Components Of Strain Tensor Research Articles

Atomistic and continuum Ascertainment of The crack tip stress fields in anisotropic elastic cubic media

The article provides a comparative analysis of stresses associated with the crack tip in an anisotropic elastic medium with cubic crystalline symmetry of elastic properties under mixed loading conditions obtained by two fundamentally different approaches: atomistic simulations and continuum fracture mechanics methods. The atomistic modelling approach is premised on the application of the molecular dynamics (MD) method implemented in the open source code program Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). The continuum mechanics approach is based on the classical elasticity theory of anisotropic media, in which the mechanical fields associated with the crack tip are represented by the crack tip stress and displacement series expansions generalizing the well-known classical asymptotical presentation of M. Williams to the case of anisotropic media. Using the MD method, a large series of computations of the loading of monocrystalline copper and aluminum plates with a face-centered cubic crystal lattice weakened by a central crack applying the embedded atom potential (EAM) was implemented. MD modeling is aimed at determining the atomic stresses and strains near the crack tip. The computed atomic stresses were compared with the stress field determined by the continuum elasticity theory of anisotropic media for crystal lattices with cubic symmetry of elastic properties. A comparative analysis was carried out for the angular dependencies of the stress and strain tensor components at various selected distances from the crack tip for the entire range of mixed deformation forms starting from pure tensile loading and ending with forms close to pure transverse shear. It is ascertained that the crack tip fields determined on the basis of two fundamentally different approaches, precisely, discrete and continuum, are completely consistent with each other. It is established that mathematical methods of continuum fracture mechanics can be applied to describe stress, strain and displacement fields at the atomic level.

Analysis of local strain fields around individual threading dislocations in GaN substrates by nanobeam x-ray diffraction

Position-dependent three-dimensional reciprocal space mapping (RSM) by nanobeam x-ray diffraction (nanoXRD) was performed to reveal the strain fields produced around individual threading dislocations (TDs) in GaN substrates. The distribution and Burgers vector of TDs for the nanoXRD measurements were confirmed by prerequisite analysis of multi-photon excited photoluminescence and etch pit methods. The present results demonstrated that the nanoXRD can identify change in the lattice plane structure for all types of TDs, i.e., edge-, screw-, and mixed TDs with the Burgers vector of b = 1a, 1c and 1m + 1c. Strain tensor components related to edge and/or screw components of the TDs analyzed from the three-dimensional RSM data showed a nearly symmetrical strained region centered on the TD positions, which were in good agreements with simulation results based on the isotropic elastic theory using a particular Burgers vector. The present method is beneficial in that it allows non-destructive analysis of screw components of TDs that tend to contribute to leakage characteristics and are not routinely accessible by conventional structural analysis. These results indicate that nanoXRD could be a powerful way to reveal three-dimensional strain fields associated with arbitrary types of TDs in semiconductor materials, such as GaN and SiC.

Surface crack effect on frequency and vibration mode switching of solar-powered aerospace structures

In the engineering design of modern wing panels and their optimal control system, the first step is to find and predict their dynamic characteristics. These characteristics are dependent on material properties, panel geometry, and boundary conditions. In addition, possible fracture existence, imperfections and environmental conditions may unpredictably change the frequency and modes shape of the panel and then the parameters of a control system. In the present paper, the effect of surface fracture depth and orientation on frequency and vibration mode switching of perovskite solar cells-based aircraft wing panels is investigated for the first time. The PSC (perovskite solar cell) panel is made of a polymer substrate reinforced with even-distributed graphene platelets (GPLs) and thin solar layers to enhance the overall electromechanical performance of the whole structure. Equations of motion of the panel are obtained by the Hamilton approach based on the properties of Reddy's third-order shear deformation theory and linear components of the Green-Lagrange strain tensor. The novel aspect is a line-spring model of surface crack by Cartmell et al. adapted to the model of the PSC panel under consideration. The discretization of the established equations of motion for the cracked panel is carried out by the element-free IMLS-Ritz method with mesh-free features. Validation of the obtained solutions is carried out by comparisons with results from previous studies for cracked laminated plates.First, material and geometric properties effect on vibration characteristics of the PSC structure was investigated. Next, partial surface crack depth and orientation angle are examined to show changes in frequency and vibration modes shape of the panel. It is shown, that there is a mode swapping between the second and third modes when transitioning from an intact panel to a cracked one. As the crack length increases, higher-order mode variations become more noticeable, potentially leading to their swapping or the excitation of new ones. Additionally, the findings demonstrate that the surface crack orientation angle significantly impacts the higher-order modes of the panel. When the orientation angle is 90°, the first six modes align with those of the intact plate. Mode shapes and frequency shifts are significant for crack orientation angles between 0° and 90°.

