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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. [ABSTRACT FROM AUTHOR]

Diatomic multilattices are congruences of simple lattices each made out of atoms of two possible chemical species. We here constitutively characterize, in three dimensions, diatomic multilattices for the geometrically and materially linear elastic regime. We give the most generic expression for the energy for (n + 1) diatomic multilattices and characterize explicitly the tensors present in such an expression for all 122 two-color point groups. For the specific case of diatomic 2- and 3-lattices, we delineate how one can solve the shift vector equation in the static case. For cases where the unique solution of the shift vector in terms of the strain tensor is not possible, we give conditions for the existence of solutions based on the standard Fredholm alternative theorem. [ABSTRACT FROM AUTHOR]

This paper presents our contribution to the a posteriori error analysis in 2D and 3D of a semi‐augmented mixed‐primal finite element method previously developed by us to numerically solve double‐diffusive natural convection problem in porous media. The model combines Brinkman‐Navier‐Stokes equations for velocity and pressure coupled to a vector advection‐diffusion equation, representing heat and concentration of a certain substance in a viscous fluid within a porous medium. Strain and pseudo‐stress tensors were introduced to establish scheme solvability and provide a priori error estimates using Raviart‐Thomas elements, piecewise polynomials and Lagrange finite elements. In this work, we derive two reliable residual‐based a posteriori error estimators. The first estimator leverages ellipticity properties, Helmholtz decomposition as well as Clément interpolant and Raviart‐Thomas operator properties for showing reliability; efficiency is guaranteed by inverse inequalities and localization strategies. An alternative estimator is also derived and analyzed for reliability without Helmholtz decomposition. Numerical tests are presented to confirm estimator properties and demonstrate adaptive scheme performance. [ABSTRACT FROM AUTHOR]

The small‐scale‐dependent nonlinear dynamical attributes of microplate‐type laminated piezoelectric systems of harvesting energy subjected to a triangular mechanical actuation are explored in the current exploration. The core of harvesting systems is supposed to be made of nanocomposites reinforced by randomly oriented carbon nanotubes (CNTs) having various degrees of agglomeration coated by piezoelectric layers. In this regard, the microplate‐type systems of harvesting energy are modeled via a quasi‐3D plate theory combined with the modified strain gradient elasticity in the presence of various microsize‐dependent strain gradient tensors. Thereafter, the solution of the developed coupled electromechanical size‐dependent nonlinear problem is obtained numerically by utilizing the meshless collocation approach employing incorporation of the polynomial and radial basis functions to remove any feasible singularity for the associated moment matrices. It comes to the conclusion that by elevating the quantity of CNTs disclosed within the clusters, the significance of the microsize dependency in the achieved voltage enhances from 22.06% to 22.73% toward the simply supported bridge‐type context, and from 33.00% to 33.92% toward the clamped bridge‐type context. Highlights: Development of a microstructural‐dependent quasi‐3D shear flexible energy harvester model.Incorporating the roles of various strain gradient tensors in the nonlinear dynamics of microharvesters.Size‐dependent time‐histories of achieved voltage from triangular mechanical‐loaded microharvesters.Influence of the CNT reinforcement clustering on the nonlinear dynamic performance. [ABSTRACT FROM AUTHOR]

In the framework of damage mechanics, we discuss a new mathematical model that describes the kinetics of the stress–strain state and damage accumulation during material degradation by the mechanism of long-term strength under complex multiaxial stress state. An experimental and theoretical technique is proposed for determination of material parameters and scalar constitutive functions for damaged media based on specially set experiments on laboratory specimens. The results of experimental studies and numerical simulations of short-term high-temperature creep of VT6 titanium alloy under uniaxial and multiaxial loading are presented. Numerical results are compared with the data obtained through experiments. Particular attention is paid to simulating the process of unsteady creep for complex deformation modes, accompanied by rotation of main areas of stress, strain and creep strain tensors. It is shown that the developed version of the constitutive relations of the damaged media enables us to describe the processes of unsteady creep and long-term strength of structural alloys under multiaxial stress with the accuracy sufficient for engineering calculations. [ABSTRACT FROM AUTHOR]

In this study, we present alternative viscoelastic models for materials developing large strains. The novelty of these models is the definition of isochoric strain components, from which we numerically calculate isochoric strain rates. The proposed models differ from the most usual frameworks present in the literature, namely Köner–Lee decomposition models, hereditary integral models and thermodynamically consistent models. The last mentioned framework also uses the Flory's multiplicative strain decomposition, but not in the same way proposed here. Our models are developed for solid mechanics and their time evolution is done through finite differences, simplifying the algorithmic tangent viscoelastic constitutive tensor and the consideration of strain rate-dependent viscosity. Using simple examples, we show that using finite difference for isochoric strain rate evolution does not introduce volumetric changes in purely distortional viscoelastic situations and vice versa. We also present examples related to experimental results, including nonlinear viscoelastic response. Finally, we present examples that show how viscoelastic materials with instantaneous response may not provide satisfactory damping for some structural sets and a comparison of our models with important classical models. [ABSTRACT FROM AUTHOR]

The aim of this paper is to decompose the displacements of thin-walled beams with rectangular cross-section. The decomposition is accompanied by estimates of all its terms with respect to the norm of the strain tensor. Korn's inequality is also given. [ABSTRACT FROM AUTHOR]

