Strain Tensor Field Research Articles

High-resolution source imaging and moment tensor estimation of acoustic emissions during brittle creep of basalt undergoing carbonation

SUMMARY 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.

An efficient system matrix factorization method for scanning diffraction based strain tensor tomography.

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.

Learning the stress-strain fields in digital composites using Fourier neural operator

Increased demands for high-performance materials have led to advanced composite materials with complex hierarchical designs. However, designing a tailored material microstructure with targeted properties and performance is extremely challenging due to the innumerable design combinations and prohibitive computational costs for physics-based solvers. In this study, we employ a neural operator-based framework, namely Fourier neural operator (FNO), to learn the mechanical response of 2D composites. We show that the FNO exhibits high-fidelity predictions of the complete stress and strain tensor fields for geometrically complex composite microstructures with very few training data and purely based on the microstructure. The model also exhibits zero-shot generalization on unseen arbitrary geometries with high accuracy. Furthermore, the model exhibits zero-shot super-resolution capabilities by predicting high-resolution stress and strain fields directly from low-resolution input configurations. Finally, the model also provides high-accuracy predictions of equivalent measures for stress-strain fields, allowing realistic upscaling of the results.

Three-dimensional thermoelectroelastic analysis of structures with distributed piezoelectric sensors and actuators with temperature-dependent material properties

In this paper, a geometrically exact hybrid-mixed four-node laminated solid-shell element with piezoelectric patches with temperature-dependent material properties through the sampling surfaces (SaS) formulation is developed. The SaS formulation is based on the choice of SaS, parallel to the middle surface and located at Chebyshev polynomial nodes within the layers, to introduce the temperatures, displacements and electric potentials of these surfaces as basic shell unknowns. The outer surfaces and interfaces are also included into a set of SaS. Such a choice of unknowns with the use of Lagrange polynomials in the through-thickness approximations of temperatures, temperature gradient, displacements, strains, electric potential and electric field leads to a very compact higher-order thermopiezoelectric shell formulation. To implement efficient analytical integration throughout the solid-shell element, the extended assumed natural strain method is employed for all components of the temperature gradient, strain tensor and electric field vector. The developed hybrid-mixed four-node laminated solid-shell element with piezoelectric sensors and actuators is based on the extended Hu-Washizu variational principle and shows superior performance in the case of coarse meshes. This can be useful for the three-dimensional temperature-dependent response of laminated composite shells with piezoelectric patches, since the SaS formulation allows one to obtain the numerical solutions with a prescribed accuracy, which asymptotically approach the exact solutions of thermopiezoelectricity as a number of SaS tends to infinity.

Recovering the second moment of the strain distribution from neutron Bragg edge data

Point by point strain scanning is often used to map the residual stress (strain) in engineering materials and components. However, the gauge volume and, hence, spatial resolution are limited by the beam defining apertures and can be anisotropic for very low and high diffraction (scattering) angles. Alternatively, wavelength resolved neutron transmission imaging has a potential to retrieve information tomographically about residual strain induced within materials through measurement in transmission of Bragg edges—crystallographic fingerprints whose locations and shapes depend on microstructure and strain distribution. In such a case, the spatial resolution is determined by the geometrical blurring of the measurement setup and the detector point spread function. Mathematically, reconstruction of the strain tensor field is described by the longitudinal ray transform; this transform has a non-trivial null-space, making direct inversion impossible. A combination of the longitudinal ray transform with physical constraints was used to reconstruct strain tensor fields in convex objects. To relax physical constraints and generalize reconstruction, a recently introduced concept of histogram tomography can be employed. Histogram tomography relies on our ability to resolve the distribution of strain in the beam direction, as we discuss in the paper. More specifically, Bragg edge strain tomography requires extraction of the second moment (variance about zero) of the strain distribution, which has not yet been demonstrated in practice. In this paper, we verify experimentally that the second moment can be reliably measured for a previously well characterized aluminum ring and plug sample. We compare experimental measurements against numerical calculation and further support our conclusions by rigorous uncertainty quantification of the estimated mean and variance of the strain distribution.

Digital volume correlation analysis of polylactic acid based fused filament fabrication printed composites.