Elucidating Piezoelectricity and Strain in Monolayer MoS2 at the Nanoscale Using Kelvin Probe Force Microscopy.

Strain engineering modifies the optical and electronic properties of atomically thin transition metal dichalcogenides. Highly inhomogeneous strain distributions in two-dimensional materials can be easily realized, enabling control of properties on the nanoscale; however, methods for probing strain on the nanoscale remain challenging. In this work, we characterize inhomogeneously strained monolayer MoS2 via Kelvin probe force microscopy and electrostatic gating, isolating the contributions of strain from other electrostatic effects and enabling the measurement of all components of the two-dimensional strain tensor on length scales less than 100 nm. The combination of these methods is used to calculate the spatial distribution of the electrostatic potential resulting from piezoelectricity, presenting a powerful way to characterize inhomogeneous strain and piezoelectricity that can be extended toward a variety of 2D materials.

Numerical Investigation of Stress-Strain State Effects on Strain Measurements with Fiber Bragg Grating Sensors

This study investigates the behaviour of resonant wavelengths of Fiber Bragg Gratings (FBG) inscribed within optically isotropic fibers under transverse loading, both in free and embedded conditions. A numerical-analytical approach is employed, utilizing the finite element method to calculate strain tensor components in the optical fiber core, followed by an analytical determination of resonant wavelengths and reflected FBG spectrum shape. The research demonstrates the influence of the ratio of host material and optical fiber elastic moduli on the birefringence level in FBG area under transversal loading. Based on analytical model of FBG spectrum simulation the discrepancy between analytically calculated and experimentally recorded resonant wavelength shifts in FBG embedded within isotropic material under varying transverse load levels is demonstrated.

ATOMISTIC AND CONTINUAL DETERMINATION OF STRESS FIELDS AT A CRACK TIP IN ANISOTROPIC ENVIRONMENT WITH CUBIC SYSTEM

The article presents a comparative analysis of stresses associated with a crack tip in an anisotropic medium with cubic symmetry of elastic properties under mixed mode loading conditions, using two different approaches: atomistic and continuum approaches. The atomistic approach is based on the use of the molecular dynamics method implemented in the open source Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). The continuum approach is based on the classical theory of elasticity of anisotropic media, in which the mechanical fields associated with the crack tip are represented using series that generalize the well-known representation of M. Williams to the case of anisotropic media. Using the method of molecular dynamics, a large series of calculations of loading of single-crystal copper and aluminum plates with a face-centered crystal lattice, weakened by a central crack, was carried out using the embedded atom potential (EAM). Molecular dynamics modeling is aimed at determining atomistic stresses and strains near the crack tip. The calculated atomistic stresses were compared with the stress field determined by the continuum theory of elasticity of anisotropic media for crystal lattices with cubic symmetry of elastic properties. A comparative analysis was carried out for the angular dependences of the stress and strain tensor components at different distances from the crack tip for the entire range of mixed strain modes: from perfect normal tension to loadings close to perfect transverse shear. It is established that the fields found on the basis of two fundamentally different approaches (discrete and continuum) are fully consistent with each other. It is demonstrated that the mathematical methods of continuum fracture mechanics can be used to describe stress, strain, and displacement fields at the atomistic level.

