Displacement Gradient Tensor Research Articles

On the role of anomalous twinning in the plasticity of magnesium

Specially oriented Mg single crystals were deformed at ambient and elevated temperatures in channel-die compression with the c-axis inclined at 45° to the compression direction and 〈101¯0〉 parallel to the lateral constraint direction. The specimens deformed by basal slip, entailing a rotation of the c-axis towards the compression direction. At room temperature, starting already during early stages of deformation, macroscopic bands comprising a mesh of anomalous {101¯2} extension twins formed parallel to the constraint direction. These twins were characterized by a negative Schmid factor and produced a strain opposite to the imposed deformation. At the final true strain of −1, a significant volume fraction of the matrix was consumed by twins, causing a reorientation of the c-axis farther away from the compression direction, and hence delaying the formation of a hard basal texture component. An analysis of the displacement gradient tensors for the six possible extension twin variants showed that the selected twin variants, of which the bands were composed, involved a lattice rotation opposite to the one caused by basal slip. Less twinning was observed at elevated temperatures with the macroscopic bands disappearing at 370 °C. The obtained results are discussed with respect to deformation heterogeneity and the role of anomalous twinning for the deformation behavior of magnesium.

An integrated method based on DInSAR, MAI and displacement gradient tensor for mapping the 3D coseismic deformation field related to the 2011 Tarlay earthquake (Myanmar)

Abstract The satellite differential interferometric synthetic aperture radar (DInSAR) technology has been widely applied for mapping the ground deformation associated with the geophysical events such as earthquakes. However, the conventional DInSAR can measure only the one-dimensional (1D) ground displacement along the radar line of sight (LOS), and the crucial displacement measurements, e.g., near the epicenter and along the surface ruptures, are usually not available or degraded in quality due to the significant deformation gradients. With availability of satellite SAR images acquired along ascending and descending orbits, this paper proposes an integrated method to map the three-dimensional (3D) coseismic deformation field by combining DInSAR for detecting LOS displacements, the multiple aperture interferometry (MAI) for detecting along-track displacements, and the displacement gradient tensor (DGT) model for characterizing spatial correlation of ground displacements. The proposed method (termed as InSAR–DGT) was first tested through an experiment of recovering 3D displacements from a simulated coseismic deformation field. We then applied the proposed method to map the 3D coseismic deformation field related to the 2011 Tarlay earthquake (Myanmar) by using the ascending and descending ALOS PALSAR images. Both the results derived for the simulated experiment and the 2011 Tarlay earthquake show that the quality of the 3D coseismic displacements can be raised efficiently by the InSAR–DGT method, and the precisions in the east–west (E–W), north–south (N–S) and up–down (U–D) displacements for the 2011 Tarlay earthquake are increased 22%, 36% and 24%, respectively. The validation indicates that the improved coseismic deformation field for the 2011 Tarlay earthquake is in good agreement with the deformation field simulated from the existing optimized fault model. The number of the missing data points in the 3D coseismic deformation field can be reduced significantly by the InSAR–DGT method. It is revealed that the causative fault (i.e., the west part of the Nam Ma Fault) for the Tarlay earthquake generated a left-lateral slip that was accompanied by a minor normal dip-slip component. The careful interpretation demonstrates that the causative fault structures include the determined main fault segment and the two suspected branched segments at east of the Tarlay town.

Symmetry conditions in strain gradient elasticity

We study the variational significance of the “order-of-differentiation” symmetry condition of strain gradient elasticity. This symmetry condition stems from the fact that in strain gradient elasticity, one can interchange the order of differentiation in the components of the second displacement gradient tensor. We demonstrate that this symmetry condition is essential for the validity of free variational formulations commonly employed for deriving the field equations of strain gradient elasticity. We show that relying on this additional symmetry condition, one can restrict consideration to strain gradient constitutive equations with a considerably reduced number of independent material coefficients. We explicitly derive a symmetry unified theory of isotropic strain gradient elasticity with only two independent strain gradient material coefficients. The presented theory has simple stability criteria and its factorized displacement form equations of equilibrium allow for expedient identification of the fundamental solutions operative in specific theoretical and application studies.