Bulk phosphorus oxynitride (PON), isoelectronic with SiO2, is confirmed to have many polymorphs as the latter. However, its 2D counterparts have not been studied. In this work, crystal structure exploration and property study are performed on 2D PON with the help of first‐principle calculation. Three 2D PON sheets are found with thermal, dynamical, and mechanical stability and all of them can be viewed as oxygen‐doped known 2D γ phosphorus nitride (γ‐PN). In the studies, it is indicated that they are indirect bandgap materials with gap values larger than 3 eV. The situations of orbital hybridization are discussed and their bonding properties are analyzed. Infrared vibrational spectra of these 2D materials are simulated and the vibrational modes at the Brillouin center are assigned by the factor group analysis. The reasons for the disappearance of some infrared peaks in the simulated spectra are revealed based on the calculated Born effective charges and vibrational eigenvectors. Their piezoelectric stress tensors are computed via density‐functional perturbation method and the piezoelectric strain tensors are also deduced via the calculated elastic constants. In the results, it is uncovered that these 2D PON sheets have considerably greater piezoelectric coefficients than the primitive 2D γ‐PN, which make them promising for nanopiezoelectric use. [ABSTRACT FROM AUTHOR]

A framework of the topology optimization incorporated with the phase field method considering the interfacial damage for optimizing the fracture resistance of inclusion-matrix composites is presented. The topology optimization was performed to redistribute the inclusion phase in order to reduce its volume while keeping the fracture resistance value of the initial design unchanged. The phase field method uses two scalar phase field variables: one is for the bulk crack and the other is for the interfacial crack. The decomposition of the strain tensor into compression and tension parts was incorporated into this phase field method to improve the mechanical behaviors of the materials. These compression and tension strain parts are orthogonal in the context of the inner product in which the tensor of elastic stiffness behaves as a metric. Moreover, in the simulation process, an investigation of the effects of the interfacial parameters on the numerical results was discussed. Through the obtained results, the method proposed is demonstrated to be accurate and efficient in eliminating spurious effects and singularity points on the behavior curves in the damage process. [ABSTRACT FROM AUTHOR]

An original approach for the determination of unit cell parameter temperature dependence using a conventional single-crystal X-ray diffractometer is proposed. The approach is based on the calibration of the goniometer equatorial circle segment using the position of several Kα1/Kα2 doublets from reference single crystals. It allows for the refinement of lattice parameters of single crystals with a relative error of less than 10−4. Its capabilities are demonstrated using an example of tetragonal single crystals. The space group, crystal structure, and composition of t-Ag0.8Li0.2InSe2 were determined using single-crystal X-ray diffraction analysis of three small (linear sizes ∼0.1 mm) fragments from a larger (linear sizes up to 10 mm) single-crystal grown using the Bridgman method. The dynamics of the crystal structure changes in the temperature range of 90–500 K were studied. The obvious anisotropy of t-Ag0.8Li0.2InSe2 thermal behavior was shown with an overall increase in the volume of the tetragonal unit cell by 0.51%—parameter a increased by 0.34, while parameter c decreased by 0.17%. The nonlinear dependence of the thermal expansion tensor elements on temperature was determined: α 11 × 106 = −4.28 + 0.08T − 1.95 × 10−4T2 and α 33 × 106 = −5.59 + 0.01T − 1.97 × 10−5T2. [ABSTRACT FROM AUTHOR]

Here, higher order models of elastic shells of revolution are developed using the variational principle of virtual power for 3-D equations of the linear theory of elasticity and generalized series in the coordinates of the shell thickness. Following the Carrera Unified Formulation (CUF), the stress and strain tensors, as well as the displacement vector, were expanded into series in terms of the coordinates of the shell thickness. As a result, all the equations of the theory of elasticity were transformed into the corresponding equations for the expansion coefficients in a series in terms of the coordinates of the shell thickness. All equations for shells of revolution of higher order are developed and presented here for cases whose middle surfaces can be represented analytically. The resulting equations can be used for theoretical analysis and calculation of the stress-strain state, as well as for modeling thin-walled structures used in science, engineering, and technology. [ABSTRACT FROM AUTHOR]

As the high-frequency analogue to field-scale earthquakes, acoustic emissions (AEs) provide a valuable complement to study rock deformation mechanisms. During the load-stepping creep experiments with CO2-saturated water injection into a basaltic sample from Carbfix site in Iceland, 8791 AE events are detected by at least one of the seven piezoelectric sensors. Here, we apply a cross-correlation-based source imaging method, called geometric-mean reverse-time migration (GmRTM) to locate those AE events. Besides the attractive picking-free feature shared with other waveform-based methods (e.g. time-reversal imaging), GmRTM is advantageous in generating high-resolution source images with reduced imaging artefacts, especially for experiments with relatively sparse receivers. In general, the imaged AE locations are found to be scattered across the sample, suggesting a complicated fracture network rather than a well-defined major shear fracture plane, in agreement with X-ray computed tomography imaging results after retrieval of samples from the deformation apparatus. Clustering the events in space and time using the nearest-neighbour approach revealed a group of 'repeaters', which are spatially co-located over an elongated period of time and likely indicate crack, or shear band growth. Furthermore, we select 2196 AE events with high signal-to-noise-ratio (SNR) and conduct moment tensor estimation using the adjoint (backpropagated) strain tensor fields at the locations of AE sources. The resulting AE locations and focal mechanisms support our previously assertion that creep of basalt at the experimental conditions is accommodated dominantly by distributed microcracking. [ABSTRACT FROM AUTHOR]

Strain‐induced crystallization (SIC) in natural rubber (NR) near crack tips significantly enhances crack growth resistance, but understanding the interplay between local strain field and crystallization remains challenging due to confined and heterogeneous characteristics. Using micro‐scale digital image correlation (DIC) and scanning wide‐angle X‐ray diffraction (WAXD, with a narrow 10 µm square beam), this study maps local strain tensor properties and SIC in the vicinity of the crack tip and its peripheral zone (≈3 mm × 1 mm area). The analysis reveals a significant correlation between these properties. In the peripheral zone, there is a noticeable deviation of both the principal strain axis and the crystal orientation from the crack opening direction. These deviations are linearly correlated, which indicates that shear strain plays a significant role in determining the crystal orientation. Crucially, the maximum tensile component in the tensor of local principal strains predominantly dictates local crystallinity. This simplicity is attributed to the limited variation in types of deformation within the SIC region, with corresponding to deformations falling between planar and uniaxial stretching. These findings pave the way for predicting crystallinity distribution using solely strain field data, offering valuable insights into the role of SIC in enhancing the crack growth resistance of NR. [ABSTRACT FROM AUTHOR]

