Strain gradients in epitaxial ferroelectrics

X-ray analysis of ferroelectric thin layers of Ba1/2Sr1/2TiO3 with different thicknesses reveals the presence of strain gradients across the films and allows us to propose a functional form for the internal strain profile. We use this to calculate the influence of strain gradient, through flexoelectric coupling, on the degradation of the ferroelectric properties of films with decreasing thickness, in excellent agreement with the observed behavior. This paper shows that strain relaxation can lead to smooth, continuous gradients across hundreds of nanometers, and it highlights the pressing need to avoid such strain gradients in order to obtain ferroelectric films with bulklike properties.

Numerical simulation of axially loaded concrete columns under transverse impact and vulnerability assessment

With a view to assessing the vulnerability of columns to low elevation vehicular impacts, a non-linear explicit numerical model has been developed and validated using existing experimental results. The numerical model accounts for the effects of strain rate and confinement of the reinforced concrete, which are fundamental to the successful prediction of the impact response. The sensitivity of the material model parameters used for the validation is also scrutinised and numerical tests are performed to examine their suitability to simulate the shear failure conditions. Conflicting views on the strain gradient effects are discussed and the validation process is extended to investigate the ability of the equations developed under concentric loading conditions to simulate flexural failure events. Experimental data on impact force–time histories, mid span and residual deflections and support reactions have been verified against corresponding numerical results. A universal technique which can be applied to determine the vulnerability of the impacted columns against collisions with new generation vehicles under the most common impact modes is proposed. Additionally, the observed failure characteristics of the impacted columns are explained using extended outcomes. Based on the overall results, an analytical method is suggested to quantify the vulnerability of the columns.

Simultaneous measurement of multiple parameters using a single fiber Bragg grating

A novel approach for simultaneous measurement of chirp (any parameter that can induce strain gradient on FBG) and temperature using a single FBG is proposed. Change in reflectivity at central wavelength of FBG reflection & Bragg wavelength shifts induced due to temperature were used for chirp & temperature measurements respectively. Theoretical resolution limit for chirp and temperature using an Optical Spectrum Analyzer (OSA) with 1pm wavelength resolution and >58dB dynamic range are 12.8fm and 1/13 degrees C respectively.

应变梯度理论进展

应变梯度理论是近10年来为解释材料在微米尺度下的尺寸效应现象而发展起来的一种新理论.首先综述了应变梯度理论近年的发展及其对材料力学行为研究方面的进展.其次主要介绍了不含高阶应力的一类应变梯度理论及其应用;最后对应变梯度理论的发展做了展望.

Effect of Particle Size on the Formation of Adiabatic Shear Band In Particle Reinforced Metal Matrix Composites

In this paper, the effect of particle size on the formation of adiabatic shear band in 2024 All matrix composites reinforced with 15% volume fraction of 3.5, 10 and 20 mum SiC particles was investigated by making use of split Hopkinson pressure bar (SHPB). The results have demonstrated that the onset of adiabatic shear banding in the composites strongly depends on the particle size and adiabatic shear banding is more readily observed in the composite reinforced with small particles than that in the composite with large particles. This size dependency phenomenon can be characterized by the strain gradient effect. Instability analysis reveals that high strain gradient is a strong driving force for the formation of adiabatic shear banding in particle reinforced metal matrix composites (MMCp).

A Study of Size-Dependent Microindentation

We recently proposed a strain gradient theory to account for the size dependence of plastic deformation at micron and submicron length scales. The strain gradient theory includes the effects of both rotation gradient and stretch gradient such that the rotation gradient influences the material character through the interaction between the Cauchy stresses and the couple stresses; the stretch gradient measures explicitly enter the constitutive relations through the instantaneous tangent modulus. Indentation tests at scales on the order of one micron have shown that measured hardness increases significantly with decreasing indent size. In the present paper, the strain gradient theory is used to model materials undergoing small-scale indentations. A strong effect of including strain gradients in the constitutive description is found with hardness increasing by a factor of two or more over the relevant range behavior. Comparisons with the experimental data for polycrystalline copper and single crystal copper indeed show an approximately linear dependence of the square of the hardness, H 2, on the inverse of the indentation depth, 1/h, I.e., H-2 proportional to 1/h, which provides an important self-consistent check of the strain gradient theory proposed by the authors earlier.