A MORE COMPLETE THERMODYNAMIC FRAMEWORK FOR SOLID CONTINUA

The Jacobian of deformation at a material point can be decomposed into the stretch tensor and the rotation tensor. Thus, varying Jacobians of deformation at the neighboring material points in the deforming volume of solid continua would yield varying stretch and rotation tensors at the material points. Measures of strain, such as Green’s strain, at a material point are purely a function of the stretch tensor, i.e. the rotation tensor plays no role in these measures. Alternatively, we could also consider decomposition of displacement gradient tensor into symmetric and skew symmetric tensors. Skew symmetric tensor is also a measure of pure rotations whereas symmetric tensor is a measure of strains, i.e. stretches. The measures of rotations in these two approaches describe the same physics but are in different forms. Polar decomposition gives the rotation matrix and not the rotation angles whereas the skew symmetric part of the displacement gradient tensor yields rotation angles that are explicitly and conveniently defined in terms of the displacement gradients. The varying rotations and rotation rates arise in all deforming solid continua due to varying deformation of the continua at neighboring material points, hence are internal to the volume of solid continua and are explicitly defined by the deformation, therefore do not require additional degrees of freedom to define them. If the internal varying rotations and their rates are resisted by the continua, then there must exist internal moments corresponding to these. The internal rotations and their rates and the corresponding moments can result in additional energy storage and dissipation. This physics is all internal to the deforming continua (hence does not require consideration of additional external degrees of freedom and associated external moments) and is neglected in the presently used continuum theories for isotropic, homogeneous solid continua. The continuum theory presented in this paper considers internal varying rotations and associated conjugate moments in the derivation of the conservation and balance laws, thus the theory presented in this paper is “a polar theory for solid continua” but is different than the micropolar theories published currently in which material points have six external degrees of freedom i.e. rotations are additional external degrees of freedom. This polar continuum theory only accounts for internal rotations and associated moments that exist as a consequence of deformation but are neglected in the present theories. We call this theory “a polar continuum theory” as it considers rotations and moments as conjugate pairs in a deforming solid continua though these are internal, hence are purely related to the deformation of the solid. It is shown that the polar continuum theory presented in this paper is not the same as the strain gradient theories reported in the literature. The differences are obviously in terms of the physics described by them and the mathematical details associated with conservation and balance laws. In this paper, we only consider polar continuum theory for small deformation and small strain. This polar continuum theory presented here is a more complete thermodynamic framework as it accounts for additional physics of internally varying rotations that is neglected in the currently used thermodynamic framework. This thermodynamic framework is suitable for isotropic, homogeneous solid matter such as thermoelastic and thermoviscoelastic solid continua with and without memory when the deformation is small. The paper also presents preliminary material helpful in consideration of the constitutive theories for polar continua.

Non-linear elastic micro-dilatation theory: Matrix exponential function paradigm

In the present paper, the micro-dilatation theory or void elasticity is extended to both large displacement and large dilatation. Firstly, the deformation gradient tensor has been freshly defined by means of the matrix exponential function. The newly defined relation for the deformation gradient has painstakingly investigated for the uniqueness, decomposition issues as well as objectivity and isotropy considerations. The relation of the displacement gradient and deformation gradient tensor is brought via the matrix logarithm function. The micro-dilatation theory constitutive laws are derived using the thermodynamic principles under the zero-centrosymmetric, weakly-centrosymmetric and fully-centrosymmetric cases. These cases have been derived and scrutinized by the numerical experiments. To achieve this assignment, the basic loadings are taken into account, e.g. the hydrostatic loading, simple traction and shear. Some conclusions and outlook pertaining to the above-mentioned cases and variable bulk density have thereafter discussed.