This work deals with the modeling of solid continua undergoing incompatible deformations due to the presence of microscopic defects like dislocations. Our approach relies on a geometrical description of the medium by the strain tensor and the representation of internal efforts using zeroth- and second-order strain gradients in an infinitesimal framework. At the same time, energetic arguments allow to monitor the corresponding moduli. We provide mathematical and numerical results to support these ideas in the framework of isotropic constitutive laws. [ABSTRACT FROM AUTHOR]

Shear stresses for a two-dimensional (2D) beam are calculated modifying the classical method developed by Jouravski, for the case the linearized strain tensor is assumed to be a nonlinear function of the Cauchy stresses. Two problems are studied, namely, the case of a cantilever beam with a point load on its free edge (considering a rectangular and a circular cross-section) and the three-point flexural test for a beam of rectangular cross-section. Numerical results are obtained for the particular case of a bimodular constitutive model for rock, and the results for the shear stresses are compared with the predictions of the classical theory of strength of materials for such problems, assuming a linearized elastic body. [ABSTRACT FROM AUTHOR]

The relationship between stress and strain tensors in soil is described by the soil constitutive equation, which depends on the soil type and deformation conditions. The development of various constitutive models has enabled a better understanding of the macromechanical properties of soil. One such model is hypoplasticity, which was discovered more than three decades ago. The purpose of this study is to determine whether hypoplasticity could accurately represent the behavior of Danube sand, a specific type of sand. The researchers conducted laboratory measurements to obtain the eight basic hypoplastic parameters of Danube sand and employed the intergranular strain concept as an extension of hypoplasticity to achieve a more precise material behavior. All the parameters were then utilized to simulate the cyclic triaxial test using the SoilTest Module of PLAXIS. The results showed that the hypoplastic constitutive simulation model for Danube sand was capable of making relatively accurate predictions for accumulative settlement. [ABSTRACT FROM AUTHOR]

The applications of nanoelectromechanical systems integrating simultaneously the mechanical and electrical functionalities at nanoscale have been gotten wider in the last decade. The main objective of this exploration is to analyze the small scale‐dependent nonlinear dynamical feedback of multilayer energy nanoharvesters containing a graded porous passive core coated by piezoelectric facesheets under a sinusoidal impulsive external actuation. For this purpose, the technique of meshless collocation is constructed at the same time the nonlocal stress and strain gradient tensors without any necessary background meshes and integration procedure. It is found that by taking the nonlocal stress tensor into account, the average of extremum points of maximum lateral deflection increases from 8.6375 to 9.6924nm, and from 4.3718 to 4.8952nm for simply supported and clamped multilayer nanoharvesters, respectively. On the contrary, through considering the strain gradient tensor, the average of extremum values of the maximum lateral deflection reduces from 8.6375 to 7.4999nm for the simply supported nanoharvester, and from 4.3718 to 3.8101nm for the clamped one. Highlights: Development a third‐order shear flexible nonlocal strain gradient (NSG)‐based plate‐type nanoharvester modelIncorporating the role of nonlocal stress and strain gradient tensors in nonlinear dynamics of nanoharvestersTime‐histories of achieved voltage of sinusoidal impulsive loaded plate‐type nanoharvestersInfluence of porosity on the NSG‐based nonlinear dynamic performance. [ABSTRACT FROM AUTHOR]

This work introduces a new family of strain tensors based on the hyperbolic sine function: The Generalized Hyperbolic Sine (GHS). This family is obtained by using any given strain measure as an argument in a normalized hyperbolic sine function. Particularly, this paper adopts the Seth-Hill family as an argument and shows how some drawbacks of the classical strain measures can be overcome by means of the proposed strain family. A broad analytical study of pure deformation modes (simple axial extension, equi-biaxial loading and simple shear) is presented to show the behavior of the proposed strain family and investigate the physical coherence of their responses. Obtained results are promising as the GHS family proved itself capable of enhancing the physical behavior of the Seth-Hill strain measures and providing versatility with the addition of just one material constant with clear physical meaning. Materials investigated in the article are isotropic and homogeneous. The proposed Hookean-type hyperelastic models can be applied for both incompressible and compressible materials without additional strain energy density function changes. [ABSTRACT FROM AUTHOR]

The effect of atomic relaxations on the temperature-dependent elastic constants (TDECs) is usually taken into account at zero temperature by the minimization of the total energy at each strain. In this paper, we investigate the order of magnitude of this approximation on a paradigmatic example: the C44 elastic constant of diamond and zincblende materials. We estimate the effect of finite-temperature atomic relaxations within the quasi-harmonic approximation by computing ab initio the internal strain tensor from the second derivatives of the Helmholtz free-energy with respect to strain and atomic displacements. We apply our approach to Si and BAs and find a visible difference between the softening of the TDECs computed with the zero-temperature and finite-temperature atomic relaxations. In Si, the softening of C44 passes from 8.6% to 4.5%, between T = 0 K and T = 1200 K. In BAs, it passes from 8% to 7%, in the same range of temperatures. Finally, from the computed elastic constant corrections, we derive the temperature-dependent Kleinman parameter, which is usually measured in experiments. [ABSTRACT FROM AUTHOR]