Size effects in the constrained deformation of metallic foams

The constrained deformation of an aluminium alloy foam sandwiched between steel substrates has been investigated. The sandwich plates are subjected to through-thickness shear and normal loading, and it is found that the face sheets constrain the foam against plastic deformation and result in a size effect: the yield strength increases with diminishing thickness of foam layer. The strain distribution across the foam core has been measured by a visual strain mapping technique, and a boundary layer of reduced straining was observed adjacent to the face sheets. The deformation response of the aluminium foam layer was modelled by the elastic-plastic finite element analysis of regular and irregular two dimensional honeycombs, bonded to rigid face sheets; in the simulations, the rotation of the boundary nodes of the cell-wall beam elements was set to zero to simulate full constraint from the rigid face sheets. It is found that the regular honeycomb under-estimates the size effect whereas the irregular honeycomb provides a faithful representation of both the observed size effect and the observed strain profile through the foam layer. Additionally, a compressible version of the Fleck-Hutchinson strain gradient theory was used to predict the size effect; by identifying the cell edge length as the relevant microstructural length scale the strain gradient model is able to reproduce the observed strain profiles across the layer and the thickness dependence of strength. © 2002 Elsevier Science Ltd. All rights reserved.

能量非局部模型和新的应变梯度理论

提出一种新的基于能量非局部模型的应变梯度理论,并应用此理论对多晶铜以及薄膜基底的微压痕硬度进行理论预测和数值分析.首先,提出了能量非局部模型,并由此模型,得出新应变梯度理论的本构关系;其次,由变分原理,得出相应的有限元公式;再次,给出了微压痕硬度的有限元分析方法;最后,将该理论预测结果与经典理论预测结果以及实验结果进行了对比.结果表明,计算结果与实验结果相符;而经典理论的预测结果远低于实验结果.

Geological evolution of two crustal scale shear zones. Part I. The Rand thrust complex, northwestern Mojave Desert, California. Part II. The Magdalena metamorphic core complex, north central Sonora, Mexico

The geology and structure of two crustal scale shear zones were studied to understand the partitioning of strain within intracontinental orogenic belts. Movement histories and regional tectonic implications are deduced from observational data. The two widely separated study areas bear the imprint of intense Late Mesozoic through Middle Cenozoic tectonic activity. A regional transition from Late Cretaceous-Early Tertiary plutonism, metamorphism, and shortening strain to Middle Tertiary extension and magmatism is preserved in each area, with contrasting environments and mechanisms. Compressional phases of this tectonic history are better displayed in the Rand Mountains, whereas younger extensional structures dominate rock fabrics in the Magdalena area.

In the northwestern Mojave desert, the Rand Thrust Complex reveals a stack of four distinctive tectonic plates offset along the Garlock Fault. The lowermost plate, Rand Schist, is composed of greenschist facies metagraywacke, metachert, and metabasalt. Rand Schist is structurally overlain by Johannesburg Gneiss (= garnet-amphibolite grade orthogneisses, marbles and quartzites), which in turn is overlain by a Late Cretaceous hornblende-biotite granodiorite. Biotite granite forms the fourth and highest plate. Initial assembly of the tectonic stack involved a Late Cretaceous? south or southwest vergent overthrusting event in which Johannesburg Gneiss was imbricated and attenuated between Rand Schist and hornblende-biotite granodiorite. Thrusting postdated metamorphism and deformation of the lower two plates in separate environments. A post-kinematic stock, the Late Cretaceous Randsburg Granodiorite, intrudes deep levels of the complex and contains xenoliths of both Rand Schist and mylonitized Johannesburg? gneiss. Minimum shortening implied by the map patterns is 20 kilometers.

