Uniform Strain Gradient Research Articles

A simple extension of FFT-based methods to strain gradient loadings - Application to the homogenization of beams and plates with linear and non-linear behaviors

Because of their simplicity, efficiency and ability for parallelism, FFT-based methods are very attractive in the context of numerical periodic homogenization, especially when compared to standard FE codes used in the same context. They allow applying to a unit-cell a uniform average strain with a periodic strain fluctuation that is an unknown quantity. Solving the problem allows to evaluate the complete stress-strain fields. The present work proposes to extend the use of the method from uniform loadings (i.e. uniform applied strain) to strain gradient loadings (i.e. strain fields with a uniform strain gradient) while keeping the algorithm as simple as possible. The identification of a subset of strain gradient loadings allows for a minimally invasive modification of the iterative algorithm so that the implementation is straightforward. In spite of a reduced subset of 9 independent loadings among the 18 available, the second part of the paper demonstrates that it is enough for considering the homogenization of beams and plates. A first application validates the approach and compares it to another FFT-based method dedicated to the homogenization of plates. The second application concerns the homogenization of beams, for the first time considered (to author's knowledge) with an FFT-based solver. The method applies to different beam cross-sections and the proposition of using composite voxels drastically improves the numerical solution when the beam cross-section is not conform with the spatial discretization, especially for torsion loading. As a result, the massively parallel AMITEX_FFTP code has been slightly modified and now offers a new functionality, allowing the users to prescribe torsions and flexions to beam or plate heterogeneous unit-cells.

Mechanical design of uniform strain-gradient schemes for transverse and longitudinal flexoelectricity

Flexoelectricity is an electro-mechanical coupling effect in which dielectric materials polarize in response to deformation gradients. Typical models—including bending beams, trapezoidal terraces, and torquing bars—are adopted to generate strain gradients for studying flexoelectricity. Also, non-uniformities are introduced, which can induce possible problems related to electric polarization induced by higher-order components of the non-uniform strain gradient. Mechanical design is proposed to solve the problem of higher order components. Specifically, we propose mechanical design of two uniform strain-gradient schemes for transverse and longitudinal flexoelectricity—a special-designed beam and a circular terrace. Theoretical and numerical analysis support the uniformity of strain gradients of these designs. Experiments were conducted to obtain the two types of flexoelectric response of uniform strain-gradient schemes. By proposing a way to achieve uniform strain gradient through mechanical design, this work promotes research on flexoelectricity by eliminating possible impacts of non-uniform strain gradients. A uniform strain-gradient beam is an ideal tool for studying uniform strain gradient, and is applicable to various avenues of research besides flexoelectricity.

Temperature Dependent Exciton Funnel Dynamics in Uniform Strain Gradient Field Observed by Time‐Resolved Photoluminescence

AbstractElastic strain engineering is an important way to reversibly tune the properties of micro/nanoscale semiconductors for promising applications in the emerging advanced nanotechnologies, such as the strain enhanced high‐mobility transistors, nanogenerators, etc. However, direct observation and control of energetic carrier dynamics with precise strain gradient fields remains a challenge. Here, observation of temperature‐dependent exciton funnel dynamics is reported in a precisely controlled strain gradient field by time‐resolved photoluminescence. An efficient exciton hopping process is revealed at picosecond time scale as the donor‐bound excitons in ZnO microwires funnel along the strain gradient, which strongly depends on the temperature. Combined experiments and simulations unravel that, in addition to the exciton funnel by the strain gradient, both the excitation efficiency and emission lifetime of the excitons increase gradually from the compressive side to the tensile side due to the role of local strain regulation. The results give a clear physical picture of the energetic carrier dynamics in the strain gradient field in semiconductors, which provides a promising paradigm for the design of high‐performance optoelectronic devices.

Localized deformation inside the Lüders front of a medium manganese steel

Medium manganese steels are promising alloys for highly demanding sheet steel applications. However, cold-rolled intercritically annealed ultrafine-grained dual-phase medium manganese steels exhibit Lüders banding, which is detrimental for the mechanical properties and for the surface finish after forming. Therefore, it is essential to prevent the formation of Lüders bands. To achieve this, it is crucial to understand the underlying micromechanisms of their formation and propagation. While the nucleation of Lüders bands was studied before in these alloys, understanding of their propagation is lacking. Here, we use electron channelling contrast imaging, with a resolution of ~8 nm to examine the defect populations within and across the Lüders front. While macroscopic digital image correlation shows a uniform strain gradient across the Lüders front, quasi in-situ tensile tests in combination with microscopic electron channelling contrast imaging investigations reveal strain localized regions (termed plastic zones) within the Lüders front. These plastic zones are aligned with the macroscopic Lüders front and parallel in nature, with less deformed regions sandwiched between them. Additionally, it is observed that the size and distribution of the plastic zones are affected by the mechanical stability of austenite. We further demonstrate the role of strain partitioning between the austenite and ferrite (austenite being the softer phase) in the formation of the plastic zones.