Magnetoelasticity theories describe the behavior of solids in response to the application of a magnetic field. This paper presents a new thermodynamically consistent constitutive theory for simulating nonlinear responses of solids for which the strain energy density depends on the deformation gradient through the Biot strain tensor and specializes the theory for application to the problem of finite deformation of a circular cylindrical tube. The derived theory appears to be attractive for applications as it provides a relatively simple mathematical expression for modeling nonlinear magneto-mechanical responses of solids. For the considered problem, it is obtained that a higher modified pressure is required to maintain a specified azimuthal stretch when a magnetic field is applied. Finally, and among other things, it is observed that the influence of the magnetic field can either decline or progressively enhance with increasing azimuthal stretch. [ABSTRACT FROM AUTHOR]

Here, higher order models of elastic composite multilayer shells of revolution are developed using the variational principle of virtual power for the 3-D linear anisotropic theory of elasticity and generalized series in the shell thickness coordinates. Following the Unified Carrera Formula (CUF), the stress and strain tensors, as well as the displacement vector, were expanded into series in terms of the coordinates of the shell thickness. The higher-order cylindrical shell supported on the edges under axisymmetric loading, is considered and solved analytically using a Navier close form solution method. Also, composite axisymmetric circular plated as well as parabolic, hyperbolic and pseudo-spheric shell fixed ate the ends are considered. Numerical calculations were performed using the computer algebra software Mathematica. The resulting equations can be used for theoretical analysis and calculation of the stress-strain state, as well as for modeling thin-walled structures used in science, engineering, and technology. The results of calculation can be used as benchmark examples for finite element analysis of the higher order elastic shells. [ABSTRACT FROM AUTHOR]

The combination of InSAR and GNSS data can provide more accurate measurements of coseismic 3D surface deformations, which are crucial in determining earthquake source parameters. Currently, the methods for inverting coseismic 3D surface deformations by integrating GNSS and InSAR data using the strain model (SM) (e.g., Simultaneous and integrated strain tensor estimation from geodetic measurements to obtain 3D displacement maps (SISTEM), SM-VCE) simply select GNSS points based on a certain number or distance, without considering the weight influence of the spatial distribution of G NSS points on the InSAR and GNSS data fusion results. To address this issue, this paper proposes a new method that uses Voronoi diagrams analyze the contribution of GNSS points to the deformation at target points. In this paper, GNSS points with high contribution can be used to construct observation value equations and optimize the internal weight of GNSS data. The proposed approach characterizes the contribution of GNSS data to target points by analyzing the changes in the Delaunay triangulation area formed before and after the target points are added to the GNSS data. Simulation experiments reveal that the proposed method outperforms conventional methods in accuracy and completeness. The proposed method is then applied to map the coseismic 3D surface deformations of the 2020 Mw6.5 Monte Cristo Range earthquake and find that the maximum deformation in the east–west, north–south, and vertical directions is about 13 cm, 11 cm, and 24 cm, respectively. [ABSTRACT FROM AUTHOR]

The problem statement is given and the algorithm for solving problems of plasticity for axisymmetric bodies made of isotropic materials with properties depending on a stress mode and plastic loosening is described. The constitutive equations of the theory of deformation processes along the trajectories of small curvature containing two nonlinear dependences are used. One of them implements the relationship between the first invariants of stress and strain tensors, and the other one implements the relationship between the second invariants of the corresponding deviators. The effect of the dependence of deformation curves on the stress mode and the plastic change in the bulk strain on the results of calculating the stress-strain state of the shell is studied. [ABSTRACT FROM AUTHOR]

• The connection between microstructural vectors and strain tensor under small elastic deformation is presented. • The evolution of microstructural vectors can be converted to deviatoric strain tensor. • Rate of inelastic deformation in microstructural vectors can be calibrated by the Hill's potential. • The rate of inelastic deformation from microstructural vectors and Hill's function, although not identical, exhibited similar trends. • Cup drawing simulation shows that the proposed equations can be practically used in metal forming analysis. The dissipation inequality provides a useful method for specifying constitutive equations of plastic deformation. There are two main ways to use the dissipation inequality to determine the plastic deformation path: by utilizing a plastic potential with the normality rule, or by not utilizing the plastic potential to determine the plastic flow according to the elastic distortion of the material coordinate axis (microstructural vectors). While both methods satisfy the inequality condition and predict experimental results well, they have been developed separately without discussion on relationship. This study presents a physics-based mathematical discussion on the relative dependence between the two methods of determining plastic flow and their point of connection, using Hill's quadratic plastic potential. For this, the conversion of the distortional length and angle changes in microstructural vectors to deviatoric strain tensor under small elastic deformation conditions is mathematically demonstrated. Using these equations, the rate of inelastic deformation derived from microstructural vectors exhibited a similar form to that derived from Hill's quadratic potential. Moreover, the model parameters of the microstructural vectors can be calibrated using the Hill's function. The plastic flow derived from the microstructural vectors and Hill's function exhibited a remarkably similar trend under the assumption of small elastic deformation, although not identical. The newly defined relationships were implemented numerically and validated through simulations. The parameters of the microstructural vector theory derived from Hill's plastic potential can be utilized in materials science and engineering. [ABSTRACT FROM AUTHOR]

Microcracks or micropores in rocks cause the elastic moduli to change with the applied strain owing to the nonlinear elasticity of the geomaterial, which causes temporal changes in the seismic wave velocity. Thus, variations in seismic wave velocity can be used as a proxy for understanding the strain or stress variations in the crust, which are crucial for figuring out the dynamics of the fault zones and volcanic domains. According to the theory of nonlinear elasticity, the second- and third-order elastic constants and strain tensors contribute to the strain derivative of seismic wave velocity changes. Although laboratory experiments have estimated third-order elastic constants for rock samples, the in situ values of those constants for the crust are difficult to obtain. In this study, seismic velocity changes were investigated in different directions of tidal deformations to provide constraints on the third-order elastic constants in the shallow crust by applying a seismic interferometry method to ambient noise records. We observed that negative velocity changes were of larger magnitude in the station-pair direction parallel to the tidal strain's direction. Nonlinear elasticity in shallow crust may cause anisotropic velocity variations in response to tidal deformations. Our results highlight the use of velocity change measurements in different directions of tidal strain to constrain nonlinear elastic parameters on a field scale. [ABSTRACT FROM AUTHOR]