Some low angle faults of the Rand Thrust Complex formed or were reactivated between Late Cretaceous and Early Miocene time. South-southwest directed mylonites derived from Johannesburg Gneiss are commonly overprinted by less penetrative north-northeast vergent structures. Available kinematic information at shallower structural levels indicates that late disturbance(s) culminated in northward transport of the uppermost plate. Persistence of brittle fabrics along certain structural horizons suggests a possible association of late movement(s) with regionally known detachment faults. The four plates were juxtaposed and significant intraplate movements had ceased prior to Early Miocene emplacement of rhyolite porphyry dikes.

In the Magdalena region of north central Sonora, components of a pre-Middle Cretaceous stratigraphy are used as strain markers in tracking the evolution of a long lived orogenic belt. Important elements of the tectonic history include: (1) Compression during the Late Cretaceous and Early Tertiary, accompanied by plutonism, metamorphism, and ductile strain at depth, and thrust driven? syntectonic sedimentation at the surface. (2) Middle Tertiary transition to crustal extension, initially recorded by intrusion of leucogranites, inflation of the previously shortened middle and upper crustal section, and surface volcanism. (3) Gravity induced development of a normal sense ductile shear zone at mid crustal levels, with eventual detachment and southwestward displacement of the upper crustal stratigraphy by Early Miocene time.

Elucidation of the metamorphic core complex evolution just described was facilitated by fortuitous preservation of a unique assemblage of rocks and structures. The "type" stratigraphy utilized for regional correlation and strain analysis includes a Jurassic volcanic arc assemblage overlain by an Upper Jurassic-Lower Cretaceous quartz pebble conglomerate, in turn overlain by marine strata with fossiliferous Aptian-Albian limestones. The Jurassic strata, comprised of (a) rhyolite porphyries interstratified with quartz arenites, (b) rhyolite cobble conglomerate, and (c) intrusive granite porphyries, are known to rest on Precambrian basement north and east of the study area. The quartz pebble conglomerate is correlated with the Glance Conglomerate of southeastern Arizona and northeastern Sonora. The marine sequence represents part of an isolated arm? of the Bisbee Basin.

Crosscutting structural relationships between the pre-Middle Cretaceous supracrustal section, younger plutons, and deformational fabrics allow the tectonic sequence to be determined. Earliest phases of a Late Cretaceous-Early Tertiary orogeny are marked by emplacement of the 78 ± 3 Ma Guacomea Granodiorite (U/Pb zircon, Anderson et al., 1980) as a sill into deep levels of the layered Jurassic series. Subsequent regional metamorphism and ductile strain is recorded by a penetrative schistosity and lineation, and east-west trending folds. These fabrics are intruded by post-kinematic Early Tertiary? two mica granites. At shallower crustal levels, the orogeny is represented by north directed thrust faulting, formation of a large intermontane basin, and development of a pronounced unconformity. A second important phase of ductile strain followed Middle Tertiary? emplacement of leucogranites as sills and northwest trending dikes into intermediate levels of the deformed section (surficial volcanism was also active during this transitional period to regional extension). Gravitational instabilities resulting from crustal swelling via intrusion and thermal expansion led to development of a ductile shear zone within the stratigraphic horizon occupied by a laterally extensive leucogranite sill. With continued extension, upper crustal brittle normal faults (detachment faults) enhanced the uplift and tectonic denudation of this mylonite zone, ultimately resulting in southwestward displacement of the upper crustal stratigraphy.

Strains associated with the two ductile deformation events have been successfully partitioned through a multifaceted analysis. R_f/Ø measurements on various markers from the "type" stratigraphy allow a gradient representing cumulative strain since Middle Cretaceous time to be determined. From this gradient, noncoaxial strains accrued since emplacement of the leucogranites may be removed. Irrotational components of the postleucogranite strain are measured from quartz grain shapes in deformed granites; rotational components (shear strains) are determined from S-C fabrics and from restoration of rotated dike and vein networks. Structural observations and strain data are compatable with a deformation path of: (1) coaxial strain (pure shear?), followed by (2) injection of leucogranites as dikes (perpendicular to the minimum principle stress) and sills (parallel to the minimum principle stress), then (3) southwest directed simple shear. Modeling the late strain gradient as a simple shear zone permits a minimum displacement of 10 kilometers on the Magdalena mylonite zone/detachment fault system. Removal of the Middle Tertiary noncoaxial strains yields a residual (or pre-existing) strain gradient representative of the Late Cretaceous-Early Tertiary deformation. Several partially destrained cross sections, restored to the time of leucogranite emplacement, illustrate the idea that the upper plate of the core complex bas been detached from a region of significant topographic relief. 50% to 100% bulk extension across a 50 kilometer wide corridor is demonstrated.