Macroscopic strains in granular materials accounting for grain rotations

Granular materials are special materials, from the continuum-mechanical viewpoint, in the sense that they possess a clear microstructure of grains and intergrain contacts. In addition, the grains have translational as well as rotational degrees of freedom. Here a micromechanical expression is formulated for the average displacement gradient tensor in terms of the grain displacements and rotations for the two-dimensional case. It is based on a tessellation of the granular assembly into closed loops, along whose boundary the displacement field is defined in terms of the grain displacements and rotations. An important consideration for this expression is that it must satisfy the surface-additivity property, according to which the average displacement gradient of a combined two-dimensional surface is determined by the average displacement gradients of the constituent surfaces (and weighted by their areas). The developed micromechanical expression is verified by comparing its results with the macroscopic deformation as determined from the displacements at the boundary. Results of DEM simulations of a biaxial test (where the average rotation is equal to zero) and a shear test (where the average rotation is not equal to zero) are employed for this verification. The developed micromechanical expression for the displacement gradient is subsequently used to study deformation patterns in a complex case of a biaxial test where a shear band is formed.

Formulation of Toupin–Mindlin strain gradient theory in prolate and oblate spheroidal coordinates

Abstract The Toupin–Mindlin strain gradient theory is reformulated in orthogonal curvilinear coordinates, and is then applied to prolate and oblate spheroidal coordinates for the first time. The basic equations, boundary conditions, the gradient of the displacement, strain and strain gradient tensors of this theory are derived in terms of physical components in these two coordinate systems, which have a potential significance for the investigation of micro-inclusion and micro-void problems. As an example, using these formulae, we formulate and discuss the boundary-value problem of a spheroidal cavity embedded in a strain gradient elastic medium subjected to uniaxial tension. In addition, the previous results given by Zhao and Pedroso (Int. J. Solids. Struct. (2008) 45, 3507–3520) in cylindrical and spherical coordinates are amended.

Variant selection during secondary and tertiary twinning in pure titanium

Three generations of twins were identified by electron backscatter diffraction techniques in pure titanium subjected to uniaxial compression at room temperature. Many primary contraction twins were observed as the initial texture was favourable for their formation. Numerous secondary extension twins formed within the primary contraction twins and some tertiary contraction twins within the secondary extension twins. The orientations of these three generations of twins were determined and their associated Schmid factors (SFs) were calculated. The formation of the twin variants selected in each generation and the absence of certain potential variants are explained by rotating the twinning displacement gradient tensor expressed in the twin system reference frame into the crystal reference frame of the relevant neighbouring grain. The presence of the observed secondary and tertiary twins is accounted for in terms of the ease of imposing the required accommodation strains on their neighbours. The results show that secondary twins can form even with low SFs as long as their accommodation takes place by prismatic or basal glide. However, the formation of certain second and third generation potential high SF twins was impeded when this would have required accommodation by the most difficult deformation mode: pyramidal glide.

Reliability of high‐resolution electron backscatter diffraction determination of strain and rotation variations using phase‐only and cross correlation

This contribution analyzes the reliability of strain and rotation variation determination using cross‐ as well as phase‐only correlation of experimental wide‐angle electron backscatter diffraction (EBSD) patterns. For both rotation and three‐point bending the resulting displacement gradient tensor components are examined in terms of magnitude and statistical scatter as a function of various correlation procedure parameters and signal‐to‐noise ratio. It is shown that on the one hand the Fourier filter width has a major impact on the strain results. At higher noise level a smaller filter width has to be applied for obtaining maximum precision. On the other hand, the influence of the degree of overlap of the regions of interest positioned in the patterns is less important. For both rotation and bending experiments the cross‐correlation variant yields a smaller standard deviation with respect to phase‐only correlation, in particular for elevated noise level. The difference is attributed to the stronger propagation of noise effects in the course of phase‐only correlation function calculation and fitting. In the rotation experiment a standard deviation of ∼1.0 × 10−4, averaged over the displacement gradient tensor components, and a rotational precision of ∼1.5 × 10−4 rad have been achieved by using optimized evaluation settings.