In this paper we consider the Navier–Stokes–Brinkman equations, which constitute one of the most common nonlinear models utilized to simulate viscous fluids through porous media, and propose and analyze a Banach spaces-based approach yielding new mixed finite element methods for its numerical solution. In addition to the velocity and pressure, the strain rate tensor, the vorticity, and the stress tensor are introduced as auxiliary unknowns, and then the incompressibility condition is used to eliminate the pressure, which is computed afterwards by a postprocessing formula depending on the stress and the velocity. The resulting continuous formulation becomes a nonlinear perturbation of, in turn, a perturbed saddle point linear system, which is then rewritten as an equivalent fixed-point equation whose operator involved maps the velocity space into itself. The well-posedness of it is then analyzed by applying the classical Banach fixed point theorem, along with a smallness assumption on the data, the Babuška–Brezzi theory in Banach spaces, and a slight variant of a recently obtained solvability result for perturbed saddle point formulations in Banach spaces as well. The resulting Galerkin scheme is momentum-conservative. Its unique solvability is analyzed, under suitable hypotheses on the finite element subspaces, using a similar fixed-point strategy as in the continuous problem. A priori error estimates are rigorously derived, including also that for the pressure. We show that PEERS and AFW elements for the stress, the velocity, and the rotation, together with piecewise polynomials of a proper degree for the strain rate tensor, yield stable discrete schemes. Then, the approximation properties of these subspaces and the Céa estimate imply the respective rates of convergence. Finally, we include two and three dimensional numerical experiments that serve to corroborate the theoretical findings, and these tests illustrate the performance of the proposed mixed finite element methods. [ABSTRACT FROM AUTHOR]

Cylindrically bent diamond single-crystal plates have a great potential for creating energy dispersive spectrometers and focusing crystal monochromators. When they are designed, it is necessary to take into account the significant stresses that appear on bending the plates. The strain tensor and the elastic stress fields in a cylindrically bent single-crystal (110) diamond plate are calculated. The calculations are based on experimental data obtained by local Laue diffraction. The calculation results can be used to design new X-ray optical devices with the ability to control their parameters. [ABSTRACT FROM AUTHOR]

Strain-based Dynamics (SBD) (Müller et al., 2014) is a well-known method to simulate the deformable models by using Green's strain tensor within the Position-based Dynamics (PBD) algorithm. Similar to the other position-based constraints, SBD performs stable and robust. However, when we transform the base algorithm from PBD to Extended Position-based Dynamics (XPBD), SBD suffers from the cumbersome parameter adjustments and uncontrollable numerical dissipation, which generates highly stiff results. To overcome these limitations, we propose Exponential Strain-based Dynamics (ESBD) which reformulates the Green's strain tensor with an exponential treatment. Our proposed method is less sensitive to the increased numerical dissipation, so it provides vivid simulation of deformable objects with fewer parameter tunings. In order to present the advantages, we compare our method with SBD by using the XPBD method and other well-known techniques. We observe that ESBD improves the hyperelastic behavior of the Green's strain tensor as a projected constraint without altering the underlying XPBD algorithm. Therefore, it is possible to adapt ESBD to the other existing numerical integration and iteration methods. Our proposed method is stable, straightforward to follow and produces high quality dynamic simulations. [Display omitted] • A novel exponential-based formulation of Green's strain tensor is introduced. • Exp. version of the mod. Green's strain tensor by (Müller et al., 2014) is provided. • Strain-based Dynamics (Müller et al., 2014) is adapted to the XPBD algorithm. • Our method produces vivid visual results with fewer parameter tunings. [ABSTRACT FROM AUTHOR]

A new class of constitutive equation is proposed for isotropic thermoelastic solids, wherein the Hencky strain tensor is assumed to be a function of the Cauchy stress tensor, via a Gibbs potential. The solid is assumed to be incompressible in the referential state, but the volume can change due to differences in the temperature relative to a reference temperature. The change in volume only depends on temperature. Some restrictions are found for the Gibbs potential, resulting in a constitutive equation for isotropic solids, wherein the volume depends on temperature. Using the resulting constitutive equation, some boundary value problems are studied, considering some relatively simple distributions for the temperature, deformations and stresses. [ABSTRACT FROM AUTHOR]

A non-ordinary state decomposition method is proposed for the bond shear form of three-body potential peridynamics (PD) model. To accomplish this formulation, the effective relative displacement between two points is first decomposed into the volumetric and deviatoric parts based on the definition of dilatation. Starting from this, we give decomposition forms of the elementary deformations and strain energy density of this PD model and finally express the PD force function in the form of the sum of dilatational and distortional parts, which enable the PD model to clearly reflect material response dependence on volume changes and shape distorsion. Combined with the computational scheme of strain and stress tensors presented in the paper, the proposed method is validated numerically. The study permits the possibility of introducing plastic incompressibility for the three-body potential PD model. [ABSTRACT FROM AUTHOR]

Diffraction‐based tomographic strain tensor reconstruction problems in which a strain tensor field is determined from measurements made in different crystallographic directions are considered in the context of sparse matrix algebra. Previous work has shown that the estimation of the crystal elastic strain field can be cast as a linear regression problem featuring a computationally involved assembly of a system matrix forward operator. This operator models the perturbation in diffraction signal as a function of spatial strain tensor state. The structure of this system matrix is analysed and a block‐partitioned factorization is derived that reveals the forward operator as a sum of weighted scalar projection operators. Moreover, the factorization method is generalized for another diffraction model in which strain and orientation are coupled and can be reconstructed jointly. The proposed block‐partitioned factorization method provides a bridge to classical absorption tomography and allows exploitation of standard tomographic ray‐tracing libraries for implementation of the forward operator and its adjoint. Consequently, RAM‐efficient, GPU‐accelerated, on‐the‐fly strain/orientation tensor reconstruction is made possible, paving the way for higher spatial resolution studies of intragranular deformation. [ABSTRACT FROM AUTHOR]