Strengthening of the railroad track for high-speed traffic on the Bukhara-Misken section

In the article the work of ballast prism and subgrade is considered, on the basis of analysis two-layer reinforcement from geotextile is offered, also strengthening by water-soluble polymer in conditions of sandy zone on high-speed sections of railroads is suggested. The principle of stationarity of the total potential energy of deformable Lagrangian systems is used to determine the stress-strain state of the earth bed. The dependences between the components of stress and strain tensors characterizing specific models of the medium have been established, and an elastic ground medium with no residual deformation from external load has been considered. And also offer designs of sleepers with concave surface, which will increase the service life and ensure the reliability of its operation on horizontal and vertical curves of railroads. A new fluoroplastic under-rail gasket, which does not change its characteristics during 5 years, contacting with the base of the rail, flat inner support surface, contacting with the sleeper, reduces load vibration from the rolling stock on the sections of high-speed traffic is offered.

CALCULATION OF SHELLS BASED ON THE MIXED FEM VARIANT WITH TENSOR-VECTOR APPROXIMATION OF THE DESIRED VALUES

An algorithm for calculating thin shells based on the mixed finite element method (FEM) in a two-dimensional formulation is proposed. To correctly take into account possible displacements of the shell as a rigid body, the algorithm implements the developed tensor-vector form of the interpolation procedure of the desired values, which were chosen as the tensors of deformations and curvature at the point of the middle surface and the displacement vectors of this point. The discretization element was a quadrangular fragment of the middle surface with unknown nodes in the form of location and their first derivatives, as well as components of strain and curvature tensors among the inner surface at the nodes of the finite element. After minimizing the modified mixed functional, a 36?36 finite element stiffness matrix is formed. In order to verify the developed algorithm, a number of test problems were solved for calculating a fragment of an elliptical cylinder with an analytical solution, as well as for calculating a shell with spring supports that allow the ellipsoidal shell to move as an absolutely rigid body. Analysis of the obtained results showed that when calculating a fragment of an elliptical cylinder, the calculation scheme of which makes it possible to obtain an analytical solution, the numerical values of bending moments and normal stresses completely coincide with the values calculated from the condition of static equilibrium. An analysis of the calculation results for an ellipsoidal shell with spring supports led to the conclusion that the developed tensor-vector interpolation procedure in the mixed FEM version allows one to correctly take into account the displacements of shells as solids and obtain an adequate assessment of their stress-strain state.

Коэффициенты электромагнитных связанностей композита "пьезоэлектрик / феррит" с учетом реальной структуры и начальных напряжений

New improved analytical solutions are obtained for tensors of effective pyro– electro–magneto–elastic properties of a piezoactive composite at initial stress-strain state (resulting from initial loading) that account for real two-point correlation functions of an irregular quasi-periodic random structure. The calculations of effective coefficients of electromagnetic and magnetoelectric coupling of the transversal isotropic unidirectional fibrous composite "PZT-4/ferrite" with an axisymmetric tensor of initial strain of the composite are presented. An increase in the absolute values of the effective coefficients is observed at negative (compressive) values of the initial strain tensor components of the composite.

Three-dimensional configuration of crystal plasticity in stainless steel assessed by high resolution digital image correlation and confocal microscopy

In polycrystalline austenitic stainless steels, a comprehensive three-dimensional configuration of crystal plasticity in relation to microstructure is required to elucidate the deformation behaviors during mechanical loading. In the present work, a combination of backscattered electron image (BSE) based high-resolution digital image correlation (HR-DIC) and laser scanning confocal microscopy (LSCM) was employed to simultaneously investigate in-plane and out-of-plane deformations at the sub-grain scale in 316 stainless steel. The high spatial resolution and precise alignment provided by the three-dimensional deformation assessment enable the identification of specific slip and twinning systems. This is achieved through the minimization of the residual L2 norm between the in-plane components of experimental and theoretical strain tensors. Dominant slip and few mechanical twinning activities are initiated at the early-stage of yielding, and their frequency and intensity increase with the applied load. The hierarchical networks formed by nanotwins, along with the presence of grain and annealing twin boundaries, serve as obstacles for slip glide, leading to substantial twin–slip interaction and grain boundary strengthening. As a consequence of wide mechanical twins formed by parallel nanotwins coalescence and strain localization at the twin boundary, slip transmission occurs, resulting in an effective accommodation of external stress. Furthermore, the consistent observation of out-of-plane roughness, in alignment with twinning Burgers vector predictions, offers valuable insights into the significant deformation gradient occurring in the vicinity of boundaries. This deformation gradient is influenced by the neighboring mechanical twinning activities, indicating strong obstacle to twin transmission and high back-stresses resulting from the highly misoriented grain boundary.