Late Cenozoic tectonics of the Magdalena region are dominated by Basin and Range style faulting. Northeast and north-northwest trending high angle normal faults have interacted to extend the crust in an east-west direction. Net extension for this period is minor (10% to 15%) in comparison to the Middle Tertiary detachment related extensional episode.

Micromechanical damage and fracture in elastomeric polymers

This thesis aims at a simple one-parameter macroscopic model of distributed damage and fracture of polymers that is amenable to a straightforward and efficient numerical implementation. The failure model is motivated by post-mortem fractographic observations of void nucleation, growth and coalescence in polyurea stretched to failure, and accounts for the specific fracture energy per unit area attendant to rupture of the material.

Furthermore, it is shown that the macroscopic model can be rigorously derived, in the sense of optimal scaling, from a micromechanical model of chain elasticity and failure regularized by means of fractional strain-gradient elasticity. Optimal scaling laws that supply a link between the single parameter of the macroscopic model, namely the critical energy-release rate of the material, and micromechanical parameters pertaining to the elasticity and strength of the polymer chains, and to the strain-gradient elasticity regularization, are derived. Based on optimal scaling laws, it is shown how the critical energy-release rate of specific materials can be determined from test data. In addition, the scope and fidelity of the model is demonstrated by means of an example of application, namely Taylor-impact experiments of polyurea rods. Hereby, optimal transportation meshfree approximation schemes using maximum-entropy interpolation functions are employed.

Finally, a different crazing model using full derivatives of the deformation gradient and a core cut-off is presented, along with a numerical non-local regularization model. The numerical model takes into account higher-order deformation gradients in a finite element framework. It is shown how the introduction of non-locality into the model stabilizes the effect of strain localization to small volumes in materials undergoing softening. From an investigation of craze formation in the limit of large deformations, convergence studies verifying scaling properties of both local- and non-local energy contributions are presented.

Structural ordering and interface morphology in symmetrically strained (GaIn)As/Ga(PAs) superlattices grown on off-oriented GaAs(100)

In this work we investigate the structural properties of symmetrically strained (GaIn)As/GaAs/Ga(PAs)/GaAs superlattices by means of x-ray diffraction, reciprocal-space mapping, and x-ray reflectivity. The multilayers were grown by metalorganic vapor-phase epitaxy on (001) GaAs substrates intentionally off-oriented towards one of the nearest [110] directions. High-resolution triple-crystal reciprocal-space maps recorded for different azimuth angles in the vicinity of the (004) Bragg diffraction clearly show a double periodicity of the x-ray peak intensity that can be ascribed to a lateral and a vertical periodicity occurring parallel and perpendicular to the growth surface. Moreover, from the intensity modulation of the satellite peaks, a lateral-strain gradient within the epilayer unit cell is found, varying from a tensile to a compressive strain. Thus, the substrate off-orientation promotes a lateral modulation of the layer thickness (ordered interface roughness) and of the lattice strain, giving rise to laterally ordered macrosteps. In this respect, contour maps of the specular reflected beam in the vicinity of the (000) reciprocal lattice point were recorded in order to inspect the vertical and lateral interface roughness correlation, A semiquantitative analysis of our results shows that the interface morphology and roughness is greatly influenced by the off-orientation angle and the lateral strain distribution. Two mean spatial wavelengths can be determined, one corresponding exactly to the macrostep periodicity and the other indicating a further interface waviness along the macrosteps. The same spatial periodicities were found on the surface by atomic-force-microscopy images confirming the x-ray results and revealing a strong vertical correlation of the interfaces up to the outer surface.