The current investigation intends, for the first time, to devise a three-dimensional plate model incorporating nonlinear Green–Lagrange strain tensor to analyze geometrical nonlinear oscillations of large-deflected functionally graded carbon nanotube (FG-CNT)-reinforced nanocomposite plates with dissimilar geometries and inconstant thickness. With the aid of isogeometric method and using desired polynomial type of B-spline basis functions, geometrical description in conjunction with large lateral deflection was accurately taken into consideration. Four different patterns of CNT reinforcement decorations were exploited, and their effectual mechanical characteristics were extracted using an extensive form of mixture rule. Additionally, it was assumed that plate thickness varied based on three different ventures including linear, convex, and concave ones. It was observed that by changing thickness variability venture from convex to concave, the role of CNT volume fraction on the nonlinear three-dimensional free oscillation feedback of FG-CNT-reinforced nanocomposite plates became more significant. This anticipation was duplicated for both types of plate boundary supports. Furthermore, it was shown that by changing thickness variability venture from linear to convex, the value of three-dimensional nonlinear frequency was reduced. However, by changing thickness variability venture from linear to concave, the three-dimensional nonlinear frequency of the system was increased. [ABSTRACT FROM AUTHOR]

We introduce a new (HHE) family of Hooke-like isotropic hyperelastic material models associated with the Doyle–Ericksen family of strain tensors with Eulerian forms of constitutive relations (CRs). This family contains the well-known Hencky and Simo–Pister (modified neo-Hookean) isotropic hyperelastic material models. The main feature of the new family of material models is that the rate counterparts of CRs for material models from this family are simultaneously CRs for Hooke-like isotropic hypoelastic-type material models based on Eulerian continuous strain-consistent convective stress rates. Models from the new family extend the only previously known Hooke-like isotropic hypoelastic material model based on the corotational logarithmic stress rate with Hooke-like hyperelastic counterpart to an infinite number of such models based on non-corotational stress rates. In addition, we develop unified Eulerian forms of CRs and specific potential strain energies for the known families of Hill (HLIH) and Kellermann–Attard (K–A) Hooke-like isotropic hyperelastic material models. For all three families (HHE, HLIH, and K–A) of material models, new explicit basis-free (eigenprojection based) expressions for the fourth-order elasticity tensors with full symmetry are obtained. Expressions for the components of the Cauchy stress tensor versus axial stretch in the simple elongation problem are derived, and their plots for integer values of the parameter n = ± 2 , ± 1 , 0 generating material models from the families considered are constructed. From an analysis of these plots, it is concluded that the Simo–Pister isotropic hyperelastic material model is the best model among those considered. [ABSTRACT FROM AUTHOR]

The present work had evaluated the effect of cryogenic treatment (233 K) on the degradation of polymeric biomaterial using a numerical model. The study on effect of cryogenic temperature on mechanical properties of cell-seeded biomaterials is very limited. However, no study had reported material degradation evaluation. Different structures of silk-fibroin-poly-electrolyte complex (SFPEC) scaffolds had been designed by varying hole distance and hole diameter, with reference to existing literature. The size of scaffolds were maintained at 5 × 5 mm2. Current study evaluates the effect of cryogenic temperature on mechanical properties (corelated to degradation) of scaffold. Six parameters related to scaffold degradation: heat transfer, deformation gradient, stress, strain, strain tensor, and displacement gradient were analyzed for three different cooling rates (− 5 K/min, − 2 K/min, and − 1 K/min). Scaffold degradation had been evaluated in the presence of water and four different concentrations of cryoprotectant solution. Heat distribution at various points (points_base, point_wall and point_core) on the region of interest (ROI) was found similar for different cooling rates of the system. Thermal stress was found developing proportional to cooling rate, which leads to minimal variation in thermal stress over time. Strain tensor was found gradually decreasing due to attenuating response of deformation gradient. In addition to that, dipping down of cryogenic temperature had prohibited the movement of molecules in the crystalline structure which had restricting the displacement gradient. It was found that uniform distribution of desired heat at different cooling rates has the ability to minimize the responses of other scaffold degradation parameters. It was found that the rates of change in stress, strain, and strain tensor were minimal at different concentrations of cryoprotectant. The present study had predicted the degradation behavior of PEC scaffold under cryogenic temperature on the basis of explicit mechanical properties. [ABSTRACT FROM AUTHOR]

Geosynthetics are commonly used in many civil engineering applications. At the same time, intensive development of these materials is being carried out. Therefore, it becomes more and more important to correctly determine the mechanical parameters of geosynthetics in laboratory tests. This is especially important for more complicated hexagonal geogrids. This article presents the results of the laboratory research carried out to evaluate the junction efficiency of hexagonal geogrid under static load by analyzing deformations of geogrid junctions and ribs. A new approach to determination the efficiency is given by the newly defined strain junction efficiency coefficient based on the true principal strains. In the laboratory tests the digital image correlation (DIC) method was used. This method allowed for the determination of displacements and strains of geogrid specimens in any direction and at any point. The results of the conducted research and analysis indicate that the proposed innovative junction effectiveness assessment method enables a better presentation of the geogrid performance at different levels of tensile load. In addition to the parameters given in the technical approval for this material, the proposed new strain junction efficiency coefficient can be an effective parameter for evaluating hexagonal geogrid performance at different strain levels. • Proper evaluation of the geosynthetic performance is an important research issue. • The ability to determine strain tensors for the entire surface of the specimen is a great advantage of the DIC method. • The newly proposed SJEC n coefficient well describes the performance of the geogrid junction at different load levels. [ABSTRACT FROM AUTHOR]

