deg)s laminate under far-field tensile or shear stress; however, it is assumed that the order of stress singularity is prescribed and