Fused filament fabrication (FFF) has rapidly begun to see implementation in industrial fields as a method of rapid manufacturing. Traditional FFF parts are made from a single thermoplastic polymer. The polymer is heated to its melting point and deposited on a work bed where a model is gradually built from the base up. While traditional FFF parts have low mechanical properties, a reinforcing phase allows for improved mechanical properties. The addition of a reinforcing material to the base polymer and complex internal microstructure of the 3 D printed party leads to anisotropic mechanical properties. Thus, these materials’ mechanical properties become challenging to characterize using traditional measurement techniques due to the previously mentioned factors. Therefore, it is essential to develop a method in which mechanical properties can be measured and analyzed. This study aims to characterize the mechanical behaviour under a uniaxial tensile load of an FFF produced polylactic acid (PLA)-copper particulate composite. The internal response of the FFF sample was imaged using micro-computed tomography at predetermined loads. The μ-CT images were input into an open-source digital volume correlation (DVC) software to measure the internal displacements and strain tensor fields. The study results show the development of different strain fields and interior features of the FFF parts.

Strain fields in twisted bilayer graphene.

Van der Waals heteroepitaxy allows deterministic control over lattice mismatch or azimuthal orientation between atomic layers to produce long-wavelength superlattices. The resulting electronic phases depend critically on the superlattice periodicity and localized structural deformations that introduce disorder and strain. In this study we used Bragg interferometry to capture atomic displacement fields in twisted bilayer graphene with twist angles <2°. Nanoscale spatial fluctuations in twist angle and uniaxial heterostrain were statistically evaluated, revealing the prevalence of short-range disorder in moiré heterostructures. By quantitatively mapping strain tensor fields, we uncovered two regimes of structural relaxation and disentangled the electronic contributions of constituent rotation modes. Further, we found that applied heterostrain accumulates anisotropically in saddle-point regions, generating distinctive striped strain phases. Our results establish the reconstruction mechanics underpinning the twist-angle-dependent electronic behaviour of twisted bilayer graphene and provide a framework for directly visualizing structural relaxation, disorder and strain in moiré materials.

Coupled thermoelectroelastic analysis of thick and thin laminated piezoelectric structures by exact geometry solid‐shell elements based on the sampling surfaces method

AbstractA hybrid‐mixed exact geometry four‐node thermopiezoelectric solid‐shell element through the sampling surfaces (SaS) formulation is proposed. The SaS formulation is based on the choice of an arbitrary number of SaS within layers parallel to the middle surface and located at Chebyshev polynomial nodes in order to introduce the temperatures, displacements and electric potentials of these surfaces as basic shell unknowns. Due to the variational formulation, the outer surfaces and interfaces are also included into a set of SaS. Such choice of unknowns with the use of Lagrange polynomials in the through‐thickness approximations of temperatures, temperature gradient, displacements, strains, electric potential, and electric field leads to a very compact higher‐order thermopiezoelectric shell formulation. To implement efficient analytical integration throughout the solid‐shell element, the extended assumed natural strain method is employed for all components of the temperature gradient, strain tensor, and electric field vector. The developed hybrid‐mixed four‐node thermopiezoelectric solid‐shell element is based on the Hu–Washizu variational principle and shows a superior performance in the case of coarse meshes. It can be useful for the three‐dimensional thermoelectroelastic analysis of thick and thin doubly curved laminated piezoelectric shells under thermal loading, since the SaS formulation allows one to obtain the numerical solutions with a prescribed accuracy, which asymptotically approach exact solutions of the theory of thermopiezoelectricity as a number of SaS tends to infinity.

Scanning electron diffraction tomography of strain

Strain engineering is used to obtain desirable materials properties in a range of modern technologies. Direct nanoscale measurement of the three-dimensional strain tensor field within these materials has however been limited by a lack of suitable experimental techniques and data analysis tools. Scanning electron diffraction has emerged as a powerful tool for obtaining two-dimensional maps of strain components perpendicular to the incident electron beam direction. Extension of this method to recover the full three-dimensional strain tensor field has been restricted though by the absence of a formal framework for tensor tomography using such data. Here, we show that it is possible to reconstruct the full non-symmetric strain tensor field as the solution to an ill-posed tensor tomography inverse problem. We then demonstrate the properties of this tomography problem both analytically and computationally, highlighting why incorporating precession to perform scanning precession electron diffraction (SPED) may be important. We establish a general framework for non-symmetric tensor tomography and demonstrate computationally its applicability for achieving strain tomography with SPED data.