Based on the spectral decomposition of the strain tensor, the tension‐compression asymmetry (TCA) in a gradient‐extended two‐surface damage‐plasticity model is considered in this study. More specifically, the gradient extension is applied in the current model as a suitable damage regularization technique to prevent pathological mesh dependence issues. Furthermore, plasticity and damage are treated as distinct but strongly coupled dissipative mechanisms by using two different types of damage and yield criteria as well as the related loading/unloading conditions, respectively. Finally, a specific material parameter htc$h_{tc}$ is adopted to control the degree of TCA which results in a unique damage evolution in compression. [ABSTRACT FROM AUTHOR]

Plane ring meshes can be produced in large dimensions for various applications, for example, in architecture or protective systems. The numerical simulation of all individual rings of ring meshes is very costly, which is why methods to abstract the structure have been sought in the past. In this paper, a method for determining the strain parameters of a representative ring mesh section under in‐plane tensile loads is presented. Methods of rigid body analysis and representative volume elements with periodic boundary conditions are used. Green Lagrangian strain tensors are determined for both an exemplary representative ring mesh section under different tensile load cases and an experimental ring mesh under one load case. The comparison of the Green Lagrangian strain tensors of the ring meshes between numerical simulations and experiments shows a very good agreement. [ABSTRACT FROM AUTHOR]

In the differential geometric formulation of nonlinear elasticity, the strain tensor can intrinsically be regarded as a measure of change of the metric of the Riemannian manifold that embodies the continuum undergoing a deformation. Here, this classical concept is revisited and complemented with two newly defined higher order deformation measures, namely the disfigurement and flexure tensors, which respectively measure the change of the Levi-Civita connexion and the Riemann curvature of the continuum. The strain, disfigurement and flexure tensors can, thereby, roughly indicate the alteration of the geometry of a deforming body in the zeroth, first and second orders, respectively. In this light, one may predict that the disfigurement and flexure tensors should be related to the first and second covariant derivatives of the strain tensor. While this is true, the exact strain-disfigurement-flexure relations, as derived in this paper, are found to be involved and non-trivial, offering a deeper understanding of the deformation. In our discussions, distance-preserving, geodesic-preserving and curvature-preserving deformations are studied as three special cases to help better interpret the above-mentioned deformation measures. Some related results have also been found as by-products that might be of current interest or prove useful in future investigations. [ABSTRACT FROM AUTHOR]

For monoatomic 2-lattices, we examine what kind of solutions for the shift vector a recently proposed (Sfyris in Math Mech Sol 28:973–988, 2023) linear theory predicts. We compare our results with that of Fadda and Zanzotto (Acta Cryst 57:492–506, 2001), Fadda and Zanzotto (Int J Nonl Mech 36:527–547, 2001), Ericksen (J Elast 55:201–218, 1999) and Ericksen (Arch Rat Mech Anal 148:145–178, 1999) taken from a fundamentally different approach. For monoatomic 3-lattices, we find conditions for ruling the shift vectors out of the analysis. For cases where the unique solution of the shift vector in terms of the strain tensor is not possible, we give conditions for the existence of solutions based on the standard Fredholm alternative theorem. Our analysis, essentially, applies a recently proposed general continuum linear theory in low-dimensional multilattices. Simultaneously, is targets to give specific meaning and tractable calculations to theories of multilattices that many issues are only studied at the theoretical level. [ABSTRACT FROM AUTHOR]

In this paper, non-linear free vibration analysis of nano-beam has been studied. The non-local strain gradient theory and curvature tensor are used to show the size effect. The length scale parameter expresses the effect of strain gradient tensor in the non-local strain gradient theory. However, the aim of this article is to show the simultaneous effect of curvature and strain gradient tensors in non-linear vibration of functionally graded porous nano-beams. The effect of curvature tensor is demonstrated with the curvature tensor dependent parameter. Considering non-linear Von Kármán strains and Euler–Bernoulli beam theory, the governing vibrational equation of FG porous nano-beams are derived using Hamilton's principle in the presence of strain gradient and curvature tensors simultaneously. The non-linear differential equation is extracted by using Galerkin's method and the non-linear natural frequency of nano-beam is obtained according to Hamiltonian approach. Results represent the simultaneous effects of the length scale and curvature tensor dependent parameters on dimensionless non-linear natural frequencies. Also effects of different parameters such as non-local parameter, length scale parameter, porosity volume index, and power-law index are discussed in the presence and absence of the curvature tensor dependent parameter. Also, the beginning points of stiffness-hardening and stiffness-softening of nano-beam are always constant values in the non-local strain gradient theory, whereas considering the curvature tensor changes the beginning points of stiffness-hardening and stiffness-softening. The results are also compared with previous researches for validation. [ABSTRACT FROM AUTHOR]

In this paper, we propose a phase-field model for electro-mechanical fracture in brittle materials and provide an open-source implementation of the proposed model using a newly developed finite element (FE) toolbox, Gridap in Julia. Here, we have considered electric potential as an additional kinematic descriptor along with the displacement and phase-field variable. Using the virtual power principle, we have derived force balances for the electro-mechanical forces and a force balance associated with the damage. To incorporate the strong electro-mechanical coupling effect into the model, we have considered the Helmholtz free energy as a function of the electric field vector, strain tensor, phase-field variable, and the gradient of the phase-field variable. The proposed model ensures that the constitutive relations of the thermodynamic fluxes are on the satisfaction of the thermodynamic laws and can readily accommodate dissipative energy effects whenever needed. The proposed model provides complex crack paths as a solution to the governing partial differential equations (PDEs), and thus bypasses the need for re-meshing, enrichment of FE shape functions, and an explicit tracking of the crack surfaces. The use of Gridap makes the FE implementation of the proposed model exceedingly compact and user-friendly requiring very low memory usage and providing a high degree of flexibility to the users in defining weak forms of the governing PDEs. We have validated the proposed model and its implementation against a compact tension test on a single edge notched plate and a set of three-point bending tests on a notched beam made of lead zirconium titanate (PZT)-4 piezoelectric ceramics. [ABSTRACT FROM AUTHOR]