Diffraction properties of a strongly bent diamond crystal used as a dispersive spectrometer for XFEL pulses

Self-amplified spontaneous emission (SASE) enables X-ray free-electron lasers (XFELs) to generate hard X-ray pulses of sub-100fs duration. However, due to the stochastic nature of SASE, the energy spectrum fluctuates from pulse to pulse. Many experiments that employ XFEL radiation require the resolution of the spectrum of each pulse. The work presented here investigates the capacity of a thin strongly bent diamond crystal to resolve the energy spectra of hard X-ray SASE pulses by studying its diffraction properties. Rocking curves of the symmetric C*(440) reflection have been measured for different bending radii. The experimental data match the theoretical modelling based on the Takagi-Taupin equations of dynamical diffraction. A uniform strain gradient has proven to be a valid model of strain deformations in the crystal.

Crystal plasticity analysis of micro-deformation, lattice rotation and geometrically necessary dislocation density

A gradient-enhanced crystal plasticity model is presented that explicitly accounts for the evolution of the densities of geometrically necessary dislocations (GNDs) on individual slip systems of deforming crystals. The GND densities are fully coupled with the crystal slip rule. Application of the model to two distinct and technologically important crystal types, namely hcp Ti and ccp Ni, is given. For the hcp crystals, slip is permitted with a -type slip directions on basal, prismatic and pyramidal planes and c + a -type slip directions on pyramidal planes. First, a single crystal under four-point bending is simulated as the uniform strain gradient expected in the central span provides a good validation of the code. Then, uniaxial deformation of a model near- α Ti polycrystal has been analysed. The resulting distributions of GND densities that develop on the various slip system types have been compared with independent experimental observations. The model predicts that GND density on the c + a systems is approximately an order of magnitude lower than that for a -type systems in agreement with experiment. For the ccp case, slip is considered to take place on the <110>{111} slip systems. Thermal loading of a single-crystal nickel alloy sample containing carbide particles of size approximately 30 μm has been analysed. Detailed comparisons are presented between model predictions and results of high-resolution electron backscatter diffraction (EBSD) measurements of the micro-deformations, lattice rotations, curvatures and GND densities local to the nickel–carbide interface. Qualitatively, good agreement is achieved between the coupled and decoupled model elastic strains with the EBSD measurements, but lattice rotations and GND densities are quantitatively well predicted by the coupled crystal model but are less well captured by the decoupled model. The GND coupling is found to lead to reduced lattice rotations and plastic strains in the region of highest heterogeneity close to the Ni matrix/particle interface, which is in agreement with the experimental measurements. The results presented provide objective evidence of the effectiveness of gradient-enhanced crystal plasticity finite element analysis and demonstrate that GND coupling is required in order to capture strains and lattice rotations in regions of high heterogeneity.

Towards a Bulk Theory of Flexoelectricity

Flexoelectricity is the linear response of polarization to a strain gradient. Here we address the simplest class of dielectrics, namely, elemental cubic crystals, and we prove that therein there is no extrinsic (i.e., surface) contribution to flexoelectricity in the thermodynamic limit. The flexoelectric tensor is expressed as a bulk response of the solid, manifestly independent of surface configurations. Furthermore, we prove that the flexoelectric responses induced by a long-wavelength phonon and by a uniform strain gradient are identical.

Characterization and modeling of strain assisted diffusion in an epoxy adhesive layer

In this paper a coupled model for strain-assisted diffusion is derived from the basic principles of continuum mechanics and thermodynamics, and material properties characterized using diffusion experiments. The proposed methodology constitutes a significant step toward modeling the synergistic bond degradation mechanism at the bonded interface between a Fiber Reinforced Polymer (FRP) and a substrate, and for predicting debond initiation and propagation along the interface in the presence of a diffusing penetrant at the crack tip and at elevated temperatures. It is now well-known that Fick’s law is frequently inadequate for describing moisture diffusion in polymers and polymer composites. Non-Fickian or anomalous diffusion is likely to occur when a polymer is subjected to external stresses and strains, as well as elevated temperatures and humidity. In this paper, a modeling methodology based on the basic principles of continuum mechanics and thermodynamics is developed which allows characterization of the combined effects of temperature, humidity, and strain on diffusion coefficients as well as on moisture saturation level, from moisture weight gain data. For tractability, the diffusion governing equations are simplified for the special case of 1-D diffusion subjected to uniaxial strain and a uniform strain gradient. A novel test specimen that introduces a uniform strain gradient is developed, and diffusion test data for an epoxy-based primer/adhesive are presented and employed for complete characterization of material constants used in the model.

X-ray interbranch resonance as a new idea to solve the diffraction problem for bent crystals

The resonance concept is extended for x-ray interbranch scattering by a bent crystal with a uniform strain gradient. Splitting each of the branches of the dispersion surface follows from these considerations. It is shown that interbranch crossover is due to interference of the waves corresponding to ‘new’ branches, making up the ‘new’ Bloch-modes.