Tensorial stress-plastic strain fields in α - ω Zr mixture, transformation kinetics, and friction in diamond-anvil cell

Various phenomena (phase transformations (PTs), chemical reactions, microstructure evolution, strength, and friction) under high pressures in diamond-anvil cell are strongly affected by fields of stress and plastic strain tensors. However, they could not be measured. Here, we suggest coupled experimental-analytical-computational approaches utilizing synchrotron X-ray diffraction, to solve an inverse problem and find fields of all components of stress and plastic strain tensors and friction rules before, during, and after α-ω PT in strongly plastically predeformed Zr. Results are in good correspondence with each other and experiments. Due to advanced characterization, the minimum pressure for the strain-induced α-ω PT is changed from 1.36 to 2.7 GPa. It is independent of the plastic strain before PT and compression-shear path. The theoretically predicted plastic strain-controlled kinetic equation is verified and quantified. Obtained results open opportunities for developing quantitative high-pressure/stress science, including mechanochemistry, synthesis of new nanostructured materials, geophysics, astrogeology, and tribology.

Numerical modeling of nonlinear deformation processes for shells of medium thickness

When modeling a nonlinear isotropic eight-node finite element, the main kinematic and physical relationships are determined. In particular, isoparametric approximations of the geometry and an unknown displacement increment vector, covariant and contravariant components of basis vectors, metric tensors, strain tensors (Cauchy - Green and Almansi) and true Cauchy stresses in the initial and current configuration are introduced. Next, a variational equation is introduced in the stress rates in the actual configuration without taking into account body forces and the Seth material is considered, where the Almansi strain tensor is used as the finite strain tensor. Linearization of this variational equation, discretization of the obtained relations (stiffness matrix, matrix of geometric stiffness) is carried out. The resulting expressions are written as a system of linear algebraic equations. Several test cases are considered. The problem of bending a strip into a ring is presented. This problem is solved analytically, based on kinematic and physical relationships. Examples of nonlinear deformation of cylindrical and spherical shells are also shown. The method proposed in this paper for constructing a three-dimensional finite element of the nonlinear theory of elasticity, using the Seth material, makes it possible to obtain a special finite element, with which it is quite realistic to calculate the stress state of shells of medium thickness using a single-layer approximation in thickness. The obtained results of test cases demonstrate the operability of the proposed technique.

An integrated methodology to estimate the effective elastic parameters of amorphous TiO2 nanostructured films, combining SEM images, finite element simulations and homogenization techniques

In this paper, a novel experimental-numerical methodology is outlined, which aims at estimating the homogenized elastic properties of thin nanostructured coatings, utilized in a variety of industrial contexts. Focus is posed on amorphous titanium dioxide (TiO2) films, a few micrometer thick, produced through Pulsed Laser Deposition in the form of columnar elements coating a Silicon flat substrate. The proposed inverse strategy rests on digital images acquired by a Scanning Electron Microscope on coated samples, with a nanometer resolution, and consists of the following steps: (i) within a representative volume element, cavities existing in between the nanostructured columns are identified in the raster images by an automatic edge detection procedure; (ii) information concerning the location of the pore borders are vectorized and passed through a Python script to a commercial finite element code; (iii) the homogenized stiffness matrix is reconstructed on the basis of finite element simulations of such a representative volume subjected to elementary loading conditions, each corresponding to a single component of the effective strain tensor. Diverse mechanical models with periodic boundary conditions have been comparatively assessed, fully three dimensional or under plane conditions, assuming for the TiO2 columns in an amorphous state (as corroborated by Raman spectroscopy) an isotropic behaviour, with a priori known Young’s modulus and Poisson’s ratio. The homogenized elastic moduli and coefficients of TiO2 thin films, at varying the deposition process parameter, are estimated by the novel procedure, resulting in an orthotropic, transversely-isotropic behaviour.