Predicting the proximity to macroscopic failure using local strain populations from dynamic in situ X-ray tomography triaxial compression experiments on rocks

Predicting the proximity of large-scale dynamic failure is a critical concern in the engineering and geophysical sciences. Here we use evolving contractive, dilatational, and shear strain deformation preceding failure in dynamic X-ray tomography experiments to examine which strain components best predict the proximity to failure. We develop machine learning models to predict the proximity to failure using time series of three-dimensional local incremental strain tensor fields acquired in rock deformation experiments under stress conditions of the upper crust. Three-dimensional scans acquired in situ throughout triaxial compression experiments provide a distribution of density contrasts from which we estimate the three-dimensional incremental strain that accumulates between each scan acquisition. Training machine learning models on multiple experiments of six rock types provides suites of feature importance that indicate the predictive power of each feature. Comparing the average importance of groups of features that include information about each strain component quantifies the ability of the contractive, dilatational and shear strain to predict the proximity of macroscopic failure. A total of 24 models of four machine learning algorithms with six rock types indicate that 1) the dilatational strain provides the best predictive power of the strain components, and 2) the intermediate values (25th-75th percentile) of the strain population provide the best predictive power of the statistics of the strain populations. In addition, the success of the predictions of models trained on one rock type and tested on other rock types quantifies the similarities and differences of the precursory strain accumulation process in the six rock types. These similarities suggest the potential existence of a unified theory of brittle rock deformation for a range of rock types.

Interaction of a deformable solid with two‐phase flows: An Eulerian‐based numerical model for fluid‐structure interaction using the level contour reconstruction method

SummaryWe describe the formulation of a method for fluid‐structure interaction involving the coupling of moving and/or flexible solid structures with multiphase flows in the framework of the Level Contour Reconstruction Method. We present an Eulerian‐based numerical procedure for tracking the motion and interaction of a liquid‐gas interface with a fluid‐solid interface in the Lagrangian frame together with the evaluation of the fluid transport equations coupled to those for the solid transport, namely the left Cauchy‐Green strain tensor field, in the Eulerian frame. To prevent excessive dissipation due to the convective nature of the solid transport equation, a simple incompressibility constraint for the strain field is enforced. A single grid structure is used for both the fluid and solid phases which allows for a simple and natural coupling of the fluid and solid dynamics. Several benchmark tests are performed to show the accuracy of the numerical method and which demonstrate accurate results compared to several of those in the existing literature. In particular we show that surface tension effects including contact line dynamics on the deforming solid phase can be properly simulated. The three‐phase interaction of a droplet impacting on a flexible cantilever is investigated in detail. The simulations follow the detailed motion of the droplet impact (and subsequent deformation, breakup, and fall trajectory) along with the motion of the deformable solid cantilever due to its own weight as well as due to the force of the droplet impact.

The mixology of precursory strain partitioning approaching brittle failure in rocks

SUMMARYWe examine the strain accumulation and localization process throughout 12 triaxial compression experiments on six rock types deformed in an X-ray transparent apparatus. In each experiment, we acquire 50–100 tomograms of rock samples at differential stress steps during loading, revealing the evolving 3-D distribution of X-ray absorption contrasts, indicative of density. Using digital volume correlation (DVC) of pairs of tomograms, we build time-series of 3-D incremental strain tensor fields as the rocks are deformed towards failure. The Pearson correlation coefficients between components of the local incremental strain tensor at each stress step indicate that the correlation strength between pairs of local strain components, including dilation, contraction and shear strain, are moderate-strong in 11 of 12 experiments. In addition, changes in the local strain components from one DVC calculation to the next show differences in the correlations between pairs of strain components. In particular, the correlation of the local changes in dilation and shear strain tends to be stronger than the correlation of changes in dilation-contraction and contraction-shear strain. In 11 of 12 experiments, the most volumetrically frequent mode of strain accommodation includes a synchronized increase in multiple strain components. Early in loading, under lower differential stress, the most frequent strain accumulation mode involves the paired increase in dilation and contraction at neighbouring locations. Under higher differential stress, the most frequent mode is the paired increase in dilation and shear strain. This mode is also the first or second most frequent throughout each complete experiment. Tracking the mean values of the strain components in the sample and the volume of rock that each component occupies reveals fundamental differences in the nature of strain accumulation and localization between the volumetric and shear strain modes. As the dilative strain increases in magnitude throughout loading, it tends to occupy larger volumes within the rock sample and thus delocalizes. In contrast, the increasing shear strain components (left- or right-lateral) do not necessarily occupy larger volumes and so involve localization. Consistent with these evolutions, the correlation length of the dilatational strains tends to increase by the largest amounts of the strain components from lower to higher differential stress. In contrast, the correlation length of the shear strains does not consistently increase or decrease with increasing differential stress.

Soft-grain compression: Beyond the jamming point.