Toward a Continuous Monitoring of the Horizontal Displacement Gradient Tensor Field in Southern California Using cGPS Observations from Plate Boundary Observatory (PBO)

Detection and characterization of strain transients provides opportunities for better understanding of crust and mantle rheology (Freed and Burgman, 2004; Freed et al. , 2007; Pollitz et al. , 2012), fault zone behavior (McGuire and Segall, 2003; Miyazaki et al. , 2011), and the mechanics of parts of the earthquake cycle (Dragert and Wang, 2011; Bartlow et al. , 2011). The relatively dense coverage of continuous GPS (cGPS) within western North America, through EarthScope Plate Boundary Observatory (PBO), enables the possibility of detecting crustal strain transient phenomena on a variety of spatial and temporal scales (Ohtani et al. , 2010; Ji and Herring, 2011). In this paper we present a geodetic network‐processing tool for detection of anomalous strain transients in Southern California, or within any region with reasonably dense cGPS coverage. The modeling procedure determines time‐dependent displacement gradient fields from cGPS time series (Hernandez et al. , 2005; Hernandez, 2007) and determines the significance of any departures of this field relative to a geodetic reference model. The derived field satisfies strain‐rate compatibility relations everywhere within the plate boundary zone on a sphere. Given sufficient spatial coverage of cGPS, the method can detect anomalous strain, either from shear or volume sources, and either on or off the known faults. In Determination of a Geodetic Reference Model, we present a brief description of the derivation of the geodetic reference model. We show from a benchmarking exercise that GPS observations in southern California (Shen et al. , 2011) are sufficiently dense to provide a reliable estimate of a reference model. We also show that a joint fitting of reference model strains, together with the sparser distribution of GPS from PBO stations, yields minimal strain‐rate artifacts. In Method for Obtaining Time‐Dependent Displacement Estimates from cGPS Observations, we introduce the method for interpolating the …

Elastic strains in polycrystalline UO2 samples implanted with He: micro Laue diffraction measurements and elastic modeling

Abstract:In the framework of the study of long-term storage of the spent nuclear fuel, polycrystalline UO2 samples have been implanted with He ions. The thin implanted layer, close to the free surface is subjected to elastic stresses which are studied by x-ray diffraction (micro Laue diffraction) and a mechanical modeling. A simple expression of the displacement gradient tensor has been evidenced; it concerns only three terms (ε3, ε4 and ε5) which strongly evolve with considered grain orientations. Finally, we show that results obtained with micro diffraction are in very good agreement with conventional x-ray diffraction measurements done in laboratory at macro scale.

Finite Element Analysis for Normal-Temperature Wet Briquetting of Straws

As one of important technologies for briquettes production from straws, the normal-temperature wet briquetting technology of straws shows important practical significance in place of fossil fuels that are progressively reduced. According to the characteristics of the normal-temperature wet briquetting of straws having large displacement and large strain, this article provided a finite element calculation method for non-linear straw problems using the large-deformation elastic-plasticity principle. Based on description of straw status by Lagrange method, analyses were performed for deformation gradient tensor, displacement gradient tensor and rigid body movement, and then finite element equations were established for the normal-temperature wet briquetting of straws according to the Green strain, Drucker-Prager criterion and balance principle, providing references for numerical simulation for the normal-temperature wet briquetting of straws with computer software.