Quantitative mapping of strain and displacement fields over HR-TEM and HR-STEM images of crystals with reference to a virtual lattice

A method for the reciprocal space treatment of high-resolution transmission electron microscopy (HR-TEM) and high-resolution scanning transmission electron microscopy (HR-STEM) images has been developed. Named “Absolute strain” (AbStrain), it allows for quantification and mapping of interplanar distances and angles, displacement fields and strain tensor components with reference to a user-defined Bravais lattice and with their corrections from the image distortions specific to HR-TEM and HR-STEM imaging. We provide the corresponding mathematical formalism. AbStrain goes beyond the restriction of the existing method known as geometric phase analysis by enabling direct analysis of the area of interest without the need for reference lattice fringes of a similar crystal structure on the same field of view. In addition, for the case of a crystal composed of two or more types of atoms, each with its own sub-structure constraint, we developed a method named “Relative displacement” for extracting sub-lattice fringes associated to one type of atom and measuring atomic columns displacements associated to each sub-structure with reference to a Bravais lattice or to another sub-structure. The successful application of AbStrain and Relative displacement to HR-STEM images of functional oxide ferroelectric heterostructures is demonstrated.

Nanoscale Mapping of the 3D Strain Tensor in a Germanium Quantum Well Hosting a Functional Spin Qubit Device.

A strained Ge quantum well, grown on a SiGe/Si virtual substrate and hosting two electrostatically defined hole spin qubits, is nondestructively investigated by synchrotron-based scanning X-ray diffraction microscopy to determine all its Bravais lattice parameters. This allows rendering the three-dimensional spatial dependence of the six strain tensor components with a lateral resolution of approximately 50 nm. Two different spatial scales governing the strain field fluctuations in proximity of the qubits are observed at <100 nm and >1 μm, respectively. The short-ranged fluctuations have a typical bandwidth of 2 × 10-4 and can be quantitatively linked to the compressive stressing action of the metal electrodes defining the qubits. By finite element mechanical simulations, it is estimated that this strain fluctuation is increased up to 6 × 10-4 at cryogenic temperature. The longer-ranged fluctuations are of the 10-3 order and are associated with misfit dislocations in the plastically relaxed virtual substrate. From this, energy variations of the light and heavy-hole energy maxima of the order of several 100 μeV and 1 meV are calculated for electrodes and dislocations, respectively. These insights over material-related inhomogeneities may feed into further modeling for optimization and design of large-scale quantum processors manufactured using the mainstream Si-based microelectronics technology.

Numerical modeling of the processes of irreversible deformation of rings under explosive loading

the quasi-static problem is discussed. The equations of the incremental theory of plasticity are used to describe fast deformation. The numerical simulation method is implemented based on the use of the Arbitrary Lagrange-Euler (ALE) approach, in which the sequence of boundary value problems in the current configuration of the solid is solved using the Finite Element Method. An axisymmetric finite element of a triangular cross-section was used for modeling. The main system of equations is presented, which, when using the Finite Element Method, describes the deformation of axisymmetric bodies in the current configuration. At each stage of the calculation in the current configuration, the initial problem is solved numerically using the finite difference method. When the limiting value of the strain tensor components is reached, according to the ALE algorithm, the finite element model is rebuilt with the implementation of new boundary conditions. To build a computational model, the RD program was used, in which a two-dimensional model of a section of an axisymmetric body is surrounded by a mesh of special elements. A presented approach for the implementation of the specified boundary conditions of an axisymmetric body in finite element modeling is discussed. Calculations of irreversible deformation were carried out in a program developed on the basis of using the FEM Creep software package in the case of finite strains. Comparison of the numerical simulation data with the experimental results of other authors, obtained in the investigations of the explosive loading of aluminum rings set on a sphere, is presented. The experimental data are processed and the constants included in the constitutive equation, using the hardening law, are determined. Satisfactory agreement between the calculated and experimentally obtained values of the radial strain of the rings at different times is shown.&#x0D; Key words: numerical simulation, axisymmetric problem, finite strains, FEM, program, explosive loading, aluminum alloy.