We present the experimental studies of highly strained soft bidisperse granular systems made of hyperelastic and plastic particles. We explore the behavior of granular matter deep in the jammed state from local field measurement from the grain scale to the global scale. By means of a dedicated digital image correlation code and an accurate image recording method, we measure for each compression step the evolution of the particle geometries and their right Cauchy-Green strain tensor fields. We analyze the evolution of the usual macroscopic observables (stress, packing fraction, coordination, fraction of nonrattlers, etc.) along the compression process through the jamming point and far beyond. Analyzing the evolution of the local strain statistics, we evidence a crossover in the material behavior deep in the jammed state for both sorts of particles. We show that this crossover is due to a competition between material compression, dilation, and shear, so its position depends on the particle material. We argue that the strain field is a reliable observable to describe the evolution of a granular system through the jamming transition and deep in the dense packing state whatever the material behavior.

On construction of SPD stress-strain curve for bronze Cu85–Pb5–Sn5–Zn5

Stress-strain (SS) curve is a necessary ingredient of any mathematical model describing metal forming processes. Severe plastic deformation (SPD) imposes additional requirements on SS-tests because some of deformation mechanisms, such as shear band sliding, micro-cracks, etc., prove to be suppressed due to specific character of stress-strain state: complex loading path, all-round high pressure, and high gradients of stress and strain tensor fields. In this work, a novel method of SPD SS-curve construction combining the classical uniaxial compression test with microhardness measurements is proposed. The barreling effect is negligibly small owing to specific experimental conditions. This renders spatially uniform the stress-strain state and allows avoiding the inverse finite element modeling. The cast leaded tin bronze alloy Cu85–Pb5 –Sn5–Zn5 has been studied during upsetting process. An experimental force-displacement diagram appears to be strictly linear up to the moment of destruction under uniaxial compression.The dendritic structure evolution in both axial and cross sections of a cylindrical sample was studied by optical microscope. Local distortions in elements of this structure are evidence of spatial fluctuations in plastic strain at macroscopic level. Microhardness in these sections was also measured. Shear bands appear at strainεp≈0.14 and they are responsible of the softening behavior. In this work, softening is contrary to the experiments on deep rolling treatment where the monotonic increase in microhardness was observed. Microhardness measurements allow expanding the total strain into dislocation sliding and shear band sliding summands. This expansion makes it possible to discard the shear band sliding item and obtain an initial segment of SS-curve (up to strain εp=0.32 for bronze Cu85–Pb5–Sn5–Zn5) available for SPD simulation.

The evolving energy budget of experimental faults within continental crust: Insights from in situ dynamic X-ray microtomography

We investigate the evolving distribution of strain produced by a sliding fault within intact crystalline rock, and the energetics of deformation that occur both on- and off-fault. We slid precut faults of differing roughness oriented at 45° to σ1 while acquiring in situ X-ray microtomograms. Digital volume correlation of tomograms provide estimates of the 3D displacement and strain fields. This characterization of the strain tensor field reveal that the differing fault roughness produced distinct slip behavior, degree of strain localization and accumulation, and energy budget partitioning. The rougher fault slipped more episodically, hosted a wider and more asymmetric damage zone, and accommodated less normal and shear strain. This fault consumed more energy in off-fault deformation (Wint) per volume and more energy in frictional slip (Wfric) as portions of the total energy input to the system (Wext) than the smoother fault. In both experiments, Wfric consumed the largest portion of the energy budget (50–100%), while Wint consumed smaller percentages (5–20%). Tracking the temporal variability of energy partitioning revealed how evolving fault architecture determined the energetic dominance of particular deformational processes, and so highlighted the importance of tracking energy partitioning through time.

Volumetric and Shear Strain Localization in Mt. Etna Basalt

AbstractTo examine the impact of preexisting weaknesses on fracture coalescence during volcanic edifice deformation, we triaxially compressed Mount Etna basalt while acquiring in situ dynamic X‐ray microtomograms and calculated the internal strain tensor fields using image correlation. Contraction localization preceded dilation and shear strain localization into the protofault zone. This onset of strain localization preceded macroscopic yielding and coincided with increases in the magnitude and volume of rock experiencing dilation, and spatial clustering of the strain populations. The exploitation of weaknesses by propagating fractures enabled the dominant shear strain to switch senses as propagating fractures lengthened along 30–60° from σ1. Scanning electron microscopy images reveal pore‐emanated fractures, and fractures linking pores. These experiments provide evidence of internal contraction preceding dilation and shear, consistent with inferences from field and laboratory observations. The transition from contraction to dilation may provide a precursory signal of volcanic flank eruption.