The role of strain accommodation during the variant selection of primary twins in magnesium

Samples of magnesium alloys AM30 and AZ31 were deformed in tension at room temperature and a strain rate of 0.1 s −1 to strains of 0.08 and 0.15. Of the numerous contraction twins that formed, the orientations of 977 were determined by electron backscatter diffraction techniques. The orientations of their host grains were also measured, so that the Schmid factors (SFs) applicable to each of the six contraction twins that could potentially form in each grain could also be calculated. About half of the observed twins were of the “high SF” (0.3–0.5) type, while nearly half had “low” SFs (0.15–0.30). Furthermore, 5% of the observed twins had associated Schmid factors of only 0.03–0.15, i.e. these were of the “very low SF” type. Of particular interest is the observation that many potential “high Schmid factor” twins did not form. The presence of the low and very low SF twins and the absence of many potential high SF twins are explained in terms of the accommodation strains that are or would be required to permit their formation. These were calculated by rotating the twinning shear displacement gradient tensor into the crystallographic reference frame of the neighboring grain. It is shown that the very high plastic anisotropy of Mg grains permits the “easy” accommodations to take place but conversely prevents accommodation of the potential twinning shears when these are “difficult” (when these involve high critical resolved shear stresses). The twins that appear require little or no “difficult” accommodation.

Accuracy assessment of elastic strain measurement by EBSD

A detailed accuracy analysis of electron backscatter diffraction (EBSD) elastic strain measurement has been carried out using both simulated and experimental patterns. Strains are determined by measuring shifts between two EBSD patterns (one being the reference) over regions of interest (ROI) using an iterative cross-correlation algorithm. An original minimization procedure over 20 regions of interests gives a unique solution for the eight independent components of the deviatoric displacement gradient tensor. It is shown that this method leads to strain measurements on simulated patterns with an accuracy better than 10(-4). The influence of the projection parameters is also investigated. The accuracy assessment is illustrated by two worked examples: (i) four-point bending of a silicon single crystal and (ii) Si(1-x)Ge(x) layers on a Si substrate. Experimental results are compared with finite-element simulations.

Combined kinematic and paleostress analysis of fault-slip data: The Multiple-slip method

We describe the Multiple-slip method for the construction of the displacement gradient tensor, which describes the faulting-related deformation of the region. Three elements must be known for each fault: (1) the fault plane orientation, (2) the slip direction, and (3) the number of parallel faults in the same size range. The data on the orientation of faults and the direction of slip along them define the geometric moment tensor for each fault, while the data on the number of parallel faults belonging to a particular fault-set constrain the weighting factor for each fault. In the Multiple-slip method, the weighting factors also depend on the stress state that produced the displacement along the faults. The stress state on the observed faults at the time of faulting can be estimated from the paleostress analysis, which finds the stress tensor capable of explaining the slip direction along the faults. Therefore, the Multiple-slip method is a combination of kinematic and paleostress techniques, in which the paleostress analysis must be performed prior to the kinematic analysis. The Multiple-slip method allows for calculating (1) the direction of kinematic axes (directions of maximum shortening and extension), (2) the direction and relative magnitude of rotation, and (3) the ratio between the principal strains.

A General Methodology for Full-Field Plastic Strain Measurements Using X-ray Absorption Tomography and Internal Markers

Probing the strain locally and throughout the bulk of various materials has long been of interest in Materials Science. This article presents a general methodology for assessing the plastic strain in terms of the displacement gradient tensor throughout the bulk of opaque samples. The method relies on a homogenous distribution of marker particles throughout the bulk of a sample, markers which are detected through the application of synchrotron X-ray tomography. Making use of the morphology of individual markers, motion of individual markers is tracked during deformation allowing the local displacement field to be determined throughout the bulk. The local displacement gradient tensor is derived from the displacement field. Spatial resolution is directly related to marker particle density in the sample, here 30 μm. The accuracy of the displacement gradient tensor calculation is dependent on the accuracy with which each marker position is determined and is shown to be in the range from 0.005 to 0.012.