Deformation of planar ring mesh under static in-plane tensile loads

A ring mesh is a large-scale manufacturable structure with versatile applications in architecture and for protective systems. However, the static and dynamic numerical simulation of a large-scale ring mesh is a resource-intensive task due to the many nonlinear contact points between the individual rings. To characterize the rigid body behavior of the ring mesh, in this paper, a representative volume element is loaded under different in-plane directions. The Green-Lagrangian strain tensor components are obtained as a result. An implicit one-step algorithm is used for this dynamic relaxation issue, modified by the use of stiff springs. Static convergence positions of a representative volume element are determined in several multiaxial tensile directions. The obtained deformation parameters can be used to simulate large deformations of large-scaled ring meshes.

Mechanics of accelerated strain hardening in harmonic-structure materials

Harmonic-structure (HS) design is one of the most efficient in the family of architected gradient-structure materials recently attracting increasing attention in global material science community due to leading structural performance characteristics. This work studies in situ the mechanics of plastic deformation in HS materials on the example of commercially pure nickel (Ni) during uniaxial tensile loading. It reveals that strain partitioning between ultrafine and coarse grain fractions in HS Ni commences already at early stages of plastic deformation. Slow accumulation of strain in the ultrafine-grain regions leads to the slow consumption of ductility resource, while accelerated accumulation of strain in the coarse-grain regions leads to an accelerated strain hardening. These effects combined with evolving variations of strain tensor components have synergetic effect leading to the unique plastic behaviour and excellent structural performance of HS materials.

High-Resolution Mapping of Strain Partitioning and Relaxation in InGaN/GaN Nanowire Heterostructures.

Growing an In x Ga1− x N/GaN (InGaN/GaN) multi‐quantum well (MQW) heterostructure in nanowire (NW) form is expected to overcome limitations inherent in light‐emitting diodes (LEDs) based on the conventional planar heterostructure. The epitaxial strain induced in InGaN/GaN MQW heterostructure can be relaxed through the sidewalls of NW, which is beneficial to LEDs because a much larger misfit strain with higher indium concentration can be accommodated with reduced piezoelectric polarization fields. The strain relaxation, however, renders highly complex strain distribution within the NW heterostructure. Here the authors show that complementary strain mapping using scanning transmission electron microscopy and dark‐field inline holography can comprehend the strain distribution within the axial In0.3Ga0.7N/GaN MQW heterostructure embedded in GaN NW by providing the strain maps which can cover the entire NW and fine details near the sidewalls. With the quantitative evaluation by 3D finite element modelling, it is confirmed that the observed complex strain distribution is induced by the strain relaxation leading to the strain partitioning between InGaN quantum disk, GaN quantum well, and the surrounding epitaxial GaN shell. The authors further show that the strain maps provide the strain tensor components which are crucial for accurate assessment of the strain‐induced piezoelectric fields in NW LEDs.

Application of Flow Theory Relations for Solving Problems of Steady-State Crack Growth

To represent local displacement fields in the problem of the steady-state growth of a crack, which contains a plate of incompressible material, the strain intensity formula is used in the form of a polynomial of the second degree. The case of plane deformation for an elastoplastic material is considered. The solution is obtained by the method of asymptotic expansions. Numerical analysis is carried out for the first term of the expansion. The aim of the work is the process of obtaining analytical solutions to applied problems of the theory of plasticity: finding the components of stress and strain tensors. The paper considers a variant of the method of asymptotic expansions and its application for the problem of the distribution of the stress-strain state in an elastoplastic specimen with a crack. The method of asymptotic expansions has some advantages over the numerical approach in studying the stress-strain state in the vicinity of a crack. It allows to establish exact quantitative relationships between the radial component, the angle, and the components of the stress and strain tensor. Another advantage of this method is the possibility of compiling the mechanical characteristics of an object at the design stage. A system of differential equations has been developed that contains V0 and its derivatives up to the third order. An example of stress distribution in the vicinity of a crack tip in a steel sample, obtained in a computer system by a numerical method, is given. The deformation diagram has been constructed for the material steel 40. The research results can be used to construct stress and strain fields in the vicinity of a crack, as well as to predict the further direction of crack development.