An ensemble of silicon vacancy ( SiV −) centers in diamond is probed using two-pulse correlation spectroscopy and multidimensional coherent spectroscopy. Two main distinct families of SiV − centers are identified, and these families are paired with two orientation groups by comparing spectra from different linear polarizations of the incident laser. By tracking the peak centers in the measured spectra, the full diamond strain tensor is calculated local to the laser spot. Measurements are made at multiple points on the sample surface, and variations in the strain tensor are observed. [ABSTRACT FROM AUTHOR]

Flexoelectricity is an electromechanical phenomenon produced in a dielectric material, with or without centrosymmetric microstructure, undergoing a non-uniform strain. It is characterized by the fourth-order flexoelectric tensor, which links the electric polarization vector with the gradient of the second-order strain tensor. Our previous work [H. Le Quang and Q.-C. He, Proc. R. Soc. A 467, 2369 (2011)] solved the fundamental theoretical problem of determining the number and types of all rotational symmetries that the flexoelectric tensor can exhibit. In the present study, compact explicit matrix representations of the flexoelectric tensor are provided so as to facilitate the use of it with any possible rotational symmetry. The number and types of all reflection symmetries that the flexoelectric tensor can have are also determined. To identify the rotational symmetry and reflection symmetry of a given flexoelectric tensor, a simple and efficient graphic method based on the concept of pole figures is presented and illustrated. [ABSTRACT FROM AUTHOR]

The pentagonal boron carbon nitride (penta-BCN) monolayer has been recently proposed as a new member of the pentagon-based two-dimensional nanosheets [Zhao et al., J. Phys. Chem. Lett. 11(9), 3501 (2020)]. By using density functional theory with the generalized gradient approximation, we have carried out detailed investigations of a hydrogenated penta-BCN sheet, where the pristine penta sheet is decorated with H atoms to the composition BCNH2. The hydrogenated penta-BCN (H-BCN) structure is mechanically, thermally, and dynamically stable. It has a wide and indirect bandgap of 4.46 eV, contrasting with the direct gap of 1.70 eV in pristine BCN. H-BCN is environmentally stable at 1 bar of H2 down to 10−10 bar; beyond this point, pristine BCN becomes more stable. Compared with penta-BCN, the components of the elastic modulus tensor C 11 and C 12 of hydrogenated penta-BCN are reduced, while C 12 and C 66 are increased. The strain tensors of piezoelectricity in H-BCN are d 21 = 0.462 , d 22 = 0.213 , and d 16 = 1.03 pm / V , which are lower than those of pristine penta-BCN. The hydrogenated BCN structure displays a higher spontaneous polarization Ps than penta-BCN (4.64 × 10−10 vs 3.38 × 10−10 C/m, respectively). The smaller in-plane Young's moduli E a and E b for H-BCN indicated that that they are softer than those for penta-BCN. Strain engineering can help tune electronic properties. In agreement with this claim, we found that the indirect gap of H-BCN was tunable from 4.46 to 3.26 eV under an applied tensile strain of 0%–16%, the range where the structure is dynamically stable throughout. Meanwhile, H-BCN is dynamically unstable under an applied compressive strain. [ABSTRACT FROM AUTHOR]

Material and geometrical nonlinearity influence on the aortic valve coaptation under the diastolic pressure is investigated. We consider the aortic leaflet as a thin Kirchhoff–Love shell made of simple hyperelastic or linear elastic materials. The main feature of chosen hyperelastic models is their simplicity, as they have minimal number of material parameters. The shear modulus is the same for all four models, and thus, the comparative study is meaningful. We find the quasi-static equilibrium of aortic leaflets coapted under the diastolic pressure by a combination of the hyperelastic nodal force method and the rotation-free shell triangle method. The numerical estimation of the coaptation zone cannot be based on linear elastic model because of large elemental rotations during the leaflets closure and the non-invariance of the linearized strain tensor to rigid rotations. Different hyperelastic material models of the leaflet with the same Young's modulus provide similar coaptation zones which are different from zones predicted by the linear elasticity model. Thus, accounting for the geometrical (large deformation of the leaflets) and material (hyperelastic material of the leaflets) nonlinearity in the assessment of the diastolic function of the aortic valve is essential. [ABSTRACT FROM AUTHOR]

The recent commissioning of a movable monochromator at the 34-ID-C endstation of the Advanced Photon Source has vastly simplified the collection of Bragg coherent diffraction imaging (BCDI) data from multiple Bragg peaks of sub-micrometre scale samples. Laue patterns arising from the scattering of a polychromatic beam by arbitrarily oriented nanocrystals permit their crystal orientations to be computed, which are then used for locating and collecting several non-co-linear Bragg reflections. The volumetric six-component strain tensor is then constructed by combining the projected displacement fields that are imaged using each of the measured reflections via iterative phase retrieval algorithms. Complications arise when the sample is heterogeneous in composition and/or when multiple grains of a given lattice structure are simultaneously illuminated by the polychromatic beam. Here, a workflow is established for orienting and mapping nanocrystals on a substrate of a different material using scanning Laue diffraction microscopy. The capabilities of the developed algorithms and procedures with both synthetic and experimental data are demonstrated. The robustness is verified by comparing experimental texture maps obtained with Laue diffraction microscopy at the beamline with maps obtained from electron back-scattering diffraction measurements on the same patch of gold nanocrystals. Such tools provide reliable indexing for both isolated and densely distributed nanocrystals, which are challenging to image in three dimensions with other techniques. [ABSTRACT FROM AUTHOR]