Experimental determination of strain partitioning among individual grains in the bulk of an aluminium multicrystal

A recently developed marker-based technique for mapping of the displacement gradient tensor and the strain throughout the bulk of optically opaque specimens is presented and applied to an aluminium alloy multicrystal. Through investigations at 4%, 10% and 14% axial strains, the internal strain field is observed to be non-homogenous with the observed patterns present throughout the range of strains investigated. The morphology of the strain field is visualized with a resolution better than 50μm and variations are tentatively associated with the grain structure as recorded by EBSD. Future applications of the technique in combination with other 3-dimensional approaches are discussed with respect to comparison with Finite Element modelling approaches.

Inhomogeneous plastic flow investigated by X‐ray absorption microtomography of an aluminium alloy containing marker particles

By combining a synchrotron X-ray source and the microtomography technique, the displacement gradient tensor components can be measured directly throughout the volume of a metal sample containing highly absorbing marker particles to detect material flow. The present article describes application of these techniques to compression of a rectangular Al-W specimen with a central hole. The results show that the externally imposed axial displacement gradient is accommodated mainly in two 45 degrees shear bands passing through the hole. A simple deformation analysis provides insight into shear band formation.

High-resolution elastic strain measurement from electron backscatter diffraction patterns: New levels of sensitivity

In this paper, we demonstrate that the shift between similar features in two electron backscatter diffraction (EBSD) patterns can be measured using cross-correlation based methods to ±0.05 pixels. For a scintillator screen positioned to capture the usual large solid angle employed in EBSD orientation mapping this shift corresponds to only ∼8.5×10 −5 rad at the pattern centre. For wide-angled EBSD patterns, the variation in the entire strain and rotation tensor can be determined from single patterns. Repeated measurements of small rotations applied to a single-crystal sample, determined using the shifts at four widely separated parts of the EBSD patterns, showed a standard deviation of 1.3×10 −4 averaged over components of the displacement gradient tensor. Variations in strains and rotations were measured across the interface in a cross-sectioned Si 1− x Ge x epilayer on a Si substrate. Expansion of the epilayer close to the section surface is accommodated by tensile strains and lattice curvature that extend a considerable distance into the substrate. Smaller and more localised shear strains are observed close to the substrate–layer interface. EBSD provides an impressive and unique combination of high strain sensitivity, high spatial resolution and ease of use.

A new finite element scheme using the Lagrangian framework for simulation of viscoelastic fluid flows

A new numerical scheme for simulation of viscoelastic fluid flows was designed, making use of finite element algorithms generally regarded as advantageous for tackling the problem. This includes the Lagrangian approach for the solution of viscoelastic constitutive equation using the co-deformational frame of reference with a possibility of analytically solving the equation along the particles trajectories, which in turn allowed eluding the solution of any system of linear equations for the stress. Then, the full ellipticity of the momentum conservation equation was utilised thanks to a possibility of accurate determination of the stress tensor independently of the velocity field at the current stage of computation. The needed independent stress was calculated at each time step on the basis of the past deformation history, which in turn was determined on the basis of the past velocity fields, all incorporated into a modified Euler time stepping algorithm. Owing to explicit inclusion of the full viscous term from the viscoelastic model into the momentum conservation equation, no stress splitting was necessary. The trajectory feet tracking was done accurately using a semi-analytic solution of the displacement gradient evolution equation and a weak formulation of the kinematics equation, the latter at the expense of solving an extra symmetric system of linear equations. The error expressed in the form of the Sobolev norms was determined using a comparison with available analytical solution for UCM fluid in the transient regime or numerically obtained steady-state stress values for the PTT fluid in Couette flow. The implementation of the PTT fluid model was done by modifying the relative displacement gradient tensor so that a new convective frame was defined. The stability of the algorithm was assessed using the well-known benchmark problem of a sphere sedimenting in a tube with viscoelastic fluid. The stable numerical results were obtained at high Weissenberg numbers, with the limit of convergence Wi=6.6, exceeding any previously reported values. The robustness of the code was proven by simulation of the Weissenberg effect (the rod-climbing phenomenon) with the use of PTT fluid.