X-ray standing waves on the surface of the crystal with the uniform strain gradient

The results of theoretical investigation of X-ray Standing Waves (XSW) on the surface of the crystal with the Uniform Strain Gradient (USG) are presented. Bent crystals are the typical examples of the crystals with the USG. The proposed theory is based on the Takagi-Taupin theory of X-ray propagation in a weakly deformed crystal. This theory is applied for calculation of the angular dependencies of reflection coefficient and XSW on the surface of the crystal with USG for the specific case of Si(400) thick crystal, CuKa radiation. The behaviour of the XSW with the different position of adsorbed atoms (different values of the phase shift) are analyzed

Fibre/matrix interface property determination

A ‘microcomposite’ single fibre pull-out test has been designed to permit the visual observation of events at the fibre/matrix interface and the collection of detailed mechanical data as interface debonding and fibre pull-out occur. The confirmation of a uniform strain gradient distribution along the embedded length of fibre by birefringence observations allows simple determination of the interfacial shear strength using the quotient F(deb)/A(int), where F(deb) is the interface debonding force and A(int) is the interfacial area given by 2πrl. Several characteristics displayed by the load/displacement measurements have been directly correlated with observed events such as fibre stick-slip and interfacial frictional effects.

X-ray topography of bent crystals

The influence of the bending of a crystal on the formation of diffracted images in Bragg section topographs as well as in Laue section and traverse topographs is studied. In the case of Bragg section topography the interferometric fringes due to the interference of waves once and twice internally reflected inside a bent crystal are described. It is established that the maximum positions of diffracted intensity satisfy the law xn = [16π(2n - 1)/5B2]1/3, where xn is the distance from the incidence slit of the X-rays, B is the uniform strain gradient and n is the fringe number. This dependence is found to be in good agreement with experimental data. The Laue section topograph of a bent crystal with a screw dislocation parallel to the diffraction vector is considered. The effects of asymmetry in the Pendellosung fringe pattern and of 'splitting' of the direct image with respect to the dislocation line in both the experimental and simulated topographs are accounted for. The variation of the contrast of dislocations with depth inside a bent crystal in Laue traverse topographs is studied by computer simulations using the reciprocity theorem.

An X-ray interferometer with an elastically deformed crystal

The diffraction of an X-ray narrow beam on a double crystal interferometer with one of crystals having a uniform strain gradient is studied. The diffracted wave amplitudes are calculated on the exit surface of the second crystal supposing that the X-ray scattering in both of the crystals takes place under the Borrmann-effect condition. Expressions for the period of the interference fringes depending on the strain degree in the concrete cases of a mechanically bent crystal and a crystal with temperature gradient are obtained. The possibility of determination of the crystal bending radius by the disappearance of the interference fringes is shown. [Russian Text Ignored].

Dynamical X-ray propagation: a theoretical approach to the creation of new wave fields

Although it is commonly invoked, the phenomenon of 'creation of new wave fields', which is responsible for some of the features visible on topographic images, has never been really explained in theoretical terms. This is done here in the case of a crystal deformed by a uniform strain gradient. The appropriate Green function is expanded in reciprocal space as a wave packet of non-plane waves, each component corresponding to a single value of the deviation parameter at the entrance surface. It is then shown that each component of this wave packet is made up of four parts, two of which can be identified as 'normal' wave fields (i.e. those predicted by the Eikonal theory); the two others are the so-called 'created wave fields'; it is shown that they correspond to interbranch scattering from one branch of the dispersion surface to the other and give rise to two extra beams. These created wave fields extract a fraction e-2π|v| out of the normal energy flow (|v| being inversely proportional to the strain gradient), in full agreement with previous computer experiments.

The dynamical theory of X-ray Bragg diffraction from a crystal with a uniform strain gradient. The Green–Riemann functions

The dynamical problem of X-ray Bragg diffraction from a thick (semi-infinite) crystal deformed by a uniform strain gradient (USG) is treated on the basis of the Green-Riemann function formalism. The rigorous solution of the problem is formulated by means of the Huygens-Fresnel principle. The exact Green functions are obtained in the form of the Laplace integrals suitable in physical applications. The quasi-classical and the Born (kinematical) asymptotic expansions of the Green functions are constructed as functions of the effective USG parameter B. Special attention is paid to the analysis of the wave-field propagation in a crystal with USG. The spatial harmonics Re(q) of the diffracted Green function, when Re(qB) < 0, as is shown, propagate within the proper 'waveguides', while the ones with Re(qB) > 0 are damped exponentially in the bulk of the crystal. The Taupin problem of the Bragg dynamical diffraction of the X-ray incident plane wave from a thick crystal, the lattice spacing being a linear function of the coordinate z (along the inward normal to the entrance surface) only is solved exactly in analytical form. In the latter case the waveguide nature of the propagation of the spatial harmonics inside such a crystal, provided that Re(qB) < 0, is clearly revealed.

The cracking of layers of brittle material by differential strains

The cracking of a brittle layer of elastic material due to differential straining has been investigated with a view to discovering what factors govern the pattern of the cracks produced by a given strain pattern. An analysis of the stress equations leads to the conclusion that two dimensionless parameters only are involved, the ratio of the applied strains to the breaking strain and the ratio of the crack spacing to the layer thickness: the shape of the curve relating these parameters is deduced by physical argument. Evidence in support of the analysis is obtained from the observed cracking of brittle beams when subjected to a uniform strain gradient by bending.