Asymptotic justification of the intrinsic equations of Koiter's model of a linearly elastic shell

We show that the intrinsic equations of Koiter's model of a linearly elastic shell can be derived from the intrinsic formulation of the three-dimensional equations of a linearly elastic shell, by using an appropriate a priori assumption regarding the three-dimensional strain tensor fields appearing in these equations. To this end, we recast in particular the Dirichlet boundary conditions satisfied by any admissible displacement field as boundary conditions satisfied by the covariant components of the corresponding strain tensor field expressed in the natural curvilinear coordinates of the shell. Then we show that, when restricted to strain tensor fields satisfying a specific a priori assumption, these new boundary conditions reduce to those of the intrinsic equations of Koiter's model of a linearly elastic shell.

Dynamic In Situ Three-Dimensional Imaging and Digital Volume Correlation Analysis to Quantify Strain Localization and Fracture Coalescence in Sandstone

Advances in triaxial compression deformation apparatus design, dynamic X-ray microtomography imaging, data analysis techniques, and digital volume correlation analysis provide unparalleled access to the in situ four-dimensional distribution of developing strain within rocks. To demonstrate the power of these new techniques and acquire detailed information about the micromechanics of damage evolution, deformation, and failure of porous rocks, we deformed 3-cm-scale cylindrical specimens of low-porosity Fontainebleau sandstone in an X-ray-transparent triaxial compression apparatus, and repeatedly recorded three-dimensional tomograms of the specimens as the differential stress was increased until macroscopic failure occurred. Experiments were performed at room temperature with confining pressure in the range of 10–20 MPa. Distinct grayscale subsets, indicative of density, enabled segmentation of the three-dimensional tomograms into intact rock matrix, pore space, and fractures. Digital volume correlation analysis of pairs of tomograms provided time series of three-dimensional incremental strain tensor fields throughout the experiments. After the yield stress was reached, the samples deformed first by dilatant opening and propagation of microfractures, and then by shear sliding via grain rotation and strain localization along faults. For two samples, damage and dilatancy occurred by grain boundary opening and then a sudden collapse of the granular rock framework at failure. For the third sample, a fault nucleated near the yield point and propagated in the sample through the development of transgranular microfractures. The results confirm findings of previous experimental studies on the same rock and provide new detailed quantifications of: (1) the proportion of shear versus dilatant strain in the sample, (2) the amount of dilatancy due to microfracture opening versus pore opening when a fault develops, and (3) the role of grain boundaries and pore walls in pinning microfracture propagation and slowing down the rate of damage accumulation as failure is approached. Our study demonstrates how the combination of high-resolution in situ dynamic X-ray microtomography imaging and digital volume image correlation analysis can be used to provide additional information to unravel brittle failure processes in rocks under stress conditions relevant to the upper crust.

Intrinsic Formulation and Lagrange Duality for Elastic Cable Networks with Geometrical Nonlinearity

The intrinsic approach to elasticity is a variational problem that characterizes the static equilibrium state and is formulated only in terms of the unknown strain tensor field. This paper establishes an intrinsic approach to elastic cable networks undergoing large deformations. In this formulation, the potential energy function attains a global minimum at strains in any equilibrium state. It is clearly explained that both the classical displacement formulation and the intrinsic strain formulation can be derived as the Lagrange dual problems of the stress formulation, and that the difference between these two formulations stems from the variety of perturbation functions used to derive the Lagrangian in the Lagrange duality theory.

The intrinsic theory of linearly elastic plates

In an intrinsic approach to a problem in elasticity, the only unknown is a tensor field representing an appropriate ‘measure of strain’, instead of the displacement vector field in the classical approach. The objective of this paper is to study the displacement traction problem in the special case where the elastic body is a linearly elastic plate of constant thickness, clamped over a portion of its lateral face. In this respect, we first explicitly compute the intrinsic three-dimensional boundary condition of place in terms of the Cartesian components of the linearized strain tensor field, thus avoiding the recourse to covariant components in curvilinear coordinates and providing an interesting example of actual computation of an intrinsic boundary condition of place in three-dimensional elasticity. Second, we perform a rigorous asymptotic analysis of the three-dimensional equations as the thickness of the plate, considered as a parameter, approaches zero. As a result, we identify the intrinsic two-dimensional equations of a linearly elastic plate modelled by the Kirchhoff–Love theory, with the linearized change of metric and change of curvature tensor fields of the middle surface of the plate as the new unknowns, instead of the displacement field of the middle surface in the classical approach.