The effect of flexoelectricity on the dielectric properties of inhomogeneously strained ferroelectric thin films

Recent experimental measurements of large flexoelectric coefficients in ferroelectric ceramics suggest that strain gradients can affect the polarization and permittivity behaviour of inhomogeneously strained ferroelectrics. Here we present a phenomenological model of the effect of flexoelectricity on the dielectric constant, polarization, Curie temperature (T-C), temperature of maximum dielectric constant (T-m) and temperature of the onset of reversible polarization (T-ferro) for ferroelectric thin films subject to substrate-induced epitaxial strains that are allowed to relax with thickness, and the qualitative and quantitative predictions of the model are compared with experimental results for (Ba0.5Sr0.5)TiO3 thin films on SrRuO3 electrodes. It is shown that flexoelectricity can play an important role in decreasing the maximum dielectric constant of ferroelectric thin films under inhomogeneous in-plane strain, regardless of the sign of the strain gradient.

Structural analysis of the Paint Lake Deformation Zone, Northern Ontario /

The Paint Lake Deformation Zone (PLDZ), located within the Superior Province of Canada, demarcates a major structural and lithological break between the Onaman-Tashota Terrane to the north and the Beardmore-Geraldton Belt to the south. The PLDZ is an east-west trending lineament, approximately 50 km in length and up to 1 km in width, comprised of an early ductile component termed the Paint Lake Shear Zone and a late brittle component known as the Paint Lake Fault. Structures associated with PLDZ development including S-, C- and C'-fabrics, stretching lineations, slickensides, C-C' intersection lineations, Z-folds and kinkbands indicate that simple shear deformation dominated during a NW-SE compressional event. Movement along the PLDZ was in a dextral sense consisting of an early differential motion with southside- down and a later strike-slip motion. Although the locus of the PLDZ may in part be lithologically controlled, mylonitization which accompanied shear zone development is not dependent on the lithological type. Conglomerate, intermediate and mafic volcanic units exhibit similar mesoscopic and microscopic structures where transected by the PLDZ. Field mapping, supported by thin section analysis, defines five strain domains increasing in intensity of deformation from shear zone boundary to centre. A change in the dominant microstructural deformation mechanism from dislocation creep to diffusion creep is observed with increasing strain during mylonitization. C'-fabric development is temporally associated with this change. A decrease in the angular relationship between C- and C'-fabrics is observed upon attaining maximum strain intensity. Strain profiling of the PLDZ demonstrates the presence of an outer primary strain gradient which exhibits a simple profile and an inner secondary strain gradient which exhibits a more complex profile. Regionally metamorphosed lithologies of lower greenschist facies outside the PLDZ were subjected to retrograde metamorphism during deformation within the PLDZ.

Reduced relative entropy techniques for a posteriori analysis of multiphase problems in elastodynamics

We give an a posteriori analysis of a semidiscrete discontinuous Galerkin scheme approximating solutions to a model of multiphase elastodynamics, which involves an energy density depending not only on the strain but also the strain gradient. A key component in the analysis is the reduced relative entropy stability framework developed in Giesselmann (2014, SIAM J. Math. Anal., 46, 3518–3539). This framework allows energy-type arguments to be applied to continuous functions. Since we advocate the use of discontinuous Galerkin methods we make use of two families of reconstructions, one set of discrete reconstructions and a set of elliptic reconstructions to apply the reduced relative entropy framework in this setting.

Reduced relative entropy techniques for a priori analysis of multiphase problems in elastodynamics

We give an a priori analysis of a semi-discrete discontinuous Galerkin scheme approximating solutions to a model of multiphase elastodynamics which involves an energy density depending not only on the strain but also the strain gradient. A key component in the analysis is the reduced relative entropy stability framework developed in Giesselmann (SIAM J Math Anal 46(5):3518–3539, 2014). The estimate we derive is optimal in the L∞(0,T;dG) norm for the strain and the L2(0,T;dG) norm for the velocity, where dG is an appropriate mesh dependent H1-like space.